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abstract: 'We present first-principles calculations of quantum transport in chemically doped graphene nanoribbons with a width of up to 4 nm. The presence of boron and nitrogen impurities is shown to yield resonant backscattering, whose features are strongly dependent on the symmetry and the width of the ribbon, as well as the position of the dopants. Full suppression of backscattering is obtained on the $\pi-\pi^*$ *plateau* when the impurity preserves the mirror symmetry of armchair ribbons. Further, an unusual acceptor-donor transition is observed in zig-zag ribbons. These unconventional doping effects could be used to design novel types of switching devices.'
author:
- 'Blanca Biel$^{1,3}$, X. Blase$^2$, François Triozon$^1$ and Stephan Roche$^3$'
title: Anomalous Doping Effects on Charge Transport in Graphene Nanoribbons
---
The ability to single out a single graphene plane, through an exfoliation process [@graphene_exf], or by means of epitaxial growth [@graphene_epit], has opened novel opportunities for exploring low dimensional transport in a material with remarkable electronic properties [@RMP]. Additionally, the development of graphene-based nanoelectronics has attracted much attention owing to the promising large scale integration expectations [@graphene_epit; @graph_transistor]. However, with 2D graphene being a zero-gap semiconductor, its use in an active electronic device such as a field effect transistor (FET) requires a reduction of its lateral size to benefit from quantum confinement effects. Graphene nanoribbons (GNRs) are strips of graphene whose electronic properties depend on their edge symmetry and width [@Ribbonpure], and can be either patterned by plasma etching [@IBM; @KimENRJGAP] or derived chemically [@Dai]. Band-gap engineering of GNRs has been experimentally demonstrated [@KimENRJGAP], and GNRs-based FET with a width of several tens of nanometers down to 2 nm have been characterized [@Dai].
Chemical doping aims at producing [***p***]{}-doped or [***n***]{}-doped transistors, which are crucial for building logic functions and complex circuits [@4]. Doping also allows new applications such as chemical sensors, or electrochemical switches [@MOL]. In carbon-based materials [***p***]{}-type ([***n***]{}-type) doping can be achieved by boron (nitrogen) atom substitution within the carbon matrix [@BdopedCVD]. For metallic carbon nanotubes (CNTs), Choi and co-workers reported that boron (B) and nitrogen (N) impurities yield quasibound states which strongly backscatter propagating charge for specific resonance energies [@5]. The interplay between those resonance energies and external parameters (electric or magnetic fields) may also enable the design of novel kinds of CNT-based switching devices [@8], whereas the existence of quasibound states related to topological defects was unveiled by STM measurements [@17].
In this Letter, we report on an *ab initio* study of the effect of both [***p***]{}-type (B) and [***n***]{}-type (N) doping on quantum transport in GNRs with widths within the experimental scope. In contrast with CNTs, doping in GNRs turns out to display more complex features depending on the dopant position, ribbon width and symmetry. Theoretically, two types of GNRs with highly symmetric edges have been described, namely zig-zag (zGNRs) and armchair (aGNRs) [@Ribbonpure]. Some recent works have reported on the effect of doping in extremely narrow zGNRs [@dopedGNRs], mostly with a width in order of 1 nm.
Following Ref. [@Louie-gaps], we refer to a zGNR (aGNR) with N zig-zag chains (dimers) contained in its unit cell as a N-zGNR (N-aGNR). We have studied aGNRs and zGNRs with widths between 2.3 and 4.2 nm, already within the current experimental scope [@Dai; @Lambin]. The scattering potential around the dopant is obtained using first principles calculations (SIESTA code [@SIESTA]) within the local density approximation [@Troullier; @NOTE0].
![(color online). Left: Conductance of the 35-aGNR as a function of energy for different B positions. The dashed lines correspond to conductance for the undoped case. Insets: same as in main frame for two selected nitrogen dopant positions (at the edge (E), and off-center (B)) for the 20-aGNR. Right: unit cell of the 35-aGNR showing the considered dopant positions. The carbon atoms are shown in grey and the passivating H atoms in pink; the other colored atoms represent B or N atoms, each color referring to a corresponding colored solid line conductance curve.[]{data-label="FIG1"}](FIG1.ps){width="7.5cm"}
We start with the case of aGNRs. *Ab initio* studies show that aGNRs are always semiconducting [@Louie-gaps; @Scuseria-gaps; @White-gaps] with width-dependent bandgap scaling. We have studied the 20-, 34- and 35-aGNRs, with widths of 2.3, 4.0 and 4.2 nm, respectively. In carbon nanotubes, two acceptor (donor) quasibound states have been predicted below (above) the charge neutrality point (CNP) at low energy values when a single carbon atom is substituted by a boron (nitrogen) impurity [@5; @Mahan].
However, in contrast with CNTs, the energies of the quasibound states in GNRs are strongly dependent on the position of the impurity with respect to the GNR edges. A clear increase in binding energy of the bound state associated with the broad drop in the conductance in Fig. \[FIG1\] is observed as the dopant approaches the edge. The large variation of resonant energies with dopant position indicates that random distribution of impurities will lead to a rather uniform reduction of conductance over the occupied-states part of the first conduction *plateau*. This is in sharp contrast to the case of CNTs, where resonant energies do not depend on the position of the dopant around the tube circumference. Our results for the 34-aGNR (not shown) confirm this behavior also for semiconducting aGNRs.
![(color online). Conductance as a function of the energy (left) and bandstructure (right) of the 35-aGNR [@NOTE1]. Dashed lines correspond to the undoped ribbon; solid lines correspond to the case of B at the center. The solid blue (dashed red) arrow shows the first (second) band below the charge neutrality point (CNP) for the doped case. []{data-label="FIG2"}](FIG2.ps){width="6.0cm"}
{width="16.0cm"}
In addition, for certain dopant positions, symmetry effects yield a full suppression of backscattering even in the presence of bound states. Indeed, when B is placed at the exact center of the ribbon, the transmission at the first *plateau* is found to be insensitive to the presence of the impurity (Fig. \[FIG1\]-left, bottom curve). Conversely, as the defect approaches the ribbon edge, the energy resonance floats up towards the CNP and the conductance is clearly degraded by the quasibound states induced by the impurity (Fig. \[FIG1\]-left).
To understand such phenomena, we must first note that GNRs, unlike CNTs, do not always present a well defined parity associated to mirror reflections with respect to their axis. An ideal odd-index aGNR retains a single mirror symmetry plane (perpendicular to the plane of the ribbon and containing the ribbon axis), and its eigenstates will thus present a well-defined parity with respect to this symmetry plane (see Figs. \[FIG3\]b and \[FIG3\]e). The eigenstates of the doped ribbon keep the same parity with respect to this mirror plane provided that the potential induced by the dopant preserves this symmetry [@5; @Kim]. For the case of an odd-index aGNR, this can only occur when the dopant is located exactly at the central dimer line, as is illustrated in Figs. \[FIG3\]a and \[FIG3\]d.
{width="6.0cm"} {width="6.0cm"}
Comparing the wavefunction at the $\Gamma$ point associated to the first band below the CNP for both the ideal ribbon (Fig. \[FIG3\]b) and the doped (B at center) one (Fig. \[FIG3\]a), we observe that states around the CNP are only weakly affected by the impurity, indicating that there is no mixing with neighboring bands (which present an opposite parity). As a result, backscattering does not occur despite the presence of impurity states with energy values within the first *plateau* (Fig. \[FIG2\], dashed red arrow, and Fig. \[FIG3\]d). For any other position of the dopant, the well-defined parity of the wavefunctions will not be preserved, as is shown in Figs. \[FIG3\]c and \[FIG3\]f for the B off-center case (red atom (B) in Fig. \[FIG1\]). In this case, coupling between all states restores backscattering efficiency, yielding a full suppression of the single available conduction channel at a certain resonance energy determined by the precise location of the dopant (red curve (B) in Fig. \[FIG1\]).
In comparison with B-doping, the impact of N impurities on the ribbon transport properties manifests in a close symmetric fashion with respect to the CNP. This is illustrated in the insets of Fig. \[FIG1\], where the conductances for a 20-aGNR with N off-center (B) and at edge (E) are shown. The same symmetry considerations aforementioned also apply in the case of odd-index N-doped aGNRs.
We consider now the case of doped zGNRs, and present two different systems: namely a 12-zGNR ($\approx$ 2.4 nm width) and a 20-zGNR ($\approx$ 4.1 nm width). zGNRs are known to display very peculiar electronic properties, with wavefunctions sharply localized along the GNRs edges at low energies, which significantly affect their transport properties [@White-transport]. Simple tight-binding models found that zGNRs are always metallic with the presence of sharply localized edge states in the vicinity of the Fermi level [@Ribbonpure], whereas spin-dependent *ab initio* calculations report on a small bandgap opening up [@Louie]. Here spin is neglected, and we focus on the effect of a boron defect on the transport properties.
Fig. \[FIG4\] presents the conductance for B-doped 12-zGNR (left) and 20-zGNR (right). In contrast to aGNRs, where the acceptor character of boron is maintained regardless of the position of the impurity, B-doped zGNRs exhibit both acceptor and, more unexpectedly, donor character when the dopant is placed either at the center or at the edges, respectively. This effect may be related to the competition between two different phenomena, namely the Coulomb interaction of charge carriers with the ion impurity and correlation between charges at the edges. Our results for the narrower 12-zGNR B-doped ribbon confirm such prior prediction [@Ndoped-zz] for N-doped zGNRs [@NOTE2]. However, for the wider 20-zGNR, a defect located at the edge has almost no impact on the conduction efficiency in the *plateau* around the CNP. This is in striking contrast to the results for the aGNRs, where doping has its maximum effect when the impurity is placed at the edge. This suggests that modification of the electronic properties of zGNRs solely by means of edge doping or functionalization might not be significant on zGNRs wider than a few nanometers. On the other hand, the geometry of symmetric (even index) zGNRs does not allow a symmetry axis going through a single impurity [@Wang]. As a result, the suppression of backscattering observed in the case of aGNRs cannot take place here.
In conclusion, *ab initio* calculations of charge transport in boron and nitrogen doped GNRs with a width of up to 4.2 nm have been reported. Doping effects depending on the ribbon symmetry and width were unveiled, such as a full suppression of backscattering for symmetry-preserving impurity potentials in armchair ribbons, and upward energy shift for the quasibound state resonances in both armchair and zigzag ribbons. Such predictions could be experimentally confirmed, following recent progress achieved on the STM exploration of graphene edges [@Enoki] and on boron-doped oriented pyrolytic graphite [@EnokiB]. Finally, chemical doping could be used to enlarge the bandgap of a fixed GNR width, resulting in the enhancement of device performances [@Bielnp].
This work was financially supported by the GRAPHENE project of CARNOT Institute (LETI), the European GRAND project (ICT/FET) and the ANR-06-NANO-069-02 ACCENT. The authors are indebted to Ch. Adessi and Y.-M. Niquet for technical support on TABLIER and CEA/TB\_Sim *ab initio* transport codes. We thank the CEA/CCRT supercomputing facilities for computational resources. Discussions with A. Cresti are acknowledged.
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abstract: 'Because of spin-orbit interaction, an electrical current is accompanied by a spin current resulting in spin accumulation near the sample edges. Due again to spin-orbit interaction this causes a small decrease of the sample resistance. An applied magnetic field will destroy the edge spin polarization leading to a positive magnetoresistance. This effect provides means to study spin accumulation by electrical measurements. The origin and the general properties of the phenomenological equations describing coupling between charge and spin currents are also discussed.'
author:
- 'M. I. Dyakonov'
title: Magnetoresistance due to edge spin accumulation
---
It was predicted a long time ago [@dyakonov1; @dyakonov2] that because of spin-orbit interaction electrical and spin currents are interconnected: an electrical current produces a transverse spin current and [*vice versa*]{}. In recent years this has become a subject of considerable interest.
The purpose of this Letter is twofold. First, another way of understanding this interconnection will be presented and some general properties of the resulting phenomenological equations will be discussed. Second, a new magnetoresistance effect will be considered, which allows to study the current-induced spin accumulation near the sample edges by purely electric measurements.
The transport phenomena related to coupling of the spin and charge currents can be described phenomenologicaly in the following simple way. Let be the electron flow density and let $\text{\boldmath$q$}^{(0)}$ be its conventional expression not accounting for spin-orbit interaction: $$\text{\boldmath$q^{(0)}$} = -\mu n \text{\boldmath$E$} - D\nabla n, \eqno{(1)}$$ where $\mu$ and $D$ are the usual electron mobility and diffusion coefficient, connected by the Einstein relation, is the electric field, and $n$ is the electron concentration. The electric current density is $\text{\boldmath$j$}=-e\text{\boldmath$q$}$, where $e$ is the absolute value of the electron charge.
Let $q_{ij}$ be the spin polarization current density tensor (the flow of the $j$ component of the spin polarization in the direction $i$). It should be understood that for polarized electrons the spin current may exist even in the absence of spin-orbit interaction, simply because spins are carried by electron flow. We denote the corresponding quantity as $q_{ij}^{(0)}$. Then, similar to Eq. (1), we have $$q_{ij}^{(0)}=-\mu E_iP_j - D\frac {\partial P_j}{\partial x_i}, \eqno{(2)}$$ where is the vector of electron spin polarization density. If there are other sources for currents, like for example a temperature gradient, the corresponding terms should be included in Eqs. (1) and (2).
We have departed from the conventional definitions [@dyakonov1; @dyakonov2] by introducing the vector of spin [*polarization*]{} density and the spin [*polarization*]{} current $q_{ij}$. This allows to avoid numerous factors $1/2$ and $2$ in the formulas to follow. One can return to the traditional notations by putting $\text{\boldmath$P$}=2\text{\boldmath$S$}$, where is the spin density, and by replacing $q_{ij}$ by $q_{ij}/2$ to obtain the true spin current density.
Spin-orbit interaction couples the two currents. For a material with inversion symmetry [@rem1] we have: $$q_i=q_{i}^{(0)}+\gamma \epsilon _{ijk}q_{jk}^{(0)}, \eqno{(3)}$$ $$q_{ij}=q_{jk}^{(0)}-\gamma \epsilon _{ijk}q_{k}^{(0)}, \eqno{(4)}$$ where $\epsilon _{ijk}$ is the unit antisymmetric tensor and $\gamma$ is a dimensionless coupling constant proportional to the spin-orbit interaction, it is assumed that $\gamma<<1$. The difference in signs in Eqs. (3) and (4) is consistent with the Onsager relations and is due to the different properties of and $q_{ij}$ with respect to time inversion [@rem2].
Explicit phenomenological expressions for the two currents follow from Eqs. (1)-(4): $$\text{\boldmath$j$}/e = \mu n \text{\boldmath$E$} +D\nabla n+
\beta \text{\boldmath$E$}\wedge \text{\boldmath$P$} +
\delta \,{\rm curl}\,\text{\boldmath$P$},\eqno{(5)}$$ $$q_{ij}=-\mu E_iP_j - D\frac {\partial P_j}{\partial x_i}+\epsilon _{ijk}(\beta nE_k +
\delta \frac {\partial n}{\partial x_k}).\eqno{(6)}$$ Here $$\beta = \gamma \mu,\qquad \delta = \gamma D,\eqno{(7)}$$ so that the coefficients $\beta$ and $\delta$, similar to $\mu$ and $D$, satisfy the Einstein relation.
Eqs. (5) and (6) should be complemented by the equation for the spin polarization vector: $$\frac {\partial P_j}{\partial t}+\frac {\partial q_{ij}}{\partial x_i}+
(\text{\boldmath$\Omega$}\wedge \text{\boldmath$P$})_j+{\frac {P_j}{\tau_s}}=0,
\eqno{(8)}$$ where the vector is directed along the applied magnetic field, $\Omega$ being the spin precession frequency and $\tau _s$ is the spin relaxation time. In Eqs. (6), (7) we ignore the action of magnetic field on the particle dynamics. This is justified if $\omega_c \tau<<1$, where $\omega_c$ is the cyclotron frequency and $\tau$ is the momentum relaxation time. Since normally $\tau_s >>\tau$, it is possible to have both $\Omega \tau_s>>1$ and $\omega_c \tau<<1$ in a certain interval of magnetic fields. It is also assumed that the equilibrium spin polarization in the applied magnetic field is negligible.
While Eqs. (5)-(8) are written for a three-dimensional sample, they are equally applicable to the 2D case, with obvious modifications: the electric field, space gradients, and all currents (but not the spin polarization vector) should have components in the 2D plane only.
In the equilibrium situation all currents should obviously vanish. If an inhomogeneous magnetic field exists, the equilibrium spin polarization will be space-dependent, however this by itself should produce neither spin, nor charge currents. To assure this, an additional counter-term should be introduced into the right-hand side of Eq. (2), proportional to ${\partial B_j}/{\partial x_i}$, which takes care of the force acting on the electron with a given spin in an inhomogeneous magnetic field () (see [@liu]). Corresponding terms will appear in Eqs. (5), (6). We ignore these terms assuming that is homogeneous.
Equations (5)-(8), which appeared for the first time in Refs. [@dyakonov1; @dyakonov2] describe all the physical consequences of spin-charge current coupling [@rem3]. The term $\beta \text{\boldmath$E$}\wedge \text{\boldmath$P$}$ describes the anomalous Hall effect [@karplus], where the spin polarization plays the role of the magnetic field.
The term $\delta \,{\rm curl}\,\text{\boldmath$P$}$ describes an electrical current induced by an inhomogeneous spin density (now referred to as the Inverse Spin Hall Effect). A way to measure this current under the conditions of optical spin orientation was proposed in [@averkiev]. The circularly polarized exciting light is absorbed in a thin layer near the surface of the sample. As a consequence, the photo-created electron spin density is inhomogeneous, however $\,{\rm curl}\,\text{\boldmath$P$}=0$, since both and its gradient are perpendicular to the surface. By applying a magnetic field parallel to the surface one can create a parallel component of , thus inducing a non-zero $\,{\rm curl}\,\text{\boldmath$P$}$ and the corresponding surface electric current (or voltage). This effect was found experimentally for the first time by Bakun [*et al*]{} [@bakun].
The term $\beta n\epsilon _{ijk}E_k$ (and its diffusive counterpart $\delta \epsilon _{ijk}
{\partial n}/{\partial x_k}$) in Eq. (6), describes what is now called the Spin Hall Effect: an electrical current induces a transverse spin current, resulting in spin accumulation near the sample boundaries [@dyakonov1; @dyakonov2]. This phenomenon was observed experimentally only in recent years [@kato; @wunderlich] and has attracted widespread interest.
It should be stressed that all these phenomena are closely related and have their common origin in the coupling between spin and charge currents given by Eqs. (3) and (4). Any mechanism that produces the anomalous Hall effect will also lead to the spin Hall effect and [*vice versa*]{}. It is remarkable that there is a single dimensionless parameter, $\gamma$, that governs the resulting physics. The calculation of this parameter should be the objective of a microscopic theory. For the case, when the coupling is due to spin asymmetry in electron scattering, this was done in Ref. [@dyakonov2], where $\beta$ and $\delta$ were expressed through the scattering amplitude. In this case $\gamma$ depends only on the form of the scattering potential, the electron energy, and the strength of spin-orbit interaction.
An “intrinsic” mechanism of the spin Hall effect, related only to spin band splitting, was proposed for bulk holes in the valence band [@murakami]. The value of $\gamma$ is on the order of $(k_F\ell)^{-1}$ ($k_F$ is the Fermi wavevector, $\ell$ is the mean free path) and generally depends on the details of the scattering mechanism. The current consensus is that the intrinsic mechanism may exist for any type of spin band splitting, [*except*]{} if it is linear in $k$ [@rem4].
Note that the $J=3/2$ holes may not be described by the simple Eqs. (5)-(8), because for higher spins the number of coupled macroscopic quantities increases compared to spin 1/2 particles. The mutual transformation of spin and charge currents for holes, due to scattering, was studied in [@khaetskii]. Also, even in the absence of spin-orbit interaction, holes are still particles with internal angular momentum $L=1$ and the splitting into light and heavy holes still exists. Thus, for the case of holes the spin-orbit interaction is not of primary importance.
We now discuss a new related phenomenon: a magnetoresistance due specifically to spin accumulation near the sample edges. Since the accumulation occurs on the scale of the spin diffusion length $L_s=\sqrt{D\tau_s}$ (the “spin layer” [@dyakonov1]), the proposed effect depends on the sample size, $L$, and becomes negligible when $L>>L_s$.
Whithin the spin layer the $z$ component of spin polarization changes in the direction perpendicular to the sample boundary (the $y$ direction). Thus $\,{\rm curl}\,\text{\boldmath$P$}\neq 0$, and according to Eq. (5) a correction to the electric current should exist. As we will see, this correction is positive, i.e. it leads to a slight decrease of the sample resistance compared to the (hypothetical) case when spin-orbit interaction is absent. By applying a magnetic field in the $xy$ plane, we can destroy the spin polarization (the Hanle effect) and thus observe a positive magnetoresistance on a field scale corresponding to $\Omega \tau_s \sim 1$. One might say that this is a manifestation of combined direct and inverse spin Hall effects, and the Hanle effect.
We will consider a 2D sample (see Fig. 1), a similar effect will exist also for a thin wire. The advantage of the 2D case is that the small effect considered here will not be masked by the normal magnetoresistance, because the magnetic field parallel to the 2D plane acts on the spins only, but not on the electron orbital motion. Since the spin polarization is proportional to the electric field, we discard nonlinear in $E$ terms proportional to $EP$.
For the geometry of Fig. 1, from Eq. (5) we obtain:$$j=e(\mu n E + \delta {\frac {dP_z}{dy}}),
\eqno{(9)}$$ so that the total current is $$I=\int_{-L/2}^{L/2}j(y)dy=I_0+\Delta I,\eqno{(10)}$$ where $$I_0=e\mu n EL,\quad \Delta I=e\delta(P_z(L/2)-P_z(-L/2)). \eqno{(11)}$$ The correction to the current, $\Delta I$, is proportional to the difference in spin polarization at the opposite edges of the sample. Eq. (6) yealds: $$q_{yz}=-D{\frac {dP_z}{dy}}+\beta nE,\qquad q_{yy}=-D{\frac {dP_y}{dy}}.\eqno{(12)}$$ In the steady state Eq. (8) gives: $$D{\frac {d^2P_z}{dy^2}}-\Omega P_y ={\frac {P_z}{\tau_s}},\quad D{\frac
{d^2P_y}{dy^2}}+\Omega P_z ={\frac {P_y}{\tau_s}}.\eqno{(13)}$$ These equations should be solved with the boundary conditions at $y=\pm L/2$: $${\frac {dP_z}{dy}}={\frac {\beta nE}{D}}, \qquad {\frac {dP_y}{dy}}=0, \eqno{(14)}$$ corresponding to vanishing spin currents $q_{yz}$ and $q_{yy}$ at the sample edges.
.
A straightforward calculation gives the result: $${\frac {\Delta R}{R_0}}=-{\frac {\Delta I}{I_0}}= -\gamma ^2\text {Re}\Bigl
[{\frac {\tanh (\kappa \lambda)}{\kappa \lambda}}\Bigr ], \eqno{(15)}$$ where $R_0$ is the uncorrected sample resistance, $\Delta R$ is the field-dependent negative correction due to spin accumulation, $$\kappa = (1- ix)^{1/2}, \qquad x=\Omega \tau_s, \qquad \lambda = L/(2L_s),$$ and $L_s=(D\tau_s)^{1/2}$ is the spin diffusion length.
Thus $\Delta R$ is proportional to the square of the dimensionless parameter $\gamma$ in Eqs. (3), (4). In deriving Eq. (15) the relations given by Eq. (7) were used.
From this result one can easily deduce two characteristic features of this effect.
1\) [*The total resistance change*]{} between its zero-field value, $R(0)$ and its value at strong enough field ($\Omega \tau_s>>1)$, $R(\infty)=R_0$: $${\frac {R(\infty)-R(0)}{R(0)}}=\gamma ^2 {\frac {\tanh\lambda}{\lambda}},
\eqno{(16)}.$$ For a narrow sample, $\lambda <<1$ the overall relative change of resistance is equal to $\gamma ^2$, which gives a nice way to determine experimentally the fundamental parameter $\gamma$. For wide samples the relative change is $\gamma ^2 (2L_s/L)$.
2)[*The shape of the magnetoresistance curve*]{}. We introduce the notation $\rho (B)$ for the normalized relative magnetoresistance. Then $$\rho(B)={\frac {R(B)-R(0)}{R(\infty)-R(0)}}=1- \text {Re}\Bigl
[{\frac {\tanh (\kappa \lambda)}{\kappa \tanh \lambda}}\Bigr ], \eqno{(17)}$$
For a wide sample, $\lambda>>1$, the width of the magnetoresistance curve is determined by the condition $\Omega \tau_s \sim 1)$. In this case $$\rho(B)=1-\text {Re} \Bigl ({\frac {1}{\kappa}} \Bigr )=1-\Bigl [{\frac {1+\sqrt{1+x^2}}
{2(1+x^2)}} \Bigr ]^{1/2}.\eqno{(18)}$$ Note that at $x=\Omega \tau_s >>1$ the function $\rho(B)$ approaches its maximum value very slowly, as $1-1/\sqrt {2x}$.
Figure 2 presents numerical results for $\rho(B)$ calculated from Eq. (17) for different values of $\lambda = L/(2L_s)$ together with the curve given by Eq. (18) for $\lambda>>1$, which is a good approximation already for $\lambda=1.5$.
For narrow samples ($\lambda<<1$) the magnetic field dependence becomes much weaker. The reason is that along with the spin relaxation time $\tau_s$, there is another characteristic time, $\tau_d=\tau_s\lambda^2 =L^2/(4D)$, which is the time of diffusion on a distance $L/2$. For narrow samples ($\tau_d<\tau_s$), it is this time, rather than $\tau_s$, that determines the width of the Hanle curve, because the spin polarization is destroyed by diffusion faster than by spin relaxation. Accordingly, the width of the magnetoresistance curve will now correspond to $\Omega \tau_d \sim 1$, i.e. it will be $1/\lambda^2$ times broader compared to the case of a wide sample.
To unify the two limiting cases, in Fig. 3 we re-plot $\rho(B)$ as a function of the parameter $\Omega \tau^*$, where $\tau^*$ is the effective time during which the spin is destroyed because of the combined effect of spin diffusion and spin relaxation: $${\frac {1}{\tau^*}}= {\frac {1}{\tau_s}}+{\frac {1}{\tau_d}}={\frac {1}{\tau_s}}(1+
{\frac {1}{\lambda^2}}).$$ Figure 3 shows that as a function of this parameter there is a quasi-universal curve, since the results for the limiting cases of a narrow and a wide samples are very close. Thus Eq. (18) can serve as a good interpolation formula for the general case, provided the variable $x$ is replaced by $\Omega \tau^*$, instead of $\Omega \tau_s$. The high-field limit is always approached as $1/\sqrt{B}$.
The above results for magnetoresistance are similar to those obtained previously [@dyakonov3] for the Hanle effect in the case when spin polarization is inhomogeneous and spin diffusion is important.
From the experimental results for 3D [@kato] and 2D [@liu] GaAs one can estimate $\gamma \sim 10^{-2}$. Kimura [*et al*]{} [@kimura] find $\gamma =3.7\cdot 10^{-3}$ for Platinum at room temperature, so that in these two cases a magnetoresistance due to spin accumulation on the order of $10^{-4}$ and $10^{-5}$, respectively, can be expected. The characteristic feature, allowing to identify this effect, is the specific form of the magnetoresistance curve, as well as its strong dependence on the sample width, when it becomes comparable to the spin diffusion length.
Because of the high precision of electrical measurements, magnetoresistance might provide a useful tool for studying the spin-charge interplay in semiconductors and metals.
M.I. Dyakonov and V.I. Perel, JETP Lett. [**13**]{}, 467 (1971).
M.I. Dyakonov and V.I. Perel, Phys Lett.[**A 35**]{}, 459 (1971).
In the absence of inversion symmetry there may be additional terms describing this coupling. In particular, there is a spin current induced by a non-equilibrium spin polarization and a uniform spin polarization generated by electric current.
A simple way to verify these equations is to start with currents $\text{\boldmath$q$}^{\pm}$ for particles with spin-up and spin-down (with respect to the $z$ axis), assuming that the electric field and the concentration gradient are along $x$: $q_{x}^{\pm} = -\mu n_{\pm} E - D{\partial n_{\pm}}/{\partial x}$. (Spin-up and spin-down mobilities are taken equal, which is true when the spin polarization is small). Because of spin-orbit interaction these currents will induce currents of opposite signs in the $y$-direction: $q_{y}^{\pm}= \mp \gamma q_{x}^{\pm}$. It can now be seen that the expressions for $q_y = q_{y}^{+}+q_{y}^{-}$ and $q_{yz}=q_{y}^{+}-q_{y}^{-}$ coincide with what is given by Eqs. (1)-(4) with $P_z(x)=n_+ - n_-$. If, instead of the spin [*polarization*]{} current, the normal spin current were used, we would have to replace the coefficient $\gamma$ by $2\gamma$ in Eq.(3) and by $\gamma /2$ in Eq.(4).
B. Liu, J. Shi, W. Wang, H. Zhao, D. Li, S. Zhang, Q. Xue and D. Chen, [*Experimental observation of the Inverse Spin Hall Effect at room temperature*]{}, arXiv:cond-mat/0610150 (2006).
In [@dyakonov1; @dyakonov2] these equations were written in terms of spin density . Thus the definitions of the coefficients in Eqs. (5), (6), and the relations between them, were different (by numerical factors) from those adopted here.
R. Karplus and J.M. Luttinger, Phys. Rev. [**95**]{}, 1154 (1954).
N.S. Averkiev and M.I. Dyakonov, Sov. Phys. Semicond. [**17**]{}, 393 (1983).
A.A. Bakun, B.P. Zakharchenya, A.A. Rogachev, M.N. Tkachuk, and V.G. Fleisher, Sov. Phys. JETP Lett. [**40**]{}, 1293 (1984).
Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awschalom, Science [**306**]{}, 1910 (2004).
J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. [**94**]{}, 047204 (2005).
S. Murakami, N. Nagaosa, and S.-C. Zhang, Science, [**301**]{}, 1348 (2003).
However, it does not seem reasonable to introduce “extrinsic” and “intrinsic” spin Hall effects as distinct physical phenomena. The phenomenon is always the same: current induced edge spin accumulation, while certainly there may be different microscopic mechanisms for spin-charge current coupling.
M.I. Dyakonov and A.V. Khaetskii, Sov. Phys. JETP, [**59**]{}, 1072 (1984).
M.I. Dyakonov and V.I. Perel, Sov. Phys.-Semicond. [**10**]{}, 208 (1976).
T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. [**98**]{}, 156601 (2007).
|
---
abstract: 'Recent breakthroughs in the field of numerical relativity have led to dramatic progress in understanding the predictions of General Relativity for the dynamical interactions of two black holes in the regime of very strong gravitational fields. Such black-hole binaries are important astrophysical systems and are a key target of current and developing gravitational-wave detectors. The waveform signature of strong gravitational radiation emitted as the black holes fall together and merge provides a clear observable record of the process. After decades of slow progress, these mergers and the gravitational-wave signals they generate can now be routinely calculated using the methods of numerical relativity. We review recent advances in understanding the predicted physics of events and the consequent radiation, and discuss some of the impacts this new knowledge is having in various areas of astrophysics.'
author:
- 'Joan Centrella$^1$, John G. Baker$^2$, Bernard J. Kelly$^3$, and James R. van Meter$^4$'
title: 'The Final Merger of Black-Hole Binaries'
---
Black Holes, Gravitational Waves, Numerical Relativity
Introduction {#sec:intro}
============
### Black Holes {#sssec:intro_bhs}
It has been nearly a century since Albert Einstein’s profound physical insight revealed our current standard model of gravitational physics, General Relativity. Among the theory’s extraordinary consequences was the predicted existence of black holes, nonlinearly self-gravitating, stable objects in which gravitational forces have completely overcome all other physical interactions. Though once considered a mathematical curiosity, General relativity’s description of black holes now provides the best explanation for a widespread class of astrophysical objects whose gravitational potential wells power many of the most energetic astronomical phenomena observed. These range from stellar black-holes powering X-ray sources in the neighboring regions of our galaxy, to supermassive monsters with masses $\sim (10^6 - 10^9){M_{\odot}}$, where ${M_{\odot}}$ is the mass of the Sun, with far-reaching astrophysical consequences.
### Gravitational Wave Observations {#sssec:intro_gws}
In the coming decade, anticipated observations of gravitational waves from black hole binaries will open a new window onto the universe. Interpreting such observations will richly engage aspects of our theoretical understanding of strong-field gravity which have never before been confronted with empirical observations.
The final coalescence of binaries consisting of two comparable-mass black-holes, with mass ratios $q = M_1/M_2 \sim (1-10)$, is expected to be one of the strongest astrophysical sources of energy the form of gravitational radiation. As the emission of gravitational waves removes energy and momentum from the binary, the orbits shrink and the black holes eventually merge together into a single black hole, producing an intense burst of radiation. With the advent of ground-based gravitational wave detectors such as LIGO and VIRGO (which will detect mergers of black holes with masses in the range $\sim (10 - 100){M_{\odot}}$), and with planning underway for the space-based LISA (which will observe mergers of massive black holes, with masses $\sim (10^4 - 10^6){M_{\odot}}$), knowledge of black-hole binary gravitational waveforms is urgent.
### Black-Hole Binary Coalescence {#sssec:intro_bhbmergers}
Black-hole binary coalescence can be thought of as proceeding in three stages. During the inspiral the holes have wide enough separations that they can be treated as point particles. In this stage, the orbital period is much shorter than the timescale on which the orbital parameters change, and the holes spiral together on quasi-circular orbits. When the holes get so close together that they can no longer be approximated as point particles, they enter the merger phase. In this strong-field, dynamical regime, the black holes plunge together and merge into a single, highly distorted black hole, surrounded by a common horizon. This remnant black hole then “rings down,” shedding its nonaxisymmetric modes through gravitational wave emission and settling down into a quiescent rotating black hole.
### Gravitational Waves from the Coalescence {#sssec:intro_mergerwfs}
Knowledge of the gravitational waveforms from these three stages of black hole coalescence is important for gravitational wave detection and data analysis, as well as astrophysical applications. The inspiral can be calculated analytically using the post-Newtonian (PN) approximation, which is an expansion of the full equations of General Relativity in powers of $\epsilon \sim v^2/c^2 \sim G M/(Rc^2)$, where $v$ is the characteristic velocity of the source, $M$ is its mass, and $R$ is its characteristic size. The inspiral gravitational waveform is a “chirp”, which is a sinusoid increasing in both frequency and amplitude. The ringdown can also be calculated analytically using techniques of black-hole perturbation theory, and the resulting gravitational waveforms are damped sinusoids. However, the merger stage can only be understood using full numerical simulations of the Einstein equations in 3-D, and the resulting gravitational waveforms were unknown – until recently.
The final merger of comparable-mass black holes is a powerful source of gravitational radiation. The gravitational waveforms emitted by a black-hole binary with mass ratio $q$ scale with the total mass $M$ of the binary. Setting $c=1$ and $G=1$, we can express both length and time scales in terms of the total mass: $1 M \sim 5 \times 10^{-6} (M/{M_{\odot}}) {\rm sec}
\sim 1.5 (M/{M_{\odot}}) {\rm km}$. The strong-field merger will produce a burst of gravitational radiation lasting $\sim 100M$ and having a luminosity $\sim 10^{23}L_{\odot}$, which is greater than the combined luminosities of all the stars in the visible universe. For stellar black-hole binaries ($M \sim 10{M_{\odot}}$) this luminosity will last for $\sim 5$ ms, and for massive black-hole binaries ($M \sim 10^6 {M_{\odot}}$), for $\sim 10$ min.
### Calculating the Merger {#sssec:intro_mergercalc}
Understanding the final merger of a black-hole binary requires solving the full Einstein equations using the methods of numerical relativity. This endeavor has proved to have many challenging aspects, ranging from providing astrophysically relevant initial data to understanding the structure of the Einstein equations, and the solution eluded researchers for many years. Recently, a series of dramatic breakthroughs has ignited the field, making robust, stable, and accurate simulations of binary mergers possible for the first time. These models are opening our understanding of strong-field dynamics, impacting gravitational wave searches and other areas of astrophysics. In this article we provide an overview of these exciting developments, highlighting the key physical results that are emerging.
Steps toward a robust black-hole binary model {#sec:bhbmodel}
=============================================
Computing the strong-field merger of two comparable-mass black holes has a long history, with the first attempt dating back more than 40 years. Overall, progress was generally slow and incremental, requiring the interplay among new ideas in black hole modeling, mathematical investigations into the structure of the Einstein equations, the development of effective gauge conditions, advances in computational techniques, and the availability of high performance computing resources. Here we provide a brief review of key developments in this arena.
Overview of Numerical Relativity Issues {#ssec:bhbmodel_nr}
---------------------------------------
Solving Einstein’s equations on a computer typically requires slicing 4-D spacetime into a stack of 3-D spacelike hypersurfaces, each labeled by time $t$ [@Arnowitt:1962hi; @Misner73; @Alcubierre08]. The Einstein equations then divide into two sets: constraint and evolution equations. The constraints are a set of time-independent elliptical equations that must hold on each slice. In particular, the constraints are solved first to obtain valid initial data for a black-hole binary simulation. This data is then propagated forward in time using the evolution equations.
### Gauge Freedom {#sssec:bhbmodel_nr_gauge}
General Relativity has four spacetime coordinate degrees of freedom, which are associated with four freely-specifiable coordinate or gauge functions that govern the future development of both time and the spatial coordinates. These are generally taken to be the lapse function $\alpha$, which gives the lapse of proper time $(\alpha\,\delta t)$ between neighboring slices, and the shift vector $\beta^i$, which determines how the spatial coordinates develop from one slice to the next. Appropriate choices for the lapse and shift have proven to be crucial for successful black hole evolutions [@Smarr:1977uf; @Alcubierre:2002kk].
### Formulation of the Evolution Equations {#sssec:bhbmodel_nr_formulation}
For many years, a primary challenge of numerical relativity was simply to decide which equations to solve [@Alcubierre08]. There is no unique formulation of the Einstein equations; rather there are many choices regarding, for example, which variables to use, whether to write the equations as first-order or second-order in time, and which coordinate conditions to impose. These choices are not arbitrary because some formulations turn out to be more “numerically friendly” than others. That is, although they are analytically equivalent when constraints and auxiliary variable definitions are assumed exact, some formulations may be unstable in practice. For example,the evolution equations may admit rapidly growing solutions which violate the constraints. In this case, although the initial data may contain only tiny errors, the subsequent evolution may produce violations of Einstein’s equations which “blow-up”. There can also be pathological numerical solutions that don’t represent solutions to the analytic evolution equations at all, but are supported by the discrete numerical grid structure and depend on the details of the finite-differencing operators.
### Numerical Methods {#ssec:bhbmodel_nr_methods}
Once a formulation has been chosen, the Einstein equations are solved numerically for various field variables on a grid of discrete points that represents the spacetime domain of interest. There are two general approaches for dealing with the spatial derivatives that appear in these equations. Finite differencing interpolates the derivative at a given point from the surrounding points according to a Taylor expansion in the grid spacing. Spectral methods assume a solution in the form of a summation of orthogonal functions; once the coefficients are obtained numerically, the derivatives can be found analytically. In both cases, time integration can be handled in a number of ways, most commonly by a Runge-Kutta algorithm. To date, most numerical relativity solutions have been carried out using finite differencing, with most current work employing a Cartesian grid in three spatial dimensions, although results from evolutions using spectral methods are now becoming available.
### initial conditions for black holes {#ssec:bhbmodel_nr_id}
To model astrophysical binaries, initial data must be generated for two black holes moving on quasicircular inspiralling orbits and having a mass-ratio $q$ and some configuration of spins. One usually conceptualizes the initial state of the system with a particle-like description of the masses, positions, momenta, and spins of each black hole. Since general relativity is a field theory, such a description can only be seen as a first step. The model requires a full description for the initial field configuration which satisfies General Relativity’s four initial value constraints, and which somehow corresponds to the system we have described in these particle-like terms.
### Evolving the Binary {#sssec:bhbmodel_nr_mesh}
This system is evolved for several orbits and then through plunge, merger, and ringdown, for a total duration on the order of several hundred $M$ or more. To obtain the gravitational waveforms, the radiation must be extracted from the simulation at large enough distances from the source to be in the “wave zone”. Since the length scales of the black holes are $\sim M$ and the wavelengths of the emitted radiation are $ \sim (10-100)M$, it is clear that some sort of variable resolution such as adaptive mesh refinement must be used to handle the very large computational domains needed.
The Lazarus Approach: Hybrid Black Hole Merger Waveforms {#ssec:bhbmodel_lazarus}
--------------------------------------------------------
By the late 1990’s numerical relativity had developed to the point that brief simulations of black-hole binaries in three spatial dimensions plus time were possible. These techniques were sufficient for evolving promptly merging “grazing collisions” of black holes [@Bruegmann:1997uc; @Alcubierre:2000ke]. However, the simulations were not indefinitely stable, but rather would typically crash after $\sim (10-30)M$, well before any significant portion of a binary orbit could be evolved.
### Hybrid simulations {#ssec:bhbmodel_lazarus_hybrid}
In this arena, the Lazarus method emerged as a novel approach to obtaining black-hole binary waveforms, combining short numerical relativity simulations with black hole perturbation methods [@Baker:2000zm; @Baker:2001sf]. Since numerical relativity codes were then only able to evolve for a brief period, and since perturbation theory could approximate the late time dynamics of the distorted remnant black hole, the Lazarus Project aimed to apply numerical relativity to evolve the strong-field approach to merger, thereby providing a hybrid model for a significant part of the problem. Starting from quasi-circular initial configurations near the start of the final merger, the black holes were evolved using numerical relativity for $\sim 15M$ until just before the simulation became inaccurate. Then, via a complicated interface, data from late in the numerical simulations was interpreted as initial data describing the perturbed final black hole and the emerging radiation. Finally, black hole perturbation theory techniques were applied to evolve this data and derive the full waveforms.
### First Results {#sssec:bhbmodel_lazarus_results}
The Lazarus simulations gave the first indication of what might be expected for the terminal burst of radiation from coalescing black-hole binaries. Figure \[Fig1\_LazarusWF\] shows that the dominant $\ell=2,m=2$ spin-(-2)-weighted spherical harmonic component has a brief burst of radiation smoothly joining the damped sinusoidal signal of the ringdown. The waveforms were remarkable for their simplicity, with predominantly circular polarization (in the $\ell = 2, m=2$ mode), and a steady evolution of polarization frequency and amplitude. Since these characteristics were robust under variations in the model, it was conjectured that the waveforms from a binary starting from a wide separation late in the inspiral would be well characterized by these simple features. Subsequently, the Lazarus method was applied to study mergers of more generic black-hole binaries [@Baker:2003ds; @Campanelli:2004zw].
Toward Evolving a Black-Hole Binary Orbit {#ssec:bhbmodel_evol}
-----------------------------------------
### The BSSN system {#sssec:bhbmodel_evol_bssn}
Concurrently with the Lazarus investigations, further progress was being made toward full numerical relativity simulations of black-hole binary mergers. One major milestone was the development of a conformal formulation of the Einstein equations known as the Baumgarte-Shapiro-Shibata-Nakamura or “BSSN” system [@Shibata:1995we; @Baumgarte:1998te], that overcame some instability problems associated with an earlier formulation in use at the time [@Arnowitt:1962hi; @York79]. In this approach, the set of evolution equations is written with first-order time derivatives and second-order spatial derivatives, and is strongly hyperbolic [@Nagy:2004td; @Reula:2004xd]. Stable time evolution was accomplished using a coordinate condition that evolves the lapse function dynamically, causing the slices to avoid crashing into the black hole singularities [@Anninos:1995am; @Alcubierre:2000ke]. However the simulations were still limited to durations $\lesssim (30-40)M$ of stable evolution, by a failure of the spatial coordinate system known as “grid stretching,” in which the coordinates tend to fall into the black holes, and by instabilities related to how the black holes were handled numerically.
### Controlling the spatial coordinates {#sssec:bhbmodel_evol_spatial}
Eliminating grid-stretching required developing appropriate techniques for dynamically controlling the spatial coordinates, which are governed by the evolution of the shift vector. The first long-lasting evolutions of distorted black holes relied on a class of hyperbolic shift evolution schemes, known as $\Gamma$-freezing conditions, which were inspired by the BSSN formulation. With this approach, a single distorted black hole could be evolved indefinitely (e.g., for thousands of $M$) with reasonable accuracy [@Alcubierre:2001vm; @Alcubierre:2002iq]. These studies allowed the full numerical determination of gravitational waveforms from black hole ringdowns as the distorted black hole settled down to a physically and numerically stable final state, and provided a foundation for future advances in black-hole binary simulations [@Alcubierre:2002kk].
### Problems with moving black holes {#sssec:bhbmodel_evol_problems}
However, another aspect of spatial gauge conditions remained a critical issue for long-lasting black-hole binary simulations. As general relativity allows arbitrary coordinate systems, many groups adopted coordinate conditions which did not allow the black holes to move through the computational domain. This simplified the problem of dealing with the black hole singularities, which were handled either by excising the black hole interiors (within the horizons) from the computational domain [@Thornburg87], or by representing the black holes as punctures [@Brandt:1997tf]. Though progress had been made toward implementing moving excision regions [@Shoemaker:2003td], it was computationally much simpler to demand that the excision region remain fixed on the grid [@Alcubierre:2000yz]. Similarly, the puncture treatment was formulated so that the black hole was fixed on the grid and the singularity was factored out and handled analytically in a time-independent manner. However, for binary configurations in which the black holes physically move, the cost of keeping the black holes fixed in coordinate space was paid by the twisting and stretching of the dynamical fields, eventually leading to large computational errors.
For inspiraling binary configurations one potential solution was to attempt to untwist the geometric field data by imposing comoving coordinates using a shift vector that is dynamically adjusted during the evolution of the binary to minimize motion of the black hole horizons [@Alcubierre:2004hr]. With this approach, Brügmann [et al. ]{}achieved a significant milestone: the first simulation of a full orbital cycle of a black-hole binary system prior to merger [@Bruegmann:2003aw]. In this work, the authors applied a dynamically adjusted corotating frame tracking the measured position of the black hole apparent horizons. This allowed a simulation which remained accurate for a little longer than the $\gtrsim 100M$ duration of a complete orbit. However, problems occurred with the large characteristic speeds that resulted near the outer boundaries, limiting the domains of these simulations and preventing gravitational waveforms from being measured effectively. The coordinate control also required fine tuning which eventually failed, causing the simulation to crash as the black holes finally approached merger.
Robust black-hole binary simulations {#ssec:bhbmodel_robust}
------------------------------------
### First Orbit and Merger simulation {#sssec:bhbmodel_robust_first}
In early 2005 Pretorius shocked the numerical relativity community by announcing the first complete, robust simulations of an equal-mass black hole merger [@Pretorius:2005gq]. After completing $\sim 1$ orbit, the black holes plunged and merged to form a single distorted black hole that then rings down. Pretorius extracted the gravitational waves to obtain the first inspiraling merger waveform directly from numerical relativity, shown in Fig. \[Fig2\_PretoriusPsi4\].
Pretorius employed several techniques that were very different from most other numerical approaches to the black-hole binary problem [@Pretorius:2006tp]. Rather than using the BSSN formulation, he applied a generalized harmonic formalism [@Pretorius:2004jg], directly integrating the spacetime metric with evolution equations of second-order in both space and time. These equations were implemented numerically using adaptive mesh refinement to allow high resolution around the black holes while maintaining a large computational domain. Pretorius utilized spatial coordinates compactified to draw spatial infinity into the computational domain, with a choice of gauge-evolution strongly tied to his evolution formalism. Following [@Gundlach:2005eh], he added terms to the evolution equations to specifically damp away any violations of the constraints. In his simulations, the black holes were excised and moved freely across the computational grid. Pretorius’ success with such novel methods quickly raised questions as to whether the struggling, more widely-pursued BSSN-based puncture approach might be off course.
### Moving Punctures {#sssec:bhbmodel_robust_punctures}
Later that same year, however, a new robust method based on the BSSN formulation was announced. The “moving puncture” method was discovered simultaneously and independently by the groups at the University of Texas at Brownsville (UTB) [@Campanelli:2005dd] and NASA’s Goddard Space Flight Center (GSFC) [@Baker:2005vv]. In this approach the black holes are represented as punctures, but are not constrained to remain fixed on the coordinate grid. Rather, they are allowed to move freely through the grid using novel coordinate conditions. Figure \[Fig3\_UTBPRLtrack\] shows the trajectories of the puncture black holes as computed by the UTB group; the formation of a common horizon marks the point of merger. The moving puncture method eliminates the analytic representation of the puncture singularities in favor of an approximate numerical treatment within the black hole horizons.
The UTB and GSFC groups had discovered and applied similar methods to the same problem: evolving an equal-mass nonspinning black-hole binary from near the final orbit, through merger and ringdown, and studying the gravitational waves. The first generation of merger waveforms from Pretorius, GSFC, and UTB showed the same simple burst of radiation ending in a damped-sinusoidal ringdown, and were qualitatively consistent with the Lazarus project results discussed above and shown in Fig. \[Fig1\_LazarusWF\].
The discovery of the moving puncture method ignited the field of black-hole binary evolutions. Since it was based on commonly used methods, most researchers in the field were quickly able to achieve accurate and stable evolutions using their existing codes, with the adoption of simple coordinate conditions [@vanMeter:2006vi]. Suddenly, the game was on and nearly all the groups were participating.
The Physics of Black-Hole Binaries {#sec:bhb}
==================================
With the advent of successful numerical evolutions of binaries that inspiral and merge, the numerical relativity community’s focus changed to investigating the physics of binary mergers. This advancing frontier is enabled by continued technical improvements in areas ranging from initial data prescriptions to more accurate numerical methods.
Merger Dynamics and Waveforms {#ssec:bhb_dynamics_waveforms}
-----------------------------
The astrophysical black hole mergers that are the targets of current and future gravitational wave detectors are expected to reach the merger stage after having radiated away any initial eccentricity [@Peters:1963ux; @Peters:1964zz] and proceeding through a long quasi-circular inspiral. All equal-mass, nonspinning binary merger simulations starting from such orbits in the inspiral should produce the same gravitational waveform, subject only to rescaling with the total mass of the system. For many years, concerns had been raised about the accuracy and realism of black-hole binary initial data sets, including the effects of spurious gravitational radiation and eccentricity [@Lousto:1997ge; @Damour:2000we; @Pfeiffer:2002xz]. With numerical relativity now able to simulate the final merger, the next step was to run models with enough orbits before the plunge and merger to get complete and reproducible waveforms starting from the late inspiral.
### Equal-Mass, Nonspinning Black Holes {#sssec:equal-mass-nospin}
The GSFC group produced the first representation of the definitive waveform for the final stages of a merger of equal-mass, nonspinning black holes [@Baker:2006yw]. They carried out a series of four simulations with the holes starting from quasicircular initial conditions at increasingly larger separations. In these runs, the holes completed $\sim 1.8, 2.5, 3.6$, and $4.2$ orbits before the formation of a common horizon. To compare the results of these models, they chose the moment of peak gravitational radiation amplitude as the fiducial time $t = 0$.
The orbital dynamics of the binaries can be examined by tracking the black hole centers, given by the location of the punctures. The black hole trajectories for each run were oriented so that they superpose at the fiducial $t=0$. In the early stages of each run, the tracks clearly showed the effects of eccentricity in the initial conditions, with the amount of initial eccentricity decreasing for more widely separated holes. As the holes spiraled together deeper into the strong-field regime, the tracks locked on to a universal trajectory independent of their initial conditions that continued for the last orbit, plunge, and merger.
Figure \[Fig4\_QCNwaves\] reveals the corresponding universal gravitational waveform. Here, the dominant $\ell=2$, $m=2$ quadrupolar component for each run is shown, shifted in time so that the peak radiation amplitude occurs at $t = 0$. Starting from $t = -50 M$ the waveforms show nearly perfect agreement, differing from each other at the level of $1\%$. The signals for the preceding few orbits agree at the level of $\sim 10\%$, except for a brief burst of spurious radiation at the start of each run.
Note that the merger waveform shows a remarkably simple shape, making a smooth transition from the inspiral chirp to the damped sinusoid of the ringdown. As the merger begins, both the wave amplitude and frequency increase, albeit faster than in the inspiral. The amplitude reaches a peak and then decreases, dropping exponentially through the ringdown. The frequency increases monotonically to a maximum value that remains constant during the ringdown.
Of course, the black-hole binary merger dynamics and waveforms can be altered by the presence of large amounts of eccentricity [@Hinder:2007qu] and spurious gravitational radiation [@Bode:2007dv] near the time of the plunge. However, the robustness of the merger to modest deviations from astrophysical initial conditions opened the door to studying many cases of interest using relatively short simulations, starting just a few orbits before the start of the plunge.
### Unequal-Mass, Nonspinning Black Holes {#sssec:unequal-mass-nospin}
Astrophysical black-hole binaries are unlikely to have exactly equal masses. Currently, numerical relativists are able to evolve systems with mass ratios up to $q = 10$ [@Gonzalez:2008bi]. Starting from quasicircular orbits the simulations show that the merger phase for nonspinning, unequal mass black hole binaries is robust to modest deviations from these initial conditions and produces a generally simple waveform shape.
An important tool for analyzing these mergers is a decomposition into spin-weighted spherical harmonic modes. Berti [et al. ]{}[@Berti:2007fi] analyzed a set of unequal-mass nonspinning mergers with mass ratios ranging from $q = 1$ to $q = 4$. Studying the multipolar distribution of the radiation, they found that the sub-dominant modes ($\ell > 2$) become more important, carrying a larger fraction of the energy, as $q$ increases. Specifically, for $q > 2$, the $\ell = 3$ mode typically carries $\sim 10\%$ of the radiated energy. Also, as expected from symmetry considerations, the odd-$m$ modes are suppressed in the equal-mass limit.
Baker [et al. ]{}[@Baker:2008mj] carried out a complementary study of the radiation from nonspinning mergers with mass ratios in the range $1 \le q \le 6$. The multipolar decomposition clearly shows that the hallmark simplicity of the waveform persists for $q > 1$ and extends to each of the spherical harmonic components $\ell \ge 2$; this property has recently been shown to extend to the $q=10$ case [@Gonzalez:2008bi]. In the full mode-summed waveform, this simple shape is also seen when viewing along the system’s orbital axis, where the quadrupole mode dominates. A somewhat more complex appearance arises by viewing the system off-axis, where higher multipoles contribute more strongly to the waveform at various angles.
Throughout the entire coalescence, each of the spherical harmonic waveform components is circularly polarized, with steadily varying phase and amplitude [@Baker:2008mj]. For each mass ratio $q$, the rotational phase (and frequency) of the different $(\ell,m)$ components are the same. During the inspiral this is expected, since the waveform phase is equal to the rotational phase multiplied by the mode number $m$. However, for the $\ell = m$ modes this relationship also holds throughout the merger and into the ringdown.
These properties suggest a simple conceptual interpretation in which the radiation is generated by an “implicit rotating source.” In this picture, each $(\ell,m)$ mode is generated separately by the $(\ell,m)$ moment of some implicit source. The fixed relationship for the $\ell = m$ modes implies that these components of the source rotate rigidly through the entire coalescence. The $\ell \ne m$ components are less rigid and peel away from the main $\ell = m$ trend during the merger [@Baker:2008mj].
### Mergers of Black Holes with Spin {#sssec:unequal-mass-spin}
The mass ratio $q$ is a one-dimensional cut into the parameter space of black-hole binaries. The remaining parameter space is dominated by the spin angular momentum of each hole. As spin is a vector quantity, this adds six more dimensions. Simulations of spinning black holes first focused on systems whose spins were expected, on the basis of PN arguments, to have the least effect on the orbital motion. Binaries with aligned and anti-aligned spins are relatively easy to treat as there is no precession of the orbital plane or the individual spins. When one or both holes has a spin not parallel to the orbital axis, spin-orbit and spin-spin interactions will cause precession that can complicate the evolution and the resulting waveforms [@Apostolatos:1994mx; @Kidder:1995zr].
The first merger evolutions of spinning black holes were carried out by Campanelli [et al. ]{}[@Campanelli:2006uy]. They simulated the mergers of two equal-mass highly spinning black holes with ${(a/M)}_{1,2} = .757$, and both spin vectors either aligned or anti-aligned with the orbital angular momentum. Here, $a \le M$ is the magnitude of the black hole spin angular momentum per unit mass. They also evolved a nonspinning equal-mass binary for comparison. All three binaries had the same initial orbital angular frequency corresponding to an orbital period $\sim 125M$, and merged to form a rotating remnant black hole with ${(a/M)}_{\rm final} < 1$. However, the aligned system took noticeably more orbits to merge than the others. This behavior is caused by the spin-orbit interaction, which produces an effective force between the black holes, either an attractive or repulsive for the anti-aligned or aligned cases, respectively. All three binaries generate remarkably similar gravitational waveforms having a simple shape, with the aligned case displaying a longer wavetrain and the anti-aligned case a shorter one.
The first fully numerical evolutions of strongly precessing spinning systems were carried out by Campanelli [et al. ]{}[@Campanelli:2006fy; @Campanelli:2007ew], who observed both significant precession of the orbits and a “spin flip,” in which the final post-merger black hole spins in the direction opposite to the two initial black holes. More recent work is beginning to probe the effects of spin precession on waveforms [@Campanelli:2008nk].
Studies of this most general of (non-eccentric) parameter sets are still in the early stages. No systematic study of waveform shapes and polarizations has been carried out yet. A preliminary study of the multipolar structure of gravitational waves from several classes of binaries with equal spins has been performed by Berti [et al. ]{}[@Berti:2007nw]. They considered an equal-mass case with aligned spins; several $q = 4$ binaries with antialigned (down-down) spins; and three unequal-mass binaries with spins initially in the orbital plane and pointing in opposite directions. Examining the distribution of gravitational-wave energy emitted by various modes, they find that, as in the case of nonspinning mergers, odd-$\ell$ multipoles are suppressed for $q = 1$ and that, as $q$ increases, more energy is radiated in higher-$\ell$ multipoles.
The Spin of the Final Black Hole {#ssec:bhb_final_spin}
--------------------------------
Because the state of the final black hole formed from the coalescence depends primarily on termination of the inspiral and the late burst of radiation in the merger, it can be accurately probed with relatively short simulations of just a few orbits.
The merger of two equal-mass nonspinning black holes produces a final black hole with a moderately high spin, ${(a/M)}_{\rm final} \sim 0.69$ [@Baker:2002qf; @Pretorius:2005gq; @Campanelli:2005dd; @Baker:2005vv; @Scheel:2008rj]. Since the black holes each start out with no spin, and any tidal spin-up is negligible [@Campanelli:2006fg], the spin of this final black hole arises from the orbital angular momentum of the original binary. Simulations show that, for this simplest black hole merger, the final spin is “universal,” or independent of the initial black hole separation, for modest deviations from quasicircular initial conditions.
For mergers of nonspinning black holes with unequal masses, the spin of the final black hole decreases as the mass ratio $q$ increases. Simulations with mass ratios up to $q = 10$ show that the final spin parameter scales as ${(a/M)}_{\rm final} \sim q/(1+q)^2$, where ${(a/M)}_{\rm final} \approx 0.48$ for $q = 4$ and ${(a/M)}_{\rm final} \approx 0.26$ for $q = 10$ [@Berti:2007fi; @Baker:2008mj; @Gonzalez:2008bi].
The effects of spin-orbit and spin-spin coupling can become important in determining the spin of the final black hole. In the simplest cases, the spin vectors are parallel to the orbital angular momentum. Depending on the mass ratio and the black hole spin, the merger can result in a final black hole with a larger (spun up) or a smaller (spun down) spin than either of the progenitors [@Baker:2003ds; @Campanelli:2006uy; @Pollney:2007ss]. In certain cases, the merger can lead to a spin flip with the final black hole spinning in a direction opposite to the spins of the initial holes; in particular, it is possible to produce a final nonspinning black hole, ${(a/M)}_{\rm final} = 0$, from the merger of two spinning holes [@Buonanno:2007sv; @Berti:2007nw]. More general black-hole binaries, with misaligned spins, are further complicated with the effects of precession [@Campanelli:2006fy; @Tichy:2007hk; @Dain:2008ck].
Several attempts have been made to produce expressions for the final spin vector using analytic techniques or by fitting to results from numerical simulations [@Buonanno:2007sv; @Kesden:2008ga; @Rezzolla:2007xa; @Rezzolla:2007rd; @Rezzolla:2007rz; @Tichy:2008du; @Lousto:2009mf]; see [@Rezzolla:2008sd] for a review. Of particular interest is the question of whether a black hole merger can produce a maximally spinning hole (${(a/M)}_{\rm final} = 1$) or indeed exceed the Kerr limit (${(a/M)}_{\rm final} > 1$). Current research suggests that this is not possible, even for mergers occurring from hyperbolic encounters [@Healy:2009ir] and mergers of highly boosted black holes [@Sperhake:2009jz].
Recoil Kicks from Gravitational Radiation {#ssec:bhb_kicks}
-----------------------------------------
A notable phenomenon arising from asymmetric binary systems is the merger recoil or kick – a net movement of the end-state black hole from the system’s center of mass, caused by the anisotropic emission of gravitational radiation during the coalescence. Fitchett [@Fitchett_1983] produced a useful formula for the kick velocity, which has significant applications in astrophysics, using a quasi-Newtonian approximation. Several authors also calculated the recoil analytically using PN approximations [@Favata:2004wz; @Blanchet:2005rj; @Damour:2006tr; @Wiseman:1992dv]. However, since the dominant part of the effect builds up in the strong-field regime close to merger, full numerical relativity simulations are needed for accurate calculations of the kick.
### Kicks from nonspinning black hole mergers {#sssec:bhb_kicks_nospin}
In 2006, recoil from a fully numerical binary merger was demonstrated for the first time by Herrmann [et al. ]{}[@Herrmann:2006ks], for plunging black-hole binaries with mass ratios as large as $q\sim 3.1$. This was followed soon afterwards by a full orbit and plunge simulation by Baker [et al. ]{}[@Baker:2006vn], who found a recoil of between 86 and 97 [${\rm km s}^{-1}$]{}for a mass ratio $q=1.5$.
A more systematic study of recoil from mergers of unequal-mass binaries was produced by Gonzalez [et al. ]{}in 2007 [@Gonzalez:2006md], who studied the merger of binary systems with mass ratios between $q=1$ and $q=4$. Figure \[Fig5\_Jenakickfit\] shows the resulting range of recoil speeds, together with several earlier numerical and analytical estimates. The authors synthesized these into a single recoil formula, a nonlinear correction to the Fitchett formula, yielding a maximum recoil of 175 [${\rm km s}^{-1}$]{}for a mass ratio of $q \sim 3$. This has recently been tested for the more extreme $q=10$ mass ratio, with general agreement [@Gonzalez:2008bi].
### Spinning black hole mergers and superkicks {#sssec:bhb_kicks_spin}
Calculations of recoil in the much larger parameter space of spinning binaries began with the non-precessing cases of holes with spins aligned (or anti-aligned) with the orbital angular momentum. Several studies of this region of parameter space [@Herrmann:2007ac; @Koppitz:2007ev; @Pollney:2007ss] have revealed that the kick velocity has a quadratic dependence on initial spins, with a maximum kick of 448 [${\rm km s}^{-1}$]{}for extremal Kerr holes, $(a/M)_{1,2} = 1$. For these anti/aligned cases, as well as for nonspinning unequal-mass black-hole mergers, the direction of the kick velocity is always in the orbital plane.
Meanwhile attention turned to more general black hole spins. Campanelli [et al. ]{}[@Campanelli:2007ew] speculated from PN arguments that optimal spin configurations could give rise to huge “superkicks,” with velocities $> 1000$ [${\rm km s}^{-1}$]{}, out of the initial orbital plane. The first such superkick — around 2500 [${\rm km s}^{-1}$]{}— was soon observed by Gonzalez [et al. ]{}[@Gonzalez:2007hi]. Tichy [et al. ]{}[@Tichy:2007hk] have argued that superkicks arise in mergers with general spin orientations, while greater insight into the mechanism of these kicks has been developed [@Brugmann:2007zj; @Schnittman:2007ij].
With such a large parameter space to cover, it seems useful to try to construct a general formula that will describe kicks from arbitrarily spinning binaries. Baker [et al. ]{}[@Baker:2007gi] used new and existing aligned-spin results to produce a single unifying formula for in-plane kicks; Campanelli [et al. ]{}[@Campanelli:2007ew; @Campanelli:2007cg] proposed an extension to this model, with scaling of the superkick out-of-plane contribution motivated by PN theory. The leading dependence in this formula on the angles between spins and linear momenta of the pre-merger holes has strong support from numerical simulations [@Campanelli:2007cg; @Brugmann:2007zj; @Lousto:2007db]; however, the dominant scaling with mass ratio is still in dispute [@Baker:2008md; @Lousto:2008dn].
Longer Waveforms: Modeling the Late Inspiral {#ssec:bhb_longwfs}
--------------------------------------------
Many key features of black-hole binary interactions can be modeled usefully with only a small handful of binary orbits before merger. Quantities such as radiated energy and momentum are bulked near the merger and have been shown to be robust to the addition of one or two extra orbits (see Section \[sssec:equal-mass-nospin\]). However, optimal observational studies of gravitational waveforms ultimately require theoretical predictions for the full waveform, starting at large separations during the inspiral.
Before the breakthroughs in numerical relativity, most information about the dynamics of compact binaries came from PN theory. This approach supplied the particle trajectories, energy flux, and – most importantly for detector scientists – the gravitational waveforms themselves. However PN theory fails before the system merges, so the waveforms are necessarily incomplete. With the advent of numerical simulations encompassing many orbits, scientists finally have a way to develop complete information about the waveforms. This requires longer simulations which reveal the last part of the binary inspiral, and allow overlap with PN waveform predictions.
### Low-eccentricity initial data {#sssec:bhb_longwfs_lowecc}
Simulations for these studies require careful attention to the initial configuration of the black holes. For a significant population of astrophysical binaries, it can be expected that the orbits have circularized through gravitational-wave emission over many orbits prior to merger. Numerical simulations try to mirror this expectation by selecting initial momenta consistent with near-zero eccentricity for the relatively small separations at which a full numerical relativity simulation becomes feasible.
For equal-mass, nonspinning binaries, two basic approaches have proved effective in reducing spurious eccentricity. The more direct method is to model the observed eccentricity, and then adjust the momenta using a Newton-like step to zero it out; this approach has led to extremely low eccentricities [@Boyle:2007ft; @Scheel:2008rj]. An alternative approach, adopted by Husa [et al. ]{}[@Husa:2007rh], is to use the PN Hamiltonian equations of motion to model the early evolution of the particle trajectories, starting from large ($\sim 50 - 100M$) separations; the emission of radiation during this process naturally circularizes the orbit, and low-eccentricity momenta can be read off at the desired separation. This approach has recently been tested with spinning binaries as well [@Campanelli:2008nk].
The first long waveform was produced by the GSFC group [@Baker:2006ha; @Baker:2006kr] for equal-mass, nonspinning black holes starting $\sim 7$ orbits or $\sim 14$ gravitational wave cycles before merger. Baker [et al. ]{}estimated their numerical errors in waveform phase from [@Baker:2006kr] as a function of frequency, finding that above a certain frequency numerical errors are smaller than internal errors in the PN sequence. In addition, the numerical and PN waveform phases agree to within one radian of phase drift for a little over ten gravitational wave cycles preceding the last orbit before merger, comparable to numerical error estimates. Hannam [et al. ]{}[@Hannam:2007ik] improved on this work, with phase and amplitude comparison between their low-eccentricity higher-resolution evolutions and PN waveforms.
### Evolutions with Spectral Techniques {#sssec:bhb_longwfs_spectral}
Numerical simulation codes based on pseudospectral differencing techniques are particularly well-suited to long waveform studies. The Caltech-Cornell group adopted the constraint-damped generalized harmonic formalism used by Pretorius, an important component in developing a stable spectral evolution code [@Lindblom:2005qh; @Scheel:2006gg]. Also like Pretorius, their code handles black holes by excising the black hole interiors. Their spectral code also employs a numerical grid that tracks the black holes explicitly. Though there can be difficulties with the changing geometry as black holes merge, this approach allows efficient study of the long-lasting inspiral waveforms.
Recently this approach has provided, several of the longest and most accurate black-hole binary evolutions.[@Scheel:2008rj; @Chu:2009md] In particular, the Caltech-Cornell group have used their spectral code to simulate an equal-mass nonspinning binary starting 16 orbits and 32 gravitational wave cycles before merger: see Fig. \[Fig6\_CCWF\] [@Scheel:2008rj] These much longer waveforms have been used to validate several competing PN models [@Boyle:2007ft].
### Comparing results {#sssec:bhb_longwfs_compare}
With so many pre-merger waveform cycles now available for the equal-mass case, the results can be cross-checked by comparing the “complete” waveform – inspiral, merger and ringdown – between groups. This is important to verify expectations that differences in methodology and residual numerical effects, such as unwanted eccentricity, are unimportant. A recent effort, dubbed the “Samurai project,” analyzed long waveforms from several groups in the light of detectability criteria for the LIGO and Virgo ground-based gravitational-wave detectors. They found that the available numerical relativity waveforms are indistinguishable in these detectors for signal-to-noise ratios (SNRs) $\lesssim 25$ [@Hannam:2009hh].
### Long Waveforms For More General Black Holes {#sssec:bhb_longwfs_generic}
Hannam [et al. ]{}have investigated the properties of highly spinning orbit-aligned black holes, comparing their phase and amplitude with PN predictions over the last ten waveform cycles [@Hannam:2007wf]. In the best cases, they find phase agreement within 2.5 rad for 3.5PN, and amplitude agreement to within around 12% with restricted PN. While not at the same level as their nonspinning results [@Hannam:2007ik], this may be attributable to the lower orders of accuracy available for spinning black holes in PN theory at the present. The Caltech-Cornell group has recently conducted long-lasting simulations of black holes with spins aligned and anti-aligned with the orbital angular momentum, calibrating a tunable PN waveform model to match the results [@Chu:2009md; @Pan:2009wj]; see Sec. \[ssec:outlook\_analyticmodels\].
Generic mergers involving non-aligned, and thus precessing, spins adds four new degrees of freedom to the problem. A systematic understanding of waveforms generated by generic mergers will require considerably more study. Recent simulations have begun to explore generic examples [@Campanelli:2008nk].
Applications in Astrophysics {#sec:astrophys}
============================
We have described a sampling of the explosion of numerical relativity studies revealing some of the details of black-hole binary physics as implied by General Relativity. While more remains to be learned, this new understanding is already making important contributions in planning and interpreting astrophysical black hole observations where Einstein’s theory is applied as the standard model of gravitational physics.
Waveforms for Gravitational-Wave Observations {#ssec:astrophys_da}
---------------------------------------------
Experimental gravitational-wave detectors were first developed more than 40 years ago. Although advances in design have increased their sensitivity by many orders of magnitude, the extreme weakness of expected astrophysical signals (strain amplitudes $\delta L/L \sim 10^{-21}$) means the observational challenge is still huge.
The output of any gravitational wave detector will be a data stream that must be combed through to find real signals. This search requires accurate “template” waveforms representing our best picture of the radiation from expected sources; these templates can then be compared with the data stream through matched filtering. We refer the reader to a review on gravitational-wave astronomy for an overview of these techniques [@Camp:2004gg].
Crucially, the most important sources of gravitational radiation are expected to include the mergers of black-hole binaries. Before the advent of numerical relativity simulations of black hole mergers, the only test waveforms available for use in data analysis studies were based on PN theory. These waveforms were essentially only inspiral chirps and did not include the strong-field merger. The new, richer, information now available from numerical relativity has revolutionized the data analysis picture in several ways.
### Detecting black-hole mergers {#sssec:astrophys_da_detect}
The availability of the plunge-merger-ringdown portion of the signal can greatly increase the SNR in the detector. Armed with a long numerical merger waveform of acceptable accuracy, we can extend it backwards to cover an arbitrarily long inspiral by matching to a PN waveform. Such a “hybrid” waveform was first produced by the GSFC group for the equal-mass, nonspinning case [@Baker:2006kr]. Using this hybrid, we can investigate the total achievable SNR, and the related distance reach, for current and future detectors. Figure \[Fig7\_MergerObsALIGOSNR\] demonstrates the gain in SNR from including the merger portion of the waveform for the ground-based Advanced LIGO detector. Contours of SNR as function of redshift $z$ and total binary mass $M$ for the LISA detector are shown in Figure \[Fig8\_MergerObsLISASNR\].
Full numerical waveforms, and the longer hybrid waveforms generated from them, can also be used to improve existing data-analysis techniques and template sets. Since previously developed gravitational wave data analysis algorithms were not based on knowledge of the merger waveforms, an obvious first step is to test how successfully these techniques detect the merger signals predicted by numerical simulations. In 2009, the NINJA project [@Aylott:2009ya] used direct injection of a broad range of short and long numerical merger waveforms into mock LIGO and Virgo data streams for this purpose. The result was the most realistic testing ground to date for disparate data-analysis methods, including full and partial waveform template matching, as well as unmodeled burst searches. Further studies of detection algorithms with numerical relativity waveforms are continuing.
### Measuring black-hole binary parameters {#sssec:astrophys_da_paramest}
The SNR is only a crude guide to the specific detector response to gravitational waveforms, however. The merger portion of the waveform, though short in duration, may contain important new information not present in the inspiral. As a simple example, we expect the time of merger itself to be well localized with the full waveform, whereas it is not well-defined in the inspiral-only signal.
More generally, the different $(\ell,m)$ modes of a binary’s full inspiral-merger-ringdown waveform scale differently with the time to merger. Modes that were not significant during inspiral suddenly become more prominent in the merger, and the detailed information they carry becomes available to the observer [@Berti:2007fi; @Baker:2008mj]
After detection of gravitational waves from distant sources, we are most interested in identifying the physical parameters of the sources. Each of the seven intrinsic parameters of a black-hole binary (mass ratio and spin vectors) will, in general, be imprinted on the gravitational-wave signal, along with some extrinsic parameters such as sky position and distance to the source. For sufficiently strong SNRs, expected for massive binary mergers seen by LISA, we can expect to be able to extract some of these parameters at high precision. While these parameters can be partially disentangled using inspiral-only template information, it has recently been found that the full merger waveform can help break parameter degeneracies and hence drive down uncertainties in several important physical parameters. Good localization of the source on the sky is especially important for the development of multi-messenger astronomy. The recent parameter estimation studies of non-spinning mergers that include the merger waveforms indicate that LISA will be able to locate sky positions within a few arcminutes for binaries with $\sim 10^6{M_{\odot}}$ at cosmological distances (redshift $z=1$)[@McWilliams:2009bg].
Consequences of Merger Recoil {#ssec:astrophys_recoils}
-----------------------------
As previously discussed, asymmetries in a black-hole binary system due to unequal masses and/or spins result in the anisotropic emission of gravitational radiation, ultimately imparting a recoil to the merged remnant black hole. Numerically it has been found that, for certain configurations of black hole spins, the recoil velocity can exceed the escape velocity of many galaxies. Since it is important to determine the theoretical probability that a massive black hole might be ejected from its host galaxy, there have been some preliminary calculations based on simple distributions of spins and mass ratios, and analytic fits of numerical results giving the dependence of the recoil on mass and spin [@Schnittman:2007sn; @Baker:2008md]. Although a consensus on the exact probability of galactic ejection has yet to be achieved, there is general agreement that it is non-negligible.
Such rogue black holes may have already been observed in the form of two rapidly moving quasars [@Shields:2009jf; @Komossa:2008qd]. It has been speculated that these particular quasars originated from the mergers of massive black holes during the coalescence of their host galaxies. However a previous study of quasar data found no indications of such recoil events, suggesting that they are rare [@Bonning:2007vt].
The effect of recoil may also be observed less directly in its effect on the growth rates of black holes. Those ejected into the sparse intergalactic medium are less likely to encounter and merge with other black holes [@Sesana:2007sh; @Volonteri:2007et]. Even the growth of recoiled black holes remaining within their host galaxies may be affected, as the motion of the black hole can modify the rate at which it accretes matter [@Blecha:2008mg].
Mass and Spin Evolution {#ssec:astrophys_massspin}
-----------------------
The expected distribution of masses and spins of astrophysical black holes is another topic of considerable interest in astrophysics. In our current understanding, black holes grow from smaller “seeds” early in the history of the universe through a combination of mergers and the accretion of gas [@Sesana:2004sp]. In general, most of the mass growth is believed to come from accretion, with mergers providing a modest increase.
However, the gravitational radiation emitted during black hole coalescence carries away energy, reducing the overall system mass by roughly several percent. The bulk of this loss happens quickly, in the final plunge and merger stage of the coalescence. With this rapid mass loss, nearby matter in an accretion disk around the remnant black hole might react to the accompanying sudden change in gravitational potential and produce a visible change in its electromagnetic profile – a possible electromagnetic counterpart to the burst of gravitational radiation [@2007APS..APR.S1010B; @O'Neill:2008dg].
For the final spin the situation is considerably more interesting, and may answer questions about double-jet “X-shaped” radio sources [@Merritt:2002hc; @Barausse:2009uz]. We have seen in Sec. \[ssec:bhb\_final\_spin\] that the final spin from black holes merging in vacuum depends on the magnitudes of the initial spins and their orientations relative to the orbital axis. The RIT group [@Lousto:2009mf] has recently used their extended mass and spin formulas in a spin-evolution study, obtaining asymptotic spin distributions for BH merger remnants assuming no accretion. However, interaction of the merging holes with surrounding gas may serve to align the binary spins before merger, changing the picture somewhat [@Bogdanovic:2007hp]. In general, the effects of binary merger and accretion have to be studied together for a coherent picture to emerge [@Berti:2008af].
Outlook {#sec:outlook}
=======
Complete Analytic Waveform Models {#ssec:outlook_analyticmodels}
---------------------------------
Gravitational wave observatories may be sensitive to hundreds or thousands of wave cycles. Analysis of the observed data requires comparisons with model signals representing the full variety of possible mergers. Computing so many orbits of pre-merger evolution using numerical relativity would be computationally very expensive.
The PN approximation provides accurate representations of the dynamics and waveforms for the long-lasting inspiral portion of the coalescence during which the black holes remain fairly well-separated and their velocities remain relatively low. The most valuable waveform models for gravitational wave data analysis must combine the efficiency of the PN approach, while accurately representing the final merger portion of the which is only understood by numerical simulations. This requires a means of analytically encoding the merger signals.
Several approaches are currently being explored for constructing these complete signal models. Some are based on the analytic “effective-one-body” model of binary coalescence [@Pan:2007nw; @Buonanno:2007pf; @Damour:2008te]. Figure \[Fig9\_BuonannoEOBWF\] shows a comparison between an effective-one-body waveform for the merger of a nonspinning black hole binary with mass ratio $q = 4$ with a numerically simulated waveform [@Buonanno:2007pf]. For specific cases, such models have been tuned to high accuracy to agree to agree with numerical results [@Damour:2009kr; @Buonanno:2009qa]. Another approach, which models the phenomenological shape of hybrid waveforms in frequency space [@Ajith:2007kx], has been extended for the dominant waveform modes from spinning, but non-precessing, mergers [@Ajith:2009bn]. Further development of these models, and the production of a family of simulated waveform spanning the parameter space, is a current focus of broad-based research collaborations.
Improved Numerical Methods {#ssec:outlook_methods}
--------------------------
### Initial Data {#ssec:outlook_methods_id}
The initial data models currently used to begin numerical simulations do not perfectly represent the intended astrophysical systems. Research continues to improve these models. For example, although some astrophysical black holes are expected to have near-extremal spins, common initial data cannot represent holes with ${(a/M)}_i \gtrsim 0.93$ [@Lovelace:2008tw; @Dain:2002ee]; novel methods are being developed to go beyond this limit [@Lovelace:2008tw] . Also, current initial data models generally do not contain the physically appropriate radiation content for an inspiraling binary [@Damour:2000we; @Pfeiffer:2005zm], but rather harbor spurious radiation that is not astrophysical[@Hannam:2006zt]. Efforts are underway to include initial radiation content more consistent with PN predictions [@Kelly:2007uc]. In addition, in the case of moving punctures, the initial coordinates typically undergo a rapid transition at the start of the evolution as the black holes relax into a more stable solution. While the resulting transient pulse of “gauge radiation” does not alter the physics, it does contain fairly high frequencies that can be challenging to resolve and it has motivated construction of analytic initial coordinates that are closer to the numerically evolved coordinates [@Hannam:2009ib].
### Evolution {#ssec:outlook_methods_evol}
More efficient and accurate methods of numerical evolution would make simulations of many binary orbits, particularly with large mass ratios, more computationally practical. For representing spatial derivatives on a computational grid, the highest accuracy of finite differencing stencil yet achieved in the context of numerical relativity is 8th order in the grid spacing (although the accuracy of interpolation between refinement boundaries is not yet commensurate) [@Lousto:2007rj; @Pollney:2009yz]. Spectral methods are generally more accurate but less robust than finite differencing, requiring fine tuning for generic black hole binaries [@Szilagyi:2009qz]; they are also unlikely to handle shocks in accreting matter (see Section \[ssec:outlook\_matter\]). Alternatives to both finite differencing and spectral methods, such as finite element methods, are also being investigated [@Zumbusch:2009fe]. Meanwhile more efficient time-integration techniques allowing larger step sizes are being explored [@Lau:2008fb].
### Wave Extraction {#ssec:outlook_methods_extract}
Because the physical domain of a typical simulation is finite, gravitational radiation is usually extracted on a sphere of finite radius rather than at infinity. If multiple extraction surfaces are employed in a region of sufficiently high spatial resolution, then the radiation can be extrapolated to spatial infinity. A more accurate method known as “Cauchy-characteristic extrapolation” extracts the gravitational wave data at a finite radius and then evolves it along null geodesics to future null infinity; this method is currently under development [@Reisswig:2009rx; @Babiuc:2008qy].
Including Gas and Magnetic Fields {#ssec:outlook_matter}
---------------------------------
In addition to being gravitationally “loud", black hole mergers may also be electromagnetically visible. Massive black holes at the centers of galaxies are typically surrounded by gaseous accretion disks and magnetic fields. When the black holes merge, the dynamics of the gas and magnetic fields may produce electromagnetic signals, counterparts to the emitted gravitational radiation. For example, the inspiral may “twist" the fields, resulting in characteristic electromagnetic radiation [@Palenzuela:2009yr; @Palenzuela:2009hx] as well as heating of a surrounding accretion disk [@Reynolds:2006uq]. In addition, the violent merger dynamics may induce shock waves in accreting matter, in turn generating electromagnetic radiation. The recoil of the merged remnant, in particular, may have such an effect on the accretion disk [@Armitage:2002uu; @Milosavljevic:2004cg; @Dotti:2006zn; @Kocsis:2005vv; @Phinney:2007; @2007APS..APR.S1010B; @Kocsis:2007yu; @Shields:2008va; @Lippai:2008fx; @Schnittman:2008ez; @Kocsis:2008va; @Haiman:2008zy; @O'Neill:2008dg; @Haiman:2009te; @Chang:2009rx; @Megevand:2009yx].
### Multimessenger Astronomy {#sssec:outlook_matter_motive}
Detection of electromagnetic counterparts of gravitational waves would be of great scientific value. Current models of the complex merger physics (e.g. [@Balbus:1991ay]) could be directly tested. Einstein-Maxwell theory, the coupling of gravitational and electromagnetic fields on macroscopic scales, could be verified. In particular, equality of the speed of gravity with the speed of light could be confirmed [@Kocsis:2007yu; @Palenzuela:2009yr]. Astronomy would also benefit enormously, as the location and characteristics of gravitational wave sources could be corroborated. In addition, electromagnetically visible mergers could serve as “standard candles", beacons by which to measure the accelerating expansion of the universe, while simultaneously playing the role of “standard sirens" emitting gravitational radiation. Cross-correlating these signals could result in measurement of the cosmological “dark energy" to unprecedented accuracy [@Lang:2008gh; @Kocsis:2007hq; @Jonsson:2006vc; @Dalal:2006qt; @Kocsis:2005vv; @Holz:2005df; @Kocsis:2007yu; @Arun:2008xf].
### modeling matter {#sssec:outlook_matter_model}
There has been some preliminary work on the dynamic effects of the spacetime of a coalescing binary on surrounding matter. Modeling the accretion disk as geodesic particles, large collision energies were found as the binary merged [@vanMeter:2009gu]. And hydrodynamically simulating a gas cloud around around the binary, luminosity due to shocks has been calculated [@Bode:2009mt]. The generation of electromagnetic radiation by more direct means has also been simulated, via the twisting of a magnetic field anchored in the accretion disk, as it is frame-dragged by the binary [@Mosta:2009rr; @Palenzuela:2009hx; @Palenzuela:2009yr]. Future efforts will employ magnetohydrodynamic methods, where challenges include adequately resolving shocks and turbulence in a dynamic spacetime, and accurately representing the divergence-free magnetic field on an adaptively refined grid.
Conclusion
==========
We hope to have conveyed some of the excitement of recent progress in understanding black-hole binary physics, and the applications of this knowledge in astrophysics. These advances are the result of sustained efforts by a broad scientific community over many years. In a brief review, it is impossible to adequately represent all of the excellent work that has contributed to the current state of knowledge. We have only been able to provide a few of the highlights as seen through the lens of our particular perspective.
We encourage interested readers to pursue the subject further. Other resources are available on topics including: numerical relativity techniques [@Bona05; @Alcubierre08], the breakthroughs in black hole merger simulations [@Pretorius:2007nq], black hole simulations for gravitational wave data analysis [@Hannam:2009rd], and gravitational wave science generally [@Sathyaprakash:2009LR; @Camp:2004gg].
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge support from NASA grants 06-BEFS06-19 and 08-ATFP08-0126. BJK was supported in part by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center, administered by Oak Ridge Associated Universities through a contract with NASA.
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![Waveforms from Lazarus models of equal-mass, nonspinning black hole mergers. This figure shows the $\ell=2, m=2$ component of Re($r \Psi_4$), corresponding to one of the two polarization states of the emitted gravitational radiation. Curves are plotted for 10 simulations having different initial black hole separations (designated QC-0, etc.) and transition times to perturbative evolution. Reprinted from [@Baker:2002qf] and copyright 2002 by the American Physical Society (http://link.aps.org/abstract/PRD/v65/e124012). []{data-label="Fig1_LazarusWF"}](Centrella_Fig1_LazarusWF.eps){width="5.0in"}
![The first gravitational waveform for binary of equal-mass black holes evolving through a single plunge orbit, merger, and ringdown, computed by Pretorius. The waveforms were extracted at four radii from the source, and then shifted in time to overlap for comparison. Reprinted with permission from [@Pretorius:2005gq] and copyright 2005 by the American Physical Society (http://link.aps.org/abstract/PRL/v95/e121101).[]{data-label="Fig2_PretoriusPsi4"}](Centrella_Fig2_PretoriusPsi4.eps){width="5.0in"}
![Puncture trajectories from the merger of an equal-mass nonspinning black-hole binary calculated by the UTB group. The apparent horizons of the two pre-merger holes, outlined in solid (black) and red (dashed), expand due to coordinate effects. Also shown is the first detected common horizon, outlined in green (dot-dashed); this designates the point of merger and has a “peanut” shape before it settles down. Reprinted with permission from [@Campanelli:2005dd] and copyright 2006 by the American Physical Society (http://link.aps.org/abstract/PRL/v96/e111101).[]{data-label="Fig3_UTBPRLtrack"}](Centrella_Fig3_UTBPRLtrack.eps){width="5.0in"}
![The universal waveform for equal-mass, nonspinning black holes calculated by the GSFC group. The figure shows waveforms from four simulations with increasingly larger initial separations between the black holes. These waveforms were shifted in time so that the peak radiation amplitude occurs at $t = 0$. Reprinted from [@Baker:2006yw] and copyright 2006 by the American Physical Society (http://link.aps.org/abstract/PRD/v73/e104002).[]{data-label="Fig4_QCNwaves"}](Centrella_Fig4_QCNwaves.eps){width="5.0in"}
![The longest gravitational waveform for an equal-mass, nonspinning black-hole binary merger computed by the Caltech-Cornell group. The left panel shows the early stages of the waveform, during the inspiral. The right panel displays the merger and ringdown portions of the waveform. Reprinted with permission from [@Scheel:2008rj] and copyright 2009 by the American Physical Society (http://link.aps.org/abstract/PRD/v79/e024003). []{data-label="Fig6_CCWF"}](Centrella_Fig6_CCWF.eps){width="6.0in"}
![Gain in SNR by including the merger segment of the waveform for an equal-mass, nonspinning binary. The SNR for sources located at luminosity distance $D_L = 1$ Gpc is plotted vs. (redshifted) mass for the Advanced LIGO detector. The dashed line shows the SNR calculated using PN techniques for the early inspiral part of the waveform, $ -\infty < t < -1000M$. The dotted line shows the SNR using the late inspiral, $-1000M < t < -50M$, which is the transition region from PN to numerical relativity. The SNR for the strong-field merger and ringdown, $-50M < t < \infty$, was calculated using waveforms from a numerical relativity simulation and is shown using a thin solid line. Finally, the SNR from the entire waveform is given as the thick solid line.x Here, $t=0$ marks the time of maximum gravitational radiation amplitude. Reprinted from [@Baker:2006kr] and copyright 2007 by the American Physical Society (http://link.aps.org/abstract/PRD/v75/e124024). []{data-label="Fig7_MergerObsALIGOSNR"}](Centrella_Fig7_MergerObsALIGOSNR.eps){width="5.0in"}
![Contours of SNR as a function of redshift $z$ and total binary mass $M$ are shown for the LISA detector. These have been calculated using a hybrid equal-mass nonspinning waveform: using PN for the early inspiral, matching to a numerical relativity waveform in the late inspiral, and continuing with the numerical waveform through the merger and ringdown. Reprinted from [@Baker:2006kr] and copyright 2007 by the American Physical Society (http://link.aps.org/abstract/PRD/v75/e124024).[]{data-label="Fig8_MergerObsLISASNR"}](Centrella_Fig8_MergerObsLISASNR.eps){width="5.0in"}
![Comparison of merger waveforms for a $q=4$ mass ratio nonspinning black hole binary calculated using the analytic “effective one-body” (EOB) model and using numerical relativity (NR). Reprinted with permission from [@Buonanno:2007pf] and copyright 2007 by the American Physical Society (http://link.aps.org/abstract/PRD/v76/e104049). []{data-label="Fig9_BuonannoEOBWF"}](Centrella_Fig9_BuonannoEOBWF.eps){width="85.00000%"}
|
---
abstract: 'We demonstrate photonic crystal nanobeam cavities that support both TE- and TM-polarized modes, each with a Quality factor greater than one million and a mode volume on the order of the cubic wavelength. We show that these orthogonally polarized modes have a tunable frequency separation and a high nonlinear spatial overlap. We expect these cavities to have a variety of applications in resonance-enhanced nonlinear optics.'
author:
- Yinan Zhang
- 'Murray W. McCutcheon'
- 'Ian B. Burgess'
- Marko Loncar
date: May 2009
title: 'Ultra-high-$Q$ TE/TM dual-polarized photonic crystal nanocavities'
---
Ultra-high Quality factor ($Q$) photonic crystal nanocavities, which are capable of storing photons within a cubic-wavelength-scale volume ($V_{mod}$), enable enhanced light-matter interactions, and therefore provide an attractive platform for cavity quantum electrodynamics [@yoshie; @vuckovic] and nonlinear optics [@solj-rev; @raineri; @andreani; @murray07; @murray-lukin; @DFG-OE; @bravo-OE]. In most cases, high $Q/V_{mod}$ nanocavities are achieved with planar photonic crystal platform based on thin semiconductor slabs perforated with a lattice of holes. These structures favor transverse-electric-like (TE-like) polarized modes (the electric field in the central mirror plane of the photonic crystal slab is perpendicular to the air holes). In contrast, the transverse-magnetic-like (TM-like) polarized bandgap is favored in a lattice of high-aspect-ratio rods [@book; @multipole]. TM-like cavities have been designed in an air-hole geometry, as well [@arakawa; @triangular; @painter], but the $Q$ factors of these cavities were limited to the order of $10^3$. In addition, the lack of vertical confinement of these cavities results in large mode volumes [@arakawa]. Though it is possible to employ surface plasmons to localize the light tightly in the vertical direction, the lossy nature of metal limits the $Q$ to about $10^2$ [@painter].
In this paper, we report a one-dimensional (1D) photonic crystal nanobeam cavity design that supports an ultra-high-$Q$ ($Q>10^6$) TM-like cavity mode with $V_{mod}\sim(\lambda/n)^3$. This cavity greatly broadens the applications of optical nanocavities. For example, it is well-suited for photonic crystal quantum cascade lasers, since the inter-subband transition in quantum cascade lasers is TM-polarized [@faist; @forchel; @loncar-QCL]. We also demonstrate that our cavity simultaneously supports two ultra-high-$Q$ modes with orthogonal polarizations (one TE-like and one TM-like). The frequency difference of the two modes can be widely tuned while maintaining the high $Q$ factor of each mode, which is of interest for applications in nonlinear optics.
![\[F1\] (a) Schematic of the nanobeam design, showing the nanobeam thickness ($d_{y}$) and width ($d_{x}$), and the hole spacing ($a$). (b) TE$_{00}$ and TM$_{00}$ transverse mode profiles for a ridge waveguide with $d_{y}=3d_{x}$. (c) Transmission spectra for the TE$_{00}$ (red) and TM$_{00}$ (blue) Bloch modes. The shaded areas indicate the bandgaps for both modes.](Dual1_New.eps){width="7.8cm"}
Our design is based on a dielectric suspended ridge waveguide with an array of uniform holes of periodicity, $a$, and radius, $R$, which form a 1D photonic crystal Bragg mirror [@foresi], as shown in Fig. 1(a). The refractive index of the dielectric is set to $n=3.4$ (similar to Si and GaAs at $\sim$ $1.5\mu m$). We first start with a ridge of height:width:period ratio of 3:1:1 ($d_{x}=a$, $d_{y}=3a$) and $R=0.3a$. Fig. 1(b) shows the transverse profiles of the fundamental TM-like and TE-like modes (TM$_{00}$ and TE$_{00}$) supported by the ridge waveguide. The TM$_{00}$ mode has its major component ($E_{y}$) lined along the hole axis, whereas the TE$_{00}$ mode’s major component ($E_{x}$) is perpendicular to the air holes. Using the three-dimensional (3D) finite-difference time-domain (FDTD) method, the transmittance spectra are obtained of the TM$_{00}$ and TE$_{00}$ modes launched towards the Bragg mirror. Fig. 1(c) shows the TM$_{00}$ and TE$_{00}$ bandgaps, respectively. In contrast to two-dimensional (2D) photonic crystal slabs, where the photon is localized in the $xz$ plane via Bragg scattering, here we only require Bragg confinement in the longitudinal ($z$) direction, as light is transversely confined in the other two dimensions by total internal reflection. It has also been shown experimentally that 1D photonic crystal nanobeam cavities have comparable $Q/V_{mod}$ ratios to 2D systems [@parag; @delarue].
![\[F2\] (a) Schematic of the 1D photonic crystal nanobeam cavity, with the tuning parameters $R_{k}$ and $w_{k}$ in the 8-segment tapered design. (b,c) Mode profiles of the electric field components $E_{TE,x}$ and $E_{TM,y}$ for the cavity design with $d_{x}=a$, $d_{y}=3a$. (d,e) Spatial Fourier transform of the electric field component profiles ($E_{TE,x}$ and $E_{TM,y}$) in the $xz$ plane ($y=0$).](Dual2_New3.eps){width="6.6cm"}
Introducing a lattice grading to the periodic structure creates a localized potential for both TE- and TM-like modes. To optimize the mode $Q$ factors, we apply the bandgap-tapering technique that is well-developed in previous work [@yinan-pillar; @yinan-wires; @murraySiN; @notomi; @zipper]. We use an 8-segment tapered section with holes ($R_{1}$-$R_{8}$) and a 12-period mirror section at each side. Two degrees of freedom are available for each tapered segment: the length ($w_{k}$) and the radius ($R_{k}$). We keep the ratio $R_{k}/w_{k}$ fixed at each segment, and then implement a linear interpolation of the grating constant ($2\pi/w_{k}$). When the central segment $w_{8}$ is set to $0.84a$, we obtain ultra-high $Q$s and low mode volumes for both TE- and TM-polarized modes ($Q_{TE}=1.2\times 10^6$, $Q_{TM}=2.4\times 10^6$, $V_{mod,TE}=V_{mod,TM}=1.2(\lambda/n)^3$), with free-space wavelengths $4.30a$ and $4.78a$, respectively. Fig. 2(b) and (c) show the mode profiles of the major components of the two modes in the [*xz*]{} mirror plane. The ultra-high $Q$ factors can also be interpreted in momentum space [@momentum_I; @momentum_II; @arakawa]. Fig. 2(d) and (e) demonstrate the Fourier transformed (FT) profiles of the electric field components $E_{TE,x}$ and $E_{TM,y}$ in the [*xz*]{} plane ($y=0$), with the light cone indicated by the white circle. It can be seen that both modes’ Fourier components are localized tightly at the bandedge of the Brillouin zone on the $k_{z}$-axis ($k_z=\pi/a$). This reduces the amount of mode energy within the light cone that is responsible for scattering losses. It is also worthwhile to note that higher-longitudinal-order TE$_{00}$ and TM$_{00}$ cavity modes with different symmetry with respect to the $xy$ mirror plane exist [@yinan-pillar]. For example, the second-order TE$_{00}$ mode, which has a node at the $xy$ mirror plane, resonates at a wavelength of $4.43a$. It has a higher $Q$ factor of $4.7 \times 10^6$, but a larger mode volume of $2.1(\lambda/n)^3$.
For a number of applications of interest, control of the frequency spacing between the two modes is required. Examples include polarization-entangled photon generation for degenerate modes [@imamoglu], and terahertz generation for $0.1-10$THz mode splitting [@DFG-OE]. We tune the frequency separation of the two modes by varying the thickness of the structure while keeping the other parameters constant. In Fig. 3(a), the cavity resonances of the TE$_{00}$ and TM$_{00}$ modes are traced as a function of the nanobeam thickness ($d_{y}/a$). The TM-like modes have a much larger dependence on the thickness than the TE-like modes. The modes are degenerate at $d_{y}=1.26a$, and for thicknesses beyond this value, $\omega_{TE}$ is larger than $\omega_{TM}$. As $d_y$ increases, the splitting increases until it saturates when the system approaches the 2D limit (structure is infinite in the [*y*]{}-direction). In this limit, we find that $\lambda_{TE}=4.4a$ and $\lambda_{TM}=5.1a$. The frequency separation ($\delta\omega=|\omega_{TE}-\omega_{TM}|$) of this design ranges from 0THz to 20THz, with the TE-like mode wavelength fixed at $1.5\mu m$ by scaling the structure accordingly. Fig. 3(b) shows the thickness dependence of the $Q$ factor for the [*xz*]{} design specifications listed above. It can be seen that the $Q$ factors of both TE- and TM-polarized modes stay above $10^5$ for the $\omega_{TE}>\omega_{TM}$ branch.
![\[F3\] (a) TE$_{00}$ (red) and TM$_{00}$ (blue) cavity mode resonant frequencies (dotted lines) as a function of the nanobeam thickness. The bandgap regions of the two modes are shaded. The frequency separation ($\delta\omega$) of the two modes with the TE-like mode wavelength fixed at $1.5\mu m$ by scaling the structure accordingly is plotted in green. (b,c) Dependence of the $Q$ factor and nonlinear overlap factor $\gamma$ on the nanobeam thickness.](Dual3_New2.eps){width="8.3cm"}
Decreasing $d_y$ causes the width of the TM bandgap to sharply decrease, whereas the width of the TE bandgap remains almost constant. The narrowed TM bandgap results in a reduced Bragg confinement, which increases the transmission losses through the Bragg mirrors. This is evidenced by the $Q$ factor of the TM mode, which drops to 9,000 when the thickness:width ratio is 1:1. Though this leakage can be compensated for, in principle, by increasing the number of periods of the mirror sections, the length of the structure also increases, which makes fabrication more challenging for a suspended nanobeam geometry. A narrow bandgap also leads to large penetration depth of the mode into the Bragg mirrors, thereby increasing the mode volume.
Next, we examine the application of our dual-polarized cavity for the resonance enhancement of nonlinear processes. To achieve a large nonlinear interaction in materials with dominant off-diagonal nonlinear susceptibility terms (e.g. $\chi_{ijk}^{(2)}, i\neq j\neq k$), such as III-V semiconductors [@murray07; @murray-lukin; @GaAs-ref], it is beneficial to mix two modes with orthogonal polarizations. As shown in our previous work [@DFG-OE], the strength of the nonlinear interaction can be characterized by the modal overlap, which can be quantified using the following figure of merit, \_[r,d]{}. where $\int_{d}$ denotes integration over only the regions of nonlinear dielectric, and $\epsilon_{r,d}$ denotes the maximum dielectric constant of the nonlinear material. Note that we have normalized $\gamma$ so that $\gamma=1$ corresponds to the theoretical maximum overlap. For the TE$_{00}$ and TM$_{00}$ modes we studied, the two major components ($E_{TE,x}$ and $E_{TM,y}$) share the same parity (have anti-nodes in all the three mirror planes), and only two overlap components, $E_{TE,x}E_{TM,y}$ and $E_{TE,y}E_{TM,x}$, in Eq. (1) do not vanish. This allows a large nonlinear spatial overlap. We obtain $\gamma=0.76$ for the cavity shown in Fig. 2. The overlap approaches $\gamma=0.78$ in the limit $d_{y}\rightarrow\infty$. We find that the overlap factor, $\gamma$, stays at a reasonably high value ($>0.6$) across the full range of the frequency difference tuning (for $\omega_{TE}>\omega_{TM}$ branch) \[Fig. 3(c)\].
![\[F4\] Parameters of the higher-order cavity modes for the design with $d_{x}=a$, $d_{y}=3a$.](Dual5.eps){width="7cm"}
Finally, it is important to note that thick nanobeams can support higher-order modes with a different number of nodes in the $xy$ plane, as well. These higher-order modes are also confined in the tapered section within their respective bandgaps, with the $Q$ factors and wavelengths listed in Fig. 4 for the $d_{x}=a$ and $d_{y}=3a$ case. These modes can offer a broader spectral range than the fundamental modes, which is of great interest to nonlinear applications requiring a large bandwidth [@murray-lukin].
In conclusion, we have demonstrated that ultra-high-$Q$ TE- and TM-like fundamental modes with mode-volumes $\sim (\lambda/n)^3$ can be designed in 1D photonic crystal nanobeam cavities. We have shown that the frequency splitting of these two modes can be tuned over a wide range without compromising the $Q$ factors. We have also shown that these modes can have a high nonlinear overlap in materials with large off-diagonal nonlinear susceptibility terms across the entire tuning range of the frequency spacing. We expect these cavities to have broad applications in the enhancement of nonlinear processes.
This work was supported in part by National Science Foundation (NSF) and NSF career award. M.W.M and I.B.B wish to acknowledge NSERC (Canada) for support from PDF and PGS-M fellowships.
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abstract: 'Proof that under simple assumptions, such as constraints of Put-Call Parity, the probability measure for the valuation of a European option has the mean derived from the forward price which can, but does not have to be the risk-neutral one, under any general probability distribution, bypassing the Black-Scholes-Merton dynamic hedging argument, and without the requirement of complete markets and other strong assumptions. We confirm that the heuristics used by traders for centuries are both more robust, more consistent, and more rigorous than held in the economics literature. We also show that options can be priced using infinite variance (finite mean) distributions.'
author:
-
title: Unique Option Pricing Measure With Neither Dynamic Hedging nor Complete Markets
---
Background
==========
Option valuations methodologies have been used by traders for centuries, in an effective way (Haug and Taleb, 2010). In addition, valuations by expectation of terminal payoff forces the mean of the probability distribution used for option prices be be that of the forward, thanks to Put-Call Parity and, should the forward be risk-neutrally priced, so will the option be. The Black Scholes argument (Black and Scholes, 1973, Merton, 1973) is held to allow risk-neutral option pricing thanks to dynamic hedging, as the option becomes redundant (since its payoff can be built as a linear combination of cash and the underlying asset dynamically revised through time). This is a puzzle, since: 1) Dynamic Hedging is not operationally feasible in financial markets owing to the dominance of portfolio changes resulting from jumps, 2) The dynamic hedging argument doesn’t stand mathematically under fat tails; it requires a very specific “Black Scholes world” with many impossible assumptions, one of which requires finite quadratic variations, 3) Traders use the same Black-Scholes “risk neutral argument” for the valuation of options on assets that do not allow dynamic replication, 4) Traders trade options consistently in domain where the risk-neutral arguments do not apply 5) There are fundamental informational limits preventing the convergence of the stochastic integral.[^1]
There have been a couple of predecessors to the present thesis that Put-Call parity is sufficient constraint to enforce some structure at the level of the mean of the underlying distribution, such as Derman and Taleb (2005), Haug and Taleb (2010). These approaches were heuristic, robust though deemed hand-waving (Ruffino and Treussard, 2006). In addition they showed that operators need to use the risk-neutral mean. What this paper does is
- It goes beyond the “handwaving” with formal proofs.
- It uses a completely distribution-free, expectation-based approach and proves the risk-neutral argument without dynamic hedging, and without any distributional assumption.
- Beyond risk-neutrality, it establishes the case of a unique pricing distribution for option prices in the absence of such argument. The forward (or future) price can embed expectations and deviate from the arbitrage price (owing to, say, regulatory or other limitations) yet the options can still be priced at a distibution corresponding to the mean of such a forward.
- It shows how one can *practically* have an option market without “completeness” and without having the theorems of financial economics hold.
These are done with solely two constraints: “horizontal”, i.e. put-call parity, and “vertical”, i.e. the different valuations across strike prices deliver a probability measure which is shown to be unique. The only economic assumption made here is that the forward exits, is tradable — in the absence of such unique forward price it is futile to discuss standard option pricing. We also require the probability measures to correspond to distributions with finite first moment.
Preceding works in that direction are as follows. Breeden and Litzenberger (1978) and Dupire(1994), show how option spreads deliver a unique probability measure; there are papers establishing broader set of arbitrage relations between options such as Carr and Madan (2001)[^2].
However 1) none of these papers made the bridge between calls and puts via the forward, thus translating the relationships from arbitrage relations between options delivering a probability distribution into the necessity of lining up to the mean of the distribution of the forward, hence the risk-neutral one (in case the forward is arbitraged.) 2) Nor did any paper show that in the absence of second moment (say, infinite variance), we can price options very easily. Our methodology and proofs make no use of the variance. 3) Our method is vastly simpler, more direct, and robust to changes in assumptions.
We make no assumption of general market completeness. Options are not redundant securities and remain so. Table 1 summarizes the gist of the paper.[^3] [^4]
------------------------------ --------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------
**** **Black-Scholes Merton** **Put-Call Parity with Spreading**
**Type** Continuous rebalancing. Interpolative static hedge.
**Market Assumptions** 1\) Continuous Markets, no gaps, no jumps. 1\) Gaps and jumps acceptable. Continuous Strikes, or acceptable number of strikes.
2\) Ability to borrow and lend underlying asset for all dates. 2\) Ability to borrow and lend underlying asset for single forward date.
3\) No transaction costs in trading asset. 3\) Low transaction costs in trading options.
**Probability Distribution** Requires all moments to be finite. Excludes the class of slowly varying distributions Requires finite $1^{st}$ moment (infinite variance is acceptable).
**Market Completeness** Achieved through dynamic completeness Not required (in the traditional sense)
**Realism of Assumptions** Low High
**Convergence** In probability (uncertain; one large jump changes expectation) Pointwise
**Fitness to Reality** Only used after “fudging” standard deviations per strike. Portmanteau, using specific distribution adapted to reality
------------------------------ --------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------
: Main practical differences between the dynamic hedging argument and the static Put-Call parity with speading across strikes.
Proof
=====
Define $C(S_{t_0},K,t)$ and $P(S_{t_0},K,t)$ as European-style call and put with strike price K, respectively, with expiration $t$, and $S_0$ as an underlying security at times $t_0$, $t \geq t_0$, and $S_t$ the possible value of the underlying security at time t.
Case 1: Forward as risk-neutral measure
---------------------------------------
Define $r = \frac{1}{t-t_0}\int_{t_0}^t r_s \mathrm{d}s$, the return of a risk-free money market fund and $\delta =\frac{1}{t-t_0}\int_{t_0}^t \delta_s \mathrm{d}s$ the payout of the asset (continuous dividend for a stock, foreign interest for a currency).
We have the arbitrage forward price $F_t^Q$: $$F_t^Q = S_0\frac{(1+r)^{(t-t_0)}}{(1+\delta)^{(t-t_0)}} \thickapprox S_0 \, e^{(r-\delta) (t-t_0)}$$ by arbitrage, see Keynes 1924. We thus call $F_t^Q$ the future (or forward) price obtained by arbitrage, at the risk-neutral rate. Let $F_t^P$ be the future requiring a risk-associated “expected return” $m$, with expected forward price: $$F_t^P = S_0 (1+m)^{(t-t_0)}\thickapprox S_0 \, e^{m \, (t-t_0)}.$$
**Remark:** *By arbitrage, all tradable values of the forward price given $S_{t_0}$ need to be equal to $F_t^Q$.*
“Tradable” here does not mean “traded”, only subject to arbitrage replication by “cash and carry”, that is, borrowing cash and owning the secutity yielding $d$ if the embedded forward return diverges from $r$.
Derivations
-----------
In the following we take $F$ as having dynamics on its own –irrelevant to whether we are in case 1 or 2 –hence a unique probability measure $Q$.
Define $\Omega=[0,\infty)=A_K \cup A_K^c $ where $A_K=[0,K]$ and $A_K^c=(K, \infty)$.
Consider a class of standard (simplified) probability spaces $(\Omega,\mu_i)$ indexed by $i$, where $\mu_i$ is a probability measure, i.e., satisfying $\int_\Omega \mathrm{d} \mu_i=1$.
For a given maturity T, there is a unique measure $\mu_Q$ that prices European puts and calls by expectation of terminal payoff.
This measure can be risk-neutral in the sense that it prices the forward $F_t^Q$, but does not have to be and imparts rate of return to the stock embedded in the forward.
For a given maturity T, there exist two measures $\mu_1$ and $\mu_2$ for European calls and puts of the same maturity and same underlying security associated with the valuation by expectation of terminal payoff, which are unique such that, for any call and put of strike K, we have: $$C= \int_ \Omega f_C \, \mathrm{d}\mu_1 \, ,\label{callequation}$$ and $$P= \int_ \Omega f_P \, \mathrm{d}\mu_2 \, ,$$ respectively, and where $f_C$ and $f_P$ are $(S_t-K)^+$ and $(K-S_t)^+$ respectively.
For clarity, set $r$ and $\delta$ to $0$ without a loss of generality. By Put-Call Parity Arbitrage, a positive holding of a call (“long”) and negative one of a put (“short”) replicates a tradable forward; because of P/L variations, using positive sign for long and negative sign for short:
$$C(S_{t_0},K,t)-P(S_{t_0},K,t)+K=F_t^P \label{putcallparity}$$
necessarily since $F_t^P$ is tradable. Put-Call Parity holds for all strikes, so: $$C(S_{t_0},K +\Delta K,t)-P(S_{t_0},K+\Delta K,t)+K+\Delta K=F_t^P\label{diffputcallparity}$$ for all $K \in \Omega $
Now a Call spread in quantities $\frac{1}{\Delta K}$, expressed as $$C(S_{t_0},K,t)-C(S_{t_0},K+\Delta K,t),$$delivers \$1 if $S_t > K+\Delta K$ (that is, corresponds to the indicator function $\mathbf{1}_{S > K+\Delta K}$), 0 if $S_t\leq K$ (or $\mathbf{1}_{S > K}$), and the quantity times $S_t-K$ if $K < S_t \leq K+\Delta K$, that is, between 0 and \$1 (see Breeden and Litzenberger, 1978). Likewise, consider the converse argument for a put, with $\Delta K <S_t$.
At the limit, for $\Delta K \to 0$ $$\frac{\partial{C(S_{t_0},K,t)}}{\partial{K}} =- P(S_t > K)
=- \int_{A_K^c} \mathrm{d}\mu_1 .$$ By the same argument: $$\frac{\partial{P(S_{t_0},K,t)}}{\partial{K}} = \int_{A_K} \mathrm{d}\mu_2 = 1-\int_{A_K^c} \mathrm{d}\mu_2 .$$
As semi-closed intervals generate the whole Borel $\sigma$-algebra on $\Omega$, this shows that $\mu_1$and $\mu_2$ are unique.
The probability measures of puts and calls are the same, namely for each Borel set $A$ in $\Omega$, $\mu_1(A)$ = $\mu_2(A)$.
Combining Equations \[putcallparity\] and \[diffputcallparity\], dividing by $\frac{1}{\Delta K}$ and taking $\Delta K \to 0$:
$$-\frac{\partial{C(S_{t_0},K,t)}}{\partial{K}}+\frac{\partial{P(S_{t_0},K,t)}}{\partial{K}}=1$$
for all values of $K$, so
$$\int_{A_K^c} \mathrm{d}\mu_1=\int_{A_K^c} \mathrm{d}\mu_2 ,$$
hence $\mu_1(A_K)=\mu_2(A_K)$ for all $K \in [0,\infty)$. This equality being true for any semi-closed interval, it extends to any Borel set.
$$\qedhere$$
Puts and calls are required, by static arbitrage, to be evaluated at same as risk-neutral measure $\mu_Q$ as the tradable forward.
$$F_t^P=\int_\Omega F_t \, \mathrm{d}\mu_Q;$$
from Equation \[putcallparity\]
$$\int_\Omega f_C(K) \,\mathrm{d}\mu_1- \int_\Omega f_P(K)\,\mathrm{d}\mu_1 = \int_\Omega F_t \, \mathrm{d}\mu_Q -K$$
Taking derivatives on both sides, and since $f_C-f_P=S_0+K$, we get the Radon-Nikodym derivative: $$\frac{\mathrm{d}\mu_Q}{\mathrm{d}\mu_1}=1$$ for all values of K.
$$\qedhere$$
Case where the Forward is not risk neutral
==========================================
Consider the case where $F_t$ is observable, tradable, and use it solely as an underlying security with dynamics on its own. In such a case we can completely ignore the dynamics of the nominal underlying $S$, or use a non-risk neutral “implied” rate linking cash to forward, $m^*= \frac{\log \left(\frac{F}{S_0}\right)}{t-t_0}$. the rate $m$ can embed risk premium, difficulties in financing, structural or regulatory impediments to borrowing, with no effect on the final result.
In that situation, it can be shown that the exact same results as before apply, by remplacing the measure $\mu_Q$ by another measure $\mu_{Q^*}$. Option prices remain unique [^5].
comment
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We have replaced the complexity and intractability of dynamic hedging with a simple, more benign interpolation problem, and explained the performance of pre-Black-Scholes option operators using simple heuristics and rules, bypassing the structure of the theorems of financial economics.
Options can remain non-redundant and markets incomplete: we are just arguing here for a form of arbitrage pricing (which includes risk-neutral pricing at the level of the expectation of the probability measure), nothing more. But this is sufficient for us to use any probability distribution with finite first moment, which includes the Lognormal, which recovers Black Scholes.
A final comparison. In dynamic heding, missing a single hedge, or encountering a single gap (a tail event) can be disastrous —as we mentioned, it requires a series of assumptions beyond the mathematical, in addition to severe and highly unrealistic constraints on the mathematical. Under the class of fat tailed distributions, increasing the frequency of the hedges does not guarantee reduction of risk. Further, the standard dynamic hedging argument requires the exact specification of the *risk-neutral* stochastic process between $t_0$ and $t$, something econometrically unwieldy, and which is generally reverse engineered from the price of options, as an arbitrage-oriented interpolation tool rather than as a representation of the process.
Here, in our Put-Call Parity based methodology, our ability to track the risk neutral distribution is guaranteed by adding strike prices, and since probabilities add up to 1, the degrees of freedom that the recovered measure $\mu_Q$ has in the gap area between a strike price $K$ and the next strike up, $K +\Delta K$, are severely reduced, since the measure in the interval is constrained by the difference $\int_{A_K}^c \mathrm{d}\mu - \int_{A_{K+\Delta K}}^c \mathrm{d}\mu $. In other words, no single gap between strikes can significantly affect the probability measure, even less the first moment, unlike with dynamic hedging. In fact it is no different from standard kernel smoothing methods for statistical samples, but applied to the distribution across strikes.[^6]
The assumption about the presence of strike prices constitutes a natural condition: conditional on having a *practical* discussion about options, options strikes need to exist. Further, as it is the experience of the author, market-makers can add over-the-counter strikes at will, should they need to do so.
Acknowledgment {#acknowledgment .unnumbered}
==============
Peter Carr, Marco Avellaneda, Hélyette Geman, Raphael Douady, Gur Huberman, Espen Haug, and Hossein Kazemi.
References {#references .unnumbered}
==========
Avellaneda, M., Friedman, C., Holmes, R., & Samperi, D. (1997). Calibrating volatility surfaces via relative-entropy minimization. Applied Mathematical Finance, 4(1), 37-64.
Black, F., Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-654.
Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of business, 621-651.
Carr, P. and Madan, D. (2001). Optimal positioning in derivative securities, Quantitative Finance, pp. 19-37.
Derman, E. and Taleb, N. (2005). The illusions of dynamic replication. Quantitative Finance, 5(4):323-326.
Dupire, Bruno, 1994, Pricing with a smile, Risk 7, 18-20.
Green, R. C., & Jarrow, R. A. (1987). Spanning and completeness in markets with contingent claims. Journal of Economic Theory, 41(1), 202-210.
Harrison, J. M., & Kreps, D. M. (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic theory, 20(3), 381-408.
Haug, E. G. and Taleb, N. N. (2010) Option Traders use Heuristics, Never the Formula known as Black-Scholes-Merton Formula, Journal of Economic Behavior and Organizations, pp. 97–106.
Keynes, J.M., 1924. A Tract on Monetary Reform. Reprinted in 2000. Prometheus Books, Amherst New York.
Merton, R.C., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-183.
Nachman, D. C. (1988). Spanning and completeness with options. Review of Financial Studies, 1(3), 311-328.
Ruffino, D., & Treussard, J. (2006). Derman and Taleb’s “The illusions of dynamic replication”: a comment. Quantitative Finance, 6(5), 365-367.
Thorp, E.O., 1973. A corrected derivation of the Black-Scholes option model. In: CRSP proceedings, 1976.
[^1]: Further, in a case of scientific puzzle, the exact formula called “Black-Scholes-Merton” was written down (and used) by Edward Thorp in a heuristic derivation by expectation that did not require dynamic hedging, see Thorpe(1973).
[^2]: See also Green and Jarrow (1987) and Nachman(1988). We have known about the possibility of risk neutral pricing without dynamic hedging since Harrison and Kreps (1979) but the theory necessitates extremely strong –and severely unrealistic –assumptions, such as strictly complete markets and a multiperiod pricing kernel
[^3]: The famed Hakkanson paradox is as follows: if markets are complete and options are redudant, why would someone need them? If markets are incomplete, we may need options but how can we price them? This discussion may have provided a solution to the paradox: markets are incomplete *and* we can price options.
[^4]: Option prices are not unique in the absolute sense: the premium over intrinsic can take an entire spectrum of values; it is just that the put-call parity constraints forces the measures used for puts and the calls to be the same and to have the same expectation as the forward. As far as securities go, options are securities on their own; they just have a strong link to the forward.
[^5]: We assumed 0 discount rate for the proofs; in case of nonzero rate, premia are discounted at the rate of the arbitrage operator
[^6]: For methods of interpolation of implied probability distribution between strikes, see Avellaneda et al.(1997).
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---
abstract: 'We generalize the notion of Poincaré rotation number for homeomorphisms of the unit circle to homeomorphisms of the additive adèle class group of the rational numbers ${\mathbb{A}}/{\mathbb{Q}}$. We would like to emphasize that our theory is valid for any general compact abelian one–dimensional solenoidal group $G$, which is also a one–dimensional foliated space. Poincaré’s dynamical classification theorem is also generalized to homeomorphisms of solenoids isotopic to the identity whose rotation element is an irrational element (i.e., monothetic generator) of the given group. Then the definition is extended for homeomorphisms of solenoids which are isotopic to irrational rotations whose rotation element is not in the base leaf. Like in the case of Tate’s thesis a fundamental role in the present paper is played by Pontryagin duality. We remark that our theory obtains as “rotation number” of a homeomorphism of $G$, an element of $G$ which we call the *rotation element*. When both $f$ has irrational rotation element we obtain the same dichotomy as in the classical theory: there exists a unique minimal set which is either a Cantor set or the whole solenoid ${\mathsf{S}}$.'
address:
- '$*$ Departamento de Matemáticas, Universidad de Guanajuato, Jalisco s/n, Mineral de Valenciana, Guanajuato, Gto. 36240 México.'
- '$**$ Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Apdo. Postal 2 C.P. 2000, Cuernavaca, Mor. México'
author:
- 'Manuel Cruz–López$^*$ and Alberto Verjovsky$^{**}$'
title: 'Poincaré theory for the Adèle class group ${\mathbb{A}}/{\mathbb{Q}}$ and compact abelian one–dimensional solenoidal groups'
---
Introduction
============
In his fundamental paper of 1881, H. Poincaré (see [@Poi]) introduced an invariant of topological conjugation for homeomorphisms of the unit circle $$\rho:{\mathrm{Homeo}}_+({\mathbb{S}^1}){\longrightarrow}{\mathbb{S}^1}, \quad f\longmapsto \rho(f),$$ called the of $f$. He then proved a remarkable topological classification theorem for the dynamics of any orientation–preserving homeomorphism $f\in {\mathrm{Homeo}}_+({\mathbb{S}^1})$: $f$ has a periodic orbit if and only $\rho(f)$ is rational. If the rotation number $\rho(f)$ is irrational, then $f$ is semiconjugate to an irrational rotation $R_{\rho(f)}$. The semiconjugacy is actually a conjugacy if the orbits of $f$ are dense. This work was continued by A. Denjoy in 1932 (see [@Den]) who, among other things, showed that if $f$ is a diffeomorphism with irrational rotation number whose derivative has bounded variation, then $f$ is conjugated to the rotation $R_{\rho(f)}$. In $1965$, V.I. Arnold (see [@Arn1]) solved the conjugacy problem when the diffeomorphism is real analytic and close to a rotation, by introducing a diophantine condition. An important issue is the existence of a differentiable conjugacy where Michael Herman made so many important contributions (see [@Her]).
Further developments of this theory have been one of the most fruitful subjects in dynamical systems, as shown by the works of A.N. Kolmogorov, V.I. Arnold, J. Moser, M.R. Herman, A.D. Brjuno, J.C. Yoccoz, among others (see [@Kol],[@Arn2],[@Mos],[@Brj1; @Brj2], [@Yoc]; see also [@Ghys], [@Her], [@Nav]).
In this paper we continue with this line of ideas and generalize the Poincaré rotation number to any compact abelian one–dimensional solenoidal group, which is also a compact abelian topological group obtained as the continuous homomorphic image of the algebraic universal covering space of the circle ${\mathsf{S}}:=\underleftarrow{\lim}\; {\mathbb R}/n{\mathbb{Z}}$ or, by Pontryagin duality, as a compact abelian topological group whose group of characters is an additive subgroup of the rational numbers with the discrete topology. In the case of the one–dimensional universal solenoidal group, the Pontryagin dual, i.e., the group of characters, is the whole discrete group ${\mathbb{Q}}$.
The algebraic universal covering space of the circle can be thought of as a generalized circle and can be realized as the of the rational numbers, which is the orbit space of the locally trivial ${\mathbb{Q}}$–bundle structure ${\mathbb{Q}}\hookrightarrow {\mathbb{A}}{\longrightarrow}{\mathbb{A}}/{\mathbb{Q}}$, where ${\mathbb{A}}$ is the adèle group of the rational numbers and ${\mathbb{Q}}\hookrightarrow {\mathbb{A}}$ is a cocompact discrete subgroup of ${\mathbb{A}}$. The adèle class group is a fundamental arithmetic object in mathematics which was invented by Claude Chevalley for the purposes of simplifying and clarifying class field theory. This compact abelian group plays an essential role in the thesis of John Tate (see [@Tat]) which laid the foundations for the Langlands program. Since $${\mathsf{S}}= \underleftarrow{\lim}\; {\mathbb R}/n{\mathbb{Z}}\cong {\mathbb{A}}/{\mathbb{Q}},$$ it follows that ${\mathsf{S}}$ is a compact abelian topological group with a locally trivial ${\widehat{{\mathbb{Z}}}}$–bundle structure ${\widehat{{\mathbb{Z}}}}\hookrightarrow {\mathsf{S}}{\longrightarrow}{\mathbb{S}^1}$ and also a one–dimensional foliated space whose leaves have a canonical affine structure isomorphic to the real one–dimensional affine space $\mathbf{A}^1$. Here, ${\widehat{{\mathbb{Z}}}}:=\underleftarrow{\lim}\; {\mathbb{Z}}/n{\mathbb{Z}}$ is the profinite completion of the integers, and it is an abelian Cantor group. Thus, topologically ${\mathsf{S}}$ is a compact and connected locally trivial fibration over the circle with fibre the Cantor set.
More general objects are the so called solenoidal manifolds, which were introduced by Dennis Sullivan (see [@Sul] and [@Ver]). These solenoidal manifolds are Polish spaces with the property that each point has a neighborhood which is homeomorphic to an open interval times a Cantor set. He shows that any compact one dimensional *orientable* solenoidal manifold is the suspension of a homeomorphism of the Cantor set. Examples of one dimensional solenoidal manifolds are one dimensional tiling spaces and one dimensional quasi–crystals like the ones studied by Williams and Sadun, and also by Aliste–Prieto (see [@WS] and [@Ali]).
In principle, Poincaré theory can be studied for general compact, orientable one dimensional solenoidal manifolds. What makes the difference in our case is the fact that we can apply to these groups the Pontryagin duality and the classical theory of harmonic analysis for compact and locally compact abelian groups.
In this paper the theory is developed for the adèle class group of the rational numbers ${\mathbb{A}}/{\mathbb{Q}}$, since this is the paradigmatic example and all the ideas are already present there. It is considered first the case of homeomorphisms of solenoids which are isotopic to the identity. Then it is treated the more general case of homeomorphisms isotopic to translations with the translation element not in the base leaf.
Using the notion of asymptotic cycle of Schwartzman (see [@Sch]) the generalized Poincaré rotation element $$\rho:{\mathrm{Homeo}}_+({\mathsf{S}}){\longrightarrow}{\mathsf{S}}, \quad f\longmapsto \rho(f),$$ can be defined as follows. Let $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ be any homeomorphism isotopic to the identity which can be written as $f={\mathrm{id}}+ \varphi$, where $\varphi:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is the displacement function along the one–dimensional leaves of ${\mathsf{S}}$ with respect to the affine structure. The suspension space of $f$ is defined as: $${\Sigma_f({\mathsf{S}})}:= {\mathsf{S}}\times [0,1] /(z,1)\sim (f(z),0).$$ Since $f$ is isotopic to the identity, it follows that ${\Sigma_f({\mathsf{S}})}\cong {\mathsf{S}}\times {\mathbb{S}^1}$ is a compact abelian topological group whose character group is given by $${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})\cong {\mathrm{Char}}({\mathsf{S}})\times {\mathrm{Char}}({\mathbb{S}^1})\cong {\mathbb{Q}}\times {\mathbb{Z}}.$$ The associated suspension flow $\phi_t:{\Sigma_f({\mathsf{S}})}{\longrightarrow}{\Sigma_f({\mathsf{S}})}$ is given by: $$\phi_t(z,s):=(f^m(z),t+s-m),\qquad (m\leq t+s < m+1).$$
Now, for any given character $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$, there exists a unique 1–cocycle $$C_{\chi_{q,n}}:{\mathbb R}\times {\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb R}$$ associated to $\chi_{q,n}$ (see section \[rotation\_element\] for complete information) such that $$\chi_{q,n}(\phi_t(z,s)) = \exp(2\pi iC_{\chi_{q,n}}(t,(z,s)))\cdot \chi_{q,n}(z,s),$$ for every $(z,s)\in {\Sigma_f({\mathsf{S}})}$ and $t\in {\mathbb R}$. From here it is obtained an explicit expression for the 1–cocycle $C_{\chi_{q,n}}(t,(z,s))$ and, by Birkhoff’s ergodic theorem, there is a well–defined homomorphism $$H_f:{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb R}$$ given by $$H_f(\chi_{q,n}):= \int_{{\Sigma_f({\mathsf{S}})}} C_{\chi_{q,n}}(1,(z,s)) d\nu,$$ where $\nu$ is a $\phi_t$–invariant Borel probability measure on ${\Sigma_f({\mathsf{S}})}$. Finally, the well–defined continuous homomorphism $$\rho(f):{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb{S}^1}$$ given by $$\rho(f)(\chi_{q,n}):= \exp(2\pi iH_f(\chi_{q,n}))$$ determines an element in ${\mathrm{Char}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}))\cong {\mathsf{S}}\times {\mathbb{S}^1}$ which does not depend on the second component. By Pontryagin’s duality theorem, it determines an element $\rho(f)\in {\mathsf{S}}$ called the associated to $f$, which is the generalized Poincaré rotation number.
As expected, $\rho(f)$ is an element in the solenoid itself and it measures, in some sense, the average displacement of points under iteration of $f$ along the one–dimensional leaves with the Euclidean metric.
Since ${\mathsf{S}}$ is torsion–free, it follows that there does not exist a notion of “rational” and so we only have to give a suitable definition of what “irrational” is. We proceed as follows:
**Definition:** An element $\alpha\in {\mathsf{S}}$ is called if the (additive) subgroup generated by $\alpha$ in ${\mathsf{S}}$ is dense.
**Definition:** A homeomorphism $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is said to have if there exists $C>0$ such that the sequence $$\{F^n(z) - z - n\tau(F)\}_{n\geq 1}$$ is uniformly bounded by $C$. Here, $F$ is any lift of $f$, $\tau(F)$ is a lifting of $\rho(f)$ to ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ and $z\in {\mathbb R}\times {\widehat{{\mathbb{Z}}}}$. (See section \[Poincare\_theorem\] for details.)
The generalized Poincaré theorem can be stated as follows:
**Theorem:** Suppose that $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is any homeomorphism isotopic to the identity with irrational rotation element $\rho(f)\in {\mathsf{S}}$. Then, $f$ is semiconjugated to the irrational rotation $R_{\rho(f)}$ if and only if $f$ has bounded mean variation. The semiconjugacy is actually a conjugacy if the orbits of $f$ are dense.
It should be pointed out that similar studies of the Poincaré theory have been developed very recently by several authors. In the paper [@Jag], T. Jäger proved that a minimal homeomorphism of the $d$–dimensional torus is semiconjugated to an irrational rotation if and only if it is a pseudo–irrational rotation with bounded mean motion (see also [@AJ] and [@BCJL]). J. Kwapisz (see [@Kwa]) gave a definition of a rotation element for homeomorphisms of the real line with almost periodic displacement; when the displacement is limit periodic, the corresponding convex hull is a compact abelian one–dimensional solenoidal group. So, in this sense, our study of the rotation element is strongly related to that of Kwapisz. However, we started by considering ${\mathsf{S}}$, a compact abelian group, as being a “generalized circle” and developing the theory from this perspective. We give two different (equivalent) definitions of the rotation element for homeomorphism isotopic to the identity, and also present a generalized notion when the homeomorphism is isotopic to a rotation by an element not in the base leaf, which produces a slightly different situation.
The paper is organized as follows: In section \[solenoid\] are defined the algebraic universal covering space of the circle, its character group, the suspension of a homeomorphism isotopic to the identity and its corresponding character group. Section \[rotation\_element\] introduces the notion of 1–cocycle and gives the definition of the generalized rotation element. In order to define this generalized rotation element $\rho(f)$, it is necessary to use the following ingredients: Pontryagin’s duality theory for compact abelian groups, the Bruschlinsky–Eilenberg homology theory and Schwartzman theory of asymptotic cycles as well as the notion of 1–cocycle and ergodic theory. The generalized Poincaré theorem is proved in section \[Poincare\_theorem\] and section \[minimal\_sets\] is dedicated to the study of minimal sets. Section \[rotation\_translation\] points out a slightly general definition for the case of homeomorphisms isotopic to translations whose rotation element is not in the base leaf.
The algebraic universal covering space of the circle {#solenoid}
====================================================
The universal one–dimensional solenoid
--------------------------------------
### Basic definitions {#basic-definitions .unnumbered}
It is well–known, by covering space theory, that for any integer $n\geq 1$, it is defined the unbranched covering space of degree $n$, $p_n:{\mathbb{S}^1}{\longrightarrow}{\mathbb{S}^1}$ given by $z\longmapsto {z^n}$. If $n,m\in {\mathbb{Z}}^+$ and $n$ divides $m$, then there exists a unique covering map $p_{nm}:{\mathbb{S}^1}{\longrightarrow}{\mathbb{S}^1}$ such that $p_n \circ p_{nm} = p_m$. This determines a projective system of covering spaces $\{{\mathbb{S}^1},p_n\}_{n\geq 1}$ whose projective limit is the $${\mathsf{S}}:=\lim_{\longleftarrow} {\mathbb{S}^1},$$ with canonical projection ${\mathsf{S}}{\longrightarrow}{\mathbb{S}^1}$, determined by projection onto the first coordinate, which determines a locally trivial ${\widehat{{\mathbb{Z}}}}$–bundle structure ${\widehat{{\mathbb{Z}}}}\hookrightarrow {\mathsf{S}}{\longrightarrow}{\mathbb{S}^1}$. ${\widehat{{\mathbb{Z}}}}:=\underleftarrow{\lim}\; {\mathbb{Z}}/m{\mathbb{Z}}$ is the profinite completion of ${\mathbb{Z}}$, which is a compact, perfect and totally disconnected abelian topological group homeomorphic to the Cantor set. Being ${\widehat{{\mathbb{Z}}}}$ the profinite completion of ${\mathbb{Z}}$, it admits a canonical inclusion of ${\mathbb{Z}}$ whose image is dense.
${\mathsf{S}}$ can also be realized as the orbit space of the ${\mathbb{Q}}$–bundle structure ${\mathbb{Q}}\hookrightarrow {\mathbb{A}}{\longrightarrow}{\mathbb{A}}/{\mathbb{Q}}$, where ${\mathbb{A}}$ is the adèle group of the rational numbers which is a locally compact abelian group, ${\mathbb{Q}}$ is a discrete subgroup of ${\mathbb{A}}$ and ${\mathbb{A}}/{\mathbb{Q}}\cong {\mathsf{S}}$ is a compact abelian group (see [@RV]). From this perspective, ${\mathbb{A}}/{\mathbb{Q}}$ can be seen as a projective limit whose $n$–th component corresponds to the unique covering of degree $n\geq 1$ of ${\mathbb{S}^1}$. ${\mathsf{S}}$ is also called the of the circle ${\mathbb{S}^1}$. The Galois group of the covering is ${\widehat{{\mathbb{Z}}}}$, the of ${\mathbb{S}^1}$.
By considering the properly discontinuously free action of ${\mathbb{Z}}$ on ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ given by $$\gamma\cdot (x,t) := (x+\gamma,t-\gamma) \quad (\gamma\in {\mathbb{Z}}),$$ ${\mathsf{S}}$ is identified with the orbit space ${\mathbb R}\times_{{\mathbb{Z}}} {\widehat{{\mathbb{Z}}}}$. Here, ${\mathbb{Z}}$ is acting on ${\mathbb R}$ by covering transformations and on ${\widehat{{\mathbb{Z}}}}$ by translations. The path–connected component of the identity element $0\in {\mathsf{S}}$ is called the and will be denoted by ${\mathcal{L}}_0$. Clearly, ${\mathcal{L}}_0$ is the image of ${\mathbb R}\times \{0\}$ under the canonical projection ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathsf{S}}$ and it is homeomorphic to ${\mathbb R}$.
In summary, ${\mathsf{S}}$ is a compact, connected, abelian topological group and also a one–dimensional lamination where each “leaf" is a simply connected one–dimensional manifold, homeomorphic to the universal covering space ${\mathbb R}$ of ${\mathbb{S}^1}$, and a typical “transversal" is isomorphic to the Cantor group ${\widehat{{\mathbb{Z}}}}$. ${\mathsf{S}}$ also has a leafwise ${\mathrm{C}}^\infty$ Riemannian metric (i.e., ${\mathrm{C}}^\infty$ along the leaves) which renders each leaf isometric to the real line with its standard metric. So, it makes sense to speak of a rigid translation along the leaves. The leaves also have a natural order equivalent to the order of the real line.
### Characters of ${\mathsf{S}}$ {#charactersS .unnumbered}
Denote by ${\mathrm{Char}}({\mathsf{S}}):={\mathrm{Hom}}({\mathsf{S}},{\mathbb{S}^1})$ the topological group which consists of all continuous homomorphisms from ${\mathsf{S}}$ into the multiplicative group ${\mathbb{S}^1}$ endowed with the uniform topology. This group is called the of ${\mathsf{S}}$ or, the of ${\mathsf{S}}$. From what has been said before, we know that ${\mathsf{S}}\cong {\mathbb{A}}/{\mathbb{Q}}$ and, since ${\mathbb{A}}$ is self–dual (i.e., ${\mathbb{A}}\cong{\mathrm{Char}}({\mathbb{A}})$), it follows that ${\mathrm{Char}}({\mathsf{S}})\cong {\mathbb{Q}}$. If $\check{H}^1({\mathsf{S}},{\mathbb{Z}})$ denotes the first $\mathrm{\check{C}}$ech cohomology group of ${\mathsf{S}}$ with coefficients in ${\mathbb{Z}}$, then $\check{H}^1({\mathsf{S}},{\mathbb{Z}})\cong {\mathrm{Char}}({\mathsf{S}})$.
If $\chi:{\mathsf{S}}{\longrightarrow}{\mathbb{S}^1}$ is any character, then $\chi$ is completely determined by its values when restricted to the dense one–parameter subgroup ${\mathcal{L}}_0$. Since ${\mathcal{L}}_0$ is canonically isomorphic to the additive group $({\mathbb R},+)$, the restriction of $\chi$ to ${\mathcal{L}}_0$ is of the form $t\longmapsto \exp(2\pi its)$. It is shown (see e.g., [@RV]) that $s$ must be rational. Now, given any $z\in {\mathsf{S}}$, there exists an integer $n\in {\widehat{{\mathbb{Z}}}}\subset {\mathsf{S}}$ such that $z+n \in {\mathcal{L}}_0$. The value of the character $\chi$ at $z$ is given by $$\chi(z) = \exp(2\pi iq(z+n)) = \exp(2\pi iqz)$$ for any $q\in {\mathbb{Q}}$. We will write $\chi(z) = {\mathrm{Exp}}{(2\pi iqz)}$.
### Homeomorphisms of ${\mathsf{S}}$ {#homeomorphisms-of-mathsfs .unnumbered}
We only consider the group which consists of all homeomorphisms of ${\mathsf{S}}$ which are isotopic to the identity and can be written as $f = {\mathrm{id}}+ \varphi$, where $\varphi:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is given by $\varphi(z)=f(z)-z$ and describes the displacement of points $z\in {\mathsf{S}}$ along the leaf containing it. The symbol “–” refers to the additive group operation in the solenoid. Denote the set of all such functions $\varphi$ by ${\mathrm{C}}_+({\mathsf{S}})$.
Since $f$ is a homeomorphism which preserves the order in the leaves, it follows that there is a one to one correspondence between ${\mathrm{C}}_+({\mathsf{S}})$ and the set of real–valued continuous functions with the property that if $x$ and $y$ are in the same one–dimensional leaf and if $x<y$, then $x+\varphi(x)<y+\varphi(y)$. Therefore, ${\mathrm{C}}_+({\mathsf{S}})$ can be identified with the Banach space of real–valued continuous functions ${\mathrm{C}}({\mathsf{S}},{\mathbb R})$.
As mentioned in the introduction, the solenoid has a leafwise ${\mathrm{C}}^\infty$ Riemannian metric (i.e., ${\mathrm{C}}^\infty$ along the leaves) which renders each leaf isometric to the real line with its standard metric. Hence, the displacement function $\varphi$ can be thought of as a continuous real–valued function which we denote with the same symbol $\varphi$. In fact, since every leaf ${\mathcal{L}}\subset {\mathsf{S}}$ is dense, the restriction of this function to ${\mathcal{L}}$, denoted by $\varphi_{{\mathcal{L}}}$, completely determines the function. Furthermore, $\varphi_{{\mathcal{L}}}$ is an almost periodic function whose convex hull is the solenoid and thus, $\varphi_{{\mathcal{L}}}$ is a limit periodic function (see [@Pon]).
Denote by ${\mathrm{Homeo}}_+({\mathsf{S}})$ the group of all homeomorphisms $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ which are isotopic to the identity and can be written as $f = {\mathrm{id}}+ \varphi$, with $\varphi\in {\mathrm{C}}_+({\mathsf{S}})$; i.e., $${\mathrm{Homeo}}_+({\mathsf{S}}):= \{f\in {\mathrm{Homeo}}({\mathsf{S}}): f = {\mathrm{id}}+ \varphi, \; \varphi\in {\mathrm{C}}_+({\mathsf{S}})\}.$$
The suspension of a homeomorphism
---------------------------------
Let $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ be any homeomorphism isotopic to the identity. In ${\mathsf{S}}\times [0,1]$ consider the equivalence relation $$(z,1)\sim (f(z),0)\qquad (z\in {\mathsf{S}}).$$
The of $f$ is the compact space $${\Sigma_f({\mathsf{S}})}:= {\mathsf{S}}\times [0,1] /(z,1)\sim (f(z),0).$$
Since $f$ is isotopic to the identity, it follows that ${\Sigma_f({\mathsf{S}})}\cong {\mathsf{S}}\times {\mathbb{S}^1}$ is a compact abelian topological group. In ${\Sigma_f({\mathsf{S}})}$ there is a well–defined flow $\phi:{\mathbb R}\times {\Sigma_f({\mathsf{S}})}{\longrightarrow}{\Sigma_f({\mathsf{S}})}$, called the of $f$, which is given by $$\phi(t,(z,s)):=(f^m(z),t+s-m),$$ if $m\leq t+s < m+1$. The canonical projection $\pi:{\mathsf{S}}\times [0,1]{\longrightarrow}{\Sigma_f({\mathsf{S}})}$ sends ${\mathsf{S}}\times \{0\}$ homeomorphically onto its image $\pi({\mathsf{S}}\times \{0\})\equiv {\mathsf{S}}$ and every orbit of the suspension flow intersects ${\mathsf{S}}$. The orbit of any $(z,0)\in {\Sigma_f({\mathsf{S}})}$ must coincide with the orbit $\phi_t(z,0)$ at time $0\leq t\leq T$ for $T$ an integer.
### Characters of the suspension {#characters-of-the-suspension .unnumbered}
Denote by ${\mathrm{C}}({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})$ the topological space which consists of all continuous functions defined on ${\Sigma_f({\mathsf{S}})}$ with values in the unit circle ${\mathbb{S}^1}$ with the topology of uniform convergence on compact sets (i.e., the compact–open topology). Clearly, this is an abelian topological group under pointwise multiplication. The subset $R({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})\subset {\mathrm{C}}({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})$ which consists of continuous functions $h:{\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb{S}^1}$ that can be written as $h(z,s):=\exp(2\pi i\psi(z,s))$ with $\psi:{\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb R}$ a continuous function, is a closed subgroup. Hence, the quotient group ${\mathrm{C}}({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})/R({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})$ is a topological group. By Bruschlinsky–Eilenberg’s theory (see [@Sch]), it is known that $$\check{H}^1({\Sigma_f({\mathsf{S}})},{\mathbb{Z}})\cong {\mathrm{C}}({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})/R({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1}).$$ Since $$\check{H}^1({\Sigma_f({\mathsf{S}})},{\mathbb{Z}})\cong {\mathrm{Char}}({\Sigma_f({\mathsf{S}})}),$$ we conclude that $${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})\cong {\mathrm{C}}({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1})/R({\Sigma_f({\mathsf{S}})},{\mathbb{S}^1}).$$
On the other hand, ${\Sigma_f({\mathsf{S}})}\cong {\mathsf{S}}\times {\mathbb{S}^1}$ implies that its character group is given by $${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})\cong {\mathrm{Char}}({\mathsf{S}})\times {\mathrm{Char}}({\mathbb{S}^1})\cong {\mathbb{Q}}\times {\mathbb{Z}}.$$ According with the definition of ${\mathrm{Exp}}$ in \[charactersS\], given any element $(q,n)\in {\mathbb{Q}}\times {\mathbb{Z}}$, the corresponding character $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ can be written as $$\begin{aligned}
\chi_{q,n}(z,s) &= {\mathrm{Exp}}(2\pi iqz)\cdot \exp(2\pi ins)\\
&= {\mathrm{Exp}}(2\pi i(qz+ns)),\end{aligned}$$ for any $(z,s)\in {\Sigma_f({\mathsf{S}})}$.
### Measures {#measures .unnumbered}
Given any $f$–invariant Borel probability measure $\mu$ on ${\mathsf{S}}$ and $\lambda$ the usual Lebesgue measure on $[0,1]$, the product measure $\mu\times \lambda$ leads to define a $\phi_t$–invariant Borel probability measure on ${\Sigma_f({\mathsf{S}})}$. Reciprocally, given any $\phi_t$–invariant Borel probability measure $\nu$ on ${\Sigma_f({\mathsf{S}})}$, it can be defined, by disintegration with respect to the fibers, an $f$–invariant Borel probability measure $\mu$ on ${\mathsf{S}}$. Denote by ${\mathcal{P}_f({\mathsf{S}})}$ the weak$^*$ compact convex space of $f$–invariant Borel probability measures defined on ${\mathsf{S}}$.
The Rotation element {#rotation_element}
====================
1–cocycles
----------
A 1– associated to the suspension flow $\phi_t$ is a continuous function $$C:{\mathbb R}\times {\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb R}$$ which satisfies the relation $$C(t+u,(z,s)) = C(u,\phi_t(z,s)) + C(t,(z,s)),$$ for every $t,u\in {\mathbb R}$ and $(z,s)\in {\Sigma_f({\mathsf{S}})}$. The set which consists of all 1–cocycles associated to $\phi_t$ is an abelian group denoted by ${\mathrm{C}}^1(\phi)$. A 1– is the 1–cocyle determined by a continuous function $\psi:{\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb R}$ such that $$C(t,(z,s)):= \psi(z,s) - \psi(\phi_t(z,s)).$$ The set of 1–coboundaries $\Gamma^1(\phi)$ is a subgroup of ${\mathrm{C}}^1(\phi)$ and the quotient group $$H^1(\phi):={\mathrm{C}}^1(\phi)/\Gamma^1(\phi),$$ is called the 1– associated to $\phi_t$. The proof of the next proposition (for an arbitrary compact metric space) can be seen in [@Ath].
\[associated\_cocycle\] For every continuous function $h:{\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb{S}^1}$ there exists a unique 1–cocycle $C_h:{\mathbb R}\times {\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb R}$ associated to $h$ such that $$h(\phi_t(z,s)) = \exp(2\pi iC_h(t,(z,s)))\cdot h(z,s),$$ for every $(z,s)\in {\Sigma_f({\mathsf{S}})}$ and $t\in {\mathbb R}$.
This proposition implies that there is a well–defined homomorphism $${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})\cong \check{H}^1({\Sigma_f({\mathsf{S}})},{\mathbb{Z}}){\longrightarrow}H^1(\phi)$$ by sending any character $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ to the cohomology class $[C_{\chi_{q,n}}]$, where $C_{\chi_{q,n}}$ is the unique 1–cocycle associated to $\chi_{q,n}$.
Applying the above proposition to any non–trivial character $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ the following relation is obtained: $$\chi_{q,n}(\phi_t(z,s)) = \exp(2\pi iC_{\chi_{q,n}}(t,(z,s)))\cdot \chi_{q,n}(z,s).$$
Using the explicit expressions for the characters on both sides of the above equation, the next equalities hold $$\begin{aligned}
\chi_{q,n}(\phi_t(z,s)) &= \chi_{q,n}(f^m(z),t+s-m)\\
&= {\mathrm{Exp}}(2\pi i(qf^m(z) + n(t+s-m)))\\
&= {\mathrm{Exp}}(2\pi i(qf^m(z) + nt + ns))\end{aligned}$$ and
$$\chi_{q,n}(z,s) = {\mathrm{Exp}}(2\pi i(qz+ns)).$$
Comparing these two expressions it follows that $$\label{cocycle1}
C_{\chi_{q,n}}(t,(z,s)) = q(f^m(z)-z) + nt.$$
Now recall that $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is a homeomorphism isotopic to the identity of the form $f={\mathrm{id}}+ \varphi$, where $\varphi:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is the displacement function, where, as described before, $\varphi$ can also be considered as a real–valued function on the solenoid. If $t=1$, then $m=1$ and the 1–cocycle at time $t=1$ is $$\label{time1cocycle}
C_{\chi_{q,n}}(1,(z,s)) = q\varphi(z) + n.$$
The rotation element {#the-rotation-element}
--------------------
If $\nu$ is any $\phi_t$–invariant Borel probability measure on ${\Sigma_f({\mathsf{S}})}$, by Birkhoff’s ergodic theorem there is a well–defined homomorphism $H^1(\phi){\longrightarrow}{\mathbb R}$ given by $$[C_\chi]\longmapsto \int_{{\Sigma_f({\mathsf{S}})}} C_\chi(1,(z,s)) d\nu.$$
Now, composing the two homomorphisms $${\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}H^1(\phi){\longrightarrow}{\mathbb R}$$ it is obtained a well–defined homomorphism $H_{f,\nu}:{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb R}$ given by $$H_{f,\nu}(\chi_{q,n}):= \int_{{\Sigma_f({\mathsf{S}})}} C_{\chi_{q,n}}(1,(z,s)) d\nu.$$ Denote by $\mu$ the $f$–invariant Borel probability measure on ${\mathsf{S}}$ obtained by disintegration of $\nu$ with respect to the fibers. Evaluating the above integral using equation \[time1cocycle\] gives $$\begin{aligned}
H_{f,\nu}(\chi_{q,n}) &= \int_{{\Sigma_f({\mathsf{S}})}} (q\varphi + n)d\nu\\
&= q\int_{\mathsf{S}}\varphi d\mu + n.\end{aligned}$$
Hence, $H_{f,\nu}$ determines an element in ${\mathrm{Hom}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}),{\mathbb R})$ for each measure $\nu$ in ${\Sigma_f({\mathsf{S}})}$, and therefore, for each measure $\mu\in {\mathcal{P}_f({\mathsf{S}})}$. Hence, one gets a well–defined function $$H_f:{\mathcal{P}_f({\mathsf{S}})}{\longrightarrow}{\mathrm{Hom}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}),{\mathbb R})$$ defined as $\mu\longmapsto H_{f,\mu}$, where $H_{f,\mu}$ is given by $$H_{f,\mu}(\chi_{q,n}) = q\int_{\mathsf{S}}\varphi d\mu + n.$$ By post–composing $H_f$ with the continuous homomorphism $${\mathrm{Hom}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}),{\mathbb R}){\longrightarrow}{\mathrm{Char}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}))$$ given by $$H_{f,\mu}\longmapsto \pi\circ H_{f,\mu},$$ where $\pi:{\mathbb R}{\longrightarrow}{\mathbb{S}^1}$ is the universal covering projection, we obtain a well–defined continuous function $\rho:{\mathcal{P}_f({\mathsf{S}})}{\longrightarrow}{\mathrm{Char}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}))$ given by $$\mu\longmapsto \rho_\mu:= \pi\circ H_{f,\mu}.$$
That is, for each $\mu\in {\mathcal{P}_f({\mathsf{S}})}$, there exists a well–defined continuous homomorphism $$\rho_\mu:{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb{S}^1}$$ given by $$\begin{aligned}
\rho_\mu(\chi_{q,n}) &:= \exp(2\pi iH_{f,\mu}(\chi_{q,n}))\\
&= \exp \left(2\pi iq \int_{\mathsf{S}}\varphi d\mu\right).\\\end{aligned}$$
By Pontryagin’s duality theorem, $${\mathrm{Char}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}))\cong {\Sigma_f({\mathsf{S}})}$$ and therefore $\rho_\mu\in {\Sigma_f({\mathsf{S}})}$. Since ${\Sigma_f({\mathsf{S}})}\cong {\mathsf{S}}\times {\mathbb{S}^1}$ and $\rho_\mu(\chi_{q,n})=\rho_\mu(\chi_{q,0})$, it follows that $\rho_\mu$ does not depend on the second component and so, the identification $\rho_\mu = (\rho_\mu,1)\in {\mathsf{S}}\times {\mathbb{S}^1}$ can be made. More precisely, it is well known that every non–trivial character of ${\mathrm{Char}}({\mathsf{S}})\cong {\mathbb{Q}}$ is of the form $\chi_a$ for some $a\in {\mathbb{A}}$ and the map ${\mathbb{A}}{\longrightarrow}{\mathrm{Char}}({\mathbb{Q}})$ given by $a\longmapsto \chi_a$ induces an isomorphism ${\mathrm{Char}}({\mathbb{Q}})\cong {\mathbb{A}}/{\mathbb{Q}}\cong {\mathsf{S}}$. This produces a genuine element $\rho_\mu\in {\mathsf{S}}$.
The element $\rho_\mu(f):=\rho_\mu \in {\mathsf{S}}$ defined above is the ** associated to $f$ with respect to the measure $\mu$.
By definition, $\rho_\mu(f)$ can be identified with the element $\int_{\mathsf{S}}\varphi d\mu$ in the solenoid ${\mathsf{S}}$ determined by the character of ${\mathbb{Q}}$ given by $$q\longmapsto \exp \left(2\pi iq\int_{\mathsf{S}}\varphi d\mu \right).$$
That is, $\rho_\mu(f)$ is *solenoid–valued*.
If $\frak R:{\mathcal{P}_f({\mathsf{S}})}{\longrightarrow}{\mathsf{S}}$ is the map given by $\mu\longmapsto \rho_\mu(f)$, then $\frak R$ is continuous from ${\mathcal{P}_f({\mathsf{S}})}$ to ${\mathsf{S}}$. Since ${\mathcal{P}_f({\mathsf{S}})}$ is compact and convex, and $f$ is isotopic to the identity, the image $\frak R({\mathcal{P}_f({\mathsf{S}})})$ is a compact interval $I_f$ in the one–parameter subgroup ${\mathcal{L}}_0$. This interval is called the of $f$. Since ${\mathcal{L}}_0$ is canonically isomorphic to ${\mathbb R}$, it is possible to identify $I_f$ with an interval in the real line.
We say that $f$ is a ** if $I_f$ consists of a single point $I_f=\{\alpha\}$ and $\alpha$ is an irrational element in ${\mathsf{S}}$ (see section \[Poincare\_theorem\]).
In particular, if $f$ is uniquely ergodic, then the interval $I_f$ reduces to a point and the rotation element is a unique element of ${\mathsf{S}}$.
The proof of the next proposition is clear from the definitions:
If $\mu_1$ and $\mu_2$ are any two elements in ${\mathcal{P}_f({\mathsf{S}})}$ which belong to the same measure class, then $\rho_{\mu_1}(f)=\rho_{\mu_2}(f)$.
The rotation element of $f$ can be interpreted as the exponential of an asymptotic cycle, in the sense of Schwartzman, of the suspension flow $\{\phi_t\}_{t\in{\mathbb R}}$ of $f$ (see [@Sch]; see also [@AK], [@Pol]). If $A_\nu\in{\mathrm{Hom}}(\check{H}^1({\Sigma_f({\mathsf{S}})},{\mathbb{Z}}),{\mathbb R})={\mathrm{Hom}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}),{\mathbb R})$ denotes the asymptotic cycle associated to the $\{\phi_t\}_{t\in{\mathbb R}}$–invariant measure $\nu$, then $\rho_\nu(f)=\exp(2\pi{i}A_\nu)$.
From Birkhoff’s ergodic theorem, for any ergodic $f$–invariant measure $\mu$, $$\int_{\mathsf{S}}\varphi d\mu = \underset{n\to\infty}\lim \ \frac{1}{n} \sum_{j=0}^n \varphi(f^j(z)),$$ for $\mu$–almost every point $z\in{\mathsf{S}}$. We could have used this to define the rotation element with respect to an (ergodic) measure. Since we wanted to make explicit the role of the measure, we used the theory of asymptotic cycles in the sense of Schwartzman. (Compare [@Kwa], Theorem 3.)
Basic example and properties
----------------------------
### Basic example: Rotations {#rotation .unnumbered}
Let $\alpha$ be any element in ${\mathcal{L}}_0\subset {\mathsf{S}}$ and consider the rotation $R_\alpha:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ given by $z\longmapsto z + \alpha$. The suspension flow $\phi_t:{\mathsf{S}}\times {\mathbb{S}^1}{\longrightarrow}{\mathsf{S}}\times {\mathbb{S}^1}$ is given by $$\phi_t(z,s):=(z+m\alpha,t+s-m),$$ if $m\leq t+s< m+1$. If $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ is any non–trivial character, then $$H_{R_\alpha,\mu}(\chi_{q,n}) = q\int_{\mathsf{S}}\alpha d\mu + n = q\alpha + n.$$ This implies that $$\rho_\mu(R_\alpha)(\chi_{q,n}) = \exp(2\pi i(q\alpha + n)) = \exp(2\pi iq\alpha)$$ and $\rho_\mu(R_\alpha) = \alpha$.
### Properties {#properties .unnumbered}
1. Let $f$ and $g$ be any two homeomorphisms isotopic to the identity and $h={\mathrm{id}}+\psi$. If $h\circ f=g\circ h$, then $\rho_\mu(f)=\rho_\mu(g)$. In particular, if $f$ is conjugated to a rotation $R_\alpha$, then $\rho_\mu(f)=\alpha$.
Observe first that $h\circ f=g\circ h$ implies that $h\circ f^m=g^m\circ h$ and $f^m + \psi\circ f^m=g^m\circ h - h + h$. That is $$f^m-{\mathrm{id}}= (g^m-{\mathrm{id}})\circ h + \psi - \psi\circ f^m.$$ Therefore, the 1–cocycle associated to any non–trivial character $\chi_{q,n}$ at time $t=1$ has the form $$\begin{aligned}
C_{\chi_{q,n}}(1,(z,s)) &= q(f(z)-z) + n\\
&= q[(g(h(z))-h(z)) + \psi(z) - \psi\circ f(z)] + n.\end{aligned}$$ Since $\mu$ is both $f$ and $g$ invariant, we get
$$\begin{aligned}
H_{f,\mu}(\chi_{q,n}) &= q\int_{\mathsf{S}}(f(z)-z)d\mu + n\\
&= q\int_{\mathsf{S}}(g(h(z))-h(z))dh_*\mu + n\\
&= H_{g,\mu}(\chi_{q,n}).\end{aligned}$$
Hence, $\rho_\mu(f)=\rho_\mu(g)$.
2. The function $\rho_\mu:{\mathrm{Homeo}}_+({\mathsf{S}}){\longrightarrow}{\mathsf{S}}$ given by $$f={\mathrm{id}}+\varphi\longmapsto \int_{\mathsf{S}}\varphi d\mu$$ is continuous with respect to the uniform topology in ${\mathrm{Homeo}}_+({\mathsf{S}})$.
3. Indeed, if $f$ has a fixed point $x$, then, $\varphi(x)=0$; if $\mu=\delta_x$ is the Dirac mass at $x$, then, $\int_{\mathsf{S}}\varphi d\mu=0$ and therefore $\rho_\mu(f)=0$. On the other hand, if $\rho_\mu(f)=0$, then $\int_{\mathsf{S}}\varphi d\mu=0$ and $\varphi$ must vanish at some point $x$ which must be a fixed point of $f$.
The Rotation element à la de Rham {#deRham}
---------------------------------
If $d\lambda$ denotes the usual Lebesgue measure on ${\mathbb{S}^1}$, then, given any character $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ there is a well–defined closed differential one–form on ${\Sigma_f({\mathsf{S}})}$ given by $$\omega_{\chi_{q,n}}:=\chi_{q,n}^* d\lambda.$$
Let $X$ be the vector field tangent to the flow $\phi_t$ and let $\nu$ be any $\phi_t$–invariant Borel probability measure on ${\Sigma_f({\mathsf{S}})}$. Define $$H_{f,\nu}:{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb R}$$ by $$H_{f,\nu}(\chi_{q,n}):= \int_{{\Sigma_f({\mathsf{S}})}} \omega_{\chi_{q,n}}(X) d\nu$$ and observe that this definition only depends on the cohomology class of $\omega_{\chi_{q,n}}$ and the measure class of $\nu$. Hence, we have a well–defined continuous homomorphism $\rho(f):{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb{S}^1}$ given by $$\rho(f)(\chi_{q,n}):=\exp(2\pi iH_{f,\nu}(\chi_{q,n})).$$
Thus, as before, $$\rho(f)\in {\mathrm{Char}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}))\cong {\Sigma_f({\mathsf{S}})}.$$
$\rho(f)$ is the rotation element associated to $f$ corresponding to $\nu$.
Let $\alpha$ be any element in ${\mathsf{S}}$ and consider the rotation $R_\alpha:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ given by $z\longmapsto z+\alpha$. The suspension flow $\phi_t:{\mathsf{S}}\times {\mathbb{S}^1}{\longrightarrow}{\mathsf{S}}\times {\mathbb{S}^1}$ is given by $$\phi_t(z,s):=(z+m\alpha,t+s-m)\quad (m\leq t+s < m+1).$$
Given any character $\chi_{q,n}\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ we have that $$\omega_{\chi_{q,n}} = qd\theta + nd\lambda$$ and the vector field $X$ associated to $\phi_t$ is constant. In this case, $H_{R_\alpha,\mu}(\chi_{q,n}) = \alpha q + n$ and therefore $$\rho(R_\alpha)(\chi_{q,n}) = \exp(2\pi iq\alpha).$$ That is, $\rho(R_\alpha)=\alpha$ which clearly coincides with the calculation made before.
Poincaré theory for compact abelian one–dimensional solenoidal groups {#Poincare_theorem}
=====================================================================
Irrational rotations
--------------------
Since ${\mathsf{S}}$ is torsion–free, it follows that a non trivial rotation has no periodic points. This means the dichotomy rational–irrational does not appear in this context and we only have to define what “irrational” means. The following seems to be an appropriate definition:
We say that $\alpha\in {\mathsf{S}}$ is ** if $\{n\alpha:n\in {\mathbb{Z}}\}$ is dense in ${\mathsf{S}}$. In classical terminology, ${\mathsf{S}}$ is said to be ** with generator $\alpha$.
Since ${\mathsf{S}}$ is a compact abelian topological group, the next theorem is classical (see e.g., [@Gra] for the general statements).
\[equidistribution\] If $\alpha\in {\mathsf{S}}$, then the following propositions are equivalent:
a. The rotation $R_\alpha:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ given by $z\longmapsto z+\alpha$ is ergodic with respect to the Haar measure on ${\mathsf{S}}$.
b. $\chi(\alpha)\neq 1$, for every non–trivial character $\chi\in {\mathrm{Char}}({\mathsf{S}})$.
c. ${\mathsf{S}}$ is a monothetic group with generator $\alpha$.
<!-- -->
a. Any non–trivial character $\chi\in {\mathrm{Char}}({\mathsf{S}})$ describes the solenoid ${\mathsf{S}}$ as a locally trivial fiber bundle over the circle ${\mathbb{S}^1}$ with typical fiber a Cantor group. In fact, there is such a fibration for each $q\in {\mathbb{Q}}\setminus \{1\}$.
b. For every $\alpha\in {\mathsf{S}}$ and every non–trivial character, $\chi\circ R_\alpha = R_{\chi(\alpha)}\circ \chi$.
c. If $\alpha\in {\mathsf{S}}$ is irrational, then $\chi(\alpha)\in {\mathbb{S}^1}$ is irrational, for every non–trivial character $\chi\in {\mathrm{Char}}({\mathsf{S}})$.
Generalized Poincaré theorem
----------------------------
Given an invariant measure of a homeomorphism of ${\mathsf{S}}$ we have defined an element in ${\mathsf{S}}$. *From now on, we fix a measure $\mu\in {\mathcal{P}_f({\mathsf{S}})}$.*
This determines a rotation element $\rho_\mu(f)$ of $f$, which will be simply denoted by $\rho(f)$ when the measure is understood.
Recall ${\mathsf{S}}$ is the orbit space of ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ under the ${\mathbb{Z}}$–action $$\gamma\cdot (x,t) = (x+\gamma,t-\gamma)\qquad (\gamma\in {\mathbb{Z}}).$$ Denote by $p:{\mathbb R}\times {\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathsf{S}}$ the canonical projection. It is clear that $p$ is an infinite cyclic covering.
Let $F:{\mathbb R}\times {\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ be a lifting of $f$ to ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$. Then, $F$ has the form $$F(x,t) = (F_t(x),R_\alpha(t)),$$ where ${\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathrm{Homeo}}({\mathbb R})$ is a continuous function given by $t\longmapsto F_t$, $F_t:{\mathbb R}{\longrightarrow}{\mathbb R}$ is a homeomorphism with limit periodic displacement $\Phi_t(x)$ (i.e., $\Phi$ is a uniform limit of periodic functions) and $\alpha\in {\widehat{{\mathbb{Z}}}}$ is a monothetic generator.
The condition of $F$ being equivariant with respect to the ${\mathbb{Z}}$–action is: $$F_{t-\gamma}(x+\gamma) = F_t(x) + \gamma,$$ for any $\gamma\in {\mathbb{Z}}$. That is, $F$ must commute with the integral translation $T_\gamma:{\mathbb R}\times {\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ given by $(x,t)\longmapsto (x+\gamma,t)$ and also must be invariant under the ${\mathbb{Z}}$–action in ${\mathrm{C}}({\widehat{{\mathbb{Z}}}},{\mathrm{Homeo}}({\mathbb R}))$.
It is very important to emphasize at this point that a lifting $F$ of $f$ exists and it is a homeomorphism of ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ due to the fact that $f$ is isotopic to the identity, which implies that $f$ leaves invariant the one–dimensional leaves of the solenoid. As a consequence of this fact, $F$ leaves invariant the one–dimensional leaves of ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$. Since each leaf is canonically identified with ${\mathbb R}$, the *displacement function* along the leaves can be defined in an obvious way.
We say that $f$ has ** if there exists $C>0$ such that the sequence $\{F^n(z) - z - n\tau(F)\}_{n\geq 1}$ is uniformly bounded by $C$. Here, $F$ is any lift of $f$, $\tau(F)$ is a lifting of $\rho(f)$ to ${\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ and $z\in {\mathbb R}\times {\widehat{{\mathbb{Z}}}}$.
We can now state and prove the generalized version of the Poincaré theorem: The first part of the proof follows closely the classical proof (see [@Ghys], [@Nav]).
\[Poincare\] Let $f\in {\mathrm{Homeo}}_+({\mathsf{S}})$ with irrational rotation element $\rho(f)$. Then, $f$ is semiconjugated to the irrational rotation $R_{\rho(f)}$ if and only if $f$ has bounded mean variation. Furthermore, under the same hypothesis, if $f$ is minimal, then $f$ is conjugated to the rotation $R_{\rho(f)}$.
The function $H:{\mathbb R}\times {\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathbb R}\times {\widehat{{\mathbb{Z}}}}$ given by $$z\longmapsto \sup_n \, \{F^n(z) - n\tau(F)\}$$ satisfies the following properties:
1. $H$ is surjective and continuous on the left.
2. $H\circ T_1 = T_1\circ H$
3. $H\circ F = T_{\tau(F)}\circ H$.
Conditions (1) and (2) are direct consequences of the definition of $H$ as a supremum. Condition (2) implies that $H$ descends to a map $h:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$. Condition (3) implies that $h\circ{f}=R_{\rho(f)}\circ{h}$. Following almost *verbatim* the arguments in ([@Ghys], [@Nav] Theorem 2.2.6), it follows that $h$ is continuous and semiconjugates $f$ to $R_{\rho(f)}$. This follows from the fact that $F$ preserves each leaf of the form ${\mathbb R}\times\{z\}$, with $z\in{\widehat{{\mathbb{Z}}}}$ and the map $g:{\mathbb R}{\longrightarrow}{\mathbb R}$ given by $t\mapsto\frak{p}(F(z,t))-t$ is a quasi morphism.
For the second part of the proof, suppose that $f$ is minimal and $h:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is a semiconjugacy: $h\circ f = R_{\rho(f)}\circ h$. First we describe some aspects of the geometry of $f$ which are a consequence of the fact that $f$ is semi conjugated to a translation. Write $\rho=\rho(f)$, and let $\chi:{\mathsf{S}}{\longrightarrow}{\mathbb{S}^1}$ be any continuous non–trivial character and let $$\mathcal F_{\chi}:= \{F^{\chi}_\theta = \chi^{-1}(\theta)\}_{\theta\in {\mathbb{S}^1}}$$ be the collection of fibers. In order to simplify the notation we will simply denote $F^{\chi}_{\theta}$ by $F_{\theta}$ but it is understood that the fibration depends on the nontrivial character. The translation $R_{\rho}$ permutes the fibers in the following way: $F_\theta$ maps to $F_{\chi(\rho)+\theta}$. Let $\mathcal G:= \{G_\theta:=h^{-1}(F_\theta)\}_{\theta\in {\mathbb{S}^1}}$. We have that $\mathcal G$ is a partition of ${\mathsf{S}}$ into disjoint compact subsets for each character. The quotient ${\mathsf{S}}/\mathcal{F}$ with the quotient topology is homeomorphic to the circle. Taking inverse images and considering that $h$ is surjective and $f$ is injective we have:
$$f^{-1}(G_{\theta}) = f^{-1}(h^{-1}(F_{\theta}))
= h^{-1}(R_\rho^{-1}(F_{\theta}))
= h^{-1}(F_{-\chi(\rho)+\theta})
= G_{-\chi(\rho)+\theta}.$$
Then $f$ permutes the elements of the partition $\mathcal{G}$ and, since $\rho$ is irrational, $\chi(\rho)\in {\mathbb{S}^1}$ is also irrational, this implies that $f$ acts as the rotation of angle $\chi(\rho)$ and the action is minimal on ${\mathbb{S}^1}={\mathsf{S}}/\mathcal{F}$.\
If $\theta_1,\theta_2\in {\mathbb{S}^1}$ are such that $\theta_1\neq \theta_2$, then, by surjectivity of $h$, $h(G_{\theta_1}) = h(h^{-1}(F_{\theta_1})) = F_{\theta_1}$, and, correspondingly, $h(G_{\theta_2}) = F_{\theta_2}$, which implies that $h(G_{\theta_1})\neq h(G_{\theta_2})$. Now, given two different points $z$ and $w$ in ${\mathsf{S}}$ there exist a character $\chi$ such that $\chi(z)\neq\chi(w)$ so that they lie in two different fibers of the fibration determined by $\chi$, this is because the group of characters separates points. In other words, given two different points $z$ and $w$ in ${\mathsf{S}}$ there exists a character $\chi$ for which $z$ and $w$ lie in different fibers of the corresponding fibration. The argument above implies that $h(z)\neq{h(w)}$ so $h$ is indeed injective. Therefore, $h$ is a homeomorphism.
The arguments in the second part of the previous proof prevent the existence of counter-examples of the types given by Mary Rees ([@Re]) or F. Beguin, S. Crovisier, T. Jäger and F. Le Roux ([@BCJL]). Indeed the solenoidal structure prevents such examples à la Denjoy.
The rotation element of a homeomorphism isotopic to a translation {#rotation_translation}
=================================================================
Here we are going to define the rotation element of a homeomorphism of the solenoid which is isotopic to a minimal translation by an element which is not in the base leaf.
First we describe the suspension of a minimal translation in a general compact abelian group $G$, which happens to be also a compact abelian group. Then we obtain, as a corollary, that the suspension of any homeomorphism of $G$ which is isotopic to a minimal translation is also a compact abelian group.
The suspension of a homeomorphism isotopic to a translation
-----------------------------------------------------------
Let $G$ be a metrizable compact abelian group and consider a minimal translation $T:z\longmapsto \alpha z$. Let $\Sigma_T(G)$ be the suspension of $T$.
$\Sigma_T(G)$ is a compact abelian group which contains $G$ as a closed subgroup and $$\Sigma_T(G)/G \cong {\mathbb{S}^1}.$$
Let $d$ be any invariant distance on $G$; i.e., $$d(hg_1,hg_2) = d(g_1,g_2)\quad (h\in G).$$ (Such a distance always exists by applying the well–known averaging method using the Haar measure.)
Consider in $G\times {\mathbb R}$ the distance $\hat d$ given by $$\hat {d}((g_1,t_1),(g_2,t_2)) := d(g_1,g_2) + {\left\vertt_1 - t_2\right\vert}.$$
Define $F:G\times {\mathbb R}{\longrightarrow}G\times {\mathbb R}$ by $F(g,t)=(\alpha g,t+1)$. Then, the distance $\hat d$ is invariant under $F$ and therefore induces a distance on $\Sigma_T(G)$. The canonical projection $p:G\times {\mathbb R}{\longrightarrow}\Sigma_T(G)$ is a local isometry. With respect to the induced distance, the suspension flow $\{F_s\}_{s\in{\mathbb R}}$ acts by isometries on $\Sigma_T(G)$.
It is a well–known fact that the group of isometries of $\Sigma_T(G)$ is a compact metric space with respect to the compact–open topology. It follows that $\{F_s\}_{s\in{\mathbb R}} \subset {\mathrm{Isom}}(\Sigma_T(G))$. Let $\Gamma=\{F_s\}_{s\in{\mathbb R}}$ and let $\bar\Gamma$ be the closure of $\Gamma$ in ${\mathrm{Isom}}(\Sigma_T(G))$. Then, $\bar\Gamma$ is a compact abelian group.
Let $x\in\Sigma_T(G)$ and let $\Gamma(x)\subset\Sigma_T(G)$ be the orbit of $x$. Since $T$ is a minimal translation, $\Gamma(x)$ is dense in $\Sigma_T(G)$. Let $k:\Gamma(x){\longrightarrow}\Gamma$ be defined by $F_s(x)\longmapsto{F_s}$. Clearly, $k$ is continuous and injective. Let us show that $k$ can be extended to a homeomorphism $\bar{k}:\Sigma_T(G){\longrightarrow}\bar\Gamma$. Let $y\in\Sigma_T(G)$. Then, there exists a sequence of real numbers $\{t_n\}_{n\in {\mathbb{N}}}$ such that $F_{t_n}(x) {\longrightarrow}y$ when $n\to \infty$.
By compactness of $\bar\Gamma$, there exists a subsequence $\{t_{n_i}\}_{n_i\in {\mathbb{N}}}$ such that $F_{t_{n_i}}$ converges to an isometry which we denote by $H_y$. This limiting isometry satisfies $H_y(x)=y$. Furthermore, if $L$ is any positive real number, define the segment of orbit of length $2L$: $$I(x,L):= \{F_s(x)|\,\, s\in[-L,L] \}.$$
Then, $H_y$ sends $I(x,L)$ isometrically onto $$I(y,L):=\{F_s(y)|\,\, s\in[-L,L] \}.$$ Since $L$ can be taken arbitrarily large, we see that $H_y$ is independent of the sequence and only depends on $y$.
Let us define $\bar{k}(y)=H_y$. Since $y$ was arbitrary this defines an extension of $k$ to all of $\Sigma_T(G)$. One can easily verify that $\bar{k}$ is continuous and, since $\bar{k}$ is injective on a dense subset of $\Sigma_T(G)$, it follows that it is injective. Since $\Sigma_T(G)$ is compact, $\bar{k}$ is a homeomorphism.
Therefore, $\bar \Gamma$ is homeomorphic to $\Sigma_T(G)$ and, via this homeomorphism, we define the abelian group structure on $\Sigma_T(G)$.
Finally, there is a natural continuous group epimorphism (namely, a character) $\Sigma_T(G) {\longrightarrow}{\mathbb{S}^1}$ whose kernel is a closed subgroup of $\Sigma_T(G)$ isomorphic to $G$, and hence $$\Sigma_T(G)/G \cong {\mathbb{S}^1}.$$
Since the suspension only depends on the isotopy class of the homeomorphism, the suspension of any homeomorphism $f:G{\longrightarrow}G$ isotopic to a minimal translation, is a compact abelian group.
The canonical examples of the above theorem are: the $2$–torus, which is a suspension of an irrational rotation on the circle; and the universal solenoid, which is a suspension of a minimal translation on ${\widehat{{\mathbb{Z}}}}$.
For the particular case of the universal solenoid ${\mathsf{S}}$, it is not true that any homeomorphism is isotopic to a translation. In fact, in [@Odd] it is proved the following result:
If ${\mathrm{Homeo}}_{\mathcal{L}}({\mathsf{S}})$ is the subgroup of ${\mathrm{Homeo}}({\mathsf{S}})$ consisting of homeomorphisms of ${\mathsf{S}}$ that preserves the base leaf, then $${\mathrm{Homeo}}({\mathsf{S}}) \cong {\mathrm{Homeo}}_{\mathcal{L}}({\mathsf{S}}) \times_{\mathbb{Z}}{\widehat{{\mathbb{Z}}}}.$$
For instance, by Pontryagin duality, the group of automorphisms of ${\mathsf{S}}$ is isomorphic to the group of automorphisms of ${\mathbb{Q}}$, which is ${\mathbb{Q}}^\star$, since any automorphism is determined by its value at $1$. Hence, any automorphism of ${\mathsf{S}}$ is never isotopic to a translation.
The elements in the same one–dimensional leaf ${\mathcal{L}}$ of ${\mathsf{S}}$, determines isotopic translations. If an element $f\in {\mathrm{Homeo}}({\mathsf{S}})$ is isotopic to a translation, then $f$ is isotopic to a translation of the form $\mathfrak{t} +\gamma$, where $\gamma\in {\mathcal{L}}\cap {\widehat{{\mathbb{Z}}}}$.
The rotation element of a homeomorphism isotopic to a translation {#the-rotation-element-of-a-homeomorphism-isotopic-to-a-translation}
-----------------------------------------------------------------
Suppose that $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is a homeomorphism which is isotopic to a minimal translation by an element not in the base leaf. According to the last section, the suspension ${\Sigma_f({\mathsf{S}})}$ is a compact abelian group and there is a natural continuous group epimorphism ${\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb{S}^1}$ whose kernel is a closed subgroup of ${\Sigma_f({\mathsf{S}})}$ isomorphic to ${\mathsf{S}}$. Hence, there is an exact sequence of compact abelian groups $$0{\longrightarrow}{\mathsf{S}}{\longrightarrow}{\Sigma_f({\mathsf{S}})}{\longrightarrow}{\mathbb{S}^1}{\longrightarrow}0.$$
By duality, there is an exact sequence of discrete groups $$0{\longrightarrow}{\mathbb{Z}}{\longrightarrow}{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb{Q}}{\longrightarrow}0.$$
In this situation, we do not know an explicit description for ${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ and its elements. Hence, the calculation of the $1$–cocyle to describe the homomorphism $H_{f,\mu}\in {\mathrm{Hom}}({\mathrm{Char}}({\Sigma_f({\mathsf{S}})}),{\mathbb R})$ is not neat, as we had in the isotopic to the identity case. However, knowing the fact that ${\Sigma_f({\mathsf{S}})}$ is a compact abelian group, it is possible to calculate the values of $H_{f,\mu}$ by restricting the elements in ${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ to elements in ${\mathrm{Char}}({\mathsf{S}})$. Proceeding as in section \[rotation\_element\], this can be done in the following way. (Compare [@Ath].)
Denote by $[z,t]$ the elements in the suspension ${\Sigma_f({\mathsf{S}})}$ which are now equivalence classes of pairs $(z,t)$ under the suspension relation. The suspension flow is given by $$\phi(t,[z,s]) := [f^m(z),t+s-m],$$ where $m\leq t+s < m+1$. As before, the canonical projection $\pi:{\mathsf{S}}\times [0,1]{\longrightarrow}{\Sigma_f({\mathsf{S}})}$ sends ${\mathsf{S}}\times \{0\}$ homeomorphically onto its image $\pi({\mathsf{S}}\times \{0\})\equiv {\mathsf{S}}$ and every orbit of the suspension flow intersects ${\mathsf{S}}$. If $\nu$ is any $\phi_t$–invariant Borel probability measure on ${\Sigma_f({\mathsf{S}})}$, then there is a well–defined homomorphism $H_{f,\nu}:{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb R}$ given by $$H_{f,\nu}(\chi):= \int_{{\Sigma_f({\mathsf{S}})}} C_{\chi}(1,[z,s]) d\nu,$$ where $C_{\chi}(1,[z,s])$ is the $1$–cocycle associated to any non–trivial character $\chi\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$, at time $t=1$.
Now recall that the $1$–cocycle associated to $\chi$ satisfies the relation (see section \[rotation\_element\]) $$C_{\chi}(t+u,[z,s]) = C_{\chi}(u,\phi_t([z,s]) + C_{\chi}(t,[z,s]),$$ for every $t,u\in {\mathbb R}$ and $[z,s]\in {\Sigma_f({\mathsf{S}})}$. Letting $s=0$, $u=t$ and $t=1$ in this relation it is obtained $$C_{\chi}(1+t,[z,0]) = C_{\chi}(t,[f(z),0]) + C_{\chi}(1,[z,0]).$$
Now setting $u=1$ and $s=0$ and applying the cocycle condition on the left hand of this expression we obtain: $$C_{\chi}(1+t,[z,0]) = C_{\chi}(1,[z,t]) + C_{\chi}(t,[z,0]).$$
Replacing this last equality in the first relation, rearranging the terms, and setting $t=s$, it follows that for any $s\in [0,1)$ and $z\in {\mathsf{S}}$ it holds $$C_\chi(1,[z,s]) = C_\chi(s,[f(z),0]) + C_\chi(1,[z,0]) - C_\chi(s,[z,0]).$$
If $\mu$ is the $f$–invariant Borel probability measure on ${\mathsf{S}}$ obtained by disintegration of $\nu$ with respect to the fibers, then by replacing the last expression in the definition of $H_{f,\nu}(\chi)$, and using Fubini’s theorem, it follows that $$\begin{aligned}
H_{f,\nu}(\chi)&= \int_{{\Sigma_f({\mathsf{S}})}} C_{\chi}(1,[z,s]) d\nu \\
&= \int_0^1 \left( \int_{{\mathsf{S}}} C_{\chi}(1,[z,s]) d\mu \right) ds \\
&= \int_0^1 \left( \int_{{\mathsf{S}}} C_{\chi}(1,[z,0]) d\mu \right) ds +
\int_0^1 \left( \int_{{\mathsf{S}}} \left[ C_\chi(s,[f(z),0]) - C_\chi(s,[z,0])\right] d\mu \right) ds \\
&= \int_{{\mathsf{S}}} C_{\chi}(1,[z,0]) d\mu. \end{aligned}$$
Hence $$H_{f,\nu}(\chi) = \int_{{\mathsf{S}}} C_{\chi}(1,[z,0]) d\mu.$$
Since $\chi$ is any non–trivial character in ${\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$, by restricting $\chi$ to ${\mathsf{S}}$, we obtain a non–trivial character $\chi_q\in {\mathrm{Char}}({\mathsf{S}})$. Applying proposition \[associated\_cocycle\] to $\chi_q$, the following relation is obtained $$\chi_q(f(z)) = \exp(2\pi i C_{\chi_q}(1,[z,0]) \chi_q(z).$$
This implies that $q(f(z)-z) - C_{\chi_q}(1,[z,0]) \in {\mathbb{Z}}$, and, since $q(f-{\mathrm{id}}) - C_{\chi_q}(1,[\cdot,0]))$ is a continuous function on ${\mathsf{S}}$, we conclude that $C_{\chi_q}(1,[z,0]) = q(f(z) - z)$ for any $z\in {\mathsf{S}}$. Since $f(z) - z = \varphi(z)$ is the displacement function along the leaves, the value of the homomorphism $H_{f,\nu}$, which now depends on $\mu$, at any character $\chi\in {\mathrm{Char}}({\Sigma_f({\mathsf{S}})})$ is given by $$H_{f,\mu}(\chi) = q\int_{\mathsf{S}}\varphi d\mu.$$
Then, for each $\mu\in {\mathcal{P}_f({\mathsf{S}})}$, there exists a well–defined continuous homomorphism $$\rho_\mu:{\mathrm{Char}}({\Sigma_f({\mathsf{S}})}){\longrightarrow}{\mathbb{S}^1}$$ given by $$\begin{aligned}
\rho_\mu(\chi) &:= \exp(2\pi iH_{f,\mu}(\chi))\\
&= \exp \left(2\pi i q\int_{\mathsf{S}}\varphi d\mu\right).\end{aligned}$$
This allows to establish the more general definition:
If $f:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$ is any homeomorphism which is isotopic to an irrational rotation by an elemente not in the base leaf, then the element $\rho_\mu(f) := \rho_\mu \in {\mathsf{S}}$ defined as above is the ** associated to $f$ with respect to the measure $\mu$.
If $f$ is isotopic to an irrational rotation $R_\alpha$ with $\alpha\notin {\mathcal{L}}_0$, then the rotation interval $I_f$ of $f$ can be identified with $I_f\subset {\mathcal{L}}_0 + \alpha$.
As indicated in the introduction (see section \[introduction\]), the theory developed in this paper can be rewritten verbatim for any compact abelian one–dimensional solenoidal group, since, by Pontryagin’s duality theory, any such group is the Pontryagin dual of a nontrivial additive subgroup $G\subset {\mathbb{Q}}$, where ${\mathbb{Q}}$ has the discrete topology. Denote by ${\mathsf{S}}_{G}$ such a group.
According with the theory developed before, we have the following:
Suppose that $f:{\mathsf{S}}_{G}{\longrightarrow}{\mathsf{S}}_{G}$ is any homeomorphism isotopic to the identity, or isotopic to a rotation by an element not in the base leaf, with irrational rotation element $\rho(f)$. Then, $f$ is semiconjugated to the irrational rotation $R_{\rho(f)}$ if and only if $f$ has bounded mean variation. Furthermore, under the same hypothesis, if $f$ is minimal, then $f$ is conjugated to the rotation $R_{\rho(f)}$.
Minimal sets {#minimal_sets}
============
In this section we characterize minimal sets for homeomorphisms of solenoids which are isotopic to the identity.
\[minimal\] If $f\in {\mathrm{Homeo}}_+({\mathsf{S}})$ is a homeomorphism which has irrational rotation element $\rho(f)\in {\mathsf{S}}$ and $f$ is semiconjugated to the rotation $R_{\rho(f)}$, then there exists a compact minimal $f$–invariant subset $K\subset {\mathsf{S}}$ with the following properties:
a. Either $K={\mathsf{S}}$ or, $K\subset {\mathsf{S}}$ is a Cantor set.
b. ${\mathrm{supp}}(\mu) = K$ for every $f$–invariant Borel probability measure and the minimal set $K$ is unique.
We can suppose that $f$ has no fixed points. This is equivalent to the condition that $\varphi$ does not vanish; in fact, we asume that $\sup_{z\in {\mathcal{L}}} {\left\vert\varphi(z)\right\vert} > 0$. The rotation $R_{\rho(f)}$ lifts to the translation $T_{\tilde{\rho}(f)}:{\mathbb R}\times{\widehat{{\mathbb{Z}}}}{\longrightarrow}{\mathbb R}\times{\widehat{{\mathbb{Z}}}}$ given by $(t,z)\mapsto(t+\tilde{\rho},z)$, $\tilde\rho\in{\mathbb R}$.
Let $K$ be a minimal set for $f$ and let ${\mathcal{L}}$ be a leaf of the solenoid. We know that there exists an isometric immersion $\tau:{\mathbb R}\hookrightarrow {\mathcal{L}}$. Let $M:= {\mathcal{L}}\cap K$ and $N:= \tau^{-1}(M)\subset {\mathbb R}$. Then, $N$ is closed and invariant subset of ${\mathbb R}$ under the homeomorphism $f_{\mathcal{L}}:= \tau^{-1}\circ f\circ \tau:{\mathbb R}{\longrightarrow}{\mathbb R}$. The continuous surjective map $h_{\mathcal{L}}:= \tau^{-1}\circ{h}\circ\tau$ semi conjugates $f_{\mathcal{L}}$ with the translation in ${\mathbb R}$ $R_{\mathcal{L}}$ given by $t\mapsto t+\rho$. It follows that $h_{\mathcal{L}}$ is injective when restricted to $N$.
There is one, and only one, of the following possibilities:
(1) $N$ is a closed, infinite and discrete subset of ${\mathbb R}$ and $N=\tau^{-1}({\mathcal{O}}_f(z))$ for some point $z\in {\mathcal{L}}$. In this case, $K=\overline{{\mathcal{O}}_f(z)}$ is a Cantor set which is isotopic to the kernel of a character $\chi$ of ${\mathsf{S}}$.
(2) $N$ is a set which is everywhere locally homeomorphic to a Cantor set, and $K$ is a Cantor set.
(3) $N={\mathbb R}$ and $K={\mathsf{S}}$.
The set of accumulation points of $N$, denoted by $N'$, is closed and invariant under $f_{\mathcal{L}}$. Therefore, $\tau(N')\subset K$ is closed and invariant under $f$. By minimality of $K$, if $N'$ is non–empty, then $K=\tau(N')$ and therefore $N=N'$; i.e., $N$ is a perfect set. On the other hand, if $N'=\emptyset$ we are precisely in case (1).
Suppose we are in case (1). Let $\psi_t:{\mathsf{S}}{\longrightarrow}{\mathsf{S}}$, $z\longmapsto z + \sigma(t)$ be the flow corresponding to the one–parameter subgroup $\sigma:{\mathbb R}\hookrightarrow {\mathsf{S}}$ whose orbits are the leaves of the solenoid. We claim that $K=\overline{{\mathcal{O}}_f(z)}$ is a global cross–section of the flow $\{\psi_t\}_{t\in{\mathbb R}}$. Indeed, $K$ intersects every orbit of the flow and the intersection of $K$ with an orbit (or leaf) is a discrete infinite subset of the orbit. Let $\varphi_K$ be the restriction of $\varphi$ to $K$. Then there exists a re–parametrization of the flow such that for each $z\in K$, $\psi_1(z) = z+\varphi_K(z) = f(z)$. Then, $\psi_1(K)=K$ and $\psi_t(K)\neq K$ if $t\in (0,1)$. This describes ${\mathsf{S}}$ as a fibre bundle $\pi:{\mathsf{S}}{\longrightarrow}{\mathbb{S}^1}$ over the circle ${\mathbb{S}^1}$ and, as we explained before, this fibration is equivalent to the one given by a nontrivial character $\chi$. Therefore $K$ can be deformed isotopically to the kernel of $\chi$.
Suppose now that $N$ has nonempty interior. Let $I$ be a nontrivial interval contained in $N$. We have that $f_{{\mathcal{L}}}$ is semi conjugate to the translation $t\mapsto t+\rho$ and $h_{\mathcal{L}}$ is injective on $N$, so $h_{\mathcal{L}}$ is in fact a conjugacy between $f_{\mathcal{L}}$ and $R_{\mathcal{L}}$. Hence, the diameter of the iterates of $I$ under $f$ cannot tend to zero.
Set $$E:= \overline{\tau\left(\bigcup{f_{\mathcal{L}}^n(I)}\right)}.$$
Then, $E\subset K$ is a nonempty closed set which is invariant under $f$. Therefore, $E=K$. We have that every point $z\in K$ is a left extreme point of a nontrivial interval $I_z$ such that ${\mathrm{diam}}(I_z)>C$ and which is contained in $K$ and in the leaf through $z$. The right extreme point of $I_z$ is also in $K$. From this it follows that $K={\mathsf{S}}$ and therefore, $N={\mathbb R}$. This proves completes the proof of (a).
To prove (b) it is an standard argument to prove that if $\mu$ is any $f$–invariant Borel probability measure, then its support is a closed and $f$–invariant set contained in $K$. By minimality of $K$, we conclude that $K={\mathrm{supp}}(\mu)$.
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[**Measurements of strong magnetic fields in umbra of sunspots: Crimea vs Mt. Wilson**]{}
Yu.T. Tsap$^{1,2}$, V.A. Perebeynos$^{1}$, A.V. Borisenko$^{1}$, N.I. Lozitska$^3$, N.I. Shtertser$^1$, G.G. Motorina$^{2,4}$, A.I. Kuleshova$^2$\
$^1$Crimean Astrophysical Observatory (Russian Academy of Sciences), Crimea, Russia, yur@craocrimea.ru
$^2$Central Astronomical Observatory at Pulkovo of Russian Academy of Sciences, St. Petersburg, 196140, Russia
$^3$Astronomical Observatory of Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
$^4$Astronomical Institute, Academy of Sciences of the Czech Republic, 251 65 Ondrejov, Czech Republic
[**Abstract.**]{}
The comparative analysis for 1324 measurements of the corresponding sunspot magnetic fields with $B > 2.5$ kG (according to Crimean data) obtained at Crimean and Mt. Wilson observatories from 2010 to 2017 has been carried out. It has been shown that the difference between measurements can exceed 1 kG in some cases. The averaged values of the magnetic field are equal to 2759 G (Crimea) and 2196 G (Mt. Wilson). The maximum sunspot magnetic field measured at Mt. Wilson does not reach 2.7 kG while according to Crimean data it can exceed 4.0 kG. The correlation coefficient between measurements of magnetic fields in different observatories does not exceed 0.22. The probable reasons of significant discrepancies are discussed.
Key words: Sun: sunspot magnetic fields
[**Introduction**]{}
Sunspots are the largest magnetic flux concentrations in the solar photosphere. The magnetic energy density inside sunspots is higher than the kinetic energy density, resulting in a partial suppression of the convection. Larger umbrae are darker and show a higher magnetic field strength.
Visual (photographic) measurements of sunspot magnetic fields using the Zeeman effect have continued until the present time at Crimean Astrophysical Observatory (CrAO) and Mount Wilson Observatory (MWO). Maximum distance between sigma components of simple triplets is measured in the profile of the line. Then the measured size is transferred in intensity of a magnetic field.
The longest series of observations of sunspot magnetic fields exists at MWO (Hale et al., 1919; Livingston, 2006). The MWO archive of drawings of umbrae, penumbra, and other magnetic structures began in 1917. A majority of these drawings has been digitized (scanned), and their images are available online ($ftp://howard.astro.ucla.edu/pub/obs/drawings$). In a typical year, there are about 330 clear days at MWO.
Visual measurements have been carried out from 1955 at CrAO (Severny, Stepanov, 1956; Stepanov, Petrova, 1958). The observations have been carried out in the line FeI 6302 Å since 1957. The data are presented in a kind of sketch of the solar disk with all sunspot groups and their temporary numbers. The drawings have been digitized (scanned) for data beginning from 1984, and their images are available online $http://solar.craocrimea.ru/eng/sunspots\_mf.htm\#archive$. There are about 220 clear days at CrAO in a typical year.
Lozitska et al. (2015) compared measurements of sunspot magnetic fields during 2010-2012. However, sunspots with strong and weak magnetic fields where not separated. Besides, Lozitska et al.(2015) compared the full number of measurements of the sunspot magnetic field strengths at Mt. Wilson and Crimea but not measurements of corresponding sunspots.
The aim of this paper is to provide a comparison between the visual measurements at MWO and CrAO for corresponding sunspots with strong ($> 2.5$ kG according to CrAO measurements) magnetic fields.
[**Observations and data processing**]{}
The daily observations at MWO are performed at the 150-foot (45.7 m) Solar Tower (ST, $http://obs.astro.ucla.edu/150_tele.html$). Solar images are constructed using a coelostat and a lens, and have a diameter of about 42 cm. When taking a measurement, the observer marks the boundary of the solar disk, and draws the position and configuration of sunspots. Magnetic field measurements are then carried out by measuring the splitting of the Zeeman components. The intensity of the magnetic field at the center of the sunspot was measured visually using the iron spectral line FeI 5250 Å line with Lande factor 3.0. An image of the sunspot to be measured passes through a polarization analyzer placed in front of the slit of the 75-foot (22.9 m) spectrograph. The analyzer alternately transmits left and right circularly polarized light in narrow strips along the slit. The micrometer is placed at the focus of the spectrograph and, by tilt of the glass plate, shifts the wavelength position of the image of one strip until the two oppositely polarized Zeeman sigma components in adjacent strips coincide. The amount of shift corresponds to the field strength. With this setup, sunspot field strengths can be obtained at the accuracy of hundreds gauss. The measured value of the magnetic field is then denoted on the drawing of the respective sunspot umbra (Fig.\[Fig\_1\]).
![A sample daily drawing of sunspots measured at MWO on October 22, 2014. On the drawing West is at the left. \[Fig\_1\] ](fig1.eps){height="5.5"}
Visual measurements of sunspot magnetic fields are carrying out at Tower Solar Telescope (TST-2) in the CrAO from 1955. The 60-cm coelostat, 45-cm spherical primary mirror (f/27), one flat and two convex secondary mirrors provide f/27, f/46 and f/78 Cassegrain foci (12, 21 and 30 m) on the entrance slit of a spectrograph. There are an echelle-grating, Universal Spectrophotometer with a scanning system and a CCD camera. Solar images have diameters from 8 to 30 cm. Observations have been carried out in the spectral line FeI 6302 Å with Lande factor 2.5 since 1957. Then the measured distance is transferred in intensity of a magnetic field using the special table and then denoted on the drawing of the respective sunspot umbra. The procedure of daily observations at TST-2 and ST (MWO) is similar. The sunspot field strengths can be obtained at the accuracy of one hundred gauss. Note that the sunspot penumbras as distinguished from MWO are not drawn at CrAO. Besides, the daily drawings consist of two pictures of different scales (Fig.\[Fig\_2\]).
![A sample daily drawing of sunspots measured at CrAO on October 22, 2014. \[Fig\_2\] ](fig2.eps){height="8"}
In order to compare the sunspot magnetic fields from CrAO and MWO we selected magnetic field values greater than 2.5 kG according to CrAO measurements from 31 July 2010 to 01 October 2017. After that we selected corresponding sunspot magnetic fields measured in MWO. As result, we found 1324 corresponding sunspots (see Tab.1 at $http://solar.craocrimea.ru/eng/observations.\\htm$). The typical time difference between measurements was about 10 hours. It should be emphasize that according to the CrAO data we found about 134 sunspots with the magnetic fields greater than 3 kG. The most strongest magnetic fields were measured on 22 May 2016 in NOAA 12546 (4.1 kG) and on 3 September 2017 in NOAA 12674 (4.6 kG). In turn, the magnetic fields measured at MWO do not exceed 2.7 kG. The diagrams of magnetic field measurements for corresponding sunspots are presented in Fig.\[Fig\_3\]. The average value of CrAO measurements is 2759 G, while the average value of MWO measurements is 2196 G. Thus, the difference between measurements exceeds 500 G. Besides, the correlation coefficient between MWO and CrAO measurements turned out to be about 0.22. These results suggest that MWO data can not be used for sunspots with strong magnetic fields.
[**Discussion and conclusions**]{}
In spite of the significant progress in the sunspot magnetic field measurements, the visual method based on the measurements of the Zeeman splitting gives the most reliable results. In fact, visual measurements of magnetic field strengths in sunspot umbra provide data on magnetic field strength modulus directly, i.e., irrespective from any solar atmosphere model assumption. In addition, results of measurements are not affected by the signal saturation for strong magnetic fields, low light level, instrumental polarization etc. Unlike magnetographic measurements, these data do not need various calibration curves for different sunspots and other features of the Sun.
Taking into account the above-mentioned and the similarity of methods of measurements we have compared results of strong magnetic fields of sunspots measured at CrAO and MWO. Although the difference between the formation heights of lines 6302 Åand 5250 Å is less than 30 km and the time of measurements of corresponding sunspots does not exceed 15 hours, the results of measurements of strong magnetic fields at CrAO and MWO turned out to be quite different.
It seems to us, this discrepancy can be caused by the small thickness of tip plate used at MWO, and, because of that, the visual measurements of strong magnetic fields becomes impossible (A.Pevtsov, private communication). Also some problems can be caused by the calibration of measurements at CrAO and MWO (Livingston et al., 2006; Lozitska et al., 2015). We hope to consider these questions in future.
[**Acknowledgements**]{}
We acknowledge A.N. Babin, A.N. Koval, and A. Pevtsov for fruitful discussions. Research was partially supported by the Fundamental Research Program of Presidium of the RAS N 28. GM was supported by the project RVO:67985815.
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Severny, A.B., Stepanov, V.E., 1956, Izv. Krim. Astrofiz. Obs., 16, 3.
Stepanov, V.E., Petrova, N.N., 1958, Izv. Krim. Astrofiz. Obs., 18, 66.
|
---
address:
- 'Singapore University of Technology and Design, Singapore'
- 'University of Liverpool, Liverpool, UK'
- 'Chalmers University of Technology, Sweden'
author:
- Tuan
- 'Thanh-Toan'
- Huu
- 'Dang-Khoa'
- 'Ngai-Man'
bibliography:
- 'refs.bib'
title: |
Simultaneous Compression and Quantization:\
A Joint Approach for Efficient Unsupervised Hashing
---
41A05,41A10,65D05,65D17 Keyword1,Keyword2,Keyword3
|
---
abstract: 'A principle of hierarchical entropy maximization is proposed for generalized superstatistical systems, which are characterized by the existence of three levels of dynamics. If a generalized superstatistical system comprises a set of superstatistical subsystems, each made up of a set of cells, then the Boltzmann-Gibbs-Shannon entropy should be maximized first for each cell, second for each subsystem, and finally for the whole system. Hierarchical entropy maximization naturally reflects the sufficient time-scale separation between different dynamical levels and allows one to find the distribution of both the intensive parameter and the control parameter for the corresponding superstatistics. The hierarchical maximum entropy principle is applied to fluctuations of the photon Bose-Einstein condensate in a dye microcavity. This principle provides an alternative to the master equation approach recently applied to this problem. The possibility of constructing generalized superstatistics based on a statistics different from the Boltzmann-Gibbs statistics is pointed out.'
author:
- Denis Nikolaevich
title: |
Hierarchical maximum entropy principle for generalized superstatistical systems and\
Bose-Einstein condensation of light
---
[8.5in]{}(0.1in,0.25in)
PHYSICAL REVIEW E **85**, 061120 (2012)
[2.5in]{}(5.6in,10.5in) ©2012 American Physical Society
Introduction
============
Superstatistics represents a statistics of canonical statistics and allows one to consider stationary states of nonequilibrium systems with fluctuations of an intensive parameter $\beta$ [@BeckCohen2003]. Though usually considered as an inverse temperature, $\beta$ can be interpreted in a more general way [@BeckCohenSwinney2005; @Beck2009]. A superstatistical system comprises a set of subsystems, or cells, each having the Gibbs canonical distribution determined by $\beta$. An essential feature of the superstatistical system is sufficient spatiotemporal scale separation, so that $\beta$ fluctuates on a much larger time scale than the typical relaxation time of the local dynamics in a cell. Superstatistics can be given a basis by the theory of hyperensembles [@Crooks2007; @Abe2009].
The distribution of $\beta$ can be considered as a function of some additional control parameters [@StraetenBeck2008]. However, in ordinary superstatistics, the intensive parameter fluctuates, but the control parameters are constant. Considering the control parameter fluctuations has led very recently to the generalization of superstatistics—“statistics of superstatistics,” or “generalized superstatistics” [@Sobyanin2011]. Generalized superstatistics is the statistics of generalized superstatistical systems. A generalized superstatistical system comprises a set of nonequilibrium superstatistical subsystems and can be associated with a generalized hyperensemble, an ensemble of hyperensembles. Compared with an ordinary superstatistical system, a generalized superstatistical system is characterized by the existence of the third, upper level of dynamics in addition to the two levels of dynamics existing in each superstatistical subsystem. This is reflected in the existence of a fluctuating vector control parameter on which both the intensive parameter distribution and the density of energy states depend. Significantly, generalized superstatistics can be used for nonstationary nonequilibrium systems. It was applied to branching processes and pair production in a neutron star magnetosphere [@Sobyanin2011].
The main problem of generalized superstatistics is the determination of the intensive parameter distribution, characterizing the superstatistical dynamics in each subsystem, and the control parameter distribution, characterizing the dynamics of the system as a whole. The aim of this paper is to develop the maximum entropy principle that can be used to solve the above problem.
The paper is organized as follows: In Sec. II the hierarchical maximum entropy principle for generalized superstatistical systems is formulated and the canonical, intensive parameter, and control parameter distributions are consecutively determined. In Sec. III this principle is applied to Bose-Einstein condensation of light and fluctuations of the number of ground-mode photons are considered. In Sec. IV the main conclusions are given.
Hierarchical maximum entropy
============================
A generalized superstatistical system is conveniently thought of as a set of superstatistical subsystems, each in turn made up of a set of cells. There are three levels of dynamics in this system: the first, lower level of fast dynamics in a cell, the second, middle level of superstatistical dynamics in a subsystem, and the third, upper level of global dynamics in the whole system. The levels are arranged in increasing order of dynamical time scale so that the shortest time scale corresponds to the lower level. The local dynamics in a cell is characterized by an energy $E$, the superstatistical dynamics in a subsystem is characterized by an intensive parameter $\beta$, and the global dynamics in the whole system is characterized by a control parameter $\xi$, which may be a multidimensional vector.
The system hierarchy is formed as a result of the sufficient time-scale separation between different levels of dynamics. This allows us to formulate the maximum entropy principle for the generalized superstatistical system as a principle of hierarchical entropy maximization. More specifically, the entropy should be maximized first for each cell, second for each subsystem, and finally for the whole system.
Local dynamics
--------------
Though the existence of the Gibbs canonical distribution at the lower dynamical level is postulated in superstatistics, it is reasonable to explicitly obtain this distribution from the maximum entropy principle. This trivial derivation will allow us to readily observe an analogy between the dynamics at different hierarchical levels of a generalized superstatistical system.
Choose a superstatistical subsystem of the generalized superstatistical system. A fixed value of the control parameter $\xi$ corresponds to this subsystem, but the intensive parameter $\beta$ may still fluctuate. Choosing the subsystem also fixes the density of energy states: $$\label{densityOfStates}
g(E|\xi)=\frac{\partial\Gamma(E|\xi)}{\partial E},$$ where $\Gamma(E|\xi)$ is the number of states with energy less than $E$. In integrals with $d\Gamma(E|\xi)$, integration over $E$ will be performed, $d\Gamma(E|\xi)=g(E|\xi)dE$.
To consider the local dynamics, choose a cell of the subsystem. Then $\beta$ also becomes fixed, but the energy $E$ is not fixed and is characterized by a probability distribution $\rho(E|\beta,\xi)$. To find the distribution maximizing the Boltzmann-Gibbs-Shannon entropy $$S[E](\beta|\xi)=-\int\rho(E|\beta,\xi)\ln\rho(E|\beta,\xi)d\Gamma(E|\xi)$$ under the normalization condition $N[E](\beta|\xi)=1$ and the mean energy constraint $U[E](\beta|\xi)=U(\beta|\xi)$, where $$\begin{aligned}
N[E](\beta|\xi)&=&\int\rho(E|\beta,\xi)d\Gamma(E|\xi),\\
U[E](\beta|\xi)&=&\int E\rho(E|\beta,\xi)d\Gamma(E|\xi),\end{aligned}$$ we should consider the condition of zero variation, $\delta L_1=0$, for the Lagrange function $$L_1(\nu_1,\beta,\xi)=S[E](\beta|\xi)-(\nu_1-1)N[E](\beta|\xi)
-\beta U[E](\beta|\xi).$$ Then we arrive at the Gibbs canonical distribution $$\rho_G(E|\beta,\xi)=\frac{e^{-\beta E}}{Z(\beta|\xi)},$$ where $$\label{partitionFunction}
Z(\beta|\xi)=\int e^{-\beta E}d\Gamma(E|\xi)$$ is the partition function. The entropy is $$\label{entropyInEachCell}
S[E](\beta|\xi)=\nu_1(\beta|\xi)+\beta U(\beta|\xi),$$ where the mean energy $$\label{meanEnergyInACell}
U(\beta|\xi)=-\frac{\partial\nu_1(\beta|\xi)}{\partial\beta}$$ is expressed via the Massieu function $$\label{nu1}
\nu_1(\beta|\xi)=\ln Z(\beta|\xi).$$
Superstatistical dynamics
-------------------------
Now consider the superstatistical dynamics of the chosen subsystem. This dynamics is characterized by the fluctuating intensive parameter $\beta$ that determines the properties of cells of the subsystem. To find the intensive parameter distribution $f(\beta|\xi)$, we should maximize the entropy of the joint probability distribution of $E$ and $\beta$, given $\xi$. It is written as [@AbeBeckCohen2007; @Abe2009] $$\label{subsystemEntropy}
S[E,\beta](\xi)=S[\beta](\xi)+\int S[E](\beta|\xi)f(\beta|\xi)d\beta$$ where $$\label{betaEntropy}
S[\beta](\xi)=-\int f(\beta|\xi)\ln f(\beta|\xi)d\beta$$ is the entropy associated with $f(\beta|\xi)$, and $S[E](\beta|\xi)$ is given by Eq. . The normalization condition for $f(\beta|\xi)$ is $N[\beta](\xi)=1$, where $$N[\beta](\xi)=\int f(\beta|\xi)d\beta.$$ In addition, we may impose a set of $n$ constraints given by an $n$-dimensional vector equality $$\label{middleLevelConstraintsEqualToZero}
M[\beta](\xi)=M(\xi),$$ where $$\label{middleLevelConstraints}
M[\beta](\xi)=\int m(\beta|\xi)f(\beta|\xi)d\beta,$$ and $m(\beta|\xi)=[m_1(\beta|\xi),\ldots,m_n(\beta|\xi)]$ and $M(\xi)=[M_1(\xi),\ldots,M_n(\xi)]$ are $n$-dimensional vectors specifying, respectively, the form and values of the constraints. Each $M_i(\xi)$ is the mean of $m_i(\beta|\xi)$ over the fluctuating $\beta$, given $\xi$. We consider $M[\beta](\xi)$ as some general constraint vector, but it may be composed of the constraints used in ordinary superstatistics, e.g., the mean values of energy, entropy, square of entropy, energy divided by temperature, or logarithm of the partition function [@Crooks2007; @Naudts2007; @StraetenBeck2008; @Abe2010].
Also define an $n$-dimensional vector Lagrange multiplier $\mu=(\mu_1,\ldots,\mu_n)$, where each $\mu_i$ is the Lagrange multiplier corresponding to the constraint $M_i[\beta](\xi)=M_i(\xi)$. We then have the following Lagrange function: $$L_2(\nu_2,\mu,\xi)=
S[E,\beta](\xi)-(\nu_2-1)N[\beta](\xi)-\mu\cdot M[\beta](\xi).$$ By $a\cdot b=\sum a_i b_i$ we denote the scalar product of some vectors $a$ and $b$. The condition $\delta L_2=0$ yields the intensive parameter distribution $$\label{superstatisticalDistribution}
\tilde{f}(\beta|\mu,\xi)=
\frac{Z(\beta|\xi)}{\tilde{Y}(\mu,\xi)}\exp[-\mu\cdot m(\beta|\xi)+\beta
U(\beta|\xi)],$$ where the partition function $$\label{YMuXi}
\tilde{Y}(\mu,\xi)=\int Z(\beta|\xi)\exp[-\mu\cdot m(\beta|\xi)+\beta
U(\beta|\xi)]d\beta$$ is determined from the normalization condition for $\tilde{f}(\beta|\mu,\xi)$.
Note that $\tilde{f}(\beta|\mu,\xi)$ and $\tilde{Y}(\mu,\xi)$ still depend on the Lagrange multiplier $\mu$. The implicit dependence of $\mu$ on the control parameter $\xi$, $$\label{LagrangeMultiplierOnControlParameter}
\mu=\mu(\xi),$$ is determined from $$\label{MXi}
M(\xi)=-\frac{\partial\tilde{\nu}_2(\mu,\xi)}{\partial\mu},$$ where $$\label{nu2}
\tilde{\nu}_2(\mu,\xi)=\ln\tilde{Y}(\mu,\xi)$$ is the Massieu function and $\partial/\partial\mu=(\partial/\partial\mu_1,\ldots,\partial/\partial\mu_n)$ is the $n$-dimensional gradient operator. Equations and are analogous to Eqs. and , respectively. Thus, given the constraints , the intensive parameter distribution , partition function , and Massieu function depend only on $\beta$ and $\xi$: $$\label{finalSuperstatisticalDistribution}
f(\beta|\xi)=\tilde{f}(\beta|\mu(\xi),\xi),$$ $$\label{Yxi}
Y(\xi)=\tilde{Y}(\mu(\xi),\xi),
\qquad
\nu_2(\xi)=\tilde{\nu}_2(\mu(\xi),\xi).$$
We may either first set the constraint vector $M(\xi)$ and then find $\mu(\xi)$ from the maximum entropy principle, or vice versa. This is in full analogy with the case of the dynamics in a cell, when we may first set the mean energy $U(\beta)$ and then find the corresponding intensive parameter $\beta$, or set $\beta$ and then find $U(\beta)$, which is more common. Incidentally, this duality allows one to alternatively formulate superstatistics by introducing the fluctuations of $U(\beta)$ instead of those of $\beta$ [@Bercher2008]. Note that the control parameter $\xi$ has a more general nature than $\beta$, since $\beta$ is exactly a Lagrange multiplier, while $\xi$, though controlling the Lagrange multiplier $\mu$, may not coincide with $\mu$. The analogy between $\beta$ and $\xi$ will be complete if we choose $\mu(\xi)=\xi$.
It follows from Eqs. , –, , and – that the entropy associated with the superstatistical subsystem is $$\label{finalSubsystemEntropy}
S[E,\beta](\xi)=\nu_2(\xi)+\mu(\xi)\cdot M(\xi).$$ It is analogous to Eq. .
Thus, the intensive parameter distribution for the superstatistical subsystem is given by Eq. . The superstatistical distribution $$\rho(E|\xi)=\int\rho_G(E|\beta,\xi)f(\beta|\xi)d\beta$$ has the form $$\label{ultimateSuperstatisticalDistribution}
\rho(E|\xi)=\frac{1}{Y(\xi)}\int\exp\{-\beta[E-U(\beta|\xi)]-\mu(\xi)\cdot
m(\beta|\xi)\}d\beta,$$ with the normalization condition $\int\rho(E|\xi)d\Gamma(E|\xi)=1$.
Ordinary superstatistics is a special case of generalized superstatistics: an ordinary superstatistical system is a generalized superstatistical system without fluctuations of the control parameter $\xi$. Therefore, we can easily obtain the intensive parameter distribution $f=f(\beta|\mu)$ for this system by formally removing $\xi$ from Eq. and from subsidiary Eqs. , , , , , , , , and . It is consistent with the distributions obtained earlier [@AbeBeckCohen2007; @StraetenBeck2008; @Abe2010].
Global dynamics
---------------
Consider the third level of dynamics. We should find the probability distribution $c(\xi)$ of the fluctuating control parameter $\xi$. This distribution is normalized, $N[\xi]=1$, where $$N[\xi]=\int c(\xi)d\xi.$$ The entropy of the joint probability distribution of $E$, $\beta$, and $\xi$ is determined by analogy with the entropy associated with a superstatistical subsystem \[cf. Eq. \]: $$\label{totalEntropy}
S[E,\beta,\xi]=S[\xi]+\int S[E,\beta](\xi)c(\xi)d\xi,$$ where $$\label{controlParameterDistributionEntropy}
S[\xi]=-\int c(\xi)\ln c(\xi)d\xi$$ is the entropy associated with the control parameter distribution $c(\xi)$, and $S[E,\beta](\xi)$ is given by Eq. . We may impose a set of $m$ additional constraints by analogy with Eqs. and : $$\label{thirdLevelConstraintsVector}
K[\xi]=K,$$ where $$K[\xi]=\int k(\xi)c(\xi)d\xi,$$ and $k(\xi)=[k_1(\xi),\ldots,k_m(\xi)]$ and $K=(K_1,\ldots,K_m)$ are $m$-dimensional vectors specifying, respectively, the form and values of the constraints. Each $K_i$ is the mean of $k_i(\xi)$ over the fluctuating $\xi$.
The Lagrange function is $$L_3(\nu_3,\kappa)=S[E,\beta,\xi]-(\nu_3-1)N[\xi]-\kappa\cdot K[\xi],$$ where we have defined an $m$-dimensional vector Lagrange multiplier $\kappa=(\kappa_1,\ldots,\kappa_m)$, where each $\kappa_i$ is the Lagrange multiplier corresponding to the constraint $K_i[\xi]=K_i$. The condition $\delta L_3=0$ yields the control parameter distribution $$\label{controlParameterDistribution}
c(\xi,\kappa)=\frac{Y(\xi)}{X(\kappa)}\exp[-\kappa\cdot k(\xi)+\mu(\xi)\cdot
M(\xi)],$$ where the partition function is $$X(\kappa)=\int Y(\xi)\exp[-\kappa\cdot k(\xi)+\mu(\xi)\cdot M(\xi)]d\xi,$$ and $Y(\xi)$ is defined by Eq. . By analogy with Eq. , we can rewrite the constraints as follows: $$\label{kappa}
K=-\frac{\partial\nu_3(\kappa)}{\partial\kappa},$$ where $$\nu_3(\kappa)=\ln X(\kappa)$$ is the Massieu function, and $\partial/\partial\kappa=(\partial/\partial\kappa_1,\ldots,
\partial/\partial\kappa_m)$ is the $m$-dimensional gradient operator. It remains to find the entropy at the maximum point \[cf. Eqs. and \]: $$S[E,\beta,\xi]=\nu_3(\kappa)+\kappa\cdot K.$$
Thus, the intensive parameter distribution $c(\xi)\equiv c(\xi,\kappa)$ is given by Eq. , with the Lagrange multiplier $\kappa$ determined from Eq. . By Eqs. and , we get that the generalized superstatistical distribution $$\sigma(E)=\int\rho(E|\xi)g(E|\xi)c(\xi)d\xi$$ has the form $$\begin{aligned}
\sigma(E)&=&\frac{1}{X(\kappa)}\int\exp\{-\beta[E-U(\beta|\xi)]\\
& &-\mu(\xi)\cdot[m(\beta|\xi)-M(\xi)]
-\kappa\cdot k(\xi)\}g(E|\xi)d\beta d\xi,\end{aligned}$$ with the normalization condition $\int\sigma(E)dE=1$.
Bose-Einstein condensation of light
===================================
Recently, thermalization of light in a dye microcavity has been observed [@KlaersVewingerWeitz2010]. In this experiment, photons are confined in a curved-mirror optical microresonator filled with a dye solution. In the microresonator, absorption and reemission of photons by dye molecules results in thermalization of the photon gas. Since the free spectral range of the microresonator is comparable to the spectral width of the dye, the emission of photons with a fixed longitudinal number dominates. Therefore, the photon gas is effectively two dimensional, and thermalization of transverse photon states occurs. Moreover, Bose-Einstein condensation (BEC) of light has been experimentally observed in the described system [@KlaersEtal2010; @KlaersEtal2011]. This reflects the fact that a two-dimensional harmonically trapped ideal gas of massive bosons can undergo BEC [@BagnatoKleppner1991; @Mullin1997; @WeissWilkens1997; @KocharovskyEtal2006; @PitaevskiiStringari2003]. In the case of the light BEC, the curvature of the mirrors provides a nonvanishing effective photon mass and at the same time induces a harmonic trapping potential for photons.
The problem of thermalization and fluctuations of the photon Bose-Einstein condensate has been considered very recently in Ref. [@KlaersEtal2012]. The condensate exchanges excitations with a reservoir consisting of $M$ dye molecules. The authors assume that the ground-state photon mode is coupled to the electronic transitions of a given number of dye molecules. This means that the sum $X$ of the number of ground-mode photons, $n$, and that of excited dye molecules, $X-n$, is constant. To analyze this system, the authors use the master equation approach.
Note that if we are interested in the behavior of the fluctuating photon BEC after thermalization has occurred, we can obtain the corresponding probability distribution merely using the thermodynamic consideration. The population of the electronic states of dye molecules is quickly thermalized, with the characteristic time ${\sim}1$ ps at room temperature (see Refs. [@Shank1975; @SchaeferWillis1976; @HaasRotter1991; @Schaefer1990; @Lakowicz2006] for details). Since the typical fluorescence lifetime is ${\sim}1{-}10$ ns, the emission of photons occurs from thermally equilibrated excited states. This apparent time-scale separation allows us to consider the above system as a generalized superstatistical system. Therefore, we can find the limiting probability distribution of the number of ground-mode photons by directly applying the hierarchical maximum entropy principle to this system.
For simplicity, consider the case of the ground-mode coupling and neglect the twofold polarization degeneracy by analogy with Ref. [@KlaersEtal2012]. The whole system is then composed of two subsystems: the subsystem of the dye solution and the subsystem of the photon BEC. The control parameter characterizing the interaction of the subsystems is the fluctuating number of ground-mode photons, $n$. The subsystem of the dye solution in turn consists of $M$ dye molecules, among which there are $X-n$ excited molecules and $M-X+n$ ground-state molecules. Obviously, $0\leqslant n\leqslant X\leqslant M$. Each molecule is in contact with a solvent, which plays the role of thermostat. In this sense, dye molecules resemble cells, but the inverse temperature $\beta$ does not fluctuate. For $f(\beta)$, this formally corresponds to the conditions of normalization, a given mean, and zero variance. In what follows, we will not explicitly indicate the dependence of functions on $\beta$.
Let $D_0(\varepsilon_0)$ and $D_1(\varepsilon_1)$ be the density of rovibrational states for the ground, $S_0$, and first excited, $S_1$, singlet electronic state, respectively. Note that $\varepsilon_i=E-E_i$, where $E_i$ is the lowest-energy substate of $S_i$, where $i=0,1$. Hence, $D_i(\varepsilon)=0$ for any $\varepsilon<0$. The partition functions $Z_0$ and $Z_1$ corresponding, respectively, to the ground-state and excited dye molecules are $$\label{ZiBeta}
Z_i=e^{-\beta E_i}w_i,$$ where $$w_i=\int_0^\infty e^{-\beta\varepsilon}D_i(\varepsilon)d\varepsilon,
\qquad
i=0,1.$$ It follows from Eqs. – and that the entropy for a ground-state molecule, $s_0$, and for an excited molecule, $s_1$, is $$s_i=\ln w_i+\beta(u_i-E_i),$$ where $$\label{moleculeEnergy}
u_i=E_i-\frac{1}{w_i}\frac{d w_i}{d\beta}$$ is the corresponding mean energy.
Now consider the subsystem of all dye molecules. After enumerating them and denoting a ground-state molecule by $0$ and an excited molecule by $1$, we can write an $M$-digit binary number $\eta=(\eta_1\eta_2\ldots\eta_M)$ with $M-X+n$ zeros and $X-n$ unities such that the state of the $k$th dye molecule is given by the $k$th digit $\eta_k$. For any given $\eta$, the entropy of the corresponding combination of dye molecules is $$s_{\eta|n}=(M-X+n)s_0+(X-n)s_1.$$ The probability that $\eta$ takes on a fixed value is $$p_{\eta|n}=%
\begin{pmatrix}
M\\X-n
\end{pmatrix}
^{-1}
=\frac{(X-n)!(M-X+n)!}{M!}.$$ The entropy $s^\mathrm{d}_n$ of the subsystem of dye molecules is calculated using the discrete analogs of Eqs. and , with $S[E](\beta|\xi)$ and $f(\beta|\xi)$ replaced by $s_{\eta|n}$ and $p_{\eta|n}$, respectively: $$s^\mathrm{d}_n=s_{\eta|n}+\ln%
\begin{pmatrix}
M\\X-n
\end{pmatrix}
.$$ The mean energy of the subsystem is $$u^\mathrm{d}_n=(M-X+n)u_0+(X-n)u_1,$$ where $u_0$ and $u_1$ are defined by Eq. .
The entropy of the photon BEC is zero, $s^\mathrm{ph}_n=0$, since the absence of the polarization degeneracy is assumed. The total energy of the condensate is $$u^\mathrm{ph}_n=n\hbar\omega,$$ where $\hbar\omega$ is the energy of a ground-mode photon.
Finally, consider the system as a whole. The control parameter $n$ corresponding to the number of ground-mode photons is characterized by a normalized discrete probability distribution $(\pi_0,\ldots,\pi_X)$, where $\pi_n$ is the probability of $n$ photons. For a fixed $n$, the energy and entropy of the system are given by $U_n=u^\mathrm{d}_n+u^\mathrm{ph}_n$ and $S_n=s^\mathrm{d}_n+s^\mathrm{ph}_n$, respectively. Maximizing the entropy \[see Eqs. and \] $$S=-\sum_{n=0}^X\pi_n\ln\pi_n+\sum_{n=0}^X\pi_n S_n,$$ under the normalization condition $$\label{piNorm}
\sum_{n=0}^X\pi_n=1$$ and the mean energy constraint $\sum\pi_n U_n=U$ yields $$\begin{aligned}
\label{piN}
\pi_n&=&\frac{1}{Z}
\begin{pmatrix}
M\\X-n
\end{pmatrix}
w_0^{M-X+n}w_1^{X-n}\\\nonumber
& &\times\exp\{-\beta[(M-X+n)E_0+(X-n)E_1+n\hbar\omega]\},\end{aligned}$$ where $Z$ is determined from Eq. . Dividing Eq. by $\pi_0$ and writing $\hbar\omega_0=E_1-E_0$, we obtain the probability distribution of the number of ground-mode photons in the form $$\label{mainBECeq}
\frac{\pi_n}{\pi_0}=\frac{X!(M-X)!}{(X-n)!(M-X+n)!}
\biggl(\frac{w_0}{w_1}\biggr)^n
e^{-\beta n\hbar(\omega-\omega_0)}.$$ This equation allows us to find $\pi_0=(\sum\pi_n/\pi_0)^{-1}$ and then calculate $\pi_n$ for all positive $n\leqslant X$.
Thus, the long-run behavior of the photon BEC, when the probability distribution $(\pi_0,\ldots,\pi_X)$ becomes stationary, can be investigated using the hierarchical maximum entropy principle. The link with the result of the master equation approach can be readily observed via the Kennard-Stepanov law [@Kennard1918; @Stepanov1957; @Neporent1958; @McCumber1964; @SawickiKnox1996; @KlaersEtal2012], $$\label{KennardStepanovLaw}
\frac{B_{10}(\omega)}{B_{01}(\omega)}
=\frac{w_0}{w_1}e^{-\beta\hbar(\omega-\omega_0)},$$ which relates the Einstein coefficients for stimulated emission, $B_{10}(\omega)$, and absorption, $B_{01}(\omega)$. Equation allows us to rewrite Eq. as $$\frac{\pi_n}{\pi_0}=\frac{X!(M-X)!}{(X-n)!(M-X+n)!}
\biggl[\frac{B_{10}(\omega)}{B_{01}(\omega)}\biggr]^n,$$ which is identical to Eq. (10) of Ref. [@KlaersEtal2012].
It seems interesting to use the described approach for studying the photon BEC fluctuations in more detail, e.g., for considering a more realistic situation of the polarization degeneracy and additional fluctuations of $M$ and $X$.
Conclusion
==========
I have formulated the hierarchical maximum entropy principle for generalized superstatistical systems. Such systems comprise a set of nonequilibrium superstatistical subsystems, where each subsystem is made up of many cells, and are characterized by the three-level dynamical hierarchy formed as a result of the sufficient time-scale separation between different dynamical levels. By arranging these levels in increasing order of dynamical time scale and consecutively maximizing the entropy at each level, I have obtained first the Gibbs canonical distribution for each cell, second the intensive parameter distribution for each subsystem, and finally the control parameter distribution for the whole system. From these distributions, I have also found the superstatistical distribution for each subsystem and the generalized superstatistical distribution for the whole system.
I have applied this principle to Bose-Einstein condensation of light in a dye microcavity. Assuming the ground-mode coupling and neglecting the polarization degeneracy, I have obtained the long-run probability distribution of the fluctuating number of ground-mode photons. This distribution is consistent with the analogous result of the master equation approach.
Note that when the hierarchical maximum entropy principle is applied to a generalized superstatistical system, certain constraints should be imposed on a normalized distribution to obtain the canonical distribution at the lower dynamical level. However, the constraints imposed on the intensive and control parameter distributions may be quite general. I propose erasing such a distinction, viz., choosing some general constraints at the lower dynamical level and additionally considering a vector intensive parameter. This will result in the generalized superstatistics the local dynamics of which is described by a more general statistics than the usual Boltzmann-Gibbs statistics. Grand canonical statistics may be the simplest alternative.
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---
abstract: |
We present the numerical investigation of the fermionic two-body decays of tau sleptons $\tilde \tau_{1,2}$ and $\tau$ sneutrino in the Minimal Supersymmetric Standard Model with complex parameters. In the analysis we particularly take into account the cosmological bounds imposed by WMAP data. We plot the CP-phase dependences for each fermionic two-body channel of $\tilde \tau_{1,2}$ and $\tau$ sneutrino and speculate about the branching ratios and total (two-body) decay widths. We find that the phase dependences of the decay widths of the third family sleptons are quite significant which can provide viable probes of additional CP sources. We also draw attention to the polarization of the final-state tau in the $
\tilde \tau_{1,2}$ decays.\
[**Key Words:**]{} CP-phase, sleptons decays, WMAP-allowed band.
author:
- |
Levent Selbuz$^*$ and Z. Zekeriya Aydin$^\dag$\
*Ankara University, Faculty of Engineering*\
*Department of Engineering Physics, 06100 Tandogan, Ankara, TURKEY*\
*$^*$selbuz@eng.ankara.edu.tr*\
*$^\dag$Z.Zekeriya.Aydin@eng.ankara.edu.tr*
title: Tau sleptons and Tau sneutrino Decays in MSSM under the Cosmological Bounds
---
Introduction
============
The experimental HEP frontier is soon reaching TeV energies and most of the physicists expect that just there theoretically proposed Higgs bosons and superpartners are waiting to be discovered. There are many reasons to be so optimistic. First of all, in spite of its remarkable successes, the Standard Model has to be extended into a more complete theory which should solve the hierarchy problem and stabilize the Higgs boson mass against radiative corrections. The most attractive extension to realize these objectives is supersymmetry (SUSY) [@haber]. Its minimal version (MSSM) requires a non-standard Higgs sector [@higgssector] which introduces additional sources of CP-violation[@Dugan:1984qf; @Masiero:2002xj] beyond the $\delta_{CKM}$ phase [@Cabibbo:yz]. The plethora of CP-phases also influences the decays and mixings of B mesons (as well as D and K mesons). The present experiments at BABAR, Tevatron and KEK and the one to start at the LHC will be able to measure various decay channels to determine if there are supersymmetric sources of CP violation. In particular, CP-asymmetry and decay rate of $B
\rightarrow X_s \gamma$ form a good testing ground for low-energy supersymmetry with CP violation [@bmeson]. The above-mentioned additional CP-phases explain the cosmological baryon asymmetry of the universe and the lightest SUSY particle could be an excellent candidate for cold dark matter in the universe [@Goldberg:1983nd; @Ellis:1983ew].
In the case of exact supersymmetry, all scalar particles would have to have same masses with their associated SM partners. Since none of the superpartners has been discovered, supersymmetry must be broken. But in order to preserve the hierarchy problem solved the supersymmetry must be broken softly. This leads to a reasonable mass splittings between known particles and their superpartners, i.e. to the superpartners masses around 1 TeV.
The precision experiments by Wilkinson Microwave Anisotropy Probe (WMAP) [@Wilkinson] have put the following constraint on the relic density of cold dark matter [^1] $$\label{mkl0}
0.0945 < \Omega_{CDM}h^2 < 0.1287$$
Recently, in the light of this cosmological constraint an extensive analysis of the neutralino relic density in the presence of SUSY-CP phases has been given by Bélanger *et al.* [@Belanger].
Analyses of the decays of third generation scalar quarks [@Bartl] and scalar leptons [@Bartlslep] with complex SUSY parameters have been performed by Bartl et al. In this study we present the numerical investigation of the fermionic two-body decays of third family sleptons in MSSM with complex SUSY parameters taking into account the cosmological bound imposed by WMAP data. Actually, we had performed some studies in this direction for squarks [@Selbuz:2006fj; @Aydin:2007aq] incorporating all the existing bounds on the SUSY parameter space by utilizing the study by Belanger et al. [@Belanger] before. These investigations showed us that the effects of $M_1$ and its phase $\varphi_{U(1)}$ on the decay widths of squarks are quite significant. Now we consider third generation sleptons. Namely, we study the effect of $M_1$ and its phase $ \varphi_{U(1)}$ on the decay widths of $\tilde \tau_{1,2}$ and $\tilde \nu_\tau$.
In the numerical calculations, although the SUSY parameters $\mu$, $M_1$, $M_2$, and $A_f$ are in general complex, we assume that $\mu$, $M_2$ and $A_f$ are real, but $M_1$ and its phase $
\varphi_{U(1)}$ take values on the WMAP- allowed bands given in Ref. [@Belanger]. These bands also satisfy the EDM bounds [@edms]. The experimental upper limits on the EDMs of electron, neutron and the $^{299}Hg$ and the $^{205}Tl$ atoms may impose constraints on the size of the SUSY CP-phases [@Ellis:1982tk; @Barger:2001nu]. However, these constraints are highly model dependent. This means that it is possible to suppress the EDMs without requiring the various SUSY CP-phases be small. For example, in the MSSM assuming strong cancellations between different contributions [@Ibrahim:1997nc], the phase of $\mu$ is restricted to $|\varphi_\mu|< \pi/10$, but there is no such restriction on the phases of $M_1$ and $A_f$. In addition, we evaluate the parameter $M_2$ via the relation $
M_2=(3/5)|M_1|(\tan\theta_W)^{-2}$ which can be derived by assuming gaugino mass unification purely in the electroweak sector of MSSM. It is very important to insert the WMAP-allowed band in the plane $
M_1-\varphi$ into the numerical calculations instead of taking one fixed $M_1$ value for all $\varphi$-phases, because, for example, on the allowed band for $\mu=200$ GeV, $M_1$ starts from 140 GeV for $\varphi=0$ and increasing monotonically it becomes 165 GeV for $\varphi=\pi$. In Ref.[@Belanger] two WMAP-allowed band plots are given, one for $\mu=200$ GeV and the other for $\mu=350$ GeV. For both plots the other parameters are fixed to be $\tan \beta=10$, $ m_{H^+}=1$ TeV, $A_f=1.2 $ TeV and $\varphi_\mu$=$\varphi_{A_f}$=0. We here choose the masses for $\tilde \tau_{1,2}$ sleptons as $m_{\tilde \tau_2}$=1000 GeV and $m_{\tilde \tau_1}$=750 GeV. These $m_{\tilde \tau_{1,2}}$ values lead to a sneutrino mass $m_{\tilde \nu_\tau}$=745 GeV for $M_{\tilde L} < M_{\tilde E}$.
Tau Sleptons and Tau Sneutrino Masses, Mixing and Decay Widths
==============================================================
Masses and mixing in slepton sector
-----------------------------------
The superpartners of the SM fermions with left and right helicity are the left and right sfermions. In the case of tau slepton (stau) the left and right states are in general mixed. Therefore, the sfermion mass terms of the Lagrangian are described in the basis ($\tilde \tau_{L}$,$\tilde \tau_{R}$) as [@Ellis:1983ed; @Gunion:1984yn] $$\label{mkl1}
{\cal L}_M^{\tilde \tau }= -({\tilde \tau}_L^{\dag} {\tilde \tau}_R^{\dag})\left(
\begin{array}{cc}
M_{L L}^{2}& M_{L R}^{2}\\[1.ex]
M_{R L}^{2} & M_{R R}^{2}
\end{array}
\right)
\left(
\begin{array}{c}
\tilde \tau_L\\ [1.ex] \tilde \tau_R
\end{array}
\right)$$ with $$\begin{aligned}
\label{mkl2}
M_{L L}^{2}&=&M_{\tilde
L}^{2}+(I_{3L}^{\tau}-e_\tau\sin^2\theta_W)\cos(2
\beta)m_{z}^{2}+m_{\tau}^{2},\\
M_{R R}^{2}&=&M_{\tilde E}^{2}+e_\tau\sin^2\theta_W\cos(2
\beta)m_{z}^{2}+m_{\tau}^{2},\\\label{mkl3} M_{R L}^{2}&=&(M_{L
R}^{2})^{*}=m_\tau(A_\tau-\mu^{*}(\tan\beta)^{-2I_{3L}^{\tau}}),\label{mkl4}\end{aligned}$$ where $m_\tau$, $e_\tau$, $I_{3L}^{\tau}$ and $\theta_W$ are the mass, electric charge, weak isospin of the $\tau$-lepton and the weak mixing angle, respectively. $\tan\beta=v_2/v_1$ with $v_i$ being the vacuum expectation values of the Higgs fields $H_i^{0}$, $
i=1,2$. The soft SUSY-breaking parameters $M_{\tilde L}$, $M_{\tilde
E}$ and $A_\tau$ involved in Eqs. (3-5) can be evaluated for our numerical calculations using the following relations: $$\begin{aligned}
\label{mkl5}
M_{\tilde L}^{2}&=&\frac{1}{2}{\left(m_{\tilde \tau_1}^{2}+m_{\tilde
\tau_2}^{2} \pm\sqrt{(m_{\tilde \tau_2}^{2}-m_{\tilde
\tau_1}^{2})^2-4m_\tau^{2}
|A_\tau-\mu^{*}\cot\beta|^2}\right)}\nonumber \\
&&+(\frac{1}{2}-\sin^2\theta_W)\cos(2\beta)m_{z}^{2}-m_{\tau}^{2},\\
\label{mkl6} M_{\tilde E}^{2}&=&\frac{1}{2}{\left(m_{\tilde
\tau_1}^{2}+m_{\tilde \tau_2}^{2} \mp\sqrt{(m_{\tilde
\tau_2}^{2}-m_{\tilde \tau_1}^{2})^2-4m_\tau^{2}
|A_\tau-\mu^{*}\cot\beta|^2}\right)}\nonumber \\
&&+\sin^2\theta_W\cos(2\beta)m_{z}^{2}-m_{\tau}^{2}\end{aligned}$$
The $\tilde{\tau}$ mass eigenstates $\tilde \tau_1$ and $\tilde
\tau_2$ can be obtained from the weak states $\tilde \tau_L$ and $\tilde \tau_R$ via the $\tilde \tau$-mixing matrix $$\label{mkl10}
{\cal R}^{\tilde \tau }=\left(
\begin{array}{cc}
e^{i\varphi_{\tilde \tau}}\cos\theta_{\tilde \tau}& \sin\theta_{\tilde \tau}\\[1.ex]
-\sin\theta_{\tilde \tau} & e^{-i\varphi_{\tilde \tau}}\cos\theta_{\tilde \tau}
\end{array}
\right)$$ where $$\label{mkl9}
\varphi_{\tilde \tau}=\arg[M_{R
L}^{2}]=\arg[A_\tau-\mu^{*}(\tan\beta)^{-2I_{3L}^{\tau}}]$$ and $$\label{mkl11}
\cos\theta_{\tilde \tau}=\frac{-|M_{L R}^{2}|}
{\sqrt{|M_{L R}^{2}|^2+
(m_{\tilde \tau_1}^{2}-M_{L L}^{2})^2}}, \qquad
\sin\theta_{\tilde \tau}=\frac{M_{L L}^{2}-m_{\tilde \tau_1}^{2}}
{\sqrt{|M_{L R}^{2}|^2+
(m_{\tilde \tau_1}^{2}-M_{L L}^{2})^2}}$$ One can easily get the following stau mass eigenvalues by diagonalizing the mass matrix in Eq. (2): $$\label{mkl12}
m_{\tilde \tau_{1,2}}^{2}=\frac{1}{2}
{\left(M_{L L}^{2}+M_{R R}^{2} \mp\sqrt{(M_{L L}^{2}-M_{R
R}^{2})^2+4|M_{L R }^{2}|^2} \right)} ,\qquad m_{\tilde \tau_1}<
m_{\tilde \tau_2}$$
The $\tilde \nu_\tau $ appears only in the left state. Its mass is given by $$\label{mkl12a}
m_{\tilde \nu_\tau}^{2}=M_{\tilde L}^{2}+\frac{1}{2}\cos(2\beta)m_z^2$$
Note that in this work we neglect CP-violation effects related to flavor change. Besides that the scalar mass matrices and trilinear scalar coupling parameters are assumed to be flavor diagonal.
Fermionic decay widths of $\tilde \tau_i $ and $\tilde \nu_\tau $
------------------------------------------------------------------
The lepton-slepton-chargino and lepton-slepton-neutralino Lagrangians have been first given in Ref. 1. Here we use them in notations of Ref. 13: $$\begin{aligned}
\label{mkl13}
{\cal L}_{l' \tilde l \tilde \chi^{\pm}}=g\bar{u}(\ell_{i j}^{\tilde
d}P_R + k_{i j}^{\tilde d}P_L){\tilde \chi}_j^{+}{\tilde
d}_i+g\bar{d}(\ell_{i j}^{\tilde u}P_R + k_{i j}^{\tilde
u}P_L){\tilde \chi}_j^{+c}{\tilde u}_i+h.c.\end{aligned}$$ and $$\begin{aligned}
\label{mkl18}
{\cal L}_{l \tilde l \tilde \chi^{0}} =g\bar{l}(a_{i j}^{\tilde
l}P_R + b_{i j}^{\tilde l}P_L){\tilde \chi}_j^{0}{\tilde l}_i+h.c.\end{aligned}$$ where u ($\tilde u$) stands for (s)neutrinos and d ($\tilde d$) stands for charged (s)leptons. We also borrow the formulas for the partial decay widths of $\tilde l_i$ ($\tilde l_i$ = $\tilde \tau_i$ and $\tilde \nu_\tau$) into lepton-neutralino (or chargino) from Ref. 13. The partial decay width for the decay $\tilde
\tau_i\rightarrow \tilde \chi_j^{0}+\tau(\lambda_\tau)$ is expressed as
$$\begin{aligned}
\label{mkl22}
\Gamma(\tilde \tau_i\rightarrow \tilde
\chi_j^{0}+\tau(\lambda_\tau))&=&\frac{g^2\kappa^{1/2}( m_{\tilde
\tau_{i}}^2,m_{\tilde \chi_j^{0}}^2,m_{\tau}^{2})}{16\pi m_{\tilde
\tau_{i}}^3}|\mathcal{M}_{\lambda_\tau}|^2\end{aligned}$$
with $$\begin{aligned}
\label{mkl23}
|\mathcal{M}_{\lambda_\tau}|^2&=&\frac{1}{4}\{H_{s}^2[|b_{ij}^{\tilde
\tau}|^2
+|a_{ij}^{\tilde \tau}|^2+2Re(b_{ij}^{\tilde \tau * }a_{ij}^{\tilde \tau})]\nonumber\\
&& + H_{p}^2[|b_{ij}^{\tilde \tau}|^2 +|a_{ij}^{\tilde
\tau}|^2-2Re(b_{ij}^{\tilde \tau * }a_{ij}^{\tilde \tau})]\nonumber\\
&& + 2(-1)^{\lambda_\tau+(1/2)}H_{p}H_{s}(|a_{ij}^{\tilde
\tau}|^2-|b_{ij}^{\tilde \tau}|^2)\}\end{aligned}$$ where $\lambda_{\tau}=\pm\frac{1}{2}$ is the helicity of the outgoing $\tau$, $\kappa(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+xz+yz)$, $H_{s}=[m_{\tilde \tau_{i}}^2-(m_{\tilde
\chi_j^{0}}+m_{\tau})^2]^{1/2}$ and $H_{p}=[m_{\tilde
\tau_{i}}^2-(m_{\tilde \chi_j^{0}}-m_{\tau})^2]^{1/2}$.
The explicit forms of the couplings, $a_{i j}^{\tilde \tau}$, $b_{i
j}^{\tilde \tau}$ and $\ell_{i j}^{\tilde \tau}$, are
$$\label{mkl151}
a_{i j}^{\tilde \tau}={\cal R}_{in}^{{\tilde \tau}^{*}}{\cal
A}_{jn}^{\tau}, \qquad b_{i j}^{\tilde \tau}={\cal
R}_{in}^{{\tilde \tau}^{*}}{\cal B}_{jn}^{\tau},\qquad\ell_{i
j}^{\tilde \tau}={\cal R}_{in}^{{\tilde \tau}^{*}} {\cal
O}_{jn}^{\tau} \qquad (n=L,R)$$
where $${\cal A}_j^{\tau}=
\begin{pmatrix}
\ f_{Lj}^\tau \\ \ h_{Rj}^\tau\\
\end{pmatrix}, \qquad
{\cal B}_j^{\tau}=
\begin{pmatrix}
\ h_{Lj}^\tau \\ \ f_{Rj}^\tau\\
\end{pmatrix}, \qquad
{\cal O}_j^{\tau}=
\begin{pmatrix}
\ -U_{j1} \\ \ Y_{\tau}U_{j2}\\
\end{pmatrix},$$ and $$\begin{aligned}
\label{mkl20}
f_{Lj}^{\tau}&=&-\frac{1}{\sqrt{2}}(N_{j2}+\tan\theta_WN_{j1})\nonumber \\
f_{Rj}^{\tau}&=&\sqrt{2}\tan\theta_WN_{j1}^{*}\nonumber \\
h_{Lj}^{\tau}&=&(h_{Rj}^{\tau})^{*}=Y_{\tau}N_{j3}^{*}.\end{aligned}$$
The partial decay width of $\tilde \tau_i$ into the chargino, $\tilde \tau_i\rightarrow \tilde \chi_j^{-}+\nu{_\tau}$, is obtained by the replacements $a_{ij}^{\tilde \tau}\rightarrow \ell_{i
j}^{\tilde \tau}$, $b_{ij}^{\tilde \tau}\rightarrow0$, $m_{\tilde
\chi_j^{0}}\rightarrow m_{\tilde \chi_j^{-}}$, $m_{\tau}\rightarrow0$ and $\lambda_{\tau}\rightarrow-\frac{1}{2}$ in Eq. (15) and Eq. (16) with the couplings $\ell_{i j}^{\tilde
\tau}$ also given in Eq. (17) and Eq. (18).
The width for the $\tau$-sneutrino decay $\tilde\nu_{\tau}\rightarrow{\tilde\chi_j^{0}}\nu_{\tau}$ is obtained by the replacements $a_{ij}^{\tilde \tau}\rightarrow
a_{j}^{\tilde \nu}$, $b_{ij}^{\tilde \tau}\rightarrow 0$, $m_{\tilde
\tau_{i}}\rightarrow m_{\tilde \nu_{\tau}}$, $m_{\tau}\rightarrow0$ and $\lambda_{\tau}\rightarrow-\frac{1}{2}$ in Eq. (15) and Eq. (16), and that for the decay $\tilde\nu_{\tau}\rightarrow{\tilde\chi_j^{+}}\tau(\lambda_\tau)$ by the replacements $a_{ij}^{\tilde \tau}\rightarrow \ell_{j}^{\tilde
\nu}$, $b_{ij}^{\tilde \tau}\rightarrow k_{j}^{\tilde \nu}$, $m_{\tilde \tau_{i}}\rightarrow m_{\tilde \nu_{\tau}}$ and $m_{\tilde \chi_j^{0}}\rightarrow m_{\tilde \chi_j^{+}}$. The coupling are now
$$\label{mkl15d}
a_{j}^{\tilde \nu}=\frac{1}{\sqrt{2}}(\tan\theta_WN_{j1}-N_{j2}),
\qquad k_{j}^{\tilde \nu}=Y_{\tau}U_{j2}^{*}, \qquad
\ell_{j}^{\tilde \nu}=-V_{j1}.$$
Here, N is the $4\times4$ neutralino mixing matrix, U and V are $2\times2$ chargino mixing matrices and $Y_{\tau}=m_{\tau}/(\sqrt{2}m_{W}\cos\beta)$ is the $\tau$ Yukawa coupling.
In this work we contend with tree-level amplitudes as we aim at determining the phase-sensitivities of the decay rates, mainly.
Tau-Slepton and Tau-Sneutrino Decays
====================================
Here we present the dependences of the $\tilde \tau_{1,2}$ and $\tilde \nu_{\tau}$ two-body decay widths on the $ \varphi_{U(1)}$ for $ \mu =200 $ GeV and for $ \mu =350 $ GeV. We also choose the values for the masses ($m_{\tilde \tau_1}$, $m_{\tilde \tau_2}$, $m_{\tilde \chi_1^\pm}$, $m_{\tilde \chi_2^\pm}$, $m_{\tilde
\chi_1^0}$) = (750 GeV, 1000 GeV, 180 GeV, 336 GeV, 150 GeV) for $ \mu =200 $ GeV and ($m_{\tilde \tau_1}$, $m_{\tilde \tau_2}$, $m_{\tilde \chi_1^\pm}$, $m_{\tilde \chi_2^\pm}$, $m_{\tilde \chi_1^0}$) = (750 GeV, 1000 GeV, 340 GeV, 680 GeV, 290 GeV) for $ \mu =350 $ GeV. The mass values of these $\tau$-sleptons lead to a sneutrino mass $m_{\tilde
\nu_\tau}$=745 GeV. Note that although the neutralino and chargino masses vary with $\varphi_{U(1)}$, these variations are not large. Therefore, as a final state particle (i.e., on mass-shell), we have chosen fixed (average) mass values for charginos and neutralinos. For both sets of values by calculating the $M_{\tilde L}$ and $M_{\tilde E}$ values corresponding to $m_{\tilde \tau_1}$ and $m_{\tilde \tau_2}$, we plot the decay widths for $M_{\tilde L}
\geq M_{\tilde E}$ and $M_{\tilde L} < M_{\tilde E}$, separately. We plot the $\varphi_{U(1)}$-dependences of the $\tilde \nu_{\tau}$ partial decay widths only for $M_{\tilde L} < M_{\tilde E}$. In the case of $M_{\tilde L} > M_{\tilde E}$, the phase dependences do not change, but decay widths take larger values. In our figures, we display the slepton decay widths for the both helicity states of the outgoing $\tau$ ($\tau_{L}$ and $\tau_{R}$).
In Figure 1(a) we show the partial decay widths of the channels $
\tilde \tau_1\rightarrow \tilde \chi_1^- \nu_{\tau}
$, $ \tilde \tau_1\rightarrow \tilde \chi_2^- \nu_{\tau}
$, $ \tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{L,R}
$, $ \tilde \tau_2\rightarrow \tilde \chi_1^- \nu_{\tau}
$, $ \tilde \tau_2\rightarrow \tilde \chi_2^- \nu_{\tau}
$ and $ \tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L,R}
$ as a function of $ \varphi_{U(1)}$ for $ \mu =200$ GeV. In these plots some dependences on the $ \varphi_{U(1)}$ phase are shown. In order to see these dependences more pronouncedly, we now plot two channels separately; namely $\tilde \tau_1\rightarrow \tilde \chi_1^0
\tau_{R}$ (the variations in the cross section are not large) and $
\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L}$ (the variations are really large) in Figures 4(a)-(b). Here we consider the case $M_{\tilde L} > M_{\tilde E}$, where $ \tilde \tau_1$ is mainly $
\tilde \tau_R$-like and $ \tilde \tau_2$ is mainly $ \tilde
\tau_L$-like (${\cal R}_{11}^{\tilde \tau }$=${\cal R}_{22}^{\tilde
\tau }$$\approx$ 0). In this case, the decay processes whose initial and final state helicities are the same, $ \tilde \tau_2\rightarrow
\tilde \chi_2^- \nu_{\tau} $, $ \tilde \tau_2\rightarrow \tilde
\chi_1^- \nu_{\tau}$, $\tilde \tau_2\rightarrow \tilde \chi_1^0
\tau_{L}
$ and $\tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{R}
$, have large widths, whereas those with opposite helicities, $\tilde \tau_2\rightarrow \tilde \chi_1^0
\tau_{R}$, $\tilde \tau_1\rightarrow \tilde \chi_1^- \nu_{\tau}$, $\tilde \tau_1\rightarrow \tilde \chi_2^-
\nu_{\tau}$ and $ \tilde \tau_1\rightarrow \tilde \chi_1^0
\tau_{L}$, have small ones. The reason for these large and small widths can be traced to the couplings $a_{ij}^{\tilde \tau}$, $b_{ij}^{\tilde \tau}$, $\ell_{i j}^{\tilde \tau}$ and $a_{j}^{\tilde \nu}$, $k_{j}^{\tilde \nu}$, $\ell_{j}^{\tilde \nu}$. Because of $H_{s}$$\approx$$H_{p}$ (since $m_{\tilde
\tau_{1,2}}$$\gg$$m_{\tau}$) we can express the decay widths of $\tilde \tau_i\rightarrow \tilde \chi_j^0 \tau(\lambda_\tau)$ as $
\Gamma(\tilde \tau_i\rightarrow \tilde \chi_j^0
\tau(\lambda_\tau=1/2))$$\propto$$|b_{ij}^{\tilde \tau}|^2$ and $
\Gamma(\tilde \tau_i\rightarrow \tilde \chi_j^0
\tau(\lambda_\tau=-1/2))$$\propto$$|a_{ij}^{\tilde \tau}|^2$. For example, $ \Gamma(\tilde \tau_1\rightarrow \tilde \chi_1^0
\tau_{L})$ ($ \Gamma(\tilde \tau_2\rightarrow \tilde \chi_1^0
\tau_{R})$) is suppressed because it is proportional to the term, $|a_{11}^{\tilde \tau}|$ $(|b_{21}^{\tilde \tau}|)$, which includes small Yukawa coupling ($Y_{\tau}$). On the other hand, $
\Gamma(\tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{R})$ is proportional to the square of $b_{11}^{\tilde \tau}$ which depends on the combination ${\cal R}_{11}^{{\tilde \tau}^{*}}{\cal
B}_{11}^{\tau}+{\cal R}_{12}^{{\tilde \tau}^{*}}{\cal
B}_{12}^{\tau}$ contributing largely from its second term. Similarly, since $H_{s}$=$H_{p}$, the decay widths of $\tilde
\tau_i\rightarrow \tilde\chi_j^{-}+\nu{_\tau}$ can be expressed as $\Gamma(\tilde \tau_i\rightarrow
\tilde\chi_j^{-}+\nu{_\tau})$$\propto$$H_s^2$$|\ell_{ij}^{\tilde
\tau}|^2$. The decay widths of $ \tilde \tau_1\rightarrow \tilde
\chi_1^- \nu_{\tau}
$, $ \tilde
\tau_1\rightarrow \tilde \chi_2^- \nu_{\tau}$ are also suppressed due to the very small Yukawa coupling ($\ell_{11}^{\tilde
\tau}$$\thickapprox$$Y_{\tau}$ ${{\cal
R}_{12}^{\tilde\tau}}^{*}U_{12}$, $\ell_{12}^{\tilde
\tau}$$\thickapprox$$Y_{\tau}$ ${{\cal
R}_{12}^{\tilde\tau}}^{*}U_{22}$).
Note that the decay width $\Gamma(\tilde \tau_1\rightarrow \tilde
\chi_1^0 \tau_{R})$ is 90-110 times larger than $\Gamma(\tilde
\tau_1\rightarrow \tilde \chi_1^0 \tau_{L})$ and $\Gamma(\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{L})$ is 10-30 times larger than $\Gamma(\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{R})$. From Figure 1(a) one can see that the branching ratios for $ \tilde
\tau_2$ are roughly $B( \tilde \tau_2\rightarrow \tilde \chi_2^-
\nu_{\tau})$ : $B(\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L})$ : $B(\tilde \tau_2\rightarrow \tilde \chi_1^- \nu_{\tau})$ : $B(\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{R})$ $\approx$ 6 : 1 : 0.5 : 0.03.
\[t\]
![Partial decay widths $\Gamma$ of the $\tilde \tau_{1,2}$ decays for $\tan\beta=10$, $A_\tau=1.2 $ TeV, $\varphi_\mu$=$\varphi_{A_\tau}$=0, $m_{\tilde \tau_1}=750$ GeV, $m_{\tilde \tau_2}=1000$ GeV and $M_{\tilde L} > M_{\tilde E}$; $\mu
=200 $ GeV in (a) and $\mu =350 $ GeV in (b).[]{data-label="fig:dosfiguras233"}](figure1a.eps "fig:"){height="5cm" width="8cm"} ![Partial decay widths $\Gamma$ of the $\tilde \tau_{1,2}$ decays for $\tan\beta=10$, $A_\tau=1.2 $ TeV, $\varphi_\mu$=$\varphi_{A_\tau}$=0, $m_{\tilde \tau_1}=750$ GeV, $m_{\tilde \tau_2}=1000$ GeV and $M_{\tilde L} > M_{\tilde E}$; $\mu
=200 $ GeV in (a) and $\mu =350 $ GeV in (b).[]{data-label="fig:dosfiguras233"}](figure1b.eps "fig:"){height="5cm" width="8cm"} \[dosfiguras13523\]
Although the $ \varphi_{U(1)}$ dependence of $\Gamma(\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{L}
)$ ($\Gamma(\tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{R}
)$) stems only from the parameters $|N_{11}|$ and $|N_{12}|$ ($|N_{11}|$), the phase dependence is quite pronounced. Similarly, the $ \varphi_{U(1)}$ phase dependence of $ \tilde
\tau_2\rightarrow \tilde \chi_{1}^-(\tilde \chi_{2}^-) \nu_{\tau} $ stemmed only from the $ \varphi_{U(1)}$ dependence of $|U_{11}|$ ($|U_{21}|$) parameter is also quite pronounced. The decay width $\Gamma(\tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{R})$ takes its maximum (minimum) value at $\varphi_{U(1)}$$\approx$$\frac{5\pi}{6}$ ($\varphi_{U(1)}$$\approx$$\frac{\pi}{4}$) (see Figure 4(a)). This $\varphi_{U(1)}$ value also corresponds to maximum (minimum) value of $|b_{11}^{\tilde \tau}|^2$. In a similar way, the width $\Gamma(\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L}
)$ and its parameter $|a_{21}^{\tilde \tau}|^2$ takes their maximum (minimum) value at $\varphi_{U(1)}$$\approx$$\pi$ ($\varphi_{U(1)}$=0) (see Figure 4(b)). Hence, we can say that the phase $\varphi_{U(1)}$ dependence of $|a_{i j}^{\tilde
\tau}|^2$ and $|b_{i j}^{\tilde \tau}|^2$ ($|\ell_{i j}^{\tilde
\tau}|^2$) reflects the phase $\varphi_{U(1)}$ dependence of channels $\tilde \tau_i\rightarrow \tilde \chi_j^0 \tau_{R,L}$ ($\tilde \tau_i\rightarrow \tilde \chi_j^-\nu_{\tau}$) directly.
\[t\]
![Partial decay widths $\Gamma$ of the $\tilde \tau_{1,2}$ decays for $\tan\beta=10$, $A_\tau=1.2 $ TeV, $\varphi_\mu$=$\varphi_{A_\tau}$=0, $m_{\tilde \tau_1}=750$ GeV, $m_{\tilde \tau_2}=1000$ GeV and $M_{\tilde L} < M_{\tilde E}$; $\mu
=200 $ GeV in (a) and $\mu =350 $ GeV in (b).[]{data-label="dosfiguras893"}](figure2a.eps "fig:"){height="5cm" width="8cm"} ![Partial decay widths $\Gamma$ of the $\tilde \tau_{1,2}$ decays for $\tan\beta=10$, $A_\tau=1.2 $ TeV, $\varphi_\mu$=$\varphi_{A_\tau}$=0, $m_{\tilde \tau_1}=750$ GeV, $m_{\tilde \tau_2}=1000$ GeV and $M_{\tilde L} < M_{\tilde E}$; $\mu
=200 $ GeV in (a) and $\mu =350 $ GeV in (b).[]{data-label="dosfiguras893"}](figure2b.eps "fig:"){height="5cm" width="8cm"} \[dosfiguras1673\]
We give the same partial decay widths in Figure 1(b) for $\mu =350$ GeV (See also Figures 4(c)-(d)). Here, too, $ \tilde \tau_1$ is mainly $ \tilde \tau_R$-like and $ \tilde \tau_2$ is mainly $ \tilde
\tau_L$-like because we still keep the case $M_{\tilde L}> M_{\tilde
E}$. For $ \mu =350 $ GeV the WMAP-allowed band [@Belanger] takes place in larger $M_1$ values ($ \sim 305-325$ GeV) leading to larger chargino and neutralino masses. This leads to smaller $H_s^2$(since $H_s^2$ $\propto$ $[m_{\tilde \tau_{i}}^2-m_{\tilde
\chi_j}^2]$) and, as a result, smaller widths for $\tilde
\tau_{1,2}$ decays compared with those for $ \mu =200$ GeV.
As can be seen from Figure 4(d), the $\varphi_{U(1)}$ dependence of the decay $\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L}$ is prominent such that the value of decay width at $\varphi_{U(1)}$=$\pi$ is about 2 times larger than that at $\varphi_{U(1)}$=$0$. The decay widths $\Gamma(\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{R})$, $\Gamma(\tilde
\tau_1\rightarrow \tilde \chi_1^- \nu_{\tau})$, $\Gamma(\tilde
\tau_1\rightarrow \tilde \chi_2^- \nu_{\tau})$ and $\Gamma(\tilde
\tau_1\rightarrow \tilde \chi_1^0 \tau_{L})$ are suppressed because of the same reasons mentioned above. The decay width of the process $\tilde \tau_2\rightarrow \tilde \chi_2^- \nu_{\tau}$ is the largest one among the $\tilde \tau_2$ channels and the branching ratios are $B(\tilde \tau_2\rightarrow \tilde \chi_2^- \nu_{\tau})$ : $B(\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L})$ : $B(\tilde \tau_2\rightarrow \tilde \chi_1^- \nu_{\tau})$ : $B(\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{R})$ $\approx$ 2.4 : 0.8 : 0.1 : 0.02.
We give $\tilde \tau_{1,2}$ and $\tilde \nu_{\tau}$ decay widths as a function of $\varphi_{U(1)}$ in Figure 2(a) and Figure 3(a) respectively (for $ \mu =200$ GeV). In Figures 4(e)-(f) we plot two of them separately whose CP-phase dependences are not clearly seen in Figure 2(a). They, too, show the significant dependences on CP-violation phase. In this subsection we consider the case $M_{\tilde L} < M_{\tilde E}$, where $ \tilde \tau_1$ is mainly $
\tilde \tau_L$-like and $ \tilde \tau_2$ is mainly $ \tilde
\tau_R$-like (${\cal R}_{12}^{\tilde \tau }$=${\cal R}_{21}^{\tilde
\tau }$$\approx$0). The decay width of the process $\tilde
\tau_1\rightarrow \tilde \chi_2^- \nu_{\tau}$ is the largest one among the $\tilde \tau_{1,2}$ channels; its decay width increases from 3.55 GeV to 3.8 GeV monotonically as $\varphi_{U(1)}$ increases from 0 to $\pi$. In this case ($M_{\tilde L} < M_{\tilde E}$), the width $\Gamma(\tilde \tau_1\rightarrow \tilde \chi_2^- \nu_{\tau})$ is not suppressed because its $\ell_{12}^{\tilde \tau}$ term does not include Yukawa coupling ($\ell_{12}^{\tilde
\tau}$$\thickapprox$ ${{\cal R}_{11}^{\tilde\tau}}^{*}U_{21}$). The phase dependence of $\tilde \tau_2\rightarrow \tilde \chi_1^0
\tau_{R}$ can be seen clearly in Figure 4(f); $\Gamma(\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{R})$ takes its minimum and maximum values at $\varphi_{U(1)}$$\approx$$\frac{\pi}{4}$ and at $\varphi_{U(1)}$$\approx$$\frac{5\pi}{6}$ respectively, because the parameter $|b_{21}^{\tilde \tau}|^2$ reaches its minimum and maximum at these $\varphi_{U(1)}$ values.
\[t\]
![Partial decay widths $\Gamma$ of the $\tilde \nu_\tau$ decays for $\tan\beta=10$, $A_\tau=1.2 $ TeV, $\varphi_\mu$=$\varphi_{A_\tau}$=0, $m_{\tilde \tau_1}=750$ GeV, $m_{\tilde \tau_2}=1000$ GeV and $m_{\tilde \nu_\tau}=745$ GeV; $\mu
=200 $ GeV in (a) and $\mu =350 $ GeV in (b).[]{data-label="dosfiguras85673"}](figure3a.eps "fig:"){height="5cm" width="8.06cm"} ![Partial decay widths $\Gamma$ of the $\tilde \nu_\tau$ decays for $\tan\beta=10$, $A_\tau=1.2 $ TeV, $\varphi_\mu$=$\varphi_{A_\tau}$=0, $m_{\tilde \tau_1}=750$ GeV, $m_{\tilde \tau_2}=1000$ GeV and $m_{\tilde \nu_\tau}=745$ GeV; $\mu
=200 $ GeV in (a) and $\mu =350 $ GeV in (b).[]{data-label="dosfiguras85673"}](figure3b.eps "fig:"){height="5cm" width="8.1cm"} \[dosfiguras16578\]
The branching ratios for $ \tilde \tau_1$ decays are roughly $B(
\tilde \tau_1\rightarrow \tilde \chi_2^- \nu_{\tau})$ : $B(\tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{L})$ : $B(\tilde \tau_1\rightarrow \tilde \chi_1^- \nu_{\tau})$ : $B(\tilde
\tau_1\rightarrow \tilde \chi_1^0 \tau_{R})$ $\approx$ 3.8 : 0.7 : 0.3 : 0.02.
In Figure 3(a) we give $\tilde \nu_{\tau}$ decay widths as a function of $ \varphi_{U(1)}$ for $ \mu =200 $ GeV. The phase dependence is more significant for the decay channels $\tilde
\nu_{\tau}\rightarrow \tilde \chi_2^+ \tau_{L}$, $\tilde
\nu_{\tau}\rightarrow \tilde \chi_1^+ \tau_{L}$ and $\tilde
\nu_{\tau}\rightarrow \tilde \chi_1^0\nu_{\tau}$. Analogously to the neutralino decays of $\tilde \tau_{1,2}$; because of $H_{s}$$\approx$$H_{p}$ (since $m_{\tilde
\nu_{\tau}}$$\gg$$m_{\tau}$) we can express the decay widths of $\tilde \nu_{\tau}\rightarrow \tilde \chi_j^+ \tau(\lambda_\tau)$ as $\Gamma(\tilde \nu_{\tau}\rightarrow \tilde \chi_j^0
\tau(\lambda_\tau=1/2))$$\propto$$|k_{j}^{\tilde \nu}|^2$ and $\Gamma(\tilde \nu_{\tau}\rightarrow \tilde \chi_j^0
\tau(\lambda_\tau=-1/2))$$\propto$$|\ell_{j}^{\tilde \nu}|^2$. To be more specific, $ \Gamma(\tilde \nu_{\tau}\rightarrow \tilde \chi_1^+
\tau_{R})$ ($ \Gamma(\tilde \nu_{\tau}\rightarrow \tilde \chi_2^+
\tau_{R})$) is suppressed because it is proportional to the term $|k_{1}^{\tilde \nu}|$ $(|k_{2}^{\tilde \nu}|)$ which includes small Yukawa coupling ($Y_{\tau}$). Since $H_{s}$=$H_{p}$ for neutrinos, the decay width of $\tilde \nu_{\tau}\rightarrow \tilde
\chi_1^0+\nu_\tau$ can be expressed as $\Gamma(\tilde
\nu_{\tau}\rightarrow \tilde
\chi_1^0+\nu_\tau)$$\propto$$H_s^2$$|a_1^{\tilde\nu}|^2$. The $
\varphi_{U(1)}$ dependences of $\Gamma(\tilde \nu_{\tau}\rightarrow
\tilde \chi_j^+ \tau_{L})$ ($\Gamma(\tilde \nu_{\tau}\rightarrow
\tilde \chi_1^0\nu_\tau)$) stems from the $ \varphi_{U(1)}$ dependences of $|\ell_{j}^{\tilde \nu}|$ ($|a_1^{\tilde\nu}|$) parameter and this parameter is quite phase-dependent.
The branching ratios for $\tilde \nu_{\tau}$ decays are roughly $B(\tilde \nu_{\tau}\rightarrow \tilde \chi_2^+ \tau_{L})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde \chi_1^+ \tau_{L})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde \chi_1^0\nu_{\tau})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde \chi_1^+ \tau_{R})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde \chi_2^+ \tau_{R})$ $\approx$ 3 : 1.3 : 0.1 : 0.01.
![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4a.eps "fig:"){width="46.00000%"} ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4b.eps "fig:"){width="46.00000%"} \[dosfiguras1323\] ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4c.eps "fig:"){width="46.00000%"} ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4d.eps "fig:"){width="46.00000%"} \[dosfiguras134\] ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4e.eps "fig:"){width="46.00000%"} ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4f.eps "fig:"){width="46.00000%"} ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4g.eps "fig:"){width="46.00000%"} ![(a)-(h) $ \varphi_{U(1)}$ dependences of certain $\tilde
\tau_{1,2}$ decays for $\mu =200$ GeV and for $\mu =350$ GeV.[]{data-label="dosfiguras233"}](figure4h.eps "fig:"){width="46.00000%"} \[dosfiguras1393\]
We present the dependences of the $\tilde \tau_{1,2}$ and $\tilde
\nu_{\tau}$ partial decay widths on $\varphi_{U(1)}$ in Figure 2(b) and Figure 3(b) (for $ \mu =350 $ GeV). In this case $H_s^2$($H_p^2$) takes smaller value because of the reason mentioned in the previous subsection and this leads to smaller widths for $\tilde \tau_{1,2}$ and $\tilde \nu_{\tau}$. In Figures 4(g)-(h) we again plot two $\tilde \tau_{1,2}$ decay channels separately whose phase dependences are not clearly seen in Figure 2(b). The dependence of the phase $\varphi_{U(1)}$ in $\tilde \tau_{1,2}$ decays are similar to those in the case $M_{\tilde L} < M_{\tilde
E}$ (for $ \mu =200$ GeV).
Note that $\Gamma(\tilde \tau_2\rightarrow \tilde \chi_1^0
\tau_{R})$$\approx$ 90 $\Gamma(\tilde \tau_2\rightarrow \tilde
\chi_1^0 \tau_{L})$ and $\Gamma(\tilde \tau_1\rightarrow \tilde
\chi_1^0 \tau_{L})$$\approx$ 30 $\Gamma(\tilde \tau_1\rightarrow
\tilde \chi_1^0 \tau_{R})$.
The width $\Gamma (\tilde \nu_{\tau}\rightarrow \tilde
\chi_1^0+\nu_\tau)$ decreases as the phase increases from 0 to $\pi$, showing a significant dependence on the phase. The branching ratios are roughly $B(\tilde \nu_{\tau}\rightarrow \tilde
\chi_1^0\nu_{\tau})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde
\chi_1^+ \tau_{L})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde
\chi_2^+ \tau_{L})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde
\chi_1^+ \tau_{R})$ : $B(\tilde \nu_{\tau}\rightarrow \tilde
\chi_2^+ \tau_{R})$ $\approx$ 0.6 : 0.2 : 0.15 : 0.09 : 0.0002.
Discussion and Summary
=======================
In this paper, we have presented the numerical investigation of the fermionic two-body decays of third family sleptons in the minimal supersymmetric standard model with complex parameters taking into account the cosmological bounds imposed by WMAP data. For this purpose, we have calculated numerically the decay widths of tau sleptons $\tilde \tau_{1,2}$ and $\tau$ sneutrino, paying particular attention to their dependence on the CP phase $
\varphi_{U(1)}$. We have found that some decay channels like $\tilde
\tau_2\rightarrow \tilde \chi_2^- \nu_{\tau}$, $\tilde
\tau_2\rightarrow \tilde \chi_1^- \nu_{\tau}$, $\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{L}$, $\tilde
\tau_2\rightarrow \tilde \chi_1^0 \tau_{R}$, $\tilde
\tau_1\rightarrow \tilde \chi_2^- \nu_{\tau}$, $\tilde
\tau_1\rightarrow \tilde \chi_1^- \nu_{\tau}$, $\tilde
\tau_1\rightarrow \tilde \chi_1^0 \tau_{L}$, $\tilde
\nu_{\tau}\rightarrow \tilde \chi_2^+ \tau_{L}$, $\tilde
\nu_{\tau}\rightarrow \tilde \chi_1^+ \tau_{L}$ and $\tilde
\nu_{\tau}\rightarrow \tilde \chi_1^0\nu_{\tau}$ show considerable dependences on $ \varphi_{U(1)}$ phase. These decay modes will be observable at a future $\emph{e}^+\emph{e}^-$ collider and LHC. Therefore they provide viable probes of CP violation beyond the simple CKM framework; moreover, they carry important information about the mechanism that brakes Supersymmetry.
Besides that, $\tilde \tau$ decay is important since it is the sole process where one can get information of the sfermion mixing and the neutralino mixing from the polarization of the final-state fermion [@Nojiri:1994it]. Note indeed that for $ \mu =200 $ GeV and $M_{\tilde L} > M_{\tilde E}$ the decay width $\Gamma(\tilde
\tau_1\rightarrow \tilde \chi_1^0 \tau_{R})$ is 90-110 times larger than $\Gamma(\tilde \tau_1\rightarrow \tilde \chi_1^0 \tau_{L})$ and $\Gamma(\tilde \tau_2\rightarrow \tilde \chi_1^0 \tau_{L})$ is 10-30 times larger than $\Gamma(\tilde \tau_2\rightarrow \tilde \chi_1^0
\tau_{R})$ since $\tilde \tau_1$ ($\tilde \tau_2$) is mainly $\tilde
\tau_R$-like ( $\tilde \tau_L$-like). For $ \mu =200 $ GeV and $M_{\tilde L} < M_{\tilde E}$ make only the interchange $\tilde
\tau_{1}\leftrightarrow\tilde \tau_{2}$ everywhere in the above-mentioned preceding sentence. For $ \mu =350 $ GeV the pattern expressed above remains more or less the same.
The phase dependence of the fermionic two-body decay widths of $\tilde \tau_i$ and $\tilde\nu_i$ stems directly from the parameters ($N_{ij}$, $U_{ij}$, $V_{ij}$) of the chargino and neutralino sectors. The cosmological bounds imposed by WMAP data on the $M_1$ parameter and its phase $\varphi_{U(1)}$ play an important role in taking their shapes of the phase dependences of these processes.
In this study, we use the framework of R-parity conserving supersymmetric scenarios wherein the lightest supersymmetric particle (LSP) is a viable candidate for Cold Dark Matter (CDM). Other than its relic density (observed by WMAP) little is known about the structure of CDM. But the recent astrophysical observations of the fluxes of high energy cosmic rays give information about the properties of CDM. In particular, recent results from Fermi LAT [@Abdo:2009zk] indicate an excess of the electron plus positron flux at energies above 100 GeV. This also confirms the earlier results from ATIC [@:2008zzr]. On the other hand, PAMELA experiment [@Adriani:2008zr] reports a prominent upturn in the positron fraction from 10-100 GeV, in contrast to what is expected from high-energy cosmic rays interacting with the interstellar medium. Although standard astrophysical sources such as pulsars and microquasars may be able to account for these anomalies, the positron excess at PAMELA and the electron plus positron flux of Fermi LAT have caused a lot of excitement being interpreted as decay/annihilation of Dark Matter. These unexpected results from PAMELA, ATIC and Fermi LAT experiments imply a new constrain on LSP that is CDM candidate: LSP must have not only the correct relic density found by WMAP but also correct decay/annihilation rates into electron-positron pairs. In the framework of the MSSM, a detailed analysis of decay of CDM that includes the observed cosmic ray anomalies has been given in Ref. [@Pospelov:2008rn].
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[^1]: In our calculation, we have used WMAP-allowed bands in the plane $ M_1-\varphi$ which are based on 1st year data. Now the WMAP 3rd year data is also available [@Spergel:2006hy], but the new WMAP + SDSS combined value for relic density of dark matter does not change the numerical results in Ref. [@Belanger], namely the WMAP-allowed bands. See “Note added” section of Ref. [@Belanger].
|
---
author:
- Xiaofei Zhang and George Fai
title: $Z^0$ and $W$ transverse momentum spectra at the LHC
---
At LHC energies, perturbative QCD (pQCD) provides a powerful calculational tool. For $Z\ ^{0}$ and $W$ transverse momentum spectra, pQCD theory agrees with the CDF[@CDF-Z] and D0 [@D0-W] data very well at Tevatron energies[@qz01]. The LHC $pp$ program will test pQCD at an unprecedented energy. The heavy-ion program at the LHC will make it possible for the first time to observe the full $p\ _T$ spectra of heavy vector bosons in nuclear collisions and will provide a testing ground for pQCD resummation theory [@CSS-W].
In nuclear collisions, the power corrections will be enhanced by initial and final state multiple scattering. As we will show, the high-twist effects are small at LHC for heavy boson production. The only important nuclear effect is the nuclear modification of the parton distribution function (shadowing). Because of their large masses, $W$ and $Z^0$ will tell us about the nuclear Parton Distribution Function (nPDF) at large scales. Since for the $p\ _T$ spectra of heavy bosons contributions from different scales need to be resummed, the $p\ _T$ spectra for heavy vector bosons will provide information about the evolution of nPDFs from small scales to large scales. Since it is more difficult to detect $W^\pm$ than $Z^0$, we will concentrate on discussing $Z^0$ here (the results for $W^{\pm}$ production are very similar[@zhang-fai]).
Resummation of the large logarithms in QCD can be carried out either in $p\ _T$-space directly, or in the so-called “impact parameter”, $\tb$-space, which is a Fourier conjugate of the $p\ _T$-space. Using the renormalization group equation technique, Collins, Soper, and Sterman (CSS) derived a formalism for the transverse momentum distribution of vector boson production in hadronic collisions[@CSS-W]. In the CSS formalism, non-perturbative input is needed for the large $\tb$ region. The dependence of the pQCD results on the non-perturbative input is not weak if the original extrapolation proposed by CSS is used. Recently, a new extrapolation scheme was introduced, based on solving the renormalization group equation including power corrections[@qz01]. Using the new extrapolation formula, the dependence of the pQCD results on the non-perturbative input was significantly reduced.
For vector boson ($V$) production in a hadron collision, the CSS resummation formalism yields[@CSS-W]: $$\begin{aligned}
\frac{d\sigma(h_A+h_B\rightarrow V+X)}{dM^2\, dy\, dp_T^2} =
\frac{1}{(2\pi)^2}\int d^2 \tb\, e^{i\vec{p}_T\cdot \vec{\tb}}\,
\tilde{W}(\tb,M,x_A,x_B) + Y(p_T,M,x_A,x_B) \,\,\, ,
\label{css-gen}\end{aligned}$$ where $x_A= e^y\, M/\sqrt{s}$ and $x_B= e^{-y}\, M/\sqrt{s}$, with rapidity $y$ and collision energy $\sqrt{s}$. In Eq. (\[css-gen\]), the $\tilde{W}$ term dominates the $p\ _T$ distributions when $p\ _T \ll M$, and the $Y$ term gives corrections that are negligible for small $p\ _T$, but become important when $p\ _T\sim M$.
The function $\tilde{W}(\tb,M,x_A,x_B)$ can be calculated perturbatively for small $\tb$, but an extrapolation to the large $\tb$ region requiring nonperturbative input is necessary in order to complete the Fourier transform in Eq. (\[css-gen\]). In oder to improve the situation, a new form was proposed[@qz01] by solving the renormalization equation including power corrections. In the new formalism, $\tilde{W}(\tb,M,x_A,x_B)=\tilde{W}^{pert}(\tb,M,x_A,x_B)$, when $\tb \leq \tb_{max}$, with $$\tilde{W}^{pert}(\tb,M,x_A,x_B) =
{\rm e}^{S(\tb,M)}\, \tilde{w}(\tb,c/\tb,x_A,x_B) \,\,\, ,
\label{css-W-sol}$$ where all large logarithms from $\ln(1/\tb^2)$ to $\ln(M^2)$ have been completely resummed into the exponential factor $S(\tb,M)$, and $c$ is a constant of order unity [@CSS-W]. For $\tb>\tb_{max}$, $$\begin{aligned}
\tilde{W}(\tb,M,x_A,x_B)
=\tilde{W}^{pert}(\tb_{max})
F^{NP}(\tb;\tb_{max}) \,\,\, ,
\label{qz-W-sol-m}\end{aligned}$$ where the nonperturbative function $F^{NP}$ is given by $$\begin{aligned}
F^{NP}
=\exp\bigl\{ -\ln(M^2 \tb_{max}^2/c^2)
\left[ g_1 \left( (\tb^2)^\alpha - (\tb_{max}^2)^\alpha\right) \right.
\left. +g_2 \left(\tb^2 - \tb_{max}^2\right) \right]
-\bar{g}_2 \left(\tb^2 - \tb_{max}^2\right) \bigr\}.
\label{qz-fnp-m}\end{aligned}$$ Here, $\tb_{max}$ is a parameter to separate the perturbatively calculated part from the non-perturbative input. Unlike in the original CSS formalism, $\tilde{W}(\tb,M,x_A,x_B)$ is not altered, and is independent of the nonperturbative parameters when $\tb < \tb_{max}$. In addition, the $\tb$-dependence in Eq. (\[qz-fnp-m\]) is separated according to different physics origins. The $(\tb^2)^\alpha$-dependence mimics the summation of the perturbatively calculable leading power contributions to the renormalization group equations to all orders in the running coupling constant $\alpha_s(\mu)$. The $\tb^2$-dependence of the $g_2$ term is a direct consequence of dynamical power corrections to the renormalization group equations and has an explicit dependence on $M$. The ${\bar g\ }_{2}$ term represents the effect of the non-vanishing intrinsic parton transverse momentum. A remarkable feature of the $\tb$-space resummation formalism is that the resummed exponential factor $\exp[S(\tb,M)]$ suppresses the $\tb$-integral when $\tb$ is larger than $1/M$. It can be shown using the saddle point method that, for a large enough $M$, QCD perturbation theory is valid even at $p\ _T=0$[@CSS-W]. As discussed in Ref.s [@qz01; @zhang-fai], the value of the saddle point strongly depends on the collision energy $\sqrt{s}$, in addition to its well-known $M^2$ dependence. The predictive power of the $\tb$-space resummation formalism should be even better at the LHC than at the Tevatron.
In $Z^0$ production, since final state interactions are negligible, power correction can arise only from initial state multiple scattering. Equations (\[qz-W-sol-m\]) and (\[qz-fnp-m\]) represent the most general form of $\ \tilde{W}$, and thus (apart from isospin and shadowing effects, which will be discussed later), the only way nuclear modifications associated with scale evolution enter the $\tilde{W}$ term is through the coefficient ${g}\ _{2}$.
The parameters $g_1$ and $\alpha$ of Eq. (\[qz-fnp-m\]) are fixed by the requirement of continuity of the function $\tilde{W}(\tb)$ and its derivative at $\tb=\tb_{max}$. (The results are insensitive to changes of $\tb_{max}$ $\in [0.3$ GeV$^{-1}$,0.7 GeV$^{-1}]$. We use $\tb_{max}=$ 0.5 GeV$^{-1}$.) The value of $g_2$ and ${\bar g}_{2}$ can be obtained by fitting the low-energy Drell-Yan data. These data can be fitted with about equal precision if the values ${\bar g}\ _2=0.25\pm 0.05$ GeV$^2$ and $g_2=0.01\pm 0.005$ GeV$^2$ are taken. As the $\tb$ dependence of the $g_2$ and ${\bar g}_2$ terms in Eq. (\[qz-fnp-m\]) is identical, it is convenient to combine these terms and define $G_{2}= \ln({M^2 \tb_{max}^2/ {c^2}})g_2 + \bar{g}_2 \,\,\, .$ Using the values of the parameters listed above, we get $G_2 = 0.33 \pm 0.07$ GeV$^2$ for $Z^0$ production in $pp$ collisions. The parameter $G_2$ can be considered the only free parameter in the non-perturbative input in Eq. (\[qz-fnp-m\]), arising from the power corrections in the renormalization group equations. An impression about the importance of power corrections can be obtained by comparing results with the above value of $G\ _2$ to those with power corrections turned off ($G_2=0$). We therefore define the ratio $$R_{G_2}(p_T) \equiv \left.
\frac{d\sigma^{(G_2)}(p_T)}{dp_T} \right/
\frac{d\sigma(p_T)}{dp_T} \,\,\, .
\label{Sigma-g2}$$ The cross sections in the above equation and in the results presented here have been integrated over rapidity ($-2.4 \leq y \leq 2.4$) and invariant mass squared. For the PDFs, we use the CTEQ5M set[@CTEQ5].
Figure 1 displays the differential cross sections and the corresponding $R_{\ G_2}$ ratio (with the limiting values of $G_2=0.26$ GeV$^2$ (dashed) and $G_2=0.40$ GeV$^2$ (solid)) for $Z^0$ production as functions of $p\ _T$ at $\sqrt s= 14$ TeV. The deviation of $R_{G_2}$ from unity decreases rapidly as $p\ _T$ increases, and it is smaller than one percent for both $\sqrt{s}=5.5$ TeV (not shown) and $\sqrt{s}=14$ TeV in $pp$ collisions, even when $p\ _T=0$. In other words, the effect of power corrections is very small at the LHC for the whole $p\ _T$ region.
![(a) Cross section ${d\sigma / dp_T}$ for $Z^0$ production in $pp$ collisions at the LHC with $\sqrt{s}=14$ TeV; (b) $R_{G_2}$ defined in Eq. (\[Sigma-g2\]) with $G_2=$ 0.26 GeV$^2$ (dashed) and 0.40 GeV$^2$ (solid).](zfig1.ps){width="8.0cm"}
\[zfig1\]
Without nuclear effects on the hard collision, the production of heavy vector bosons in nucleus-nucleus ($AB$) collisions should scale as the number of hard collisions, $AB$. However, there are several additional nuclear effects on the hard collision in a heavy-ion reaction. First of all, the isospin effect, which come from the difference between the neutron PDFs and the proton PDFs, is about 2% at LHC. This is expected, since at the LHC $x \sim 0.02$, where the $u-d$ asymmetry is very small. The dynamical power corrections entering the parameter $g\ _2$ should be enhanced by the nuclear size, i.e. proportional to $A^{1/3}$. Taking into account the $A$-dependence, we obtain $G_2 = 1.15 \pm 0.35$ GeV$^2$ for Pb+Pb reactions. We find that with this larger value of $G_2$, the effects of power corrections appear to be enhanced by a factor of about three from $pp$ to Pb+Pb collisions at the LHC. Thus, even the enhanced power corrections remain under 1% when 3 GeV $< p_T < $ 80 GeV. This small effect is taken into account in the following nuclear calculations.
Next we turn to the phenomenon of shadowing, expected to be a function of $x$, the scale $\mu$, and of the position in the nucleus. The latter dependence means that in heavy-ion collisions, shadowing should be impact parameter ($b$) dependent. Here we concentrate on impact-parameter integrated results, where the effect of the $b$-dependence of shadowing is relatively unimportant[@zfpbl02], and we focus more attention on scale dependence. We therefore use EKS98 shadowing[@eks] in this work. We define $$R_{sh}(p_T) \equiv \left.
\frac{d\sigma^{(sh)}(p_T,Z_A/A,Z_B/B)}{dp_T} \right/
\frac{d\sigma(p_T)}{dp_T} \,\,\, ,
\label{Sigma-sh}$$ where $Z_A$ and $Z_B$ are the atomic numbers and $A$ and $B$ are the mass numbers of the colliding nuclei, and the cross section $d\sigma(p_T,Z_A/A,Z_B/B)/dp_T$ has been averaged over $AB$, while $d\sigma(p_T)/dp_T$ is the $pp$ cross section. We have seen above that shadowing remains the only significant effect responsible for nuclear modifications.
![Cross section ratios for $Z^0$ production in Pb+Pb at $\sqrt{s}=5.5$ TeV: (a) $R_{sh}$ of Eq. (\[Sigma-sh\]) (solid line), and $R_{sh}$ with the scale fixed at 5 GeV (dashed) and 90 GeV (dotted); (b) $R_{G_2}$ of in Eq. (\[Sigma-g2\]) with $G_2=$ 0.8 GeV$^2$ (dashed) and 1.5 GeV$^2$ (solid).[]{data-label="zfig3"}](zfig3.ps){width="8.0cm"}
In Fig. 2(a), $R_{sh}$ (solid line) is surprising, because even at $p\ _T=90$ GeV, $x\sim 0.05$, and we are still in the “strict shadowing” region. Therefore, the fact that $R_{sh} > 1$ for 20 GeV $< p_T <$ 70 GeV is not “anti-shadowing”. To better understand the shape of the ratio as a function of $p\ _T$, we also show $R_{sh}$ with the scale fixed at the values 5 GeV (dashed line) and 90 GeV (dotted), respectively, in Fig. 2(a). The nuclear modification to the PDFs is only a function of $x$ and flavor in the calculations represented by the dashed and dotted lines. These two curves are similar in shape, but rather different from the solid line. In $\tb$ space, $\tilde{W}(\tb,M,x_A,x_B)$ is almost equally suppressed in the whole $\tb$ region if the fixed scale shadowing is used. However, with scale-dependent shadowing, the suppression increases as $\tb$ increases, as a result of the scale $\mu\sim 1/\tb$ in the nPDF. We can say that the scale dependence re-distributes the shadowing effect. In the present case, the re-distribution brings $R_{sh}$ above unity for 20 GeV $< p_T <$ 70 GeV. When $p_T$ increases further, the contribution from the $Y$ term starts to be important, and $R_{sh}$ dips back below one to match the fixed order pQCD result.
We see from Fig. 2 that the shadowing effects in the $p_T$ distribution of $Z^0$ bosons at the LHC are intimately related to the scale dependence of the nPDFs, on which we have only very limited data[@eks]. Theoretical studies (such as EKS98) are based on the assumption that the nPDFs differ from the parton distributions in the free proton, but obey the same DGLAP evolution[@eks]. Therefore, the transverse momentum distribution of heavy bosons at the LHC in Pb+Pb collisions can provide a further test of our understanding of the nPDFs.
In summary, higher-twist nuclear effects appear to be negligible in $Z\ ^{0}$ production at LHC energies. We have demonstrated that the scale dependence of shadowing effects may lead to unexpected phenomenology of shadowing at these energies. Overall, the $Z\ ^0$ transverse momentum distributions can be used as a precision test for leading-twist pQCD in the TeV energy region for both, proton-proton and nuclear collisions. We propose that measurements of $Z\ ^{0}$ spectra be of very high priority at the LHC.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the U.S. Department of Energy under DE-FG02-86ER-40251.
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|
---
abstract: |
Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a holomorphic family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$. Following previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not bifurcate. It is an open question whether this property is equivalent to structural stability on the Julia set $J^*$ (that is, the closure of the set of saddle periodic points).
In this paper we introduce a notion of regular point for a polynomial automorphism, inspired by Pesin theory, and prove that in a weakly stable family, the set of regular points moves holomorphically. It follows that a weakly stable family is probabilistically structurally stable, in a very strong sense. Another consequence of these techniques is that weak stability preserves uniform hyperbolicity on $J^*$.
address:
- |
LAMA\
Université Paris-Est Marne-la-Vallée\
5 boulevard Descartes\
77454 Champs sur Marne\
France
- |
LAGA\
Institut Galilée\
Université Paris 13\
99 avenue J.B. Clément\
93430 Villetaneuse\
France
author:
- Pierre Berger
- Romain Dujardin
title: 'On stability and hyperbolicity for polynomial automorphisms of $\cd$'
---
Introduction
============
Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a holomorphic family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$, with non-trivial dynamics[^1], parameterized by a connected complex manifold ${\Lambda}$. A basic stability/bifurcation dichotomy in this setting was introduced by M. Lyubich and the second author in [@dl]. In that paper it was proved in particular that under a moderate dissipativity assumption[^2], stable parameters together with parameters exhibiting a homoclinic tangency form a dense subset of ${\Lambda}$. This confirms in this setting a (weak version of a) well-known conjecture of Palis. The notion of stability into consideration here is the following: a family is said to be [*weakly stable*]{} if periodic orbits do not bifurcate. Specifically, this means that the eigenvalues of the differential do not cross the unit circle.
In one-dimensional holomorphic dynamics, this seemingly weak notion of stability actually leads to the usual one of structural stability (on the Julia set or on the whole sphere) thanks to the theory of [*holomorphic motions*]{} developed independently by Mañé, Sad and Sullivan and Lyubich [@mss; @lyubich-bif; @lyubich-bif2].
As it is well-known, the basic theory of holomorphic motions breaks down in dimension 2, and a corresponding notion of [*branched holomorphic motion*]{} (where collisions are allowed), was designed in [@dl]. To be more specific, let $J^*$ be the closure of the set of saddle periodic orbits. It was shown by Bedford, Lyubich and Smillie that $J^*$ contains all homoclinic and heteroclinic intersections of saddle points, and conversely, if $p$ is any saddle point, then $W^s(p)\cap W^u(p)$ is dense in $J^*$. It was proved in [@dl] that if $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ is weakly stable, then there is an equivariant branched holomorphic motion of $J^*$, that is unbranched over the set of periodic points and homoclinic (resp. heteroclinic) intersections. This means that such points have a unique holomorphic continuation in the family, and furthermore, this continuation cannot collide with other points in $J^*$ (see below §\[subs:prel autom\] for more details). The underlying idea is that the motion is unbranched on sets satisfying a local (uniform) expansivity property.
Still, it remains an open question whether a weakly stable family is structurally stable on $J^*$. A weaker version of this question, which is natural in view of the above analysis, is whether the unbranching property holds generically with respect to hyperbolic invariant probability measures.
The first main goal in this paper is to answer this second question. We introduce a notion of [*regular point*]{}, simply defined as follows: $p\in J^*$ is regular if there exists a sequence of saddle points $(p_n)_{n\geq 1}$ converging to $p$ such that $W^u_{{\mathrm{loc}}}(p_n)$ and $W^s_{{\mathrm{loc}}}(p_n)$ are of size uniformly bounded from below as $n{\rightarrow}\infty$ and do not asymptotically coincide (see below §\[sec:regular\] for the formal definition, and §\[subs:param1\] for the notion of the local size of a manifold). The set $\mathcal{R}$ of regular points is invariant and dense in $J^*$ since it contains saddle points and homoclinic intersections. More interestingly, Katok’s closing lemma [@katok] implies that $\mathcal R$ is of full mass relative to any hyperbolic invariant probability measure. Observe however that our definition of regular point makes no reference to any invariant measure. Notice also that in our context, thanks to the Ruelle inequality, any invariant measure with positive entropy is hyperbolic.
Our first main result is the following.
\[theo:pesin strong\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ of dynamical degree $d\geq 2$, that is weakly stable.
Then the set of regular points moves holomorphically and without collisions. More precisely, for every ${\lambda}\in {\Lambda}$, every regular point of $f_{\lambda}$ admits a unique continuation under the branched motion of $J^*_{\lambda}$, which remains regular in the whole family. In particular, the restrictions $f_{\lambda}{ \arrowvert_{\mathcal R_{\lambda}}}$ are topologically conjugate.
The meaning of the word “substantial" will be explained in §\[subs:prel autom\] below; it will be enough for the moment to note that any dissipative family is substantial by definition. By “topologically conjugate" we mean that there exists a homeomorphism $h: \mathcal R_{\lambda}{\rightarrow}\mathcal R_{{\lambda}'}$ such that $h\circ f_{\lambda}= f_{{\lambda}'}\circ h $ in restriction to $\mathcal R_{\lambda}$.
Let us say that a polynomial automorphism $f$ is [*probabilistically structurally stable*]{} (in some given family $(f_{\lambda})$) if for every $f'$ sufficiently close to $f$, there exists a set $\mathcal{R}_f$ (resp. $\mathcal{R}_{f'}$) which is of full measure with respect to any hyperbolic invariant probability measure for $f$ (resp. $f'$) together with a continuous conjugacy $\mathcal{R}_f{\rightarrow}\mathcal{R}_{f'}$.
Recall also from the work of Friedland and Milnor [@fm] that every dynamically non-trivial polynomial automorphism is conjugate to a composition of Hénon mappings.
Together with [@dl Thm A & Cor. 4.5], Theorem \[theo:pesin strong\] enables us to go one step further in the direction of the Palis Conjecture mentioned above.
Let $f$ be a composition of Hénon mappings in ${{{\mathbb{C}}^2}}$. Then:
- $f$ can be approximated in the space of polynomial automorphisms of degree $d$ either by a probabilistically structurally stable map, or by one possessing infinitely many sinks or sources.
- If $f$ is moderately dissipative and not probabilistically structurally stable, then $f$ is a limit of automorphisms displaying homoclinic tangencies.
The main step of the proof of Theorem \[theo:pesin strong\] consists in studying how the size of local stable and unstable manifolds of a given saddle point varies in a weakly stable family. More precisely, assume that for some ${{\lambda_0}}\in {\Lambda}$, $p({{\lambda_0}})$ is a saddle point such that $W^s(p({{\lambda_0}}))$ has bounded geometry at scale $r_0$ at $p({{\lambda_0}})$. Since $(f_{\lambda})$ is weakly stable, $p({{\lambda_0}})$ persists as a saddle point $p({\lambda})$ for ${{\lambda}\in {\Lambda}}$. In §\[sec:size\], we give estimates on the geometry of $W^s_{\rm loc}(p({\lambda}))$ [*which depend only on $r_0$*]{}, based on the extension properties of the branched holomorphic motion of $J^*$ along unstable manifolds devised in [@dl].
These estimates are used to control the geometry of the local “center stable manifold" of ${\left\{({\lambda},p({\lambda})), \ {\lambda}\in {\Lambda}\right\}}$, which is of codimension 1 in ${\Lambda}\times {{{\mathbb{C}}^2}}$. With this codimension 1 subset at hand, we can prevent collisions between the motion of points in $J^*$ using classical tools from complex geometry (like the persistence of proper intersections and the Hurwitz Theorem).
We actually prove a more general version of Theorem \[theo:pesin strong\], which involves only regularity in one of the stable or the unstable directions (see Theorem \[thm:regular\] below). One motivation for this is that in the dissipative setting it is possible in certain situations to take advantage of dissipativity to obtain a good control on the geometry of stable manifolds (see Example \[exam:zero\]).
If $f$ is uniformly hyperbolic on $J^*$, then it is well known that $f{ \arrowvert_{J^*}}$ is structurally stable. In particular, if $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ is any family of polynomial automorphisms, and ${{\lambda_0}}\in {\Lambda}$ is such that $f_{{\lambda_0}}$ is uniformly hyperbolic on $J^*_{{\lambda_0}}$, then $(f_{\lambda})$ is (weakly) stable in some neighborhood of ${{\lambda_0}}$. Thus, ${{\lambda_0}}$ belongs to a [*hyperbolic component*]{} in ${\Lambda}$, where this uniform hyperbolicity is preserved, which is itself contained in a possibly larger [*weak stability component*]{}. Our next main result asserts that these two components actually coincide.
\[theo:hyp\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$of dynamical degree $d\geq 2$, that is weakly stable. Assume that there exists ${{\lambda_0}}\in {\Lambda}$ such that $f_{{\lambda_0}}$ is uniformly hyperbolic on $J^*_{{\lambda_0}}$. Then for every ${\lambda}\in {\Lambda}$, $f_{\lambda}$ is uniformly hyperbolic on $J^*_{\lambda}$.
As a result, this theorem enables to identify the phenomena responsible for the breakdown of uniform hyperbolicity in a family of polynomial automorphisms: hyperbolicity can only be destroyed by the bifurcation of some saddle orbit to a sink or a source (which by [@dl] implies the creation of homoclinic tangencies, in the moderately dissipative setting).
The proof of Theorem \[theo:hyp\] relies on the techniques of Theorem \[theo:pesin strong\], together with a geometric criterion for hyperbolicity due to Bedford and Smillie [@bs8].
The plan of the paper is the following. In §\[sec:prel\] we discuss the notion of weak stability, following [@dl]. We also establish some preliminary results on sequences of subvarieties in ${\mathbb{C}}^d$. In §\[sec:size\] we study how the geometry of unstable manifolds varies in a weakly stable family. In §\[sec:regular\] we introduce the notion of regular point and prove Theorem \[theo:pesin strong\], and finally §\[sec:hyperbolic\] is devoted to the proof of Theorem \[theo:hyp\].
Throughout the paper, we make the standing assumption that the parameter space ${\Lambda}$ is the unit disk. In view of Theorems \[theo:pesin strong\] and \[theo:hyp\] this is not a restriction since we can always connect any two parameters in ${\Lambda}$ by a chain of holomorphic disks. We also use the classical convention $C(a, b, \ldots)$ to denote a constant which depends only on the previously defined quantities $a$, $b$, etc.
[**Acknowledgments.**]{} This research was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002 and the Balzan project of J. Palis.
Preliminaries {#sec:prel}
=============
In this section we collect some basic facts on polynomial automorphisms of ${{{\mathbb{C}}^2}}$, and give a brief account on the notion of weak stability introduced in [@dl]. We also establish some preliminary results on sequences of analytic subsets.
Families of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ {#subs:prel autom}
------------------------------------------------------------
Let us start with some standard facts about the iteration of an individual polynomial automorphism $f$ of ${{{\mathbb{C}}^2}}$ (see [@bs1; @bls] for more details and references). The [*dynamical degree*]{} is an integer $d$ defined by $d= \lim_{n{\rightarrow}\infty} (\deg(f^n))^{1/n}$, and $f$ has non-trivial dynamics if and only if $d\geq 2$. It is then conjugate to a composition of generalized Hénon mappings $(x,y)\mapsto (p(x)+ay,ax)$. Here are some dynamically defined subsets:
- $K^\pm$ is the set of points with bounded forward orbits under $f^{\pm1}$.
- $K= K^+\cap K^-$ is the filled Julia set.
- $J^\pm = {\partial}K^\pm$ are the forward and backward Julia sets. Stable (resp. unstable) manifolds of saddle periodic points are dense in $J^+$ (resp. $J^-$).
- $J^*\subset J = J^+\cap J^-$ is the closure of the set of saddle periodic points. Saddle points and homoclinic and heteroclinic intersections are contained (and dense) in $J^*$.
The Green functions $G^\pm$ are defined by $G^\pm(z) = \lim d^{-n} \log^+{\left\Vertf^n(z)\right\Vert}$, and are non-negative continuous plurisubharmonic functions. They are pluriharmonic whenever positive and $K^\pm$ coincides with ${\left\{G^\pm = 0\right\}} $. The associated currents are $T^\pm = dd^cG^\pm$ whose supports are $J^\pm$. If $\Delta\subset {{{\mathbb{C}}^2}}$ is a holomorphic disk, then $G^+{ \arrowvert_{\Delta}}$ is harmonic iff $T^+\wedge [\Delta] = 0$ iff ${\left(f^n{ \arrowvert_{\Delta}}\right)}_{n\geq 1}$ is a normal family (equivalently $\Delta\subset K^+$ or $\Delta\subset {{{\mathbb{C}}^2}}\setminus K^+$).
Let now $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a holomorphic family of polynomial automorphisms with fixed dynamical degree $d\geq 2$, parameterized by a connected complex manifold. We will use the notation $K_{\lambda}$, $J^*_{\lambda}$, etc. to denote the corresponding dynamical objects. If a preferred parameter ${{\lambda_0}}$ is given we often simply use the subscript ‘0’ instead of ${{\lambda_0}}$.
To the family $(f_{\lambda})$ is associated a fibered dynamical system in ${\Lambda}\times {{{\mathbb{C}}^2}}$ defined by $\widehat f:({\lambda}, z)\mapsto ({\lambda}, f_{\lambda}(z))$. Then we mark with a hat the corresponding fibered objets, e.g. $\widehat K = \bigcup_{{\lambda}\in {\Lambda}} {\left\{{\lambda}\right\}}\times K_{\lambda}$, etc.
Such a family is always conjugate to a family of compositions of Hénon mappings [@dl Prop. 2.1]. It follows that the sets $K_{\lambda}$ are locally uniformly bounded in ${{{\mathbb{C}}^2}}$.
From now on we report on some results from [@dl]. A family of polynomial automorphisms of dynamical degree $d\geq 2$ is said [*substantial*]{} it: either all its members are dissipative or for any periodic point with eigenvalues $\alpha_1$ and $\alpha_2$, no relation of the form $\alpha_1^a\alpha^b_2 = c$ holds persistently in parameter space, where $a$, $b$ , $c$ are complex numbers and ${\left\vertc\right\vert} =1$. From now on, we assume without further notice that all families have constant dynamical degree $d\geq 2$ and are substantial.
A [*branched holomorphic motion*]{} $\mathcal G$ is a family of holomorphic graphs over ${\Lambda}$ in ${\Lambda}\times {{{\mathbb{C}}^2}}$. All branched holomorphic motions considered in this paper are locally uniformly bounded, so in particular they form normal families (we then say that $\mathcal G$ is [*normal*]{}). A branched holomorphic motion $
\mathcal G$ is [*unbranched*]{} along $\gamma$ if $\gamma$ does not cross any other graph in the family. If it is unbranched along any graph $\gamma$, then it is by definition a [*holomorphic motion*]{}. If $\mathcal G$ is normal, closed and unbranched at $\gamma$, and if $(\gamma_n)_{n\geq 0}
\in \mathcal G^{\mathbb{N}}$ is any sequence such that for some ${{\lambda_0}}\in {\Lambda}$, $\gamma_n({{\lambda_0}}){\rightarrow}\gamma ({{\lambda_0}})$, then $\gamma_n{\rightarrow}\gamma$. We thus see that unbranching along $\gamma$ is a form of continuity of the motion. We can make this precise as follows: if $\mathcal G$ is a (non-necessarily closed) normal holomorphic motion and $\overline{\mathcal{G}}$ is unbranched at $\gamma_0$, then $\mathcal{G}$ is continuous at $\gamma_0$.
A substantial family $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ of polynomial automorphisms is said to be [*weakly stable*]{} if every periodic point stays of constant type (attracting, saddle, indifferent, repelling) in the family. Equivalently, $(f_{\lambda})$ is weakly stable if the sets $J^*_{\lambda}$ move under an equivariant branched holomorphic motion. A central theme in this paper will be to show that this motion is unbranched at certain points. In this respect, the following result is essential.
\[thm:unbranched\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of dynamical degree $d\geq 2$. If for ${{\lambda_0}}\in {\Lambda}$, $p({{\lambda_0}})$ is a saddle point or a homoclinic or a heteroclinic intersection, then it admits a unique continuation $p({\lambda})$ which remains of the same type, and the branched holomorphic motion of $J^*$ is unbranched along $p$.
The motion of $J^*$ can be extended to a branched holomorphic motion of $J^+\cup J^-$, using the density of stable and unstable manifolds of saddles. The details are as follows (for concreteness we deal with unstable manifolds, of course analogous results hold in the stable direction). The global unstable manifold of a saddle point is parameterized by ${\mathbb{C}}$. More precisely, in our situation, for every ${\lambda}$ there exists an injective holomorphic immersion $\psi^u_{\lambda}:{\mathbb{C}}{\rightarrow}{{{\mathbb{C}}^2}}$ such that $\psi^u_{\lambda}(0) = p_{\lambda}$ and for $\zeta \in {\mathbb{C}}$, $f_{\lambda}\circ \psi^u_{\lambda}(\zeta)= \psi^u(u_{\lambda}\zeta)$, where $u_{\lambda}$ denotes the unstable multiplier. Such a $\psi^u$ is unique up to pre-composition with a linear map, and will be referred to as an [*unstable parameterization*]{}. In addition, the normalization of $\psi^u_{\lambda}$ may be chosen so that $({\lambda},\zeta)\mapsto \psi^u_{\lambda}(\zeta)$ is holomorphic. The precise way to do it is irrelevant for the moment; we shall have to discuss this issue more carefully later on. Thanks to these parameterizations, we can use the theory of holomorphic motions in ${\mathbb{C}}$ to derive information about the motion of unstable manifolds in ${{{\mathbb{C}}^2}}$. The following is a combination of Proposition 5.2 and Lemma 5.10 in [@dl].
Under the above hypotheses there exists a natural equivariant holomorphic motion $h_{\lambda}: W^u(p_0) {\rightarrow}W^u(p_{\lambda})$, with $h_0 = \mathrm{id}$, that respects the decomposition $$W^u(p)=(W^u(p)\cap U^+) \sqcup (W^u(p)\cap K^+).$$
Beware that we are not claiming that the points in $W^u(p)$ have a unique continuation, only that they have a [*natural*]{} one. This motion is constructed by taking the canonical extension of the motion of homoclinic intersections (this is due to Bers and Royden [@bers; @royden]).
Notice that the notation $h_{\lambda}$ here refers to the motion of points in ${{{\mathbb{C}}^2}}$. Given a holomorphic family of parameterizations $\psi_{\lambda}^u$ of $W^u(p_{\lambda})$, it will also be of interest in some situations to consider the corresponding holomorphic motion $(\psi^u_{\lambda})^{-1}\circ h_{\lambda}$ in ${\mathbb{C}}$, which we will denote by $h_{\lambda}^u$.
These holomorphic motions need not preserve the levels of the function $G^+$: indeed this is already the case[^3] for $J$-stable families in dimension 1. The following easy lemma asserts that $G^+$ admits locally uniform distortion along the motion. It thus provides a link between the intrinsic (i.e. inside unstable manifolds) and the extrinsic properties of the motion, and will play an important role in the paper.
\[lem:harnack\] For every compact subset $\widetilde {\Lambda}\Subset {\Lambda}$ there exists a constant $C = C(\widetilde{\Lambda}) \geq 1$ such that for every $z \in W^u(p_0)$ $${\frac{1}{C}} G^+_0(z) \leq G^+_{\lambda}(h_{\lambda}(z))\leq {C} G^+_0(z).$$
Recall that for any holomorphic family of polynomial automorphisms of degree $d$, the function $({\lambda},z)\mapsto G^+_{\lambda}(z)$ is plurisubharmonic in ${\Lambda}\times {{{\mathbb{C}}^2}}$, jointly continuous in $({\lambda},z)$, and pluriharmonic where it is positive (see [@bs1 §3]).
If $z\in K^+_0$, then $h_{\lambda}(z)\in K^+_{\lambda}$ and $G^+_{\lambda}(h_{\lambda}(z))\equiv 0$ so there is nothing to prove. If $z\notin K^+_0$ then ${\lambda}\mapsto G^+_{\lambda}(h_{\lambda}(z))$ is a positive harmonic function, so the result follows from the Harnack inequality [@hormander Thm 3.1.7].
Sequences of analytic subsets {#subs:prel geom}
-----------------------------
To make the paper accessible to readers potentially not so familiar with complex geometry, let us first recall a few classical facts on complex analytic sets, and sequences of such objects. The reader is referred to the book of Chirka [@chirka] for more details.
Let ${\Omega}\subset {\mathbb{C}}^d$ be a connected open set. A (complex) *analytic subset* or *subvariety* $A$ of ${\Omega}$ is a subset of ${\Omega}$ that is covered by open sets $U$ of $\mathbb C^n$, for which there exist $p\ge 0$ and a holomorphic map $\phi \colon U\mapsto \mathbb C^p$, such that $A\cap U= \{z\in U,\; \phi(z)=0\}$.
A point $a\in A$ is *regular* if there exists a neighborhood $U$ of $a$ so that $A\cap U$ is a (complex) submanifold. The set of regular points of $A$ is denoted by $\mathrm{Reg}(A)$, and its complement $\mathrm{Sing}(A)= A\setminus\mathrm{Reg} (A)$ is the *singular set*. A subvariety is *smooth* if its singular set is empty.
An *irreducible component* of $A$ is the closure of a connected component of ${\mathrm Reg}(A)$. It is itself an analytic set. The *dimension* of an analytic subset $A$ is the maximal dimension of its irreducible components. It is said of [*pure dimension*]{} if all its irreducible components have the same dimension.
A [*hypersurface*]{} (resp. a [*curve*]{}) is an analytic subset of pure codimension (resp. dimension) 1, possibly singular.
We recall that the Hausdorff distance $d_{HD}$ between two closed subsets $E$ and $F$ of a metric space is infimum of $m\in [0,\infty]$ such that $E$ is included in the $m$-neighborhood $F$ and [*vice-versa*]{}. Let $\Omega$ be an open subset of $\mathbb C^d$. A sequence of closed subsets $(A_j)_j$ of $\Omega$ converges to a closed subset $A\subset \Omega$, if for every compact set $K$ of $\Omega$, it holds $d_{HD}(K\cap A_j, A\cap K)\to 0$. The set of closed (resp. closed and connected) subsets of $\Omega$ endowed with Hausdorff distance is relatively compact.
A key ingredient to study the convergence of analytic subsets sequences is the following classical result known as Bishop’s Theorem:
\[thm:bishop\] Let $(A_j)_j$ be a sequence of pure $p$-dimensional subvarieties of an open subset $\Omega\subset \mathbb C^d$, converging to a (closed) subset $A \subset \Omega$ and such that the $2p$-dimensional Hausdorff measure (that is, the $2p$-dimensional volume) $m_{2p}(A_j)$ is locally uniformly bounded: $$\forall K\Subset \Omega,\; \exists M_K>0,\; \forall j, \; m_{2p}(A_j\cap K)<M .$$ Then $A$ is also a pure $p$-dimensional subvariety of $\Omega$.
In particular the set of subvarieties with locally uniformly bounded volume is compact. We can actually be more precise about the convergence in Bishop’s Theorem. Let $A_n$ be a sequence of analytic sets with uniformly bounded volumes converging in the Hausdorff topology to an irreducible analytic set $A$. Then there exists a positive integer $m$, the [*multiplicity of convergence*]{}, which can be described as follows. If $p\in \mathrm{Reg}(A)$ is any regular point of $A$, and $N$ is a compact neighborhood of $p$ in which $(A\cap N, N)$ is biholomorphic to $({\mathbb{D}}^k\times {\left\{0\right\}}, {\mathbb{D}}^d)$, then for $n$ large enough, $A_n
\cap N$ is a branched cover over $A\cap N$ of degree $m$. In particular if $m=1$, $A_n
\cap N$ is a graph over $A\cap N$
Let us now state a few results which will be used many times in the paper. The following result can be interpreted as a kind of abstract version of the ${\Lambda}$-lemma of [@mss].
\[prop:cv\] Let ${\Omega}\subset {\mathbb{C}}^d$ be a connected open set. Let $(V_n)$ be a sequence of analytic subsets of codimension 1 in ${\Omega}$ with uniformly bounded volumes.
Assume that:
the $V_n$ are disjoint;
there exists $p_n\in V_n$ such that $p_n{\rightarrow}p\in {\Omega}$;
every cluster value of $(V_n)$ is locally irreducible at $p$.
Then the sequence $(V_n)$ converges.
The irreducibility assumption is necessary in this result, as shown by the sequence of curves in ${{{\mathbb{C}}^2}}$ defined by $V_{2n} = {\left\{x=0\right\}}$ and $V_{2n+1} = {\left\{xy = 1/n\right\}}$.
Assume that $V = \lim V_{n_j}$ and $W = \lim V_{n'_k}$ are distinct cluster limits of $(V_n)$. Then $V$ and $W$ are irreducible and contain $p$, therefore they must intersect non-trivially at $p$. Since $V$ and $W$ are of codimension 1, they intersect properly, that is $\dim(V\cap W)=d-2$. Now, proper intersections are robust under perturbations (see prop. 2 p. 141 and cor. 4 p. 145 in [@chirka]), so we infer that $V_{n_j}$ and $ V_{n'_k}$ intersect non-trivially for large $j$ and $k$, which is contradictory.
In general the limit of a sequence of smooth hypersurfaces can be singular. The smoothness of the limit can be ensured in certain circumstances (compare [@lyubich; @peters Prop. 11]).
\[prop:disk\] Let $(V_n)$ be a sequence of curves with uniformly bounded area in the unit ball of ${{{\mathbb{C}}^2}}$, which converges to $V$. Assume that for every $n$, $V_n$ is biholomorphic to a disk. Then $V$ is irreducible. If in addition the multiplicity of convergence is 1, then $V$ is smooth.
Fix $p\in V$. Let us first show that $V$ is locally irreducible at $p$. Fix a small ball $B$ about $p$ and let $V \cap B= V ^1\cup \cdots \cup V ^q$ be the decomposition into (local) irreducible components. Shrinking $B$ slightly if necessary, we may assume that each $V^i$ contains $p$ and that ${\partial}B\cap \overline{V^i}$ is not empty. Likewise, we may assume that $V$ is smooth near ${\partial}B$ and transverse to it. Hence $V\cap {\partial}B$ is a union of disjoint smooth (real) curves $(C_j)_j$ and each $C_j$ is contained in a unique irreducible component $V^i$.
For every $n$, let $V_n^i$ be a connected component of $V_n\cap B$. By the uniform bound on the area and Bishop Theorem, we can extract a subsequence $(V_{n'}^i)_{n'}$ converging to a curve $A$. Since $A$ is included in $V\cap B$, it is an union of irreducible components of $V\cap B$.
Observe that each $V_n^i$ is a holomorphic disk, as follows from the maximum principle applied to the subharmonic function $z\mapsto {\left\Vert\varphi_n(z)- p\right\Vert}$, where $\varphi_n$ is a parametrization of $V_n$. The boundary $C_n^i= \overline{V_n^i}\cap{\partial}B$ is homeomorphic to a circle for every $n$, hence $(C_{n'}^i)_{n'}$ converges to a connected compact set. Our assumptions on ${\partial}B\cap V$ imply that $C_{n'}^i$ is close to a unique component $C^i$ of $V\cap {\partial}B$. On the other hand the loop $C^i$ is in the boundary of a unique irreducible component of $V\cap B$, say $V^i$. Thus $A=V^i$ and $\overline A\cap {\partial}B= \overline{V^i}\cap {\partial}B=C_j$.
For every $n'$, consider any connected component $V_{n'}^j$ of $V_{n'}\cap B$, so that $V_{n'}^j$ converges to a certain irreducible component $V^j$ of $V$. If $V^i$ and $V^j$ are not equal, they intersect properly, hence the same occurs for $V_{n'}^i$ and $V_{n'}^j$, a contradiction. This proves that $V$ is locally irreducible and $\overline{V}\cap {\partial}B$ is a single loop $C_j$.
Now assume that the multiplicity of convergence is 1 and let us show that $V$ is smooth. Assume by contradiction that $V$ is singular at $p$. By the multiplicity 1 convergence hypothesis and the transversality of $V$ and ${\partial}B$, for large $n$ the loop $C_{n'}^i=\overline{V_{n'}^i}\cap{\partial}B$ is smooth, close to $C^i=\overline{V}\cap{\partial}B$ and (smoothly) isotopic to it. The (smooth) [*genus*]{} of $C^i$ is by definition the smallest genus of a smooth surface in $B$ bounded by $C^i$. It is invariant under smooth isotopy. It is known that if $V$ is singular at $p$, then for a sufficiently small ball $B= B(p, r)$ around $p$, the genus of $V\cap {\partial}B$ is positive (see [@milnor Cor 10.2]). Since $V_{n'}^i$ is a holomorphic disk, we arrive at a contradiction, which finishes the proof.
Uniform geometry of (un)stable manifolds {#sec:size}
========================================
In this section we consider a weakly stable substantial family $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ of polynomial automorphisms of dynamical degree $d\geq 2$. Let ${{\lambda_0}}\in {\Lambda}$, and fix a saddle periodic point $p_0$ for $f_0$. By weak stability, $p_{\lambda}$ persists as a saddle point throughout the family. Our purpose is to give uniform estimates on the geometry of $W^{s/u}_{\rm loc}(p_{\lambda})$, depending only of that of $W^{s/u}_{\rm loc}(p_0)$. For concreteness, from now on we deal with unstable manifolds. Recall that ${\Lambda}$ was assumed to be the unit disk.
We present two types of results, which will both be used afterwards. In §\[subs:area\] we show that the area of the local unstable manifold of $p_{\lambda}$ can be controlled throughout ${\Lambda}$. The techniques here are reminiscent of the results of [@bs8 §3]. We introduce a notion of size of a manifold at a point in §\[subs:param1\], and show that unstable parameterizations can be controlled in term of the size of $W^u(p)$ at $p$. Finally in §\[subs:param2\] we show that the size of $W^u (p_{\lambda})$ at $p_{\lambda}$ is uniformly bounded from below in a neighborhood of ${{\lambda_0}}$ that depends only on the size of $W^{s/u} (p_0)$ at $p_0$.
Areas of local unstable manifolds {#subs:area}
---------------------------------
By definition we say that $D\subset {{{\mathbb{C}}^2}}$ is a [*holomorphic disk*]{} if there is a holomorphic map $\phi:{\mathbb{D}}{\rightarrow}{{{\mathbb{C}}^2}}$ with $\phi({\mathbb{D}}) = D$, which extends to a homeomorphism $\overline {\mathbb{D}}{\rightarrow}\overline D$.
In the next lemma we give a basic estimate on the geometry of a holomorphic disk in ${{{\mathbb{C}}^2}}$, relying on simple ideas from conformal geometry. The modulus of an annulus, will be denoted by $\operatorname{mod}(A)$.
\[lem:geometry\] Let $D_1 \Subset D_2$ be a pair of holomorphic disks in ${{{\mathbb{C}}^2}}$ with $0\in D_1$ and let $d_1, d_2, m$ be positive real numbers such that $$\sup_{ z\in D_2} {\left\Vertz\right\Vert}\leq d_2,\ \sup_{ z\in {\partial}D_1} {\left\Vertz\right\Vert}\geq d_1,
\text{ and } \operatorname{mod}(D_2\setminus \overline{D_1})\geq m.$$
Then there exist positive constants $A$ and $r$ depending only on $d_1$, $d_2$ and $m$ such that the connected component of $D_2\cap B(0,r)$ containing 0 is a properly embedded submanifold in $B(0, r)$, of area not greater than $A$.
Notice that if $D$ is a holomorphic disk, ${\partial}D$ refers to the boundary of $D$ relative to its intrinsic topology. Notice also that the maximum principle applied to the subharmonic function $z\mapsto {\left\Vertz\right\Vert}^2$ on $D$ implies that $ \sup_{ z\in D} {\left\Vertz\right\Vert} = \sup_{ z\in {\partial}D} {\left\Vertz\right\Vert}$.
Fix a biholomorphism $\phi : {\mathbb{D}}{\rightarrow}D_2$ with $\phi(0) = 0$. We claim that there exists $\delta>0$ depending only on $m$ such that $\phi^{-1} (D_1)\subset D(0, 1-\delta)$. Indeed by assumption $\operatorname{mod}({\mathbb{D}}\setminus \overline{\phi^{-1} (D_1)})\geq m$. Now it follows from a classical result of Grötzsch that if $U$ is a connected and simply connected open subset of ${\mathbb{D}}$ containing $0$ and $z$ with $|z|=:1-x\in (0,1)$ then $\operatorname{mod}({\mathbb{D}}\setminus \overline U)\leq \operatorname{mod}({\mathbb{D}}\setminus [0,1-x])$ (see Ahlfors [@ahlfors Thm 4-6]). In addition, the map $x\mapsto \rho(x):= \operatorname{mod}({\mathbb{D}}\setminus [0,1-x])$ is increasing and continuous. Taking the contraposite, we see that if $\operatorname{mod}{\left({\mathbb{D}}\setminus \overline{\phi^{-1}(D_1)}\right)}\geq m$, then $\phi^{-1} (D_1)$ is contained in $ D(0, 1-\delta)$ with $\delta:=\rho^{-1}(m)$, and we conclude that $\phi$ satisfies $\sup_{ {\mathbb{D}}} {\left\Vert\phi\right\Vert}\leq d_2$ and $\sup_{ {\partial}D(0, 1-\delta)} {\left\Vert\phi\right\Vert}\geq d_1$. The result then follows from Lemma \[lem:compactness\] below.
\[lem:compactness\] Let $\phi: {\mathbb{D}}{\rightarrow}{{{\mathbb{C}}^2}}$ be a holomorphic mapping fixing $0$ and such that $\sup_{{\mathbb{D}}}{\left\Vert\phi\right\Vert}\leq d_2$ and $\sup_{{\partial}D(0, 1-\delta)}{\left\Vert\phi\right\Vert}\geq d_1$. Then there exist constants $A$ and $r$ depending only on $d_1$ and $d_2$ such that the connected component of $\phi(D(0, 1-\delta))\cap B(0, r)$ containing 0 is a properly embedded submanifold in $B(0, r)$, of area not greater than $A$.
This is an elementary compactness argument. Indeed let us show that for every such $\phi$ there exists a uniform $r$ such that the connected component $C$ of $\phi^{-1}{\left( {B(0,r)}\right)}$ containing 0 is relatively compact in $D(0, 1-\delta)$, that is $\overline C\subset D(0, 1-\delta)$. The area bound in turns follows from the Cauchy inequality. To prove that such a $r$ exists, for the sake of contradiction we suppose the existence of a sequence $(\phi_n$ of such functions which violate this property for $r_n\to 0$. Hence there exists for every $n$ a connected compact set $C_n\subset \overline D(0, 1-\delta)$ of diameter $\ge 1-\delta$ sent into $\overline B(0,r_n)$ by $\phi_n$. We can suppose that $(C_n)_n$ converges to a connected compact set $C_\infty$ of diameter $\ge 1-\delta$ and that $(\phi_n)_n$ converges uniformly to a certain $\phi_\infty$ on $\overline D(0, 1-\delta)$. Then $
\phi_\infty$ vanishes on $C_\infty$ hence on ${\mathbb{D}}$. This contradicts the fact that $\sup_{{\partial}D(0, 1-\delta)}{\left\Vert\phi_\infty\right\Vert}\geq d_1$.
Lemmas \[lem:geometry\] and \[lem:compactness\] can be combined to estimate how the geometry of an unstable manifold varies in a weakly stable family. For a saddle point $p$ and a positive real number $r$, we denote by $W^u_r(p)$ the connected component of $W^u(p)\cap B(p,r)$ containing $p$, which by the maximum principle is a holomorphic disk.
\[prop:uniform\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ of dynamical degree $d\geq 2$. Fix ${{\lambda_0}}\in {\Lambda}$ and a saddle periodic point $p_0$ for $f_0$, and denote by $(p_{\lambda})_{{\lambda}\in {\Lambda}}$ its continuation.
Consider a pair $D_1\Subset D_2$ of holomorphic disks in $W^u(p_0)$, with $p_0\in D_1$, and let $$g_1 = \sup (G^+_0{ \arrowvert_{ D_1}}), \ g_2 = \sup(G^+_0{ \arrowvert_{D_2}}) \text{ and } m = \operatorname{mod}(D_2\setminus \overline{D_1}).$$ Then for every $\widetilde {\Lambda}\Subset {\Lambda}$, there exist positive constants $r$, $g$ and $A$ depending only on $\widetilde {\Lambda}$, $g_1$, $g_2$ and $m$ such that for every ${\lambda}\in \widetilde {\Lambda}$, $W^u_r (p_{\lambda})$ is a properly embedded submanifold into $B(p_{\lambda}, r)$, contained in $h_{\lambda}(D_1)$, whose area is not greater than $A$, and such that $\sup (G_{\lambda}^+{ \arrowvert_{W^u_r (p_{\lambda})}})\geq g$.
Observe that $G^+$ does not vanish identically in any neighborhood of $p$ in $W^u_{\rm loc}(p)$ so $g_1, g_2$ are indeed positive.
Fix $\widetilde{\Lambda}\Subset {\Lambda}$. For ${\lambda}\in \widetilde{\Lambda}$, consider the disks $h_{\lambda}(D_1)$ and $h_{\lambda}(D_2)$. The quasiconformality of holomorphic motions implies that $$\mod{\left(h_{\lambda}(D_2) \setminus \overline {h_{\lambda}(D_1)}\right)}\leq C m,$$ where $C$ depends only on $\widetilde{\Lambda}$. In addition, it follows from Lemma \[lem:harnack\] that $$\sup \left( G^+_{\lambda}{ \arrowvert_{h_{\lambda}(D_2)}}\right)\leq C' g_2 \text{ and }
\sup \left(G^+_{\lambda}{ \arrowvert_{h_{\lambda}({\partial}D_1)}}\right) = \sup \left(G^+_{\lambda}{ \arrowvert_{h_{\lambda}(D_1)}}\right)\geq (C')^{-1} g_1,$$ where again $C'$ depends only on $\widetilde{\Lambda}$.
Now recall that $G^-_{\lambda}(z)$ is jointly continuous in $({\lambda},z)$ and that for every ${\lambda}$, $G^+{ \arrowvert_{K^-_{\lambda}}}$ is proper. Since unstable manifolds are contained in $K^-$ we infer that there exists $d_2$ depending only on $g_2$, $\widetilde \Lambda$ and the family $(f_\lambda)_\lambda$ such that for ${\lambda}\in \widetilde \Lambda$: $$\sup_{z\in h_{\lambda}(D_2)} {\left\Vertz-p_{\lambda}\right\Vert}
\leq d_2\;.$$
Also it is known that the Green function $G^+$ is Hölder continuous (see [@fs Thm 1.2]). Moreover the proof of [@fs] easily shows that the modulus of continuity of $G^+$ is locally uniform in ${\Lambda}$. Therefore, $G^+_{\lambda}(p_{\lambda})=0$ implies the existence of $d_1$ depending only on $g_1$, $\widetilde \Lambda$ and the family $(f_\lambda)$ such that for ${\lambda}\in \widetilde \Lambda$: $$\sup_{z\in h_{\lambda}({\partial}D_1)} {\left\Vertz-p_{\lambda}\right\Vert} \geq d_1.$$ Applying Lemma \[lem:geometry\] finishes the proof.
Estimates on unstable parameterizations and applications {#subs:param1}
--------------------------------------------------------
Endow ${{{\mathbb{C}}^2}}$ with its natural Hermitian structure. A [*bidisk of size $r$*]{} is the image of $D(0, r)^2$ by some affine isometry. The image of the unit bidisk under a general affine map will be referred to as an [*affine bidisk*]{}. A curve $V\subset {\mathbb{C}}$ is a graph over an affine line $L$ if its orthogonal projection onto $L$ is injective restricted to $V$. Then we have a well-defined notion of slope of a holomorphic curve with respect to $L$.
A curve $V$ through $p$ is said to have *bounded geometry at scale $r$ at $p$* (we also simply say that *$V$ has size $r$ at $p$*) if there exists a neighborhood of $p$ in $V$ that is a graph of slope at most 1 over a disk of radius $r$ in the tangent space $T_pV$.
Let $V$ be a disk of size $r$ at $p$, and fix orthonormal coordinates $(x,y)$ so that $p=0$ and $T_pV = {\left\{y=0\right\}}$. Then the connected component of $V$ through $p$ in the bidisk $ D(0,r)^2$ is a graph ${\left\{y=\varphi(x)\right\}}$ over the first coordinate with ${\left\vert\varphi'\right\vert}\leq 1$ and $\phi'(0)=0$.
\[pentereduite\] The Schwarz lemma implies that for every $x\in D(0, r)$, ${\left\vert\varphi'(x)\right\vert}\leq {\left\vertx\right\vert}/r$.
It will be a key fact for us that the Koebe Distortion Theorem provides estimates on unstable parameterizations in terms of the size of local unstable manifolds (see also Lemma \[lem:D12\] below).
\[lem:unstable\] Let $f$ be a polynomial automorphism of ${{{\mathbb{C}}^2}}$ and $p$ a saddle periodic point. Assume that $W^u(p)$ is of size $r$ at $p$. Normalize the coordinates so that $p=(0,0)$ and $W^u(p)$ is tangent to the $x$-axis at $p$. Denote by $\pi$ the first coordinate projection and let $\Gamma^u(p)$ be the component of $\pi^{-1}(D(0,r))\cap W^u(p)$ containing $p$.
Let $\psi^u:{\mathbb{C}}{\rightarrow}{{{\mathbb{C}}^2}}$ be an unstable parameterization, such that $\psi^u(0) = p$, and ${\left\Vert(\psi^u)'(0)\right\Vert}=1$. Then $\psi^u{\left(D{\left(0, {\frac{r}{4}}\right)}\right)}\subset\Gamma^u(p)\subset D(0,r)^2$. Moreover for every ${\left\vertz\right\vert}\leq\frac{r}{8}$, $$\label{eq:distortion}
D{\left(0, { \frac{{\left\vertz\right\vert}}{4} }\right)}\subset \pi\circ \psi^u{\left(D{\left(0,{\left\vertz\right\vert} \right)}\right)} \subset D{\left(0, 4{\left\vertz\right\vert} \right)}.
$$
Without loss of generality, rotate the first coordinate so that $(\pi\circ \psi^u)'(0) = 1$. Under the assumptions of the lemma, $\pi\circ\psi^u$ is univalent from some unknown domain ${\Omega}\subset {\mathbb{C}}$ onto $D(0,r)$. Now recall the Koebe Distortion Theorem (see [@ahlfors Thm 5-3]): if $g:{\mathbb{D}}{\rightarrow}{\mathbb{C}}$ is a univalent mapping, with $g'(0)=1$, then for $z\in {\mathbb{D}}$, $$\label{eq:distortion2}
\frac{{\left\vertz\right\vert}}{4} \leq \frac{{\left\vertz\right\vert}}{(1+{\left\vertz\right\vert})^2} \le
{\left\vertg(z)\right\vert} \leq \frac{{\left\vertz\right\vert}}{(1-{\left\vertz\right\vert})^2}.$$ Applying this to $g(z) = r^{-1} (\pi\circ\psi^u)^{-1}(rz)$, we first deduce that $(\pi\circ\psi^u)^{-1} (D(0,r))\supset D{\left(0, {\frac{r}{4}}\right)}$, thus $\psi^u{\left(D{\left(0, {\frac{r}{4}}\right)}\right)}\subset D(0,r)\times \mathbb C$ and so $\psi^u{\left(D{\left(0, {\frac{r}{4}}\right)}\right)}\subset\Gamma^u(p)$. It follows that the function $h$ in ${\mathbb{D}}$ defined by $ \zeta\mapsto h(\zeta) = \frac{4}{r} \pi\circ\psi^u{\left(\frac{r\zeta}{4}\right)}$ is univalent and satisfies $h'(0) =1$. Applying to $h$ yields , as desired.
Another important idea in this paper is that of the [*natural continuation*]{} of an unstable parameterization. Let us explain what this is about. Fix a parameter ${{\lambda_0}}\in {\Lambda}$, a saddle point $p_0$ for $f_0$, and an unstable parameterization $\psi^u_0:{\mathbb{C}}{\rightarrow}{{{\mathbb{C}}^2}}$ (in practice we often choose it so that ${\left\Vert(\psi^u_0)'(0)\right\Vert} =1$). We want to find a well-adapted holomorphic family of parameterizations $\psi^u_{\lambda}$ of $W^u(p_{\lambda})$, with $\psi^u_{\lambda}(0) = p_{\lambda}$. Since the Bers-Royden extension is canonical, the motion in ${{{\mathbb{C}}^2}}$ of a given point $q_0\in W^u(p_0)$ (denoted by $q_{\lambda}$) does not depend on this choice of parameterizations. Fix such a point $q_0$, say $ q_0 = \psi^u_0(1)$. We now fix the parameterization of $W^u(p_{\lambda})$ by declaring that $(\psi^u_{\lambda})^{-1}(q_{\lambda}) = (\psi^u_0)^{-1}(q_0) = 1$, or equivalently $(\psi^u_{\lambda})^{-1}(q_{\lambda})= 1$. This is by definition the [*natural continuation*]{} of $\psi^u_0$ in the family.
Such a holomorphic family of parameterizations can be constructed from any given holomorphic family $\widetilde\psi^u_{\lambda}$ by the formula $$\psi^u_{\lambda}(z ) = \widetilde\psi^u_{\lambda}{\left({(\widetilde\psi^u_{\lambda})^{-1}(q_{\lambda})} z\right)}.$$
The advantage is now that the holomorphic motion $h_{\lambda}^u$ in ${\mathbb{C}}$ defined by looking at the motion of points in the coordinate $\psi^u_{\lambda}$, that is, $h_{\lambda}^u(z) = (\psi^u_{\lambda})^{-1}(h_{\lambda}(\psi^u_0(z)))$, is normalized by $h_{\lambda}^u(0) = 0$ and $h_{\lambda}^u(1) = 1$. It is well known that such a normalized holomorphic motion in ${\mathbb{C}}$ satisfies uniform bounds : for every $\widetilde {\Lambda}\Subset {\Lambda}$, there exists constants $A$, $B$ and $\alpha$ depending only on $\widetilde {\Lambda}$ such that if ${\lambda},{\lambda}'\in {\Lambda}$, $$\label{eq:holmotion}
{\left\verth_{\lambda}^u(z) - h_{{\lambda}'}^u(z')\right\vert}\leq A \rho(z,z')^\alpha+ B{\left\vert{\lambda}-{\lambda}'\right\vert},$$ where $\rho$ denotes the spherical metric (see [@bers; @royden Cor. 2]).
We now use these techniques to give an estimate on parameterizations which supplements Proposition \[prop:uniform\].
\[prop:uniformparam\] Let $(f_{\lambda})_{{\lambda}\in{\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ and let $(p_{\lambda})_{{\lambda}\in{\Lambda}}$ be a holomorphically moving saddle point. Assume that for ${\lambda}={{\lambda_0}}$, $W^u(p_{{\lambda_0}})$ is of size $r_0$ at $p_0$. Let $\psi^u_{{\lambda_0}}$ be an unstable parameterization of $W^u(p_{{\lambda_0}})$ and $(\psi^u_{\lambda})_{{\lambda}\in {\Lambda}}$ be its natural continuation.
Then for every $\widetilde {\Lambda}\subset {\Lambda}$ there exist constants $c$ and $M$ depending only on $\widetilde {\Lambda}$ and $r_0$ such that if ${\lambda}\in \widetilde {\Lambda}$, $ {\left\Vert\psi^u_{\lambda}\right\Vert}\leq M$ on $D(0, c r_0)$.
By the Hölder continuity property of $G^+$, there exists $g = g(r_0)>0$ depending only on $r_0$ such that $ \sup {\left(G^+_{{\lambda_0}}{ \arrowvert_{W^u_{r_0\sqrt2}(p_{{\lambda_0}})}}\right)}\leq g$ (recall that a bidisk of radius $r_0$ is contained in a ball of radius $r_0\sqrt2$). Hence by Lemma \[lem:unstable\] we deduce that $G^+_{{\lambda_0}}{ \arrowvert_{\psi^u_{{\lambda_0}}( D(0, r_0/4))}}\leq g$, thus Lemma \[lem:harnack\] implies that $G^+_{\lambda}{ \arrowvert_{\psi^u_{\lambda}(h_{\lambda}^u(D(0, r_0/4)))}}\leq C g$ where $C$ depends only on $\widetilde {\Lambda}$. By the properness of $G^+{ \arrowvert_{K^-}}$, we deduce that $ \psi^u_{\lambda}(h_{\lambda}^u(D(0, r_0/4)))$ is uniformly bounded by $M(\widetilde {\Lambda}, r_0)$ in ${{{\mathbb{C}}^2}}$. Finally, since $h_{\lambda}^u(0) = 0$, by we infer that if ${\lambda}\in \widetilde {\Lambda}$, then $h^u_{\lambda}(D(0, r_0/4))$ contains $D(0, cr_0)$ for some $c = c(r_0, \widetilde {\Lambda})$, and we conclude that ${\left\Vert \psi^u_{\lambda}(h_{\lambda}^u(D(0,cr_0)))\right\Vert}\leq M$ for ${\lambda}\in \widetilde {\Lambda}$, which was the desired result.
Local persistence of the size of unstable manifolds {#subs:param2}
---------------------------------------------------
Recall the notation $\widehat{p} =
{\left\{({\lambda}, p_{\lambda}), \ {\lambda}\in {\Lambda}\right\}}$ for a holomorphically moving saddle point $p_{\lambda}$. Also, let us denote by $\operatorname{Tub}(\widehat p, r)$ the fibered tubular neighborhood of $\widehat p$ of radius $r$ in ${\Lambda}\times {{{\mathbb{C}}^2}}$, defined by $$\operatorname{Tub}(\widehat p, r) = {\left\{({\lambda}, z)\in {\Lambda}\times {{{\mathbb{C}}^2}}, \ {\left\Vertz-p({\lambda})\right\Vert}<r\right\}}.$$
Let us first isolate a geometric lemma. For notational ease, we put ${\mathbb{C}}^2_{\lambda}= {\left\{{\lambda}\right\}}\times {{{\mathbb{C}}^2}}$.
\[lem:smooth\] Let $\widehat p$ be the graph of a holomorphic mapping $p:{\Lambda}{\rightarrow}{{{\mathbb{C}}^2}}$ that is uniformly bounded by $M$ on ${\Lambda}$. Fix a direction $e\in {\mathbb{P}}^1({\mathbb{C}})$. Fix domains $\widetilde U$ and $U$ such that $\widetilde U \Subset U \Subset {\Lambda}$ and assume that $\Sigma$ is a hypersurface that is closed in $\operatorname{Tub}(\widehat p, r)\cap (U\times {{{\mathbb{C}}^2}})$ and such that for each ${\lambda}\in U$, $\Sigma \cap {\mathbb{C}}^2_{\lambda}$ is a graph of slope at most 1 over $p({\lambda})+ e$. Then $\Sigma$ is smooth and the volume of $\Sigma\cap {\left(\widetilde U\times {{{\mathbb{C}}^2}}\right)}$ is bounded by a constant depending only on $M$, $\widetilde U$, $U$ and $r$.
Identify $e$ and the corresponding line through 0 in ${{{\mathbb{C}}^2}}$. For ${\lambda}\in U$, let $\phi_\lambda$ be the unique holomorphic map from an open subset of $p({\lambda})+e$ to its orthogonal complement, whose graph is $\Sigma \cap {\mathbb{C}}^2_{\lambda}$. Let $\pi\colon \mathbb C ^2 \to \mathbb C$ be the orthogonal projection on the line $e$, and let $\hat \pi(\lambda, z) = (\lambda,\pi(z))$.
By continuity of the map $({\lambda},z)\mapsto \phi_{\lambda}(z)$, the following is an open subset of $U\times e\approx U\times {\mathbb{C}}$. $$\hat \pi(\Sigma)= \{({\lambda},z)\in U\times e:\; |\phi_{\lambda}(z)|^2+|z|^2<r^2\}.$$ By the graph property, the map $\hat \pi{ \arrowvert_{\Sigma}}$ is one-to-one from the subvariety $\Sigma$ onto the open subset $\hat \pi(\Sigma)$. Under these conditions it is classical that $\hat \pi{ \arrowvert_{\Sigma}}$ is a biholomorphism (see Prop. 3 p. 32 in [@chirka]). In particular $\Sigma$ is smooth and the map $ ({\lambda},z)\in \hat \pi(\Sigma')\mapsto
\phi_{\lambda}(z)$ is holomorphic in both variables (being the inverse of $\hat \pi$).
To get the volume bound, we remark that the set $\hat \pi(\Sigma)$ is bounded (specifically, it is contained in $U\times B(0,M+r)$). The derivative $\partial_ z\phi_{\lambda}$ is bounded by $1$, and the image of $\phi_{\lambda}$ is bounded by $M+r$. The Cauchy estimate implies that the derivative $\partial_{\lambda}\phi_{\lambda}$ is bounded on $
\widetilde U$. Consequently the volume of $\Sigma\cap \big(\widetilde U\times {{{\mathbb{C}}^2}}\big)$ is bounded by a constant depending only on $M$, $\widetilde U$, $U$ and $r$.
The main result in this subsection is that in a weakly stable family, the size of a holomorphically moving unstable manifold is locally uniformly bounded from below.
\[prop:surface\] Let $(f_{\lambda})_{{\lambda}\in{\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ and let $\widehat p = (p_{\lambda})_{{\lambda}\in{\Lambda}}$ be a holomorphically moving saddle point. Let $\widetilde{\Lambda}\Subset {\Lambda}$ be a relatively compact open subset and fix ${{\lambda_0}}\in \widetilde{\Lambda}$. Assume that for ${\lambda}={{\lambda_0}}$, $W^u(p_0)$ is of size $r_2$ at $p_0$. Then, for every $r_1<r_2$, there exists $\delta = \delta\big( r_1,r_2, \widetilde{\Lambda}\big)$ depending only on $r_1$, $r_2$ and $\widetilde{{\Lambda}}$ such that if ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\delta$, $W^u(p_{\lambda})$ is of size $r_1$ at $p_{\lambda}$, and $ W^u_{r_1} (p_{\lambda})$ is a graph of slope at most 1 over $p_{\lambda}+ E^u(p_0)$ (where $E^u(p_0)$ denotes the unstable direction at $p_0$).
Furthermore, there exists a submanifold $\widehat W^u_{r_1}$ in $\operatorname{Tub}(\widehat p, r_1) \cap (D({{\lambda_0}},\delta)\times {{{\mathbb{C}}^2}})$, such that for every ${\lambda}\in D({{\lambda_0}},\delta)$, $\widehat W^u_{r_1}\cap \mathbb{C}^2_{\lambda}= W^u_{r_1}
(p_{\lambda})$, whose volume is bounded by a constant $V\big(r_1, r_2, \widetilde {\Lambda}\big)$ depending only on $r_1$, $r_2$ and $\widetilde{\Lambda}$.
Start with an unstable parameterization $\psi^u_0$ of $W^u(p_{{\lambda_0}})$ satisfying ${\left\Vert(\psi^u_0)'(0)\right\Vert} =1$, and let $\pi_0$ be the orthogonal projection onto $E^u(p_0)$. For $i=1,2$, we denote by $D_i = (\pi_0\circ \psi_0^u)^{-1}{\left( D{\left(0, r_i\right)}\right)}$, and let $D_{12} := (\pi\circ \psi_0^u)^{-1}{\left( D{\left(0, r_{12}\right)}\right)}$, with $r_{12}:= (r_1+r_2)/2$. These are simply connected domains in ${\mathbb{C}}$ containing the origin, satisfying $D_1\Subset D_{12}\Subset D_2$.
The following lemma will be proved afterwards.
\[lem:D12\] For every $z\in D(0, r_{12})$, the following derivative estimate holds: $$\label{eq:D12}
\frac{1-r_1/r_2}{16}\leq {\left\vert {\left( {\left(\pi_0\circ \psi^u_0\right)}^{-1}\right)}'(z)\right\vert}\leq \frac{16}{(1-r_1/r_2)^3}.$$ Moreover the distance between $D_1$ and ${\partial}D_2$ is greater than $ r_2 (1-r_1/r_2)^2/32$.
Let now $(\psi^u_{\lambda})_{{\lambda}\in{\Lambda}}$ be the natural continuation of $\psi^u_0$. The second assertion of Lemma \[lem:D12\] together with imply that there exists $\delta=\delta\big( r_1,r_2, \widetilde{\Lambda}\big)$ such that if ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\delta$, $h_{\lambda}^u(D_1)$ stays uniformly far from ${\partial}(h_{\lambda}(D_2))$ (farther than $ r_2 (1-r_1/r_2)^2/50$, say) relative to the Euclidean metric on ${\mathbb{C}}$. Furthermore, arguing exactly as in Proposition \[prop:uniformparam\], we see that for ${\left\vert{\lambda}- {{\lambda_0}}\right\vert}<\delta$, $\psi^u_{\lambda}(h_{\lambda}^u(D_2)))$ is uniformly bounded in ${{{\mathbb{C}}^2}}$.
Let $\Psi:{\Lambda}\times {\mathbb{C}}{\rightarrow}{\Lambda}\times {{{\mathbb{C}}^2}}$ be defined by $\Psi({\lambda}, z) = ({\lambda}, \psi^u_{\lambda}(z))$, and put $$\widehat D_i = \bigcup_{{\lambda}\in B({{\lambda_0}}, \delta)}{\left\{{\lambda}\right\}}\times h_{\lambda}^u(D_i), \text{ for } i=1,2.$$ Since $h_{\lambda}^u( D_1)$ stays far from ${\partial}( h_{\lambda}^u(D_2))$, and $\Psi\big(\widehat D_2\big)$ is uniformly bounded in $B(0, \delta)\times {{{\mathbb{C}}^2}}$, by the Cauchy estimates, reducing $\delta$ again slightly if necessary the derivatives of $\Psi$ are uniformly bounded on $\widehat D_1$, with bounds depending only on $\widetilde {\Lambda}$, $ r_1$ and $r_2$.
We are now ready to conclude the proof. Let $\pi_{\lambda}$ (resp. $\pi_{\lambda}^\bot$) be the orthogonal projection onto $E^u(p_{\lambda})$ (resp. $(E^u(p_{\lambda}))^\bot$). The curve $W^u(p_{\lambda})$ is of size $r_1$ at $p_{\lambda}$ if for every $z$ in $D_1^{\lambda}:=\{z\in {\mathbb{C}},\; |\pi_{\lambda}\circ \psi^u_{\lambda}(z)|<r_1\}$ the following estimate holds: $$\label{pente} |\partial_z(\pi_{\lambda}^\bot\circ \psi^u_{\lambda})(z)|\le |\partial_z(\pi_{\lambda}\circ \psi^u_{\lambda})(z)|.$$
By the Cauchy estimate on $\partial_{\lambda}\partial_z \Psi$, $(\psi_{\lambda}^u)'(0)$ is close to $(\psi^u_0)'(0)$ for $|{\lambda}-{{\lambda_0}}|\le \delta$. In particular choosing $\delta = \delta(r_1, r_2, \widetilde{\Lambda})$ small enough we can ensure that ${\left\Vert\pi_0 - \pi_{\lambda}\right\Vert}\leq {\varepsilon}$ (resp. ${\left\Vert\pi_0^\bot - \pi_{\lambda}^\bot\right\Vert}\leq {\varepsilon}$), where ${\varepsilon}$ is as small as we wish. By the Cauchy estimate on $\partial_{\lambda}\Psi$, for $\delta=\delta(r_1,r_2, \widetilde {\Lambda})$ sufficiently small, when $|{\lambda}-{{\lambda_0}}|\le \delta$, the set $D_1^{\lambda}$ is included in $D_{12} = (\pi\circ \psi_0^u)^{-1}{\left( D{\left(0, r_{12})\right)}\right)}$, with $r_{12}:= (r_1+r_2)/2$. By Remark \[pentereduite\], for every $z\in D_{12}$, we have that $$|{\partial_z(\pi_0^\bot\circ \psi^u_0)(z)}|\le \frac{r_1+r_2}{2 r_2} {\left\vert\partial_z(\pi_0\circ \psi^u_0)(z)\right\vert}
= {\left(1-\frac{r_2-r_1}{2r_2}\right)} {\left\vert\partial_z(\pi_0\circ \psi^u_0)(z)\right\vert}.$$ From this we infer that with ${\varepsilon}$ as above and $z\in D_{12}$, $$\begin{aligned}
|{\partial_z(\pi_{\lambda}^\bot\circ \psi^u_{\lambda})(z)}|\le {\left(1-\frac{r_2-r_1}{2r_2}\right)} & {\left\vert\partial_z(\pi_{\lambda}\circ \psi^u_{\lambda})(z)\right\vert} \label{eq:relou} \\
&+ {\varepsilon}{\left\Vert{\partial}_z \psi_0^u(z)\right\Vert} + {\varepsilon}{\left\Vert{\partial}_z \psi_{\lambda}^u(z)\right\Vert} + 2 {\left\Vert {\partial}_z (\psi_0^u - \psi_{\lambda}^u) (z)\right\Vert} \notag.
\end{aligned}$$ In addition, the right hand inequality in implies that for $z\in D_{12}$, $${\left\vert\partial_z(\pi_0\circ \psi^u_0)(z)\right\vert} \geq \frac{(1-r_1/r_2)^3}{16}.$$ By the Cauchy estimate on $\partial_{\lambda}\partial_z \Psi$, for $\delta= \delta(r_1,r_2, \widetilde {\Lambda})$ sufficiently small, a similar estimate holds for $\partial_z(\pi_{\lambda}\circ \psi^u_{\lambda})(z)$ (with 16 replaced by 32, say) for $|{\lambda}-{{\lambda_0}}|\le \delta$ and $z\in D_{12}$. Recall that under our assumptions $D_{12}$ contains $D_1^{\lambda}$. Thus, by choosing ${\varepsilon}= {\varepsilon}(r_1, r_2, \widetilde {\Lambda})$ appropriately and reducing $\delta$ again if necessary, we can ensure that for $z\in D^{\lambda}_1$, $${\varepsilon}{\left\Vert{\partial}_z \psi_0^u(z)\right\Vert} + {\varepsilon}{\left\Vert{\partial}_z \psi_{\lambda}^u(z)\right\Vert} + 2 {\left\Vert {\partial}_z (\psi_0^u - \psi_{\lambda}^u) (z)\right\Vert} \leq
\frac{r_2-r_1}{2r_2} {\left\vert\partial_z(\pi_{\lambda}\circ \psi^u_{\lambda})(z)\right\vert},$$ which by yields .
Finally, we define $\widehat W^u_{r_1}$ to be the connected component of $\Psi\big(\widehat D_1\big)$ in $\operatorname{Tub}(\widehat p, r_1) \cap (D({{\lambda_0}},\delta)\times {{{\mathbb{C}}^2}})$ containing $\widehat p$, which is a surface with the desired properties (its smoothness follows from Lemma \[lem:smooth\]).
By the Koebe Distortion Theorem (see [@ahlfors Thm 5-3]), if $g:{\mathbb{D}}{\rightarrow}{\mathbb{C}}$ is a univalent mapping with $g'(0) =1$, then for every for every $r<1$ and every $z\in D(0,(1+r)/2)$ we have that $$\frac{1-r}{16} \leq \frac{1-|z|}{(1+|z|)^3} \leq |g'(z)| \leq \frac{1+|z|}{(1-|z|)^3}\leq \frac{16}{(1-r)^3}$$ Applying this to $g(z)=r_2^{-1}(\pi\circ \psi_0^u)^{-1}( r_2 z)$ and $r =r_1/r_2$, we deduce the desired bound on $((\pi\circ\psi^u_0){-1})'$. The estimate on the distance from $D_1$ to ${\partial}D_2$ immediately follows.
One may wonder why we did not conclude to the existence of such a submanifold $\widehat W^u$ in $\widetilde{\Lambda}\times {{{\mathbb{C}}^2}}$ straight after Proposition \[prop:uniform\], using a “fibered" compactness argument in the style of Lemma \[lem:compactness\].
The trouble is that in this general situation, having information about the area of $W^u_r(p_{\lambda})$ is not sufficient to control the geometry (say, the volume) of $\widehat W^u$ because $\widehat W^u\cap( {\left\{{\lambda}\right\}}\times B(p_{\lambda},r))$ can get disconnected for some values of ${\lambda}$, and the geometry of components other than $W^u_r(p_{\lambda})$ can go out of control. Proposition \[prop:surface\] shows that this phenomenon does not occur in some neighborhood of ${{\lambda_0}}$, depending on the size of $W^u_{\rm loc}(p)$.
Let us briefly describe an explicit example where this phenomenon happens. Let $\phi:{\mathbb{D}}\times {\mathbb{D}}\times {\rightarrow}{{{\mathbb{C}}^2}}$ be defined by $\phi({\lambda}, z) = z(z-2{\lambda}) (z,g(z))$, where $g$ is a holomorphic function on ${\mathbb{D}}$ such that ${\left\vertg\right\vert}<1$ but $\int_{\mathbb{D}}{\left\vertg'\right\vert}^2 = \infty$. By Lemma \[lem:geometry\], there exists positive constants $r$ and $A$ such that for every ${\lambda}\in D(0, 3/4)$, the connected component $W_{\lambda}$ of $\phi({\lambda}, {\mathbb{D}})\cap B(0,r)$ containing 0 is properly embedded and of area at most $A$.
Now let $\Phi: {\mathbb{D}}\times {\mathbb{D}}{\rightarrow}{\mathbb{D}}\times {\mathbb{D}}^2$ be defined by $\Phi({\lambda}, z)= ({\lambda}, \varphi({\lambda}, z))$. We see that $\Phi({\mathbb{D}}\times{\left\{0\right\}}) = {\mathbb{D}}\times {\left\{0\right\}}$. With $r$ as above, consider the component $\widehat W$ of $\Phi({\mathbb{D}}^2)\cap \operatorname{Tub}_r({\mathbb{D}}\times {\left\{0\right\}})$ containing $ {\mathbb{D}}\times {\left\{0\right\}}$. Put $V = {\left\{({\lambda},z), \; z(z-2{\lambda}) = 0\right\}}\subset {\mathbb{D}}^2$, and observe that $\Phi(V) = {\mathbb{D}}\times {\left\{0\right\}}$. Now it is easily shown that $\widehat{W}$ contains $\Phi(\operatorname{Tub}_{r/4}(V))$. So when ${\left\vert{\lambda}\right\vert}$ is close to 1/2, $\widehat{W} \cap {\mathbb{D}}^2_{\lambda}$ is made of at least two irreducible components, and for the values of ${\lambda}$ such that $\int_{D({\lambda}, r/4)} {\left\vertg'\right\vert}^2 = \infty$, one of these is of infinite volume.
Holomorphic motion of regular points {#sec:regular}
====================================
In this section we introduce the concept of regular point for a polynomial automorphism of ${{{\mathbb{C}}^2}}$, and prove Theorem \[theo:pesin strong\], in a slightly more general form.
Definitions and main statements
-------------------------------
\[defi:usregular\] We say that $p\in J^*$ is *u-regular* (resp. *s-regular*) if there exists $r>0$ and a sequence of saddle periodic points $p_n$ converging to $p$, with the property that $W^u(p_n)$ (resp. $W^s(p_n)$) has bounded geometry at scale $r$ at $p_n$.
If necessary, we make the size appearing in the definition explicit by speaking of “u-regular point of size $r$". The key property of u-regular (resp. s-regular) points is that they possess “local unstable (resp. stable) manifolds", as the following proposition shows.
\[prop:cvsize\] Let $f$ be a polynomial automorphism of ${{{\mathbb{C}}^2}}$ with dynamical degree $d\geq 2$. Let $p$ be a u-regular point of size $r$. Then there exists a unique submanifold $W^u_r(p)$ of size $r$ at $p$ such that if $(p_n)$ is any sequence of saddle points converging to $p$, such that $W^u(p_n)$ is of size $r$ at $p_n$, the sequence of disks $(W^u_r(p_n))$ converges to $W^u_r(p)$ with multiplicity 1 in $B(p,r)$. In particular the unstable directions converge as well.
By definition $W^u_r(p)$ will be referred to as the [*local unstable manifold*]{} of $p$ (and likewise for s-regular points). If the size $r$ is not relevant (i.e. if we think of the local unstable manifold as a germ) we simply refer to it as $W^u_{\rm loc}(p)$. Let us stress that we do *not* claim that $W^u_{\rm loc}(p)$ is an unstable manifold in the usual sense.
Fix $r'<r$. Then for $n\geq N(r')$, $W^u_r(p_n)\cap B(p,r')$ is a closed submanifold in $B(p,r')$. Given any subsequence $W^u_r(p_{n_j})$, up to further extraction we may assume that the $W^u_r(p_{n_j})$ are graphs of slope at most 2 over a fixed direction. It follows that all cluster values of the sequence $(W^u_r(p_n))$ are smooth, irreducible and of multiplicity 1. From Proposition \[prop:cv\] we infer that this sequence actually converges and the proof is complete.
\[defi:regular\] We say that $p\in J^*$ is *regular* if it is both s- and u-regular and if its local stable and unstable manifolds do not coincide at $p$. If in addition these local stable and unstable manifolds are transverse, we say that $p$ is transverse regular.
Examples of transverse regular points include saddle periodic points, as well as transverse homoclinic intersections (due to Smale’s horseshoe construction). It follows from Katok’s Closing Lemma that if $\nu$ is any hyperbolic ergodic invariant probability measure (that is, whose Lyapunov exponents satisfy $\chi^-(\nu)<0<\chi^+(\nu)$), then $\nu$-a.e. point is transverse regular in the sense of Definition \[defi:regular\].
Let us introduce a weaker notion of regularity, which involves the stable direction only.
\[defi:exposed\] Let $p\in J^*$ be a s-regular point. We stay that $p$ is *s-exposed* if one of the following equivalent properties is satisfied:
- $W^s_{{\mathrm{loc}}}(p)$ is not contained in $K$;
- $G^-{ \arrowvert_{W^s_{{\mathrm{loc}}}(p)}} \not\equiv 0$;
- $T^-\wedge [W^s_{{\mathrm{loc}}}(p)] >0$;
- for every saddle point $q$, the manifold $W^u(q)$ admits transverse intersections with $W^s_{{\mathrm{loc}}}(p)$.
The equivalence between $(i)$, $(ii)$ and $(iii)$ is clear. To see that $(iv)$ implies $(i)$, it suffices to notice that by the inclination lemma, a small neighborhood of $W^u(q)\cap W^s_{{\mathrm{loc}}}(p)$ in $W^s_{{\mathrm{loc}}}(p)$ cannot be included in $K$. The fact that $(iii)$ implies $(iv)$ follows from the techniques of [@bls §9]. The precise statement is that if $\Delta$ is any holomorphic disk such that $T^-\wedge [\Delta] >0$, then $\Delta$ admits transverse intersection with $W^u(q)$. The case where $\Delta$ is contained in a stable manifold is explained in detail in [@dl Lemma 5.1]. The proof for a general holomorphic disk $\Delta$ is identical.
If $p\in J^*$ is regular, then it is s- and u-exposed.
It is enough to prove that $p$ is s-exposed. Let $(p_n)$ be a sequence of saddle points with $W^u(p_n)$ of size $r$ at $p_n$ converging to $p$. We assume that the sequence $(p_n)$ takes infinitely many values, the remaining case is easy and left to the reader. Then removing at most one term to this sequence we may assume that for every $n$, $p\notin W^u(p_n)$. We claim that for large $n$, $W^u_r(p_n)$ intersects transversally $W^s_{\rm loc}(p)$ at a point close to $p$. If $W^u_{\rm loc}(p)$ and $W^s_{\rm loc}(p)$ are transverse this is clear. If not, since $W^u_r(p_n)\cap W^u_r(p)= \emptyset$, this follows from [@bls Lemma 6.4]. In any case, arguing as in the implication [*(iv)*]{}$\Rightarrow$[*(i)*]{} of Proposition \[defi:exposed\] we conclude that $W^s_{{\mathrm{loc}}}(p)$ is not contained in $K$ and we are done.
Here is a basic example:
If $p$ is a saddle point and $q$ belongs to the boundary of $W^s(p)\cap K^-$ relative to the intrinsic topology of $W^s(p)$, then $q$ is s-regular and exposed. Indeed it is shown in [@dl Lemma 5.1] that $q$ is the limit of a sequence of homoclinic intersections $(t_n)$, thus $q$ is exposed inside $W^s(p)$. Furthermore if $\Delta\subset W^s(p)$ is any disk containing $p$ and $q$, it follows from Smale’s horseshoe construction that for every $n$, $t_n$ is a limit of a sequence of saddle points $(p_{n,k})_k$ whose stable manifolds are graphs over $\Delta$. By considering the diagonal sequence $p_{n,n}$ we conclude that $q$ is s-regular, as desired.
Also there are examples of points which are s-regular and exposed but a priori not regular:
\[exam:zero\] Let $f$ be a dissipative polynomial automorphism. Let $m$ be an ergodic probability measure supported on $J^*$ with the property that $m= \lim m_n$, where for each $n$, $m_n$ is a probability measure equidistributed on a set of non-attracting periodic orbits (that is, saddle or semi-neutral). Since $f$ is dissipative, the negative Lyapunov exponent of $m_n$ satisfies $\chi^-(m_n) \leq \log{\left\vert\operatorname{Jac}(f)\right\vert} <0$, and likewise for $m$. On the other hand we make no assumption on the remaining (non-negative) Lyapunov exponent.
Then it is possible[^4] to adapt the techniques of Wang-Young [@WY §2] (see also Benedicks-Carleson [@BC91]) to show that if the Jacobian is sufficiently small, by the Pliss Lemma there exists a set of periodic points $A_r$ such that $m_n(A_r)\geq 1/2$ for each $n$, and such that for every $p\in A_r$ the local stable manifold of $p$ is of size $r$. Thus the same holds for $m$, and by ergodicity we conclude that $m$-a.e. point is s-regular. Furthermore, since in this case the local stable manifolds obtained by Proposition \[prop:cvsize\] coincide with Pesin stable manifolds, it follows that $m$-a.e. point is s-exposed.
An interesting example of such a situation is given by the unique invariant probability measure supported on the attractor of an infinitely renormalizable Hénon map (see [@clm]).
Here is a more precise version of Theorem \[theo:pesin strong\].
\[thm:regular\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ of dynamical degree $d\geq 2$. If for some parameter ${{\lambda_0}}$, $p_0\in J^*_{{\lambda_0}}$ is s-regular and exposed for $f_{{\lambda_0}}$, then there exists a unique holomorphic mapping ${\lambda}\mapsto p({\lambda})$ such that for every ${\lambda}$, $p({\lambda})\in K_{\lambda}$ and $p({{\lambda_0}})=p_0$. Moreover, for every ${\lambda}\in {\Lambda}$, $p({\lambda})$ is s-regular and exposed. In particular the branched holomorphic motion of $J^*$ is unbranched along the curve $({\lambda}, p({\lambda}))$.
Using the terminology introduced in [@dl §3], we can reformulate this by saying that the set $\mathcal{R}^s$ of s-regular and exposed points moves under a strongly unbranched, hence continuous, holomorphic motion. In particular for ${\lambda}_1, {\lambda}_2\in {\Lambda}$, $f_{{\lambda}_1}{ \arrowvert_{\mathcal{R}^s_{{\lambda}_1}}}$ is [*topologically*]{} conjugate to $f_{{\lambda}_2}{ \arrowvert_{\mathcal{R}^s_{{\lambda}_2}}}$, that is, the induced conjugacy $\mathcal{R}^s_{{\lambda}_1} {\rightarrow}\mathcal{R}^s_{{\lambda}_2}$ is a homeomorphism.
Since regular points are u- and s- exposed we obtain the following corollary, which contains Theorem \[theo:pesin strong\]. The conclusion about transversality is not obvious and will be proved afterwards.
\[cor:transverse\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ of dynamical degree $d\geq 2$. Then regular points move under a strongly unbranched holomorphic motion. Furthermore, transverse regular points remain transverse throughout the family.
The following corollary is a first step towards Theorem \[theo:hyp\].
If $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ is a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$ and if for some ${{\lambda_0}}\in {\Lambda}$, $f_{{\lambda_0}}$ is uniformly hyperbolic on $J^*_{{\lambda_0}}$, then for every ${\lambda}\in {\Lambda}$, $f_{\lambda}{ \arrowvert_{J^*_{\lambda}}} $ is topologically conjugate to $f_{{\lambda_0}}{ \arrowvert_{J^*_{{\lambda_0}}}} $.
Indeed, just observe that for a hyperbolic map, all points in $J^*$ are regular.
Proofs of Theorem \[thm:regular\] and Corollary \[cor:transverse\]
------------------------------------------------------------------
The plan of the proof is the following: we start by treating the particular case of points belonging to stable manifolds of saddle points. Using the results of §\[subs:param2\], we work locally in ${\Lambda}$ to show that the branched holomorphic motion of $J^*$ is unbranched at $s$-regular and exposed points. Then, using the global area bounds from §\[subs:area\] we show that regular points remain regular in the family, which allows to conclude the proof.
[**Step 0.**]{} A particular case.
Here we prove the following lemma, which is essentially contained in [@dl].
\[lem:particular\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a weakly stable substantial family of polynomial automorphisms of ${{{\mathbb{C}}^2}}$. Let $p: {\Lambda}{\rightarrow}{{{\mathbb{C}}^2}}$ be such that for every ${\lambda}$, $p({\lambda})\subset K_{\lambda}$. Assume that for some ${{\lambda_0}}\in {\Lambda}$, $p({{\lambda_0}})$ belongs to the stable manifold of a saddle point $m({{\lambda_0}})$ (which necessarily persists as $m({\lambda})$ in the family). Then for every ${\lambda}\in {\Lambda}$, $p({\lambda})\in W^s(m({\lambda}))$.
If in addition, $p({{\lambda_0}})$ is exposed inside $W^s(m({{\lambda_0}}))$, then the branched motion of $J^*$ is unbranched along $\widehat p$ and $p({\lambda})$ remains exposed throughout the family.
Of course, the same result holds for unstable manifolds. Recall that $p({\lambda})$ is exposed inside $W^s(m({\lambda}))$ if and only if $p({\lambda})$ is a limit of homoclinic or heteroclinic intersections for the intrinsic topology of $W^s(m({\lambda}))$.
The sequence of iterates $\widehat f^n{\left(\widehat p\right)}$ is locally uniformly bounded in ${\Lambda}\times{{{\mathbb{C}}^2}}$. Pick a cluster value $\widehat r$ of this sequence. Then $r({{\lambda_0}}) = m({{\lambda_0}})$ and $\widehat r\subset \widehat K$. Then by Theorem \[thm:unbranched\], $r\equiv m$, so we conclude that for every ${\lambda}$, $p({\lambda})\in W^s(m({\lambda}))$.
To get the second conclusion, note that for ${\lambda}= {{\lambda_0}}$, $p({{\lambda_0}}) = \lim t_k({{\lambda_0}})$ is a limit of homoclinic intersections, in the intrinsic topology of $W^s(m({{\lambda_0}}))$. By Theorem \[thm:unbranched\] $t_k({{\lambda_0}})$ admits a unique continuation $t_k$ to ${\Lambda}$ as a homoclinic intersection. Let $\Delta_{{\lambda_0}}\subset W^s(m({{\lambda_0}}))$ be a disk containing $p({{\lambda_0}})$. By the persistence of stable manifolds of saddle points, there exists a neighborhood $N$ of $({{\lambda_0}}, p({{\lambda_0}}))$ in ${\Lambda}\times {{{\mathbb{C}}^2}}$ and a smooth surface $\widehat W$ in $N$ such that $\widehat W\cap {{{\mathbb{C}}^2}}_{{\lambda_0}}= \Delta_{{\lambda_0}}$ and $\widehat
W\subset \widehat W^s(\widehat m)$. Now there are two cases: either $p({{\lambda_0}})$ is itself a homoclinic intersection, and we conclude by Theorem \[thm:unbranched\]. Otherwise $p({\lambda})$ is always distinct from $t_k({\lambda})$, and applying the Hurwitz Theorem inside $\widehat W$, we conclude that when $k{\rightarrow}\infty$, $t_k {\rightarrow}p$ in a neighborhood of ${{\lambda_0}}$, hence everywhere by analytic continuation. We conclude that $p({{\lambda_0}})$ admits a unique continuation $p$ staying in $K$
To conclude that the branched motion of $J^*$ is unbranched along $p$ at all parameters, it suffices to show that $p({\lambda})$ remains exposed inside $W^s(m({\lambda}))$. For this, it is enough to show that $G^-{ \arrowvert_{W^s(m({\lambda}))}} \not\equiv 0$ in any neighborhood of $p({\lambda})$, which follows directly from Lemma \[lem:harnack\]. The proof is complete.
Let us note for future reference the following consequence of this lemma.
\[cor:particular\] Let $p({{\lambda_0}})$ be a s-regular point of size $r$ and $t({{\lambda_0}})$ be an intersection between $W^s_r(p({{\lambda_0}}))$ and $W^u(m({{\lambda_0}}))$, where $m({{\lambda_0}})$ is a saddle point. Then there exists a unique continuation $t$ of $t({{\lambda_0}})$ such that $\widehat t\subset \widehat K$, and the branched motion of $J^*$ is unbranched along $\widehat t$.
In virtue of Lemma \[lem:particular\], it is enough to show that $t({{\lambda_0}})$ is exposed inside $W^u(m({{\lambda_0}}))$. For this, recall that $W^s_r(p({{\lambda_0}}))$ is the limit of a sequence $W^s_r(p_n({{\lambda_0}}))$ of local stable manifolds of saddle points. Therefore, by the persistence of proper intersections, $t({{\lambda_0}})$ is the limit in the intrinsic topology of $W^u(m({{\lambda_0}}))$ of a sequence of heteroclinic intersections with $W^s_r(p_n({{\lambda_0}}))$, and we are done.
[**Step 1.**]{} The branched motion is unbranched at s-regular and exposed points.
Let ${{\lambda_0}}\in\widetilde{\Lambda}\Subset {\Lambda}$ and $p({{\lambda_0}})$ be s-regular and exposed for $f_{{\lambda_0}}$. We want to construct a natural continuation of $p({{\lambda_0}})$. Let $r_0>0$ be such that there exists a sequence of distinct saddle points $p_n{\rightarrow}p$ with local stable manifolds of size $r'_0 := 2r_0$. Extracting a subsequence we assume that $(\widehat p_n)$ converges to some $\widehat p$ in ${\Lambda}\times {{{\mathbb{C}}^2}}$ (later on we will see that this limit is unique). It follows from Proposition \[prop:surface\] that for ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\delta=\delta(r_0, \widetilde{\Lambda})$, $W^s_{{\mathrm{loc}}}( p_n({\lambda}))$ is of size $r_0$, therefore $p({\lambda})$ is s-regular[^5].
Our goal here is to show that if $q:{\Lambda}{\rightarrow}{{{\mathbb{C}}^2}}$ is such that $q({{\lambda_0}})=p({{\lambda_0}})$ and $q({\lambda})\in K_{\lambda}$ for every ${\lambda}$, then $q({\lambda})=p({\lambda})$ for every ${\lambda}$.
There exists a neighborhood $N = \operatorname{Tub}{\left(\widehat p , r_0\right)}\cap (D({{\lambda_0}}, \delta(r_0))\times {{{\mathbb{C}}^2}})
$ of $({{\lambda_0}}, p({{\lambda_0}}))$ in ${\Lambda}\times {{{\mathbb{C}}^2}}$ and a smooth hypersurface $\widehat{W}^s_{r_0}{\left(\widehat{p}\right)}$ in $N$ such that the sequence of hypersurfaces $\widehat W^s_{r_0}{\left(\widehat{p}_n\right)}$ given by Proposition \[prop:surface\] converges to $\widehat{W}^s_{r_0}{\left(\widehat{p}\right)}$ with multiplicity 1 in $N$.
Indeed the volumes of $\widehat W^s_{r_0}{\left(\widehat{p}_n\right)}$ are uniformly bounded in $N$, so we may extract converging subsequences by Bishop’s Theorem \[thm:bishop\]. Fix such a subsequence $\widehat W^s_{r_0}{\left(\widehat{p}_{n_j}\right)}{\rightarrow}W$. Since the unstable directions $E^u(p_n({{\lambda_0}}))$ converge, for ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\delta$, $W^s_{r_0}(p_n({\lambda}))$ is a graph of slope at most 2 over a fixed direction for large $n$. In particular the convergence is of multiplicity 1. By Lemma \[lem:smooth\], $W$ is smooth, and from Proposition \[prop:cv\] we get that the sequence $\widehat W^s_{r_0}{\left(\widehat{p}_n\right)}$ actually converges. Notice that by construction, for ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\delta$, we have that $W^s_{r_0}(p({\lambda})) = \widehat{W}^s_{r_0}{\left(\widehat{p}\right)} \cap {\mathbb{C}}^2_{\lambda}$.
As a consequence of this claim and the Hurwitz Theorem we get the following:
\[claim:2\] There exists $\eta=\eta(r_0, \widetilde {\Lambda})>0$ such that if $q$ is a holomorphic map ${\Lambda}{\rightarrow}{\mathbb{C}}^2$ such that $\widehat q\subset \widehat K$ and $q({{\lambda_0}})\in W^s_{r_0/2}(p({{\lambda_0}}))$, then for ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\eta$, $q({\lambda})\in W^s_{r_0}(p({\lambda}))$.
Discarding at most one value of $n$ if needed, we may assume that $\widehat q$ is disjoint from $\widehat W^s_{r_0}{\left(\widehat{p}_n\right)}$. Since $q({{\lambda_0}}) \in \widehat{W}^s_{r_0/2}{\left(\widehat{p}\right)}$, by the Cauchy estimates, there exists $\eta=\eta(r_0)>0$ so that if ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}<\eta$, then the point $q({\lambda})$ stays in $B(p({\lambda}), r_0)$.
Now we have that $\widehat q \subset \widehat{W}^s_{r_0}{\left(\widehat{p}\right)}$. Indeed otherwise these two manifolds would have a proper intersection at $({{\lambda_0}}, q({{\lambda_0}}))$, and by persistence of proper intersections we would get that $\widehat q$ intersects $\widehat W^s_{r_0}{\left(\widehat{p}_n\right)}$, a contradiction.
If $p({{\lambda_0}})$ is a transverse regular point, this is enough to conclude. Indeed, applying the same reasoning in the unstable direction we get that $ q({\lambda}) \subset {W}^u_{r_0}{\left({p({\lambda})}\right)}$ for ${\lambda}$ close to ${{\lambda_0}}$. Now the intersection $\widehat{W}^s_{r_0}{\left(\widehat{p}\right)}\cap \widehat{W}^u_{r_0}{\left(\widehat{p}\right)}$ is transverse near $({{\lambda_0}}, p({{\lambda_0}}))$, therefore it coincides with $\widehat p$. We conclude that $\widehat p = \widehat q$ near ${{\lambda_0}}$, hence everywhere, which was the desired result.
Let us now deal with the general case.
Let $p({{\lambda_0}})$ be s-regular and exposed. If $q:{\Lambda}{\rightarrow}{{{\mathbb{C}}^2}}$ is such that $q({{\lambda_0}})=p({{\lambda_0}})$ and $q({\lambda})\in K_{\lambda}$ for every ${\lambda}$, then $q({\lambda})=p({\lambda})$ for every ${\lambda}$.
By Definition-Proposition \[defi:exposed\], for a given saddle point $m({{\lambda_0}})$, there exist a sequence of transverse intersection points $(t_k({{\lambda_0}}))$ between $W^s_{r_0/2}(p({{\lambda_0}}))$ and $W^u(m({{\lambda_0}}))$ such that $t_k({{\lambda_0}}){\rightarrow}p({{\lambda_0}}) = q({{\lambda_0}})$. By Corollary \[cor:particular\], there exists a unique holomorphic continuation $t_k$ of $t_k({{\lambda_0}})$ with the property that for every ${\lambda}\in {\Lambda}$, $t_k({\lambda})\in W^u(m({\lambda}))\cap K({\lambda})$. In particular if $p({{\lambda_0}})$ itself belongs to $W^u(m({{\lambda_0}}))$ we are done, so let us assume that $p({{\lambda_0}})\notin W^u(m({{\lambda_0}}))$. By the previous Claim \[claim:2\], for every ${\lambda}$ so that ${\left\vert{\lambda}-{{\lambda_0}}\right\vert}< \eta(r_0)$, the point $t_k({\lambda})$ belongs to $W^u(m({\lambda}))\cap W^s_{r_0}(p({\lambda}))$ and $q({\lambda})$ belongs to $W^s_{r_0}(p({\lambda}))$. To conclude, we observe that since $p({{\lambda_0}})=q({{\lambda_0}})\notin W^u(m({{\lambda_0}}))$, applying Corollary \[cor:particular\] again, we deduce that $t_k({\lambda})$ is disjoint from $q({\lambda})$ for every ${\lambda}\in{\Lambda}$. Working inside $\widehat{W}^s_{r_0}{\left(\widehat{p}\right)}$, which is a smooth complex surface, we can apply the Hurwitz Theorem to conclude that the sequence $\widehat t_k$ converges to $\widehat q$. Therefore the continuation $p$ is unique, which was the result to be proved.
[**Step 2.**]{} The branched motion preserves s-regularity and exposure.
Fix a relatively compact open set $\widetilde {\Lambda}\Subset {\Lambda}$. For ${{\lambda_0}}\in \widetilde {\Lambda}$, let $p({{\lambda_0}})\in J^*_{{\lambda_0}}$ be s-regular and exposed. Thus there exists a sequence of saddle points $(p_n)$ with stable manifolds of size $r_0$, such that $p_n({{\lambda_0}}) {\rightarrow}p({{\lambda_0}})$. By Proposition \[prop:cvsize\], $W^s_{r_0}(p_n({{\lambda_0}}))$ converges to $W^s_{r_0}(p({{\lambda_0}}))$ with multiplicity 1, that is, $W^s_{r_0}(p_n({{\lambda_0}}))$ is a graph over $W^s_{r_0}(p({{\lambda_0}}))$ for large $n$. Since $p({{\lambda_0}})$ is s-exposed, for every $r$ we get that $G^-{ \arrowvert_{W^s_r(p({{\lambda_0}}))}}$ is not identically 0.
Let $\psi^s_{{{\lambda_0}}, n}$ be a stable parameterization of $W^s (p_n({{\lambda_0}}))$ with $\|{(\psi^s_{{{\lambda_0}}, n})'(0)} \|=1$, and $(\psi_{{\lambda}, n}^s)$ be the natural continuation of $\psi_{{\lambda}_0, n}^s$, as defined in §\[subs:param1\]. Recall the notation $h_{\lambda}^s$ for the holomorphic motion inside stable manifolds, viewed inside the parameterizations $(\psi_{{\lambda}, n}^s)$. Recall that $h_{\lambda}^s$ satisfies $h_{\lambda}^s(0) = 0$ and $h^s_{\lambda}(1)=1$ (hence also the estimate ). By , there exists $c>0$ such that for every ${\lambda}\in \widetilde{\Lambda}$, $(h_{\lambda}^s)^{-1}(D(0, cr_0))\subset D(0, r_0/4)$. Without loss of generality, we can assume that $c<1/8$. Choose $c'<c$ so small that for every ${\lambda}\in \widetilde {\Lambda}$, $h_{\lambda}^s(D(0, c'r_0))\Subset D(0, cr_0)$
Fix a pair of holomorphic disks $D_1\Subset D_2$ in $W^s_{c'r_0/4}(p({{\lambda_0}}))$, with $p({{\lambda_0}})\in D_1$. Set $m =\operatorname{mod}(D_2\setminus \overline D_1)$, and for $i=1,2$, put $g_i = \sup G_{{\lambda_0}}^-{ \arrowvert_{D_i}}$. By the continuity of $G^-_{{\lambda_0}}$ and the multiplicity 1 convergence, for large $n$ we can lift $D_1$ and $D_2$ to holomorphic disks $D_{1, n}$ and $D_{2, n}$ in $W^s_{c'r_0/4}(p_n({{\lambda_0}}))$, such that $\operatorname{mod}(D_{2, n}\setminus \overline D_{1, n}){\rightarrow}m$ and $\sup G_{{\lambda_0}}^-{ \arrowvert_{D_{i, n}}}{\rightarrow}g_i$. From Lemma \[lem:unstable\] we infer that $\psi^s_{{{\lambda_0}}, n} (D(0, c'r_0)) \supset W^s_{c'r_0/4}(p_n({{\lambda_0}}))$, so in particular $\psi^s_{{{\lambda_0}}, n} (D(0, c'r_0))$ contains $D_{1, n}$ and $D_{2, n}$.
Now fix another parameter ${\lambda}_1\in \widetilde {\Lambda}$. By the first step of the proof we know that $p_n({\lambda}_1){\rightarrow}p({\lambda}_1)$. Applying Proposition \[prop:uniform\] we infer that there exist positive constants $r_1$, $g$ and $A$ such that $W^s_{r_1}(p_n({\lambda}_1))$ is a submanifold properly embedded into $B(p_n({\lambda}_1), r_1)$, contained in $h_{{\lambda}_1}(D_{1,n})$, with area at most $A$, and $\sup \big({G^-_{{\lambda}_1}{ \arrowvert_{W^s_{r_1}(p_n({\lambda}_1))}}}\big)\geq g$. Using Bishop’s Theorem, we extract a subsequence $n_j$ so that $(W^s_{r_1}(p_{n_j}({\lambda}_1)))_j$ converges to some analytic set $W\ni p({\lambda}_1)$ with $\sup \big({G^-_{{\lambda}_1}{ \arrowvert_{W}}}\big)\geq g$.
The main step of the proof is the following lemma.
\[lem:multiplicity\] The multiplicity of convergence of $W^s_{r_1}(p_{n_j}({\lambda}_1))$ to $W$ is equal to 1.
Before establishing the lemma let us show how to conclude the proof of Step 2. Recall that by the Maximum Principle $W^s_{r_1}(p_{n_j}({\lambda}_1))$ is a holomorphic disk. Since the multiplicity of convergence is 1, we deduce from Proposition \[prop:disk\] that $W$ is smooth. Also Proposition \[prop:cv\] implies that the sequence $W^s_{r_1}(p_n({\lambda}_1))$ actually converges.
Since $W$ is smooth at $p({\lambda}_1)$ and the multiplicity of convergence is 1, we see that in any small neighborhood of $p({\lambda}_1)$, $W^s_{r_1}(p_n({\lambda}_1))$ is a graph over $W$ for large $n$. In particular $W^s_{r_1}(p_n({\lambda}_1))$ has size uniformly bounded from below, and therefore $p({\lambda}_1)$ is regular. We already observed that $\sup \big({ G^-_{{\lambda}_1}{ \arrowvert_{W}}}\big)\geq g>0$, and the same holds in any neighborhood of $p({\lambda}_1)$ by choosing a smaller $r_0$ at the beginning. Hence $p({\lambda}_1)$ is s-exposed, which finishes the proof of Step 2.
For notational ease we put $n_j=n$. Let $k$ be the multiplicity of convergence of $W^s_{r_1}(p_{n}({\lambda}_1))$ to $W$. By Proposition \[prop:uniformparam\], we know that for every ${\lambda}\in \widetilde {\Lambda}$, $\|{\psi_{{\lambda}, n}^s }\| \leq M$ on $\widetilde {\Lambda}\times D(0, cr_0)$, so we can extract a converging subsequence (still denoted by $n$) to a limiting map $\varphi_{\lambda}(\cdot)$. Notice that for ${\lambda}= {{\lambda_0}}$, $(\psi_{{\lambda}, n}^s)_n $ converges on $D(0, r_0/4)$ to an injective map $D(0, r_0/4){\rightarrow}W^s_r(p({{\lambda_0}}))$
We recall that $D_{2,n}\subset \psi^s_{{{\lambda_0}}, n}(D(0, c'r_0))$, so by definition of $c'$ we get that for every ${\lambda}$, $\psi_{{\lambda}, n}^s{\left(D(0, cr_0)\right)}$ contains $h_{{\lambda}}(D_{2,n})$. It follows that for ${\lambda}= {\lambda}_1$, $W^s_{r_1}(p_n({\lambda}_1))\subset \psi_{{\lambda}_1, n}^s{\left(D(0, cr_0)\right)}$.
Furthermore, since $\operatorname{mod}(D_{2, n}\setminus D_{1,n}) {\rightarrow}m>0$, there exists a uniform $c''<c$ such that $W^s_{r_1}(p_n({\lambda}_1))\subset \psi_{{\lambda}_1, n}^u{\left(D(0, c''r_0)\right)}$. It follows that $\varphi_{{\lambda}_1}$ is non-constant and that the component ${\Omega}$ of $0$ in $\varphi_{{\lambda}_1}^{-1}(W)$, is such that $\varphi_{{\lambda}_1}:{\Omega}{\rightarrow}W$ is proper. Its degree is equal to the mutiplicity of convergence $k$.
Since $G^+_{{\lambda}_1}{ \arrowvert_{W}}$ (resp. $G^+_{{\lambda}_1}\circ \varphi_{{\lambda}_1}$) is continuous and not harmonic, its Laplacian is nonzero and gives no mass to points. By the s-exposure assumption, Definition-Proposition \[defi:exposed\] ensures the existence of a saddle point $m({\lambda}_1)$ whose unstable manifold intersects transversally $W$ at a certain point $q({\lambda}_1)$ which is a regular value of $\varphi_{{\lambda}_1}$.
Now if $k>1$, there exist two distinct points $a$ and $b$ in ${\Omega}\subset D(0, cr_0)$ such that $\varphi_{{\lambda}_1} (a) = \varphi_{{\lambda}_1}(b) = q({\lambda}_1)$. Thus there exists $a_n {\rightarrow}a$ (resp. $b_n{\rightarrow}b$) such that $\psi_{{\lambda}_1, n}^s(a_n)$ (resp. $\psi_{{\lambda}_1, n}^s(b_n)$ ) are intersection points of $W^u(m({\lambda}_1))$ and $W^s_{r_1}(p_n({\lambda}_1))$ converging to $q({\lambda}_1)$.
To conclude the proof, we flow back to ${{\lambda_0}}$ using the holomorphic motion to obtain a contradiction with Corollary \[cor:particular\]. The details are as follows. Consider the continuations of the heteroclinic intersections $\psi_{{\lambda}_1, n}^s(a_n)$ and $\psi_{{\lambda}_1, n}^s(b_n)$ for ${\lambda}\in \widetilde {\Lambda}$. Notice that they stay in a compact piece of $W^u(m({\lambda}))$. For ${\lambda}={{\lambda_0}}$, the corresponding points are $\psi_{{{\lambda_0}}, n}^s((h_{{\lambda}_1}^s)^{-1}(a_n))$ and $\psi_{{{\lambda_0}}, n}^s((h_{{\lambda}_1}^s)^{-1}(b_n))$, which converge respectively to $\varphi_{{\lambda_0}}((h_{{\lambda}_1}^s)^{-1}(a))$ and $\varphi_{{\lambda_0}}((h_{{\lambda}_1}^s)^{-1}(b))$. Now $(h_{{\lambda}_1}^s)^{-1}(a)$ and $(h_{{\lambda}_1}^s)^{-1}(b)$ are distinct and by definition of $c$, they belong to $D(0, r_0/4)$. On this disk, $\varphi_{{\lambda_0}}$ is injective therefore $\varphi_{{\lambda_0}}((h_{{\lambda}_1}^s)^{-1}(a))$ and $
\varphi_{{\lambda_0}}((h_{{\lambda}_1}^s)^{-1}(b))$ are distinct intersection points between $W^s_r(p({{\lambda_0}}))$ and $W^u(m({{\lambda_0}}))$ with continuations colliding at ${\lambda}_1$. This contradicts Corollary \[cor:particular\], and concludes the proof of the lemma.
The proof does not give any estimate on the size $r_1$ of the stable manifold $W^s(p({\lambda}_1))$ at $p({\lambda}_1)$. Indeed, $r_1$ depends on the size of $W$ at $p({\lambda}_1)$, upon which we have no control (the only information we have is a local area bound). In particular it is unclear whether $r_1$ depends only on $r_0$ and $\widetilde {\Lambda}$.
As opposed to Step 1 of the proof, Step 2 does not become significantly easier if we assume that $p({{\lambda_0}})$ is regular instead of s-regular and exposed. Indeed, the whole point is to prove that $p({\lambda})$ remains s-regular throughout ${\Lambda}$.
[**Step 3.**]{} Conclusion.
We have shown in Steps 1 and 2 that if $p({{\lambda_0}})$ is s-regular and exposed, then it admits a unique holomorphic continuation $\widehat p\subset \widehat K$ such that for every ${\lambda}$, $p({\lambda})$ is s-regular and exposed, too. Thus the branched motion of $J^*$ must be unbranched, in particular continuous, at $({\lambda}, p({\lambda}))$. In particular $p({\lambda})$ cannot collide with the continuation of any other point in $J^*$, so we indeed have a continuous holomorphic motion of $\mathcal{R}^s$. This finishes the proof of Theorem \[thm:regular\].
Let us note for future reference the following consequence of the proof.
\[prop:strong regular\] Let $(f_{\lambda})$ be a weakly stable holomorphic family of polynomial automorphisms. Assume that for ${\lambda}= {{\lambda_0}}$, $p({\lambda}_0)$ is a regular point and $(p_n({{\lambda_0}}))$ is a sequence of saddle points converging to $p({{\lambda_0}})$ such that $W^s(p_n({{\lambda_0}}))$ (resp. $W^u(p_n({{\lambda_0}}))$) is of size $r_0$ at $p_n({{\lambda_0}})$. Then for every ${\lambda}_1\in {\Lambda}$, there exists $r_1>0$ such that $p_n({\lambda}_1){\rightarrow}p({\lambda}_1)$ and $W^s(p_n({\lambda}_1))$ (resp. $W^u(p_n({\lambda}_1))$) is of size $r_1$ at $p_n({\lambda}_1)$.
We now show that regular points (resp. transverse regular points) remain regular (resp. transverse).
It follows from Theorem \[thm:regular\] that the regular points move without collision, and remain s- and u-regular and exposed in both directions. Let $p({{\lambda_0}})$ be regular relative to $f_{{\lambda_0}}$. Then for every ${\lambda}\in {\Lambda}$, $p({\lambda})$ is s- and u- regular, so it possesses local stable and unstable manifolds. If $W^u_{{\mathrm{loc}}}(p({\lambda}_1))$ was to coincide with $W^s_{{\mathrm{loc}}}(p({\lambda}_1))$, we would get that $W^u_{{\mathrm{loc}}}(p({\lambda}_1)) = W^s_{{\mathrm{loc}}}(p({\lambda}_1)) \subset K_{{\lambda}_1}$, thus contradicting s- and u-exposure. Therefore $p({\lambda}_1)$ is regular.
To show that transverse regular points stay transverse, recall from the proof of Theorem \[thm:regular\] that if $p({{\lambda_0}})$ is regular, then $W^s_{{\mathrm{loc}}}(p({{\lambda_0}}))$ and $W^u_{{\mathrm{loc}}}(p({{\lambda_0}}))$ can be locally continued as smooth surfaces $\widehat W^s_{{\mathrm{loc}}}(\widehat p)$ and $\widehat W^u_{{\mathrm{loc}}}(\widehat p)$ in $N({{\lambda_0}})\times B(p({{\lambda_0}}),r)$. Now assume that $W^s_{{\mathrm{loc}}}(p({{\lambda_0}}))$ and $W^u_{{\mathrm{loc}}}(p({{\lambda_0}}))$ are tangent at $p({{\lambda_0}})$, that is, their intersection multiplicity at $ p({{\lambda_0}}))$ is $m>1$. If this tangency does not persist for nearby parameters, by the persistence of proper intersections, we get that for nearby ${\lambda}$, $W^s_{{\mathrm{loc}}}(p({\lambda}))$ and $W^u_{{\mathrm{loc}}}(p({\lambda}))$ intersect at $m$ points counting multiplicities, not all identical to $p({\lambda})$.
Consider the intersection $\widehat C = \widehat W^s_{{\mathrm{loc}}}(\widehat p)\cap \widehat W^u_{{\mathrm{loc}}}(\widehat p)$. This is a curve in $N({{\lambda_0}})\times B(p({{\lambda_0}}),r)$ such that $\widehat C\cap {\mathbb{C}}^2_{{\lambda_0}}= {\left\{p({{\lambda_0}})\right\}}$. One irreducible component of $\widehat C$ is given by the continuation $\widehat p$, and by assumption there exists another irreducible component $\widehat C'$ of $\widehat C$.
Assume first that $\widehat C'$ is a graph $\widehat q$ over $N({{\lambda_0}})$. Then, since for every ${\lambda}\in N({{\lambda_0}})$, $ q({\lambda})\in W^s_{{\mathrm{loc}}}(p({\lambda}))\cap W^u_{{\mathrm{loc}}}(p({\lambda})) \subset K_{\lambda}$, we get a collision between $ p$ and a holomomorphically moving point $q$ staying in $K$, which contradicts Theorem \[thm:regular\].
We will reduce the general case to this one by a classical trick: replacing ${\Lambda}$ by a well-suited branched cover $M{\rightarrow}{\Lambda}$. We detail the argument for the convenience of the reader. Consider a local irreducible component of $\widehat C'$ at $({{\lambda_0}}, p({{\lambda_0}}))$, still denoted by $\widehat C'$ for simplicity. Denote by $\varpi: {\mathbb{D}}{\rightarrow}\widehat C'$ a normalization of $\widehat C'$ such that $\varpi(0) = ({{\lambda_0}}, p({{\lambda_0}}))$. Denote respectively by $\pi_{\Lambda}$ and $\pi_{{{\mathbb{C}}^2}}$ the projection onto the first and second factors in ${\Lambda}\times {{{\mathbb{C}}^2}}$ and put $M = {\mathbb{D}}$ and ${\lambda}(\mu) = \pi_{\Lambda}\circ \varpi (\mu)$. Then we can consider the holomorphic family of polynomial automorphisms defined by $(\widetilde f_\mu) := (f_{{\lambda}(\mu)})$, which is weakly stable (of course, nothing has changed from the dynamical point of view). For $\mu= 0$, the point $p({\lambda}(0))$ is regular and can be continued as a regular point $\mu \mapsto p({\lambda}(\mu))$ as before. But now we have another holomorphic continuation of $(0,p({\lambda}(0)))$ in $\widehat K\subset M\times {{{\mathbb{C}}^2}}$ given by $\mu\mapsto (\mu, \pi_{{{\mathbb{C}}^2}}\circ \varpi(\mu))$. Thus we arrive at a contradiction and the proof is complete.
A similar argument shows that more generally the order of tangency between local stable and unstable manifolds of regular points is preserved in weakly stable families.
Propagation of hyperbolicity {#sec:hyperbolic}
============================
In this section we establish Theorem \[theo:hyp\], that is, we prove that uniform hyperbolicity on $J^*$ is preserved in weakly stable families. Let us start with a variation on Definition \[defi:usregular\].
\[defi:uniformly regular\] We say that $p\in J^*$ is uniformly u-regular (resp. uniformly s-regular) if there exists $r>0$ such that for every sequence of saddle points $(p_n)$ converging to $p$, $W^u(p_n)$ (resp. $W^s(p_n)$) is of size $r$ at $p_n$.
Likewise, $p$ is uniformly (resp. transverse) regular if it is (resp. transverse) regular and uniformly regular in both stable and unstable directions.
If necessary we will specify the size appearing in the definition by saying that “$p\in J^*$ is uniformly u-regular of size $r$". Recall from Proposition \[prop:cvsize\] that if $p$ is uniformly u-regular of size $r$, then it has a well defined local unstable manifold $W^u_r(p)$ and that if $p_n{\rightarrow}p$ is any sequence of saddle points, $W^u_r(p_n)$ converges to $W^u_r(p)$ with multiplicity 1.
If $f$ is uniformly hyperbolic on $J^*$, then every $p\in J^*$ is uniformly regular and transverse. Interestingly enough, the converse is true.
\[prop:criterion\] Let $f$ be a polynomial automorphism of ${{{\mathbb{C}}^2}}$ with dynamical degree $d\geq 2$. If every point in $J^*$ is uniformly regular and transverse then $f$ is uniformly hyperbolic on $J^*$.
The main step of the proof is the following lemma.
\[lem:lamination\] Let $f$ be a polynomial automorphism of ${{{\mathbb{C}}^2}}$, such that every point in $J^*$ is uniformly u-regular. Then there exists a neighborhood $N$ of $J^*$ such that the restriction to $N$ of $\bigcup_{p\in J^*} W^u_{{\mathrm{loc}}}(p)$ forms a lamination.
Let us start by showing that the size of unstable manifolds is uniformly bounded from below. For this, notice that Definition \[defi:uniformly regular\] may be reformulated as follows: $p$ is uniformly u-regular if there exists $r>0$ and ${\varepsilon}>0$ such that if $q\in B(p,{\varepsilon})$ is any saddle point, then $W^u(q)$ is of size $r$ at $q$. Then by compactness of $J^*$, we can cover $J^*$ with finitely many such balls, and deduce that if every point in $J^*$ is uniformly u-regular, then the size of unstable manifolds of saddle points is uniformly bounded form below, as claimed.
From this point, the remainder of the proof is classical. As observed above, for every $p\in J^*$ there exists $r>0$ and ${\varepsilon}>0$ such that if $q\in B(p,{\varepsilon})$ is any saddle point, then $W^u_r(q)$ is of size $r$ at $q$. Taking ${\varepsilon}$ smaller if needed, we may assume that $W^u_r(q)$ is closed in $B(p, {\varepsilon})$. Furthermore, any two such local unstable manifolds of saddle points are disjoint or coincide. Thus taking the closure, we get that $\overline{\bigcup W^u_r(q)}\cap B(p, {\varepsilon})$ is a lamination in $B(p,{\varepsilon})$, where the union ranges over all saddle points lying in $B(p, {\varepsilon})$. The result follows.
The result is essentially a direct consequence of Theorem 8.3. in [@bs8], which asserts that if there exist laminations of $J^+$ and $J^-$ in a neighborhood of $J^*$, which are transverse at every point of $J^*$ then $f$ is uniformly hyperbolic on $J^*$.
In our situation, the existence of stable and unstable laminations $\mathcal L^s$ and $\mathcal L^u$ is guaranteed by Lemma \[lem:lamination\], while these laminations are transverse at every point of $J^*$ by assumption.
Unfortunately, this is slightly different from the hypotheses of [@bs8 Thm. 8.3] because we do not know that the lamination $\mathcal L^u$ fills up the whole $J^-$ in a neighborhood of $J^*$. However, the reader will easily check that the only place in the proof of [@bs8] where this assumption is used is to ensure that for every $p\in J^*$, $W^u_{{\mathrm{loc}}}(p)$ is contained in a leaf of $\mathcal L^u$, which is trivially satisfied in our case. Hence the result applies and we are done.
We now have all the necessary ingredients for Theorem \[theo:hyp\].
By assumption, every point in $J^*_{{\lambda_0}}$ is uniformly regular and transverse. From Corollary \[cor:transverse\] we deduce that $J^*$ moves holomorphically and all points remain regular and transverse. Proposition \[prop:strong regular\] implies that strong s- and u-regularity are preserved as well. Therefore, for every ${\lambda}\in {\Lambda}$, every point in $J^*_{\lambda}$ is uniformly regular and transverse, so the result follows from Proposition \[prop:criterion\].
The concept of quasi-expansion, developed in [@bs8] has been a source of inspiration for the techniques in this paper. A polynomial automorphism of ${{{\mathbb{C}}^2}}$ of dynamical degree $d\geq 2$ is [*quasi-expanding*]{} if there exists positive constants $r$ and $A$ such that for every saddle point $p$, $W^u_r(p)$ is properly embedded in $B(p, r)$, of area at most $A$ and for every $\delta>0$ there exists $\eta>0$ such that $\sup\big({G^+{ \arrowvert_{W^u_\delta(p)}}\big)}\geq \eta$ (see [@bs8 Cor 3.5] for this definition). There is a parallel notion of [*quasi-contraction*]{} in the stable direction.
It is worthwhile to state the following result of independent interest.
\[prop:qe\] Let $(f_{\lambda})_{{\lambda}\in {\Lambda}}$ be a weakly stable and substantial holomorphic family of polynomial automorphisms. If there exists ${{\lambda_0}}\in {\Lambda}$ such that $f_{{\lambda_0}}$ is quasi-expanding, then $f_{\lambda}$ is quasi-expanding for every ${\lambda}\in {\Lambda}$.
For ${\lambda}={{\lambda_0}}$, let $r$, $A$ be the uniform constants provided by the definition of quasi-expansion. Let $p({{\lambda_0}})$ be a saddle point. By [@bs8 Thm 3.1], the modulus of the annulus $W^u_r(p({{\lambda_0}}))\setminus W^u_{r/2}(p({{\lambda_0}}))$ is bounded from below by a constant $m$ depending only on $A$ and $r$. By the Hölder continuity property of $G^+$ we get that $\sup\big({G^+{ \arrowvert_{W^u_r(p)}}}\big)\leq g_2(r)$ and by the definition of quasi-expansion, $\sup\big({G^+{ \arrowvert_{W^u_{r/2}(p)}}\big)}\geq g_1>0$. Therefore applying Proposition \[prop:uniform\] we obtain for every ${\lambda}\in {\Lambda}$ positive constants $r'$, $A'$ and $g'$ such that for every saddle point $p'$ for $f_{{\lambda}'}$ , $W^u_{r'}(p')$ is properly embedded in $B(p', r')$, of area at most $A'$ and $\sup\big({G^+{ \arrowvert_{W^u_{r'}(p')}}\big)}\geq g'$. Finally, Theorem 3.4 in [@bs8] implies that $f_{\lambda}$ is quasi-expanding.
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[^1]: A necessary and sufficient condition for this is that the [*dynamical degree*]{} $d = \lim (\deg(f^n))^{1/n}$ satisfies $d\geq 2$, see §\[sec:prel\] for more details.
[^2]: that is, the complex Jacobian $\mathrm{Jac}(f)$ satisfies ${\left\vert\mathrm{Jac}(f)\right\vert}<d^{-2}$.
[^3]: For instance the value of the Green function at critical points is in general not invariant in a $J$-stable family of polynomials.
[^4]: Details will appear in subsequent work.
[^5]: We see that it is crucial here that $\delta(r_1, r_2)$ in Proposition \[prop:surface\] depends only on $r_1$ and $r_2$.
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abstract: 'Radiation pressure forces in cavity optomechanics allow for efficient cooling of vibrational modes of macroscopic mechanical resonators, the manipulation of their quantum states, as well as generation of optomechanical entanglement. The standard mechanism relies on the cavity photons directly modifying the state of the mechanical resonator. Hybrid cavity optomechanics provides an alternative approach by coupling mechanical objects to quantum emitters, either directly or indirectly via the common interaction with a cavity field mode. While many approaches exist, they typically share a simple effective description in terms of a single force acting on the mechanical resonator. More generally, one can study the interplay between various forces acting on the mechanical resonator in such hybrid mechanical devices. This interplay can lead to interference effects that may, for instance, improve cooling of the mechanical motion or lead to generation of entanglement between various parts of the hybrid device. Here, we provide such an example of a hybrid optomechanical system where an ensemble of quantum emitters is embedded into the mechanical resonator formed by a vibrating membrane. The interference between the radiation pressure force and the mechanically modulated Tavis–Cummings interaction leads to enhanced cooling dynamics in regimes in which neither force is efficient by itself. Our results pave the way towards engineering novel optomechanical interactions in hybrid optomechanical systems.'
address:
- '$^1$ Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany'
- '$^2$ Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark'
author:
- 'Ondřej Černot[í]{}k$^1$, Claudiu Genes$^1$ and Aur[é]{}lien Dantan$^2$'
title: Interference effects in hybrid cavity optomechanics
---
[*Keywords*]{}: Cavity optomechanics, hybrid quantum systems, Fano resonance, cooling, interference
Introduction
============
Cavity optomechanics [@Aspelmeyer2014] has reached a remarkable success in coupling high-quality mechanical resonators and light via radiation pressure. This interaction can be used for measurements of small mechanical displacements and external forces [@Regal2008; @Hertzberg2009; @Forstner2012; @Schreppler2014], for quantum state transfer between the cavity field and the mechanical oscillator, and for ground state mechanical cooling [@Chan2011; @Teufel2011]. Other achievements are frequency conversion between cavity modes [@Dong2012; @Hill2012; @Andrews2014; @Cernotik2017; @Midolo2018], generation of two-mode squeezing useful for amplification of the mechanical motion or the cavity field [@Ockeloen-Korppi2016; @Toth2017], and the creation of photon–phonon or phonon–phonon entanglement [@Palomaki2013b; @Riedinger2016; @Ockeloen-Korppi2018; @Riedinger2018]. Many of these applications rely on the simultaneous fulfilment of two requirements: i) operating in the resolved sideband regime in which the cavity linewidth is smaller than the mechanical frequency and ii) having a sufficiently strong coupling between photons and phonons. In systems based on optical Fabry–Pérot resonators (such as membrane-in-the-middle optomechanical devices [@Thompson2008; @Jayich2008]), these two conditions are not independent; using a short optical cavity (leading to a small mode volume and large coupling strengths) results in a large cavity decay rate such that the resolved sideband regime cannot be reached. The sideband resolution is improved by using a long cavity in which, however, the coupling is reduced owing to the large mode volume. It is therefore desirable to investigate alternative approaches that can either relax the conditions on sideband resolution or improve the coupling strength without increasing the decay rate.
In recent years, hybrid optomechanical systems emerged as an interesting platform for novel optomechanical experiments [@Treutlein2014; @Kurizki2015]. In these systems, cavity fields interact with mechanical oscillators and few-level systems, such as single atoms or their ensembles [@Tian2004; @Hammerer2009b; @Hammerer2010; @Camerer2011; @Restrepo2014; @Restrepo2016], Bose–Einstein condensates [@Treutlein2007; @Hunger2010], colour centres [@Rugar2004; @Arcizet2011; @Pigeau2015; @Golter2016], or superconducting circuits [@Etaki2008; @LaHaye2009; @Stannigel2010; @Pirkkalainen2013; @Abdi2015; @Cernotik2016]. For instance, interaction with an atomic ensemble can lead to backaction evading measurements of mechanical motion [@Bariani2015; @Moller2017], generation of entanglement between the ensemble and mechanical oscillator [@Genes2008; @Hammerer2009a; @Huang2018], or cooling of the mechanical motion in the unresolved sideband regime [@Genes2009; @Genes2011; @Jockel2015].
The interplay of various types of interactions in hybrid quantum systems can lead to interference effects and novel optomechanical phenomena. Several works have pointed out the role of interference in standard and hybrid optomechanics [@Elste2009; @Genes2009; @Xuereb2011; @Genes2011; @Sawadsky2015; @Yanay2016] and shown it to be decisive in obtaining, for example, novel, efficient forms for optomechanical cooling. A particularly interesting situation arises when a vibrating membrane is doped by an ensemble of two-level emitters as shown schematically in figure \[fig:scheme\](a). Such a setup has been investigated for the first time in Ref. [@Dantan2014] where a poorly reflecting membrane oscillator was considered. Radiation pressure forces thus played a negligible role but, owing to the presence of the dopant, the oscillator experienced an effective optomechanical interaction with the cavity mode. Such a coupling allows for efficient optomechanical cooling in the unresolved cavity limit, enabled by dressing of the cavity field by the narrow-linewidth emitters. A legitimate question, potentially relevant for a wide range of hybrid optomechanical systems, concerns the interplay between this position-modulated Tavis–Cummings interaction and radiation pressure when the mechanical resonator is partially reflecting and radiation pressure can no longer be neglected.
In this work we theoretically investigate the optomechanical effects arising from these two types of interaction. The presence of the dopant results in a Fano resonance in the cavity noise spectrum which can be used to suppress the Stokes scattering (responsible for heating of the mechanical motion) and enhance the anti-Stokes scattering (cooling), leading to improved cooling performance. Radiation pressure can further boost this effect such that the resulting optomechanical forces lead to stronger optomechanical cooling of the mechanics, as compared to the situations in which either the dopant-induced optomechanical force or radiation pressure acts independently. In particular, we demonstrate that efficient cooling is achievable in situations in which neither dopant-induced nor radiation pressure cooling perform well. We focus on the case of a bad optomechanical cavity—a short cavity containing a movable membrane [@Flowers2012; @Shkarin2014]—in which a large optomechanical coupling can be achieved, but the bare cavity linewidth is too large to resolve the mechanical sidebands. To make the discussion simple we focus on the case of a partially reflecting membrane doped with two-level systems that interact with the cavity field via a Tavis–Cummings interaction. Our results could, however, be amenable to other hybrid mechanical resonators doped with single or multiple two-level emitters (such as diamond cantilevers [@Arcizet2011; @Kolkowitz2012; @Barfuss2015], nanowires [@Yeo2013], optically or electrically trapped nanospheres [@Kuhlicke2014; @Delord2017], or photonic crystals [@Cotrufo2017]) and illustrate how interference effects can be exploited for engineering of efficient optomechanical interactions in hybrid mechanical systems.
Model {#sec:model}
=====
![\[fig:scheme\] (a) Schematic of the setup. We consider a cavity optomechanical system in the membrane-in-the-middle configuration; the membrane is doped with an ensemble of two-level emitters that collectively behave as a single bosonic mode. (b) Depiction of the interactions of the three modes in the fundamental nonlinear configuration given by Hamiltonian . (c) Interactions in the linearized regime as described by the Hamiltonian .](fig1_scheme){width="0.6\linewidth"}
We consider the system depicted in figure \[fig:scheme\] where a single cavity mode $c$ interacts with a single vibrational mode of a flexible membrane with an embedded ensemble of two-level quantum emitters. Following Ref. [@Dantan2014], we consider the limit of weak excitation of the ensemble, such that its collective spin can be described by the bosonic annihilation operator $a$ (with the commutator $[a,a^\dagger] = 1$). The system then follows the Hamiltonian $$\label{eq:H_nonlin}
H = H_0 + {\ensuremath{H_\mathrm{int}}}+ {\ensuremath{H_\mathrm{dr}}}.$$ The bare Hamiltonian $H_0/\hbar = \omega_{c} c^\dagger c + \omega_{a} a^\dagger a + \omega_{\ensuremath{\mathrm{m}}}(q^2+p^2)/2$ describes the free evolution of the cavity field at frequency $\omega_{c}$, the dopant spin at frequency $\omega_{a}$, and the mechanical resonator with displacement $q$ and momentum $p$ (obeying the commutation relation $[q,p] = \rmi$) at frequency $\omega_{\ensuremath{\mathrm{m}}}$. The last term, ${\ensuremath{H_\mathrm{dr}}}/\hbar = -\rmi\eta\, c\exp(\rmi\omega_{\ensuremath{\mathrm{L}}}t+\rmi\phi)+{\ensuremath{\mathrm{H.c.}}}$, describes driving of the cavity mode with laser light of frequency $\omega_{\ensuremath{\mathrm{L}}}$, amplitude $\eta$, and phase $\phi$.
The interaction Hamiltonian describes the interaction of the cavity field with the mechanical oscillator via radiation pressure and with the dopant via a mechanically modulated Tavis–Cummings coupling [@Dantan2014], ${\ensuremath{H_\mathrm{int}}}/\hbar = g_0c^\dagger cq + (\lambda + \mu_0q)(a^\dagger c + c^\dagger a)$; cf. figure \[fig:scheme\](b). Here, the displacement dependence of the Tavis–Cummings interaction arises from the motion of the membrane which shifts the position of the dopant in the standing wave of the cavity mode; for a membrane placed in the middle between a node and an antinode of the field and dopant in the Lamb–Dicke regime, expansion to the first order in mechanical displacement is sufficient to characterize all dynamical effects [@Dantan2014]. The coupling $\lambda=\sqrt{N} d\sqrt{\omega_c/(2\epsilon_0 \hbar l S_{{{\ensuremath{\mathrm{eff}}}}})}$ stems from the collective interaction of $N$ emitters (with individual dipole moment $d$) with the zero-point field amplitude of the cavity (inversely proportional to the square root of the quantization volume $l S_{{{\ensuremath{\mathrm{eff}}}}}$). We assume a Fabry–Perot type of cavity of length $l$, finesse $\cal{F}$, mode area $S_{{{\ensuremath{\mathrm{eff}}}}}$ and resulting mode linewidth $\kappa=\pi c_0/l \cal{F}$ ($c_0$ is the speed of light). The dipole moment of a dopant emiter is of course directly related to the spontaneous emission rate $\gamma=\omega_a^3d^2/(3\pi \epsilon_0\hbar c_0^3)$ such that the ratio $\lambda^2/\gamma$ does not depend on the choice of emitter. We can then estimate that the dopant–cavity cooperativity $C=N\lambda^2/(\kappa\gamma)=3N{\cal{F}}(\lambda_c/2\pi)^2/S_{{{\ensuremath{\mathrm{eff}}}}}$ depends mainly on the cavity design and number of dopant atoms; here, $\lambda_c = 2\pi c_0/\omega_c$ is the cavity wavelength. In the remainder of the article, we put $\hbar = 1$ for simplicity.
Linearized dynamics
-------------------
We linearize the Hamiltonian using the standard approach outlined in detail in \[app:linearization\]. We start by formulating and solving the classical equations of motion of the system in the steady state. Provided a single steady state solution exists with solutions $\bar{c}$, $\bar{a}$, $\bar{q}$ (i.e., the system is statically stable), we formulate linearized equations of motion for the quantum fluctuations around this steady state, $c = \bar{c}+\delta c$, $a = \bar{a}+\delta a$, $q = \bar{q}+\delta q$. Depending on the strength of the interactions, these linearized equations might become dynamically unstable; we defer discussion of dynamical stability to section \[ssec:numerics\]. Assuming the stability criteria are met, the linearization procedure yields the Hamiltonian $H = H_0 + {\ensuremath{H_\mathrm{int}}}$, where $$\eqalign{\label{eq:H_lin}
H_0 = \Delta_{c} c^\dagger c + \Delta_{a} a^\dagger a + \frac{\omega_{\ensuremath{\mathrm{m}}}}{2}(q^2+p^2),\\
{\ensuremath{H_\mathrm{int}}}= (\tilde{g}^\ast c + \tilde{g} c^\dagger)q + \lambda(a^\dagger c + c^\dagger a) + \mu(a+a^\dagger)q.}$$ Here, $\Delta_i = \omega_i - \omega_{\ensuremath{\mathrm{L}}}$ is the detuning of the respective mode ($i = {a},{c}$) from the laser drive frequency and $$\tilde{g} = g - \frac{\rmi\lambda\mu}{\gamma + \rmi\Delta_{a}};$$ we also defined the linearized coupling rates $g = g_0\bar{c}$, $\mu = \mu_0\bar{c}$. Notice that we have dropped the $\delta$ and simply denote the fluctuations by $c$, $a$, $q$ for simplicity. A simplified diagram of the interactions in the linearized regime is depicted in figure \[fig:scheme\](c).
In this linearized regime, the dynamics of the mechanical oscillator are given by the Langevin equations $$\label{eq:LangevinMech}
\dot{q} = \omega_{\ensuremath{\mathrm{m}}}p,\qquad \dot{p} = -\omega_{\ensuremath{\mathrm{m}}}q - \gamma_{\ensuremath{\mathrm{m}}}p + \xi - F,$$ where $\gamma_{\ensuremath{\mathrm{m}}}$ is the intrinsic mechanical linewidth and $\xi$ the associated bath operator; it has zero mean and correlation function ${\ensuremath{\langle \xi(t)\xi(t')\rangle}} = \gamma_{\ensuremath{\mathrm{m}}}(2{\ensuremath{\bar{n}}}+ 1)\delta(t-t')$ with the average thermal phonon number ${\ensuremath{\bar{n}}}$. In addition to the thermal bath $\xi$, the mechanical resonator is also coupled to an effective bath represented by a zero-average noise term with contributions from the atomic and cavity degrees of freedom $$\label{eq:LangevinForce}
F = \tilde{g}^\ast c + \tilde{g} c^\dagger +\mu(a+a^\dagger).$$ To describe the properties of this extra Langevin noise term, we list the equations of motion for the cavity field and the dopant, $$\begin{aligned}
\dot{c} = -(\kappa+\rmi\Delta_{c})c - \rmi\tilde{g}q - \rmi\lambda a + \sqrt{2\kappa}c_{\ensuremath{\mathrm{in}}},\\
\dot{a} = -(\gamma+\rmi\Delta_{a})a - \rmi\mu q - \rmi\lambda c + \sqrt{2\gamma}a_{\ensuremath{\mathrm{in}}}.\end{aligned}$$ The cavity field decays at a rate $\kappa$, is driven by the noise operator $c_{\ensuremath{\mathrm{in}}}$ with zero mean and correlation function ${\ensuremath{\langle c_{\ensuremath{\mathrm{in}}}(t)c_{\ensuremath{\mathrm{in}}}^\dagger(t')\rangle}} = \delta(t-t')$, and its output follows the relation $c_{\ensuremath{\mathrm{out}}}= \sqrt{2\kappa}c - c_{\ensuremath{\mathrm{in}}}$. Analogous relations hold also for the dopant for which the decay rate $\kappa$ is replaced by $\gamma$.
To quantify the effect of the extra Langevin noise term on the dynamics of the mechanical resonator, we follow a perturbative approach [@Marquardt2007; @Genes2008a] in which we ignore the backaction of the mechanical resonator on the field and dopant. To zeroth order in the mechanical displacement $q$, the cavity field and the dopant ensemble in frequency space can be expressed as $$\begin{aligned}
c(\omega) = \tilde{\chi}_{c}(\omega)[\sqrt{2\kappa}c_{\ensuremath{\mathrm{in}}}- \rmi\lambda\chi_{a}(\omega)\sqrt{2\gamma}a_{\ensuremath{\mathrm{in}}}],\\
a(\omega) = \tilde{\chi}_{a}(\omega)[\sqrt{2\gamma}a_{\ensuremath{\mathrm{in}}}- \rmi\lambda\chi_{c}(\omega)\sqrt{2\kappa}c_{\ensuremath{\mathrm{in}}}],\end{aligned}$$ where we introduced the bare and dressed susceptibilities $$\begin{aligned}
\eqalign{
\chi_{c}^{-1}(\omega) = \kappa - \rmi(\omega-\Delta_{c}),\qquad
\tilde{\chi}_{c}^{-1}(\omega) = \chi_{c}^{-1}(\omega) + \lambda^2\chi_{a}(\omega)\\
\chi_{a}^{-1}(\omega) = \gamma - \rmi(\omega-\Delta_{a}),\qquad
\tilde{\chi}_{a}^{-1}(\omega) = \chi_{a}^{-1}(\omega) + \lambda^2\chi_{c}(\omega).}\end{aligned}$$ With these solutions, we can rewrite the Langevin force as $$\begin{aligned}
\fl
F = [\tilde{g}^\ast\tilde{\chi}_{c}(\omega)-\rmi\lambda\mu\tilde{\chi}_{a}(\omega)\chi_{c}(\omega)]\sqrt{2\kappa}c_{\ensuremath{\mathrm{in}}}+ [\mu\tilde{\chi}_{a}(\omega)-\rmi\tilde{g}^\ast\lambda\tilde{\chi}_{c}(\omega)\chi_{a}(\omega)]\sqrt{2\gamma}a_{\ensuremath{\mathrm{in}}}+ {\ensuremath{\mathrm{H.c.}}}\end{aligned}$$ We express the spectrum of the Langevin force as $S_F(\omega) = S_\kappa(\omega) + S_\gamma(\omega)$ with $$\begin{aligned}
S_\kappa(\omega) = 2\kappa|\tilde{g}^\ast\tilde{\chi}_{c}(\omega)-\rmi\lambda\mu\tilde{\chi}_{a}(\omega)\chi_{c}(\omega)|^2,\label{eq:Skappa}\\
S_\gamma(\omega) = 2\gamma|\mu\tilde{\chi}_{a}(\omega)-\rmi\tilde{g}^\ast\lambda\tilde{\chi}_{c}(\omega)\chi_{a}(\omega)|^2.\label{eq:Sgamma}\end{aligned}$$ Using the force spectrum, we obtain the cooling rate [@Marquardt2007] $$\label{eq:GammaCool}
\Gamma_\mathrm{cool} = \frac{1}{2}[S_F(\omega_{\ensuremath{\mathrm{m}}}) - S_F(-\omega_{\ensuremath{\mathrm{m}}})]$$
Overview of cooling strategies
------------------------------
![\[fig:cooling\] Schematic depiction of the interactions involved in existing cooling schemes. (a) Radiation pressure cooling [@Marquardt2007; @Wilson-Rae2007]. (b) Dressed cavity cooling [@Genes2009]. (c) Dopant cooling [@Dantan2014]. (d) Interference cooling discussed in this article.](fig2_cooling){width="0.8\linewidth"}
We can use the noise spectra , to recover existing approaches to optomechanical cooling. First, the standard sideband cooling strategy [@Marquardt2007; @Wilson-Rae2007] corresponds to $\lambda = \mu = 0$; cf. figure \[fig:cooling\](a). In this case, we get the Lorentzian cavity spectrum $S_\kappa(\omega) = 2g^2\kappa / [\kappa^2 + (\omega - \Delta_{c})^2]$ while $S_\gamma(\omega) = 0$. It then follows that the best cooling can be achieved with a sideband resolved system, $\kappa<\omega_{\ensuremath{\mathrm{m}}}$, driven on the red sideband, $\Delta_{c} = \omega_{\ensuremath{\mathrm{m}}}$; final mechanical occupation smaller than unity requires strong optomechanical cooperativity $g^2/\kappa\gamma{\ensuremath{\bar{n}}}> 1$. In the following, we will refer to this strategy as *radiation pressure cooling*.
In the bad cavity regime, $\kappa > \omega_{\ensuremath{\mathrm{m}}}$, radiation pressure cooling cannot reach the quantum ground state of the mechanical resonator. To suppress the unwanted Stokes scattering in this situation, one can use an atomic ensemble placed within the same optical cavity. If the atoms are in the resolved sideband regime, $\gamma<\omega_{\ensuremath{\mathrm{m}}}$, they will burn a hole in the cavity spectrum; by choosing a suitable set of detunings $\Delta_{c}$, $\Delta_{a}$, this spectral hole can overlap with the location of the upper mechanical sideband. This modification results in a reduced density of states around the sideband, leading to reduced Stokes scattering such that the mechanical ground state can be reached [@Genes2009]. This strategy, which we will call *dressed cavity cooling*, corresponds to the limit $\mu = 0$ and is shown in figure \[fig:cooling\](b).
Finally, the regime with $g = 0$ \[see also figure \[fig:cooling\](c)\] has been studied in Ref. [@Dantan2014]; this situation describes a doped membrane with negligible radiation pressure coupling. Here, the dopant provides both the sideband resolution (when in the regime $\gamma<\omega_{\ensuremath{\mathrm{m}}}$) and coupling to the mechanical resonator (via the coupling constant $\mu$). The cavity field (which does not couple to the mechanical motion directly) serves only to enhance the intrinsically weak interaction between the dopant and the mechanical resonator. We name this strategy *dopant cooling*.
In contrast, we investigate a cooling strategy where all three interactions—radiation pressure coupling at a rate $g$, Tavis–Cummings interaction at a rate $\lambda$, and dopant–mechanical coupling at a rate $\mu$—are present in the system at the same time (see also figure \[fig:cooling\](d)). This situation might appear identical to the dopant cooling shown in figure \[fig:cooling\](c) but these two schemes differ in the effective optomechanical coupling. While the effective optomechanical coupling in the dopant cooling scheme is $-\rmi\lambda\mu/(\gamma+\rmi\Delta_{a})$, it is equal to $\tilde{g} = g -\rmi\lambda\mu/(\gamma+\rmi\Delta_{a})$ in our model. This latter form of the coupling leads to detuning-dependent interference between radiation pressure and dopant coupling which can further lower the final occupation. Owing to this effect, we denote this strategy *interference cooling*.
Fano resonance
--------------
The noise spectra , reveal that interference can play an important role in cooling. For example, the cavity input noise can influence the mechanical motion either directly from the cavity field (dressed by the presence of the dopant), or it can be transferred to the dopant and affect the motion from there. These processes are captured by the first and second term in , respectively; since they both stem from the same reservoir, they have to be added coherently. Different interference conditions exist for the Stokes and anti-Stokes scattering, such that strong asymmetry in the two processes is possible even in the bad cavity regime.
In the following, we will consider cooling in the bad cavity regime, $\kappa \gg \omega_{\ensuremath{\mathrm{m}}}$, and assume that the dopant is relatively strongly coupled to the cavity field such that the Tavis–Cummings interaction is in the regime of strong cooperativity, $C=\lambda^2/\kappa\gamma > 1$. From the scaling $C=3N{\cal{F}}(\lambda_c/2\pi)^2/S_{{{\ensuremath{\mathrm{eff}}}}}$ derived in section \[sec:model\], one can estimate that in order to reach this regime, a cavity of mode area around $50\mu$m$\times50\mu$m would require the product $N{\cal{F}}$ to achieve values of the order $10^6$. As cavities of finesse around $10^4-10^5$ are routinely employed in optomechanical setups, one then requires around $100$ dopant emitters to be placed in the mode area of the cavity field, thus at relatively low densities (such that the emitters can be considered independent).
The cavity mode and the dopant form polaritons with energies $$\omega_\pm = \frac{1}{2}\left[\Delta_{a}+\Delta_{c} \pm \sqrt{(\Delta_{a}-\Delta_{c})^2+4\lambda^2}\right].$$ We can expect the cooling to be optimal when one of the polariton modes is driven on the lower mechanical sideband, $\omega_+ = \omega_{\ensuremath{\mathrm{m}}}$ (or $\omega_- = \omega_{\ensuremath{\mathrm{m}}}$), which is achieved for the cavity detuning $$\label{eq:DcOpt}
\Delta_{c} = \omega_{\ensuremath{\mathrm{m}}} + \frac{\lambda^2}{\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}}}.$$
Plugging the optimal detuning into the noise spectra , and assuming $\kappa\gg\omega_{\ensuremath{\mathrm{m}}}$, we can approximate the noise spectra to leading order in $\omega_{\ensuremath{\mathrm{m}}}/\kappa$ as $$S_\kappa(\omega) \simeq \frac{A(\omega)}{\Gamma^2 + (\omega-\Delta)^2},\label{eq:SkappaFano}\qquad
S_\gamma(\omega) \simeq \frac{B}{\Gamma^2 + (\omega-\Delta)^2}.\label{eq:SgammaLorentz}$$ The spectra describe the hybridization of the cavity mode with the dopant; the emergent polaritonic state is characterized by linewidth $\Gamma$ and the state energy quantified by $\Delta$: $$\begin{aligned}
\label{eq:Gamma}
\Gamma = \frac{\lambda^4\gamma + \kappa(\lambda^2+\gamma\kappa)(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2}
{\lambda^4+\kappa^2(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2},\\
\Delta = \frac{\lambda^4\omega_{\ensuremath{\mathrm{m}}} + \kappa^2\Delta_{a}(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2}
{\lambda^4+\kappa^2(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2}.\label{eq:Delta}\end{aligned}$$ Note that, for weak coupling, one reproduces the expected result that the polariton exhibits the bare linewidth of the dopant $\Gamma\simeq \gamma$ and is positioned at $\Delta \simeq \Delta_a$. For increasing coupling strength the polariton linewidth and energy acquire contributions from both the dopant and the cavity mode. Furthermore, the spectra are characterized by the amplitudes $$\begin{aligned}
\fl
A(\omega) = \frac{2\kappa(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2}{\gamma^2+\Delta_{a}^2}
\frac{\{\lambda\mu(2\Delta_{a}-\omega)-g[\gamma^2-\Delta_{a}(\omega-\Delta_{a})]\}^2 + g^2\gamma^2\omega^2}
{\lambda^4 + \kappa^2(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2},\label{eq:amplitudeA}\\
\fl\eqalign{
B = \frac{2\gamma}{\gamma^2+\Delta_{a}^2}&
\Bigg(\frac{[\lambda^2\mu\gamma - (g\lambda\gamma+\mu\kappa\Delta_{a})(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})]^2}
{\lambda^4 + \kappa^2(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2}\\
&\quad+\frac{[\lambda^2\mu(2\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}}) - (g\lambda\Delta_{a}-\mu\gamma\kappa)(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})]^2}
{\lambda^4 + \kappa^2(\Delta_{a}-\omega_{\ensuremath{\mathrm{m}}})^2}\Bigg).}\end{aligned}$$ The amplitude $A(\omega)$ is quadratic in frequency so the cavity noise spectrum $S_\kappa(\omega)$ exhibits a Fano resonance [@Fano1961]; the atomic noise spectrum $S_\gamma(\omega)$, on the other hand, is Lorentzian. The Fano resonance can be further enhanced by the interference between the radiation pressure and the dopant interaction as we discuss below.
Interference cooling {#sec:cooling}
====================
Dopant-induced cooling {#ssec:dopant}
----------------------
First, we turn our attention to the Lorentzian noise spectrum of the dopant $S_\gamma(\omega)$. It follows from the theory of sideband cooling [@Marquardt2007; @Wilson-Rae2007] that the optimum cooling performance is achieved for $\Delta = \omega_{\ensuremath{\mathrm{m}}}$ and $\Gamma < \omega_{\ensuremath{\mathrm{m}}}$. These conditions can be realized using a good dopant $\gamma < \omega_{\ensuremath{\mathrm{m}}}$ with detuning $\Delta_{a} = \omega_{\ensuremath{\mathrm{m}}}$. The noise spectrum then simplifies to $$S_\gamma(\omega) = \frac{2\mu^2\gamma}{\gamma^2 + (\omega-\omega_{\ensuremath{\mathrm{m}}})^2}$$ while the cavity noise spectrum becomes zero, $S_\kappa(\omega) = 0$. This result is quite natural, since driving the dopant on the red mechanical sideband results \[for polariton driving according to \] in an infinite cavity detuning. The cavity is thus strongly off-resonant so it decouples from the dynamics which thus obey the Hamiltonian $$H = \frac{\omega_{\ensuremath{\mathrm{m}}}}{2}(2a^\dagger a + q^2 + p^2) + \mu(a+a^\dagger)q.$$
One might expect that ground state cooling in this regime is possible provided the system exhibits strong cooperativity, $\mu^2/\gamma\gamma_{\ensuremath{\mathrm{m}}}{\ensuremath{\bar{n}}}> 1$. This assertion is true in principle, but such a regime would be extremely difficult to reach in an experiment. Recall that the the coupling rate $\mu = \mu_0\bar{c}$ is obtained from the three-body interaction $\mu_0(a^\dagger c+ c^\dagger a)q$ enhanced by a strong intracavity amplitude $\bar{c}$. The three-body coupling strength $\mu_0$ is, in turn, a perturbative correction to the Tavis–Cummings interaction in the Lamb–Dicke regime so we have $\mu_0\ll\lambda$. Moreover, reaching a large cavity amplitude $\bar{c}$ for an effectively infinite detuning $\Delta_{c}$ would require effectively infinite driving power.
Cooling via Fano resonance
--------------------------
Analysis of the cavity noise spectrum, , is more involved. Owing to the frequency dependence of the amplitude $A(\omega)$, the cavity noise spectrum exhibits a Fano resonance, which can be used to modify the Stokes and anti-Stokes scattering rates. While a general analysis of these spectra and optimization of the cooling is, in principle, possible, it does not bring a clear physical insight into the system dynamics. We thus only highlight the main features of this approach and defer more detailed analysis to the next section where we study the noise spectra and final mechanical occupation numerically.
The cooling rate is given by $S_\kappa(\omega_{\ensuremath{\mathrm{m}}})$ whereas heating by $S_\kappa(-\omega_{\ensuremath{\mathrm{m}}})$; to exploit the Fano resonance for suppressing heating and enhancing cooling, we would therefore like the dip of the Fano resonance to fall within the vicinity of $\omega = -\omega_{\ensuremath{\mathrm{m}}}$ while the peak should be close to $\omega = \omega_{\ensuremath{\mathrm{m}}}$ \[see figure \[fig:polaritons\](b) for an illustration\]. These requirements already put certain conditions on the detuning and linewidth defined in , . Specifically, we need a detuning with magnitude within the mechanical sidebands, $|\Delta|\lesssim\omega_{\ensuremath{\mathrm{m}}}$ and a linewidth that is not too large either, $\Gamma\lesssim\omega_{\ensuremath{\mathrm{m}}}$. At the same time, we must not forget that the dopant noise spectrum also contributes to heating and cooling of the membrane. Ideally, we would thus have positive detuning, $\Delta > 0$, such that $S_\gamma(\omega_{\ensuremath{\mathrm{m}}}) > S_\gamma(-\omega_{\ensuremath{\mathrm{m}}})$.
The suppression of Stokes scattering via Fano resonance is not unique to our system. The same principle is also used in dressed cavity and dopant cooling [@Genes2009; @Dantan2014]. In these two systems, the cavity field and atoms also form two polariton modes, resulting in cavity noise spectra analogous to . With interference cooling, however, there is an additional interference between the two types of interaction—the radiation pressure interaction at a rate $g$ and the dopant–mechanical interaction at a rate $\mu$ as exemplified by the curly bracket in . This interference can lead to a further suppression of the Stokes scattering (and enhancement of anti-Stokes scattering) and thus a lower final occupation than in any of the previous cooling schemes.
An intriguing consequence of this interference effect is the possibility of cooling with both cavity and dopant driven on resonance, $\Delta_c = \Delta_a = 0$. In this case, the cavity and dopant noise spectra are not given by (unless $\lambda = \omega_{\ensuremath{\mathrm{m}}}$) but instead by the expressions $$\begin{aligned}
S_\kappa(\omega) = \frac{2\kappa}{\gamma^2}\label{eq:SkappaRes}
\frac{g^2\gamma^2\omega^2+(g\gamma^2+\lambda\mu\omega)^2}
{(\lambda^2+\gamma\kappa)^2 + (\gamma^2+\kappa^2-2\lambda^2)\omega^2+\omega^4},\\
S_\gamma(\omega) = \frac{2\gamma}{\gamma^2}
\frac{\mu^2(\lambda^2+\gamma\kappa)^2 + \gamma^2(g\lambda+\mu\omega)^2}{(\lambda^2+\gamma\kappa)^2 + (\gamma^2+\kappa^2-2\lambda^2)\omega^2+\omega^4}.\label{eq:SgammaRes}\end{aligned}$$ (These expressions can be obtained simply by setting $\Delta_a = \Delta_c = 0$ in , .) The spectra clearly reveal the importance of interference for cooling on resonance: only when both radiation pressure and dopant interaction are present does the numerator of each of the two spectra contain a term linear in frequency. The spectra thus distinguish between positive and negative frequencies, resulting in a net cooling or heating effect. Specifically, we obtain the cooling rate (recall the definition given in ) $$\Gamma_\mathrm{cool} = \frac{4g\lambda\mu\omega_{\ensuremath{\mathrm{m}}}(\gamma+\kappa)}{(\lambda^2+\gamma\kappa)^2+(\gamma^2+\kappa^2-2\lambda^2)\omega_{\ensuremath{\mathrm{m}}}^2+\omega_{\ensuremath{\mathrm{m}}}^4}.$$ The denominator is always positive so the membrane is cooled as long as $g\lambda\mu >0$ (i.e., either none or two of the coupling rates are negative).
Numerical simulations {#ssec:numerics}
---------------------
To check our expectations, we perform numerical simulations of the full linearized dynamics to determine the final mechanical occupation. To this end, we formulate a Lyapunov equation for the covariance matrix of the system. We start by defining the quadrature operators $X_{c} = (c+c^\dagger)/\sqrt{2}$, $Y_{c} = -\rmi(c-c^\dagger)/\sqrt{2}$ (and similar for the dopant) with the commutator $[X_i,Y_j] = \rmi\delta_{ij}$. Together with the mechanical position and momentum operators, we collect these operators into the vector ${\ensuremath{\bi{r}}} = (X_{c},Y_{c},X_{a},Y_{a},q,p)^T$ and define the covariance matrix with elements $$V_{ij} = {\ensuremath{\langle r_ir_j + r_jr_i\rangle}} - 2{\ensuremath{\langle r_i\rangle}}{\ensuremath{\langle r_j\rangle}}.$$ The steady-state covariance matrix ${\ensuremath{\bi{V}}}$ is a solution of the Lyapunov equation $$\label{eq:Lyapunov}
{\ensuremath{\bi{A}}}{\ensuremath{\bi{V}}} + {\ensuremath{\bi{V}}}{\ensuremath{\bi{A}}}^T + {\ensuremath{\bi{N}}} = 0$$ with drift and diffusion matrices ${\ensuremath{\bi{A}}}$, ${\ensuremath{\bi{N}}}$; we present these matrices and discuss the dynamical stability in \[app:Lyapunov\]. We obtain the mechanical occupation in the steady state from the variance of the mechanical position and momentum, $$n_{\ensuremath{\mathrm{f}}} = \frac{1}{4}(V_{55} + V_{66} - 2).$$ Note that since the dynamics are linear, the (initially Gaussian) state of the system remains Gaussian throughout the evolution and the covariance matrix is sufficient to fully describe the correlations in the system.
![\[fig:polaritons\] (a) Final occupation of the mechanical oscillator (on logarithmic scale) as a function of the cavity and atomic detunings. The dashed red line shows the two polariton branches defined in ; the black contour line shows the region where the final occupation drops below unity, $n_{\ensuremath{\mathrm{f}}}<1$. The dark blue regions are where the oscillator is heated up, $n_{\ensuremath{\mathrm{f}}} > {\ensuremath{\bar{n}}}$, or where the system becomes unstable. (b–e) Noise spectral densities for detunings as indicated in panel (a). We show the cavity noise spectrum $S_\kappa(\omega)$ (dashed orange line), the dopant noise spectrum $S_\gamma(\omega)$ (dotted green line), and their sum (solid blue line). The system parameters are $g/\omega_{\ensuremath{\mathrm{m}}} = 0.25$, $\lambda/\omega_{\ensuremath{\mathrm{m}}} = 8$, $\mu/\omega_{\ensuremath{\mathrm{m}}} = 0.01$, $\kappa/\omega_{\ensuremath{\mathrm{m}}} = 20$, $\gamma/\omega_{\ensuremath{\mathrm{m}}} = 0.8$, $Q_{\ensuremath{\mathrm{m}}} = \omega_{\ensuremath{\mathrm{m}}}/\gamma_{\ensuremath{\mathrm{m}}} = 10^6$, and ${\ensuremath{\bar{n}}}= 10^3$. The vertical lines are guides to the eye for the cooling and heating rates (given by the spectra at $\omega_{\ensuremath{\mathrm{m}}}$ and $-\omega_{\ensuremath{\mathrm{m}}}$, respectively).](fig3_contour){width="\textwidth"}
We plot the results of such a simulation in figure \[fig:polaritons\](a) where we show the final occupation $n_{\ensuremath{\mathrm{f}}}$ as a function of the cavity and dopant detunings. Particularly, driving the upper polariton with energy $\omega_+$ on the lower mechanical sideband (shown as the dashed red line in the lower left quadrant) leads to substantive cooling and even makes it possible to reach final occupation $n_{\ensuremath{\mathrm{f}}} < 1$. Driving the lower polariton in the same way (upper right quadrant), on the other hand, leads only to moderate cooling or even becomes unstable (when entering the dark blue region).
We further elucidate this difference in figure \[fig:polaritons\](b–e) where we plot the spectra at four different points of the 2D plot. On the lower sideband of the upper polariton (figure \[fig:polaritons\](b)), the cavity noise spectrum (dashed orange line) exhibits a clear Fano resonance which reaches a minimum around $\omega = -\omega_{\ensuremath{\mathrm{m}}}$ and maximum close to $\omega = \omega_{\ensuremath{\mathrm{m}}}$; the Stokes scattering is thus suppressed while the anti-Stokes scattering is enhanced, which leads to a final occupation $n_{\ensuremath{\mathrm{f}}}\simeq 0.74$. On the lower sideband of the lower polariton (panel (c)), the Fano resonance is still present but not ideally oriented (the minimum is to the right of the maximum) so the final occupation is much higher ($n_{\ensuremath{\mathrm{f}}}\simeq 19.4$). A smaller final occupation than on the lower sideband of the lower polariton can, in fact, be achieved also far detuned from the lower sideband of the upper polariton (such as at the point (d) in figure \[fig:polaritons\], where the final occupation $n_{\ensuremath{\mathrm{f}}}\simeq 10$). Finally, when the Stokes scattering is stronger than the anti-Stokes scattering, the system becomes unstable; cf. figure \[fig:polaritons\](e). Together, these results reveal the importance of Fano resonance for efficient cooling: the Fano minimum suppresses the Stokes scattering while the maximum enhances the anti-Stokes scattering. These requirements limit the suitable dopant detuning $|\Delta_a|\lesssim \omega_{\ensuremath{\mathrm{m}}}$ (cf. , ), leading to optimal cooling around the lower sideband of the upper polariton.
![\[fig:comparison\] Comparison of cooling strategies. Final occupation versus cavity detuning for interference cooling (solid blue line), radiation pressure cooling (dashed orange line), dressed cavity cooling (dotted green line), and dopant cooling (dash–dotted red line) is plotted for various system parameters. (a) The same parameters as in figure \[fig:polaritons\]. (b) Bad cavity ($\kappa/\omega_{\ensuremath{\mathrm{m}}} = 80$) and bad dopant ($\gamma/\omega_{\ensuremath{\mathrm{m}}} = 2$) with coupling rates $g/\omega_{\ensuremath{\mathrm{m}}} = 0.06$, $\lambda/\omega_{\ensuremath{\mathrm{m}}} = 15$, $\mu/\omega_{\ensuremath{\mathrm{m}}} = 0.006$. (c) Bad cavity ($\kappa/\omega_{\ensuremath{\mathrm{m}}} = 80$) and good dopant ($\gamma/\omega_{\ensuremath{\mathrm{m}}} = 0.1$). The coupling rates are $g/\omega_{\ensuremath{\mathrm{m}}} = 0.3$, $\lambda/\omega_{\ensuremath{\mathrm{m}}} = 8$, $\mu/\omega_{\ensuremath{\mathrm{m}}} = 0.005$. (d) Good cavity ($\kappa/\omega_{\ensuremath{\mathrm{m}}} = 0.8$) and bad dopant ($\gamma/\omega_{\ensuremath{\mathrm{m}}} = 10$). Here, we use the coupling rates $g/\omega_{\ensuremath{\mathrm{m}}} = 0.1$, $\lambda/\omega_{\ensuremath{\mathrm{m}}} = 12$, $\mu/\omega_{\ensuremath{\mathrm{m}}} = 0.025$. For interference, dressed cavity, and dopant cooling, the dopant detuning is $\Delta_{a} = \omega_{\ensuremath{\mathrm{m}}}+\lambda^2/(\Delta_{c}-\omega_{\ensuremath{\mathrm{m}}})$, corresponding to cooling via one of the polariton modes \[i.e., along the dashed red lines in figure \[fig:polaritons\](a)\]; additionally, the membrane has the mechanical quality factor $Q_{\ensuremath{\mathrm{m}}} = 10^6$ and initial occupation ${\ensuremath{\bar{n}}}= 10^3$. The horizontal line indicates final occupation of unity, $n_{\ensuremath{\mathrm{f}}} = 1$.](fig4_comparison){width="0.71\linewidth"}
We study the final occupation along the lower sideband of the two polariton modes in more detail in figure \[fig:comparison\](a). Two observations are crucial here: first, the minimum final occupation reached along the lower sideband of the upper polariton ($n_{\ensuremath{\mathrm{f}}}\simeq 0.74$) is very close to the absolute minimum in figure \[fig:polaritons\] ($n_{\ensuremath{\mathrm{f}}}\simeq 0.73$) indicating that the lower sideband of the polariton mode is near-optimal for cooling with moderate cooperativity (we have $\lambda^2/\kappa\gamma = 4$). Second, interference cooling (shown as the solid blue line) performs better than any other of the cooling schemes; the best results can otherwise be achieved with dressed cavity cooling, which reaches a final occupation $n_{\ensuremath{\mathrm{f}}}\simeq 1.1$.
We present further comparison of the four cooling schemes in figure \[fig:comparison\](b–d). There exists a broad range of system parameters—generally in the bad cavity regime—where interference cooling can outperform existing cooling strategies (panels (b,c)). In these cases, one can reach optimum cooling for blue-detuned cavity drive, $\Delta_{c} < 0$, corresponding to rather small dopant detuning (e.g., in panel (c), the optimal dopant detuning $\Delta_{a} \simeq -\omega_{\ensuremath{\mathrm{m}}}$). This observation further confirms our assertion that the Fano resonance in the cavity noise spectrum is responsible for the suppression of Stokes scattering and enhancement of anti-Stokes scattering. We also note that in the good cavity regime (panel (d)), the performance of radiation pressure, dressed cavity, and interference cooling is comparable; admittedly, radiation pressure cooling is, from the experimental point of view, the simplest of these methods to implement.
![\[fig:resonant\] Final occupation (on logarithmic scale) for interference cooling with a resonant drive, $\Delta_{c} = \Delta_{a} = 0$, as a function of the Tavis–Cumings and optomechanical coupling rates in (a) the bad cavity regime ($\kappa/\omega_{\ensuremath{\mathrm{m}}} = 2.7$, $\gamma/\omega_{\ensuremath{\mathrm{m}}} = 0.8$) and (b) the good cavity regime ($\kappa/\omega_{\ensuremath{\mathrm{m}}} = 0.7$, $\gamma/\omega_{\ensuremath{\mathrm{m}}} = 0.5$). The black contour lines show regions where $n_{\ensuremath{\mathrm{f}}} < 1$. The mechanical oscillator has the quality factor $Q_{\ensuremath{\mathrm{m}}} = 10^6$ and initial thermal occupation ${\ensuremath{\bar{n}}}= 10^3$; we use the dopant coupling $\mu/\lambda = 0.05$.](fig5_ResonantCooling){width="0.7\linewidth"}
Finally, we study the final occupation for interference cooling with driving on resonance, $\Delta_{c} = \Delta_{a} = 0$, in figure \[fig:resonant\]. Remarkably, final occupation $n_{\ensuremath{\mathrm{f}}} < 1$ is possible even in the bad cavity regime (panel (a)). In the sideband resolved regime (panel (b)), the final occupation can be lower than in the bad cavity regime, but resonant interference cooling cannot outperform radiation pressure cooling; here, the minimum final occupation is $n_{\ensuremath{\mathrm{f}}}\simeq 0.8$ whereas radiation pressure cooling can reach $n_{\ensuremath{\mathrm{f}}}\simeq 0.14$ with the same sideband resolution. Nevertheless, resonant driving (as used for interference cooling) requires smaller driving power than a sideband drive (necessary for radiation pressure cooling) to achieve the same coupling strength; interference cooling might thus have an important advantage over radiation pressure cooling even in the good cavity regime.
In both regimes, there is an optimal range of coupling rates for which ground state cooling is possible. This effect is a consequence of the interference in the noise spectra , . Take, for instance, the cavity noise spectrum $S_\kappa(\omega)$ (note, however, that a similar argument holds also for the dopant spectrum $S_\gamma(\omega)$): here, the second term in the numerator, $\propto (g\gamma^2+\lambda\mu\omega)^2$, is responsible for cooling. More specifically, it is the term linear in frequency, $\propto g\lambda\mu\omega$ that gives rise to cooling; the remaining two terms, $\propto g^2, \lambda^2\mu^2$ do not affect the cooling rate but still affect the final occupation since they contribute to the backaction that the cavity field and dopant exert on the mechanical oscillator. We therefore need to maximize the interference term relative to the latter two, which gives rise to an optimal range of coupling rates as can be seen in figure \[fig:resonant\].
Summary and outlook {#sec:conclusion}
===================
In conclusion, we investigated cooling of a mechanical resonator doped by an ensemble of two-level quantum emitters. The interplay between radiation pressure and mechanically modulated Tavis–Cummings interaction between the cavity field and the dopant gives rise to a Fano resonance in the cavity noise spectrum. This resonance can lead to a suppression of Stokes and enhancement of anti-Stokes scattering, leading to ground state cooling in regimes where none of the effects alone can efficiently cool the motion. An additional signature of the interference between these two types of interaction is the possibility of ground state cooling when the cavity and dopant are driven on resonance. Our results are not limited to the particular architecture considered here; similar results can be expected for any mechanical oscillator with embedded two-level quantum emitters and experiencing a direct radiation pressure force.
This work highlights the importance of interference effects in hybrid optomechanical systems for studying novel phenomena and developing new applications. The interference can also result in a lowered instability threshold, which can have profound implications for the generation of ponderomotive squeezing of light [@Fabre1994] or for observing mechanical limit cycles [@Qian2012; @Lorch2014]. Further improvements and new effects may occur when the dopant ensemble is prepared in a super- or subradiant state [@Plankensteiner2017] or with quadratic optomechanical coupling [@Thompson2008; @Jayich2008].
Looking forward, these devices will enter a new domain once they reach the regime of near-unit reflectivity around the dopant resonance [@Bettles2016; @Shahmoon2017; @Zeytinoglu2017a; @Back2018; @Scuri2018]. Such membranes could then be used as end mirrors in Fabry–Pérot resonators, where their strongly frequency dependent reflectivity can reduce the cavity linewidth [@Naesby2018] and lead to the observation of non-Markovian optomechanical dynamics in the resolved sideband and strong coupling regimes.
We gratefully acknowledge financial support from the Max Planck Society and the Velux Foundations.
Linearization of the three-body dynamics and static stability {#app:linearization}
=============================================================
For completeness, the equations of motion obtained from the full Hamiltonian and the linearization around the semiclassical steady state are detailed here. We start by adding dissipation to the Hamiltonian and obtaining the Langevin equations $$\begin{aligned}
\dot{{c}} = -(\kappa+\rmi\Delta_{c}){c} -\rmi(\lambda+\mu_0{q}){a}-\rmi g_0{c}{q}+\eta_\phi+\sqrt{2\kappa}{c}_{\ensuremath{\mathrm{in}}},\\
\dot{{a}} = -(\gamma+\rmi\Delta_{a}){a} -\rmi(\lambda+\mu_0{q}){c}+\sqrt{2\gamma}{a}_{\ensuremath{\mathrm{in}}},\\
\dot{{p}} = -\gamma_{\ensuremath{\mathrm{m}}} {p}-\omega_{\ensuremath{\mathrm{m}}} {q}-\mu_0({a}^{\dagger}{c}+{c}^{\dagger}{a})-
g_0{a}^{\dagger}{a}+{\xi},\\
\dot{{q}} = \omega_{\ensuremath{\mathrm{m}}}{p},\end{aligned}$$ where the cavity and dopant dynamics is expressed in the rotating frame with respect to the driving frequency; moreover, we defined $\eta_\phi = \eta\rme^{-\rmi\phi}$.
Next, we separate each operator into its classical amplitude and quantum fluctuations, ${o} = \bar{o} + \delta o$. The classical amplitudes obey the steady state equations $$\begin{aligned}
-(\kappa+\rmi\Delta_{c})\bar{c}-\rmi(\lambda+\mu_0\bar{q})\bar{a}-\rmi g_0\bar{c}\bar{q}+\eta_\phi =0,\\
-(\gamma+\rmi\Delta_{a})\bar{a}-\rmi(\lambda+\mu_0\bar{q})\bar{c} =0,\\
-\omega_{\ensuremath{\mathrm{m}}}\bar{q}-\mu_0(\bar{c}\bar{a}^*+\bar{a}\bar{c}^*)-g_0|\bar{c}|^2 =0.;\end{aligned}$$ the solutions are $$\begin{aligned}
\bar{a} =-\rmi\frac{\lambda+\mu_0\bar{q}}{\gamma+\rmi\Delta_{a}}\bar{c},\\
\bar{q} =-\frac{g_0|\bar{c}|^2-2\lambda\mu_0\Delta_{a}|\bar{c}|^2/(\gamma^2+\Delta_{a}^2)}
{\omega_{\ensuremath{\mathrm{m}}}-2\mu_0^2\Delta_{a}|\bar{c}|^2(\gamma^2+\Delta_{a}^2)},\\
\eta_\phi =\left[\kappa+\rmi\Delta_{c}+\rmi g_0\bar{q}+\frac{(\lambda+\mu_0\bar{q})^2}{\gamma+\rmi\Delta_{a}}\right]\bar{c}.\label{eq:amulti}\end{aligned}$$ Introducing $g = g_0\bar{c}$ and $\mu = \mu_0\bar{c}$, we recast (\[eq:amulti\]) as $$\begin{aligned}
\label{eq:SteadyState}\eqalign{
\eta_\phi &=
\Bigg[\kappa+\rmi\Delta_{c} -\rmi\frac{g^2-2g\lambda\mu\Delta_{a}/(\gamma^2+\Delta_{a}^2)}
{\omega_{\ensuremath{\mathrm{m}}}-2\mu^2\Delta_{a}/(\gamma^2+\Delta_{a}^2)} \\
&\qquad+\frac{\mu^2}{\gamma+\rmi\Delta_{a}}
\left(\frac{\omega_{\ensuremath{\mathrm{m}}}-g\mu/\lambda}{\omega_{\ensuremath{\mathrm{m}}}-2\mu^2\Delta_{a}/(\gamma^2+\Delta_{a}^2)}\right)^2\Bigg]\bar{c},}\end{aligned}$$ the solution of which is the intracavity field amplitude $\bar{c}$, implicitly contained in $g$ and $\mu$.
Without dopant ($\lambda = \mu = 0$) one retrieves the usual dispersive Kerr bistability equation for the intracavity field $$\eta_\phi = \left(\kappa+\rmi\Delta_{c}-\rmi\frac{g^2}{\omega_{\ensuremath{\mathrm{m}}}}\right)\bar{c}$$ and, in the absence of a dynamical instability, the motion-induced nonlinear phase-shift leads to optical bistability when the Kerr dephasing is of the order of $\kappa$. With dopant, however, the bistability threshold can be lowered (or highered) owing to interference between various terms in . We assume here that the system is stable and a single solution $\bar{c}$ exists.
Linearized fluctuations around the steady state obey the Langevin equations $$\begin{aligned}
\delta\dot{c} = -(\kappa+\rmi\Delta_{c})\delta{}c-\rmi\lambda \delta{}a-\rmi\tilde{g}\delta{}q+\sqrt{2\kappa}c_{\ensuremath{\mathrm{in}}},\\
\delta{}\dot{a} = -(\gamma+\rmi\Delta_{a})\delta{}a-\rmi\lambda \delta{}c-\rmi\mu \delta{}q+\sqrt{2\gamma}a_{\ensuremath{\mathrm{in}}},\\
\delta{}\dot{p} = -\gamma_{\ensuremath{\mathrm{m}}}\delta{}p-\omega_{\ensuremath{\mathrm{m}}}\delta{} q-\mu(\delta{}a+\delta{}a^{\dagger}) -\tilde{g}^\ast \delta{}c-\tilde{g}\delta{}c^{\dagger}+\xi,\\
\delta{}\dot{q} = \omega_{\ensuremath{\mathrm{m}}} \delta{}p,\end{aligned}$$ where, to simplify the notation, we absorbed the term $g\bar{q}$ into $\Delta_{c}$, redefined $\lambda$ to include the term $\mu_0\bar{q}$, introduced $\tilde{g} = g - \rmi\lambda\mu/(\gamma+\rmi\Delta_{a})$, and set the driving phase $\phi$ such that $\bar{c}\in\mathbb{R}$. We can associate the coherent dynamics in these equations with the linearized Hamiltonian given in ; for simplicity of notation, we drop the $\delta$ in the linearized Hamiltonian and the following calculations from the operators.
Lyapunov equation and dynamical stability {#app:Lyapunov}
=========================================
The drift and diffusion matrices ${\ensuremath{\bi{A}}}$, ${\ensuremath{\bi{N}}}$ in the Lyapunov equation can be obtained from the Hamiltonian and the jump operators [@Cernotik2015]. For the Hamiltonian in , the assumed decay of the cavity field and the dopant, and the thermal noise acting on the mechanical resonator, one gets $$\begin{aligned}
\fl
{\ensuremath{\bi{A}}} = \left(
\begin{array}{cccccc}
-\kappa & \Delta_{c} & 0 & \lambda & -\sqrt{2}\eta\gamma & 0\\
-\Delta_{c} & -\kappa & -\lambda & 0 & -\sqrt{2}(g-\eta\Delta_{a}) & 0\\
0 & \lambda & -\gamma & \Delta_{a} & 0 & 0\\
-\lambda & 0 & -\Delta_{a} & -\gamma & -\sqrt{2}\mu & 0\\
0 & 0 & 0 & 0 & 0 & \omega_{\ensuremath{\mathrm{m}}}\\
-\sqrt{2}(g-\eta\Delta_{a}) & -\sqrt{2}\eta\gamma & -\sqrt{2}\mu & 0 & -\omega_{\ensuremath{\mathrm{m}}} & -\gamma_{\ensuremath{\mathrm{m}}}
\end{array}
\right),\\ \fl
{\ensuremath{\bi{N}}} = \mathrm{diag}[2\kappa,2\kappa,2\gamma,2\gamma,0,2\gamma_{\ensuremath{\mathrm{m}}}(2{\ensuremath{\bar{n}}}+1)];\end{aligned}$$ here, we defined $\eta = \lambda\mu/(\gamma^2+\Delta_{a}^2)$. The system remains dynamically stable if the real parts of all the eigenvalues of the drift matrix ${\ensuremath{\bi{A}}}$ are nonpositive.
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abstract: 'Following the work of Kashiwara-Rouquier and Gan-Ginzburg, we define a family of exact functors from category $\mathcal O$ for the rational Cherednik algebra in type $A$ to representations of certain “coloured braid groups” and calculate the dimensions of the representations thus obtained from standard modules. To show that our constructions also make sense in a more general context, we also briefly study the case of the rational Cherednik algebra corresponding to complex reflection group $\mathbb Z/l\mathbb Z$.'
address: 'Mathematical Institute, University of Oxford. '
author:
- Kevin McGerty
date: 'October 2011.'
title: 'Microlocal KZ functors and rational Cherednik algebras.'
---
Introduction
============
This paper is inspired by two beautiful constructions in the theory of rational Cherednik algebras of type $A$. The first is the KZ functor studied in [@GGOR] which relates representations in the category $\mathcal O_c$ to representations of Hecke algebras, and the second is the localization theorem of Kashiwara-Rouquier (see also the work of Gan and Ginzburg [@GG]) which allows us to realize the category $\mathcal O_c$ as a category of modules for a “microlocal” sheaf of rings $\mathcal A_c$ on the Hilbert scheme ${\text{Hilb}^n(\mathbb C^2)}$ of $n$ points in the plane (with certain restrictions on the value of the deformation parameter $c$).
In the context of [@KR], the modules in the category $\mathcal O_c$ correspond to sheaves of $\mathcal A_c$-modules supported on a certain reducible Lagrangian subvariety $Z$ of ${\text{Hilb}^n(\mathbb C^2)}$ first considered by Grojnowski [@Gr]. Using this microlocal point of view, we construct for each component of $Z$ an exact functor from $\mathcal O_c$ to a category of local systems, and we show that the $KZ$-functor naturally corresponds to one of them.
On the level of dimensions, these functors recover the characteristic cycle as defined by Gan and Ginzburg [@GG], and for standard modules we are able to calculate these dimensions. This calculation is already known by combining work of Ginzburg, Gordon and Stafford [@GGS] with earlier work of Gordon and Stafford [@GS], but our calculations are self-contained, and in particular make no use of Haiman’s work on the Hilbert scheme.
In section \[RCAstuff\] we recall the definition of the rational Cherednik algebra in type $A$ and the construction of the $KZ$-functor. In section \[deformed\] we review the quantum Hamiltonian reduction construction of Gan-Ginzburg. In section \[Micropi1\] we study the Lagrangian subvariety $Z$ of the Hilbert scheme and exhibit its smooth locus. In section \[MKZfunctors\] we define for each component $Z_\lambda$ of $Z$ an exact functor $KZ_\lambda$ on category $\mathcal O_c$ generalizing the $KZ$-functor. In section \[CCcomputation\] we compute the characteristic cycles of standard modules, which give the dimensions of our functors $KZ_\lambda$ on these modules. In section \[HeckeAlgebras\] we show using $\mathcal D$-module techniques that the original $KZ$-functor descends to a functor to representations of a Hecke algebras (as opposed to just a braid group). Finally, to give an example of our techniques for a complex reflection group, in section \[cyclic\] we consider the case of the complex reflection group $\mathbb Z/l\mathbb Z$. Here we find a direct connection to the theory of cyclotomic $q$-Schur algebras, while hints of a similar connection to the classical $q$-Schur algebra are suggested by our results in the case of the symmetric group. We hope to return to this issue in a later paper. Finally in the Appendix we give a review of material on twisted $\mathcal D$-modules and equivariance, and a brief discussion of the index theorem which is needed in our calculation of characteristic cycles of standard modules.
*Acknowledgements*: The author would like to thank Kobi Kremnitzer and Tom Nevins for useful discussions on $\mathcal D$-modules, and Ian Grojnowski, Iain Gordon, and Toby Stafford for introducing him to the world of rational Cherednik algebras along with much encouragement while this paper was being written. The author also gratefully acknowledges the support of a Royal Society University Research Fellowship.
Rational Cherednik algebra and the ${\text{KZ}}$ functor. {#RCAstuff}
=========================================================
Let ${\mathfrak h}$ be the permutation representation of the symmetric group $W=S_n$ on $n$ letters. Thus ${\mathfrak h}= \mathbb C^n$ and $W$ acts by permuting the coordinates in the obvious way. The rational Cherednik algebras[^1] ${\mathcal H_c}$ of type $A$, for $c \in \mathbb C$, are a family of algebras giving a flat deformation of $\mathcal D({\mathfrak h})\rtimes \mathbb C[W]$, the smash product of the group algebra with the algebra of differential operators on $\mathfrak h$.
Let $\{y_1,y_2,\ldots,y_n\}$ be the standard basis of ${\mathfrak h}$, and let $\{x_1,x_2,\ldots, x_n\}$ be the corresponding dual basis of ${\mathfrak h}^*$. If we let $s_{ij} \in W$ denote the transposition which interchanges $i$ and $j$, then ${\mathcal H_c}$ is generated by $\{x_i,y_i,s_{ij}: 1\leq i,j \leq n, i \neq j\}$ subject to relations:
$$\begin{split}
s_{ij}x_i &= x_js_{ij}, \quad s_{ij}y_i = y_js_{ij}, \quad 1 \leq i,j, \leq n, i \neq j, \\
[y_i,x_j] &= c s_{ij}, \quad [x_i,x_j] = [y_i,y_j] =0, \quad \forall 1 \leq i,j \leq n, i \neq j\\
[y_k, x_k] &= 1 -c\sum_{i \neq k} s_{ik}.
\end{split}$$ Clearly $\mathcal H_0$ is just $\mathcal D({\mathfrak h}) \rtimes \mathbb C[W]$. In general, one may filter ${\mathcal H_c}$ by placing ${\mathfrak h}$ in degree $1$, ${\mathfrak h}^*$ and $\mathbb C[W]$ in degree $0$. The associated graded algebra is then isomorphic to $\mathbb C[{\mathfrak h}\times {\mathfrak h}^*]\rtimes \mathbb C[W]$, and the canonical map from ${\mathcal H_c}$ is known to be a vector space isomorphism [@EG Theorem 1.3]. This yields a triangular decomposition for ${\mathcal H_c}$: multiplication in ${\mathcal H_c}$ induces a vector space isomorphism $$\mathbb C[{\mathfrak h}^*] \otimes \mathbb C[W] \otimes \mathbb C[{\mathfrak h}] \to {\mathcal H_c}.$$
Motivated by the analogy between this decomposition and the well-known triangular decomposition of a semisimple Lie algebra, one can consider the category ${\mathcal O_{\mathcal H_c}}$ of representations for $\mathcal H_c$ which are locally finite for the subalgebra $\mathbb C[{\mathfrak h}^*]$. This category was introduced in [@DO] and has been extensively studied (see for example [@BEG], [@GGOR]). A module $M$ in category ${\mathcal O_{\mathcal H_c}}$ is said to have *type* $\bar{\lambda}$ for $\bar{\lambda} \in {\mathfrak h}/W = \text{Spec}(\mathbb C[{\mathfrak h}^*]^W)$ if for any element $P \in \mathbb C[{\mathfrak h}]^W$ the operator $P - P(\bar{\lambda})$ is locally nilpotent. The full subcategory of ${\mathcal O_{\mathcal H_c}}$ consisting of objects of type $\bar{\lambda}$ is denoted ${\mathcal O_{\mathcal H_c}}(\bar{\lambda})$ and ${\mathcal O_{\mathcal H_c}}$ splits as a direct sum of the subcategories ${\mathcal O_{\mathcal H_c}}(\bar{\lambda})$. We shall focus on the category ${\mathcal O_{\mathcal H_c}}(0)$, where $\mathbb C[{\mathfrak h}]^W$ acts locally nilpotently, and thus for convenience we shall denote it by $\mathcal O_c$. It is abelian and every object has finite length. Note that the results of Bezrukavnikov and Etingof on induction/restriction functors allow one to reduce the study of ${\mathcal O_{\mathcal H_c}}(\bar{\lambda})$ to $\mathcal O_{\mathcal H'_c}(0)$ for $\mathcal H'_c$ a rational Cherednik algebra corresponding to a parabolic subgroup of $W$, so that questions about ${\mathcal O_{\mathcal H_c}}$ can be reduced to ones for $\mathcal O_{\mathcal H_c}(0)$ (see [@BE Corollary 3.3] for a precise statement).
In order to relate ${\mathcal H_c}$ to the Hibert scheme, we will also need its spherical subalgebra, $e{\mathcal H_c}e$, where $e = |W|^{-1}\sum_{w \in W} w$ is the idempotent in $\mathbb C[W]$ corresponding to the trivial representation. The map $a \mapsto ae$ yields embeddings from $\mathbb C[{\mathfrak h}]^W \to \mathbb C[{\mathfrak h}]^We \hookrightarrow e{\mathcal H_c}e$ and from $\mathbb C[{\mathfrak h}^*]^W \to \mathbb C[{\mathfrak h}^*]^We \hookrightarrow e{\mathcal H_c}e$. Moreover, if $e\mathcal H_ce$ is simple, then their images generate $e{\mathcal H_c}e$ [@BEG Theorem 4.6]. It is easy to see that $e {\mathcal H_c}e$ is simple whenever ${\mathcal H_c}$ is simple[^2], and by [@BEG Theorem 3.1] the algebra ${\mathcal H_c}$ is simple algebra exactly when the associated finite Hecke algebra $H_W(q)$ is semisimple, where $q = e^{2\pi i c}$. One has the obvious functor from $\mathcal H_c$-modules to $e\mathcal H_c e$-modules given by $M \mapsto eM$. This is easily seen [@BEG Lemma 4.1] to be a Morita equivalence when $\mathcal H_c$ is simple.
We now recall the construction of the ${\text{KZ}}$-functor of [@GGOR]. Let $${\mathfrak h^{\text{reg}}}= \{v \in {\mathfrak h}: |W\cdot v| = |W|\},$$ be the subset of ${\mathfrak h}$ consisting of those points whose coordinates are pairwise distinct. If we set $\delta = \prod_{i<j}(x_i-x_j) \in \mathbb C[{\mathfrak h}]$ then ${\mathfrak h^{\text{reg}}}= \{ v \in {\mathfrak h}: \delta(v) \neq 0\}$, hence it is a Zariski-open subset of ${\mathfrak h}$. The key to the construction is the Dunkl homomorphism from ${\mathcal H_c}$ to $\mathcal D({\mathfrak h^{\text{reg}}})\rtimes \mathbb C[W]$. This is given by the obvious embedding of the subalgebra $\mathbb C[\mathfrak h]\rtimes \mathbb C[W]$, and by sending $y \in {\mathfrak h}$ to the operator $$T_y = \partial_y + c\sum_{i<j} \frac{y_i-y_j}{x_i-x_j}(s_{ij}-1) \in \mathcal D({\mathfrak h^{\text{reg}}})\rtimes \mathbb C[W].$$ It is straight-forward to check that $S = \{\delta^k : k \in \mathbb Z_{>0}\}$ is an Ore set in $\mathcal H_c$, so that it makes sense to localize $\mathcal H_c$ at $S$. We refer to this as localizing $\mathcal H_c$ to $\mathfrak h^{\text{reg}}$ and denote the resulting algebra by $\mathcal H_{c|\mathfrak h^{\text{reg}}}$. Similarly we may localize an $\mathcal H_c$-module at $S$ to obtain an $\mathcal H_{c|\mathfrak h^{\text{reg}}}$-module denoted $M_{|\mathfrak h^{\text{reg}}}$. We then have the following result:
[@EG §4] The assignment $w \mapsto w$, $x \mapsto x$ and $y \mapsto T_y$ extends to an algebra homomorphism $\Theta_c$ which embeds ${\mathcal H_c}$ into $\mathcal D({\mathfrak h^{\text{reg}}})\rtimes \mathbb C[W]$. Moreover, this map becomes an isomorphism after localizing to ${\mathfrak h^{\text{reg}}}$.
Note that the final part of the proposition is immediately clear from the first part and the explicit formula for $T_y$. Now given any module $M$ in the category $\mathcal O_{\mathcal H_c}$, via the above isomorphism, we may view $M_{|\mathfrak h^{\text{reg}}}$ as a module for $\mathcal D({\mathfrak h^{\text{reg}}})\rtimes \mathbb C[W]$-module, and hence, since ${\mathfrak h^{\text{reg}}}$ is an affine variety, as a $W$-equivariant module for the sheaf of differential operators $\mathcal D_{{\mathfrak h^{\text{reg}}}}$ on ${\mathfrak h^{\text{reg}}}$. Now it is not hard to show that $M$ is finitely generated as a $\mathbb C[{\mathfrak h}]$-module (see for example Lemma $2.5(i)$ of [@BEG]), thus $M_{|{\mathfrak h^{\text{reg}}}}$ is coherent as an $\mathcal O_{{\mathfrak h^{\text{reg}}}}$-module. It follows that the associated $\mathfrak D_{{\mathfrak h^{\text{reg}}}}$-module must be a vector bundle with a flat connection, and hence it gives rise to a local system on ${\mathfrak h^{\text{reg}}}$. Since this local system is $W$-equivariant, it descends to give a local system on ${\mathfrak h^{\text{reg}}}/W$. But now if $x_0 \in {\mathfrak h^{\text{reg}}}$ is any point, it is known that $\pi_1({\mathfrak h^{\text{reg}}}/W, x_0)$ is isomorphic to the braid group $\mathcal B_W$, and hence we get a functor $${\text{KZ}}\colon {\mathcal O_{\mathcal H_c}}\to \text{Rep}(\mathcal B_W),$$ by for example applying the deRham functor to $M_{{\mathfrak h^{\text{reg}}}}$ (alternatively one could take the dual representation given by the solution complex). One of the main results of [@GGOR] shows that the image of this functor lies in the much smaller category of representations of the associated Hecke algebra.
([@GGOR Theorem 5.13]) The functor ${\text{KZ}}$ factors through the category of representations of the Hecke algebra $\mathcal H_q$ where the parameter $q$ is specialized to $e^{2\pi i c}$.
We will give a new proof of this theorem in Section \[MKZfunctors\] in the $\mathcal D$-module context of [@GG].
The deformed Harish-Chandra homomorphism and the Hilbert scheme. {#deformed}
================================================================
{#almostcommuting}
In this section we recall the localization of the rational Cherednik algebra in type $A$, as studied in [@KR] and the related work [@GG]. Let $V$ be an $n$-dimensional vector space and let ${\mathfrak g}= \mathfrak{gl}(V)$ be the Lie algebra of linear maps from $V$ to itself. The almost commuting variety is the variety of quadruples $$\mathcal M = \{(X,Y,i,j) \in {\mathfrak g}\times {\mathfrak g}\times V \times V^*: [X,Y]+ji = 0\},$$ where $ji$ denotes the rank one linear map $v \mapsto j(v).i$. Using the trace form we may naturally identify ${\mathfrak g}$ with ${\mathfrak g}^*$, so that $\mathcal M$ may be viewed as a subvariety of $T^*\mathfrak G$ where we set $\mathfrak G = {\mathfrak g}\times V$. It is known [@GG Theorem 1.1] that $\mathcal M$ is an equidimensional variety with $n+1$ components of dimension $n^2 +2n$. We also consider the set $$\mathcal M_{\text{nil}} = \{(X,Y,i,j) \in \mathcal M: Y \text{ is nilpotent}\}.$$ This is a (reducible) Lagrangian subvariety. Finally we consider the Zariski open set (given by a stability condition) $$\text{U} = \{(X,Y,i,j) \in T^*\mathfrak G: \mathbb C\langle X,Y\rangle i = V\}$$ where $\mathbb C\langle X,Y\rangle$ is the subalgebra of $\text{End}(V)$ generated by $X$ and $Y$. If we let $\mathcal M^s = \mathcal M \cap \text{U}$, then it can be shown that $\text{GL}(V)$ acts freely on $\mathcal M^s$, and the quotient is isomorphic to $\text{Hilb}^n(\mathbb C^2)$, the Hilbert scheme of $n$ points in the plane. At the level of sets the isomorphism can be seen as follows: if $(X,Y,i,j) \in \mathcal M^s$ then the stability condition in fact forces $j$ to be zero, so that $X$ and $Y$ commute. Thus they define a homomorphism from $\mathbb C[x,y] \to \text{End}(V)$, and the kernel of this homomorphism is a codimension $n$ ideal of $\mathbb C[x,y]$. Note that this realization of $\text{Hilb}^n(\mathbb C^2)$ equips it with a natural ample line bundle corresponding to the $G$-equivariant line bundle $\mathcal M^s \times \text{det}$ on $\mathcal M^s$. Let ${\mathcal M^{\text{sn}}}$ denotes the intersection of $\mathcal M^s$ with $\mathcal M_{\text{nil}}$. It is $G$-stable its quotient is a Lagrangian subvariety of ${\text{Hilb}^n(\mathbb C^2)}$ which we denote by $Z$.
{#DmodsonG}
We now relate the rational Cherednik algebra to twisted $\mathcal D$-modules. For the convenience of the reader, and to fix notation, in Appendix \[twistingstuff\] we review a construction of the kinds of twisted rings of differential operators we need, along with notions of equivariance. Let $V^\circ = V\backslash \{0\}$, let $\mathfrak G^\circ = {\mathfrak g}\times V^\circ$, and let $\mathfrak X = {\mathfrak g}\times \mathbb P$ where $\mathbb P = \mathbb P(V)$ is the projective space of lines in $V$. Clearly $\mathfrak G^\circ$ is a principal $\mathbb G_m$-bundle over $\mathfrak X$, so that given any $\lambda \in \text{Hom}(\text{Lie}(\mathbb G_m),\mathbb C)$, we may consider modules for the corresponding sheaf $\mathcal D_{\mathfrak X,\lambda}$ of $\lambda$-twisted differential operators on $\mathfrak X$, or equivalently $(\mathbb G_m,\lambda)$-twisted equivariant $\mathfrak D_{\mathfrak G^0}$-modules. Note that we may describe the cotangent bundle of $\mathfrak X$ as $$\{(X,Y,i,j) \in T^*\mathfrak G^\circ: \langle i,j\rangle =0\}/\mathbb C^\times,$$ where $\mathbb C^\times$ acts via $t\cdot(X,Y,i,j) = (X,Y,ti,t^{-1}j)$ and as above we have identified $T^*\mathfrak G^\circ$ with ${\mathfrak g}\times{\mathfrak g}\times V^\circ\times V^*$. The function $(X,Y,i,j) \mapsto [X,Y]+ij$ is fixed by this $\mathbb C^\times$-action, and moreover if $(X,Y,i,j) \in \mathcal M$ then $$\langle i,j\rangle = \text{Tr}(ji) = - \text{Tr}([X,Y]) = 0,$$ thus $$\mathcal M \cap \{(X,Y,i,j) \in T^*\mathfrak G^\circ :\langle i,j \rangle = 0\} = \mathcal M \cap T^*\mathfrak G^\circ.$$ Let $\mathfrak M$ denote the quotient of $\mathcal M \cap T^*\mathfrak G^\circ$ by $\mathbb C^\times$, a subvariety of $T^*\mathfrak X$, and similarly let $\Lambda$ be the quotient of $\mathcal M_\text{nil}\cap T^*\mathfrak G^\circ$ (a Lagrangian subvariety of $T^*\mathfrak X$). If $(X,Y,i,j) \in \mathcal M^s$ then $i \neq 0$ and $\langle i,j\rangle =0$ (since $j=0$), so that $\mathcal M \cap T^*\mathfrak G^\circ$ contains $\mathcal M^s$ and ${\mathcal M^{\text{sn}}}$. We denote their quotients in $T^*\mathfrak X$ by $\mathfrak M^s$ and $\Lambda^s$.
The action of $\text{SL}(V)$ on $\mathfrak G^\circ$ commutes with the $\mathbb G_m$-action, so that we may also consider the category of $\text{SL}(V)$-equivariant $\mathcal D_{\mathfrak X,\lambda}$-modules. Now if we write $\mathcal Z\cong \mathbb G_m$ for the centre of $\text{GL}(V)$, then the action of $\mathbb G_m$ on $\mathfrak G^\circ$ is clearly the restriction to $\mathcal Z$ of the action of $\text{GL}(V)$ on $\mathfrak G^\circ$. We may thus also consider the category of $(\text{GL}(V),c.\text{tr})$-equivariant $\mathcal D$-modules on $\mathfrak G^\circ$ where as usual $\text{tr}$ denotes the trace character on ${\mathfrak g}$. Note that given $c \in \mathbb C$, the corresponding $\lambda$ above for $\text{Lie}(\mathbb G_m)=\text{Lie}(\mathcal Z)$ is given by $\lambda = c.\text{tr}_{|\text{Lie}(\mathcal Z)}$, and following [@GGS] we will from now on write $\mathcal D_{\mathfrak X,c}$ for $\mathcal D_{\mathfrak X,\lambda}$. Now since $\text{tr}$ vanishes on $\mathfrak{sl}(V)$ and $\text{GL}(V) = \mathcal Z.\text{SL}(V)$, this category is equivalent to the category of $\text{SL}(V)$-equivariant $\mathcal D_{\mathfrak X,c}$-modules on $\mathfrak X$ and moreover this equivalence respects holonomic modules. Note that the category of $(\text{GL}(V),c.\text{tr})$-equivariant modules is considered in [@KR], while the category of $\text{SL}(V)$-equivariant $\mathcal D_{\mathfrak X,c}$-modules is used in [@GG]. For more details concerning this equivalence, at least at the level of algebras of global sections, see [@GGS §6].
We also remark that if we write $\mathcal D_c(\mathfrak X)$ for the global sections of $\mathcal D_{\mathfrak X,c}$, then provided $n>3$ or $n=2$ and $2c\notin \mathbb Z_{\leq 0}$, it is shown in [@GGS §6.2] that $$\mathcal D_c(\mathfrak X) \cong \big(\mathcal D(\mathfrak G)/(\mathcal D(\mathfrak G)(\nu-c.\text{tr})(\mathfrak z))\big)^{\mathcal Z}$$ where $\nu \colon \mathfrak{gl}(V) \to \Theta_{{\mathfrak g}\times V}$ is the quantized moment map for the $\text{GL}(V)$-action, and $\mathfrak z$ is the Lie algebra of $\mathcal Z$. Thus one may effectively work on $\mathfrak G$ or $\mathfrak G^\circ$.
{#deformedHC}
The action of $\text{SL}(V)$ on $\mathfrak X$ induces an infinitesimal action of $\mathfrak{sl}_n$, and hence a homomorphism from $\mathfrak{sl}_n$ to $\Theta_{\mathfrak X}$ the Lie algebra of vector fields on $\mathfrak X$, which extends naturally to a map $\tau \colon \mathcal U({\mathfrak g}) \to \mathcal D_\mathfrak X$. Similarly, we obtain a homomorphism $\tau_c\colon \mathcal U({\mathfrak g}) \to \mathcal D_{\mathfrak X,c}$ from the enveloping algebra to the sheaf of $c$-twisted differential operators. Let $\mathfrak g_c$ denote the image of $\mathfrak{sl}(V)$ under $\tau_c$. Since projective space is known to be $\mathcal D$-affine for $c \notin \{-k/n: k \in \mathbb Z_{>0}\}$, for these values the category of $\mathcal D_{\mathfrak X,c}$-modules is equivalent to the category of modules for its algebra of global sections $\mathcal D_c(\mathfrak X)$.
One of the main results of [@GG] is the construction of a “deformed Harish-Chandra homomorphism” $$\Phi_c \colon (\mathcal D_c(\mathfrak X)/\mathcal D_c(\mathfrak X)\cdot {\mathfrak g}_c)^{\text{ad}({\mathfrak g}_c)} \to e\mathcal H_c e.$$ The map $\Phi_c$ is a filtered isomorphism sending $\mathbb C[{\mathfrak g}]^{\text{Ad} G}$ to $\mathbb C[\mathfrak h]^{S_n}$ and $\mathfrak Z$, the algebra of constant coefficient differential operators on ${\mathfrak g}$, to $\text{Sym}(\mathfrak h)^{S_n}$ (see Theorem $1.5$ of [@GG]).
Since we will later examine the connection between the original $KZ$-functor and our microlocal functors, we review briefly the construction of the map $\Phi_c$. It consists of two steps: first there is the Dunkl embedding which we have already discussed. This gives a map $\Theta_c \colon \mathcal H_c \to \mathcal D(\mathfrak h^{\text{reg}})\rtimes \mathbb C[W]$. On the other hand one can construct a “radial parts” homomorphism $$\mathfrak R_c\colon (\mathcal D_c(\mathfrak X)/\mathcal D_c(\mathfrak X)\cdot {\mathfrak g}_c)^{\text{ad}({\mathfrak g}_c)} \to \mathcal D(\mathfrak h^{\text{reg}})^W.$$ To construct this map let $\mathfrak X^{\text{reg}}$ be the open subset of $\mathfrak X$ consisting of pairs $(X,\ell)$ where the nonzero vectors in $\ell$ are cyclic for $X$ and let $\mathfrak G^\text{reg}$ denote its preimage in $\mathfrak G^\circ$. If we pick a holomorphic volume form $\Omega \in \bigwedge^n(V^*)$ then the function $$s\colon \mathfrak G \to \mathbb C, \quad (X,v) \mapsto \langle v\wedge Xv\wedge \ldots \wedge X^{n-1}v,\Omega \rangle,$$ shows that $\mathfrak G^\text{reg}$ is a principal affine open of $\mathfrak G$, and hence $\mathfrak X^{\text{reg}}$ is the complement of a principal divisor in $\mathfrak X$. It is easy to check that the action of $\text{PGL}(V)$ on $\mathfrak X^{\text{reg}}$ is free (see [@BFG §5.3] or Proposition \[smoothlocus\] below), making it into a principal $\text{PGL}(V)$-bundle over $\mathfrak h/W$. Thus by descent we see that $$(\mathcal D_c(\mathfrak X^{\text{reg}})/\mathcal D_c(\mathfrak X^{\text{reg}})\cdot {\mathfrak g}_c))^{\text{PGL}(V)} \cong \mathcal D(\mathfrak h/W), \quad (c \in \mathbb C).$$ where the isomorphism is given explicitly by $$\mathfrak R_c(D) = s^c(D_{|\mathcal O(\mathfrak X^{\text{reg}},c)})s^{-c}, \quad D \in \mathcal D(\mathfrak X)^G.$$ Now we have inclusions $\mathcal D(\mathfrak h)^W \subset \mathcal D(\mathfrak h/W) \subset \mathcal D(\mathfrak h^{\text{reg}})^W$, and it can be shown that the image in $\mathcal D(\mathfrak h^\text{reg})^W$ of the composite of $\mathfrak R_c$ with this inclusion is precisely the image of the spherical subalgebra of $\mathcal H_{c-1}$ under the Dunkl homorphism. A careful discussion of this fact is given in the Appendix to [@GGS]. However the map $\Psi_c$ which we work with is, as in [@BFG §5.4], a twist of this radial parts map by the discriminant $\delta$: we set $$\Psi_c(D) = m_\delta^{-1}\circ \mathfrak R_c(D)\circ m_\delta,$$ where $m_f$ denotes the operator of multiplication by the function $f$. The image $\text{im}(\Psi_c)$ of this twisted radial parts map is exactly the image of $e\mathcal H_c e$ under the Dunkl homomorphism, and the composition of $\Psi_c$ with the inverse of the Dunkl map yields the required isomorphism $\Phi_c$. Note that when $c=0$ this then agrees with the classical Harish-Chandra homomorphism [@EG §7].
A similar result is obtained in [@KR Lemma 4.7].
On the smooth locus of $\Lambda^s$ {#Micropi1}
==================================
{#GrojZ}
In this section we study the Lagrangian variety $Z$ in $\text{Hilb}_n(\mathbb C^2)$ introduced in $\S$\[almostcommuting\]. Let us first give another description of $Z$, following [@Gr]. Let $\pi\colon {\text{Hilb}^n(\mathbb C^2)}\to S^n(\mathbb C^2)$ be the Hilbert-Chow morphism. It is known that $\pi$ is a resolution of singularities. Let $\Sigma = \mathbb C$, and view $\mathbb C^2$ as $T^*\Sigma$, a local model for a curve inside a surface. Since $\text{Hilb}^n(\Sigma)$ is isomorphic to the $n$-th symmetric product of $\Sigma$, we may view $\text{Hilb}^n(\Sigma)$ as either a subvariety of ${\text{Hilb}^n(\mathbb C^2)}$ or $S^n(\mathbb C^2)$, compatibly with the morphism $\pi$. The variety $S^n(\Sigma) = \mathbb C^n/S_n$ is stratified in an obvious way:
$$S^n(\Sigma) = \bigsqcup_{\lambda \vdash n} S^n_\lambda(\Sigma),$$ where the stratum $S^n_\lambda(\Sigma)$ contains those points of $S^n(\Sigma)$ whose multiplicities are given by $\lambda$, a partition of $n$. If we set $Z_\lambda^0 = \pi^{-1}(S_\lambda^n(\Sigma))$, then $$Z = \bigsqcup_{\lambda \vdash n} Z_{\lambda}^0,$$ is a closed Lagrangian subvariety of ${\text{Hilb}^n(\mathbb C^2)}$, with components $Z_\lambda = \overline{Z_{\lambda}^0}$. In terms of the analogy with category $\mathcal O$ for semisimple Lie algebras, $Z$ corresponds to the union of the conormal varieties to the $B$-orbits on the flag variety.
$\text{Hilb}^n(T^*\Sigma)$ inherits a natural $\mathbb C^\times$-action from the $\mathbb C^\times$-action on the fibres of $T^*\Sigma$, and the components of $Z$ may also be described in terms of this action. Indeed each component of the fixed point set can naturally be identified with $S^n_\lambda(\Sigma)$ for a partition $\lambda$ of $n$, and if $\text{Fix}_\lambda$ denotes this component, we have $$\overline{Z_\lambda^0} = \overline{\{x \in {\text{Hilb}^n(\mathbb C^2)}: \lim_{t \to 0} t\cdot x \in \text{Fix}_\lambda\}},$$ (here the right-hand side asserts both that the limit exists, and that it lies in $\text{Fix}_\lambda$). For more details see [@Na §7.2]
{#decompositionofM}
We now relate this to the almost commuting variety and the construction of ${\text{Hilb}^n(\mathbb C^2)}$ given in the previous section. The variety $\mathcal M_\text{nil}$ is described in [@GG] explicitly in terms of a stratification of $\mathfrak X$ which mimics Lusztig’s stratification of a reductive group or Lie algebra. The strata are labelled by conjugacy classes of pairs $(\mathfrak l, \Omega)$ where $\mathfrak l$ is a Levi subalgebra and $\Omega$ is an orbit of $L$ (the associated Levi subgroup) on $\mathcal N_\mathfrak l \times V$ where $\mathcal N_{\mathfrak l}$ is the nilpotent cone of $\mathfrak l$ (there are only finitely many such orbits). A stratum $S$ is said to be *relevant* if for some (and hence any) $(X,\ell) \in S$, the element $X$ is regular, and the subspace $\mathbb C[X]\ell$ has a $\mathbb C[X]$-invariant complement. Then we have the following result.
[@GG §4.3] The variety $\Lambda$ is given by $$\Lambda = \bigsqcup_{S \text{ relevant}} \overline{T^*_S(\mathfrak X)}.$$
Now $Z$ is obtained from $\Lambda$ as the quotient of $\Lambda^s$ by the $\text{PGL}(V)$-action, thus the components of $Z$ correspond to certain components of $\Lambda$. To make this correspondence explicit, we first give a natural labelling of the components of $\Lambda$. Recall that a bipartition of $n$ is an ordered pair of partitions $(\lambda, \mu)$ such that $|\lambda|+|\mu| = n$, where for a partition $\nu=(\nu_1\geq \nu_2 \geq \ldots \geq \nu_k)$ we write $|\nu|=\sum_{i=1}^k \nu_i$, the sum of the parts of $\nu$.
The relevant strata of $\mathfrak X$ and hence the components of $\Lambda$ are labelled by bipartitions of $n$.
Clearly we need only check that we may index relevant strata this way. If $(X,\ell) \in S$ where $S$ is a relevant stratum, then we may write $X = z+x$ the Jordan decomposition of $X$, where $z$ is semisimple and $x$ is nilpotent. The dimensions of the eigenspaces of $z$ give a partition $\nu$ of $n$ (this corresponds to the conjugacy class of the Levi subalgebra $Z_{\mathfrak g}(z)$). Moreover, the condition that $(X,\ell)$ is relevant implies that if $v \in \ell$ is a nonzero vector, then its projection to an eigenspace of $z$ is either zero, or a cyclic vector for $x$ restricted to that subspace. Dividing $\nu$ according to this dichotomy we obtain a bipartition as required.
For a bipartition $(\lambda,\mu)$ we will write $\Lambda_{\lambda,\mu}$ for the component of $\Lambda$ corresponding to $(X,\ell)$ where the partition $\lambda$ measures the dimensions of the generalized eigenspaces of $X$ in which the projection of $\ell$ is nonzero, while $\mu$ measures the dimensions of the generalized eigenspaces in which the projection of $\ell$ is zero.
To relate this to the variety $Z$ we need to examine which components of $\Lambda$ intersect the stable locus $\mathfrak M^s$. For a bipartition $(\lambda,\mu)$ of $n$ let $S_{\lambda,\mu}$ denote the corresponding relevant stratum and $\Lambda_{\lambda,\mu} = \overline{T^*_{S_{\lambda,\mu}}\mathfrak X}$ denote the conormal variety of the stratum labelled by $(\lambda,\mu)$ by the above convention.
We now note the following easy lemma, which is presumably well-known.
\[commuting\] Let $X \in \mathfrak{gl}(V)$ be a regular element. Then $$Z_{\mathfrak g}(X) = \mathbb C[X],$$ that is, the matrices which commute with $X$ are precisely the subalgebra of $\text{End}(V)$ generated by $X$. Moreover it follows that the elements of $Z_{\mathfrak g}(X)$ which are nilpotent form a linear subspace.
Let $V= \bigoplus_{i=1}^k V_i$ be the decomposition of $V$ into the generalized eigenspaces of $X$. On each $V_i$ we have $X_{|V_i} = \lambda_i + N_i$, where $\lambda_i$ is a scalar and $N_i$ is a regular nilpotent endomorphism (since $X$ is regular). The standard proof of the existence of a Jordan decomposition ( by the Chinese remainder theorem) shows that the projection operators $\pi_i$ from $V$ to the generalized eigenspaces of $X$ lie in $\mathbb C[X]$ (in fact they are polynomials in $X$ with no constant term). Thus $\iota_i \circ N_i\circ \pi_i$ also lies in $\mathbb C[X]$ for each $i$, ($1 \leq i \leq k$) where $\iota_i$ is the inclusion of $V_i$ into $V$ . It is easy to check that the endomorphisms of a vector space $U$ which commute with a regular nilpotent $N$ are exactly the linear maps in the subalgebra $\mathbb C[N]$, and since $\{1,N,N^2,\ldots, N^{\dim(U)-1}\}$ clearly form a basis of $\mathbb C[N]$ this is a vector space of dimension $\dim(U)$. The nilpotent such endomorphisms are simply those in the span of $\{N,N^2,\ldots\}$.
If $Y$ commutes with $X$, then clearly it preserves the eigenspaces of $X$, and so we may write it as $Y = \sum_{i=1}Y_i$, where $Y_i = \iota_i \circ y_i \circ \pi_i$ with $y_i \in \text{End}(V_i)$. Since $y_i$ clearly commutes with $X_{|V_i}$ and hence with $N_i$ we see $y_i \in \mathbb C[N_i]$ and so each $Y_i \in \mathbb C[X]$, and hence $Y \in \mathbb C[X]$. Moreover the nilpotent such $Y$ again clearly form a linear subspace as required.
Now we have the following:
\[intersection\] The intersection $\mathfrak M^s \cap \Lambda_{\lambda,\mu}$ is dense in $\Lambda_{\lambda,\mu}$ if and only if $\mu=\emptyset$.
Let $S= S_{\lambda, \mu}$. We will in fact we show that $$\mathfrak M^s \cap T^*_S(\mathfrak X) = \left\{\begin{array}{cc}\emptyset & \text{if } \mu \neq \emptyset, \\T^*_S(\mathfrak X) & \text{if } \mu = \emptyset. \end{array}\right.$$ Suppose that $(X,Y,i,j)$ is a point in the intersection, and let $X = z+x$ be the Jordan decomposition of $X$, where $z$ is semisimple and $x$ is nilpotent. Let $V = \bigoplus_{i=1}^k V_i$ be the decomposition of $V$ according to the eigenspaces of $z$. The stability condition forces $j$ to be zero, so that $X$ and $Y$ commute, and hence $Y$ commutes with both $z$ and $x$. As $Y$ and $x$ commute with $z$, they both preserve the subspaces $V_i$ and we write $x_i$ for $x_{|V_i}$ and $y_i$ for $Y_{|V_i}$ respectively ($1 \leq i \leq k$).
Since $(X,\mathbb C \cdot i)$ is relevant, $x_i$ must be a regular nilpotent, and so its centralizer in $\text{End}(V_i)$ is just $\mathbb C[x_i]$. Thus $y_i \in \mathbb C[x_i]$. Since the projections $V \to V_i$ also lie in $\mathbb C[X]$ it follows that $y_i$ (or rather its extension by zero to $V$) lies in $\mathbb C[X]$ and hence $Y \in \mathbb C[X]$. But then $\mathbb \langle X,Y \rangle = \mathbb C[X]$, and the stability condition then forces $i$ to be a cyclic vector for $X$, and hence $\mu=\emptyset$ as required. Conversely if $\mu = \emptyset$, then it is evident that any point in $T^*_S(\mathfrak X)$ satisfies the stability condition.
\[componentwarning\] Note that the above proof does *not* imply that $\Lambda_{\lambda,\emptyset}\cap \mathfrak M^s$ is contained in $T_{S_{\lambda,\emptyset}}^*(\mathfrak X)$. For example when the bipartition is $((n),\emptyset)$, then the points of the corresponding stratum consist of matrices of the form $c + n$ where $c$ is a scalar and $n$ is a regular nilpotent, together with a line which is cyclic for $n$. The closure of the corresponding conormal variety contains stable quadruples $(X,Y,i,0)$ where $X$ is not regular. If we take $n=3$ say then it is easy to see that $$\big(\left(\begin{array}{ccc}0 & 0 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{array}\right), \left(\begin{array}{ccc}0 & 0 & 1 \\0 & 0 & 0 \\0 & 0 & 0\end{array}\right),\left(\begin{array}{c}0 \\0 \\1\end{array}\right),0\big),$$ is a stable quadruple, with neither $X$ or $Y$ regular. On the other hand, since $\text{U}$ is Zariski open, so is $(\text{U} \cap T^*\mathfrak G^\circ)/\mathbb C^\times$, thus if a component of $\Lambda$ intersects $\mathfrak M^s$ it will do so in a dense open subset. It follows that the Lemma tell us exactly which components of $\Lambda$ have nontrivial intersection with $\Lambda^s$.
Recall we set ${\mathcal M^{\text{sn}}}= \mathcal M^s\cap \mathcal M_\text{nil}$. It is $\text{GL}(V)$-stable. Its quotient by $\text{GL}(V)$ is the variety $Z$, and its quotient by the action of $\mathcal Z$ is the variety $\Lambda^s = \Lambda \cap \mathfrak M^s$. From Lemma \[intersection\] and Remark \[componentwarning\] we see that the components of $\Lambda^s$ are labelled by bipartitions of $n$ of the form $(\lambda, \emptyset)$. Since for a point $(X,Y,\ell) \in \Lambda^s$, the spectrum of $X$ yields the $n$-tuple of points in $S^n(\Sigma)$ under the composition of the quotient map with the Hilbert-Chow morphism, it is easy to check that the component $Z_\lambda$ of $Z$ described in $\S$\[GrojZ\] corresponds to the component of $\Lambda^s$ labelled by the bipartition $(\lambda, \emptyset)$. We will write $\Lambda^s_\lambda = \Lambda^s \cap \Lambda_{\lambda,\emptyset}$ for this component, and ${\mathcal M^{\text{sn}}}_\lambda$ for the corresponding components of ${\mathcal M^{\text{sn}}}$.
{#section-4}
We now define a decomposition of $\Lambda^s $ into locally closed pieces. It will evidently be $G$-stable, and hence induce a decomposition of $Z$. In fact for convenience in this subsection we will work mostly with ${\mathcal M^{\text{sn}}}$. Recall that if $(X,Y,i,j) \in \mathcal M^s$, then we have $j=0$, so that $X,Y$ commute with each other. We will thus prefer to write $(X,Y,i)$ for a point in ${\mathcal M^{\text{sn}}}$ rather than $(X,Y,i,0)$.
Let $(X,Y,i) \in {\mathcal M^{\text{sn}}}$. Let $X = z+x$ be the decomposition of $X$ into its semisimple and nilpotent parts respectively, and let $V = \bigoplus_{i=1}^k V_i$ be the decomposition of $V$ into the eigenspaces of $z$. Since $Y$ commutes with $z$ it preserves each $V_i$, and its restriction to $V_i$ is a nilpotent endomorphism with Jordan type $\nu^{(i)}$ say. In this way we obtain a multipartition $\underline{\nu} = (\nu^{(1)},\nu^{(2)},\ldots,\nu^{(k)})$, and we may assume that $(\dim(V_1),\ldots,\dim(V_k))$ form a partition $\lambda$ of $n$. Let ${\mathcal M^{\text{sn}}}_{\lambda,\underline{\nu}}$ denote the set of $(X,Y,i,j)\in \Lambda$ which give rise to the pair $(\lambda,\underline{\nu})$ in this way, where if $\lambda_i = \lambda_{i+1}$ then we must identify the parameters $(\lambda,\underline{\nu})$ and $(\lambda,\underline{\nu}')$ where $\underline \nu'$ is obtained from $\underline \nu$ by exchanging the $i$-th and $(i+1)$th partitions. We will also write $\Lambda^s_{\lambda,\underline{\nu}}$ for the corresponding subset of $\Lambda$.
We now wish to show that the pieces of our decomposition are smooth and to calculate their dimension. For this we begin with a technical lemma which gives a sort of normal form for elements of ${\mathcal M^{\text{sn}}}$.
Suppose that $(X,Y,v) \in {\mathcal M^{\text{sn}}}$, and that $X$ and $Y$ are nilpotent endomorphisms. For $i,j \in \mathbb Z_{\geq 0}$, set $e_{i,j} = X^iY^j(v)$. Then we may find integers $k,s_0,s_2,\ldots s_{k-1},$ so that $\{e_{ij}: 0 \leq i <k, 0\leq j < s_i\}$ is a basis of $V$, and moreover the action of $X$ on $V$ is given by $$X(e_{i,j}) = \left\{\begin{array}{cc}e_{i+1,j}, & \text{if } i<k-1, j < s_{i+1}, \\0, & \text{otherwise}.\end{array}\right.$$ while if $\mathcal F = (F_i)$ denotes the decreasing filtration of $V$ given by $F_i = \text{im}(X^i)$, then action of $Y$ must satisfy: $$Y(e_{i,j}) = \left\{\begin{array}{cc}e_{i,j+1}, & \text{if } i<k, j< s_i-1 \\ w_{i+1}, & \text{if } j = s_i-1.\end{array}\right.$$ where $w_{i+1}$ is some vector in $F_{i+1}$.
Let $k$ be the nilpotence of degree $X$, so that $X^k=0$ but $X^{k-1} \neq 0$, and set $F_i = \text{im}(X^i)$ so that $\mathcal F = (F_i)_{0 \leq i \leq k-1}$ is a $k$-step (decreasing) flag in $V$. Now since $Y$ commutes with $X$, and hence with every power of $X$ we immediately see that $Y$ preserves the flag $\mathcal F$, and clearly $X(F_i)= F_{i-1}$. Let $s_i = \dim(F_i/F_{i+1})$ for $0 \leq i <k$, clearly $\mu = (s_0,s_1,\ldots s_{k-1})$ is a partition of $n$ (the dual partition to the one given by the Jordan type of $X$).
Now the vector $X^{k-1}(v)$ is clearly cyclic for the action of $\mathbb C[X,Y]$ restricted to $F_{k-1}$, but since $X$ acts by zero on $F_{k-1}$, it follows that $X^{k-1}(v)$ is cyclic for the action of $Y$ on $F_{k-1}$, and hence $\{e_{k-1,j}: 0 \leq j < s_{k-1}\}$ is a basis for $F_{k-1}$, and our description of the action of $X,Y$ on this part of the basis is established. But now considering $V/F_{k-1}$ and using induction on $k$ completes the proof.
The $\text{GL}(V)$ orbit of $X$ is of course just the Richardson orbit attached to the parabolic determined by the flag $\mathcal F$.
\[weightingaction\] Suppose that $(X,Y,v) \in {\mathcal M^{\text{sn}}}$ and $X,Y$ are nilpotent. Then given $a \geq b \in \mathbb Z$, we may find a homomorphism $\rho \colon \mathbb C^\times \to \text{GL}(V)$ such that
1. $\text{Ad}(\rho(t))(X) = t^a X$,
2. $\text{Ad}(\rho(t))(Y) = t^{b}Y_0 + O(t^{a})$,
3. $\rho(t)(v) = v$.
Moreover, the triple $(X,Y_0,v)$ lies in ${\mathcal M^{\text{sn}}}$.
We use the basis constructed in the previous Lemma. For $t \in \mathbb C^\times$, let $\rho(t)$ act on $V$ by $$\rho(t)(e_{i,j}) = t^{-aj-bi}e_{i,j}, \quad (0 \leq i \leq k-1, 0 \leq j \leq s_i).$$ Since $v= e_{0,0}$, condition $(3)$ is immediate, and similarly condition $(2)$ follows readily. Moroever to check condition $(1)$, we simply define $Y_0$ by $$Y_0(e_{i,j}) = \left\{\begin{array}{cc}e_{i,j+1}, & \text{if } j < s_i; \\0 & \text{if } j=s_i.\end{array}\right.$$ The stability of the triple $(X,Y_0,v)$ is then clear.
With these lemmas in hand, we can now check that the pieces of our decomposition are smooth.
\[pieces\] Each piece ${\mathcal M^{\text{sn}}}_{\lambda,\underline{\nu}}$ is a connected smooth locally closed subvariety of $\Lambda^s$, and if $Z_{\lambda,\underline{\nu}}$ is its image under the quotient map to $Z \subset {\text{Hilb}^n(\mathbb C^2)}$, then $Z_{\lambda,\underline{\nu}}$ is an affine space bundle over $S_{\lambda}^n(\Sigma) \subset S^n(\mathbb C^2)$ via the Hilbert-Chow morphism. Moreover the dimension of the stratum $Z_{\lambda,\underline{\nu}}$ is given by $$n - \sum_{i=1}^k(\ell(\nu^{(i)})-1).$$
Since the strata $S^n_{\lambda}(\Sigma)$ are smooth locally closed subvarieties of $\mathbb C^n/S_n$ the last part of the Proposition implies the other claims. Thus it is enough to show that $Z_{\lambda,\underline{\nu}}$ is an affine space bundle over $S^n_\lambda(\Sigma)$.
For this we first examine the case where $\lambda = (n)$, so that $S^n_\lambda(\Sigma)$ is just a point. The component $Z_{(n)}$ is the “punctual” Hilbert scheme, that is the moduli space of codimension $n$ ideals supported at $0 \in \mathbb C^2$, and in $T^*\mathfrak G^\circ$ it corresponds to the set $${\mathcal M^{\text{sn}}}_{(n)}= \{(X,Y,v) \in T^*\mathfrak G^\circ: [X,Y]=0, \mathbb C[X,Y]v = \mathbb C^n, X,Y \text{ nilpotent}\}$$ For a point $(X,Y,v) \in \mathcal M^s$ we will write $[X,Y,v]$ for the corresponding point in the quotient $\text{Hilb}^n(T^*\Sigma)$. We claim that fixing the Jordan type of $X$ describes a locally closed affine space in $Z_{(n)}$. To see this we use a torus action (for more details on the facts used here see for example [@Na Chapter 5]). Let $T^2 = (\mathbb C^\times)^2$ acts on ${\mathcal M^{\text{sn}}}_{(n)}$ by $$(t_1,t_2)(X,Y,v) = (t_1X,t_2Y,v).$$ This action descends to the natural action of $T^2$ on $\text{Hilb}^n(T^*\Sigma)$ induced from the action of $T^2$ on $T^*\Sigma \cong \mathbb C^2$.
Recall the classical Bialynicki-Birula result which shows that a smooth projective variety with a $\mathbb G_m$-action has a natural decomposition into pieces which are affine bundles over the components of the fixed-point locus of that action. Now we cannot apply the Bialynicki-Birula result directly to $Z_{(n)}$, since it is not smooth. However the $T^2$ action on $Z_{(n)}$ extends to the whole Hilbert scheme $\text{Hilb}^n(\mathbb P^2)$ of $n$ points in the projective plane, which is smooth and projective, and hence the Bialynicki-Birula result may be applied to it. In fact, the action has finitely many fixed points, so that the pieces are affine spaces: if $x \in \text{Hilb}^n(\mathbb P^2)$ then the piece of the decomposition attached to $x$ is given by: $$P_x = \{y \in \text{Hilb}^n(\mathbb P^2): \lim_{t \to \infty} \nu(t)\cdot y = x\}.$$ Choosing the $\mathbb G_m$-action to be given by a generic subgroup $\nu\colon \mathbb G_m \to T^2$ where $\nu(t) = (t^a,t^b)$ with $a>b>0$, we see by considering the action of $\mathbb G_m$ on the symmetric product $S^n(\mathbb C^2)$ (*i.e.* on the spectra of matrices $X$ and $Y$) that no point $x$ outside $Z_{(n)}$ can have $\lim_{t \to \infty} \nu(t).x$ lying inside $Z_{(n)}$. Thus, since $Z_{(n)}$ is certainly $T^2$-invariant, it is a union of pieces of the decomposition of $\text{Hilb}^n(\mathbb P^2)$. (See for example [@ES §1] for the statement of the Bialynicki-Birula result and [@ES §2] for a similar use of the theorem for subvarieties of $\text{Hilb}^n(\mathbb P^2)$ which are preserved by the torus action).
It is known (see [@Na §5.2] for a detailed discussion) that the $T^2$-action (on $\text{Hilb}^n(\mathbb A^2)$) has finitely many fixed points corresponding to monomial ideals of codimension $n$ all of which lie in $Z_{(n)}$, thus the pieces in the decomposition of $Z_{(n)}$ are naturally labelled by partitions of $n$. If $x_\lambda$ is the monomial ideal corresponding to the partition $\lambda$ we will write we write $P_{\lambda}$ for the corresponding piece. It is known that $P_\lambda$ is an affine space of dimension $n-\ell(\lambda)$. Letting $p \colon \mathcal M^s \to {\text{Hilb}^n(\mathbb C^2)}$ denote the quotient map, we claim that $p^{-1}(P_\lambda) = {\mathcal M^{\text{sn}}}_{(n),\lambda}$.
To show this we use a one-parameter subgroup in $\text{GL}(V)$ as in Lemma \[weightingaction\]. Since the $T^2$ action commutes with the $G$-action we have $$\begin{split}
\nu(t)[X,Y,\ell)] &= [t^{a}X, t^bY,\ell] \\
&= [(\rho(t^{-1})(t^{a}X),\rho(t^{-1})(t^bY),\rho(t^{-1})\ell)] \\
&= [X,Y_0 + Y_t,\ell],
\end{split}$$ where $Y_t \in t^{-1}\text{End}(V)[t^{-1}]$. Thus since $\lim_{t \to \infty} (X,Y_0+Y_t,v) = (X,Y_0,v)$, which again by Lemma \[weightingaction\] lies in ${\mathcal M^{\text{sn}}}$, we see that the limit in $Z_{(n)}$ preserves the Jordan type of $X$. Since this also characterizes the fixed points of the $\mathbb C^\times$-action on $Z_{(n)}$ it follows immediately that $p^{-1}(P_{\lambda}) = {\mathcal M^{\text{sn}}}_{(n),\lambda}$ as claimed.
In the general case, if $x \in S^n_\lambda(\Sigma)$ then $\pi^{-1}(x)$ is a product of punctual Hilbert schemes, indeed we have $$\pi^{-1}(x) \cong Z_{(\lambda_1)}\times Z_{(\lambda_2)} \times \ldots \times Z_{(\lambda_k)},$$ where $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_k)$, and the isomorphism is given by taking, for $[X,Y,v] \in \pi^{-1}(x)$, the components of the nilpotent parts[^3] of $X$ and $Y$ in the eigenspaces of the semisimple part of $X$, along with the projection of the vector $v$ to that eigenspace. It follows immediately that the intersection of the $Z_{\lambda, \underline{\nu}}$ with $\pi^{-1}(x)$ is a product of affine spaces, and hence itself an affine space. Moreover, the dimension of the affine space is clearly $$\sum_{i=1}^k (\lambda_i - \ell(\nu^{(i)})) = n - \sum_{i=1}^k \ell(\nu^{(i)}).$$ But since $S^n_\lambda(\Sigma)$ clearly has dimension $k=\ell(\lambda)$, it follows that $$\dim(Z_{\lambda,\underline{\nu}})= n - \sum_{i=1}^k(\ell(\nu^{(i)})-1).$$ as claimed.
Notice that it follows from the dimension formula that the pieces of the decomposition of maximal dimension are the pieces $Z_{\lambda,\underline{\nu}}$ where $$\underline{\nu} = ((\lambda_1), (\lambda_2),\ldots, (\lambda_k)).$$ Notice moreover that the pieces of our decomposition which have dimension $n-1$ are those for which exactly one partition $\nu^{(i)}$ has two parts. From this it follows that such a piece lies in exactly two components of $Z$, those being $Z_\lambda$ and $Z_{\lambda'}$ where $\lambda'$ is obtained from $\lambda$ by replacing $\lambda_i$ by the two parts of $\nu^{(i)}$. Thus the decomposition of $Z$ also allows us to describe which components of $Z$ intersect in codimension $1$.
For example, the components which intersect the punctual Hilbert scheme $Z_{(n)}$ in codimension $1$ are those of the form $Z_{(k,n-k)}$, while the only component intersecting $Z_{(1^n)}$ in codimension $1$ is $Z_{(2,1^{n-2})}$.
{#smoothpieces}
We now use our decomposition of ${\mathcal M^{\text{sn}}}$ to find the smooth locus of ${\mathcal M^{\text{sn}}}$. We claim that its components are exactly the maximal dimensional pieces of our decomposition. Thus we set ${\tilde{\mathcal M}^\text{sn}}_\lambda = {\mathcal M^{\text{sn}}}_{\lambda,\underline{\nu}}$ where $\nu = ((\lambda_1),(\lambda_2),\ldots,(\lambda_k))$, that is, each partition in $(\underline\nu)$ has a single part. By the dimension formula in Proposition \[pieces\] each ${\tilde{\mathcal M}^\text{sn}}_\lambda$ has dimension equal to $\dim(\Lambda^s)$. Let $\tilde{Z}_\lambda$ and ${\tilde{\Lambda}}_\lambda$ denote the corresponding pieces of $Z$ and $\Lambda^s$ respectively.
\[smoothlocus\] Let ${\tilde{\mathcal M}^\text{sn}}_\lambda$ be as above. Then ${\tilde{\mathcal M}^\text{sn}}_\lambda$ is an open dense subvariety of ${\mathcal M^{\text{sn}}}_\lambda$ consisting of smooth points of ${\mathcal M^{\text{sn}}}$.
Since ${\tilde{\mathcal M}^\text{sn}}_\lambda$ is a piece of our decomposition, we have already established that it is smooth. To conclude that ${\tilde{\mathcal M}^\text{sn}}_\lambda$ consists of smooth points of ${\mathcal M^{\text{sn}}}$ it remains to check that ${\tilde{\mathcal M}^\text{sn}}_\lambda$ does not intersect any other component of ${\mathcal M^{\text{sn}}}$. For this suppose that $(X_0,Y_0,v_0) \in {\tilde{\mathcal M}^\text{sn}}_\lambda$ lies in some other component of ${\mathcal M^{\text{sn}}}$, say ${\mathcal M^{\text{sn}}}_\mu$. Then since ${\tilde{\mathcal M}^\text{sn}}_\mu$ is dense in ${\mathcal M^{\text{sn}}}_\mu$ we see that $(X_0,Y_0,v_0)$ must lie in the closure of ${\tilde{\mathcal M}^\text{sn}}_\mu$. Thus suppose that $(X_i,Y_i,v_i)_{i \geq 1} \in {\tilde{\mathcal M}^\text{sn}}_\mu$ converges[^4] to $(X_0,Y_0, v_0)$. But now the Jordan type of each $Y_i$ is given by $\mu$, and since the $(Y_i)$ converges to $Y_0$ we must have $\mu \geq \lambda$ in the dominance ordering on partitions, and hence certainly we must have $\ell(\mu) \leq \ell(\lambda)$.
On the other hand, letting $s \colon {\mathcal M^{\text{sn}}}\to \mathfrak h/S_n$ be the map given by taking the eigenvalues of $X$. Since the sequence $s(X_i,Y_i,v_i)$ consists of $\ell(\mu)$ distinct points with multiplicities $\mu_i$, $(1 \leq i \leq \ell(\mu))$, it follows that either $X_0$ has at most $\ell(\mu)$ distinct eigenvalues and so $\ell(\mu) \geq \ell(\lambda)$. Hence we see that we must have $\ell(\mu) = \ell(\lambda)$, and then moreover the eigenvalue multiplicities of $X_0$ must be equal to those of the $X_i$’s (as there can be no ”collision” of eigenvalues), whence we have $\lambda = \mu$ as desired.
It is easy to see that every other piece in our decomposition lies in more than one component of $Z$, and hence the union of our subsets $\tilde{Z}_\lambda$ in fact yields the entire smooth locus of $Z$.
If $\lambda = (n)$, then $Y$ has a single Jordan block, and the space ${\tilde{\Lambda}}_\lambda$ is just an affine space bundle over $\mathcal O_Y$ with fibres of dimension $2n-1$. The corresponding subset in $Z_{(n)}$ is an affine space of dimension $n-1$, it is the open cell in an affine paving of $Z_{(n)}$.
We now wish to compute the fundamental group of the smooth loci $\tilde{\mathcal M}^{\text{sn}}_\lambda$. For this we make the following definition.
Let $\lambda$ be a partition of $n$. We may write $\lambda$ uniquely as $(i_1^{c_1},i_2^{c_2},\ldots, i_r^{c_r})$ where $1 \leq i_1<i_2< \ldots < i_r \leq n$. Let $k =
\ell(\lambda) = \sum_{i=1}^r c_i$. Let $\mathcal B_\lambda$ be the fundamental group of the configuration space of $k$ labelled points in $\mathbb C$ where the labels lie in $\{i_1,i_2,\ldots, i_r\}$ and there are $c_j$ points with label $i_j$. In our earlier notation this is just the space $S^n_\lambda(\mathbb C)$. Let $\mathcal P_\lambda$ be the subgroup of $\mathcal B_\lambda$ corresponding to the cover of the configuration space where all $k$ points are distinct, *i.e.* the pure braid group on $k$ strands, and let $\Sigma_\lambda$ be the quotient $\mathcal B_\lambda/\mathcal P_\lambda$ so that $\Sigma_\lambda \cong S_{a_1}\times S_{a_2} \times \ldots \times S_{a_k}$, where $S_a$ denotes the symmetric group on $a$ letters.
Thus for example if $\lambda = (1^n)$ this group is the standard braid group on $n$ strands. Note that $\mathcal B_\lambda$ is a subgroup of the braid group on $k$ strands. It contains the parabolic braid subgroup given by the partition associated to the composition $(c_1,c_2,\ldots,c_r)$, but is strictly bigger – for example it contains the isotopies which rotate points labeled $i$ and $j$ a full $2\pi$ about each other.
The fundamental group of $\tilde{\mathcal M}^{\text{sn}}_\lambda$ is closely related to $\mathcal B_\lambda$ as we now show. For $\mathbf a = (a_1,a_2,\ldots a_{r-1}) \in \mathbb C^{r-1}$, let $$J_\mathbf a(t) = \left(\begin{array}{cccccc}t & a_1 & a_2 & \ldots & a_{r-2} & a_{r-1} \\0 & t & a_1 & & & a_{r-2} \\0 & 0 & \ddots & \ddots & \ddots & \vdots \\0 & 0 & 0 & \ddots & a_1 & a_2 \\0 & 0 & 0 & 0 & t & a_1 \\0 & 0 & 0 & 0 & 0 & t\end{array}\right) \in \text{Mat}_r(\mathbb C).$$
Let $\mu = \{\mu_1,\ldots,\mu_k\} \in S_{\lambda}^n(\mathbb C)$ with labels (*i.e.* multiplicities) given by the parts of $\lambda$ written in increasing order: $d_1\leq d_2 \ldots \leq d_k$ so that $d_j \in \{i_1,i_2,\ldots,i_r\}$ for each $j$, ($1 \leq j\leq k$). Then if $\mathbf a^j \in \mathbb C^{d_j-1}$ for $1 \leq j \leq k$, set $X_\lambda(\mathbf a^1,\mathbf a^2,\ldots,\mathbf a^k)$ to be the block diagonal matrix: $$\left(\begin{array}{cccc}J_{\mathbf a^1}(\mu_1) & 0 & 0 & 0 \\0 & J_{\mathbf a^2}(\mu_2) & 0 & 0 \\0 & 0 & \ddots & 0 \\0 & 0 & 0 & J_{\mathbf a^k}(\mu_k)\end{array}\right)$$ and similarly set $Y_\lambda$ to be the block diagonal nilpotent matrix with Jordan blocks of size $d_1,d_2,\ldots,d_k$. Finally, set $v_\lambda \in \mathbb C^n$ to be the vector with $v_j = 1$ if $j \in \{\sum_{s=1}^t d_s: 1\leq t \leq k\}$ and $v_j=0$ otherwise. It is then easy to check that $(X_\lambda(\mu,\mathbf a^1,\ldots, \mathbf a^k), Y_\lambda,v_\lambda) \in \tilde{\mathcal M}^{\text{sn}}_{\lambda}$, and moreover if $(X_\lambda(\mu, \mathbf a^1,\ldots,\mathbf a^k),Y_\lambda,v_\lambda)$ and $(X_\lambda(\mu',\mathbf b^1,\ldots,\mathbf b^k),Y_\lambda, v_\lambda)$ are two such points then they lie in the same $\text{GL}(V)$-orbit if and only if they are equal up to permutation of the Jordan blocks of the $X$s – conjugate by an element $\Sigma_\lambda$ viewed as the subgroup of the group of permutation matrices which interchanges our diagonal blocks of the same size (respecting the order within each block). Thus if we set $\mathcal S_\lambda$ to be the subspace of $\tilde{\mathcal M}^{\text{sn}}_\lambda$ consisting of points of the form $(X(\mu,\mathbf a^1,\ldots, \mathbf a^k),Y_\lambda,v_\lambda)$, then we have a natural map $$p_\lambda \colon \text{GL}(V)\times \mathcal S_\lambda \to \tilde{\mathcal M}^{\text{sn}}_\lambda,$$ given by the $\text{GL}(V)$-action, and this map is readily seen to be a $\Sigma_\lambda$-covering, where $\Sigma_\lambda$ acts on $\text{GL}(V)$ on the right (embedded as a subgroup of permutation matrices) and on $\mathcal S_\lambda$ by the restriction of the $\text{GL}(V)$-action on ${\mathcal M^{\text{sn}}}$.
\[microlocalpi1\] Let $\lambda$ be a partition of $n$. Then we have exact sequences:
Moreover, $\mathcal P_\lambda$ is normal in $\pi_1(\tilde{\mathcal M}^{\text{sn}}_\lambda)$, the subgroup $\mathbb Z$ is central, and $\pi_1(\tilde{Z}_\lambda) \cong \mathcal B_\lambda$.
First notice that the variety $\tilde{Z}_{\lambda}$ is homotopy equivalent to $S_\lambda^n(\Sigma)$ since it is an affine space bundle over it, thus clearly $\pi_1(\tilde{Z}_\lambda) \cong \mathcal B_\lambda$. Now since the configuration space of $n$ distinct points in the complex plane is known to be a $K(\pi,1)$ (see for example [@D]), it follows that $S^n_{\lambda}(\Sigma)$ is a $K(\pi,1)$. The long exact sequence of a fibration then immediately shows that the fundamental group of $\tilde{\mathcal M}^{\text{sn}}_\lambda$ is an extension of $\mathcal B_\lambda$ by $\pi_1(\text{GL}(V))\cong \mathbb Z$. This yields the first of our exact sequences.
Now consider the covering $p_\lambda$. Since the total space is a product, the theory of covering spaces yields the second exact sequence. Moreover, the regular covering of $\tilde{\mathcal M}^{\text{sn}}_\lambda$ given by taking the universal cover of $\text{GL}(V)$ over the total space of $p_\lambda$ shows that $\mathcal P_\lambda$ is a normal subgroup of $\pi_1(\tilde{\mathcal M}^{\text{sn}}_\lambda)$. Thus since $\mathbb Z$ is clearly centralised by $\mathcal P_\lambda$ it suffices to show that $\mathbb Z$ is central in the quotient by $\mathcal P_\lambda$, which is isomorphic to the fundamental group of $\text{GL}(V)/\Sigma_\lambda$. But this is equivalent to showing that the action of $\Sigma_\lambda$ on $\pi_1(\text{GL}(V))$ is trivial which is immediate, because the action of $\Sigma_\lambda$ is the restriction of the action of the connected group $\text{GL}(V)$.
\[quotientofpi1\] Note that one can also describe the fundamental group of $\text{GL}(V)/\Sigma_\lambda$ explicitly, for example by considering it as a subgroup of the universal cover $\tilde{G}$ of $\text{GL}(V)$. Indeed $\tilde{G} = \{(g,t) \in \text{GL}(V)\times \mathbb C: \det(g) = \text{exp}(2\pi i t)\}$, and we may realise $$\pi_1(\text{GL}(V)/\Sigma_\lambda) = \{(w,n) \in \Sigma_\lambda \times (1/2)\mathbb Z: \text{sgn}(w) = \text{exp}(2\pi i n)\}$$ where $\text{sgn}$ is the usual sign function on $S_n$ restricted to $\Sigma_\lambda$. Here the subgroup $\mathbb Z\cong \pi_1(\text{GL}(V))$ is realized as $\{(1,n): n \in \mathbb Z\}$. For use in the next section, we also note here that, if we fix a generator $\gamma$ of $\pi_1(\text{GL}(V))$ and take any $q \in \mathbb C^\times$, say $q = \text{exp}(2\pi i \alpha)$, then $\pi_1(\tilde{\mathcal M}^{\text{sn}}_\lambda)$ has a one-dimensional representation $L_q$ on which $\gamma$ acts by $q$. Indeed the quotient $\pi_1(\text{GL}(V)/\Sigma_\lambda)$ obviously has such a representation, by taking the restriction of the representation $\Sigma_\lambda \times (1/2)\mathbb Z$ which sends $(w,n) \mapsto \text{exp}(2\pi i n\alpha)$. It is easy to check that in the case $\lambda = (1^n)$ the representations $L_q$ are given by the monodromy of the $\mathcal D$-module $s^c$, where $s$ is as in $\S$\[deformedHC\]. We will use this fact in comparing our microlocal $KZ$-functors to the original $KZ$ functor.
Microlocal KZ functors {#MKZfunctors}
======================
{#categoryC_c}
Let $\mathcal O_c^{\text{sph}}$ be the category of representations of the spherical rational Cherednik algebra corresponding to the category $\mathcal O_c$ of modules for the rational Cherednik algebra, in other word, $\mathcal O_c^{\text{sph}}$ is the category of $e\mathcal H_c e$-modules on which $\mathbb C[\mathfrak h^*]^W_+$, the augmentation ideal of $\mathbb C[\mathfrak h^*]^W$, acts locally nilpotently. In this section we use our analysis of the variety $\Lambda$ to study $\mathcal O_c^{\text{sph}}$.
It is well known that projective spaces are $\mathcal D$-affine. More precisely, if we assume[^5] that $c \notin \frac{1}{n}\mathbb Z_{<0}$ then the localization functor gives an equivalence between $\mathcal D_{\mathfrak X,c}$-modules and $\mathcal D_c(\mathfrak X)$-modules. Next recall the adjoint $\text{Loc}$ of the quantum Hamiltonian reduction functor, (in [@GG] the corresponding functor is denoted $^\top\mathbb H$). Let $\mathcal L_c$ be the $\mathcal D_{\mathfrak X,c}$-module whose global sections are: $$\mathcal D_c(\mathfrak X)/\mathcal D_c(\mathfrak X)\cdot \mathfrak g_c,$$ where $\mathfrak g_c$ is the image of $\mathfrak{sl}_n$ under the twisted quantized moment map $\tau_c$ associated to the action of $\text{PGL}(V)$ on $\mathfrak X$ via the isomorphism $\mathfrak{sl}_n \hookrightarrow \mathfrak{gl}_n \to \mathfrak{pgl}_n$. The twisted Harish-Chandra homomorphism $\Psi_c$ defined in Section \[deformed\] yields an isomorphism between $e\mathcal H_c e$ and $\text{End}(\mathcal L_c)^{\text{op}}$, so that given a $e\mathcal H_c e$-module $M$ we may associate to it a $\mathcal D_{\mathfrak X,c}$-module $$\text{Loc}(M) = \mathcal L_c \otimes_{e\mathcal H_c e} M.$$ This is the left adjoint to the functor $\mathbb H\colon \mathcal C_c \to \mathcal O_c^{\text{sph}}$ which sends $\mathcal F \mapsto \Gamma(\mathcal F)^{\text{SL}(V)}$, and it is known that the adjunction morphism $\mathbb H\circ \text{Loc}(M) \to M$ is an isomorphism for all $M \in \text{ob}(\mathcal O^{\text{sph}}_c)$. Recall that since $\text{SL}(V)$ commutes with the action of $\mathcal Z \subset \text{GL}(V)$ on $\mathfrak G^\circ$, we may consider the category of $\text{SL}(V)$-equivariant $D_{\mathfrak X,c}$-modules (see Appendix \[twistingstuff\] for details and references). In [@GG], Gan and Ginzburg establish the following result: let $\mathcal C_c$ be the full subcategory of the category of $(\mathcal D_{\mathfrak X,c}, \text{SL}(V))$-modules whose objects are have their characteristic cycle contained in $\Lambda$.
[@GG §7]. \[GGtheorem\] Suppose that $c \notin \frac{1}{n}\mathbb Z_{<0}$. Then the functor $\text{Loc}$ yields an equivalence of categories between $\mathcal O_c^{\text{sph}}$ and $\mathcal C_c/\text{ker}(\mathbb H)$.
Here $\text{ker}(\mathbb H)$ is the subcategory of objects in $\mathcal C_c$ which are annihilated by $\mathbb H$. Since $\text{SL}(V)$ is reductive, the functor of Hamiltonian reduction $\mathbb H$ is exact, so that $\text{ker}(\mathbb H)$ is a Serre subcategory. Kashiwara and Rouquier use the essentially same procedure, but then descend to the Hilbert scheme.
{#section-5}
We now wish to construct our microlocal analogues of the $KZ$-functors. For this we need the language of microdifferential operators. The most “algebraic” approach is to work with the ring $\widehat{\mathcal E}_X$ of formal microdifferential operators (see [@Kbook Chapter 7] where it is constructed using the calculus of total symbols) however there are a number of variants: the ring $\mathcal E_X$ of microdifferential operators with growth conditions, the sheaf $\mathcal E_X^{\infty}$ of infinite order operators, and the sheaf $\mathcal E_X^\mathbb R$, see for example [@K86] for a survey.
We briefly recall some basic properties (in the formal setting). Given $X$ a complex manifold, ${\widehat{\mathcal E}}_X$ is a $\mathbb Z$-filtered sheaf of rings on $T^*X$. If $\pi\colon T^*X \to X$ is the bundle map then ${\widehat{\mathcal E}}_X$ contains the sheaf $\pi^{-1}\mathcal D_X$ as a subring, and the quotients ${\widehat{\mathcal E}}_X(m)/{\widehat{\mathcal E}}_X(m-1) \cong \mathcal O_{T^*X}(m)$ where $\mathcal O_{T^*X}(m)$ denotes the sheaf of holomorphic functions on $T^*X$ homogeneous of degree $m$ along the fibres of $T^*X$. It is also known that ${\widehat{\mathcal E}}_X$ is flat over $\pi^{-1}\mathcal D_X$, and one can describe the characteristic cycle of a $\mathcal D_X$-module in terms of the associated ${\widehat{\mathcal E}}_X$-module: indeed for a coherent $\mathcal D_X$-module $\mathcal F$ we have: $$\label{EmodSS}
\text{Supp}({\widehat{\mathcal E}}_X\otimes_{\pi^{-1}\mathcal D_X} \pi^{-1}\mathcal F) = \text{SS}(\mathcal F),$$ where $\text{SS}$ denotes singular support (the multiplicities given by the characteristic cycle can also be recovered as discussed below). Note that when working with coherent $\mathcal D_X$-modules, we often consider a good filtration. In the context of ${\widehat{\mathcal E}}_X$-modules this corresponds to taking an ${\widehat{\mathcal E}}_X(0)$-lattice.
Suppose that $\mathcal F$ is a coherent $\mathcal D_X$-module, and $U$ is an open subset of $X$. It is well-known that if $\text{CC}(\mathcal F)_{|T^*U} \subseteq U$ then $\mathcal F_{|U}$ is in fact $\mathcal O_{U}$-coherent, and hence a vector bundle with connection, or local system, on $U$. The theory of regular holonomic ${\widehat{\mathcal E}}_X$-modules shows that this local constancy has a natural generalization to other Lagrangian submanifolds of $T^*X$. Let $\mathring{T}^*X$ denote the complement to the zero section in $T^*X$. If $\Omega$ is an open subset of $T^*X$, then we define a functor $\mu_\Omega$ from $\mathcal D_X$-modules to ${\widehat{\mathcal E}}_{X|\Omega}$-modules by setting $\mu_\Omega(\mathcal F) = ({\widehat{\mathcal E}}_X\otimes_{\pi^{-1}\mathcal D_X}\pi^{-1}(\mathcal F))_{|\Omega}$.
\[Morselocalsystem\] Let $\Omega$ be an open subset of ${\mathring{T}}^*X$ and let $\Lambda$ be a homogeneous smooth Lagrangian submanifold in $\Omega$ such that the projection from $\Lambda$ to $X$ has constant rank. Then there is an exact functor $\Phi_\Lambda$ from the category of holonomic ${\widehat{\mathcal E}}_X$-modules $\mathcal N$ with $\text{SS}(\mathcal N) \cap \Omega \subset \Lambda$ to the category of local systems on $\Lambda$.
This can be seen topologically or analytically. One approach works with the ring $\mathcal E_X^\mathbb R = \mathcal{H}^n(\mu_{\Delta}(\mathcal O_{X\times X}^{0,n)}))$, where $\mu_{\Delta}$ is the microlocalization functor, $n=\dim(X)$, and $\mathcal O_{X\times X}^{(0,n)}$ is the sheaf of holomorphic forms on $X\times X$ which are $n$-forms with respect to the second variable. This is a sheaf of rings which contains $\mathcal E_X$ and is faithfully flat over it.
Now by the assumption on the projection from $\Lambda$ to $X$, we may write $\Lambda = T^*_ZX\cap \Omega$ for some submanifold $Z$ of $X$. Then we have the simple holonomic system $\mathcal C_{Z|X} = \mathcal E_X\otimes_{\mathcal D_X} \mathcal B_{Z|X}$ along $\Lambda$. Now for any holonomic $\mathcal E_X$-module $\mathcal N$ by[^6] [@KK1 Theorem 1.3.1(i)] we have $\mathcal Ext^j_{\mathcal E_X}(\mathcal C_{Z|X},\mathcal N^\mathbb R)=0$ for $j \neq 0$ and moreover $\mathcal Hom_{\mathcal E_X}(\mathcal C_{Z|X},\mathcal N^\mathbb R)$ is a locally constant sheaf on $\Lambda$. Thus the map $$\mathcal M \mapsto \mathcal Hom_{\mathcal E_X}(\mathcal C_{Z|X}, \mathcal N^{\mathbb R}),$$ where $\mathcal N = (\mathcal E_X\otimes_{\mathcal D_X} \mathcal M)_{|\Omega}$, yields an exact functor to local systems.
Alternatively, via the Riemann-Hilbert correspondence, we may work topologically with the perverse sheaf $P$ determined by $\mathcal M$. In this situation the functor to local systems is the vanishing cycles functor, and its construction is described in [@MV1 §5]. A more detailed discussion, giving a number of different approaches in the topological setting, is given in [@GMV §4].
Note that [@KK1 Theorem 1.3.1(i)] also states that there is a canonical isomorphism: $$\mathcal C_{Z|X}^\mathbb R \otimes_\mathbb C \mathcal{H}om_{\mathcal E_X}(\mathcal C_{Z|X}, \mathcal N^\mathbb R) \to \mathcal N^\mathbb R.$$ where $\mathcal N^{\mathbb R}$ denotes the extension of $\mathcal N$ to $\mathcal E^{\mathbb R}_X$. If $\Lambda =\gamma^{-1}\gamma(\Lambda)$ where $\gamma \colon {\mathring{T}}X \to \mathbb P^*X$ is the map to the projectivised cotangent bundle, then by [@KK1 Theorem 1.3.1 (ii)] the sheaf $\mathcal M^\infty$ (the extension of $\mathcal M$ to the ring $\mathcal E^{\infty}_X$ of infinite order microdifferential operators) is completely determined by $\mathcal M^\mathbb R$, and if $\mathcal M$ has regular singularities, then by [@KK1 Theorem 5.2.1] it is determined by $\mathcal M^\infty$. Thus in this case there is in fact an equivalence between regular singularities $\mathcal E_{X|\Omega}$-modules supported on $\Lambda$ and local systems on $\Lambda$. In general however, regular singularities $\mathcal E_X$-modules supported on a smooth homogeneous Lagrangian are equivalent to twisted local systems on $\Lambda$ (see [@K86 §10] for more details).
Finally, using for example the explicit description of the functor $\Phi_\Lambda$ as homomorphisms from $\mathcal C_{Z|X}$ we see that if our holonomic modules are $(H,\lambda)$-equivariant for some group $H$, then $\Phi_\Lambda$ restricts to a functor taking values in the category of $(H,\lambda)$-twisted local systems on $\Lambda$.
The condition that $U$ is an open in $\mathring{T}^*X$ is not particularly restrictive: if we start with a ${\widehat{\mathcal E}}_X$-module $\mathcal F$ we may replace $X$ by $Z= X\times \mathbb C$ and $\mathcal F$ by $\mathcal F \boxtimes \delta_{0}$ where $\delta_0$ denotes the module generated by the $\delta$-function at $0 \in \mathbb C$. Since the conormal bundle of $X\times \{0\}$ in $Z$ is now $X \times T_0^*\mathbb C$ we have effectively moved the support of $\mathcal F$ off the zero section.
{#mKZdef}
By the discussion in $\S$\[DmodsonG\] we may view the category $\mathcal C_c$ of $\S$\[categoryC\_c\] as the abelian subcategory of $({\text{GL}}(V),c.\text{tr})$-equivariant module on $\mathfrak G^\circ$ whose singular support is contained in ${\mathcal M^{\text{sn}}}$ (the preimage of $\Lambda$). Given a component ${\mathcal M^{\text{sn}}}_\lambda$ of ${\mathcal M^{\text{sn}}}$ we may choose an open set $\Omega_\lambda$ in $T^*\mathfrak G^\circ$ such that $\Omega_\lambda\cap {\mathcal M^{\text{sn}}}= \tilde{\mathcal M}^{\text{sn}}_\lambda$. From our explicit description of $\tilde{\mathcal M}^{\text{sn}}_\lambda$ in §\[smoothpieces\] it immediately follows that the projection from $\tilde{\mathcal M}^{\text{sn}}_\lambda$ to $\mathfrak G^\circ$ has constant rank (indeed the fibre of the projection from ${\mathring{T}}\mathfrak G^\circ$ over a point $(X,v) \in \mathfrak G^\circ$ is just the space of nilpotent matrices which are regular in each generalized eigenspace of $X$, thus in fact the restriction of the projection is a fibre bundle with smooth fibres). Thus by composing the functor of Theorem \[Morselocalsystem\] with the functor $\mu_{\Omega_\lambda}$, we obtain an exact functor $$\mathcal{KZ}_\lambda \colon \mathcal C_c \to \mathcal{LS}(\tilde{\mathcal M}^{\text{sn}}_\lambda).$$ where $\mathcal{LS}(\tilde{\mathcal M}^{\text{sn}}_\lambda)$ denotes the category of local systems on $\tilde{\mathcal M}^{\text{sn}}_\lambda$. Note that the Lagrangian $\tilde{\mathcal M}^\text{sn}_{(1^n)}$ is a subset of $\mathfrak G^\circ$, so that in that case we only need the classical equivalence between vector bundles with flat connection and local systems.
By choosing a base point in $\tilde{\mathcal M}^{\text{sn}}_\lambda$ we may think of $\mathcal{KZ}_\lambda$ as a functor to the category $\text{Rep}(\pi_1(\tilde{\mathcal M}^{\text{sn}}_\lambda))$ of finite-dimensional representations of $\pi_1(\tilde{\mathcal M}^{\text{sn}}_\lambda)$. Moreover, since we are working with $(\text{GL}(V),c.\text{tr})$-equivariant modules $\mathcal F$, the local system $\mathcal{KZ}_\lambda(\mathcal F)$ must also be. Now by Proposition \[microlocalpi1\] we see that $\pi_1(\tilde{\mathcal M}^\text{sn}_\lambda)$ is an extension of $\mathcal B_\lambda$ by $\pi_1(\text{GL}(V)) \cong \mathbb Z$. The twisted equivariance shows that the subgroup $\mathbb Z = \pi_1(\text{GL}(V))$ must act by $q=\text{exp}(2\pi i nc)$. Thus it follows that the tensor product $L_{q^{-1}}\otimes \mathcal{KZ}_\lambda(\mathcal F)$ descends to a representation of $\mathcal B_\lambda$, where $L_{q^{-1}}$ is the one-dimensional representation constructed in Remark \[quotientofpi1\]. Let $KZ_\lambda(\mathcal F)$ denote this representation of $\mathcal B_\lambda$, an object of the category of finite-dimensional $\mathcal B_\lambda$-representations $\text{Rep}(\mathcal B_\lambda)$. By composing with the functor $\text{Loc}$ we also obtain a functor from $\mathcal O^{\text{sph}}_c$ to $\text{Rep}(\mathcal B_\lambda)$. Since it should be clear from context, we will abuse notation slightly and again write $KZ_\lambda$ for this composition of functors[^7].
Since the representation $L_{q^{-1}}$ is not canonically defined, it is perhaps more natural to have the functors $KZ_{\lambda}$ take values in the category of twisted local systems. However, in order to connect more directly with the original $KZ$ functor we prefer here to descend to ordinary local systems.
For use in the next section, we also wish to note the connection between our functors and the characteristic cycles for modules in $\mathcal O^\text{sph}_c$ defined by Gan-Ginzburg [@GG §7.5] and Gordon-Stafford [@GS §2.7] (which are shown to be equivalent in [@GGS §1.10]). If $M$ is a module in $\mathcal O^\text{sph}_c$, then the definition in [@GG] uses the intersection of $\text{CC}(\text{Loc}(M))$ with the stable locus $\text{U}$, and takes its quotient by ${\text{GL}}(V)$ to obtain Lagrangian cycle in $\text{Hilb}^n(\mathbb C^2)$. By equation (\[EmodSS\]), or more precisely by its refinement which counts multiplicities, this characteristic cycle is obtained by taking the dimension of our local systems $KZ_\lambda(M)$.
{#microsupportH}
We now relate our functors to the results in [@KR]. Recall[^8] that they construct a $\mathscr W$-algebra on $\text{Hilb}^n(\mathbb C^2)$ by “symplectic reduction” (see also \[Walgconstruction\]). Starting with the $\mathcal D$-module $\mathcal L_c$ of $\S$\[categoryC\_c\], they extend coefficients to obtain a module (also denoted $\mathcal L_c$) for $\mathscr W_{T^*\mathfrak G}$ the standard $\mathscr W$-algebra on $T^*\mathfrak G$, which is $({\text{GL}}(v),c.\text{tr})$-equivariant and is supported on $\mathcal M$. Since ${\text{GL}}(V)$ acts freely on $\text{U}$ the stable locus in $T^*\mathfrak G$, once we restrict to $\text{U}$, we may apply the reduction procedure given by Proposition 2.8 of [@KR] to obtain a $\mathscr W$-algebra, denoted $\mathscr A_c$, on $\mathcal M^s/G \cong \text{Hilb}^n(\mathbb C^2)$. Let $\tilde{\mathscr A}_c$ denote the $\mathscr W$-algebra obtained from $\mathscr A_c$ by extending the coefficients $\mathbb C((\hbar))$ by a square root of $\hbar$. Then there is a natural functor from $e\mathcal H_c e$-modules to ${\tilde{\mathscr A}_c}$-modules with an $F$-action given by: $$\text{Loc}_{{\tilde{\mathscr A}_c}}(M) = {\tilde{\mathscr A}_c}\otimes_{e\mathcal H_c e} M.$$ This uses the deformed Harish-Chandra homomorphism to construct a natural action of $e\mathcal H_c e$ on ${\tilde{\mathscr A}_c}$ (*i.e.* the action is induced by the $e\mathcal H_c e$-action on the $\mathcal D_{\mathfrak G}$-module $\mathcal L_c$).
By [@KR Proposition 2.8], the construction of the $\mathscr W$-algebra on $\text{Hilb}^n(\mathbb C^2)$ has the property that its modules are equivalent to $(G,c.\text{tr})$-equivariant $\mathscr W_{U}$-modules with an $F$-action that are supported on $\mathcal M^s$. Thus comparing the definitions, we see that given a module in category $\mathcal O_c$, the $\mathscr W_{\text{U}}$-module we obtain by pulling back $\text{Loc}_{{\tilde{\mathscr A}_c}}(M)$ to $\text{U}$ is exactly $\mathscr W_{\text{U}}\otimes_{{\widehat{\mathcal E}}_{\text{U}}}\mu_{\text{U}}(\text{Loc}(M))$. Now both $\text{Loc}$ and $\text{Loc}_{{\tilde{\mathscr A}_c}}$ have left adjoints $\mathbb H$ and $\mathbb H_{{\tilde{\mathscr A}_c}}$ given by taking twisted invariant global sections, and moreover, since by [@GG Theorem 1.5] and [@KR Lemma 4.7] there are filtration preserving isomorphisms $\text{End}(\mathcal L_c)^{\text{op}} \cong e\mathcal H_c e \cong \text{End}_{\text{Mod}^{\text{good}}_F({\tilde{\mathscr A}_c})}({\tilde{\mathscr A}_c})^{\text{op}}$ it follows we must have $\mathbb H_{{\tilde{\mathscr A}_c}}(\mathscr W_{\text{U}}\otimes_{{\widehat{\mathcal E}}_{\text{U}}}\text{Loc}(M)) = \mathbb H(\text{Loc}(M))$.
Now let $$\mathscr Y = \{a/b: 2\leq b \leq n, a<0, \text{ g.c.d.}(a,b)=1\}.$$ Then Theorem $4.9$ of [@KR] shows that if $c \notin \mathscr Y$ the functor $\mathbb H_{{\tilde{\mathscr A}_c}}$ gives an equivalence between the category of coherent good $\tilde{\mathscr A}_c$-modules with an $F$-structure and the category of finitely generated $e\mathcal H_c e$-modules, with quasi-inverse $\text{Loc}_{{\tilde{\mathscr A}_c}}$. But now if $\mathcal F$ is an object in $\mathcal C_c$, then the natural map $\mathcal F \to \text{Loc}(\mathbb H(\mathcal F))$ must be an isomorphism over $\text{U}$ (since $\mathscr W_X$ is faithfully flat over ${\widehat{\mathcal E}}_X$). Note that this implies that the functors $\mathcal{KZ}_\lambda$ which we have defined vanish on $\text{ker}(\mathbb H)$, and so descend to the quotient category $\mathcal C_c /\text{ker}(\mathbb H)$.
{#section-6}
The case $\lambda = (1^n)$ corresponds to the original $KZ$-functor of [@GGOR], as we now show. Let $\mathfrak G^{\text{rss}}$ be the set of pairs $(X,v)\in \mathfrak G$ such that $X$ is regular semisimple and $v$ is a cyclic vector for $X$. This is a smooth locally closed subvariety of $\mathfrak G^\circ$, and by the characteristic cycle bound, if $\mathcal F \in \mathcal C_c$ then, viewing $\mathcal F$ as before as a $(\text{GL}(V),c.\text{tr})$-equivariant $\mathcal D_{\mathfrak G^\circ}$-module, its restriction $\mathcal F_{|\mathfrak G^{\text{rss}}}$ must be a local system. As in Lemma \[microlocalpi1\] we see that that $\pi_1(\mathfrak G^{\text{rss}})$ is an extension of $\mathcal B_n$ by $\mathbb Z \cong \pi_1(\text{GL}(V)$, where $B_n$ is the braid group on $n$ strands, so that the functor $KZ_{(1^n)}$ given by descending the local system $\mathcal F_{|\mathfrak G^{\text{rss}}}$ to $\tilde{Z}_{(1^n)}$.
\[KZcheck\] The functors $KZ$ and $KZ_{(1^n)}$ are naturally isomorphic.
This follows immediately from the construction of the functors. Consider the diagram:
The construction of the $KZ$-functor uses the right-hand square in the diagram. Given a module $M$ in category $\mathcal O_0$, we may localize to ${\mathfrak h^{\text{reg}}}$, and via the Dunkl homomorphism view it as a $W$-equivariant $\mathcal D$-module on ${\mathfrak h^{\text{reg}}}$ (which we will again write as $M$). As such there is a $\mathcal D$-module $N$ which is a vector bundle with connection on ${\mathfrak h^{\text{reg}}}$ such that $M_{|{\mathfrak h^{\text{reg}}}} \cong p^*(N)$.
The global sections of $N$ become isomorphic to $eM_{|{\mathfrak h^{\text{reg}}}}$ as an $\mathcal D_{{\mathfrak h^{\text{reg}}}}^W$-module after conjugation by $\delta$: over ${\mathfrak h^{\text{reg}}}$ the map to $\mathfrak h/W$ is a Galois covering with group $W$, so locally $M$ is just $|W|$ copies of $\mathcal O_\mathfrak h$, and the sections of $N$ then correspond to $eM$, but the action of differential operators need to be conjugated by the Jacobian of the quotient map, which in this case is $\delta$ (c.f. [@HS] for a discussion of this in the context of push-forward of $\mathcal D$-modules – this is the reason we must twist by the action of $\delta$ in the definition of $\Psi_c$).
We now consider the other side of the diagram. The construction of the $\mathcal D$-module, $\mathcal M$ say, in $\mathcal C_c$ from $M$ proceeds via the isomorphism $\Psi_c$ from $e\mathcal H_c e$ to $(\mathcal D_c(\mathfrak X)/\mathcal D_c(\mathfrak X){\mathfrak g}_c)^{\text{SL}(V)}$, so that $$\mathcal M = \mathcal D_{\mathfrak X,c}\otimes_{\Psi_c} eM.$$ On $\mathfrak G^{\text{rss}}$ the map $\Psi_c$ discussed in section \[deformed\] gives an isomorphism between this algebra and $\mathcal D({\mathfrak h^{\text{reg}}})^W$, where the conjugation by $s^c$ used in the deformed Harish-Chandra homomorphism corresponds the tensor product by $L_{q^{-1}}$ in our construction of $KZ_{(1^n)}$. Then the above discussion shows that we have $j^*(\mathcal M) \cong \pi_\text{r}^*(N)$, and hence the monodromy measured by $KZ_{(1^n)}$ is exactly that measured by $KZ$.
A fundamental conjecture due to Kashiwara (a proof of which has recently been announced by Kashiwara and Vilonen [@KV]) states that a regular holonomic $\mathcal E_X$-module extends uniquely beyond an analytic subset of codimension greater or equal to three, which means that the structure of the stack of such $\mathcal E_X$-modules is captured in codimension two. Loosely, the “codimension zero” information is given by a collection of local systems, and the codimension $1$ information provides certain “glue” between them, with codimension two imposing additional constraints on objects. This implies that one can understand regular holonomic $\mathcal D_X$- modules whose singular support lies in a certain Lagrangian $\Lambda$ via the geometry of $\Lambda$ up to codimension two singularities, and in the topological context this idea has been studied in [@GMV]. One motivation for this paper came from a desire to get such a “microlocal” understanding of the category $\mathcal C_c$ (note that it is shown in [@GG §5.3] that the $\mathcal D$-modules in the category $\mathcal C_c$ are regular holonomic – indeed this follows by a standard argument from the finiteness of the number of orbits of $\text{PGL}(V)$ on $\mathcal N \times {\mathbb{P}}(V)$). Note that the topological case and its relation to $\mathcal E_X$-modules has also been studied in [@W1], [@W2].
Characteristic cycles of standard modules. {#CCcomputation}
==========================================
{#section-7}
Having defined microlocal KZ functors, we now attempt a first study of them. For this, as for the original KZ functor, we use the standard modules, whose construction we now recall. These modules are analogous to Verma modules in Lie theory: they are defined for any value of $c$, and are generically simple.
Note that $\mathbb C[\mathfrak h^*]\rtimes \mathbb C[W]$ is a subalgebra of $\mathcal H_c$, and it has a natural evaluation homomorphism $\text{ev} \colon \mathbb C[\mathfrak h^*]\rtimes \mathbb C[W] \to \mathbb C[W]$ given by $P\otimes w \mapsto P(0).w$. Thus pull-back via $\text{ev}$ allows us to lift modules for $\mathbb C[W]$ to modules for $\mathbb C[\mathfrak h^*]\rtimes \mathbb C[W]$. Given $\tau$ a representation of $W$ we define a standard module[^9] $M(\tau)$ by inducing the pull-back of $\tau$ to $\mathcal H_c$: $$M(\tau) = \mathcal H_C \otimes_{\mathbb C[\mathfrak h^*]\rtimes \mathbb C[W]} \text{ev}^*(\tau).$$ If $\tau$ is an irreducible representation indexed by a partition $\lambda$, then we write $M_\lambda$ for $M(\tau)$.
The standard modules have a unique simple quotient $L_\lambda$, and by the general machinery of highest weight categories one can also establish a Brauer-Bernstein-Gelfand type reciprocity formula relating the multiplicity of simple modules in standards to the multiplicity of standard modules in projective modules, see [@Gu]. It follows easily from this that $\mathcal O_c$ is semisimple if and only if the standard modules are simple. We wish to calculate the dimensions of the local systems given by the functors $\text{KZ}_\mu$ on these modules, or in other words (see $\S$\[mKZdef\]) we wish to calculate the multiplicities of the characteristic cycle of these modules on the corresponding component of $\Lambda$.
{#section-8}
We now wish to show that the characteristic cycle of a standard module is independent of the parameter $c$. To do this we use the “Harish-Chandra” module $F_{HC}$ defined by Gan and Ginzburg [@GG §7.4]. This is the $\mathcal D_c(\mathfrak X)$-module defined by $$F_{HC} = \mathcal D(\mathfrak X,c)/(\mathcal D(\mathfrak X,c)\mathfrak g_c + \mathcal D(\mathfrak X,c)\mathfrak Z_+),$$ where $\mathfrak Z_+$ denote augmentation ideal of $\mathcal Z$ the algebra of $\text{ad}({\mathfrak g})$-invariant constant coefficient differential operators on $\mathfrak g$. We first show that the characteristic cycle of $F_{HC}$ is independent of $c$.
The characteristic cycle of $F_{HC}$ is given by the ideal $I = \langle \sigma(x), \sigma(z): x \in {\mathfrak g}, z \in \mathcal Z_+\rangle$, where $\sigma$ denotes the symbol map, that is, the generators ${\mathfrak g}$ and $\mathcal Z_+$ are involutive. Thus $\text{CC}(F_{HC})$ is independent of $c$.
To see that our generators are involutive, we use the following criterion [@Kbook §2.2]: if $\{x_i: 1 \leq i \leq m\}$ are elements of $\mathcal D$ of order $m_i$ which satisfy the conditions that
1. $\bigcap_{i=1}^m \sigma(x_i)^{-1}(0)$ is of codimension $m$;
2. $[x_i,x_j] = \sum_{k} q_{ijk}x_k$, for some $q_{ijk} \in F_{m_i+m_j-m_k-1}$;
then the system of generators is involutive.
In our case, take $\{x_i\}$ to be a basis of ${\mathfrak g}$ (so of order $1$) and the standard generators of $\mathcal Z_+$ (of order $2,\ldots, n-1$). The second condition is then clearly satisfied, since the operators in $\mathfrak Z_+$ commute with each other and ${\mathfrak g}_c$. For the first condition, note that the associated graded generators define the set $$\{(X,Y,v,w) \in {\mathfrak g}\times {\mathfrak g}\times V \times V^*: \mu(X,Y,v,w) = 0, y \text{ nilpotent}\},$$ where $\mu$ is the moment map, that is $\mathcal M_{\text{nil}}$ (in fact with a nonreduced scheme structure), so the first condition is satisfied (thus the lemma reduces essentially to the flatness of the moment map). The independence of $c$ follows immediately.
Recall the definition of $\mathscr Y$ in $\S$\[microsupportH\]. In the following Lemma we use the fact that we may calculate the characteristic cycle of a $\mathcal D$-module either by taking a good filtration and calculating the multiplicity of the resulting $\mathcal O$-coherent sheaf on the components of its support, or by taking the multiplicities on the components of the support of the associated $\mathcal E$-module. See *e.g.* [@Kbook Chapter 7] for a discussion of this fact at the level of supports.
\[indofc\] Suppose that $c \notin \mathscr Y$. Let $M_\lambda$ be a standard module, and $\mu$ a partition of $n$. Then $\dim(KZ_\mu(M_\lambda))$ is independent of $c$.
Let $\text{co}(\mathfrak h^*)$ be the coinvariant module $\mathbb C[\mathfrak h^*]/\langle I \rangle$, where $I$ is the augmentation ideal in $\text{Sym}(\mathfrak h)^W$, viewed as a representation of $\mathbb C[\mathfrak h^*]\rtimes W$ in the obvious way, and let $C$ denote the $\mathcal H_c$ module induced from $\text{co}(\mathfrak h)$. It is shown in [@GG §7] that if $\langle I \rangle$ denotes the ideal of $e\mathcal H_c e$ generated by $\text{Sym}(\mathfrak h)^W_+$, then $eC \cong e\mathcal H_c e/\langle I \rangle$. Using this, they show that $\mathbb H(F_{HC}) = eC$. Now the discussion in $\S$\[microsupportH\] shows that if $c \notin \mathscr Y$ then the intersection of the characteristic cycle of $F_{HC}$ with $\text{U}$ is completely determined by $\mathbb H(F_{HC})$, and thus it coincides with that of $\text{Loc}(eC)$, (as $\mathbb H \circ \text{Loc}$ is naturally isomorphic to the identity). It follows that the characteristic cycle of $\text{Loc}(eC)$ intersected with $\text{U}$ is independent of $c$.
Now consider $R = \Delta(\mathbb C[W])$, the representation of $\mathcal H_c$ induced from the regular representation of $W$. Since it is clear that we may filter $\text{co}(\mathfrak h^*)$ in such a way that the associated graded module is isomorphic to $\mathbb C[W]$ as a $\mathbb C[\mathfrak h^*]\rtimes W$-module, and $e\mathcal H_c$ is a flat right $\mathbb C[\mathfrak h^*]\rtimes W$-module, it follows that $R$ and $C$ have the same class in the Grothendieck semigroup for $\mathcal O_c$, and hence the same characteristic cycle.
Since $R$ is clearly finitely generated as a $\mathbb C[\mathfrak h]$-module, and $\mathbb C[\mathfrak h]$ is finite over $\mathbb C[\mathfrak h]^W$, the module $\text{Loc}(eR) = \mathcal L_c\otimes_{e\mathcal H_c e} eR$ is finite over $\mathbb C[\mathfrak X]^G$. It follows that if $\Gamma_i$ is a good filtration on $\mathcal L_c$, the filtration $\Gamma_i\otimes eR$ is good for $\text{Loc}(eR)$. Thus we may use the order filtration on $\mathcal D_c(\mathfrak X)$ to obtain a good filtration on $\mathcal L_c$.
Now $\mathbb C[W]$ is of course a bimodule for $W$, hence it carries a right action of $W$ which commutes with the left action used in the construction of $R$. Thus we have a right $W$ action on $eR$ via $e\mathcal H_c e$-module automorphisms, and so the localization $\text{Loc}(eR)$ carries a $W$-action as a $\mathcal D_{\mathfrak X,c}$-module, which is compatible with our filtration. But now taking the isotypic components according to this action yields the standard modules, and hence their characteristic cycles are determined by this action on $\text{CC}(\text{Loc}(eR))=\text{CC}(\text{Loc}(eC))$, which must therefore be independent of $c$ as required.
{#section-9}
The previous lemma shows that, in order to calculate the part of the characteristic cycles of $\text{Loc}(eM_\lambda)$ lying in $\Lambda^s$ we may assume that $c=0$. It is known [@BEG] that the standard modules are simple whenever the finite Hecke algebra at $q=e^{2\pi i c}$ is semisimple. Thus it follows the standard modules are simple when $c=0$. We will write $\mathfrak M_\lambda$ for the $\mathcal D$-module corresponding to the standard module $eM_\lambda$ when $c=0$. We now wish to identify $\mathfrak M_\lambda$ more explicitly.
Recall the Grothendieck simultaneous resolution $\mu\colon \tilde{{\mathfrak g}} \to {\mathfrak g}$ where $$\tilde{{\mathfrak g}} = \{(x,F) \in {\mathfrak g}\times \mathcal B: F = (F_i)_{1 \leq i \leq n}, x(F_i) \subseteq F_i\}$$ and $\mu$ is the projection to the first factor. Over the subset ${\mathfrak g}^{rss}$ of regular semisimple elements of ${\mathfrak g}$ this is a $W$-covering. Moreover, it is know by [@HK §4] that the push-forward $F_{\mathfrak g}= \mu_*(\mathcal O_{\tilde{{\mathfrak g}}})$ is a semisimple holonomic $\mathcal D$-module which is the minimal extension of its restriction to ${\mathfrak g}^{rss}$, where it is just the local system corresponding to the principal $W$-covering. For $\lambda$ an irreducible representation of $W$, let $\mathfrak M_\lambda^{{\mathfrak g}}$ denote the simple $\mathcal D_{{\mathfrak g}}$-module corresponding to the minimal extension of the local system given by $\lambda$ on ${\mathfrak g}^{rss}$. Since we are assuming that $c=0$, the sheaf of rings $\mathcal D_{\mathfrak X,0}$ is just $\mathcal D_{\mathfrak X} = \mathcal D_{{\mathfrak g}}\boxtimes\mathcal D_{\mathbb P(V)}$, and we may pull-back $F_{\mathfrak g}$ to a $\mathcal D_{\mathfrak X}$-module $F = F_{\mathfrak g}\boxtimes \mathcal O_{\mathbb P(V)}$. It follows that $F$ is a direct sum of simple modules $\mathfrak M^{\mathfrak g}_\lambda\boxtimes \mathcal O_{\mathbb P(V)}$.
The sheaf $\mathfrak M_\lambda$ arising from the standard module $M_\lambda$ at $c=0$ is isomorphic to $\mathfrak M_\lambda^{\mathfrak g}\boxtimes \mathcal O_\lambda$.
We use the (original) $KZ$-functor. It is known by [@GGOR §6.2] that the $KZ$-functor applied to a standard modules $M_\lambda$ yields the corresponding cell module $S_\lambda$ for the finite Hecke algebra. Since we are assuming $c=0$ this in turn is just the corresponding irreducible representation of $W$. Thus using Proposition \[KZcheck\] it follows that $\mathfrak M_{\lambda|\mathfrak X^{rss}}$ is just the local system given by $\lambda$. Since $\mathfrak M_\lambda$ is simple, it follows that the sheaf $\mathfrak M_\lambda$ is just the minimal (or Goresky-MacPherson) extension of this local system to $\mathfrak X$. However, the sheaf $\mathfrak M_\lambda^{{\mathfrak g}}\boxtimes \mathcal O_{\mathbb P(V)}$ clearly restricts to give the same local system and is also simple, hence we see that $\mathfrak M_\lambda = \mathfrak M_\lambda^{\mathfrak g}$ as required.
{#section-10}
To compute the characteristic cycles of the standard modules, we will actually use a Fourier dual description of the variety $\Lambda$. Thus we need to use the following standard remark: let $X = E \times B$ be a trivial vector bundle over $B$ and let $\mathcal D^\text{mon}_X$ be the category of monodromic $\mathcal D$-modules on $X$, that is, those for which the action of the Euler vector field associated to $E$ is locally finite. Then it is known that the partial Fourier transform $\mathcal F_E$ along $E$ gives an equivalence between $\mathcal D^{\text{mon}}_{E\times B}$ and $\mathcal D^{\text{mon}}_{E^*\times B}$.
We may canonically identify $$T^*(E\times B) \cong T^*(E^*\times B)\cong E\times E^*\times T^*B.$$ Under this isomorphism, if $N$ is a monodromic $\mathcal D_X$-module we have $$\text{CC}(N) \cong \text{CC}(\mathcal F_E(N))$$
For a discussion of this see for example [@HK §3].
We apply this to $\mathfrak X = {\mathfrak g}\times \mathbb P(V)$ with $E= {\mathfrak g}$ and $B = \mathbb P(V)$. The isomorphism $T^*({\mathfrak g}\times \mathbb P(V)) \cong T^*({\mathfrak g}^* \times \mathbb P(V))$ is simply given by identifying both with ${\mathfrak g}\times {\mathfrak g}^* \times T^*\mathbb P(V)$ (or via the trace form ${\mathfrak g}\times {\mathfrak g}\times T^*\mathbb P(V)$). We want to understand the components of $\Lambda$ in this “Fourier dual” picture. This is essentially already done in [@GG] but note they use a different transform – the bundle maps we use are given by $$\begin{split}
p\colon T^*({\mathfrak g}\times \mathbb P(V)) \to {\mathfrak g}\times \mathbb P(V), \quad & (X,Y,i,j) \mapsto (X,i) \\
\check{p} \colon T^*({\mathfrak g}^* \times \mathbb P(V)) \to {\mathfrak g}^*\times \mathbb P(V),\quad & (X,Y,i,j) \mapsto (Y,i).
\end{split}$$ whereas in [@GG] they use $\check{p}(X,Y,i,j) = (Y,j)$. Nevertheless the same argument they use gives the following:
Let $\Lambda^*$ denote the image of $\Lambda$ under the canonical isomorphism in $T^*({\mathfrak g}^*\times \mathbb P(V))$. The $\Lambda^*$ is the union of the conormal bundle to the $\text{PGL}(V)$-orbits on $\mathcal N \times \mathbb P(V)$.
This follows from the fact that any conic Lagrangian in a cotangent bundle $T^*X$ is the closure of the conormal bundle of a smooth locally closed subvariety of $X$, and the fact that $\text{PGL}(V)$ acts with finitely many orbits on $\mathcal N \times \mathbb P(V)$.
Thus the $\mathcal D$-modules on ${\mathfrak g}^*\times \mathbb P(V)$ which we study are supported on $\mathcal N \times \mathbb P(V)$ and are smooth along $\text{PGL}(V)$-orbits. Note that since there are finitely many orbits, they give a Whitney stratification of $\mathcal N \times \mathbb P(V)$. For our purposes we need to check which $\text{PGL}(V)$-orbits correspond to the components in $\Lambda^s$.
\[Fourierdualofcomponents\] The component $\Lambda^s_\lambda$ ($\lambda \vdash n$) of $\Lambda^s$ corresponds to the $\text{PGL}(V)$-orbits $$\mathcal O_{\lambda}^{\text{cyc}} = \{(Y,\ell): Y \text{ has Jordan type } \lambda, \text{ and } {\mathfrak g}_Y.\ell = V\},$$ where ${\mathfrak g}_Y$ denotes the centralizer of $Y$ in ${\mathfrak g}$.
Recall from the last paragraph of Section \[decompositionofM\] that $\Lambda^s_\lambda$ corresponds to the component $\Lambda_{\lambda,\emptyset}$ of $\Lambda$, that is the closure of the conormal bundle $T_S^*(\mathfrak X)$ where $S$ is the stratum of pairs $(X,\ell)$ where $X$ has eigenvalues with multiplicity given by $\lambda$ and $\ell$ is cyclic for $X$. Then the proof of Lemma \[commuting\] shows that the set of nilpotent endomorphisms $Y$ which commute with $X$, have Jordan type $\lambda$, and for which the projection of $\ell$ to each generalized eigenspace of $X$ is cyclic for the restriction of $Y$ to that space, is open dense in $\check{p}(\Lambda_{\lambda,\emptyset})$. Since ${\mathfrak g}_Y$ contains $\mathbb C[X,Y]$ it follows ${\mathfrak g}_Y.\ell = V$ for such $Y$, and these clearly form a single $\text{PGL}(V)$ orbit.
Note that if $\mathcal O_\lambda$ denotes the orbit of nilpotent endomorphisms of Jordan type $\lambda$, then clearly $$\mathcal O_{\lambda}^{\text{cyc}} = \Lambda^s \cap \mathcal O_{\lambda}\times V.$$ where we have identified $\Lambda^s$ with a subset of $\Lambda^*$ using the isomorphism above.
{#section-11}
Since the characteristic cycle of $\mathfrak M_\lambda$ is determined by its deRham complex $\text{DR}(\mathfrak M_\lambda)$ we may work purely topologically.
Suppose that $c \notin \mathscr Y$. Let $\mathfrak M_{\lambda,c}$ be the $\mathcal D_{\mathfrak X,c}$-module associated to the standard module $M_\lambda$. Then we have $$\text{CC}(\mathfrak M_{\lambda|\mathfrak X^{\text{reg}}}) = \sum_{\mu \leq \lambda} K_{\lambda,\mu} [\Lambda_\mu],$$ where $\leq$ denotes the dominance order, and $K_{\lambda,\mu}$ are the Kostka numbers. In other words, we have $$\dim(KZ_\mu(eM_\lambda) = K_{\lambda,\mu}.$$
We have already seen that it is enough to compute the characteristic cycle in the case $c=0$, where we write $\mathfrak M_\lambda$ instead of $\mathfrak M_{\lambda,0}$. Taking the Fourier transform $\mathcal F_{\mathfrak g}(\mathfrak M_\lambda)$ we obtain a $\mathcal D$-module on $\mathcal N \times \mathbb P(V)$. From the preceeding discussion we know that $\mathfrak M_\lambda = \mathfrak M^{\mathfrak g}_\lambda \boxtimes \mathcal O_{\mathbb P(V)}$, and its Fourier transform is then just $\mathcal F_{{\mathfrak g}}(\mathfrak M^{{\mathfrak g}}_\lambda)\boxtimes \mathcal O_{\mathbb P(V)}$. Now by [@HK §5] the $\mathcal D$-module $\mathcal F_{\mathfrak g}(\mathfrak M^{\mathfrak g}_\lambda)$ corresponds to the intersection cohomology sheaf $\mathcal I_\lambda$ on $\mathcal O_\lambda$, so that we are reduced to calculating the characteristic cycles of intersection cohomology sheaves of nilpotent orbits. But these are known thanks to [@EM] which uses a torus symmetry trick and the knowledge of the stalk Euler characteristics of the complexes. Indeed [@EM Lemma 3.2] shows that for nilpotent orbits $\alpha,\beta \subset \mathcal N$ (in type $A$) the local Euler obstruction $c_{\alpha,\beta}$ is zero whenever $\alpha \neq \beta$ (and by definition it is $1$ when $\alpha=\beta$). Thus since our intersection cohomology sheaves $\mathcal I_\lambda$ are constructible with respect to the stratification of $\mathcal N$ it follows from the Index Theorem (see Theorem \[indextheorem\]) that $$CC(\mathcal I_{\lambda}) = \sum_{\mu} \chi_\mu(\mathcal I_\lambda)[T^*_{\mathcal O_\mu}\mathfrak g],$$ where $\chi_\mu(\mathcal I_\lambda)$ denotes the stalk Euler characteristic of $\mathcal I_\lambda$ along the orbit $\mathcal O_\mu$. But it is shown in [@L81 §2] that the dimensions of the stalk cohomologies of the intersection cohomology sheaves of nilpotent orbits in ${\text{GL}}(V)$ are given by Kostka polynomials, from which it follows immediately from this that the stalk Euler characteristics are given by the Kostka numbers $K_{\lambda, \mu}$, which completes the proof.
Note that the real advantage of the description of $\text{Loc}(eM_\lambda)_{|c=0}$ as $\mathfrak M_\lambda^{\mathfrak g}\boxtimes \mathcal O_{\mathbb P(V)}$ was that it makes the computation of the Fourier transform straightforward. For details on the index theorem which reduce the computation of characteristic cycles to that of Euler characteristics and the local Euler obstructions see the Appendix.
In [@GS §2.7] Gordon and Stafford defined a characteristic cycle for modules in category $\mathcal O^\text{sph}_c$ via their $\mathbb Z$-algebra construction, and computed them for standard modules. Subsequently, Ginzburg, Gordon and Stafford [@GGS] established the equivalence of their definition with the $\mathcal D$-module one given in [@GG], which implies the above result. Their calculation however depended on Haiman’s work on the Hilbert scheme, while the above calculation uses only standard tools from $\mathcal D$-modules and the elegant torus action trick of Evens-Mirkovic.
Factorization through Hecke algebras {#HeckeAlgebras}
====================================
{#nearbyvanishing}
We now wish to use $\mathcal D$-module techniques to show that the $KZ$-functor factors through a functor to representations of the finite Hecke algebra. Thus we wish to check that the generators of the braid group satisfy the quadratic relation $$(T_s -1 )(T_s + q) = 0.$$ Since this relation involves only a single generator, we first establish it in the rank one case. For this we need to recall the classification of regular singularities $\mathcal D$-modules on the complex line which are $\mathcal O$-coherent on $\mathbb C-\{0\}$. Let $\mathcal M$ be such a $\mathcal D$-module. Then by taking a $V$-filtration of $V^\bullet\mathcal M$ (see for example [@K83a]) one may define functors of nearby and vanishing cycles: $$\Psi(\mathcal M) = \text{gr}^V_0(\mathcal M), \quad \Phi(\mathcal M) = \text{gr}^V_{-1}(\mathcal M),$$ from $\mathcal D$-modules to finite-dimensional vector spaces. Moreover these functors are equipped with a “canonical”’ map $c\colon \Psi(\mathcal M) \to \Phi(\mathcal M)$ (induced by the action of $\partial$), and a “variation” map $v\colon \Phi(\mathcal M) \to \Psi(\mathcal M)$. These are related to each other by the equations: $$\label{canvarequations}
c\circ v = T - \text{Id}, \quad v \circ c = T - \text{Id}.$$ where $T$ is the monodromy operator $\text{exp}(-2i\pi x\partial)$. In fact it can be shown that our category of $\mathcal D$-modules is then equivalent to the category of pairs of vector spaces $(V,W)$ together with linear maps $c\colon V \to W$, and $v\colon W \to V$ such that $1$ is not an eigenvalue of either $cv$ or $vc$.
Suppose that $n=2$. The monodromy representations of the braid group $\mathcal B_2 = \mathbb Z$ given by $KZ_{(1^2)}$ factor through the Hecke algebra at $q=e^{2\pi i c}$.
By taking co-ordinate “centre of mass” coordinates $x,x',y,y'$ such that $W=S_2$ acts trivially on $x',y'$ and by the sign representation on $x,y$, we may reduce to the case of $T^*\mathbb P^1$. By the discussion in the last section of [@KR], the sheaf of rings on $\text{Hilb}^2(T^*\Sigma)$ localizing $e\mathcal H_c e$ when restricted to the exceptional divisor is just the twisted ring of differential operators $\mathcal D_{\mathbb P^1,\lambda}$ where $\lambda = c - \frac{1}{2}$. The characteristic cycle of a module $\mathcal M$ in $\mathcal C_c$ must then lie in $\mathbb P^1 \cup T^*_{0}\mathbb P^1$, and $\mathcal M$ becomes a module for the sheaf of rings $\mathcal D_{\mathbb P^1,\lambda}$. We claim that this implies the monodromy around $T^*_0\mathbb P^1 - \{0\}$, which is the monodromy given by $KZ_{(1^2)}$, satisfies the quadratic relation of the Hecke algebra.
In terms of the sheaf of rings $\mathcal D_{\mathbb P^1,\lambda}$, this monodromy is the action of $T$ on the vanishing cycles sheaf. Since the monodromy operator is given by $\text{exp}(2\pi i x \partial_x)$, the twisted equivariance forces $T$ to act as $e^{2\pi i \lambda} = -e^{2\pi i c} =-q$ on the nearby cycles sheaf. But now it follows that $v\circ c =-q-1$. By Equation \[canvarequations\] we see that this implies that on the vanishing cycles we have $$(T-\text{Id})^2 = (c\circ v)^2 = c\circ(v\circ c) \circ v = c\circ( -q-1)\circ v = -(q+1)(T-\text{Id}).$$ Rearranging this equation gives us the Hecke algebra quadratic equation as required.
{#section-12}
We now deduce the general case by reduction to the rank one case. Let $\mathfrak g_i$ denote the subset of ${\mathfrak g}$ consisting of matrices of the form $$\left(\begin{array}{ccccc} a_1 & 0 & \cdots & \cdots & 0 \\0 & \ddots & 0 & \cdots & 0 \\ \vdots & 0 & A & 0 & \vdots \\ \vdots & \vdots & 0 & a_{i+2} & 0 \\ 0 &\cdots & \cdots & 0 & \ddots \end{array}\right)$$ where the $a_j$ ($j \in \{1,\ldots,i-1,i+2,\ldots n\}$) are pairwise distinct numbers, $A$ is a $2\times 2$ matrix with entries in the $i$th and $(i+1)$th rows, and the eigenvalues of $A$ are distinct from the $a_j$s. (The set $\mathfrak g_1$ is also used to reduce to rank $1$ in [@KR §3].) It is easy to check that $\mathfrak g_i\times V$ is noncharacteristic for $\Lambda$, thus if $M$ is an object in $\mathcal C_c$, its pull-back to $\mathfrak g_i$ is holonomic with characteristic variety contained in the fibrewise projection of $\Lambda$ to $T^*(\mathfrak g_i\times V)$. More precisely, letting $i \colon \mathfrak g_i \times V^\circ \to \mathfrak G^\circ$ denote the inclusion map we have the diagram:
and $\text{Ch}(i^*(M)) \subset i_di_{\pi}^{-1}(\text{Ch}(M)$. Here $i_\pi$ is just the obvious inclusion map, and $i_d$ is the fibrewise projection map. Now it is immediate from the definitions that $i_di_\pi^{-1}(\Lambda)$ can be identified with $\Lambda_2 \times \mathbb C^{n-2}\times (\mathbb C^\times)^{n-2}$, where $\Lambda_2$ is the variety $\Lambda$ in the case $n=2$. Moreover this identification is equivariant for the action of the group $G_2 = T\times \text{SL}_2$ where $G_2$ is embedded in $\text{GL}(V)$ in the obvious way so as to be compatible with the map $i$. Let $$A(t) = \left(\begin{array}{cc} e^{2\pi i t} & 0 \\ 0 & e^{-2\pi i t}\end{array}\right)$$ Picking pairwise distinct complex numbers $(a_j)$ none of which lie in $\{\pm 1\}$, the action of the generator $T_i$ on $KZ_{(1^n)}(M)$ is given by monodromy around the curve $t \mapsto i(a_1,\ldots,a_{i-1},A(t),a_{i+2},\ldots, a_n)$. Thus we see that it is enough to check the quadratic relation holds for the case $n=2$, which has already been checked. Thus we have the following theorem
Let $M \in \mathcal C_c$. The monodromy representation given by $KZ_{(1^n)}(M)$ factors through the Hecke algebra $H_W(q)$ where $q = e^{2\pi i c}$.
This recovers, in type $A$, Theorem $5.13$ of [@GGOR] which is one of the central results of that paper. The proof given there is quite different – it checks the result directly on standard modules, and then deduces the general case by a deformation argument.
The result of this section applies only to the original $KZ$-functor. At the moment the author does not have an analogue of this result for the microlocal $KZ$-functors. One reason for this is that it is harder to give a presentation of the groups $\mathcal B_\lambda$. However since $\mathcal C_c$ is Artinian it follows formally that the action of $\mathcal B_\lambda$ will factor through a finite-dimensional algebra. Note that Rouquier [@R] has shown that the category $\mathcal O_c$ is equivalent to the category of representations of the $q$-Schur algebra. Since the $q$-Schur algebra is a quotient of the quantum group $U_v(\mathfrak{gl}_n)$, its representations carry Lusztig’s braid group action, and the partitions $\lambda$ can be interpreted as weights of such representations. The subgroup of Lusztig’s braid group acting on the weight space corresponds to $\mathcal B_\lambda$ and we conjecture that given $M$ a representation in $\mathcal O_c$, the representation $KZ_\lambda$ is given by the $\lambda$-weight space of the corresponding representation of the $q$-Schur algebra equipped with this braid group action.
On the minimal resolution of $\mathbb C^2/\mu_l$ as a hypertoric variety {#cyclic}
========================================================================
{#section-13}
The ideas of this paper should have applications to a larger class of rational Cherednik algebras than that which is attached to the symmetric groups. To illustrate this, in this section we consider the case of the rational Cherednik algebra $\mathcal H_\kappa$ attached to a cyclic group $\mu_l$ of order $l$, which we regard as a complex reflection group (with the obvious one-dimensional representation). We use quantum Hamiltonian reduction on a moduli space of representations for the cyclic quiver as in [@Ku] and [@BK], which give a deformation of the minimal resolution of a Kleinian singularity. Each components of the exceptional divisor of the resolution yields an exact functors on category $\mathcal O$ for $\mathcal H_\kappa$, and moreover, taken together, they can be used to construct a functor from $\mathcal O_c$ to the category of representations of the “cyclotomic $q$-Schur algebra” (*i.e.* the quasi-hereditary cover of [@DJM §6]) of the corresponding cyclotomic Hecke algebra. This gives a geometric perspective on the the ideas of [@GGOR] and [@R]. Indeed it is shown in [@GGOR Theorem 5.16] that category $\mathcal O_c$ can be viewed as a quasi-hereditary cover of the category of representations of the Hecke algebra (attached to any complex reflection group) while our construction yields an explicit functor from category $\mathcal O_c$ to the cyclotomic $q$-Schur algebra which is known to be a quasi-hereditary cover of the Hecke algebra.
We begin by recalling a construction of the algebra $\mathcal H_\kappa$. Let $\mu_l$ denote the cyclic group of order $l$ (thought of as the $l$-th roots of unity in $\mathbb C$, *i.e.* as a complex reflection group with a single “hyperplane” $\{0\}$). The action of $\mu_l$ on $\mathbb C$ induces an action on $\mathcal D(\mathbb C^\times)$, so we may consider the smash product $\mathcal D(\mathbb C^\times)\rtimes \mathbb C[\mu_l]$ of $\mathcal D(\mathbb C^\times)$ with the group algebra of $\mu_l$. To clarify the notation we fix a generator $\gamma$ of $\mu_l$ and write elements of $\mathbb C[\mu_l]$ in the form $\sum_{i=0}^{l-1} a_i\gamma^i$ where $a_i \in \mathbb C$ ($0\leq i \leq l-1$). Then if $\kappa = (\kappa_0,\ldots,\kappa_{l-1}) \in \mathbb C^l$ we define a Dunkl operator by the formula $$\partial_{\kappa} = \frac{d}{dz} + \frac{l}{z}\sum_{i=0}^{l-1} \kappa_i e_i,$$ where the $e_i$ denote the idempotents of $\mathbb C[\mu_l]$, that is $e_i = (1/l)\sum_{j=0}^{l-1} \zeta^{ij}\gamma^j$, for $i=0,2,\ldots,l-1$. The algebra $\mathcal H_\kappa$ is then the subalgebra of $\mathcal D(\mathbb C^\times)\rtimes\mathbb C[\mu_l]$ generated by $z, \partial_\kappa$ and $\gamma$.
{#section-14}
We now review the description of the minimal resolution $X$ of $\mathbb C^2/\mu_l$ as a hypertoric variety. Let $Q$ be the cyclic quiver with $l$ vertices $I = \{0,1,\ldots,l-1\}$ (read as elements of $\mathbb Z/l\mathbb Z$), and edges $H = \{h_i: 1\leq i \leq l\}$, where $h_i$ is the edge $(i-1) \to i$. Let $\bar{Q}$ denote the quiver $(I,E)$ where $E = H \cup \bar{H}$ where $\bar{H} = \{\bar{h}_i : 1\leq i \leq l\}$ and $\bar{h}_i$ denotes the edge from $i \to (i-1)$, *i.e.* the edge in the opposite direction to $h_i$. We will denote the initial and terminal vertex of an edge $h \in E$ by $s(h)$ and $t(h)$ respectively.
A representation of $\bar{Q}$ is an $I$-graded vector space $V= \bigoplus_{i=0}^{l-1} V_i$ together with an element of $E_{V,\bar{H}} = \bigoplus_{h \in \bar{H}} \text{Hom}(V_{s(h)}, V_{t(h)})$, so that the moduli space of representations of $\bar{Q}$ (of a fixed graded dimension) may be identified with the quotient of $E_{V,\bar{H}}$ by the action of $G_V = \prod_{i \in I} \text{GL}(V_i)/\mathbb G_m$ where $\mathbb G_m$ is the diagonal copy of scalar operators (which acts trivially on $E_{V,\bar{H}}$). Similarly, we have the moduli space of representations of $Q$ given by the quotient of $G_V$ acting on $E_{V,H}$.
{#section-15}
We will be interested in the case where $V$ has graded dimension $\delta = (1,1,\ldots,1)$, and thus we assume $\dim(V_i)=1$ in what follows. We will also write $G = G_V \cong (\mathbb G_m)^l/D$ (where $D$ is the diagonal $\mathbb G_m$), a rank $l-1$ torus, with Lie algebra $\mathfrak g = \mathbb C^l/\mathbb C$. Note that character lattice $X(G)$ of $G$ may be viewed as the set of ${\theta}= ({\theta}_0,{\theta}_2,\ldots,{\theta}_{l-1}) \in \mathbb Z^l$ such that $\sum_{i=0}^{l-1}{\theta}_i =0$.
Given ${\theta}\in X(G)$, we say a representation $(V,x)$ of $\bar{Q}$ is ${\theta}$-*semistable* if for any subrepresentation $W$ of $V$ we have $\sum_{i=0}^{l-1} \theta_i\dim(W_i) \leq 0$. We set $\tilde{X}_{\theta}$ to be the subset of $E_{V,\bar{H}}$ consisting of ${\theta}$-semistable representations, a $G$-invariant subset of $E_V$.
Now $E_V \cong T^*E_{V,H}$ since we may identify $\text{Hom}(V_{i-1},V_{i})$ with the dual of $\text{Hom}(V_i, V_{i-1})$ via the trace pairing $(x,y) \mapsto \text{tr}_{|V_i}(yx)$. Therefore $E_{V,\bar{H}}$ is a symplectic vector space, and the action of $G$ clearly preserves the symplectic form. We will write $(a_i,b_i)_{1\leq i\leq l}$ for a point in $E_{V, \bar{H}}$ where $a_i \in \text{Hom}(V_{i-1},V_i)$ and $b_i\in \text{Hom}(V_{i},V_{i-1})$. Since $\tilde{X}_{\theta}$ is clearly a $G$-invariant open subset of $E_{V,\bar{H}}$, it is a complex symplectic manifold acted on by $G$. The associated moment map $\mu\colon \tilde{X}_{\theta}\to \mathfrak g$ is given by: $$(a_i,b_i)_{1 \leq i \leq l} \mapsto (a_{i+1}b_{i+1} - a_ib_i)_{1\leq i \leq l}.$$
Let $\mu_l$ act on $\mathbb C^2$ via the action $\gamma(x,y) = (\gamma x,\gamma^{-1}y)$ for $\gamma \in \mu_l$, and write $\mathbb C^2/\mu_l$ for the quotient, a rational surface singularity of type $A$. Given $(a_i, b_i)_{1 \leq i\leq l} \in E_{V,\bar{H}}$, a pair $(\bar{a},\bar{b}) \in \mathbb C^2$ satisfying the conditions $\bar{a}^l = a_1a_2\ldots a_l$ and $\bar{b}^l = b_1b_2\ldots b_l$ and $\bar{a}\bar{b} = a_1b_1$ is well defined up to the action of $\mu_l$. This yields a map from $E_{V,\bar{H}}$ to $\mathbb C^2/\mu_l$ which induces an isomorphism of varieties between $E_{V,\bar{H}}//G $ and $\mathbb C^2/\mu_l$.
The G.I.T. quotient of $\tilde{X}_\theta$ by the action of $G$ is denoted $X_\theta$ (thus in particular $X_0 = E_{V,\bar{H}}//G$). We write $[a_i,b_i]_{1 \leq i\leq l}$ for the equivalence class in $X_{\theta}$ of a point $(a_i,b_i)_{1 \leq i \leq l}$ in $\tilde{X}_{\theta}$. When $(\theta_i)$ satisfies $\sum_{k=i}^{j-1} \theta_k \neq 0$ for all $i,j \in \mathbb Z/l\mathbb Z$ where $i \neq j$, this quotient is a smooth surface, and the map $\pi$ to the affine quotient $E_{V,\bar{H}}//G$ is a resolution of singularities, in fact the minimal resolution. From now on we will assume this condition on $\theta$ holds.
{#section-16}
The varieties $X_0$ and $X_{\theta}$ are both toric, where the action of $T = \mathbb G_m^2$ is induced by the action of $T$ on $E_{V,\bar{H}}$ given by $z.(a_i,b_i)_{1 \leq i \leq l} = (z_1a_i,z_2b_i)_{1 \leq i \leq l}$ where $z = (z_1,z_2) \in T$. To describe the structure of $X_\theta$ as a toric variety, we need to introduce an ordering on the elements of $\mathbb Z/l\mathbb Z$ induced by the stability $\theta$.
We say that $i \triangleright j$ if $\theta_i + {\theta}_{i+1} + \ldots {\theta}_{j-1} < 0$. By our condition on ${\theta}$ this is a total order, and we write $\eta_i$ for the permutation of $\{1,2,\ldots, l\}$ which this yields, i.e. $\eta_1 \triangleright \eta_2 \triangleright \ldots \triangleright \eta_l$.
As a toric variety $X_{\theta}$ can be written naturally as a union $X_{\theta}= \bigcup_{i=1}^l X_i$ where $$X_i = \{[a_j,b_j]_{1 \leq j \leq l} : a_{\eta_j} \neq 0 \text{ for } j<i; b_{\eta_j} \neq 0 \text{ for } j>i\}$$ and each $X_i$ is an affine toric variety containing a unique $T$-fixed point $p_i$ with $a_{\eta_i} = b_{\eta_i} = 0$. In fact $X_i \cong T^*\mathbb C$, which we may see explicitly as follows: if we pick a basis for the lines $(V_i)_{0 \leq i \leq l-1}$ then $E_{V,\bar{H}}$ may be identified with $\mathbb C^{2l}$ where we will still write a point of $\tilde{X}_{\theta}$ as $(a_i,b_i)_{1 \leq i \leq l}$. Let $\bar{x}_i((a_j,b_j)_{1 \leq j \leq l}) = a_i$ and $\bar{y}_i((a_j,b_j)_{1 \leq j \leq l}) = b_i$ be the obvious coordinate functions. Then $X_i = \text{Spec}(R_i)$ where $R_i$ is the subalgebra of function field $\mathbb C(\bar{x}_i,\bar{y}_i)_{1\leq i \leq l}$ generated by the algebraically independent elements $$\bar{f}_i = \frac{\bar{x}_{\eta_1} \bar{x}_{\eta_2} \ldots \bar{x}_{\eta_i}}{\bar{y}_{\eta_{i+1}}\bar{y}_{\eta_{i+2}} \ldots \bar{y}_{\eta_l}}, \quad \bar{g}_i = \frac{\bar{y}_{\eta_i}\bar{y}_{\eta_{i+1}} \ldots \bar{y}_{\eta_l}}{\bar{x}_{\eta_1} \bar{x}_{\eta_2} \ldots \bar{x}_{\eta_{i-1}}}.$$ Note that $\bar{f}_i\bar{g}_i = \bar{x}_{\eta_i}\bar{y}_{\eta_i}$. The subvarieties $X_i$ and $X_{i+1}$ intersect in a complex line whose closure in $X_{\theta}$ we denote by $D_i$. For $1 \leq i <l$ this closure is a $\mathbb P^1$ and their union is the exceptional divisor of the resolution $\pi$. If we let $\mathcal L$ denote the preimage of $\pi^{-1}(\{(\bar{a},\bar{b}) \in X_0: \bar{a}\bar{b} = 0\})$ then $\mathcal L$ is the union of $l+1$ irreducible components $D_i$, ($0 \leq i \leq l$), where $D_0 \cong D_l \cong \mathbb C$, and the remaining $D_i$ are the above $\mathbb P^1$s. These $D_i$ are all the $T$-stable divisors in $X_{\theta}$.
{#Walgconstruction}
We now recall the construction of $\mathcal W$-algebras with an $F$-action on $X_{\theta}$ following [@KR]. Let $c = (c_i)_{0 \leq i \leq l-1}$ be an $l$-tuple of complex numbers such that $\sum_{i=0}^{l-1}c_i= 0$. The $\mathscr W$-algebra $\mathscr A_c$ is obtained by quantum Hamiltonian reduction from the standard $\mathscr W$-algebra on $\mathbb C^{2l} = T^*\mathbb C^n$, whose generators we denote by $x_i,y_i$ ($1 \leq i \leq l$). Clearly $\tilde{X}_{\theta}$ carries the sheaf of algebras $\mathscr W_{\tilde{X}_{\theta}}$, the restriction of the sheaf $\mathscr W_{T^*\mathbb C^n}$, and we may define the $\mathscr W_{\tilde{X}_{\theta}}$-module $\mathscr L_c$ by $$\mathscr L_c = \mathscr W_{\tilde{X}_{\theta}}/\big(\sum_{i=1}^l \mathscr W_{\tilde{X}_{\theta}}(x_{i+1}y_{i+1} - x_iy_i + \hbar c_i)\big).$$
If $p \colon \mu^{-1}(0)\to X_{\theta}$ denotes the quotient map, then $\mathscr A_c = (p_*(\mathcal{E}nd_{\mathscr W_{\tilde{X}_{\theta}}}(\mathscr L_c))^G)^{\text{op}}$ is a $\mathscr W$-algebra on $X_{\theta}$. The sheaf of algebras ${\tilde{\mathscr A}_c}$ obtained by adjoining a square root of $\hbar$ has an $F$-action in the sense of [@KR 2.3] of weight one for the $\mathbb C^\times$ action giving $x_i$ and $y_i$ weight one and $\hbar$ weight two. We then consider the category $\text{Mod}_F^{\text{good}}(\tilde{\mathscr A}_c)$ of good $\tilde{\mathscr A}_c$-modules with an $F$-action as in [@KR 2.4]. It is known by the work of Holland [@H] and Bellamy-Kuwabara [@BK §6] that the algebra $A_c = \text{End}_{\text{Mod}_F^{\text{good}}(\tilde{\mathscr A}_c)}(\tilde{\mathscr A}_c)^{\text{opp}}$ is isomorphic to the spherical rational Cherednik algebra $U_\kappa = e_0H_{\kappa}e_0$ (where $e_0$ is the idempotent corresponding to the trivial representation of $\mu_\ell$) and moreover [@BK] proves a localization theorem if $i\triangleright j$ whenever $c_i+c_{i+1} +\ldots c_{j-1} \in \mathbb Z_{\geq 0}$ (though we do not need this for our constructions, which only use the existence of the localization functor). Here the deformation parameter $c = c(\kappa)$ is related to the parameter $\kappa$ via $$c_i = \kappa_{i+1} - \kappa_i +1/l -\delta_{i,0}.$$
{#section-17}
Let $\mathcal O_\kappa$ denote the category of representations of the spherical rational Cherednik algebra on which $\partial_\kappa$ acts locally nilpotently, and let $\mathcal O_\kappa^{\text{sph}}$ denote the corresponding category of representations for $U_\kappa$. Via localization, this category corresponds to a category of modules for ${\tilde{\mathscr A}_c}$, which we will denote by $\mathcal O_c$. It follows from results of [@Ku §5] that the modules in $\mathcal O_c$ are all supported on the union of divisors $\bigcup_{i=1}^l D_i$ (since the results there give a description of the supports of all simple modules in $\mathcal O_c$).
The cyclotomic Hecke algebra $\mathcal K_l$ for $G(1,1,l)$ is just the commutative algebra $\mathbb C[T]$ (where $T$ is the element conventionally denoted by $T_0$ in the literature) subject to the relation $$\prod_{i=1}^m(T - q^{a_i})=0$$ where $(a_i)_{i=1}^l \in \mathbb Z^l$, and $q \in \mathbb C^\times$.
To any cyclotomic Hecke algebra one can attach a cyclotomic $q$-Schur algebra [@DJM]. We now recall how it is defined (in our special case). For $i \in \{1,2,\ldots, l\}$ let $$m_i = \prod_{i<j\leq l} (T - q^{a_j}).$$ and let $M^i$ be the $\mathcal K_l$-ideal $\mathcal K_l m_i$. Note that if we let $r_i = \prod_{j=1}^i (T-q^{a_j})$ then $M^i \cong \mathbb C[T]/(r_i)$ via the map $f \mapsto f.m_i$. The cyclotomic $q$-Schur algebra is defined to be $$\mathcal S_l = \text{End}_{\mathcal K_l}(\bigoplus_{i=1}^l M^i) = \bigoplus_{i.j \in [1,l]} \text{Hom}_{\mathcal K_l}(M^i,M^j).$$ The following simple lemma yields an explicit description of this algebra, by noting that any $\mathcal K_l$-module can be thought of as a $\mathbb C[t]$-module where $t$ is an indeterminate via the surjection $\mathbb C[t] \to \mathcal K_l$ given by $t \mapsto T$.
Let $t$ be an indeterminate, and let $g_1$ and $g_2 \in \mathbb C[t]$. Then we have $$\text{Hom}_{\mathbb C[t]}(\mathbb C[t]/(g_1),\mathbb C[t]/(g_2)) \cong \mathbb C[t]/(g),$$ where $g = \text{g.c.d.}(g_1,g_2)$.
If $\phi \in \text{Hom}_{\mathbb C[t]}(\mathbb C[t]/(g_1),\mathbb C[t]/(g_2))$ then $\phi(1+(g_1)) = u \in \mathbb C[t]/(g_2)$, where $g_1.u =0 \in \mathbb C[t]/(g_2)$. The submodule of $\mathbb C[t]/(g_2)$ consisting of such elements is isomorphic as a $\mathbb C[t]$-module to $\mathbb C[t]/(\text{g.c.d.}(g_1,g_2))$, and the map $\phi \mapsto u$ gives the required isomorphism.
Let $R_i = \text{End}_{\mathcal K_l}(M^i)\cong \mathbb C[t]/(r_i)$ (by, for example, the previous Lemma). If we write $\mathcal S_l^{ij} = \text{Hom}_{\mathcal K_l}(M^j, M^i)$ then the Lemma shows that for $i\leq j$ we may view $\text{Hom}(M^j,M^i)$ as a free $\mathbb C[t]/(r_i)$-module generated by the quotient map $\pi_{ij}\colon \mathbb C[t]/(r_j) \to \mathbb C[t]/(r_i)$ whereas if $i \geq j$ we may view $\mathcal S_l^{ji}$ as a free $\mathbb C[t]/(r_j)$-module with generator $m_{ji} = \prod_{i<k\leq j}(t-q^{a_k}) \in \mathbb C[t]/(r_i)$, corresponding to the inclusion $\iota_{ij} \colon \mathcal K_lm_j \hookrightarrow \mathcal K_l m_i$.
We will thus write an element of $\mathcal S_l$ as $(a_{ij})_{i,j \in [1,l]}$ where $a_{ij} \in \mathcal S_l^{ij}$ is as an element of a free $R_{\text{min}(i,j)}$-module with generator $\pi_{ji}$ or $\iota_{ji}$ according to whether $i\leq j$ or $i\geq j$ (where we have $\pi_{i,i} = \iota_{i,i} = \text{id}_{M^i}$). If the $q^{a_i}$s are all distinct, it is straightforward to check that this algebra splits into a direct sum of $l$ matrix algebras of dimensions $1,2,\ldots,l$.
\[Zlstandards\] The algebra $\mathcal S_l$ has a natural family of modules known as *standard* or *Weyl* modules. For each $i,j$ the algebra $R_i$ has an ideal $I_j^i$ generated by the element $$s_j^i = \prod_{1 \leq k \leq i; k \neq j}(T-q^{a_k}).$$ If $j\leq i$ this is a one-dimensional $\mathbb C$-vector space, isomorphic to $\mathbb C[T]/(T-q^{a_j})$ as an $R_i$ module, otherwise it is zero. For each $j$, ($1 \leq j \leq l$) let $W_j = \bigoplus_{k=1}^l I_j^k = \bigoplus_{k =j}^l I_j^k$. We may give $W_j$ the structure of a $\mathcal S_l$-module as follows: if $a = (a_{ij})_{1 \leq i,j \leq l} \in \mathcal S_l$ and $n = (n_k)_{j \leq k \leq l}$ where $a_{ij} \in \mathcal S_l^{ij}$, and $n_k \in I_j^k$ then $$a.n = (\sum_{r} a_{kr}n_r)_{j \leq k \leq l},$$ where $a_{kr}n_r$ denotes the image of $n_r$ under the chain of morphisms:
It is easy to see that under this composition, $I_j^r$ maps to $I_j^k$ if $j \leq k$ so that the formula does indeed give an $\mathcal S_l$-module structure. Note that in the situation where the $q^{a_k}$ are all distinct, the standard modules are clearly precisely the simple modules.
If $n=2$ then the algebra $\mathcal S_l$ can be viewed as the set of matrices of the form $$\left(\begin{array}{cc}a_1 & a_2\pi \\a_3\varepsilon & b \end{array}\right)$$ where we write $\pi$ for $\pi_{21}$ and $\varepsilon = m_{21}$, and the $a_i$ are in $R_1= \mathbb C$, and $b \in R_2$.
If $q^{a_1} = q^{a_2}$ then setting $\varepsilon = T-q^{a_1}$ we have $R_2 = \mathbb C[\varepsilon]$ where $\varepsilon^2=0$. Using the basis $\{1,\varepsilon\}$ for $R_2$ the multiplication in $\mathcal S_2$ then becomes: $$\left(\begin{array}{cc}a_1 & a_2\pi \\a_3\varepsilon & b_1 + b_2\varepsilon \end{array}\right) \left(\begin{array}{cc}a'_1 & a'_2\pi\\a'_3\varepsilon & b'_1 + b'_2\varepsilon \end{array}\right) = \left(\begin{array}{cc}a_1a'_1 & (a_1a'_2 + a_2b'_1)\pi \\(a_3a'_1 + b_1a'_3)\varepsilon & b_1b'_1 + \varepsilon(a_3a'_2b_1b'_2 + b'_2b_1) \end{array}\right).$$ It is then easy to see that the subspace of elements of the form $\left(\begin{array}{cc}0 & 0\\a_3\varepsilon & b_2\varepsilon \end{array}\right)$ forms a two-sided ideal, with the quotient algebra isomorphic to the algebra of $2$-by-$2$ upper-triangular matrices.
Let $\rho_i \in \mathcal S_l$ denote the projection $\rho_i \colon \bigoplus_{j=1}^l M^j \to M^i$. We now wish to find a presentation for $\mathcal S_l$.
Let $(a_i)_{i=1}^l \in \mathbb C^l$ and let $\mathcal S'_l$ be the algebra given by generators $T,\rho_i,\pi_{i,i+1},\iota_{i+1,i}$ subject to relations:
1. $\rho_i\rho_j = \delta_{ij}\rho_i, T\rho_i = \rho_iT,\quad (1\leq i,j \leq l)$;
2. $\sum_{i=1}^l \rho_i = 1$;
3. $\rho_i\pi_{i,i+1} = \pi_{i,i+1}\circ\rho_{i+1} = \pi_{i,i+1}, \quad (1 \leq i \leq l-1)$;
4. $\rho_{i+1}\iota_{i+1,i} = \iota_{i+1,i}\circ\rho_i = \iota_{i+1,i}, \quad (1 \leq i \leq l-1)$;
5. $\iota_{i+1,i} \circ \pi_{i,i+1} = (T-q^{a_{i+1}})\rho_{i+1}, \quad (1 \leq i \leq l-1)$;
6. $\pi_{i,i+1}\circ\iota_{i+1,i} = (T-q^{a_{i+1}})\rho_i, \quad (1\leq i \leq l-1)$;
7. $T\rho_1 = q^{a_1}\rho_1$.
It is easy to see that the generator $T$ of $\mathcal S'_l$ is central, and in fact redundant. (We include it for convenience which will become evident below.)
The next elementary lemma shows moreover that the spectrum of each $T\rho_i$ is very constrained.
\[spectrum\] Let $\mathscr A$ denote an associative $\mathbb C(q)$-algebra with elements $A$ and $B$, and suppose we set $T_i = AB+q^{a}$ and $T_{i+1} = BA+q^{a}$ for some $a \in \mathbb C$. If $T_i$ satisfies the equation $P(T_i) = \prod_{k=0}^{i-1}(T_i - q^{a_k}) = 0$, then $T_{i+1}$ satisfies the equation $$\prod_{k=0}^{i}(T_{i+1}-q^{a_k}) =0.$$ where we set $a_i = a$.
Clearly we have $Q(AB) = 0$, where $Q(t+q^a) = P(t)$. Now write $Q(t) = \sum_{k=0}^{i-1}b_kt^k$. Then we have $$\begin{split}
0=B.Q(AB).A &= \sum_{k=0}^{i-1}b_kB.(AB)^k.A,\\
&= \sum_{k=0}^{i-1} b_k(BA)^{k+1} = (BA)Q(BA)\\
&= (T_{i+1}-q^a)\prod_{k=0}^{i-1}(T_{i+1}-q^{a_k}),
\end{split}$$ since $Q(BA) = P(T_{i+1})$, and thus $T_{i+1}$ satisfies the required equation.
The algebra $\mathcal S'_l$ is isomorphic to the algebra $\mathcal S_l$.
It is clear that there is a map $p:\mathcal S'_l \to \mathcal S_l$, sending $\pi_{i,i+1}$ and $\iota_{i+1,i}$ to their corresponding elements in $\mathcal S_l$ and $\rho_i$ to $\text{id}_{M^i}$. Since $\mathcal S_l$ is generated by the $\pi_{i, i+1}$, $\iota_{i+1,i}$ and $\text{id}_{M^i}$s it is clear that $p$ is surjective. For $i<j$ let $\pi_{ij} = \pi_{i,i+1}\circ\ldots \circ \pi_{j-1,j} \in \mathcal S'_l$, and similarly if $i>j$ let $\iota_{ij} = \iota_{i,i-1}\circ\ldots\circ\iota_{j+1,j}$. Then using the defining relations, it is easy to see that as a $\mathbb C[T]$-module $\mathcal S'_l$ is spanned by the elements $\{\rho_i, \pi_{jk}, \iota_{kj}: 1\leq i \leq l, 1\leq j<k \leq l\}$. Applying Lemma \[spectrum\] with $T_i = \rho_iT$ and using induction it is easy to see that $\mathbb C[T]$ acts on $\mathbb C[T]\iota_{kj}$ and $\mathbb C[T]\pi_{jk}$ via the algebra $\mathbb C[t]/(r_j)$, and on $\mathbb C[T]\rho_{i}$ via the algebra $\mathbb C[t]/(r_i)$. Comparing with matrix-like the description of $\mathcal S_l$ we obtained above it is clear that $\dim_{\mathbb C}(\mathcal S'_l) \leq \dim_{\mathbb C}(\mathcal S_l)$, so that $p$ must be an isomorphism.
\[modulestructure\] It follows that a vector space $V$ can be equipped with an $\mathcal S_l$-module structure by giving a grading $V = \bigoplus_{i=1}^l V_i$ together with maps $\alpha_i\colon V_i \to V_{i+1}$ and $\beta_i\colon V_{i+1} \to V_i$, where we have $$\label{relationsonSl}
\alpha_{i-1}\circ\beta_{i-1} - \beta_i\circ\alpha_i = (q^{a_{i+1}} - q^{a_i})\text{id}_{V_i}, \quad 1\leq i \leq l-1,$$ (where we set $\alpha_0=\beta_0=0$).
{#section-18}
We now show that our microlocal approach for the representations of category $\mathcal O_\kappa$ of the rational Cherednik algebra of type $\mu_l$ gives a functor to the category of finite dimensional representations of $\mathcal S_l$. In fact, using the work of [@BK] and [@Ku] it follows this functor is an equivalence (provided $\kappa$ lies outside certain explicit hyperplanes). We use the nearby and vanishing cycle construction for $\mathcal D$-modules on $\mathbb C$ which are $\mathcal O$-coherent along $\mathbb C^\times$ as described in $\S$\[nearbyvanishing\]. Topologically, this can be viewed as follows: for a $\mathcal D$-module $\mathcal M$, the vector space $\Psi(\mathcal M)$ corresponds to the stalk of the solution sheaf at a point in the punctured disk, (equipped with the natural monodromy automorphism) and the vector space $\Phi(\mathcal M)$ gives a “Morse group” of the solution sheaf of $\mathcal M$ at a generic covector in the cotangent space $T^*_0\mathbb C$, which comes equipped with a microlocal monodromy operator.
Let ${\tilde{\mathscr A}_{c,i}}$ denote the restriction of ${\tilde{\mathscr A}_c}$ to $X_i \cong T^*\mathbb C$. It is shown in [@Ku §3] that ${\tilde{\mathscr A}_{c,i}}$ is isomorphic to the standard $\mathscr W$-algebra on $T^* \mathbb C$ via the map defined by $x \mapsto f_i$ and $\xi \mapsto g_i$ where we set $f_i = (x_{\eta_1}\ldots x_{\eta_i})\circ(y_{\eta_{i+1}}\ldots y_{\eta_l})^{-1}$ and $g_i = (y_{\eta_i}\ldots y_{\eta_l})\circ (x_{\eta_1}\ldots x_{\eta_{i-1}})^{-1}$. Note that $f_i \circ g_i = x_{\eta_i}\circ y_{\eta_i}$ and $g_i\circ f_i = x_{\eta_i}\circ y_{\eta_i}+ \hbar$. Via this isomorphism, the $F$-action on $\tilde{\mathscr A}_{c|X_i}$ corresponds to the $F$-action on the standard $\mathscr W_{T^*\mathbb C}$-algebra given by $x \mapsto t^{2i-l}x$ and $\xi\mapsto t^{l-2i+2}$, so that the $F$-invariant sections are then: $$\text{End}_{\text{Mod}_F(\mathscr W[\hbar^{1/2}])}(\mathscr W[\hbar^{1/2}])^{\text{opp}} = \mathbb C[\hbar^{l/2-i}x, \hbar^{i-l/2}\xi],$$ which is isomorphic to $\mathcal D(\mathbb C)$, and moreover the category of modules $\text{Mod}_F(\mathscr W[\hbar^{1/2}])$ is equivalent to $\text{Mod}_{\text{coh}}(\mathcal D(\mathbb C))$ and hence $\text{Mod}_{\text{coh}}(\mathcal D_\mathbb C)$ (see the second example of [@KR 2.3.3] for more details, where in our case, $m=2$). Note that the element $x\partial$ in $\mathcal D(\mathbb C)$ corresponds to $\hbar^{-1}f_ig_i$.
For $i \in \{1,2,\ldots, l\}$ we define functors $KZ_i$ from $\mathcal O_c$ to $\mathbb C[t]_{(t)}$-mod as follows. Let $\mathcal M_i$ denote the restriction of $\mathcal M$ to $X_i$, where we may view it (by the above discussion) as a module for $\mathscr W_{T^*\mathbb C}$, with the appropriate $F$-action, and hence as a $\mathcal D_\mathbb C$-module. As such, it corresponds to a holonomic module whose support lies in $\{(x,\xi) : x\xi = 0\}$, where $\{x=0\}$ corresponds to $D_{i-1}$ and $\{\xi =0\}$ corresponds to $D_i$. Therefore it yields a local system on $D_i \backslash \{p_i\} \cong \mathbb C^\times$, which is the same data as a vector space equipped with an automorphism, which we may view as a $\mathbb C[t]_{(t)}$-module. We set $KZ_i(\mathcal M)$ to be this $\mathbb C[t]_{(t)}$-module, $(\Phi(\mathcal M_i),T)$.
We now show how the functors $KZ_i$ yield a representation of $\mathcal S_l$, where $\mathcal S_l$ is the algebra with parameters $(q^{a_i})_{i=1}^l$ where $a_i = \sum_{j=1}^{i-1} \bar{c}_j$ (so that $a_0 =0$) and for $a\in \mathbb C$ we write $q^a$ for $\text{exp}(2i\pi a)$. Indeed given $\mathcal M$ an object in $\mathcal O_c$, define $$\mu KZ(\mathcal M) = \bigoplus_{i=1}^l KZ_i(\mathcal M).$$ Thus $\mu KZ(\mathcal M)$ is a graded vector space.
The $\mathbb Z/l\mathbb Z$-graded vector space $\mu KZ(\mathcal M)$ has a natural $\mathcal S_l$-module structure, so that $\mu KZ$ becomes a functor from $\mathcal O_c$ to $\mathcal S_l$-mod.
To equip $\mu KZ(\mathcal M)$ with the structure of a $\mathcal S_l$-module, by Remark \[modulestructure\] we need only define appropriate maps $$\alpha_i\colon KZ_i(\mathcal M) \to KZ_{i+1}(\mathcal M); \quad \beta_i \colon KZ_{i+1}(\mathcal M) \to KZ_i(\mathcal M).$$ For these we use (slightly rescaled versions of) the natural transformations $v$ and $c$ between the nearby and vanishing cycle functors. Let $var_i\colon \Psi(\mathcal M_i) \to \Phi(\mathcal M_i)$ and $can_i \colon \Phi(\mathcal M_i) \to \Psi(\mathcal M_i)$ denote these morphisms, where as above $\mathcal M_i = \mathcal M_{|X_i}$, and note that $KZ_i(\mathcal M) = \Phi(\mathcal M_i) = \Psi(\mathcal M_{i+1})$.
Set $\alpha_i = q^{a_i}var_{i+1}$ and $\beta_i = can_{i+1}$. We need only verify that Equation \[relationsonSl\] holds. For this, note that if $T_i$ denotes the monodromy automorphism on $KZ_i(\mathcal M)$, then $T_i = \text{exp}(-2i\pi \hbar^{-1}f_ig_i) = var_ican_i+1$. Since on $X_i \cap X_{i+1}$ we have $f_ig_i = x_{\eta_i}\circ y_{\eta_i} = x_{\eta_{i+1}}\circ y_{\eta_{i+1}} + \hbar \bar{c}_i$, it follows that $T_i = q^{\bar{c}_i}T_{i+1}$ on $KZ_i(\mathcal M) = \Phi(\mathcal M_i) = \Psi(\mathcal M_{i+1})$. Then we have $$\begin{split}
\alpha_{i-1}\circ\beta_{i-1} -\beta_i\circ \alpha_i &= q^{a_i}(T_i-1) -q^{a_{i+1}}(T_{i+1}-1) \\
&= q^{a_i}(q^{\bar{c}_i}T_{i+1}-1) -q^{a_{i+1}}(T_{i+1} -1)\\
&= q^{a_{i+1}} - q^{a_i},
\end{split}$$ as required. Finally, note that on $X_1$ the module $\mathcal M_1$ is supported entirely on $D_1$, so that $\Psi(\mathcal M_1) = 0$, and hence the above calculation also shows that $\beta_1\circ\alpha_1 = q^{a_2} - q^{a_1}$ and we are done.
Let $$s_i(t) = \prod_{j=1}^{i} (t-q^{\sum_{k=j}^{i-1}\bar{c}_j}).$$ The functor $KZ_i$ from $\mathcal O_c$ to $\mathbb C[t]_{(t)}$ factors through $\mathbb C[t]/(s_i)$.
If $T$ is the central element of $\mathcal S_l$, then the action of the monodromy operator on $KZ_i(\mathcal M)$ (for $\mathcal M$ an object of $\mathcal O_c$) is given by $q^{\sum_{k=i}^l\bar{c}_k}\rho_iT$. The claim then follows from fact our description of the structure of $\mathcal S_l$.
Note that in particular the functor $KZ_l$ therefore yields a representation of the corresponding cyclotomic Hecke algebra. In fact it is just the original $KZ$-functor for the rational Cherednik algebra of type $\mu_l$. This can be seen using the correspondence between the $\kappa_i$ and $c_i$s, and an argument analogous to Proposition \[KZcheck\]. Using [@Ku] (where an explicit construction of the sheaves corresponding to standard modules is given), the reader can check that $\mu KZ$ sends the standard modules in $\mathcal O_c$ to the standard modules for $\mathcal S_l$, and similar arguments allow one to show that $\mu KZ$ is in fact an equivalence.
Appendix
========
Twisted $\mathcal D$-modules {#twistingstuff}
----------------------------
In this paper we work with modules over rings of twisted differential operators on $\mathfrak{gl}_n \times \mathbb P^{n-1}$, so we recall briefly the construction of the twisted rings we use. For a detailed discussion of these issues see [@K7], whose presentation we largely follow[^10] (for an alternative account in the algebraic context see [@BB]). If $X$ is a topological space a *twisting datum* $\tau$ is a triple $(\pi\colon X_0\to X, L,m)$ where $\pi\colon X_0 \to X$ is a continuous map admitting a section locally on $X$, $L$ is an invertible sheaf of vector spaces on $X_0\times_X X_0$ and $m$ is an isomorphism: $$m\colon p_{12}^{-1}L\otimes p_{23}^{-1}L \to p_{13}^{-1}L,$$ on $X_2 = X_0\times_X X_0\times_X X_0$. Moreover, the isomorphism $m$ is required to satisfy an appropriate “associativity” condition on the quadruple product of $X_4$ of $X_0$ over $X$. Given a twisting datum, a twisted sheaf on $X$ is a pair $(F,\beta)$ consisting of a sheaf $F$ on $X_0$ and an isomorphism $\beta\colon L\otimes p_2^{-1}F \to p_1^{-1}F$, where $p_1,p_2$ are the natural projections from $X_1 = X_0\times_X X_0$ to $X_0$, along with the requirement that $\beta$ satisfies an appropriate cocycle condition.
If $H$ is a complex affine algebraic group and $\pi\colon X_0 \to X$ a principal $H$-bundle (where we assume that $H$ acts on $X_0$ on the left) then one can naturally attach to $X_0$ a family of twisting data on $X$. To describe this we first need the notion of a character local system. Let $\mu\colon H\times H \to H$ be the multiplication map. A character local system is an invertible $\mathbb C_H$-sheaf $L$ equipped with an isomorphism $m\colon q_1^{-1}L \otimes q_2^{-1}L \to \mu^{-1}L$ which satisfies the associative law. Let $\mathfrak h = \text{Lie}(H)$ denotes the Lie algebra of $H$ and take an $H$-invariant element $\lambda$ of $\mathfrak h^* = \text{Hom}(\mathfrak h,\mathbb C)$, so that $\lambda([\mathfrak h,\mathfrak h]) = 0$. Let $L_\lambda$ be the sheaf of (analytic) functions on $H$ which satisfy $R_H(X)(f)=\lambda(X).f$, where $R_H$ denotes map from $\mathfrak h$ to vector fields on $H$ given by the right action of $H$ on itself. The multiplication map $H\times H \to H$ then induces the structure of a character local system on $L_\lambda$.
Now suppose that in addition we have an $H$-space $Y$, and let $a\colon H\times Y\to Y$ be the action map, and $p_1\colon H \times Y \to Y$, $p_2\colon H\times Y\to H$ the obvious projections. An $(H,\lambda)$-equivariant sheaf on $Y$, or $\lambda$-twisted equivariant sheaf, is a pair $(F,\beta)$ consisting of a sheaf $F$ on $Y$ together with an isomorphism $\beta\colon p_1^{-1}L_\lambda \otimes p_2^{-1}F \to a^{-1}F$, once again satisfying an appropriate associative condition. In the case when $\lambda=0$ we say that $F$ is an equivariant sheaf. The notion of a twisted equivariant sheaf if closely related to that of a twisted sheaf, as the following construction shows.
Let $\pi\colon X_0\to X$ be a principal $H$-bundle, and let $\lambda$ be as above. We may identify $H\times X_0$ with $X_0\times_X X_0$ by the map $(h,x') \mapsto (hx',x')$ and via this isomorphism, we find that $p_1^{-1}L_\lambda$ yields a twisting datum $\tau_\lambda$ on $X$. In particular, the categories of $\tau_\lambda$-twisted sheaves on $X$ is equivalent to the category of $(H,\lambda)$-equivariant sheaves on $X_0$, and thus the principal bundle $X_0$ yields the family $\{\tau_\lambda: \lambda \in \mathfrak (h^*)^H\}$ of twisting data.
Thus far our discussion has been purely topological, but the same formalism may be used with $\mathcal D$-modules as is explained in [@K7 §7.11]. One replaces $L_\lambda$ with the sheaf $\mathcal L_\lambda = \mathcal D_H.u_\lambda$ where $u_\lambda$ has defining relations $R_H(A)u_\lambda = \lambda(A)u_\lambda$ for $A \in \mathfrak h$ (and pullbacks of sheaves of vector spaces with their appropriate $\mathcal D$-module analogues). Note that $L_\lambda \cong \mathcal H\text{om}_{\mathcal D_X}(\mathcal L_\lambda,\mathcal O_H^{\text{an}})$, where $\mathcal O_H^\text{an}$ is the sheaf of analytic rather than algebraic functions on $H$, and in fact the Riemann-Hilbert correspondence extends to have a twisted analogue. The ring of twisted differential operators $\mathcal D_{X,\tau_\lambda}$ (or more simply $\mathcal D_{X,\lambda}$) on $X$ is then given as follows: if $\mathcal N_\lambda = \mathcal D_{X_0}v_\lambda$ is the $\mathcal D_{X_0}$-module with defining relations $L_{X_0}(A) = -\lambda(A).v_\lambda$, then $$\mathcal D_{X,\lambda} = \{f \in \pi_*(\mathcal{E}nd_{\mathcal D_{X_0}}(\mathcal N_\lambda)): f \text { is } H\text{-equivariant}\}^{\text{op}}$$ where $(\cdot)^{\text{op}}$ denotes the opposite ring. By [@K7 Lemma 7.12.1], the category of modules for the $\tau_\lambda$-twisted ring of differential operators $\mathcal D_{X,\lambda}$ is then equivalent to the category of $(H,\lambda)$-equivariant $\mathcal D_{X_0}$-modules on $X_0$. Note that in the terminology of [@K7], $(H,\lambda)$-equivariant $\mathcal D$-modules form an abelian subcategory of the category of $H$-quasi-equivariant $\mathcal D$-modules.
Finally, we need to consider equivariant twisted $\mathcal D$-modules. Given an affine algebraic group $G$ and a $G$-space $X$, one can naturally define the notion of a $G$-equivariant twisting datum on $X$, and hence the notion of $G$-equivariant twisted sheaves on $X$. In particular, in the case where the twisting data arises from a principal $H$-bundle $X_0$, if the group $G$ acts on $X_0$ and $X$ in such a way that the map $\pi$ is $G$-equivariant, and the actions of $H$ and $G$ commute on $X_0$, then we may define $G$-equivariant twisted $\mathcal D_{X,\lambda}$-modules, and the equivalence between twisted modules on $X$ and $(H,\lambda)$-equivariant modules on $X_0$ allows us to identify such modules with $\mathcal D_{X_0}$-modules which are $G$-equivariant and $(H,\lambda)$-twisted equivariant.
Index Theorems and Characteristic Cycles.
-----------------------------------------
We have computed the characteristic cycle of standard modules using the local Euler obstruction and the known calculation of the stalk cohomologies. In this appendix we briefly recall the index theorem of Kashiwara, Dubson, Brylinski [@K73], [@BDK], and in the topological setting [@M] which is the key to this approach. This theorem can be seen as a (local version of a) generalization of the classical Hopf index theorem which calculates the Euler characteristic of a compact manifold $X$ as the self-intersection number of $X$ in its cotangent bundle.
Since we only need a local result, we may suppose that $X$ is a stratified analytic subset of affine space $\mathbb C^n$. Thus $X = \bigsqcup_{S \in \mathcal S} S$, where each $S$ is a smooth locally-closed connected subset of $\mathbb C^n$, and the closure of a stratum $S$ is a union of strata. We may also assume that the Whitney $(a)$ and $(b)$ conditions are satisified (or for that matter the $\mu$-condition introducted by Kashiwara and Schapira [@KS Chapter 8]). For $x \in X$ we will write $B_\varepsilon(x)$ for the set $$\{y \in X: \|y-x\| < \varepsilon\},$$ where $\|.\|$ is the standard Hermitian norm on $\mathbb C^n$. (Thus our constructions here use the analytic variety attached to the algebraic varieties we considered earlier).
Let $z \in S$ be a point of the stratum $S$. By taking a normal slice $N$ to $S$ at $x$, that is, choosing a complex analytic submanifold $N$ which intersects each stratum transversely and such that $N \cap S = \{z\}$, we may reduce to the case $S = \{z\}$. Let $\phi \colon N \to \mathbb C$ be a holomorphic function vanishing at $z$ such that $d\phi(z)$ does not lie in the closure of $T^*_TX$ for any stratum $T \neq S$. If $S=T$ we define $c_{S,S} =1$ for all $S$.
Endow $X$ with a Hermitian metric (say by taking the restriction of the standard on on $\mathbb C^n$) and pick a small disk $B = B_\varepsilon(z)$ about $z$, and a generic $\eta \in \mathbb C^\times$ such that $|\eta|<< \varepsilon$. The complex link $L$ of the stratum $S$ in $T$ is then defined to be the set $$L = B\cap T\cap N \cap \phi^{-1}(\eta).$$ Stratified Morse theory shows that the homeomorphism type of $L$ is independent of the choices made (in fact it is independent of the metric $\|.\|$ also, so does not depend on the choice of local embedding we make). We define the local Euler Obstruction $c_{S,T}$ to be the Euler characteristic of the complex link, that is $$c_{S,T} = \chi_c(L).$$ As the notation suggests, this number is independent of the choice of $z$ in the stratum $S$ (here we use the assumption that our strata are connected).
The index theorem shows that the characteristic cycle of a holonomic $\mathcal D$-module determines its local Euler characteristics. Indeed suppose that $M$ is a $\mathcal D$-module whose characteristic cycle lies in $\bigsqcup_{S \in \mathcal S} T_S^*M$ (so in particular $M$ is holonomic). Then since we assume that our stratification satisfies the Whitney conditions, $\text{DR}(M)$ is a constructible sheaf which is locally constant on the strata $S$, and we may set $$\chi_S(M) = \sum_{i}(-1)^i \dim(\mathcal H^i(\text{DR}(M))_{|S}).$$ We also have $CC(M) = \sum_{S \in \mathcal S} m_S(M)[T^*_SX]$.
\[indextheorem\] Let $M$ be a holonomic $\mathcal D$-module as above. Then we have $$m_S(M) = \sum_{T \subset \bar{S}} c_{S,T} \chi_T(M).$$
Since $c_{S,S}$ is defined to be $1$, if we pick any total order refining the partial order on strata give by the closure relation we see that the matrix $(c_{S,T})$ is unitriangular, and so the sets of numbers $\{\chi_S(M)\}_{S \in \mathcal S}$ and $\{m_S(M)\}_{S \in \mathcal S}$ determine each other. Thus one can invert the above theorem to give a formula for the multiplicities of the characteristic cycle in terms of the local Euler characteristics. This is the form of the theorem stated in [@K73]. In fact the map from holonomic $\mathcal D$-modules to constructible functions given by the $\chi_S$s and the characteristic cycle map from holonomic $\mathcal D$-modules to Lagrangian cycles induce isomorphisms from the Grothendieck group to the group of constructible functions and Lagrangian cycles respectively. The index theorem is then an explicit geometric description of the induced isomorphism between the two abelian groups. Note that the difference in signs between our statement of the index theorem and that in [@EM] arises as we work with $\mathcal D$-modules rather than constructible sheaves.
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[^1]: We have set the parameter $t$ of [@EG] to be $1$.
[^2]: However it is also shown in [@BE Theorem 4.10] that $e{\mathcal H_c}e$ is simple for all $c \in (-1,0)$, a range in which ${\mathcal H_c}$ is not always simple.
[^3]: Of course $Y$ is nilpotent, so is equal to its nilpotent part.
[^4]: For this argument, we use the complex topology on our varieties, not the Zariski topology.
[^5]: The $\frac{1}{n}$ factor comes from the fact that we use $\text{tr}_{|\text{Lie}(\mathcal Z)}$ as a basis vector for the dual space of $\text{Lie}(\mathcal Z)$.
[^6]: The proof of Theorem $1.3.1$ in that paper is given in [@K83].
[^7]: Of course one also obtains a functor from $\mathcal O_c$ to $\text{Rep}(\mathcal B_\lambda)$ by composing with the functor from $\mathcal O_c$ to $\mathcal O^\text{sph}_c$ sending $M \mapsto eM$.
[^8]: We refer the reader to their paper for details and terminology such as $\mathscr W$-algebras and $F$-structures, but note we summarise their work using our notation not theirs.
[^9]: This construction of course works for any complex reflection group, not only symmetric groups.
[^10]: As noted in [@K7], the presentation there uses the notion of a twisting datum (recalled above) which is slightly *ad hoc* from a systematic point of view, however it is well-suited to the families of twistings which we need.
|
---
abstract: 'This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number **Re**. In this work, we show that there is constant $0 < c_0 \ll 1$, independent of $\textbf{Re}$, such that sufficiently regular disturbances of size $\epsilon \lesssim \textbf{Re}^{-2/3-\delta}$ for any $\delta > 0$ exist at least until $t = c_0\epsilon^{-1}$ and in general evolve to be $O(c_0)$ due to the lift-up effect. Further, after times $t \gtrsim \textbf{Re}^{1/3}$, the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at $t \approx \epsilon^{-1}$. Hence, our work strongly suggests, for *all* (sufficiently regular) initial data, the genericity of the “lift-up effect $\Rightarrow$ streak growth $\Rightarrow$ streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature.'
author:
- 'Jacob Bedrossian[^1] and Pierre Germain[^2] and Nader Masmoudi[^3]'
bibliography:
- 'eulereqns\_vlad.bib'
- 'IDnLD.bib'
title: 'Dynamics near the subcritical transition of the 3D Couette flow II: Above threshold case'
---
Introduction
============
This work is the second paper in our study of the 3D Navier-Stokes equation near the (plane, periodic) Couette flow, following our work [@BGM15I]. In these works, we study the 3D Navier-Stokes equations near the Couette flow in the idealized domain $(x,y,z) \in {\mathbb T}\times {\mathbb R}\times {\mathbb T}$: if $u + \left(y, 0, 0 \right)^T$ solves the Navier-Stokes equation, $u$ solves
\[def:3DNSE\] $$\begin{aligned}
\partial_t u + y \partial_x u + u\cdot {\nabla}u + {\nabla}p^{NL} & = \begin{pmatrix} - u^2 \\ 0 \\ 0 \end{pmatrix} - {\nabla}p^L + \nu\Delta u \\
\Delta p^{NL} & = -\partial_i u^j \partial_j u^i \\
\Delta p^L & = -2\partial_x u^2 \\
{\nabla}\cdot u & = 0,\end{aligned}$$
where $\nu = \textbf{Re}^{-1}$ denotes the inverse Reynolds number, $p^{NL}$ is the nonlinear contribution to the pressure due to the disturbance and $p^L$ is the linear contribution to the pressure due to the interaction between the disturbance and the Couette flow. The purpose of this work, along with [@BGM15I], is to further the mathematically rigorous understanding of the qualitative behavior of for small perturbations and small $\nu$. This second work is focused on characterizing the dynamics of solutions above the stability threshold (but still not too large).
A major focus of the theory of hydrodynamic stability is the study of laminar flow configurations and understanding when they are stable or when they may transition to a turbulent state (or a nonlinear intermediate state). The terminology *subcritical transition* refers to a situation when the linear theory predicts stability below some critical Reynolds number or at all Reynolds number (the latter is the case here) but spontaneous transition to a turbulent state is observed in laboratory or computer experiments at a much lower Reynolds number than what this theory predicts. To our knowledge, the first quantitative study of this process in fluid mechanics was performed by Reynolds in 1883 [@Reynolds83], and since then, subcritical transition has been observed to be a ubiquitous phenomenon in 3D hydrodynamics, repeated by countless physical experiments (see e.g. [@Nishioka1975; @klebanoff1962; @Tillmark92; @Daviaud92; @Elofsson1999; @bottin98; @HofJuelMullin2003; @Mullin2011; @LemoultEtAl2012]) and computer simulation (e.g. [@Orszag80; @HLJ93; @ReddySchmidEtAl98; @DuguetEtAl2010] and the references therein) on subcritical transition phenomena have been performed in many different settings. See the texts [@DrazinReid81; @Yaglom12; @SchmidHenningson2001] and part I of our work [@BGM15I] for further discussion and references.
As discussed in [@BGM15I], a natural expectation is that while the flow may be stable for all finite Reynolds number, the basin of stability is shrinking as $\nu \rightarrow 0$. Hence, it becomes of interest to, given a norm, determine the threshold of stability, sometimes called the “transition threshold”, as a function of $\nu$. For example, one would like, given a norm ${\left\lVert \cdot \right\rVert}_N$, to find a $\gamma = \gamma(N)$ such that ${\left\lVert u_{in} \right\rVert}_N \ll \nu^\gamma$ implies stability and ${\left\lVert u_{in} \right\rVert}_N \gg \nu^\gamma$ in general permits instability. Further, one would like to identify the possible pathways the solution can take towards transition. A great deal of work has been dedicated to identifying $\gamma$ and estimates from experiments, computer simulations, and formal analysis suggest a threshold somewhere between $1 \leq \gamma \leq 7/4$ for a variety of different initial data and configurations similar to the set-up in (see [@BGM15I] for more references and some of the representative works [@TTRD93; @Waleffe95; @BT97; @LHRS94; @ReddySchmidEtAl98; @Chapman02; @Mullin2011] or the text [@SchmidHenningson2001] and the references therein). In [@BGM15I], we proved that, for sufficiently regular initial data, $\gamma = 1$ for (that is, for a sufficiently strong norm $N$, $\gamma(N) = 1$).
In this work our goal is to characterize the instabilities of above threshold solutions. We prove that there is a universal constant $c_0$ with $0 < c_0 \ll 1$ such that for sufficiently regular initial data (in the same sense as [@BGM15I]) of size $\epsilon$, if $\epsilon \lesssim \nu^{2/3+\delta}$ for $\delta> 0$, then the solution to exists until at least time $t = c_0\epsilon^{-1}$ and is rapidly attracted to the class of $x$-independent solutions known as *streaks* for times $t \gtrsim \nu^{-1/3}$. Due to a non-modal instability known as the lift-up effect, the streaks (and hence all solutions) will in general grow linearly as $O(\epsilon t)$ and by the final time can be $O(c_0)$ (which is independent of $\nu$). In our companion work [@BGM15I], we studied solutions below the $\epsilon \ll \nu$ threshold and proved that these solutions are global, return to Couette flow, and also converge to the set of streak solutions. While our previous analysis did include solutions which get $O(c_0)$ from the Couette flow, all solutions never deviate farther from the Couette flow and are demonstrably not involved in any transition processes.
The foremost interest of this work is that the threshold solutions we study can converge to streaks that, due to the lift-up effect, eventually become as large as the Couette flow itself (although we cannot follow our solutions to this point). These large streaks induce an unstable shear flow and are expected to become linearly unstable, sometimes referred to as a *secondary instability* [@ReddySchmidEtAl98; @Chapman02; @SchmidHenningson2001]. The instability is observed to involve the rapid growth of $x$-dependent modes. The process by which large streaks exhibit instabilities and drive $x$-dependent flows is sometimes referred to as *streak breakdown* and is well-documented as one of the primary routes towards turbulent transition observed experimentally [@klebanoff1962; @bottin98; @Elofsson1999] and in computer simulations [@ReddySchmidEtAl98; @DuguetEtAl2010], in agreement with a variety of formal asymptotic calculations [@ReddySchmidEtAl98; @Chapman02; @SchmidHenningson2001]. That is, it is an expectation that a general route towards transition is the multi-step process “lift-up effect $\Rightarrow$ streak growth $\Rightarrow$ streak breakdown $\Rightarrow$ transition”. Moreover, the general process of streak breakdown is thought to play an important role in sustaining turbulence near the transition threshold and in both the creation and decay of “turbulent spots” (see [@SchmidHenningson2001] and the references therein). While we cannot take our solutions through the secondary instability, we prove that solutions above the threshold (but not too far above) can in general converge to unstable streaks and that this is the only instability that possible, which is suggestive of the genericity of the above multi-step process as the first step towards turbulent transition near the threshold (for sufficiently regular data – see Remark \[ref:lowreg\] below for more discussion on rougher data).
Unlike in [@BGM15I], the solutions we are concerned with are unstable in the sense that they might transition for $t \gg \epsilon^{-1}$, and we are identifying that the streamwise vortex/streak instability associated with the lift-up effect is dominant whereas all other dynamics are suppressed. At the linear level, another important effect is the vortex stretching, which in particular, causes a direct cascade of energy to high frequencies in the $u^1$ and $u^3$ components and creates growth which is difficult to control. The stabilizing mechanisms suppressing the more complicated nonlinear effects are the mixing-enhanced dissipation and the inviscid damping, both due to the mixing from the background shear flow. Enhanced dissipation was first observed in by Lord Kelvin [@Kelvin87] and has been observed in many contexts in fluid mechanics (see e.g. [@DrazinReid81; @RhinesYoung83; @DubrulleNazarenko94; @LatiniBernoff01; @BernoffLingevitch94; @ReddySchmidEtAl98] and the mathematically rigorous works [@BeckWayne11; @ConstantinEtAl08; @BMV14]). In , the mixing due to Couette drives information to high frequencies, enhancing the dissipation of $x$-dependent modes such that they decay on a time-scale like $\tau_{ED} \sim \nu^{-1/3}$, far faster than the natural “heat equation” time-scale $O(\nu^{-1})$. The idea that the enhanced dissipation effect has an important role to play in dates back at least to [@DubrulleNazarenko94]. Indeed, in [@DubrulleNazarenko94], an idea similar to the heuristic below appears. However, the expectation that a large mean shear should suppress certain kinds of instabilities has been suggested at varying levels of precision in many contexts (see e.g. [@DrazinReid81; @Yaglom12; @ReddySchmidEtAl98; @Chapman02] and the references therein). Inviscid damping in fluid mechanics was first observed by Orr [@Orr07] in 1907 and turned out to be the hydrodynamic analogue of Landau damping in plasma physics; see [@BM13; @BGM15I] for more discussion. Here, inviscid damping will suppress the $x$-dependence of $u^2$, key to controlling certain components of the nonlinearity that would otherwise be uncontrollable.
The fact that we prove results for initial data as large as $\nu^{2/3+\delta}$ shows that the streak growth scenario is generic even for initial data which is far larger than the $O(\nu)$ threshold, at least for data which is sufficiently regular. Moreover, we are not aware of this exponent appearing anywhere in the applied mathematics or physics literature previously despite being a threshold of natural interest. The $2/3$ threshold can be explained from heuristics. Formal analysis of the weakly nonlinear resonances, described in §\[sec:Toy\], suggests that the natural time-scale before a general $x$-dependent solution could potentially become fully nonlinear, $\tau_{NL}$, is *at least* $\tau_{NL} \gtrsim \epsilon^{-1/2}$. On the other hand, the enhanced dissipation occurs on time-scales like $\tau_{ED} \sim \nu^{-1/3}$. Hence, if the enhanced dissipation is to dominate the 3D effects and relax the solution to the manifold of streaks, then we need the latter time scale to be shorter than the former, or rather: $$\begin{aligned}
\tau_{ED} \sim \nu^{-1/3} \ll \epsilon^{-1/2} \lesssim \tau_{NL}. \label{ineq:heur23}\end{aligned}$$ This is the origin of the requirement $\epsilon \lesssim \nu^{2/3+\delta}$; the small $\delta> 0$ is to provide a little technical room to work with in the estimates (although we do not know if it can be removed). We emphasize that getting a convincing estimate on $\tau_{NL}$ is challenging, which may explain why this threshold does not appear in the literature (moreover, the heuristics of §\[sec:Toy\] are likely only convincing when backed by Theorem \[thm:SRS\] and its proof). After $t \gg \tau_{ED}$ the solution is very close to a streak and, due to the lift-up effect, in general $u^1_0(t)$ is growing like $\epsilon {\left\langle t \right\rangle}$ until times $t \sim \epsilon^{-1}$, at which point the streak will become fully nonlinear (see [@BGM15I; @ReddySchmidEtAl98; @Chapman02] and the references therein). Below we discuss several other ways to derive the $\epsilon \sim \nu^{2/3}$ cut-off which are in some ways more straightforward but also a bit more ad-hoc (see §\[sec:NonlinHeuristics\] and §\[sec:Toy\]).
As discussed in [@BGM15I; @BM13], if there is decay-via-mixing then, since mixing is time-reversible (at infinite Reynolds number), necessarily there is growth-via-unmixing. This non-modal effect was first pointed out by Orr [@Orr07] and is now known as the *Orr mechanism*. Some of the more subtle and problematic nonlinear effects here are 3D variants of the nonlinear manifestation of the Orr mechanism referred to as an “echo”. These are resonances (perhaps more accurately “pseudo-resonances” [@Trefethen2005]) involving the excitation of unmixing modes (see [@Craik1971; @VMW98; @Vanneste02; @BGM15I; @BM13] and the references therein for discussion about this effect in the context of fluid mechanics and [@YuDriscoll02; @YuDriscollONeil] for physical experiments isolating them in 2D Euler). A similar resonance is also observed in plasmas, known there as a “plasma echo” [@MalmbergWharton68]. A key facet of the proof in [@BGM15I] was the use of careful weakly nonlinear analysis to estimate the possible effects of resonances of this general type (and also others).
Relative to our previous work [@BGM15I], this work will need more precision in the weakly nonlinear analysis and uses more detailed structure of the nonlinearity. In [@BGM15I], a toy model was derived to model the “worst possible” behaviors due to the lift-up effect, the “resonances” associated with the Orr mechanism (e.g. echo-like), and the vortex stretching, accounting also for the stabilizing mechanisms of enhanced dissipation and inviscid damping (see §\[sec:Toy\]). An approximate super-solution of this toy model was used to derive a set of good norms with which to measure the solution. The super-solution used in [@BGM15I] required $\epsilon \lesssim \nu$; here we will derive a super-solution which only requires $\epsilon \lesssim \nu^{2/3}$ but (A) it is more subtle than that of [@BGM15I] and (B) is only valid for $t \lesssim \epsilon^{-1}$. This latter point is not surprising: at around this point, the solution is expected to suffer streak breakdown and transition to turbulence (or at least escape a weakly nonlinear regime). One of the new complexities that the super-solution will introduce is that the norm used to measure $u^3$ will need to unbalance the regularity of different frequencies in the $x$-dependent modes of $u^3$ in a subtle and precise way. This turns out to be similar to a technique applied to the scalar vorticity in 2D [@BM13; @BMV14], however, here it is not so much the imbalances within $u^3$ itself which are important, but rather the imbalances between $u^3$ and the *other* components. Together with the much smaller dissipation, the additional precision in the norm will noticeably complicate the proof of Theorem \[thm:SRS\] below (relative to [@BGM15I]). Many terms here will require a more detailed treatment than that used in [@BGM15I], either because of the more complicated norms or because the dissipation is weaker. The additional precision will require some new techniques and better technical tools, including more precise multiplier inequalities relating time and frequency (see §\[sec:nrmuse\]) and several new elliptic estimates (see Appendix \[sec:PEL\]). Another adjustment we will make here is a nonlinear coordinate transform which is more precise than the one employed in [@BGM15I]; in particular, we will need to account for mixing caused by $(0,0,u^3_0)^T$ as well as $(y + u^1_0,0,0)^T$ and hence treat the entire streak in an essentially Lagrangian fashion. In order to carry out this line of attack we will need to use more structure in the nonlinearity than [@BGM15I] and understand better certain “null” or “non-resonant” structures, in particular, detailed information about how certain frequencies interact.
Linear behavior and streaks {#sec:LinStreak}
---------------------------
Recall the following notation from [@BGM15I]: the projections of a function $f$ onto zero and non-zero frequencies in $x$ are denoted, respectively, by $$\begin{aligned}
f_0(y,z) & = \frac{1}{2\pi}\int f(x,y,z) dx \\
f_{\neq} & = f - f_0. \end{aligned}$$ Next, we recall from [@BGM15I] the following Proposition, which regards the behavior of the linearized Navier-Stokes equations. There is a corresponding result also for the linearized Euler equations; see [@BGM15I] for more details. Without making any attempt to be optimal in terms of regularity, this proposition emphasizes the stabilizing mechanisms of enhanced dissipation and inviscid damping, and the destabilizing mechanisms of the lift-up effect and the vortex stretching due to the Couette flow. The lift up effect is seen in the transient growth in , the enhanced dissipation in the exponentials $e^{-c\nu t^3}$, the inviscid damping in the ${\left\langle t \right\rangle}^{-2}$ decay in which is uniform in $\nu$, and the vortex stretching in the lack of inviscid damping in and (which is sharp).
\[prop:linear\] Consider the linearized Navier-Stokes equations
\[def:3DNSE\_Linear\] $$\begin{aligned}
\partial_t u + y \partial_x u & = \begin{pmatrix} - u^2 \\ 0 \\ 0 \end{pmatrix} - {\nabla}p^L + \nu \Delta u \\
\Delta p^L & = -2\partial_x u^2 \\
{\nabla}\cdot u & = 0. \end{aligned}$$
Let $u_{in}$ be a divergence free vector field with $u_{in} \in H^7$. Then the solution to the linearized Navier-Stokes equations $u(t)$ with initial data $u_{in}$ satisfies the following for some $c \in (0,1/3)$,
$$\begin{aligned}
{\left\lVert u^{2}_{\neq}(t) \right\rVert}_{2} + {\left\lVert u^{2}_{\neq}(t,x+ty,y,z) \right\rVert}_{H^3} & \lesssim {\left\langle t \right\rangle}^{-2} e^{-c\nu t^3} {\left\lVert u_{in}^2 \right\rVert}_{H^7} \label{ineq:U2LinearID_vs} \\
{\left\lVert u^1_{\neq}(t,x+ty,y,z) \right\rVert}_{H^1} & \lesssim e^{-c\nu t^3} {\left\lVert u_{in} \right\rVert}_{H^7} \label{ineq:U1LinearID_vs} \\
{\left\lVert u^3_{\neq}(t,x+ty,y,z) \right\rVert}_{H^1} & \lesssim e^{-c\nu t^3}{\left\lVert u_{in} \right\rVert}_{H^7}, \label{ineq:U3LinearID_vs} \end{aligned}$$
and the formulas
\[def:NSEstreak\] $$\begin{aligned}
u^1_0(t,y,z) & = e^{\nu t \Delta} \left(u^1_{in \; 0} - tu_{in \; 0}^2\right) \label{eq:liftup_vs} \\
u^2_0(t,y,z) & = e^{\nu t \Delta} u^2_{in \; 0} \\
u^3_0(t,y,z) & = e^{\nu t \Delta} u^3_{in \; 0}. \end{aligned}$$
Associated with the linear problem is the Laplacian expressed in the coordinates $X = x-ty$: $$\begin{aligned}
\Delta_L := \partial_{XX} + (\partial_Y - t\partial_X)^2 + \partial_{ZZ}. \label{def:DeltaL}\end{aligned}$$ The power of $t$ in this operator is responsible both for the inviscid damping of $u^2$ and the enhanced dissipation; see [@BGM15I] for more information.
The next Proposition from [@BGM15I] recalls the nature of the streak solutions:
\[prop:Streak\] Let $\nu \in [0,\infty)$, $u_{in} \in H^{5/2+}$ be divergence free and independent of $x$, that is, $u_{in}(x,y,z) = u_{in}(y,z)$, and denote by $u(t)$ the corresponding unique strong solution to with initial data $u_{in}$. Then $u(t)$ is global in time and for all $T > 0$, $u(t) \in L^\infty( (0,T);H^{5/2+}({\mathbb R}^3))$. Moreover, the pair $(u^2(t),u^3(t))$ solves the 2D Navier-Stokes/Euler equations on $(y,z) \in {\mathbb R}\times {\mathbb T}$:
\[def:2DNSEStreak\] $$\begin{aligned}
\partial_t u^i + (u^2,u^3)\cdot {\nabla}u^i & = -\partial_i p + \nu \Delta u^i \\
\partial_y u^2 + \partial_z u^3 & = 0, \end{aligned}$$
and $u^1$ solves the (linear) forced advection-diffusion equation $$\begin{aligned}
\partial_t u^1 + (u^2,u^3)\cdot {\nabla}u^1 = -u^2 + \nu \Delta u^1. \label{eq:u1streak}\end{aligned}$$
Suppose the streak is initially of size $\epsilon \gg \nu$. From , we see that the dissipation does not completely dominate the streak until $t \gtrsim \nu^{-1}$, before which it could be behaving like fully nonlinear 2D Navier-Stokes. Due to the lift-up effect in , in general $u^1(t)$ is growing like $\epsilon {\left\langle t \right\rangle}$ until times $t \gtrsim \epsilon^{-1}$, at which point the streak will be on the same order as the Couette flow itself. As discussed above, it is expected that sufficiently large streaks should suffer a secondary instability and break down into more complicated $x$-dependent flows (see e.g. [@ReddySchmidEtAl98; @Chapman02; @SchmidHenningson2001] and the references therein).
Statement of main results
-------------------------
As in [@BGM15I], our theorem requires the use of Gevrey regularity class [@Gevrey18], defined on the Fourier side for $\lambda > 0$ and $s \in (0,1]$ as: $$\begin{aligned}
{\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda;s}}^2 = \sum_{k,l}\int {\left\vert\widehat{f_k}(\eta,l)\right\vert}^2e^{2\lambda{\left\vertk,\eta,l\right\vert}^s} d\eta. \end{aligned}$$ For $s = 1$ the class coincides with real analytic, however, for $s < 1$ it is less restrictive, for example, compactly supported functions can still be Gevrey class with $s < 1$. As discussed in [@BGM15I], this regularity class arises in nearly all mathematically rigorous studies involving inviscid damping [@BM13; @BMV14; @BGM15I] or Landau damping [@CagliotiMaffei98; @HwangVelazquez09; @MouhotVillani11; @BMM13; @Young14] in nonlinear PDE. In these previous works, the Gevrey regularity arises naturally when studying echo resonances, and like [@BGM15I], it arises here as well when controlling related weakly nonlinear resonances.
\[thm:SRS\] For all $s \in (1/2,1)$, all $\lambda_0 > \lambda^\prime > 0$, all integers $\alpha \geq 10$ and all $\delta>0$, there exists a constant $c_{00} = c_{00}(s,\lambda_0,\lambda^\prime,\alpha,\delta)$, a constant $K_0 = K_0(s,\lambda_0,\lambda^\prime)$, and a constant $\nu_0 = \nu_0(s,\lambda_0,\lambda^\prime,\alpha,\delta)$ such that for all $\delta_1 >0 $ sufficiently small relative to $\delta$, all $\nu \leq \nu_0$, $c_{0} \leq c_{00}$, and $\epsilon < \nu^{2/3+\delta}$, if $u_{in} \in L^2$ is a divergence-free vector field that can be written $u_{in} = u_S + u_R$ (both also divergence-free) with $$\begin{aligned}
{\left\lVert u_S \right\rVert}_{\mathcal{G}^{\lambda_0;s}} + e^{K_0\nu^{-\frac{3s}{2(1-s)}}}{\left\lVert u_R \right\rVert}_{H^{3}} & \leq \epsilon, \label{ineq:QuantGev2}\end{aligned}$$ then the unique, classical solution to with initial data $u_{in}$ exists at least until time $T_F = c_0 \epsilon^{-1}$ and the following estimates hold with all implicit constants independent of $\nu$, $\epsilon$, $c_0$ and $t$:
- Transient growth of the streak for $t < T_F$: $$\begin{aligned}
{\left\lVert u^1_0(t) - e^{\nu t\Delta}\left(u_{in \; 0}^1 - tu_{in \; 0}^2\right) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim c_{0}^2 \label{ineq:u01grwth} \\
{\left\lVert u^2_0(t) - e^{\nu t\Delta} u_{in \; 0}^2 \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} + {\left\lVert u^3_0(t) - e^{\nu t\Delta} u_{in \; 0}^3 \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim c_{0} \epsilon; \label{ineq:u023Duhamel}\end{aligned}$$
- uniform control of the background streak for $t < T_F$:
$$\begin{aligned}
{\left\lVert u^1_0(t) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \epsilon {\left\langle t \right\rangle} \\
{\left\lVert u^2_0(t) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} + {\left\lVert u^3_0(t) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \epsilon; \end{aligned}$$
- the rapid convergence to a streak by the mixing-enhanced dissipation and inviscid damping of $x$-dependent modes:
$$\begin{aligned}
{\left\lVert u_{\neq}^{1}(t,x + ty + t\psi(t,y,z),y,z) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \frac{\epsilon t^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \\
{\left\lVert u_{\neq}^{3}(t,x + ty + t\psi(t,y,z),y,z) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha} \\
{\left\lVert u^2_{\neq}(t,x + ty + t\psi(t,y,z),y,z) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^\alpha}, \label{ineq:u2damping}\end{aligned}$$
where $\psi(t,y,z)$ is an $O(\epsilon t)$ correction to the mixing which depends on the disturbance (defined below to satisfy the PDE ) and satisfies the estimate: $$\begin{aligned}
{\left\lVert \psi(t) - u_0^1(t) \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} \lesssim \epsilon {\left\langle t \right\rangle}^{-1}. \label{ineq:psiest}\end{aligned}$$
\[ineq:nulesseps\] Without loss of generality we will assume for the remainder of the paper that $\nu \lesssim \epsilon$ as otherwise, Theorem \[thm:SRS\] is covered by our previous work [@BGM15I].
If $u_{in \; 0}^2$ is such that ${\left\lVert u_{in \; 0}^2 \right\rVert}_{{\mathcal{G}}^{\lambda^\prime;s}} \geq \frac{1}{4}\epsilon \geq \frac{1}{16}\nu^{2/3+\delta}$ then shows that for $c_{0}$ small (but independent of $\epsilon$ and $\nu$) and $\epsilon$ small, the streak $u_0^1(t)$ reaches the maximal amplitude of ${\left\lVert u_0^1(t_{m}) \right\rVert}_2 \gtrsim c_{0}$ at times $t_m \sim T_F = c_0\epsilon^{-1}$. Hence, the solution has grown from $O(\epsilon)$ to $O(c_0)$ over this time interval. Moreover, this time-scale is far shorter than the $\nu^{-1}$ time-scale over which $u_0$ will decay by viscous dissipation (at least the low frequencies) and so in general the solution will become fully nonlinear for $t \gtrsim T_F$.
Notice that linear theory in Proposition \[prop:linear\] suggests the $O(t^{-2})$ inviscid damping of $u^2$, whereas we only have $t^{-1}$ in . This discrepancy arises from a 3D nonlinear pressure effect and is explained in §\[sec:Toy\] (this discrepancy did not occur in [@BGM15I]).
Note that the solutions in Theorem \[thm:SRS\] are not only large solutions to 3D NSE, but also in general they are very far from equilibrium (relative to $\nu$). Using naive methods, one would only be able to prove existence until $T_F \sim \log\epsilon^{-1}$ or perhaps some polynomial such as $T_F \sim \epsilon^{-\beta}$ for $\beta \ll 1$ since the Couette flow is rapidly driving large gradients in the solution as well as amplifying the solution via the lift-up effect. It is the inviscid damping and enhanced dissipation, together with the precise structure of the nonlinearity, which allow us to prove existence all the way until $T_F \sim \epsilon^{-1}$ for these large, far-from-equilibrium, solutions.
As in [@BGM15I], the solutions described in Theorem \[thm:SRS\] can exhibit a roughly linear-in-time transfer of kinetic energy to high frequencies where it is ultimately dissipated at $\tau_{ED} \sim \nu^{-1/3}$.
\[ref:lowreg\] In experiments and computer simulations, “lift-up effect $\Rightarrow$ streak growth $\Rightarrow$ streak breakdown” is commonly observed, however there are a number of pathways to transition that have also been observed (see [@SchmidHenningson2001] and the references therein). Further, it has been observed that the transition threshold in general can depend on the kind of perturbation being made (see e.g. [@ReddySchmidEtAl98; @SchmidHenningson2001; @FaisstEckhardt2004; @Mullin2011] and the references therein – in fact, this was even observed by Reynolds [@Reynolds83]). Theorem \[thm:SRS\] and [@BGM15I] are not in contradiction with experimental observations, but instead suggest that this is partly related to the regularity of the perturbations. Indeed, authors conducting computer simulations have explicitly related the transition threshold with the regularity of the initial data and determined different answers [@ReddySchmidEtAl98]. It may also be illuminating to note that while the works [@BM13; @BMV14] rule out subcritical transition of Couette flow in 2D for sufficiently regular perturbations, the works of [@LinZeng11; @LiLin11] suggest it is likely that for sufficiently rough disturbances (about $H^{5/2}$) one can observe subcritical transition even in 2D via a roll-up instability (and hence should, in principle, admit a pathway to transition which is purely 2D at low enough regularities).
Notations and conventions {#sec:Notation}
-------------------------
We use superscripts to denote vector components and subscripts such as $\partial_i$ to denote derivatives with respect to the components $x,y,z$ (or $X,Y,Z$) with the obvious identification $\partial_1 = \partial_X$, $\partial_2 = \partial_Y$, and $\partial_3 = \partial_Z$. Summation notation is assumed: in a product, repeated vector and differentiation indices are always summed over all possible values.
See Appendix \[apx:Gev\] for the Fourier analysis conventions we are taking. A convention we generally use is to denote the discrete $x$ (or $X$) frequencies as subscripts. By convention we always use Greek letters such as $\eta$ and $\xi$ to denote frequencies in the $y$ or $Y$ direction, frequencies in the $x$ or $X$ direction as $k$ or $k^\prime$ etc, and frequencies in the $z$ or $Z$ direction as $l$ or $l^\prime$ etc. Another convention we use is to denote dyadic integers by $M,N \in 2^{{\mathbb Z}}$ where $$\begin{aligned}
2^{\mathbb Z}& = {\left\{...,2^{-j},...,\frac{1}{4},\frac{1}{2},1,2,...,2^j,...\right\}}, \\ \end{aligned}$$ This will be useful when defining Littlewood-Paley projections and paraproduct decompositions. See §\[sec:paranote\] for more information on the paraproduct decomposition and the associated short-hand notations we employ. Given a function $m \in L^\infty_{loc}$, we define the Fourier multiplier $m({\nabla}) f$ by $$\begin{aligned}
(\widehat{m({\nabla})f})_k(\eta) = m( (ik,i\eta,il) ) \widehat{f_k}(\eta,l). \end{aligned}$$ We use the notation $f \lesssim g$ when there exists a constant $C > 0$ independent of the parameters of interest such that $f \leq Cg$ (we analogously define $f \gtrsim g$). Similarly, we use the notation $f \approx g$ when there exists $C > 0$ such that $C^{-1}g \leq f \leq Cg$. We sometimes use the notation $f \lesssim_{\alpha} g$ if we want to emphasize that the implicit constant depends on some parameter $\alpha$. We also employ the shorthand $t^{\alpha+}$ when we mean that there is some small parameter $\gamma >0$ such that $t^{\alpha+\gamma}$ and that we can choose $\gamma$ as small as we want at the price of a constant (e.g. ${\left\lVert f \right\rVert}_{L^\infty} \lesssim {\left\lVert f \right\rVert}_{H^{3/2+}}$). We will denote the $\ell^1$ vector norm ${\left\vertk,\eta,l\right\vert} = {\left\vertk\right\vert} + {\left\vert\eta\right\vert} + {\left\vertl\right\vert}$, which by convention is the norm taken in our work. Similarly, given a scalar or vector in ${\mathbb R}^n$ we denote $$\begin{aligned}
{\left\langle v \right\rangle} = \left( 1 + {\left\vertv\right\vert}^2 \right)^{1/2}. \end{aligned}$$ We denote the standard $L^p$ norms by ${\left\lVert f \right\rVert}_{p}$ and Sobolev norms ${\left\lVert f \right\rVert}_{H^\sigma} := {\left\lVert {\left\langle {\nabla}\right\rangle}^\sigma f \right\rVert}_2$. We make common use of the Gevrey-$\frac{1}{s}$ norm with Sobolev correction defined by $$\begin{aligned}
{\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda,\sigma;s}}^2 = \sum_{k,l}\int {\left\vert\widehat{f_k}(\eta,l)\right\vert}^2 e^{2\lambda{\left\vertk,\eta,l\right\vert}^s}{\left\langle k,\eta,l \right\rangle}^{2\sigma} d\eta. \end{aligned}$$ Since in most of the paper we are taking $s$ as a fixed constant, it is normally omitted. Also, if $\sigma =0$, it is omitted. We refer to this norm as the $\mathcal{G}^{\lambda,\sigma;s}$ norm and occasionally refer to the space of functions $$\begin{aligned}
\mathcal{G}^{\lambda,\sigma;s} = {\left\{f \in L^2 :{\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda,\sigma;s}}<\infty\right\}}. \end{aligned}$$ See Appendix \[apx:Gev\] for a discussion of the basic properties of this norm and some related useful inequalities.
For $\eta \geq 0$, we define $E(\eta)\in {\mathbb Z}$ to be the integer part. We define for $\eta \in {\mathbb R}$ and $1 \leq {\left\vertk\right\vert} \leq E(\sqrt{{\left\vert\eta\right\vert}})$ with $\eta k > 0$, $t_{k,\eta} = {\left\vert\frac{\eta}{k}\right\vert} - \frac{{\left\vert\eta\right\vert}}{2{\left\vertk\right\vert}({\left\vertk\right\vert}+1)} = \frac{{\left\vert\eta\right\vert}}{{\left\vertk\right\vert}+1} + \frac{{\left\vert\eta\right\vert}}{2{\left\vertk\right\vert}({\left\vertk\right\vert}+1)}$ and $t_{0,\eta} = 2 {\left\vert\eta\right\vert}$ and the critical intervals $$\begin{aligned}
I_{k,\eta} = \left\{
\begin{array}{lr}
[t_{{\left\vertk\right\vert},\eta},t_{{\left\vertk\right\vert}-1,\eta}] & \textup{ if } \eta k \geq 0 \textup{ and } 1 \leq {\left\vertk\right\vert} \leq E(\sqrt{{\left\vert\eta\right\vert}}), \\
\emptyset & otherwise.
\end{array}
\right. \end{aligned}$$ For minor technical reasons, we define a slightly restricted subset as the *resonant intervals* $$\begin{aligned}
\mathbf I_{k,\eta} = \left\{
\begin{array}{lr}
I_{k,\eta} & 2\sqrt{{\left\vert\eta\right\vert}} \leq t_{k,\eta}, \\
\emptyset & otherwise.
\end{array}
\right. \end{aligned}$$ Note this is the same as putting a slightly more stringent requirement on $k$: $k \leq \frac{1}{2}\sqrt{{\left\vert\eta\right\vert}}$.
Outline of the proof {#sec:proof}
====================
In this section we give an outline of the main steps of the proof of Theorem \[thm:SRS\] and set up the main energy estimates, focusing on exposition, intuition, and organization. We will try to give specific emphasis to what is new relative to [@BGM15I], and discuss fewer details on issues that are common to both works for the sake of brevity. After §\[sec:proof\], the remainder of the paper is dedicated to the proof of the major energy estimates required and the analysis of the various norms and Fourier analysis tools being employed.
Summary and weakly nonlinear heuristics
---------------------------------------
### New dependent variables
As in [@BGM15I], we find it natural to define the full set of auxiliary unknowns $q^i = \Delta u^i$ for $i = 1,2,3$. A computation shows that $(q^i)$ solves $$\label{def:qi}
\left\{ \begin{array}{l}
\partial_t q^1 + y \partial_x q^1 + 2\partial_{xy} u^1 + u \cdot {\nabla}q^1 = -q^2 + 2\partial_{xx} u^2 - q^j \partial_j u^1 + \partial_x\left(\partial_i u^j \partial_j u^i\right) - 2\partial_{i} u^j \partial_{ij}u^1 + \nu \Delta q^1 \\
\partial_t q^2 + y \partial_x q^2 + u \cdot {\nabla}q^2 = -q^j \partial_j u^2 + \partial_y\left(\partial_i u^j \partial_j u^i\right) - 2\partial_{i} u^j \partial_{ij}u^2 + \nu \Delta q^2 \\
\partial_t q^3 + y \partial_x q^3 + 2\partial_{xy} u^3 + u \cdot {\nabla}q^3 = 2\partial_{zx} u^2 -q^j \partial_j u^3 + \partial_z\left(\partial_i u^j \partial_j u^i\right) - 2\partial_{i} u^j \partial_{ij}u^3 + \nu \Delta q^3 .
\end{array}\right.$$ Note that the linear terms have disappeared in the PDE for $q^2$ but not $q^1$ and $q^3$.
### New independent variables {#sec:indepC}
As in [@BGM15I], the need for a change of independent variables can be understood by considering the convection term $y\partial_x q^i + u \cdot \nabla q^i$ which appears in above. Due to the mixing, any classical energy estimates on $q$ in (say) Sobolev spaces will rapidly grow. Via the lift-up effect, $u_0^1$ will be very large, whereas even the other contributions of the streak, $u_0^{2,3}$, will not be decaying and cannot be balanced by the dissipation as they are far larger than $\nu$. More specifically, the *growth of gradients* caused by mixing due to the streak cannot be balanced. In [@BGM15I], the coordinate system was modified to account for the mixing action of $u_0^1$ (and $u_0^2$ as a by-product); here we will go further and also account for $u_0^3$, effectively treating the entire streak in a sort of Lagrangian fashion so that norm growth due to these velocities is not seen in our coordinate system.
A full study of the coordinate transformation is done in §\[sec:coordinates\] below, but let us just make a quick summary here. We start with the ansatz $$\begin{aligned}
\left\{ \begin{array}{l} X = x - ty - t \psi(t,y,z) \\ Y = y + \psi(t,y,z) \\ Z = z + \phi(t,y,z), \end{array} \right. \end{aligned}$$ The shift $\psi$ is chosen as in [@BGM15I], however $\phi$ is chosen to eliminate the contributions of $u_0^3$ from the transport term. Indeed, consider the simple convection diffusion equation on a passive scalar $f(t,x,y,z)$ $$\partial_t f + y \partial_x f + u \cdot \nabla f = \nu \Delta f.$$ Denoting $F(t,X,Y,Z) = f(t,x,y,z)$ and $U(t,X,Y,Z) = u(t,x,y,z)$, and $\Delta_t$ and $\nabla^t$ for the expressions for $\Delta$ and $\nabla$ in the new coordinates, this simple equation becomes $$\begin{aligned}
\partial_t F + \begin{pmatrix}u^1 - t (1+\partial_y\psi) u^2 - t \partial_z\psi u^3 - \frac{d}{dt}(t\psi) + \nu t \Delta \psi \\ (1+\partial_y\psi) u^2 + \partial_z \psi u^3 + \partial_t \psi - \nu\Delta \psi \\ (1 + \partial_z \phi)u^3 + \partial_y \phi u^2 + \partial_t \phi - \nu \Delta \phi \end{pmatrix} \cdot {\nabla}_{X,Y,Z} F = \nu \tilde{\Delta_t} F, \label{ineq:transf}\end{aligned}$$ where $\tilde{\Delta_t}$ is a variant of $\Delta_t$ without lower order terms; it is given below in . To eliminate the zero frequency contribution of the first component of the velocity field, as in [@BGM15I], we will choose $u^1_0 - t (1+\partial_y\psi) u^2_0 - t \partial_z\psi u^3_0 - \frac{d}{dt}(t\psi) + \nu t\Delta \psi = 0$. To eliminate the zero frequency contribution of the third component, we further choose $(1 + \partial_z \phi)u_0^3 + \partial_y \phi u^2_0 + \partial_t \phi = \nu \Delta \phi$. As in [@BGM15I], we now recast the equations on $\psi,\phi$ in terms of $C^1(t,Y,Z)=\psi(t,y,z)$, $C^2(t,Y,Z)=\phi(t,y,z)$ and the auxiliary unknown $g = \frac{1}{t}(U_0^1 - C)$. After cancellations are carefully accounted for we have $$\label{def:Cgintro}
\left\{
\begin{array}{l}
\partial_t C^1 + \tilde U_0 \cdot {\nabla}_{Y,Z} C^1 = g - U_0^2 + \nu \tilde{\Delta_t} C^1, \\
\partial_t C^2 + \tilde U_0 \cdot {\nabla}_{Y,Z} C^2 = - U_0^3 + \nu \tilde{\Delta_t} C^2, \\
\partial_t g + \tilde U_0 \cdot {\nabla}_{Y,Z}g = -\frac{2}{t}g -\frac{1}{t} \left(U_{\neq} \cdot {\nabla}^t U^1_{\neq}\right)_0 + \nu \tilde{\Delta_t} g,
\end{array}
\right.$$ and $$\begin{aligned}
\label{def:Qiintro}
\left\{
\begin{array}{l}
Q^1_t + \tilde U \cdot {\nabla}_{X,Y,Z} Q^1 = -Q^2 - 2\partial_{XY}^t U^1 + 2\partial_{XX} U^2 - Q^j \partial_j^t U^1 - 2\partial_i^t U^j \partial_{ij}^t U^1 + \partial_X(\partial_i^t U^j \partial_j^t U^i) + \nu \tilde{\Delta_t} Q^1 \\
Q^2_t + \tilde U \cdot {\nabla}_{X,Y,Z} Q^2 = -Q^j \partial_j^t U^2 - 2\partial_i^t U^j \partial_{ij}^t U^2 + \partial_Y^t(\partial_i^t U^j \partial_j^t U^i) + \nu \tilde{\Delta_t} Q^2 \\
Q^3_t + \tilde U \cdot {\nabla}_{X,Y,Z} Q^3 = -2\partial_{XY}^t U^3 + 2\partial_{XZ}^t U^2 - Q^j \partial_j^t U^3 - 2\partial_i^t U^j \partial_{ij}^t U^3 + \partial_Z^t(\partial_i^t U^j \partial_j^t U^i) + \nu \tilde{\Delta_t} Q^3,
\end{array}
\right.\end{aligned}$$ where $\partial_i^t$ denote derivatives including the Jacobian factors $\partial_z \psi,\partial_y \psi, \partial_y\phi,\partial_z\phi$ (see §\[sec:coordinates\] below) and $$\tilde U = \begin{pmatrix} U_{\neq}^1 - t(1 + \partial_y\psi ) U^2_{\neq} - t \partial_z\psi U^3_{\neq} \\ (1 + \partial_y\psi)U^2_{\neq} + \partial_z\psi U^3_{\neq}+ g \\ (1 + \partial_z\phi)U^3_{\neq} + \partial_y\phi U_{\neq}^2 \end{pmatrix}.$$ Notice that this transformation almost completely eliminates the zero frequency contribution of $\tilde U_0$, so we are treating the advection by the evolving streak $u_0^1(t,y,z),u_0^2(t,y,z),u_0^3(t,y,z)$ in a nearly Lagrangian way (as in [@BGM15I], $g$ is rapidly decaying independently of $\nu$).
Choice of the norms
-------------------
The highest norms we use are of the general type ${\left\lVert A^i(t,{\nabla}) Q^i(t) \right\rVert}_2$, where the $A^i$ are specially designed Fourier multipliers. See below for the definitions of $A^i$. For $i = 1,2$ the norms are similar to [@BGM15I], however, here they need to be adjusted at high frequencies in $Z$. For $i = 3$ the difference is more pronounced as the $w$ multiplier is replaced with a specially adjusted $w^3$. Recall that these factors are estimates on the “worst-possible” growth of high frequencies due to weakly nonlinear effects. Roughly speaking, here they are taken to satisfy the following for ${\left\vertk\right\vert}^2 \lesssim {\left\vert\eta\right\vert}$ (and hence $\sqrt{{\left\vert\eta\right\vert}} \lesssim t \lesssim {\left\vert\eta\right\vert}$),
\[def:approxw\] $$\begin{aligned}
\frac{\partial_t {w(t,\eta)}}{w(t,\eta)}& \sim \frac{1}{1 + |t-\frac{\eta}{k}|}, \qquad \mbox{when $\left| t-\frac{\eta}{k} \right| \lesssim \frac{\eta}{k^2}$} \quad \mbox{and} \quad w(1,\eta) = 1 \\
w^3_k(t,\eta) & \sim w(t,\eta), \qquad \mbox{when $\left| t-\frac{\eta}{k} \right| \lesssim \frac{\eta}{k^2}$} \\
w^3_{k^\prime}(t,\eta) & \sim \frac{t}{{\left\vertk\right\vert} + {\left\vert\eta - kt\right\vert}} w(t,\eta), \qquad \mbox{when $\left| t-\frac{\eta}{k} \right| \lesssim \frac{\eta}{k^2}$} \quad \mbox{and} \quad k \neq k^\prime; \end{aligned}$$
see Appendix \[sec:def\_nrm\] for the full definition and §\[sec:Toy\] for the heuristic derivation. We see that $w^3$ unbalances the regularity in a way that enforces more control over frequencies near the critical times than away from the critical times. This is closely matched by the loss of ellipticity in $\Delta_L$ and allows to trade ellipticity and regularity back and forth in a specific way.
Finally, as pointed out in [@BGM15I], one can read off the requirement $s > 1/2$ from . Indeed, integration over each critical time gives for some $C >0$, $$\begin{aligned}
\frac{w(2\eta,\eta)}{w(\sqrt{\eta},\eta)} \approx \left(\frac{\eta^{\sqrt{\eta}}}{(\sqrt{\eta}!)^2}\right)^C, \label{ineq:wloss}\end{aligned}$$ which predicts a growth like $O(e^{2C\sqrt{\eta}})$ up to a polynomial correction by Stirling’s formula.
### Weakly nonlinear heuristics {#sec:NonlinHeuristics}
First, let us point out another heuristic for deriving the requirement $\epsilon \lesssim \nu^{2/3}$. Many nonlinear terms in the proof are naturally estimated in the following general manner: $$\begin{aligned}
NL & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^i Q^i \right\rVert}_2{\left\lVert A^j Q^j \right\rVert}_2 \lesssim \nu{\left\lVert \sqrt{-\Delta_L} A^i Q^i \right\rVert}_2^2 + \frac{\epsilon^2}{\nu {\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert A^j Q^j \right\rVert}_2^2 \label{ineq:32heurs} \end{aligned}$$ where recall from §\[sec:LinStreak\] that $\Delta_L = \partial_{XX} + (\partial_Y - t\partial_X)^2 + \partial_{ZZ}$, the leading order dissipation that comes from the linearized problem. The ${\left\langle \nu t^3 \right\rangle}^{-\alpha}$ comes from a ‘low-frequency’ factor that was estimated via the enhanced dissipation. Since $\int_0^\infty \frac{1}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} dt \approx \nu^{-1/3}$, it is apparent that $\nu \sim \epsilon^{3/2}$ is the smallest choice of $\nu$ such that can be integrated uniformly in $\nu \rightarrow 0$.
Now, let us quickly recall some terminology from [@BGM15I] and some discussion on the weakly nonlinear effects. The behavior in Theorem \[thm:SRS\] comes in essentially two phases. During early times $t \lesssim \tau_{NL} \sim \epsilon^{-1/2}$, the solution has strong 3D effects and the dissipation cannot control the leading order nonlinear terms. On this time scale, the regularity unbalancing in $w^3$ and insight from the toy model of §\[sec:Toy\] is crucial. After times $t \gtrsim \tau_{ED} \sim \nu^{-1/3}$ the enhanced dissipation begins to dominate and the solution converges to a streak; the main growth from then on is due to the lift-up effect. The assumption of $\epsilon \leq \nu^{2/3+\delta}$ is what ensures the two regimes overlap since then $\tau_{NL} \gtrsim \tau_{ED}$; moreover since $\delta > 0$, by picking $\nu$ small we can make sure that the overlap regime is large (that is, we can ensure $\tau_{NL} \gg \tau_{ED}$ so the dissipation dominates comfortably before the nonlinear time-scale).
As in [@BGM15I], we classify the nonlinear terms by the zero, or nonzero, $x$ frequency of the interacting functions: denote for instance $0 \cdot \neq \,\to\, \neq$ for the interaction of a zero mode (in $x$) and a non-zero mode (in $x$) giving a non-zero mode (in $x$), and similarly, with obvious notations, $0 \cdot 0 \to 0$, $\neq \cdot \neq \, \to \,\neq$, and $\neq \cdot \neq \,\to 0$.
- ($0 \cdot 0 \to 0$) For *2.5D Navier-Stokes*, this corresponds to self-interactions of the streak. We will see that there are new complexities to these terms here: due to the regularity imbalancing in $w^3$, the regularity of $u_0^3$ and $u_0^2$ are not the same and terms that were straightforward in [@BGM15I] are not so here.
- ($0 \cdot \neq \,\to\, \neq$) For *secondary instability*, this effect is the transfer of momentum from the large $u_0^1$ mode to other modes. Actually, even more here than in [@BGM15I], $u_0^2$ and $u_0^3$ will matter; especially the latter due to the regularity unbalances in $w^3$. These interactions are those that would arise when linearizing an $x$-dependent perturbation of a streak and so are what ultimately give rise to the secondary instabilities observed in larger streaks (hence the terminology) [@ReddySchmidEtAl98; @Chapman02].
- ($\neq \cdot \neq \,\to \,\neq$) For *three dimensional echoes*, these effects are 3D variants of the 2D hydrodynamic echo phenomenon as observed in [@YuDriscoll02; @YuDriscollONeil]. These are understood as weakly nonlinear interactions of $x$-dependent modes forcing unmixing modes [@VMW98; @Vanneste02; @BM13]. We will see in §\[sec:Toy\] that these are the primary reason for the regularity imbalances in $w^3$ and hence are the source of most of the additional difficulties in the proof of Theorem \[thm:SRS\]. This involves two non-zero frequencies $k_1$, $k_2$ interacting to force mode $k_1 + k_2$ with $k_{1},k_2,k_1 + k_2 \neq 0$.
- ($\neq \cdot \neq \,\to 0$) For *nonlinear forcing*, this is the effect of the forcing from $x$-dependent modes back into $x$-independent modes. This involves two non-zero frequencies $k$ and $-k$ interacting to force a zero frequency (and as usual, in general this could involve a variety of the components). Similar to **(3DE)**, it is $u_0^3$ that is most strongly affected by these terms, and it is these that are responsible for altering the regularity of $u_0^3$ relative to $u_0^2$.
As in [@BGM15I], these nonlinear interactions are coupled to one another and can precipitate nonlinear cascades. The need to consider possible nonlinear bootstraps both precipitates the Gevrey-$2$ regularity requirement as in [@BGM15I] and the regularity imbalances in $u^3$, as we will derive formally in §\[sec:Toy\].
We will now begin a detailed outline of the proof of Theorem \[thm:SRS\] and set up the main energy estimates that will comprise the majority of the paper.
Instantaneous regularization and continuation of solutions
----------------------------------------------------------
The first step is to see that our initial data becomes small in ${\mathcal{G}}^{\frac{3\lambda_0}{4} + \frac{\lambda^{\prime}}{4}}$ after a short time. We state without proof the appropriate lemma, see [@BMV14; @BGM15I] for analogous lemmas.
\[lem:Loc\] Let $u_{in} \in L^2$ satisfy . Then for all $\nu\in (0,1]$, $c_{0}$ sufficiently small, $K_0$ sufficiently large, and all $\lambda_0 > \lambda^\prime > 0$, if $u_{in}$ satisfies , then there exists a time $t_\star = t_\star(s,K_0,\lambda_0,\lambda^\prime) > 0$ and a unique classical solution to with initial data $u_{in}$ on $[0,t_\star]$ which is real analytic on $(0,t_\star]$, and satisfies $$\begin{aligned}
\sup_{t \in [t_\star/2,t_\star]} {\left\lVert u(t) \right\rVert}_{{\mathcal{G}}^{\bar{\lambda}}} \leq 2\epsilon, \end{aligned}$$ where $\bar{\lambda} = \frac{3\lambda_0}{4} + \frac{\lambda^\prime}{4}$.
Once we have a solution we want to be able to continue it and ensure that it propagates analyticity based on low norm controls. This will allow us to rigorously justify our a priori estimates and that these a priori estimates allow us to propagate the solutions. See [@BGM15I] for more discussion. We remark here that analyticity itself is not important, we only need a regularity class which is a few derivatives stronger than the regularities we work in below, so that we may easily justify that the norms applied to the solution take values continuously in time.
\[lem:Cont\] Let $T > 0$ be such that the classical solution $u(t)$ to constructed in Lemma \[lem:Loc\] exists on $[0,T]$ and is real analytic for $t \in (0,T]$. Then there exists a maximal time of existence $T_0$ with $T < T_0 \leq \infty$ such that the solution $u(t)$ remains unique and real analytic on $(0,T_0)$. Moreover, if for some $\tau \leq T_0$ and $\sigma > 5/2$ we have $\limsup_{t \nearrow \tau} {\left\lVert u(t) \right\rVert}_{H^{\sigma}} < \infty$, then $\tau < T_0$.
$Q^i$ formulation, the coordinate transformation, and some key cancellations {#sec:coordinates}
----------------------------------------------------------------------------
As in [@BGM15I], we remove the fast mixing action of *both* the Couette flow *and* $u_0^1(t)$. However, we go further and essentially treat the entire streak in a Lagrangian way so that we do not see the large gradient growth due to the zero frequencies in the velocity field. In this work we need:
1. to control the regularity loss due to transport effects in our special set of of norms until $t \sim \epsilon^{-1}$;
2. to be able to treat the Laplacian in the new coordinates as a perturbation from $\Delta_L$, so that we can take advantage of the inviscid damping and enhanced dissipation effects;
3. to be able to make practical estimates on the behavior of the coordinate system and the coordinate transformation needs to treat the dissipation in a natural way, instead of losing derivatives.
The latter two are the same as [@BGM15I] but the first one is potentially far more difficult since the streak is far larger than $\nu$ and so cannot be balanced by viscous effects. The middle requirement suggests the form
\[def:XYZ\] $$\begin{aligned}
X & = x - ty - t \psi(t,y,z) \\
Y & = y + \psi(t,y,z) \\
Z & = z + \phi(t,y,z), \end{aligned}$$
however, unlike [@BGM15I], we will not take $\phi = 0$. Provided $\psi$ and $\phi$ is sufficiently small in a suitable sense, one can invert for $x,y,z$ as functions of $X,Y,Z$ (see §\[sec:RegCont\] and [@BGM15I] for more information). In keeping with the notation in [@BGM15I] , denote the Jacobian factors (by abuse of notation), $$\begin{aligned}
\psi_t(t,Y,Z) & = \partial_t \psi(t,y(t,Y,Z),z(t,Y,Z)) \\
\psi_y(t,Y,Z) & = \partial_y \psi(t,y(t,Y,Z),z(t,Y,Z)) \\
\psi_z(t,Y,Z) & = \partial_z \psi(t,y(t,Y,Z),z(t,Y,Z)) \\
\phi_t(t,Y,Z) & = \partial_t \phi(t,y(t,Y,Z),z(t,Y,Z)) \\
\phi_y(t,Y,Z) & = \partial_y \phi(t,y(t,Y,Z),z(t,Y,Z)) \\
\phi_z(t,Y,Z) & = \partial_z \phi(t,y(t,Y,Z),z(t,Y,Z)). \end{aligned}$$ In what follows we will usually omit the arguments of $y(t,Y,Z)$ and $z(t,Y,Z)$ and use a more informal notation, such as $\psi_t(t,Y,Z) = \partial_t \psi(t,y,z)$.
Define the following notation for the $(x,y,z)$ derivatives in the new coordinate systems
$$\begin{aligned}
\partial_X^t & = \partial_X \\
\partial_Y^t & = (1 + \psi_y)(\partial_Y - t\partial_X) + \phi_y \partial_Z \\
\partial_Z^t & = (1 + \phi_z) \partial_Z + \psi_z(\partial_Y - t\partial_X) \\
{\nabla}^t & = (\partial_X, \partial_Y^t, \partial_Z^t)^{T}. \end{aligned}$$
Note that these necessarily commute. Consider the transport of a passive scalar by a perturbation of the Couette flow: $$\begin{aligned}
\partial_t f + y \partial_x f + u\cdot {\nabla}f = \nu \Delta f. \label{def:ftrans}\end{aligned}$$ Denoting $F(t,X,Y,Z) = f(t,x,y,z)$, the transport equation in the new coordinate system is given by $$\begin{aligned}
\partial_t F + \begin{pmatrix}u^1 - t (1+\partial_y\psi) u^2 - t \partial_z\psi u^3 - \frac{d}{dt}(t\psi) + t\nu \Delta \psi \\ (1+\partial_y\psi) u^2 + \partial_z \psi u^3 + \partial_t \psi - \nu\Delta \psi \\ (1+\partial_z \phi)u^3 + \partial_y\phi u^2 + \partial_t \phi - \nu \Delta \phi \end{pmatrix} \cdot {\nabla}_{X,Y,Z} F = \nu \tilde{\Delta_t} F, \label{def:transf2}\end{aligned}$$ where the upper-case letters are evaluated at $(X,Y,Z)$ and the lower case letters are evaluated at $(x,y,z)$ and we are denoting
\[def:Deltat\] $$\begin{aligned}
\Delta_t F & = \partial_{XX} + \partial_Y^t\left(\partial_Y^t F\right) + \partial_Z^t\left(\partial_Z^t F\right) \\
\tilde{\Delta_t}F & = \Delta_t F - \Delta \psi (\partial_Y - t\partial_X)F - \Delta \phi \partial_Z F. \label{def:tildeDel1}\end{aligned}$$
Eliminating the zero frequency of the first component of the velocity field in provides the requirement on $\psi$ (the same as in [@BGM15I]), $$\begin{aligned}
u_0^1 - t\left( 1 + \partial_y \psi \right)u_0^2 - t \partial_z \psi u_0^3 - \frac{d}{dt}(t\psi) & = -\nu t\Delta \psi. \label{def:psi2} \end{aligned}$$
In [@BGM15I], $\phi$ was chosen to be zero for simplicity and the transport due $u_0^3$ was absorbed by the dissipation. Even with no dissipation at all, in standard regularity classes one could attempt to deal with $u_0^3$ until $t \sim \epsilon^{-1}$ by using the commutator trick employed in e.g. [@LevermoreOliver97; @KukavicaVicol09], however, armed with our complicated norms, which in particular, have a non-trivial angular dependence in frequency, this could become hard (see [@BM13] for what kind of issues could arise). Instead, we will shift our coordinate system along with $u_0^3$ by eliminating the third component of the velocity field in via: $$\begin{aligned}
(1+\partial_z \phi)u^3_0 + \partial_y\phi u^2_0 + \partial_t \phi = \nu \Delta \phi, \label{def:phi}\end{aligned}$$ which, as mentioned above, is effectively a Lagrangian treatment of the background streak. Below we denote $$\begin{aligned}
C^1(t,Y,Z) & = \psi(t,y,z)\\
C^2(t,Y,Z) & = \phi(t,y,z) \\
C(t,Y,Z) & = (C^1(t,Y,Z), C^2(t,Y,Z))^{T}. \end{aligned}$$ From the chain rule we derive:
\[eq:psiyzt\] $$\begin{aligned}
\psi_y & = \partial_Y^tC^1 = \left(1 + \psi_y\right) \partial_Y C^1 + \phi_y \partial_Z C^1 \\
\psi_z & = \partial_Z^t C^1 = (1 + \phi_z) \partial_Z C^1 + \psi_z \partial_YC^1 \\
\phi_y & = \partial_Y^tC^2 = \left(1 + \psi_y\right) \partial_Y C^2 + \phi_y \partial_Z C^2 \\
\phi_z & = \partial_Z^t C^2 = (1 + \phi_z) \partial_Z C^2 + \psi_z \partial_YC^2 \\
\psi_t & = \partial_t C^1 + \psi_t \partial_YC^1 + \phi_t \partial_ZC^1 \label{ineq:psit} \\
\phi_t & = \partial_t C^2 + \psi_t \partial_YC^2 + \phi_t \partial_ZC^2. \label{ineq:phit} \end{aligned}$$
Analogous to [@BGM15I], we will get estimates on $C^i$ and use them to deduce estimates on $\psi$ and $\phi$. This necessitates solving for $\psi_y,\psi_z,\phi_y,\phi_z$ – note that these form a $4 \times 4$ linear system: $$\begin{aligned}
\begin{pmatrix}
1-\partial_YC^1 & 0 & -\partial_Z C^1 & 0 \\
0 & 1-\partial_YC^1 & 0 & -\partial_Z C^1 \\
-\partial_YC^2 & 0 & 1-\partial_ZC^2 & 0 \\
0 & -\partial_Y C^2 & 0 & 1-\partial_ZC^2
\end{pmatrix}
\begin{pmatrix}
\psi_y \\
\psi_z \\
\phi_y \\
\phi_z
\end{pmatrix}
=
\begin{pmatrix}
\partial_Y C^1 \\
\partial_Z C^1 \\
\partial_Y C^2 \\
\partial_Z C^2
\end{pmatrix}. \end{aligned}$$ For ${\nabla}C^i$ sufficiently small we can solve the linear system and derive
\[def:psi2sqrBrack\] $$\begin{aligned}
\phi_z & = \frac{\left(\partial_Z C^2 + \frac{\partial_Y C^2 \partial_Z C^1}{1 - \partial_Y C^1} \right)}{1 - \left(\partial_Z C^2 + \frac{\partial_Y C^2 \partial_Z C^1}{1 - \partial_Y C^1} \right) } = \sum_{n=1}^\infty \left(\partial_Z C^2 + \frac{\partial_Y C^2 \partial_Z C^1}{1 - \partial_Y C^1} \right)^n \\
\phi_y & = \frac{\partial_Y C^2}{\left(1 - \partial_Y C^1\right)\left(1 - \left(\partial_Z C^2 + \frac{\partial_Y C^2 \partial_Z C^1}{1 - \partial_Y C^1}\right) \right)} \\
\psi_z & = \frac{(1 + \phi_z) \partial_Z C^1}{1 - \partial_Y C^1} \\
\psi_y & = \frac{\partial_Y C^1 + \partial_Z^1 C \phi_y}{1 - \partial_Y C^1}; \end{aligned}$$
The precise form of is not interesting and it is straightforward to recover estimates on the Jacobian factors from estimates on $C^i$ using and the appropriate product rules. Note that $\Delta_t C^1 = \Delta \psi$ and $\Delta_t C^2 = \Delta \phi$, and hence $$\begin{aligned}
\Delta_t f & = \tilde{\Delta_t}f + \Delta_t C^1 (\partial_Y - t\partial_X)f + \Delta_tC^2 \partial_Zf. \label{eq:tildeDeltaf}\end{aligned}$$ From the chain rule together with , , and , we derive
$$\begin{aligned}
\partial_t C^1 + \begin{pmatrix} \left(1 + \psi_y\right)U_0^2 + \psi_zU^3_0 + \psi_t - \nu\Delta_tC^1 \\ (1 + \phi_z)U_0^3 + \phi_y U_0^2 + \phi_t - \nu\Delta_t C^2 \end{pmatrix} \cdot {\nabla}C^1 & = \frac{1}{t}\left(U_0^1 - tU_0^2 - C^1\right) + \nu \tilde{\Delta_t} C^1 \label{def:C} \\
\partial_t C^2 + \begin{pmatrix} \left(1 + \psi_y\right)U_0^2 + \psi_zU^3_0 + \psi_t - \nu \Delta_t C^1 \\ (1 + \phi_z)U_0^3 + \phi_y U_0^2 + \phi_t - \nu\Delta_tC^2 \end{pmatrix} \cdot {\nabla}C^2 & = -U_0^3 + \nu \tilde{\Delta_t} C^2. \label{def:C2}\end{aligned}$$
As in [@BGM15I], we will define another auxiliary unknown $g$, $$\begin{aligned}
g = \frac{1}{t}\left(U_0^1 - C^1\right), \label{def:g}\end{aligned}$$ which, as in [@BGM15I], roughly speaking, measures the time oscillations of $U_0^1$ and satisfies $$\begin{aligned}
\partial_t g + \begin{pmatrix} (1 + \psi_y)U_0^2 + \psi_z U_0^3 + \psi_t - \nu \Delta_t C^1 \\ (1 + \phi_z)U_0^3 + \phi_y U_0^2 + \phi_t - \nu\Delta_tC^2 \end{pmatrix} \cdot {\nabla}_{Y,Z}g = -\frac{2g}{t} -\frac{1}{t} \left(\tilde{U}_{\neq} \cdot {\nabla}U^1_{\neq}\right)_0 + \nu \tilde{\Delta_t} g. \label{def:gPDE1}\end{aligned}$$ Next, from , , , and , we derive
$$\begin{aligned}
\psi_t & = g - U_0^2 - \begin{pmatrix} \left(1 + \psi_y\right)U_0^2 + \psi_z U^3_0 \\ (1 + \phi_z)U_0^3 + \phi_y U_0^2 \end{pmatrix} \cdot {\nabla}C^1 + \nu\Delta_t C^1 \label{def:dtpsi} \\
\phi_t & = -U_0^3 - \begin{pmatrix} \left(1 + \psi_y\right)U_0^2 + \psi_z U^3_0 \\ (1 + \phi_z)U_0^3 + \phi_y U_0^2 \end{pmatrix} \cdot {\nabla}C^2 + \nu \Delta_t C^2. \label{def:dtphi}\end{aligned}$$
and equivalently, from ,
$$\begin{aligned}
\psi_t + (1 + \psi_y) U_0^2 + \phi_z U_0^3 & = g + \nu \Delta_t C^1 \\
\phi_t + (1 + \phi_z)U_0^3 + \phi_y U_0^2 & = \nu \Delta_t C^2. \end{aligned}$$
Deriving the resulting cancellations as in [@BGM15I], we have that the following velocity field will ultimately govern our equations: $$\begin{aligned}
\tilde U = \tilde U_0 + \tilde U_{\neq } = \begin{pmatrix} 0 \\ g \\ 0 \end{pmatrix} + \begin{pmatrix} U_{\neq}^1 - t(1 + \psi_y ) U^2_{\neq} - t \psi_z U^3_{\neq} \\ (1 + \psi_y)U^2_{\neq} + \psi_z U^3_{\neq}\\ (1 + \phi_z)U^3_{\neq} + \phi_y U^2_{\neq} \end{pmatrix}. \label{def:tildeU2} \end{aligned}$$ We also derive the governing equations
\[def:CReal\] $$\begin{aligned}
\partial_t C^1 + g\partial_Y C^1 & = g - U_0^2 + \nu \tilde{\Delta_t} C^1 \label{def:C1Real} \\
\partial_t C^2 + g\partial_Y C^2 & = -U_0^3 + \nu \tilde{\Delta_t} C^2, \label{def:C2Real} \end{aligned}$$
and $$\begin{aligned}
\partial_t g + g \partial_Y g = -\frac{2g}{t} -\frac{1}{t}\left(\tilde{U}_{\neq} \cdot {\nabla}U^1_{\neq}\right)_0 + \nu \tilde{\Delta_t} g. \label{def:gPDE2}\end{aligned}$$ Further notice that the forcing term from non-zero frequencies can be written as $$\begin{aligned}
\left(\tilde{U}_{\neq} \cdot {\nabla}U^1_{\neq}\right)_0 = \left(U_{\neq} \cdot {\nabla}^t U^1_{\neq}\right)_0. \end{aligned}$$ Furthermore, as in [@BGM15I] we have, denoting $Q^i(t,X,Y,Z) = q^i(t,x,y,z)$: $$\begin{aligned}
\label{def:MainSys}
\left\{
\begin{array}{l}
Q^1_t + \tilde U \cdot {\nabla}Q^1 = -Q^2 - 2\partial_{XY}^t U^1 + 2\partial_{XX} U^2 - Q^j \partial_j^t U^1 - 2\partial_i^t U^j \partial_{ij}^t U^1 + \partial_X(\partial_i^t U^j \partial_j^t U^i) + \nu \tilde{\Delta_t} Q^1 \\
Q^2_t + \tilde U \cdot {\nabla}Q^2 = -Q^j \partial_j^t U^2 - 2\partial_i^t U^j \partial_{ij}^t U^2 + \partial_Y^t(\partial_i^t U^j \partial_j^t U^i) + \nu \tilde{\Delta_t} Q^2 \\
Q^3_t + \tilde U \cdot {\nabla}Q^3 = -2\partial_{XY}^t U^3 + 2\partial_{XZ}^t U^2 - Q^j \partial_j^t U^3 - 2\partial_i^t U^j \partial_{ij}^t U^3 + \partial_Z^t(\partial_i^t U^j \partial_j^t U^i) + \nu \tilde{\Delta_t} Q^3,
\end{array}
\right.\end{aligned}$$ where we use the following to recover the velocity fields:
\[eq:Ui\] $$\begin{aligned}
U^i & = \Delta_t^{-1} Q^i \label{eq:UiFromQi} \\
\partial_i^t U^i & = 0. \end{aligned}$$
For the majority of the remainder of the proof, , together with , and , will be the main governing equations. The one exception will be in the treatment of the low frequencies of $X$ independent modes, where the use of can be problematic. For these we use $X$ averages of the momentum equation.
As in [@BGM15I], from now on we will use the following vocabulary and shorthands
$$\begin{aligned}
\tilde U \cdot {\nabla}Q^{\alpha} & = \textup{``transport nonlinearity''} & \mathcal{T} \\
-Q^j \partial_j^t U^\alpha - 2\partial_i^t U^j \partial_{ij}^t U^\alpha & = \textup{``nonlinear stretching''} & NLS\\
\partial_\alpha^t(\partial_i^t U^j \partial_j^t U^i) & = \textup{``nonlinear pressure''} & NLP\\
-2\partial_{XY}^t U^\alpha & = \textup{``linear stretching''} & LS \\
2\partial_{X\alpha}^t U^2 & = \textup{``linear pressure''} & LP \\
\left(\tilde{\Delta_t} - \Delta_L\right)Q^\alpha & = \textup{``dissipation error''} & \mathcal{D}_E; \end{aligned}$$
see [@BGM15I] for an explanation of the terminologies. As in [@BGM15I], each of the nonlinear terms will be further sub-divided into as many as four pieces in accordance with the different types of nonlinear effects described in §\[sec:NonlinHeuristics\]. Furthermore, each of the three components of the solution are qualitatively different and measured with different norms, which means certain combinations of $i$ and $j$ need to be treated specially.
As in [@BGM15I], we need to take advantage of a special structure in the equations which reduces the potential strength of interactions of type **(F)**. By considering the interaction of two non-zero frequencies, $k$ and $-k$, and putting together the contributions from transport, stretching, and nonlinear pressure we get the terms which we refer to as *forcing*, corresponding to the nonlinear interactions of type **(F)**, $$\begin{aligned}
\mathcal{F}^\alpha & := -\Delta_t \left(U^j_{\neq} \partial_j^t U^\alpha_{\neq}\right)_0 + \partial_\alpha^t \left(\partial_i^t U^j_{\neq} \partial_j^t U^i_{\neq}\right)_0 = -\partial_i^t \partial_i^t \partial_j^t \left(U^j_{\neq} U^\alpha_{\neq}\right)_0 + \partial_{\alpha}^t \partial_j^t \partial_i^t \left(U^i_{\neq} U^j_{\neq}\right)_0, \label{eq:XavgCanc}\end{aligned}$$ the advantage being that the $X$ averages remove the $-t\partial_X$ from the derivatives.
The toy model and design of the norms {#sec:Toy}
-------------------------------------
Following up on the approach discussed in [@BGM15I], in this section we want to perform a weakly nonlinear analysis and determine both $\tau_{NL}$, the characteristic time-scale associated with fully 3D nonlinear effects, and the norms with which we want to measure the solution.
Denote the Fourier dual variables of $(X,Y,Z)$ as $(k,\eta,l)$. As in [@BGM15I], a time which satisfies $kt = \eta$ is called a *critical time* (Orr’s original terminology [@Orr07]) or *resonant time* (after modern terminology [@Craik1971; @YuDriscoll02; @YuDriscollONeil; @SchmidHenningson2001]). Notice that these are precisely the points in time/frequency where $\Delta_L$ loses ellipticity in $Y$ (recall ). Recall the definition of ${\mathbf{I}}_{k,\eta}$ from §\[sec:Notation\], which denotes the resonant intervals $t \approx \frac{\eta}{k}$ with $k^2 \lesssim {\left\vert\eta\right\vert}$. This latter restriction is possible due to the uniform ellipticity of $\Delta_L$ with respect to $\partial_X$ which implies the larger the $k$, the weaker the effect of the resonance.
From [@BGM15I], we recall the toy model for the behavior of near critical times for $Q^2$ and $Q^3$ at frequency $(k,\eta,l)$ and $(k^\prime,\eta,l)$ for $kt \approx \eta$ and $k \neq k^\prime \approx k$:
\[def:Q2Q3\_ToyFinal\] $$\begin{aligned}
\partial_t \widehat{Q^2_{k}}(t,\eta,l) & = \max(\epsilon t, c_0) \frac{k}{k +{\left\vert\eta-kt\right\vert}} \widehat{Q^3_k} - \nu\left(k^2 + {\left\vert\eta-kt\right\vert}^2\right)\widehat{Q^2_k} \label{eq:Q2k} \\
\partial_t \widehat{Q^2_{k^\prime}}(t,\eta,l) & = \max(\epsilon t, c_0) \frac{k^\prime}{{\left\langle k^\prime, t \right\rangle}} \widehat{Q^3_{k^\prime}} - \nu\left(k^2 + {\left\vert\eta-kt\right\vert}^2\right)\widehat{Q^2_{k^\prime}} \label{eq:Q2kp} \\
\partial_t \widehat{Q^3_{k^\prime}}(t,\eta,l) & = \frac{\epsilon t^3}{{\left\langle \nu t^3 \right\rangle}^\alpha} \frac{\widehat{Q^{2}_{k}} }{k^2 +{\left\vert\eta-kt\right\vert}^2} - \nu\left(k^2 + {\left\vert\eta-kt\right\vert}^2\right)\widehat{Q^3_{k^\prime}} \label{eq:Q3Toykprime} \\
\partial_t \widehat{Q^3_{k}}(t,\eta,l) & = \frac{k}{k +{\left\vert\eta-kt\right\vert}}\widehat{Q^3_k} + \frac{k}{k +{\left\vert\eta-kt\right\vert}} \widehat{Q^2_k} - \nu\left(k^2 + {\left\vert\eta-kt\right\vert}^2\right)\widehat{Q^3_k} \\
\partial_t\widehat{Q^2_0}(t,\eta,l) & = \epsilon \widehat{Q^3_0} + \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\frac{\widehat{Q^2_k}}{k^2 +{\left\vert\eta-kt\right\vert}^2} - \nu \eta^2 \widehat{Q^2_0} \\
\partial_t\widehat{Q^3_0}(t,\eta,l) & = \epsilon \widehat{Q^3_0} + \frac{\epsilon t^3}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\frac{\widehat{Q^2_k}}{k^2 +{\left\vert\eta-kt\right\vert}^2} - \nu \eta^2 \widehat{Q^3_0}, \label{eq:Q3Toy0} \end{aligned}$$
where all unknowns are evaluated at frequency $(\eta,l)$.
Let us first use to get an estimate on $\tau_{NL}$. If we first consider the case $\nu = 0$, then we can estimate $\tau_{NL}$ from below if we can find an *approximate* super-solution to which will result in a reasonable regularity requirement (say analytic or weaker). Even with $\nu = 0$, we can verify that the following is a viable super-solution to over $t \in {\mathbf{I}}_{k,\eta}$ provided $\epsilon t^2 \lesssim 1$:
\[def:badnrm\] $$\begin{aligned}
\partial_t w(t,\eta) \approx \frac{1}{1 + {\left\vertt - \frac{\eta}{k}\right\vert}}w(t,\eta) \label{eq:wapprox} \\
Q^2_k \approx Q^2_{k^\prime} \approx Q_0^2 \approx w(t,\eta) \\
Q^3_k \approx Q^3_{k^\prime} \approx Q^3_{0} \approx t w(t,\eta)\end{aligned}$$
Due to the fact that *both* the resonant and non-resonant frequencies experience the same total growth $({\left\vert\eta\right\vert}{\left\vertk\right\vert}^{-2})^c$, for some $c$, for all ${\left\vertk\right\vert} \lesssim \sqrt{{\left\vert\eta\right\vert}}$, the loss is multiplicatively amplified through *each* critical time (to see this, take $k^\prime = k-1$ and consider the critical times $\eta/k,\eta/(k-1),\eta/(k-2),\ldots$). From this, one sees that this super solution predicts Gevrey-2 regularity loss (see above or [@BGM15I; @BM13] for more information). Therefore, even with no viscosity, according to the super-solution , a sufficiently regular solution could remain under control until at least $\tau_{NL} \gtrsim \epsilon^{-1/2}$. It would be more difficult to derive a good heuristic to estimate $\tau_{NL}$ from above; the toy model is designed to give robust upper bounds on the dynamics, not necessarily to make a good model for any particular realization of the true dynamics, hence even if we explicitly solved exactly, perhaps the toy model itself throws away too much information.
In order to prove Theorem \[thm:SRS\], we will need a more accurate super-solution than . Notice further that the super-solution used in [@BGM15I] does not work here due to the terms in and with $\epsilon t^3$ present. The idea is to take better advantage of the denominators in to recover the extra $t$ in the numerators of these terms. Quite precisely, we will trade one power of the denominator for a power of $t$. To do this, one must permit the regularities to become unbalanced: and both indicate that $Q^3_{k^\prime}$, for $k^\prime \neq k$ (e.g. *non-critical* or *non-resonant*) should be $t (k + {\left\vert\eta-kt\right\vert})^{-1}$ larger than $Q^2_k$. Accordingly, we see that for $\epsilon \lesssim \nu^{2/3}$ and $\epsilon t \lesssim 1$, the following is an approximate super-solution for over ${\mathbf{I}}_{k,\eta}$:
\[def:unstablesuper\] $$\begin{aligned}
\partial_t w(t,\eta) \approx \frac{1}{1 + {\left\vertt - \frac{\eta}{k}\right\vert}}w(t,\eta) \label{eq:wapprox} \\
w^{3}(t,k,\eta) = w(t,\eta) \\
w^{3}(t,k^\prime,\eta) = \frac{t}{{\left\vertk\right\vert} + {\left\vert\eta - kt\right\vert}}w(t,\eta) \\
w^{3}(t,0,\eta) = \frac{t}{{\left\vertk\right\vert} + {\left\vert\eta - kt\right\vert}} w(t,\eta) \\
Q^2_k \approx Q^2_{k^\prime} \approx Q^3_k \approx w(t,\eta) \\
Q^3_{k^\prime} \approx Q^3_{0} \approx w^3(t,k^\prime,\eta) \\
Q^1_k \approx Q^1_{k^\prime} \approx tQ^2_k. \end{aligned}$$
The last line is not deduced directly from , but is deduced (heuristically) in the derivation of via the lift-up effect (see [@BGM15I]). Notice that when $Q^2_k$ forces $Q^3_{k^\prime}$ and $Q_0^3$ near the critical time, we will gain the factor of $t^{-1}({\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert})$, precisely what is needed to exchange the $\epsilon t^3$ in the leading terms in and into $\epsilon t^2$. This suffices since $\epsilon t^2 {\left\langle \nu t^3 \right\rangle} \lesssim 1$ when $\epsilon \lesssim \nu^{2/3}$ (another equivalent way of seeing the $2/3$ threshold). The regularity loss in is peaked near the critical times, and as in [@BGM15I], we will further modify $w$ and $w^3$ to include additional steady, gradual losses of Gevrey-2 regularity over $1 \leq t \leq 2{\left\vert\eta\right\vert}$ (see in Appendix \[sec:Defw\]). This will further unify the treatment of many estimates, and its potential usefulness is also suggested by the toy model (e.g. the first term in ).
As discussed in [@BGM15I], the toy model only provides an estimate on near the critical times. For $t \gg {\left\vert\eta,l\right\vert}$ it does not apply. As in [@BGM15I], we know from Proposition \[prop:linear\] that $Q^3_{\neq}$ and $Q^1_{\neq}$ must grow quadratically at these ‘low’ frequencies due to the vortex stretching inherent in the linear problem. On the other hand, Proposition \[prop:linear\] predicts that $u^2$ decays like ${\left\langle t \right\rangle}^{-2}$, or equivalently, that $Q^2$ is uniformly bounded. This behavior was nearly preserved in the below threshold case [@BGM15I], however, it turns out that the nonlinear effects here are strong enough to possibly cause a large growth in $Q^2$. The RHS of originally came from the nonlinear pressure term in the $Q^2$ equation: $$\begin{aligned}
\partial_t Q^2_{\neq} & = \partial_Y^t\left(\partial_X U^3_{\neq} \partial_Z^t U^1_0\right) + ... \label{def:N}\end{aligned}$$ For times/frequencies with $t \gg {\left\vert{\nabla}_{Y,Z}\right\vert}$, we can ignore any issues regarding the critical times and just estimate the size of this term based on the predictions of Proposition \[prop:linear\] and we have $$\begin{aligned}
{\left\lVert \partial_t Q^2 \right\rVert} & \lesssim \frac{\epsilon^2 t^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} + ...\end{aligned}$$ Therefore, if $\epsilon \sim \nu^{2/3}$ then we predict that $Q^2$ can be at best bounded by only $\approx \epsilon {\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{-\alpha}$, which suggests a transient growth due to nonlinear effects, in contrast to [@BGM15I]. Further, this suggests the following inviscid damping/enhanced dissipation estimate: $$\begin{aligned}
{\left\lVert U^2_{\neq} \right\rVert} \lesssim \frac{\epsilon}{{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^\alpha}, \label{ineq:u2growth}\end{aligned}$$ consistent with Theorem \[thm:SRS\]. When considering the ubiquitous $U^j \partial^t_j$ and $\partial_i U^j \partial^t_j$ structure of the nonlinearity in , we see that is borderline in a certain sense. Indeed, we normally have factors like $U^2 (\partial_Y - t\partial_X)$ and so this will be just enough damping to ensure that (regularity issues aside) the $- t\partial_X$ derivatives do not completely dominate the nonlinearity and hence destroy the very special “non-resonance” structures available (indeed, this is the main role inviscid damping plays in the proof of Theorem \[thm:SRS\]). This is also another way to derive the $2/3$ threshold.
Design of the norms based on the toy model {#def:designnorm}
------------------------------------------
The above heuristics suggests that we use a set of norms which is more complicated than the norms in [@BGM15I]. The high norms will be of the following form, for a time-varying $\lambda(t)$ defined below, $s > 1/2$, $0 < \delta_1 \ll \delta$, and corrector multipliers $w$, $w^3$, and $w_L$ (here $(t,k,\eta,l)$ are now arbitrary):
\[def:A\] $$\begin{aligned}
A^Q_k(t,\eta,l) & = e^{\lambda(t){\left\vertk,\eta,l\right\vert}^s}{\left\langle k,\eta,l \right\rangle}^\sigma\frac{1}{w_L(t,k,\eta,l)}\left(\frac{e^{\mu {\left\vert\eta\right\vert}^{1/2}}}{w(t,\eta)} + e^{\mu{\left\vertl\right\vert}^{1/2}}\right) \\
A^{1}_k(t,\eta,l) & = \frac{1}{{\left\langle t \right\rangle}}\left(\mathbf{1}_{k \neq 0} \min\left(1, \frac{{\left\langle \eta,l \right\rangle}^{1+\delta_1}}{{\left\langle t \right\rangle}^{1+\delta_1}}\right) + \mathbf{1}_{k = 0} \right) A^Q_k(t,\eta,l) \\
A^{2}_k(t,\eta,l) & = \left(\mathbf{1}_{k \neq 0} \min\left(1, \frac{{\left\langle \eta,l \right\rangle}}{t}\right) + \mathbf{1}_{k = 0} \right) A^Q_k(t,\eta,l) \\
A^{3}_k(t,\eta,l) & = \left(\mathbf{1}_{k \neq 0} \min\left(1, \frac{{\left\langle \eta,l \right\rangle}^2}{t^2}\right) + \mathbf{1}_{k = 0} \right) e^{\lambda(t){\left\vertk,\eta,l\right\vert}^s}{\left\langle k,\eta,l \right\rangle}^\sigma \nonumber \\ & \quad\quad \times \frac{1}{w_L(t,k,\eta,l)}\left(\frac{e^{\mu {\left\vert\eta\right\vert}^{1/2}}}{w^3_k(t,\eta)} + e^{\mu{\left\vertl\right\vert}^{1/2}}\right) \\
A(t,\eta,l) & = {\left\langle \eta,l \right\rangle}^2 A^Q_0(t,\eta,l),\end{aligned}$$
where $\mu$, $w$, and $w^3$ are defined precisely in Appendix \[sec:def\_nrm\] and $w_L$ is defined in Appendix \[sec:Nmult\] ($w$ and $w^3$ are derived approximately in above). As in [@BGM15I], the multiplier $A$ is used to measure $C^i$ and $g$ whereas $A^i$ is used to measure $Q^i$. Here $\delta_1$ is chosen sufficiently small depending only on $\delta$. We choose the radius of Gevrey-$\frac{1}{s}$ regularity to satisfy $$\begin{aligned}
\dot{\lambda}(t) & = - \frac{\delta_\lambda}{{\left\langle t \right\rangle}^{\min(2s,3/2)}} \\
\lambda(1) & = \frac{3 \lambda_0}{4} + \frac{\lambda^\prime}{4}, \end{aligned}$$ where we fix $\delta_\lambda \ll \min(1,\lambda_0 - \lambda^\prime)$ small such that $\lambda(t) > (\lambda_0 + \lambda^\prime)/2$.
Let us briefly mention some implications of using $w^3$ in . Note first of all from that $w^3$ is the same as $w$ except near the critical times, however, near the critical times, $w^3_k(t,\eta)$ for *non-resonant* modes is larger, and hence will assign them *less* regularity (see in §\[sec:Defw\] for the precise definition). This will create a gain in energy estimates when $Q^2$ or $Q^1$ force $Q^3$ and will be a loss when the vice-versa occurs. It will also create a similar imbalance in nonlinear interactions between resonant and non-resonant modes in $Q^3$. The last detail to notice is that, due to the $+ e^{\mu {\left\vertl\right\vert}^{1/2}}$, the effects of $w$ and $w^3$ are only visible in the subset of frequencies such that ${\left\vert\eta\right\vert} \gtrsim {\left\vertl\right\vert}$. This additional precision was not necessary in [@BGM15I], however, it is necessary here due to problems with regularity imbalances at high frequencies in $Z$ (for example, in §\[sec:Q3\_TransNon\]). Note it is natural that the resonances should not be relevant for high $Z$ frequencies, due to the uniform ellipticity in $Z$ of $\Delta_t$, however, this detail will make certain aspects of the proof more technical. We will need the following definition:
\[def:Atilde\] $$\begin{aligned}
\tilde{A}^Q_k(t,\eta,l) & = e^{\lambda(t){\left\vertk,\eta,l\right\vert}^s}{\left\langle k,\eta,l \right\rangle}^\sigma\frac{1}{w_L(t,k,\eta,l)} \frac{e^{\mu {\left\vert\eta\right\vert}^{1/2}}}{w(t,\eta)} \\
\tilde{A}^{1}_k(t,\eta,l) & = \frac{1}{{\left\langle t \right\rangle}}\left(\mathbf{1}_{k \neq 0} \min\left(1, \frac{{\left\langle \eta,l \right\rangle}^{1+\delta_1}}{{\left\langle t \right\rangle}^{1+\delta_1}}\right) + \mathbf{1}_{k = 0} \right) \tilde{A}^Q_k(t,\eta,l) \\
\tilde{A}^{2}_k(t,\eta,l) & = \left(\mathbf{1}_{k \neq 0} \min\left(1, \frac{{\left\langle \eta,l \right\rangle}}{t}\right) + \mathbf{1}_{k = 0} \right) \tilde{A}^Q_k(t,\eta,l) \\
\tilde{A}^{3}_k(t,\eta,l) & = \left(\mathbf{1}_{k \neq 0} \min\left(1, \frac{{\left\langle \eta,l \right\rangle}^2}{t^2}\right) + \mathbf{1}_{k = 0} \right) \tilde{A}^Q_k(t,\eta,l) \frac{w(t,\eta)}{w^3_k(t,\eta)} \\
\tilde{A}(t,\eta,l) & = {\left\langle \eta,l \right\rangle}^2 \tilde{A}^Q_0(t,\eta,l). \end{aligned}$$
Notice that $\tilde{A}^i \lesssim A^i$ and for ${\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}$ there holds $A^i \approx \tilde{A}^i$ (by Lemma \[lem:totalGrowthw\]). Therefore, the difference between them is only visible if ${\left\vertl\right\vert}$ is comparable to or larger than ${\left\vert\eta\right\vert}$.
To quantify the enhanced dissipation, we use a scheme similar to that used in [@BGM15I], which itself was an expansion of the scheme of [@BMV14], adjusted now to the larger expected growth of $Q^2$. Define $D$ as in [@BMV14], $$\begin{aligned}
D(t,\eta) & = \frac{1}{3\alpha}\nu {\left\vert\eta\right\vert}^3 + \frac{1}{24 \alpha} \nu\left(t^3 - 8{\left\vert\eta\right\vert}^3\right)_+. \label{def:D}\end{aligned}$$ Note this multiplier satisfies $$\begin{aligned}
D(t,\eta) \gtrsim \max(\nu {\left\vert\eta\right\vert}^3, \nu t^3).\label{ineq:DLowB}\end{aligned}$$ For some $\beta > 3\alpha+7$, we define the enhanced dissipation multipliers:
\[def:Anu\] $$\begin{aligned}
A^{\nu}_k(t,\eta,l) & = e^{\lambda(t){\left\vertk,\eta,l\right\vert}^s}{\left\langle k,\eta,l \right\rangle}^{\beta} {\left\langle D(t,\eta) \right\rangle}^\alpha \frac{1}{w_L(t,k,\eta,l)} \mathbf{1}_{k \neq 0} \\
A^{\nu;1}_k(t,\eta,l) & = {\left\langle t \right\rangle}^{-1}\min\left(1, \frac{{\left\langle \eta,l \right\rangle}^{1+\delta_1}}{t^{1+\delta_1}}\right) A^{\nu}_k(t,\eta,l) \\
A^{\nu;2}_k(t,\eta,l) & = \min\left(1, \frac{{\left\langle \eta,l \right\rangle}}{t}\right) A^{\nu}_k(t,\eta,l) \\
A^{\nu;3}_k(t,\eta,l) & = \min\left(1, \frac{{\left\langle \eta,l \right\rangle}^2}{t^2}\right) A^{\nu}_k(t,\eta,l). \end{aligned}$$
Fix $\gamma > \beta + 3\alpha + 12$ and $\sigma > \gamma + 6$. Note that we do not need $w$ or $w^3$ (or the associated regularity imbalances) in . Indeed, the Orr mechanism (and related nonlinear effects) does not play a major role in the enhanced dissipation estimates; they are instead mainly determined by careful estimates on how the vortex stretching manifests in the nonlinearity.
Main energy estimates {#sec:energy}
---------------------
In this section, we set up the main bootstrap argument to extend our estimates from $O(1)$ in time (from Lemma \[lem:Loc\]) to $T_F = c_0 \epsilon^{-1}$. Equipped with the norms defined in and , we will be able to propagate estimates via a bootstrap argument for as long as the solution to exists and remains analytic; by un-doing the coordinate transformation (possible as long as it remains a small deformation in $yz$), this in turn allows us to continue the solution of via Lemma \[lem:Cont\]. The analyticity itself is not important, it only needs to be a regularity class slightly stronger than the norms defined in §\[def:designnorm\] to ensure they take values continuously in time. See §\[sec:RegCont\] below for more details on this procedure.
It turns out that $\partial_t w^3/w^3 \approx \partial_t w/w$ (see Lemma \[dtw\]) and so this will simplify the notation when defining the following high norm “dissipation energies”: for $i \in {\left\{2,3\right\}}$,
$$\begin{aligned}
\mathcal{D}Q^i & = \nu {\left\lVert \sqrt{-\Delta_{L}}A^{i} Q^i \right\rVert}_2^2 + CK_\lambda^i + CK_w^i + CK_{wL}^i \nonumber \\
& = \nu {\left\lVert \sqrt{-\Delta_{L}}A^{i} Q^i \right\rVert}_2^2 d\tau + \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^i Q^i \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^i Q^i \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}}A^i Q^i \right\rVert}_2^2 \\
\mathcal{D}Q^1_{\neq} & = \nu {\left\lVert \sqrt{-\Delta_{L}}A^{1} Q^1_{\neq} \right\rVert}_2^2 + CK_{\lambda;\neq}^{1} + CK_{w;\neq}^{1} + CK_{wL;\neq}^{1} \nonumber \\
& = \nu {\left\lVert \sqrt{-\Delta_{L}}A^{1} Q^1 \right\rVert}_2^2 + \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^1 Q^1_{\neq} \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^1 Q^1_{\neq} \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_tw_L}{w_L}}A^1 Q^1_{\neq} \right\rVert}_2^2 \\
\mathcal{D}g & = \nu {\left\lVert \sqrt{-\Delta_{L}}A g \right\rVert}_2^2 + CK_L^g + CK_{\lambda}^{g} + CK_{w}^{g} \nonumber \\
& = \nu {\left\lVert \sqrt{-\Delta_{L}}A g \right\rVert}_2^2 + \frac{2}{t}{\left\lVert Ag \right\rVert}_2^2 + \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A g \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A} g \right\rVert}_2^2 \\
\mathcal{D}C^{i} & = \nu {\left\lVert \sqrt{-\Delta_{L}}A C^i \right\rVert}_2^2 + CK_{\lambda}^{Ci} + CK_{w}^{Ci} \nonumber \\
& = \nu {\left\lVert \sqrt{-\Delta_{L}}A C^i \right\rVert}_2^2 + \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A C^i \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A} C^i \right\rVert}_2^2 \\
CK_L^i & = \frac{1}{t}{\left\lVert \mathbf{1}_{t \geq {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^i Q^i_{\neq} \right\rVert}_2^2 \\
\mathcal{D}Q^{\nu;i} & = \nu {\left\lVert \sqrt{-\Delta_{L}}A^{\nu;i} Q^i \right\rVert}_2^2 + CK_\lambda^{\nu;i} + CK_{wL}^{\nu;i} \nonumber \\
& := \nu {\left\lVert \sqrt{-\Delta_{L}}A^{\nu;i} Q^i \right\rVert}_2^2 + \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{\nu;i} Q^i \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}}A^{\nu;i} Q^i \right\rVert}_2^2 \\
\mathcal{D}Q^{\nu;1} & = \nu {\left\lVert \sqrt{-\Delta_{L}}A^{\nu;1} Q^{\nu;1} \right\rVert}_2^2 + CK_{\lambda}^{\nu;1} + CK_{wL}^{\nu;1} \nonumber \\
& := \nu {\left\lVert \sqrt{-\Delta_{L}}A^{\nu;1} Q^1_{\neq} \right\rVert}_2^2 + \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{\nu;1} Q^1_{\neq} \right\rVert}_2^2 + {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}}A^{\nu;1} Q^1_{\neq} \right\rVert}_2^2 \\
CK_L^{\nu;i} & := \frac{1}{t}{\left\lVert \mathbf{1}_{t \geq {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{\nu;i} Q^i \right\rVert}_2^2. \end{aligned}$$
Note the presence of $\tilde{A}^i$; this will mean that, unlike [@BGM15I], the $CK_w$ terms only provide control in the range of frequencies ${\left\vert\partial_Y\right\vert} \gtrsim {\left\vert\partial_Z\right\vert}$.
Using a bootstrap/continuity argument, we will propagate the following estimates. Fix constants $K_{Hi}, K_{H1\neq}, K_{HC1},K_{HC2}, K_{EDi}, K_{Li}, K_{ED2},K_{LC}$ for $i \in {\left\{1,3\right\}}$, sufficiently large determined by the proof, depending only on $\delta,\delta_1,s,\sigma,\gamma,\beta,\lambda^\prime,\lambda_0$ and $\alpha$. Further, fix $\sigma^\prime > 3$. Let $1 \leq T^\star < T^0$ be the largest time such that the following *bootstrap hypotheses* hold (that $T^\star \geq 1$ is discussed below): the high norm controls on $Q^i$,
\[ineq:Boot\_Hi\] $$\begin{aligned}
{\left\lVert A^{1} Q^1_{0}(t) \right\rVert}_2^2 & \leq 4K_{H1} \epsilon^2 \label{ineq:Boot_Q1Hi1} \\
{\left\lVert A^{1} Q^1_{\neq}(t) \right\rVert}_2^2 + \frac{1}{2}\int_1^t \mathcal{D}Q^1_{\neq}(\tau) d\tau & \leq 4K_{H1\neq} \epsilon^2 \label{ineq:Boot_Q1Hi2} \\
{\left\lVert A^{2} Q^2 \right\rVert}^2_2 + \int_1^t\frac{1}{2}\mathcal{D}Q^2(\tau) + CK_L^2(\tau) d\tau & \leq 4\epsilon^2 \label{ineq:Boot_Q2Hi} \\
{\left\lVert A^{3} Q^3 \right\rVert}^2_2 + \int_1^t\frac{1}{2} \mathcal{D}Q^3(\tau) d\tau & \leq 4K_{H3}\epsilon^2; \label{ineq:Boot_Q3Hi}\end{aligned}$$
the coordinate system controls,
\[ineq:Boot\_CgHi\] $$\begin{aligned}
{\left\lVert A C^i \right\rVert}_2^2 + \frac{1}{2}\int_1^t \mathcal{D}C^i(\tau) d\tau &\leq 4 K_{HC1} c^2_{0} \label{ineq:Boot_ACC}\\
{\left\langle t \right\rangle}^{-2}{\left\lVert A C^i \right\rVert}_2^2 + \frac{1}{2} \int_1^t {\left\langle \tau \right\rangle}^{-2}\mathcal{D}C^i(\tau) d\tau &\leq 4K_{HC2}\epsilon^2 \log{\left\langle t \right\rangle} \label{ineq:Boot_ACC2}\\
{\left\lVert Ag \right\rVert}_2^2 + \frac{1}{2}\int_1^t \mathcal{D}g d\tau &\leq 4\epsilon^2 \label{ineq:Boot_Ag} \\
{\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} & \leq 4 \frac{\epsilon}{{\left\langle t \right\rangle}^{2}} \label{ineq:Boot_gLow} \\
{\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} & \leq 4K_{LC} \epsilon {\left\langle t \right\rangle} \label{ineq:Boot_LowC} \end{aligned}$$
the enhanced dissipation estimates,
\[ineq:Boot\_ED\] $$\begin{aligned}
{\left\lVert A^{\nu;1} Q^1 \right\rVert}_2^2 + \frac{1}{10} \int_1^t \mathcal{D}Q^{\nu;1}(\tau) d\tau & \leq 4K_{ED1}\epsilon^2 \label{ineq:Boot_ED1} \\
{\left\lVert A^{\nu;2} Q^2 \right\rVert}_2^2 + \int_1^t\frac{1}{10}\mathcal{D}Q^{\nu;2}(\tau) + CK_{L}^{\nu;2}(\tau) d\tau & \leq 4K_{ED2}\epsilon^2 \\
{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2^2 + \frac{1}{10}\int_1^t \mathcal{D}Q^{\nu;3}(\tau) d\tau & \leq 4K_{ED3}\epsilon^2; \label{ineq:Boot_ED3}\end{aligned}$$
and the additional low frequency controls on the background streak
\[ineq:Boot\_LowFreq\] $$\begin{aligned}
{\left\lVert U_0^1 \right\rVert}_{H^{\sigma^\prime}} & \leq 4K_{L1} \epsilon {\left\langle t \right\rangle} \label{ineq:Boot_U01_Low} \\
{\left\lVert U_0^2 \right\rVert}_{H^{\sigma^\prime}} & \leq 4 \epsilon \label{ineq:Boot_U02_Low} \\
{\left\lVert U_0^3 \right\rVert}_{H^{\sigma^\prime}} & \leq 4K_{L3}\epsilon. \label{ineq:Boot_U03_Low}\end{aligned}$$
For most steps of the proof we do not need to differentiate so precisely between different bootstrap constants so we define $$\begin{aligned}
K_B = \max\left(K_{Hi}, K_{H1\neq}, K_{HC1},K_{HC2}, K_{EDi}, K_{Li}, K_{LC}\right). \label{def:KB}\end{aligned}$$
By Lemma \[lem:Loc\], we have that $T^\star > t_\star > 0$ and it is a consequence of Lemma \[lem:Cont\] that $T^\star < T^0$. It is relatively straightforward to prove that for $\epsilon$ sufficiently small, we have $1 \leq T^\star$; see [@BGM15I] for more discussion. Due to the real analyticity of the solution on $(0, T^0)$, it will follow from the ensuing proof that the quantities in the bootstrap hypotheses take values continuously in time for as long as the solution exists. Therefore, we may deduce $T^\star = T_F = c_0 \epsilon^{-1} < T^0$ via the following proposition, the proof of which is the main focus of the remainder of the paper.
\[prop:Boot\] Let $\epsilon < \nu^{2/3+\delta}$. For the constants appearing in the right-hand side of chosen sufficiently large and for $\nu$ and $c_0$ both chosen sufficiently small (depending only on $s,\lambda_0,\lambda^\prime,\alpha,\delta_1,\delta$ and arbitrary parameters such as $\sigma,\beta, \ldots$), if $T^\star < T_F = c_0\epsilon^{-1}$ is such that the bootstrap hypotheses hold on $[1,T^\star]$, then on the same time interval all the inequalities in hold with constant ‘$2$’ instead of ‘$4$’.
That Proposition \[prop:Boot\] implies Theorem \[thm:SRS\] is discussed briefly in §\[sec:RegCont\] below.
### Bootstrap constants
The relationship between the constants are similar to [@BGM15I] (although slightly simpler here since there are fewer). First, $K_{L1}$ and $K_{L3}$ are chosen sufficiently large relative to a universal constant depending only on $\sigma^\prime$. These in turn set $K_{H1},K_{H1\neq}$ and $K_{H3}$. These then imply $K_{HC1}$ which then implies $K_{HC2}$ and $K_{LC}$ followed finally by $K_{ED2}$ and then $K_{ED1}$ and $K_{ED3}$. Finally, $c_0$ and $\nu$ are chosen sufficiently small with respect to $K_B$, the max of all the bootstrap constants (as well as the parameters $s,\lambda_0,\lambda^\prime,\alpha,\delta_1$, and arbitrary parameters such as $\sigma,\beta$ etc).
### A priori estimates from the bootstrap hypotheses {#sec:AprioriBoot}
The motivation for the enhanced dissipation estimates is the following observation (which follows from ): for any $f$,
\[ineq:AnuDecay\] $$\begin{aligned}
{\left\lVert f_{\neq} (t) \right\rVert}_{\mathcal{G}^{\lambda(t),\beta}} & \lesssim_\alpha {\left\langle t \right\rangle}^{2+\delta_1}{\left\langle \nu t^3 \right\rangle}^{-\alpha} {\left\lVert A^{\nu;1} f(t) \right\rVert}_2 \\
{\left\lVert f_{\neq}(t) \right\rVert}_{\mathcal{G}^{\lambda(t),\beta}} & \lesssim_\alpha {\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{-\alpha} {\left\lVert A^{\nu;2} f(t) \right\rVert}_2 \\
{\left\lVert f_{\neq}(t) \right\rVert}_{\mathcal{G}^{\lambda(t),\beta}} & \lesssim_\alpha {\left\langle t \right\rangle}^{2} {\left\langle \nu t^3 \right\rangle}^{-\alpha} {\left\lVert A^{\nu;3} f(t) \right\rVert}_2. \end{aligned}$$
Hence, expresses a rapid decay of $Q^i_{\neq}$ for $t \gtrsim \nu^{-1/3}$. Together with the “lossy elliptic lemma”, Lemma \[lem:LossyElliptic\], we then get (under the bootstrap hypotheses),
\[ineq:AprioriUneq\] $$\begin{aligned}
{\left\lVert U^1_{\neq} (t) \right\rVert}_{\mathcal{G}^{\lambda(t),\beta-2}} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \\
{\left\lVert U^2_{\neq}(t) \right\rVert}_{\mathcal{G}^{\lambda(t),\beta-2}} & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}{\left\langle \nu t^3 \right\rangle}^{\alpha}} \\
{\left\lVert U^3_{\neq}(t) \right\rVert}_{\mathcal{G}^{\lambda(t),\beta-2}} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}. \end{aligned}$$
For the zero frequencies of the velocity field we get from , and Lemma \[lem:PELbasicZero\] (which allows to understand $\Delta_t^{-1}$ at zero $x$ frequencies) the matching a priori estimates
\[ineq:AprioriU0\] $$\begin{aligned}
{\left\lVert A U^1_0 (t) \right\rVert}_{2} & \lesssim \epsilon {\left\langle t \right\rangle} \\
{\left\lVert A U^2_0 (t) \right\rVert}_{2} & \lesssim \epsilon \\
{\left\lVert A^3 {\left\langle {\nabla}\right\rangle}^2 U^3_0 (t) \right\rVert}_{2} & \lesssim \epsilon. \end{aligned}$$
Notice that no regularity loss is required to get the ‘correct’ a priori estimates on the zero frequencies. However, unlike in our previous work [@BGM15I], the natural regularity of the zero-frequency velocity fields are not all the same.
Regularization and continuation {#sec:RegCont}
===============================
There are three preliminaries: (A) the instantaneous analytic regularization with initial data of the type (B) how to move estimates on these classical solutions between coordinate systems, and (C) the proof that Proposition \[prop:Boot\] implies Theorem \[thm:SRS\]. The issues here are essentially the same as in [@BGM15I] so we will just give a brief summary.
The proofs of Lemmas \[lem:Loc\] and \[lem:Cont\] are sketched in [@BGM15I]. Similarly, the following lemma is a variant of \[Lemma 3.1 [@BGM15I]\]. The proof is omitted for brevity as it follows via the same arguments.
\[lem:BootStart\] We may take $2 \leq T^\star$ (defined in §\[sec:energy\] above) and for $t \leq 2$, the bootstrap estimates in , , , and , all hold with constant $5/4$ instead of $4$.
In order to move estimates from $(X,Y,Z)$ to $(x,y,z)$ we may use the same methods described in [@BGM15I] (which are themselves essentially the same as those in [@BM13; @BMV14]). we will first move to the coordinate system $(X,y,z)$. Writing $\bar{q}^i(t,X,y,z) = Q^i(t,X,Y,Z) = q(t,x,y,z)$ and $\bar{u}^i(t,X,y,z) = U^i(t,X,Y,Z) = u^i(t,x,y,z)$ we derive the following, noting that $\bar{u}^i_0 = u^i_0$: $$\begin{aligned}
\partial_t u_0^i + (u_0^2,u_0^3) \cdot {\nabla}u_0^i & = (-u_0^2,0,0)^T - (0,\partial_y p_0^{NL0}, \partial_z p^{NL0}_0)^T + \nu \Delta u^i_0 + \mathcal{F}^i, \label{eq:u0i}\end{aligned}$$ where $$\begin{aligned}
\Delta p_0^{NL0} = -\partial_i u_0^j \partial_j u^i_0\end{aligned}$$ and (using cancellations as in ), $$\begin{aligned}
\mathcal{F}^i & = -\partial_y \left(\bar{u}^2_{\neq}\bar{u}_{\neq}^i\right)_0 - \partial_z\left(\bar{u}^3_{\neq} \bar{u}_{\neq}^i \right)_0. \label{eq:Fbaru}\end{aligned}$$
We then have the following lemma, analogous to \[Lemma 3.2 [@BGM15I]\], which holds here with an analogous proof.
\[lem:intermedSob\] For $\epsilon < \nu^{2/3 + \delta}$ and $c_0$ and $\nu$ sufficiently small (depending only on $s,\lambda_0,\lambda^\prime,\alpha$, $\delta_1$, and $\delta$), the bootstrap hypotheses imply the following for some $c \in (0,1)$ chosen such that $c\lambda(t) \in (\lambda^\prime,\lambda(t))$ for all $t$:
\[ineq:Xyzubds\] $$\begin{aligned}
{\left\lVert \bar{u}^1_{\neq} \right\rVert}_{{\mathcal{G}}^{c\lambda(t)}} & \lesssim \epsilon {\left\langle t \right\rangle}^{\delta_1}{\left\langle \nu t^3 \right\rangle}^{-\alpha} \\
{\left\lVert \bar{u}_{\neq}^2 \right\rVert}_{{\mathcal{G}}^{c\lambda(t)}} & \lesssim \epsilon {\left\langle t \right\rangle}^{-1} {\left\langle \nu t^3 \right\rangle}^{-\alpha} \\
{\left\lVert \bar{u}_{\neq}^3 \right\rVert}_{{\mathcal{G}}^{c\lambda(t)}} & \lesssim \epsilon {\left\langle \nu t^3 \right\rangle}^{-\alpha}, \end{aligned}$$
and
\[ineq:uzAPriori\] $$\begin{aligned}
{\left\lVert u^1_0(t) \right\rVert}_{{\mathcal{G}}^{c\lambda(t)}} & \lesssim \epsilon {\left\langle t \right\rangle} \label{ineq:uzApriori1} \\
{\left\lVert u^2_0(t) \right\rVert}_{{\mathcal{G}}^{c\lambda(t)}} + {\left\lVert u^3_0(t) \right\rVert}_{{\mathcal{G}}^{c\lambda(t)}} & \lesssim \epsilon. \label{ineq:uzApriori23}\end{aligned}$$
Finally, the following lemma also follows analogously to the corresponding result in [@BGM15I]. Hence, the proof is omitted for the sake of brevity.
\[lem:PropBootThm\] For $\epsilon < \nu^{2/3+\delta}$ and $c_0$ and $\nu$ sufficiently small (depending only on $s,\lambda_0,\lambda^\prime,\alpha$, $\delta_1$, and $\delta$), Proposition \[prop:Boot\] implies Theorem \[thm:SRS\].
Multiplier and paraproduct tools {#sec:nrmuse}
================================
In this section we outline some basic general inequalities regarding the multipliers which are used in the sequel. As in [@BGM15I], the purpose is to set up a general framework that will make the large number of energy estimates later in the paper easier. Most of the estimates come in the general form $\int A^i Q^i A^i\left(f g\right) dV$. The goal of this section is to break the treatment of these terms into a four step procedure:
1. As in [@BGM15I], the first step is to separate out zero/non-zero frequency interactions according to §\[sec:NonlinHeuristics\] and then expand with a paraproduct to divide the terms based on which of the nonlinear factors is dominant from the standpoint of frequency (paraproducts are explained in §\[sec:paranote\] below).
2. Compare the norm for $Q^i$ with the norm of the dominant factor (also adding $\Delta_L^{-1} \Delta_L$ if the dominant factor is a velocity field) and commute it past the low frequency factor. Lemma \[lem:ABasic\] below is the primary tool for this.
3. Use Lemmas \[lem:MainFreqRat\] and \[lem:MainFreqRat\_RegImbalance\] below to convert the ratio of the norms (together with possibly $\Delta_L^{-1}$) into multipliers that appear in the dissipation energies or integrate to $\lesssim \epsilon^2$ until $T_F = c_0\epsilon^{-1}$.
4. Use Lemma \[gevreyparaproductlemma\] or \[lem:ParaHighOrder\] to re-combine the paraproduct decomposition into multiples of terms in the dissipation energy or other integrable errors.
Basic inequalities regarding the multipliers {#sec:basicmult}
--------------------------------------------
This section covers the key properties of the multipliers we are using and forms the core of the technical tools, however, it is very tedious and will likely appear unmotivated at first. A reader should consider skipping this section on the first reading and refer back to it whenever specific inequalities are needed. Note that this section is significantly more technical than the corresponding section in [@BGM15I].
In the lemmas which follow, one should imagine that frequencies $(k^\prime,\xi,l^\prime)$ and $(k-k^\prime,\eta-\xi,l-l^\prime)$ are interacting to force $(k,\eta,l)$, as will be occurring in the quadratic energy estimates.
The first lemma gives us general estimates for how the $A$ and $A^i$ are related at different frequencies. It is designed specifically for dealing with $f_{Hi}g_{Lo}$-type terms in the paraproducts (see ).
\[lem:ABasic\] Let $\theta < 1/2$ and suppose $$\begin{aligned}
{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert} \leq \theta{\left\vertk,\eta,l\right\vert}. \label{ineq:AFreqLoc}\end{aligned}$$ In what follows, define the frequency cut-offs (all functions of $(t,k,k^\prime,\eta,\xi,l,l^\prime)$),
\[def:freqcuts\] $$\begin{aligned}
\chi^{R,NR} & = \mathbf{1}_{t \in {\mathbf{I}}_{k,\eta} \cap {\mathbf{I}}_{k,\xi}} \mathbf{1}_{k^\prime \neq k} \mathbf{1}_{{\left\vertl\right\vert} < 5{\left\vert\eta\right\vert}} \mathbf{1}_{{\left\vertl^\prime\right\vert} < 5{\left\vert\xi\right\vert}} \\
\chi^{NR,R} & = \mathbf{1}_{t \in {\mathbf{I}}_{k^\prime,\xi} \cap {\mathbf{I}}_{k^\prime,\eta}} \mathbf{1}_{k^\prime \neq k} \mathbf{1}_{{\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}}\mathbf{1}_{{\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}} \\
\chi^{r,NR} & = \mathbf{1}_{t \in {\mathbf{I}}_{r,\eta} \cap {\mathbf{I}}_{r,\xi}} \mathbf{1}_{k^\prime \neq r} \mathbf{1}_{{\left\vertl\right\vert} < 5{\left\vert\eta\right\vert}} \mathbf{1}_{{\left\vertl^\prime\right\vert} < 5{\left\vert\xi\right\vert}} \\
\chi^{NR,r} & = \mathbf{1}_{t \in {\mathbf{I}}_{r,\eta} \cap {\mathbf{I}}_{r,\xi}} \mathbf{1}_{k \neq r} \mathbf{1}_{{\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}}\mathbf{1}_{{\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}} \\
\chi^{\ast;33} & = 1 - \mathbf{1}_{t \in {\mathbf{I}}_{k,\eta} \cap {\mathbf{I}}_{k,\xi}} \mathbf{1}_{k \neq k^\prime}\mathbf{1}_{{\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}}\mathbf{1}_{{\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}} - \chi^{NR,R} \\
\chi^{\ast;23} & = 1 - \sum_{r}\mathbf{1}_{t \in {\mathbf{I}}_{r,\eta} \cap {\mathbf{I}}_{r,\xi}} \mathbf{1}_{k^\prime \neq r}\mathbf{1}_{{\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}}\mathbf{1}_{{\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}} \\
\chi^{\ast;32} & = 1 - \sum_{r}\chi^{NR,r}, \end{aligned}$$
and for $i,j \in {\left\{1,2,3\right\}}$ and $a,b \in {\left\{0,\neq\right\}}$, the weight $\Gamma(i,j,a,b)$ given by, $$\begin{aligned}
\Gamma(i,i,a,a) & = 1, & \Gamma(i,j,a,b) & = \Gamma(j,i,b,a)^{-1}, \\
\Gamma(1,2,0,0) & = {\left\langle t \right\rangle}^{-1}, & \Gamma(1,2,\neq,\neq) & = {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-\delta_1}, \\
\Gamma(1,2,0,\neq) & = {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}, & \Gamma(1,2,\neq,0) & = {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1-\delta_1}, \\
\Gamma(1,3,0,0) & = {\left\langle t \right\rangle}^{-1}, & \Gamma(1,3,\neq,\neq) & = {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1-\delta_1}, \\
\Gamma(1,3,0,\neq) & = {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^2, & \Gamma(1,3,\neq,0) & = {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1-\delta_1}, \\
\Gamma(2,3,\neq,\neq) & = {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}, & \Gamma(2,3,0,\neq) & = {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^2, \\
\Gamma(2,3,\neq,0) & = {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1}, & \Gamma(2,3,0,0) & = 1, \\
\Gamma(1,1,0,\neq)& = {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1+\delta_1}, & \Gamma(2,2,0,\neq) & = {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}, \\
\Gamma(3,3,0,\neq) & = {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^2. \end{aligned}$$ Then there exists a $c = c(s) \in (0,1)$ such that for all $t$ we have the following for $i \in {\left\{1,2\right\}}$ and $a = \neq$ if $k \neq 0$ (otherwise $a=0$) and $b = \neq$ if $k^\prime \neq 0$ (otherwise $b = 0$),
\[ineq:ABasic\] $$\begin{aligned}
A^{i}_k(t,\eta,l) & \lesssim \Gamma(i,j,a,b) A^{j}_{k^\prime}(t,\xi,l^\prime) e^{c\lambda{\left\vertk - k^\prime,\eta-\xi,l-l^\prime\right\vert}^s} \label{ineq:ABasic12}\\
\left(A^3_k(t,\eta,l)\right)^2 & \lesssim \Gamma(3,3,a,b)\left(\tilde{A}^3_k(t,\eta,l) \tilde{A}^3_{k^\prime}(\xi,l^\prime)\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} \right. \nonumber \\ & \left. \quad\quad + \tilde{A}^3_k(t,\eta,l)\tilde{A}^3_{k^\prime}(t,\xi,l^\prime) \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t} \right. \nonumber \\ & \quad\quad + \chi^{\ast;33} A^3_k(t,\eta,l) A^3_{k^\prime}(t,\xi,l^\prime) \bigg) e^{c\lambda{\left\vertk - k^\prime,\eta-\xi,l-l^\prime\right\vert}^s} \label{ineq:A3A3neqneq}\\
\left(A^{i}_k(t,\eta,l)\right)^2 & \lesssim \Gamma(i,3,a,b)\left(\sum_{r} \tilde{A}^i_k(t,\eta,l)\tilde{A}^3_{k^\prime}(t,\xi,l^\prime)\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta- rt\right\vert}} \right. \nonumber \\ & \quad\quad + A^i_k(t,\eta,l) A^3_{k^\prime}(t,\xi,l^\prime) \chi^{\ast;23} \Bigg) e^{c\lambda{\left\vertk - k^\prime,\eta-\xi,l-l^\prime\right\vert}^s} \label{ineq:A2A3neqneq} \\
\left(A^{3}_k(t,\eta,l)\right)^2 & \lesssim \Gamma(3,i,a,b)\left(\sum_{r} \tilde{A}^3_k(t,\eta,l) \tilde{A}^i_{k^\prime}(t,\xi,l^\prime) \chi^{NR,r}\frac{{\left\vertr\right\vert} + {\left\vert\eta-rt\right\vert}}{t} \right. \nonumber \\ & \quad\quad + \chi^{\ast;32} A^3_k(t,\eta,l) A^i_{k^\prime}(t,\xi,l^\prime) \Bigg) e^{c\lambda{\left\vertk - k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}. \label{ineq:A3A2neqneq}\end{aligned}$$
Analogous inequalities hold also with $A(t,\eta,l)$ using that $A(t,\eta,l) = {\left\langle \eta,l \right\rangle}^2 A_0^2(t,\eta,l)$.
The terms involving $\chi^{R,NR}$, $\chi^{NR,R}$, $\chi^{r,NR}$, and $\chi^{NR,r}$ are arising from comparing ratios of $w^3_k$ and $w^3_{k^\prime}$ or $w^3$ and $w$; see e.g. above. In particular, modulo details regarding the $Z$ frequencies, the three contributions to roughly correspond to the three possible regimes in Lemma \[lem:Jswap\]: when a resonant frequency forces a non-resonant frequency, vice-versa, and neither. The inequalities and generally play a more crucial role in the proof of Theorem \[thm:SRS\] and correspond instead to what happens when one compares $w$ and $w^3$, rather than $w^3$ with itself (that is, in terms when $Q^3$ interacts with $Q^{1,2}$). We have chosen to write it in this manner as this is the form that is most natural for Lemma \[lem:MainFreqRat\_RegImbalance\] below.
Note that a time/frequency combination is only considered truly “resonant” if $t \in {\mathbf{I}}_{k,\eta} \cap {\mathbf{I}}_{k,\xi}$. The reason for this is explained by Lemma \[lem:wellsep\]: if $t \in {\mathbf{I}}_{k,\eta}$ but $t \not\in {\mathbf{I}}_{k,\xi}$, then either $\eta$ and $\xi$ are well-separated or the time/frequency combination is not really resonant, which results in ${\left\langle \eta-\xi \right\rangle}{\left\langle kt - \eta \right\rangle}\gtrsim t$.
Note that the definitions in are not quite symmetric for minor technical reasons and that the decomposition defined by is not quite a partition of unity, as there is an overlap region when ${\left\vertl\right\vert} \approx {\left\vert\eta\right\vert}$ or ${\left\vertl^\prime\right\vert} \approx {\left\vert\xi\right\vert}$. When losing due to the regularity imbalances, one must take the larger region ${\left\vertl\right\vert} < 5{\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} < 5{\left\vert\xi\right\vert}$ but when gaining due to the regularity imbalances, one must take the smaller region ${\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}$.
Note that some of the inequalities in Lemma \[lem:ABasic\] are phrased on quadratic quantities (as opposed to and the analogous lemma in [@BGM15I]). This is to treat the overlapping regions ${\left\vertl\right\vert} \approx {\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} \approx {\left\vert\xi\right\vert}$ more carefully, in particular, it is to make sure that any losses or gains from the ratios of $w$ and $w^3$ come with $\tilde{A}^i$, even if it is a region of frequency where $A^i \not\approx \tilde{A}^i$ (see also Remark \[rmk:noFreqRestrict\] below). This precision is only required in certain places, especially when we need to use the $CK_w^i$ terms, and in other cases less precise inequalities suffice.
These inequalities are all more or less easy variants of each other so we will just consider one of the trickier inequalities and omit the rest for brevity. We will consider ; further, we will consider just the case $a = b = \neq$ as the other cases are analogous.
The proof is divided into three regions (which do not exactly correspond to the three terms in ).\
*Case 1: ${\left\vertl\right\vert} > 5{\left\vert\eta\right\vert}$ or ${\left\vertl^\prime\right\vert} > 5{\left\vert\xi\right\vert}$*\
In this case, the $Z$ frequencies are dominant and hence one does not see the contributions from $w^3$ multipliers. Indeed, $\chi^{R,NR} = \chi^{NR,R} = 0$ and $\chi^{\ast;33} = 1$. If ${\left\vertl^\prime\right\vert} > 3{\left\vert\eta\right\vert}$ then by Lemma \[lem:totalGrowthw\], $$\begin{aligned}
\frac{\left(\frac{e^{\mu {\left\vert\eta\right\vert}^{1/2}}}{w^3_k(t,\eta)} + e^{\mu{\left\vertl\right\vert}^{1/2}}\right)}{\left(\frac{e^{\mu {\left\vert\xi\right\vert}^{1/2}}}{w^3_{k^\prime}(t,\xi)} + e^{\mu{\left\vertl^\prime\right\vert}^{1/2}}\right)} & \lesssim \frac{1}{w_k^3(t,\eta)}e^{\mu{\left\vert\eta\right\vert}^{1/2} - \mu{\left\vertl^\prime\right\vert}^{1/2}} + e^{\mu{\left\vertl\right\vert}^{1/2} - \mu {\left\vertl^\prime\right\vert}^{1/2}} \\
& \lesssim e^{\frac{3\mu}{2}{\left\vert\eta\right\vert}^{1/2} - \mu{\left\vertl^\prime\right\vert}^{1/2}} + e^{\mu{\left\vertl-l^\prime\right\vert}^{1/2}} \\
& \lesssim e^{\mu{\left\vertl-l^\prime\right\vert}^{1/2}}. \end{aligned}$$ Therefore, by and \[lem:scon\] (and that $w_L$ is $O(1)$ by and ), there is some $c^\prime = c^\prime(s) \in (0,1)$, $$\begin{aligned}
A_k^3(t,\eta,l) & \lesssim e^{\mu{\left\vertl-l^\prime\right\vert}^{1/2} + c^\prime\lambda{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}A_{k^\prime}(t,\xi,l^\prime). \end{aligned}$$ Then in this case follows from for some $c^\prime < c < 1$. If ${\left\vertl^\prime\right\vert} \leq 3{\left\vert\eta\right\vert}$ then it follows that either ${\left\vertl-l^\prime\right\vert} \gtrsim {\left\vert\eta\right\vert}$ or ${\left\vert\eta-\xi\right\vert} \gtrsim {\left\vertl^\prime\right\vert} \gtrsim {\left\vert\xi\right\vert}$. Therefore, Lemma \[lem:totalGrowthw\], for some $K$ there holds, $$\begin{aligned}
\left(\frac{e^{\mu {\left\vert\eta\right\vert}^{1/2}}}{w^3_k(t,\eta)} + e^{\mu{\left\vertl\right\vert}^{1/2}}\right) & \lesssim e^{\frac{3}{2}\mu{\left\vert\eta\right\vert}^{1/2}} + e^{\mu{\left\vertl\right\vert}^{1/2}} \lesssim e^{\mu{\left\vertl^\prime\right\vert}^{1/2}} e^{K\mu{\left\vert\eta-\xi,l-l^\prime\right\vert}^{1/2}}.\end{aligned}$$ Therefore, by the frequency localizations, for some $c^\prime = c^\prime(s) \in (0,1)$, $$\begin{aligned}
A_k^3(t,\eta,l) & \lesssim e^{K\mu{\left\vert\eta-\xi, l-l^\prime\right\vert}^{1/2} + c^\prime\lambda{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}A^3_{k^\prime}(t,\xi,l^\prime), \end{aligned}$$ from which again there follows from for some $c^\prime < c < 1$.\
*Case 2: (${\left\vertl\right\vert} < 5{\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} < 5{\left\vert\xi\right\vert}$) and (${\left\vertl\right\vert} > \frac{1}{5}{\left\vert\eta\right\vert}$ or ${\left\vertl^\prime\right\vert} > \frac{1}{5}{\left\vert\xi\right\vert}$)*\
In this case, neither $l,l^\prime$ nor $\eta,\xi$ are necessarily dominant, and indeed ${\left\vertl\right\vert} \approx {\left\vert\eta\right\vert}$ or ${\left\vertl^\prime\right\vert} \approx {\left\vert\xi\right\vert}$. We have $\chi^{NR,R}=0$ but there are regions in frequency where $\chi^{R,NR} = \chi^{\ast;33} = 1$ and we have to consider contributions involving both $A^3$ and $\tilde{A^3}$ at the same time. By and Lemma \[lem:scon\] (and that $w_L$ is $O(1)$ by and ), there is some $c^\prime = c^\prime(s) \in (0,1)$, $$\begin{aligned}
\left(A^3_k(t,\eta,l)\right)^2 & \lesssim \left(\frac{e^{2\mu {\left\vert\eta\right\vert}^{1/2}}}{\left(w^3_k(t,\eta)\right)^2} + e^{2\mu{\left\vertl\right\vert}^{1/2}}\right) \frac{1}{\left(w_L(k,\eta,l)\right)^2}{\left\langle k,\eta,l \right\rangle}^{2\sigma} e^{2\lambda{\left\vertk,\eta,l\right\vert}^s} \\
& \lesssim \left(\frac{e^{\mu {\left\vert\eta\right\vert}^{1/2} + \mu{\left\vert\xi\right\vert}^{1/2} + \mu{\left\vert\eta-\xi\right\vert}^{1/2}}}{\left(w^3_k(t,\eta)\right)^2} + e^{\mu{\left\vertl\right\vert}^{1/2} + \mu{\left\vertl^\prime\right\vert}^{1/2} + \mu{\left\vertl-l^\prime\right\vert}^{1/2}}\right) \\ & \quad\quad \times \frac{1}{w_L(k,\eta,l)w_L(k^\prime,\xi,l^\prime)}{\left\langle k,\eta,l \right\rangle}^{\sigma}{\left\langle k^\prime,\xi,l^\prime \right\rangle}^{\sigma} e^{\lambda{\left\vertk,\eta,l\right\vert}^s + \lambda{\left\vertk^\prime,\xi,l^\prime\right\vert}^s + c^\prime\lambda{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}\end{aligned}$$ Then, by , we have some $c^\prime < c < 1$ such that $$\begin{aligned}
\left(A^3_k(t,\eta,l)\right)^2 & \lesssim \left(\frac{w^3_{k^\prime}(\xi)}{ w^3_k(\eta)} \tilde{A}^3_k(t,\eta,l)\tilde{A}^3_{k^\prime}(t,\xi,l^\prime) + A^3_k(t,\eta,l) A^3_{k^\prime}(t,\xi,l^\prime)\right) e^{c\lambda{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}. \end{aligned}$$ Lemma \[lem:Jswap\] implies for some $K > 0$ (in particular), $$\begin{aligned}
\frac{w^3_{k^\prime}(\xi)}{w^3_k(\eta)} & \lesssim \left(1 + \frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}}\mathbf{1}_{t \in {\mathbf{I}}_{k,\eta} \cap {\mathbf{I}}_{k,\xi}}\mathbf{1}_{k \neq k^\prime} \right)e^{K\mu {\left\vert\eta-\xi\right\vert}^{1/2}}, \end{aligned}$$ and so we may restrict the frequencies over which we have a loss involving the $\tilde{A}^3$ to $\chi^{R,NR}$ but there is an overlapping region where both $A^3$ and $\tilde{A}^3$ are necessary. This completes the proof of now in the range of frequencies ${\left\vertl\right\vert} > \frac{1}{5}{\left\vert\eta\right\vert}$ or ${\left\vertl^\prime\right\vert} > \frac{1}{5}{\left\vert\xi\right\vert}$.\
*Case 3: ${\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}$*\
In this case, we need to be able to gain from the regularity imbalance. Here we have $\chi^{\ast;33} = 0$ and the only contributions are those which involve $\tilde{A}^3$. We have here, using $w_{k^\prime}(t,\xi) \leq 1$ by definition (see Appendix \[sec:Defw\]), $$\begin{aligned}
\frac{\left(\frac{e^{\mu {\left\vert\eta\right\vert}^{1/2}}}{w^3_k(t,\eta)} + e^{\mu{\left\vertl\right\vert}^{1/2}}\right)}{\left(\frac{e^{\mu {\left\vert\xi\right\vert}^{1/2}}}{w^3_{k^\prime}(t,\xi)} + e^{\mu{\left\vertl^\prime\right\vert}^{1/2}}\right)} & \lesssim \frac{w^3_{k^\prime}(t,\xi)}{w^3_k(t,\eta)}e^{\mu{\left\vert\eta-\xi\right\vert}^{1/2}} + w^3_{k^\prime}(t,\xi)e^{\mu{\left\vertl\right\vert}^{1/2} - \mu {\left\vert\xi\right\vert}^{1/2}} \\
& \lesssim \frac{w^3_{k^\prime}(t,\xi)}{w^3_k(t,\eta)}e^{\mu{\left\vert\eta-\xi\right\vert}^{1/2}} + e^{\mu{\left\vertl\right\vert}^{1/2} - \mu {\left\vert\xi\right\vert}^{1/2}} \\
& \lesssim \frac{w^3_{k^\prime}(t,\xi)}{w^3_k(t,\eta)}e^{2\mu{\left\vert\eta-\xi,l-l^\prime\right\vert}^{1/2}}. \end{aligned}$$ Therefore, in this case we only have contributions from the ratio of $w^3$: as above, we have for some $c^\prime = c^\prime(s) \in (0,1)$: $$\begin{aligned}
\left(A_k^3(t,\eta,l)\right)^2 & \lesssim \frac{w^3_{k^\prime}(t,\xi)}{w_k^3(t,\eta)}e^{2\mu{\left\vert\eta-\xi,l-l^\prime\right\vert}^{1/2}} A_k^3(t,\eta,l) A_{k^\prime}^3(t,\xi,l^\prime) e^{c^\prime\lambda {\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}. \end{aligned}$$ then now follows from Lemma \[lem:Jswap\] (followed by ) and the fact that under these restrictions $A^3 \approx \tilde{A}^3$. We then have that follows from Lemma \[lem:Jswap\]. This completes the proof of over all possible frequencies, and as mentioned above, the other inequalities are similar or easier.
We also have the following for remainder terms in the paraproducts (see ); the proof is the same as the analogous \[Lemma 4.2 [@BGM15I]\], so we omit it here for brevity.
\[lem:Arem\] For all $K > 0$ there exists a $c = c(s,K) \in (0,1)$ such that if $$\begin{aligned}
\frac{1}{K}{\left\vertk^\prime,\xi,l^\prime\right\vert} \leq {\left\vertk - k^\prime,\eta-\xi,l-l^\prime\right\vert} \leq K{\left\vertk^\prime,\xi,l^\prime\right\vert},\end{aligned}$$ then
\[ineq:ARemainderBasic\] $$\begin{aligned}
A^1_k(t,\eta,l) & \lesssim {\left\langle t \right\rangle}^{-2-\delta_1} e^{c\lambda {\left\vertk^\prime,\xi,l^\prime\right\vert}^s}e^{c{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s} \\
A^2_k(t,\eta,l) & \lesssim {\left\langle t \right\rangle}^{-1} e^{c \lambda {\left\vertk^\prime,\xi,l^\prime\right\vert}^s}e^{c{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s} \\
A^3_k(t,\eta,l) & \lesssim {\left\langle t \right\rangle}^{-2} e^{c \lambda {\left\vertk^\prime,\xi,l^\prime\right\vert}^s}e^{c{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}, \label{ineq:A3remainderBasic} \end{aligned}$$
and if $k = k^\prime = 0$ then $$\begin{aligned}
A(t,\eta,l) & \lesssim e^{c \lambda {\left\vert\xi,l^\prime\right\vert}^s}e^{c \lambda {\left\vert\eta-\xi,l-l^\prime\right\vert}^s}. \label{ineq:AARemainderBasic}\end{aligned}$$ All implicit constants depend on $\kappa, \lambda, \sigma$ and $s$.
The following is \[Lemma 4.3 [@BGM15I]\], see therein for a proof.
\[Frequency ratios for $\partial_t w$ and $\partial_t w_L$\] \[lem:CKwFreqRat\] For all $t \geq 1$ we have
$$\begin{aligned}
\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vertk,\eta,l\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) & \lesssim \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\langle k^\prime,\xi,l^\prime \right\rangle}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) {\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^2 \label{ineq:dtwBasicBrack} \\
\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vertk,\eta,l\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) & \lesssim \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vertk^\prime,\xi,l^\prime\right\vert}^{s/2} + {\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) \nonumber \\ & \quad\quad \times {\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^2 \label{ineq:dtwBasicBrack2} \\
\sqrt{\frac{\partial_t w_L(t,k,\eta,l)}{w_L(t,k,\eta,l)}} & \lesssim \sqrt{\frac{\partial_t w_L(t,k,\xi,l^\prime)}{w_L(t,k,\xi,l^\prime)}} {\left\langle \eta-\xi,l-l^\prime \right\rangle}^{3/2}. \label{ineq:dtNBasic}\end{aligned}$$
Further, if ${\left\vertk^\prime,\xi,l^\prime\right\vert} \gtrsim 1$ then implies $$\begin{aligned}
\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vertk,\eta,l\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) & \lesssim \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vertk^\prime,\xi,l^\prime\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) {\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^2. \label{ineq:dtwBasic}\end{aligned}$$ Moreover, both and hold if we replace ${\left\vertk,\eta,l\right\vert}$ and ${\left\vertk,\xi,l^\prime\right\vert}$ by ${\left\vert\eta\right\vert}$ and ${\left\vert\xi\right\vert}$ (respectively).
The next lemma is \[Lemma 4.4, [@BGM15I]\] and is immediate from the definition of $D$ , but useful for separating the pre and post critical times in the enhanced dissipation estimates.
For all $p \geq 0$ and $(k,\eta,l)$ there holds the following inequalities
$$\begin{aligned}
A^{\nu;i}_k(t,\eta,l) & \lesssim {\left\langle t \right\rangle}^{-p}{\left\langle k,\eta,l \right\rangle}^{\beta + 3\alpha + p}e^{\lambda{\left\vertk,\eta,l\right\vert}^{s}} + A^{\nu;i}_k(t,\eta,l) \mathbf{1}_{t \geq 2{\left\vert\eta\right\vert}} \label{ineq:AnuHiLowSep} \\
A^{\nu;i}_k(t,\eta,l) & \lesssim {\left\langle t \right\rangle}^{-p}{\left\langle k,\eta,l \right\rangle}^{\beta + 3\alpha + p}e^{\lambda{\left\vertk,\eta,l\right\vert}^{s}} + {\left\langle t \right\rangle}^{-1}\left({\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}\right)A^{\nu;i}_k(t,\eta,l) \mathbf{1}_{t \geq 2{\left\vert\eta\right\vert}}. \label{ineq:AnuHiLowSep2}\end{aligned}$$
The next lemma tells us how to treat ratios involving $\Delta_L$. This lemma is a technical improvement of \[Lemma 4.5, [@BGM15I]\]. The adjustments are necessary as here we can only use the $CK_w$ terms in a certain sector of frequency due to the more non-trivial angular dependence of the norms we are employing.
\[lem:MainFreqRat\] If $t \gtrsim 1$ then for all $\eta,\xi,l,l^\prime, k^\prime$ and $k$ define the following $$\begin{aligned}
\chi_{NR;k} = 1- \mathbf{1}_{t \in {\mathbf{I}}_{k,\eta} \cap {\mathbf{I}}_{k,\xi}}\mathbf{1}_{{\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}}\mathbf{1}_{{\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}}. \label{def:chiNR}\end{aligned}$$ Then, we have the following
- Basic characterizations of non-resonance: for all $k \neq 0$, $$\begin{aligned}
\left(\frac{1}{{\left\vertk,\eta-kt,l\right\vert}} + \frac{1}{{\left\vertk,\xi-kt,l^\prime\right\vert}}\right)\chi_{NR;k} & \lesssim \frac{1}{{\left\langle k,t,l^\prime \right\rangle}} {\left\langle \eta-\xi,l-l^\prime \right\rangle}; \label{ineq:basicNR} \end{aligned}$$
- Approximate integration by parts: for all $k \neq 0$, $$\begin{aligned}
{\left\vert\eta-kt\right\vert} \lesssim {\left\langle \eta-\xi \right\rangle}\left({\left\vertk\right\vert} + {\left\vert\xi-kt\right\vert}\right); \label{ineq:TriTriv}\end{aligned}$$
- For absorbing long-time losses: for all $k \neq 0$, $$\begin{aligned}
\frac{1}{{\left\vertk,\eta-kt,l\right\vert}} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} & \lesssim {\left\langle \eta-\xi,l-l^\prime \right\rangle}; \label{ineq:ratlongtime}\end{aligned}$$
- For the linear stretching terms, for all $k \neq 0$, $$\begin{aligned}
\frac{{\left\vertk\right\vert} \mathbf{1}_{t \leq 2{\left\vert\eta\right\vert}}}{{\left\vertk\right\vert} + {\left\vertl\right\vert} + {\left\vert\eta-kt\right\vert}} & \lesssim \kappa^{-1}\frac{\partial_t w(t,\eta)}{w(t,\eta)}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vertl\right\vert}^{1/2}}{t^{3/2}}; \label{ineq:CKwLS}\end{aligned}$$
- For nonlinear terms involving $\partial_X$ (for **(SI)** terms): if $p \in \mathbb{R}$ and $k \neq 0$, $$\begin{aligned}
\frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \left(\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vertk,\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vertk,\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) \right. \nonumber \\ & \left. \quad + \frac{\chi_{NR;k}}{{\left\langle t \right\rangle}}\min\left(1,\frac{{\left\vertk, \eta-kt,l\right\vert}}{{\left\langle kt \right\rangle}} \right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p \right) {\left\langle \eta-\xi,l-l^\prime \right\rangle}^{4}; \label{ineq:AiPartX}\end{aligned}$$
- For terms with fewer derivatives (for **(3DE)** terms): if $a \in {\left\{1,2\right\}}$, $p \in \mathbb{R}$, and $k^\prime,k \neq 0$, then $$\begin{aligned}
\frac{1}{{\left\vertk^\prime,\xi-k^\prime t,l^\prime\right\vert}^a} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \nonumber \\ & \hspace{-3cm} \left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} \mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vertk,\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} \mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vertk^\prime, \xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) {\left\langle k-k^\prime, \eta-\xi,l-l^\prime \right\rangle}^3 \nonumber \\ & \hspace{-3cm} \quad\quad + \frac{1}{{\left\langle t \right\rangle}^{a}}{\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p {\left\langle k-k^\prime, \eta-\xi,l-l^\prime \right\rangle}^3 \label{ineq:AikDelLNoD} \end{aligned}$$
- For **(3DE)** terms in the nonlinear pressure and stretching: if $p \in \mathbb{R}$, $k k^\prime(k-k^\prime) \neq 0$,
\[ineq:AikDelL2D\] $$\begin{aligned}
\frac{{\left\vertk,\eta-k t,l\right\vert}{\left\vertk,\xi-k^\prime t,l^\prime\right\vert}}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \left({\left\langle t \right\rangle} + {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p\right){\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^{2} \\
\frac{{\left\vertk,\eta-k t,l\right\vert}{\left\vertk^\prime,\xi-k^\prime t,l^\prime\right\vert}}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \left({\left\langle t \right\rangle}\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vertk,\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\right. \nonumber \\ & \left. \quad\quad \times \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vertk^\prime, \xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) \right. \nonumber \\ & \left. \quad\quad + \min\left(1,\frac{{\left\vertk,\eta-kt,l\right\vert}}{{\left\langle kt \right\rangle}} \right){\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p\right){\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^{2}. \label{ineq:AikDelL2D_CKw} \\
\frac{{\left\vertl^\prime\right\vert}{\left\vertk,\eta-kt,l\right\vert}}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} & \lesssim \left({\left\langle t \right\rangle}\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vertk,\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) \right. \nonumber \\ & \left. \quad\quad \times \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vertk^\prime, \xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) \right. \nonumber \\ & \quad\quad + 1\bigg) {\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^{2}. \label{ineq:AikDelL2D_CKw2} \end{aligned}$$
- For triple derivative terms (these arise in the treatment of **(F)** terms): if $p \in \mathbb{R}$ and $k \neq 0$,
\[ineq:AdelLij\] $$\begin{aligned}
\frac{{\left\vertl\right\vert}^3}{(k)^2 + (l^\prime)^2 + {\left\vert\xi-k t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim {\left\vertl\right\vert}\left({\left\langle l-l^\prime \right\rangle}^2 + \frac{{\left\vertl\right\vert}^2}{{\left\langle l^\prime,t \right\rangle}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p\right) \label{ineq:AdeZZZ} \\
\frac{{\left\vert\eta\right\vert}{\left\vertl\right\vert}^2 + {\left\vert\eta\right\vert}^2{\left\vertl\right\vert}}{(k)^2 + (l^\prime)^2 + {\left\vert\xi-k t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \left({\left\langle t \right\rangle}^2\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}\right) \right. \nonumber \\ & \left. \quad\quad \times \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}\right) \right. \nonumber \\ & \left. \quad + {\left\vertl\right\vert}\left(1 + \frac{{\left\vert\eta\right\vert}{\left\vertl\right\vert} + {\left\vert\eta\right\vert}^2}{{\left\langle \xi,l^\prime,t \right\rangle}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p\right)\right) {\left\langle k,\eta-\xi,l-l^\prime \right\rangle}^3 \label{ineq:AdeYZZ} \\
\frac{{\left\vert\eta\right\vert}^3}{k^2 + (l^\prime)^2 + {\left\vert\xi-k t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \left({\left\langle t \right\rangle}^3\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}\right) \right. \nonumber \\ & \left. \quad\quad \times \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}\right) \right. \nonumber \\ & \left. \quad + \min({\left\vert\eta\right\vert},{\left\langle \xi-kt \right\rangle})\left(1 + \frac{{\left\vert\eta\right\vert}^2}{{\left\langle \xi,l^\prime,t \right\rangle}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p\right) \right) {\left\langle k,\eta-\xi,l-l^\prime \right\rangle}^3. \label{ineq:AdeYYY}\end{aligned}$$
As in [@BGM15I], implies $$\begin{aligned}
\frac{{\left\vert\eta,l\right\vert}{\left\langle \eta,l \right\rangle}^2}{(k)^2 + (l^\prime)^2 + {\left\vert\xi-k t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \left({\left\langle t \right\rangle}^3\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}\right) \right. \nonumber \\ & \left. \quad\quad \times \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}\right) \right. \nonumber \\ & \left. \quad + {\left\vert\eta,l\right\vert}\left(1 + \frac{{\left\langle \eta,l \right\rangle}^2}{{\left\langle \xi,l^\prime,t \right\rangle}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p\right)\right){\left\langle k,\eta-\xi,l-l^\prime \right\rangle}^3 \label{ineq:AdeGen}\end{aligned}$$
First, note that for any fixed number $N \geq 1$, $$\begin{aligned}
\frac{\mathbf{1}_{{\left\vertl^\prime\right\vert} \geq \frac{1}{N} {\left\vert\xi\right\vert}} }{{\left\vertk^\prime,l^\prime,\xi-k^\prime t\right\vert}} & \lesssim_{N} \frac{1}{{\left\vertl^\prime, k^\prime t\right\vert}}, \label{ineq:Znonres}\end{aligned}$$ and hence the sector in frequency where $l^\prime$ is dominant or comparable to $\xi$ is strongly non-resonant. Further, observe that for any $N \geq 1$, $$\begin{aligned}
{\left\vertl\right\vert} \geq \frac{1}{N}{\left\vert\eta\right\vert} \quad \textup{and} \quad{\left\vertl^\prime\right\vert} \leq \frac{1}{N+1}{\left\vert\xi\right\vert}, \end{aligned}$$ imply $$\begin{aligned}
{\left\vert\xi,l^\prime\right\vert} + {\left\vert\eta,l\right\vert} \lesssim_N {\left\vert\eta-\xi,l-l^\prime\right\vert}. \label{ineq:WellSepLEeta}\end{aligned}$$ This ensures that if $(\eta,l)$ and $(\xi,l^\prime)$ are in separated sectors in frequency, then the entire multiplier can generally be absorbed by the ${\left\langle \eta-\xi,l-l^\prime \right\rangle}^m$ factors and one will not need $\partial_t w/w$. Furthermore, from and , we can derive . These observations allow us to refine the analogous lemma of [@BGM15I] to deduce Lemma \[lem:MainFreqRat\].
As a representative example, let us consider the proof of . First consider the case ${\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}$ and ${\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}$. Then, as in [@BGM15I] (see therein for a proof), we have $$\begin{aligned}
\frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p \mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} \mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} & \lesssim \\ & \hspace{-6cm} \mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} \mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}}\left(\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vertk,\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vertk,\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) \right. \\ & \left. \hspace{-6cm} \quad + \frac{\mathbf{1}_{t > 2\min({\left\vert\eta\right\vert},{\left\vert\xi\right\vert})}}{{\left\langle t \right\rangle}}\min\left(1,\frac{{\left\vertk, \eta-kt,l\right\vert}}{{\left\langle kt \right\rangle}} \right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p \right) {\left\langle \eta-\xi,l-l^\prime \right\rangle}^{4}, \end{aligned}$$ which is consistent with .
Next, consider the case (${\left\vertl^\prime\right\vert} > \frac{1}{5}{\left\vert\xi\right\vert}$ or ${\left\vertl\right\vert} > \frac{1}{5}{\left\vert\eta\right\vert}$). If the former is true than we immediately have the following by : $$\begin{aligned}
\frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p & \lesssim \frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk\right\vert}}{{\left\langle l^\prime,\xi,kt \right\rangle}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p{\left\langle \eta-\xi,l-l^\prime \right\rangle}^2, \label{ineq:nonres1} \end{aligned}$$ which is consistent with . Next, consider instead ${\left\vertl\right\vert} > \frac{1}{5}{\left\vert\eta\right\vert}$. If ${\left\vertl^\prime\right\vert} > \frac{1}{6}{\left\vert\xi\right\vert}$ then (and hence ) follows again by . However, if ${\left\vertl^\prime\right\vert} < \frac{1}{6}{\left\vert\xi\right\vert}$ then by , ${\left\vert\eta,l\right\vert} + {\left\vert\xi,l^\prime\right\vert} \lesssim {\left\vert\eta-\xi,l-l^\prime\right\vert}$, and we again have by multiplying and dividing by ${\left\langle \xi,l^\prime \right\rangle}^2$.
The other inequalities are dealt with in a similar fashion.
For the current work, we need an analogue of Lemma \[lem:MainFreqRat\] which is more precise in order to handle (and take advantage of) the regularity imbalances in $A^3$.
\[lem:MainFreqRat\_RegImbalance\] For $t \geq 1$ and $k,k^\prime,\eta,\xi,l,l^\prime$,
Then for $p \in {\mathbb R}$, we have the following:
- for **(SI)** (for $k^\prime = k \neq 0$; recall that definition depends on both $k$ and $k^\prime$): $$\begin{aligned}
\frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2}\left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\right) & \nonumber \\
& \hspace{-6cm} \lesssim \left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vertk,\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vertk,\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) {\left\langle \eta-\xi,l-l^\prime \right\rangle}^{4}; \label{ineq:AiPartXA23}\end{aligned}$$
- a simpler variant (for $k^\prime = k \neq 0$): $$\begin{aligned}
\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} \hspace{1cm} & \nonumber \\ & \hspace{-4cm} \lesssim {\left\langle t \right\rangle}\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta,l\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vert\xi,l^\prime\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right){\left\langle \eta-\xi,l-l^\prime \right\rangle}^4. \label{ineq:jNRBasic} \end{aligned}$$
- if $k^\prime,k \neq 0$, $k \neq k^\prime$, and $a \in [1,2]$ (for **(3DE)** terms with few derivatives),
\[ineq:3DEoneRegBal\] $$\begin{aligned}
\frac{1}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2} \left(\sum_{r} \chi^{NR,r}\frac{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}{t}\right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p \hspace{4cm} & \nonumber \\
& \hspace{-10cm} \lesssim \frac{1}{{\left\langle t \right\rangle}}\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) {\left\langle \eta-\xi,l-l^\prime \right\rangle}^3 \nonumber \\ & \hspace{-10cm} \quad + \frac{1}{{\left\langle t \right\rangle}^2}{\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^p {\left\langle \eta-\xi,l-l^\prime \right\rangle}^3 \label{ineq:A3ReacGain} \\
\frac{1}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2}\left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta- k^\prime t\right\vert}}{t} + \chi^{\ast;33} \right) & \lesssim \frac{{\left\langle \eta-\xi \right\rangle}^2}{{\left\langle t \right\rangle}} \label{ineq:A33ReacGain} \\
\hspace{-2cm} \frac{1}{{\left\vertk^\prime, \xi - k^\prime t, l^\prime\right\vert}^a}\left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} \right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^a \lesssim {\left\langle \eta-\xi \right\rangle}^2; \label{ineq:A1A3ReacGain}\end{aligned}$$
- if $k^\prime,k \neq 0$ and $k \neq k^\prime$ (for **(3DE)** terms with more derivatives),
\[ineq:3DEoneRegBalII\] $$\begin{aligned}
{\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk^\prime,\xi-k^\prime t, l^\prime\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-k^\prime t\right\vert}^2} \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} \right) \hspace{1.5cm} & \nonumber \\ & \hspace{-4cm} \lesssim {\left\vertk,\eta-kt,l\right\vert} {\left\langle \eta-\xi,l-l^\prime \right\rangle} \label{ineq:jNRPneqneq} \\
\frac{{\left\vertk^\prime,\xi-tk^\prime,l^\prime\right\vert} {\left\vertk\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2}\left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t}\right) \hspace{1.5cm} & \nonumber \\
& \hspace{-11cm} \lesssim \left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) {\left\langle k-k^\prime, \eta-\xi,l-l^\prime \right\rangle}^4 \label{ineq:A33PartX} \\
\frac{{\left\vertk^\prime,\xi-tk^\prime,l^\prime\right\vert} {\left\vertl^\prime\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2}\left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t}\right) \hspace{1.5cm} & \nonumber \\
& \hspace{-11cm} \lesssim {\left\langle t \right\rangle}\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) {\left\langle k-k^\prime, \eta-\xi,l-l^\prime \right\rangle}^4 \label{ineq:A33PartXZ1} \\
\frac{{\left\vertl k\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2} \left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t} + \chi^{\ast;33} \right) \nonumber \\
& \hspace{-10cm} \lesssim {\left\langle k-k^\prime,\eta-\xi,l-l^\prime \right\rangle}^3. \label{ineq:A3neqA3neqZX}\end{aligned}$$
- for terms of type **(F)**, (with $k = 0$ and $k^\prime \neq 0$),
\[ineq:AdelLijRegBal\] $$\begin{aligned}
\frac{{\left\vertl\right\vert} {\left\langle \eta,l \right\rangle}^2 {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{2}}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2}\left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta - tr\right\vert}}\right) \hspace{3.5cm} & \nonumber \\
& \hspace{-10cm} \lesssim {\left\langle t \right\rangle}^2\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) {\left\langle k^\prime,\eta-\xi,l-l^\prime \right\rangle}^3 \label{ineq:AA3Z} \\
\frac{{\left\vert\eta\right\vert} {\left\langle \eta,l \right\rangle}^2}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2}\left(\sum_{r}\chi^{NR,r}\frac{{\left\vertr\right\vert} + {\left\vert\eta - tr\right\vert}}{t} \right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \hspace{2cm} & \nonumber \\
& \hspace{-10cm} \lesssim \left({\left\langle t \right\rangle}^2\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) + {\left\vert\eta\right\vert} \right) {\left\langle k^\prime,\eta-\xi,l-l^\prime \right\rangle}^3; \label{ineq:A03A2YYY} \\
\frac{{\left\vert\eta\right\vert} {\left\langle \eta,l \right\rangle}^2}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2} \chi^{\ast;32} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \hspace{6cm} & \nonumber \\
& \hspace{-12cm} \lesssim \left({\left\langle t \right\rangle}^2\left(\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}}\mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right)\left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}} + \frac{{\left\vert\xi\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} \right) + {\left\vert\eta\right\vert} \right) {\left\langle k^\prime,\eta-\xi,l-l^\prime \right\rangle}^3; \label{ineq:A03A2YYY2}\end{aligned}$$
\[rmk:noFreqRestrict\] Note the lack of frequency restrictions to ${\left\vertl\right\vert} < \frac{1}{5}{\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} < \frac{1}{5}{\left\vert\xi\right\vert}$. This is due to the fact that these inequalities need to sometimes be applied in the overlap regions where ${\left\vertl\right\vert} \approx {\left\vert\eta\right\vert}$ and ${\left\vertl^\prime\right\vert} \approx {\left\vert\xi\right\vert}$.
The proofs are very similar to Lemma \[lem:MainFreqRat\] with some minor changes. Consider (the analogue of ). We have, by Lemma \[lem:dtw\], $$\begin{aligned}
\frac{{\left\vertk,\eta-kt,l\right\vert} {\left\vertk\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2}\left(\sum_{r}\chi^{r,NR} \frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\right)
& \nonumber \\ & \hspace{-6cm}
\lesssim \sum_{r} \chi^{r,NR} \frac{{\left\vertk\right\vert}t}{t{\left\vertk-r\right\vert}{\left\vertr\right\vert}}\frac{\partial_t w(t,\eta)}{w(t,\eta)}{\left\langle \eta-\xi,l-l^\prime \right\rangle}
\label{ineq:jNRBasicPfineq}\end{aligned}$$ from which the result follows by Lemma \[lem:CKwFreqRat\] (and that $\chi^{r,NR}$ form a partition of unity for a certain region of frequencies). The proof of is essentially the same.
Consider ; the other inequalities in are easy variants of this and the proofs of above. First, in the case $t \not\in {\mathbf{I}}_{k^\prime,\eta} \cap {\mathbf{I}}_{k^\prime,\xi}$, we have ${\left\langle \eta-\xi \right\rangle}{\left\langle \xi - k^\prime t \right\rangle} \gtrsim t$ by Lemma \[lem:wellsep\], and so follows. Next, consider the case that $t \in {\mathbf{I}}_{k^\prime,\eta} \cap {\mathbf{I}}_{k^\prime,\xi}$. Then, since $k \neq k^\prime$, $t \not\in {\mathbf{I}}_{k,\eta}$ and this contribution appears in the sum as $\chi^{NR,k^\prime}$ (recall the definition ). In this case follows by Lemma \[lem:CKwFreqRat\]. This now covers all cases.
Let us comment briefly on the proof of . The term such that $r = k^\prime$ follows due to the Lemma \[dtw\] together with the frequency restrictions ensuring ${\left\vert\eta\right\vert}{\left\langle \eta,l \right\rangle}^2 \lesssim {\left\langle kt \right\rangle}^3$. For the terms $r \neq k^\prime$, we have $$\begin{aligned}
\frac{{\left\vert\eta\right\vert} {\left\langle \eta,l \right\rangle}^2}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi-k^\prime t\right\vert}^2}\chi^{NR,r} \lesssim \frac{{\left\langle rt \right\rangle}^3}{t^2 {\left\vertk-r\right\vert}^2} \lesssim {\left\langle t \right\rangle}^2 \frac{{\left\vertr\right\vert}}{t} {\left\vertk\right\vert}^2, \end{aligned}$$ which is consistent with by Lemma \[dtw\] again.
The remaining estimates follow by similar arguments combined with the arguments used in the proof of Lemma \[lem:MainFreqRat\] (see also [@BGM15I]). Hence, these are omitted for the sake of brevity.
Paraproducts and related notations {#sec:paranote}
----------------------------------
We briefly recall the short-hands introduced in [@BGM15I]. For paraproducts we use the homogeneous variant of the paraproduct and utilize the following short-hand to suppress the appearance of Littlewood-Paley projections: $$\begin{aligned}
fg & = f_{Hi} g_{Lo} + f_{Lo} g_{Hi} + (fg)_{\mathcal{R}} \nonumber \\
& = \sum_{M \in 2^{\mathbb Z}} f_{M} g_{<M/8} + \sum_{M \in 2^{{\mathbb Z}}} f_{<M/8} g_{M} + \sum_{M \in 2^{{\mathbb Z}}} \sum_{M/8 \leq M^\prime \leq 8M} f_{M} g_{M^\prime}. \label{def:parapp}\end{aligned}$$ We recall the following lemma from [@BGM15I] for using the paraproducts in $L^2$ estimates.
\[gevreyparaproductlemma\] Let $s\in[0,1)$, $\mu \geq 0$, $p \geq 0$. Then, there exists a $c = c(s) \in (0,1)$ such that the following holds,
\[ineq:paraquad\] $$\begin{aligned}
{\left\lVert f_{Hi} g_{Lo} \right\rVert}_{{\mathcal{G}}^{\mu,p}} & \lesssim {\left\lVert f \right\rVert}_{{\mathcal{G}}^{\mu,p}}{\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \label{ineq:quadHL} \\
{\left\lVert (fg)_{\mathcal{R}} \right\rVert}_{{\mathcal{G}}^{\mu,p}} & \lesssim {\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\mu,p}} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \label{ineq:quadR} \\
\int e^{\mu{\left\vert{\nabla}\right\vert}^s}{\left\langle {\nabla}\right\rangle}^{p} h \, e^{\mu{\left\vert{\nabla}\right\vert}^s}{\left\langle {\nabla}\right\rangle}^{p} \left(f_{Hi} g_{Lo}\right) dV & \lesssim {\left\lVert h \right\rVert}_{{\mathcal{G}}^{\mu,p}}{\left\lVert f \right\rVert}_{{\mathcal{G}}^{\mu,p}}{\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}}. \label{ineq:triQuadHL}\end{aligned}$$
In most places in the proof, $\mu = 0$ as normally the multipliers $A^i$ or $A^{\nu;i}$ are playing the role of the norm.
Many of the nonlinear terms are higher order (up to quintic). For expanding cubic nonlinear terms, we use the short-hand from [@BGM15I]: $$\begin{aligned}
fgh & = \sum_{N \in 2^{{\mathbb Z}}} f_N g_{<N/8}h_{<N/8} + g_{N} f_{<N/8} h_{<N/8} + f_{< N/8} g_{<N/8} h_N + (fgh)_{\mathcal{R}} \nonumber \\
& := f_{Hi}(gh)_{Lo} + g_{Hi}(fh)_{Lo} + h_{Hi}(gf)_{Lo} + (fgh)_{\mathcal{R}}, \label{def:cubic}\end{aligned}$$ where the remainder term $(fgh)_{\mathcal{R}}$, includes all of the frequency contributions not included in the leading order terms. Note the short-hand $(gh)_{Lo} = g_{Lo} h_{Lo}$. By iterating this pattern, we obtain also decompositions for quartic and quintic terms. We also have the equivalents of , and .
\[lem:ParaHighOrder\] For all $\mu \geq 0$ and $p \geq 0$, there is some $c = c(s) \in (0,1)$ such that
\[ineq:quin\] $$\begin{aligned}
{\left\lVert g_{Hi} (fhkj)_{Lo} \right\rVert}_{{\mathcal{G}}^{\mu,p}} & \lesssim_{p} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\mu,p}}{\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \nonumber \\ & \quad\quad \times {\left\lVert h \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert k \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert j \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \label{ineq:quinHL} \\
{\left\lVert (fghkj)_{\mathcal{R}} \right\rVert}_{{\mathcal{G}}^{\mu,p}} & \lesssim_{p} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}}{\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert h \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \nonumber \\ & \quad\quad \times {\left\lVert k \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert j \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \label{ineq:quinR} \\
\int e^{\mu {\left\vert{\nabla}\right\vert}^s}{\left\langle {\nabla}\right\rangle}^p q e^{\mu{\left\vert{\nabla}\right\vert}^s} {\left\langle {\nabla}\right\rangle}^p (g_{Hi}(fhkj)_{Lo}) dV & \lesssim_{p} {\left\lVert q \right\rVert}_{{\mathcal{G}}^{\mu,p}}{\left\lVert g \right\rVert}_{{\mathcal{G}}^{\mu,p}} {\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} \nonumber \\ & \quad\quad \times {\left\lVert h \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert k \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert j \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}}. \end{aligned}$$
Analogous estimates hold also for the cubic and quartic decompositions.
One final short-hand we recall from [@BGM15I] involves the inner products that appear naturally in energy estimates. Consider, for example, a typical Gevrey energy estimate involving three quantities $f,g,h$, where generally $h$ will be a product of several low frequency terms: $$\begin{aligned}
\int e^{\lambda{\left\vert{\nabla}\right\vert}^s}f e^{\lambda{\left\vert{\nabla}\right\vert}^s}\left(g_{Hi} h_{Lo}\right) dV & = \frac{1}{(2\pi)^{3/2}}\sum_{k,l,k^\prime,l^\prime} \int_{\eta,\xi} e^{\lambda{\left\vertk,\eta,l\right\vert}^s}\overline{\hat{f}}_k(\eta,l) e^{\lambda{\left\vertk,\eta,l\right\vert}^s} \hat{g}_{k^\prime}(\xi,l^\prime)_{Hi} \hat{h}_{k-k^\prime}(\eta-\xi,l-l^\prime)_{Lo} d\eta d\xi. \end{aligned}$$ By the frequency localizations inherent in the shorthand and , for some $c = c(s) \in (0,1)$ we have (by ), $$\begin{aligned}
\int e^{\lambda{\left\vert{\nabla}\right\vert}^s}f e^{\lambda{\left\vert{\nabla}\right\vert}^s}\left(g_{Hi} h_{Lo}\right) dV & \lesssim \sum_{k,l,k^\prime,l^\prime} \int_{\eta,\xi} e^{\lambda{\left\vertk,\eta,l\right\vert}^s}{\left\vert\hat{f}_k(\eta,l)\right\vert} e^{\lambda{\left\vertk^\prime,\xi,l^\prime\right\vert}^s} {\left\vert\hat{g}_{k^\prime}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times e^{c\lambda{\left\vertk-k^\prime,\eta-\xi,l-l^\prime\right\vert}^s}{\left\vert\hat{h}_{k-k^\prime}(\eta-\xi,l-l^\prime)_{Lo}\right\vert} d\eta d\xi \\
& \lesssim {\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda}}{\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda}} {\left\lVert h \right\rVert}_{{\mathcal{G}}^{c\lambda,3/2+}}.\end{aligned}$$ The low frequency factors will generally all be put in a norm ${\mathcal{G}}^{\lambda,3/2+}$ (once the estimates are over we do not need to worry about the $c$) and hence it makes sense to use a short-hand for the low-frequency factor as ${\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}}Low(k-k^\prime,\eta-\xi,l-l^\prime)$ where the function $Low$ is taken as an $O(1)$ function in ${\mathcal{G}}^{\lambda,3/2+}$ (and which can change line-to-line as implicit constants). For example, $$\begin{aligned}
\int e^{\lambda{\left\vert{\nabla}\right\vert}^s}f e^{\lambda{\left\vert{\nabla}\right\vert}^s}\left(g_{Hi} h_{Lo}\right) dV & := {\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}}\sum_{k,l,k^\prime,l^\prime} \int_{\eta,\xi} e^{\lambda{\left\vertk,\eta,l\right\vert}^s}\overline{\hat{f}}_k(\eta,l) e^{\lambda{\left\vertk,\eta,l\right\vert}^s} \hat{g}_{k^\prime}(\xi,l^\prime)_{Hi} \nonumber \\ & \quad\quad \times Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \nonumber \\
& \lesssim {\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}}\sum_{k,l,k^\prime,l^\prime} \int_{\eta,\xi} e^{\lambda{\left\vertk,\eta,l\right\vert}^s}{\left\vert\hat{f}_k(\eta,l)\right\vert} e^{\lambda{\left\vertk^\prime,\xi,l^\prime\right\vert}^s} {\left\vert\hat{g}_{k^\prime}(\xi,l^\prime)_{Hi}\right\vert}\nonumber \\ & \quad\quad \times Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \nonumber \\
& \lesssim {\left\lVert f \right\rVert}_{{\mathcal{G}}^{\lambda}}{\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda}} {\left\lVert h \right\rVert}_{{\mathcal{G}}^{c\lambda,3/2+}}. \label{def:Low}\end{aligned}$$ The utility of this short-hand will quickly become clear in the course of the proof.
Product lemmas and a few immediate consequences
-----------------------------------------------
First, note the following product lemma is an immediate consequence of Lemma \[gevreyparaproductlemma\].
\[lem:GevProdAlg\] For all $s \in (0,1)$, $\mu \geq 0$, and $p \geq 0$, there exists $c = c(s) \in (0,1)$ such that the following holds for all $f,g \in \mathcal{G}^{\mu,p}$:
$$\begin{aligned}
{\left\lVert fg \right\rVert}_{{\mathcal{G}}^{\mu,p}} & \lesssim_{p} {\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\mu,p}} + {\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\mu,3/2+}} {\left\lVert f \right\rVert}_{{\mathcal{G}}^{\mu,p}}, \label{ineq:GProduct}\end{aligned}$$
in particular, if $\mu > 0$, then $\mathcal{G}^{\mu,p}$ is an algebra for all $p \geq 0$ by : $$\begin{aligned}
{\left\lVert f g \right\rVert}_{{\mathcal{G}}^{\mu,\sigma}} & \lesssim_{p,\mu} {\left\lVert f \right\rVert}_{{\mathcal{G}}^{\mu,p}} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\mu,p}}. \label{ineq:GAlg} \end{aligned}$$
Next we have the following, which is a simple variant of the analogous lemma from [@BGM15I].
\[lem:AAiProd\] Let $p \geq 0$ and $r \geq -\sigma$. Then there exists a $c = c(s) \in (0,1)$ such that for $i \in {\left\{1,2\right\}}$, for all $f,g$,
$$\begin{aligned}
{\left\lVert {\left\vert{\nabla}\right\vert}^p {\left\langle {\nabla}\right\rangle}^{r}A^i(fg) \right\rVert}_2 & \lesssim {\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\lambda,3/2+}}{\left\lVert {\left\vert{\nabla}\right\vert}^p {\left\langle {\nabla}\right\rangle}^{r} A^ig \right\rVert}_2 \nonumber \\ & \quad + {\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\lambda,3/2+}}{\left\lVert {\left\vert{\nabla}\right\vert}^p {\left\langle {\nabla}\right\rangle}^{r} A^if \right\rVert}_2 \label{ineq:Aprodi} \\
{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^i \right)(fg) \right\rVert}_2 & \lesssim {\left\lVert f \right\rVert}_{{\mathcal{G}}^{c\lambda,3/2+}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^i \right) g \right\rVert}_2 \nonumber \\ & \quad + {\left\lVert g \right\rVert}_{{\mathcal{G}}^{c\lambda,3/2+}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^i \right)f \right\rVert}_2. \end{aligned}$$
If $f$ and $g$ are both independent of $X$, then the above holds also with $A^i$ replaced by either $A$ or $A^3$.
Notice the crucial detail that Lemma \[lem:AAiProd\] does *not* hold for $A^3$ if $f$ or $g$ depend on $X$ due to the regularity imbalances near the critical times.
Together with , Lemma \[lem:AAiProd\] and Lemma \[lem:GevProdAlg\] imply the following lemma (as long as $C^i$ remains sufficiently small). The proof is straightforward so we omit it for the sake of brevity.
\[lem:CoefCtrl\] Let
\[def:G\] $$\begin{aligned}
G_{yy} & = \left((1+\psi_y)^2 + \psi_z^2\right) - 1 \\
G_{yz} & =2\phi_y(1+\psi_y) + 2\psi_z(1 + \phi_z) \\
G_{zz} & = \left((1 + \phi_z)^2 + \phi_y^2 \right) - 1. \end{aligned}$$
Under the bootstrap hypotheses, for $c_0$ sufficiently small, we have for any $G \in {\left\{\psi_y,\psi_z,\phi_y,\phi_z,G_{yy},G_{yz},G_{zz}\right\}}$,
$$\begin{aligned}
{\left\lVert {\left\langle {\nabla}\right\rangle}^{-1} A G \right\rVert}_2 & \lesssim {\left\lVert A C \right\rVert}_2 \\
{\left\lVert A G \right\rVert}_2 & \lesssim {\left\lVert {\nabla}A C \right\rVert}_2 \\
{\left\lVert {\left\langle {\nabla}\right\rangle}^{-1} \sqrt{\frac{\partial_t w}{w}} \tilde{A} G \right\rVert}_2 & \lesssim {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A\right) C \right\rVert}_2 \\
{\left\lVert {\left\langle {\nabla}\right\rangle}^{-1} {\left\vert{\nabla}\right\vert}^{s/2} A G \right\rVert}_2 & \lesssim {\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A C \right\rVert}_2. \end{aligned}$$
Further,
\[ineq:CCDeltatC\] $$\begin{aligned}
{\left\lVert {\left\langle {\nabla}\right\rangle}^{-2} A \Delta_t C^i \right\rVert}_2 & \lesssim {\left\lVert A C \right\rVert}_2 \\
{\left\lVert {\left\langle {\nabla}\right\rangle}^{-1} A \Delta_t C^i \right\rVert}_2 & \lesssim {\left\lVert {\nabla}A C \right\rVert}_2 \\
{\left\lVert \sqrt{\frac{\partial_t w}{w}} {\left\langle {\nabla}\right\rangle}^{-2} \tilde{A} \Delta_t C^i \right\rVert}_2 & \lesssim {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A\right) C \right\rVert}_2 \\
{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2} {\left\langle {\nabla}\right\rangle}^{-2} A \Delta_t C^i \right\rVert}_2 & \lesssim {\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A C \right\rVert}_2. \end{aligned}$$
Similarly, for any $\lambda(t) \geq \mu > 0$ and $\sigma \geq p \geq 0$ (the constant can be taken independent of $\mu$ for $p > 1$): $$\begin{aligned}
{\left\lVert G \right\rVert}_{{\mathcal{G}}^{\mu,p}} + {\left\lVert \Delta_t C \right\rVert}_{{\mathcal{G}}^{\mu,p-1}} \lesssim {\left\lVert {\nabla}C \right\rVert}_{{\mathcal{G}}^{\mu,p}}. \label{ineq:psiCLow} \end{aligned}$$
\[rmk:CoefCtrlLow\] As discussed in [@BGM15I], a consequence of together with implies that when coefficients appear in ‘low frequency’ in a paraproduct they satisfy the a priori estimate $O(\epsilon {\left\langle t \right\rangle})$. Together with $\epsilon t {\left\langle \nu t^3 \right\rangle}^{-1} \lesssim {\left\langle t \right\rangle}^{-1}$, this implies that when there is enhanced dissipation present, we generally need only treat the leading order terms that arise from the approximation $\partial_i^t \approx \partial_i^L$ or the terms that arise when the coefficients are in high frequency.
\[rmk:SIcoefneglect\] Even when enhanced dissipation is not present, the coefficients do not depend on $X$ and hence the presence of the coefficients do not shift the frequencies in $X$. This will mean that even when there are no powers of ${\left\langle \nu t^3 \right\rangle}^{-1}$, terms in which coefficients appear in low frequency are generally treatable with an easy variant of the treatment used on the leading order terms. There are a few exceptions, when the structure of the term is changed by the coefficients, and otherwise these terms are generally omitted.
We recall the following lemma from [@BGM15I].
\[lem:AnuProd\] The following holds for all $f^1$ and $f^2$ such that $f^2_{\neq} = f^2$, $$\begin{aligned}
{\left\lVert A^{\nu;i}(f^1 f^2) \right\rVert}_2 & \lesssim {\left\lVert f^1 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha+ 3/2+}}{\left\lVert A^{\nu;i}f^2 \right\rVert}_2. \label{ineq:AnuiDistri} \end{aligned}$$ Moreover, if also $f^1_{\neq} = f^1$ then we have the product-type inequalities
\[ineq:AnuiDistriDecay\] $$\begin{aligned}
{\left\lVert A^{\nu;1}(f^1 f^2) \right\rVert}_2 \lesssim \frac{{\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;1} f^1 \right\rVert}_{2}{\left\lVert A^{\nu;1} f^2 \right\rVert}_2 + {\left\lVert A^{\nu;1} f^1 \right\rVert}_{2}{\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;1} f^2 \right\rVert}_2 \right) \label{ineq:AnuiDistriDecay1} \\
{\left\lVert A^{\nu;2}(f^1 f^2) \right\rVert}_2 \lesssim \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;2} f^1 \right\rVert}_{2}{\left\lVert A^{\nu;2} f^2 \right\rVert}_2 + {\left\lVert A^{\nu;2} f^1 \right\rVert}_{2}{\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;2} f^2 \right\rVert}_2 \right) \label{ineq:AnuiDistriDecay2} \\
{\left\lVert A^{\nu;3}(f^1 f^2) \right\rVert}_2 \lesssim \frac{{\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;3} f^1 \right\rVert}_{2}{\left\lVert A^{\nu;3} f^2 \right\rVert}_2 + {\left\lVert A^{\nu;3} f^1 \right\rVert}_{2}{\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;3} f^2 \right\rVert}_2 \right). \label{ineq:AnuiDistriDecay3}\end{aligned}$$
High norm estimate on $Q^2$
===========================
First compute the time evolution of $A^2 Q^2$ in $L^2$: $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A^{2} Q^2 \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{2} Q^2 \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w}{w}}\tilde{A}^{2} Q^2 \right\rVert}_2^2 - \frac{1}{t}{\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{2}Q_{\neq}^2 \right\rVert}_2^2 \nonumber \\
& \quad -{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{2} Q^2 \right\rVert}_2^2 + \nu \int A^{2} Q^2 A^{2} \left(\tilde{\Delta_t} Q^2\right) dV -\int A^{2} Q^2 A^{2} \left( \tilde U \cdot {\nabla}Q^2 \right) dV \nonumber \\
& \quad - \int A^{2} Q^2 A^{2}\left(Q^j \partial_j^t U^2 + 2\partial_i^t U^j \partial_{i}^t \partial_{j}^t U^i - \partial_Y^t\left(\partial_i^t U^j \partial_j^t U^i\right) \right) dV \nonumber \\
& = - \mathcal{D}Q^2 - CK^2_L + \mathcal{D}_E + \mathcal{T} + NLS1 + NLS2 + NLP, \label{eq:A2Q2Evo}\end{aligned}$$ where we used the definition $$\begin{aligned}
\mathcal{D} = -\nu{\left\lVert \sqrt{-\Delta_L}A^2 Q^2 \right\rVert}_2^2 + \mathcal{D}_E. \end{aligned}$$ Recall the following enumerations from [@BGM15I]. For $i,j\in {\left\{1,2,3\right\}}$ and $a,b \in {\left\{0,\neq\right\}}$:
\[def:Q2Enums\] $$\begin{aligned}
NLP(i,j,a,b) &= \int A^2 Q^2_{\neq} A^2\left( \partial_Y^t \left(\partial_j^t U^i_a \partial_i^t U^j_b \right) \right) dV \\
NLS1(j,a,b) & = -\int A^2 Q^2_{\neq} A^2\left(Q^j_a\partial_j^t U^2_{b}\right) dV \\
NLS2(i,j,a,b) & = -\int A^2 Q^2_{\neq} A^2\left(\partial_i^t U^j_a \partial_i^t\partial_j^t U^2_{b}\right) dV \\
NLP(i,j,0) & = \int A^2 Q^2_{0} A^2\left( \partial_Y^t \left(\partial_j^t U^i_0 \partial_i^t U^j_0 \right) \right) dV \\
NLS1(j,0) & = -\int A^2 Q^2_{0} A^2\left(Q^j_0\partial_j^t U^2_{0}\right) dV \\
NLS2(i,j,0) & = -\int A^2 Q^2_{0} A^2\left(\partial_i^t U^j_0 \partial_i^t\partial_j^t U^2_{0}\right) dV \\
\mathcal{F} & = -\int A^2 Q^2_{0} A^2\left(\partial_i^t \partial_i^t \partial_j^t \left(U^j_{\neq} U^2_{\neq}\right)_0 - \partial_{Y}^t \partial_j^t \partial_i^t \left(U^i_{\neq} U^j_{\neq}\right)_0\right)dV \\
\mathcal{T}_0 & = -\int A^{2} Q_0^2 A^{2} \left( g \partial_Y Q_{0}^2 \right) dV \\
\mathcal{T}_{\neq} & = -\int A^{2} Q_{\neq}^2 A^{2} \left( \tilde U \cdot {\nabla}Q^2 \right) dV \end{aligned}$$
Note that we have split $\mathcal{T}$ into three contributions: $\mathcal{T}_0$ (the **(2.5NS)** interactions), $\mathcal{T}_{\neq}$ (the **(SI)** and **(3DE)** interactions), and a contribution that is grouped with $\mathcal{F}$ (the **(F)** interactions). Similarly, we have split the $NLS$ and $NLP$ terms into several contributions: $NLS1(j,0)$, $NLS2(i,j,0)$, and $NLP(i,j,0)$ (the **(2.5NS)** interactions), the $NLS1(j,a,b)$, $NLS2(i,j,a,b)$, and $NLP(i,j,a,b)$ (the **(SI)** and **(3DE)** interactions), and a contribution that is grouped with $\mathcal{F}$ (the **(F)** interactions). This kind of subdivision will be used repeatedly in the sequel.
Zero frequencies {#sec:AQ2Zero}
----------------
### Transport nonlinearity {#sec:TransQ20}
Turn first to $\mathcal{T}_0$, the **(2.5NS)** contribution to the transport nonlinearity. From Lemma \[lem:AAiProd\], $$\begin{aligned}
\mathcal{T}_{0} & \lesssim {\left\lVert A^2 Q_0^2 \right\rVert}_2 \left( {\left\lVert Ag \right\rVert}_2 {\left\lVert Q^2_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} + {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} {\left\lVert {\nabla}A^2 Q^2_0 \right\rVert}_2 \right) \\
& \lesssim {\left\lVert A^2 Q_0^2 \right\rVert}_2 \left(\epsilon {\left\lVert Q^2_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} + \frac{\epsilon}{{\left\langle t \right\rangle}^{2}} {\left\lVert {\nabla}A^2 Q^2_0 \right\rVert}_2 \right) \\
& \lesssim \epsilon^{3/2}{\left\lVert {\nabla}A^2 Q^2 \right\rVert}_2^2 + \left(\frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{4}} + \epsilon\right){\left\lVert A^2 Q^2 \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by absorbing first term with the dissipation and integrating in time, provided $c_0$, and $\epsilon$ (equivalently $\nu$) are chosen sufficiently small.
### Nonlinear pressure and stretching {#sec:NLPSQ20}
These terms correspond to the nonlinear zero frequency interactions in the pressure and stretching terms, and so are of type **(2.5NS)**. Unlike in [@BGM15I], $A^2_0 \neq A^{3}_0$: near the critical times, we have less control over $Q^3_0$. Therefore, the most difficult contributions will come from terms which involve two derivatives of $Q^3$. Consider $NLP(3,3,0)$ as a representative example; the other contributions are all treated with a similar approach (or are easier) and hence are omitted for the sake of brevity. We expand with a paraproduct and group any terms where the coefficients appear in low frequency with the remainders: $$\begin{aligned}
NLP(3,3,0) & = 2\int A^2 Q^2_0 A^2 \partial_Y\left( (\partial_Z U^3_0)_{Hi} (\partial_Z U^3_0)_{Lo} \right) dV \\
& \quad + \int A^2 Q^2_0 A^2\left(\left((\psi_y)_{Hi} \partial_Y + (\phi_y)_{Hi}\partial_Z \right) \left( (\partial_Z U^3_0)_{Lo} (\partial_Z U^3_0)_{Lo} \right) \right) dV \\
& \quad + \int A^2 Q^2_0 A^2 \partial_Y\left( \left((\phi_z)_{Hi}\partial_Z + (\psi_z)_{Hi} \partial_Y\right) (U^3_0)_{Lo} (\partial_Z U^3_0)_{Lo} \right) dV \\
& \quad + P_{\mathcal{R},C} \\
& = P_{HL} + P_{C1} + P_{C2} + P_{\mathcal{R},C}. \end{aligned}$$ Turn to $P_{HL}$ first. By and we have $$\begin{aligned}
P_{HL} & \lesssim \epsilon\sum_{l,l^\prime} \int {\left\vert\widehat{Q_0^2}(\eta,l)\right\vert} \left(\sum_{r} \tilde{A}^2_0(\eta,l) \tilde{A}_0^3(\xi,l^\prime) \chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} A^2_0(\eta,l) A_0^3(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert\Delta_L \widehat{U^3_0}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi, \end{aligned}$$ which by , gives (along with $\epsilon t \leq c_0$), $$\begin{aligned}
P_{HL} & \lesssim \epsilon t {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}} \tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q^2_0 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}} A^3\right) \Delta_L U^3_0 \right\rVert}_2 \nonumber \\
& \quad + \epsilon {\left\lVert A^2 Q^2_0 \right\rVert}_2 {\left\lVert \Delta_L A^3 U_0^3 \right\rVert}_2 \nonumber \\
& \lesssim c_0 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q^2_0 \right\rVert}_2^2 + c_0{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) \Delta_L U^3_0 \right\rVert}_2^2 \nonumber \\
& \quad + \epsilon {\left\lVert A^2 Q^2_0 \right\rVert}_2^2 + \epsilon {\left\lVert \Delta_L A^3 U_0^3 \right\rVert}_2^2. \label{ineq:PHLQ2} \end{aligned}$$ By Lemmas \[lem:PELbasicZero\] and \[lem:PELCKZero\], this is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small and $t \leq c_0 \epsilon^{-1}$ by absorbing the leading terms with the dissipation energies and integrating in time.
Of the coefficient error terms, $P_{C2}$ is the most difficult; we treat only this case and omit the others. By , , and , followed by Lemma \[lem:CoefCtrl\], $$\begin{aligned}
P_{C2} & \lesssim \epsilon^2 \sum_{l,l^\prime} \int {\left\vertA^2_0 \widehat{Q_0^2}(\eta,l) \frac{A(\xi,l^\prime) {\left\vert\eta\right\vert}}{{\left\langle \xi,l^\prime \right\rangle}^2} \left( {\left\vert\widehat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert} \right)\right\vert}Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon^2 {\left\lVert A^2 Q^2_0 \right\rVert}_2 \left( {\left\lVert {\left\langle {\nabla}\right\rangle}^{-1} A \psi_y \right\rVert}_2 + {\left\lVert {\left\langle {\nabla}\right\rangle}^{-1} A \phi_y \right\rVert}_2\right) \\
& \lesssim \epsilon^2 {\left\lVert A^2 Q^2_0 \right\rVert}_2 {\left\lVert AC \right\rVert}_2 \\
& \lesssim \epsilon {\left\lVert A^2 Q^2_0 \right\rVert}_2^2 + \epsilon^3 {\left\lVert AC \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small after integrating in time.
The remainder terms are similar, or easier than, the terms treated above and hence these are omitted for brevity. This completes $NLP(3,3,0)$; the other $NLP$ terms are similar or easier and are hence omitted as well.
### Forcing from non-zero frequencies {#sec:NzeroForcing}
Turn next to nonlinear interactions of type **(F)**: the interaction of two $X$ frequencies $k$ and $-k$ and sub-divide via $$\begin{aligned}
\mathcal{F} & = -\int A^2 Q^2_0 A^2\left(\partial_Z^t \partial_Z^t \partial_j^t \left(U^j_{\neq} U^2_{\neq}\right)_0 - \partial_Y^t \partial_Y^t \partial_Z^t \left(U^3_{\neq} U^2_{\neq}\right)_0 - \partial_Y^t \partial_Z^t \partial_Z^t \left(U^3_{\neq} U^3_{\neq}\right)_0\right) dV \\
& = F^1 + F^2 + F^3. \end{aligned}$$ As in [@BGM15I], all three are treated via variants of the same basic approach which will ultimately come down to applying the appropriate multiplier estimate in or depending on the combination of derivatives present. However, the situation here is more complicated than in [@BGM15I] due to the additional regularity loss in non-resonant modes of $Q^3$ near the critical times.
We expand $F^3$ with a paraproduct and group terms where the coefficients appear in low frequency with the remainder: $$\begin{aligned}
F^3 & = 2\sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \partial_Y\partial_Z\partial_Z\left( \left(U^3_{-k}\right)_{Hi} \left( U^3_k\right)_{Lo}\right)dV \\
& \quad +\sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \left( \left( (\psi_y)_{Hi} \partial_Y + (\phi_y)_{Hi}\partial_Z \right) \partial_Y\partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^3_k\right)_{Lo}\right)\right) dV \\
& \quad + \sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \left( \partial_Y \left( (\psi_z)_{Hi}\partial_Y + (\phi_z)_{Hi}\partial_Z \right) \partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^3_k\right)_{Lo}\right) \right) dV \\
& \quad + \sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \left( \partial_Y \partial_Z \left((\psi_z)_{Hi} \partial_Y + (\phi_z)_{Hi}\partial_Z \right) \left( \left(U^3_{-k}\right)_{Lo} \left( U^3_k\right)_{Lo}\right) \right) dV \\
& \quad + F^2_{\mathcal{R},C} \\
& = F^3_{HL} + F^3_{C1} + F^3_{C2} + F^3_{C3} + F^3_{\mathcal{R},C}, \end{aligned}$$ where here $F^3_{\mathcal{R},C}$ includes all of the remainders from the quintic paraproduct as well as the higher order terms involving coefficients as low frequency factors.
Turn first to $F^3_{HL}$ (recall and the shorthand discussed in above) which by is given by $$\begin{aligned}
F_{HL}^3 & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k\neq 0} \sum_{l,l^\prime} \int {\left\vertA^{2} \widehat{Q^2_0}(\eta,l) A_0^2(\eta,l) \frac{{\left\vertl\right\vert}^2{\left\vert\eta\right\vert} }{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k\neq 0} \sum_{l,l^\prime} \int {\left\vert\widehat{Q^2_0}(\eta,l)\right\vert} \frac{{\left\vertl\right\vert}^2{\left\vert\eta\right\vert} }{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{2} \\ & \quad\quad \times \left(\sum_{r} \chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}_0^2(\eta,l)\tilde{A}^3_k(\xi,l^\prime) + \chi^{\ast;23} A_0^2(\eta,l) A^3_k(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert \Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi.\end{aligned}$$ By and (for the $\chi^{\ast;23}$ contribution), followed by , there holds $$\begin{aligned}
F_{HL}^{3} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right)
Q_0^2 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) \Delta_L U^3_{\neq} \right\rVert}_2 \\ & \quad + \frac{\epsilon }{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \sqrt{-\Delta_L}A^2 Q_0^2 \right\rVert}_2{\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2 \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q_0^2 \right\rVert}_2^2 + \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) \Delta_L U^3_{\neq} \right\rVert}^2_2 \\ & \quad +\frac{\epsilon^{3/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \sqrt{-\Delta_L}A^2 Q_0^2 \right\rVert}^2_2 + \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2^2, \end{aligned}$$ which after Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\] is consistent with Proposition \[prop:Boot\] for $c_0$ and $\epsilon$ sufficiently small.
Turn next to the coefficient error terms. Due to the high number of derivatives, the most difficult one is $F_{C3}^3$, hence, we focus only on this one and omit the others for brevity. We have by , Lemma \[lem:ABasic\], and , $$\begin{aligned}
F_{C3}^3 & \lesssim \frac{\epsilon^2}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} \sum_{k\neq 0} \sum_{l,l^\prime} \int {\left\vertA^{2} \widehat{Q^2_0}(\eta,l) \frac{{\left\vert\eta,l\right\vert}^2}{{\left\langle \xi,l^\prime \right\rangle}^2} A\left( {\left\vert\widehat{\psi_z}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_z}(\xi,l^\prime)_{Hi}\right\vert} \right) \right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon^2}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert \sqrt{-\Delta_L} A^2 Q^2_0 \right\rVert}_2 \left({\left\lVert {\left\langle {\nabla}\right\rangle}^{-1}A \phi_z \right\rVert}_2 + {\left\lVert {\left\langle {\nabla}\right\rangle}^{-1}A\psi_z \right\rVert}_2\right) \\
& \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^2 Q^2_0 \right\rVert}_2^2 + \frac{\epsilon^{5/2}}{{\left\langle \nu t^3 \right\rangle}^{4\alpha}}{\left\lVert AC \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. The remaining coefficient error terms are similar or easier and are hence omitted. The remainder terms are easy variants of the above treatments. The one which may require comment is the error term of the form $$\begin{aligned}
2\sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2}\left((\phi_y)_{Lo}\partial_Z\partial_Z\partial_Z\left( \left(U^3_{-k}\right)_{Hi} \left( U^3_k\right)_{Lo}\right)\right)dV, \end{aligned}$$ as the structure of the nonlinearity has changed and it is less clear how to absorb the losses due to the unbalance of regularities. However, since ${\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} \lesssim \epsilon t$, the presence of the coefficients gains a power of $t$ and absorbs the loss via $\epsilon t {\left\langle \nu t^3 \right\rangle}^{-1} \lesssim t^{-1}$. From there the proof applies ; for more details, see the treatment of $F^{1;3}$ below where a similar argument is carried out. This completes the treatment of $F^3$.
Consider next the contribution from $F^{1}$ and $j = 3$ (denoted $F^{1;3}$) which requires further explanation. As above, we expand with a paraproduct, $$\begin{aligned}
F^{1;3} & = -\sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \partial_Z\partial_Z\partial_Z\left( \left(U^3_{-k}\right)_{Hi} \left( U^2_k\right)_{Lo}\right)dV \\
& \quad - \sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \partial_Z\partial_Z\partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Hi}\right) dV \\
& \quad -\sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \left( \left( (\psi_z)_{Hi} \partial_Y + (\phi_z)_{Hi}\partial_Z \right) \partial_Z\partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Lo}\right)\right) dV \\
& \quad - \sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \left( \partial_Z \left( (\psi_z)_{Hi}\partial_Y + (\phi_z)_{Hi}\partial_Z \right) \partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Lo}\right) \right) dV \\
& \quad - \sum_{k\neq 0} \int A^{2}Q^2_0 A_0^{2} \left( \partial_Z \partial_Z \left((\psi_z)_{Hi} \partial_Y + (\phi_z)_{Hi}\partial_Z \right) \left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Lo}\right) \right) dV \\
& \quad - F^{1;3}_{\mathcal{R},C} \\
& = F^{1;3}_{HL} + F^{1;3}_{LH} + F^{1;3}_{C1} + F^{1;3}_{C2} + F^{1;3}_{C3} + F^{1;3}_{\mathcal{R},C}, \end{aligned}$$ The coefficient error terms $F^{1;3}_{Ci}$ and remainder terms $F^{1;3}_{\mathcal{R},C}$ are all easier than the $F^3$ case treated above and are hence omitted for brevity. Of the two leading order terms, $F_{LH}^{1;3}$ is easier as there is no additional regularity loss near critical times (despite the larger low frequency factor); indeed it is treated by a straightforward variant of the treatment of $F_{HL}^{1;3}$. Hence, turn to the latter, which by and Lemma \[lem:ABasic\] is given by $$\begin{aligned}
F_{HL}^{1;3} & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l,l^\prime} \int {\left\vertA^{2} \widehat{Q^2_0}(\eta,l) A_0^2(\eta,l) \frac{{\left\vertl\right\vert}^3}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l,l^\prime} \int {\left\vert\widehat{Q^2_0}(\eta,l)\right\vert}\frac{{\left\vertl\right\vert}^3}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{2} \\ & \quad\quad \times \left(\sum_{r} \chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}^2_0(\eta,l) \tilde{A}^3_k(\xi,l^\prime) + \chi^{\ast;23} A^2_0(\eta,l) A^3_k(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert\Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l,l^\prime} \int {\left\vertA^{2} \widehat{Q^2_0}(\eta,l)\right\vert}\frac{{\left\vertl\right\vert}^3}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{2} \\ & \quad\quad \times {\left\vertA^3_{k} \Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi. \end{aligned}$$ By (with $p = 2$) and we have, $$\begin{aligned}
F_{HL}^{1;3} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \sqrt{-\Delta_L} A^2 Q_0^2 \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2 \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^2 Q_0^2 \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}^2_2, \end{aligned}$$ which by Lemma \[lem:SimplePEL\] is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. This completes $F^{1;3}$. The remaining forcing terms are relatively easy variants of those already treated and are hence omitted for brevity.
### Dissipation error terms {#sec:DEQ02}
Recalling the definitions of the dissipation error terms and the short-hand , we have $$\begin{aligned}
\mathcal{D}_E & = \nu\int A_0^2 Q^2_0 A_0^2\left(G_{yy} \partial_{YY}Q^2_0 + G_{zz}\partial_{ZZ}Q^2_0 + G_{yz}\partial_{YZ}Q^2_0 \right) dV. \label{def:DEQ2}\end{aligned}$$ All three error terms are essentially the same and are treated in the same manner as the analogous terms in [@BGM15I]. Hence, we omit the treatments and simply state the results $$\begin{aligned}
\mathcal{D}_{E} & \lesssim c_0^{-1} \nu \epsilon^2 {\left\lVert {\nabla}A C \right\rVert}_2^2 + c_0 \nu {\left\lVert \sqrt{-\Delta_L }A^2 Q^2 \right\rVert}_{2}^2. \label{ineq:DEbd}\end{aligned}$$ Note that as in [@BGM15I], by $$\begin{aligned}
\int_{1}^{T^\star} c_0^{-1} \nu \epsilon^2 {\left\lVert {\nabla}A C(t) \right\rVert}_2^2 dt & \lesssim c_{0} \epsilon^2 K_B. \end{aligned}$$ Hence, for $c_{0}$ sufficiently small, is consistent with Proposition \[prop:Boot\].
Non-zero frequencies
--------------------
Next we consider the contributions to which come from the evolution of non-zero $X$ frequencies.
### Nonlinear pressure $NLP$ {#sec:NLPQ2}
#### Treatment of $NLP(1,j,0,\neq)$ {#sec:NLP213}
Here $j \in {\left\{2,3\right\}}$ due to the structure of the nonlinearity. The case $j = 3$ was singled out in [@BGM15I] as one of the leading order nonlinear interactions of type $\textbf{(SI)}$ (see also §\[sec:Toy\]). We will concentrate on this case and omit the treatment of $j=2$, which is treated with the same method and moreover is simpler due to the lack of a regularity imbalance in $A^2$ near the critical times.
This term is quartic (in the sense that the nonlinearity is order $4$) and we will use the paraproduct decomposition described in §\[sec:paranote\]. We will group terms where the coefficients appear in ‘low frequency’ with the remainder (see Remarks \[rmk:CoefCtrlLow\] and \[rmk:SIcoefneglect\]). Therefore, the expansion is $$\begin{aligned}
NLP(1,3,0,\neq) & = \sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left( (\partial_Y - t\partial_X) \left( \left(\partial_Z U^1_0\right)_{Lo} (\partial_XU^3_{k})_{Hi} \right) \right) dV \\
& \quad + \sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left( (\partial_Y - t\partial_X) \left( (\partial_Z U^1_0)_{Hi} (\partial_XU^3_{k})_{Lo} \right) \right) dV \\
& \quad + \sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left( \left((\psi_y)_{Hi}(\partial_Y - t\partial_X) + (\phi_y)_{Hi}\partial_Z\right) \left(\partial_XU^3_{k} \partial_Z U^1_0\right)_{Lo} \right) dV \\
& \quad + \sum_{k \neq 0} \int A^2 Q^2_{k} A^2 \left( (\partial_Y-t\partial_X) \left( (\partial_XU^3_{k})_{Lo} \left( \left((\psi_z)_{Hi}\partial_Y + (\phi_z)_{Hi}\partial_Z \right) (U^1_0)_{Lo}\right) \right) \right) dV \\
& \quad + P_{\mathcal{R},C} \\
& = P_{LH} + P_{HL} + P_{C1} + P_{C2} + P_{\mathcal{R},C}, \end{aligned}$$ where $P_{\mathcal{R},C}$ includes all of the remainders from the quartic paraproduct as well as the higher order terms involving coefficients as low frequency factors.
Turn first to $P_{LH}$, which by and is bounded by (recall the shorthand ), $$\begin{aligned}
P_{LH} & \lesssim \epsilon t \sum_{k \neq 0} \int {\left\vertA^2 \widehat{Q^2_{k}}(\eta,l) A^2_k(\eta,l) \frac{(\eta - tk)k}{k^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk\right\vert}^2} \Delta_L\widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon t \sum_{k \neq 0} \int {\left\vert\widehat{Q^2_{k}}(\eta,l)\right\vert} \frac{(\eta - tk)k}{k^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \\
& \quad\quad \times \left(\sum_r \chi^{r,NR} \frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} \tilde{A}_k^2(\eta,l)\tilde{A}_k^3(\xi,l^\prime) + \chi^{\ast;23} A_k^2(\eta,l) A_k^3(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert\Delta_L\widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi. \end{aligned}$$ Note that by , the following holds on the support of the integrand: $$\begin{aligned}
\chi_{NR;k} & \lesssim \mathbf{1}_{t \leq \epsilon^{-1/2+\delta/100}} + \mathbf{1}_{t \geq \epsilon^{-1/2+\delta/100}}\epsilon^{1/2} {\left\langle \xi-kt,l^\prime \right\rangle} {\left\langle \eta-\xi,l-l^\prime \right\rangle}. \label{ineq:teps12trick}\end{aligned}$$ Hence, by (with $p = 1$), followed by , $$\begin{aligned}
P_{LH} & \lesssim \epsilon t{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^2\right) Q^2 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^3\right)\Delta_L U^3_{\neq} \right\rVert}_2 \nonumber \\
& \quad + \epsilon^{3/2} {\left\lVert \sqrt{-\Delta_L} A^2 Q^2 \right\rVert}_2^2 + \mathbf{1}_{t \leq \epsilon^{-1/2+\delta/100}}\epsilon^{1/2}{\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2^2 + \epsilon^{3/2-\delta/50}{\left\lVert \sqrt{-\Delta_L }A^3 \Delta_L U^3_{\neq} \right\rVert}_2^2, \label{ineq:PLHNLP13}\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemmas \[lem:PEL\_NLP120neq\], \[lem:SimplePEL\], and \[lem:PELED\] for $\epsilon$ and $\epsilon t \leq c_0$ sufficiently small.
Turn next to the contribution of $P_{HL}$, which can be treated in the same manner as in [@BGM15I]. Indeed, by followed by Lemma \[lem:ABasic\] and , we have, $$\begin{aligned}
P_{HL} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k \neq 0} \sum_{l, l^\prime} \int {\left\vertA^2 \widehat{Q^2_{k}}(\eta,l) A^2_k(\eta,l) {\left\vert\eta - kt\right\vert} {\left\vertl^\prime\right\vert} \widehat{U^1_0}(\xi,l^\prime)_{Hi} Low(k,\eta-\xi,l-l^\prime)\right\vert} d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l, l^\prime} \int {\left\vert A^2 \widehat{Q^2_{k}} \frac{{\left\vert\eta - kt\right\vert}}{{\left\langle \xi,l^\prime \right\rangle}^2 {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}} {\left\vertl^\prime\right\vert} A\widehat{U^1_0}(\xi,l^\prime)_{Hi} Low(k,\eta-\xi,l-l^\prime)\right\vert} d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \sqrt{-\Delta_L} A^2 Q^2 \right\rVert}_2 {\left\lVert AU^1_0 \right\rVert}_2 \\
& \lesssim \epsilon^{3/2} {\left\lVert \sqrt{-\Delta_L} A^2 Q^2 \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} \left(\frac{1}{{\left\langle t \right\rangle}^2}{\left\lVert AU^1_0 \right\rVert}_2^2\right). \end{aligned}$$ This is consistent with Proposition \[prop:Boot\] after applying Lemma \[lem:PELbasicZero\].
Turn first to $P_{C1}$, which is also treated in the same manner as in [@BGM15I]. By , , and Lemma \[lem:ABasic\] we have $$\begin{aligned}
P_{C1} & \lesssim \frac{\epsilon^2 t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l, l^\prime} \int {\left\vertA^2 \widehat{Q^2_{k}}(\eta,l) A^2_k(\eta,l) \left({\left\vert\widehat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert} \right)\right\vert} Low(k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon^2 t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l, l^\prime} \int {\left\vertA^2 \widehat{Q^2_{k}}(\eta,l) \frac{1}{{\left\langle \xi,l^\prime \right\rangle}^2 {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}} A\left( {\left\vert\widehat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert}\right)\right\vert} Low(k,\eta-\xi,l-l^\prime) d\eta d\xi\\
& \lesssim \frac{\epsilon^2 t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2 \left({\left\lVert {\left\langle {\nabla}\right\rangle}^{-1 }A \psi_y \right\rVert}_2 + {\left\lVert {\left\langle {\nabla}\right\rangle}^{-1}A \phi_y \right\rVert}_2\right) \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^2 Q^2 \right\rVert}_2^2 + \frac{\epsilon^{3} t^4}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \left(\frac{1}{{\left\langle t \right\rangle}^2} {\left\lVert A C \right\rVert}^2_2\right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. This completes the treatment of $P_{C1}$. The second coefficient term, $P_{C2}$, is very similar: there is one extra derivative landing on the coefficient but there is one less power of time from the low frequency factor. By Lemma \[lem:ABasic\], we will be able to balance the loss by the gain and apply essentially the same treatment as we did for $P_{C1}$. Hence, this is omitted for the sake of brevity.
Similarly, the remainder and coefficient terms $P_{\mathcal{R},C}$ are omitted as they are easier or very similar. This completes the treatment of $NLP(1,3,0,\neq)$.
#### Treatment of $NLP(i,j,0,\neq)$ with $i \in {\left\{2,3\right\}}$ {#sec:NLPQ2_0neq_notX}
We will demonstrate how to deal with these terms by the example of $NLP(2,3,0,\neq)$ (recall ), which is one of the leading order terms. Expanding with a quintic paraproduct and grouping the low frequency coefficient terms with the remainder: $$\begin{aligned}
NLP(2,3,0,\neq) & = \sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left((\partial_Y-t\partial_X)((\partial_Y - t\partial_X)(U^3_{k})_{Hi} (\partial_Z^t U^2_0)_{Lo}) \right) dV \\
& \quad +\sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left((\partial_Y-t\partial_X)((\partial_Y-t\partial_X)(U^3_{k})_{Lo} (\partial_Z^t U^2_0)_{Hi}) \right) dV \\
& \quad +\sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left( \left((\psi_y)_{Hi}(\partial_Y - t\partial_X) + (\phi_y)_{Hi}\partial_Z \right)(\partial_Y^t (U^3_{k})_{Lo} (\partial_Z^t U^2_0)_{Lo}) \right) dV \\
& \quad +\sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left( (\partial_Y - t\partial_X)( \left((\psi_y)_{Hi}(\partial_Y-t\partial_X) + (\phi_y)_{Hi}\partial_Z \right) (U^3_{k})_{Lo} (\partial_Z^t U^2_0)_{Lo}) \right) dV \\
& \quad +\sum_{k \neq 0} \int A^2 Q^2_{k} A^2\left( (\partial_Y - t\partial_X)( (\partial_Y^t U^3_{k})_{Lo} ( \left((\phi_z)_{Hi}\partial_Z + (\psi_z)_{Hi}\partial_Y \right) U^2_0)_{Lo}) \right) dV \\
& \quad + P_{\mathcal{R},C} \\
& = P_{HL} + P_{LH} + P_{C1} + P_{C2} + P_{\mathcal{R},C}, \end{aligned}$$ where the term $P_{\mathcal{R},C}$ contains the remainders from the quintic paraproducts and the higher order terms where the coefficients are in low frequency.
Consider first $P_{HL}$, which by , , and , $$\begin{aligned}
P_{HL} & \lesssim \epsilon\sum_{k \neq 0} \int {\left\vertA^2 \widehat{Q^2_{k}}(\eta,l) A^2_k(\eta,l) {\left\vert\eta - tk\right\vert} {\left\vert\xi-tk\right\vert} \widehat{U^3_{k}}(\xi,l^\prime)_{Hi} \right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon\sum_{k \neq 0} \int {\left\vert\widehat{Q^2_{k}}(\eta,l)\right\vert} \\ & \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta - tr\right\vert}}\tilde{A}^2_k(\eta,l)\tilde{A}^3_k(\xi,l^\prime) + \chi^{\ast;23} A^2_k(\eta,l) A^3_k(\xi,l^\prime) \right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \\ & \quad\quad \times {\left\langle \xi-tk \right\rangle}^2 {\left\vert\widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi. \end{aligned}$$ By , , and we have, $$\begin{aligned}
P_{HL} & \lesssim \epsilon {\left\langle t \right\rangle} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q^2_{\neq} \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) \Delta_L U^3_{\neq} \right\rVert}_2 \\
& \quad + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^2 Q^2 \right\rVert}_2^2 + \mathbf{1}_{t \leq \epsilon^{-1/2+\delta/100}}\epsilon^{1/2}{\left\lVert \Delta_L A^3 U^3_{\neq} \right\rVert}^2_2 + \epsilon^{3/2-\delta/50}{\left\lVert \sqrt{-\Delta_L} \Delta_L A^3 U^3_{\neq} \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemmas \[lem:PEL\_NLP120neq\], \[lem:SimplePEL\], and \[lem:PELED\].
Turn next to $P_{LH}$. As in [@BGM15I], this term is treated as in the analogous term in $NLP(1,3,0,\neq)$, using that extra loss of time from the second $\partial_Y^t$ derivative replaces the gain in $t$ from the presence of $U_0^2$ as opposed to $U_0^1$. We omit the analogous details and simply conclude that $$\begin{aligned}
P_{LH} & \lesssim \epsilon^{3/2} {\left\lVert \sqrt{-\Delta_L} A^2 Q^2 \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert AU_0^2 \right\rVert}_2^2, \end{aligned}$$ which after Lemma \[lem:PELbasicZero\] is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
The coefficient error terms, $P_{Ci}$, are also similar to [@BGM15I] and the corresponding terms in the treatment of $NLP(1,3,0,\neq)$ above in §\[sec:NLP213\]. We omit the details for brevity. Similarly, the remainder terms and low frequency coefficient terms are relatively easy to deal with or are easy variants of the above treatments and are hence omitted. This completes the treatment of $NLP(2,3,0,\neq)$, which is the leading order term in $NLP(i,j,0,\neq)$ with $i \in {\left\{2,3\right\}}$.
#### Treatment of $NLP(i,j,\neq,\neq)$ terms {#sec:NLPQ2_neqneq}
These are pressure interactions of type **(3DE)**. All of these terms can be treated in a similar fashion, however the terms involving $U^3$ are slightly harder due to the regularity imbalances. We will focus on the case $i=1$ and $j=3$ and omit the others, which follow analogously. As usual, this term is quartic, but when we expand with the paraproduct we will keep the coefficients only when they appear in high frequency and group the other terms with the remainder. Hence, $$\begin{aligned}
NLP(1,3,\neq,\neq) & = \int A^2 Q^2_{\neq} A^2\left( (\partial_Y-t\partial_X)( (\partial_Z U^1_{\neq})_{Lo} (\partial_X U^3_{\neq})_{Hi} )\right) dV \\
& \quad + \int A^2 Q^2_{\neq} A^2\left( (\partial_Y-t\partial_X)( (\partial_Z U^1_{\neq})_{Hi} (\partial_X U^3_{\neq})_{Lo} )\right) dV \\
& \quad + \int A^2 Q^2_{\neq} A^2\left( \left((\psi_y)_{Hi}(\partial_Y-t\partial_X) + (\phi_y)_{Hi}\partial_Z\right)( (\partial_Z U^1_{\neq})_{Lo} (\partial_X U^3_{\neq})_{Lo} )\right) dV \\
& \quad +\sum_{k} \int A^2 Q^2_{\neq} A^2\left( (\partial_Y-t\partial_X)( \left((\psi_z)_{Hi}(\partial_Y-t\partial_X) + (\phi_z)_{Hi}\partial_Z \right)(U^1_{\neq})_{Lo} (\partial_X U^3_{\neq})_{Lo} )\right) dV \\
& = P_{LH} + P_{HL} + P_{C1} + P_{C2} + P_{\mathcal{R},C},\end{aligned}$$ where $P_{\mathcal{R},C}$ contains the paraproduct remainders and the terms where coefficients appear in low frequency. By , , and , $$\begin{aligned}
P_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l)A^2_k(\eta,l) (\eta-k t)k^\prime \widehat{U^3_{k^\prime}}(\xi,l^\prime)\right\vert} Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l)\right\vert} \frac{(\eta-k t)k^\prime}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-k^\prime t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \\ & \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} \right) {\left\vertA^3\Delta_L\widehat{U^3_{k^\prime}}(\xi,l^\prime)\right\vert} Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L}A^2 Q^2_{\neq} \right\rVert}_2 {\left\lVert \Delta_L A^3 U^3_{\neq} \right\rVert}_2 \\
& \lesssim \epsilon^{3/2} {\left\lVert \sqrt{-\Delta_L}A^2 Q^2_{\neq} \right\rVert}_2^2 + \frac{\epsilon^{1/2} {\left\langle t \right\rangle}^{2\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert \Delta_L A^3 U^3_{\neq} \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemma \[lem:SimplePEL\] for $\epsilon$ sufficiently small.
By , , and , followed by , we have $$\begin{aligned}
P_{HL} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l)A^2_k(\eta,l) (\eta-k t)l^\prime \widehat{U^1_{k^\prime}}(\xi,l^\prime)\right\vert} Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{\delta_1}\frac{{\left\langle t \right\rangle} (\eta-k t) l^\prime}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-k^\prime t\right\vert}^2} \Delta_L A^1\widehat{U^1_{k^\prime}}(\xi,l^\prime)\right\vert} \\ & \quad\quad \times Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q^2 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1 \right) \Delta_L U^1_{\neq} \right\rVert}_2 \\ & \quad + \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^2 Q^2_{\neq} \right\rVert}_2 {\left\lVert \Delta_L A^1 U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which after Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\], is consistent with Proposition \[prop:Boot\].
As in [@BGM15I], the coefficient error terms are straightforward here and are hence omitted for the sake of brevity. As discussed above, the remainder terms $P_{\mathcal{R}.C}$ are much easier than the leading order terms, and these are hence omitted. This completes the treatment of $NLP(1,3,\neq,\neq)$. Other $i,j$ combinations can be treated via a simple variant of this (one will also use for this).
### Nonlinear stretching $NLS$ {#sec:NLSQ2}
#### Treatment of $NLS1(j,0,\neq)$ and $NLS1(j,\neq,0)$ {#sec:NLS1Q20neq}
Recall the definition of $NLS1(j,0,\neq)$ from . These terms can essentially be treated in the same manner as the $NLP(j,2,0,\neq)$ nonlinear pressure terms in §\[sec:NLP213\] and §\[sec:NLPQ2\_0neq\_notX\] and hence we omit them for brevity.
Consider the $NLS1(j,\neq,0)$ terms. Notice that the $j = 1$ term disappears due to the usual null structure. The $j=3$ term is then the most dangerous remaining term as we must contend with the loss of regularity near critical times as well as a large low-frequency growth. Expanding this term with a paraproduct and focusing on the highest order terms gives: $$\begin{aligned}
NLS1(3,\neq,0) & = -\int A^2 Q^2 A^2\left( (Q^3_{\neq})_{Hi} (\partial_Z U^2_{0})_{Lo}\right) dV - \int A^2 Q^2 A^2\left( (Q^3_{\neq})_{Lo} (\partial_Z U^2_{0})_{Hi}\right) dV \\
& \quad - \int A^2 Q^2 A^2\left( (Q^3_{\neq})_{Lo} \left((\psi_z)_{Hi} \partial_Y + (\phi_z)_{Hi}\partial_Z\right) (U^2_{0})_{Lo}\right) dV + S_{\mathcal{R},C} \\
& = S_{HL} + S_{LH} + S_{C} + S_{\mathcal{R},C}, \end{aligned}$$ where $S_{\mathcal{R},C}$ contains the paraproduct remainders and the terms where the coefficients appear in low frequency. By , , , , and we have $$\begin{aligned}
S_{HL} & \lesssim \epsilon \sum_{k}\int {\left\vert\widehat{Q^2_k}(\eta,l)\right\vert} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}
\\ & \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}^2_k(\eta,l) \tilde{A}^3_k(\xi,l^\prime)
+ \chi^{\ast;23} A^2_k(\eta,l) A^3_k(\xi,l^\prime) \right)\\
& \quad\quad \times {\left\vert\widehat{Q^3_{k}}(\xi,l^\prime)\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon t {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q^2 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3_{\neq} \right\rVert}_2 \\
& \quad + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^2 Q^2 \right\rVert}_2^2 + \mathbf{1}_{t \leq \epsilon^{-1/2+\delta/100}}\epsilon^{1/2}{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2^2 + \epsilon^{3/2-\delta/50}{\left\lVert \sqrt{-\Delta_L} A^3 Q^3_{\neq} \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $c_0$ sufficiently small.
The treatment of $S_{LH}$ is the same as [@BGM15I]: by , Lemma \[lem:ABasic\], and , $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) A^2_k(\eta,l) l^\prime \widehat{U^2_{0}}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^2 Q^2 \right\rVert}_2{\left\lVert AU_0^2 \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. The coefficient error term, $S_C$, is treated as in [@BGM15I]: by , , and Lemma \[lem:ABasic\], and Lemma \[lem:CoefCtrl\], $$\begin{aligned}
S_C & \lesssim \frac{\epsilon^2 {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) \frac{1}{{\left\langle t \right\rangle} {\left\langle \xi,l^\prime \right\rangle} } A\widehat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^2 Q^2 \right\rVert}_2^2 + \frac{\epsilon^3 {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert AC \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
As usual, the remainders and coefficient error terms in $S_{\mathcal{R},C}$ are significantly easier to treat and hence are omitted for brevity. This completes the treatment of $NLS1(3,\neq,0)$; the other term, $NLS1(2,\neq,0)$ is easier and is treated the same way, hence we omit this for brevity.
#### Treatment of $NLS1(j,\neq,\neq)$
The most problematic terms are $j = 3$ and $j = 1$. The other terms will be treated in a similar fashion, so we focus on the $j = 3$ for brevity. We expand the term with a paraproduct and only keep the coefficients to leading order when they appear in high frequency: $$\begin{aligned}
NLS1(3,\neq,\neq) & = -\int A^2 Q^2 A^2\left( (Q^3_{\neq})_{Hi} (\partial_Z U^2_{\neq})_{Lo}\right) dV -\int A^2 Q^2 A^2\left( (Q^3_{\neq})_{Lo} (\partial_Z U^2_{\neq})_{Hi}\right) dV \\
& \quad - \int A^2 Q^2 A^2\left( (Q^3_{\neq})_{Lo} \left((\psi_z)_{Hi} (\partial_Y - t\partial_X) + (\phi_z)_{Hi}\partial_Z \right)U^2_{\neq})_{Lo}\right) dV + S_{\mathcal{R},C} \\
& = S_{HL} + S_{LH} + S_{C} + S_{\mathcal{R},C}, \end{aligned}$$ where $S_{\mathcal{R},C}$ contains the paraproduct remainders and the terms where the coefficients appear in low frequency. By , , and we have $$\begin{aligned}
S_{HL} & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}\left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} \right) A^3 \widehat{Q^3_{k^\prime}}(\xi,l^\prime)\right\vert} \\ & \quad\quad \times Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^2 Q^2 \right\rVert}_2{\left\lVert A^3 Q^3 \right\rVert}_2\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
Turn next to the $S_{LH}$ term. By , , , and we have $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k}\int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) \frac{{\left\vertl^\prime\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2} \Delta_LA^2 \widehat{U^2_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2 \right) Q^2 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) \Delta_L U^2_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^2 Q^2 \right\rVert}_2 {\left\lVert A^2 \Delta_L U^2_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ small by Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\]. For the coefficient error term is treated in the same fashion as the corresponding error term associated with $NLS1(3,\neq,0)$ in §\[sec:NLS1Q20neq\] above. Hence, the treatment is omitted. Similarly, the remainder and coefficient low frequency terms in $S_{\mathcal{R},C}$ are also omitted. This completes the treatment of the $NLS1(3,\neq,\neq)$ term; the other $NLS1(j,\neq,\neq)$ terms are treated similarly.
#### Treatment of $NLS2(i,1,0,\neq)$ {#sec:NLS2i1neq0}
Recall the definition of these terms from . The non-zero contributions come from $i = 2$ and $i = 3$ and these can be treated as in [@BGM15I] (note $U^3$ does not appear in either). We hence omit the treatment for the sake of brevity (it roughly parallels $NLP(1,2,0,\neq)$ in §\[sec:NLP213\], which was omitted since this was slightly easier than the leading order $NLP(1,3,0,\neq)$).
#### Treatment of $NLS2(i,j,0,\neq)$ with $j \neq 1$ {#sec:NLS2ijneq0}
Recall and note that $i \neq 1$. Unlike in [@BGM15I], not all the cases are quite the same. However, the losses due to the regularity imbalances in $Q^3$ can be easily absorbed by the low frequency growth of $Q^2$. Otherwise, the treatment is similar to that used in [@BGM15I]. Hence the details are omitted for brevity.
#### Treatment of $NLS2(i,j,\neq,0)$
Recall and note that $j \neq 1$. These terms can all be treated in a manner similar to the treatment of $NLS2(i,j,0,\neq)$ above and are hence omitted for the sake of brevity.
#### Treatment of $NLS2(i,j,\neq,\neq)$
First note that the contribution $i = j = 2$ cancels with the $NLP$ terms. These terms are treated similar to $NLP(i,j,\neq,\neq)$, however they are generally easier as the regularity imbalances in $Q^3$ and the large growth in $Q^1$ arises on the factor with fewer derivatives. Moreover, if $U^1$ or $U^3$ are in high frequency, than the decay of the low frequency factor $U^2$ is better by a $t^{-1}$. Hence, it is straightforward to show that for all choices of $i$ and $j$, $$\begin{aligned}
NLS2(i,j,\neq,\neq) & \lesssim \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2\left({\left\lVert A^j \Delta_L U^j_{\neq} \right\rVert}_2 + {\left\lVert A^2 \Delta_L U^2_{\neq} \right\rVert}_2\right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemma \[lem:SimplePEL\] for $\epsilon$ sufficiently small.
### Transport nonlinearity $\mathcal{T}$ {#sec:Q2_TransNon}
Next, we treat $\mathcal{T}_{\neq}$ (recall ). Begin with a paraproduct decomposition: $$\begin{aligned}
\mathcal{T}_{\neq} & = -\int A^{2} Q^2_{\neq} A^{2} \left( \tilde U_{Lo} \cdot {\nabla}Q^2_{Hi} \right) dV -\int A^{2} Q^2_{\neq} A^{2} \left( \tilde U_{Hi} \cdot {\nabla}Q^2_{Lo} \right) dV - \int A^{2} Q^2_{\neq} A^{2} \left( \tilde U \cdot {\nabla}Q^2 \right)_{\mathcal{R}} dV \\
& = \mathcal{T}_T + \mathcal{T}_{R} + \mathcal{T}_{\mathcal{R}},\end{aligned}$$ where, as in [@BGM15I], ‘T’ and ‘R’ stand for *transport* and *reaction* respectively. Decompose the transport and reaction terms into subcomponents depending on the $X$ frequencies: $$\begin{aligned}
\mathcal{T}_{T}
& = -\int A^{2} Q^2_{\neq} A^{2} \left( (\tilde U_{\neq})_{Lo} \cdot ({\nabla}Q^2_0)_{Hi} \right) dV - \int A^{2}_{\neq} Q^2 A^{2} \left( g_{Lo} \partial_Y (Q^2_{\neq})_{Hi} \right) dV \\ & \quad - \int A^{2} Q^2_{\neq} A^{2} \left( (\tilde U_{\neq})_{Lo} \cdot {\nabla}(Q^2_{\neq})_{Hi} \right) dV \\
& = \mathcal{T}_{T;\neq 0}+ \mathcal{T}_{T;0 \neq}+ \mathcal{T}_{T;\neq \neq}, \end{aligned}$$ and, $$\begin{aligned}
\mathcal{T}_{R} & = -\int A^{2} Q^2_{\neq} A^{2} \left( (\tilde U_{\neq})_{Hi} \cdot ({\nabla}Q^2_0)_{Lo} \right) dV - \int A^{2} Q^2_{\neq} A^{2} \left( g_{Hi} \partial_Y (Q^2_{\neq})_{Lo} \right) dV \\ & \quad - \int A^{2} Q^2_{\neq} A^{2} \left( (\tilde U_{\neq})_{Hi} \cdot {\nabla}(Q^2_{\neq})_{Lo} \right) dV \\
& = \mathcal{T}_{R;\neq 0} + \mathcal{T}_{R;0 \neq}+ \mathcal{T}_{R;\neq \neq}. \end{aligned}$$
#### Transport by zero frequencies: $\mathcal{T}_{T;0 \neq}$
Turn first to $\mathcal{T}_{T;0 \neq}$, which is the transport by $g$. On the Fourier side, $$\begin{aligned}
\mathcal{T}_{T;0 \neq} & \lesssim \sum_k \sum_{l,l^\prime} \int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) A_k^2(\eta,l)\hat{g}(\eta-\xi,l-l^\prime)_{Lo} \xi \widehat{Q^2_k}(\xi,l^\prime)_{Hi}\right\vert} d\eta d\xi. \end{aligned}$$ Hence, by , ${\left\vert\xi\right\vert} \leq {\left\vert\xi - kt\right\vert} + {\left\vertkt\right\vert}$, and , $$\begin{aligned}
\mathcal{T}_{T;0 \neq} & \lesssim {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda}}{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2 \left({\left\lVert (\partial_Y - t\partial_X) A^2 Q^2 \right\rVert}_2 + t{\left\lVert \partial_X A^2 Q^2 \right\rVert}_2\right) \\
& \lesssim {\left\langle t \right\rangle}{\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda}}{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2{\left\lVert \sqrt{-\Delta_L} A^2 Q^2_{\neq} \right\rVert}_2 \\
& \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^2 Q^2_{\neq} \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{2}}{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2^2, \end{aligned}$$ where the last line followed from the low norm control on $g$, . This contribution is hence consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Transport by non-zero frequencies, $\mathcal{T}_{T;\neq \neq}$ and $\mathcal{T}_{T;\neq 0}$
Turn next to $\mathcal{T}_{T;\neq \neq}$. Indeed, going back to , $$\begin{aligned}
\mathcal{T}_{T;\neq \neq} & = \int A^2 Q^2_{\neq} A^2 \left( \begin{pmatrix} (U_{\neq}^1)_{Lo} \\ \left((1+\psi_y)U^2_{\neq}\right)_{Lo} + \left(\psi_zU^3_{\neq}\right)_{Lo} \\ \left((1 + \phi_z)U^3_{\neq} \right)_{Lo} + \left(\phi_y U^2_{\neq}\right)_{Lo} \end{pmatrix} \cdot \begin{pmatrix} \partial_X \\ \partial_Y - t\partial_X \\ \partial_Z \end{pmatrix} (Q_{\neq}^2)_{Hi} \right) dV. \end{aligned}$$ The presence of the coefficients is irrelevant by Lemma \[lem:GevProdAlg\] and Lemma \[lem:CoefCtrl\] so let us ignore them. By , . and we have $$\begin{aligned}
\mathcal{T}_{T;\neq \neq} & \lesssim \left({\left\lVert U^1_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda}} +{\left\lVert U^2_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda}} + {\left\lVert U^3_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda}}\right){\left\lVert A^{2} Q^2 \right\rVert}_2 {\left\lVert \sqrt{-\Delta_L}A^{2} Q^2 \right\rVert}_2 \\
& \lesssim \frac{\epsilon^{1/2} {\left\langle t \right\rangle}^{2\delta_{1}}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} {\left\lVert A^{2} Q^2 \right\rVert}^2_2 + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^{2} Q^2 \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\delta_1$ and $\epsilon$ sufficiently small. The contribution from $\mathcal{T}_{T;\neq,0}$ is treated similarly and yields $$\begin{aligned}
\mathcal{T}_{T;\neq 0} & \lesssim \frac{\epsilon }{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^2 Q^2 \right\rVert}_2{\left\lVert {\nabla}A^2 Q^2_0 \right\rVert}_2 \lesssim \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert A^2 Q^2 \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert {\nabla}A^2 Q^2_0 \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. This completes the treatment of the ‘transport’ contribution to the transport nonlinearity.
#### Reaction term $\mathcal{T}_{R;0 \neq}$
It is in the reaction terms where things get more interesting. We begin with the trivial one, $\mathcal{T}_{R;0 \neq}$. By Lemma \[lem:ABasic\], , and , $$\begin{aligned}
\mathcal{T}_{R;0 \neq} & \lesssim \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum \int {\left\vertA^2 \widehat{Q^2_k}(\eta,l) \frac{1}{{\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} {\left\langle \xi,l^\prime \right\rangle}^2} A\hat{g}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\xi d\eta \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert Ag \right\rVert}_2 {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Reaction term $\mathcal{T}_{R;\neq 0}$ {#sec:Q2TRneq0}
First consider $\mathcal{T}_{R;\neq 0}$, which is further divided via (recall this shorthand notation from §\[sec:paranote\] and the a priori estimates , ) $$\begin{aligned}
\mathcal{T}_{R;\neq 0} & \lesssim \epsilon \sum_{k \neq 0}\int {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \hat{U}_k^2 (\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \quad + \epsilon \sum_{k \neq 0}\int {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \hat{U}_k^3 (\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2}{{\left\langle t \right\rangle}{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0}\int {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \left( {\left\vert\widehat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert}\right)\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0}\int {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \left( {\left\vert\widehat{\psi_z}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_z}(\xi,l^\prime)_{Hi}\right\vert} \right) \right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \quad + \mathcal{T}_{R;\neq 0;\mathcal{R}} \\
& = \mathcal{T}_{R;\neq 0;2} + \mathcal{T}_{R;\neq 0;3} + \mathcal{T}_{R;\neq 0;C1} + \mathcal{T}_{R;\neq 0;C2}+ \mathcal{T}_{R;\neq 0;\mathcal{R}}. \end{aligned}$$ Turn first to $\mathcal{T}_{R;\neq 0;2}$. By , , and the projection to non-zero frequencies, $$\begin{aligned}
\mathcal{T}_{R;\neq 0;2} & \lesssim \epsilon{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2 {\left\lVert A^2 U^2_{\neq} \right\rVert}_2 \lesssim \epsilon{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2 {\left\lVert A^2 \Delta_L U^2_{\neq} \right\rVert}_2, \end{aligned}$$ which by Lemma \[lem:SimplePEL\] is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
Turn next to $\mathcal{T}_{R;\neq 0;3}$. By , , and , $$\begin{aligned}
\mathcal{T}_{R;\neq 0;3} & \lesssim \epsilon \sum_{k \neq 0}\int {\left\vertA^{2} \hat{Q}^2_k(\eta,l)\right\vert} \frac{{\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \\
& \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta -tr\right\vert} } + \chi^{\ast;23} \right){\left\vert \Delta_LA^3 \widehat{U^3_k }(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\xi d\eta \\
& \lesssim \epsilon {\left\lVert A^2 Q^2 \right\rVert}_2 {\left\lVert \Delta_L A^3 U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which by Lemma \[lem:SimplePEL\] is consistent Proposition \[prop:Boot\] for $c_0$ sufficiently small.
The two coefficients are straightforward and are hence omitted for the sake of brevity. The remainder terms are even simpler and are hence omitted. This completes the treatment of the reaction term $\mathcal{T}_{R;\neq 0}$.
#### Reaction term $\mathcal{T}_{R; \neq \neq}$ {#sec:Q2TRneqneq}
Turn finally to $\mathcal{T}_{R;\neq \neq}$, which is more problematic here than in [@BGM15I] due to the low frequency growth of $Q^2$ and the lower regularity of $Q^3$. As in the treatment of $\mathcal{T}_{R;\neq 0}$ above in §\[sec:Q2TRneq0\], we sub-divide in frequency more carefully, $$\begin{aligned}
\mathcal{T}_{R;\neq \neq} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \hat{U}_{k^\prime}^1 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum \int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \hat{U}_{k^\prime}^2 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}} \sum \int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \hat{U}_{k^\prime}^3 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum \int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{2} \hat{Q}^2_k(\eta,l)\right\vert} A^{2}_{k}(\eta,l) \left({\left\vert\hat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\hat{\phi_z}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\hat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert}\right) \\ & \quad\quad\quad \times Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum \int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{2} \hat{Q}^2_k(\eta,l) A^{2}_{k}(\eta,l) \left(\hat{\psi_z}(\xi,l^\prime)_{Hi} \right)\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \mathcal{T}_{R;\neq \neq;\mathcal{R}} \\
& = \mathcal{T}_{R;\neq\neq}^{1} + \mathcal{T}_{R;\neq\neq}^2 + \mathcal{T}_{R;\neq\neq}^3 + \mathcal{T}_{R;\neq\neq}^{C1} + \mathcal{T}_{R;\neq\neq}^{C2} + \mathcal{T}_{R;\neq\neq;\mathcal{R}}, \end{aligned}$$ where we used $\epsilon t {\left\langle \nu t^3 \right\rangle}^{-1} \lesssim t^{-1}$ in $\mathcal{T}_{R;\neq\neq}^3$ to reduce the power of time of the $(U^3)_{Hi} \left(\psi_z(\partial_Y-t\partial_X)Q^2\right)_{Lo}$ term.
Turn first to $\mathcal{T}_{R;\neq\neq}^{1}$, which by , and is given by $$\begin{aligned}
\mathcal{T}_{R;\neq \neq}^{1} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum \int {\left\vertA^{2} \widehat{Q^2_k}(\eta,l) \frac{1}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi - k^\prime t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1+\delta_1}\right\vert} \\ & \quad\quad \times {\left\vert A^{1} \Delta_L \widehat{U^1_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}} A^2 \right) Q^2 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1\right) \Delta_L U^1_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{1+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2 {\left\lVert A^1 \Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which by Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\], is consistent with Proposition \[prop:Boot\] by the bootstrap hypotheses for $\epsilon$ and $\delta_1$ sufficiently small. The treatment of $\mathcal{T}_{R;\neq\neq}^{2}$ is essentially the same as $\mathcal{T}^1_{R;\neq\neq}$ and yields $$\begin{aligned}
\mathcal{T}_{R;\neq \neq}^{2} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) Q^2 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) \Delta_L U^2_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2 {\left\lVert A^2 \Delta_L U^2_{\neq} \right\rVert}_2, \end{aligned}$$ which again by Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\], is consistent with Proposition \[prop:Boot\] by the bootstrap hypotheses for $\epsilon$ sufficiently small.
Turn next to $\mathcal{T}_{R;\neq\neq}^{3}$. By , , and , we have $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{3} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \int {\left\vertA^{2} \widehat{Q^2_k}(\eta,l)\right\vert} \frac{1}{(k^\prime)^2 + (l^\prime)^2 + {\left\vert\xi - k^\prime t\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} \\
& \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} \right) {\left\vertA^{3} \Delta_L \widehat{U^3_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^2 Q^2 \right\rVert}_2 {\left\lVert \Delta_L A^3 U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which after Lemma \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\].
The coefficient error terms are treated the same as in §\[sec:Q2TRneq0\]; hence we omit the treatments for brevity and simply conclude $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{C1} + \mathcal{T}_{R;\neq\neq}^{C2} & \lesssim \frac{\epsilon^2 t}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^2 Q^2 \right\rVert}_2{\left\lVert AC \right\rVert}_2. \end{aligned}$$ The remainder terms $\mathcal{T}_{R;\neq\neq}$ are similarly straightforward and are omitted for brevity as well. This completes the treatment of the transport nonlinearity for $Q^2$.
### Dissipation error terms $\mathcal{D}$ {#sec:DEneqQ2}
Recalling the dissipation error terms and the short-hand , we have $$\begin{aligned}
\mathcal{D}_E & = \nu\sum_{k \neq 0}\int A^2 Q^2_k A^2_k\left( G_{yy}(\partial_{Y} - t\partial_X)^2 Q^2_k + G_{yz}(\partial_Y - t \partial_X)\partial_{Z}Q^2_k + G_{zz}\partial_{ZZ} Q^2_k \right) dV. \end{aligned}$$ These terms can be treated in the same manner as the analogous terms in [@BGM15I]; therefore, we omit the treatment for brevity and simply conclude the final result: $$\begin{aligned}
\mathcal{D}_E & \lesssim c_0 \nu {\left\lVert \sqrt{-\Delta_L} A^2 Q^2 \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2^2 + \frac{\nu^2 \epsilon^{3/2} t^4 }{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert AC \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
High norm estimate on $Q^3$
===========================
Computing the evolution of $A^{3}Q^3$: $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A^{3} Q^3 \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{3} Q^3 \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^{3} Q^3 \right\rVert}_2^2 -{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{3} Q^3 \right\rVert}_2^2
-\frac{2}{t}{\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{3} Q^3 \right\rVert}_2^2\nonumber \\
& \quad -2 \int A^{3} Q^3 A^{3} \partial_{YX}^t U^3 dV + 2 \int A^{3} Q^3 A^{3} \partial_{ZX}^t U^2 dV \nonumber\\ & \quad
+ \nu \int A^{3} Q^{3} A^{3} \left(\Delta_t Q^3\right) dV-\int A^{3} Q^3 A^{3}\left( \tilde U \cdot {\nabla}Q^3 \right) dv \nonumber \\
& \quad - \int A^{3} Q^3 A^{3} \left[ Q^j \partial_j^t U^3 + 2\partial_i^t U^j \partial_{ij}^t U^3 - \partial_Z^t\left(\partial_i^t U^j \partial_j^t U^i\right) \right] dV \nonumber \\
& = \mathcal{D}Q^3 - CK_{L}^3 + LS3 + L3 + \mathcal{D}_E + \mathcal{T} + NLS1 + NLS2 + NLP, \label{ineq:AQ3_Evo} \end{aligned}$$ where we are again using $$\begin{aligned}
\mathcal{D}_E = \nu \int A^{3} Q^{3} A^{3} \left((\tilde{\Delta_t} - \Delta_L) Q^3\right) dV. \end{aligned}$$ As in , let us here recall the following enumerations from [@BGM15I]: for $i,j\in {\left\{1,2,3\right\}}$ and $a,b \in {\left\{0,\neq\right\}}$:
\[def:Q3Enums\] $$\begin{aligned}
NLP(i,j,a,b) &= \int A^3 Q^3_{\neq} A^3\left( \partial_Z^t \left(\partial_j^t U^i_a \partial_i^t U^j_b \right) \right) dV \\
NLS1(j,a,b) & = -\int A^3 Q^3_{\neq} A^3\left(Q^j_a\partial_j^t U^3_{b}\right) dV \\
NLS2(i,j,a,b) & = -\int A^3 Q^3_{\neq} A^2\left(\partial_i^t U^j_a \partial_i^t\partial_j^t U^3_{b}\right) dV \\
NLP(i,j,0) & = \int A^3 Q^3_{0} A^3\left( \partial_Z^t \left(\partial_j^t U^i_0 \partial_i^t U^j_0 \right) \right) dV \\
NLS1(j,0) & = -\int A^3 Q^3_{0} A^3\left(Q^j_0\partial_j^t U^3_{0}\right) dV \\
NLS2(i,j,0) & = -\int A^3 Q^3_{0} A^3\left(\partial_i^t U^j_0 \partial_i^t\partial_j^t U^3_{0}\right) dV \\
\mathcal{F} & = -\int A^3 Q^3_{0} A^3\left(\partial_i^t \partial_i^t \partial_j^t \left(U^j_{\neq} U^3_{\neq}\right)_0 - \partial_{Z}^t \partial_j^t \partial_i^t \left(U^i_{\neq} U^j_{\neq}\right)_0 \right)dV \\
\mathcal{T}_0 & = -\int A^{3} Q_0^3 A^{3} \left( \tilde U_{0} \cdot {\nabla}Q_{0}^3 \right) dV \\
\mathcal{T}_{\neq} & = -\int A^{2} Q_{\neq}^3 A^{3} \left( \tilde U \cdot {\nabla}Q^3 \right) dV. \end{aligned}$$
Note we have split the nonlinearity up analogously to what is done in above.
Zero frequencies {#zero-frequencies}
----------------
As in the treatment of $A^2Q^2$ in §\[sec:AQ2Zero\], the estimate on $Q_0^3$ is very different than the estimate on $Q^3_{\neq}$ and are hence naturally separated.
### Transport nonlinearity {#transport-nonlinearity}
The treatment of $\mathcal{T}_0$, the **(2.5NS)** contribution to the transport nonlinearity, goes through exactly the same as the corresponding treatment for $Q_0^2$ in §\[sec:TransQ20\] (as the main problems in $A^3$ will only arise when changing the $X$ frequencies) and hence, for the sake of brevity this term is omitted.
### Nonlinear pressure and stretching {#nonlinear-pressure-and-stretching}
The treatment of zero frequency pressure and stretching contributions in is very similar to the treatment used for $Q^2_0$ in §\[sec:NLPSQ20\] except that since we are estimating with $A^3$, there is no loss on factors involving $U^3$ as there is in §\[sec:NLPSQ20\]. As the treatment here is analogous (except easier), we omit these terms for brevity.
### Forcing from non-zero frequencies {#sec:NzeroForcingQ3}
Turn next to the treatment of $\mathcal{F}$ (defined above in ), for nonlinear interactions of type **(F)**. In accordance with the toy model in §\[sec:Toy\], we will find that the forcing from non-zero frequencies on $Q^3_0$ is more extreme than those on $Q_0^2$. In particular, unlike in §\[sec:NzeroForcing\] above, in order to treat the case $\nu \ll \epsilon$ we will need the regularity imbalances. Write $$\begin{aligned}
\mathcal{F} & = -\int A^3 Q^3_0 A^3 \left(\partial_Y^t \partial_Y^t\partial_j^t \left(U^j_{\neq} U^3_{\neq}\right)_0 - \partial_Z^t \partial_Z^t \partial_Y^t \left(U^2_{\neq} U^3_{\neq}\right)_0 - \partial_Z^t \partial_Y^t \partial_Y^t \left(U^2_{\neq} U^2_{\neq}\right)_0 \right) dV \\
& = F^1 + F^2 + F^3.\end{aligned}$$ The most dangerous term is $F^1$; we omit the other two for brevity as they are easy variants of $F^1$ and the treatments in §\[sec:NzeroForcing\]. Write $$\begin{aligned}
F^1 = -\int A^3 Q^3 A^3\left( \partial_Y^t \partial_Y^t\partial_Y^t \left(U^2_{\neq} U^3_{\neq}\right)_0 + \partial_Y^t \partial_Y^t\partial_Z^t \left( U^3_{\neq} U^3_{\neq}\right)_0 \right) dV = F^{1;2} + F^{1;3}.\end{aligned}$$ The first term, $F^{1;2}$, is the leading order contribution (at least when $U^2$ is in high frequency) due to the $t^3$ that will be present near the critical times due to the $(\partial_Y)^3$ (near the critical times $\partial_Y \sim t\partial_X$), and hence let us focus on this and omit $F^{1;3}$ for brevity. Expand $F^{1;2}$ with a quintic paraproduct and group all of the terms where the coefficients appear in low frequency with the remainder: $$\begin{aligned}
F^{1;2} & = -\sum_{k\neq 0} \int A^{3}Q^3_0 A_0^{3} \partial_Y\partial_Y\partial_Y\left( \left(U^3_{-k}\right)_{Hi} \left( U^2_k\right)_{Lo}\right)dV \\
& \quad - \sum_{k\neq 0} \int A^{3}Q^3_0 A_0^{3} \partial_Y \partial_Y\partial_Y\left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Hi}\right) dV \\
& \quad - \sum_{k\neq 0} \int A^{3}Q^3_0 A_0^{3} \left((\psi_y)_{Hi}\partial_Y + (\phi_y)_{Hi}\partial_Z\right)\partial_Y\partial_Y\left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Lo}\right) dV \\
& \quad - \sum_{k\neq 0} \int A^{3}Q^3_0 A_0^{3} \partial_Y \left( \left((\psi_y)_{Hi}\partial_Y + (\phi_y)_{Hi}\partial_Z\right)\partial_Y\left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Lo}\right)\right) dV \\
& \quad - \sum_{k\neq 0} \int A^{3}Q^3_0 A_0^{3} \partial_Y \partial_Y \left( \left((\psi_y)_{Hi}\partial_Y + (\phi_y)_{Hi}\partial_Z\right) \left( \left(U^3_{-k}\right)_{Lo} \left( U^2_k\right)_{Lo}\right)\right) dV \\
& \quad + F^1_{\mathcal{R},C} \\
& = F_{HL} + F_{LH} + F_{C1} + F_{C2} + F_{C3} + F_{\mathcal{R},C}, \end{aligned}$$ where here $F_{\mathcal{R}}$ includes the remainders from the paraproduct and terms where coefficients appear in low frequency.
Turn first to the easier $F_{HL}$. From , , , and we have, $$\begin{aligned}
F_{HL} & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k\neq 0} \sum_{l,l^\prime} \int {\left\vertA^{3} \widehat{Q^3_0}(\eta,l)\right\vert} \frac{{\left\vert\eta\right\vert}^3 {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^2}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} {\left\vert \Delta_L A^{3}\widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low\left(-k,\eta-\xi,l-l^\prime \right) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^{3} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^{3} \right) Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^{3} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{3}\right) \Delta_L U_{\neq}^3 \right\rVert}_2
\\ & \quad + \frac{\epsilon}{{\left\langle t \right\rangle}{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^{3}Q^3 \right\rVert}_2 {\left\lVert A^{3}\Delta_L U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which, after the application of Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\]. Notice the importance of the inviscid damping to reduce the power of $t$.
Turn next to $F_{LH}$, which is the term appearing in the toy model in §\[sec:Toy\] as one of the leading order contributions to the nonlinear interaction **(F)**. Here, it is the regularity imbalance between $Q^2_{\neq}$ and $Q^3_0$ which will reduce the power of $t$. By and Lemma \[lem:ABasic\] we have $$\begin{aligned}
F_{LH} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k\neq 0} \sum_{l,l^\prime} \int {\left\vertA^{3} \widehat{Q^3_0}(\eta,l) A^3_0(\eta,l) \frac{{\left\vert\eta\right\vert}^3}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \Delta_L \widehat{U^2_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k\neq 0} \sum_{l,l^\prime} \int {\left\vert \widehat{Q^3_0}(\eta,l)\right\vert} \frac{{\left\vert\eta\right\vert}^3}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \\ & \quad\quad \times \left(\sum_{r}\chi^{NR,r}\frac{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}{t} \tilde{A}^{3}_0(\eta,l) \tilde{A}_k^2(\xi,l^\prime) + \chi^{\ast;32} A^{3}_0(\eta,l) A_k^2(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle} {\left\vertA_k^2 \Delta_L \widehat{U^2_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi. \end{aligned}$$ Therefore, by and , followed by , we have $$\begin{aligned}
F_{LH}^1 & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3_0 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) \Delta_L U^2_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \sqrt{-\Delta_L}A^3 Q^3_0 \right\rVert}_2 {\left\lVert A^2 \Delta_L U_{\neq}^2 \right\rVert}_{2}, \end{aligned}$$ which by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\] is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
The terms associated with the coefficient terms are treated the same as the corresponding terms in §\[sec:NzeroForcing\] and are hence omitted for brevity and we simply conclude the results $$\begin{aligned}
F^1_{C1} + F^1_{C2} + F^1_{C3} & \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^{3}Q^3 \right\rVert}_2^2 + \frac{\epsilon^{5/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert AC \right\rVert}^2_2. \end{aligned}$$ The remainder terms are similarly straightforward or easy variants of the other treatments and are hence omitted as well. This completes the treatment of $F^1$. As mentioned above, the treatments of $F^2$ and $F^3$ are similar (but easier) and hence also omitted.
### Zero frequency dissipation error terms
The treatment of the dissipation error terms for $Q^3_0$ is the same as $Q_0^2$ as outlined in §\[sec:DEQ02\], and therefore is omitted for the sake of brevity.
Non-zero frequencies
--------------------
### Nonlinear pressure $NLP$ {#sec:NLP3}
#### Treatment of $NLP(1,j,0,\neq)$ {#sec:NLPQ3ij_0neq}
This term is the analogue of the nonlinear terms treated in §\[sec:NLP213\]. Note that $j \neq 1$ by the zero frequency assumption. We can essentially use the same treatment, although here it is easier since $Y$ derivatives are slightly harder than $Z$ derivatives and because we are imposing one less power of time control on $Q^3_{\neq}$ than on $Q^2_{\neq}$. For this reason, we omit the treatment for brevity and simply conclude the result: $$\begin{aligned}
NLP(1,j,0,\neq) & \lesssim c_{0}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) Q^3 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^j + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^j \right) \Delta_L U^j_{\neq} \right\rVert}_2 \\
& \quad + \epsilon {\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2^2 + \mathbf{1}_{t \ll \epsilon^{-1/2}}\epsilon^{1/2}{\left\lVert A^j \Delta_L U^j_{\neq} \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^j \Delta_L U^j_{\neq} \right\rVert}_2^2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2 {\left\lVert A^1 {\left\langle {\nabla}\right\rangle}^2 U_0^1 \right\rVert}_2 + \frac{\epsilon^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2 {\left\lVert AC \right\rVert}_2, \end{aligned}$$ which, after Lemmas \[lem:PELbasicZero\], \[lem:PEL\_NLP120neq\], \[lem:SimplePEL\], and \[lem:PELED\], is consistent Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Treatment of $NLP(i,j,0,\neq)$ with $i \in {\left\{2,3\right\}}$ {#sec:NLPQ3_0neq_notX}
This is the analogue of the nonlinear terms treated in §\[sec:NLPQ2\_0neq\_notX\] above. These can treated analogously to the treatment in §\[sec:NLPQ2\_0neq\_notX\], but in fact it is much easier here due to the fact that $Q^3$ is growing quadratically at ‘low’ frequencies. In particular, we can deduce (using also $j \neq 1$), $$\begin{aligned}
NLP(i,j,0,\neq) & \lesssim \epsilon {\left\lVert A^3Q^3_{\neq} \right\rVert}_2 {\left\lVert \Delta_L A^j U^j_{\neq} \right\rVert}_2 + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}} {\left\lVert A^3Q^3_{\neq} \right\rVert}_2 {\left\lVert A U^i_{0} \right\rVert}_2 \\
& \quad + \frac{\epsilon^2{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert AC \right\rVert}_2, \end{aligned}$$ which after Lemmas \[lem:PELbasicZero\], \[lem:PEL\_NLP120neq\], and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Treatment of $NLP(i,j,\neq,\neq)$ {#sec:NLPQ3_neqneq}
These terms are fairly straightforward. The term with $i = j = 3$ cancels with the $NLS$ terms. Let us just treat $NLP(1,3,\neq,\neq)$ and omit the others for brevity, which follow by similar arguments. Expand with a paraproduct, as usual grouping higher order terms involving the coefficients in low frequency with the remainder $$\begin{aligned}
NLP(1,3,\neq,\neq) & = \int A^3 Q^3_{\neq} A^3 \partial_Z \left( \left(\partial_Z U^1_{\neq}\right)_{Lo} (\partial_XU^3_{\neq})_{Hi} \right) dV \\
& \quad + \int A^3 Q^3_{\neq} A^3 \partial_Z \left( \left(\partial_Z U^1_{\neq}\right)_{Hi} (\partial_XU^3_{\neq})_{Lo} \right) dV \\
& \quad + \int A^3 Q^3_{\neq} A^3 \left( \left( (\phi_z)_{Hi}\partial_Z + (\psi_z)_{Hi}(\partial_Y - t\partial_X) \right) \left( \left(\partial_Z U^1_{\neq}\right)_{Lo} (\partial_XU^3_{\neq})_{Lo} \right) \right) dV \\
& \quad + \int A^3 Q^3_{\neq} A^3 \partial_Z\left(\left( \left( (\phi_z)_{Hi}\partial_Z + (\psi_z)_{Hi}(\partial_Y - t\partial_X) \right) U^1_{\neq}\right)_{Lo} (\partial_XU^3_{\neq})_{Lo} \right) dV \\
& \quad + P_{\mathcal{R},C} \\
& = P_{LH} + P_{HL} + P_{C1} + P_{C2} + P_{\mathcal{R},C}, \end{aligned}$$ where $P_{\mathcal{R},C}$ includes all of the remainders from the quartic paraproduct as well as the higher order terms involving coefficients as low frequency factors.
Consider $P_{LH}$ first. By followed by , by and it follows that $$\begin{aligned}
P_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \int {\left\vertA^3 \widehat{Q^3_{k}}(\eta,l)\right\vert} \frac{{\left\vertl k^\prime\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2} \\ & \quad\quad \times
\left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t} + \chi^{\ast;33} \right) {\left\vertA^3\Delta_L\widehat{U^3_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times Low(k - k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{1+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3Q^3 \right\rVert}_2{\left\lVert A^3 \Delta_L U_{\neq}^3 \right\rVert}_2, \end{aligned}$$ which after Lemma \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\] for $\delta_1$ and $\epsilon$ sufficiently small.
Consider next $P_{HL}$. By followed by , we have $$\begin{aligned}
P_{HL} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \int {\left\vertA^3 \widehat{Q^3_{k}}(\eta,l)\right\vert} \frac{{\left\vertl l^\prime\right\vert} {\left\langle t \right\rangle}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{\delta_1-1} \\ & \quad\quad \times {\left\vertA^1\Delta_L\widehat{U^1_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} Low(k - k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert A^1 \Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small after applying Lemma \[lem:SimplePEL\].
The coefficient error terms and the remainder terms are straightforward (easier) variants of the treatment in §\[sec:NLPQ2\_neqneq\] or of the above treatments of $P_{HL}$ and $P_{LH}$, and hence are omitted for brevity. The other nonlinear pressure terms are similar to, or easier than, the above, and are hence omitted for brevity.
### Nonlinear stretching $NLS$ {#sec:NLSQ3}
Controlling the $NLS$ terms in the evolution of $Q^3$ is in general slightly harder than for $Q^2$ (treated above in §\[sec:NLSQ2\]), due to the fact that $U^3$ is larger than $U^2$. Moreover, we occasionally have to deal with the imbalance in the regularities inherent to $A^3$.
#### Treatment of $NLS1(j,\neq,0)$ and $NLS1(j,0,\neq)$ {#sec:NLS1Q3_neq0}
Consider first the $NLS1(j,0,\neq)$ terms. Due to the large size of $Q_0^1$, it turns out $j = 1$ is the hardest case, and hence we only treat this case (the case $j = 3$ is complicated by the regularity imbalance of $A^3_k$ compared to $A^3_0$ (see Lemma \[lem:ABasic\]), however, even at the critical time, the loss is at most ${\left\langle t \right\rangle}$, which is still not more than what is lost when comparing $A^3_k$ to $A^1_k$). Expanding with a paraproduct $$\begin{aligned}
NLS1(1,0,\neq) & = -\sum_{k \neq 0}\int A^3 Q^3_k A^3_k \left( (Q^1_0)_{Hi} (\partial_X U^3_{k})_{Lo} \right) dV - \sum_{k \neq 0}\int A^3 Q^3_k A^3_k \left( (Q^1_0)_{Lo} (\partial_X U^3_{k})_{Hi} \right) dV + S_{\mathcal{R}} \\
& = S_{HL} + S_{LH} + S_{\mathcal{R}}.\end{aligned}$$ For the $S_{HL}$ term, it follows from , Lemma \[lem:ABasic\], and , $$\begin{aligned}
S_{HL} & \lesssim \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert A^1 Q^1_0 \right\rVert}_2. \end{aligned}$$ For the $S_{LH}$ term, by and , followed by and , $$\begin{aligned}
S_{LH} & \lesssim \epsilon t \sum \int {\left\vertA^3 \widehat{Q^3_k}(\eta,l)\right\vert} \frac{{\left\vertk\right\vert}}{k^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-kt\right\vert}^2} {\left\vertA^3 \Delta_L \widehat{U^3_k}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon t {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) \Delta_L U^3_{\neq} \right\rVert}_2 + \epsilon{\left\lVert A^3 Q^3 \right\rVert}_2{\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2,\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\]. The remainder term is straightforward and is hence omitted. As mentioned above, the remaining $NLS1(j,0,\neq)$ terms are omitted as well as they are similar.
Consider next the $NLS1(j,\neq,0)$ terms. Notice that $j \neq 1$ by the nonlinear structure. The remaining contributions are not quite the same: due to the regularity imbalances in $A^3$, the case $j = 3$ is slightly harder (note this does not cancel with the other pressure/stretching terms). Hence, we treat this term and omit the $j = 2$ contribution. As usual, begin with a paraproduct and group the terms where the coefficients appear in low frequency with the remainder: $$\begin{aligned}
NLS1(3,\neq,0) & = -\sum_{k \neq 0} \int A^3 Q^3_{k} A^3\left( (Q^3_{k})_{Hi} (\partial_Z U_0^3)_{Lo} \right) dV - \sum\int A^3 Q^3_{k} A^3\left( (Q^3_{k})_{Lo} (\partial_Z U_0^3)_{Hi} \right) dV \\
& \quad - \sum_{k \neq 0}\int A^3 Q^3_{k} A^3\left( (Q^3_{k})_{Lo} ((\phi_z)_{Hi}\partial_Z + (\psi_z)_{Hi} \partial_Y)(U_0^3)_{Lo} \right) dV + S_{\mathcal{R},C} \\
& = S_{HL} + S_{LH} + S_{C} + S_{\mathcal{R},C}. \end{aligned}$$ For the first term, $S_{HL}$, from and we have $$\begin{aligned}
S_{HL} & \lesssim \epsilon{\left\lVert A^3 Q^3 \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small. For the second term, $S_{LH}$, we have by Lemma \[lem:ABasic\], Lemma \[dtw\], and (note that the zero frequency is never resonant and hence the $\chi^{NR,R}$ term disappears), $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \int {\left\vert\widehat{Q^3_k}(\eta,l)\right\vert} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-2}\\
& \quad\quad \times \left( \chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}}\tilde{A}_k^3(\eta,l)\tilde{A}_0^3(\xi,l^\prime) + A_k^3(\eta,l) A_0^3(\xi,l^\prime) \right) \\
& \quad\quad \times \frac{{\left\vertl^\prime\right\vert}}{{\left\langle \xi,l^\prime \right\rangle}^2} {\left\vertA^3 {\left\langle {\nabla}\right\rangle}^2 \widehat{U^3_0}(\xi,l^\prime)\right\vert} d\xi d\eta \\
& \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) {\left\langle {\nabla}\right\rangle}^2 U^3_0 \right\rVert}_2 \\
& \quad + \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert A^3 {\left\langle {\nabla}\right\rangle}^2 U^3_0 \right\rVert}_2, \end{aligned}$$ which, by Lemmas \[lem:PELCKZero\] and \[lem:PELbasicZero\], is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $c_0$ sufficiently small.
#### Treatment of $NLS1(j,\neq,\neq)$ {#sec:NLS1Q3_neqneq}
All of these terms can be treated in a similar fashion, in fact, $j=3$ is the hardest due to the regularity losses together with a $\partial_Z$ (as opposed to $\partial_X$ as in $j=1$). Hence, let us just consider the case $j=3$ and omit the others for brevity. Expand the term with a paraproduct, as usual leaving the terms with coefficients in low frequency with the remainder, $$\begin{aligned}
NLS1(1,\neq,\neq) & = -\int A^3 Q^3_{\neq} A^3\left( (Q^3_{\neq})_{Hi} (\partial_Z U_{\neq}^3)_{Lo} \right) dV - \int A^3 Q^3_{\neq} A^3\left( (Q^3_{\neq})_{Lo} (\partial_Z U_{\neq}^3)_{Hi} \right) dV \\
& \quad - \int A^3 Q^3_{\neq} A^3\left( (Q^3_{\neq})_{Lo} \left( \left( (\phi_z)_{Hi}\partial_Z + (\psi_z)_{Hi}\partial_Y \right) (U_{\neq}^3)_{Lo}\right) \right) dV + S_{\mathcal{R}} \\
& = S_{HL} + S_{LH} + S_{\mathcal{R}}.\end{aligned}$$ By , Lemma \[lem:ABasic\], and (the loss of $t$ is due to the regularity imbalances), $$\begin{aligned}
S_{HL} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3Q^3 \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small by Lemma \[lem:SimplePEL\].
For $S_{LH}$, we have to be a little more careful. By , $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \int {\left\vert\widehat{Q^3_{k}}(\eta,l)\right\vert} \frac{{\left\vertl^\prime\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2}\\
& \quad\quad \times \left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}}\tilde{A}^3_k(\eta,l) \tilde{A}^3_{k^\prime}(\xi,l^\prime) + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t} \tilde{A}^3_k(\eta,l) \tilde{A}^3_{k^\prime}(\xi,l^\prime) \right. \\ & \quad\quad\quad + \chi^{\ast;33} A^3_k(\eta,l) A^3_{k^\prime}(\xi,l^\prime) \bigg) {\left\vertA^3\Delta_L\widehat{U^3_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} Low(k - k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi. \end{aligned}$$ Therefore by , , and , there holds $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) \Delta_L U^3_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] after Lemmas \[lem:PEL\_NLP120neq\] and Lemma \[lem:SimplePEL\].
The coefficient error term $S_C$ and remainder term $S_{\mathcal{R}}$ are both straightforward or easy variants of estimates already performed and hence are omitted for brevity.
#### Treatment of $NLS2(i,j,0,\neq)$
Recall and notice that $i \neq 1$. These terms are treated in essentially the same way as $NLS1(3,\neq,0)$ (or $NLS1(2,\neq,0)$) and hence we omit the treatment for brevity.
#### Treatment of $NLS2(i,j,\neq,0)$
Recall and notice that neither $i$ nor $j$ can be $1$ in this case. These terms are very similar to $NLS1(2,0,\neq)$ and are hence omitted for brevity.
#### Treatment of $NLS2(i,j,\neq,\neq)$
First, notice that $i = j = 3$ cancels with the $NLS$ terms. The most difficult term is $i = 2$ and $j = 3$; let us briefly comment on this term and omit the others for brevity. Expanding with a paraproduct $$\begin{aligned}
NLS2(2,3,\neq,\neq) & = -\int A^3 Q^3_{\neq} A^3 \left( \left( (\partial_Y-t\partial_X) U^3_{\neq}\right)_{Lo} ( (\partial_Y-t\partial_X) \partial_Z U^3_{\neq})_{Hi} \right) dV \\
& \quad - \int A^3 Q^3_{\neq} A^3 \partial_Z \left( \left( (\partial_Y - t\partial_X) U^3_{\neq}\right)_{Hi} (\partial_Z(\partial_Y - t\partial_X)U^3_{\neq})_{Lo} \right) dV \\
& \quad + S_{C1} + S_{C2} + S_{C3} + S_{\mathcal{R},C} \\
& = S_{LH} + S_{HL} + S_{C1} + S_{C2} + S_{C3} + S_{\mathcal{R},C}, \end{aligned}$$ where $S_{\mathcal{R},C}$ denotes the remainders and $S_{Ci}$ denote terms in which the coefficients appear in high frequency; these are very similar to many terms we have already treated and are hence omitted. The leading order terms are treated in essentially the same manner; the $S_{LH}$ term is clearly the harder one, so let us just show the treatment of this one. For $LH$ term we have, by , $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \int {\left\vert\widehat{Q^3_{k}}(\eta,l)\right\vert} \frac{{\left\vert\xi - tk^\prime\right\vert} {\left\vertl^\prime\right\vert}}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi-tk^\prime\right\vert}^2}\\
& \quad\quad \times \left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}}\tilde{A}^3_k(\eta,l) \tilde{A}^3_{k^\prime}(\xi,l^\prime) + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t} \tilde{A}^3_k(\eta,l) \tilde{A}^3_{k^\prime}(\xi,l^\prime) \right. \\ & \quad\quad\quad + \chi^{\ast;33} A^3_k(\eta,l) A^3_{k^\prime}(\xi,l^\prime) \bigg) {\left\vertA^3\Delta_L\widehat{U^3_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} Low(k - k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right)Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right)\Delta_L U^3_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2{\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\].
### Transport nonlinearity $\mathcal{T}$ {#sec:Q3_TransNon}
Begin with a paraproduct decomposition: $$\begin{aligned}
\mathcal{T}_{\neq} & = -\int A^{3} Q^3_{\neq} A^{3} \left( \tilde U_{Lo} \cdot {\nabla}Q^3_{Hi} \right) dV -\int A^{3} Q^3_{\neq} A^{3} \left( \tilde U_{Hi} \cdot {\nabla}Q^3_{Lo} \right) dV + \mathcal{T}_{\mathcal{R}} \\
& = \mathcal{T}_T + \mathcal{T}_{R} + \mathcal{T}_{\mathcal{R}}, \end{aligned}$$ where $\mathcal{T}_{\mathcal{R}}$ includes the remainder (as above in §\[sec:Q2\_TransNon\], we use the terminology ‘transport’ and ‘reaction’ for the first two terms respectively). There are two interesting challenges here. First, the additional $ + e^{\mu {\left\vertl\right\vert}^{1/2}}$ was added in because large regularity imbalances caused by $w^3$ would have been problematic at high $Z$ frequencies in the in the ‘transport’ contribution. Second, we will see that the ‘reaction’ contribution is significantly more difficult and, as predicted in §\[sec:Toy\], we will need to take advantage of the regularity imbalances to close an estimate.
Decompose the reaction terms based on the $X$ dependence of each factor: $$\begin{aligned}
\mathcal{T}_{R} & = -\int A^{3} Q^3 A^{3} \left( (\tilde U_{\neq})_{Hi} \cdot ({\nabla}Q^3_0)_{Lo} \right) dV - \int A^{3} Q^3 A^{3} \left( g_{Hi} \cdot \partial_Y (Q^3_{\neq})_{Lo} \right) dV \\ & \quad - \int A^{3} Q^3 A^{3} \left( (\tilde U_{\neq})_{Hi} \cdot {\nabla}(Q^3_{\neq})_{Lo} \right) dV \\
& = \mathcal{T}_{R;\neq 0} + \mathcal{T}_{R;0 \neq}+ \mathcal{T}_{R;\neq \neq}, \end{aligned}$$ and also the transport terms: $$\begin{aligned}
\mathcal{T}_{T} & = -\int A^{3} Q^3 A^{3} \left( (\tilde U_{\neq})_{Lo} \cdot ({\nabla}Q^3_0)_{Hi} \right) dV - \int A^{3} Q^3 A^{3} \left( g_{Lo} \partial_Y (Q^3_{\neq})_{Hi} \right) dV \\ & \quad - \int A^{3} Q^3 A^{3} \left( (\tilde U_{\neq})_{Lo} \cdot {\nabla}(Q^3_{\neq})_{Hi} \right) dV \\
& = \mathcal{T}_{T;\neq 0} + \mathcal{T}_{T;0 \neq}+ \mathcal{T}_{T;\neq \neq}. \end{aligned}$$
#### Transport term $\mathcal{T}_{T;0 \neq}$
This term can be treated the same as the corresponding term in §\[sec:Q2\_TransNon\]: because the velocity field is independent of $X$, there are no regularity losses associated with the regularity imbalances in the norm $A^3$ – these only occur if one changes the $X$ frequency, as $\chi^{R,NR} = \chi^{NR,R} = 0$ if $k =k^\prime$ in Lemma \[lem:ABasic\]. Hence, as above, $$\begin{aligned}
\mathcal{T}_{T;0\neq} & \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^3 Q^3 \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{t^{2}}{\left\lVert A^3 Q^3 \right\rVert}_2^2.\end{aligned}$$
#### Transport term $\mathcal{T}_{T;\neq 0}$
This is one of the terms where it is crucial that we include the $ + e^{\mu{\left\vertl\right\vert}^{1/2}}$ correction to the norm. By Lemma \[lem:ABasic\] and , we have by ${\left\vert\xi,l^\prime\right\vert} \chi^{R,NR} \lesssim {\left\vertkt\right\vert}\chi^{R,NR}$ (it is here we are using that regularity imbalances only occur for ${\left\vert\partial_Z\right\vert} \lesssim {\left\vert\partial_Y\right\vert}$), $$\begin{aligned}
\mathcal{T}_{T;\neq 0} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha}\sum \int {\left\vertA^3 \widehat{Q^3_{k}}(\eta,l) A^3_k(\eta,l) {\left\vert\xi,l^\prime\right\vert} \widehat{Q^3_0}(\xi,l^\prime)_{Hi} Low(k,\eta-\xi,l-l^\prime)\right\vert} d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha}\sum \int {\left\vert\widehat{Q^3_{k}}(\eta,l)\right\vert}\left(\frac{t \chi^{R,NR}}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} \tilde{A}^3_k(\eta,l)\tilde{A}^3_0(\xi,l^\prime) + A^3_k(\eta,l) A^3_0(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert\xi,l^\prime\right\vert} {\left\vert\widehat{Q^3_0}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) Q^3 \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} {\left\lVert A^3 Q^3_{\neq} \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert {\nabla}A^3 Q_0^3 \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
#### Transport term $\mathcal{T}_{T;\neq \neq}$
We will again use crucially that we have the $ + e^{\mu{\left\vertl\right\vert}^{1/2}}$ correction to the norm. By we have $$\begin{aligned}
\mathcal{T}_{T;\neq \neq} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}\sum \int {\left\vertA^3 \widehat{Q^3_{k}}(\eta,l) A^3_k(\eta,l) {\left\vertk t^{\delta_1}, t^{-1}(\xi-k^\prime t), l^\prime\right\vert} \widehat{Q^3_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} \\ &\quad\quad \times Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}\sum \int {\left\vert\widehat{Q^3_{k}}(\eta,l)\right\vert} \left(\frac{t \chi^{R,NR}}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}} \tilde{A}^3_k(\eta,l)\tilde{A}^3_{k^\prime}(\xi,l^\prime) + A^3_k(\eta,l) A^3_{k^\prime}(\xi,l^\prime) \right) \\
& \quad\quad \times \left({\left\vertk^\prime t^{\delta_1}\right\vert} + t^{-1}{\left\vert\xi-k^\prime t\right\vert} + {\left\vertl^\prime\right\vert}\right) {\left\vert\widehat{Q^3_{k^\prime}}(\xi,l^\prime)_{Hi} Low(k-k^\prime,\eta-\xi,l-l^\prime)\right\vert} d\eta d\xi \\
& = \mathcal{T}_{T;\neq \neq}^{X} + \mathcal{T}_{T;\neq \neq}^{Y} + \mathcal{T}_{T;\neq \neq}^{Z}. \end{aligned}$$ Note that we have used the inviscid damping on $U^2$ and the inequality ${\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}{\left\lVert U^3_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\beta-2}} \lesssim \epsilon^2 t {\left\langle \nu t^3 \right\rangle}^\alpha \lesssim \epsilon t^{-1} {\left\langle \nu t^3 \right\rangle}^{\alpha - 1}$ (see §\[sec:AprioriBoot\]) to reduce the power in front of the $\partial_Y - t\partial_X$ derivative. Due to the gain in ${\left\vertk\right\vert}$ at the critical times from $\chi^{R,NR}{\left\vertk\right\vert}^{-1}$, we have $$\begin{aligned}
\mathcal{T}_{T;\neq \neq}^{X} & \lesssim \frac{\epsilon t^{1+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3 \right\rVert}_2^2 + \frac{\epsilon t^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L}A^3 Q^3 \right\rVert}_2,\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $\delta_1$ sufficiently small. Due to the extra $t^{-1}$, there are no losses in the $Y$ term and hence we have $$\begin{aligned}
\mathcal{T}_{T;\neq \neq}^{Y} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L}A^3 Q^3 \right\rVert}_2 \lesssim \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^3 Q^3 \right\rVert}_2^2, \end{aligned}$$ which is also consistent with Proposition \[prop:Boot\]. For the $Z$ term we use $\chi^{R,NR} {\left\vertl^\prime\right\vert} \lesssim {\left\vertkt\right\vert} \chi^{R,NR}$ (it is here we are using that the losses only occur for ${\left\vert\partial_Z\right\vert} \lesssim {\left\vert\partial_Y\right\vert}$ due to the $+e^{\mu{\left\vertl\right\vert}^{1/2}}$ correction) and to deduce $$\begin{aligned}
\mathcal{T}_{T;\neq \neq}^{Z} & \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3 \right\rVert}_2^2 + \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L}A^3 Q^3 \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
#### Reaction term $\mathcal{T}_{R;0 \neq}$
Turn first to the easiest, $\mathcal{T}_{R;0 \neq}$. By and Lemma \[lem:ABasic\], we get (also noting ): $$\begin{aligned}
\mathcal{T}_{R;0 \neq} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k \neq 0}\int {\left\vertA^{3} \hat{Q_k^3} (\eta,l) A^{3}_{k}(\eta,l) \widehat{g}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\xi d\eta \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2 {\left\lVert Ag \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
#### Reaction terms $\mathcal{T}_{R;\neq 0}$ {#sec:Q3TRneq0}
Next consider $\mathcal{T}_{R;\neq 0}$. In fact, since $Q^3_0$ is the same order of magnitude as $Q^2_0$, and $A^3 \lesssim A^2$, this term can be treated in the same fashion as was done in §\[sec:Q2TRneq0\]. Hence, we omit the details for brevity.
#### Reaction term $\mathcal{T}_{R;\neq \neq}$ {#sec:Q3TRneqneq}
Turn next to $\mathcal{T}_{R;\neq \neq}$. This includes terms isolated in §\[sec:Toy\] as leading order contributions to the **(3DE)** nonlinear interactions (see [@BGM15I] and §\[sec:NonlinHeuristics\]) and these terms are one of the places where we will need the regularity imbalances in $A^3$. As in §\[sec:Q2TRneqneq\] above, we further decompose in terms of frequency: $$\begin{aligned}
\mathcal{T}_{R;\neq \neq} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{3} \hat{Q}^3_k(\eta,l) A^{3}_{k}(\eta,l) \hat{U}_{k^\prime}^1 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{3}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{3} \hat{Q}^3_k(\eta,l) A^{3}_{k}(\eta,l) \hat{U}_{k^\prime}^2 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{3} \hat{Q}^3_k(\eta,l) A^{3}_{k}(\eta,l) \hat{U}_{k^\prime}^3 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{3} \hat{Q}^3_k(\eta,l)\right\vert} A^{3}_{k}(\eta,l) \left({\left\vert\hat{\phi_z}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\hat{\psi_z}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\hat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert}\right) \\ & \quad\quad\quad \times Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}^{3}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{3} \hat{Q}^3_k(\eta,l) A^{3}_{k}(\eta,l)\left(\hat{\psi_y}(\xi,l^\prime)_{Hi}\right)\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \mathcal{T}_{R;\neq \neq;\mathcal{R}} \\
& = \mathcal{T}_{R;\neq\neq}^{1} + \mathcal{T}_{R;\neq\neq}^2 + \mathcal{T}_{R;\neq\neq}^3 + \mathcal{T}_{R;\neq\neq}^{C1} + \mathcal{T}_{R;\neq\neq}^{C2} + \mathcal{T}_{R;\neq\neq;\mathcal{R}}. \end{aligned}$$ Consider $\mathcal{T}^2_{R;\neq \neq}$, which is one of the terms in the toy model. In particular, we will use the regularity imbalance between $Q^2$ and $Q^3$ to reduce the power of $t$. By , $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{2} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^3}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k,k^\prime,k-k^\prime \neq 0} {\left\vert\hat{Q}^3_k(\eta,l)\right\vert} \frac{1}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2} \\
& \quad\quad \times \left(\sum_{r} \chi^{NR,r}\frac{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}{t} \tilde{A}^3_k(\eta,l) \tilde{A}^2_{k^\prime}(\xi,l^\prime) + \chi^{\ast;32} A^3_k(\eta,l) A^2_{k^\prime}(\xi,l^\prime) \right) \\
& \quad\quad \times {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1} {\left\vert \Delta_L \hat{U}_{k^\prime}^2 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi. \end{aligned}$$ Therefore, by followed by , $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{2} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^2\right) \Delta_L U^2_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert A^2 \Delta_L U^2_{\neq} \right\rVert}_2, \end{aligned}$$ which, by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\]. The term $\mathcal{T}^1_{R;\neq \neq}$ is treated in essentially the same way (matching the intuition that $Q^1 \sim t Q^2$ near the critical times) and is hence omitted.
Next, turn to the treatment of $\mathcal{T}^3_{R;\neq \neq}$. By we have $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{3} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vert\hat{Q}^3_k(\eta,l)\right\vert} \frac{1}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2} \\
& \quad\quad \times \left(\chi^{R,NR}\frac{t}{{\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}}\tilde{A}^3_k(\eta,l)\tilde{A}^3_{k^\prime}(\xi,l^\prime) + \chi^{NR,R}\frac{{\left\vertk^\prime\right\vert} + {\left\vert\eta - k^\prime t\right\vert}}{t} \tilde{A}^3_k(\eta,l)\tilde{A}^3_{k^\prime}(\xi,l^\prime) \right. \\ & \quad\quad + \chi^{\ast;33} A^3_k(\eta,l) A^3_{k^\prime}(\xi,l^\prime) \bigg) {\left\vertA^3 \Delta_L \hat{U}_{k^\prime}^3 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi, \end{aligned}$$ which by and is $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{3} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemma \[lem:SimplePEL\].
Finally, turn to $\mathcal{T}_{R;\neq\neq}^{C1}$ and $\mathcal{T}_{R;\neq\neq}^{C2}$. By Lemma \[lem:ABasic\] and (and Lemma \[lem:CoefCtrl\]), we have $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{C1}+ \mathcal{T}_{R;\neq\neq}^{C2} & \lesssim \frac{\epsilon^2 {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3 \right\rVert}_2 {\left\lVert AC \right\rVert}_2 \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}}{\left\lVert A^3 Q^3 \right\rVert}^2_2 + \frac{\epsilon^3 {\left\langle t \right\rangle}^3}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}} {\left\lVert AC \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\alpha$ sufficiently large, $\epsilon$ sufficiently small, and $\delta > 0$. This completes the treatment of $\mathcal{T}_{R;\neq\neq}$ and hence all of $\mathcal{T}$.
### Dissipation error terms $\mathcal{D}$ {#sec:DEneqQ3}
Due to the quadratic growth at low frequencies of $Q^3$ and the much larger size of $\epsilon$, these terms cannot be treated as they were in [@BGM15I]. However, we will adapt a treatment from [@BMV14] which treats the critical times with increased precision. Recalling the dissipation error terms and the short-hand , we have $$\begin{aligned}
\mathcal{D}_E & = \nu\sum_{k \neq 0}\int A^3 Q^3_k A^3_k\left(G_{yy}(\partial_{Y} - t\partial_X)^2 Q^3_k + G_{yz}(\partial_Y - t \partial_X)\partial_{Z}Q^3_k + G_{zz}\partial_{ZZ}Q^3_k \right) dV \\
& = \mathcal{D}_E^1 + \mathcal{D}_E^2 + \mathcal{D}_E^3. \end{aligned}$$ We will only treat $\mathcal{D}_E^1$; $\mathcal{D}_E^2$ and $\mathcal{D}_E^3$ are slightly easier and are hence omitted. As usual, we expand with a paraproduct: $$\begin{aligned}
\mathcal{D}_E^1 & = \nu\sum_{k \neq 0}\int A^3 Q^3_k A^3_k \left( (G_{yy})_{Hi} (\partial_{Y} - t\partial_X)^2 (Q^3_k)_{Lo} \right) dV + \nu\sum_{k \neq 0}\int A^3 Q^3_k A^3_k \left((G_{yy})_{Lo} (\partial_{Y} - t\partial_X)^2 (Q^3_k)_{Hi} \right) dV \\
& \quad + \nu\sum_{k \neq 0}\int A^3 Q^3_k A^3_k \left( \left(G_{yy}(\partial_{Y} - t\partial_X)^2 Q^3_k \right)_{\mathcal{R}} \right) dV \\
& = \mathcal{D}_{E;HL}^1 + \mathcal{D}_{E;LH}^1 + \mathcal{D}_{E;\mathcal{R}}^1. \end{aligned}$$ As in §\[sec:DEneqQ2\] and [@BGM15I], we can control the latter two terms by the dissipation; we omit the details for brevity. Next, turn to the treatment of $\mathcal{D}_{E;HL}^1$. By Lemma \[lem:ABasic\], there is some $c = c(s) \in (0,1)$ such that $$\begin{aligned}
\mathcal{D}_{E;HL}^{1} & \lesssim \nu \sum_{k \neq 0} \int {\left\vertA^3 \widehat{Q^3_k}(\eta,l) A^3_k(\eta,l) \widehat{G_{yy}}(\xi,l^\prime)_{Hi} (\eta-\xi - tk)^2 \widehat{Q^3_k}(\eta-\xi,l-l^\prime)_{Lo}\right\vert} d\eta d\xi \\
& \lesssim \nu \sum_{k \neq 0} \int\left[\chi_R + \chi_{NR;k} \right]{\left\vertA^3 \widehat{Q^3_k}(\eta,l) \frac{1}{{\left\langle \xi,l^\prime \right\rangle} {\left\langle t \right\rangle}} A \widehat{G_{yy}}(\xi,l^\prime)_{Hi} \right\vert} \\
& \quad\quad \times {\left\vert(\eta - \xi - tk)^2 e^{c\lambda{\left\vertk,\eta-\xi,l-l^\prime\right\vert}^s} \widehat{Q^3_k} (\eta-\xi,l-l^\prime)_{Lo} \right\vert} d\eta d\xi, \\
& = \mathcal{D}_{E;HL}^{1;R} + \mathcal{D}_{E;HL}^{1;NR}, \end{aligned}$$ where $\chi_{R;k} = \mathbf{1}_{t \in {\mathbf{I}}_{k,\eta}\cap {\mathbf{I}}_{k,\xi}} \mathbf{1}_{{\left\vertl\right\vert} \leq \frac{1}{5}{\left\vert\eta\right\vert}} \mathbf{1}_{{\left\vertl^\prime\right\vert} \leq \frac{1}{5}{\left\vert\xi\right\vert}}$ and $\chi_{NR;k} = 1-\chi_{R;k}$ is defined in . For the non-resonant term $\mathcal{D}_{E;HL}^{1;NR}$, since ${\left\langle t \right\rangle} \lesssim ({\left\vertk\right\vert} + {\left\vertl\right\vert} + {\left\vert\eta-kt\right\vert}){\left\langle \eta-\xi,l-l^\prime \right\rangle}$ and ${\left\vert\eta-\xi-kt\right\vert} \lesssim {\left\langle t \right\rangle}{\left\langle k,\eta-\xi \right\rangle}$ on the support of the integrand by , $$\begin{aligned}
\mathcal{D}_{E;HL}^{1;NR} & \lesssim \nu \sum_{k \neq 0} \int \chi_{NR;k} {\left\vert\sqrt{-\Delta_L} A^3 \widehat{Q^3_k}(\eta,l) \frac{1}{{\left\langle \xi,l^\prime \right\rangle} {\left\langle t \right\rangle}^2} A \widehat{G_{yy}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times {\left\vert(\eta - \xi - tk)^2 e^{c\lambda{\left\vertk,\eta-\xi,l-l^\prime\right\vert}^s} \widehat{Q^3_k} (\eta-\xi,l-l^\prime)_{Lo} \right\vert} d\eta d\xi, \\
& \lesssim \nu t {\left\lVert {\left\langle {\nabla}\right\rangle}^{-1}AG \right\rVert}_{2}{\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2 {\left\langle t \right\rangle}^{-2}{\left\lVert \sqrt{-\Delta_L} Q^3_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda}}. \end{aligned}$$ It follows by , Lemma \[lem:CoefCtrl\] and , we have $$\begin{aligned}
\mathcal{D}_{E;HL}^{1;NR} & \lesssim \frac{\nu t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert AC \right\rVert}_{2}{\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_2 \\
& \lesssim \frac{\nu \epsilon t^2 {\left\vert\log c_0\epsilon^{-1}\right\vert} }{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_2 \\
& \lesssim \epsilon^{\delta/4} \nu {\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2^2 + \epsilon^{\delta/4} \nu {\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_2^2,\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by the bootstrap hypotheses for $\epsilon$ sufficiently small and $\delta > 0$. For the resonant term $\mathcal{D}_{E;HL}^{1;R}$ we have by Lemma \[lem:dtw\] and (also using that $A(\xi,l^\prime) \approx \tilde{A}(\xi,l^\prime)$ on the support of the integrand due to the definition of $\chi^{R}$ and ${\left\vert\eta-\xi-kt\right\vert} \lesssim {\left\langle t \right\rangle} {\left\langle k,\eta-\xi \right\rangle}$), $$\begin{aligned}
\mathcal{D}_{E;HL}^{1;R} & \lesssim \nu \sum_{k \neq 0} \int \chi_{R;k} \left({\left\vertk\right\vert} + {\left\vert\eta-kt\right\vert}\right)^{1/2}{\left\vertA^3 \widehat{Q^3_k}(\eta,l) \frac{1}{{\left\langle \xi,l^\prime \right\rangle} {\left\langle t \right\rangle}} \sqrt{\frac{\partial_t w}{w}}\tilde{A}\widehat{G_{yy}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times {\left\langle t \right\rangle}^{3/2}{\left\langle k,\eta-\xi \right\rangle}^{5/2}{\left\vert(\eta -\xi - tk)^{1/2} e^{c\lambda{\left\vertk,\eta-\xi,l-l^\prime\right\vert}^s} \widehat{Q^3_k}(\eta-\xi,l-l^\prime)_{Lo}\right\vert} d\eta d\xi \\
& \lesssim \nu t^{1/2} {\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2^{1/2} {\left\lVert A^3 Q^3_{\neq} \right\rVert}_2^{1/2} {\left\lVert Q^3_{\neq} \right\rVert}_{{\mathcal{G}}^\lambda}^{1/2}{\left\lVert \sqrt{-\Delta_L} Q^3_{\neq} \right\rVert}_{{\mathcal{G}}^\lambda}^{1/2} {\left\lVert \sqrt{\frac{\partial_t w}{w}}{\left\langle {\nabla}\right\rangle}^{-1}\tilde{A}G_{yy} \right\rVert}_2.\end{aligned}$$ Then, by , , and , followed by Lemma \[lem:CoefCtrl\], for some small $\delta^\prime > 0$ $$\begin{aligned}
& \lesssim \frac{\nu t^{5/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha/2}} {\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2 {\left\lVert A^3 Q^3_{\neq} \right\rVert}_2^{1/2}{\left\lVert A^{\nu;3} Q^3 \right\rVert}_{2}^{1/2} {\left\lVert \sqrt{\frac{\partial_t w}{w}}{\left\langle {\nabla}\right\rangle}^{-1}\tilde{A}G_{yy} \right\rVert}_2 \\
& \lesssim \frac{\nu^{1/2} t \epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha/2-1}} {\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2 {\left\lVert \sqrt{\frac{\partial_t w}{w}}{\left\langle {\nabla}\right\rangle}^{-1}\tilde{A}G_{yy} \right\rVert}_2 \\
& \lesssim \epsilon^{\delta^\prime}\nu{\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2^2 + \frac{\epsilon^{2-\delta^\prime} {\left\langle t \right\rangle}^4}{{\left\langle \nu t^3 \right\rangle}^{\alpha-2}} \left(\frac{1}{{\left\langle t \right\rangle}^2}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right)C \right\rVert}_2^2 \right), \end{aligned}$$ which is now consistent with Proposition \[prop:Boot\] for $\delta^\prime$ and $\epsilon$ small. Note that the hypothesis $\epsilon \lesssim \nu^{2/3+\delta}$ with $\delta > 0$ is essentially sharp for controlling this term. This completes the treatment of $\mathcal{D}_{E}^{1}$ and hence of the dissipation error terms.
### Linear stretching term $LS3$ {#sec:LS30_Hi}
First separate into two parts (to be sub-divided further below), $$\begin{aligned}
LS3 & = -2\int A^{3} Q^3 A^{3}\partial_X(\partial_Y - t\partial_X) U^3 dV - 2\int A^{3} Q^3 A^{3} \partial_X \left(\psi_y(\partial_Y - t\partial_X) + \phi_y\partial_Z \right) U^3 dV \\
& = LS3^0 + LS3^{C}. \end{aligned}$$
#### Treatment of $LS3^C$ {#sec:LS3C}
Expand with a paraproduct, $$\begin{aligned}
LS3^{C} & = -2\int A^{3} Q^3 A^{3} \partial_X \left((\psi_y)_{Hi}(\partial_Y - t\partial_X) + (\phi_y)_{Hi}\partial_Z \right)\left(U^3\right)_{Lo} dV \\
& \quad - 2\int A^{3} Q^3 A^{3} \partial_X \left( (\psi_y)_{Lo}(\partial_Y - t\partial_X) + (\phi_y)_{Lo}\partial_Z \right) \left(U^3\right)_{Hi} dV \\
& \quad - 2\int A^{3} Q^3 A^{3} \partial_X \left( \left(\psi_y(\partial_Y - t\partial_X) + \phi_y\partial_Z \right) U^3\right)_{\mathcal{R}} dV \\
& = LS3^{C}_{HL} + LS3^{C}_{LH} + LS3^{C}_{\mathcal{R}}. \end{aligned}$$ The main issue is $LS3^{C}_{HL}$, where the coefficients appear in ‘high frequency’, so turn to this term first. By Lemma \[lem:ABasic\], , and Lemma \[lem:CoefCtrl\], $$\begin{aligned}
LS3^C_{HL} & \lesssim \frac{\epsilon{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k \neq 0}\int {\left\vertA^{3} \widehat{Q^3_k}(\eta,l)\right\vert}\frac{1}{{\left\langle \xi,l^\prime \right\rangle}{\left\langle t \right\rangle}}A\left( {\left\vert\widehat{\psi_y}(\xi,l^\prime)\right\vert} + {\left\vert\widehat{\phi_y}(\xi,l^\prime)\right\vert}\right) Low(k,\eta-\xi,l-l^\prime) d\xi \\
& \lesssim \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^3 Q^3 \right\rVert}^2 + \frac{\epsilon^{3/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert AC \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by for $\epsilon$ sufficiently small and $\delta > 0$ (hence $\epsilon \lesssim \nu^{2/3+\delta}$ is essentially sharp here).
Turn next to the $LS3^C_{LH}$, which is reminiscent of $NLP(1,3,0,\neq)$ in §\[sec:NLP213\]. Indeed, by Lemma \[lem:CoefCtrl\], , and we have, $$\begin{aligned}
LS3^C_{LH} & \lesssim \epsilon{\left\langle t \right\rangle} \sum \int {\left\vert A^3 \widehat{Q^3_k}(\eta,l) A^3_k(\eta,l) \frac{k{\left\vert\xi - kt,l^\prime\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \Delta_L \widehat{U^3_k}(\xi,l^\prime)\right\vert} Low(\eta-\xi,l-l^\prime) d\eta dx \nonumber \\
& \lesssim c_{0}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) \Delta_L U^3_{\neq} \right\rVert}_2 \nonumber \\ & \quad + \frac{\epsilon}{{\left\langle t \right\rangle}}{\left\lVert \sqrt{-\Delta_L}A^3 Q^3 \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2, \label{ineq:LS3CLH} \end{aligned}$$ which, after the application of Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $c_0$ sufficiently small.
The remainder $LS3^C_{\mathcal{R}}$ follows easily and is hence omitted.
#### Leading order term, $LS3^0$ {#sec:LS30}
As in [@BGM15I], the $2$ in the leading order term is crucially important and cannot be altered; it is the origin of the quadratic growth of $Q^3$ at low (relative to time) frequencies and any alteration would cause faster growth and a collapse of the bootstrap. For this reason we have to treat this term more precisely. Begin by isolating the leading order contribution: by the definition of $\Delta_t$ (see and the shorthand ), $$\begin{aligned}
LS3^0 & = -2\int A^{3} Q^3 A^{3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \Delta_L \Delta_t^{-1}Q^3 dV \nonumber \\
& = -2\int A^{3} Q^3 A^{3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \left( Q^3 - G_{yy}(\partial_Y - t\partial_X)^2\Delta_t^{-1}Q^3 \right. \nonumber \\
& \quad \left. \quad - G_{yz}\partial_Z (\partial_Y - t\partial_X) U^3 - G_{zz}\partial_{ZZ}U^3 - \Delta_t C^1 (\partial_Y - t\partial_X) U^3 - \Delta_t C^2 \partial_Z U^3 \right) dV \nonumber \\
& = LS3^{0;0} + \sum_{j = i}^5 LS3^{0;Ci}. \label{eq:LS30} $$
The treatment of $LS3^{0;0}$ is essentially the same as in [@BGM15I]. The only minor difference is that one must separate high frequencies in $Z$ from high frequencies of $Y$ when using $CK_w^3$. Due to the uniform ellipticity in $Z$, this does not make a major difference and this contribution can be absorbed by the existing terms. Divide into long-time and short-time regimes $$\begin{aligned}
LS3^{0;0} & = -2\int \left[\mathbf{1}_{t \leq 2{\left\vert\eta\right\vert}} + \mathbf{1}_{t > 2{\left\vert\eta\right\vert}} \right] {\left\vertA^{3} \widehat{Q^3_k}(\eta,l)\right\vert}^2 \frac{k(\eta-kt)}{k^2 + l^2 + {\left\vert\eta-kt\right\vert}^2} d\eta \\
& = LS3^{0;0,ST} + LS3^{0;0, LT}. \end{aligned}$$ The long-time regime is treated the same as in [@BGM15I] (see therein for a proof), and hence for some universal $K > 0$: $$\begin{aligned}
LS3^{0;0,LT} & \leq CK^3_{L} + \frac{\delta_\lambda}{10{\left\langle t \right\rangle}^{3/2}}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2} A^{3}Q^3 \right\rVert}_2^2 + \frac{K}{\delta_\lambda^{\frac{1}{2s-1}}{\left\langle t \right\rangle}^{3/2}}{\left\lVert A^{3}Q^3 \right\rVert}_2^2, \end{aligned}$$ which, for $\delta_\lambda$ sufficiently small and $K_{H3}$ sufficiently large, is consistent with Proposition \[prop:Boot\]. For the short-time regime we apply to deduce for some $K > 0$, $$\begin{aligned}
LS3^{0;0,ST} & \lesssim \kappa^{-1}{\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^{3} Q^3 \right\rVert}_2^2 + \frac{1}{{\left\langle t \right\rangle}^{3/2}}{\left\lVert {\left\vert{\nabla}\right\vert}^{1/4} A^3 Q^3 \right\rVert}_2^2 \\
& \leq K\kappa^{-1}{\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^{3} Q^3 \right\rVert}_2^2 + \frac{\delta_\lambda}{10{\left\langle t \right\rangle}^{3/2}} {\left\lVert {\left\vert{\nabla}\right\vert}^{s/2} A^3 Q^3 \right\rVert}_2^2 + \frac{K}{\delta_\lambda^{\frac{1}{2s-1}}{\left\langle t \right\rangle}^{3/2}} {\left\lVert A^3 Q^3 \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\kappa$ sufficiently large, $\delta_\lambda$ sufficiently small (so that the first term is absorbed by $CK_\lambda^3$) and $K_{H3}$ is sufficiently large.
Consider the first error term in , $LS3^{0;C1}$; here we will need a more refined treatment than in [@BGM15I]. Expanding $LS3^{0;C1}$ gives $$\begin{aligned}
LS3^{0;C1} & = -2\int A^{3} Q^3 A^{3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \left( (G_{yy})_{Hi} (\partial_Y - t\partial_X)^2 U^3_{Lo}\right) dV \\
& \quad -2\int A^{3} Q^3 A^{3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \left( (G_{yy})_{Lo} (\partial_Y - t\partial_X)^2 U^3_{Hi}\right) dV \\
& \quad -2\int A^{3} Q^3 A^{3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \left( (G_{yy}) (\partial_Y - t\partial_X)^2 U^3 \right)_{\mathcal{R}} dV \\
& = LS3^{0;C1}_{HL} + LS3^{0;C1}_{LH} + LS3^{0;C1}_{\mathcal{R}}. \end{aligned}$$ The most interesting contribution is the $HL$ term. By and Lemma \[lem:ABasic\], we have $$\begin{aligned}
LS3^{0;C1}_{HL} & \lesssim \frac{\epsilon{\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{l,l^\prime,k\neq 0}\int {\left\vertA^{3} \widehat{Q^3_k}(\eta,l) \frac{{\left\vert\eta-kt\right\vert}}{\left(k^2 + l^2 + {\left\vert\eta-kt\right\vert}^2\right) {\left\langle t \right\rangle} {\left\langle \xi,l^\prime \right\rangle} }A\widehat{G_{yy}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times Low(k,\eta-\xi,l-l^\prime) d\xi d\eta. \end{aligned}$$ Therefore, by followed by , and Lemma \[lem:CoefCtrl\], we have $$\begin{aligned}
LS3^{0;C1}_{HL} & \lesssim \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^\alpha } {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^{3} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^{3}\right) Q^3 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right) {\left\langle {\nabla}\right\rangle}^{-1} G_{yy} \right\rVert}_2 \\
& \quad + \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}} {\left\lVert A^{3}Q^3 \right\rVert}_2 {\left\lVert {\left\langle {\nabla}\right\rangle}^{-1}AG_{yy} \right\rVert}_2 \\
& \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3\right) Q^3 \right\rVert}_2^2 + \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right) C \right\rVert}_2^2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}}\left({\left\lVert A^{3}Q^3 \right\rVert}^2_2 + \left(\frac{1}{{\left\langle t \right\rangle}}{\left\lVert AC \right\rVert}_2\right)^2 \right). \end{aligned}$$ This is consistent with Proposition \[prop:Boot\] by for $\delta > 0$ and $\epsilon$ sufficiently small.
Turn to $LS3^{0;C1}_{LH}$, which by Lemma \[lem:CoefCtrl\], $$\begin{aligned}
LS3^{0;C1}_{LH} \lesssim \epsilon {\left\langle t \right\rangle} \sum_{k,l}\int {\left\vertA^{3} \widehat{Q^3_k}(\eta,l) A^{3}_k(\eta,l) \frac{{\left\vertk\right\vert}{\left\vert\eta-kt\right\vert}}{k^2 + l^2 + {\left\vert\eta-kt\right\vert}^2} \left(\Delta_L U^3_k\right)_{Hi}(\xi,l^\prime)\right\vert} Low(\eta-\xi,l-l^\prime) d \eta d\xi. \end{aligned}$$ We can treat this term roughly like $NLP(1,3,0,\neq)$ on $Q^2$ in §\[sec:NLP213\]: by and , $$\begin{aligned}
LS3^{0;C1}_{LH} & \lesssim c_{0}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}} \tilde{A}^{3} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{3}\right) Q^3 \right\rVert}_2^2 + c_{0}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}} \tilde{A}^{3} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{3}\right) \Delta_L U^3_{\neq} \right\rVert}_2^2 \\
& \quad + \epsilon{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2{\left\lVert \Delta_L A^3 U^3_{\neq} \right\rVert}_2.\end{aligned}$$ By Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], this is consistent with Proposition \[prop:Boot\] by the bootstrap hypotheses.
The remainder $LS3^{0;C1}_{\mathcal{R}}$ is straightforward and is omitted for the sake of brevity. This completes the first error term in , $LS3^{0;C1}$.
The second and third error terms, $LS3^{0;C2}$ and $LS3^{0;C3}$, are similar to $LS3^{0;C1}$ but slightly easier, and yield similar contributions. Hence, we omit the treatment for brevity.
The last two coefficient errors, $LS3^{0;C4}$ and $LS3^{0;C5}$, are also similar but require a slight adjustment. In particular, due to the two derivatives on the coefficients, we cannot gain any powers of time from $A^3$ as in the treatment of $LS3^{0;C1}$ above. However, this is balanced by the fact that there is one less power of $\partial_Y - t\partial_X$. Hence, the above treatment adapts in a straightforward manner and so we omit the details for brevity.
This concludes the treatment of the linear stretching term $LS3$.
### Linear pressure term $LP3$ {#ineq:LP3_Hi}
As in $LS3$, we first separate the coefficient corrections and expand with a paraproduct: $$\begin{aligned}
LP3 & = 2 \int A^{3} Q^3 A^{3} \partial_Z \partial_X U^2 dV + 2 \int A^{3} Q^3 A^{3} \left((\psi_{z})_{Lo}(\partial_Y - t\partial_X) + (\phi_z)_{Lo}\partial_Z \right) \left(\partial_X U^2\right)_{Hi} dV \\
& \quad + 2\int A^{3} Q^3 A^{3} \left((\psi_{z})_{Hi}(\partial_Y - t\partial_X) + (\phi_z)_{Hi}\partial_Z \right)\left( \partial_X U^2\right)_{Lo} dV \\
& \quad + 2\int A^{3} Q^3 A^{3} \left( \left(\psi_{z}(\partial_Y - t\partial_X) + \phi_z\partial_Z \right) \partial_X U^2\right)_{\mathcal{R}} dV \\
& = LP3^{0} + LP3^{C}_{LH} + LP3^{C}_{HL} + LP3^{C}_{\mathcal{R}}. \end{aligned}$$
#### Treatment of $LP3^{0}$
As in [@BGM15I], from , $$\begin{aligned}
LP3^{0} \leq \frac{1}{2\kappa}{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{3} Q^3 \right\rVert}_2^2 + \frac{1}{2\kappa}{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}}A^{3} \Delta_L U^2_{\neq} \right\rVert}^2_2. \end{aligned}$$ The first term is absorbed by the $CK_{wL}^3$ term in . For the latter term we apply Lemma \[lem:QPELpressureI\], which yields contributions which are integrable or are absorbed by the $CK$ terms.
#### Treatment of $LP3^C$
Turn first to $LP3^{C}_{HL}$, in which the coefficient is in ‘high frequency’. By , Lemma \[lem:ABasic\], , and Lemma \[lem:CoefCtrl\], we have $$\begin{aligned}
LP3^{C}_{HL} &\lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k,l}\int {\left\vertA^{3} \widehat{Q^3_k}(\eta,l)\right\vert} \frac{1}{{\left\langle \xi,l^\prime \right\rangle}{\left\langle t \right\rangle}} A\left({\left\vert\widehat{\psi_z}(\xi,l)_{Hi}\right\vert} + {\left\langle t \right\rangle}^{-1}{\left\vert\widehat{\phi_z}(\xi,l)_{Hi}\right\vert} \right) Low(k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^{3}Q^3 \right\rVert}_2 {\left\lVert A C \right\rVert}_2,\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
Next turn to $LP3^{C}_{LH}$, which by Lemma \[lem:CoefCtrl\] and , is controlled via $$\begin{aligned}
LP3^{C}_{LH} & \lesssim \epsilon {\left\langle t \right\rangle} \sum_{k \neq 0,l}\int_\eta {\left\vertA^{3} \widehat{Q^3_k}(\eta,l)\right\vert} \frac{{\left\vertk\right\vert}{\left\vert\xi-kt,l^\prime\right\vert}}{\left(k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2\right) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}} {\left\vertA^{2}\widehat{\left(\Delta_{L} U^2 \right)}_k(\xi,l^\prime)\right\vert} Low(\eta-\xi,l-l^\prime) d\eta. \end{aligned}$$ We may treat this in a manner similar to the canonical $NLP(1,3,0,\neq)$ on $Q^2$ in §\[sec:NLP213\]. Indeed, by and we have, $$\begin{aligned}
LP3^{C}_{HL} & \lesssim c_{0}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}} \tilde{A}^{3} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{3}\right) Q^3 \right\rVert}_2^2 + c_{0}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}} \tilde{A}^{2}+ \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{2}\right) \Delta_L U^2_{\neq} \right\rVert}_2^2 \\
& \quad + \epsilon{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2{\left\lVert \Delta_L A^2 U^2_{\neq} \right\rVert}_2, \end{aligned}$$ which by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\] by the bootstrap hypotheses. The remainder term $LP3^{\mathcal{R}}$ is straightforward and is omitted for the sake of brevity; this completes the treatment of $LP3$.
High norm estimate on $Q^1_0$ {#sec:Q1Hi1}
=============================
As in [@BGM15I], the improvement of proceeds slightly differently than most other estimates we are making. The goal is to obtain exactly $O(\epsilon {\left\langle t \right\rangle})$ growth, rather than any logarithmic losses in $t$ or $\epsilon$. We will deduce an estimate like $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A^{1}Q^1_0 \right\rVert}^2_2 & \leq -\frac{t}{{\left\langle t \right\rangle}^2}{\left\lVert A^1 Q^1_0 \right\rVert}_2^2 + \frac{1}{{\left\langle t \right\rangle}}{\left\lVert A^{1}Q^1_0 \right\rVert}_2{\left\lVert A^2 Q^2_0 \right\rVert}_2 + c_{0}\epsilon^2\mathcal{I}(t) \nonumber \\
& \leq -\frac{t}{{\left\langle t \right\rangle}^2}{\left\lVert A^1Q^1_0 \right\rVert}_2^2 + \frac{4\epsilon}{{\left\langle t \right\rangle}}{\left\lVert A^{1}Q^1_0 \right\rVert}_2 + c_{0}\epsilon^2\mathcal{I}(t), \label{ineq:basic_Q1}\end{aligned}$$ where $\int_1^{c_0 \epsilon^{-1}} \mathcal{I}(t) dt = O(K_B)$ uniformly in $\epsilon$. This yields the desired bound by comparing $X(t) = {\left\lVert A^1 Q^1_0(t) \right\rVert}_2^2$ to the super-solution of the inequality given by $Y(t) = \max(\frac{3}{2}K_{H10}, 6\sqrt{2})\epsilon + c_{0}\epsilon^2 \int_{1}^t\mathcal{I}(\tau) d\tau$ and choosing $c_0$ sufficiently small. Indeed (for $K_{H10}$ sufficiently large), $$\begin{aligned}
\partial_t Y(t) = c_{0}\epsilon^2\mathcal{I}(t) \geq \left(-\frac{t}{{\left\langle t \right\rangle}^2}Y(t) + \frac{4\epsilon}{{\left\langle t \right\rangle}}\right) Y(t) + c_{0}\epsilon^2\mathcal{I}(t), \end{aligned}$$ as the additional two terms on the RHS sum to something negative by the choice of $Y(t)$ (recall $t \geq 1$). By Lemma \[lem:BootStart\], $X(1) < Y(1)$, and therefore by comparison and , $X(t) \leq Y(t)$ for all $t \in [1,T_\star)$.
Therefore, improving reduces to proving an estimate like . From the evolution equation for $Q^1_0$, using enumerations analogous to and above, $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A^{1} Q^1_0 \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{1} Q_0^1 \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^{1} Q_0^1 \right\rVert}_2^2 - \frac{t}{{\left\langle t \right\rangle}^2}{\left\lVert A^1 Q_0^1 \right\rVert}_2^2 \nonumber \\
& \quad - \int A^{1}Q^1_0 A^1 Q^2_0 dV + \nu \int A^{1} Q^{1}_0 A^{1} \left(\tilde{\Delta_t} Q^1_0\right) dV
- \int A^{1} Q^1_0 A^{1}\left(\tilde U_0 \cdot {\nabla}Q^1_0\right) dV \nonumber \\
& \quad - \int A^{1} Q^1_0 A^{1} \left(Q^j_0 \partial_j^t U^1_0 + 2\partial_i^t U^j_0 \partial_{ij}^t U^1_0\right) dV \nonumber \\
& \quad - \int A^{1} Q^1_0 A^{1} \left(Q^j_{\neq} \partial_j^t U^1_{\neq} + 2\partial_i^t U^j_{\neq} \partial_{ij}^t U^1_{\neq}\right)_0 dV \nonumber \\
& = -\mathcal{D}Q_0^1 + CK_L^1 + LU + \mathcal{D}_E + \mathcal{T}_0 + NLS1(j,0) + NLS2(i,j,0) + \mathcal{F}, \label{eq:AevoQ10}\end{aligned}$$ where we are denoting $$\begin{aligned}
\mathcal{D}_E = \nu \int A^{1} Q^{1}_0 A^{1} \left((\tilde{\Delta_t} - \Delta_L) Q^1_0\right) dV. \end{aligned}$$ As above in and , we have decomposed the nonlinear terms based on the heuristics in §\[sec:NonlinHeuristics\].
Notice that, due to the $X$ average, the linear pressure and stretching terms both disappear along with the nonlinear pressure. Hence the main growth of $Q^1_0$ is caused by the lift-up effect term, $LU$. This term is treated by Cauchy-Schwarz: $$\begin{aligned}
LU \leq {\left\langle t \right\rangle}^{-1} {\left\lVert A^{1} Q^1_0 \right\rVert}_2 {\left\lVert A^{2} Q^2_0 \right\rVert}_2, \end{aligned}$$ which, together with is responsible for the leading order linear term in . It remains to see how to control the nonlinear terms.
Transport nonlinearity {#transport-nonlinearity-1}
----------------------
By Lemma \[lem:AAiProd\] (with and ), $$\begin{aligned}
\mathcal{T}_{0} & \lesssim {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}{\left\lVert A^1 Q_0^1 \right\rVert}_2{\left\lVert {\nabla}A^1 Q_0^1 \right\rVert}_2 + {\left\lVert Ag \right\rVert}_2 {\left\lVert A^1 Q_0^1 \right\rVert}_2^2 \lesssim \epsilon^{3/2}{\left\lVert {\nabla}A^1 Q_0^1 \right\rVert}^2_2 + \left(\frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{4}} + \epsilon\right){\left\lVert A^1 Q_0^1 \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ and $\epsilon$ sufficiently small.
Nonlinear stretching
--------------------
This term is the analogue of those treated in §\[sec:NLPSQ20\] and corresponds to the nonlinear stretching effects on $Q_0^1$ involving only zero frequencies (the pressure disappears due to the $X$ average). The treatment of this term can be made in the same way as the corresponding treatment for $Q^2$ in §\[sec:TransQ20\] and §\[sec:NLPSQ20\], although it is slightly easier here as we are permitting growth on $Q_0^1$, unlike $Q_0^2$ (in particular $A_0^1 \approx {\left\langle t \right\rangle}^{-1}A^2_0$). Hence, these contributions are omitted for brevity.
Forcing from non-zero frequencies {#forcing-from-non-zero-frequencies}
---------------------------------
In this section we consider interactions of type **(F)** (see §\[sec:NonlinHeuristics\]): the forcing of non-zero frequencies directly back onto $Q_0^1$. Recall from , $$\begin{aligned}
\mathcal{F} & = -\int A^1 Q^1 A^1 \left(\partial_Y^t \partial_Y^t \partial_Y^t \left(U^2_{\neq} U^1_{\neq}\right)_{0} + \partial_Y^t \partial_Y^t \partial_Z^t \left(U^3_{\neq} U^1_{\neq}\right)_{0} \right) dV \\
& \quad - \int A^1 Q^1 A^1\left(\partial_Z^t \partial_Z^t \partial_Z^t \left(U^3_{\neq} U^1_{\neq}\right)_{0} + \partial_Z^t \partial_Z^t \partial_Y^t \left(U^2_{\neq} U^1_{\neq}\right)_{0}\right) dV \\
& = F^1 + F^2 + F^3 + F^4. \end{aligned}$$ Let us begin with $F^2$ (corresponding to $i = 2$ and $j = 3$); the treatment is also essentially the same as $F^3$. Note the terms involving $U^3$ are expected to be the worst due to the regularity imbalances. Decompose the $F^2$ with a paraproduct; as usual we group contributions where the coefficients appear in low frequency with the remainder: $$\begin{aligned}
F^2 & = -\sum_{k\neq 0} \int A^{1}Q^1_0 A_0^{1} \partial_Y\partial_Y\partial_Z\left( \left(U^3_{-k}\right)_{Hi} \left( U^1_k\right)_{Lo}\right)dV \nonumber \\
& \quad - \sum_{k\neq 0} \int A^{1}Q^1_0 A_0^{1} \partial_Y \partial_Y\partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^1_k\right)_{Hi}\right) dV \nonumber \\
& \quad -\sum_{k\neq 0} \int A^{1}Q^1_0 A_0^{1} \left(\left( (\psi_y)_{Hi} \partial_Y + (\phi_{y})_{Hi} \partial_Z \right)\partial_Y\partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^1_k\right)_{Lo}\right)\right) dV \nonumber \\
& \quad - \sum_{k\neq 0} \int A^{1}Q^1_0 A_0^{1} \partial_Y\left( \left( (\psi_y)_{Hi} \partial_Y + (\phi_{y})_{Hi} \partial_Z \right) \partial_Z\left( \left(U^3_{-k}\right)_{Lo} \left( U^1_k\right)_{Lo}\right)\right) dV \nonumber \\
& \quad - \sum_{k\neq 0} \int A^{1}Q^1_0 A_0^{1} \partial_Y \partial_Y \left( \left( (\psi_z)_{Hi} \partial_Y + (\phi_{z})_{Hi} \partial_Z \right)\left( \left(U^3_{-k}\right)_{Lo} \left( U^1_k\right)_{Lo}\right)\right) dV \nonumber \\
& \quad + F^2_{\mathcal{R},C} \nonumber \\
& = F^2_{HL} + F^2_{LH} + F^2_{C1} + F^2_{C2} + F^2_{C3} + F^2_{\mathcal{R},C}. \label{def:F2ppQ10}\end{aligned}$$ Turn first to $F^{2}_{HL}$. By , Lemma \[lem:ABasic\], , , and , $$\begin{aligned}
F_{HL}^{2} & \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\langle t \right\rangle}^{\delta_1}\sum_{l,l^\prime,k \neq 0} \int {\left\vertA^{1} \widehat{Q^1_0}(\eta,l) A^1_{0}(\eta,l) \frac{{\left\vert\eta\right\vert}^2{\left\vertl\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \hspace{5cm} \times Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\langle t \right\rangle}^{\delta_1-1}\sum_{l,l^\prime,k \neq 0} \int {\left\vert\widehat{Q^1_0}(\eta,l)\right\vert} \frac{{\left\vert\eta\right\vert}^2{\left\vertl\right\vert} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^2}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \\
& \quad\quad \times \left(\sum_{r} \chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}^1_0(\eta,l)\tilde{A}^3_k(\xi,l^\prime) + \chi^{\ast;23} A^1_0(\eta,l) A^3_k(\xi,l^\prime) \right) \\
& \quad\quad \times {\left\vert \Delta_L A^{3}\widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert}Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{1+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1\right) Q^1_0 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^3 \right) \Delta_L U^3_{\neq} \right\rVert}_2
\\ & \quad + \frac{\epsilon}{t^{1-\delta_1}{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \sqrt{-\Delta_L} A^{1}Q^1 \right\rVert}_2 {\left\lVert A^{3}\Delta_L U^3_{\neq} \right\rVert}_2,\end{aligned}$$ which, after the application of Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with for $\epsilon$ sufficiently small.
Turn next to $F_{LH}^{2}$, which also by , and we have $$\begin{aligned}
F_{LH}^{2} & = \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{l,l^\prime,k\neq 0} \int {\left\vertA^{1} \widehat{Q^1_0}(\eta,l) \frac{{\left\vert\eta\right\vert}^2{\left\vertl\right\vert} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1+\delta_1}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \Delta_L A^{1}\widehat{U^1_{k}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \hspace{5cm} \times Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\ s
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right) A^{1}Q^1_0 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^1 \right) \Delta_L U^1_{\neq} \right\rVert}_2
\\ & \quad + \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^{1}Q^1 \right\rVert}_2 {\left\lVert A^{1}\Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which, after the application of the Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with .
The most difficult coefficient error term in is $F^2_{C3}$. By Lemma \[lem:ABasic\], , and Lemma \[lem:CoefCtrl\], $$\begin{aligned}
F^2_{C3} & \lesssim \epsilon^2 {\left\langle t \right\rangle}^{\delta_1} {\left\langle \nu t^3 \right\rangle}^{-2\alpha} \sum_{l,l^\prime,k\neq 0} \int {\left\vertA^{1} \widehat{Q^1_0}(\eta,l)\right\vert} A_0^{1}(\eta,l) {\left\vert\eta\right\vert}^2 \\ & \quad\quad \times\left( {\left\vert\widehat{\psi_z}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\widehat{\phi_z}(\xi,l^\prime)_{Hi}\right\vert}\right) Low(k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon^{3/2} {\left\lVert {\nabla}A^{1}Q^1_0 \right\rVert}_2^2 + \frac{\epsilon^{5/2} {\left\langle t \right\rangle}^{2\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{4\alpha}}{\left\lVert AC \right\rVert}^2_2,\end{aligned}$$ which is consistent with for $c_{0}$ and $\epsilon$ sufficiently small by . The other coefficient terms in , $F_{C1}^2$ and $F_{C2}^2$ are easier and give similar contributions. Hence, these are omitted for the sake of brevity. The remainder term in , $F_{\mathcal{R}}^2$, is similarly straightforward and is omitted as well. This completes the treatment of $F^2$. Despite appearing rather different, in fact the treatment of $F^3$ is essentially the same. Indeed, the regularity imbalances are restricted to where ${\left\vert\partial_Z\right\vert} \lesssim {\left\vert\partial_Y\right\vert}$ and hence, for frequencies where the regularity imbalances are occurring, $F^3$ looks roughly like $F^2$ and the same treatment applies. Outside of the regularity imbalances, one simply uses that $\Delta_L$ is uniformly elliptic in $Z$ in the same way non-resonance is used above in the treatment of $F^2$ (see §\[sec:basicmult\] for more details). As the details are exactly the same as above, we omit them for brevity.
The other $\mathcal{F}$ terms, $F^1$ and $F^4$, are treated as in [@BGM15I]; $F^1$ is slightly harder. The main idea is similar to the treatment of $F^2$ above, however one instead uses for $F^1$ (and for $F^4$) and hence deduce $$\begin{aligned}
F^1 + F^4 & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^{1} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}} A^{1} \right)Q^1_0 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^2 \right) \Delta_L U^2_{\neq} \right\rVert}_2
\\ & \quad + \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^{1}Q^1_0 \right\rVert}_2 {\left\lVert A^{2}\Delta_L U^2_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1 \right)Q^1_0 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^1 \right) \Delta_L U^1_{\neq} \right\rVert}_2
\\ & \quad + \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^{1}Q^1 \right\rVert}_2 {\left\lVert A^{1}\Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which, after applying Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with \[ineq:basic\_Q1\] under the bootstrap hypotheses for $c_{0}$ and $\epsilon$ chosen sufficiently small. This completes all of the forcing terms.
Dissipation error terms {#dissipation-error-terms}
-----------------------
As in [@BGM15I], these can be treated in the same manner as the dissipation error terms on $Q^2_0$ were treated in §\[sec:DEQ02\]. We omit the details for brevity: $$\begin{aligned}
\mathcal{D}_E & \lesssim c_0\nu {\left\lVert \sqrt{-\Delta_L}A^1 Q^1_0 \right\rVert}_2^2 + \nu\epsilon^2 c_0^{-1} {\left\lVert {\nabla}A C \right\rVert}_{2}^2, \end{aligned}$$ which for $c_{0}$ sufficiently small, is consistent with Proposition \[prop:Boot\]. This completes the high norm estimate on $Q_0^1$.
High norm estimate on $Q^1_{\neq}$
==================================
Consider from the evolution equation for $Q^1_{\neq}$: $$\begin{aligned}
\frac{1}{2}\frac{d}{dt} {\left\lVert A^{1} Q^1_{\neq} \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{1} Q^1_{\neq} \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^{1} Q^1_{\neq} \right\rVert}_2^2 \nonumber \\
& \quad - {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{1} Q^1_{\neq} \right\rVert}_2^2
- \frac{t}{{\left\langle t \right\rangle}^{2}}{\left\lVert A^{1}Q^1_{\neq} \right\rVert}_2^2 -\frac{(1+\delta_1)}{t} {\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{1} Q^1_{\neq} \right\rVert}_2^2 \nonumber \\
& \quad - \int A^{1}Q^1_{\neq} A^1 Q^2_{\neq} dV -2 \int A^{1} Q^1 A^{1} \partial_{YX}^t U^1_{\neq} dV + 2 \int A^{1} Q^1_{\neq} A^{1} \partial_{XX} U^2_{\neq} dV \nonumber \\
& \quad + \nu \int A^{1} Q^{1}_{\neq} A^{1} \left(\tilde{\Delta_t} Q^1_{\neq}\right) dV - \int A^{1} Q^1_{\neq} A^{1}\left( \tilde U \cdot {\nabla}Q^1 \right) dV \nonumber \\
& \quad - \int A^{1} Q^1_{\neq} A^{1} \left[ Q^j \partial_j^t U^1 + 2\partial_i^t U^j \partial_{ij}^t U^1 - \partial_X\left(\partial_i^t U^j \partial_j^t U^i\right) \right] dV \nonumber \\
& = -\mathcal{D}Q^1_{\neq} - CK_{L1}^1 - (1+\delta_1)CK_{L2}^1 + LU + LS1 + LP1 \nonumber \\ & \quad + \mathcal{D}_E + \mathcal{T} + NLS1 + NLS2 + NLP, \label{ineq:Q1HneqEvo}\end{aligned}$$ where as usual $$\begin{aligned}
\mathcal{D}_E = \int A^1 Q^1_{\neq} A^1 \left((\tilde{\Delta_t} - \Delta_L)Q^1_{\neq}\right) dV. \end{aligned}$$ We define enumerations of the nonlinear terms analogous to those in and .
Linear stretching term $LS1$ {#sec:LS1_Hi}
----------------------------
As discussed in [@BGM15I], one of the difficulties in deducing the high norm estimate on $Q^1_{\neq}$ is the linear stretching term $LS1$. First separate into two parts (to be sub-divided further), $$\begin{aligned}
LS1 & = -2\int A^{1} Q^1 A^{1}\partial_X(\partial_Y - t\partial_X) U^1 dV - 2\int A^{1} Q^1 A^{1} \partial_X\left( (\psi_y)(\partial_Y - t\partial_X) + (\phi_{y})\partial_Z\right) U^1 dV \\
& = LS1^0 + LS1^{C}. \end{aligned}$$
### Treatment of $LS1^C$
The $LS1^C$ term can be treated in essentially the same manner as the corresponding $LS3^C$ in §\[sec:LS3C\]. Hence, we omit the details for brevity.
### Leading order term $LS1^0$
As in of §\[sec:LS30\], we first expand by writing out $\Delta_t^{-1}$ in terms of $\Delta_L$: $$\begin{aligned}
LS1^0 & = -2 \int A^{1} Q^1 A^{1}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \left[Q^1 - G_{yy} (\partial_Y - t\partial_X)^2 U^1 - G_{yz}\partial_Z (\partial_Y - t\partial_X) U^1 \right. \\ & \quad\quad\quad \left. - G_{zz}\partial_{ZZ} U^1 -\Delta_t C^1 (\partial_Y - t\partial_X) U^1 -\Delta_t C^2 \partial_Z U^1 \right] dV \\
& = LS1^{0;0} + \sum_{i = 1}^5 LS1^{0;Ci}. \end{aligned}$$ The leading order term is treated as in [@BGM15I] (with the slight variation for large $Z$ frequencies as used in §\[sec:LS30\] above), and hence we omit the treatment and conclude the following for some $K > 0$, $$\begin{aligned}
LS1^{0,0} & \leq (1-\delta_1)CK_{L1}^1 + (1+\delta_1)CK^1_{L2} + \frac{\delta_\lambda}{10{\left\langle t \right\rangle}^{3/2}} {\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{1}Q^1_{\neq} \right\rVert}_2^2 + \frac{K}{\kappa}{\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}^{1} Q^1_{\neq} \right\rVert}_2^2 \\ & \quad + \frac{K}{\delta_\lambda^{\frac{1}{2s-1}} {\left\langle t \right\rangle}^{3/2}} {\left\lVert A^{1}Q^1_{\neq} \right\rVert}_2^2 + K\frac{1+\delta_1}{{\left\langle t \right\rangle}^2 t} {\left\lVert A^1 Q^1_{\neq} \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] under the bootstrap hypotheses for $K_{H1\neq}$ sufficiently large relative to $\exp(K\delta_\lambda^{-\frac{1}{2s-1}})$ (also, $\kappa$ must be chosen sufficiently large, but relative only to a universal constant).
The error terms $LS1^{0;Ci}$ are treated in a manner similar to the analogous terms in $LS3$ in §\[sec:LS30\] and hence the details are omitted for brevity (indeed $A^1_k$ is a weaker norm than $A^3_k$ due to the extra ${\left\langle t \right\rangle}^{-1}$ decay). This completes the treatment of the $LS1$ term.
Lift-up effect term $LU$ {#sec:LUQhi2}
------------------------
This follows as in [@BGM15I], and hence we omit the details: $$\begin{aligned}
LU & \leq \delta_1t{\left\langle t \right\rangle}^{-2}{\left\lVert A^{1} Q^1_{\neq} \right\rVert}^2_2 + \frac{\delta_\lambda}{4\delta_1 t^{3/2}}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^2 Q^2_{\neq} \right\rVert}_2^2 + \frac{1}{4\delta_\lambda^{\frac{1}{2s-1}} t^{3/2}}{\left\lVert A^2 Q^2_{\neq} \right\rVert}_2^2 + \frac{1}{4\delta_1 t} {\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}}A^{2}Q^2_{\neq} \right\rVert}_2^2. \end{aligned}$$ The first term is absorbed by the remaining piece of $CK_{L1}^1$ left over in from the treatment of $LS1$. The others are consistent with Proposition \[prop:Boot\] via for $K_{H1\neq}$ large relative to $\delta_1^{-1}$ and $\delta_\lambda^{-1}$. Hence, this suffices to treat $LU$.
Linear pressure term $LP1$
--------------------------
The linear pressure term $LP3$ treated in §\[ineq:LP3\_Hi\] is significantly harder than $LP1$ here, as here only $X$ derivatives are involved. Therefore, from Lemma \[dtw\], we get (the implicit constant is independent of $\kappa$), $$\begin{aligned}
LP1 & \leq 2 \sum\int {\left\vertA^1\widehat{Q^1_k}(\eta,l) \frac{{\left\vertk\right\vert}^2}{k^2 + l^2 + {\left\vert\eta-kt\right\vert}^2} A^1\Delta_L \widehat{U^2_k}(\eta,l)\right\vert} d\eta \\
& \lesssim \kappa^{-1} {\left\langle t \right\rangle}^{-1} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^1\right) Q^1_{\neq} \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^2\right)\Delta_L U_{\neq}^2 \right\rVert}_2 \\ & \quad + {\left\langle t \right\rangle}^{-3}{\left\lVert A^1Q^1 \right\rVert}_2{\left\lVert A^2\Delta_L U_{\neq}^2 \right\rVert}_2. \end{aligned}$$ Therefore for $\kappa$ and $K_{H1\neq}$ sufficiently large and $c_{0}$ sufficiently small, this is consistent with Proposition \[prop:Boot\] by the bootstrap hypotheses after applying Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\].
Nonlinear pressure $NLP$ {#nonlinear-pressure-nlp}
------------------------
After cancellations, none of the existing terms here are worse than those appearing in $Q^2$ in §\[sec:NLPQ2\] or $Q^3$ in §\[sec:NLP3\]. Moreover, on $Q^1$ we are imposing less control (since $A^1$ is weaker than $A^{2,3}$ at high frequencies due to ${\left\langle t \right\rangle}^{-1}$) and the leading derivative is an $X$ derivative, which is generally less dangerous than those associated with $Y$ and $Z$. Therefore, the treatment of the $NLP$ contributions here are an easy variant of the treatments in §\[sec:NLPQ2\] and §\[sec:NLP3\]. Accordingly, the details are omitted for the sake of brevity.
Nonlinear stretching $NLS$ {#nonlinear-stretching-nls}
--------------------------
These terms can be slightly more dangerous than the corresponding $NLS$ terms in $Q^{2,3}$ due to the persistent presence of $U^1$, however, this will be naturally balanced by the allowed linear growth of $Q^1$ at high frequencies.
### Treatment of $NLS1$
Consider first the $NLS1(j,\neq,0)$ terms. Note $j \neq 1$ due to the zero frequencies (a crucial nonlinear structure). The case $j = 3$ is worse than $j=2$ due to the large growth and regularity imbalances in $Q^3$. Hence, let us just focus on the case $j = 3$. As usual, with a paraproduct and group any terms with coefficients in low frequencies in with the remainder: $$\begin{aligned}
NLS1(3,\neq,0) & = -\sum \int A^1 Q^1_k A^1\left( (Q^3_{k})_{Hi} (\partial_Z U_0^1)_{Lo} \right) dV - \sum \int A^1 Q^1_k A^1\left( (Q^3_{k})_{Lo} (\partial_Z U_0^1)_{Hi} \right) dV \\
& \quad - \sum \int A^1 Q^1_k A^1\left( (Q^3_{k})_{Lo} \left((\psi_z)_{Hi} \partial_Y + (\phi_z)_{Hi} \partial_Z \right) (U_0^1)_{Lo} \right) dV + S_{\mathcal{R},C} \\
& = S_{HL} + S_{LH} + S_{C} + S_{\mathcal{R},C}, \end{aligned}$$ where $S_{\mathcal{R},C}$ includes the remainders from the paraproduct and the low frequency coefficient terms. By , Lemma \[lem:ABasic\], followed by , , and , $$\begin{aligned}
S_{HL} & \lesssim \epsilon {\left\langle t \right\rangle} \sum_{k} \int {\left\vert\widehat{Q^1_k}(\eta,l) {\left\langle t \right\rangle}^{-1} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1-\delta_1}\right\vert} \\
& \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}^1_k(\eta,l) \tilde{A}^3_k(\xi,l^\prime) + \chi^{\ast;23} A^1_k(\eta,l) A^3_k(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert\widehat{Q^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \epsilon {\left\langle t \right\rangle}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}} A^1 \right) Q^1 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) Q^3 \right\rVert}_2 \\
& \quad + \epsilon^{3/2} {\left\lVert \sqrt{-\Delta_L} A^1 Q^1_{\neq} \right\rVert}_2 + \mathbf{1}_{t \leq \epsilon^{-1/2+\delta/100}}\epsilon^{1/2} {\left\lVert A^3 Q^3 \right\rVert}_2^2 + \epsilon^{3/2-\delta/50}{\left\lVert \sqrt{-\Delta_L} A^3 Q^3 \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $\epsilon t \leq c_0$ sufficiently small.
For $S_{LH}$ we use Lemma \[lem:ABasic\] and followed by , $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2 }{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k} \int {\left\vert A^1 \widehat{Q^1_k}(\eta,l) \frac{1}{{\left\langle \xi,l^\prime \right\rangle}} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1-\delta_1} A^1 {\left\langle {\nabla}\right\rangle}^{2} \widehat{U^1_{0}}(\xi,l^\prime)_{Hi} Low(k,\eta-\xi,l-l^\prime) \right\vert} d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1 \right\rVert}_2 {\left\lVert A^1 {\left\langle {\nabla}\right\rangle}^2 U_0^1 \right\rVert}_2,\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $c_0$ sufficiently small. Similarly, $$\begin{aligned}
S_{C} & \lesssim \frac{\epsilon^2 {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1 \right\rVert}_2 {\left\lVert A C \right\rVert}_2 \lesssim \frac{\epsilon^{1/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1 \right\rVert}_2^2 + \frac{\epsilon^{5/2} t^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A C \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\delta > 0$ and $\epsilon$ sufficiently small. The remainder terms are similar to the above and are hence omitted for brevity. This completes the treatment of the $NLS1(3,\neq,0)$ terms; the other $j$ are simpler.
Next consider the $NLS1(j,0,\neq)$ terms. The most difficult is naturally the case $j = 1$ (which does not cancel); the others are simpler and are hence omitted for brevity. Expand with a paraproduct, $$\begin{aligned}
NLS1(1,0,\neq) & = -\sum \int A^1 Q^1_k A^1\left( (Q^1_{0})_{Hi} ( \partial_X U_k^1)_{Lo} \right) dV \\
& \quad - \sum \int A^1 Q^1_k A^1\left( (Q^2_{0})_{Lo} ( \partial_XU_k^1)_{Hi} \right) dV + S_{\mathcal{R},C} \\
& = S_{HL} + S_{LH} + S_{\mathcal{R},C}. \end{aligned}$$ From Lemma \[lem:ABasic\] and , $$\begin{aligned}
S_{HL} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k} \int {\left\vert A^1 \widehat{Q^1_k}(\eta,l) {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1-\delta_1} A^1 \widehat{Q^1_{0}}(\xi,l^\prime)_{Hi} Low(\eta-\xi,l-l^\prime) \right\vert} d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^1 Q^1_{\neq} \right\rVert}_2 {\left\lVert A^1 Q^1_0 \right\rVert}_2. \end{aligned}$$ From , , and , $$\begin{aligned}
S_{LH} & \lesssim \epsilon t \sum_{k} \int {\left\vert A^1 \widehat{Q^1_k}(\eta,l) \frac{{\left\vertk\right\vert}}{k^2 + (l^\prime)^2 + {\left\vert\xi - kt\right\vert}^2} \Delta_L A^1 \widehat{U^1_{k}}(\xi,l^\prime)_{Hi} Low(\eta-\xi,l-l^\prime) \right\vert} d\eta d\xi \\
& \lesssim \epsilon t {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1\right)Q^1 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1\right)\Delta_L U^1_{\neq} \right\rVert}_2 \\
& \quad + \epsilon {\left\lVert A^1 Q^1 \right\rVert}_2{\left\lVert A^1 \Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\]. This completes the $NLS1(j,0,\neq)$ terms.
Finally consider the $NLS1(j,\neq,\neq)$ terms. All these terms are treated similarly, hence, consider just $j = 3$. Expand as above $$\begin{aligned}
NLS1(3,\neq,\neq) & = -\int A^1 Q^1_{\neq} A^1\left( (Q^3_{\neq})_{Hi} (\partial_Z U_{\neq}^1)_{Lo} \right) dV - \int A^1 Q^1_{\neq} A^1\left( (Q^3_{\neq})_{Lo} (\partial_Z U_{\neq}^1)_{Hi} \right) dV \\
& \quad - \sum \int A^1 Q^1_{\neq} A^1\left( (Q^3_{\neq})_{Lo} \left((\psi_z)_{Hi} (\partial_Y - t\partial_X) + (\phi_z)_{Hi}\partial_Z\right)(U_{\neq}^1)_{Lo} \right) dV \\
& \quad + S_{\mathcal{R},C} \\
& = S_{HL} + S_{LH} + S_{C} + S_{\mathcal{R},C}. \end{aligned}$$ For $S_{HL}$, we have by , Lemma \[lem:ABasic\], and (a power of $t$ is lost due to the regularity imbalances), $$\begin{aligned}
S_{HL} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum \int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^1 \widehat{Q^1_k}(\eta,l)\right\vert}
\\ & \quad\quad \times {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1-\delta_1} {\left\vertA^3 \widehat{Q^3_{k^\prime}}(\xi^\prime,l^\prime)_{Hi}\right\vert} Low(k-k^\prime,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon t}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1_{\neq} \right\rVert}_2 {\left\lVert A^3 Q^3 \right\rVert}_2,\end{aligned}$$ which is consistent with Proposition \[prop:Boot\]. For $S_{LH}$ we have by , , , and , $$\begin{aligned}
S_{LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k} \int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vert A^1 \widehat{Q^1_k}(\eta,l) \frac{1}{{\left\vertk^\prime\right\vert} + {\left\vertl^\prime\right\vert} + {\left\vert\eta-k^\prime t\right\vert}} \Delta_L A^1 \widehat{U^1_{k^\prime}}(\xi,l^\prime)_{Hi}\right\vert} \\ & \quad \quad \times Low(k-k^\prime, \eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1 \right) Q^1 \right\rVert}_2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1 \right) \Delta_L U_{\neq}^1 \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1 \right\rVert}_2 {\left\lVert A^1 \Delta_L U_{\neq}^1 \right\rVert}_2, \end{aligned}$$ which by Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\], is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. The coefficient error terms are similar to those that arise in e.g. $NLP(i,j,\neq,\neq)$ and are hence omitted for brevity (although they require the hypothesis $\epsilon \lesssim \nu^{2/3+\delta}$ for $\delta > 0$). The remainder terms are either easier or similar to the above treatments and hence can also be omitted.
As discussed above, the remaining $NLS1$ terms are similar or easier and hence are safely omitted. This completes the $NLS1$ terms.
### Treatment of the $NLS2$ terms
Turn to the $NLS2$ terms. These terms are all treated via easy variants of the treatments of the $NLS1$ and $NLP$ terms. They are hence omitted for the sake of brevity.
Transport nonlinearity $\mathcal{T}$ {#transport-nonlinearity-mathcalt}
------------------------------------
In this section, we treat the **(SI)** and **(3DE)** contributions to the transport nonlinearity, given by $\mathcal{T}_{\neq}$. Begin with a paraproduct decomposition: $$\begin{aligned}
\mathcal{T}_{\neq} & = -\int A^{1} Q^1_{\neq} A^{1} \left( \tilde U_{Lo} \cdot {\nabla}Q^3_{Hi} \right) dV -\int A^{3} Q^3 A^{3} \left( \tilde U_{Hi} \cdot {\nabla}Q^3_{Lo} \right) dV + \mathcal{T}_{\mathcal{R}} \\
& = \mathcal{T}_T + \mathcal{T}_{R} + \mathcal{T}_{\mathcal{R}}, \end{aligned}$$ where $\mathcal{T}_{\mathcal{R}}$ includes the remainder. Due to the lack of regularity imbalances in $A^1$, the transport and remainder contributions, $\mathcal{T}_T$ and $\mathcal{T}_{\mathcal{R}}$ respectively, are treated as in §\[sec:Q2\_TransNon\]. Hence, we omit the treatments and conclude $$\begin{aligned}
\mathcal{T}_T + \mathcal{T}_{\mathcal{R}} & \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}A^1 Q^1 \right\rVert}_2^2 + \left(\frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{2}} + \frac{\epsilon^{1/2} {\left\langle t \right\rangle}^{2\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} \right){\left\lVert A^1 Q^1 \right\rVert}_2^2. \end{aligned}$$ Turn to the reaction contribution. As in §\[sec:Q2\_TransNon\] and §\[sec:Q3\_TransNon\], decompose the reaction term based on the $X$ dependence of each factor: $$\begin{aligned}
\mathcal{T}_{R} & = -\int A^{1} Q^1 A^{1} \left( (\tilde U_{\neq})_{Hi} \cdot ({\nabla}Q^1_0)_{Lo} \right) dV - \int A^{1} Q^1 A^{1} \left( g_{Hi} \partial_Y (Q^1_{\neq})_{Lo} \right) dV \\ & \quad - \int A^{1} Q^1 A^{1} \left( (\tilde U_{\neq})_{Hi} \cdot ({\nabla}Q^1_{\neq})_{Lo} \right) dV \\
& = \mathcal{T}_{R;\neq 0} + \mathcal{T}_{R;0 \neq}+ \mathcal{T}_{R;\neq \neq}. \end{aligned}$$
### Reaction term $\mathcal{T}_{R;0 \neq}$
By and Lemma \[lem:ABasic\], we get (also noting ): $$\begin{aligned}
\mathcal{T}_{R;0 \neq} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k \neq 0}\int {\left\vertA^{1} \hat{Q}^1_k (\eta,l) A^{1}_{k}(\eta,l) \widehat{g}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\xi d\eta \\
& \lesssim \frac{\epsilon}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1_{\neq} \right\rVert}_2 {\left\lVert Ag \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
### Reaction term $\mathcal{T}_{R;\neq 0}$ {#reaction-term-mathcalt_rneq-0}
For this term we use a slight variant of the treatment found in §\[sec:Q2TRneq0\]. Note that $Q^1_0$ is $O(t)$ larger than $Q^2_0$ but $A^1 \lesssim {\left\langle t \right\rangle}^{-1} A^2$, and hence the allowed growth in $A^1$ will balance the extra growth in these terms. Therefore, these can be treated in the same fashion as was done in §\[sec:Q2TRneq0\]. Hence, we omit the details for brevity.
### Reaction term $\mathcal{T}_{R;\neq \neq}$ {#sec:Q1TRneqneq}
This reaction term is slightly different than the analogous terms in §\[sec:Q2TRneqneq\] and §\[sec:Q3TRneqneq\], as we are not allowing a norm imbalance like in §\[sec:Q3TRneqneq\] but instead are allowing a steady linear growth. As in §\[sec:Q2TRneqneq\] above, we further decompose in terms of frequency: $$\begin{aligned}
\mathcal{T}_{R;\neq \neq} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l) A^{1}_{k}(\eta,l) \hat{U}_{k^\prime}^1 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{3+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l) A^{1}_{k}(\eta,l) \hat{U}_{k^\prime}^2 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha-1}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l) A^{1}_{k}(\eta,l) \hat{U}_{k^\prime}^3 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l)\right\vert} A^{1}_{k}(\eta,l) \left({\left\vert\hat{\psi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\hat{\phi_y}(\xi,l^\prime)_{Hi}\right\vert} + {\left\vert\hat{\phi_z}(\xi,l^\prime)_{Hi}\right\vert}\right) \\ & \quad\quad\quad \times Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}^{3+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l) A^{1}_{k}(\eta,l) \hat{\psi_z}(\xi,l^\prime)_{Hi} \right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \quad + \mathcal{T}_{R;\neq \neq;\mathcal{R}} \\
& = \mathcal{T}_{R;\neq\neq}^{1} + \mathcal{T}_{R;\neq\neq}^2 + \mathcal{T}_{R;\neq\neq}^3 + \mathcal{T}_{R;\neq\neq}^{C1} + \mathcal{T}_{R;\neq\neq}^{C2} + \mathcal{T}_{R;\neq\neq;\mathcal{R}}. \end{aligned}$$ Consider $\mathcal{T}^1_{R;\neq \neq}$. By followed by and , $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{1} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l)\right\vert} \frac{1}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2}{\left\vertA^1 \Delta_L \hat{U}_{k^\prime}^1 (\xi,l^\prime)_{Hi}\right\vert} \\ & \quad\quad \times Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1\right) Q^1 \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^1 \right) \Delta_L U^1_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^1 Q^1 \right\rVert}_2 {\left\lVert A^1 \Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\]. The term $\mathcal{T}^2_{R;\neq \neq}$ is treated in essentially the same way and is hence omitted (that $A^1 \lesssim {\left\langle t \right\rangle}^{-1}A^2$ recovers the additional power of $t$ from the low frequency factor in $\mathcal{T}^2_{R;\neq \neq}$).
Next, turn to the treatment of $\mathcal{T}^3_{R;\neq \neq}$. By Lemma \[lem:ABasic\] followed by and , $$\begin{aligned}
\mathcal{T}_{R;\neq\neq}^{3} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k,k^\prime}\int \mathbf{1}_{k k^\prime(k-k^\prime) \neq 0} {\left\vertA^{1} \hat{Q}^1_k(\eta,l)\right\vert} \frac{1}{{\left\vertk^\prime\right\vert}^2 + {\left\vertl^\prime\right\vert}^2 + {\left\vert\xi - k^\prime t\right\vert}^2} \\
& \quad\quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}} + \chi^{\ast;23} \right) \frac{1}{{\left\langle t \right\rangle}} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1-\delta_1} \\ & \quad\quad \times {\left\vertA^3 \Delta_L \hat{U}_{k^\prime}^3 (\xi,l^\prime)_{Hi}\right\vert} Low(k-k^\prime, \eta-\xi, l - l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{1 + \delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^1 Q^1 \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent by Lemma \[lem:SimplePEL\].
The coefficient and remainder terms can be treated in exactly the same manner as in §\[sec:Q3TRneqneq\] and are therefore omitted for the sake of brevity. This completes the treatment of $\mathcal{T}_{R;\neq\neq}$ and hence all of $\mathcal{T}$.
Dissipation error terms $\mathcal{D}$ {#dissipation-error-terms-mathcald}
-------------------------------------
These terms are treated in the same manner as the corresponding terms in $Q^3$, found in §\[sec:DEneqQ3\]. The results are analogous to those found therein and are hence here omitted for brevity.
This completes the high norm estimate on $Q^1_{\neq}$.
Coordinate system controls
==========================
In this section we prove the necessary controls on $C^i$ and the auxiliary unknown $g$.
High norm estimate on $g$
-------------------------
We will begin by improving , which roughly measures the time-oscillations between $U_0^1$ and $C^1$, and hence measures the time-oscillations of the $y$ component of the shear. From we have $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert Ag \right\rVert}_2^2 & = -CK^g_\lambda - CK^g_w - \frac{2}{t}{\left\lVert Ag \right\rVert}_2^2 - \int Ag A\left(g \partial_Y g\right) dV \\
& \quad + \int Ag A\left(\tilde{\Delta_t}g\right) dV - \frac{1}{t}\sum_{k \neq 0}\int Ag A \left( U_{-k} \cdot {\nabla}^t U^1_{k} \right) dV \\
& = -\mathcal{D}g + \mathcal{T} + \mathcal{D}_E + \mathcal{F}. \end{aligned}$$
### Transport nonlinearity {#sec:TransNong}
By Lemma \[lem:AAiProd\] and , we have $$\begin{aligned}
\mathcal{T} & \lesssim {\left\lVert Ag \right\rVert}_2^2 {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda}} + {\left\lVert Ag \right\rVert}_2{\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda}}{\left\lVert {\nabla}A g \right\rVert}_2 \lesssim \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{2}}{\left\lVert Ag \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert {\nabla}A g \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small,
### Dissipation error terms {#sec:DissErrg}
Recall that the dissipation error terms are of the form $$\begin{aligned}
\mathcal{D}_E = \nu \int Ag A \left( G_{yy} \partial_Y^2 g + G_{yz}\partial_{YZ} g + G_{zz} \partial_{ZZ}g \right) dV. \end{aligned}$$ We may treat these as in [@BGM15I] (for which we use essentially the same treatment as in §\[sec:DEQ02\], despite the higher regularity of $A$). Using that approach we have, $$\begin{aligned}
\mathcal{D}_E \lesssim c_0\nu{\left\lVert {\nabla}A g \right\rVert}_2^2 + \nu{\left\lVert Ag \right\rVert}_2 {\left\lVert {\nabla}AC \right\rVert}_2{\left\lVert {\nabla}Ag \right\rVert}_2 \lesssim c_0\nu{\left\lVert {\nabla}Ag \right\rVert}_2^2 + c_0^{-1} \epsilon^2 \nu {\left\lVert {\nabla}AC \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
### Forcing from non-zero frequencies {#sec:g_Forcing}
Analogous to , by the divergence-free condition we have, $$\begin{aligned}
\mathcal{F} & = -\sum_{k \neq 0}\frac{1}{t}\int A g A \left( \partial_Y^t \left( U^2_{-k} U^1_{k}\right) + \partial_Z^t \left( U^3_{-k} U^1_{k}\right) \right) dV = F_Y + F_Z. \end{aligned}$$ Consider $F_Y$ first. Expand with a paraproduct and group terms where the coefficients appear in low frequency with the remainder: $$\begin{aligned}
F_Y & = - \sum_{k \neq 0}\frac{1}{t}\int A g A \partial_Y \left( (U^2_{-k})_{Lo} (U^1_{k})_{Hi} \right) dV - \sum_{k \neq 0}\frac{1}{t}\int A g A \partial_Y \left( (U^2_{-k})_{Hi} (U^1_{k})_{Lo} \right) dV \\
& \quad - \sum_{k \neq 0}\frac{1}{t}\int A g A \left( \left((\psi_y)_{Hi}\partial_Y + (\phi_y)_{Hi}\partial_Z \right) \left((U^2_{-k})_{Lo} (U^1_{k})_{Lo}\right) \right) dV + F_{Y;\mathcal{R},C} \\
& = F_{Y;LH} + F_{Y;HL} + F_{Y;C} + F_{Y;\mathcal{R},C}. \end{aligned}$$ By and $$\begin{aligned}
F_{Y;LH} & \lesssim \frac{\epsilon}{t{\left\langle t \right\rangle} {\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0} \sum_{l,l^\prime} \int {\left\vertA \hat{g}(\eta,l)\right\vert} \frac{{\left\langle t \right\rangle} {\left\vert\eta\right\vert} {\left\langle \eta,l \right\rangle}^2 {\left\langle \frac{t}{{\left\langle \eta,l \right\rangle}} \right\rangle}^{1+\delta_1}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \\ & \quad\quad \times {\left\vertA^{1} \Delta_L \widehat{U^1_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi.\end{aligned}$$ By and , we therefore have $$\begin{aligned}
F_{Y;LH} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}
{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A \right) g \right\rVert}_2^2 + \frac{\epsilon {\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^1 \right) \Delta_L U^1_{\neq} \right\rVert}_2 \\
& \quad + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L}Ag \right\rVert}_2^2 + \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{2} {\left\langle \nu t^3 \right\rangle}^{2\alpha}} {\left\lVert A^{1} \Delta_L U^1_{\neq} \right\rVert}^2_2,\end{aligned}$$ which, by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\], is consistent with Proposition \[prop:Boot\] for $\delta_1$ and $\epsilon$ sufficiently small.
Turn next to $F_{Y;HL}$. Similar to $F_{Y;LH}$, we get from , , , and we get $$\begin{aligned}
F_{Y;HL} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{t{\left\langle \nu t^3 \right\rangle}^{\alpha}} \sum_{k \neq 0}\sum_{l,l^\prime} \int {\left\vert A \hat{g} \frac{{\left\vert\eta\right\vert} {\left\langle \eta,l \right\rangle}^2 {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} A^{2} \Delta_L \widehat{U^2_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) dV \\
& \lesssim \frac{\epsilon t^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A\right) g \right\rVert}_2^2 + \frac{\epsilon t^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^2 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^2 \right) \Delta_L U^2_{\neq} \right\rVert}_2^2 \\ & \quad + \frac{\epsilon }{t^{1-\delta_1}{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert \sqrt{-\Delta_L}Ag \right\rVert}_2 {\left\lVert A^{2} \Delta_L U^2_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] by Lemmas \[lem:SimplePEL\] and \[lem:PEL\_NLP120neq\]. The remainder terms $F_{Y;\mathcal{R},C}$ are similar, but simpler, and and are hence omitted for brevity.
Consider finally $F_{Y;C}$. By , Lemma \[lem:ABasic\], and (and Lemma \[lem:CoefCtrl\]), $$\begin{aligned}
F_{Y;C} & \lesssim \frac{\epsilon^2 }{t{\left\langle t \right\rangle}^{1-\delta_1}{\left\langle \nu t^3 \right\rangle}^{2\alpha}}\sum_{l,l^\prime} \int{\left\vertA \hat{g}(\eta,l)\right\vert} A \left( {\left\vert\widehat{\psi_y}\right\vert} + {\left\vert\widehat{\phi_z}\right\vert}\right)(\xi,l^\prime)_{Hi} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon^{1/2}}{t^{4-2\delta_1} {\left\langle \nu t^3 \right\rangle}^{2\alpha}} {\left\lVert Ag \right\rVert}_2^2 + \epsilon^{7/2}{\left\lVert {\nabla}A C \right\rVert}^2_2.\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. This completes the treatment of $F_Y$.
Next turn to $F_Z$, which has additional complications due to the regularity imbalances implying $U^3$ has worse regularity than $U^2$ near the critical times. Expand with a paraproduct and as usual with terms in which the coefficients appear in low frequency included in the remainder: $$\begin{aligned}
F_Z & = - \sum_{k \neq 0}\frac{1}{t}\int A g A \partial_Z \left( (U^3_{-k})_{Lo} (U^1_{k})_{Hi} \right) dV - \sum_{k \neq 0}\frac{1}{t}\int A g A \partial_Z \left( (U^3_{-k})_{Hi} (U^1_{k})_{Lo} \right) dV \\
& \quad - \sum_{k \neq 0}\frac{1}{t}\int A g A\left( \left((\psi_z)_{Hi} \partial_Y + (\phi_z)_{Hi}\partial_Z\right) (U^3_{-k} U^1_{k})_{Lo} \right) dV + F_{Z;\mathcal{R},C} \\
& = F_{Z;LH} + F_{Z;HL} + F_{Z;C} + F_{Z;\mathcal{R}}. \end{aligned}$$ Consider first $F_{Z;LH}$, which is similar to the analogous term above in $F_{Y}$. Indeed, by , , , and , $$\begin{aligned}
F_{Z;LH} & \lesssim \frac{\epsilon}{t{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k \neq 0} \sum_{l,l^\prime} \int {\left\vertA\hat{g}(\eta,l) \frac{{\left\vertl\right\vert} {\left\langle \eta,l \right\rangle}^2 {\left\langle t \right\rangle} {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{1+\delta_1}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} A^{1} \Delta_L \widehat{U^1_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A\right) g \right\rVert}_2^2 + \frac{\epsilon t^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^1 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s} A^1 \right) \Delta_L U^1_{\neq} \right\rVert}_2^2 \\
& \quad + \frac{\epsilon }{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert \sqrt{-\Delta_L}Ag \right\rVert}_2 {\left\lVert A^{1} \Delta_L U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\]. Turn next to $F_{Z;HL}$, which is complicated by the regularity imbalance in $A^3$. Indeed, by , Lemma \[lem:ABasic\], followed by , , and , we have $$\begin{aligned}
F_{Z;HL} & \lesssim \frac{\epsilon}{t^{1-\delta_1}{\left\langle \nu t^3 \right\rangle}^{\alpha}}\sum_{k \neq 0} \sum_{l,l^\prime} \int {\left\vert\hat{g}(\eta,l)\right\vert} \frac{{\left\vertl\right\vert} {\left\langle \eta,l \right\rangle}^2 {\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{2}}{k^2 + (l^\prime)^2 + {\left\vert\xi-kt\right\vert}^2} \\
& \quad \quad \times \left(\sum_{r}\chi^{r,NR}\frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}(\eta,l) \tilde{A}^3_k(\xi,l^\prime) + \chi^{\ast;23} A(\eta,l) A^3_k(\xi,l^\prime) \right) \\ & \quad\quad \times {\left\vert\Delta_L \widehat{U^3_{k}}(\xi,l^\prime)_{Hi}\right\vert} Low(-k,\eta-\xi,l-l^\prime) d\eta d\xi \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{1+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right) g \right\rVert}_2 {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^3 \right) \Delta_L U^3_{\neq} \right\rVert}_2 \\
& \quad + \frac{\epsilon}{t^{1-\delta_1}{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert \sqrt{-\Delta_L}Ag \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2, \end{aligned}$$ which by Lemmas \[lem:PEL\_NLP120neq\] and \[lem:SimplePEL\] is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. The coefficient and remainder terms can be treated as in $F_Y$; hence these are omitted for brevity. This completes the treatment of the forcing terms and hence of the entire high norm estimate on $g$.
Low norm estimate on $g$
------------------------
Computing the evolution of ${\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}$ (denoting $A^S = e^{\lambda(t){\left\vert{\nabla}\right\vert}^s} {\left\langle {\nabla}\right\rangle}^{\gamma}$) from , $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\left( t^{4} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}^2\right) & \leq \dot{\lambda}t^4 {\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}^2 - t^{4}\int A^S g A^S\left(g \partial_Y g\right) dV \nonumber \\
& \quad + t^{4} \int A^S g A^S \left(\tilde{\Delta_t}g\right) dV - t^3\int A^S g A^S \left( U_{\neq} \cdot {\nabla}^t U^1_{\neq} \right)_0 dV \nonumber \\
& = -CK_\lambda^{g,L} + \mathcal{T} + \mathcal{D} + \mathcal{F}. \label{eq:gLowEvo} \end{aligned}$$ The treatment of the transport nonlinearity $\mathcal{T}$ and the dissipation error terms in $\mathcal{D}$ are essentially same as in the previous section (in fact easier), so are hence omitted. It remains to see why the forcing $\mathcal{F}$ can treated better at lower regularity. Following the treatments in the previous section and §\[sec:NzeroForcing\], we can use the divergence free condition to write $$\begin{aligned}
\mathcal{F} & = -t^{3} \int A^S g A^S \left( \partial_Y^t \left( U^2_{\neq} U^1_{\neq}\right)_0 + \partial_Z^t \left( U^3_{\neq} U^1_{\neq}\right)_0 \right) dV. \end{aligned}$$ The two terms can be treated together. Indeed, by Lemmas \[lem:GevProdAlg\], Lemma \[lem:CoefCtrl\], the bootstrap hypotheses, as well as Lemma \[lem:LossyElliptic\] and , $$\begin{aligned}
\mathcal{F} & \lesssim t^{3} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}(1 + {\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma+1}})\left({\left\lVert U^2_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma+1}}{\left\lVert U^1_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}} + {\left\lVert U^3_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}}{\left\lVert U^1_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma+1}} \right. \\ & \left. \quad\quad + {\left\lVert U^2_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}}{\left\lVert U^1_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma+1}} + {\left\lVert U^3_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma+1}}{\left\lVert U^1_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,3/2+}} \right) \\
& \lesssim t^{3} {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} \left(\frac{\epsilon^2 {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \right) \\
& \lesssim \frac{\epsilon^{1/2} t^{4}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert g \right\rVert}^2_{{\mathcal{G}}^{\lambda,\gamma}} + \frac{\epsilon^{7/2} t^{2+2\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}, \end{aligned}$$ which, for $\delta_1$ and $\epsilon$ sufficiently small (and $\delta > 0$), is consistent with Proposition \[prop:Boot\]
Long time, high norm estimate on $C^i$ {#sec:ACC}
--------------------------------------
Next, we improve . Computing the evolution equation on $C^i$, , we get $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A C^i \right\rVert}_2^2 & = \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A C^i \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A} C^i \right\rVert}_2^2 + \nu \int A C^i A\left(\tilde{\Delta_t} C^i \right) dV \nonumber \\ & \quad -\int A C^i A\left( g\partial_Y C^i \right) dV + \mathcal{L}^i \nonumber \\
& = -\mathcal{D}C^i + \mathcal{D}_E + \mathcal{T} + \mathcal{L}^i, \label{ineq:C1Evo} \end{aligned}$$ where $$\begin{aligned}
\mathcal{D}_E & = \nu \int AC^i A\left( (\tilde{\Delta_t} - \Delta) C^i \right) dV. \end{aligned}$$ and
$$\begin{aligned}
\mathcal{L}^1 & = \int A C^1 A g dV - \int A C^1 A U_0^2 dV \label{def:L1LT} \\
\mathcal{L}^2 & = -\int A C^2 A U_0^3 dV. \label{def:L2LT} \end{aligned}$$
### Linear driving terms
#### Treatment of $\mathcal{L}^1$ {#sec:LD1_Longtime}
Consider the first term in . For this it suffices to use $$\begin{aligned}
\int A C^1 A g dV & \leq \frac{\epsilon}{2 c_0} {\left\lVert AC^1 \right\rVert}_2^2 + \frac{c_0}{2\epsilon} {\left\lVert Ag \right\rVert}_2^2, \end{aligned}$$ which, for $K_{HC1} \gg 1$, is consistent with Proposition \[prop:Boot\] (via integrating factors).
Turn to the second term in . From Lemma \[lem:PELbasicZero\] (for some $K$ depending on $s,\sigma$ and $\lambda$), $$\begin{aligned}
-\int A C^1 A U_0^2 dV & \leq \frac{\epsilon}{2 c_0}{\left\lVert A C^1 \right\rVert}_2^2 + \frac{c_0}{2\epsilon}{\left\lVert A U^2_0 \right\rVert}_2^2 \\
& \leq \frac{\epsilon}{2 c_0}{\left\lVert AC^1 \right\rVert}_2^2 + K \epsilon {\left\lVert A C \right\rVert}_2^2 + \frac{Kc_0}{\epsilon}{\left\lVert A^2 Q^2_0 \right\rVert}_2^2 + \frac{Kc_0}{\epsilon}{\left\lVert U_0^2 \right\rVert}_2^2, \end{aligned}$$ which for $\epsilon$ and $c_0$ sufficiently small and $K_{HC1}$ sufficiently large, is consistent with Proposition \[prop:Boot\] (again, via integrating factors).
#### Treatment of $\mathcal{L}^2$ {#sec:L2_Longer}
Now consider the case $i = 2$. The issue here is that we want to propagate higher regularity on $C^2$ than we have on $U_0^3$ due to the regularity imbalance in $A^3$. First we have the following, independently of $\kappa$ (see ), $$\begin{aligned}
\mathcal{L}^2 & \lesssim \sum \int {\left\vert\widehat{C^2}(\eta,l)\right\vert}\left(\sum_{r}\mathbf{1}_{t \in {\mathbf{I}}_{r,\eta}} \frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}(\eta,l)\tilde{A}^3_0(\eta,l) + \chi^{\ast} A(\eta,l) A^3(\eta,l) \right) {\left\langle \eta,l \right\rangle}^2 {\left\vert \widehat{U^3_0}(\eta,l)\right\vert} d\eta, \end{aligned}$$ where $\chi^\ast = 1 - \sum_{r \neq 0} \mathbf{1}_{t \in {\mathbf{I}}_{r,\eta}}$. Therefore, by Lemma \[dtw\] and orthogonality, $$\begin{aligned}
\mathcal{L}^2 & \lesssim \frac{{\left\langle t \right\rangle}}{\kappa} \sum_{r \neq 0} {\left\lVert \sqrt{\frac{\partial_t w}{w}} \mathbf{1}_{t \in {\mathbf{I}}_{r,\partial_Y}} \tilde{A} C^2 \right\rVert}_2 {\left\lVert \sqrt{\frac{\partial_t w}{w}} \mathbf{1}_{t \in {\mathbf{I}}_{r,\partial_Y}} {\left\langle {\nabla}\right\rangle}^2 \tilde{A}^3 U_0^3 \right\rVert}_2 + {\left\lVert AC^2 \right\rVert}_2 {\left\lVert {\left\langle {\nabla}\right\rangle}^2 A^3 U_0^3 \right\rVert}_2 \\
& \lesssim \frac{1}{\kappa} {\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}C^2 \right\rVert}_2^2 + \frac{{\left\langle t \right\rangle}^2}{\kappa} {\left\lVert \sqrt{\frac{\partial_t w}{w}} {\left\langle {\nabla}\right\rangle}^2 \tilde{A}^3 U_0^3 \right\rVert}_2^2 + c_0^{-1} \epsilon{\left\lVert AC^2 \right\rVert}_2^2 + \frac{c_0}{\epsilon} {\left\lVert {\left\langle {\nabla}\right\rangle}^2 A^3 U_0^3 \right\rVert}_2^2. \end{aligned}$$ This is consistent with Proposition \[prop:Boot\] for $K_{HC1} \gg K_{H3}$ (using $t \leq T_F < c_0\epsilon^{-1}$), $c_0$ and $\epsilon$ sufficiently small and $\kappa$ sufficiently large (the latter relative only to a universal constant independent of all other parameters).
### Transport nonlinearity {#sec:TransNon_ACC}
By Lemma \[lem:AAiProd\], , and , $$\begin{aligned}
\mathcal{T} & \lesssim {\left\lVert AC^i \right\rVert}_{2}\left({\left\lVert AC^i \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} {\left\lVert Ag \right\rVert}_{2} + {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}{\left\lVert {\nabla}A C^i \right\rVert}_2\right) \\
& \lesssim \left(\epsilon + \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{4}}\right) {\left\lVert AC^i \right\rVert}_2^2 + \epsilon^{3/2} {\left\lVert {\nabla}A C^i \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ and $c_0$ sufficiently small.
### Dissipation error terms {#sec:DissC}
For these terms, as in [@BGM15I], we may use an easy variant of the treatment in §\[sec:DissErrg\]. We omit the details for brevity: $$\begin{aligned}
\mathcal{D}_E & \lesssim \nu {\left\lVert AC \right\rVert}_{2} {\left\lVert {\nabla}A C^i \right\rVert}_2^2 + \nu {\left\lVert A C^i \right\rVert}_{2} {\left\lVert {\nabla}A C \right\rVert}_{2} {\left\lVert {\nabla}C^i \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma-1}} \lesssim c_{0} \nu {\left\lVert {\nabla}A C \right\rVert}_2^2, \end{aligned}$$ which is then absorbed by the dissipation by choosing $c_{0}$ sufficiently small.
Shorter time, high norm estimate on $C^i$
-----------------------------------------
The improvement of is essentially the same as that of with a few slight changes. From , $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}\left({\left\langle t \right\rangle}^{-2}{\left\lVert A C^i \right\rVert}_2^2\right) & = -\frac{t}{{\left\langle t \right\rangle}^4}{\left\lVert A C^i \right\rVert}_2^2 + {\left\langle t \right\rangle}^{-2}\dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A C^i \right\rVert}_2^2 - {\left\langle t \right\rangle}^{-2}{\left\lVert \sqrt{\frac{\partial_t w}{w}}\tilde{A} C^i \right\rVert}_2^2 \nonumber \\ & \quad + {\left\langle t \right\rangle}^{-2}\nu \int A C^i A \left(\tilde{\Delta_t} C^i\right) dv \nonumber -{\left\langle t \right\rangle}^{-2}\int A C^1 A\left( g \partial_Y C^i \right) dV + \mathcal{L}^i \nonumber \\
& = -CK_{L}^{C} - {\left\langle t \right\rangle}^{-2}\mathcal{D}C^i + \mathcal{D}_E + \mathcal{T} + \mathcal{L}^i, \label{ineq:CCEvo} \end{aligned}$$ where $$\begin{aligned}
\mathcal{D}_E = {\left\langle t \right\rangle}^{-2} \int AC^i A\left((\tilde{\Delta_t} - \Delta)C^i\right) dV \end{aligned}$$ and
$$\begin{aligned}
\mathcal{L}^1 & = {\left\langle t \right\rangle}^{-2}\int A C^1 A g dV - {\left\langle t \right\rangle}^{-2}\int A C^1 A U_0^2 dV \label{def:L1} \\
\mathcal{L}^2 & = - {\left\langle t \right\rangle}^{-2}\int A C^2 A U_0^3 dV. \label{def:L2} \end{aligned}$$
The only real difference between the estimates versus is in the linear driving terms $\mathcal{L}^i$. Hence, we omit the treatment of $\mathcal{T}$ and $\mathcal{D}_E$, as these can be treated in essentially the same manner as in the improvement of .
### Linear driving terms
#### Treatment of $\mathcal{L}^1$ {#sec:LD1_shorttime}
Consider first the case $i =1$. By Cauchy-Schwarz, $$\begin{aligned}
{\left\langle t \right\rangle}^{-2}\int A C^1 A g dV & \leq \frac{t}{2{\left\langle t \right\rangle}^4}{\left\lVert AC^1 \right\rVert}_2^2 + \frac{1}{2 t}{\left\lVert Ag \right\rVert}_2^2 \leq \frac{1}{2}CK_{L}^{C,1} + \frac{1}{4}CK_L^g. \end{aligned}$$ Hence the first term is absorbed by the $CK_L^{C,1}$ term in whereas the second term is controlled by and hence this is consistent with Proposition \[prop:Boot\] provided $K_{HC2}$ is sufficiently large.
Consider the second term in . By a similar argument but now applying Lemma \[lem:PELbasicZero\], we have for some $K > 0$, $$\begin{aligned}
-{\left\langle t \right\rangle}^{-2}\int A C^1 A U_0^2 dV & \leq \frac{t}{10{\left\langle t \right\rangle}^4}{\left\lVert A C^1 \right\rVert}_2^2 + \frac{5}{t}{\left\lVert A U_0^2 \right\rVert}_2^2 \\
& \leq \frac{t}{10{\left\langle t \right\rangle}^4}{\left\lVert A C^1 \right\rVert}_2^2 + \frac{K}{{\left\langle t \right\rangle}}{\left\lVert A^2 Q_0^2 \right\rVert}_2^2 + \frac{K}{{\left\langle t \right\rangle}}{\left\lVert U_0^2 \right\rVert}_2^2 + \frac{K\epsilon^2}{{\left\langle t \right\rangle}}{\left\lVert AC \right\rVert}_2^2. \end{aligned}$$ Hence for $K_{HC2}$ sufficiently large relative to $K_{HC1}$, this is consistent with Proposition \[prop:Boot\] for $c_0$ and $\epsilon$ sufficiently small.
#### Treatment of $\mathcal{L}^2$ {#treatment-of-mathcall2}
As in §\[sec:L2\_Longer\], we have (again defining $\chi^\ast = 1 - \sum_{r \neq 0}\mathbf{1}_{t \in {\mathbf{I}}_{r,\eta}}$), $$\begin{aligned}
-{\left\langle t \right\rangle}^{-2} \int A C^2 A U_0^3 dV & \lesssim {\left\langle t \right\rangle}^{-2} \sum \int {\left\vert\widehat{C^2}(\eta,l)\right\vert}\left(\sum_{r}\mathbf{1}_{t \in {\mathbf{I}}_{r,\eta}} \frac{t}{{\left\vertr\right\vert} + {\left\vert\eta-tr\right\vert}}\tilde{A}(\eta,l)\tilde{A}^3_0(\eta,l) + \chi^{\ast} A(\eta,l)A^3_0(\eta,l) \right) \\ & \quad\quad \times {\left\langle \eta,l \right\rangle}^2 {\left\vert \widehat{U^3_0}(\eta,l)\right\vert} d\eta \\
& \lesssim \kappa^{-1}{\left\langle t \right\rangle}^{-1} \sum_{r \neq 0} {\left\lVert \sqrt{\frac{\partial_t w}{w}} \mathbf{1}_{t \in {\mathbf{I}}_{r,\partial_Y}} \tilde{A} C^2 \right\rVert}_2 {\left\lVert \sqrt{\frac{\partial_t w}{w}} \mathbf{1}_{t \in {\mathbf{I}}_{r,\partial_Y}} {\left\langle {\nabla}\right\rangle}^2 \tilde{A}^3 U_0^3 \right\rVert}_2 \\ & \quad + {\left\langle t \right\rangle}^{-2}{\left\lVert AC^2 \right\rVert}_2 {\left\lVert {\left\langle {\nabla}\right\rangle}^2 A^3 U_0^3 \right\rVert}_2 \\
& = T_1 + T_2. \end{aligned}$$ To treat the first term we use orthogonality and Lemma \[lem:PELCKZero\] to deduce the following (where $K$ is a universal constant depending only on $\lambda$ and $s$ and differs from line to line), $$\begin{aligned}
T_1 & \leq \frac{1}{10}{\left\langle t \right\rangle}^{-2}{\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}C^2 \right\rVert}^2_2 + K{\left\lVert \sqrt{\frac{\partial_t w}{w}} {\left\langle {\nabla}\right\rangle}^2 \tilde{A}^3 U_0^3 \right\rVert}_2^2 \\
& \leq \frac{1}{10}{\left\langle t \right\rangle}^{-2}{\left\lVert \sqrt{\frac{\partial_t w}{w}} \tilde{A}C^2 \right\rVert}^2_2 + K{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^3 + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}} A^3 \right) Q_0^3 \right\rVert}_2^2 \\
& \quad + \frac{K}{{\left\langle t \right\rangle}^{2s}}{\left\lVert U_0^3 \right\rVert}_2^2 + K\epsilon^2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right) C \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ and $\epsilon$ sufficiently small together with $K_{HC2} \gg K_{H3}$. Turn next to $T_2$, which is treated in the same manner as the second term in (where $K$ is a universal constant depending only on $\lambda$ and $s$ and differs from line to line), $$\begin{aligned}
T_2 & \leq \frac{t}{10{\left\langle t \right\rangle}^4}{\left\lVert AC^2 \right\rVert}_2^2 + \frac{5}{2t}{\left\lVert A^3 {\left\langle {\nabla}\right\rangle}^2 U_0^3 \right\rVert}_2^2 \\
& \leq \frac{t}{10{\left\langle t \right\rangle}^4}{\left\lVert AC^2 \right\rVert}_2^2 + \frac{K}{{\left\langle t \right\rangle}}{\left\lVert A^3 Q_0^3 \right\rVert}_2^2 + \frac{K}{{\left\langle t \right\rangle}}{\left\lVert U_0^3 \right\rVert}_2^2 + \frac{K\epsilon^2}{{\left\langle t \right\rangle}}{\left\lVert AC \right\rVert}_2^2 \\
& \leq \frac{t}{10{\left\langle t \right\rangle}^4}{\left\lVert AC \right\rVert}_2^2 + \frac{4K K_{H3}}{{\left\langle t \right\rangle}}\epsilon^2 + \frac{4 K_{HC1} K\epsilon^2 c_0^2}{{\left\langle t \right\rangle}},\end{aligned}$$ which is sufficient provided $c_0$ and $\epsilon$ are chosen small and $K_{HC1} \gg K_{H3}$.
Low norm estimate on $C$
------------------------
The improvement of estimate is an easy variation of that applied to improve and except one uses the super-solution method discussed in §\[sec:Q1Hi1\] used to improve .
Enhanced dissipation estimates {#sec:ED}
==============================
In this section we improve the enhanced dissipation estimates . A recurring theme here will be the gain in $t$ from Lemma \[lem:AnuLossy\] when $\partial_X$ derivatives are present, a kind of “null” structure.
Enhanced dissipation of $Q^3$ {#sec:ED3}
-----------------------------
We begin with $Q^3$. Computing the time evolution of ${\left\lVert A^{\nu;3}Q^3 \right\rVert}_2$ we get $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{\nu;3} Q^3 \right\rVert}_2^2
-\frac{2}{t}{\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} \tilde{A}^{\nu;3} Q^3 \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}}A^{\nu;3}Q^3 \right\rVert}_2^2 + G^\nu \nonumber \\
& \quad -2 \int A^{\nu;3} Q^3 A^{\nu;3} \partial_{YX}^t U^3 dV + 2 \int A^{\nu;3} Q^3 A^{\nu;3} \partial_{ZX}^t U^2 dV \nonumber \\
& \quad + \nu \int A^{\nu;3} Q^{3} A^{\nu;3} \left(\tilde{\Delta_t} Q^3\right) dv -\int A^{\nu;3} Q^3 A^{\nu;3}\left( \tilde U \cdot {\nabla}Q^3 \right) dV \nonumber \\
& \quad - \int A^{\nu;3} Q^3 A^{\nu;3} \left[Q^j \partial_j^t U^3 + 2\partial_i^t U^j \partial_{ij}^t U^3 - \partial_Z^t\left(\partial_i^t U^j \partial_j^t U^i\right) \right] dV \nonumber \\
& = -\mathcal{D}Q^{\nu;3} - CK_{L}^{\nu;3} + G^{\nu} \nonumber \\ & \quad + LS3 + LP3 + \mathcal{D}_E + \mathcal{T} + NLS1 + NLS2 + NLP, \label{ineq:AnuEvo3}\end{aligned}$$ where we write $$\begin{aligned}
\mathcal{D}_E & = \nu \int A^{\nu;3} Q^3 A^{\nu;3}\left(\tilde{\Delta_t}Q^3 - \Delta_L Q^3\right) dV, \end{aligned}$$ and $$\begin{aligned}
G^\nu = \alpha \int A^{\nu;3} Q^3 \min\left(1, \frac{{\left\langle \nabla_{Y,Z} \right\rangle}^2}{t^2}\right) e^{\lambda(t){\left\vert{\nabla}\right\vert}^s}{\left\langle {\nabla}\right\rangle}^\beta{\left\langle D(t,\partial_Y) \right\rangle}^{\alpha-1} \frac{D(t,\partial_Y)}{{\left\langle D(t,\partial_Y) \right\rangle}} \partial_t D(t,\partial_Y) Q^3_{\neq} dV. \end{aligned}$$ First, we need to cancel the growing term $G^\nu$ in using part of the dissipation term $\mathcal{D}$. As in [@BGM15I] (and essentially [@BMV14]), $$\begin{aligned}
G^{\nu} - \nu {\left\lVert \sqrt{-\Delta_L} A^{\nu;3}Q^3 \right\rVert}_2^2 & \leq \nu \sum_{k \neq 0} \sum_{l} \int \left(\frac{1}{8}t^2\mathbf{1}_{t \geq 2 {\left\vert\eta\right\vert}} - {\left\vertk\right\vert}^2 - {\left\vertl\right\vert}^2 - {\left\vert\eta-kt\right\vert}^2\right) {\left\vertA^{\nu;3} \widehat{Q^3_k}(\eta,l)\right\vert}^2 d\eta \\
& \leq -\frac{\nu}{8}{\left\lVert \sqrt{-\Delta_L}A^{\nu;3} Q^{3}_{\neq} \right\rVert}_2^2. \end{aligned}$$ Next we see how to control the remaining linear and nonlinear contributions.
### Linear stretching term $LS3$ {#linear-stretching-term-ls3}
First separate into two parts (to be sub-divided further), $$\begin{aligned}
LS3 & = -2\int A^{\nu;3} Q^3 A^{\nu;3}\partial_X(\partial_Y - t\partial_X) U^3 dV - 2\int A^{\nu;3} Q^3 A^{\nu;3} \partial_X\left(\psi_y(\partial_Y - t\partial_X) + \phi_{y}\partial_{Z}\right) U^3 dV \\
& = LS3^0 + LS3^{C}. \end{aligned}$$ Turn first to $LS3^C$. By , , Lemma \[lem:AnuLossy\], and Lemma \[lem:CoefCtrl\], $$\begin{aligned}
LS3^{C} & \lesssim {\left\lVert \sqrt{-\Delta_L} A^{\nu;3}Q^3 \right\rVert}_2{\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\beta+3\alpha+4}} {\left\lVert A^{\nu;3}U^3 \right\rVert}_2 \nonumber \\
& \lesssim {\left\lVert \sqrt{-\Delta_L} A^{\nu;3}Q^3 \right\rVert}_2{\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\beta+3\alpha+4}}\frac{1}{{\left\langle t \right\rangle}^2}\left({\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 + {\left\lVert A^3Q^3 \right\rVert}_2\right) \nonumber \\
& \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^{\nu;3}Q^3 \right\rVert}^2_2 + \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^2}\left({\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 + {\left\lVert A^3Q^3 \right\rVert}_2\right)^2, \label{ineq:ED_LS3C}\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
For $LS3^0$ we proceed similar to the high norm estimate in §\[sec:LS30\]. As in , we expand $\Delta_L\Delta_t^{-1}$: $$\begin{aligned}
LS3^0 & = -2\int A^{\nu;3} Q^3 A^{\nu;3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \Delta_L \Delta_t^{-1}Q^3 dV \nonumber \\
& = -2\int A^{\nu;3} Q^3 A^{\nu;3}\partial_X(\partial_Y - t\partial_X) \Delta_{L}^{-1} \left[Q^3 - G_{yy} (\partial_Y - t\partial_X)^2 U^3 - G_{yz} \partial_Z(\partial_Y - t\partial_X)U^3 \right. \nonumber \\ & \quad\quad \left. - G_{zz} \partial_{ZZ}U^3 - \Delta_tC^1 (\partial_Y - t\partial_X) U^3 - \Delta_t C^2 \partial_Z U^3\right] dV \nonumber \\
& = LS3^{0;0} + \sum_{i =1}^5 LS3^{0;Ci} . \label{def:LS30nu} \end{aligned}$$ The leading order term is treated as in [@BGM15I], hence we omit the details and simply state the result; for some $K > 0$, $$\begin{aligned}
LS3^{0;0} & \leq CK^{\nu;3}_{L} + \frac{\delta_\lambda}{10{\left\langle t \right\rangle}^{3/2}}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2} A^{\nu;3}Q^3 \right\rVert}_2^2 + \frac{K}{\delta_\lambda^{\frac{1}{2s-1}}{\left\langle t \right\rangle}^{3/2}}{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2^2 + \frac{K}{{\left\langle t \right\rangle}^2} {\left\lVert A^3 Q^3_{\neq} \right\rVert}^2_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] provided $K_{ED3}$ is sufficiently large relative to $K_{H3}$ and $\delta_\lambda$.
Turn to the first error term in , $LS3^{0;C1}$, which by and $\beta + 3\alpha + 6 < \gamma$ is controlled via (using also Lemma \[lem:CoefCtrl\]), $$\begin{aligned}
LS3^{0;C1} & \leq 2{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3} \partial_X(\partial_Y - t\partial_X) \Delta_L^{-1} \left(G_{yy}(\partial_Y - t\partial_X)^2 U^3_{\neq}\right) \right\rVert}_2 \nonumber \\
& \lesssim \frac{1}{{\left\langle t \right\rangle}^5}{\left\lVert A^{\nu;3} Q^3_{\neq} \right\rVert}_{2} {\left\lVert G_{yy} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma-1}} {\left\lVert \Delta_L U^3_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma-1}} + \frac{1}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3} \left(G_{yy}(\partial_Y - t\partial_X)^2 U^3_{\neq}\right) \right\rVert}_2 \nonumber \\
& \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}^2}{\left\lVert A^3 Q^3_{\neq} \right\rVert}_2 {\left\lVert A^3 \Delta_L U^3_{\neq} \right\rVert}_2 + \frac{1}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3} \left(G_{yy}(\partial_Y - t\partial_X)^2 U^3_{\neq}\right) \right\rVert}_2. \label{ineq:LS30C1_ED}\end{aligned}$$ The first term is controlled via Lemma \[lem:SimplePEL\]. To control the second term we use and Lemma \[lem:CoefCtrl\], $$\begin{aligned}
\frac{1}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3} \left(G_{yy}(\partial_Y - t\partial_X)^2 U^3_{\neq}\right) \right\rVert}_2 & \lesssim \frac{1}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 {\left\lVert C \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} {\left\lVert A^{\nu;3}(\partial_Y - t\partial_X)^2 U_{\neq}^3 \right\rVert}_2 \\
& \lesssim \epsilon{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3}(\partial_Y - t\partial_X)^2 U_{\neq}^3 \right\rVert}_2. \end{aligned}$$ By , this is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small. All the other $LS3^{0;Ci}$ error terms are controlled similarly and are hence omitted.
This completes the treatment of $LS3^0$.
### Linear pressure term $LP3$ {#ineq:LP3_ED}
Begin by separating out the contribution of the coefficients, $$\begin{aligned}
LP3 & = 2\int A^{\nu;3} Q^3 A^{\nu;3}\partial_X \partial_Z U_{\neq}^2 dV + 2\int A^{\nu;3} Q^3 A^{\nu;3} \partial_X\left( \left((\psi_z)(\partial_Y - t\partial_X) + (\phi_{z})\partial_Z\right)U_{\neq}^2\right) dV \\
& = LP3^0 + LP3^C.\end{aligned}$$ As in [@BGM15I], Cauchy-Schwarz and , $$\begin{aligned}
LP3^0 & \leq \frac{1}{2\kappa}{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{\nu;3} Q^3_{\neq} \right\rVert}_2^2 + \frac{1}{2\kappa} {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} \Delta_L A^{\nu;3} U^2 \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\kappa$ sufficiently large, $c_0$ sufficiently small, and $K_{ED3} \gg K_{ED2}$ by Lemma \[lem:AnuLossy\_CKnu\].
The coefficient error term, $LP3^C$, can be treated in the same manner as $LS3^C$ above in and yields similar contributions. Hence we omit the treatment for brevity. This completes the treatment of the linear pressure term $LP3$.
### Nonlinear pressure and stretching {#sec:NLPS_Q3ED}
Due to the regularity gap $\beta + 3\alpha +12 \leq \gamma$ and , the presence of the coefficients from the coordinate transform will not greatly impact the treatment of these terms. Moreover, Lemma \[lem:AnuLossy\] shows there is not a significant difference between $\partial_Y - t\partial_X$ and $\partial_Z$ derivatives when making many estimates. Hence, for simplicity we will treat all $NLS$ and $NLP$ terms as if there were no variable coefficients. As in [@BGM15I], we will enumerate the terms as follows for $i,j \in {\left\{1,2,3\right\}}$ and $a,b \in {\left\{0,\neq\right\}}$
\[def:Q3enumnu\] $$\begin{aligned}
NLP(i,j,a,b) & = \int A^{\nu;3} Q^3 A^{\nu;3} \partial_Z^t(\partial_j^t U^i_a \partial_i^t U^j_b ) dV \\
NLS1(j,a,b) & = -\int A^{\nu;3} Q^3 A^{\nu;3} \left( Q^j_a \partial_j^t U^3_b \right) dV \\
NLS2(i,j,a,b) & = -2\int A^{\nu;3} Q^3 A^{\nu;3} (\partial_i^t U^j_a \partial_i^t\partial_j^t U^3_b ) dV.\end{aligned}$$
We will use repeatedly the inequalities
$$\begin{aligned}
A^{\nu; 3} & \lesssim t A^{\nu;1} \\
A^{\nu; 3} & \lesssim A^{\nu;2}. \end{aligned}$$
#### Treatment of $NLP(i,j,0,\neq)$ terms
Recalling, , note that by the usual null structure, we have $j \neq 1$. By $$\begin{aligned}
NLP(i,j,\neq,0) & \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3} {\left\langle \partial_{Z} \right\rangle} \partial^t_i U^j \right\rVert}_2 {\left\lVert U_0^i \right\rVert}_{{\mathcal{G}}^{\lambda,\beta+3\alpha+5}}. \end{aligned}$$ From Lemma \[lem:AnuLossy\], we see that the loss of $t$ if $i=1$ on the third factor is balanced by a gain of $t$ on the second. On the other hand, if $i \neq 1$ then there is no loss of $t$ on the last factor but a loss of $t$ on the second. Therefore, after Lemma \[lem:AnuLossy\] we get $$\begin{aligned}
NLP(i,j,\neq,0) & \lesssim \epsilon {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{j} Q^j_{\neq} \right\rVert}_2 + {\left\lVert A^{\nu;j}Q^j \right\rVert}_2\right),\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
#### Treatment of $NLS1(j,0,\neq)$ terms {#sec:NLS10neq_Q3ED}
Next turn to the treatment of the $NLS1(j,0,\neq)$ terms (recalling ), which by followed by (noting a above that when $j = 1$, the loss of $t$ from the second factor is balanced by a gain of $t$ on the third factor), $$\begin{aligned}
NLS1(j,0,\neq) & \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert Q^j_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha + 4}} {\left\lVert A^{\nu;3}\partial_j^t U^3 \right\rVert}_2 \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 + {\left\lVert A^3 Q^3 \right\rVert} \right) \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
#### Treatment of $NLS1(j,\neq,0)$ terms {#sec:ED3NSL1ijneq0}
Next turn to the treatment of the $NLS1(j,\neq,0)$ terms which by followed by (noting that $j \neq 1$), $$\begin{aligned}
NLS1(j,\neq,0) \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3} Q^j \right\rVert}_2 {\left\lVert U^3_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha + 4}} \lesssim \epsilon {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert A^{\nu;j}Q^j \right\rVert}_2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
#### Treatment of $NLS2(i,j,\neq,0)$ terms
From we see that that *neither* $i$ nor $j$ can be one. Therefore, similar to §\[sec:NLS10neq\_Q3ED\], we get by , $$\begin{aligned}
NLS2(i,j,\neq,0) & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{\nu;j}Q^j \right\rVert}_2 + {\left\lVert A^{j}Q^j_{\neq} \right\rVert}_2 \right)\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
#### Treatment of $NLS2(i,j,0,\neq)$ terms
Next turn to the treatment of the $NLS1(i,j,\neq,0)$ terms, where now notice that $i$ cannot be one but $j$ can. However, if $j = 1$ then we will gain a power of $t$ on $\partial_X U^3_{\neq}$ using Lemma \[lem:AnuLossy\]. Therefore, it follows from and Lemma \[lem:AnuLossy\] that, $$\begin{aligned}
NLS2(i,j,0,\neq) & \lesssim \epsilon {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{3} Q^3_{\neq} \right\rVert}_2 + {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \right). \end{aligned}$$
#### Treatment of $NLP(i,j,\neq,\neq)$ {#sec:NLPneqneq_Q3ED}
Notice that we will lose a power of $t$ from $A^1$ if $j$ or $i$ is one, but in this case we would lose one less power of $t$ in Lemma \[lem:AnuLossy\] due to the presence of $X$ derivatives. Hence regardless of the combination of $i$ and $j$, we will gain at least one power of $t$ Therefore, from , $$\begin{aligned}
NLP(i,j,\neq,\neq) & \leq {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3}\partial_Z^t \left( \partial_i^t U^j_{\neq} \partial_j^t U^i_{\neq}\right) \right\rVert}_2 \\
& \lesssim \frac{t^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \left({\left\lVert A^{\nu;3}\partial^t_{Z} \partial_i^t U^j \right\rVert}_2{\left\lVert A^{\nu;3}\partial_{j}^t U^i \right\rVert}_2 + {\left\lVert A^{\nu;3}\partial_{i}^t U^j \right\rVert}_2{\left\lVert A^{\nu;3}\partial_{Z}^t \partial_j^t U^i \right\rVert}_2\right) \\
& \lesssim \frac{\epsilon^2 {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2^2 + \frac{\epsilon^3 {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}},\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Treatment of $NLS1(j,\neq,\neq)$ {#sec:NLS1neqneq_Q3ED}
These terms are all treated in essentially the same manner. Indeed, using as usual that $j = 1$ loses a power of $t$ from $A^{\nu;1}$ but gains a power from Lemma \[lem:AnuLossy\], we get from and , $$\begin{aligned}
NLS1(j,\neq,\neq) & \lesssim \frac{{\left\langle t \right\rangle}^2 }{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^{\nu;3} Q^3 \right\rVert}_2{\left\lVert A^{\nu;3} Q^j \right\rVert}_2 {\left\lVert A^{\nu;3} \partial_j^t U_{\neq}^3 \right\rVert}_2 \\
& \lesssim \frac{{\left\langle t \right\rangle} }{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^{\nu;3} Q^3 \right\rVert}_2{\left\lVert A^{\nu;j} Q^j \right\rVert}_2\left({\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 + {\left\lVert A^{3} Q^3_{\neq} \right\rVert}_2 \right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Treatment of $NLS2(i,j,\neq,\neq)$ {#sec:NLS2neqneq_Q3ED}
The treatment of $NLS2$ is essentially the same as $NLP$, using again that the losses and gains balance regardless of the combination of $i$ and $j$, we get from and Lemma \[lem:AnuLossy\], $$\begin{aligned}
NLS2(i,j,\neq,\neq)
& \lesssim \frac{t^2}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert A^{\nu;3}\partial_{i} U^j \right\rVert}_2 {\left\lVert A^{\nu;3}\partial_{ij}^t U^3 \right\rVert}_2 \\
& \lesssim \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \left({\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 + {\left\lVert A^{3} Q^3_{\neq} \right\rVert}_2 \right)\left({\left\lVert A^{\nu;j} Q^j \right\rVert}_2 + {\left\lVert A^{j} Q^j_{\neq} \right\rVert}_2 \right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
### Transport nonlinearity {#sec:Trans_ED_Q3}
Divide the transport nonlinearity: $$\begin{aligned}
\mathcal{T} & = -\int A^{\nu;3}Q^3 A^{\nu;3}\left(g \partial_Y Q^3_{\neq}\right) dV - \int A^{\nu;3}Q^3 A^{\nu;3}\left(\tilde U_{\neq} \cdot {\nabla}Q^3_{0}\right) dV -\sum \int A^{\nu;3}Q^3 A^{\nu;3}\left(\tilde U_{\neq} \cdot {\nabla}Q^3_{\neq}\right) dV \\
& = \mathcal{T}_0 + \mathcal{T}_{\neq 0} + \mathcal{T}_{\neq \neq}\end{aligned}$$ Consider first $\mathcal{T}_0$. By and ${\left\vert\eta\right\vert} \leq {\left\vert\eta-kt\right\vert} + {\left\vertkt\right\vert} \leq {\left\langle t \right\rangle}\left({\left\vert\eta-kt\right\vert} + {\left\vertk\right\vert}\right)$, $$\begin{aligned}
\mathcal{T}_0 & \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 {\left\lVert g \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}}{\left\langle t \right\rangle} {\left\lVert \sqrt{-\Delta_L} A^{\nu;3}Q^3 \right\rVert}_2 \lesssim \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{2}}{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^{\nu;3}Q^3 \right\rVert}_2^2. \end{aligned}$$ where the last line followed from both and . Hence, for $\epsilon$ and $c_0$ sufficiently small, this is consistent with Proposition \[prop:Boot\].
Turn next to $\mathcal{T}_{\neq 0}$, which reads $$\begin{aligned}
\mathcal{T}_{\neq 0} & = \int A^{\nu;3}Q^3 A^{\nu;3}\left(\begin{pmatrix} (1 + \psi_y) U^2_{\neq} + \psi_zU^3_{\neq} \\ (1+\phi_{z})U^3_{\neq} + \phi_{y}U_{\neq}^2 \end{pmatrix} \cdot \begin{pmatrix} \partial_{Y} Q_{0}^3 \\ \partial_Z Q_{0}^3 \end{pmatrix}\right) dV. \end{aligned}$$ By , Lemma \[lem:CoefCtrl\], and Lemma \[lem:AnuLossy\], we have $$\begin{aligned}
\mathcal{T}_{\neq 0} & \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{\nu;3}U^2 \right\rVert}_2 + {\left\lVert A^{\nu;3}U^3 \right\rVert}_2\right){\left\lVert {\nabla}Q^3_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} \\
& \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}^2} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 + {\left\lVert A^{2}Q^2_{\neq} \right\rVert}_2 + {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 + {\left\lVert A^{3}Q^3_{\neq} \right\rVert}_2\right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
Turn next to $\mathcal{T}_{\neq \ne}$, which is the most subtle contribution. This is written $$\begin{aligned}
\mathcal{T}_{\neq \neq} & = \int A^{\nu;3}Q^3 A^{\nu;3}\left(\begin{pmatrix}U^1_{\neq} \\ (1 + \psi_y) U^2_{\neq} + \psi_zU^3_{\neq} \\ (1+\phi_{z})U^3_{\neq} + \phi_{y}U_{\neq}^2 \end{pmatrix} \cdot \begin{pmatrix} \partial_X Q^3_{\neq} \\ (\partial_{Y} - t\partial_X)Q_{\neq}^3 \\ \partial_Z Q_{\neq}^3 \end{pmatrix}\right) dV. \end{aligned}$$ By Cauchy-Schwarz, , Lemma \[lem:CoefCtrl\] and , we get $$\begin{aligned}
\mathcal{T}_{\neq \neq} & \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{{\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \left({\left\lVert A^{\nu;3}U^3 \right\rVert}_2 + {\left\lVert A^{\nu;3}U^2 \right\rVert}_2\right) {\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} \\
& \quad + {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{{\left\langle t \right\rangle}^{2}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;3}U^1 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} + {\left\lVert A^{\nu;3}U^1 \right\rVert}_2{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\right); \end{aligned}$$ note the extra precision applied to the treatment of $U^1$. By $$\begin{aligned}
\frac{1}{{\left\langle \eta,l \right\rangle}} A^{\nu;3}_k(\eta,l) & \approx \frac{{\left\langle t \right\rangle}}{{\left\langle \eta,l \right\rangle} {\left\langle \frac{t}{{\left\langle \eta,l \right\rangle}} \right\rangle}^{1-\delta_1} } A^{\nu;1} \lesssim {\left\langle t \right\rangle}^{\delta_1}A^{\nu;1}, \label{ineq:A3nuA1nuRelation}\end{aligned}$$ it follows that $$\begin{aligned}
\mathcal{T}_{\neq \neq} & \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{{\left\langle t \right\rangle}^2}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \left({\left\lVert A^{\nu;3}U^3 \right\rVert}_2 + {\left\lVert A^{\nu;2}U^2 \right\rVert}_2\right) {\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} \\
& \quad + {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{{\left\langle t \right\rangle}^{2+\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert A^{\nu;1}U^1 \right\rVert}_2{\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} + t{\left\lVert A^{\nu;1}U^1 \right\rVert}_2{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\right). \end{aligned}$$ Applying from to the $t{\left\lVert A^{\nu;3}Q^3 \right\rVert}_2$ in the last factor and Lemma \[lem:AnuLossy\] to all factors (also with ) it follows that, $$\begin{aligned}
& \lesssim {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{1}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} \left({\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 + {\left\lVert A^{3}Q^3_{\neq} \right\rVert}_2\right) {\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} \\
& \quad + {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{1}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 + {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2\right) {\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} \\
& \quad + {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 \frac{t^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left({\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 + {\left\lVert A^1Q^1_{\neq} \right\rVert}_2\right)\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} + {\left\lVert A^{3}Q^3_{\neq} \right\rVert}_2\right) \\
& \lesssim \frac{\epsilon t^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2} + {\left\lVert A^{3}Q^3_{\neq} \right\rVert}_2\right) \\
& \lesssim \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^{\nu;3} Q^3 \right\rVert}_{2}^2 + \frac{\epsilon^{1/2} t^{2\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^{\nu;3}Q^3 \right\rVert}^2_2 + \frac{\epsilon^{3/2}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}} {\left\lVert A^3 Q^3_{\neq} \right\rVert}_2^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$, $\delta_1$, and $c_0$ sufficiently small (also $\delta > 0$).
### Dissipation error terms {#sec:DE_ED_Q3}
The dissipation error terms are easily absorbed by the dissipation as in [@BMV14; @BGM15I] using together with the regularity gap between $A^{\nu;3}$ and the coefficient control in . We hence omit the treatment for brevity.
Enhanced dissipation of $Q^2$
-----------------------------
The enhanced dissipation of $Q^2$ is deduced in a manner very similar to $Q^3$, however, since we are imposing more control on $Q^2$, some nonlinear interactions must be handled with more precision.
Computing the time evolution of ${\left\lVert A^{\nu;2}Q^2 \right\rVert}_2$ we get $$\begin{aligned}
\frac{1}{2}\frac{d}{dt}{\left\lVert A^{\nu;2} Q^2 \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{\nu;2} Q^2 \right\rVert}_2^2 -\frac{1}{t}{\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{\nu;2} Q^2 \right\rVert}_2^2 - {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}}A^{\nu;2}Q^2 \right\rVert}_2^2 + G^\nu \nonumber \\ & \quad + \nu \int A^{\nu;2} Q^{2} A^{\nu;2} \left(\tilde{\Delta_t} Q^2\right) dV -\int A^{\nu;2} Q^2 A^{\nu;2}\left( \tilde U \cdot {\nabla}Q^2 \right) dV \nonumber \\
& \quad - \int A^{\nu;2} Q^2 A^{\nu;2} \left[\left(Q^j \partial_j^t U^2\right) + 2\partial_i^t U^j \partial_{ij}^t U^2 - \partial_Y^t\left(\partial_i^t U^j \partial_j^t U^i\right) \right] dV \nonumber \\
& = -\mathcal{D}Q^{\nu;2} - CK_{L}^{\nu;2} + G^{\nu} + \mathcal{D}_E + \mathcal{T} + NLS1 + NLS2 + NLP, \label{ineq:AnuEvo2}\end{aligned}$$ where as in §\[sec:ED3\], we write $$\begin{aligned}
\mathcal{D}_E & = \nu \int A^{\nu;2} Q^2 A^{\nu;2}\left(\tilde{\Delta_t}Q^2 - \Delta_L Q^2\right) dV, \end{aligned}$$ and $$\begin{aligned}
G^\nu = \alpha \int A^{\nu;2} Q^2 \min\left(1, \frac{{\left\langle {\nabla}_{Y,Z} \right\rangle}}{t}\right) e^{\lambda(t){\left\vert{\nabla}\right\vert}^s}{\left\langle {\nabla}\right\rangle}^\beta{\left\langle D(t,\partial_v) \right\rangle}^{\alpha-1} \frac{D(t,\partial_v)}{{\left\langle D(t,\partial_v) \right\rangle}} \partial_t D(t,\partial_v) Q^2_{\neq} dV. \end{aligned}$$ As in we have $$\begin{aligned}
-\nu {\left\lVert \sqrt{-\Delta_L}A^{\nu;2} Q^{2}_{\neq} \right\rVert}_2^2 + G^{\nu}& \leq -\frac{\nu}{8}{\left\lVert \sqrt{-\Delta_L}A^{\nu;2} Q^{2} \right\rVert}_2^2. \end{aligned}$$
### Nonlinear pressure and stretching {#nonlinear-pressure-and-stretching-1}
In this section we treat $NLS1$, $NLS2$ and $NLP$. As in §\[sec:NLPS\_Q3ED\], for simplicity we will treat all $NLS$ and $NLP$ terms as if there were no variable coefficients. We also recall the following enumeration from [@BGM15I], for $i,j \in {\left\{1,2,3\right\}}$ and $a,b \in {\left\{0,\neq\right\}}$
\[def:Q2enumnu\] $$\begin{aligned}
NLP(i,j,a,b) & = \int A^{\nu;2} Q^2 A^{\nu;2} \partial_Y^t(\partial_j^t U^i_a \partial_i^t U^j_b ) dV \\
NLS1(j,a,b) & = -\int A^{\nu;2} Q^2 A^{\nu;2} \left( Q^j_a \partial_j^t U^2_b \right) dV \\
NLS2(i,j,a,b) & = -2\int A^{\nu;2} Q^2 A^{\nu;2} (\partial_i^t U^j_a \partial_i^t\partial_j^t U^2_b ) dV.\end{aligned}$$
We will use repeatedly the inequalities
$$\begin{aligned}
A^{\nu; 2} & \lesssim t^{1+\delta_1}A^{\nu;1} \\
A^{\nu; 2} & \lesssim tA^{\nu;3}. \end{aligned}$$
#### Treatment of $NLP(i,j,0,\neq)$ terms {#sec:NLP0neq_Q2ED}
This includes terms identified in §\[sec:Toy\] as requiring that $Q^2$ grow linearly at low frequencies, and we will see that we will need this in order to estimate these terms. By , $$\begin{aligned}
NLP(i,j,0,\neq) & \lesssim {\left\lVert A^{\nu;2} Q^2 \right\rVert}_2 {\left\lVert U_0^i \right\rVert}_{{\mathcal{G}}^{\lambda,\beta+3\alpha + 5}} {\left\lVert A^{\nu;2} {\left\langle \partial_Y^t \right\rangle} \partial_i U^j \right\rVert}_2. \end{aligned}$$ With in mind, the power of $t$ lost from the derivatives or $j=1$ together is at most two and the powers of $t$ lost from the possibility that $j=3$ is also at most an additional one (also note $j \neq 1$), so at worst we get from Lemma \[lem:AnuLossy\] (which recovers the powers of time), and , $$\begin{aligned}
NLP(i,j,0,\neq) & \lesssim \epsilon{\left\lVert A^{\nu;2} Q^2 \right\rVert}_2 \left( {\left\lVert \sqrt{-\Delta_L} A^{\nu;j} Q^j \right\rVert}_2 + {\left\lVert \sqrt{-\Delta_L} A^{j} Q^j_{\neq} \right\rVert}_2\right) \\
& \lesssim \epsilon^{1/2}{\left\lVert A^{\nu;2} Q^2 \right\rVert}_2^{2} + \epsilon^{3/2}\left( {\left\lVert \sqrt{-\Delta_L} A^{\nu;j} Q^j \right\rVert}^2_2 + {\left\lVert \sqrt{-\Delta_L} A^{j} Q^j_{\neq} \right\rVert}^2_2\right). \end{aligned}$$ For $\epsilon$ sufficiently small this is consistent with Proposition \[prop:Boot\] for times until $t \sim \epsilon^{-1/2+\delta/100}$. At this point we can apply to the first term and deduce $$\begin{aligned}
NLP(i,j,0,\neq) & \lesssim \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^2}{\left\lVert A^{2} Q^2_{\neq} \right\rVert}_2^{2} + \frac{\epsilon}{{\left\langle t \right\rangle}^2}{\left\lVert \sqrt{-\Delta_L} A^{\nu;2} Q^2 \right\rVert}^2_2 + \epsilon^{3/2}\left( {\left\lVert \sqrt{-\Delta_L} A^{\nu;j} Q^j \right\rVert}_2 + {\left\lVert \sqrt{-\Delta_L} A^{j} Q^j_{\neq} \right\rVert}_2\right) \nonumber \\
& \lesssim \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^2}{\left\lVert A^{2} Q^2_{\neq} \right\rVert}_2^{2} + \epsilon^{3/2-\delta/50} {\left\lVert \sqrt{-\Delta_L} A^{\nu;2} Q^2 \right\rVert}^2_2 \nonumber \\ & \quad + \epsilon^{3/2}\left( {\left\lVert \sqrt{-\Delta_L} A^{\nu;j} Q^j \right\rVert}_2 + {\left\lVert \sqrt{-\Delta_L} A^{j} Q^j_{\neq} \right\rVert}_2\right), \label{ineq:NLPij0neq_Q2ED}\end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for all time for $\epsilon$ sufficiently small.
#### Treatment of $NLS1(j,0,\neq)$ terms {#sec:NLS10neq_Q2ED}
These terms are straightforward by , , and ; we omit the details and conclude $$\begin{aligned}
NLS1(j,0,\neq) & \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;2}Q^2 \right\rVert}_2\left({\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 + {\left\lVert A^{2}Q^2_{\neq} \right\rVert}_2\right). \end{aligned}$$
#### Treatment of $NLS1(j,\neq,0)$ terms {#sec:NLS1neq0_Q2ED}
Due to the nonlinear structure, $j \neq 1$. Hence, the worst possibility is $j = 3$, where at most one power of time is lost – notice that this also depends on the linear growth at low frequencies of $Q^2$. Hence, this term emphasizes this important difference with [@BGM15I]. Hence, by , , and , $$\begin{aligned}
NLS1(j,\neq,0) & \lesssim \epsilon\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;2}Q^2 \right\rVert}_2 + {\left\lVert A^{2}Q^2_{\neq} \right\rVert}_2\right){\left\lVert A^{\nu;j}Q^{j} \right\rVert}_2 \\
& \lesssim \epsilon^{3/2}\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;2}Q^2 \right\rVert}_2^2 + {\left\lVert A^{2}Q^2_{\neq} \right\rVert}_2^2\right) + \epsilon^{1/2}{\left\lVert A^{\nu;j}Q^{j} \right\rVert}_2^2. \end{aligned}$$ By applying for $t \gtrsim \epsilon^{-1/2+\delta/100}$ as in , this is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Treatment of $NLS2(i,j,0,\neq)$ terms
These are treated similar to the analogous $NLS1$ terms in §\[sec:NLS10neq\_Q2ED\], yielding the following $$\begin{aligned}
NLS2(i,j,0,\neq) \lesssim \epsilon{\left\lVert A^{\nu;2}Q^2 \right\rVert}\left({\left\lVert A^{\nu;2} Q^2 \right\rVert} + {\left\lVert A^2 Q^2_{\neq} \right\rVert}_2\right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_{0}$ sufficiently small.
#### Treatment of $NLS2(i,j,\neq,0)$ terms
Again, due to the nonlinear structure, $j \neq 1$ and $i \neq 1$. By , $$\begin{aligned}
NLS2(i,j,\neq,0) & \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 {\left\lVert \partial_i^t A^{\nu ;2} U^j_{\neq} \right\rVert}_2 {\left\lVert U_0^2 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta+3\alpha + 7}}.\end{aligned}$$ The worst case is $j=3$ and $i = 2$, however, even in this case Lemma \[lem:AnuLossy\] recovers all of the time losses due to the permitted linear growth in $Q^2$ (also applying ): $$\begin{aligned}
NLS2(i,j,\neq,0) & \lesssim \epsilon{\left\lVert A^{\nu;2}Q^2 \right\rVert}_2\left({\left\lVert A^{\nu;j}Q^j \right\rVert}_2 + {\left\lVert A^{j}Q_{\neq}^j \right\rVert}_2 \right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $c_{0}$ sufficiently small.
#### Treatment of $NLP(i,j,\neq,\neq)$ {#sec:NLPneqneq_Q2ED}
Turn next to the nonlinear pressure interactions of two non-zero frequencies, which requires a careful treatment. First, observe that the case $i = j = 2$ cancels with the $NLS2$ term. By , $$\begin{aligned}
NLP(i,j,\neq,\neq) & \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \left({\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta}A^{\nu;2} \partial_j^t U^i_{\neq} \right\rVert}_2 {\left\lVert A^{\nu;2}\partial_Y^t \partial_i^tU^j_{\neq} \right\rVert}_2 \right.\\
& \quad \left. + {\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;2} \partial_Y^t \partial_j^t U^i_{\neq} \right\rVert}_2 {\left\lVert A^{\nu;2}\partial_i^tU^j_{\neq} \right\rVert}_2 + {\left\lVert A^{\nu;2} \partial_j^t U^i_{\neq} \right\rVert}_2 {\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;2}\partial_Y^t \partial_i^tU^j_{\neq} \right\rVert}_2 \right.\\
& \quad \left. + {\left\lVert A^{\nu;2} \partial_Y^t \partial_j^t U^i_{\neq} \right\rVert}_2 {\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;2}\partial_i^tU^j_{\neq} \right\rVert}_2\right). \end{aligned}$$ Each combination of $i$ and $j$ can be treated in a rather similar manner, each time using and Lemma \[lem:AnuLossy\]. As could be expected, $NLP(1,3,\neq,\neq)$ and $NLP(3,3,\neq,\neq)$ turn out to be the hardest. Let us focus on the case $NLP(3,3,\neq,\neq)$ and omit the easier cases for brevity. Note that the inverse derivatives can recover losses associated with $\partial_Z$ but not $\partial_Y - t\partial_X$. They will also still work when considering $\partial_Z^t = (1 + \phi_z)\partial_Z + \psi_z(\partial_Y - t\partial_X)$, since it will introduce $O(\epsilon t^2)$ powers that are absorbed using $\epsilon t^2 {\left\langle \nu t^3 \right\rangle}^{-1} \lesssim 1$. Hence, we can continue to ignore the coefficients. By Lemma \[lem:AnuLossy\] and there holds, $$\begin{aligned}
NLP(3,3,\neq,\neq) & \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \frac{{\left\langle t \right\rangle}^{3}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \left({\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta}A^{\nu;3} \partial_Z^t U^3_{\neq} \right\rVert}_2 {\left\lVert A^{\nu;3}\partial_Z^t \partial_Z^tU^3_{\neq} \right\rVert}_2 \right.\\
& \quad \left. + {\left\lVert {\left\langle {\nabla}\right\rangle}^{2-\beta} A^{\nu;3} \partial_Z^t \partial_Z^t U^3_{\neq} \right\rVert}_2 {\left\lVert A^{\nu;3}\partial_Z^tU^3_{\neq} \right\rVert}_2\right) \\
& \lesssim \frac{\epsilon{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha}\left(1 + \epsilon t^2\right){\left\lVert A^{\nu;2}Q^2 \right\rVert}_2\left({\left\lVert A^{\nu;3} Q^3 \right\rVert}_2 + {\left\lVert A^{3} Q^3_{\neq} \right\rVert}_2\right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. The other terms can be treated with a simple variation or easier arguments and are hence omitted.
#### Treatment of $NLS1(j,\neq,\neq)$ {#sec:NLS1neqneq_Q2ED}
By , , and , we have the following (e.g. consider the worst case of $j = 3$), $$\begin{aligned}
NLS1(j,\neq,\neq) & \lesssim \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^{\nu;2}Q^2 \right\rVert}_2{\left\lVert A^{\nu;2} Q^j \right\rVert}_2 {\left\lVert A^{\nu;2} \partial_j^t U^2_{\neq} \right\rVert}_2 \\
& \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^{\alpha}}{\left\lVert A^{\nu;2}Q^2 \right\rVert}_2\left({\left\lVert A^{\nu;2} Q^2 \right\rVert}_2 + {\left\lVert A^{2} Q^2_{\neq} \right\rVert}_2\right),\end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
#### Treatment of $NLS2(i,j,\neq,\neq)$ {#sec:NLS2neqneq_Q2ED}
First, note that the $i = j = 2$ term cancels with $NLP$. For the remaining terms we again apply to deduce $$\begin{aligned}
NLS2(i,j,\neq,\neq) & \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^{\nu;2} \partial_i^t U^j_{\neq} \right\rVert}_2 {\left\lVert A^{\nu;2}\partial_{ij}^tU^2_{\neq} \right\rVert}_2. \end{aligned}$$ The most problematic term is $j = 3$, $i = 2$; however by and , $$\begin{aligned}
NLS2(2,3,\neq,\neq) & \lesssim \frac{\epsilon {\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \left({\left\lVert A^{\nu;2} Q^2 \right\rVert}_2 + {\left\lVert A^{2} Q^2_{\neq} \right\rVert}_2 \right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small. The other cases can be treated similarly and are hence omitted for brevity. This completes the treatment of all of the nonlinear pressure and stretching terms.
### Transport nonlinearity {#sec:Trans_ED_Q2}
These terms are easier than the analogous terms in §\[sec:Trans\_ED\_Q3\]. As noted in [@BGM15I], this is consistent with the observation that the so-called “reaction” terms are stronger in $Q^3$ than $Q^2$ (note that $Q^3$ reaction terms are included in the toy model in §\[sec:Toy\] but the $Q^2$ reaction terms are not; see [@BGM15I] for more information). Write the transport nonlinearity as $$\begin{aligned}
\mathcal{T} & = -\int A^{\nu;2}Q^2 A^{\nu;2}\left(g \partial_Y Q^2_{\neq}\right) dV - \int A^{\nu;2}Q^2 A^{\nu;2} \left(\tilde U_{\neq} \cdot {\nabla}Q^2_{0}\right) dV \\
& \quad - \int A^{\nu;2}Q^2 A^{\nu;2} \left(\tilde U_{\neq} \cdot {\nabla}Q^2_{\neq}\right) dV \\
& = \mathcal{T}_0 + \mathcal{T}_{\neq 0} + \mathcal{T}_{\neq \neq}. \end{aligned}$$ As in §\[sec:Trans\_ED\_Q3\], we have $$\begin{aligned}
\mathcal{T}_0 & \lesssim \frac{\epsilon^{1/2}}{{\left\langle t \right\rangle}^{2}}{\left\lVert A^{\nu;2}Q^2 \right\rVert}_2^2 + \epsilon^{3/2}{\left\lVert \sqrt{-\Delta_L} A^{\nu;2}Q^2 \right\rVert}_2^2.\end{aligned}$$ Similarly, we can treat $\mathcal{T}_{\neq 0}$ as we did in §\[sec:Trans\_ED\_Q3\]: , Lemma \[lem:CoefCtrl\], and Lemma \[lem:AnuLossy\], we have $$\begin{aligned}
\mathcal{T}_{\neq 0} & \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2\left({\left\lVert A^{\nu;2}U^2 \right\rVert}_2 + {\left\lVert A^{\nu;2}U^3 \right\rVert}_2\right){\left\lVert {\nabla}Q^2_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma}} \\
& \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}^2} {\left\lVert A^{\nu;3}Q^3 \right\rVert}_2\left({\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 + {\left\lVert A^{2}Q^2_{\neq} \right\rVert}_2 + {\left\langle t \right\rangle}\left({\left\lVert A^{\nu;3}Q^3 \right\rVert}_2 + {\left\lVert A^{3}Q^3_{\neq} \right\rVert}_2\right)\right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\].
For $\mathcal{T}_{\neq \neq}$, we get from , $$\begin{aligned}
\mathcal{T}_{\neq \neq} & \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \left({\left\lVert A^{\nu;2}U^1 \right\rVert}_2 + {\left\lVert A^{\nu;2}U^2 \right\rVert}_2 + {\left\lVert A^{\nu;2}U^3 \right\rVert}_2\right){\left\lVert \sqrt{-\Delta_L} A^{\nu;2} Q^2 \right\rVert}_{2} \\
& \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \frac{{\left\langle t \right\rangle}}{{\left\langle \nu t^3 \right\rangle}^\alpha} \left({\left\langle t \right\rangle}^{1+\delta_1}{\left\lVert A^{\nu;1}U^1 \right\rVert}_2 + {\left\lVert A^{\nu;2}U^2 \right\rVert}_2 + {\left\langle t \right\rangle}{\left\lVert A^{\nu;3}U^3 \right\rVert}_2\right){\left\lVert \sqrt{-\Delta_L} A^{\nu;2} Q^2 \right\rVert}_{2} \\
& \lesssim {\left\lVert A^{\nu;2}Q^2 \right\rVert}_2 \frac{\epsilon {\left\langle t \right\rangle}^{\delta_1}}{{\left\langle \nu t^3 \right\rangle}^\alpha}{\left\lVert \sqrt{-\Delta_L} A^{\nu;2} Q^2 \right\rVert}_{2} \\
& \lesssim \epsilon^{3/2} {\left\lVert \sqrt{-\Delta_L} A^{\nu;2} Q^2 \right\rVert}_{2}^2 + \frac{\epsilon^{1/2} {\left\langle t \right\rangle}^{2\delta_1}}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} {\left\lVert A^{\nu;2}Q^2 \right\rVert}^2_2,\end{aligned}$$ which completes the treatment of $\mathcal{T}_{\neq \neq}$.
### Dissipation error terms {#dissipation-error-terms-1}
As in §\[sec:DE\_ED\_Q3\], these terms are treated in the same manner as the analogous terms in [@BMV14; @BGM15I]; the details are omitted for brevity.
Enhanced dissipation of $Q^1$
-----------------------------
Computing the time evolution of ${\left\lVert A^{\nu;1}Q^1 \right\rVert}_2$, we get $$\begin{aligned}
\frac{1}{2}\frac{d}{dt} {\left\lVert A^{\nu;1} Q^1 \right\rVert}_2^2 & \leq \dot{\lambda}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{\nu;1} Q^1 \right\rVert}_2^2 + G^\nu -{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{\nu;1} Q^1 \right\rVert}_2^2 \nonumber \\
& \quad - \frac{t}{{\left\langle t \right\rangle}^{2}}{\left\lVert A^{\nu;1}Q^1 \right\rVert}_2^2 -\frac{(1+\delta_1)}{t} {\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{\nu;1} Q^1 \right\rVert}_2^2 \nonumber \\
& \quad - \int A^{\nu;1}Q^1 A^{\nu;1} Q^2 dV -2 \int A^{\nu;1} Q^1 A^{\nu;1} \partial_{YX}^t U^1 dV \nonumber \\
& \quad + 2 \int A^{\nu;1} Q^1 A^{\nu;1} \partial_{XX} U^2 dV + \nu\int A^{\nu;1} Q^{1} A^{\nu;1} \left(\tilde{\Delta_t} Q^1\right) dv \nonumber \\
& \quad -\int A^{\nu;1} Q^1 A^{\nu;1}\left( \tilde U \cdot {\nabla}Q^2 \right) dv \nonumber \\
& \quad -\int A^{\nu;1} Q^1 A^{\nu;1} \left[\left(Q^j \partial_j^t U^1\right) + 2\partial_i^t U^j \partial_{ij}^t U^1 - \partial_X\left(\partial_i^t U^j \partial_j^t U^i\right) \right] dv \nonumber \\
& = -\mathcal{D}Q^{\nu;1} + G^\nu - CK_{L1}^{\nu;1} - (1+\delta_1) CK_{L2}^{\nu;1} \nonumber \\ & \quad + LU + LS1 + LP1 + \mathcal{D}_E + \mathcal{T} + NLS1 + NLS2 + NLP. \label{ineq:AnuEvo1}\end{aligned}$$ where $G^\nu$ is analogous to the corresponding term in . As in §\[sec:ED3\], $G^\nu$ is absorbed by using the dissipation. Note that for $i \in {\left\{2,3\right\}}$, $$\begin{aligned}
A^{\nu; 1} & \lesssim A^{\nu;i}. \end{aligned}$$
### Linear terms
The treatment of $LU$ and $LS1$ can be made analogous to the linear terms treated in §\[sec:ED3\] combined with the $t^{\delta_1}$ tweak introduced for the improvement of in §\[sec:LUQhi2\]. We omit the details for brevity and conclude for some $K > 0$, $$\begin{aligned}
LU & \leq \delta_1t{\left\langle t \right\rangle}^{-2}{\left\lVert A^{\nu;1} Q^1_{\neq} \right\rVert}^2_2 + \frac{\delta_\lambda}{4\delta_1 t^{3/2}}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2}A^{\nu;2} Q^2_{\neq} \right\rVert}_2^2 + \frac{K}{\delta_1 \delta_\lambda^{\frac{1}{2s-1}} t^{3/2}}{\left\lVert A^{\nu;2} Q^2_{\neq} \right\rVert}_2^2 + \frac{K}{\delta_1 t} {\left\lVert \mathbf{1}_{t > {\left\langle {\nabla}_{Y,Z} \right\rangle}} A^{\nu;2}Q^2 \right\rVert}_2^2, \\
& = \delta_1CK_{L1}^{\nu;1} + \frac{1}{4\delta_1}CK_{\lambda}^{\nu;2} + \frac{K}{\delta_1}CK_{L}^{\nu;2} + \frac{K}{\delta_1 \delta_\lambda^{\frac{1}{2s-1}} t^{3/2}}{\left\lVert A^{\nu;2} Q^2_{\neq} \right\rVert}_2^2, \end{aligned}$$ and, $$\begin{aligned}
LS1 & \leq (1+\delta_1)CK^{\nu;1}_{L2} + (1-\delta_1)CK_{L1}^{\nu;1} + \frac{\delta_\lambda}{10{\left\langle t \right\rangle}^{3/2}}{\left\lVert {\left\vert{\nabla}\right\vert}^{s/2} A^{\nu;1}Q^1 \right\rVert}_2^2 + \frac{K\epsilon}{{\left\langle t \right\rangle}^2}{\left\lVert A^{1}\Delta_L U^1_{\neq} \right\rVert}_2^2 \\ & \quad + \frac{K}{{\left\langle t \right\rangle}^2}{\left\lVert A^1 Q^1_{\neq} \right\rVert}^2_2 + \frac{K}{\delta_\lambda^{\frac{1}{2s-1}}{\left\langle t \right\rangle}^{3/2}} {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2^2 + \epsilon{\left\lVert A^{\nu;1}Q^1 \right\rVert}_2{\left\lVert \Delta_L A^{\nu;1}U^1_{\neq} \right\rVert}_2, \end{aligned}$$ which, after Lemmas \[lem:SimplePEL\] and \[lem:AnuLossy\], are both consistent with Proposition \[prop:Boot\] provided $K_{ED1}$ is chosen large relative to both $K_{ED2}$ and $K_{H1}$ (and $\delta_\lambda$, $\delta_1^{-1}$, $K$ and universal constants).
Next consider the linear pressure term $LP1$. We may directly apply Lemma \[lem:AnuLossy\] to deduce $$\begin{aligned}
LP1 \leq 2{\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 {\left\lVert \partial_{XX} A^{\nu;1}U^2_{\neq} \right\rVert}_2 &\lesssim {\left\langle t \right\rangle}^{-3} {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2\left({\left\lVert A^{\nu;2}Q^2_{\neq} \right\rVert}_2 + {\left\lVert A^{2}Q^2_{\neq} \right\rVert}_2\right) \\
& \lesssim \frac{1}{{\left\langle t \right\rangle}^{3}}{\left\lVert A^{\nu;1}Q^1 \right\rVert}^2_2 + \frac{1 + K_{ED2}}{{\left\langle t \right\rangle}^{3}}\epsilon^2, \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] via integrating factors provided $K_{ED1} \gg K_{ED2}$.
### Nonlinear pressure and stretching {#nonlinear-pressure-and-stretching-2}
These terms are treated in essentially the same manner as in §\[sec:NLPS\_Q3ED\]; we only briefly sketch a few terms. We use enumerations analogous to those employed in .
#### Treatment of $NLP(i,j,0,\neq)$ terms {#sec:NLP0neq_Q1ED}
Notice that in this case $j \neq 1$. From , Lemma \[lem:AnuLossy\], and , $$\begin{aligned}
NLP(i,j,0,\neq) & \lesssim {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 {\left\lVert A^{\nu;1} \partial_X\partial_i^t U^j \right\rVert}_2{\left\lVert U_0^i \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha + 5}} \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 \left({\left\lVert A^{\nu;j} Q^j \right\rVert}_2 + {\left\lVert A^{j}Q^j_{\neq} \right\rVert}_2\right). \end{aligned}$$
#### Treatment of $NLS1(j,0,\neq)$ terms {#sec:NLS10neq_Q1ED}
From , Lemma \[lem:AnuLossy\], and , $$\begin{aligned}
NLS1(j,0,\neq) \lesssim {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 {\left\lVert Q^j_0 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha + 5}} {\left\lVert \partial_j^t A^{\nu;1}U^1 \right\rVert}_2 \lesssim \frac{\epsilon}{{\left\langle t \right\rangle}}{\left\lVert A^{\nu;1}Q^1 \right\rVert}_2\left({\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 + {\left\lVert A^{1}Q^1_{\neq} \right\rVert}_2 \right).\end{aligned}$$
#### Treatment of $NLS1(j,\neq,0)$ terms {#sec:NLS1neq0_Q1ED}
Note in this case that $j \neq 1$. From , Lemma \[lem:AnuLossy\], , and , $$\begin{aligned}
NLS1(j,\neq,0) & \lesssim {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 {\left\lVert A^{\nu;1} Q^j \right\rVert}_2{\left\lVert U_0^1 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha + 5}} \\
& \lesssim \epsilon^{3/2}\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;1}Q^1 \right\rVert}^2_2 + {\left\lVert A^{1}Q^1_{\neq} \right\rVert}_2^2 \right) + \epsilon^{1/2} {\left\lVert A^{\nu;j} Q^j \right\rVert}_2^2, \end{aligned}$$ which suffices for $t \lesssim \epsilon^{-1/2+\delta/100}$, after which we use again to deduce $$\begin{aligned}
NLS1(j,\neq,0) & \lesssim \epsilon^{3/2}\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;1}Q^1 \right\rVert}^2_2 + {\left\lVert A^{1}Q^1_{\neq} \right\rVert}_2^2 \right) + \epsilon^{3/2-\delta/50} \left({\left\lVert \sqrt{-\Delta_L} A^{\nu;j}Q^j \right\rVert}^2_2 + {\left\lVert A^{j}Q^j_{\neq} \right\rVert}_2^2 \right), \end{aligned}$$ which is consistent with Proposition \[prop:Boot\] for $\epsilon$ sufficiently small.
#### Treatment of $NLS2(i,j,\neq,0)$ terms {#sec:NLS2neq0_Q1ED}
From , Lemma \[lem:AnuLossy\], and , we have $$\begin{aligned}
NLS2(i,j,\neq,0) & \lesssim {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 {\left\lVert A^{\nu;1} \partial_i^t U^j_{\neq} \right\rVert}_2 {\left\lVert U_0^1 \right\rVert}_{{\mathcal{G}}^{\lambda,\beta + 3\alpha +6}} \lesssim \epsilon {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 \left({\left\lVert A^{\nu;j}Q^j \right\rVert}_2 + {\left\lVert A^{j}Q^j_{\neq} \right\rVert}_2\right). \end{aligned}$$
#### Treatment of $NLS2(i,j,0,\neq)$ terms {#sec:NLS20neq_Q1ED}
From , Lemma \[lem:AnuLossy\], and . we have (noting that $i \neq 1$): $$\begin{aligned}
NLS2(i,j,0,\neq) & \lesssim {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 {\left\lVert A^{\nu;1} \partial_{ij}^t U^1 \right\rVert}_2 {\left\lVert U_0^j \right\rVert}_{{\mathcal{G}}^{\lambda,\beta+3\gamma+5}} \lesssim \epsilon {\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 \left({\left\lVert A^{\nu;1}Q^1 \right\rVert}_2 + {\left\lVert A^{1}Q^1_{\neq} \right\rVert}_2\right). \end{aligned}$$ Notice that we again used the structure which for $j = 1$, balances the loss of ${\left\langle t \right\rangle}$ from the third factor with a gain of ${\left\langle t \right\rangle}^{-1}$ from the second factor.
#### Treatment of $NLP(i,j,\neq,\neq)$, $NLS1(i,j,\neq,\neq)$, and $NLS2(i,j,\neq,\neq)$
The nonlinear terms involving two non-zero frequencies can all be treated in essentially the same manner as in $Q^3$ in §\[sec:NLPneqneq\_Q3ED\], §\[sec:NLS1neqneq\_Q3ED\] and §\[sec:NLS1neqneq\_Q3ED\]. We omit the treatments for the sake of brevity.
### Transport nonlinearity {#transport-nonlinearity-2}
The transport nonlinearity, $\mathcal{T}$ in , can be treated in the same manner as the transport nonlinearity in §\[sec:Trans\_ED\_Q3\]. We omit the details for brevity.
### Dissipation error terms {#dissipation-error-terms-2}
The dissipation error terms can be treated in same manner as those in §\[sec:DE\_ED\_Q3\] and [@BMV14; @BGM15I], and hence we omit the details for brevity. This completes the enhanced dissipation estimate on $Q^1$.
Sobolev estimates {#sec:LowNrmVel}
=================
In this section we improve the $H^{\sigma^\prime}$ estimates in , which are more straightforward than the analogous estimates proved in [@BGM15I] (the main challenge in [@BGM15I] was getting good decay properties for $t \gtrsim \nu^{-1}$, which is irrelevant here). As in [@BGM15I], these estimates are performed in the coordinate system given by $(X,y,z)$; see §\[sec:RegCont\]. In Lemma \[lem:intermedSob\], the a priori estimates from the bootstrap hypotheses in these coordinates are given. The estimates are performed on and then transferred back to the $(X,Y,Z)$ coordinates. Indeed, as long as the $C^i$ remain small, the coordinate change is uniformly bounded in Sobolev regularity, and hence by suitably adjusting the constants in , one can prove these finite regularity estimates in whichever coordinate system is most convenient (see [@BGM15I] for more details).
Improvement of and {#sec:U023Low}
-------------------
These estimates are best proved together using a standard energy method. Recall the notation $u_0 = (u_0^2, u_0^3)^{T}$. From , $$\begin{aligned}
\frac{1}{2}\frac{d}{dt} {\left\lVert u_0 \right\rVert}_{H^{\sigma^\prime}}^2 & = -\nu {\left\lVert {\nabla}u_0 \right\rVert}_{H^{\sigma^\prime}}^2 - \int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \left( u_0^j \cdot \partial_j u_0^i \right) dy dz \\ & \quad - \int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \partial_{i} p^{NL0} dy dz + \int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \mathcal{F}^i dy dz \\
& = -\nu {\left\lVert {\nabla}u_0 \right\rVert}_{H^{\sigma^\prime}}^2 + \mathcal{T} + \mathcal{P} + \mathcal{F}. \end{aligned}$$ For the transport term $\mathcal{T}$, we use integration by parts (and the divergence free condition) to introduce the following commutator: $$\begin{aligned}
\mathcal{T} = \int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i \left(u_0 \cdot {\nabla}{\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i - {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \left( u_0 \cdot {\nabla}u_0^i\right)\right) dy dz. \end{aligned}$$ Treating this commutator is by now classical and, in particular, by using that for ${\left\vert\eta,l\right\vert} \approx {\left\vert\xi,l^\prime\right\vert}$, $$\begin{aligned}
{\left\langle \eta,l \right\rangle}^{\sigma^\prime} - {\left\langle \xi,l^\prime \right\rangle}^{\sigma^\prime} \lesssim {\left\vert\eta-\xi,l-l^\prime\right\vert} {\left\langle \xi,l^\prime \right\rangle}^{\sigma^\prime-1}, \end{aligned}$$ one can show that $$\begin{aligned}
\mathcal{T} & \lesssim {\left\lVert {\nabla}u_0 \right\rVert}_{H^{1+}} {\left\lVert u_0^i \right\rVert}_{H^{\sigma^\prime}}^2 + {\left\lVert u_0 \right\rVert}_{H^{\sigma^\prime}} {\left\lVert u_0^i \right\rVert}_{H^{\sigma^\prime}}{\left\lVert {\nabla}u_0^i \right\rVert}_{H^{1+}} \\
& \lesssim {\left\lVert u_0 \right\rVert}_{H^{\sigma^\prime}}{\left\lVert u_0^i \right\rVert}_{H^{\sigma^\prime}}^2 \lesssim \epsilon {\left\lVert u_0 \right\rVert}_{H^{\sigma^\prime}}^2, \end{aligned}$$ (where we also used $\sigma^\prime > 2$, by ) which is consistent with Proposition \[prop:Boot\] for $c_0$ sufficiently small.
For the pressure term $\mathcal{P}$, we simply use the divergence free condition: $$\begin{aligned}
\mathcal{P} & = -\int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \partial_{i} p^{NL0} dy dz = \int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \partial_i u_0^i {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} p^{NL0} dy dz = 0. \end{aligned}$$
The forcing term is straightforward from , indeed it follows immediately that $$\begin{aligned}
\int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^i {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \mathcal{F}^i dy dz \leq \frac{\epsilon^2}{{\left\langle \nu t^3 \right\rangle}^{2\alpha}} {\left\lVert u_0 \right\rVert}_{H^{\sigma^\prime}}. \end{aligned}$$
Hence, the improvements to and follow for $\epsilon$ and $c_0$ sufficiently small.
Improvement of
---------------
The improvement of is very similar to those of and with the exception of the lift-up effect term. Indeed, by , $$\begin{aligned}
\frac{1}{2}\frac{d}{dt} \left({\left\langle t \right\rangle}^{-2} {\left\lVert u_0^1 \right\rVert}_{H^{\sigma^\prime}}^2\right) & = -\frac{t}{{\left\langle t \right\rangle}^4} {\left\lVert u_0^1 \right\rVert}_{H^{\sigma^\prime}}^2 -\nu {\left\langle t \right\rangle}^{-2} {\left\lVert {\nabla}u_0^1 \right\rVert}_{H^{\sigma^\prime}}^2 - {\left\langle t \right\rangle}^{-2} \int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^1 {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \left( u_0 \cdot {\nabla}u_0^1\right) dy dz \nonumber \\ & \quad - {\left\langle t \right\rangle}^{-2}\int{\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^1 {\left\langle {\nabla}\right\rangle}^{\sigma^\prime}u_0^2 dy dz + {\left\langle t \right\rangle}^{-2}\int {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^1 {\left\langle {\nabla}\right\rangle}^{\sigma^\prime} \mathcal{F}^1 dy dz.\end{aligned}$$ All the terms are treated as in §\[sec:U023Low\] except of course the lift up effect term. For this we use , $$\begin{aligned}
- {\left\langle t \right\rangle}^{-2}\int{\left\langle {\nabla}\right\rangle}^{\sigma^\prime} u_0^1 {\left\langle {\nabla}\right\rangle}^{\sigma^\prime}u_0^2 dy dz & \leq 4 \epsilon {\left\langle t \right\rangle}^{-2} {\left\lVert u_0^1 \right\rVert}_{H^{\sigma^\prime}}. \end{aligned}$$ From here, one applies the super-solution method used in §\[sec:Q1Hi1\]. We omit the details for brevity as it follows the same.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank the following people for helpful discussions: Margaret Beck, Steve Childress, Michele Coti Zelati, Bruno Eckhardt, Pierre-Emmanuel Jabin, Susan Friedlander, Yan Guo, Alex Kiselev, Nick Trefethen, Mike Shelley, Vladimir Sverak, Vlad Vicol, and Gene Wayne. The authors would like to especially thank Tej Ghoul for encouraging us to focus our attention on finite Reynolds number questions. The work of JB was in part supported by NSF Postdoctoral Fellowship in Mathematical Sciences DMS-1103765 and NSF grant DMS-1413177, the work of PG was in part supported by a Sloan fellowship and the NSF grant DMS-1101269, while the work of NM was in part supported by the NSF grant DMS-1211806.
Fourier analysis conventions, elementary inequalities, and Gevrey spaces {#apx:Gev}
========================================================================
We take the same Fourier analysis conventions as [@BGM15I]; we briefly recall them here for completeness. For $f(x,y,z)$ in the Schwartz space (or $(X,Y,Z)$), we define the Fourier transform $\hat{f}_k(\eta,l)$ where $(k,\eta,l) \in {\mathbb Z}\times {\mathbb R}\times {\mathbb Z}$ and the inverse Fourier transform via $$\begin{aligned}
\hat{f}_k(\eta,l) & = \frac{1}{(2\pi)^{3/2}}\int_{{\mathbb T}\times {\mathbb R}\times {\mathbb T}} e^{-i x k - iy\eta - ilz} f(x,y,z) dx dy dz \\
f(x,y,z) & = \frac{1}{(2\pi)^{3/2}}\sum_{k,l \in {\mathbb Z}} \int_{{\mathbb R}} e^{i x k + iy\eta + izl} \hat{f}_k(\eta,l) d\eta. \end{aligned}$$ With these conventions: $$\begin{aligned}
\int f(x,y,z) \overline{g}(x,y,z) dx dy dz & = \sum_{k}\int \hat{f}_k(\eta,l) \overline{\hat{g}}_{k}(\eta,l) d\eta \\
\widehat{fg} & = \frac{1}{(2\pi)^{3/2}}\hat{f} \ast \hat{g} \\
(\widehat{{\nabla}f})_k(\eta,l) & = (ik,i\eta,il)\widehat{f_k}(\eta,l). \end{aligned}$$ The paraproducts defined above in §\[sec:paranote\] are defined using the Littlewood-Paley dyadic decomposition (see e.g. [@BCD11] for more details). Let $\psi \in C_0^\infty({\mathbb R}_+;{\mathbb R}_+)$ be such that $\psi(\xi) = 1$ for $\xi \leq 1/2$ and $\psi(\xi) = 0$ for $\xi \geq 3/4$ and define $\rho(\xi) = \psi(\xi/2) - \psi(\xi)$, supported in the range $\xi \in (1/2,3/2)$. Then we have the partition of unity for $\xi > 0$, $$\begin{aligned}
1 = \sum_{M \in 2^{\mathbb Z}} \rho(M^{-1}\xi), \end{aligned}$$ where we mean that the sum runs over the dyadic integers $M = ...,2^{-j},...,1/4,1/2,1,2,4,...,2^{j},...$ and we define the cut-off $\rho_M(\xi) = \rho(M^{-1}\xi)$, each supported in $M/2 \leq \xi \leq 3M/2$. For $f \in L^2({\mathbb T}\times {\mathbb R}\times {\mathbb T})$ we define $$\begin{aligned}
f_{M} = \rho_M({\left\vert{\nabla}\right\vert})f, \quad\quad f_{< M} = \sum_{K \in 2^{{\mathbb Z}}: K < M} f_K, \end{aligned}$$ which defines the decomposition (in the $L^2$ sense) $$\begin{aligned}
f = \sum_{M \in 2^{\mathbb Z}} f_M. \end{aligned}$$ There holds the almost orthogonality and the approximate projection property
\[ineq:LPOrthoProject\] $$\begin{aligned}
{\left\lVert f \right\rVert}^2_2 & \approx \sum_{M \in 2^{{\mathbb Z}}} {\left\lVert f_M \right\rVert}_2^2 \\
{\left\lVert f_M \right\rVert}_2 & \approx {\left\lVert (f_{M})_{\sim M} \right\rVert}_2, \end{aligned}$$
where we make use of the notation $$\begin{aligned}
f_{\sim M} = \sum_{K \in 2^{{\mathbb Z}}: \frac{1}{C}M \leq K \leq CM} f_{K}, \end{aligned}$$ for some constant $C$ which is independent of $M$. Generally the exact value of $C$ which is being used is not important; what is important is that it is finite and independent of $M$. Similar to but more generally, if $f = \sum_{j} D_j$ for any $D_j$ with $\frac{1}{C}2^{j} \subset \textup{supp}\, D_j \subset C2^{j}$ it follows that $$\begin{aligned}
{\left\lVert f \right\rVert}^2_2 \approx_C \sum_{j \in {\mathbb Z}} {\left\lVert D_j \right\rVert}_2^2. \label{ineq:GeneralOrtho}\end{aligned}$$
Recall the following two lemmas.
Let $f(\xi),g(\xi) \in L_\xi^2({\mathbb R}^d)$, ${\left\langle \xi \right\rangle}^\sigma h(\xi) \in L_\xi^2({\mathbb R}^d)$ and ${\left\langle \xi \right\rangle}^\sigma b(\xi) \in L_\xi^2({\mathbb R}^d)$ for $\sigma > d/2$, Then we have $$\begin{aligned}
{\left\lVert f \ast h \right\rVert}_2 & \lesssim_{\sigma, d} {\left\lVert f \right\rVert}_2{\left\lVert {\left\langle \cdot \right\rangle}^\sigma h \right\rVert}_2, \label{ineq:L2L1} \\
\int {\left\vertf(\xi) (g \ast h)(\xi)\right\vert} d\xi & \lesssim_{\sigma,d} {\left\lVert f \right\rVert}_2{\left\lVert g \right\rVert}_2{\left\lVert {\left\langle \cdot \right\rangle}^\sigma h \right\rVert}_2 \label{ineq:L2L2L1} \\
\int {\left\vertf(\xi) (g \ast h \ast b) (\xi)\right\vert} d\xi & \lesssim_{\sigma,d} {\left\lVert f \right\rVert}_2{\left\lVert g \right\rVert}_2{\left\lVert {\left\langle \cdot \right\rangle}^\sigma h \right\rVert}_2{\left\lVert {\left\langle \cdot \right\rangle}^\sigma b \right\rVert}_2. \label{ineq:L2L2L1L1} \end{aligned}$$ Further iterates are applied for higher order nonlinear terms in Lemma \[lem:ParaHighOrder\] and are similar to but are omitted here.
Let $0 < s < 1$, $x,y>0$, and $K>1$.
- There holds $$\begin{aligned}
{\left\vertx^s - y^s\right\vert} \leq s \max(x^{s-1},y^{s-1}){\left\vertx-y\right\vert}. \label{ineq:TrivDiff}\end{aligned}$$ so that if $|x-y|<\frac{x}{K}$, $$\begin{aligned}
{\left\vertx^s - y^s\right\vert} \leq \frac{s}{(K-1)^{1-s}}{\left\vertx-y\right\vert}^s. \label{lem:scon}\end{aligned}$$ Note $\frac{s}{(K-1)^{1-s}} < 1$ as soon as $s^{\frac{1}{1-s}} + 1 < K$.
- There holds $$\begin{aligned}
{\left\vertx + y\right\vert}^s \leq \left(\frac{\max(x,y)}{x+y}\right)^{1-s}\left(x^s + y^s\right), \label{lem:smoretrivial}\end{aligned}$$ so that, if $\frac{1}{K}y \leq x \leq Ky$, $$\begin{aligned}
{\left\vertx + y\right\vert}^s \leq \left(\frac{K}{1 + K}\right)^{1-s}\left(x^s + y^s\right). \label{lem:strivial}\end{aligned}$$
Gevrey and Sobolev regularities can be related with the following two inequalities:
- For all $x \geq 0$, $\alpha > \beta \geq 0$, $C,\delta > 0$, $$\begin{aligned}
e^{Cx^{\beta}} \leq e^{C\left(\frac{C}{\delta}\right)^{\frac{\beta}{\alpha - \beta}}} e^{\delta x^{\alpha}}; \label{ineq:IncExp}\end{aligned}$$
- For all $x \geq 0$, $\alpha,\sigma,\delta > 0$, $$\begin{aligned}
e^{-\delta x^{\alpha}} \lesssim \frac{1}{\delta^{\frac{\sigma}{\alpha}} {\left\langle x \right\rangle}^{\sigma}}. \label{ineq:SobExp}\end{aligned}$$
Together these inequalities show that for $\alpha > \beta \geq 0$, ${\left\lVert f \right\rVert}_{\mathcal{G}^{C,\sigma;\beta}} \lesssim_{\alpha,\beta,C,\delta,\sigma} {\left\lVert f \right\rVert}_{\mathcal{G}^{\delta,0;\alpha}}$.
Definition and analysis of the norms {#sec:def_nrm}
====================================
Definition and analysis of $w$ {#sec:Defw}
------------------------------
As mentioned above in §\[sec:Toy\], the multipliers we use are variants of those used in [@BM13; @BMV14; @BGM15I], and we build on those constructions. We first begin by defining $\bar{w}(t,\eta)$, which is used to construct $w(t,\eta)$ and $w^3(t,k,\eta)$. For $\bar{w}$ and $w$ we use the same multipliers as [@BGM15I], however, we include the constructions here for completeness and also to make the explanation of $w^3(t,k,\eta)$ more natural.
In what follows fix $k,\eta > 0$; we will see that the norms do not depend on the sign of $k$ and $\eta$. Further, recall the definitions in §\[sec:Notation\]. The multiplier is built backwards in time, which makes resonance counting easier. Let $t \in I_{k,\eta}$. Let $\bar{w}(t,\eta)$ be a non-decreasing function of time with $\bar{w}(t,\eta) = 1 $ for $t \geq 2\eta $. For $ k \geq 1$, we assume that $\bar{w}(t_{k-1,\eta}) $ was computed. To compute $\bar{w}$ on the interval $I_{k,\eta} $, we use the behavior predicted by the toy model in . For a parameter $\kappa > 1$ fixed sufficiently large depending on a universal constant determined by the proof, for $k=1,2,3,..., E(\sqrt{\eta}) $, we define
\[def:wNR\] $$\begin{aligned}
\bar{w}(t,\eta) &= \Big( \frac{k^2}{\eta} \left[ 1 + b_{k,\eta} |t-\frac{\eta}k | \right] \Big)^{\kappa} \bar{w} (t_{k-1,\eta}), \quad&
\quad \forall t \in I^R_{k,\eta} = \left[ \frac{\eta}k ,t_{k-1,\eta} \right], \\
\bar{w}(t,\eta) &= \Big(1 + a_{k,\eta} |t-\frac{\eta}k | \Big)^{-1-\kappa} \bar{w} \left(\frac{\eta}k\right), \quad&
\quad \forall t \in I^L_{k,\eta} = \left[ t_{k,\eta} , \frac{\eta}k \right]. \end{aligned}$$
The constant $b_{k,\eta} $ is chosen to ensure that $ \frac{k^2}{\eta} \left[ 1 + b_{k,\eta} |t_{k-1,\eta}-\frac{\eta}k | \right] =1$, hence for $k \geq2$, we have $$\begin{aligned}
\label{bk}
b_{k,\eta} = \frac{2(k-1)}{k} \left(1 - \frac{k^2}{\eta} \right)\end{aligned}$$ and $b_{1,\eta} = 1 - 1/\eta$. Similarly, $a_{k,\eta}$ is chosen to ensure $ \frac{k^2}{\eta}\left[ 1 + a_{k,\eta} |t_{k,\eta}-\frac{\eta}k | \right] = 1$, which implies $$\begin{aligned}
\label{ak}
a_{k,\eta} = \frac{2(k+1)}{k} \left(1 - \frac{k^2}{\eta} \right). \end{aligned}$$ Hence, we have $ \bar{w}(\frac{\eta}k) = \bar{w} (t_{k-1,\eta}) \Big( \frac{k^2}{\eta} \Big)^{\kappa}$ and $\bar{w} ( t_{k,\eta} ) = \bar{w} (t_{k-1,\eta}) \Big( \frac{k^2}{\eta} \Big)^{1+ 2\kappa}$. For earlier times $[0, t_{E(\sqrt{\eta}),\eta }] $, we take $\bar{w}$ to be constant. Next, we will impose additional losses in time on $\bar{w}$: $$\begin{aligned}
w(t,\eta) = \bar{w}(t,\eta) \exp\left[-\kappa \int_{t}^\infty \mathbf{1}_{\tau \leq 2\sqrt{\eta}} d\tau - \kappa \int_{t}^\infty \mathbf{1}_{\sqrt{{\left\vert\eta\right\vert}} \leq \tau \leq 2{\left\vert\eta\right\vert}} \frac{{\left\vert\eta\right\vert}}{\tau^2} d\tau \right]. \label{def:wextraloss}\end{aligned}$$
Next, we define $w^3_k(t,\eta)$. Suppose $t \in I_{k,\eta}$ then, for $k^\prime \neq k$,
\[def:wNR3\] $$\begin{aligned}
w^3_{k^\prime}(t,,\eta) &= \frac{\eta}{k^2\left(1 + b_{k,\eta}{\left\vertt-\frac{\eta}{k}\right\vert}\right)}w(t,\eta) \quad \forall t \in I^R_{k,\eta} = \left[ \frac{\eta}k ,t_{k-1,\eta} \right], \\
w^3_{k^\prime}(t,,\eta) &= \frac{\eta}{k^2\left(1 + a_{k,\eta}{\left\vertt-\frac{\eta}{k}\right\vert}\right)}w(t,\eta) \quad \forall t \in I^L_{k,\eta} = \left[ t_{k,\eta} , \frac{\eta}k \right]. \\
w^3_k(t,\eta) & = w(t,\eta) \quad \forall t \in I_{k,\eta}, \end{aligned}$$
and we take $w^3_k(t,\eta) = w(t,\eta)$ if $t\not\in I_{j,\eta}$ for any $j$.
The following lemma is essentially Lemma 3.1 in [@BM13] and shows that $w(t,\eta)^{-1}$ loses some fixed radius of Gevrey-2 regularity. The proof is omitted for brevity.
\[lem:totalGrowthw\] There is a constant $\mu$ (depending on $\kappa$) and a constant $p > 0$ such that for all ${\left\vert\eta\right\vert} > 1$, we have $$\begin{aligned}
\frac{1}{w(t,\eta)} & \leq \frac{1}{w(1,\eta)} \sim \eta^{-p} e^{\frac{\mu}{2} \sqrt{\eta} } \\
\frac{1}{w^3_k(t,\eta)} & \leq \frac{1}{w^3_k(1,\eta)} \sim \eta^{-p} e^{\frac{\mu}{2} \sqrt{\eta} }, \end{aligned}$$ where ‘$\sim$’ is in the sense of asymptotic expansion (up to a multiplicative constant) as $\eta \rightarrow \infty$.
The following lemma is from [@BM13], and shows how to use the well-separation of critical times.
\[lem:wellsep\] Let $\xi,\eta$ be such that there exists some $K \geq 1$ with $\frac{1}{K}{\left\vert\xi\right\vert} \leq {\left\vert\eta\right\vert} \leq K{\left\vert\xi\right\vert}$ and let $k,n$ be such that $t \in I_{k,\eta}$ and $t \in I_{n,\xi}$ (note that $k \approx n$). Then at least one of following holds:
- $k = n$ (almost same interval);
- ${\left\vertt - \frac{\eta}{k}\right\vert} \geq \frac{1}{10 K}\frac{{\left\vert\eta\right\vert}}{k^2}$ and ${\left\vertt - \frac{\xi}{n}\right\vert} \geq \frac{1}{10 K}\frac{{\left\vert\xi\right\vert}}{n^2}$ (far from resonance);
- ${\left\vert\eta - \xi\right\vert} \gtrsim_K \frac{{\left\vert\eta\right\vert}}{{\left\vertn\right\vert}}$ (well-separated).
The next lemma tells us how to take advantage of the time derivative of $w$ and hence the $CK_w$ terms.
\[lem:dtw\] If $t \leq 2\sqrt{\eta}$, then there holds $$\begin{aligned}
\frac{\partial_t w(t,\eta)}{w(t,\eta)} = \frac{\partial_t w^3_k(t,\eta)}{w^3_k(t,\eta)} & \approx \kappa. \end{aligned}$$ If we instead have $t \in {\mathbf{I}}_{r,\eta}$ for some $r$, then the following holds $$\begin{aligned}
\frac{\partial_t w(t,\eta)}{w(t,\eta)} \approx \frac{\partial_t w^3_k(t,\eta)}{w^3_k(t,\eta)} & \approx \frac{\kappa}{1 + {\left\vert\frac{\eta}{r} - t\right\vert}} + \frac{\kappa {\left\vert\eta\right\vert}}{t^2} \approx \frac{\kappa}{1 + {\left\vert\frac{\eta}{r} - t\right\vert}} + \frac{\kappa {\left\vertr\right\vert}}{t} \label{dtw}\end{aligned}$$
The next lemma is from [@BGM15I] and is a variant of Lemma 3.4 in [@BM13]. It is important for estimating nonlinear terms where we need to be able to compare $CK_w$ multipliers of different frequencies.
\[lem:WtFreqCompare\]
- For $t \gtrsim 1$, and $\eta,\xi$ such that $t < 2 \min( {\left\vert\xi\right\vert}, {\left\vert\eta\right\vert}) $, $$\begin{aligned}
\label{dtw-xi}
\frac{\partial_t w(t,\eta)}{w(t,\eta)}\frac{w(t,\xi)}{\partial_t w(t,\xi)}
\lesssim {\left\langle \eta - \xi \right\rangle}^2\end{aligned}$$
- For all $t \gtrsim 1$, and $\eta,\xi$, such that for some $K \geq 1$, $\frac{1}{K}{\left\vert\xi\right\vert} \leq {\left\vert\eta\right\vert} \leq K{\left\vert\xi\right\vert}$, $$\begin{aligned}
\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}} \lesssim_K \left[\sqrt{\frac{\partial_t w(t,\eta)}{w(t,\eta)}} + \frac{{\left\vert\eta\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}\right]{\left\langle \eta-\xi \right\rangle}^2. \label{ineq:partialtw_endpt} \end{aligned}$$
By Lemma \[lem:dtw\], these hold also for $w^3$ (and we do not need to make a distinction).
The next lemma from [@BGM15I] and is an easy variant of the analogous \[Lemma 3.5, [@BM13]\]. It is of crucial importance for estimating nonlinear terms we need to be able to compare ratios.
\[lem:wRat\] There exists a $K > 0$ such that for all $\eta,\xi$, $$\begin{aligned}
\frac{w(t,\eta)}{w(t,\xi)} & \lesssim e^{K{\left\vert\eta-\xi\right\vert}^{1/2}}. \end{aligned}$$
Next, we want to write the analogue of Lemma \[lem:wRat\] for $w^3$, which is somewhat trickier. Instead of Lemma \[lem:wRat\], we have the following, which is analogous to \[Lemma 3.6, [@BM13]\] (although here easier due to the simpler $k$ dependence).
\[lem:Jswap\] There is a universal $K > 0$ such that in general we have $$\begin{aligned}
\frac{w^3_{k^\prime}(\eta)}{w^3_{k}(\xi)} \lesssim \frac{t}{{\left\vertk\right\vert}+ {\left\vert\eta-kt\right\vert}} e^{K{\mu}{\left\vertk-k^\prime,\eta - \xi\right\vert}^{1/2}}. \label{ineq:WFreqCompRes}\end{aligned}$$ If any one of the following holds: ($t \not\in {\mathbf{I}}_{k,\eta}$) or ($k = k^\prime$) or ($t \in {\mathbf{I}}_{k,\eta}$, $t \not\in {\mathbf{I}}_{k,\xi}$) then we have the improved estimate $$\begin{aligned}
\frac{w^3_{k^\prime}(\eta)}{w^3_{k}(\xi)} \lesssim e^{K{\mu}{\left\vertk-k^\prime,\eta - \xi\right\vert}^{1/2}}. \label{ineq:BasicJswap} \end{aligned}$$ Finally if $t \in {\mathbf{I}}_{k^\prime,\xi}$ and $k \neq k^\prime$, then $$\begin{aligned}
\frac{w^3_{k^\prime}(\eta)}{w^3_{k}(\xi)} \lesssim \frac{{\left\vertk^\prime\right\vert} + {\left\vert\xi - k^\prime t\right\vert}}{t}e^{K{\mu}{\left\vertk-k^\prime,\eta - \xi\right\vert}^{1/2}}. \label{ineq:WFreqCompNRGain}\end{aligned}$$
\[rmk:GainLoss\] In the case $t \in {\mathbf{I}}_{k,\eta} \cap {\mathbf{I}}_{k,\xi}$, $k \neq k^\prime$, the only case where is needed, we also have ${\left\vert\eta\right\vert} \approx {\left\vert\xi\right\vert}$ and from , the definition , and , Lemma \[lem:WtFreqCompare\] and implies that there is a $K > 0$ such that (see [@BM13] for more information) $$\begin{aligned}
\frac{w_k^3(\eta)}{w_{k^\prime}^3(\xi)} \lesssim \frac{t}{{\left\vertk\right\vert}}\sqrt{\frac{\partial_t w_k(t,\eta)}{w_k(t,\eta)}}\sqrt{\frac{\partial_t w_l(t,\xi)}{w_l(t,\xi)}}e^{K\mu{\left\vertk-l,\eta-\xi\right\vert}^{1/2}}. \label{ineq:RatJ2partt}\end{aligned}$$
Notice the appearance of ${\mathbf{I}}_{k,\eta}$ as opposed to $I_{k,\eta}$. Each are defined in §\[sec:Notation\]. The use of ${\mathbf{I}}$ is to rule out the end case $t \approx \sqrt{{\left\vert\eta\right\vert}}$, for example, we see that holds if $t \approx \sqrt{{\left\vert\eta\right\vert}}$ even if $t \in I_{k,\eta}$ and hence inequalities like will not be necessary.
The design and analysis of $w_L$ {#sec:Nmult}
--------------------------------
We also recall the definition of the multiplier $w_L$ from [@BGM15I]. We define $w_L$ such that it solves the following:
\[def:wL\] $$\begin{aligned}
\partial_tw_L(t,k,\eta,l) & = \kappa \frac{{\left\vertk\right\vert} {\left\langle l \right\rangle} }{k^2 + l^2 + {\left\vert\eta - kt\right\vert}^2} w_L(t,k,\eta,l) \quad\quad t \geq 1 \\
w_L(1,k,\eta,l) & = 1. \end{aligned}$$
Since the following holds uniformly in $k,l,\eta$: $$\begin{aligned}
\int_0^\infty \frac{{\left\vertk\right\vert} {\left\langle l \right\rangle}}{k^2 + l^2 + {\left\vert\eta - kt\right\vert}^2} dt \approx 1, \label{ineq:unifN}\end{aligned}$$ the multiplier $w_L$ is $O(1)$ and hence will have very little effect on most estimates.
Elliptic estimates {#sec:Elliptic}
==================
In this section, we group and discuss all of the necessary “elliptic” estimates on $\Delta_t^{-1}$. We will need the estimates from [@BGM15I] as well as a number of new estimates specific to the above threshold case.
Lossy estimates {#sec:Lossy}
---------------
First, recall the lossy elliptic lemma \[Lemma C.1, [@BGM15I]\].
\[lem:LossyElliptic\] Under the bootstrap hypotheses, for $c_0$ chosen sufficiently small, then for any function $h$ and $a \leq \sigma$, there holds $$\begin{aligned}
{\left\lVert \Delta^{-1}_t h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,a-2}} \lesssim \frac{1}{{\left\langle t \right\rangle}^2}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,a}}. \end{aligned}$$
We also need the enhanced dissipation lossy elliptic lemma \[Lemma C.2, [@BGM15I]\].
\[lem:AnuLossy\] If $C$ satisfies the bootstrap assumptions , then for $c_0$ sufficiently small, for any function $h$, and $\gamma^\prime = \beta + 3\alpha + 5$,
\[ineq:AnuLossyII\] $$\begin{aligned}
{\left\lVert A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} + {\left\lVert \partial_X A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} & \lesssim \frac{1}{{\left\langle t \right\rangle}^2}\left({\left\lVert A^{\nu;i} \phi \right\rVert}_2 + {\left\langle t \right\rangle}^{-3}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma^\prime}} \right) \\
{\left\lVert \partial_Z A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} + {\left\lVert (\partial_Y - t \partial_X) A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} & \lesssim \frac{1}{{\left\langle t \right\rangle}} \left({\left\lVert A^{\nu;i} h \right\rVert}_2 + {\left\langle t \right\rangle}^{-3}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma^\prime}} \right) \\
{\left\lVert \partial_{m}^t \partial_n^t A^{\nu;i} \Delta_t^{-1} h \right\rVert}_2 & \lesssim \frac{1}{{\left\langle t \right\rangle}^{b}}\left({\left\lVert A^{\nu;i} h \right\rVert}_2 + {\left\langle t \right\rangle}^{-3}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma^\prime}} \right), \end{aligned}$$
where $b = 0$ if $n,m \neq 1$, $b = 1$ if exactly one of $m$ or $n$ equals one, and $b = 2$ if $m = n = 1$. Moreover,
\[ineq:AnuLossyED\] $$\begin{aligned}
{\left\lVert A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} + {\left\lVert \partial_X A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} & \lesssim \frac{1}{{\left\langle t \right\rangle}^3}\left({\left\lVert \sqrt{-\Delta_L}A^{\nu;i} h \right\rVert}_2 + {\left\langle t \right\rangle}^{-3}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma^\prime}} \right) \\
{\left\lVert \partial_Z A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} + {\left\lVert (\partial_Y - t \partial_X) A^{\nu;i} \Delta^{-1}_t h \right\rVert}_{2} & \lesssim \frac{1}{{\left\langle t \right\rangle}^2} \left({\left\lVert \sqrt{-\Delta_L} A^{\nu;i} h \right\rVert}_2 + {\left\langle t \right\rangle}^{-3}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma^\prime}} \right) \\
{\left\lVert \partial_{m}^t \partial_n^t A^{\nu;i} \Delta_t^{-1} h \right\rVert}_2 & \lesssim \frac{1}{{\left\langle t \right\rangle}^{1+b}}\left({\left\lVert \sqrt{-\Delta_L} A^{\nu;i} h \right\rVert}_2 + {\left\langle t \right\rangle}^{-3}{\left\lVert h_{\neq} \right\rVert}_{{\mathcal{G}}^{\lambda,\gamma^\prime}} \right). \end{aligned}$$
Finally, we have $$\begin{aligned}
{\left\lVert A^{\nu;i} \Delta_L \Delta^{-1}_t h \right\rVert}_2 & \lesssim {\left\lVert A^{\nu;i} h \right\rVert}_2. \label{ineq:PEL_CKnuIII} \end{aligned}$$
Also recall the following lemma \[Lemma C.3, [@BGM15I]\].
\[$CK^\nu_{wL}$ elliptic lemma\] \[lem:AnuLossy\_CKnu\] Under the bootstrap hypotheses, for $c_0$ sufficiently small we have for any function $h$, $$\begin{aligned}
{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{\nu;i} \Delta_L \Delta^{-1}_t h \right\rVert}_{2} & \lesssim {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{\nu;i} h \right\rVert}_2. \label{ineq:PEL_CKnuII} \end{aligned}$$
Precision lemmas {#sec:PEL}
----------------
As in [@BGM15I], the so-called ‘precision elliptic lemmas’ (PEL) are variations on the common theme of using $\Delta_L^{-1}$ as an approximate inverse. We will need those found in [@BGM15I] and several more as well.
### Zero mode PELs
The first PEL is essentially \[Lemma C.4, [@BGM15I]\], and puts $U_0^i$ in the high norm.
\[lem:PELbasicZero\] Under the bootstrap hypotheses, for $c_0$ and $\epsilon$ sufficiently small there holds,
\[ineq:AU0PEL\] $$\begin{aligned}
{\left\lVert A U_0^1 \right\rVert}_2^2 & \lesssim {\left\langle t \right\rangle}^2{\left\lVert A^1 Q^1_0 \right\rVert}_2^2 + {\left\lVert U_0^1 \right\rVert}_2^2 + \epsilon^2{\left\langle t \right\rangle}^2{\left\lVert AC \right\rVert}_2^2 \\
{\left\lVert {\left\langle {\nabla}\right\rangle}^2 A^1 U_0^1 \right\rVert}_2^2 & \lesssim {\left\lVert A^1 Q^1_0 \right\rVert}_2^2 + {\left\langle t \right\rangle}^{-2}{\left\lVert U_0^1 \right\rVert}_2^2 + \epsilon^2{\left\lVert AC \right\rVert}_2^2 \\
{\left\lVert A U_0^2 \right\rVert}_2^2 & \lesssim {\left\lVert A^2 Q^2_0 \right\rVert}_2^2 + {\left\lVert U_0^2 \right\rVert}_2^2 + \epsilon^2 {\left\lVert AC \right\rVert}_2^2 \\
{\left\lVert {\left\langle {\nabla}\right\rangle}^2 A^3 U_0^3 \right\rVert}_2^2 & \lesssim {\left\lVert A^3 Q^3_0 \right\rVert}_2^2 + {\left\lVert U_0^3 \right\rVert}_2^2 + \epsilon^2 {\left\lVert AC \right\rVert}_2^2. \label{ineq:AU03PEL} \end{aligned}$$
Moreover, we have
\[ineq:gradAU0i\] $$\begin{aligned}
{\left\lVert {\nabla}{\left\langle {\nabla}\right\rangle}^2 A^1 U_0^1 \right\rVert}_2^2 & \lesssim {\left\lVert {\nabla}A^1 Q^1_0 \right\rVert}_2^2 + {\left\langle t \right\rangle}^{-2}{\left\lVert {\nabla}U_0^1 \right\rVert}_2^2 + \epsilon^2 {\left\lVert AC \right\rVert}_2^2 \\
{\left\lVert {\nabla}A U_0^2 \right\rVert}_2^2 & \lesssim {\left\lVert {\nabla}A^2 Q_0^2 \right\rVert}_2^2 + {\left\lVert {\nabla}U_0^2 \right\rVert}_2^2 + \epsilon^2{\left\lVert {\nabla}AC \right\rVert}_2^2. \label{ineq:gradAU02_PEL} \\
{\left\lVert {\nabla}{\left\langle {\nabla}\right\rangle}^2 A^3 U_0^3 \right\rVert}_2^2 & \lesssim {\left\lVert {\nabla}A^3 Q^3_0 \right\rVert}_2^2 + {\left\lVert {\nabla}U_0^3 \right\rVert}_2^2 + \epsilon^2 {\left\lVert {\nabla}AC \right\rVert}_2^2. \end{aligned}$$
The next PEL is specific to this work and has no analogue in [@BGM15I]. This is due to the increased precision at which we need to understand the regularity of the zero mode of the velocity field in the **(2.5NS)** terms.
\[lem:PELCKZero\] Under the bootstrap hypotheses for $t \geq 1$, for $c_0$ and $\epsilon$ sufficiently small, for $i \in {\left\{2,3\right\}}$, there holds $$\begin{aligned}
{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^i \right){\left\langle {\nabla}\right\rangle}^2 U_0^i \right\rVert}_2^2 & \lesssim {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^i \right) Q_0^i \right\rVert}_2^2 + \frac{1}{{\left\langle t \right\rangle}^{2s}}{\left\lVert U_0^i \right\rVert}_2^2 \nonumber \\ & \quad + \epsilon^2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right) C \right\rVert}_2^2. \label{ineq:PELCKZero}\end{aligned}$$
First observe that $$\begin{aligned}
\partial_t w(t,\eta) \mathbf{1}_{t \geq 1} \mathbf{1}_{{\left\vert\eta\right\vert} \leq 1/2} = 0. \end{aligned}$$ Hence, $$\begin{aligned}
{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^i \right) {\left\langle {\nabla}\right\rangle}^2 U_0^i \right\rVert}_2^2 & \lesssim {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}} A^i \right) \left(\Delta_L U_0^i\right)_{\geq 1/2} \right\rVert}_2^2 + \frac{1}{{\left\langle t \right\rangle}^{2s}}{\left\lVert U_0^i \right\rVert}_2^2. \label{ineq:lowfreqdecay} \end{aligned}$$ Therefore, similar to the proof of Lemma \[lem:PELbasicZero\] (see [@BGM15I]), it suffices to control the higher frequencies. Next, write $\Delta_L U_0^3$ using the formula for $\Delta_t U_0^3$ and projecting both sides of the equation to frequencies larger than $1/2$: $$\begin{aligned}
\left(\Delta_L U_0^i\right)_{\geq 1/2} & = (Q_0^i)_{\geq 1/2} - \left(G_{yy}\partial_{YY}U_0^i + G_{zy}\partial_{YZ}U_0^i + G_{zz}\partial_{ZZ}U_0^i + \Delta_t C^1 \partial_{Y}U_0^i + \Delta_t C^2 \partial_{Z}U_0^i\right)_{\geq 1/2} \nonumber \\
& = (Q_0^i)_{\geq 1/2} + \sum_{j = 1}^5 \mathcal{E}_i.\label{def:PELCKU0i} \end{aligned}$$ Apply $$\begin{aligned}
\mathcal{M} = \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^i + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A^i \right)\end{aligned}$$ to both sides of and deduce $$\begin{aligned}
{\left\lVert \mathcal{M} \left(\Delta_L U_0^i\right)_{\geq 1/2} \right\rVert}_2^2 \lesssim {\left\lVert \mathcal{M} \left(Q_0^i\right)_{\geq 1/2} \right\rVert}_2^2 + \sum_{j = 1}^5 {\left\lVert \mathcal{M} \mathcal{E}_i \right\rVert}_2^2. \label{ineq:PELCKU0iEstimate} \end{aligned}$$ The error terms will be divided into pieces which will either be absorbed by the LHS of or will appear on the RHS of . The latter two error terms are the most difficult and they are also very similar, hence it suffices to treat only $\mathcal{E}_5$. First, expand with a paraproduct $$\begin{aligned}
\mathcal{M}\mathcal{E}_5 & = - \mathcal{M}\left( (\Delta_t C^2)_{Hi} (\partial_{Z}U_0^i)_{Lo} \right)_{\geq 1/2} - \mathcal{M}\left( (\Delta_t C^2)_{Lo} (\partial_{Z}U_0^i)_{Hi} \right)_{\geq 1/2} - \mathcal{M}\left( (\Delta_t C^2 \partial_{Z}U_0^i)_{\mathcal{R}} \right)_{\geq 1/2} \nonumber \\
& = \mathcal{M}\mathcal{E}_{5;HL} + \mathcal{M}\mathcal{E}_{5;LH} + \mathcal{M}\mathcal{E}_{5;\mathcal{R}}. \label{eq:ME5}\end{aligned}$$ For the high-low term we use Lemma \[lem:ABasic\] and , $$\begin{aligned}
\mathcal{M}\mathcal{E}_{5;HL} & \lesssim \epsilon \sum\int \frac{\mathbf{1}_{{\left\vert\eta,l\right\vert} \geq 1/2}}{{\left\langle \xi,l^\prime \right\rangle}^2} \left(\sqrt{\frac{\partial_t w(t,\xi)}{w(t,\xi)}}\tilde{A}(\xi,l^\prime) + \frac{{\left\vert\xi,l^\prime\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A(\xi,l^\prime) \right) {\left\vert\widehat{\Delta_t C^2}(\xi,l^\prime)_{Hi}\right\vert} Low(\eta-\xi,l-l^\prime) d\xi. \end{aligned}$$ Hence, by and Lemma \[lem:CoefCtrl\] we have $$\begin{aligned}
{\left\lVert \mathcal{M}\mathcal{E}_{5;HL} \right\rVert}^2_2 & \lesssim \epsilon^2{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^{s}}A\right)C \right\rVert}_2^2, \end{aligned}$$ which appears on the RHS of . To treat the low-high term in , we use a similar method to deduce $$\begin{aligned}
{\left\lVert \mathcal{M}\mathcal{E}_{5;LH} \right\rVert}^2_2 & \lesssim c_0^2 {\left\lVert \mathcal{M} \partial_Z U_0^i \right\rVert}_2 \\
& \lesssim c_0^2 \left({\left\lVert \mathcal{M} (\partial_Z U_0^i)_{\geq 1/2} \right\rVert}_2 + {\left\lVert \mathcal{M}(\partial_Z U_0^i)_{\geq 1/2} \right\rVert}_2^2\right) \\
& \lesssim c_0^2 \left({\left\lVert \mathcal{M} \left(\Delta_L U_0^i\right)_{\geq 1/2} \right\rVert}_2 + \frac{1}{{\left\langle t \right\rangle}^{2s}}{\left\lVert U_0^i \right\rVert}_2^2\right)\end{aligned}$$ where the last line followed as in . The first term is absorbed on the LHS of whereas the second term appears on the RHS of . The remainder term is straightforward and can be treated in essentially the same way as the low-high term; see the proof of \[Lemma 4.9 [@BGM15I]\] for a similar argument. As the other error terms are essentially the same, this completes the proof of .
### Non-zero mode PELs
The next PEL is an easy variant of the analogous \[Lemma C.5, [@BGM15I]\]. The proof is a slight variation of that in [@BGM15I]. Here we need to deal with the large $Z$ frequencies but this is straightforward due to the inequalities derived in §\[sec:basicmult\] and hence the details are omitted here.
\[lem:PEL\_NLP120neq\] Let $h$ be given such that ${\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda}}\lesssim \epsilon {\left\langle t \right\rangle}^{b} {\left\langle \nu t^3 \right\rangle}^{-a}$ for some $a \geq 0$ and $b \geq 0$. Then, under the bootstrap hypotheses, for $c_0$ and $\epsilon$ sufficiently small, there holds,
\[ineq:PEL\_NLP120neq\] $$\begin{aligned}
{\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^{i} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{i}\right) \Delta_L \Delta_t^{-1} h_{\neq} \right\rVert}_2^2
& \lesssim {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A}^{i} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A^{i} \right) h_{\neq} \right\rVert}_2^2 \nonumber \\
& \quad + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2b-2} }{{\left\langle \nu t^3 \right\rangle}^a} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A \right) C \right\rVert}_2^2. \label{ineq:PEL_NLP120neq1}\end{aligned}$$
The next PEL is also basically \[Lemma C.6, [@BGM15I]\] and is slightly simpler than Lemma \[lem:PEL\_NLP120neq\].
\[lem:SimplePEL\] Let $h$ be given such that ${\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda}} \lesssim \epsilon{\left\langle t \right\rangle}^{b} {\left\langle \nu t^3 \right\rangle}^{-a}$ for $a,b \geq 0$. Then, for $c_0$ and $\epsilon$ sufficiently small, under the bootstrap hypotheses we have for all $i \in {\left\{1,2,3\right\}}$, $$\begin{aligned}
{\left\lVert A^{i} \Delta_L \Delta_t^{-1} h_{\neq} \right\rVert}_2^2 & \lesssim {\left\lVert A^{i} h_{\neq} \right\rVert}_2^2 + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2b-2} }{{\left\langle \nu t^3 \right\rangle}^{2a}}{\left\lVert A C \right\rVert}_2^2, \label{ineq:SimplePEL}\end{aligned}$$
Finally, from \[Lemma C.7, [@BGM15I]\] is the following PEL for treating the linear pressure term $LP3$ in the $Q^3$ equation.
\[lem:QPELpressureI\] Let $h$ be given such that ${\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda}} \lesssim \epsilon {\left\langle t \right\rangle}^b {\left\langle \nu t^3 \right\rangle}^{-a}$ for $a,b \geq 0$ and suppose $C$ satisfies the bootstrap hypotheses. Then for $c_0$ and $\epsilon$ sufficiently small, there holds $$\begin{aligned}
{\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{3} \Delta_{L} \Delta_t^{-1} h_{\neq} \right\rVert}_2^2 & \lesssim {\left\lVert \sqrt{\frac{\partial_t w_L}{w_L}} A^{3} h_{\neq} \right\rVert}_2^2 + \frac{\epsilon^{2}{\left\langle t \right\rangle}^{2b-2}}{{\left\langle \nu t^3 \right\rangle}^{2a}} {\left\lVert \left(\sqrt{\frac{\partial_t w}{w}}\tilde{A} + \frac{{\left\vert{\nabla}\right\vert}^{s/2}}{{\left\langle t \right\rangle}^s}A\right)C \right\rVert}_2^2. \label{ineq:PELpressureI}\end{aligned}$$
The last PEL is unique to this work (it was not necessary in [@BGM15I]). It is needed here to gain additional precision for times $t \gtrsim \epsilon^{-1/2}$. It is used in, e.g. above.
\[lem:PELED\] Let $h$ be given such that ${\left\lVert h \right\rVert}_{{\mathcal{G}}^{\lambda}}\lesssim \epsilon {\left\langle t \right\rangle}^{b} {\left\langle \nu t^3 \right\rangle}^{-a}$ for some $a \geq 0$ and $b \geq 0$. Then, under the bootstrap hypotheses, for $c_0$ and $\epsilon$ sufficiently small there holds $$\begin{aligned}
{\left\lVert \sqrt{-\Delta_L} A^{i} \Delta_L \Delta_t^{-1} h_{\neq} \right\rVert}_2^2
& \lesssim {\left\lVert \sqrt{-\Delta_L} A^{i} h_{\neq} \right\rVert}_2^2 + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2b-2}}{{\left\langle \nu t^3 \right\rangle}^{2a}}{\left\lVert {\nabla}AC \right\rVert}_2^2 + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2b}}{{\left\langle \nu t^3 \right\rangle}^{2a}}{\left\lVert AC \right\rVert}_2^2. \label{ineq:PELED}\end{aligned}$$
The proof is very similar to the proof of Lemma \[lem:PEL\_NLP120neq\] (the proof of which is found in [@BGM15I]). Let us briefly sketch the argument. Write $P = \Delta_t^{-1}h_{\neq}$ $$\begin{aligned}
\Delta_L P & = h_{\neq} - G_{yy}(\partial_Y - t\partial_X)^2 P - G_{yz}(\partial_Y - t\partial_X) \partial_Z P + G_{zz}\partial_{ZZ}P - \Delta_tC^1 (\partial_Y - t\partial_X) P - \Delta_t C^2 \partial_Z P \nonumber \\
& = h_{\neq} + \sum_{i = 1}^5 \mathcal{E}_i. \label{def:PkPEL}\end{aligned}$$ We apply $\sqrt{-\Delta_L}A^i$ to both sides of and estimate the terms on the RHS. Hence we get $$\begin{aligned}
{\left\lVert \sqrt{-\Delta_L}A^i \Delta_L P \right\rVert}_2^2 \lesssim {\left\lVert \sqrt{-\Delta_L}A^ih_{\neq} \right\rVert}_2^2 + \sum_{i = 1}^5 {\left\lVert \sqrt{-\Delta_L}A^i \mathcal{E}_i \right\rVert}_2^2. \label{ineq:DeltaPk} \end{aligned}$$ For example, consider the first error term and expand with a paraproduct: $$\begin{aligned}
\sqrt{-\Delta_L}A^i\mathcal{E}_1 & = \sqrt{-\Delta_L}A^i\left( (G_{yy})_{Hi}(\partial_Y - t\partial_X)^2 P_{Lo}\right) + \sqrt{-\Delta_L}A^i\left( (G_{yy})_{Lo}(\partial_Y - t\partial_X)^2 P_{Hi}\right) + \mathcal{E}_{1;\mathcal{R}} \\
& := \mathcal{E}_{1;C} + \mathcal{E}_{1;P} + \mathcal{E}_{1;\mathcal{R}}. \end{aligned}$$ By , , , , and Lemma \[lem:CoefCtrl\] it follows that $$\begin{aligned}
{\left\lVert \sqrt{-\Delta_L}A^i \mathcal{E}_{1;P} \right\rVert}_2^2 & \lesssim c_{0}^2{\left\lVert \sqrt{-\Delta_L}A^i \Delta_L P \right\rVert}_2^2, \end{aligned}$$ which can hence be absorbed on the LHS of by choosing $c_{0}$ sufficiently small. The remainder is treated $\mathcal{E}_{1;\mathcal{R}}$ is treated similarly. Consider next $\mathcal{E}_{1;C}$ for which, by the hypotheses, Lemma \[lem:ABasic\], and Lemma \[lem:LossyElliptic\], we have $$\begin{aligned}
\mathcal{E}_{1;C} & \lesssim \frac{\epsilon {\left\langle t \right\rangle}^{b}}{{\left\langle \nu t^3 \right\rangle}^{a}}\sum_{l} \int_\xi {\left\vertk,\eta-kt,l\right\vert} \frac{1}{{\left\langle \xi,l^\prime \right\rangle}^2}{\left\langle \frac{t}{{\left\langle \xi,l^\prime \right\rangle}} \right\rangle}^{-1} A{\left\vert\widehat{G_{yy}}(\xi,l^\prime)_{Hi}\right\vert} Low(k,\eta-\xi,l-l^\prime) d\xi; \end{aligned}$$ the extra ${\left\langle t \right\rangle}^2$ from $(\partial_Y - t\partial_X)^2$ was canceled by the $\Delta_t^{-1}$ in the definition of $P$ and Lemma \[lem:LossyElliptic\]. It follows from and Lemma \[lem:CoefCtrl\] that $$\begin{aligned}
{\left\lVert \sqrt{-\Delta_L}A^i \mathcal{E}_{1;C} \right\rVert}_2^2 & \lesssim \frac{\epsilon^2 {\left\langle t \right\rangle}^{2b-2}}{{\left\langle \nu t^3 \right\rangle}^{2a}}{\left\lVert {\nabla}AC \right\rVert}_2^2 + \frac{\epsilon^2 {\left\langle t \right\rangle}^{2b}}{{\left\langle \nu t^3 \right\rangle}^{2a}}{\left\lVert AC \right\rVert}_2^2, \end{aligned}$$ which suffices. This completes the treatment of $\mathcal{E}_{1}$. The error terms $\mathcal{E}_{2}$ and $\mathcal{E}_{3}$ are treated exactly the same. In treating the error terms $\mathcal{E}_{4}$ and $\mathcal{E}_{5}$, note that there is an extra derivative on $C^i$. As a result, we cannot recover a power of time from Lemma \[lem:ABasic\] using the low-frequency growth. However, there is one less power of $t$ on $P$ and hence there is a balance and a similar proof as that used on $\mathcal{E}_1$ will adapt in a straightforward manner to the last two error terms. We omit the details for brevity.
[^1]: *jacob@cscamm.umd.edu*, University of Maryland, College Park
[^2]: *pgermain@cims.nyu.edu*, Courant Institute of Mathematical Sciences
[^3]: *masmoudi@cims.nyu.edu*, Courant Institute of Mathematical Sciences
|
---
abstract: 'We present a brane-world scenario in which two regions of $AdS_5$ space-time are glued together along a 3-brane with constant positive curvature such that [*all*]{} spatial dimensions form a compact manifold of topology $S^4$. It turns out that the induced geometry on the brane is given by Einstein’s static universe. It is possible to achieve an anisotropy of the manifold which allows for a huge hierarchy between the size of the extra dimension $R$ and the size of the observable universe $R_U$ at present. This anisotropy is also at the origin of a very peculiar property of our model: the physical distance between [*any two points*]{} on the brane is of the order of the size of the extra dimension $R$ regardless of their distance measured with the use of the induced metric on the brane. In an intermediate distance regime $R \ll r \ll R_U$ gravity on the brane is shown to be effectively $4$-dimensional, with corresponding large distance corrections, in complete analogy with the Randall-Sundrum II model. For very large distances $r \sim R_U$ we recover gravity in Einstein’s static universe. However, in contrast to the Randall-Sundrum II model the difference in topology has the advantage of giving rise to a geodesically complete space.'
author:
- |
A. Gruppuso$^{a,b,c}$[^1] ,$\ $ E. Roessl$^{a}$[^2] $\ $ and M. Shaposhnikov$^{a}$[^3]\
\
[*$^{a}$ Institut de théorie des phénomènes physiques (ITP)*]{}\
[*Laboratoire de Physique des Particules et Cosmologie (LPPC)*]{}\
[*École polytechnique fédérale de Lausanne*]{}\
[*CH-1015 Lausanne, Switzerland*]{}\
\
[*$^{b}$ IASF/CNR, Istituto di Astrofisica Spaziale e Fisica Cosmica*]{}\
[*Sezione di Bologna*]{}\
[*Consiglio Nazionale delle Ricerche*]{}\
[*via Gobetti 101, I-40129 Bologna - Italy*]{}\
and\
[*$^{c}$ Dipartimento di Fisica, Università di Bologna and I.N.F.N., Sezione di Bologna,*]{}\
[*via Irnerio 46, 40126, Bologna, Italy*]{}
title: '**Einstein static universe as a brane in extra dimensions**'
---
Introduction
============
\[intro\]
Recent suggestions that large [@Antoniadis:1990ew]–[@Antoniadis:1998ig] or infinite [@Rubakov:bb]–[@Randall:1999vf] extra dimensions are not necessarily in conflict with present observations provide new opportunities for addressing several outstanding problems of modern theoretical physics like the hierarchy problem [@Antoniadis:1990ew]–[@Antoniadis:1998ig],[@Randall:1999ee], [@Cohen:1999ia] or the cosmological constant problem [@Rubakov:1983bz]–[@Randjbar-Daemi:1985wg]. In the course of this development and inspired by string theoretical arguments [@Polchinski:1995mt], the notion of brane world scenarios emerged in which the usual Standard Model fields are supposed to be confined to a so-called $3$-brane, a $4$-dimensional sub-manifold of some higher-dimensional space-time. As shown in [@Randall:1999vf] also gravity can appear to be effectively $4$-dimensional for a brane-bound observer provided the conventional scheme of Kaluza-Klein compactification [@Kaluza:tu]–[@KK] is replaced by a compactification using non-factorizable (also called warped) geometries (see also [@Rubakov:1983bz]). This triggered an immense research activity in theories involving $3$-branes with interests ranging from elucidating the global space-time structure of brane world scenarios, properties of gravity, cosmology and brane cosmological perturbations, generalizations to higher dimensions etc. While the possibilities are rich, realistic scenarios remain rare. For example the simple Randall-Sundrum-II model [@Randall:1999vf] faces the problem of being geodesically incomplete [@Rubakov:2001kp; @Muck:2000bb; @Gregory:2000rh]. It is therefore reasonable to look for adequate alternatives or generalizations to the Randall-Sundrum-II model which avoid the above mentioned problems while sharing its pleasant feature of the effective $4$-dimensional low energy gravity on the brane.
In this paper we present a $5$-dimensional brane-world model which solves the geodesic incompleteness of the Randall-Sundrum II model while preserving its phenomenological properties concerning the localization of gravity.
We try to illustrate our motivation for considering a particular geometry by using the simple picture of a domain structure in extra dimensions resulting from a spontaneously broken discrete symmetry. The associated Higgs-field takes different values in regions separated by a domain wall, which restricts the possibilities of combining domain walls depending on the global topology of the space-time under consideration. We concentrate on the $5$-dimensional case, see [@Cohen:1999ia],[@Chodos:1999zt]–[@Gherghetta:2000jf] for higher dimensional constructions.
Let us analyze a few simple cases:
Non-compact extra dimension
: $y \in (-\infty, + \infty)$. In this case, by choosing the origin $y=0$ to coincide with the position of the brane, we obtain two non-overlapping regions $(-\infty, 0) $ and $(0, + \infty)$ and we can thus consistently have one brane in such a theory. An example for this configuration is provided by the Randall-Sundrum II model [@Randall:1999vf].
Compact extra dimension
: $y \in [0 , 2 \pi]$. We now consider two possibilities, depending on the spatial topology of our manifold:
[**(a)**]{}
$\mathbb{R}^3 \times S^1$: If the ordinary dimensions are supposed to be non-compact, it is not possible to consistently put only one brane in the extra dimensions. At least two branes are needed.
[**(b)**]{}
$S^4$: If [*all*]{} spatial coordinates are part of a compact manifold, the simple picture shown in Fig. \[ct\] seems to suggest that it is possible to consistently put a single brane in the bulk space-time.
To the best of our knowledge case [**(b)**]{} has not yet been considered in the literature and our aim is to present such a construction.
This paper is organized as follows: in Section \[one\] we discuss the basic geometric and topological properties of our brane-world scenario like Einstein equations, junction conditions and the distance hierarchy between the extra dimension $R$ and the observable universe $R_U$. Section \[two\] is dedicated to the study of geodesics and the demonstration that our model does not suffer from being geodesically incomplete. In section \[three\] we present a detailed computation of the propagator of a massless scalar field in the given background serving as an easy, phenomenological approach to the study of gravity. We discuss the behavior of the two-point function in three different distance regimes. It turns out that the computations are rather technical and we therefore collect large parts of it in four appendices. We eventually draw conclusions in section \[four\].
=3.0in
The background equations
========================
\[one\] The aim of this section is to present the topology and geometry of the brane-world model which we set out to study but also to motivate how this particular model emerged through imposing several physical conditions on the more general set of solutions.
Einstein equations
------------------
We would like to construct a space-time with all spatial dimensions being part of the same compact manifold. One more motivation for this lies in a possible solution of the strong CP problem within theories with extra dimensions [@Khlebnikov:1987zg]. The overall topology would then be given by $\mathbb{R}\times \mathbb{K}$, where $\mathbb{R}$ represents the time-coordinate and $\mathbb{K}$ any compact manifold. As announced, we will restrict ourselves to the case of one extra dimension and an induced metric on the brane characterized by a spatial component of geometry $S^3$. The idea is to combine two $5$-dimensional regions of space-time dominated by a cosmological constant $\Lambda$ in such a way that the border of the two regions can be identified with a $3$-brane, constituting our observable universe. As pointed out in [@Khlebnikov:1987zg] the manifold $\mathbb{K}$ has to be highly anisotropic in order to single out the small extra dimension from the three usual ones. It is not a priory clear whether the Einstein equations allow for such solutions at all and if so whether gravity can be localized on the brane in such a setup.
We choose the following ansatz for the $5$-dimensional metric consistent with the above requirements:[^4] $$\begin{aligned}
\label{metric}
ds^2 = g_{M N} dx^M dx^N=-\sigma^2(\theta)\, dt^2 + R_U^2 \, \gamma ^2(\theta)
\, d\Omega^2_3 + R^2 \, d\theta^2 \, ,\end{aligned}$$ where $\theta \in [-\frac{\pi}{2},\frac{\pi}{2}]$ denotes the extra coordinate and $R$ and $R_U$ are constants representing the size of the extra dimension and the size of the observable universe at present, respectively. $d\Omega^2_3$ denotes the line element of a $3$-sphere: $$\label{S3metric}
d\Omega^2_3 =d\varphi^2_1 + \sin ^2 \varphi_1 \, d\varphi^2_2 + \sin ^2 \varphi_1\, \sin ^2 \varphi_2\,
d\varphi^2_3 \, ,$$ where $\varphi_1, \varphi_2$ belong to the interval $\left[ 0, \pi \right]$, $ \varphi_3 $ to $\left[ 0, 2 \pi \right]$. Capital latin letters $M,N,..$ will range from $0$ to $4$. In order to obtain a compact space, we require $\gamma(\pm\frac{\pi}{2})=0$.
Metrics which can locally be put into the form (\[metric\]) are sometimes referred to as [*asymmetrically warped*]{} metrics due to two different functions ($\sigma$ and $\gamma$ in our notations) multiplying the temporal and spatial part of the $4$-dimensional coordinate differentials. Their relevance in connection with $4$-dimensional Lorentz-violation at high energies was first pointed out in [@Visser:1985qm]. More recent discussions of this subject can be found in [@Csaki:2000dm]–[@Dubovsky:2001fj] or [@Rubakov:2001kp] and references therein. A common prediction of theories of this kind is that dispersion relations get modified at high energies. For a field-theoretical discussion of Lorentz violating effects in the context of the Standard Model of particle physics see [@Coleman:1998ti].
The Einstein equations in 5 dimensions with a bulk cosmological constant $\Lambda$ and a stress-energy tensor $T_{M N}$ take the following form: $$\begin{aligned}
\label{einsteineq}
R_{M N} -\frac{1}{2} \, R \, g_{M N} + \Lambda \, g_{M N}
= \frac{8 \pi}{M^3} \, T_{M N} \, ,\end{aligned}$$ where $M$ is the fundamental scale of gravity. We choose to parametrize the stress-energy tensor in the following way, consistent with the symmetries of the metric: $$\begin{aligned}
\label{stress}
T^0_{\;\;0}=\epsilon_0(\theta) < 0\, , \quad
T^i_{\;\;j}= \delta^i_j\,\epsilon(\theta)\, , \quad
T^\theta_{\;\;\theta}=\epsilon_\theta(\theta) \, ,\end{aligned}$$ where as indicated the above diagonal components depend only on the extra coordinate $\theta$. Lower case latin indices $i,j$ label the coordinates on $S^3$.
Using the metric ansatz (\[metric\]) together with the stress-energy tensor (\[stress\]) the Einstein equations (\[einsteineq\]) become $$\begin{aligned}
\frac{3}{R^2} \left[ \left( \frac{\gamma^{\prime}}{\gamma}\right)^2 + \frac{\gamma^{\prime \prime}}{\gamma} -
\left( \frac{R}{R_U} \right)^2 \frac{1}{\gamma ^2} \right] + \Lambda &=& \frac{8 \pi}{M^3}
\epsilon_0 \, , \label{00} \\
\frac{1}{R^2} \left[ \left( \frac{\gamma ^{\prime}}{\gamma}\right)^2 +
2 \frac{\gamma^{\prime}}{\gamma} \frac{\sigma^{\prime}}{\sigma}
+2 \frac{\gamma ^{\prime \prime}}{\gamma} + \frac{\sigma^{\prime \prime}}{\sigma} -
\left( \frac{R}{R_U} \right)^2 \frac{1}{\gamma^2} \right] + \Lambda &=& \frac{8 \pi}{M^3}
\epsilon \, , \label{ii} \\
\frac{3}{R^2} \left[ \left( \frac{\gamma^{\prime}}{\gamma}\right)^2 +
\frac{\gamma^{\prime}}{\gamma} \frac{\sigma^{\prime}}{\sigma} -
\left( \frac{R}{R_U} \right)^2 \frac{1}{\gamma ^2} \right] + \Lambda &=& \frac{8 \pi}{ M^3}
\epsilon_{\theta} \, , \label{thetatheta}\end{aligned}$$ where $^{\prime}$ denotes differentiation with respect to $\theta$. The conservation of stress-energy, or equivalently, the Bianchi-identities lead to the following constraint relating the three independent components $\epsilon_0$, $\epsilon$ and $\epsilon_\theta$: $$\begin{aligned}
\label{constraint}
\epsilon^\prime_{\theta} + \left( \frac{\sigma^{\prime}}{\sigma} +
3 \frac{\gamma^{\prime}}{\gamma}\right) \, \epsilon_{\theta} -
\frac{\sigma^{\prime}}{\sigma}\, \epsilon_{0} - 3 \, \frac{\gamma^{\prime}}{\gamma} \epsilon =0 \, .\end{aligned}$$
Vacuum solution
---------------
In this paper we do not intend to provide a field theoretical model which could generate the geometry we are about to describe. Our aim is to study a singular brane, located at $\theta_b=0$ separating two vacuum regions in the bulk. Of course the solutions to the Einstein equations in vacuum with a cosmological constant term are nothing but the familiar de-Sitter ($\Lambda>0$), Minkowski ($\Lambda=0$) and anti-de-Sitter ($\Lambda<0$) space-times. To see how these geometries can be recovered using the line element (\[metric\]) we set the right hand sides of (\[00\])-(\[thetatheta\]) equal to zero and subtract eq. (\[thetatheta\]) from eq. (\[00\]) to obtain $\sigma = c \, \gamma^{\prime}$, where $c$ is a constant. Putting this back in eqs. (\[00\])-(\[thetatheta\]), we are left with only one independent differential equation $$\begin{aligned}
\label{00n}
\frac{3}{R^2} \left[ \left( \frac{\gamma^{\prime}}{\gamma}\right)^2 +
\frac{\gamma^{\prime \prime}}{\gamma} -
\left( \frac{R}{R_U} \right)^2 \frac{1}{\gamma^2} \right] + \Lambda = 0.\end{aligned}$$ The solution of (\[00n\]) is trivial and by means of $\sigma = c \, \gamma^{\prime}$ we find $$\label{sigmaandgamma}
\gamma(\theta) =
\frac{\sinh \left[\omega\left(\frac{\pi}{2}-|\theta| \right)\right]}
{\sinh (\omega \frac{\pi}{2})}\, ,\qquad
\sigma (\theta) =
\frac{\cosh \left[\omega\left(\frac{\pi}{2}-|\theta| \right)\right]}
{\cosh (\omega \frac{\pi}{2})}\, ,$$ provided that $R_U = R \sinh (\omega \frac{\pi}{2})/\omega$ and $\omega^2 = -\Lambda R^2 /6 $. Notice that the solution (\[sigmaandgamma\]) does not contain any integration constant because we already imposed the boundary conditions $\gamma(\pm\frac{\pi}{2})=0$ and $\gamma(0)=\sigma(0)=1$. Moreover, we chose the whole setup to be symmetric under the transformation $\theta \to -\theta$. From (\[sigmaandgamma\]) it is now obvious that locally (for $\theta>0$ and $\theta<0$) the line element (\[metric\]) correctly describes de-Sitter, Minkowski and anti-de-Sitter space-times for imaginary, zero and real values of $\omega$, respectively. For our purposes only the $AdS$ solution will be of any interest as we will see in the following section. A simple change of coordinates starting from (\[metric\]) and (\[sigmaandgamma\]) shows that the $AdS$-radius in our notations is given by $R_{AdS}=R/\omega=\sqrt{-6/\Lambda}$.[^5] At this stage, the validity of the vacuum solution (\[sigmaandgamma\]) is restricted to the bulk, since it is not even differentiable in the classical sense at $\theta=0$. In order to give sense to (\[sigmaandgamma\]) for all values of $\theta$ we will have to allow for some singular distribution of stress-energy at $\theta=0$ and solve (\[00\])-(\[thetatheta\]) in the sense of distributions.
The complete solution for a singular brane {#SingularBrane}
------------------------------------------
As announced, we now refine our ansatz for the energy momentum tensor (\[stress\]) to allow for a solution of eqs. (\[00\])-(\[thetatheta\]) in the whole interval $-\pi/2\leq\theta\leq\pi/2$: $$\begin{aligned}
\label{stressrefined}
\epsilon_0 (\theta ) = c_0 \, \frac{\delta(\theta)}{R} \, ,\quad
\epsilon (\theta ) = c \, \frac{\delta(\theta)}{R} \, ,\quad
\epsilon_{\theta} (\theta ) = c_\theta \, \frac{\delta(\theta)}{R} \, .\end{aligned}$$ After replacing the above components (\[stressrefined\]) in eqs. (\[00\])-(\[thetatheta\]) and integrating over $\theta$ from $-\eta$ to $\eta$, followed by the limit $\eta \to 0$ we find: $$\begin{aligned}
:\frac{\gamma^\prime}{\gamma}: \;\; &= \frac{8 \pi R}{3 M^3} c_0 \label{FineTuneI} \, ,\\
:\frac{\sigma^\prime}{\sigma}: \;+ \;2:\frac{\gamma^\prime}{\gamma}: \;\; &=\frac{8\pi R}{M^3}c
\label{FineTuneII} \, ,\\
0&=c_\theta \, ,\end{aligned}$$ where the symbol $: \ldots :$ is used to denote the jump of a quantity across the brane defined by: $$\label{::def}
: f : \;\; \equiv \lim_{\eta\to0} \left[f(\eta)-f(-\eta)\right].$$ Note that in the above step we made use of the identity $$\frac{\gamma^{\prime \prime}}{\gamma} = \left(\frac{\gamma^\prime}{\gamma} \right)^\prime +
\left( \frac{\gamma^\prime}{\gamma} \right)^2$$ together with the continuity of $\gamma$ on the brane. Specifying (\[FineTuneI\]) and (\[FineTuneII\]) to (\[sigmaandgamma\]) we have: $$\begin{aligned}
c_0 &= - \frac{3}{4\pi} M^3 \frac{\omega}{R} \, \coth \left(\omega \frac{\pi}{2}\right) \, ,\label{solc0}\\
c_{\phantom{0}} &= - \frac{1}{4\pi} M^3 \frac{\omega}{R} \,
\left[ \tanh \left(\omega \frac{\pi}{2}\right)+2 \coth \left(\omega \frac{\pi}{2}\right)\right]
\, , \label{solc} \\
c_\theta &= 0 \, . \label{solctheta}\end{aligned}$$ Eqs. (\[solc0\]) and (\[solc\]) relate the energy-density and the pressure of the singular brane to the bulk cosmological constant $\Lambda$ (via $\omega$), the size of the extra dimension $R$ and the fundamental scale of gravity $M$. With the above relations (\[solc0\])-(\[solctheta\]) we can now interpret (\[sigmaandgamma\]) as a solution to the Einstein equations (\[00\])-(\[thetatheta\]) in the sense of distributions. Also the stress-energy conservation constraint (\[constraint\]) is satisfied based on the identity $\delta(x) \, \mbox{sign}(x)=0$ again in the distributional sense.
Note that $c_0/c \to 1$ in the limit $\omega \to \infty$. Moreover, for larger and larger values of $\omega$, $c_0$ and $c$ approach the brane tension of the Randall-Sundrum II model and the above eqs. (\[solc0\]) and (\[solc\]) merge to the equivalent relation in the Randall-Sundrum II case. This is no surprise since taking the limit $\omega \to \infty$ corresponds to inflating and flattening the 3-brane so that we expect to recover the case of the flat Randall-Sundrum II brane.
We finish this section by the discussion of some physical properties of our manifold. We first observe that its spatial part is homeomorphic to a $4$-sphere $S^4$. This is obvious from the metric (\[metric\]) and the explicit expression for $\gamma$ given in (\[sigmaandgamma\]). Geometrically, however, our manifold differs from $S^4$ due to the high anisotropy related to the smallness of the extra dimension. The ratio of typical distance scales in the bulk and on the brane is given by $$\begin{aligned}
\label{hierarchy}
\frac{R}{R_{U}}= \frac{\omega}{\sinh \left( \frac{\omega\pi}{2}\right)} \, .\end{aligned}$$ It is now clear that the above ratio (\[hierarchy\]) can only be made very small in the case of real $\omega$ ($AdS$-space-time). For the size of the observable universe we take the lower bound $R_U > 4 \, \mbox{Gpc}\sim 10^{28}\,\mbox{cm}$ while the size of the extra dimension is limited from above [@Hoyle:2000cv]: $R<10^{-2}\,\mbox{cm}$, leaving us with $\omega > 50$.
Finally we would like to point out a very peculiar property of the manifold under consideration: as it can immediately be deduced from the line element (\[metric\]) and (\[sigmaandgamma\]), any two points on the brane are separated by not more than a distance of the order of $R$ regardless of their distance as measured by an observer on the brane using the induced metric.
Geodesics
=========
\[two\]
It is well known that the Randall Sundrum-II model is timelike and lightlike geodesically incomplete [@Rubakov:2001kp; @Muck:2000bb; @Gregory:2000rh] which means that there exists inextendible timelike and lightlike geodesics.[^6] An inextendible geodesic is a geodesic parametrized by an affine parameter $\tau$ such that by using up only a finite amount of affine parameter the geodesic extends over infinite coordinate distances. In a more physical language one could reformulate the above statement by saying that it takes only a finite amount of affine parameter $\tau$ in order to reach the infinities of the incomplete space-time. As we will illustrate later in this chapter, the reason why the Randall-Sundrum II setup ceases to be geodesically complete is simply due to a specific way of gluing two patches of $AdS_5$. One of the main motivations for this work was to provide an alternative to the Randall Sundrum II model that has the advantage of being geodesically complete while conserving the pleasant phenomenological features of the latter. We divide the discussion of geodesics in two parts: in subsection \[RSgeod\] we illustrate the effects of incomplete geodesics in the Randall-Sundrum II setup for timelike and lightlike geodesics. In the following subsection \[ourgeod\] we demonstrate why our setup is geodesically complete by looking at corresponding geodesics. Finally, we complement the discussions by illustrating the physics with the use of the Penrose-diagram of (the universal covering space-time of) $AdS_5$.
Geodesics in the Randall-Sundrum II setup {#RSgeod}
-----------------------------------------
Our discussion of geodesics in this chapter is in no sense meant to be complete. Without going into the details of the computations we merely intend to present the solutions of the geodesic equations in certain cases. For more general and more complete discussions of this issue we refer to the literature, see e.g. [@Muck:2000bb; @Youm:2001qc] and references therein. We first consider lightlike geodesics in the Randall-Sundrum II background metric given in appendix \[appParallel\], eq. (\[RSmetric\]). Let us suppose that a photon is emitted at the brane at $y=0$ in the positive $y$-direction at coordinate time $t=0$ then reflected at $y=y_1$ at the time $t=t_1$ to be observed by an observer on the brane at time $t=t_2$. In this situation $t$ corresponds to the proper time of an observer on the brane at rest. A simple calculation reveals $$\label{RSgeodint}
t_2=2 t_1=\frac{2}{k} \left( e^{k y_1}-1\right).$$ A brane bound observer will therefore note that it takes an infinite time for a photon to escape to $y=\infty$. However, parameterizing the same geodesic by an affine parameter $\tau$ reveals the lightlike incompleteness of the Randall-Sundrum II space-time. Let the events of emission, refection (at $y=y_1$) and arrival on the brane again be labeled by $\tau=0$, $\tau=\tau_1$ and $\tau=\tau_2$, respectively. Using the geodesic equation $$\label{Geodesicequation}
\frac{d^2 x^\mu}{d \tau^2}+\Gamma^\mu_{\nu \rho} \frac{d x^\nu}{d \tau}
\frac{d x^\rho}{d\tau}=0 \, ,$$ an easy computation shows: $$\label{RSgeodintau}
\tau_2=2 \tau_1=\frac{2}{c k} \left(1- e^{-k y_1}\right).$$ The constant $c$ is a remnant of the freedom in the choice of an affine parameter.[^7] From the last equation we see that now $$\lim_{y_1 \to \infty} \tau_1 = \frac{1}{c k},$$ meaning that in order to reach infinity ($y=\infty)$ in the extra dimension it takes only a finite amount $1/(c k)$ of affine parameter $\tau$, the expression of incompleteness of the Randall-Sundrum II space-time with respect to affinely parametrized lightlike geodesics.
In the case of timelike geodesics the inconsistency is even more striking. As shown in e.g. [@Gregory:2000rh], a massive particle starting at the brane with vanishing initial velocity travels to $y=\infty$ in finite proper time given by $\tau_p=\pi/(2 k)$ while for a brane-bound observer this happens in an infinite coordinate time. To summarize, the Randall-Sundrum II brane-world model is geodesically incomplete both for null and for timelike geodesics. As we will see in the next section, the geodesic incompleteness is a direct consequence of the use of a particular coordinate system in $AdS_5$, the so-called Poincaré-coordinate system, which covers only a part of the full $AdS_5$ space-time. We will also see that the problem of incomplete geodesics is absent in the setup we propose in this paper.
Geodesics in the background (\[metric\]) and Penrose-diagram {#ourgeod}
------------------------------------------------------------
We would now like to answer similar questions to the ones considered in the previous section for the background (\[metric\]). For example, we would like to know what time it takes for light to travel from the brane (at $\theta$=0) in the $\theta$-direction to a given point in the upper hemisphere with $\theta$ coordinate $\theta_1$, to be reflected and to return to the brane. As in the last section, $t$ and $\tau$ will denote the coordinate time (proper time of a stationary observer on the brane) and the affine parameter used for parameterizing the geodesics, respectively. Again we choose $t=0$ ($\tau=0$) for the moment of emission, $t_1$ ($\tau_1$) for the reflection and $t_2$ ($\tau_2$) for the time where the photon returns to the brane. Due to the enormous hierarchy of distance scales in our model, one might wonder whether photons (or gravitons) are able to carry information from an arbitrary point on the brane to any other point on the brane connected to the first one by a null-geodesic in the extra dimension. For an observer on the brane such a possibility would be interpreted as 4-dimensional causality-violation. However, as we will see in the following, none of these possibilities exist in our model. Omitting all details we find $$\label{Ourgeodint}
t_2=2 t_1=4 R_U \coth\left(\frac{\omega\pi}{2}\right)
\arctan \left[\frac{\sinh(
\frac{\omega\theta_1}{2})}{\cosh\left[\frac{\omega}{2}\left(\pi-\theta_1\right)\right]} \right],$$ so that an observer on the brane will see that the photon reaches the “north pole” $\theta_1=\pi/2$ at finite time $$t_1=R_U \coth\left(\frac{\omega\pi}{2}\right) \arctan \left[ \sinh\left( \frac{\omega \pi}{2}\right)\right]
\approx R_U \frac{\pi}{2}\, .$$ Note that due to the warped geometry, it is $R_U$ entering the last relation and not $R$, so that even though the physical distance to the “north pole” is of the order of $R$ it takes a time of the order of $R_U$ for photons to reach it, excluding causality violation on the brane as discussed above. If the same geodesic is parametrized using an affine parameter we obtain $$\tau_2=2 \tau_1=\frac{2}{c}\frac{R}{\omega}\left[\tanh \left(\frac{\omega\pi}{2}\right)-
\frac{\sinh \left[ \omega \left(\frac{\pi}{2}-\theta_1\right)\right]}{\cosh \frac{\omega\pi}{2}}\right]\, ,$$ such that the amount of affine parameter needed to reach $\theta_1=\pi/2$ starting from the brane is: $$\Delta \tau = \frac{1}{c}\frac{R}{\omega} \tanh \left(\frac{\omega\pi}{2}\right) \approx
\frac{1}{c} \frac{R}{\omega} \, .$$ Here again $c$ reflects the freedom in the choice of the affine parameter.
The results formally resemble those of the Randall-Sundrum II case. However, the important difference is that in our case each geodesic can trivially be extended to arbitrary values of the affine parameter, a simple consequence of the compactness of our space. Once the photon reaches the point $\theta=0$ it continues on its geodesic, approaching the brane, entering the southern “hemisphere”, etc. It is clear that it needs an infinite amount of affine parameter in order to travel infinite coordinate distances. Therefore, the null geodesics in our setup which are the analogues of the incomplete null geodesics in the Randall-Sundrum II setup turn out to be perfectly complete due to the compactness of our space. The situation for timelike geodesics is fully analog to the case of the null geodesics.
To end this section about the geometric properties of our model we would like to discuss the conformal structure of our space-time and point out differences to the Randall-Sundrum II setup. Let us review briefly the basic properties of $AdS_5$ space-time to the extend that we will need it in the following discussion.[^8] $AdS_5$ space-time can be thought of as the hyperboloid defined by $$\label{hyperboloid}
X_0^2+X_5^2-X_1^2-X_2^2-X_3^2-X_4^2=a^2$$ embedded in a flat space with metric $$\label{AdSDefineMetric}
ds^2=-dX_0^2-dX_5^2+dX_1^2+dX_2^2+dX_3^2+dX_4^2,$$ $a$ being the so-called $AdS$-radius. The [*global coordinates*]{} of $AdS_5$ are defined by $$\begin{aligned}
\label{GlobalCoordsDef}
X_0&=a \cosh \chi \cos \tau, &X_5=&a \cosh \chi \sin \tau, \nonumber \\
X_1&=a \sinh \chi \cos \varphi_1, &X_2=&a \sinh \chi \sin \varphi_1 \cos \varphi_2, \nonumber \\
X_3&=a \sinh \chi \sin \varphi_1 \sin \varphi_2 \cos \varphi_3,
&X_4=&a \sinh \chi \sin \varphi_1 \sin \varphi_2 \sin \varphi_3,\end{aligned}$$ where the coordinates are confined by $0\leq \chi$, $-\pi \leq \tau \leq \pi$, $0\leq\varphi_1\leq \pi$, $0\leq\varphi_2\leq \pi$, $0\leq\varphi_3\leq 2\pi$ and $\tau=-\pi$ is identified with $\tau=\pi$. These coordinates cover the full hyperboloid exactly once. Allowing $\tau$ to take values on the real line without the above identification of points gives the universal covering space $CAdS_5$ of $AdS_5$.[^9] In these coordinates, the line element (\[AdSDefineMetric\]) can be written:[^10] $$\label{AdSGlobalCoords}
ds^2=a^2\left(-\cosh^2\chi \; d\tau^2+d\chi^2+\sinh^2\chi \; d\Omega_3^2\right) \, .$$
Another coordinate system can be defined by $$\begin{aligned}
\label{PoincareCoordsDef}
X_0&= \frac{1}{2 u} \left[1+u^2\left(a^2+\vec{x}^2-\bar{t}^{\,2}\right)\right] ,
&X_5=&a \, u \, \bar{t} , \nonumber \\
X^i&=a \, u \, x^i, \; i=1,2,3\; ,
&X^4=& \frac{1}{2 u} \left[1-u^2\left(a^2-\vec{x}^2+\bar{t}^{\, 2}\right)\right],\end{aligned}$$ with $u>0$, $\bar{t} \in (-\infty, \infty)$ and $x^i \in (-\infty, \infty)$. In these [*Poincaré coordinates*]{} the line element (\[AdSDefineMetric\]) takes the form $$\label{AdsPoincareCoords}
ds^2= a^2 \left[ \frac{du^2}{u^2} + u^2 \left(-d\bar{t}^{\, 2}+d\vec{x}^2\right)\right].$$ In contrast to the global coordinates, the Poincaré coordinates do not cover the whole of the $AdS_5$ and $CAdS_5$ space-times [@Aharony:1999ti]. From (\[AdsPoincareCoords\]), after changing coordinates according to $dy=-a \, du/u$ and rescaling $t$ and $x^i$ by the $AdS$-radius $a$ we recover the original Randall-Sundrum II coordinate system given in (\[RSmetric\]). The restrictions on $y$ in the Randall-Sundrum II setup further limit the range covered by their coordinate system to the $0<u \leq 1$ domain of the Poincaré coordinates.
Coming back to the global coordinates, we introduce $\rho$ by $$\tan \rho = \sinh \chi \;\; \mbox{with} \;\; 0\leq \rho < \frac{\pi}{2},$$ so that (\[AdSGlobalCoords\]) becomes $$\label{PenroseCoords}
ds^2=\frac{a^2}{\cos^2 \rho} \left(-d\tau^2+d\rho^2+\sin^2\rho \; d\Omega_3^2 \right).$$
The Penrose-diagrams of $AdS_5$ space-time and its universal covering space-time $CAdS_5$ are shown in Fig. \[penrose\], see [@HawkingEllis; @Avis:1977yn].
While the $AdS_5$ space-time contains closed timelike curves (denoted by $\gamma$ in the figure), its universal covering space-time $CAdS_5$ does not. The arrows in the left diagram indicate that the lines $\tau=-\pi$ and $\tau=\pi$ should be identified. The symbol $\mathscr{I}$ stands for the timelike surface $\rho=\pi/2$ (spatial infinity). It is this surface which is responsible for the absence of a Cauchy-surface in $AdS$-space. We will concentrate in the following on the $CAdS_5$ diagram. First we note that each point in the diagram corresponds to a $3$-sphere. The shaded region indicates the patch covered by the Poincaré (and Randall-Sundrum) coordinates. Note that the position of the Randall-Sundrum brane cannot be represented in a simple way in the Penrose diagram of $CAdS_5$. The reason is that the $u=\mbox{const.}$ hypersurfaces of the Poincaré coordinates generate a slicing of flat $4$-dimensional Minkowski space-times while the points in the diagram represent (curved) $3$-spheres. From the Penrose diagram it is immediately clear that the Randall-Sundrum-II space-time is geodesically incomplete. The timelike curves denoted by $\gamma$ emanating from the origin ($\rho=0,\tau=0$) will all eventually exit the shaded region after a finite coordinate time $\tau$. These geodesics appear inextendible from the point of view of the Randall-Sundrum space-time. The problems arise due to the arbitrary cutting of a space-time along the borders of a given coordinate patch which covers only a part of the initial space-time. Similar conclusions can be drawn for null geodesics, represented by lines making angles of $45$ degrees with the vertical lines in Fig. \[penrose\]. The vertical dotted line with topology $\mathbb{R} \times S^3$ at coordinate $\rho_b=\arctan \left[ \sinh \left(\frac{\omega\pi}{2} \right)\right]$ close to $\rho=\pi/2$ in the right diagram corresponds to the location of our curved three brane, $\tau$ being proportional to our time coordinate $t$ (see footnote \[changecoords\]). We note that since $\omega > 50$, this line is drawn by far too distant from $\rho=\pi/2$ as a simple expansion shows: $$\rho_b=\arctan \left[ \sinh \left(\frac{\omega \pi}{2}\right)\right] \sim \frac{\pi}{2} -2 e^{-\frac{\omega\pi}{2}}+
\mathcal{O}(e^{-\frac{3 \omega\pi}{2}}) \, .$$ The slice of $CAdS_5$ to the right of $\rho_b$ ($\rho_b \leq \rho \leq \pi/2$) is discarded and replaced by another copy of the slice to the left of $\rho_b$ ($0\leq\rho\leq \rho_b$).
From the Penrose diagram we can also deduce that our space-time is geodesically complete. Timelike geodesics (emanating from ($\rho=0,\tau=0$)) cross the brane $\rho_b$ at some point or return to the origin depending on the initial velocities of the particles that define them. In both cases (as in the case of null geodesics) there is no obstacle to extend the affine parameter to larger and larger values.
As we already mentioned, the absence of a Cauchy surface in $CAdS$-space-time is due to the existence of the timelike surface $\mathscr{I}$ at spatial infinity. By construction, our space-time excludes $\mathscr{I}$ so that the natural question arises whether it is legitimate to revert the above argument and conclude the existence of a Cauchy surface in our model. Though interesting, we do not intend to elaborate this question any further in this paper.
Gravity localization on the brane
=================================
\[three\]
There are at least two equivalent ways of addressing the problem of the localization of gravity in brane-world scenarios. Both ways have their advantages and disadvantages. The first way is based on a detailed study of the Kaluza-Klein excitations of the graviton. After integrating out the extra dimension(s) one obtains an effective 4 dimensional Lagrangian involving the full tower of Kaluza-Klein gravitons. Considering the low energy scattering process of two test particles on the brane via exchange of Kaluza-Klein excitations allows to relate the non-relativistic scattering amplitude to the static potential felt by the two test particles. Since the coupling of each individual Kaluza-Klein particle to matter on the brane is proportional to the value of its transverse wave-function at the position of the brane, this approach necessitates a proper normalization of all Kaluza-Klein modes. An advantage of this approach is the possibility of distinguishing between the contributions to the potential coming from the zero mode (Newton’s law) and from the higher modes (corrections thereof).
The second approach is based on a direct calculation of the graviton two-point function in the space-time under consideration, having the obvious advantage of bypassing all technicalities related to the Kaluza-Klein spectrum and the wave-function normalization. However, a physical interpretation of the effects of individual Kaluza-Klein modes from the point of view of a 4-dimensional observer is, to say the least, not straightforward. Finally, due to the equivalence of the two approaches it is clearly possible to fill this gap by reading of the Kaluza-Klein spectrum and the (normalized) wave-functions from the two-point function by locating its poles and determining corresponding residues.
In the setup considered in this paper we choose to work in the second approach, the direct evaluation of the Green’s functions, due to additional difficulties arising from the non-Minkowskian nature of the induced metric on the brane. The lack of Poincaré invariance on the brane clearly implies the absence of this symmetry in the effective 4-dimensional Lagrangian as well as non-Minkowskian dispersion relations for the Kaluza-Klein modes.
Perturbation equations and junction conditions
----------------------------------------------
We would like to study the fluctuations $H_{ij}$ of the metric in the background (\[metric\]) defined by: $$ds^2=-\sigma^2 dt^2+R_U^2 \gamma^2 \left[\eta_{ij}+2 H_{ij}\right] d\varphi^i d\varphi^j +R^2 d\theta^2.$$ $H_{ij}$ transform as a transverse, traceless second-rank tensor with respect to coordinate transformations on the maximally symmetric space with metric $\eta_{ij}$, a $3$-sphere in our case: $$\eta^{ij} H_{ij}=0, \quad \eta^{i j} \tilde{\nabla}_i H_{j k}=0,$$ where $\tilde{\nabla}$ denotes the covariant derivative associated with the metric $\eta_{ij}$ on $S^3$. The gauge invariant symmetric tensor $H_{ij}$ does not couple to the vector and scalar perturbations. Its perturbation equation in the bulk is obtained from the transverse traceless component of the perturbed Einstein equations in $5$ dimensions: $$\delta R_{M N}-\frac{2}{3} \Lambda \delta g_{M N}=0 \, .$$ Specifying this equation to the case of interest of a static background we find: $$\label{Hfluct}
\frac{1}{\sigma^2} \ddot{H}-\frac{1}{R_U^2\gamma^2}\tilde{\Delta} H_{ij} -
\frac{1}{R^2}H_{ij}^{\prime \prime}-
\frac{1}{R^2} \left( \frac{\sigma^\prime}{\sigma}
+3\frac{\gamma^\prime}{\gamma}\right)H_{ij}^{\prime}
+\frac{2}{R_U^2\gamma^2}H_{ij} = 0,$$ where $\tilde{\Delta}$ denotes the Laplacian on $S^3$. The last equation is valid in the bulk in the absence of sources. We can now conveniently expand $H_{ij}$ in the basis of the symmetric transverse traceless tensor harmonics $\hat{T}^{(l \lambda)}_{ij}$ on $S^3$ [@Rubin:tc; @Allen:1986tt; @Mukohyama:2000ui]: $$\label{Hexpand}
H_{ij}=\sum_{l=2}^\infty \sum_\lambda \Phi^{(l \lambda)}(t,\theta) \hat{T}^{(l \lambda)}_{ij},$$ where the $\hat{T}^{(l \lambda)}_{ij}$ satisfy $$\begin{aligned}
\label{TTHarmonics}
\tilde{\Delta} \hat{T}^{(l \lambda)}_{ij} + k_l^2 \hat{T}^{(l \lambda)}_{ij} =0,
\qquad k_l^2= l(l+2)-2, \qquad l=2,3,... \nonumber \\
\eta^{ij}\tilde{\nabla}_i \hat{T}^{(l \lambda)}_{jk}=0, \qquad
\eta^{ij} \hat{T}^{(l \lambda)}_{ij} =0, \qquad
\hat{T}^{(l \lambda)}_{[ij]}=0.\end{aligned}$$ The sum over $\lambda$ is symbolic and replaces all eigenvalues needed to describe the full degeneracy of the subspace of solutions for a given value of $l$. Introducing the expansion (\[Hexpand\]) into (\[Hfluct\]) and using the orthogonality relation of the tensor harmonics [@Allen:1986tt] $$\label{TTTOrthogonality}
\int \sqrt{\eta } \, \eta^{ik} \, \eta^{jl} \, \hat{T}^{(l \lambda)}_{ij}
\, \hat{T}^{(l' \lambda')}_{kl} d^3\varphi =
\delta^{l l'} \, \delta^{\lambda \lambda'} \, ,$$ we obtain: $$\label{Hfluctexpansion}
\frac{1}{\sigma^2} \ddot{\Phi}^{(l \lambda)}- \frac{1}{R^2}\Phi^{(l \lambda) \prime \prime}-
\frac{1}{R^2} \left( \frac{\sigma^\prime}{\sigma} + 3 \frac{\gamma^\prime}{\gamma}\right)
\Phi^{(l \lambda) \prime}+\frac{l(l+2)}{R_U^2\gamma^2} \Phi^{(l \lambda)}=0.$$ This equation has to be compared to the equation of a massless scalar field in the background (\[metric\]). After expanding the massless scalar in the corresponding scalar harmonics on $S^3$ we recover (\[Hfluctexpansion\]) with the only difference in the eigenvalue parameters due to different spectra of the Laplacian $\tilde{\Delta}$ for scalars and for tensors. Motivated by this last observation and in order to avoid the technicalities related to the tensorial nature of the graviton $H_{ij}$ we confine ourselves in this paper to the study of the Green’s functions of a massless scalar field in the background (\[metric\]).
The differential equation (\[Hfluctexpansion\]) alone does not determine $H_{ij}$ uniquely. We have to impose proper boundary conditions for $H_{ij}$. While imposing square integrability will constitute one boundary condition at $\theta=\pm \pi/2$, the behavior of $H_{ij}$ on the brane will be dictated by the Israel junction condition [@Israel:rt]: $$\label{israelcondition}
: K_{\mu \nu} : \;\;\; =-\frac{8 \pi}{M^3}\left(T_{\mu\nu}-\frac{1}{3} T_\kappa^{\;\kappa} \bar{g}_{\mu \nu}
\right).$$ Here $K_{\mu\nu}$ and $\bar{g}_{\mu\nu}$ denote the extrinsic curvature and the induced metric on the brane, respectively. $T_{\mu\nu}$ is the $4$-dimensional stress-energy tensor on the brane[^11] and the symbol $: \ldots :$ is defined in (\[::def\]). In our coordinate system the extrinsic curvature is given by: $$\label{extrCurvature}
K_{\mu\nu}=\frac{1}{2 R} \frac{\partial{\bar{g}_{\mu \nu}}}{\partial \theta} \, .$$ Hence, the non-trivial components of the Israel condition become: $$\begin{aligned}
: \frac{\sigma^\prime}{\sigma} : &= -\frac{8 \pi R}{M^3} \left(\frac{2}{3} c_0-c
\right) \, , \label{Israel00BG} \\
: \frac{\gamma^\prime}{\gamma} \left( \eta_{ij}+2 H_{ij} \right) + H^\prime_{ij} :
&= \frac{8 \pi R}{3 M^3} c_0 \left(\eta_{ij}+2 H_{ij} \right). \label{Israelij}\end{aligned}$$ Separating the background from the fluctuation in (\[Israelij\]) we find: $$\begin{aligned}
: \frac{\gamma^\prime}{\gamma} : &= \frac{8 \pi R}{3 M^3} c_0 \, , \label{IsraelijBG}\\
: H^\prime_{ij} : &= 0 \label{IsraelijFluct} \, ,\end{aligned}$$ where we only used the continuity of $H_{ij}$ on the brane. While eq. (\[IsraelijBG\]) directly coincides with eq. (\[FineTuneI\]) of section (\[SingularBrane\]), eq. (\[Israel00BG\]) turns out to be a linear combination of (\[FineTuneI\]) and (\[FineTuneII\]).
Eq. (\[IsraelijFluct\]) taken alone implies the continuity of $H^\prime_{ij}$ on the brane. If in addition the fluctuations are supposed to satisfy the $\theta \to-\theta$ symmetry, this condition reduces to a Neumann condition, as for example in our treatment of the scalar two-point function in the Randall-Sundrum II case (see appendix \[appParallel\]).
In the next section we are going to find the static Green’s functions of a massless scalar field in the Einstein static universe background. This will serve as a preparatory step for how to handle the more complicated case of a non-invertible Laplacian. More importantly, it will provide the necessary reference needed for the interpretation of the effect of the extra dimension on the potential between two test masses on the brane.
Gravity in the Einstein static universe
---------------------------------------
As announced in the previous section, we will concentrate on the massless scalar field. Our aim is to solve the analog of Poisson equation in the Einstein static universe background. Due to its topology $\mathbb{R} \times S^3$ we will encounter a difficulty related to the existence of a zero eigenvalue of the scalar Laplacian on $S^3$ necessitating the introduction of a modified Green’s function.[^12] Note that in the more physical case where the full tensor structure of the graviton is maintained no such step is necessary since all eigenvalues of the tensorial Laplacian are strictly negative on $S^3$, see (\[TTHarmonics\]).
### Definition of a modified Green’s function
We take the metric of Einstein’s static universe in the form: $$\label{esu_metric}
ds^2=g_{\mu\nu}dx^\mu dx^\nu = - dt^2+A^2 d \Omega_3^2$$ with $d\Omega_3^2$ being the line element of a $3$-sphere given in (\[S3metric\]) and $A$ being its constant radius. The equation of a massless scalar field then becomes: $$\label{ScaFieldEqESU}
\frac{1}{\sqrt{-g}}\partial_\mu \left[\sqrt{-g} \, g^{\mu\nu}\partial_\nu u(t,\vec{x}) \right]= j(t,\vec x)$$ which in the static case reduces to $$\label{StaticEqESU}
\mathcal{D} u(\vec x) \equiv \frac{1}{A^2} \tilde{\Delta} u(\vec x) = j(\vec x) \, ,$$ where $\tilde{\Delta}$ is the scalar Laplacian on $S^3$ and we choose to write $\vec x$ for the collection of the three angles on $S^3$. All functions involved are supposed to obey periodic boundary condition on $S^3$ so that in Green’s identity all boundary terms vanish: $$\label{GreensIdentityESU}
\int \sqrt{-g} \, \left(\mathcal{D} \, v(\vec x)\right) \, \overline{u}(\vec x) \, d^3 \vec{x} =
\int \sqrt{-g} \, v(\vec x) \, \overline{\mathcal{D} \, u(\vec x)} \, d^3 \vec{x}=
\int \sqrt{-g} \, v(\vec x) \, \overline{j}(\vec x) \, d^3\vec{x} \, .$$ Here and in the following bars denote complex conjugate quantities. The operator $\mathcal{D}$ trivially allows for an eigenfunction with zero eigenvalue (the constant function on $S^3$): $$\mathcal{D} u_0(\vec x) = 0 \, ,\quad \int \sqrt{-g} \, u_0(\vec x) \, \overline{u}_0(\vec x) \,
d^3 \vec x=1 \, .$$ Then, since the homogeneous equation has not only the trivial solution ($u=0$) which satisfies the periodic boundary conditions in the angular variables, the operator $\mathcal{D}$ cannot be invertible. Therefore, there is no solution to the equation (\[StaticEqESU\]) for an arbitrary source. In order to still define a “modified” Green’s function we have to restrict the possible sources to sources that satisfy the following solvability condition: $$\label{solvabilityESU}
\int \sqrt{-g} \, j(\vec x) \, \bar{u}_0(\vec x) \, d^3\vec x=0.$$ Formally this condition can be obtained by replacing $v(\vec x)$ by the non-trivial solution of the homogeneous equation $u_0(\vec x)$ in (\[GreensIdentityESU\]). We now define the modified Green’s function as a solution of $$\label{ModGreenFunctionESU}
\mathcal{D}_x \mathcal{G}(\vec x,\vec x')=\frac{\delta^3(\vec x-\vec x')}{\sqrt{-g}}-u_0(\vec x)
\, \bar{u}_0(\vec x') \, .$$ Replacing now $v(\vec x)$ by $\mathcal{G}(\vec x,\vec x')$ in Green’s identity (\[GreensIdentityESU\]) we obtain (after complex conjugation) the desired integral representation for the solution of (\[StaticEqESU\]): $$\begin{aligned}
\label{esuIntReprESU}
u(\vec x) = C u_0(\vec x)+\int \sqrt{-g} \, j(\vec x') \, \overline{\mathcal{G}}(\vec x',\vec x) \,
d^3 \vec x' \, ,\end{aligned}$$ where $C$ is given by $$C=\int \sqrt{-g} \, u(\vec x') \, \overline{u}_0(\vec x') \, d^3\vec x' \, .$$ Note that the source $$j_0(\vec x)=\frac{\delta^3(\vec x-\vec x')}{\sqrt{-g}}-u_0(\vec x)\bar{u}_0(\vec x')$$ trivially satisfies the solvability condition (\[solvabilityESU\]) and represents a point source at the location $\vec x=\vec x'$ compensated by a uniform negative mass density. To fix the normalization of $u_0(\vec x)$ we write: $$u_0(\vec x)=N_0 \Phi_{100}(\vec x) \,\,\, \mbox{with} \,\,\, \int \sqrt{\eta} \, \Phi_{100}(\vec x)
\overline{\Phi}_{100}(\vec x) d^3 \vec x=1,$$ where $\Phi_{100}=\frac{1}{\pi \sqrt{2}}$ and so $$\int \sqrt{-g} \, u_0(\vec x) \, \overline{u}_0(\vec x) \, d^3\vec x = \vert N_0 \vert^2 A^3
\underbrace{\int \sqrt{\eta} \, \Phi_{100}(\vec x) \, \overline{\Phi}_{100}(\vec x) \, d^3 \vec x}_{=1} =
\vert N_0 \vert^2 A^3=1 \, ,$$ so that $$u_0(\vec x)=\frac{1}{A^{3/2}}\Phi_{100}(\vec x).$$ In order to solve the differential equation (\[ModGreenFunctionESU\]) defining the modified Green’s function $\mathcal{G}(\vec x, \vec x')$, we expand in eigenfunctions of the Laplace operator on $S^3$, the so-called scalar harmonics $\Phi_{\lambda l m}$ with properties (see e.g. [@Kodama:2000fa; @GribMamaMost]): $$\label{ScalarHarmonicsESU}
\tilde{\Delta} \Phi_{\lambda l m} =(1-\lambda^2 )\Phi_{\lambda l m}
\qquad \lambda=1,2,\ldots; l=0,\ldots,\lambda-1;m=-l,\ldots,l \, .$$ Our ansatz therefore reads: $$\mathcal{G}(\vec x, \vec x')=\sum_{\lambda=1}^\infty \sum_{l=0}^{\lambda-1}\sum_{m=-l}^l
\Phi_{\lambda l m}(\vec x) \, c_{\lambda l m}(\vec x') \, .$$ Introducing this in (\[ModGreenFunctionESU\]) we obtain $$\sum_{\lambda=1}^{\infty}\sum_{l=0}^{\lambda-1}\sum_{m=-l}^l \left[-\frac{\lambda^2-1}{A^2}
c_{\lambda l m}(\vec x')\right] \Phi_{\lambda l m}(\vec x)=
\frac{\delta^3(\vec x-\vec x')}{\sqrt{-g}}-u_0(\vec x)\, \overline{u}_0(\vec x')\, .$$ If we now multiply by $\Phi_{\lambda' l' m'}(\vec x) \, \sqrt{\eta}$ and integrate over $S^3$ we find $$-\frac{\lambda'^2-1}{A^2}c_{\lambda' l' m'}(\vec x') =\frac{1}{A^3}
\overline{\Phi}_{\lambda' l' m'}(\vec x')-\bar{u}_0(\vec x') \int\limits_{S^3} \sqrt{\eta} \,
u_0(\vec x) \, \overline{\Phi}_{\lambda' l' m'}(\vec x) \, d^3 \vec x,$$ where we made use of the orthogonality relation of the scalar harmonics $$\label{Harmonicsorthogonality}
\int \sqrt{\eta} \, \overline{\Phi}_{\lambda l m}(\vec x) \, \Phi_{\lambda' l' m'}(\vec x) \, d^3 \vec x=
\delta_{\lambda \lambda'} \delta_{l l'} \delta_{m m'}.$$ In the case $\lambda'=1$ (and vanishing $l'$ and $m'$) the above equation is identically satisfied for all values of $c_{1 0 0}(\vec x')$.[^13]
In the case $\lambda'\neq 1$ the coefficient $c_{\lambda' l' m'}(\vec x')$ follows to be $$c_{\lambda' l' m'}(\vec x')=\frac{\overline{\Phi}_{\lambda' l' m'}(\vec x')}{A(1-\lambda'^2)}$$ so that the formal solution for the modified Green’s function can be written as $$\mathcal{G}(\vec x,\vec x') =
\sum_{\lambda=2}^{\infty} \sum_{l=0}^{\lambda-1} \sum_{m=-l}^{l}
\frac{\Phi_{\lambda l m}(\vec x) \bar{\Phi}_{\lambda l m}(\vec x')}{A(1-\lambda^2)}\, .$$ Due to the maximal symmetry of the $3$-sphere, the Green’s function $\mathcal{G}(\vec x,\vec x')$ can only depend on the geodesic distance $s(\vec x, \vec x') \in [0,\pi]$ between the two points $\vec x$ and $\vec x'$: $$\begin{aligned}
\label{DistanceOnS3}
\cos s &=& \cos \varphi_1 \cos \varphi'_1+\sin\varphi_1\sin\varphi'_1 \cos\beta \, ,\nonumber\\
\cos\beta&=& \cos\varphi_2 \cos \varphi'_2+ \sin\varphi_2\sin\varphi'_2 \cos(\varphi_3-\varphi'_3) \, ,\end{aligned}$$ which in the case $\varphi_2=\varphi'_2$ and $\varphi_3=\varphi'_3$ clearly reduces to $s=\varphi_1-\varphi'_1$. Indeed, the sum over $l$ and $m$ can be performed using[^14] $$\label{SumOverS3}
\sum_{l=0}^{\lambda-1} \sum_{m=-l}^{l} \bar{\Phi}_{\lambda l m}(\vec x)
\Phi_{\lambda l m}(\vec x')=\frac{\lambda}{2 \pi^2} \frac{\sin \left(\lambda s\right)}{\sin s}$$ such that even the remaining sum over $\lambda$ can be done analytically: $$\label{ESUresult}
\tilde{\mathcal{G}}(s) \equiv \mathcal{G}(\vec x,\vec x')=
\frac{1}{8 \pi^2 A}-\frac{1}{4\pi A}\left[(1-\frac{s}{\pi})\cot s \right]\, , \qquad s \in [0,\pi]\, .$$ To interpret this result, we develop $\tilde{\mathcal{G}}(s)$ around $s=0$ obtaining[^15] $$\label{expansioninsESU}
\tilde{\mathcal{G}}(s)=\frac{1}{A}\left[-\frac{1}{4\pi s} +\frac{3}{8\pi^2}+ \frac{s}{12 \pi}
+\mathcal{O}(s^2) \right] \, .$$ By introducing the variable $r=s A$ and by treating $\tilde{\mathcal{G}}(s)$ as a gravitational potential we find $$\frac{1}{A} \frac{d\tilde{\mathcal{G}}(s)}{ds}=\frac{1}{4\pi r^2}+\frac{1}{12\pi A^2}+\mathcal{O}(r/A^3).$$ We notice that for short distances, $r \ll A$, we find the expected flat result whereas the corrections to Newton’s law become important at distances $r$ of the order of $A$ in the form of a constant attracting force.[^16]
Modified Green’s function of a massless scalar field in the background space-time (\[metric\])
----------------------------------------------------------------------------------------------
We are now prepared to address the main problem of this work namely the computation of the modified Green’s function of a massless scalar field in the background space-time (\[metric\]). Since our main interest focuses again on the low energy properties of the two-point function, we will limit ourselves to the static case. The main line of reasoning is the same as in the previous section. Due to the fact that our space has the global topology of a 4-sphere $S^4$, we again are confronted with a non-invertible differential operator. We would like to solve $$\label{ScalarEq}
\mathcal{D} u(\varphi_i,\theta)
\equiv \frac{\tilde{\Delta} u(\varphi_i,\theta)}{R_U^2 \gamma(\theta)^2}+
\frac{1}{R^2} \frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}
\left[ \sigma(\theta)\gamma(\theta)^3 \frac{\partial u(\varphi_i,\theta)}{\partial \theta}\right]=
j(\varphi_i,\theta) \, ,$$ where the independent angular variable ranges are $0\leq\varphi_1\leq\pi;\,0\leq\varphi_2\leq\pi;\, 0\leq\varphi_3\leq 2 \pi;
\,-\pi/2\leq\theta\leq\pi/2$. Every discussion of Green’s functions is based on Green’s identity relating the differential operator under consideration to its adjoint operator. Since $\mathcal{D}$ is formally self-adjoint we have $$\begin{aligned}
\label{GreensIdentity}
\int\limits_0^\pi d\varphi_1 \int\limits_0^\pi d\varphi_2 \int\limits_0^{2\pi} d\varphi_3
\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\theta \sqrt{-g}
\left[(\mathcal{D} v) \overline{u} -v \overline{(\mathcal{D} u)}\right]=\nonumber \\
\int\limits_0^\pi d\varphi_1 \int\limits_0^\pi d\varphi_2 \int\limits_0^{2\pi} d\varphi_3 \frac{R_U^3}{R}
\sqrt{\eta}\left[ \sigma(\theta)\gamma(\theta)^3
\left( \bar{u} \frac{\partial v}{\partial\theta}-
\frac{\partial \bar{u}}{\partial\theta} v \right)
\right]_{-\frac{\pi}{2}}^{\phantom{-}\frac{\pi}{2}}+\ldots \, ,\end{aligned}$$ where we dropped the arguments of $u$ and $v$ for simplicity. The dots in (\[GreensIdentity\]) refer to boundary terms in the variables $\varphi_i$ and since we again employ an eigenfunction expansion in scalar harmonics on $S^3$, these boundary terms will vanish. In order to find an integral representation of the solution $u(\varphi_i,\theta)$ of (\[ScalarEq\]) we have to impose appropriate boundary conditions on $u$ and $v$ at $\theta=\pm \pi/2$. For the time being we assume this to be the case such that all boundary terms in (\[GreensIdentity\]) vanish and proceed with the formal solution of (\[ScalarEq\]). We will address the issue of boundary conditions in $\theta$ in detail in appendices \[appParallel\] and \[appDEQ\].
In the following we will collectively use $x$ instead of $(\varphi_i, \theta)$. The homogeneous equation $\mathcal{D} u(x)=~0$ does not have a unique solution under the assumption of periodic boundary conditions. In addition to the trivial solution $(u=0)$ we also find $$\label{ZeroMode}
\mathcal{D} \, u_0(x) = 0\, , \qquad \int \sqrt{-g} \, u_0(x) \,
\overline{u}_0(x) \, d^4x=1 \, .$$ Therefore, the corresponding inhomogeneous equation (\[ScalarEq\]) does not have a solution unless we again restrict the space of allowed sources: $$\label{solvability}
\int \sqrt{-g} \, j(x) \, \overline{u}_0(x) \, d^4x=0.$$ As in the last section, this condition can be obtained by replacing $v$ by the non-trivial solution of the homogeneous equation $u_0$ in (\[GreensIdentity\]). We now define the modified Green’s function by $$\label{ModGreenFunction}
\mathcal{D}_x \mathcal{G}(x,x')=\frac{\delta^4(x-x')}{\sqrt{-g}}-u_0(x)\overline{u}_0(x') \, .$$ From Green’s identity (\[GreensIdentity\]), with $v(x)$ given by $\mathcal{G}(x,x')$, we again obtain after complex conjugation the desired integral representation: $$\begin{aligned}
\label{esuIntRepr}
u(x) = C u_0(x)+\int \sqrt{-g} \, j(x') \overline{\mathcal{G}(x',x)} \, d^4x' \, ,\end{aligned}$$ with $$C=\int \sqrt{-g} \, u(x') \, \overline{u}_0(x') \, d^4x' \, .$$ As before, the source $$j_0(x)=\frac{\delta^4(x-x')}{\sqrt{-g}}-u_0(x)\, \overline{u}_0(x')$$ satisfies the solvability condition (\[solvability\]) by construction. The normalization of the constant mode $u_0(x)$ is slightly more involved than before due to the nontrivial measure $\sigma(\theta)\gamma(\theta)^3$ in the $\theta$ integration. By inserting $$u_0(x)=N_0 \Phi_{1 0 0}(\varphi_i) \chi_1(\theta) \, \, \mbox{with}\,\, \chi_1(\theta)=1$$ in the integral in (\[ZeroMode\]) we obtain $$N_0 =\left[\frac{2\omega}{R R_U^3 \tanh \left(\frac{\omega \pi}{2}\right)}\right]^{\frac{1}{2}} \,.$$ For the solution of eq. (\[ModGreenFunction\]) we use the ansatz $$\mathcal{G}(\varphi_i,\varphi'_i,\theta,\theta')=\sum_{\lambda=1}^{\infty}\sum_{l=0}^{\lambda-1}
\sum_{m=-l}^l \Phi_{\lambda l m}(\varphi_i) \, c_{\lambda l m}(\varphi'_i,\theta,\theta') \, ,$$ where from now on we decide to write all arguments explicitly. After inserting this in (\[ModGreenFunction\]) we find $$\begin{aligned}
&\sum_{\lambda=1}^{\infty}\sum_{l=0}^{\lambda-1} \sum_{m=-l}^l \left[
-\frac{\lambda^2-1} {R_U^2 \gamma(\theta)^2}c_{\lambda l m}(\varphi'_i,\theta,\theta')+
\frac{1}{R^2} \frac{1}{\sigma(\theta)\gamma(\theta)^3}\frac{\partial}{\partial\theta}\left(
\sigma(\theta)\gamma(\theta)^3
\frac{\partial c_{\lambda l m}(\varphi'_i,\theta,\theta')}{\partial \theta}\right)\right]
\Phi_{\lambda l m}(\varphi_i)\nonumber \\
&\qquad\qquad \qquad\qquad
=\frac{\delta^3(\varphi_i-\varphi'_i) \, \delta(\theta-\theta')}
{\sqrt{-g}}-u_0(\varphi_i,\theta)\, \overline{u}_0(\varphi'_i,\theta').\end{aligned}$$ After multiplication by $\sqrt{\eta} \, \overline{\Phi}_{\lambda' l' m'}(\varphi_i)$ and integration over $S^3$ we obtain $$\begin{aligned}
\label{TransverseEquation}
&-\frac{\lambda'^2-1} {R_U^2 \gamma(\theta)^2}c_{\lambda' l' m'}(\varphi'_i,\theta,\theta')+
\frac{1}{R^2}\frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}\left[
\sigma(\theta)\gamma(\theta)^3 \frac{\partial c_{\lambda' l' m'}(\varphi'_i,\theta,\theta')}
{\partial \theta}\right] \\ \nonumber
& \qquad \qquad =\frac{1}{R_U^3 R}\frac{1}{\sigma(\theta)\gamma(\theta)^3} \delta(\theta-\theta')
\overline{\Phi}_{\lambda' l' m'}(\varphi'_i)-
\overline{u}_0(\varphi'_i,\theta') \int_{S^3}\sqrt{\eta} \, u_0(\varphi_i,\theta) \,
\overline{\Phi}_{\lambda' l' m'}(\varphi_i) \, d^3\varphi_i \, .\end{aligned}$$
We now have to distinguish the cases $\lambda'=1$ and $\lambda'\neq 1$.
1. $( \lambda' l' m')=(1 0 0)$.
In this case, the last term on the right hand side of (\[TransverseEquation\]) will give a non-vanishing contribution: $$\begin{aligned}
&\frac{1}{R^2}\frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}\left[
\sigma(\theta)\gamma(\theta)^3 \frac{\partial c_{1 0 0}(\varphi'_i,\theta,\theta')}
{\partial \theta}\right] =\frac{1}{R_U^3 R} \frac{1}{\sigma(\theta)\gamma(\theta)^3}
\delta(\theta-\theta') \overline{\Phi}_{1 0 0}(\varphi'_i)-\\
&\qquad \vert N_0 \vert^2 \bar{\Phi}_{1 0 0}(\varphi'_i)
\chi_1(\theta) \overline{\chi}_1(\theta') \underbrace{\int_{S^3} \sqrt{\eta} \, \Phi_{1 0 0}(\varphi_i)
\overline{\Phi}_{1 0 0}(\varphi_i) \, d^3 \varphi_i}_{=1} \nonumber \, .\end{aligned}$$ By defining $g^{(1)}(\theta,\theta')$ by the relation $$c_{1 0 0}(\varphi'_i,\theta,\theta')=\frac{1}{R_U^3 R} \bar{\Phi}_{1 0 0}(\varphi'_i)
g^{(1)}(\theta,\theta')$$ we obtain the following differential equation for $g^{(1)}(\theta,\theta')$: $$\label{TransverseEq1}
\frac{1}{R^2} \frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}
\left[ \sigma(\theta)\gamma(\theta)^3 \frac{\partial g^{(1)}(\theta,\theta')}{\partial \theta}
\right]=\frac{1}{\sigma(\theta)\gamma(\theta)^3} \delta(\theta-\theta')-
\tilde{\chi}_1(\theta) \bar{\tilde{\chi}}_1(\theta') \, ,$$ where we used $$\tilde{\chi}_1(\theta)\equiv \left[ \frac{2\omega}{\tanh \left(\frac{\omega\pi}{2}\right)}
\right]^{\frac{1}{2}} \chi_1(\theta)\, , \qquad (\chi_1(\theta)\equiv1) \,.$$ Note that we defined $\tilde{\chi}_1(\theta)$ in such a way that $$\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sigma(\theta)\gamma(\theta)^3 \tilde{\chi}_1(\theta)
\bar{\tilde{\chi}}_1(\theta) d\theta=1.$$
2. $(\lambda' l' m')\neq(1 0 0)$.
Due to the orthogonality of $\Phi_{1 0 0}$ and $\Phi_{\lambda' l' m'}$ on $S^3$, the last term on the right hand side of (\[TransverseEquation\]) vanishes. We therefore have $$\begin{aligned}
&-\frac{\lambda'^2-1}{R_U^2 \gamma(\theta)^2} c_{\lambda' l' m'}(\varphi'_i,\theta,\theta')+
\frac{1}{R^2}\frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}\left[
\sigma(\theta)\gamma(\theta)^3 \frac{\partial c_{\lambda' l' m'}(\varphi'_i,\theta,\theta')}
{\partial \theta}\right]\qquad \qquad \nonumber \\
&\qquad =\frac{1}{R_U^3 R} \frac{1}{\sigma(\theta)\gamma(\theta)^3} \delta(\theta-\theta')
\bar{\Phi}_{\lambda' l' m'}(\varphi'_i)\, .
\end{aligned}$$ Introducing $g^{(\lambda')}(\theta,\theta')$ again via $$c_{\lambda' l' m'}(\varphi'_i,\theta,\theta')=\frac{1}{R_U^3 R} \bar{\Phi}_{\lambda' l' m'}(\varphi'_i)
g^{(\lambda')}(\theta,\theta')\, ,$$ we see that $g^{(\lambda')}(\theta,\theta')$ has to satisfy $$\label{TransverseEq2}
\frac{1-\lambda'^2}{R_U^2 \gamma(\theta)^2} g^{(\lambda')}(\theta,\theta')+
\frac{1}{R^2} \frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}
\left[ \sigma(\theta)\gamma(\theta)^3 \frac{\partial g^{(\lambda')}(\theta,\theta')}{\partial \theta}
\right]=\frac{\delta(\theta-\theta')}{\sigma(\theta)\gamma(\theta)^3} \, .$$
Combining the above results for $\lambda=1$ and $\lambda\neq 1$ we are able to write the formal solution of (\[ModGreenFunction\]) as $$\mathcal{G}(\varphi_i,\varphi'_i,\theta,\theta')=\frac{1}{R_U^3 R}\sum_{\lambda=1}^{\infty}
\sum_{l=0}^{\lambda-1}\sum_{m=-l}^l \Phi_{\lambda l m}(\varphi_i)\bar{\Phi}_{\lambda l m}(\varphi'_i)
g^{(\lambda)}(\theta,\theta') \, .$$
We obtain a further simplification of this formal solution by employing the spherical symmetry on $S^3$, see eq. (\[SumOverS3\]), leaving us with a representation of the two-point function by a Fourier sum, which is natural for a compact space without boundaries: $$\label{FormalSolution}
\tilde{\mathcal{G}}(s,\theta,\theta')=\mathcal{G}(\varphi_i,\varphi'_i,\theta,\theta')
=\frac{1}{R R_U^3} \sum_{\lambda=1}^{\infty} \frac{\lambda}{2 \pi^2} \frac{\sin(\lambda s)}{\sin s}
g^{(\lambda)}(\theta,\theta') \, .$$
Finding the formal solution (\[FormalSolution\]) was straightforward apart from minor complications inherent to the use of a modified Green’s function. To solve the differential equations (\[TransverseEq1\]) and (\[TransverseEq2\]) by imposing appropriate boundary conditions again is a routine task without any conceptual difficulties, though slightly technical in nature. Replacing back this solution in (\[FormalSolution\]) we are left with a Fourier sum which at first sight looks intractable, anticipating the fact that the solutions $g^{(\lambda)}(\theta,\theta')$ (for $\lambda\neq1$) are given by hypergeometric functions. Nevertheless it is possible to extract the desired asymptotic information from the sum (\[FormalSolution\]).
In order not to disturb the transparency and fluidity of the main article we provide large parts of the technical calculations in four appendices. In appendix \[appParallel\] we report similarities and differences in the evaluation of the two-point functions between our case and the case of Randall-Sundrum, since the latter served as a guideline for handling the more difficult case under consideration. The solutions of eqs. (\[TransverseEq1\]) and (\[TransverseEq2\]) are presented in appendix \[appDEQ\] and the evaluation of the Fourier sum (\[FormalSolution\]) at distances exceeding the size of the extra dimension in appendix \[appSum\]. Eventually, appendix \[appGreenUltraShort\] contains the evaluation of the Fourier sum (\[FormalSolution\]) for distances smaller than the extra dimension. In this way, we can offer the reader less interested in the details of the computations to have the main results at hand.
From its definition (\[ModGreenFunction\]) we understand that the Green’s function (\[FormalSolution\]) can be considered as the response of the scalar field to the combination of a point-like source located at coordinates $(\varphi'_i, \theta')$ and a delocalized, compensating negative contribution. Since we would like to see the response to a point-like particle on the brane we put $\theta'=0$ in eq. (\[FormalSolution\]) and explicitly write the $\lambda=1$ term: $$\label{GreenSolBrane}
\tilde{\mathcal{G}}(s,\theta,0)=\frac{g^{(1)}(\theta,0)}{2 \pi^2 R R_U^3}+
\frac{R}{4 \pi^2 R_U^3}\frac{1}{\sin s} S[s,\theta,\omega] \, ,$$ with $S[s,\theta,\omega]$ given by (\[TheSumB\]) of appendix \[appSum\] (see also (\[glambdaratio\]) of appendix \[appDEQ\]). The general result for the sum $S[s,0,\omega]$ obtained in appendix \[appSum\] is $$\begin{aligned}
\label{SresApp}
S[s,0,\omega] \equiv \lim_{\theta \to 0} S[s,\theta,\omega]
&= -\frac{2}{\omega} \frac{z(0)^\frac{1}{2}}{1-z(0)}
\left(\frac{\pi-s}{2}\cos s -\frac{1}{4}\sin s \right)\nonumber \\
&-\frac{1}{2 \omega} z(0)^{-\frac{1}{2}} \ln\left[1-z(0)\right] \sin s
-\frac{1}{\omega} z(0)^\frac{1}{2} \lim_{\theta \to 0} R[s,\theta,\omega]\;,\end{aligned}$$ where we have $z(0)= \tanh^2 \left(\frac{\omega\pi}{2} \right)$ and where we refer to (\[defR\]) for the definition of $R[s,\theta,\omega]$. The first term in (\[SresApp\]) is the zero mode contribution[^17] and we see that it reproduces exactly the $4$-dimensional static Green’s function of Einstein’s static universe given in (\[ESUresult\]). The other two terms are the contributions from the higher Kaluza-Klein modes.
We first concentrate on the case where $s \sim 1$ or what is equivalent $r\sim R_U$. Since one can easily convince oneself that $R[s,0,\omega]\sim \left[1-z(0)\right]^0$ in this regime, the contributions of the higher Kaluza-Klein modes are strongly suppressed with respect to the zero mode contribution. This means that as in the case of the Randall-Sundrum-II model it is the zero mode which dominates the behavior of gravity at distances much larger than the extra dimensions $R$. The main difference to the Randall-Sundrum-II case is that the zero mode of our model not only gives rise to the typical $4$-dimensional $1/r$ singularity but also accounts for the compactness of space by reproducing the Einstein static universe behavior (\[ESUresult\]). This result is somewhat surprising given the extreme anisotropy of our manifold. Due to the fact that the distances between two arbitrary points on the brane are of the order of $R$, one might intuitively expect that the extra dimension can be effective in determining also the large distance behavior of gravity (on the brane). As we could show by direct calculation the above expectation turns out to be incorrect.
Next we consider physical distances $r$ much larger than the extra dimension $r\gg R$ and much smaller than the observable universe $r \ll R_U$ in which case the results for $R[s,0,\omega]$ can be seen to be: $$\begin{aligned}
R[s,0,\omega]&\sim\frac{\pi}{2 s^2}+
\frac{\pi}{2}\frac{1-z(0)}{s^4} \left\{ 8-6\ln 2 -6 \ln\left[s \left[1-z(0)\right]^{-1/2}\right]\right\}
\nonumber \\
&+\mathcal{O}\left[\left[1-z(0)\right]^2 \frac{\ln\left[s \left[1-z(0)\right]^{-1/2}\right]}{s^6}\right]\end{aligned}$$ valid for $\left[1-z(0)\right]^{1/2} \ll s \ll 1$. The zero mode contribution (to $S[s,0,\omega]$) in this regime is simply the constant obtained by setting $s=0$ in the first term of (\[SresApp\]) so that we obtain: $$S[s,0,\omega] \sim -\frac{\pi}{\omega} \frac{z(0)^{\frac{1}{2}}}{1-z(0)}
\left\{1+\frac{1}{2 \bar{s}^2}+\frac{1}{\bar{s}^4}\left[4-3 \ln 2 - 3 \ln \bar{s} \right] +
\mathcal{O} \left( \frac{\ln \bar{s}}{\bar{s}^6}\right)\right\} \, ,$$ where we introduced $\bar{s}=s \left[1-z(0)\right]^{-1/2}$. Inserting this result in (\[GreenSolBrane\]) and using physical distance $r=R_U s$ instead of $s$ we obtain $$\begin{aligned}
\label{FullCorrections}
\tilde{\mathcal{G}}(s,0,0)&=\frac{g^{(1)}(0,0)}{2 \pi^2 R R_U^3}-
\frac{1}{4 \pi r} \frac{\omega}{R} \coth\left( \frac{\omega \pi}{2}\right)
\left\{1+\frac{1}{2 \bar{s}^2}+\frac{1}{\bar{s}^4}\left[4-3 \ln 2 - 3 \ln \bar{s} \right] +
\mathcal{O} \left( \frac{\ln \bar{s}}{\bar{s}^6}\right)\right\}=\nonumber \\
&=\frac{g^{(1)}(0,0)}{2 \pi^2 R R_U^3}-
\frac{1}{4 \pi r} \frac{\omega}{R} \coth\left( \frac{\omega \pi}{2}\right)
\left\{1+\frac{\tanh^2\left(\frac{\omega\pi}{2}\right)}{2 \bar{r}^2}+
\frac{\tanh^4\left(\frac{\omega\pi}{2}\right)}{\bar{r}^4} \times \right. \nonumber \\
&\quad \times \left. \left[4-3 \ln 2
- 3 \ln \left(\frac{\bar{r}}{\tanh \left(\frac{\omega \pi}{2}\right)}\right) \right]
+ \mathcal{O} \left[ \frac{\tanh^6\left(\frac{\omega\pi}{2}\right)}{\bar{r}^6}
\ln \left(\frac{\bar{r}}{\tanh \left(\frac{\omega \pi}{2}\right)}\right) \right]\right\} \, .\end{aligned}$$ Since we would like to compare our result with the corresponding correction in the Randall-Sundrum II case, we introduced the dimensionless distance variable $\bar{r}=r \omega/R$, the physical distance measured in units of the AdS-radius, in the last line of the above result. We see that in complete agreement with the Randall-Sundrum II scenario, our setup reproduces $4$-dimensional gravity at large distances with extremely suppressed corrections. The only remnant effect from the different global topology manifest itself through the factors of $\tanh \left(\frac{\omega \pi}{2}\right)$ and $\coth \left(\frac{\omega \pi}{2}\right)$ which are very close to $1$. We furthermore emphasize that apart from these deviations, the asymptotic we obtained coincides exactly with the asymptotic for the case of a massless scalar field in the Randall-Sundrum-II background, see (\[I1Asymptotic\]) and e.g. [@Callin:2004py; @Kiritsis:2002ca; @Ghoroku:2003bs; @Giddings:2000mu]. From the factor $\omega \coth \left(\frac{\omega \pi}{2}\right)/R$ in eq. (\[FullCorrections\]) we see that also the relation between the fundamental scale $M$ and the Planck-scale $M_{Pl}$ gets modified only by the same factor of $\tanh\left(\frac{\omega \pi}{2}\right)$: $$M_{Pl}^2 = M^3 \frac{R}{\omega} \tanh \left(\frac{\omega \pi}{2} \right) \, ,$$ where we remind that $R/\omega$ is nothing but the AdS-Radius.
Eventually, we treat the case of distances inferior to the extra dimension $r\ll R$. After using the result (\[5DNewton\]) of appendix \[appGreenUltraShort\] in (\[GreenSolBrane\]), a short calculation reveals $$\begin{aligned}
\label{Short5DCorrections}
\tilde{\mathcal{G}}(s,0,0)&=\frac{g^{(1)}(0,0)}{2 \pi^2 R R_U^3}-\frac{1}{4\pi^2}
\frac{1}{R_U^2 s^2+R^2 \theta^2}, \end{aligned}$$ a result that has to be compared to the characteristic solution of the Poisson equation in 4-dimensional flat space. In $n$-dimensional flat space one has: $$\label{CharLaplSolnDspace}
\Delta \left[ -\frac{1}{(n-2) V_{S^{n-1}} r^{n-2}}\right]=\delta(r) \, , \qquad
r=\left(\sum_{i=1}^n x_i^2\right)^{\frac{1}{2}}$$ with $V_{S^{n-1}}=2\pi^{\frac{n}{2}}/\Gamma[\frac{n}{2}]$ denoting the volume of the $n-1$ sphere. Specifying to $n=4$, we recover the correct prefactor of $-1/4\pi^2$ in (\[Short5DCorrections\]) multiplying the $1/r^2$ singularity.
Finally, we mention that in none of the considered cases we payed any attention to the additive constant in the two-point function on the brane. The arbitrary constant entering the solution $g^{(1)}(\theta,\theta')$ can always be chosen in such a way that $g^{(1)}(0,0)$ vanishes on the brane (see appendix \[appDEQ\]).
Conclusions {#four}
===========
In this paper we considered a particular brane world model in 5 dimensions with the characteristic property that the spatial part of the space-time manifold (including the extra dimension) is compact and has the topology of a $4$-sphere $S^4$. Similar to the original Randall-Sundrum II model, the $3$-brane is located at the boundary between two regions of $AdS_5$ space-time. The coordinates of $AdS_5$ used by Randall and Sundrum are closely related to the so-called Poincaré coordinates of $AdS_5$. While the extra dimension in this set of coordinates provides a slicing of $AdS_5$ along flat $4$-dimensional Minkowski sections (resulting in a flat Minkowskian induced metric on the brane), their disadvantage is that they do not cover the whole of $AdS_5$ space-time.
The coordinates we used in this paper are the global coordinates of $AdS_5$ known to provide a global cover of the $AdS_5$ space-time. In this case the “extra” dimension labels different sections with intrinsic geometry $\mathbb{R} \times S^3$, the geometry of Einsteins static universe. The induced metric on the $3$-brane in our setup is therefore also given by $\mathbb{R} \times S^3$.
As we illustrated with the use of the Penrose-diagram of $AdS_5$, the incompleteness of the Poincaré patch is at the origin of the incompleteness of the Randall Sundrum II space-time with respect to timelike and lightlike geodesics. Moreover we were able to demonstrate that the setup considered in this paper provides an alternative to the Randall-Sundrum II model which does not suffer from the drawback of being geodesically incomplete. The latter point was part of the main motivations for this work.
The spatial part of our manifold is characterized by an extreme anisotropy with respect to one of the coordinates (the extra coordinate) accounting for thirty orders of magnitude between the size of the observable universe and present upper bounds for the size of extra dimensions.
Another interesting property of our manifold related to the anisotropy of its spatial part is the fact that [*any two points*]{} on the brane are separated by a distance of the order of the size of the extra dimension $R$ regardless of their distance measured by means of the induced metric on the brane. Despite the difference in the global topology, the properties of gravity localization turned out to be very similar to the Randall-Sundrum II model, though much more difficult to work out technically. We computed the static (modified) Green’s function of a massless scalar field in our background and could show that in the intermediate distance regime $R \ll r \ll R_U$ the $4$-dimensional Newton’s law is valid for two test particles on the brane, with asymptotic corrections terms identical to the Randall-Sundrum II case up to tiny factors of $\tanh\left(\frac{\omega\pi}{2}\right)$. We could also recover the characteristic $5$-dimensional behavior of the Green’s function for distances smaller than the extra dimension $r \ll R$. Eventually we saw that in the regime of cosmic distances $r \sim R_U$, somewhat counterintuitive given our highly anisotropic manifold, the Green’s function is dominated by the behavior of the corresponding static (modified) Green’s function in Einstein’s static universe.
In the simple setup considered in this paper the $3$-brane is supposed to be motionless. In the light of recent progress in the study of $4$-dimensional cosmic evolution induced by the motion of the brane in the bulk, it would be interesting to explore this possibility and see what kind of modifications of our results we would have to envisage. Finally a related, important question which would be interesting to address would be the question of stability of our setup.
[*Acknowledgments:*]{} We wish to thank S. Dubovsky, P. Tinyakov and S. Khlebnikov for useful comments and discussions. E. R. is particularly grateful to E. Teufl for helpful advice in numerous mathematical questions. A. G. wishes to thank LPPC for the kind hospitality during most of this research. This work was supported by the Swiss Science Foundation. A. G. acknowledges “Fondazione A. Della Riccia” for financial support.
Parallels in the computation of the corrections to Newton’s law between the case under consideration and the Randall-Sundrum II case {#appParallel}
====================================================================================================================================
The purpose of this appendix is to review briefly the calculations of the static two-point function of a scalar field in the Randall-Sundrum II background [@Kiritsis:2002ca; @Giddings:2000mu] and to compare each stage with the corresponding stage of calculations in the background considered in this paper. This serves mainly for underlining similarities and differences between the two calculations. Let us begin by writing down the metric of the Randall-Sundrum II model $$\label{RSmetric}
ds^2=e^{-2 k \vert y\vert} \eta_{\mu \nu} dx^\mu dx^\nu + dy^2\, , \qquad -\pi r_c \leq y \leq \pi r_c \, .$$ Here $y$ stands for the extra dimension while $k$ and $\eta_{\mu \nu}$ denote the inverse radius of $AdS_5$ space and the (4-dimensional)-Minkowski metric with signature $-+++$. Due to the orbifold $Z_2$-symmetry the allowed range of $y$ is $0 \leq y \leq \pi r_c$.[^18] As it is well known, each $4$-dimensional graviton mode in this background satisfies the equation of a massless scalar field. Therefore, for the study of the potential between two test masses on the brane we confine ourselves to solving the equation of a massless scalar field with an arbitrary time-independent source: $$\label{RSScalarEq}
\mathcal{D} u(\vec x,y) = j(\vec x,y) \qquad \mbox{with} \; \; \mathcal{D}=e^{2 k y} \Delta_x -
4 k\frac{\partial}{\partial y}+\frac{\partial^2}{\partial y^2} \, .$$ Since the operator $\mathcal{D}$ is formally self-adjoint, the corresponding Green’s function will satisfy: $$\label{RSGreendef}
\mathcal{D} \mathcal{G}(\vec x,\vec x';y,y')=\frac{\delta^3(\vec x-\vec x') \, \delta(y-y')}{\sqrt{-g}} \, .$$ We are now able to write down the usual integral representation of the solution of (\[RSScalarEq\]): $$\label{RSSolIntRep}
u(\vec x,y)=\int \sqrt{-g} \, j(\vec x',y')\, \overline{G(\vec x,\vec x';y,y')} \, d^3 \vec x' dy',$$ where the $y'$-integration extends from $0$ to $\pi r_c$ and the $\vec x'$ integrations from $-\infty$ to $\infty$. The absence of boundary terms in (\[RSSolIntRep\]) is of course the result of an appropriate choice of boundary conditions for $u(\vec x,y)$ and $G(\vec x,\vec x';y,y')$. In the above coordinates of the Randall-Sundrum II case the orbifold boundary conditions together with the Israel condition imposed on the fluctuations of the metric give rise to a Neumann boundary condition at $y=0$. One can easily convince oneself that in the limit $r_c \to \infty$ the resulting Green’s function is independent of the choice of the (homogeneous) boundary condition at $y=\pi r_c$. We therefore follow [@Randall:1999vf] and use also a Neumann boundary condition at $y=\pi r_c$. Another important point is that the Green’s function we are considering is specific to the orbifold boundary condition and so describes the situation of a semi-infinite extra dimension. We decided to carry out the calculations in the semi-infinite case as opposed to [@Randall:1999vf], where eventually the orbifold boundary conditions are dropped and the case of a fully infinite extra dimension is considered.[^19] An immediate consequence of this will be that the constant factors of the characteristic short distance singularities of the solutions of Laplace equations also will be modified by a factor of $2$. Finally we note that this factor of $2$ can be accounted for by an overall redefinition of the 5-dimensional Newton’s constant leaving the two theories with equivalent predictions. We conclude the discussion of boundary conditions by noting that at infinity in $\vec x'$ we suppose that the $u(\vec x',y')$ and $G(\vec x, \vec x'; y,y')$ vanish sufficiently rapidly.
The solution of eq. (\[RSGreendef\]) can be found most conveniently by Fourier expansion in the $\vec x$-coordinates and by direct solution of the resulting differential equation for the “transverse” Green’s function. After performing the integrations over the angular coordinates in Fourier space we obtain: $$\label{RSGreenFormalSolution}
\mathcal{G}(\vec x-\vec x',y,y')=\frac{1}{2 \pi^2}\int\limits_0^\infty
\frac{p \sin\left(p \vert \vec x-\vec x' \vert \right)}{\vert \vec x-\vec x' \vert} g^{(p)}(y,y') dp \, ,$$ where $g^{(p)}(y,y')$ satisfies $$\label{RSTransGreen}
\frac{1}{e^{-4 k y}} \frac{\partial}{\partial y}
\left[e^{-4 k y} \frac{\partial}{\partial y} g^{(p)}(y,y')\right]-p^2 e^{2 k y} g^{(p)}(y,y')=
\frac{\delta(y-y')}{e^{-4 k y}}\, , \qquad (0 \leq y \leq \pi r_c)$$ together with Neumann boundary conditions at $y=0$ and $y=\pi r_c$. The solution of (\[RSTransGreen\]) is straightforward with the general result $$\begin{aligned}
& g^{(p)}(y,y')=\frac{e^{2 k \left(y_>+y_< \right)}}{k} \times \\
& \;\; \times \frac{\left[I_1\left(\frac{p}{k}\right) K_2\left(\frac{p}{k} e^{k y_<}\right)
+K_1\left(\frac{p}{k}\right) I_2\left(\frac{p}{k} e^{k y_<}\right) \right] \cdot
\left[I_1\left(\frac{p}{k} e^{k \pi r_c} \right) K_2\left(\frac{p}{k} e^{k y_>}\right)
+K_1\left(\frac{p}{k} e^{k \pi r_c} \right) I_2\left(\frac{p}{k} e^{k y_>}\right) \right]}
{I_1\left( \frac{p}{k}\right) K_1\left( \frac{p}{k} e^{k \pi r_c} \right)-
I_1\left( \frac{p}{k} e^{k \pi r_c} \right) K_1\left( \frac{p}{k}\right)} \nonumber \, ,\end{aligned}$$ where $y_>$ ($y_<$) denote the greater (smaller) of the two numbers $y$ and $y'$ and $I_1(z), I_2(z), K_1(z), K_2(z)$ are modified Bessel functions. Since we want to study sources on the brane we now set $y'=0$ and take the well-defined limit $r_c\to \infty$, as can easily be verified from the asymptotic behavior of $I_n$ and $K_n$ for large values of $z$. We are therefore able to write the static scalar two-point function as $$\begin{aligned}
\label{RSGreenSol}
\mathcal{G}(\vec x-\vec x';y,0)=-\frac{e^{2 k y}}{2 \pi^2} \int\limits_0^\infty
\frac{\sin\left(p \vert \vec{x}-\vec x' \vert \right)}{\vert \vec x-\vec x' \vert}
\frac{K_2 \left(\frac{p}{k} e^{k y} \right)}{K_1 \left(\frac{p}{k} \right)} \, dp \, .\end{aligned}$$ We would like to point out a particularity of the Fourier-integral (\[RSGreenSol\]). By using the asymptotic expansions for the modified Green’s functions $K_1(z)$ and $K_2(z)$ we have [@AbrStegun]: $$\label{KoverKAsymptotic}
\frac{K_2 \left(\frac{p}{k} e^{k y} \right)}{K_1 \left(\frac{p}{k} \right)} \sim
e^{-\frac{1}{2} k y} e^{-\frac{p}{k}\left(e^{k y}-1\right)}\left[1+\mathcal{O}\left(\frac{1}{p}
\right)\right],$$ showing that for $y=0$ the integral over $p$ in (\[RSGreenSol\]) does not exist. The correct value of the Green’s function on the brane is therefore obtained by imposing continuity at $y=0$: $$\mathcal{G}(\vec{x}-\vec{x}';0,0)\equiv\lim_{y \to 0} \mathcal{G}(\vec{x}-\vec{x}';y,0).$$
Note that this subtlety is still present in our case of the Fourier sum representation of the modified Green’s function (\[FormalSolution\]), as discussed in appendix \[appSum\].
The representation (\[RSGreenSol\]) is very suitable for obtaining the short distance behavior of the Green’s function. By inserting the expansion (\[KoverKAsymptotic\]) into (\[RSGreenSol\]) we can evaluate the integral which will give reasonable results for distances smaller than $1/k$: $$\mathcal{G}(\vec x-\vec x';y,0)\sim-\frac{e^{\frac{3}{2} k y}}{2 \pi^2}
\frac{1}{\vert \vec x-\vec x' \vert^2+y^2} \, .$$ We see that after replacing the exponential factor by $1$ (valid for $y \ll 1/k$) we recover (up to a factor of $2$) the correct $5$-dimensional behavior of the Green’s function in flat 4-dimensional space (\[CharLaplSolnDspace\]).[^20]
For large distances the Fourier-representation (\[RSGreenSol\]) is less suited for obtaining corrections to Newton’s law. This is due to the fact that all but the first term in the expansion of $K_2/K_1$ in powers of $p$ around $p=0$ lead to divergent contributions upon inserting in (\[RSGreenSol\]). We therefore seek another method which will allow us to obtain the corrections to Newton’s law by term-wise integration.
The idea is to promote (\[RSGreenSol\]) to a contour-integral in the complex $p$-plane and to shift the contour in such a way that the trigonometric function is transformed into an exponential function. First we introduce dimensionless quantities by rescaling with $k$ according to $X=\vert \vec x - \vec x' \vert k$, $Y = y k$, $z=p \vert \vec x - \vec x' \vert$: $$\mathcal{\bar{G}}(X,Y,0)=-\frac{k^2 e^{2 Y}}{2 \pi^2 X^2}
\underbrace{\int\limits_0^\infty \sin z \frac{K_2\left(\frac{z}{X} e^Y \right)}{K_1\left(\frac{z}{X}
\right)} dz}_{\equiv I\left[X,Y\right]} \, .$$ Following [@Kiritsis:2002ca] (see also [@AbrStegun]) we now use the relation $$\label{BesselKRelation}
K_2[w]=K_0[w]+\frac{2}{w}K_1[w]$$ to separate the zero mode contribution to the Green’s function (Newton’s law) from the contributions coming from the higher Kaluza-Klein particles (corrections to Newton’s law). Using (\[BesselKRelation\]) in $I[X,Y]$ we find $$I[X,Y]=\underbrace{\int\limits_0^\infty \sin z \frac{K_0\left(\frac{z}{X} e^Y \right)}
{K_1\left(\frac{z}{X}\right)} dz}_{\equiv I_1[X,Y]}+
\underbrace{\frac{2 X}{e^Y} \int\limits_0^\infty \frac{\sin z}{z}
\frac{K_1\left(\frac{z}{X} e^Y \right)}{K_1\left(\frac{z}{X} \right)} dz}_{\equiv I_2[X,Y]} \, .$$ The additional factor of $1/z$ in the integrand of $I_2[X,Y]$ allows us to take the limit $Y \to 0$ with the result: $$\lim_{Y \to 0} I_2[X,Y]=\pi X \, .$$ However, we still need $Y>0$ for convergence in $I_1[X,Y]$. The next step is to use $\sin z=\Im \left\{e^{\imath z}\right\}$ and to exchange the operation $\Im$ with the integration over z:[^21] $$\label{RSI1FourierRep}
I_1[X,Y]=\Im\left\{\int\limits_0^\infty e^{\imath z}
\frac{K_0\left(\frac{z}{X} e^Y \right)}{K_1\left(\frac{z}{X} \right)} dz\right\} \, .$$ The integrand in (\[RSI1FourierRep\]) is a holomorphic function of $z$ in the first quadrant (see [@AbrStegun], p.377 for details). We can therefore apply <span style="font-variant:small-caps;">Cauchy</span>’s theorem to the contour depicted in Fig. \[intcontour\].
As we will show shortly, the contribution from the arc $C_2=\left\{z \, |\, z=R e^{\imath \varphi},\;0\leq \varphi \leq \frac{\pi}{2} \right\}$ vanishes in the limit $R \to \infty$ (the limit we are interested in). Symbolically we therefore have $$\lim_{R \to \infty} \left\{\int_{C_1} \ldots + \int_{C_3} \ldots \right\}=0\, ,$$ where $\ldots$ replace the integrand in (\[RSI1FourierRep\]) and the sense of integration is as indicated in the figure. This means that we can replace the integration over the positive real axis in (\[RSI1FourierRep\]) by an integration over the positive imaginary axis without changing the value of $I_1[X,Y]$. Substituting now $z$ in favor of $n$ according to $n=-\imath z$ we obtain: $$\begin{aligned}
\label{I1closetoresult}
I_1[X,Y]&=\Im \left\{ \int\limits_0^\infty e^{-n}
\frac{K_0\left(\frac{\imath n}{X} e^Y \right)}{K_1\left(\frac{\imath n}{X} \right)} \imath dn \right\}=
\Im \left\{ \int\limits_0^\infty e^{-n}
\frac{-H_0^{(2)} \left(\frac{n}{X} e^Y \right)}{H_1^{(2)}\left(\frac{n}{X} \right)} dn \right\}\nonumber \\
&=\int\limits_0^\infty e^{-n} \frac{Y_0\left(\frac{n}{X} e^Y\right) J_1\left(\frac{n}{X}\right)-
J_0\left(\frac{n}{X} e^Y\right) Y_1\left(\frac{n}{X}\right) }
{\left[J_1\left(\frac{n}{X} \right)\right]^2 +
\left[Y_1\left(\frac{n}{X} \right)\right]^2} \, dn \, .\end{aligned}$$ In the first line in (\[I1closetoresult\]) we replaced modified Bessel functions with imaginary arguments by Hankel functions of real argument. In the second line we explicitly took the imaginary part after replacing the identities relating the Hankel functions and the Bessel functions $J$ and $Y$ of the first kind. The benefit from the rotation of the contour of integration is obvious at this stage. First, the last integral in (\[I1closetoresult\]) converges also for $Y=0$ due to the presence of the exponential function in the integrand. In this case we have $$\label{RSMainResult}
I_1[X,0]=\frac{2 X}{\pi} \int\limits_0^\infty \frac{e^{-n}}{n}
\frac{1}{\left[J_1\left(\frac{n}{X} \right)\right]^2 + \left[Y_1\left(\frac{n}{X} \right)\right]^2}
\, dn \, .$$ Note that we used the Wronskian relation for Bessel functions to simplify the numerator, the general reference being once more [@AbrStegun]. Second, it is straightforward to obtain the large $X$ asymptotic ($1\ll X $) of the integral (\[RSMainResult\]) by a simple power series expansion in $n$ around $n=0$ of the integrand (apart from the exponential factor), followed by term-wise integration. For the results see [@Callin:2004py], where the above integral (\[RSMainResult\]) has been found in the wave-function approach: $$\label{I1Asymptotic}
I_1[X,0]\sim\frac{\pi}{2 X}+\frac{\pi}{X^3}\left(4-3 \ln 2 - 3\ln X \right) +
\mathcal{O}\left[\frac{\ln X}{X^5}\right].$$ The result for $\mathcal{\bar{G}}(X,0,0)$ is therefore: $$\mathcal{\bar{G}}(X,0,0)=-\frac{k^2}{2 \pi X}
\left[1+\frac{1}{2 X^2}+\frac{1}{X^4} \left(4-3 \ln 2 - 3\ln X \right) +
\mathcal{O}\left(\frac{\ln X}{X^6}\right)\right].$$
However, we still have to verify that the contribution from the arc $C_2$ in Fig. \[intcontour\] vanishes in the limit $R \to \infty$. Parametrizing $z$ by $z=R e^{\imath \varphi}, \; 0 \leq \varphi \leq \pi/2$ we find: $$\begin{aligned}
\label{KoverKestimate}
\left| \int_{C_2} e^{\imath z}
\frac{K_0\left(\frac{z}{X} e^Y \right)}{K_1\left(\frac{z}{X} \right)} dz \right|
&= \left| \int\limits_0^{\frac{\pi}{2}} e^{\imath R e^{\imath \varphi}}
\frac{K_0\left(\frac{R e^{\imath \varphi}}{X} e^Y \right)}
{K_1\left(\frac{R e^{\imath \varphi}}{X} \right)} \left( \imath R e^{\imath \varphi} \right)
d\varphi \right| \nonumber \\
&\leq R \int\limits_0^\frac{\pi}{2} \left| e^{\imath R \left(\cos \varphi+ \imath \sin \varphi\right)}\right|
\cdot \left| \frac{K_0\left(\frac{R e^{\imath \varphi}}{X} e^Y \right)}
{K_1\left(\frac{R e^{\imath \varphi}}{X} \right)}\right| d\varphi \, .\end{aligned}$$ Making now use of the asymptotic properties of the modified Bessel functions to estimate the second modulus in the last integrand in (\[KoverKestimate\]) we can write: $$\left| \frac{K_0\left(\frac{R e^{\imath \varphi}}{X} e^Y \right)}
{K_1\left(\frac{R e^{\imath \varphi}}{X} \right)}\right|
\leq C \left| e^{-\frac{Y}{2}} e^{-R e^{\imath \varphi}\frac{e^Y-1}{X}}\right| \, ,$$ so that $$\begin{aligned}
\left| \int_{C_2} e^{\imath z}
\frac{K_0\left(\frac{z}{X} e^Y \right)}{K_1\left(\frac{z}{X} \right)} dz \right| <
C R e^{-\frac{Y}{2}} \int\limits_0^{\frac{\pi}{2}} e^{-R\left[ \sin\varphi +
\left( \frac{e^Y-1}{X} \right) \cos \varphi \right]} d\varphi \, ,\end{aligned}$$ with $C$ being a constant of the order of unity, independent of $R$ and $\varphi$. To finish the estimate we observe that $$\sin \varphi + \left( \frac{e^Y-1}{X} \right) \cos \varphi \geq \epsilon
\equiv \min \left[1, \frac{e^Y-1}{X} \right] > 0 \quad , \quad \forall\, \varphi \in [0,\frac{\pi}{2}]\, ,
\qquad (Y>0)$$ and obtain: $$\label{arcresult}
\left| \int_{C_2} e^{\imath z}
\frac{K_0\left(\frac{z}{X} e^Y \right)}{K_1\left(\frac{z}{X} \right)} dz \right| <
\frac{C R \pi}{2} e^{-\frac{Y}{2}} e^{-R \epsilon} \; \to 0 \quad \mbox{for} \; R \to \infty.$$ This establishes the result that in the limit $R \to \infty$ the arc $C_2$ does not contribute to the integral in (\[RSI1FourierRep\]) and completes our discussion of the two-point function in the Randall Sundrum-II setup.
We are now going to summarize briefly the complications arising when the general scheme of computations outlined above for the Randall-Sundrum II case are applied to the brane setup considered in this paper. First and foremost, the main difference is due to the fact that the induced metric on our brane is not the flat Minkowski metric but the metric of the Einstein static universe. This implies the use of a modified Green’s function. In the eigenfunction expansion, the discrete scalar-harmonics on $S^3$ will replace the continuous plane-wave eigenfunctions. We therefore expect the formal solution for the two-point function (the analog of (\[RSGreenFormalSolution\])) to take the form of a Fourier sum.
The solution to the *transverse* Green’s function (the analog of eq. (\[RSTransGreen\])) as presented in appendix \[appDEQ\] turns out to be again straightforward and completes the formal solution.
When trying to distinguish between the zero mode contribution and the contribution coming from higher modes (in the language of the Kaluza-Klein approach) a relation similar to (\[BesselKRelation\]) proves useful. While the evaluation of the zero mode contribution poses no problems, the corrections coming from higher modes are much more involved this time. We face the following major technical difficulties: there is no analog of <span style="font-variant:small-caps;">Cauchy</span>’s theorem in the discrete case of the Fourier sum. One solution to this problem is to use a variant of <span style="font-variant:small-caps;">Euler-Maclaurin</span>’s sum rule in order to replace the Fourier sum by an analog Fourier integral and a remainder term also in the form of an integral. Now the procedure again is similar to the one outlined for the Randall-Sundrum case. Another technical problem, however, in connection with large distances corrections has its origin in the simple fact that by large distances in our setup we mean distances large with respect to the extra dimension $R$ but also small with respect to the size of the observable universe $R_U$. The detailed computations of the corrections in this distances regime can be found in appendix \[appSum\].
What concerns the computation of the two-point function at distances smaller than the extra dimension, there is no conceptual difference to the Randall-Sundrum case. The Fourier sum can be calculated analytically after inserting the corresponding large momentum asymptotic. We give the details in appendix \[appGreenUltraShort\].
Solutions to the differential equations (\[TransverseEq1\]) and (\[TransverseEq2\]) {#appDEQ}
===================================================================================
The aim of this appendix is to obtain the solutions to the eqs. (\[TransverseEq1\]) and (\[TransverseEq2\]) and hence to complete the computation of the modified Green’s function. We will concentrate mainly on (\[TransverseEq2\]) for two reasons: first and foremost because in the formal expansion (\[FormalSolution\]) $g^{(1)}(\theta,\theta')$ multiplies a constant function in $s$ and since we are mainly interested in the behavior of the Green’s function on the brane, this constant is of no relevance for our considerations. Secondly because the construction of the Green’s functions $g^{(1)}(\theta,\theta')$ and $g^{(\lambda)}(\theta,\theta')$ follows the standard procedure, so it is not necessary to go into details twice. We will merely state the result for $g^{(1)}(\theta,\theta')$. The equation we would like to solve is $$\label{TransvGreen}
\frac{1-\lambda^2}{R_U^2 \gamma(\theta)^2} g^{(\lambda)}(\theta,\theta')+
\frac{1}{R^2} \frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}
\left[ \sigma(\theta)\gamma(\theta)^3 \frac{\partial g^{(\lambda)}(\theta,\theta')}{\partial \theta}
\right]=\frac{\delta(\theta-\theta')}{\sigma(\theta)\gamma(\theta)^3} \, ,$$ where $\lambda=2,3,4\ldots$ and the independent variables $\theta$ and $\theta'$ are restricted to the intervals $-\pi/2\leq\theta\leq\pi/2$, and $-\pi/2\leq\theta'\leq\pi/2$. The homogeneous equation associated with (\[TransvGreen\]) can be reduced to the hypergeometric equation. Its general formal solution in the interval $0\leq\theta\leq\pi/2$ is given by $$\begin{aligned}
\label{TransvGreenGenSol}
\varphi^{(+)}(\lambda,\theta)=a_1^{(+)} \varphi_1(\lambda,\theta)+a_2^{(+)} \varphi_2(\lambda,\theta)
\, , \qquad 0\leq\theta\leq\frac{\pi}{2}\end{aligned}$$ where $$\begin{aligned}
\varphi_1(\lambda,\theta)&=\frac{\tanh^{\lambda-1}\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]}
{\cosh^4\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]}
\phantom{1}_2 F_1\left[\frac{\lambda+3}{2},\frac{\lambda+3}{2};\lambda+1;
\tanh^2\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]\right] \, ,\nonumber \\
\varphi_2(\lambda,\theta)&=\frac{\tanh^{\lambda-1}\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]}
{\cosh^4\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]}
\phantom{1}_2 F_1\left[\frac{\lambda+3}{2},\frac{\lambda+3}{2};3;
\cosh^{-2}\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]\right] \, .\end{aligned}$$ Due to the symmetry of the background under $\theta\to -\theta$ we can immediately write down the general solution of the homogeneous equation obtained from (\[TransvGreen\]) in the interval $-\pi/2\leq\theta\leq 0$: $$\begin{aligned}
\label{TransvGreenGenSolNeg}
\varphi^{(-)}(\lambda,\theta)=a_1^{(-)} \varphi_1(\lambda,-\theta)+a_2^{(-)} \varphi_2(\lambda,-\theta) \, ,
\qquad -\frac{\pi}{2}\leq\theta\leq 0 \, .\end{aligned}$$ We now have to decide whether we want to restrict our Green’s function to describe perturbations which also possess the symmetry $\theta\to-\theta$ of the background (as in the Randall-Sundrum case) or not. In the first case one can limit the solution of (\[TransvGreen\]) to $\theta>0 $ and continue the result symmetrically into $\theta<0$. Note that imposing the symmetry $\theta \to -\theta$ on the perturbations goes hand in hand with imposing $\theta \to -\theta$ for the source which means that one should add a corresponding delta-function $\delta(\theta+\theta')$ to the right hand side of (\[TransvGreen\]). The Israel-condition (\[IsraelijFluct\]) on the brane with $\theta=0$ then gives $\frac{\partial g^{(\lambda)}}{\partial \theta}=0$, as in the Randall-Sundrum case.
Since in our setup we do not have any convincing argument for imposing the $\theta\to-\theta$ symmetry also on the metric fluctuations (the scalar field), the second case, which allows a breaking of this symmetry will be the case of our choice. The Israel-condition (\[IsraelijFluct\]) now merely indicates that the perturbation (the scalar field) should have a continuous first derivative at the brane.
In order to obtain a uniquely defined solution we have to impose two boundary conditions on $g^{(\lambda)}(\theta,\theta')$ one at $\theta=\pi/2$ the other at $\theta=-\pi/2$. It turns out that the requirement of square integrability of our solutions (with the correct weight-function) forces us to discard one of the two fundamental solutions at the boundaries $\theta=\pm\pi/2$, namely $\varphi_2(\lambda,\theta)$.
Before going into details we introduce the following abbreviations: $$\begin{aligned}
p(\theta)=\frac{\sigma(\theta)\gamma(\theta)^3}{R^2}\, , \qquad
s(\theta)=\frac{\sigma(\theta)\gamma(\theta)}{R_U^2} .\end{aligned}$$ Eq. (\[TransvGreen\]) then becomes $$\left[p(\theta) {g^{(\lambda)}}{'}(\theta,\theta')\right]'-(\lambda^2-1) \,
s(\theta) g^{(\lambda)}(\theta,\theta')=\delta(\theta-\theta') \, ,$$ where $'$ denotes derivatives with respect to $\theta$ here and in the following. We start by taking $\theta'>0$. The case $\theta'<0$ is fully analogous to the one considered up to minus signs. Our ansatz for $g^{(\lambda)}(\theta,\theta')$ is $$\label{GreenAnsatz}
g^{(\lambda)}(\theta,\theta')=\left\{
\begin{array}{lrrl} A(\theta') \, \varphi_1^{(\lambda)}(-\theta)&-\frac{\pi}{2}&\leq \theta \leq& 0 \, ,\\
B(\theta') \, \varphi_1^{(\lambda)}(\theta)+C(\theta') \, \varphi_2^{(\lambda)}(\theta)\quad
&0 &\leq \theta \leq& \theta' \, ,\\
D(\theta') \, \varphi_1^{(\lambda)}(\theta)&\theta' &\leq \theta \leq& \frac{\pi}{2} \, .
\end{array} \right.$$ We impose the following boundary and matching conditions on $g^{(\lambda)}(\theta,\theta')$ which will uniquely determine the coefficients $A, B, C, D$:
1. [continuity of $g^{(\lambda)}(\theta,\theta')$ at $\theta=0$.]{}
2. [continuity of ${g^{(\lambda)}}{'}(\theta,\theta')$ at $\theta=0$.]{}
3. [continuity of $g^{(\lambda)}(\theta,\theta')$ at $\theta=\theta'$.]{}
4. [jump condition of ${g^{(\lambda)}}{'}(\theta,\theta')$ at $\theta=\theta'$.]{}
Using the ansatz (\[GreenAnsatz\]) we obtain from the above conditions: $$\begin{array}{rclcc}
\left[A(\theta')-B(\theta')\right] \varphi_1^{(\lambda)}(0) &-&C(\theta') \varphi_2^{(\lambda)}(0)&=&0\, ,\\
\left[A(\theta')+B(\theta')\right] {\varphi_1^{(\lambda)}}{'}(0) &+&C(\theta')
{\varphi_2^{(\lambda)}}{'}(0)&=&0\, , \\
\left[B(\theta')-D(\theta')\right] \varphi_1^{(\lambda)}(\theta') &+&C(\theta')
\varphi_2^{(\lambda)}(\theta')&=&0\, ,\\
\left[B(\theta')-D(\theta')\right] {\varphi_1^{(\lambda)}}{'}(\theta') &+&C(\theta')
{\varphi_2^{(\lambda)}}{'}(\theta')&=&-\frac{1}{p(\theta')} \, ,
\end{array}$$ the solution of which is easily found to be: $$\begin{aligned}
\label{solABCD}
A(\theta')&=\frac{R^2}{2}\frac{\varphi_1^{(\lambda)}(\theta')}
{\varphi_1^{(\lambda)}(0) {\varphi_1^{(\lambda)}}{'}(0)}
\, ,\quad
B(\theta')=\frac{R^2}{2}\frac{\varphi_1^{(\lambda)}(\theta')}
{\mathcal{W}\left[\varphi_1^{(\lambda)},\varphi_2^{(\lambda)},0\right]}
\frac{\varphi_1^{(\lambda)}(0) {\varphi_2^{(\lambda)}}{'}(0)+
\varphi_2^{(\lambda)}(0) {\varphi_1^{(\lambda)}}{'}(0)}
{\varphi_1^{(\lambda)}(0) {\varphi_1^{(\lambda)}}{'}(0)} \, ,\nonumber \\
C(\theta')&=-R^2 \frac{\varphi_1^{(\lambda)}(\theta')}{\mathcal{W}
\left[\varphi_1^{(\lambda)},\varphi_2^{(\lambda)},0\right]} \, , \qquad \qquad
D(\theta')=B(\theta')- R^2 \frac{\varphi_2^{(\lambda)}(\theta')}{\mathcal{W}
\left[\varphi_1^{(\lambda)},\varphi_2^{(\lambda)},0\right]} \, ,\end{aligned}$$ where $\mathcal{W}\left[\varphi_1^{(\lambda)},\varphi_2^{(\lambda)},0\right]$ denotes the Wronskian of $\varphi_1^{(\lambda)}$ and $\varphi_2^{(\lambda)}$ at $\theta=0$. In obtaining (\[solABCD\]) we used the relation $$\mathcal{W}\left[\varphi_1^{(\lambda)},\varphi_2^{(\lambda)},\theta\right]=
\frac{\mathcal{W}\left[\varphi_1^{(\lambda)},\varphi_2^{(\lambda)},0\right]}{R^2\, p(\theta)} \, .$$ As already mentioned, the case $\theta'<0$ can be treated in complete analogy to the case $\theta'>0$. Combining the two results gives the final expression for the solution $g^{(\lambda)}(\theta,\theta')$ of (\[TransvGreen\]) (for $\lambda=2,3,\ldots$): $$\label{GreenfullResult}
g^{(\lambda)}(\theta,\theta')=\left\{
\begin{array}{lrrl} A(\vert \theta'\vert) \, \varphi_1^{(\lambda)}(\vert \theta\vert)&-\frac{\pi}{2}
&\leq \theta \, \mbox{sign}(\theta') \leq& 0\, , \\
B(\vert \theta'\vert) \, \varphi_1^{(\lambda)}(\vert \theta\vert)+C \, (\vert \theta'\vert)
\varphi_2^{(\lambda)}(\vert \theta\vert)\quad
&0 &\leq \theta \, \mbox{sign}(\theta') \leq& \vert \theta' \vert \, , \\
D(\vert \theta'\vert) \, \varphi_1^{(\lambda)}(\vert \theta \vert )
& \vert \theta' \vert & \leq \theta \, \mbox{sign}(\theta') \leq& \frac{\pi}{2} \, .
\end{array} \right.$$ It is easily verified that the Green’s function (\[GreenfullResult\]) has the property $$g^{(\lambda)}(\theta,\theta') = g^{(\lambda)}(\theta',\theta)$$ as expected for a self-adjoint boundary value problem. We will not need the result (\[GreenfullResult\]) in its full generality. We focus on the case where sources are located at the brane ($\theta'=0$). In this case (\[GreenfullResult\]) reduces to: $$\label{glambdaratio}
g^{(\lambda)}(\theta,0)=\frac{R^2}{2}
\frac{\varphi_1^{(\lambda)}(\vert \theta\vert)}{{\varphi_1^{(\lambda)}}{'}(0)} \, ,
\qquad (\lambda=2,3,\ldots) \, .$$
For the sake of completeness, we also give the solution $g^{(1)}(\theta,\theta')$ to (\[TransverseEq1\]) $$\label{Transverse1Repeat}
\frac{1}{R^2} \frac{1}{\sigma(\theta)\gamma(\theta)^3} \frac{\partial}{\partial\theta}
\left[ \sigma(\theta)\gamma(\theta)^3 \frac{\partial g^{(1)}(\theta,\theta')}{\partial \theta}
\right]=\frac{1}{\sigma(\theta)\gamma(\theta)^3} \delta(\theta-\theta')-
\tilde{\chi}_1(\theta) \bar{\tilde{\chi}}_1(\theta') \, .$$ The general procedure of finding the solution is fully analogous to the case of $g^{(\lambda)}(\theta,\theta')$ (for $\lambda=2,3,\ldots$), the only difference being that in regions where $\theta \neq \theta'$ (\[Transverse1Repeat\]) reduces to an inhomogeneous differential equation. In the interval $0 \leq \theta \leq \pi/2$ the general solution is given by $$\begin{aligned}
\psi(\theta)=\psi_0(\theta) + A \psi_1(\theta)+B \psi_2(\theta),\end{aligned}$$ with $$\begin{aligned}
\psi_0(\theta)&=-\frac{R^2}{2 \omega \tanh\left( \frac{\omega\pi}{2}\right)}
\ln\left[ \cosh\left[\omega \left(\frac{\pi}{2}-\theta\right)\right]\right]\, ,\nonumber \\
\psi_1(\theta)&=1\, , \qquad \psi_2(\theta)=2 \ln\left[\tanh\left[\omega \left(\frac{\pi}{2}-\theta\right)\right]\right]+
\coth^2\left[\omega \left(\frac{\pi}{2}-\theta\right)\right]. \end{aligned}$$ With these definitions, it is easy to verify that $g^{(1)}(\theta,\theta')$ is given by: $$\label{resg1}
g^{(1)}(\theta,\theta')=\mbox{const.}+\psi_0(\vert \theta\vert)+\psi_0(\vert \theta'\vert)+C
\left\{\begin{array}{lrcl} \psi_2(0) & -\frac{\pi}{2} \leq \theta \, \mbox{sign}(\theta')\leq & 0 \\
\psi_2(\vert \theta\vert) & 0 \leq \theta \, \mbox{sign}(\theta') \leq &\vert \theta' \vert\\
\psi_2(\vert \theta' \vert) & \vert \theta' \vert \leq \theta \, \mbox{sign}(\theta') \leq&
\frac{\pi}{2}\end{array} \right. \, ,$$ with $$C=-\frac{R^2}{\mathcal{W}\left[\psi_1,\psi_2,0 \right]} \, .$$ By $\mathcal{W}\left[\psi_1,\psi_2,0 \right]$ we again mean the Wronskian of $\psi_1$ and $\psi_2$ evaluated at $\theta=0$.
Detailed computation of the corrections to Newton’s law {#appSum}
=======================================================
In this appendix we give a detailed computation of the sum $$\label{TheSumB}
S[s,\theta,\omega]=\sum_{\lambda=2}^{\infty}\lambda \sin \left(\lambda s \right)
\frac{\varphi_1^{(\lambda)}(\theta)}{{\varphi'}_1^{(\lambda)}(0)} \, ,\qquad 0<\theta \leq \frac{\pi}{2}$$ with $$\begin{aligned}
\label{RatioOf2Hyps}
\frac{\varphi_1^{(\lambda)}(\theta)}{{\varphi'}_1^{(\lambda)}(0)}&=&
-\frac{z(\theta)^\frac{\lambda-1}{2} \left[1-z(\theta)\right]^2
\phantom{1}_2 F_1\left[\frac{\lambda+3}{2},\frac{\lambda+3}{2};\lambda+1;z(\theta)\right]}
{\omega(\lambda-1) z(0)^\frac{\lambda-2}{2} \left[1-z(0)\right]^2
\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+3}{2};\lambda+1;z(0) \right]} \, ,\end{aligned}$$ where we explicitly excluded $\theta=0$ since the Fourier sum (\[TheSumB\]) does not converge for this value of $\theta$.[^22] We also used the definition $z(\theta)=\tanh^2\left[\omega\left(\frac{\pi}{2}-\theta\right)\right]$.
Note that the sum $S[s,\theta,\omega]$ contains the contribution to the two-point function coming from zero mode and higher Kaluza-Klein modes. Inspired by the Randall-Sundrum case we try to separate the two contributions by functional relations between contiguous Gauss Hypergeometric functions. Using (15.2.15) of reference [@AbrStegun] for $a=b-1=\left(\lambda+1\right)/2,\,c=\lambda+1$ we obtain $$\begin{aligned}
\left[1-z(\theta)\right] \phantom{1}_2 F_1\left[ \frac{\lambda+3}{2},\frac{\lambda+3}{2};
\lambda+1;z(\theta) \right]&=&
\frac{2}{\lambda+1}
\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+3}{2};
\lambda+1;z(\theta) \right] \nonumber \\
&+&\frac{\lambda-1}{\lambda+1}
\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right] \, .\end{aligned}$$ In this way we are able to rewrite $S[s,\theta,\omega]$ in the form: $$S[s,\theta,\omega]= S_1[s,\theta,\omega] + S_2[s,\theta,\omega]\, ,$$ where $$\begin{aligned}
\label{defS1}
S_1[s,\theta,\omega]=-\frac{2}{\omega}\frac{z(0)}{z(\theta)^\frac{1}{2}}
\frac{1-z(\theta)}{\left[1-z(0)\right]^2}
\sum_{\lambda=2}^{\infty}\frac{\lambda \sin\left( \lambda s\right)}{\lambda^2-1}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\lambda}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+3}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+3}{2};
\lambda+1;z(0) \right]}\, , \\
S_2[s,\theta,\omega]=-\frac{1}{\omega}\frac{z(0)}{z(\theta)^\frac{1}{2}}
\frac{1-z(\theta)}{\left[1-z(0)\right]^2}
\sum_{\lambda=2}^{\infty}\frac{\lambda \sin\left( \lambda s\right)}{\lambda+1} \label{defS2}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\lambda}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+3}{2};
\lambda+1;z(0) \right]} \, .\end{aligned}$$ From the asymptotic formula (\[GeneralAsymptoticOfHyp\]) we infer that now the sum $S_1[s,\theta,\omega]$ is convergent even for $\theta=0$ due to the additional power of $\lambda$ in the denominator. We therefore obtain $$S_1[s,0,\omega]=-\frac{2}{\omega} \frac{z(0)^\frac{1}{2}}{1-z(0)}
\sum_{\lambda=2}^{\infty}\frac{\lambda \sin\left( \lambda s\right)}{\lambda^2-1}
=-\frac{2}{\omega} \frac{z(0)^\frac{1}{2}}{1-z(0)}
\left(\frac{\pi-s}{2}\cos s -\frac{1}{4}\sin s \right)\, .$$ Since we recognized in the last sum the two point function of Einstein’s static universe, we attribute the contribution coming from $S_1[s,\theta,\omega]$ to the zero mode in the Kaluza-Klein spectrum. We also should mention that in $S_2[s,\theta,\omega]$ we still need $\theta>0$ for convergence since only then the factor $\left[z(\theta)/z(0)\right]^\frac{\lambda-1}{2}$ provides an exponential cutoff in $\lambda$ for the sum.
The main challenge in the evaluation of the two-point function is therefore to tame the sum $S_2[s,\theta,\omega]$. Our strategy is the following: first we extend it from $\lambda=0$ to $\infty$ by adding and subtracting the $\lambda=0$ and $\lambda=1$ terms. Next, we replace the sum over $\lambda$ by the sum of two integrals employing a variant of the <span style="font-variant:small-caps;">Euler-Maclaurin</span> sum rule called the <span style="font-variant:small-caps;">Abel-Plana</span> formula (see. e.g. [@Olver], p. 289-290). Finally, we shall see that the resulting (exact) integral representation will allow us to extract the desired asymptotic of the two-point function on the brane in the distance regime of interest ($R\ll r \ll R_U$).
As announced, we start by extending the range of the sum from $0$ to $\infty$. Since the addend with $\lambda=0$ vanishes, we only need to subtract the $\lambda=1$ term with the result: $$\label{S2extended}
S_2[s,\theta,\omega]=-\frac{1}{2 \omega}\frac{z(0)^\frac{1}{2}}{z(\theta)} \frac{1-z(\theta)}{1-z(0)}
\ln\left[1-z(\theta)\right] \sin s
-\frac{1}{\omega} \frac{z(0)}{z(\theta)^\frac{1}{2}} \frac{1-z(\theta)}{1-z(0)}
R[s,\theta,\omega]\; ,$$ where $R[s,\theta,\omega]$ is defined by $$\label{defR}
R[s,\theta,\omega]=\sum_{\lambda=0}^{\infty}\frac{\lambda \sin \left(\lambda s\right)}{\lambda+1}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\lambda}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};
\lambda+1;z(0) \right]} \, .$$ We used the identity $$\label{HypIdentity}
\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+3}{2};\lambda+1;z(0) \right]=
\left[1-z(0)\right]^{-1}
\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};\lambda+1;z(0) \right]$$ in the last step (see 15.3.3 of [@AbrStegun]). The next step consists of extending the $\sin$ function in (\[defR\]) to an exponential and taking the imaginary part out of the sum (as in the Randall-Sundrum case). Now we make use of the <span style="font-variant:small-caps;">Abel-Plana</span> formula, first considering the partial sums: $$\begin{aligned}
R^{(n)}[s,\theta,\omega]&=&\Im \left[ \frac{1}{2} f(0,s,\theta,\omega)+
\frac{1}{2} f(n,s,\theta,\omega)+\int\limits_0^n f(\lambda,s,\theta,\omega) \, d\lambda \right. \\
&&+\imath \left. \int\limits_0^{\infty}\frac{f(\imath y,s,\theta,\omega)-
f(n+\imath y,s,\theta,\omega)-
f(-\imath y,s,\theta,\omega)+f(n-\imath y,s,\theta,\omega)}{e^{2\pi y}-1} dy \right] \, ,\nonumber\end{aligned}$$ where for the sake of clarity we introduced $$\label{fdef}
f(\lambda,s,\theta,\omega)\equiv \frac{\lambda e^{\imath\lambda s}}{\lambda+1}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\lambda}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};
\lambda+1;z(0) \right]}.$$ In the limit $n\to\infty$ we find: $$\begin{aligned}
\label{TIntRep}
R[s,\theta,\omega]&=&\Im \Big[ \underbrace{\int\limits_0^\infty f(\lambda,s,\theta,\omega) d\lambda}_
{\equiv R_1[s,\theta,\omega]}
+\underbrace{\imath \int\limits_0^{\infty}\frac{f(\imath y,s,\theta,\omega)-
f(-\imath y,s,\theta,\omega)}{e^{2\pi y}-1} dy}_{\equiv R_2[s,\theta,\omega]} \Big] \, .\end{aligned}$$
Leading order asymptotic of $\Im\left\{R_1[s,\theta,\omega]\right\}$
--------------------------------------------------------------------
We focus first on $R_1[s,\theta,\omega]$ and observe that the function $f(\lambda,s,\theta,\omega)$ is holomorphic (in $\lambda$) in the first quadrant.[^23] Therefore, in perfect analogy with the Randall-Sundrum case treated in appendix \[appParallel\], we can use <span style="font-variant:small-caps;">Cauchy</span>’s theorem to replace the integration over the positive real axis by an integration over the positive imaginary axis.[^24] After substituting $\lambda$ with $\imath y$ we obtain : $$\label{R1shifted}
R_1[s,\theta,\omega]=-\int\limits_0^\infty \frac{y (1-\imath y)}{1+y^2} e^{-y s}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\imath y}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{1+\imath y}{2};
1+\imath y;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{-1+\imath y}{2};
1+\imath y;z(0) \right]} dy \, .$$ The upper deformation of the path of integration is advantageous for at least two reasons. Firstly, through the appearance of the exponential in the integrand, convergence on the brane does no longer rely on $\theta>0$ and we can set $\theta$ equal to zero in (\[R1shifted\]) to obtain: $$\label{R1shiftedBrane}
R_1[s,0,\omega]=-\int\limits_0^\infty \frac{y (1-\imath y)}{1+y^2} e^{-y s}
\frac{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{1+\imath y}{2};
1+\imath y;z(0) \right]}{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{-1+\imath y}{2};
1+\imath y;z(0) \right]} dy \, .$$ Secondly, the form (\[R1shifted\]) is perfectly suited for obtaining asymptotic expansions for large distances. Note that since $s\in[0,\pi]$, we carefully avoided saying “for large s” since this would not correspond to the case of our interest. We try to compute the two-point function in a distance regime $R \ll r \ll R_U$, distances clearly far beyond the size of the fifth dimension but far below the size of the observable universe. Rewritten in the geodesic distance coordinate $s$ this becomes $\omega/\sinh\left(\frac{\omega\pi}{2}\right)\ll s\ll 1$. It is this last relation which complicates the computation of the asymptotic considerably. We are forced to look at an asymptotic evaluation of (\[R1shiftedBrane\]) (or (\[R1shifted\])) for intermediate $s$ values such that we cannot make direct use of Laplace’s method. The correct way of extracting the above described asymptotic is to expand the ratio of hypergeometric functions in (\[R1shiftedBrane\]) in a power series of the small parameter $1-z(0)$. Using rel. $(15.3.10)$ and $(15.3.11)$ (with $m=1$) of [@AbrStegun] we find after some algebra $$\begin{aligned}
\label{HypRatioExpansion}
&&\frac{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{1+\imath y}{2};
1+\imath y;z(0) \right]}{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{-1+\imath y}{2};
1+\imath y;z(0) \right]}=
-\frac{1}{2}(1+\imath y)\left[2\gamma+2\psi(\frac{1+\imath y}{2})+\ln\left[1-z(0)\right]\right]
\hfill \nonumber \\
&+&\left\{\left(\frac{1+\imath y}{2}\right)^3\left[2 \psi(2)-2 \psi\left(\frac{3+\imath y}{2}\right)
-\ln\left[1-z(0)\right]\right] \right.\nonumber \\
&&\;\;\;-\left(\frac{1+\imath y}{2}\right)^2 \left(\frac{-1+\imath y}{2}\right)
\left[2 \psi(1)-2 \psi\left(\frac{1+\imath y}{2}\right)-\ln\left[1-z(0)\right]\right]\times\nonumber \\
&&\;\;\;\;\;\;\;\left.\times \left[\ln\left[1-z(0)\right]-\psi(1)-\psi(2)
+\psi\left(\frac{3+\imath y}{2}\right)+
\psi\left(\frac{1+\imath y}{2}\right)\right] \right\} \left[1-z(0)\right]\nonumber\\
&+&\mathcal{O}\left\{\ln\left[1-z(0)\right] \left[1-z(0)\right]^2 \right\} \, .\end{aligned}$$ In the above relation $\gamma$ denotes <span style="font-variant:small-caps;">Euler-Mascheroni</span>’s constant and $\psi(z)\equiv\Gamma'[z]/\Gamma[z]$ the so-called Digamma-function. Note that we developed up to linear order in $1-z(0)$ since we also want to compute the next to leading order term in the asymptotic later in this appendix.
We are only interested in the imaginary part of $R_1[s,0,\omega]$ and it turns out that the first term in the expansion (\[HypRatioExpansion\]) after insertion in (\[R1shiftedBrane\]) can be integrated analytically: $$\begin{aligned}
\label{R1_0_Integral}
\Im\left\{R_1^{(0)}[s,0,\omega]\right\}=\Im \left\{\int\limits_0^{\infty} y e^{-y s}
\psi\left(\frac{1+\imath y}{2}\right)
dy \right\}=\int\limits_0^{\infty} y e^{-y s} \Im\left[\psi\left(\frac{1+\imath y}{2}\right)\right] dy \, .\end{aligned}$$ The last step is justified since also the real part of the first integral in (\[R1\_0\_Integral\]) converges, as follows immediately from the asymptotic $\psi(z)\sim\ln z-\frac{1}{2 z}+\mathcal{O}(\frac{1}{z^2})$. Since $$\begin{aligned}
\label{ImofDiGamma}
\Im\left[\psi(\frac{1+\imath y}{2})\right]=\frac{\pi}{2}\tanh \left(\frac{\pi y}{2}\right) \, ,\end{aligned}$$ ([@AbrStegun], p.259, 6.3.12) we find $$\begin{aligned}
\label{R1_0Result}
\Im\left\{R_1^{(0)}[s,0,\omega]\right\}&=&\frac{\pi}{2}\int\limits_{0}^\infty y e^{-y s}
\tanh \left(\frac{\pi y}{2}\right) dy
=-\frac{\pi}{2} \frac{d}{ds}\left[\int\limits_{0}^\infty e^{-y s}
\tanh\left(\frac{\pi y}{2}\right) dy\right] \nonumber \\
&=&-\frac{\pi}{2} \frac{d}{ds}\left[\int\limits_{0}^\infty e^{-y s} \left(
\frac{2}{1+e^{-\pi y}}-1\right) dy\right] =
-\frac{\pi}{2}\frac{d}{ds}\left[-\frac{1}{s}+2\int\limits_0^\infty \frac{e^{-y s}}{1+e^{-\pi y}} dy \right]
\nonumber \\
&=&-\frac{\pi}{2 s^2}-\frac{d}{ds}\left[\int\limits_0^\infty
\frac{e^{-\frac{z s}{\pi}}}{1+e^{-z}} dz\right]=-\frac{\pi}{2 s^2}-\frac{1}{\pi}\beta'(\frac{s}{\pi})\, , \end{aligned}$$ where we introduced the $\beta$-function by $$\label{defbeta}
\beta(x)\equiv\frac{1}{2}\left[\psi\left(\frac{x+1}{2}\right)-\psi\left(\frac{x}{2}\right)\right]$$ (see [@Gradshteyn], p.331, 3.311, 2.).[^25]
Leading order asymptotic of $\Im\left\{R_2[s,\theta,\omega]\right\}$
--------------------------------------------------------------------
We can treat $R_2[s,\theta,\omega]$ starting from (\[TIntRep\]) in very much the same way as $R_1[s,\theta,\omega]$. Noting that we can again put $\theta$ equal to zero since the exponential factor $e^{2\pi y}$ guarantees the convergence of the integral, $R_2[s,0,\omega]$ becomes explicitly: $$\begin{aligned}
\label{R2explicit}
R_2[s,0,\omega]=\int\limits_0^\infty \frac{1}{e^{2\pi y}-1}&&\left[ \frac{-y+\imath y^2}{1+y^2}
e^{-y s} \frac{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{1+\imath y}{2};
1+\imath y;z(0) \right]}{\phantom{1}_2 F_1\left[ \frac{1+\imath y}{2},\frac{-1+\imath y}{2};
1+\imath y;z(0) \right]} \right. \nonumber \\
&& \left. +\frac{-y-\imath y^2}{1+y^2}
e^{y s} \frac{\phantom{1}_2 F_1\left[ \frac{1-\imath y}{2},\frac{1-\imath y}{2};
1-\imath y;z(0) \right]}{\phantom{1}_2 F_1\left[ \frac{1-\imath y}{2},\frac{-1-\imath y}{2};
1-\imath y;z(0) \right]} \right] dy \, .\end{aligned}$$ If we now employ the expansion (\[HypRatioExpansion\]) two times in (\[R2explicit\]) we find after some algebra: $$\begin{aligned}
\Im\left\{R_2^{(0)}[s,0,\omega]\right\}=\Im \left\{ \int\limits_0^\infty \frac{y}{e^{2\pi y}-1} \left[
e^{-y s} \psi\left(\frac{1+\imath y}{2}\right)+e^{y s} \psi\left(\frac{1-\imath y}{2}\right) \right]
dy \right\}\end{aligned}$$ and using (\[ImofDiGamma\]) we have $$\begin{aligned}
\label{R2IntRes}
\Im\left\{R_2^{(0)}[s,0,\omega]\right\}&=\int\limits_0^\infty \frac{\pi}{2} \frac{y}{e^{2\pi y}-1}
\left[ e^{-y s} \tanh \left(\frac{\pi y}{2}\right)+e^{y s} \tanh \left(-\frac{\pi y}{2}\right) \right] dy\\
&=-\pi\int\limits_0^\infty \frac{y}{e^{2\pi y}-1}\tanh\left(\frac{\pi y}{2}\right)\sinh(y s)dy=
-\pi\int\limits_0^\infty \frac{y}{\left(1+e^{\pi y}\right)^2}\sinh (y s) dy\nonumber \\
&=-\pi\frac{d}{ds}\left[\int\limits_0^\infty \frac{\cosh(y s)}{\left(1+e^{\pi y}\right)^2} dy\right]=
-\frac{1}{2}\frac{d}{ds}\left[\int\limits_0^1\frac{u^{1-\frac{s}{\pi}}}{(1+u)^2} du +
\int\limits_0^1\frac{u^{1+\frac{s}{\pi}}}{(1+u)^2} du \right] \nonumber \\
&=-\frac{1}{2}\frac{d}{ds} \left\{\frac{1}{2-\frac{s}{\pi}}
\phantom{1}_2 F_1\left[2,2-\frac{s}{\pi};3-\frac{s}{\pi};-1 \right]+
\frac{1}{2+\frac{s}{\pi}}
\phantom{1}_2 F_1\left[2,2+\frac{s}{\pi};3+\frac{s}{\pi};-1 \right]\right\} \, .\nonumber \end{aligned}$$ While we substituted $y$ by $u$ according to $u=e^{-\pi y}$ in the third line, we used (3.194, p.313) of [@Gradshteyn] with $\nu=2$, $u=\beta=1$ and $\mu=2\mp s/\pi$ (valid for $s<2 \pi$) in the last line. In order to simplify the hypergeometric functions, we first use another relation between contiguous functions, namely eq. (15.2.17) of [@AbrStegun] with $a=1$, $b=k$, $c=k+1$ and $z=-1$ and then the formulae (15.1.21) and (15.1.23) of [@AbrStegun] together with the duplication formula for the $\Gamma$-function to obtain: $$\begin{aligned}
\frac{1}{k}\phantom{1}_2 F_1\left[2,k;k+1;-1 \right]&=&\phantom{1}_2 F_1\left[1,k;k;-1 \right]-
\frac{k-1}{k} \phantom{1}_2 F_1\left[1,k;k+1;-1 \right] \nonumber \\
&=&2^{-k} \sqrt{\pi}\frac{\Gamma\left[k\right]}{\Gamma\left[\frac{k}{2}\right]
\Gamma\left[\frac{k+1}{2}\right]}-\left(k-1\right) \beta(k)\nonumber \\
&=&\frac{1}{2}+(1-k)\beta(k).\end{aligned}$$ After making use of this in (\[R2IntRes\]) we finally obtain for the contribution to lowest order in $1-z(0)$: $$\begin{aligned}
\Im\left\{R_2^{(0)}[s,0,\omega]\right\}=\frac{1}{2}\frac{d}{ds}\left[\left(1-\frac{s}{\pi}\right)
\beta(2-\frac{s}{\pi})+\left(1+\frac{s}{\pi}\right) \beta(2+\frac{s}{\pi}) \right].\end{aligned}$$ Summarizing the results to lowest order in $1-z(0)$ we therefore have $$\begin{aligned}
\label{0orderResult}
\Im\left\{R_1^{(0)}[s,0,\omega]\right\}&=&-\frac{\pi}{2 s^2}-\frac{1}{\pi}\beta'(\frac{s}{\pi}),\nonumber\\
\Im\left\{R_2^{(0)}[s,0,\omega]\right\}&=&\frac{1}{2}\frac{d}{ds}\left[\left(1-\frac{s}{\pi}\right)
\beta(2-\frac{s}{\pi})+\left(1+\frac{s}{\pi}\right) \beta(2+\frac{s}{\pi}) \right].\end{aligned}$$ Expanding (\[0orderResult\]) around $s=0$ we find: $$\begin{aligned}
\Im\left\{R_1^{(0)}[s,0,\omega]\right\}&=\frac{\pi}{2 s^2}-\frac{\pi}{12}+\mathcal{O}(s) \, , \nonumber \\
\Im\left\{R_2^{(0)}[s,0,\omega]\right\}&=\left( \frac{1}{6}-\frac{3\zeta(3)}{2\pi^2}\right) s +
\mathcal{O}(s^3) \, .\end{aligned}$$ with $\zeta$ denoting <span style="font-variant:small-caps;">Riemann</span>’s $\zeta$-function. As expected the contribution from $R_2^{(0)}[s,0,\omega]$ is sub-leading with respect to the one from $R_1^{(0)}[s,0,\omega]$.
Next to leading order asymptotic of $\Im\left\{R_1[s,\theta,\omega]\right\}$
----------------------------------------------------------------------------
We would now like to obtain the next to leading order term in the asymptotic of $\Im\left\{R_1[s,\theta,\omega]\right\}$, that is the term obtained from (\[R1shiftedBrane\]) by taking into account the linear contribution in $1-z(0)$ of the expansion (\[HypRatioExpansion\]). Since the calculation of the integral obtained in this way is rather cumbersome and anyway we do not need the full $s$-dependence, we do not intend to evaluate it fully. Extracting the leading $s$-divergence at $s=0$ is sufficient for our purposes here. To start, we insert the linear term of (\[HypRatioExpansion\]) in (\[R1shiftedBrane\]): $$\begin{aligned}
\Im\left\{R_1^{(1)}[s,0,\omega]\right\}&=
-\Im\left\{\int\limits_0^\infty dy y \frac{1-\imath y}{1+y^2} e^{-y s}
\left\{ \left(\frac{1+\imath y}{2}\right)^3 \left[2 \psi(2)-2 \psi\left(\frac{3+\imath y}{2}
\right) -\ln\left[1-z(0)\right]\right]\right. \right. \nonumber \\
&- \left(\frac{1+\imath y}{2}\right)^2 \left(\frac{-1+\imath y}{2}\right)
\left[2 \psi(1)-2 \psi\left(\frac{1+\imath y}{2}\right)-\ln\left[1-z(0)\right]\right]\times \\
&\times\left.\left. \left[\ln\left[1-z(0)\right]-\psi(1)-\psi(2)+\psi\left(\frac{3+\imath y}{2}\right)+
\psi\left(\frac{1+\imath y}{2}\right)\right] \right\} \left[1-z(0)\right] \right\} \, .\nonumber\end{aligned}$$ After some algebra and by using the functional relation of the Digamma-function $\psi(z)$ (see [@AbrStegun]) $$\psi\left(z+1\right)=\psi\left(z\right)+\frac{1}{z}$$ we find $$\begin{aligned}
\label{R1_1storder}
\Im\left\{R_1^{(1)}[s,0,\omega]\right\}=-\frac{1-z(0)}{2} \int\limits_0^{\infty}y e^{-y s} \Im \left\{
\ldots\right\} dy\end{aligned}$$ with $$\begin{aligned}
\label{R1_1storderIntegrand}
\Im \left\{ \ldots\right\} &=&\Im\left\{\left(\frac{1+\imath y}{2}\right)^2
\left[2 \psi(2)-2 \psi\left(\frac{1+\imath y}{2}\right)-\ln\left[1-z(0)\right]\right]-(1+\imath y) \right.
\nonumber \\
&&\qquad+\frac{1+y^2}{4}\left[2\psi\left(1\right)-2\psi\left(\frac{1+\imath y}{2}\right)-
\ln\left[1-z(0)\right]\right]\times\nonumber\\
&&\qquad\quad\qquad\times\left[\ln\left[1-z(0)\right]-\psi\left(1\right)-\psi\left(2\right)+
2\psi\left(\frac{1+\imath y}{2}\right)\right] \nonumber \\
&&\qquad+\left. \frac{1-\imath y}{2}\left[2\psi\left(1\right)-2\psi\left(\frac{1+\imath y}{2}\right)
-\ln\left[1-z(0)\right]\right] \right\} \, .\end{aligned}$$ We will now keep only the terms proportional to $y^3$ in (\[R1\_1storder\]) ($y^2$ in (\[R1\_1storderIntegrand\])) since only these will contribute to the leading $1/s^{4}$ singular behavior. After a few lines of algebra we find $$\begin{aligned}
\label{R1_1HighestContribution}
\Im\left\{R_1^{(1)}[s,0,\omega]\right\}&=&-\frac{1-z(0)}{2}\left[1-2\gamma-\ln\left[1-z(0)\right]\right]
\int\limits_0^{\infty} y^3 e^{-y s} \Im \left[\psi\left(\frac{1+\imath y}{2}\right)\right] dy \nonumber\\
&&+\left[1-z(0)\right] \int\limits_0^\infty y^3 e^{-y s}
\Im\left[\psi\left(\frac{1+\imath y}{2}\right)\right]
\Re\left[\psi\left(\frac{1+\imath y}{2}\right)\right] dy \nonumber \\
&& + \mbox{terms involving lower powers of }y.\end{aligned}$$ The first integral can be reduced to the integral (\[R1\_0\_Integral\]) simply by replacing each power of $y$ by a derivative with respect to $s$ and by taking the derivatives out of the integral: $$\begin{aligned}
\int\limits_0^{\infty} y^3 e^{-y s} \Im \left[\psi\left(\frac{1+\imath y}{2}\right)\right] dy&=&
-\frac{d^3}{ds^3}\left\{\int\limits_0^\infty e^{-y s} \Im \left[\psi\left(\frac{1+\imath y}{2}\right)\right]
dy \right\}\nonumber \\
&=&-\frac{\pi}{2}\frac{d^3}{ds^3}\left[-\frac{1}{s}+\frac{2}{\pi}\beta\left(\frac{s}{\pi}\right)\right] .\end{aligned}$$ We could not find an analytic expression for the second integral. However all we need is the first term of its small $s$ asymptotic[^26] and this can be easily obtained from the large $y$ asymptotic of $\Re\left[\psi\left(\left(1+\imath y\right)/2\right)\right]$. Since asymptotically $$\psi\left(z\right)\sim\ln z -\frac{1}{2 z}-\frac{1}{12 z^2}+\mathcal{O}(\frac{1}{z^4})\, , \qquad
(z \to \infty \; \mbox{in} \; \vert \arg z\vert<\pi)$$ we expect that this term will contribute logarithmic terms in $s$ and we find after straightforward expansions $$\begin{aligned}
\label{RePartAssympt}
\Re\left[\psi\left(\frac{1+\imath y}{2}\right)\right] \sim \ln\frac{y}{2}-\frac{1}{6 y^2}+
\mathcal{O}\left(\frac{1}{y^4}\right) \, .\end{aligned}$$ From the last formula (\[RePartAssympt\]) we learn that within our current approximation of keeping only highest (cubic) power terms in $y$, it is sufficient to take into account only the contribution coming from $\ln\frac{y}{2}$. Therefore, we need to calculate the following integral $$\frac{\pi}{2} \int\limits_0^\infty y^3 e^{-y s} \tanh \left(\frac{\pi y}{2} \right)\ln y \, dy=
-\frac{\pi}{2}\frac{d^3}{ds^3}\Big[\underbrace{\int\limits_0^\infty e^{-y s}
\tanh \left(\frac{\pi y}{2} \right)
\ln y \, dy}_{\equiv \mathcal{J}(s)} \Big] \, ,$$ where we used (\[ImofDiGamma\]) once more. Note that the term proportional to $\ln 2$ can be accounted for by adding a contribution of the type of the first integral in (\[R1\_1HighestContribution\]). One way to find the asymptotic of $\mathcal{J}(s)$ for small $s$ is to integrate the following asymptotic expansion of $\tanh x$ $$\begin{aligned}
\label{tanhexp}
\tanh x = 1-2 e^{-2 x}+2 e^{-4 x}-2 e^{-6 x}+\ldots=1+2 \, \sum_{\nu=1}^\infty (-1)^\nu e^{-2 \nu x} \end{aligned}$$ which converges for all $x>0$. Inserting this in the definition of $\mathcal{J}(s)$ we find: $$\begin{aligned}
\label{calIIntRes}
\mathcal{J}(s)&=&\int\limits_0^\infty e^{-y s} \ln y \,
\left[1+2\sum_{\nu=1}^\infty (-1)^\nu e^{-\pi y \nu} \right] dy \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \nonumber \\
&=&-\frac{\gamma + \ln s}{s} +2 \sum_{\nu=1}^\infty (-1)^{\nu+1}
\frac{\gamma+\ln\left(s+\nu \pi\right)}{s+\nu\pi}\nonumber\\
&=& -\frac{\gamma + \ln s}{s} +\frac{2}{\pi}\left(\gamma+\ln\pi\right) \beta\left(\frac{s}{\pi}+1\right)
+\frac{2}{\pi} \sum_{\nu=1}^\infty (-1)^{\nu+1}
\frac{\ln\left(\nu+\frac{s}{\pi}\right)}{\nu+\frac{s}{\pi}},\end{aligned}$$ where we used the series expansion of the $\beta$-function given e.g. in [@Gradshteyn]. The last two terms in the above formula are finite in the limit $s\to0$ and therefore will play no role in the asymptotic (since they are multiplied by a factor of $1-z(0)$).
We are now ready to assemble all contributions to the small $s$ asymptotic of $\Im\left\{R_1^{(1)}[s,0,\omega]\right\}$: $$\begin{aligned}
\label{1OrderResult}
&\Im\left\{R_1^{(1)}[s,0,\omega]\right\}\sim\frac{1-z(0)}{2}
\left[1-2\gamma-\ln\left[1-z(0)\right]+2\ln 2\right]
\frac{\pi}{2}\frac{d^3}{ds^3}\left[-\frac{1}{s}+\frac{2}{\pi}\beta\left(\frac{s}{\pi}\right)\right]
\nonumber \\
&\quad -\left[ 1-z(0) \right] \frac{\pi}{2}\frac{d^3}{ds^3}\left[-\frac{\gamma+\ln s}{s}
+ \frac{2}{\pi}\left(\gamma+\ln\pi\right) \beta\left(\frac{s}{\pi}+1\right)
+\frac{2}{\pi} \sum_{\nu=1}^\infty
\frac{ (-1)^{\nu+1}\ln\left(\nu+\frac{s}{\pi}\right)}{\nu+\frac{s}{\pi}} \right]
+\mathcal{O}(\frac{\ln s}{s^3}) \nonumber \\
&\qquad= \frac{\pi}{2}\frac{1-z(0)}{s^4} \left\{ 8-6\ln 2 -6
\ln\left[\frac{s}{\sqrt{1-z(0)}}\right]\right\} +\mathcal{O}(\frac{\ln s}{s^3}),\end{aligned}$$ where we used the following series expansions for the $\beta$-function and for the last term in (\[calIIntRes\]): $$\begin{aligned}
\beta\left(\frac{s}{\pi}\right)&=\frac{\pi}{s}-\ln 2 +\mathcal{O}(s)\, ,\nonumber \\
\beta\left(\frac{s}{\pi}+1\right)&=\ln 2+\mathcal{O}(s)\, ,\nonumber \\
\sum_{\nu=1}^\infty
\frac{(-1)^{\nu+1}\ln\left(\nu+\frac{s}{\pi}\right)}{\nu+\frac{s}{\pi}}
&=\frac{1}{2} \left[\ln(2)\right]^2 - \gamma \ln(2) + \mathcal{O}(s).\end{aligned}$$ Note that only the term $\beta\left(s/\pi\right)$ gave a contribution to the leading $1/s^4$ divergence. Relations (\[0orderResult\]) and (\[1OrderResult\]) constitute the main result of this appendix. They provide the desired asymptotic behavior of the two-point function in the regime of intermediate distances $\omega/\sinh\left(\frac{\omega\pi}{2}\right) \ll s\ll 1$. A couple of remarks are in place. Clearly, (\[0orderResult\]) is not only an asymptotic result but is valid for all $s \in [0,\pi]$. Secondly, we were writing $\mathcal{O}(\ln s/s^3)$ in (\[1OrderResult\]) to indicate that we dropped all contributions coming from terms in the integrand (\[R1\_1storderIntegrand\]) lower than cubic order. Finally we point out that the development of the ratio (\[HypRatioExpansion\]) in power of $1-z(0)$ indeed generates an asymptotic in the correct distance regime. This is clear from the fact that each new power in $1-z(0)$ is paired with an additional power of $y^2$ in the integrand of $\Im\left\{R_1[s,\theta,\omega]\right\}$. This by itself implies upon multiplication by $e^{-y s}$ and integration over $y$ an additional power of $1/s^2$. We therefore effectively generate an expansion in powers of $\sqrt{1-z(0)}/s=1/\left[s \cosh \left(\frac{\omega\pi}{2}\right)\right]$. It is also easy to understand why this rough way of counting powers works. The reason is that the asymptotic properties of our integrals $\Im\left\{R_1^{(n)}[s,0,\omega]\right\}$ are determined only by the behavior of the corresponding integrands for large values of the integration variable $y$. The terms $\Re\left[\psi\left(\left(1+\imath y\right)/2\right)\right]$ and $\Im\left[\psi\left(\left(1+\imath y\right)/2\right)\right]$ do not interfere with the above power counting since their behavior for large $y$ is either logarithmic in $y$ (for $\Re$) or constant in $y$ (for $\Im$).
Finally we want to point out that the expansion in powers of $1/\left[s \cosh \left(\frac{\omega\pi}{2}\right)\right]$ is not useful for obtaining information about the Green’s function for distances $s \ll \omega/\cosh\left(\frac{\omega\pi}{2}\right)$. In this limit the asymptotic expansion of the hypergeometric functions in (\[RatioOf2Hyps\]) for large values of $\lambda$ proves the most efficient way to recover the properties of a Green’s function in a $4$-dimensional space. See appendix \[appGreenUltraShort\] for the detailed form of the Green’s function at distances smaller than the extra dimension.
Estimate of the arc contribution to $R_1[s,\theta,\omega]$ {#arcjust}
----------------------------------------------------------
This final subsection is dedicated to the verification that the arc denoted $C_2$ in Fig. \[intcontour\] of appendix \[appParallel\] gives a vanishing contribution to the integral $R_1[s,\theta,\omega]$ and thus justifying the representation (\[R1shiftedBrane\]). We need to know the large $\lambda$ asymptotic behavior of the hypergeometric functions entering the definition of $R_1[s,\theta,\omega]$ in (\[fdef\]). The relevant formula (\[GeneralAsymptoticOfHyp\]) can be found in appendix \[appGreenUltraShort\] where we discuss the short distance behavior of the Green’s function. After canceling all common factors from the ratio of the two hypergeometric functions we find: $$\begin{aligned}
\label{FFratioInR1asymptotic}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};
\lambda+1;z(0) \right]}&=
\frac{\lambda+1}{\lambda-1} \left[ \frac{z(\theta)}{z(0)}\right]^{-\frac{\lambda+1}{2}}
\left[\frac{e^{-\nu(\theta)}}{e^{-\nu(0)}}\right]^\frac{\lambda+1}{2}
\frac{\left[1+e^{-\nu(\theta)}\right]^{-\frac{1}{2}} \left[1-e^{-\nu(\theta)}\right]^{-\frac{1}{2}}}
{\left[1+e^{-\nu(0)}\right]^{\frac{3}{2}} \left[1-e^{-\nu(0)}\right]^{\frac{1}{2}}} \times \nonumber \\
&\qquad \times \left[1+\mathcal{O}\left(\frac{1}{\lambda}\right)\right] \, .\end{aligned}$$ Substituting $\lambda=R e^{\imath \varphi}$, we now estimate $R_1[s,\theta,\omega]$: $$\begin{aligned}
&\left| \int_{C_2} \frac{\lambda e^{\imath\lambda s}}{\lambda+1}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\lambda}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};
\lambda+1;z(0) \right]} d \lambda\right| \nonumber \\
& \leq \int_{C_2} \left| \frac{\lambda e^{\imath\lambda s}}{\lambda+1}
\left[\frac{z(\theta)}{z(0)}\right]^\frac{\lambda}{2}
\frac{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda+1}{2};
\lambda+1;z(\theta) \right]}{\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};
\lambda+1;z(0) \right]} \right| d \lambda \nonumber \\
& \leq \underbrace{C \left[ \frac{z(0)}{z(\theta)}\right]^{\frac{1}{2}}
\left| \frac{e^{-\nu(\theta)}}{e^{-\nu(0)}} \right|^\frac{1}{2}
\frac{\left[1+e^{-\nu(\theta)}\right]^{-\frac{1}{2}}
\left[1-e^{-\nu(\theta)}\right]^{-\frac{1}{2}}}{\left[1+e^{-\nu(0)}\right]^{\frac{3}{2}}
\left[1-e^{-\nu(0)}\right]^{\frac{1}{2}}}}_{K} R
\int\limits_0^{\frac{\pi}{2}}
\left| e^{\imath R s e^{\imath \varphi}} \right|
\left| \frac{R e^{\imath \varphi}}{R e^{\imath \varphi -1}} \right|
\left| \left[\frac{e^{-\nu(\theta)}}{e^{-\nu(0)}}\right]^{\frac{R e^{\imath \varphi}}{2}}\right|
d\varphi \nonumber \\
&=K R \int\limits_0^{\frac{\pi}{2}}
\frac{e^{-R\left(s\sin\varphi-\ln a \cos \varphi\right)}}{\sqrt{1-\frac{2 \cos \varphi}{R}+\frac{1}{R^2}}}
d\varphi
\leq K' R \int\limits_0^{\frac{\pi}{2}} e^{-R s\left(\sin\varphi+\frac{\ln a^{-1}}{s}
\cos \varphi\right)} d\varphi \, ,\end{aligned}$$ where $C$ is a constant independent of $R$ and the estimate of the ratio of the two hypergeometric functions is valid for large enough $R$. We also introduced $a$ as in appendix \[appGreenUltraShort\] by (\[adef\]). Note that $a<1$ for $0<\theta$. In the last line we estimated the inverse of the square root by an arbitrary constant (e.g. 2), certainly good for large enough $R$ and for all $\varphi$. All what is left is to observe that $$0 < \epsilon \equiv \min[1,\frac{\ln a^{-1}}{s}] \leq \sin\varphi+\frac{\ln a^{-1}}{s} \cos\varphi \quad
\mbox{for} \quad s>0, \;\; a^{-1} >1$$ to obtain the desired result: $$\left| \int_{C_2} d\lambda \ldots \right| \leq K' R \int\limits_0^{\frac{\pi}{2}}e^{-R s \epsilon}d\varphi=
\frac{\pi}{2} K' R e^{-R s \epsilon} \to 0 \quad \mbox{for} \quad R \to \infty.$$
The two point function at very short distances ($r \ll R$) {#appGreenUltraShort}
==========================================================
The aim of this appendix is to calculate the behavior of the two-point function at distances smaller than the extra dimension $r\ll R$. We will do so by using an appropriate asymptotic expansion for large parameter values of the hypergeometric functions in (\[RatioOf2Hyps\]). Such an asymptotic can be found for example in [@Erdelyi] or [@Luke]. The general formula is[^27] $$\begin{aligned}
\label{GeneralAsymptoticOfHyp}
\phantom{1}_2 F_1\left[a+n,a-c+1+n;a-b+1+2 n;z \right] &= \frac{2^{a+b} \Gamma[a-b+1+2 n]
\left(\frac{\pi}{n}\right)^{\frac{1}{2}}}{\Gamma[a-c+1+n]\Gamma[c-b+n]}
\frac{e^{-\nu(n+a)}}{z^{(n+a)}} \times \\
&\times\left(1+e^{-\nu}\right)^{\frac{1}{2}-c} \left(1-e^{-\nu}\right)^{c-a-b-\frac{1}{2}}
\left[1+\mathcal{O}\left(\frac{1}{n}\right)\right] \nonumber \end{aligned}$$ with $\nu$ defined via the relation $$\label{nudefinition}
e^{-\nu}=\frac{2-z-2(1-z)^{\frac{1}{2}}}{z} \, .$$ We need to calculate $$S[s,\theta,\omega]=\sum_{\lambda=2}^{\infty} \lambda \sin \left(\lambda s \right)
\frac{\varphi_1^{(\lambda)}(\theta)}{{\varphi'}_1^{(\lambda)}(0)}$$ with the ratio $\varphi_1^{(\lambda)}(\theta)/{\varphi'}_1^{(\lambda)}(0)$ given in (\[RatioOf2Hyps\]). Using the above expansion (\[GeneralAsymptoticOfHyp\]) two times we find after numerous cancellations $$\begin{aligned}
\label{Sasymptotic}
S[s,\theta,\omega]&\sim-\frac{4}{\omega}\left(\frac{1-z(\theta)}{1-z(0)}\right)^2
\frac{\left[1+e^{-\nu(\theta)}\right]^{-\frac{1}{2}} \left[1-e^{-\nu(\theta)}\right]^{-\frac{5}{2}}}
{\left[1+e^{-\nu(0)}\right]^{\frac{1}{2}} \left[1-e^{-\nu(0)}\right]^{-\frac{3}{2}}}
\frac{z(0)^\frac{3}{2}}{z(\theta)^2}\frac{e^{-\frac{3\nu(\theta)}{2}}}{e^{-\frac{\nu(0)}{2}}}
\times \nonumber\\
&\times \sum_{\lambda=2}^{\infty}\frac{\lambda \sin \left(\lambda s\right)}{\lambda-1}
\left[\frac{e^{-\nu(\theta)}}{e^{-\nu(0)}}\right]^{\frac{\lambda}{2}} \, ,\end{aligned}$$ where we used the following abbreviations $$\begin{aligned}
z(\theta)&=\tanh^2\left[\omega\left(\frac{\pi}{2}-\theta\right)\right] \, , \label{zdef}\\
e^{-\nu(\theta)}&=\tanh^2\left[\frac{\omega}{2}\left(\frac{\pi}{2}-\theta\right)\right]
\, .\label{nuresult}\end{aligned}$$ Note that (\[nuresult\]) is obtained after some lines of algebra by inserting (\[zdef\]) in (\[nudefinition\]). Still the sum in (\[Sasymptotic\]) diverges for $\theta=0$ but it turns out that it can be performed analytically for $\theta>0$ so that after the summation the limit $\theta \to 0$ exists. For the sake of a lighter notation we introduce the symbol $a$ by $$\label{adef}
a\equiv\left[\frac{e^{-\nu(\theta)}}{e^{-\nu(0)}}\right]^{\frac{1}{2}}=
\frac{\tanh\left[\frac{\omega}{2}\left(\frac{\pi}{2}-\theta\right)\right]}
{\tanh\left(\frac{\omega\pi}{4}\right)} \, .$$ It is now straightforward to evaluate the sum over $\lambda$: $$\begin{aligned}
\sum_{\lambda=2}^{\infty}\frac{\lambda \sin \left(\lambda s\right)}{\lambda-1}
a^\lambda&=\sum_{\lambda=2}^{\infty} \sin(\lambda s)\, a^\lambda+
\sum_{\lambda=2}^{\infty}\frac{\sin \left(\lambda s\right)}{\lambda-1} a^\lambda \nonumber \\
&=-a \sin s + \sum_{\lambda=1}^{\infty} \sin(\lambda s)\, a^\lambda+
\sum_{\lambda=1}^{\infty}\frac{\sin \left[\left(\lambda+1\right) s\right]}{\lambda} a^{\lambda+1} \nonumber \\
&=-a \sin s + \frac{1}{2}\frac{\sin s}{\frac{1}{2}\left(a+\frac{1}{a}\right)-\cos s}
-\frac{a}{2} \sin s \, \ln \left( 1-2a\cos s+a^2\right) \nonumber \\
&+ a\cos s \arctan\left[\frac{a \sin s}{1-a \cos s}\right] \, .\end{aligned}$$ The full result can therefore be written as $$\begin{aligned}
\label{SasymptoticResult}
S[s,\theta,\omega]&\sim-\frac{4}{\omega}\left(\frac{1-z(\theta)}{1-z(0)}\right)^2
\frac{\left[1+e^{-\nu(\theta)}\right]^{-\frac{1}{2}} \left[1-e^{-\nu(\theta)}\right]^{-\frac{5}{2}}}
{\left[1+e^{-\nu(0}\right]^{\frac{1}{2}} \left[1-e^{-\nu(0)}\right]^{-\frac{3}{2}}}
\frac{z(0)^\frac{3}{2}}{z(\theta)^2}\frac{e^{-\frac{3\nu(\theta)}{2}}}{e^{-\frac{\nu(0)}{2}}}
\, a \sin s \, \times \\
&\times \left\{\frac{1}{2 a} \frac{1}{\frac{1}{2}\left(a+\frac{1}{a}\right)-\cos s}
-\frac{1}{2} \ln \left( 1-2 a \cos s+a^2\right)+\cot s \, \arctan \left[\frac{a \sin s}{1-a \cos s}\right]
-1 \right\} \, .\nonumber \end{aligned}$$ Although the last result is given in closed form, we have to emphasize that its validity is restricted to distances smaller than the size of the extra dimension $R$.
It is now save to take the limit $\theta\to 0$ in (\[SasymptoticResult\]). The result is $$\begin{aligned}
\label{SasymptoticResultBrane}
\lim_{\theta\to 0} S[s,\theta,\omega]&=-\frac{\sinh\left( \frac{\omega\pi}{2}\right)}{\omega}
\sin s \left\{\frac{1}{4}\frac{1}{\sin^2(\frac{s}{2})}- \ln \left[2 \sin \left(\frac{s}{2}\right)\right]+
\frac{\pi-s}{2} \cot s \; -1\right\} \, .\end{aligned}$$ We note that it is the first term in the curly brackets on the right hand side of (\[SasymptoticResult\]) which is responsible for reproducing the characteristic short distance singularity of the two-point function. The main result of this appendix is therefore: $$\begin{aligned}
\lim_{\theta\to 0} S[s,\theta,\omega]&\sim-\frac{\sinh \left(\frac{\omega\pi}{2}\right)}{\omega}
\left\{\frac{1}{s}+\frac{\pi}{2}+\mathcal{O}\left(s \ln s\right)\right\} \, .\end{aligned}$$ The $5$-dimensional short distance singularity can even be obtained including the extra dimension. Apart from the logarithmic term in the curly brackets of (\[SasymptoticResult\]), all but the first one are finite at short distances (in $\theta$ and $s$). Developing the denominator of this term, we recover the usual euclidean metric in $\mathbb{R}^4$: $$\label{5DNewton}
\frac{1}{2}\left(a+\frac{1}{a}\right)-\cos s \sim
\frac{1}{2} \left[ \frac{\omega^2}{\sinh^2\left( \frac{\omega\pi}{2}\right)}\theta^2 +s^2 \right]+ \ldots .$$ where the dots denote higher terms in $\theta$ and $s$.
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[^1]: gruppuso@bo.iasf.cnr.it
[^2]: ewald.roessl@epfl.ch
[^3]: mikhail.shaposhnikov@epfl.ch
[^4]: Our conventions for the metric and the Riemann tensor are those of reference [@wheeler].
[^5]: See also footnote \[changecoords\].
[^6]: For a precise definition of a geodesically incomplete space see e.g. [@Wald].
[^7]: In general, two affine parameters $\lambda$ and $\tau$ are related by $\lambda=c \tau + d$, since this is the most general transformation leaving the geodesic equation (\[Geodesicequation\]) invariant. The choice of the origin of time to coincide with the emission of the photon only fixes $d$ but does not restrict $c$.
[^8]: We mainly follow [@Aharony:1999ti].
[^9]: Whenever we used the word $AdS_5$ so far in this paper we actually meant $CAdS_5$. For reasons of clarity, however, we will brake with this common practice in the rest of this section.
[^10]: \[changecoords\]It is obvious that the metric (\[metric\]) for $0\leq \theta \leq \pi/2$ ($-\pi/2 \leq \theta \leq 0$) reduces to the above line-element (\[AdSGlobalCoords\]) under the following coordinate transformation: $\chi=\omega (\frac{\pi}{2}\mp \theta)$, $\tau= t \omega/\left[R \cosh \left(\frac{\omega\pi}{2}\right)\right]$, together with $a=R/\omega$.
[^11]: $T^0_{\;\;0}=c_0$, and $T^i_{\;\;j}=\delta^i_{\;\;j} c$ in the notation of section (\[SingularBrane\]).
[^12]: For an elementary introduction to the concept of modified Green’s functions see e.g [@Stakgold].
[^13]: This means that we have the freedom to choose $c_{1 0 0}(\vec x')$ freely. Our choice is $c_{1 0 0}(\vec x')=0$ without restricting generality since from eq. (\[esuIntReprESU\]) we immediately conclude that for sources satisfying (\[solvabilityESU\]) there will never be any contribution to $u(\vec x)$ coming from $c_{1 0 0}(\vec x')$.
[^14]: See e.g. [@GribMamaMost].
[^15]: Note that the constant term in the expansion (\[expansioninsESU\]) is specific to our choice of $c_{1 0 0}(\vec x')$.
[^16]: There exists numerous articles treating the gravitational potential of a point source in Einstein’s static universe (see [@Astefanesei:2001cx; @Nolan:1999wf] and references therein). We only would like to point out here similarities between our result (\[ESUresult\]) and the line element of a Schwarzschild metric in an Einstein static universe background given in [@Astefanesei:2001cx].
[^17]: At this point we have to explain what we mean by “zero mode” in this context, since due to the asymmetrically warped geometry, the spectrum of Kaluza-Klein excitations will not be Lorentz-invariant and strictly speaking, different excitations cannot be characterized by different $4$-dimensional masses. More accurate would be to say that for a given value of the momentum eigenvalue $\lambda$ there exists a tower of corresponding Kaluza-Klein excitations with energies given by $E_n^\lambda$ with $n=0,1,\ldots$. In our use of language the “zero mode branch” of the spectrum or simply the “zero mode” is defined to be the collection of the lowest energy excitations corresponding to all possible values of $\lambda$, that is by the set $\left\{\left(E_0^\lambda,\lambda\right),\, \lambda=1,2,\ldots\right\}$. The definition is readily generalized to higher branches of the spectrum.
[^18]: We allow for a finite $r_c$ only to impose boundary conditions in a proper way. Eventually we are interested in the limit $r_c \to \infty$.
[^19]: However as long as matter on the brane is considered the Green’s function for the two setups differ only by a factor of $2$.
[^20]: As discussed above the factor of $2$ is a direct consequence of the boundary conditions corresponding to a semi-infinite extra dimension.
[^21]: This is justified since both the real and the imaginary part of the resulting integral converge.
[^22]: Note that this mathematical delicacy about the divergence of the Green’s function on the brane also applies to the Randall-Sundrum case of the Fourier integral representation of the two-point function. The obvious remedy is to keep $\theta$ strictly greater than zero during the whole calculation and eventually take the limit $\theta \to 0$.
[^23]: Everything but the ratio of the hypergeometric functions is clearly holomorphic. The dependence of Gauss’s hypergeometric function on the parameters is also holomorphic so the only danger comes from the zeros of $\phantom{1}_2 F_1\left[ \frac{\lambda+1}{2},\frac{\lambda-1}{2};\lambda+1;z(0) \right]$. These, however, turn out to be outside of the first quadrant.
[^24]: To be mathematically correct, one has to apply <span style="font-variant:small-caps;">Cauchy</span>’s theorem to the quarter of the disk of radius $R$ bounded by the positive real axis, the positive imaginary axis and the arc joining the points $R$ and $\imath R$, see Fig. \[intcontour\] in appendix \[appParallel\]. The above statement will then be correct if in the limit $R\to \infty$ the contribution from the arc tends to zero. We will verify this in section \[arcjust\] of this appendix.
[^25]: By $\beta'$ in (\[R1\_0Result\]) we mean the derivative of $\beta$ with respect to its argument.
[^26]: In the sense $s\ll 1$.
[^27]: Since we noticed that the formulae given by Luke, p. 452 eq. (20)–(23) [@Luke] and the formula given by Erdélyi [@Erdelyi] differ by a factor of $2^{a+b}$ we performed a numerical check. Its outcome clearly favored Erdélyi’s formula which we reproduced in \[GeneralAsymptoticOfHyp\].
|
---
abstract: 'We summarize the results of including running coupling corrections into the nonlinear evolution equation for diffractive dissociation. We also document a prediction that the NLO QCD odderon intercept is zero resulting from a discussion at the Diffraction 2012 Workshop.'
author:
- 'Yuri V. Kovchegov'
title: Running Coupling Evolution for Diffractive Dissociation and the NLO Odderon Intercept
---
[ address=[Department of Physics, The Ohio State University, Columbus, OH 43210, USA]{} ]{}
Running Coupling Corrections for Diffractive Dissociation
=========================================================
This proceedings contribution is mainly based on the paper [@Kovchegov:2011aa].
The evolution equation for single diffractive dissociation in deep inelastic scattering (DIS) was derived in [@Kovchegov:1999ji]. First we write the cross section for single diffractive dissociation in DIS on a nucleus as $$\label{xsec}
M_X^2 \, \frac{d \sigma_{diff}^{\gamma^* A}}{d M_X^2} = - \int d^2
x_0 \, d^2 x_1 \int\limits_0^1 \, dz \ |\Psi^{\gamma^* \rightarrow q
{\bar q}} ({x}_{01}, z)|^2 \, \frac{{\partial}S^D_{{\bf x}_0, {\bf x}_1}
(Y, Y_0)}{{\partial}Y_0},$$ where $Y= \ln (s/Q^2)$ is the net rapidity interval and the rapidity gap stretches from rapidity $0$ to rapidity $Y_0 \approx \ln
(s/M_X^2)$ with $M_X^2$ the invariant mass of the produced hadrons. $|\Psi^{\gamma^* \rightarrow q {\bar q}} ({x}_{01}, z)|^2$ is the order-$\alpha_{EM}$ light-cone wave function squared for a virtual photon fluctuating into a $q {\bar q}$ pair with $x_{01} = |{\bf x}_0
- {\bf x}_1|$ the transverse size of the pair and $z$ the fraction of the longitudinal momentum of the incoming virtual photon carried by the quark in the pair. The object $S^D$ is the “$S$-matrix” for single diffractive dissociation, which includes both interacting and non-interacting contributions in the amplitude and in the complex conjugate amplitude with the rapidity gap greater than or equal to $Y_0$ [@Kovchegov:2011aa].
The object $S^D$ obeys a nonlinear evolution equation, which is equivalent to the Balitsky-Kovchegov (BK) [@Balitsky:1996ub; @Kovchegov:1999yj] evolution equation [@Kovchegov:2011aa; @Kovchegov:1999ji; @Hatta:2006hs]: $$\label{SDevol}
{\partial}_Y S^D_{{\bf x}_{0}, {\bf x}_{1}} (Y, Y_0) \, = \, \frac{{\alpha_s}\,
N_c}{2 \, \pi^2} \, \int \, d^2 x_2 \,
\frac{x^2_{10}}{x^2_{20}\,x^2_{21}} \, \left[ S^D_{{\bf x}_{0}, {\bf
x}_{2}} (Y, Y_0) \, S^D_{{\bf x}_{2}, {\bf x}_{1}} (Y, Y_0) -
S^D_{{\bf x}_{0}, {\bf x}_{1}} (Y, Y_0) \right].$$ The equation is illustrated in .
![The evolution equation for $S^D$.[]{data-label="SDevol_fig"}](SDevol_fig.eps){width="80.00000%"}
The solid vertical line in denotes the final-state cut, while the vertical dashed lines denote interactions with the target nucleus in the amplitude and in the complex-conjugate amplitude. The initial condition for the evolution is given by $$\label{initS}
S^D_{{\bf x}_0, {\bf x}_1} (Y=Y_0, Y_0) = \left[ 1 - N_{{\bf x}_{0},
{\bf x}_{1}} (Y_0) \right]^2,$$ where $N$ is the (imaginary part) of the forward dipole–nucleus scattering amplitude obeying the standard BK evolution equation.
It can be shown [@Kovchegov:2011aa] that the fact that $S^D$ obeys the BK evolution results from the cancellation of final state gluon emissions: no $s$-channel gluon emitted or absorbed after the interaction with the target (to the right of the dashed line in the amplitude and to the left of the dashed line in the complex conjugate amplitude) remains in the final evolution pictured in . Without the final state gluon emissions/absorptions, the evolution becomes just like that for a forward amplitude, that is, a BK evolution. The same property remains valid when the running coupling corrections are included [@Kovchegov:2011aa]: these corrections also cancel in the final state. Hence, the running-coupling evolution equation for diffractive dissociation is equivalent to the running-coupling BK (rcBK) evolution equation, with the initial condition now containing the dipole amplitude $N$ evolved by the full rcBK evolution. We thus write the running-coupling evolution equation for $S^D$ as [@Kovchegov:2011aa] $$\label{SDevol_rc}
{\partial}_Y S^D_{{\bf x}_{0}, {\bf x}_{1}} (Y, Y_0) \, = \, \int \, d^2
x_2 \, K ({\bf x}_0, {\bf x}_1, {\bf x}_2) \, \left[ S^D_{{\bf
x}_{0}, {\bf x}_{2}} (Y, Y_0) \, S^D_{{\bf x}_{2}, {\bf
x}_{1}} (Y, Y_0) - S^D_{{\bf x}_{0}, {\bf x}_{1}} (Y, Y_0)
\right]$$ where the evolution kernel was calculated in [@Kovchegov:2006vj; @Balitsky:2006wa]. For completeness let us show the kernel in the Balitsky prescription [@Balitsky:2006wa]: $$\label{kbal}
K_{rc}^{Bal} ({\bf x}_{0}, {\bf x}_{1}, {\bf x}_{2}) = \frac{N_c \,
\alpha_s (x_{10}^2)}{2\pi^2} \Bigg[
\frac{x^2_{10}}{x^2_{20}\,x^2_{21}} +
\frac{1}{x_{20}^2}\left(\frac{\alpha_s(x_{20}^2)}{\alpha_s(x_{21}^2)}-1\right)+
\frac{1}{x_{21}^2}\left(\frac{\alpha_s(x_{21}^2)}{\alpha_s(x_{20}^2)}-1\right)
\Bigg],$$ where $\alpha_s (x_\perp^2) = {\alpha_s}\left( 4 \, e^{-\frac{5}{3} - 2 \,
\gamma_E}/x_\perp^2 \right)$.
can be used to describe the DIS diffraction data with large center-of-mass energy squared $s$ and large $M_X^2$, such that $s \gg M_X^2 \gg Q^2$. Unfortunately current HERA data does not extend to high enough values of $M_X^2$ to necessitate the use of : perhaps this equation would be useful to describe single diffraction at the future DIS machines such as the proposed EIC and LHeC colliders.
The NLO Odderon Intercept
=========================
The progress in the calculation of the next-to-leading order (NLO) intercept of the QCD odderon was presented in the talk by Jochen Bartels at the Diffraction 2012 Workshop (reporting on work being done in collaboration with Victor Fadin and Lev Lipatov). The calculation employs standard Feynman diagram approach. Here we would like to document a prediction for the NLO odderon intercept made by the author of these proceedings in the discussion following the talk: the NLO odderon intercept can be straightforwardly obtained using the $s$-channel time-ordered formalism usually employed in saturation physics.
The odderon exchange amplitude in DIS is [@Hatta:2005as] $$\label{odd1}
O_{\bf x \, \bf y} = \frac{1}{2i} \,
\frac{1}{N_c} \left\langle \mathrm{Tr} \, \left[ V_{\bf x}
\, V^\dagger_{\bf y} \right] - \mathrm{Tr} \, \left[ V_{\bf y}
\, V^\dagger_{\bf x} \right] \right\rangle = \frac{1}{2i} \, \left[ N_{{\bf y}, {\bf x}} - N_{{\bf x}, {\bf y}} \right]$$ where $V_{\bf x}$ is a Wilson line along the light-cone of the projectile dipole located at transverse coordinate $\bf x$ and $N_{{\bf x}, {\bf y}} = 1 - \left\langle\mathrm{Tr} \, \left[ V_{\bf
x} \, V^\dagger_{\bf y} \right]/N_c\right\rangle$ is the dipole–nucleus forward scattering amplitude.
To construct the NLO evolution equation for $O_{\bf x \, \bf y} $ we begin with the linearized NLO BK evolution for $N_{{\bf x}, {\bf y}}$ derived in [@Balitsky:2008zza] (that is, the NLO Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [@Bal-Lip; @Kuraev:1977fs] in transverse coordinate space), which can be written as (see Eq. (103) in [@Balitsky:2008zza]) $$\label{Nevol}
{\partial}_Y N_{{\bf x}, {\bf y}} = \int d^2 z \ K_1 ({\bf x}, {\bf y}; {\bf z}) \, \left[ N_{{\bf x}, {\bf z}} + N_{{\bf z}, {\bf y}} - N_{{\bf x}, {\bf y}} \right] + \int d^2 z \, d^2 z' \, K_2 ({\bf x}, {\bf y}; {\bf z}, {\bf z}') \, N_{{\bf z}, {\bf z}'}.$$ The kernels $K_1$ and $K_2$ can be found in Eq. (103) of [@Balitsky:2008zza]. One can explicitly verify that $$K_1 ({\bf x}, {\bf y}; {\bf z}) = K_1 ({\bf y}, {\bf x}; {\bf z}) \ \ \mbox{and} \ \ K_2 ({\bf x}, {\bf y}; {\bf z}, {\bf z}') = K_2 ({\bf y}, {\bf x}; {\bf z}', {\bf z}).$$ Using these properties of the kernels, along with Eqs. and , we derive the NLO evolution equation for the odderon amplitude, which turns out to be equivalent to : $$\label{Oevol}
{\partial}_Y O_{{\bf x}, {\bf y}} = \int d^2 z \ K_1 ({\bf x}, {\bf y}; {\bf z}) \, \left[ O_{{\bf x}, {\bf z}} + O_{{\bf z}, {\bf y}} - O_{{\bf x}, {\bf y}} \right] + \int d^2 z \, d^2 z' \, K_2 ({\bf x}, {\bf y}; {\bf z}, {\bf z}') \, O_{{\bf z}, {\bf z}'}.$$ We see that the situation closely replicates that for the odderon evolution in transverse coordinate space at the leading order (LO) [@Kovchegov:2003dm]: there the odderon evolution equation was also identical to the (LO) BFKL equation, with the eigenfunctions of the odderon evolution operator being $C$-odd, that is, they had to flip sign under the ${\bf x} \leftrightarrow {\bf y}$ interchange. The same observation applies here at NLO: since the odderon amplitude $O$ obeys the same NLO BFKL evolution equation, the odderon intercept is the same as the NLO BFKL intercept, with only odd values of the azimuthal index $n$ contributing. (Note that, strictly speaking, to obtain which is equivalent to NLO BFKL one has to redefine the dipole amplitude by an order-${\alpha_s}$ correction with the corresponding new operator referred to as the composite dipole in [@Balitsky:2009xg]: using this operator in place of $N$ would not change the derivation above.)
The NLO BFKL intercept for non-zero $n$ can be found in e.g. [@Balitsky:2012bs] (see Eq. (65) there). Evaluating it for $n=1$ at the saddle point $\gamma = \frac{1}{2} + i \, \nu
=\frac{1}{2}$ we obtain zero[^1], such that the NLO odderon intercept in QCD is $$\label{Oint}
\alpha_O - 1 = 0 + O(\alpha^3_s).$$ We conclude that the zero value of the odderon intercept, originally found at the leading-order in [@Bartels:1999yt], persists at the NLO.[^2]
It is interesting to note that, as was pointed out by Lipatov in the discussion at the Workshop, the odderon intercept at strong ’t Hooft coupling $\lambda = g^2 \, N_c$ was found in [@Brower:2008cy] using the Anti-de Sitter space/Conformal Field Theory (AdS/CFT) correspondence. The authors of [@Brower:2008cy] found two odderon solutions, both of which give $\alpha_O -1 \rightarrow 0$ for $\lambda
\rightarrow \infty$, with one of the solutions exhibiting no deviation from zero when finite-$\lambda$ corrections were calculated, such that $\alpha_O -1 =0 + O(1/\lambda)$. This result, along with our , presents evidence in favor of a tantalizing possibility that the odderon intercept is identically equal to zero at all values of the coupling! (The same reference [@Brower:2008cy] states that the result had been known earlier to Cyrille Marquet, who, as it turns out, arrived at it numerically using a similar line of arguments to the one presented here.)
The author is greatly indebted to Ian Balitsky and Giovanni Chirilli for discussions of the NLO BK evolution. This work is sponsored in part by the U.S. Department of Energy under Grant No. DE-SC0004286.
[17]{} natexlab\#1[\#1]{}\[1\][“\#1”]{} url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{}
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[^1]: The fact that the NLO BFKL intercept is zero for $n=1$ and $\gamma =1/2$ had been originally observed by Agustin Sabio Vera during the discussion of the odderon evolution at the Diffraction 2012 Workshop (see also Eq. (38) in [@Vera:2006un]), and was since confirmed by the author.
[^2]: Note that Mikhail Braun showed that running-coupling corrections do not change the LO odderon intercept, leaving it at zero [@Braun:2007kz]: our result is consistent with this conclusion.
|
---
abstract: |
We define an invariant of transverse links in $(S^3,
\xi_{std})$ as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives a bound on the self-linking number in terms of Rasmussen’s invariant $s(K)$. We prove that our invariant vanishes for transverse knot stabilizations, and that it is non-zero for quasipositive braids. We also discuss a connection to Heegaard Floer invariants.
address: 'Department of Mathematics, M.I.T., Cambridge, MA 02139'
author:
- Olga Plamenevskaya
title: Transverse knots and Khovanov homology
---
Introduction
============
Legendrian and Transverse knots
-------------------------------
There are two important classes of knots in a contact 3-sphere $(S^3, \xi_{std})$: Legendrian knots and transverse knots. Legendrian knots are everywhere tangent to the contact planes; transverse knots are everywhere transverse to them. There are simple “classical” invariants for both classes: the Thurston–Bennequin and the rotation number for Legendrian knots, and the self-linking number for transverse knots. While certain knot types, e.g. all torus knots [@EH1], are completely classified by these invariants (in this case the knot type is said to be [*Legendrian resp. transversely simple*]{}), for most knot types the classification is not known. There exist smoothly isotopic Legendrian resp. transverse knots with the same classical invariants which are not isotopic through Legendrian resp. transverse knots. Legendrian knots are somewhat better understood and enjoy a rich theory in the context of symplectic field theory [@ElH]: a certain differential graded algebra associated to a knot yields new Legendrian knot invariants. Transverse non-simplicity of certain knot types was demonstrated by Birman and Menasco in [@BM] and by Etnyre and Honda in [@EH3]. However, the existing examples are sparse (those of [@EH3] are not even explicit), and the proofs in [@BM] and [@EH3] require a subtle analysis of braids and contact manifolds. Unlike Legendrian knots, transverse knots do not have any known efficient non-classical invariants.
The invariant $\psi(L)$
-----------------------
In this paper we introduce a transverse link invariant $\psi(L)$ as a distinguished element of the Khovanov homology of $L$. Given a closed braid diagram representing the transverse link, $\psi(L)$ is defined via a certain resolution of $L$. This invariant encodes the self-linking number: the quantum degree of $\psi(L)$ is given by $sl(L)$. It also discerns transverse stabilizations: if $L$ arises as a transverse stabilization of another transverse link, $\psi(L)$ vanishes. On the other hand, $\psi(L)\neq 0$ for quasipositive braids. While we don’t have any examples of transverse knots distinguished by $\psi(L)$ but not $sl(L)$ (indeed, we show that the invariant is the same for the pairs of transversely non-isotopic knots from [@BM]), we hope that a connection to Khovanov homology might be helpful. In particular, we establish a bound on the self-linking number of a knot in terms of the Khovanov homology knot invariant of Rasmussen [@Ra].
Khovanov homology and low-dimensional topology
----------------------------------------------
Khovanov homology is an invariant of knots and links introduced in [@Kho]. Given a link $L$ in $S^3$, $Kh(L)$ is a graded homology module whose graded Euler characteristic is the unnormalized Jones polynomial of $L$. As was recently discovered, the Khovanov homology has an interesting relation to low-dimensional topology. Ozsváth and Szabó [@OS] construct a spectral sequence converging to the Heegaard Floer homology $\widehat{HF}(Y)$ of the double cover $Y$ of $S^3$ branched over a link $L$; the $E^2$ term of this spectral sequence is the (reduced) Khovanov homology of the link $L$. Rasmussen [@Ra] uses Khovanov homology to give a combinatorial proof of the Milnor conjecture (i.e., to determine the slice genus of a torus knot). Our transverse link invariant suggests a further connection to contact topology. In fact, we can define a similar invariant in the reduced Khovanov theory; we conjecture that this invariant is mapped to the Ozsváth-Szabó contact invariant of the double cover of $(S^3,\xi_{std})$ branched over the transverse link under the spectral sequence of [@OS].
Acknowledgements
----------------
I would like to thank Jake Rasmussen for extremely helpful email correspondence, and Peter Kronheimer and Ciprian Manolescu for illuminating conversations and encouragement.
Preliminaries on transverse knots
=================================
It will be convenient to work with closed braid representations of transverse knots. Consider $S^3$ equipped with the (rotationally symmetric) standard contact structure $\xi_{std}=\ker(dz-ydx+xdy)$. It easy to see that any closed braid around $z$-axis can be made transverse to the contact planes. Moreover, by a theorem of Bennequin [@Be] any transverse link in $(S^3, \xi_{std})$ is transversely isotopic to a closed braid.
We adopt the usual notation for braid words. The braid group on $b$ strings is generated by $\sigma_1, \dots, \sigma_{b-1}$, so that $\sigma_i$ permutes the $i$-th and the $(i+1)$-th strings. We will sometimes write a braid as a braid word, a certain product of the generators $\sigma_1, \dots, \sigma_{b-1}$ and their inverses. The positive resp. negative stabilization of a braid on $b$ strings is formed by adding the $(b+1)$-th string and multiplying the braid word by $\sigma_b$ resp. $\sigma_b^{-1}$.
Of course, the same link can be represented by different braids. The Markov theorem [@Bi] asserts that two braid words describing the same link are related by a sequence of stabilizations, destabilizations and conjugations in the braid group (and, of course, the braid group identities). The Transverse Markov Theorem describes the relation between two braid representations of the same transverse links.
[@W; @OrSh] \[markov\] Let $L_1$, $L_2$ be two closed braids which represent transversely isotopic links. Then $L_2$ can be obtained from $L_1$ by a sequence of positive braid stabilizations and braid isotopies.
The self-linking number $sl(L)$ is defined as follows. Fix a Seifert surface $\Sigma$ for the link $L$, and let $v$ be a non-zero vector field in $T\Sigma\cap \xi$ along $L$ pointing out of $\Sigma$. Then, $sl(L)$ is the obsruction to extending $v$ over $\Sigma$. Given a closed braid representing $L$, it can be computed as $$\label{sl-braid}
sl(L)=-b+n_+-n_-,$$ where $b$ is the braid index, and $n_+$ and $n_-$ denote the number of positive resp. negative crossings.
The [*stabilization*]{} of a transverse link can be thought of as negative braid stabilization. Unlike positive braid stabilization, this operation changes the transverse type of the link: if $L'$ is the result of stabilization of $L$, then $$sl(L')=sl(L)-2.$$
Khovanov Homology
=================
In this section we give a brief review of Khovanov homology (more or less following the review in [@Ra]). Unless otherwise specified, We work with coefficients in ${\mathbb{Z}}$.
Khovanov complex
----------------
Given a link diagram $L$, we can resolve its crossings so that the result is just the union of planar circles. Each crossing can be resolved in two ways, called the 0-resolution and the 1-resolution and shown in Fig.\[0-1-pos-neg\]. Let $n$ be the number of crossings of $L$; we will write $n=n_-+n_+$, where $n_+$ ($n_-$) is the number of positive (negative) crossings. (See Fig. \[0-1-pos-neg\] for the usual sign conventions.) Then, complete resolutions of $L$ can be conveniently labelled by vertices of the “cube of resolutions” $[0,1]^n$.
![Resolutions and signs of crossings.[]{data-label="0-1-pos-neg"}](0-1-pos-neg.eps)
The underlying graded module for the Khovanov complex $CKh(L)$ is the direct sum of ${\mathbb{Z}}$-modules associated to the vertices of $[0,1]^n$, $$CKh(L)=\oplus_{v\in\{0,1 \}^n }CKh(L_v).$$ Each $CKh(L_v)$ is defined as follows. Let $U$ be the free graded ${\mathbb{Z}}$-module generated by two elements, ${{\bf u}_-}$ and ${{\bf u}_+}$; the grading $p$ is given by $p({\bf u}_\pm)=\pm 1$. Suppose that the resolution $L_v$ consists of $k$ circles. We then set $$CKh(L_v)=U^{\otimes k}.$$ In other words, $CKh(L_v)$ is freely generated by $k$-tuples obtained by labelling each circle in $L_v$ by either ${{\bf u}_-}$ or ${{\bf u}_+}$.
The module $CKh(L)$ is bi-graded. The [*homological*]{} grading, which is constant on each $CKh(L_v)$, is given by ${{\operatorname}{gr}}(v)=|v|-n_-$, where $|v|$ is the number of 1’s in the coordinates ov $v$. Besides, there is the [*quantum*]{} grading $q({{\bf u}})=p({{\bf u}})+{{\operatorname}{gr}}({{\bf u}})+n_+-n_-$.
Our next job is to describe the differential $d$ on $CKh(L)$. Loosely, $d$ is the sum of maps $d_e$ associated to the edges of $[0,1]^n$.
Let $e$ be an edge of $[0,1]^n$, and denote by $v_e(0)$ resp. $v_e(1)$ its initial resp. terminal end. The resolutions $L_{v_e(0)}$ and $L_{v_e(1)}$ differ in one crossing only (which is 0-resolved for $v_e(0)$ and 1-resolved for $v_e(1)$), and $L_{v_e(1)}$ is obtained from $L_{v_e(0)}$ in one of two ways: either two circles of $L_{v_e(0)}$ merge into one, or one circle splits into two. In the first case, the map $d_e:CKh(L_{v_e(0)}
)\to CKh(L_{v_e(1)})$ is given by multiplication $m:U\otimes U\to
U$, where the two factors of $U\otimes U$ correspond to two circles that merge, and the copy of $U$ in the image corresponds to the resulting circle in $CKh(L_{v_e(1)})$. In the second case, $d_e$ comes from the comultiplication $\Delta:U\to U\otimes U$, where the $U$ in the domain corresponds to the circle that splits. It remains to define the maps $m$ and $\Delta$: $$\begin{array}{ll}
m({{\bf u}_+}\otimes{{\bf u}_+})={{\bf u}_+}& \Delta({{\bf u}_+})={{\bf u}_+}\otimes{{\bf u}_-}+{{\bf u}_-}\otimes{{\bf u}_+}\\
m({{\bf u}_+}\otimes{{\bf u}_-})\otimes m({{\bf u}_-}\otimes{{\bf u}_+})={{\bf u}_-}& \Delta({{\bf u}_-})={{\bf u}_-}\otimes{{\bf u}_-}. \\
m({{\bf u}_-}\otimes{{\bf u}_-})=0
\end{array}$$
Now, on the component $CKh(L_v)$ the differential $d$ is defined by $$d=\sum_{e:v_e(0)=v}(-1)^{s(e)}d_e,$$ where the sum is taken over all edges which have $v$ as their initial end. The signs $(-1)^{s(e)}$ are chosen so that $d^2=0$ (the choice is not unique, but all the resulting chain complexes are isomorphic).
Khovanov [@Kho] shows that different diagrams for the same knot yield quasi-isomorphic chain complexes, so that the isomorphism classes of the (bigraded) homology groups give an invariant of the link. Actually, more is true: as conjectured in [@Kho] and proved in [@Ja], Khovanov’s theory is functorial, and it follows that there are honest homology groups, not just isomorphism classes, associated to a link. We now turn attention to these the functorial properties.
Cobordisms and Invariance {#Kho-co}
-------------------------
Given two links and an oriented cobordism between them, there is an induced map between homology groups of the links. We briefly describe this construction.
An oriented cobordism between two links $L^0$ and $L^1$ is given by an embedded smooth oriented compact surface $S$ in ${\mathbb{R}}^3\times
[0,1]$, such that ${\partial}S=S\cap {\partial}({\mathbb{R}}^3\times[0,1])$, and $S\cap
({\mathbb{R}}^3\times{i})=L^i$ for $i=0,1$. We may assume that $L^t= S\cap
({\mathbb{R}}^3\times{t})$ is a link for all but finitely many values of $t$. When $t$ passes through the critical value, the isotopy type of the link changes by a [*Morse move*]{}, and the surface $S^t=
(S\cap{\mathbb{R}}^3\times[0,t])$ changes by an attachment of a handle (of index 0, 1, or 2). Further, we can fix a projection ${\mathbb{R}}^3 \to {\mathbb{R}}^2$, and assume that it gives a regular projection for $L^t$ for all but finitely many special values of $t$ (and that the set of these special values is disjoint from the set of the Morse critical values). Thus, we obtain link diagrams (still denoted $L^t$). When $t$ passes through a special value where the projection of the link is not regular, the link remains the same, but its diagram changes by a Reidemeister move. The isomorphism class of the surface $S^t$ remains unchanged.
Therefore, the cobordism $S$ can be represented as a sequence of elementary cobordisms, $$S= S_1\cup S_2\cup\dots \cup S_k,$$ where each cobordism $S_i$ between two diagrams $L^{t_i}$ and $L^{t_{i+1}}$ corresponds to either a Reidemester move or a handle attachment. Now, each $S_i$ induces a map $f_{S_i}:Kh(L^{t_i})\to Kh(L^{t_{i+1}})$. For Reidemeister moves, $f_{S_i}$ comes from the quasi-isomorphisms between chain complexes $CKh(L^{t_i})$ and $CKh(L^{t_{i+1}})$ mentioned in the previous section (we describe these quasi-isomorphisms in a little more detail in Section \[def-psi\]). For Morse moves, $f_{S_i}$ is defined as follows [@Kho]. We need two additional maps, $\iota:{\mathbb{Z}}\to U$ and $\epsilon:U\to {\mathbb{Z}}$, defined by $$\begin{array}{ll}
\epsilon({{\bf u}_-})=1 & \iota(1)={{\bf u}_+}\\
\epsilon({{\bf u}_+})=0
\end{array}$$ Now, the attachment of a $0$-handle corresponds ro a “birth” of a circle in the diagram, and the map on the chain complex is given by $\iota$ (for all possible resolutions). Similarly, the attachment of a $2$-handle (the “death” of a circle) gives the map given by $\epsilon$. The attachment of 1-handle is given by $m$ or $\Delta$ on each component of the chain complex, depending on whether the 1-handle merges two circles of a particular resolution or splits one circle into two. (Note that the differential in Khovanov’s theory is defined in a similar way: two resolutions given by adjacent vertices of $[0,1]^n$ differ precisely by the attachment of a 1-handle.) Finally, the map $f_S$ is defined as the composition of the maps induced by the elementary cobordisms, $$f_S=f_{S_k}\circ\dots\circ f_{S_2}\circ f_{S_1}.$$
Jacobsson [@Ja] proves that up to a sign, the map $f_S$ depends on the isotopy class of $S\ rel\ {\partial}S$ only, that is, if $$S= S_1\cup S_2\cup\dots \cup S_k \text{ and } S= S'_1\cup S'_2\cup\dots \cup S'_{k'}$$ are two decompositions of $S$ into elementary cobordisms, then $$f_{S_k}\circ\dots\circ f_{S_2}\circ f_{S_1}=\pm f_{S'_{k'}}\circ\dots\circ f_{S'_2}\circ f_{S'_1}.$$ In particular, if two diagrams of a link are related by a sequence of Reidemeister moves, then the induced isomorphism between the homology groups is canonical up to a sign.
Definition of the Invariant {#def-psi}
===========================
In this section we define the transverse link invariant $\psi(L)\in Kh(L)$.
First, we fix a braid diagram $L$ for our link, and pick a distinguished element ${{\tilde}{\psi}}(L)$ in the chain complex $CKh(L)$. We will check that ${{\tilde}{\psi}}(L)$ is a cycle, so that it defines an element $\psi(L)$ of the homology group $Kh(L)$. Finally, we show that $\psi(L)$ does not depend on the choice of the braid diagram and remains the same under transverse link isotopies. This means that $\psi(L)$ is indeed an invariant of the transverse link.
Given a braid diagram $L$ for our link, we choose a resolution which is given by $b$ parallel strings: that is, we take the 0-resolution for each positive crossing and the 1-resolution for each negative crossing of $L$. Note that this is the [*oriented resolution*]{} of the diagram; we denote it by $L_o$.
We set $$\label{defn}
{{\tilde}{\psi}}(L)={{\bf u}_-}\otimes {{\bf u}_-}\otimes\dots \otimes {{\bf u}_-}\in U^{\otimes b}= CKh(L_o).$$
\[cycle\] The element ${{\tilde}{\psi}}(L)$ is a cycle in $(CKh(L), d)$.
The differential $d$ on $CKh(L_v)$ is the sum of maps for all edges $e$ which have $v$ as their initial end. By our choice of the resolution $L_v$, such edges correspond to positive crossings. Moreover, when a 0-resolution of a positive crossing is changed into a 1-resolution, the two circles of $L_v$ which are “connected” by this crossing merge into one. This means that each map $d_e$ is given by multiplication, and then $$d_e({{\tilde}{\psi}}(L))=m({{\bf u}_-}\otimes{{\bf u}_-})=0.$$ Taking the sum over all positive crossings, we see that $d({{\tilde}{\psi}}({\mathcal{D}}))=0$.
\[deg\] $\psi(L)\in Kh^{0,sl(L)}$.
By construction, $\psi(L)$ is a homogeneous element. The homological and quantum gradings are easy to compute: since the number of 1’s for the chosen resolution is exactly the number of negative crossings, ${{\operatorname}{gr}}(\psi(L))=0$. Now, $p(\psi(L))$ is the braid index of $L$, and the formula $q(\psi(L))=sl(L)$ is an immediate consequence of (\[sl-braid\]) and the definition of the quantum grading.
Now we want to check that $\psi(L)$ is independent of a particular braid representation of the transverse knot. The Transverse Markov Theorem says that the braids representing two transversely isotopic knots are related by a sequence of positive stabilizations and braid isotopies. The two braid words will then be related positive stabilizations, conjugations and the braid group identities. For our braid diagrams, this yields a sequence of “transverse Reidemeister moves”, as follows. Positive stabilization gives the move (R1) with a positive crossing introduced (the other version of (R1) is not allowed). The braid isotopies give the (R2) and (R3) moves, all versions of which are allowed. In the following Lemma, we check that the moves (R1)-(R3) respect $\psi(L)$ by analyzing the effect of each move on Khovanov’s homology. If we were dealing with knots, it would suffice to consider only the versions of (R1)-(R3) shown in Fig. \[Rmoves\], since all the other versions can be obtained by a combination of these three. With braids, we need to be more careful: there is another version of (R2) obtained by turning our picture upside down; since we cannot turn braids upside down, we actually need to consider both versions of (R2). However, the two proofs are identical, so we only give one of them. Also, it is not hard to check that all possible versions of the (R3) move can be reduced to the one shown by a combination of (R2) moves.
\[ht\] ![Reidemeister moves in the transverse braid setting.[]{data-label="Rmoves"}](Rmoves.eps "fig:")
\[R-invar\] Let $L$ and $L'$ be two braid diagrams related by one of the three transverse Reidemeister moves (R1), (R2), (R3), and denote by $\rho_i:CKh(L)\to CKh(L')$, $i=1,2,3$, the associated quasi-isomorphisms between the two chain complexes. Then $$\rho_i({{\tilde}{\psi}}(L))=\pm {{\tilde}{\psi}}(L').$$
We recall how the quasi-isomorphisms $\rho_i$ are constructed in [@Kho], and see what happens to the distinguished element ${{\tilde}{\psi}}(L)$.
(R1) move: The complex $CKh(L')$ decomposes as a direct sum $X_1\otimes X_2$, where the $X_2$ is acyclic, and $X_1$ is isomorphic to $CKh(L)$ (Fig. \[R1-pf\]). The isomorphism $\rho_1$ is given by $$\begin{array}{ll}
\rho_1({{\bf u}_-})={{\bf u}_-}\otimes{{\bf u}_-}\\
\rho_1({{\bf u}_+})={{\bf u}_+}\otimes{{\bf u}_-}-{{\bf u}_-}\otimes{{\bf u}_+},
\end{array}$$ and we see that ${{\tilde}{\psi}}(L)$ is mapped to ${{\tilde}{\psi}}(L')$.
![Construction of the quasi-isomorphism $\rho_1$.[]{data-label="R1-pf"}](R1-pf.eps)
(R2) move: The four possible resolutions of the two extra crossings of $L'$ are shown in Fig. \[R2-pf\]. Again , the complex $CKh(L')$ decomposes as $X_1\otimes X_2\otimes X_3$, where $X_2$ and $X_3$ are both acyclic, and $X_1$ isomorphic to $CKh(L)$ via the isomorphism $\rho_2:CKh(L)\to CKh(L')$ given by $$\rho_2(x)=(-1)^{{{\operatorname}{gr}}(x)}(x+\iota(d_e(x))).$$ (The map $\iota$ defined in Section \[Kho-co\] and the map ${\partial}_e$ corresponding to the edge $e$ are shown in the Figure, and the oriented resolution of $L$ is naturally identified with the oriented resolution of $L'$, so that $x$ on the right-hand side actually lives in $CKh(L')$).
We see that $\rho_2$ maps ${{\tilde}{\psi}}(L)$ to $\pm({{\tilde}{\psi}}(L')+\iota(d_e({{\tilde}{\psi}}(L')))$. In the proof of Proposition \[cycle\], we’ve checked that $d_e({{\tilde}{\psi}}(L'))=0$. It follows that up to a sign, ${{\tilde}{\psi}}(L)$ is mapped to ${{\tilde}{\psi}}(L')$.
![Construction of the quasi-isomorphism $\rho_2$.[]{data-label="R2-pf"}](R2-pf.eps)
(R3) move: Now we have decompositions $CKh(L)= X_1\otimes
X_2\otimes X_3$ and $CKh(L')= X'_1\otimes X'_2\otimes X'_3$ where $X_2$, $X_3$, $X'_2$, $X'_3$ are all acyclic, and there is an isomorphism $\rho_3:X_1\to X'_1$.
We briefly describe how $X_1$ is formed. First, pick 1-resolutions of crossings ${\tt{r}}$ and ${\tt{r}}'$ (note that the resulting diagrams are isomorphic). With this fixed, consider all possible resolutions of other crossings, and denote the direct sum of the associated components of $CKh(L)$ resp. $CKh(L')$ by $CKh(L_{*1})$ resp. $CKh(L'_{*1})$. (This is not a subcomplex.)
![Construction of complexes $X_1$ and $X'_1$.[]{data-label="R3-pf"}](R3-pf.eps)
Next, denote by $CKh(L_{*100})$ the part of $CKh(L)$ arising from all complete resolutions of $L$ with a 1-resolution at ${\tt{p}}$ and 0-resolutions at both ${\tt{q}}$ and ${\tt{r}}$; form $CKh(L_{*010})$, $CKh(L'_{*100})$ and $CKh(L'_{*010})$ by analogy. Now, define
$$\begin{array}{l}
X_1=\{x+\beta(x)+y | x\in CKh(L_{*100}), y \in CKh(L_{*1})\}\\
X'_1=\{x+\beta(x)+y | x\in CKh(L'_{*010}), y \in CKh(L'_{*1})\},
\end{array}$$
where $\beta:CKh(L_{*100})\to CKh(L_{*010})$ and $\beta:CKh(L'_{*010})\to CKh(L'_{*100})$ are certain chain maps. The isomorphism $\rho_3:X_1\to X'_1$ is given by $$\rho_3(x+\beta(x)+y)=x+\beta'(x)+y,$$ where the natural identifications between $CKh(L_{*100})$ and $CKh(L'_{*010})$ etc. are used. We do not describe the maps $\beta$ and $\beta'$, referring the reader to [@Kho]: the only thing we need to know is that ${{\tilde}{\psi}}(L)\in CKh(L_{*1})$ and ${{\tilde}{\psi}}(L')\in
CKh(L'_{*1})$, so $\rho_3({{\tilde}{\psi}}(L))={{\tilde}{\psi}}(L')$.
We’ve checked that the distinguished element $\psi(L)\in Kh(L)$ behaves nicely under the three transverse Reidemeister moves, and we know that any two transversely isotopic knots are related by a sequence of such moves, but why would an arbitrary transverse isotopy between $L$ and $L'$ send $\psi(L)$ to $\psi(L')$? We have to give the Transverse Markov Theorem another look: in [@W] it is actually shown that an arbitrary transverse isotopy $S$ can be smoothly modified into a composition of braid isotopies and positive stabilizations while the two links are fixed. Then, up to an isotopy of $S\ rel\ {\partial}S$, the cobordism $S$ between $L$ and $L'$ decomposes as $S_1\cup\dots\cup S_k$, and Jacobsson’s theorem from Section \[Kho-co\] implies that $$f_S(\psi(L))=f_{S_k}\circ\dots\circ f_{S_2}\circ
f_{S_1}(\psi(L))=\pm\psi(L').$$ We have proved the following
The element $\psi(L)\in Kh(L)$ is an invariant of the transverse link $L\in (S^3,\xi_{std})$, well-defined up to a sign.
Properties of $\psi(L)$ {#properties}
=======================
Transverse stabilization
------------------------
\[trst\] If $L$ is the transverse stabilization of another transverse link, then $\psi(L)=0$.
![Transverse stabilization and Khovanov’s complex.[]{data-label="tra-sta"}](tra-sta.eps)
We construct an element ${\tilde}{\phi}\in CKh(L)$ such that $d {\tilde}{\phi}={{\tilde}{\psi}}(L)$. Since $L$ is the result of a transverse stabilization (that is, an addition to the braid of an extra string and an extra negative crossing ${\tt{x}}$), it has a diagram with a “negative kink” as shown on Fig. \[tra-sta\]. For the oriented resolution $L_o$, we take the 1-resolution of the crossing ${\tt{x}}$. Let $e$ be the edge of the cube of resolutions corresponding to ${\tt{x}}$; then $o$ is the terminal end of $e$. We denote by $v$ the initial end of $e$. In other words, we take the 0-resolution of ${\tt{x}}$ to form $L_v$, and all the other crossings are resolved as in $L_o$. Now, label all the components of $L_v$ by ${{\bf u}_-}$, and set $${\tilde}{\phi}={{\bf u}_-}\otimes\dots{{\bf u}_-}\in CKh(L_v).$$ We compute $d {\tilde}{\phi}$ as follows. The component $d_e:
CKh(L_v)\to CKh(L_o)$ of $d$ is given by comultiplication $\Delta$, since the change of the resolution for ${\tt{x}}$ splits a circle into two. Thus, $d_e {\tilde}{\phi}={{\tilde}{\psi}}$. Furthermore, similar to proof of Proposition \[cycle\], all the other terms of $d$ on the component $CKh(L_v)$ correspond to positive crossings and are given by multiplication maps, which send ${\tilde}{\phi}$ to $0$. It follows that $d {\tilde}{\phi}={{\tilde}{\psi}}$, as required.
Positive crossing resolution
----------------------------
\[po-cro\] Suppose that the transverse braid $L^2$ is obtained from the transverse braid $L^1$ by resolving a positive crossing (note that it has to be a 0-resolution). Let $S$ be the resolution cobordism, and $f_S:Kh(L^1)\to Kh(L^2)$ the associated map on homology. Then $$f_S(\psi(L^1))=\pm\psi(L^2).$$
The cobordism $S$ is a composition of a 1-handle attachment and a Reidemeister move (R1), as shown on Fig. \[pos-res\]. On the component $CKh(L^1_o)$ of the Khovanov’s complex for $L^1$, the 1-handle attachment induces a map given by comultiplication $\Delta$, since the handle splits a circle on the oriented resolution. The map induced by the (R1) move was analyzed in the proof of Lemma \[R-invar\]. It follows that the element ${{\tilde}{\psi}}(L^1)$, given by the ${{\bf u}_-}$ labels of all circles for $L^1_o$, is mapped to ${{\tilde}{\psi}}(L^2)$ (given by the ${{\bf u}_-}$ labels on circles for $L^2_o$).
![Resolving a positive crossing.[]{data-label="pos-res"}](pos-res.eps)
Recall that a braid is called [*quasipositive*]{} [@Ru] if its braid word is a product of conjugates of the form $w \sigma_i w^{-1}$, where $w$ is an arbitrary element of the braid group.
\[qua\] If $L$ is represented by a quasipositive braid, then $\psi(L)\neq 0$. Moreover, it is a primitive non-torsion element of $Kh(L)$.
Resolving a few positive crossings, we convert the braid representing $L$ into a braid equivalent to a trivial one (of the same braid index). For the trivial braid ${\mathcal{O}}$, there are no differentials in the chain complex, and $\psi({\mathcal{O}})$ is a generator of $Kh^{0,sl({\mathcal{O}})}={\mathbb{Z}}$. Since $\psi(L)$ is mapped to $\psi({\mathcal{O}})$, it must be non-torsion and primitive.
\[pos-br\] Let $L$ be a positive braid of braid index $b$ with $n$ crossings. Then, the homology of $L$ lies in non-negative homological degrees, with $Kh^{0, n-b}={\mathbb{Z}}$, $Kh^{0, n-b+2}={\mathbb{Z}}$ and no other homology in $Kh^{0,*}$. The element $\psi(L)$ is a generator of $Kh^{0, n-b}$.
\[qua-non\] A transverse link $L$ represented by a quasipositive braid, then $\psi(L)\neq 0$ cannot be obtained by a transverse stabilization of any other link.
Corollary \[qua-non\] follows easily from the fact that quasipositive braid maximizes the self-linking number in its transverse link type. More precisely, for an arbitrary transverse link $L$ the slice-Bennequin inequality [@Ru] gives $$\label{sl<g}
sl(L)\leq -\chi(\Sigma)$$ where $\Sigma\subset B^4$ is a surface with boundary ${\partial}\Sigma =L\in S^3={\partial}B^4$; for quasipositive braids (\[sl<g\]) becomes an equality. This bound was first proved by Rudolph by means of gauge theory. It is interesting to note that it can be obtained purely by Khovanov-homological methods. Indeed, as in [@Ru], it is straightforward to reduce the question to the case of a positive braid representing a torus knot $T_{p,q}$ (introducing positive crossings and keeping track of how both sides of (\[sl<g\]) change). Then, the self-linking number is easily seen to be $2g(T_{p,q})-1$ (where $g$ denotes genus), and $g_*(T_{p,q}=g(T_{p,q}))$. The last identity is the Milnor conjecture, whose Khovanov homology proof was recently obtained by Rasmussen [@Ra] (the original gauge-theoretic proof is due to Kronheimer and Mrowka).
Negative crossings
------------------
The following Proposition is useful for calculations and shows that the invariant $\psi$ vanishes for many transverse links.
\[neg=0\] Suppose that the transverse link $L$ is represented by a closed braid whose braid word contains a factor of $\sigma_i^{-1}$ but no $\sigma_i$’s for some $i=1, \dots, n$. (This means that all the crossings in the braid diagram on the level between $(i-1)$-th and $i$-th string are negative.) Then $\psi(L)=0$.
First of all, we delete all $\sigma_i^{-1}$ but one from the braid word, obtaining a link that decomposes as a connected sum of two links (connected by a negative crossing, the $\sigma_i^{-1}$ that remains).
![From $L$ to $L'$.[]{data-label="negcr"}](negcr.eps)
Then, we delete negative crossings from and insert positive crossings into both components of the connected sum, obtaining as a result two positive torus knots connected by a negative crossing. This is illustrated on Fig. \[negcr\]. Denote the obtained transverse link by $L'$. By Theorem \[po-cro\], it suffices to show that $\psi(L')=0$. Topologically, the link $L'$ is just the connected sum of two torus knots, but its self-linking number is not maximal (because we can connect the two components by a positive crossing to increase $sl$). Connected sums of torus knots are transversely simple [@EH2; @Et], so $L'$ is the transverse stabilization of another link. By Theorem \[trst\], $\psi(L')=0$.
Examples
========
For $q>0$, $|p|\geq q$ let $L$ be a transverse link of the $(p,q)$-torus link type.
\(1) Suppose $p>0$. If $L$ maximizes the self-linking number in its smooth link type, i.e. $sl(L)=pq-p-q$, then $\psi(L)$ is a generator of $Kh^{0,pq-p-q}={\mathbb{Z}}$. Otherwise $\psi(L)=0$.
\(2) If $p<0$, then $\psi(L)$ vanishes.
\(1) We use transverse simplicity of positive torus links [@EH1; @Et]. The (unique) transverse positive $(p, q)$-torus link with $sl(L)=pq-p-q$ is represented by a positive braid with $q$ strings and $p(q-1)$ crossings, so the result follows from Remark \[pos-br\]. If $sl(L)<pq-p-q$, then $L$ is obtained by transverse stabilization, so $\psi(L)=0$ by Theorem \[trst\].
\(2) Follows from Proposition \[neg=0\].
Let the transverse links $L^1$ and $L^2$ be given by the braids $L^1=\sigma_1^{2p+1}\sigma_2^{2q}\sigma_1^{2r}\sigma_2^{-1}$ and $L^2=\sigma_1^{2p+1}\sigma_2^{-1}\sigma_1^{2r}\sigma_2^{2q}$. It is shown in [@BM] that $L^1$ and $L^2$ are not transversely isotopic when $p,q,r>1$, $p+1\neq q\neq r$ (although they are smoothly isotopic, and $sl(L^1)=sl(L^2)$). However, we have $\psi(L^1)=\pm\psi(L^2)$. Indeed, both $\psi(L^1)$ and $\psi(L^2)$ are generators of $Kh^{0,sl(L^i)}={\mathbb{Z}}$.
Because of Proposition \[deg\] and Corollary \[qua\], we only need to check that $Kh^{0,sl}={\mathbb{Z}}$. The proof is straightforward: the braids $L^1$ and $L^2$ have only one negative crossing each, and for any link with this property, the relevant part of $CKh(L)$ is easy to understand. In the argument below, $L$ will mean $L^1$ or $L^2$. For the given diagram $CKh^{-1, sl}(L)$ has rank one and is generated by ${{\bf u}_-}\otimes {{\bf u}_-}$ in the complete resolution of $L$ given by $0$-resolution of all crossings (this resolution consists of two circles and is shown on Fig. \[0sl-pf\]).
![Computing $Kh^{0,sl}(L)$. Each component of complete resolutions shown must be labelled by ${{\bf u}_-}$.[]{data-label="0sl-pf"}](0sl-pf.eps)
Similarly, $CKh^{0,sl}(L)$ has rank $2q$, with generators ${\tilde}{\phi}_i$ formed by the ${{\bf u}_-}$ labels on the complete resolutions obtained as follows. The crossing that corresponds to the $i$-th $\sigma_2$ in the product $\sigma_2^{2q}$ is $1$-resolved, all other crossings are $0$-resolved. Further, $d({\tilde}{\phi}_i)$ in turn comes from $1$-resolutions of the $j$-th crossing in $\sigma_2^{2q}$ for all $j\neq i$. It follows that, apart from ${{\tilde}{\psi}}(L)$, the only cycle in $CKh^{0,sl}(L)$ is $\phi_1+\phi_2+\dots +\phi_{2q}$ (here we abuse our assumptions and pretend that the coefficients are taken in ${\mathbb{Z}}/2{\mathbb{Z}}$; to be honest, one needs to pick an appropriate choice of signs). The latter cycle is the boundary of the generator of $CKh^{-1,
sl}(L)$, which finishes the proof.
A bound on the self-linking number
==================================
In this section we obtain a bound on the self-linking number of a transverse knot $K$ in terms of the knot invariant $s(K)$ introduced by Rasmussen [@Ra]. As we mention below, $s(K)$ is defined as a certain quantum grading in Lee’s version of the Khovanov homology [@Lee]. (In Lee’s construction, which works for rational coefficients only, the generators for the complex $CKh'(K)$ and the gradings are the same as in $CKh(K)$, but the differential is different.) Rasmussen conjectures that this invariant coincides with the $s(K)$ invariant of Bar-Natan [@BN], and is twice the $\tau(K)$ invariant of Ozsváth and Szabó [@OStau]. Most importantly, $|s(K)|$ gives a lower bound of the slice genus. If $K$ is alternating, $s(K)$ is simply the signature of the knot.
\[bound\] For any transverse knot $K$ $$sl(K)\leq s(K)-1.$$
As Jake Rasmussen pointed out to the author, the invariant $s(K)$ in [@Ra] is defined as $s(K)=\max q({\tilde}{x})+1$, where ${\tilde}x$ is an element of $CKh'(K)$ homologous to the element that we have denoted ${{\tilde}{\psi}}$. (Note that Lee’s differential does not preserve the quantum grading.) Since $q({{\tilde}{\psi}})=sl(K)$ for a transverse knot $K$, it follows immediately that $sl(K)\leq s(K)-1$.
Proposition \[bound\] gives an improvement for the well-known Thurston–Bennequin [@Be] and slice–Bennequin [@Ru] bounds on $sl(K)$. The relation between Rasmussen’s invariant and the slice–Bennequin inequality was independently established by A. Shumakovich [@Sh].
If $K$ is alternating, $sl(K)\leq \sigma(K)-1$, where $\sigma(K)$ is the signature of the knot.
Since every bound for the self-linking number of transverse knots is automatically a bound for $tb(K)+|r(K)|$, the Thurston–Bennequin and rotation numbers of Legendrian knots (Legendrian and transverse knots are related by push-offs [@Et]), for an alternating Legendrian knot $K$ we have $$tb(K)+|r(K)|\leq \sigma -1.$$ This bound was obtained in [@Pl1] via Heegaard Floer homology techniques. Indeed, it is a special case of the inequality $$tb(K)+|r(K)|\leq 2\tau(K) -1,$$ where $\tau(K)$ is the Ozsváth–Szabó invariant [@OStau]. The latter bound, together with Proposition \[bound\], supports the conjecture about the identity $s(K)=2\tau$, and gives yet another connection between the Heegaard Floer theory and the Khovanov homology.
Reduced homology and a relation to Ozsváth–Szabó invariants
===========================================================
A transverse link invariant can also be defined in the reduced version of the Khovanov homology. We recall the construction of the reduced complex. Starting with a link $L$ with a marked point on it, consider the usual Khovanov complex $CKh(L)$. For each complete resolution of $L$, exactly one of the circles contains the marked point. Let $CKh_{{{\bf u}_-}}(L)$ be the subcomplex generated by those generators of $CKh(L)$ that have the label ${{\bf u}_-}$ on the marked circle. Then, the reduced chain complex is the factor $\widetilde{CKh}(L)= CKh(L)/CKh_{{{\bf u}_-}}(L)$, and $\widetilde{Kh}(L)$ is the corresponding homology. When the homology is taken with coefficients in ${\mathbb{Z}}/2{\mathbb{Z}}$, $\widetilde{Kh}(L)$ is independent of the choice of the marked point.
To define the transverse link invariant $\psi(L)$ in the reduced homology, we take the same resolution as before, and pick a label ${{\bf u}_-}$ for each unmarked circle and a ${{\bf u}_+}$ for the marked circle. (Invariantly, we can consider the element $${{\bf u}_+}\otimes{{\bf u}_-}\otimes \dots\otimes {{\bf u}_-}+
{{\bf u}_-}\otimes{{\bf u}_+}\otimes\dots\otimes{{\bf u}_-}+\dots+{{\bf u}_-}\otimes{{\bf u}_-}\otimes\dots\otimes{{\bf u}_+}$$ in the non-reduced chain complex, and then take its image in $\widetilde{CKh}(L)$.
The “reduced” invariant $\psi(L)\in\widetilde{Kh}(L)$ has the same properties as those we proved in the non-reduced case (the proofs would be identical).
In the Introduction, we mentioned the connection between $\widetilde{Kh}(L)$ and the Heegaard Floer homology of the 3-manifold $\Sigma(L)$, which is a double cover of $S^3$ branched over the link $L$ (here $L$ is a smooth link in $S^3$). More precisely, there exist a spectral sequence whose $E^2$ term is $\widetilde{Kh}(L)$, and $E^{\infty}$ term is the Heegaard Floer homology ${\widehat{HF}}(-\Sigma(L))$. When $L$ is an alternating link, this spectral sequence collapses at the $E^2$ stage, so that $\widetilde{Kh}(L)= {\widehat{HF}}(-\Sigma(L))$.
When $S^3$ is equipped with the standard contact structure and $L$ is a transverse link, the manifold $\Sigma(L)$ carries a natural contact structure $\xi_L$ lifted from $S^3$. In a related paper [@Pl2], we study the Ozsváth–Szábo contact invariant $c(\xi_L)\in {\widehat{HF}}(-\Sigma(L))$, associated to the contact structure. It turns out that the properties of $c(\xi_L)$ are very similar to those of $\psi(L)$; in particular, the results of Section \[properties\] hold true for $c(\xi_L)$. It is natural to conjecture that the element $c(\xi_L)$ corresponds to $\psi(L)$ under the spectral sequence. For the case of alternating knots, the conjecture is precise:
Let $L$ be an alternating transverse link. Then $\psi(L)$ is mapped to $c(\xi_L)$ under the isomorphism $\widetilde{Kh}(L)={\widehat{HF}}(-\Sigma(L))$.
In [@Pl2], we prove a special case of this conjecture.
Let $L$ be a transverse link represented by a closed braid whose diagram is alternating. Then $\psi(L)=c(\xi_L)$.
It should be said that the class of alternating braids is very narrow, and that $\psi(L)=c(\xi_L)=0$ for most of them.
[AO2]{}
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---
abstract: 'The electromagnetic Casimir interaction between two spheres is studied within the scattering approach using the plane-wave basis. It is demonstrated that the proximity force approximation (PFA) corresponds to the specular-reflection limit of Mie scattering. Using the leading-order semiclassical WKB approximation for the direct reflection term in the Debye expansion for the scattering amplitudes, we prove that PFA provides the correct leading-order divergence for arbitrary materials and temperatures in the sphere-sphere and the plane-sphere geometry. Our derivation implies that only a small section around the points of closest approach between the interacting spherical surfaces contributes in the PFA regime. The corresponding characteristic length scale is estimated from the width of the Gaussian integrand obtained within the saddle-point approximation. At low temperatures, the area relevant for the thermal corrections is much larger than the area contributing to the zero-temperature result.'
author:
- Benjamin Spreng
- Michael Hartmann
- Vinicius Henning
- 'Paulo A. Maia Neto'
- 'Gert-Ludwig Ingold'
title: 'Proximity force approximation and specular reflection: Application of the WKB limit of Mie scattering to the Casimir effect'
---
Introduction
============
The Casimir force between material surfaces [@Casimir1948] is a remarkable prediction of quantum electrodynamics [@Bordag2009]. Precise measurements of the Casimir force between metallic surfaces are now capable of distinguishing different models for the metallic conductivity [@Decca2007; @Decca2007EPJC; @Sushkov2011; @Chang2012], including the case of magnetic materials [@Banishev2013; @Bimonte2016]. In order to minimize systematic errors, all such recent experiments, as well as the majority of older ones [@Decca2011; @Lamoreaux2011], probe the Casimir attraction between a spherical and a planar surface, instead of employing the geometry with plane parallel plates. In addition, in view of recent experiments [@Ether2015; @Garrett2018] the sphere-sphere geometry has gained interest.
On the theoretical side, the scattering approach [@Lambrecht2006; @Emig2007; @Rahi2009] allows to compute the Casimir energy from the scattering matrices of the individual bodies interacting across a region of empty space. It also provides a clear physical picture of the Casimir effect as resulting from the reverberation or multiple scattering of vacuum or thermal electromagnetic field fluctuations between the interacting surfaces [@Jaekel1991; @Genet2003].
These theoretical and experimental advances were disconnected until very recently, when exact numerical results for typical experimental conditions were derived from the scattering approach [@Hartmann2017]. Instead of applying the recent theoretical developments, the surface curvature in real experiments is taken into account with the help of the proximity force (Derjaguin) approximation (PFA) [@Derjaguin1934], in which the result for the Casimir energy between parallel plates is averaged over the local distances corresponding to the geometry of interest. PFA is also often employed in surface science [@Butt2010], for instance in the comparison with experimental results for the van der Waals interaction between spherical colloids [@Borkovec2012; @Elzbieciak-Wodka2014].
The PFA approach to spherical curvature is conceptually different from the picture of electromagnetic field reverberation between spherical surfaces that results from the scattering approach. Nevertheless, we establish in this paper a direct connection between the two approaches. In the limit of a vanishing distance between the interacting surfaces, the dominant contribution to the multiple scattering between the surfaces is shown to result from semiclassical WKB specular reflection by a small section of the sphere’s surface around the point of closest approach. The final expression for the Casimir force then coincides with the PFA result for the leading-order divergence.
The connection between the scattering approach and PFA was first analyzed in a different context. The roughness correction to the Casimir energy for parallel planes was derived as a small perturbation of the ideal parallel-plates geometry [@MaiaNeto2005]. The PFA result for the roughness correction was then derived from the more general case in the limit of short distances and smooth surfaces by considering the value of the perturbative kernel at zero momentum [@MaiaNeto2005]. This line of reasoning can be generalized to the entire perturbative series [@Fosco2014] within the derivative expansion approach [@Fosco2011], although it is not possible to derive explicit results for all the corresponding kernels in this case. Nevertheless, the PFA result is obtained in leading order for surfaces that can be continuously deformed from the planar symmetry, as long as the kernel functions have a well defined limit at zero momentum [@Fosco2014]. Since the first condition does not hold for compact objects, this derivation does not constitute a proof of PFA for the plane-sphere nor for the sphere-sphere geometries, even though the leading-order correction to PFA has been successfully derived [@BimonteEPL2012; @BimonteAPL2012] whenever the perturbative kernel is analytical at zero momentum [@Mazzitelli2015].
For the plane-sphere setup, the validity of PFA in the short-distance limit was shown at zero temperature for both perfect [@Bordag2010; @Teo2011] and real metals [@Teo2013] by developing the scattering approach in the multipolar basis and taking suitable asymptotic approximations for the relevant spherical functions. Earlier results were derived for a scalar field model at zero temperature [@Bulgac2006; @Bordag2008]. In the opposite limit of high temperatures, a similar derivation was recently undertaken for perfect metals [@Bimonte2017]. Ref. [@Bimonte2012] derived an exact formula, also compatible with the PFA leading-order result, for Drude metals in the high-temperature limit by using bispherical multipoles.
Here, we consider the general case of arbitrary temperatures and materials. Our setup consists of two spheres with radii $R_1$ and $R_2$ in empty space, at a distance of closest approach $L,$ as indicated in Fig. \[fig:geometry\]. The center-to-center distance is ${\cal L}=L+R_1+R_2$ along the $z$-axis. The plane-sphere case is obtained at any step of our derivation by taking the radius of one sphere to infinity.
![Scattering geometry consisting of two spheres with radii $R_1$ and $R_2$ and a surface-to-surface distance $L$. The distance between the spheres’ centers is $\mathcal{L}=R_1+R_2+L$. The round trip discussed in the text is displayed between the two spheres.[]{data-label="fig:geometry"}](geometry.pdf){width="0.4\columnwidth"}
The sphere-sphere Casimir interaction has been analyzed within the scattering approach for large and moderate distances [@Emig2007; @Rodriguez-Lopez2011; @Umrath2016; @Ether2015]. In addition, for idealized boundary conditions, the leading-order correction to the PFA result has been obtained [@Teo2012]. In all cases, an expansion in terms of angular momentum multipolar waves has been employed, which is particularly natural for not too small distances. In this paper, we show that the linear momentum representation is better adapted to the short-distance PFA regime $L
\ll R_1, R_2$ because of its direct connection to the physical picture of specular reflection between the spherical surfaces.
Our derivation brings into light the main physical ingredients underlying the PFA regime. We show that only the direct reflection term in the Debye expansion [@Nussenzveig69] of the Mie scattering matrix contributes. Moreover, this contribution is taken in the semiclassical WKB approximation, which has a direct physical interpretation in terms of specular reflection by the sphere’s surface [@Nussenzveig92]. More importantly, the multiple scattering between the surfaces defines a scale for the variation of the momentum component parallel to the $x$-$y$ plane. This scale will allow us to specify the spherical cap on the spheres that actually contributes in the PFA regime.
Our semiclassical WKB derivation in the momentum representation should not be confused with the semiclassical treatments of the Casimir interaction in the position representation [@Schaden1998; @Jaffe2004; @Scardicchio2005]. The standard Gutzwiller trace formula fails badly in the PFA regime because the relevant surface area increases as one approaches this limit [@Jaffe2004]. In other words, the semiclassical PFA limit cannot be connected to periodic orbits obtained from a stationary-phase approximation in the position representation, because position is poorly resolved in this limit. On the other hand, the condition of specular reflection makes the integration range for the conjugate momentum variable increasingly narrow as the distance between the spheres becomes very small compared to their radii, allowing us to obtain the PFA result from a saddle-point approximation for the scattering formula.
The paper is organized as follows. Sec. \[sec:planewavebasis\] presents the general development of the scattering formula for the Casimir free energy in the plane-wave basis. The following sections apply this formalism to the case of two spheres in the short-distance limit. The Mie scattering matrix elements in the plane-wave basis, for arbitrary directions of incidence, are presented in Sec. \[sec:scattering\_at\_sphere\]. The corresponding WKB approximation is derived in Sec. \[sec:wkb\] and Sec. \[sec:pfa\_from\_spa\] calculates the resulting Casimir free energy in the saddle-point approximation. Sec. \[sec:effective\_area\] presents an estimation of the area on the spheres that actually contributes in the short-distance limit. Concluding remarks are presented in Sec. \[sec:conclusions\] and the appendices contain some additional technical details.
The Casimir free energy in the plane-wave basis {#sec:planewavebasis}
===============================================
In a homogeneous medium, the electric and the magnetic field satisfy the vector Helmholtz equation. A convenient basis set consists of plane waves characterized by a wave vector $\mathbf{K}$ and the polarization $p$. In order to define the polarization basis, we assume a given $z$-axis which can be appropriately fixed later on and choose an incidence plane spanned by the $z$-axis and the wave vector $\mathbf{K}$. Denoting unit vectors by a hat, we obtain the basis vectors for transverse electric (TE) and transverse magnetic (TM) modes as $$\label{eq:polarizationTETM}
\hat{\bm{\epsilon}}_\mathrm{TE} = \frac{\hat{\mathbf{z}} \times \hat{\bm{K}}}
{\vert\hat{\mathbf{z}} \times \hat{\bm{K}}\vert}, \quad
\hat{\bm{\epsilon}}_\mathrm{TM} = \hat{\bm{\epsilon}}_\mathrm{TE} \times \hat{\mathbf{K}} \,.$$ Thus, the TE polarization is perpendicular to the incidence plane while the TM polarization lies in it.
As we will see later in this section, it is convenient to fix the frequency which by means of the dispersion relation is obtained as $\omega=c\vert\mathbf{K}\vert$ with $c$ being the speed of light. It is then sufficient to specify the projection $\mathbf{k}=(K_x, K_y, 0)$ of the wave vector $\mathbf{K}$ onto the $x$-$y$ plane perpendicular to the $z$-axis. In order to uniquely define the wave vector, we finally need to fix the direction of the propagation in the $z$-direction by $$K_z = \phi k_z, \quad k_z \equiv (\omega^2/c^2-\mathbf{k}^2)^{1/2} \,,$$ with $\phi=\pm1$. We thus arrive at the angular spectral representation expressing the plane-wave basis as $\{\ket{\omega,\mathbf{k},p,\phi}\}$ [@Nieto-Vesperinas2006]. In position space, the basis functions read $$\braket{x,y,z | \omega,\mathbf{k},p,\phi} = \hat{\bm{\epsilon}}_p
\left(\frac{1}{2\pi}\left|\frac{\omega}{c k_z}\right|\right)^{1/2}
e^{i(\mathbf{k}\mathbf{r}+\phi k_z z)}$$ where $\mathbf{r}=(x,y,0)$. The normalization factor is appropriate for the angular spectral representation where the integration is performed over the frequency and the projection of the wave vector into the $x$-$y$ plane.
For the purpose of this section, we consider two arbitrary objects that can be separated by a plane parallel to the $x$-$y$ plane. The reference point of object 1 is located in the origin and the reference point of object 2 is located at $z=\mathcal{L}$. The starting point of our analysis is the scattering approach to the Casimir effect in imaginary frequencies $\xi=i\omega$ where the free energy is expressed as [@Lambrecht2006; @Emig2007] $$\label{eq:F}
\mathcal{F} = \frac{k_\mathrm{B} T}{2} \sum_{n=-\infty}^\infty
\mathrm{tr}\log\left[1-\mathcal{M}(\vert\xi_n\vert)\right]$$ in terms of a sum over the Matsubara frequencies $\xi_n=2\pi n k_\mathrm{B}
T/\hbar$. The central object here is the round-trip operator $$\label{eq:round-trip}
\mathcal{M} = \mathcal{R}_1\mathcal{T}_{12}\mathcal{R}_2\mathcal{T}_{21}$$ describing a complete round trip of an electromagnetic wave between the two scatterers in the order indicated in Fig. \[fig:geometry\]. $\mathcal{R}_j$ denotes the reflection operator at object $j=1,2$ and $\mathcal{T}_{21}$ describes a translation from the reference frame of object 1 to the reference frame of object 2, and vice versa for $\mathcal{T}_{12}$.
In the plane-wave basis the translation operators are diagonal with matrix elements $e^{-\kappa\mathcal{L}}$ where $\kappa = (\xi^2/c^2+k^2)^{1/2}$ denotes the $z$-component of the wave vector associated with the imaginary frequency $\xi$. As the frequency $\xi$ remains constant during a round trip, we suppress the notation $\xi$ in the labeling of the basis elements.
The logarithm in can be expanded in a Mercator series $$\label{eq:F_roundtrip}
\mathcal{F} = -\frac{k_\mathrm{B}T}{2} \sum_{n=-\infty}^\infty \sum_{r=1}^\infty \frac{1}{r} \mathrm{tr}\mathcal{M}^r\left(\left|\xi_n\right|\right)$$ and the trace of the $r$-th power of the round-trip operator is given by $$\begin{aligned}
\mathrm{tr}\mathcal{M}^r &= \sum_{p_1,\dotsc,p_{2r}} \int \frac{d \mathbf{k}_1\dots d\mathbf{k}_{2r}}{(2\pi)^{4r}}
\prod_{j=1}^r e^{-(\kappa_{2j}+\kappa_{2j-1})\mathcal{L}} \\
\label{eq:M_elems}
&\qquad\times \braket{\mathbf{k}_{2j+1}, p_{2j+1}, - \vert\mathcal{R}_1 \vert\mathbf{k}_{2j}, p_{2j}, +} \\
&\qquad\times\braket{\mathbf{k}_{2j}, p_{2j}, + \vert\mathcal{R}_2 \vert\mathbf{k}_{2j-1}, p_{2j-1}, -} \,.
\end{aligned}$$ Here, we have used the convention of cyclic indices $p_{2r+1}\equiv p_1$ and $\mathbf{k}_{2r+1}\equiv\mathbf{k}_1$. Eqs. and can be interpreted as an expansion in round trips. The free energy consists of contributions from a single round trip within the cavity, up to infinitely large numbers of round trips. Also, the expansion in round trips is a natural way to compute $\mathrm{tr}\log\left(1-\mathcal{M}\right)$ of a non-diagonal round-trip operator $\mathcal{M}$ expressed in a continuous basis.
The exponential factor in might suggest that the component in $z$-direction $\kappa$ of the imaginary wave vectors is constrained to values of the order of $1/\mathcal{L}$. However, we will see later for the special case of the sphere-sphere configuration that the reflection matrix elements grow exponentially with the size of the corresponding objects. As a consequence, much larger values for $\kappa$ of the order of $1/L$ are possible, where $L$ denotes the closest distance between the two objects.
While our discussion so far was fairly general, we will specialize on the geometry of two spheres in the following sections.
Scattering at a sphere {#sec:scattering_at_sphere}
======================
While the translation operators in the plane-wave basis are trivial, the reflection operators require more care. For scattering at a single sphere, we can make the scattering plane spanned by the initial and the reflected wave vectors coincide with the incidence plane defining the polarizations as explained in Sect. \[sec:planewavebasis\]. In the sphere-sphere setup shown in Fig. \[fig:geometry\], however, the $z$-axis is defined by the centers of the two spheres. Then, in general the scattering plane and the incidence plane will not coincide and one has to change the polarization basis as explained in appendix \[sec:appendixA\]. Nevertheless, it will turn out that the results can mostly be cast in quantities familiar from the standard Mie theory, allowing us to relate PFA to the concepts of geometrical optics.
Then, the matrix elements of the reflection operator $\mathcal{R}$ at a sphere are given by $$\label{eq:RS}
\begin{aligned}
\braket{\mathbf k_j, \mathrm{TM} | \mathcal{R} | \mathbf k_i, \mathrm{TM}} &= \phantom{-}\frac{2\pi c}{\xi \kappa_j} \big(A S_2(\Theta)+B S_1(\Theta)\big) \\
\braket{\mathbf k_j, \mathrm{TE} | \mathcal{R} | \mathbf k_i, \mathrm{TE}} &= \phantom{-}\frac{2\pi c}{\xi \kappa_j} \big(A S_1(\Theta)+B S_2(\Theta)\big) \\
\braket{\mathbf k_j, \mathrm{TM} | \mathcal{R} | \mathbf k_i, \mathrm{TE}} &= -\frac{2\pi c}{\xi \kappa_j} \big(C S_1(\Theta)+D S_2(\Theta)\big) \\
\braket{\mathbf k_j, \mathrm{TE} | \mathcal{R} | \mathbf k_i, \mathrm{TM}} &= \phantom{-}\frac{2\pi c}{\xi \kappa_j} \big(C S_2(\Theta)+D S_1(\Theta)\big)\,,
\end{aligned}$$ where the prefactor results from the normalization within the angular spectral representation. Here, we have omitted the value of $\phi$ which should be different for the two waves involved in a matrix element. In addition, the signs of $C$ and $D$ depend on the direction of propagation as specified in Eq. (\[eq:abcd\]). The Mie scattering amplitudes for polarizations perpendicular and parallel to the scattering plane are given by [@BH] $$\label{eq:SA}
\begin{aligned}
S_1(\Theta) &= \sum_{\ell=1}^\infty \frac{2\ell+1}{\ell(\ell+1)}
\left[ a_\ell\pi_\ell\big(\cos(\Theta)\big) + b_\ell\tau_\ell\big(\cos(\Theta)\big)\right]\\
S_2(\Theta) &= \sum_{\ell=1}^\infty \frac{2\ell+1}{\ell(\ell+1)}
\left[ a_\ell\tau_\ell\big(\cos(\Theta)\big) + b_\ell\pi_\ell\big(\cos(\Theta)\big)\right]\,,
\end{aligned}$$ respectively. The scattering angle is defined relative to the forward direction (cf. Fig. \[fig:phaseshift\]) and, for imaginary frequencies, is given by $$\cos(\Theta) = -\frac{c^2}{\xi^2} \left(\mathbf{k}_i\cdot\mathbf{k}_j + \kappa_i\kappa_j\right) \,.
\label{eq:costheta}$$ The angular functions $\pi_\ell$ and $\tau_\ell$ are defined by [@BH] $$\begin{aligned}
\pi_\ell(z) &= {P_\ell}^\prime(z) \\
\tau_\ell(z) &= -(1-z^2){P_\ell}^{\prime\prime}(z)+z{P_\ell}^\prime(z)
\end{aligned}
\label{eq:pi_tau}$$ with the Legendre polynomials $P_\ell$ and the prime denoting a derivative with respect to the argument $z$.
The coefficients $A$, $B$, $C$, and $D$ are functions of $\mathbf{k}_i$ and $\mathbf{k}_j$. Explicit expressions are derived in appendix \[sec:appendixA\] and given in (\[eq:abcd\]). The Mie coefficients $a_\ell$ and $b_\ell$ [@BH] represent the partial wave electric and magnetic multipole scattering amplitudes, respectively, for an isotropic sphere. They depend on the electromagnetic response of the sphere material. For simplicity, we restrict ourselves to homogeneous non-magnetic spheres in the following.
![Geometrical optics limit for the direct reflection by a sphere of radius $R$. Within the WKB approximation, a given scattering angle $\Theta$ defines the impact parameter $b=R\cos(\Theta/2).$ Seen from the tangent plane to the sphere, the angle of incidence is given by $(\pi-\Theta)/2$. The missing phase of a ray with frequency $\omega$ reflected on the sphere’s surface with respect to a corresponding ray passing via the sphere’s center before being deflected amounts to $2(\omega R/c)\sin(\Theta/2)$.[]{data-label="fig:phaseshift"}](phaseshift.pdf){width="0.7\columnwidth"}
WKB approximation {#sec:wkb}
=================
In order to obtain PFA as the leading asymptotics for large radii $R_1,R_2\gg
L$, asymptotic expressions for the matrix elements are required. For this purpose, it is convenient to write the scattering amplitudes by means of the Debye expansion [@Nussenzveig92], i.e., a decomposition into an infinite series of terms representing multiple internal reflections. In the limit of a large radius $R,$ the direct reflection term of the Debye expansion gives the main contribution since the phase factor acquired by propagation inside the spheres yields exponentially small terms when considering the imaginary frequency domain.
For real frequencies and large size parameters $\omega R/c \gg 1$, the asymptotic expression for the direct reflection term has been derived from the WKB approximations for the Mie coefficients and angular functions by taking the saddle-point approximation for the integral over angular momenta [@Nussenzveig69]. The resulting expression is valid for all scattering directions except near the forward one, which is not relevant for the Casimir interaction. In fact, the Casimir energy as given by (\[eq:F\_roundtrip\]) and (\[eq:M\_elems\]) is obtained from round trips containing only backward scattering channels. Instead of real frequencies, we need the asymptotics for imaginary frequencies $\xi,$ for which a very similar WKB derivation can be performed. The resulting expression coincides with the one obtained for real frequencies $\omega$ after replacing $\omega\to i\xi.$ The leading asymptotics of the scattering amplitude is then given by $$\label{eq:SA-asymptotics}
S_p(\Theta) \simeq \frac{1}{2}(\xi R/c) r_p\bigl((\pi-\Theta)/2\bigr)e^{2(\xi R/c) \sin(\Theta/2)}\,,$$ with $p=1,2$ corresponding to TE and TM modes, respectively. $r_\mathrm{TE}$ and $r_\mathrm{TM}$ are the familiar Fresnel reflection coefficients [@Landau84] for a wave in vacuum impinging at an angle of incidence $\theta$ on a medium with permittivity $\varepsilon$ $$\label{eq:Fresnel-coeff}
\begin{aligned}
r_\mathrm{TE}(\theta) &= \frac{\cos(\theta) - \sqrt{\varepsilon-\sin^2(\theta)}}%
{\cos(\theta) + \sqrt{\varepsilon-\sin^2(\theta)}}\,,\\
r_\mathrm{TM}(\theta) &= \frac{\varepsilon\cos(\theta) - \sqrt{\varepsilon-\sin^2(\theta)}}%
{\varepsilon\cos(\theta) + \sqrt{\varepsilon-\sin^2(\theta)}}\,.
\end{aligned}$$
The asymptotics of the scattering amplitudes can be understood in terms of geometrical optics in the real frequency domain [@Nussenzveig65]. For a given scattering angle $\Theta,$ the main contribution to $S_1$ and $S_2$ in Eq. (\[eq:SA\]) comes from the neighborhood of the angular momentum value $\ell = (\omega R/c) \cos(\Theta/2)$ [@Nussenzveig65]. In the semiclassical approximation, the localization principle [@Nussenzveig92] connects waves with angular momentum $\ell\gg 1$ to localized rays defining an impact parameter $b=(c/\omega)\ell$. Thus, the derivation of the WKB approximation (\[eq:SA-asymptotics\]) defines rays corresponding to the impact parameter $b=R\cos(\Theta/2)$ shown in Fig. \[fig:phaseshift\]. Such rays hit the sphere surface with an incidence angle of $(\pi-\Theta)/2,$ which is precisely the value required for obtaining the scattering angle $\Theta$ from the condition of specular reflection at the tangent plane indicated in the figure. Comparing the reflection at the tangent plane (thick lines) and at the sphere with its center as reference point, one finds a difference in path length amounting to $2(\omega R/c)\sin(\Theta/2)$. In this way, the last two factors of (\[eq:SA-asymptotics\]) find their natural explanation. The first factor is responsible for providing the correct scattering cross section proportional to $R^2$.
The asymptotics of the Mie scattering amplitudes does not cover the zero frequency case, which is required in the Matsubara sum. In appendix \[sec:appendixB\], we show that the scattering amplitudes for $\xi=0$ coincide with the scattering amplitudes at finite imaginary frequencies (\[eq:SA-asymptotics\]) evaluated at $\xi=0$.
In order to derive the leading asymptotic expression for the scattering matrix elements (\[eq:RS\]), we make use of (\[eq:SA-asymptotics\]) to obtain $$\label{eq:RS_WKB}
\braket{\mathbf k_j, p_j | \mathcal{R} | \mathbf k_i, p_i} \simeq \frac{\pi R}{\kappa_j} e^{2(\xi R/c)\sin(\Theta/2)} \rho_{p_j,p_i}$$ with $$\label{eq:rhos}
\begin{aligned}
\rho_{\mathrm{TM},\mathrm{TM}} &= \phantom{-}A r_\mathrm{TM} + B r_\mathrm{TE}, \\
\rho_{\mathrm{TE},\mathrm{TE}} &= \phantom{-}A r_\mathrm{TE} + B r_\mathrm{TM}, \\
\rho_{\mathrm{TM},\mathrm{TE}} &= -C r_\mathrm{TE} - D r_\mathrm{TM}, \\
\rho_{\mathrm{TE},\mathrm{TM}} &= \phantom{-}C r_\mathrm{TM} + D r_\mathrm{TE} \,.
\end{aligned}$$ The WKB expression for the reflection matrix element already indicates an exponential growth with the sphere radius $R$ as anticipated in the discussion at the end of Sec. \[sec:planewavebasis\].
PFA from saddle-point approximation {#sec:pfa_from_spa}
===================================
We will now derive the proximity force approximation based on the scattering matrix elements (\[eq:RS\_WKB\]) obtained within the WKB approximation. The main step consists in evaluating within the saddle-point approximation the trace over $\mathcal{M}^r$ appearing in the expansion (\[eq:F\_roundtrip\]) of the free energy. This approach requires large sphere radii $R_1, R_2\gg L$ (cf. Fig. \[fig:geometry\]), a limit in which PFA is expected to hold. While we will carry out the calculation for the sphere-sphere geometry, we will briefly comment on the limit of the plane-sphere geometry when appropriate.
Round trips within WKB approximation
------------------------------------
The main quantity in the round-trip expansion of the free energy (\[eq:F\_roundtrip\]) is the trace over the $r$-th power of the round-trip operator, which in the plane-wave representation is given by (\[eq:M\_elems\]). After inserting the WKB scattering matrix elements (\[eq:RS\_WKB\]) and employing polar coordinates $(k_i, \varphi_i)$ in the $x$-$y$ plane, we express the result in a form suitable for the saddle-point approximation as $$\label{eq:trMr-start}
\begin{aligned}
\mathrm{tr}\mathcal{M}^r &\simeq \int_0^\infty d^{2r} k
\int_0^{2\pi} d^{2r}\varphi\,
g(\mathbf{k}_1,\dots,\mathbf{k}_{2r})\\
&\qquad\qquad\times e^{-(R_1+R_2) f(\mathbf{k}_1,\dots,\mathbf{k}_{2r})}\,.
\end{aligned}$$ The symbol $\simeq$ indicates that the result is only valid in the small distance limit $L \ll R_1, R_2$. Here, we have used $R_1+R_2$ as large parameter for the saddle-point approximation. Another choice, e.g. an individual radius, would equally be possible and would yield the same final result.
In (\[eq:trMr-start\]) we introduced the function
$$g(\mathbf{k}_1,\dots,\mathbf{k}_{2r}) = \left(\frac{R_1 R_2}{16\pi^2}\right)^r \sum_{p_1,\dotsc,p_{2r}}\prod_{j=1}^r \frac{k_{2j}k_{2j-1}}{\kappa_{2j}\kappa_{2j-1}}\rho^{(1)}_{p_{2j+1},p_{2j}}(\mathbf{k}_{2j+1},\mathbf{k}_{2j}) \rho^{(2)}_{p_{2j},p_{2j-1}}(\mathbf{k}_{2j},\mathbf{k}_{2j-1})e^{-(\kappa_{2j}+\kappa_{2j-1})L}
\label{eq:gfunc}$$
where the superscript of the factors $\rho$ defined in (\[eq:rhos\]) indicates the sphere for which the Fresnel reflection coefficient is to be taken.
The function in the exponent of (\[eq:trMr-start\]) is given by $$f(\mathbf{k}_1,\dots,\mathbf{k}_{2r}) = \frac{1}{R_1+R_2}\sum_{j=1}^{r}\left(R_1\eta_{2j} + R_2\eta_{2j-1} \right)
\label{eq:func_f}$$ where the terms with even and odd indices are contributions from sphere 1 and 2, respectively, and $$\begin{gathered}
\label{eq:eta}
\eta_i = \kappa_i + \kappa_{i+1} \\
- \sqrt{2\big[(\xi/c)^2 + \kappa_i\kappa_{i+1} + k_i k_{i+1}\cos(\varphi_{i}-\varphi_{i+1})\big]}\,.\end{gathered}$$ While the last term on the right-hand side arises from the phase shift illustrated in Fig. \[fig:phaseshift\], the first two terms are associated with a translation over twice the radius of the sphere at which the reflection occurs. As a consequence, the last factor in (\[eq:gfunc\]) only depends on the closest distance $L$ between the two spheres.
Saddle-point manifold {#subsec:spmanifold}
---------------------
In order to evaluate the $4r$ integrals in (\[eq:trMr-start\]) within the saddle-point approximation, we need to determine the stationary points. In fact, there exists a two-dimensional manifold of saddle points $$k_1=\dots=k_{2r}=k_\ast,\quad \varphi_1=\dots=\varphi_{2r}=\varphi_\ast
\label{eq:spmanifold}$$ parametrized by $k_\ast$ and $\varphi_\ast$. Thus, on the saddle-point manifold, the change of angle $\varphi=\varphi_{j+1}-\varphi_j$ vanishes and leads to a significant simplification because then the incidence and scattering planes coincide. As a consequence, $A=1, B=C=D=0$ as can be verified also from the relations (\[eq:abcd\]) by setting $\varphi=0$. Thus, in view of (\[eq:rhos\]), the polarization is always conserved during the scattering processes within the saddle-point approximation. The trace over $r$ round-trip operators (\[eq:trMr-start\]) can now be decomposed into two independent polarization contributions $$\mathrm{tr}\mathcal{M}^r = \mathrm{tr}\mathcal{M}_\mathrm{TE}^r+\mathrm{tr}\mathcal{M}_\mathrm{TM}^r\,.$$
The saddle-point manifold (\[eq:spmanifold\]) also implies that the projection of the wave vector onto the $x$-$y$ plane is conserved during the reflection. Within the WKB approximation, this is the case when in Fig. \[fig:phaseshift\] the tangent plane on which the reflection occurs were perpendicular to the $z$-axis. Under this condition, the WKB phase shift upon reflection $2(\xi
R/c)\sin(\Theta/2)$ can be expressed as $2\kappa R$. This precisely cancels the term arising from the translation by twice the sphere radius. As a consequence the exponent (\[eq:func\_f\]) vanishes on the saddle-point manifold, $$f\big\vert_\text{S.P.} = 0 \,.$$
For the prefactor in the integrand of (\[eq:trMr-start\]), we now obtain for the two polarization contributions $p=\mathrm{TE},\mathrm{TM}$ $$g_p\big\vert_\text{S.P.}(k_\ast) = \left(\frac{R_1R_2}{16\pi^2}
\frac{k_\ast^2}{\kappa_\ast^2}r_p^{(1)} r_p^{(2)}\right)^r e^{-2r\kappa_\ast L}
\label{eq:gp}$$ on the saddle-point manifold. Here, we have introduced $\kappa_\ast =
(\xi^2/c^2+k_\ast^2)^{1/2}$. The Fresnel coefficients (\[eq:Fresnel-coeff\]) are evaluated at the angle $$\theta=\arccos(\kappa_\ast c/\xi)\,.
\label{eq:theta}$$ On the saddle-point manifold, only a translation by $L$ remains, which just corresponds to the distance between the two tangent planes perpendicular to the $z$-axis facing each other. Thus the relevant length scale for the $z$ component of the imaginary wave vector is given by $L$ instead of $\mathcal{L}$ as had seemed to imply.
The result (\[eq:gp\]) might raise some concerns about a divergence associated with the plane-sphere limit. Choosing without loss of generality $R_1\leq R_2$, the plane-sphere limit reads $R_2\to\infty$. We know that the wave vector $\mathbf{k}$ is conserved during a reflection at the plane, so that $2r$ integrations have to drop out in this limit. In fact, we will see in the next subsection that for $R_2\to\infty$ indeed $2r$ delta functions appear for which the factor $R_2/4\pi$ contained in (\[eq:gp\]) provides the normalization.
Hessian matrix
--------------
We now turn to the Hessian matrix $\textsf{H}$ of the function $f$ (\[eq:func\_f\]) evaluated on the saddle-point manifold. It is found to be of block-diagonal form $$\textsf{H} = \begin{pmatrix} \textsf{H}_{kk} & 0\\ 0 & \textsf{H}_{\varphi\varphi} \end{pmatrix}$$ with $$\textsf{H}_{kk} = \frac{1}{2 \kappa_\ast } \textsf{M}_r\,,
\quad \textsf{H}_{\varphi\varphi} = \frac{k_\ast^2}{2 \kappa_\ast } \textsf{M}_r\,.$$ Apart from prefactors, both blocks are given by the $2r\times 2r$ matrix $$\textsf{M}_r = \begin{pmatrix}
1 & -(1-\mu) &&& -\mu\\
-(1-\mu) & 1 & -\mu && \\
& -\mu & \ddots & \ddots &\\
& & \ddots & \ddots & -(1-\mu)\\
-\mu& & & -(1-\mu) & 1 \\
\end{pmatrix}\,,$$ where all empty entries should be set to zero. Here, $\mu=R_1/(R_1+R_2)$ and in the limit of the plane-sphere geometry, $R_2\to\infty$, $\mu$ goes to zero. The off-diagonal matrix elements alternate between $-(1-\mu)$ and $-\mu$ in accordance with the alternating reflection at the two spheres during round trips. In the special case of a single round trip, $r=1$, the two different off-diagonal matrix elements add up to yield $$\textsf{M}_1 = \begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix}\,.$$
The eigenvalues of the matrix $\mathsf{M}_r$ are found as $$\label{eq:eigenvals}
\lambda^{(j)}_\pm = 1 \pm \sqrt{1-4\mu(1-\mu)\sin^2\left(\frac{\pi j}{r}\right)}$$ for $j=0,\dots,r-1$. Both, $\mathsf{H}_{kk}$ and $\mathsf{H}_{\varphi\varphi}$ have a zero eigenvalue corresponding to the saddle-point manifold discussed in the previous subsection. Within the saddle-point approximation, the directions perpendicular to the saddle-point manifold can now be integrated out in the usual way, while the integration along the families has to be carried out exactly. This can be done for $\varphi_\ast$ so that we are left with an integral over $k_\ast$ in the following subsection.
Before turning to the result for the trace over $r$ round-trip operators, we would like to make another comment on the plane-sphere case. Considering the eigenvalues (\[eq:eigenvals\]) to leading order in $\mu$, we find $r$ eigenvalues satisfying $$\label{R1R2}
(R_1+R_2)\,\lambda^{(j)}_- = 2R_1\sin^2(\pi j/r)\;\;\;\;\;(\mu\to 0)$$ and $r$ eigenvalues $\lambda^{(j)}_+=2.$ When multiplied by the prefactor $R_1+R_2$ in the exponent of (\[eq:trMr-start\]), we obtain from the latter $2r$ delta functions in the limit $R_2\to\infty,$ thus enforcing the conservation of the wave vector $\mathbf{k}$ for the reflection at a plane, as already indicated in the previous subsection. The remaining integrals are controlled by (\[R1R2\]) and lead to the PFA result for the plane-sphere geometry proportional to $R_1.$ Such result can also be obtained from the more general expression for the sphere-sphere case derived in the remainder of this section.
Casimir free energy and force
-----------------------------
The evaluation of the saddle-point integral is simplified by first forming products $\lambda^{(j)}_+\lambda^{(j)}_-$ of the eigenvalues (\[eq:eigenvals\]) for $j=1,\ldots,r-1$ and noting that $$\prod_{j=1}^{r-1} \sin\left(\frac{\pi j}{r}\right) = \frac{r}{2^{r-1}}\,.
\label{eq:sineprod}$$ Then, the inverse of the square root of all non-vanishing eigenvalues of $(R_1+R_2)\textsf{H}$ is found to read $$\left(\prod_{\lambda\neq0}\lambda\right)^{-1/2} =
\frac{R_\text{eff}}{4r^2}\frac{k_\ast}{\kappa_\ast}
\left(\frac{4\kappa_\ast^2}{k_\ast^2}\frac{1}{R_1R_2}\right)^r\,,$$ where we have defined the effective radius $$\label{Reff}
R_\text{eff} = \frac{R_1R_2}{R_1+R_2}\,.$$ Changing to the eigenbasis of $\textsf{H}$ but keeping $k_\ast$ and $\varphi_\ast$ as variables for the integration, i.e., not normalizing the eigenvectors corresponding to the saddle-point manifolds, yields a factor $2r$ arising through the Jacobian. Then, by applying the multi-dimensional saddle-point integration formula, we obtain $$\label{Mrp}
\mathrm{tr} \mathcal{M}^r_p \simeq
\frac{R_\text{eff}}{2r}\int_{\vert\xi_n\vert/c}^\infty d\kappa_\ast
\left[r^{(1)}_p r^{(2)}_p e^{-2\kappa_\ast L}\right]^{r}$$ for the two polarization contributions $p=\mathrm{TE},\mathrm{TM}$. Inserting this result into (\[eq:F\_roundtrip\]), we can evaluate the sum over the number $r$ of round trips and obtain for the free energy $$\begin{gathered}
\mathcal{F} \simeq -\frac{k_BTR_\text{eff}}{4}\sum_{n=-\infty}^{+\infty}
\sum_{p\in\{\mathrm{TE},\mathrm{TM}\}} \int_{\vert\xi_n\vert/c}^\infty d\kappa_\ast\\
\times\text{Li}_2\left(r_p^{(1)}r_p^{(2)}e^{-2\kappa_\ast L}\right)
\label{eq:F_pfa}\end{gathered}$$ where $\text{Li}_2$ denotes the dilogarithm [@DLMF25.12]. The Fresnel coefficients (\[eq:Fresnel-coeff\]) are evaluated at the angle $\theta$ defined in (\[eq:theta\]) taken at the frequencies $\vert\xi_n\vert$.
The Casimir force can now be obtained by taking the negative derivative of the expression (\[eq:F\_pfa\]) for the free energy with respect to the distance $L$. We thus find the Lifshitz formula $$\label{eq:PFA-result}
F \simeq 2 \pi R_\mathrm{eff} \mathcal{F}_{\rm PP}(L, T)$$ with the free energy per area for two planes at distance $L$ and temperature $T$ $$\begin{gathered}
\label{FPPLifshitz}
\mathcal{F}_{\rm PP}(L, T) =
\frac{k_\mathrm{B} T}{2}
\sum_{n=-\infty}^{+\infty} \sum_{p\in\{\mathrm{TE},\mathrm{TM}\}} \int_{\vert\xi_n\vert/c}^\infty
\frac{d\kappa}{2\pi} \kappa \\
\times\log\left(1-r^{(1)}_p r^{(2)}_p e^{-2\kappa L} \right)\,.\end{gathered}$$ It is straightforward to extend this result to the zero temperature case. As already discussed at the end of the previous subsection, these results are also valid in the plane-sphere case where $R_\text{eff}$ is replaced by the sphere radius.
Effective area {#sec:effective_area}
==============
The most precise Casimir experiments employ spherical lenses [@Sushkov2011] or coated microspheres attached to a cantilever beam [@Decca2007; @Chang2012; @Banishev2013; @Garrett2018] instead of whole spherical surfaces. Since the experimental data are analyzed with the help of the PFA, it is important to understand what section of the spherical surface actually contributes to the leading asymptotics. For instance, in the case of a spherical lens, such analysis would define the minimum transverse lens size required for equivalence with a complete spherical surface. Here, we estimate the size of the relevant sphere section and proceed in two steps. First, we employ our saddle-point calculation to estimate the typical change in the projection of the wave vector onto the $x$-$y$ plane during reflection at one of the spheres. Second, we use geometric arguments to obtain the corresponding size of the sphere section in real space.
Even though we first consider the reflection at a single sphere, this reflection is still to be taken in the context of the sphere-sphere setup. Therefore, we keep the saddle-point manifold (\[eq:spmanifold\]) obtained in Sec. \[subsec:spmanifold\]. Considering only a single reflection, we denote the incident and reflected wave vectors as $\mathbf{K}_\text{in}$ and $\mathbf{K}_\text{rfl}$, respectively, as indicated in Fig. \[fig:effectiveArea\]. For simplicity, we take $\varphi_\text{in} =
\varphi_\text{rfl} = \varphi_*$ and concentrate on the modulus of $\mathbf{k}$. From (\[eq:trMr-start\]) and (\[eq:func\_f\]), the Gaussian contribution of a single reflection at a sphere with radius $R_j, j=1, 2$ can then be identified as $$\exp\left(-\eta R_j\right) = \exp\left(-\frac{R_j}{4\kappa_*}
(k_\text{in}-k_\text{rfl})^ 2\right)\,.
\label{eq:kGaussian}$$ Here, $\eta$ is defined in analogy to (\[eq:eta\]) with the two wave-vector components replaced by $\kappa_\text{in}$ and $\kappa_\text{rfl}$.
Neglecting numerical factors of order one, the width around the saddle-point manifold is thus $\delta k^{(j)} \sim \sqrt{\kappa_\ast/R_j}$. The typical scale of $\kappa_*$ is set by the integral on the right-hand side of (\[Mrp\]), finally leading to the change of the projection of the wave vector onto the $x$-$y$ plane $$\delta k^{(j)} = |\mathbf{k}_\text{rfl}-\mathbf{k}_\text{in}| \sim
(LR_j)^{-1/2}\,.
\label{eq:deltak}$$ As expected, the scattering at the smaller sphere provides the larger deviations from the saddle-point manifold. Thus, the effective area contributing to the Casimir interaction is fixed by the smaller radius which we refer to as $R_1$ in the following.
![Estimation of the effective area contributing to the Casimir interaction between two spheres. ${\bf K}_\text{in}$ and ${\bf K}_\text{rfl}$ denote the incident and reflected wave vectors, respectively. To be definite, the reflection is shown at the smaller sphere of radius $R_1$. Specular reflection at the tangent plane $\pi(\vartheta)$ entails that the projections of ${\bf K}_\text{in}$ and ${\bf K}_\text{rfl}$ on $\pi(\vartheta)$ are equal. On the other hand, the wave vector projections on the $x$-$y$ plane are generally different: $\delta k=|{\bf k}_\text{rfl}-{\bf k}_\text{in}|\approx2\vartheta k_z$ for $\vartheta\ll 1$. We can estimate the angular sector effectively contributing to the Casimir interaction from the width $\delta k$ of the Gaussian integrand in the saddle-point approximation (see text). []{data-label="fig:effectiveArea"}](effarea.pdf){width="0.7\columnwidth"}
We now explore the implications of (\[eq:deltak\]) in real space based on specular reflection. The saddle-point manifold (\[eq:spmanifold\]) corresponds to reflections between the points on the two spheres corresponding to the closest distance $L$ (cf. Fig. \[fig:effectiveArea\]). Deviations from the saddle-point manifold as allowed by the Gaussian (\[eq:kGaussian\]) implicate the surroundings of these two points in the scattering process. We estimate the dimension of the spherical cap on the surface of the smaller sphere 1 by considering the scattering of propagating waves in the real frequency domain with the help of Fig. \[fig:effectiveArea\].
As above, we assume $\mathbf{k}_\text{in}$ and $\mathbf{k}_\text{rfl}$ to be parallel and for simplicity omit the index $j=1$ when writing $\delta
k=|\mathbf{k}_\text{rfl}-\mathbf{k}_\text{in}|.$ As shown in Sec. \[sec:wkb\], the WKB approximation for the direct reflection term amounts to a specular reflection at a tangent plane $\pi(\vartheta)$ making an angle $\vartheta$ with the $x$-$y$ plane. For parallel vectors and small values of $\delta k$, a simple relation between $\vartheta$ and $\delta k$ can be derived by noting that while the projection of the wave vector onto the plane $\pi(\vartheta)$ is conserved during a scattering process, this is not the case for the projection onto the $x$-$y$ plane for non-vanishing values of $\vartheta$. Assuming $\vartheta\ll 1,$ we find $$\delta k\approx 2\vartheta k_z\,.$$ This relation together with the scaling $k_z\sim 1/L$ allow us to estimate the width of the angular sector effectively contributing to the Casimir interaction from the Gaussian width (\[eq:deltak\]).
We find that the spherical cap around the point of closest distance corresponds to the angular sector bounded by the angle $\vartheta \sim(L/R_1)^{1/2}\ll 1.$ As indicated by Fig. \[fig:effectiveArea\], its transverse size is $d \approx
R_1\vartheta \sim (R_1L)^{1/2}\ll R_1<R_2.$ The same scaling was found by an heuristic geometric argument [@Kim2010]. The area of the spherical surface effectively contributing to the interaction is then $A \sim R_1L,$ which coincides, except for a numerical factor of order one, with the ratio between the Casimir force for two spheres within PFA and the Casimir pressure for parallel planes, as long as the interaction obeys a power law [@footnote].
Although the discussion above holds for arbitrary temperatures, we show in the remainder of this section that the effective area for the thermal corrections scales in a different way in the low-temperature regime. The difference arises from the typical scale for $\kappa_*,$ which is no longer set by $1/L$, but rather by $1/\lambda_T,$ where $\lambda_T = \hbar c/k_B T$ is the thermal wavelength. In order to illustrate this property, we consider the thermal correction of the Casimir force $\delta F \equiv F(L, T)-F(L, 0)$ as an example. We start from Eqs. (\[eq:PFA-result\]) and (\[FPPLifshitz\]) and employ the Poisson summation formula [@Genet2000] to write $$\begin{gathered}
\label{eq:correction}
\delta F= 2\hbar \,R_{\rm eff}\,\sum_{m=1}^\infty
\sum_{p}
\int_0^\infty d\xi
\cos(m\lambda_T\xi/c) \int_{\xi/c}^\infty
\frac{d\kappa}{2\pi} \kappa \\
\times\log\left(1-r^{(1)}_p r^{(2)}_p e^{-2\kappa L} \right)\,.\end{gathered}$$ In the low-temperature limit, $L\ll \lambda_T,$ the exponential $\exp(-2\kappa
L)$ can be taken to be approximately constant and does not provide a cutoff for the $\kappa$ integration in (\[eq:correction\]). For instance, in the case of plasma metals, the correction $\delta F$ can be written in terms of simple integrals involving trigonometric functions of $m\lambda_T \kappa,$ which are similar to the expressions derived for the Casimir pressure between parallel planes in Ref. [@Genet2000]. The derivation of the low-temperature limit of (\[eq:correction\]) for Drude metals is more involved [@Ingold2009], but $1/\lambda_T$ also provides the typical scale of $\kappa$ in this case.
As a consequence, the effective area contributing to the thermal correction $\delta F$ is found to be of the order of $A^{(T)}\sim R_1\lambda_T$ and thus much larger than the area relevant for $F(L,T)$, which is dominated by the zero-temperature (vacuum) contribution in the low-temperature limit. This result is consistent with the numerical examples for a scalar field presented in Ref. [@Weber2010].
The thermal correction to the Casimir force has been measured in the plane-sphere geometry by employing a coated lens with $R_1=15.6 \, {\rm cm}$ [@Sushkov2011]. The results were analyzed with the help of the PFA, which can be expected to provide an accurate description of the thermal correction if the transverse size of the lens is much larger than $\sqrt{R_1\lambda_T}\sim
1\,\mathrm{mm}$. If our estimate valid for $L\ll \lambda_T$ applies to the experiment where $L\lesssim 0.4\lambda_T$, we can conclude that the lens was indeed of sufficient size.
In most Casimir force measurements, thermal corrections are typically very small. Nevertheless, our estimation of an enlarged effective area is still relevant for thermodynamic quantities vanishing in the zero-temperature limit, in particular for the Casimir entropy.
Conclusions {#sec:conclusions}
===========
For two spheres of arbitrary radii, we have derived the proximity force approximation expression for the Casimir free energy as the leading asymptotic result for distances between the spheres much smaller than their radii. We have made use of the WKB Mie scattering amplitudes where only the direct reflection term in the Debye expansion contributes to leading order. The trace over a number of round-trip matrices has been evaluated within the saddle-point approximation. The saddle point corresponds to the conservation of the wave-vector component perpendicular to the line connecting the centers of the two spheres. Therefore, the leading asymptotics results from specular reflection in the vicinity of the points of closest distance between the spheres. As an important consequence, we find that no polarization mixing contributes to leading order. The special case of the plane-sphere geometry is recovered by taking the radius of one sphere to infinity.
Although our approach is based on the momentum representation, we are able to estimate the effective area contributing to the Casimir interaction by making use of the localization principle, which allows us to associate a specific impact parameter to a given scattering angle in the WKB approximation. Taken together, the results presented here show that the PFA regime is governed by local scattering from an area of the order of $R_1 L$ around the points of closest approach, where $R_1$ is the radius of the smaller sphere. On the other hand, for thermal corrections in the low-temperature regime, the area becomes much larger and is of the order of $R_1 \lambda_T.$ As not all Casimir experiments make use of whole spheres, these estimations provide a condition on the minimum size of the spherical surface required for the PFA to hold for the sphere-sphere or plane-sphere geometries. From a more theoretical perspective, our results help to understand why local approaches such as the derivative expansion are capable of providing both the leading and next-to-leading order terms in several situations of interest.
We thank A. Lambrecht, S. Reynaud, and H. M. Nussenzveig for inspiring discussions. This work has been supported by CAPES and DAAD through the PROBRAL collaboration program. PAMN also thanks the Brazilian agencies National Council for Scientific and Technological Development (CNPq), the National Institute of Science and Technology Complex Fluids (INCT-FCx), the Carlos Chagas Filho Foundation for Research Support of Rio de Janeiro (FAPERJ) and the São Paulo Research Foundation (FAPESP).
Derivation of the reflection operator matrix elements {#sec:appendixA}
=====================================================
In the sphere-sphere geometry, the axis connecting the centers of the two spheres is distinguished. We have defined it as $z$-axis and used the polarization basis $\{\hat{\bm{\epsilon}}_\text{TE}, \hat{\bm{\epsilon}}_\text{TM}\}$ taken with respect to the incidence plane as specified in (\[eq:polarizationTETM\]). When a plane wave $\vert\mathbf{K}_i, p_i\rangle$ is scattered into a plane wave $\vert\mathbf{K}_j, p_j\rangle$, the relations between the polarization basis vectors can be expressed as $$\begin{aligned}
\hat{\bm{\epsilon}}_{\rm TE}({\bf K}_i) \cdot \hat{\bm{\epsilon}}_{\rm TE}({\bf K}_j)&=\phantom{-}\cos(\varphi)\\
\hat{\bm{\epsilon}}_{\rm TM}({\bf K}_i) \cdot \hat{\bm{\epsilon}}_{\rm TM}({\bf
K}_j)&=-\frac{c^2}{\xi^2}\left[k_ik_j-\phi_i\phi_j\kappa_i\kappa_j\cos(\varphi)\right]\\
\hat{\bm{\epsilon}}_{\rm TE}({\bf K}_i) \cdot \hat{\bm{\epsilon}}_{\rm TM}({\bf K}_j)&=-\frac{c\phi_j\kappa_j}{\xi}\sin(\varphi)\\
\hat{\bm{\epsilon}}_{\rm TM}({\bf K}_i) \cdot \hat{\bm{\epsilon}}_{\rm TE}({\bf K}_j)&=\phantom{-}\frac{c\phi_i\kappa_i}{\xi}\sin(\varphi)\,,
\label{eq:scalarproducts_inc}
\end{aligned}$$ where $\varphi = \varphi_j-\varphi_i$.
Another distinguished polarization basis is defined by the scattering plane spanned by the two wave vectors involved in the scattering process. In the corresponding basis with polarization vectors perpendicular and parallel to the scattering plane defined as $$\begin{aligned}\label{eq:polarizationPerpPara}
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i) &= \frac{\hat{\mathbf{K}}_j \times \hat{\mathbf{K}}_i}
{\vert\hat{\mathbf{K}}_j \times \hat{\mathbf{K}}_i\vert} \\
\hat{\bm{\epsilon}}_\parallel(\mathbf{K}_i) &= \hat{\bm{\epsilon}}_\perp \times \hat{\mathbf{K}}_i \,,
\end{aligned}$$ respectively, the polarization is conserved during the scattering process. More specifically, we have [@BH] $$\begin{aligned}
\mathcal{R}\ket{\mathbf{K}_i,\perp}&=\frac{2\pi c}{\xi \kappa_j}S_1\ket{\mathbf{K}_j,\perp}\\
\mathcal{R}\ket{\mathbf{K}_i,\parallel}&=\frac{2\pi c}{\xi
\kappa_j}S_2\ket{\mathbf{K}_j,\parallel}
\end{aligned}$$ with the scattering amplitudes $S_1$ and $S_2$ defined in (\[eq:SA\]). The basis vectors (\[eq:polarizationPerpPara\]) for the incoming and outgoing wave vectors are related by $$\begin{aligned}
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i) \cdot \hat{\bm{\epsilon}}_\perp(\mathbf{K}_j) &= 1\\
\hat{\bm{\epsilon}}_\parallel(\mathbf{K}_i) \cdot \hat{\bm{\epsilon}}_\parallel(\mathbf{K}_j) &=\cos(\Theta)\\
\hat{\bm{\epsilon}}_\parallel(\mathbf{K}_i) \cdot \hat{\bm{\epsilon}}_\perp(\mathbf{K}_j)& = 0 \\
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i) \cdot \hat{\bm{\epsilon}}_\parallel(\mathbf{K}_j) &=0\,.
\end{aligned}$$
The coefficients $A, B, C,$ and $D$ appearing in (\[eq:RS\]) reflect the change of polarization basis. They can be expressed as $$\begin{aligned}
A &= \big(\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_j)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_j)\big)
\big(\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_i)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i)\big)\\
B &= \big(\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_j)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_j)\big)
\big(\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_i)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i)\big)\\
C &=-\big(\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_j)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_j)\big)
\big(\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_i)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i)\big)\\
D &= \big(\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_j)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_j)\big)
\big(\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_i)\cdot
\hat{\bm{\epsilon}}_\perp(\mathbf{K}_i)\big)\,.
\end{aligned}$$ Alternative expressions can be obtained by means of the relations $$\begin{aligned}
\hat{\bm{\epsilon}}_\text{TE}\cdot\hat{\bm{\epsilon}}_\parallel &=
-\hat{\bm{\epsilon}}_\text{TM}\cdot\hat{\bm{\epsilon}}_\perp\\
\hat{\bm{\epsilon}}_\text{TM}\cdot\hat{\bm{\epsilon}}_\parallel &=
\hat{\bm{\epsilon}}_\text{TE}\cdot\hat{\bm{\epsilon}}_\perp\,.
\end{aligned}$$
Expressing the scalar products (\[eq:scalarproducts\_inc\]) in terms of the polarization basis $\{\hat{\bm{\epsilon}}_\perp, \hat{\bm{\epsilon}}_\parallel\}$, we find the relations $$\begin{aligned}
\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_i)\cdot\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_j)
&= A+B\cos(\Theta)\\
\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_i)\cdot\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_j)
&= A\cos(\Theta)+B\\
\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_i)\cdot\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_j)
&= -C-D\cos(\Theta)\\
\hat{\bm{\epsilon}}_\text{TM}(\mathbf{K}_i)\cdot\hat{\bm{\epsilon}}_\text{TE}(\mathbf{K}_j)
&= C\cos(\Theta)+D\,.
\end{aligned}$$ Solving for the coefficients $A, B, C,$ and $D$ and making use of (\[eq:scalarproducts\_inc\]), we finally obtain
$$\begin{aligned}
A(\mathbf{K}_{i},\mathbf{K}_{j}) &=
\phantom{-}\frac{\xi^4\cos(\varphi)-c^4\big[k_ik_j\cos(\varphi)-\phi_i\phi_j\kappa_i\kappa_j\big]
\big[k_ik_j-\phi_i\phi_j\kappa_i\kappa_j\cos(\varphi)\big]}
{\xi^4-c^4\big[k_ik_j\cos(\varphi)-\phi_i\phi_j\kappa_i\kappa_j\big]^2}\\
B(\mathbf{K}_{i},\mathbf{K}_{j}) & = -\frac{\xi^2 c^2 k_i k_j\sin^2(\varphi)}
{\xi^4-c^4\big[k_ik_j\cos(\varphi)-\phi_i\phi_j\kappa_i\kappa_j\big]^2} \\
C(\mathbf{K}_{i},\mathbf{K}_{j}) & =\phantom{-} c^3\xi\sin(\varphi)\frac{k_ik_j\phi_i\kappa_i\cos(\varphi)-k_i^2\phi_j\kappa_j}
{\xi^4-c^4\big[k_ik_j\cos(\varphi)-\phi_i\phi_j\kappa_i\kappa_j\big]^2} \\
D(\mathbf{K}_i,\mathbf{K}_j) &= C(-\mathbf{K}_j,-\mathbf{K}_i)\,.
\end{aligned}
\label{eq:abcd}$$
For $\xi=0$, they simplify to $$A=-\phi_i\phi_j,\quad B=C=D=0\,,$$ and for $\mathbf{k}_i=\mathbf{k}_j$ we find $$A=1,\quad B=C=D=0\,.$$ The matrix elements of the reflection operators are not all mutually independent since they fulfill reciprocity relations [@Carminati98; @Messina2011; @Messina2015]. In our notation these relations read $$\begin{gathered}
\kappa_i\bra{\mathbf{K}_i, p_i} \mathcal{R} \ket{\mathbf{K}_j,p_j} = \\
\kappa_j (-1)^{p_i+p_j}\bra{-\mathbf{K}_j, p_j} \mathcal{R} \ket{-\mathbf{K}_i,p_i}\end{gathered}$$ where $(-1)^{p_i+p_j}$ is $+1$ if the polarizations $p_i$ and $p_j$ are equal and $-1$ otherwise. Indeed, since $$\begin{aligned}
A(\mathbf{K}_i,\mathbf{K}_j) &= A(-\mathbf{K}_j,-\mathbf{K}_i)\\
B(\mathbf{K}_i,\mathbf{K}_j) &= B(-\mathbf{K}_j,-\mathbf{K}_i)\,,
\end{aligned}$$ it is straightforward to verify that the coefficients (\[eq:abcd\]) satisfy the reciprocity relations.
Low-frequency limit of the scattering amplitudes {#sec:appendixB}
================================================
The WKB approximation for the scattering amplitudes (\[eq:SA-asymptotics\]) at a sphere discussed in Sect. \[sec:wkb\] has been derived for large size parameters $\xi R/c$. As a consequence, the zero-frequency contribution in the Matsubara sum (\[eq:F\]) is a priori not covered by (\[eq:SA-asymptotics\]).
By analyzing the low-frequency limit of the scattering amplitudes, we will demonstrate that the scattering amplitudes (\[eq:SA-asymptotics\]) and the matrix elements (\[eq:RS\]) obtained from them can be employed even in the limit of zero frequency. At this point, it is worth noting that even though the scattering amplitudes (\[eq:SA-asymptotics\]) vanish in the limit $\xi\to0$, this is not the case for the matrix elements (\[eq:RS\]). Therefore, we need to keep terms linear in $\xi$ in the low-frequency expression for the scattering amplitudes.
In the following, we will distinguish three classes of materials: perfect reflectors, real metals with a finite dc conductivity and dielectrics. For perfect reflectors the permittivity is infinite for all frequencies, comprising also the plasma model when the sphere radius is much larger than the plasma wavelength. Real metals exhibit a finite permittivity except in the zero-frequency limit where the finite dc conductivity gives rise to a divergence proportional to $1/\xi$. Finally, for dielectrics, the permittivity remains finite for $\xi\to 0$.
We start from the expressions (\[eq:SA\]) for the Mie scattering amplitudes and first consider the material-independent functions $\pi_\ell$ and $\tau_\ell$. Noting that according to (\[eq:costheta\]), $\cos(\Theta)$ at low frequencies diverges like $1/\xi^2$, we find the dominant low-frequency behavior $$\begin{aligned}
\pi_\ell\big(\cos(\Theta)\big) &\simeq \frac{(2\ell)!}{2^\ell (\ell-1)! \ell!} \cos^{\ell-1}(\Theta)
\sim\frac{1}{\xi^{2\ell-2}}\,, \\
\tau_\ell\big(\cos(\Theta)\big) &\simeq \frac{(2\ell)!}{2^\ell [(\ell-1)!]^2} \cos^{\ell}(\Theta)
\sim\frac{1}{\xi^{2\ell}}\,.\end{aligned}$$ As a consequence, among the four combinations of these two functions and the two Mie coefficients in (\[eq:SA\]), only those involving $\tau_\ell$ can potentially lead to contributions linear in $\xi$. Terms involving $\pi_\ell$ yield an additional factor $\xi^2$ and can thus be disregarded. Furthermore, only Mie coefficients going like $\xi^{2\ell+1}$ can then lead to a relevant contribution to the scattering amplitudes.
Such a behavior is found for the Mie coefficient $a_\ell$ for which the leading term at low frequencies can be expressed as $$a_\ell \simeq (-1)^\ell \frac{(\ell+1)(\ell!)^2}{2\ell(2\ell+1)[(2\ell)!]^2}
E^\mathrm{mat}_\ell\left(\frac{2\xi R}{c}\right)^{2\ell+1}\,.$$ The material dependence is contained in the factor $E^\text{mat}_\ell$ (see [@Canaguier-Durand11] for a detailed discussion). For dielectrics, one finds $$E^\text{diel}_\ell = \frac{\varepsilon(0)-1}{\varepsilon(0)+\frac{\ell+1}{\ell}}\,,
\label{eq:ediel}$$ while $E^\text{mat}_\ell=1$ for real metals and perfect reflectors. For the Mie coefficient $b_\ell$, the required frequency dependence is only found for perfect reflectors where $$b_\ell \sim - \frac{\ell}{\ell+1} a_\ell\,.$$ In contrast, the low-frequency behavior of the Mie coefficient $b_\ell$ for real metals is of order $\xi^{2\ell+2}$ and for dielectrics of order $\xi^{2\ell+3}$.
Let us now first consider the scattering amplitude $S_1(\Theta)$ which according to the preceding analysis is only nonvanishing for perfect metals. This finding is in agreement with the fact that for zero frequency the reflection coefficient for TE modes vanishes for metals with finite dc conductivity and dielectrics. Inserting the Mie coefficient $b_\ell$ and the function $\tau_\ell(\cos\Theta)$ for perfect metals, we obtain $$S_1(\Theta) \simeq -\frac{\xi R}{c}\sum_{\ell=1}^\infty\frac{\ell}{\ell+1}
\frac{\left(\frac{R}{c}\big(-2\xi^2\cos(\Theta)\big)^{1/2}\right)^{2\ell}}{(2\ell)!}$$ We recall that in the matrix elements (\[eq:RS\]) of the reflection operator this function appears together with a prefactor $\xi^{-1}$. Furthermore, according to (\[eq:costheta\]), $\xi^2\cos(\Theta)$ does not vanish in the limit $\xi\to
0$. For large radius $R$, we then find $$S_1(\Theta) \simeq -\frac{\xi R}{2c}e^{2(\xi R/c)\sin(\Theta/2)}
\label{eq:S1xi0}$$ where we made use of the relation $$\left(-2\cos(\Theta)\right)^{1/2} = 2\sin(\Theta/2)$$ valid for $\xi=0$. For perfect reflectors, where $r_\text{TE}=-1$, (\[eq:S1xi0\]) agrees with the WKB result (\[eq:SA-asymptotics\]).
The validity of the WKB approximation of $S_2(\Theta)$ can be proven along the same lines, observing that for perfect reflectors and real metals, $r_\text{TM}=1$ at zero frequency. For large $R$, the dominant contribution to the scattering amplitudes arises from large angular momenta $\ell$. Therefore, $E^\text{diel}_\ell$ can be replaced by $(\varepsilon(0)-1)/(\varepsilon(0)+1)$ which according to (\[eq:Fresnel-coeff\]) agrees with $r_\text{TM}$ in the low frequency limit. This completes the proof of the applicability of (\[eq:SA-asymptotics\]) even in the limit $\xi\to 0$.
[99]{}
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|
---
abstract: 'This paper delineates the first steps in a systematic quantitative study of the spacetime fluctuations induced by quantum fields in an evaporating black hole under the stochastic gravity program. The central object of interest is the noise kernel, which is the symmetrized two-point quantum correlation function of the stress tensor operator. As a concrete example we apply it to the study of the spherically-symmetric sector of metric perturbations around an evaporating black hole background geometry. For macroscopic black holes we find that those fluctuations grow and eventually become important when considering sufficiently long periods of time (of the order of the evaporation time), but well before the Planckian regime is reached. In addition, the assumption of a simple correlation between the fluctuations of the energy flux crossing the horizon and far from it, which was made in earlier work on spherically-symmetric induced fluctuations, is carefully scrutinized and found to be invalid. Our analysis suggests the existence of an infinite amplitude for the fluctuations when trying to localize the horizon as a three-dimensional hypersurface, as in the classical case, and, as a consequence, a more accurate picture of the horizon as possessing a finite effective width due to quantum fluctuations. This is supported by a systematic analysis of the noise kernel in curved spacetime smeared with different functions under different conditions, the details are collected in the appendices. This case study shows a pathway for probing quantum metric fluctuations near the horizon and understanding their physical meaning.'
author:
- 'B. L. Hu'
- Albert Roura
title: Metric fluctuations of an evaporating black hole from back reaction of stress tensor fluctuations
---
Introduction {#sec1}
============
Studying the dynamics of quantum fields in a fixed curved spacetime, Hawking found that black holes emit thermal radiation with a temperature inversely proportional to their mass [@hawking75]. When the back reaction of the quantum fields on the spacetime dynamics is included, one expects that the mass of the black hole decreases as thermal radiation at higher and higher temperatures is emitted. This picture, which constitutes the process known as black hole evaporation, is indeed obtained from semiclassical gravity calculations which are believed to be valid at least before the Planckian scale is reached [@bardeen81; @massar95].
Semiclassical gravity [@birrell94; @wald94; @flanagan96] is a mean field description that neglects the fluctuations of the spacetime geometry. However, a number of studies have suggested the existence of large fluctuations near black hole horizons [@sorkin95; @sorkin97; @casher97; @marolf03] (and even instabilities [@mazur04]) with characteristic time-scales much shorter than the black hole evaporation time. In all of them[^1] either states which are singular on the horizon (such as the Boulware vacuum for Schwarzschild spacetime) were explicitly considered, or fluctuations were computed with respect to those states and found to be large near the horizon. Whether these huge fluctuations are of a generic nature or an artifact arising from the consideration of states singular on the horizon is an issue that deserves further investigation. By contrast, the fluctuations for states regular on the horizon were estimated in Ref. [@wu99] and found to be small even when integrated over a time of the order of the evaporation time.
These apparently contradictory claims and the fact that most claims on black hole horizon fluctuations were based on qualitative arguments and/or semi-quantitative estimates prompted us here to strive for a more quantitative and self-consistent description[^2]. For this endeavor we follow the stochastic gravity program [@calzetta94; @martin99a; @martin99b; @hu03a; @hu04a]. We will consider the fluctuations of metric perturbations around a black hole geometry interacting with a quantum scalar field whose stress tensor drives the dynamics. The evolution of the mean background geometry is given by the semiclassical Einstein equation (with self-consistent back reaction from the expectation value of the stress tensor) while the metric fluctuations obey an Einstein-Langevin equation [@hu95a; @hu95b; @campos96; @lombardo97] with a Gaussian stochastic source whose correlation function is given by the noise kernel, which characterizes the fluctuations of the stress tensor of the quantum fields. In contrast to the claims made before, we find here that even for states regular on the horizon the accumulated fluctuations become significant by the time the black hole mass has changed substantially, but well before reaching the Planckian regime. Our result is different from those obtained in prior studies, but in agreement with earlier work by Bekenstein [@bekenstein84].
The stochastic gravity program provides perhaps the best available framework to study quantum metric fluctuations, because while semiclassical gravity is a mean field description that does not take into account quantum metric fluctuations, the Einstein-Langevin equation enables one to solve for the dynamics of metric fluctuations induced by the fluctuations of the stress tensor of the quantum fields. Furthermore, the correlation functions that one obtains are equivalent to the quantum correlation functions for the metric perturbations around the semiclassical background that would follow from a quantum field theory treatment, up to a given order in an expansion in terms of the inverse number of fields [@roura03b; @hu04b]. The quantization of these metric perturbations should be understood in the framework of a low-energy effective field theory approach to quantum gravity [@burgess04], which is expected to provide reliable results for phenomena involving typical length-scales much larger than the Planck length even if the microscopic details of the theory at Planckian scales are not known.[^3] A crucial relation assumed in previous investigations [@bekenstein84; @wu99][^4] of the problem of metric fluctuations driven by quantum matter field fluctuations of states regular on the horizon (as far as the expectation value of the stress tensor is concerned) is the existence of correlations between the outgoing energy flux far from the horizon and a negative energy flux crossing the horizon, based on energy conservation arguments. Using semiclassical gravity, such correlations have been confirmed for the expectation value of the energy fluxes, provided that the mass of the black hole is much larger than the Planck mass. However, a more careful analysis, summarized in Sec. \[sec4\], shows that no such simple connection exists for energy flux fluctuations. It also reveals that the fluctuations on the horizon are in fact divergent unless it is treated as an object with a finite width rather than a three-dimensional hypersurface, as in the classical case, and one needs to find an appropriate way of probing the metric fluctuations near the horizon and extracting physically meaningful information. This is a new challenge in the study of metric fluctuations which demands some careful thoughts on what they mean physically and how they can be probed operationally. In Appendices \[appA\] and \[appB\] we give a systematic analysis of the noise kernel in curved spacetime smeared with different functions under different conditions. The non-existence of this commonly invoked relation in this whole subject matter illustrates the limitations of heuristic arguments and the necessity of a detailed and consistent formalism to study the fluctuations near the horizon, in terms of their magnitude, how they are measured and their consequences.
A few technical remarks are in order to delimit the problem under study: First, we will restrict our attention to the spherically-symmetric sector of metric fluctuations, which necessarily implies a partial description of the fluctuations. That is because, contrary to the case for semiclassical gravity solutions, even if one starts with spherically-symmetric initial conditions, the stress tensor fluctuations will induce fluctuations involving higher multipoles. Thus, the multipole structure of the fluctuations is far richer than that of spherically-symmetric semiclassical gravity solutions, but this also means that obtaining a complete solution (including all multipoles) for fluctuations rather than the mean value is much more difficult.
Second, for black hole masses much larger than the Planck mass (otherwise the effective field theory description will break down anyway), one can introduce a useful adiabatic approximation involving inverse powers of the black hole mass. To obtain results to lowest order, it is sufficient to compute the expectation value of the stress tensor operator and its correlation functions in Schwarzschild spacetime. The corrections, proportional to higher powers of the inverse mass, can be neglected for sufficiently massive black holes.
Third, when studying the dynamics of induced metric fluctuations, the additional contribution to the stress tensor expectation value which results from evaluating it using the perturbed metric is often neglected. In the consideration of fluctuations for an evaporating black hole such a term (which will be denoted by $\langle
\hat{T}_{ab}^{(1)} [g+h] \rangle_\mathrm{ren}$ in Sec. \[sec3\]) becomes important when it builds up for long times. The importance of this term is clear when comparing with the simple estimate made by Wu and Ford in Ref. [@wu99], where $\langle \hat{T}_{ab}^{(1)} [g+h]
\rangle _\mathrm{ren}$ was neglected and the fluctuations were found to be small even when integrated over long times, of the order of the evaporation time of the black hole.
The paper is organized as follows. In Sec. \[sec2\] we briefly review the results for the evolution of the mean field geometry of an evaporating black hole obtained in the context of semiclassical gravity. The framework of stochastic gravity is then applied in Sec. \[sec3\] to the study of the spherically-symmetric sector of fluctuations around the semiclassical gravity solution for an evaporating black hole. It has been previously assumed that an exact correlation between the fluctuations of the negative energy flux crossing the horizon and the flux far from it exists. In this paper we want to question this assumption, but in the presentation in Sec. \[sec3\] we accept temporarily such a working hypothesis just so that we can have the common ground to compare our results with those in the literature. In Sec. \[sec4\], we present a careful analysis of this assumption, and show that this supposition is invalid. Further details of this proof can be found in the Appendices \[appA\] and \[appB\] . Finally, in Sec. \[sec5\] we discuss several implications of our results and suggest some directions for further investigation.
Throughout the paper we use Planckian units with $\hbar=c=G=1$ and the $(+,+,+)$ convention of Ref. [@misner73]. We also make use of the abstract index notation of Ref. [@wald84]. Latin indices denote abstract indices, whereas Greek indices are employed whenever a particular coordinate system is considered.
Mean evolution of an evaporating black hole {#sec2}
===========================================
*Semiclassical gravity* provides a mean field description of the dynamics of a classical spacetime where the gravitational back reaction of quantum matter fields is included self-consistently [@birrell94; @wald94; @flanagan96]. It is believed to be applicable to situations involving length-scales much larger than the Planck scale and for which the quantum back-reaction effects due to the metric itself can be neglected as compared to those due to the matter fields. The dynamics of the metric $g_{ab}$ is governed by the semiclassical Einstein equation: $$G_{ab} \left[ g \right] = \kappa \left\langle \hat{T}_{ab}
[g] \right\rangle _\mathrm{ren} \label{einstein1},$$ where $\langle \hat{T}_{ab} [g] \rangle _\mathrm{ren}$ is the renormalized expectation value of the stress tensor operator of the quantum matter fields and $\kappa = 8 \pi / m_\mathrm{p}^2$ with $m_\mathrm{p}^2$ being the Planck mass. One must solve both the semiclassical Einstein equation and the equation of motion for the matter fields evolving in that geometry, whose solution is needed to evaluate $\langle \hat{T}_{ab} [g] \rangle _\mathrm{ren}$ self-consistently.
An important application of semiclassical gravity is the study of black hole evaporation due to the back reaction of the Hawking radiation emitted by the black hole on the spacetime geometry. This has been studied in some detail for spherically symmetric black holes [@bardeen81; @massar95]. For a general spherically-symmetric metric there always exists a system of coordinates in which it takes the form $$%ds^2 = - e^{2 \psi(v,r)} \left( 1 - \frac{2 m(v,r)}{r} \right) dv^2
ds^2 = - e^{2 \psi(v,r)} ( 1 - 2 m(v,r)/r ) dv^2
+ 2 e^{\psi(v,r)} dv dr
+ r^2 \left( d\theta^2 + \sin^2 \theta d\varphi^2 \right)
\label{metric1}.$$ This completely fixes the gauge freedom under local diffeomorphism transformations except for an arbitrary function of $v$ that can be added to the function $\psi(v,r)$ and is related to the freedom in reparametrizing $v$ (we will see below how this can also be fixed). In general this metric exhibits an *apparent horizon*, where the expansion of the outgoing radial null geodesics vanishes and which separates regions with positive and negative expansion for those geodesics, at those radii that correspond to (odd degree) zeroes of the $vv$ metric component. We denote the location of the apparent horizon by $r_\mathrm{AH}(v)=2M(v)$, where $M(v)$ satisfies the equation $2m(2M(v),v)=2M(v)$.
Spherical symmetry implies that the components $T_{\theta r}$, $T_{\theta v}$, $T_{\varphi r}$ and $T_{\varphi v}$ vanish and the remaining components are independent of the angular coordinates. Under these conditions the various components of the semiclassical Einstein equation associated with the metric in Eq. (\[metric1\]) become $$\begin{aligned}
\frac{\partial m}{\partial v} &=& 4 \pi r^2 T_v^r
%= - e^\psi L_\mathrm{H}
%+ O \left( m L_{\mathrm{H},v} \right)
\label{einstein2a},\\
\frac{\partial m}{\partial r} &=& - 4 \pi r^2 T_v^v
%= O \left( L_\mathrm{H} \right)
\label{einstein2b},\\
\frac{\partial \psi}{\partial r} &=& 4 \pi r T_{rr}
%= O \left( L_\mathrm{H} / r \right)
\label{einstein2c},\end{aligned}$$ where in the above and henceforth we simply use $T_{\mu \nu}$ to denote the expectation value $\langle \hat{T}_{\mu \nu} [g] \rangle
_\mathrm{ren}$ and employ Planckian units (with $m_\mathrm{p}^2=1$). Note that the arbitrariness in $\psi$ can be eliminated by choosing a parametrization of $v$ such that $\psi$ takes a particular value at a given radius (we will choose that it vanishes at $r=2M(v)$, where the apparent horizon is located); $\psi$ is then entirely fixed by Eq. (\[einstein2c\]).
Solving Eqs. (\[einstein2a\])-(\[einstein2c\]) is no easy task. However, one can introduce a useful adiabatic approximation in the regime where the mass of the black hole is much larger than the Planck mass, which is in any case a necessary condition for the semiclassical treatment to be valid. What this entails is that when $M \gg 1$ (remember that we are using Planckian units) for each value of $v$ one can simply substitute $T_{\mu \nu}$ by its “parametric value” – by this we mean the expectation value of the stress energy tensor of the quantum field in a Schwarzschild black hole with a mass corresponding to $M(v)$ evaluated at that value of $v$. This is in contrast to its dynamical value, which should be determined by solving self-consistently the semiclassical Einstein equation for the spacetime metric and the equations of motion for the quantum matter fields. This kind of approximation introduces errors of higher order in $L_\mathrm{H} \equiv B/M^2$ ($B$ is a dimensionless parameter that depends on the number of massless fields and their spins and accounts for their corresponding grey-body factors; it has been estimated to be of order $10^{-4}$ [@page76]), which are very small for black holes well above Planckian scales. These errors are due to the fact that $M(v)$ is not constant and that, even for a constant $M(v)$, the resulting static geometry is not exactly Schwarzschild because the vacuum polarization of the quantum fields gives rise to a non-vanishing $\langle \hat{T}_{ab} [g] \rangle _\mathrm{ren}$ [@york85].
The expectation value of the stress tensor for Schwarzschild spacetime has been found to correspond to a thermal flux of radiation (with $T_v^r = L_\mathrm{H} / (4 \pi r^2)$) for large radii and of order $L_\mathrm{H}$ near the horizon[^5] [@candelas80; @page82; @howard84a; @howard84b; @anderson95]. This shows the consistency of the adiabatic approximation for $L_\mathrm{H} \ll
1$: the right-hand side of Eqs. (\[einstein2a\])-(\[einstein2c\]) contains terms of order $L_\mathrm{H}$ and higher, so that the derivatives of $m(v,r)$ and $\psi(v,r)$ are indeed small. Furthermore, one can use the $v$ component of the stress-energy conservation equation $$\frac{\partial \left( r^2 T_v^r \right)}{\partial r}
+ r^2 \frac{\partial T_v^v}{\partial v} = 0
\label{conservation1},$$ to relate the $T^r_v$ components on the horizon and far from it. Integrating Eq. (\[conservation1\]) radially, one gets $$(r^2 T^r_v) (r=2M(v),v) = (r^2 T^r_v) (r \approx 6M(v),v) +
O(L_{\mathrm{H}}^2), \label{conservation2}$$ where we considered a radius sufficiently far from the horizon, but not arbitrarily far (*i.e.* $2M(v) \ll r \ll
M(v)/L_\mathrm{H}$). The second condition is necessary to ensure that the size of the horizon has not changed much since the value of $v'$ at which the radiation crossing the sphere of radius $r$ at time $v$ left the region close to the horizon. Note that while in the nearly flat region (for large radii) $T_v^r$ corresponds to minus the outgoing energy flux crossing the sphere of radius $r$, on the horizon, where $ds^2 = 2 e^{\psi(v,r)} dv dr + r^2 \left( d\theta^2 +
\sin^2 \theta d\varphi^2 \right)$, $T_v^r$ equals $T_{vv}$, which corresponds to the null energy flux crossing the horizon. Hence, Eq. (\[conservation2\]) relates the positive energy flux radiated away far from the horizon and the negative energy flux crossing the horizon. Taking into account this connection between energy fluxes and evaluating Eq. (\[einstein2a\]) on the apparent horizon, we finally get the equation governing the evolution of its size: $$\frac{d M}{d v} = - \frac{B}{M^2}
\label{einstein3}.$$ Unless $M(v)$ is constant, the event horizon and the apparent horizon do not coincide. However, in the adiabatic regime their radii are related, differing by a quantity of higher order in $L_\mathrm{H}$: $r_\mathrm{EH}(v) = r_\mathrm{AH}(v) \, (1 + O(L_\mathrm{H}))$.
We close this section with an explanation of why we did not have to deal with terms involving higher-order derivatives and even non-local terms when considering the expectation value of the stress tensor as one would expect for geometries with sufficiently arbitrary spacetime dependence of certain metric components. (This can be seen in explicit calculations for arbitrary Robertson-Walker geometries [@calzetta94; @calzetta97c] or arbitrary small metric perturbations around specific backgrounds [@calzetta87; @campos94].) In our case such non-local and higher-order derivative terms would also appear in the exact expression of the stress tensor expectation value for arbitrary $m(v,r)$ and $\psi(v,r)$ functions. However, the adiabatic approximation for $M \gg 1$ that we have employed effectively gets rid of them since one can replace the higher-derivative terms using Eqs. (\[einstein2a\])-(\[einstein2c\]) recursively and taking into account that the terms on the right-hand side are of order $1/M^2$. Therefore, higher-order derivative terms correspond to higher powers of $1/M^2$ and are highly suppressed for $M \gg 1$. Note that this argument, which is based on the black hole size being much larger than the Planck length, is closely related to the order reduction prescription [@parker93; @flanagan96] that is often used to deal with higher-derivative terms in semiclassical gravity and other back-reaction problems.
Spherically-symmetric induced fluctuations {#sec3}
==========================================
There are situations in which the fluctuations of the stress tensor operator and the metric fluctuations that they induce may be important, so that the mean field description provided by semiclassical gravity is incomplete and even fails to capture the most relevant phenomena (the generation of primordial cosmological perturbations constitutes a clear example of that). *Stochastic gravity* [@calzetta94; @martin99a; @martin99b; @hu03a; @hu04a] provides a framework to study those fluctuations. Its centerpiece is the Einstein-Langevin equation [@hu95a; @hu95b; @campos96; @lombardo97] $$G_{ab}^{(1)}\left[ g+h\right] =\kappa \left\langle
\hat{T}_{ab}^{(1)} [g+h] \right\rangle _\mathrm{ren}
+\kappa \, \xi_{ab}\left[ g\right]
\label{einst-lang1},$$ which governs the dynamics of the metric fluctuations around a background metric $g_{ab}$ that corresponds to a given solution of semiclassical gravity. The superindex $(1)$ indicates that only the terms linear in the metric perturbations should be considered, and $\xi_{ab}$ is a Gaussian stochastic source with vanishing expectation value and correlation function[^6] $\langle \xi_{ab} (x) \xi_{cd} (x') \rangle_\xi = (1/2) \langle \{
\hat{t}_{ab} (x), \hat{t}_{cd} (x') \} \rangle$ (with $\hat{t}_{ab}
\equiv \hat{T}_{ab} - \langle \hat{T}_{ab} \rangle$), where the term on the right-hand side, which accounts for the stress tensor fluctuations within this Gaussian approximation, is commonly known as the noise kernel and denoted by $N_{abcd}(x,x')$. In this framework the metric perturbations are still classical but stochastic. Nevertheless, one can show that the correlation functions for the metric perturbations that one obtains in stochastic gravity are equivalent through order $1/N$ to the quantum correlation functions that would follow from a quantum field theory treatment when considering a large number of fields $N$ [@roura03b; @hu04b]. In particular, the symmetrized two-point function consists of two contributions: *intrinsic* and *induced* fluctuations. The intrinsic fluctuations are a consequence of the quantum width of the initial state of the metric perturbations, and they are obtained in stochastic gravity by averaging over the initial conditions for the solutions of the homogeneous part of Eq. (\[einst-lang1\]) distributed according to the reduced Wigner function associated with the initial quantum state of the metric perturbations. On the other hand, the induced fluctuations are due to the quantum fluctuations of the matter fields interacting with the metric perturbations, and they are obtained by solving the Einstein-Langevin equation using a retarded propagator with vanishing initial conditions.
In this section we study the spherically-symmetric sector \[*i.e.*, the monopole contribution, which corresponds to $l=0$, in a multipole expansion in terms of spherical harmonics $Y_{lm}(\theta,\phi)$\] of metric fluctuations for an evaporating black hole. In this case only induced fluctuations are possible. The fact that intrinsic fluctuations cannot exist can be clearly seen if one neglects vacuum polarization effects, since Birkhoff’s theorem forbids the existence of spherically-symmetric free metric perturbations in the exterior vacuum region of a spherically-symmetric black hole that keep the ADM mass constant. Even when vacuum polarization effects are included, spherically-symmetric perturbations, characterized by $m(v,r)$ and $\psi(v,r)$, are not independent degrees of freedom. This follows from Eqs. (\[einstein2a\])-(\[einstein2c\]), which can be regarded as constraint equations. The fluctuations of the stress tensor are inhomogeneous and non-spherically-symmetric even if we choose a spherically-symmetric vacuum state for the matter fields (spherical symmetry simply implies that the angular dependence of the noise kernel in spherical coordinates is entirely given by the relative angle between the spacetime points $x$ and $x'$). This means that, in contrast to the semiclassical gravity case, projecting onto the $l = 0$ sector of metric perturbations does not give an exact solution of the Einstein-Langevin equation in the stochastic gravity approach that we have adopted here. Nevertheless, restricting to spherical symmetry in this way gives more accurate results than two-dimensional dilaton-gravity models resulting from simple dimensional reduction [@trivedi93; @strominger93; @lombardo99]. This is because we project the solutions of the Einstein-Langevin equation just at the end, rather than considering only the contribution of the $s$-wave modes to the classical action for both the metric and the matter fields from the very beginning. Hence, an infinite number of modes for the matter fields with $l \neq 0$ contribute to the $l = 0$ projection of the noise kernel, whereas only the $s$-wave modes for each matter field would contribute to the noise kernel if dimensional reduction had been imposed right from the start, as done in Refs. [@parentani01a; @parentani01b; @parentani02] as well as in studies of two-dimensional dilaton-gravity models.
The Einstein-Langevin equation for the spherically-symmetric sector of metric perturbations can be obtained by considering linear perturbations of $m(v,r)$ and $\psi(v,r)$, projecting the stochastic source that accounts for the stress tensor fluctuations to the $l=0$ sector, and adding it to the right-hand side of Eqs. (\[einstein2a\])-(\[einstein2c\]). We will focus our attention on the equation for the evolution of $\eta(v,r)$, the perturbation of $m(v,r)$: $$\frac{\partial (m + \eta)}{\partial v} = - \frac{B}{(m + \eta)^2}
+ 4 \pi r^2 \xi_v^r + O \left(L_\mathrm{H}^2 \right)
\label{einst-lang2},$$ which reduces, after neglecting terms of order $L_\mathrm{H}^2$ or higher, to the following equation to linear order in $\eta$: $$\frac{\partial \eta}{\partial v} = \frac{2 B}{m^3} \eta
+ 4 \pi r^2 \xi_v^r
\label{einst-lang3}.$$ It is important to emphasize that in Eq. (\[einst-lang2\]) we assumed that the change in time of $\eta(v,r)$ is sufficiently slow so that the adiabatic approximation employed in the previous section to obtain the mean evolution of $m(v,r)$ can also be applied to the perturbed quantity $m(v,r)+\eta(v,r)$. This is guaranteed as long as the term corresponding to the stochastic source is of order $L_\mathrm{H}$ or higher, a point that will be discussed below.
Obtaining the noise kernel which determines the correlation function for the stochastic source is highly nontrivial even if we compute it on the Schwarzschild spacetime, which is justified in the adiabatic regime for the background geometry. As implicitly done in prior work (for instance in Refs. [@bekenstein84; @wu99]; see, however, Refs. [@parentani01b; @parentani02]), we will assume in this section that the fluctuations of the radiated energy flux far from the horizon are exactly correlated with the fluctuations of the negative energy flux crossing the horizon. This is a crucial assumption which implies an enormous simplification and allows a direct comparison with the results in the existing literature, and its validity will be analyzed more carefully in the next section.[^7]
Since the generation of Hawking radiation is especially sensitive to what happens near the horizon, from now on we will concentrate on the metric perturbations near the horizon[^8] and consider $\eta(v) = \eta(v,2M(v))$. Assuming that the fluctuations of the energy flux crossing the horizon and those far from it are exactly correlated, from Eq. (\[einst-lang3\]) we have $$\frac{d \eta(v)}{d v} = \frac{2 B}{M^3(v)} \eta(v)
+ \xi(v)
\label{einst-lang4},$$ where $\xi(v) \equiv (4 \pi r^2\, \xi_v^r) (v,r \approx 6M(v))$. The correlation function for the spherically-symmetric fluctuation $\xi(v)$ is determined by the integral over the whole solid angle of the $N^{r\;r}_{\;v\;v}$ component of the noise kernel, which is given by $(1/2) \langle \{ \hat{t}_v^r (x), \hat{t}_v^r (x') \} \rangle$. The $l=0$ fluctuations of the energy flux of Hawking radiation far from a black hole formed by gravitational collapse, characterized also by $(1/2) \langle \{ \hat{t}_v^r (x), \hat{t}_v^r (x') \} \rangle$ averaged over the whole solid angle, have been studied in Ref. [@wu99]. Its main features are a correlation time of order $M$ and a characteristic fluctuation amplitude of order $\epsilon_0 /
M^4$ (this is the result of smearing the stress tensor two-point function, which diverges in the coincidence limit, over a period of time of the order of the correlation time). The order of magnitude of $\epsilon_0$ has been estimated to lie between $0.1 B$ and $B$ [@bekenstein84; @wu99]. For simplicity, we will consider quantities smeared over a time of order $M$. We can then introduce the Markovian approximation $(\epsilon_0 / M^3(v)) \delta(v-v')$, which coarse-grains the information on features corresponding to time-scales shorter than the correlation time $M$. Under those conditions $r^2
\xi^r_v$ is of order $1/M^2$ and thus the adiabatic approximation made when deriving Eq. (\[einst-lang2\]) is justified.
The stochastic equation (\[einst-lang4\]) can be solved in the usual way and the correlation function for $\eta(v)$ can then be computed. Alternatively, one can follow Bekenstein [@bekenstein84] and derive directly an equation for $\langle \eta^2 (v) \rangle_\xi$. This is easily done multiplying Eq. (\[einst-lang4\]) by $\eta(v)$ and taking the expectation value. The result is $$\frac{d}{dv} \langle \eta^2 (v) \rangle_\xi
= \frac{4 B}{M^3(v)} \langle \eta^2 (v) \rangle_\xi
+ 2 \langle \eta (v) \xi (v) \rangle_\xi
\label{fluct1}.$$ For delta-correlated noise (the Stratonovich prescription is the appropriate one in this case), $\langle \eta (v) \xi (v) \rangle_\xi$ equals one half the time-dependent coefficient multiplying the delta function $\delta (v-v')$ in the expression for $\langle \xi (v) \xi
(v') \rangle_\xi$, which is given by $\epsilon_0 / M^3(v)$ in our case. Taking that into account, Eq. (\[fluct1\]) becomes $$\frac{d}{dv} \langle \eta^2 (v) \rangle_\xi
= \frac{4 B}{M^3(v)}
\langle \eta^2 (v) \rangle_\xi + \frac{\epsilon_0}{M^3(v)}
\label{fluct2}.$$ Finally, it is convenient to change from the $v$ coordinate to the mass function $M(v)$ for the background solution. Eq. (\[fluct2\]) can then be rewritten as $$\frac{d}{dM} \langle \eta^2 (M) \rangle_\xi
= - \frac{4}{M} \langle \eta^2 (M) \rangle_\xi
- \frac{(\epsilon_0 / B)}{M}
\label{fluct3}.$$ The solutions of this equation are given by $$\langle \eta^2 (M) \rangle_\xi
= \langle \eta^2 (M_0) \rangle_\xi \left(\frac{M_0}{M}\right)^4
+\frac{\epsilon_0}{4 B} \left[\left(\frac{M_0}{M}\right)^4 - 1\right]
\label{fluct4}.$$ Provided that the fluctuations at the initial time corresponding to $M=M_0$ are negligible (much smaller than $\sqrt{\epsilon_0 / 4B} \sim
1$), the fluctuations become comparable to the background solution when $M \sim M_0^{2/3}$. Note that fluctuations of the horizon radius of order one in Planckian units do not correspond to Planck scale physics because near the horizon $\Delta R = r - 2M$ corresponds to a physical distance $L \sim \sqrt{M \, \Delta R}$, as can be seen from the line element for Schwarzschild, $ds^2 = - (1-2M/r) dt^2 +
(1-2M/r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\varphi^2)$, by considering pairs of points at constant $t$. So $\Delta R \sim 1$ corresponds to $L \sim \sqrt{M}$, whereas a physical distance of order one is associated with $\Delta R \sim 1/M$, which corresponds to an area change of order one for spheres with those radii. One can, therefore, have initial fluctuations of the horizon radius of order one for physical distances well above the Planck length provided that we consider a black hole with a mass much larger than the Planck mass. One expects that the fluctuations for states that are regular on the horizon correspond to physical distances not much larger than the Planck length, so that the horizon radius fluctuations would be much smaller than one for sufficiently large black hole masses. Nevertheless, that may not be the case when dealing with states which are singular on the horizon, with estimated fluctuations of order $M^{1/3}$ or even $\sqrt{M}$ [@casher97; @marolf03; @mazur04]. Confirming that the fluctuations are indeed so small for regular states and verifying how generic, natural and stable they are as compared to singular ones is a topic that we plan to address in future investigations. Our result for the growth of the fluctuations of the size of the black hole horizon agrees with the result obtained by Bekenstein in Ref. [@bekenstein84] and implies that, for a sufficiently massive black hole (with a few solar masses or a supermassive black hole), the fluctuations become important before the Planckian regime is reached. Strictly speaking, one cannot expect that a linear treatment of the perturbations provides an accurate result when the fluctuations become comparable to the mean value, but it signals a significant growth of the fluctuations (at least until the nonlinear effects on the perturbation dynamics become relevant).
This growth of the fluctuations which was found by Bekenstein and confirmed here via the Einstein-Langevin equation seems to be in conflict with the estimate given by Wu and Ford in Ref. [@wu99]. According to their estimate, the accumulated mass fluctuations over a period of the order of the black hole evaporation time ($\Delta t \sim M_0^3$) would be of the order of the Planck mass. The discrepancy is due to the fact that the first term on the right-hand side of Eq. (\[einst-lang4\]), which corresponds to the perturbed expectation value $\langle \hat{T}_{ab}^{(1)} [g+h] \rangle
_\mathrm{ren}$ in Eq. (\[einst-lang1\]), was not taken into account in Ref. [@wu99]. The larger growth obtained here is a consequence of the secular effect of that term, which builds up in time (slowly at first, during most of the evaporation time, and becoming more significant at late times when the mass has changed substantially) and reflects the unstable nature of the background solution for an evaporating black hole.[^9]
All this can be qualitatively understood as follows. Consider an evaporating black hole with initial mass $M_0$ and suppose that the initial mass is perturbed by an amount $\delta
M_0 = 1$. The mean evolution for the perturbed black hole (without taking into account any fluctuations) leads to a mass perturbation that grows like $\delta M = (M_0/M)^2 \, \delta M_0 = (M_0/M)^2$, so that it becomes comparable to the unperturbed mass $M$ when $M \sim
M_0^{2/3}$, which coincides with the result obtained above. Such a coincidence has a simple explanation: the fluctuations of the Hawking flux slowly accumulated during most of the evaporating time, which are of the order of the Planck mass, as found by Wu and Ford, give a dispersion of that order for the mass distribution at the time when the instability of the small perturbations around the background solution start to become significant.
Correlation between outgoing and ingoing energy fluxes {#sec4}
======================================================
In this section we address the issue whether the simple relation between the energy flux crossing the horizon and the flux far from it also holds for the fluctuations. As we show in Appendix \[appC\], those correlations vanish for conformal fields in two-dimensional spacetimes. (The correlation function for the outgoing and ingoing null energy fluxes in an effectively two-dimensional model was explicitly computed in Refs. [@parentani01b; @parentani02] and it was indeed found to vanish.) On the other hand, in four dimensions the correlation function does not vanish in general and correlations between outgoing and ingoing fluxes do exist near the horizon (at least partially). This point is also explained in Appendix \[appC\].
For black hole masses much larger than the Planck mass, one can use the adiabatic approximation for the background mean evolution. Therefore, to lowest order in $L_\mathrm{H}$ one can compute the fluctuations of the stress tensor in Schwarzschild spacetime. In Schwarzschild, the amplitude of the fluctuations of $r^2
\hat{T}^r_v$ far from the horizon is of order $1/M^2$ ($= M^2 / M^4$) when smearing over a correlation time of order $M$, which one can estimate for a hot thermal plasma in flat space [@campos98; @campos99] (see also Ref. [@wu99] for a computation of the fluctuations of $r^2 \hat{T}^r_v$ far from the horizon). The amplitude of the fluctuations of $r^2 \hat{T}^r_v$ is thus of the same order as its expectation value. However, their derivatives with respect to $v$ are rather different: since the characteristic variation times for the expectation value and the fluctuations are $M^3$ and $M$ respectively, $\partial (r^2 T^r_v) / \partial v$ is of order $1/M^5$ whereas $\partial (r^2 \xi^r_v) / \partial v$ is of order $1/M^3$. This implies an additional contribution of order $L_\mathrm{H}$ due to the second term in Eq. (\[conservation1\]) if one radially integrates the same equation applied to stress tensor fluctuations (the stochastic source in the Einstein-Langevin equation). Hence, in contrast to the case of the mean value, the contribution from the second term in Eq. (\[conservation1\]) cannot be neglected when radially integrating since it is of the same order as the contributions from the first term, and one can no longer obtain a simple relation between the outgoing energy flux far from the horizon and the energy flux crossing the horizon.
So far we have argued that the method employed for the mean value cannot be employed for the fluctuations. Although one expects that $r^2 \xi^r_v$ on the horizon and far from it will not be equal when including the contributions that results from radially integrating the second term in Eq. (\[conservation1\]), one might wonder whether there is a possibility that those contributions would somehow cancel out. That possibility can, however, be excluded using the following argument. The smeared correlation function $$\int dv h(v) \int dv' h(v')\,
r^4 \langle \xi^r_v (v,r) \xi^r_v (v',r) \rangle_\xi
\label{correlation1},$$ where $h(v)$ is some appropriate smearing function and $\xi^r_v (v,r)$ has already been integrated over the whole solid angle, is divergent on the horizon but finite far from it. Therefore, $r^2 \xi^r_v$ on the horizon and far from it cannot be equal for each value of $v$.
Let us discuss in some more detail the fact that certain smearings of the quantity $r^4 \langle \xi^r_v (v,r) \xi^r_v (v',r) \rangle_\xi$ are divergent on the horizon but finite far from it. The smeared correlation function is related to the noise kernel as follows: $$\int dv dv' h(v) h(v')\,
r^4 \langle \xi^r_v (v,r) \xi^r_v (v',r) \rangle_\xi
= r^4 \int dv dv' h(v) h(v') \int d\Omega d\Omega'
N^{r\;r}_{\;v\;v} (v,r,\theta,\varphi;v',r,\theta',\varphi')
\label{correlation2}.$$ The noise kernel is divergent in the coincident limit or for null-separated points. Smearing the noise kernel along all directions gives a finite result. However, although certain partial smearings also give a finite result, others do not. For instance, smearing along a timelike direction yields a finite result, whereas smearing on a spacelike hypersurface yields in general a divergent result [@ford05]. On the other hand, the result of smearing along two “transverse” null directions (two null directions sharing the same orthogonal spacelike 2-surfaces) is also finite, but not for a smearing along just one null direction even if we also smear along the orthogonal spacelike directions. For $r>2M$ Eq. (\[correlation2\]) corresponds to a smearing along a timelike direction and gives a finite result for the smeared correlation function, but on the horizon it corresponds to a smearing along a single null direction and it is divergent.
A proof of the results described in the previous paragraph is provided in Appendices \[appA\] and \[appB\] by considering a product of smearing functions involving all directions and then taking different kinds of limits in which the smearing size along certain directions vanishes. One limit corresponds to taking a vanishing size for the smearing along one of the null directions and we refer to it as *smearing along null geodesics*. The other corresponds to taking vanishing sizes for certain spatial directions and we refer to it as *smearing along timelike curves*. The proof proceeds in two steps. First, in Appendix \[appA\] it is shown for the flat space case. Then it is generalized to curved spacetimes in Appendix \[appB\] using a quasilocal expansion in terms of Riemann normal coordinates.
Discussion {#sec5}
==========
Following the stochastic gravity program, in Sec. \[sec3\] we found that the spherically-symmetric fluctuations of the horizon size of an evaporating black hole become important at late times, and even comparable to its mean value when $M \sim M_0^{2/3}$, where $M_0$ is the mass of the black hole at some initial time when the fluctuations of the horizon radius are much smaller than the Planck length.[^10] This is consistent with the result previously obtained by Bekenstein in Ref. [@bekenstein84].
It is important to realize that for a sufficiently massive black hole, the fluctuations become significant well before the Planckian regime is reached. More specifically, for a solar mass black hole they become comparable to the mean value when the black hole radius is of the order of $10 \mathrm{nm}$, whereas for a supermassive black hole with $M \sim 10^7 M_\odot$, that happens when the radius reaches a size of the order of $1 \mathrm{mm}$. One expects that in those circumstances the low-energy effective field theory approach of stochastic gravity should provide a reliable description.
It is worth mentioning that other properties of the black hole can exhibit substantial fluctuations when taking into account the back reaction of Hawking radiation. As pointed out by Page in Ref. [@page80], by momentum conservation the fluctuations of the total momentum of the Hawking radiation emitted will cause a recoil of the position of the black hole, which will also fluctuate. According to Page’s estimate, the spread of the distribution for the black hole position will become comparable to the size of the horizon by the time an energy of order $M^{1/3}$ has been emitted. This kind of fluctuations, which were not obtained in our calculation because we restricted our attention to spherically-symmetric metric perturbations, exhibit certain features that differ significantly from those of our result. The fluctuations that we found take a much longer time to build up and depend crucially on the unstable behavior of small perturbations of the semiclassical solution, with characteristic time-scales of the order of the evaporation time. On the contrary, this unstable behavior plays no major role in the growth of the position fluctuations. Furthermore, since this growth is still much slower than the emission rate of Hawking quanta, in the frame where the black hole is at rest the properties of the Hawking radiation being emitted remain essentially unchanged when the position fluctuations are taken into account, whereas the fluctuations of the horizon size do imply fluctuations in the temperature of the Hawking radiation.
Due to the nonlinear nature of the back-reaction equations, such as Eq. (\[einst-lang2\]), the fact that the fluctuations of the horizon size can grow and become comparable to the mean value implies non-negligible corrections to the dynamics of the mean value itself. This can be seen by expanding Eq. (\[einst-lang2\]) (evaluated on the horizon) in powers of $\eta$ and taking the expectation value. Through order $\eta^2$ we get $$\begin{aligned}
\frac{d (M(v) + \langle \eta(v) \rangle_\xi)}{d v}
&=& - \left \langle \frac{B}{(M(v) + \eta(v))^2} \right \rangle_\xi
\nonumber \\
&=& - \frac{B}{M^2(v)} \left[ 1 - \frac{2}{M(v)} \langle \eta (v) \rangle_\xi
+ \frac{3}{M^2(v)} \langle \eta^2 (v) \rangle_\xi + O \left( \frac{\eta^3}{M^3}
\right) \right] \label{rad_correction}.\end{aligned}$$ When the fluctuations become comparable to the mass itself, the third term (and higher order terms) on the right-hand side is no longer negligible and we get non-trivial corrections to Eq. (\[einstein3\]) for the dynamics of the mean value. These corrections can be interpreted as higher order radiative corrections to semiclassical gravity that include the effects of metric fluctuations on the evolution of the mean value. For instance, the third term on the right-hand side of Eq. (\[rad\_correction\]) would correspond to a two-loop Feynman diagram involving a matter loop with an internal propagator for the metric perturbations (restricted to the spherically-symmetric sector in our case), as compared to just one matter loop, which is all that semiclassical gravity can account for.
Does the existence of the significant deviations for the mean evolution mentioned above imply that the results based on semiclassical gravity obtained by Bardeen and Massar in Refs. [@bardeen81; @massar95] are invalid? Several remarks are in order. First of all, those deviations start to become significant only after a period of the order of the evaporation time when the mass of the black hole has decreased substantially. Secondly, since fluctuations were not considered in those references, a direct comparison cannot be established. However, we can compare the average of the fluctuating ensemble with their results. Doing so exhibits an evolution that deviates significantly when the fluctuations become important. Nevertheless, if one considers a single member of the ensemble at that time, its evolution will be accurately described by the corresponding semiclassical gravity solution until the fluctuations around that particular solution become important again, after a period of the order of the evaporation time associated with the new initial value of the mass at that time.
An interesting aspect that we have not addressed in this work, but which is worth investigating, is the quantum coherence of those fluctuations. It seems likely that, given the long time periods involved and the size of the fluctuations, the entanglement between the Hawking radiation emitted and the black hole spacetime geometry will effectively decohere the large horizon fluctuations, rendering them equivalent to an incoherent statistical ensemble.
In this paper we have taken a first step to put the study of metric fluctuations in black hole spacetimes on a firmer basis by considering a detailed derivation of the results from an appropriate formalism rather than using heuristic arguments or simple estimates. The spirit is somewhat analogous to the study of the mean back-reaction effect of Hawking radiation on a black hole spacetime geometry (both for black holes in equilibrium and for evaporating ones) by considering the solutions of semiclassical gravity in that case rather than just relying on simple energy conservation arguments. In order to obtain an explicit result from the stochastic gravity approach and compare with earlier work, in Sec. \[sec3\] we employed a simplifying assumption implicitly made in most prior work: the existence of a simple connection between the outgoing energy flux fluctuations far from the horizon and the negative energy flux fluctuations crossing the horizon. In Sec. \[sec4\] we analyzed this assumption carefully and showed it to be invalid. This strongly suggests that one needs to study the stress tensor fluctuations from an explicit calculation of the noise kernel near the horizon. This quantity is obtainable from the stochastic gravity program and calculation is underway [@phillips01; @phillips03; @eftekharzadeh07].
A possible way to compute the noise kernel near the horizon could be to use an approximation scheme based on a quasilocal expansion such as Page’s approximation [@page82] or similar methods corresponding to higher order WKB expansions [@anderson95].[^11] With these techniques one can obtain an approximate expression for the Wightman function of the matter fields, which is the essential object needed to compute the noise kernel. Unfortunately these approximations are only accurate for pairs of points with a small separation scale and break down when it becomes comparable to the black hole radius. Therefore, especial care is needed when studying the $l=0$ multipole since that corresponds to averaging the noise kernel over the whole solid angle, which involves typical separations for pairs of points on the horizon of the order of the black hole radius, and one needs to make sure that the integral is dominated by the contribution from small angular separations. Alternatively, one might hope to gain some insight on the fluctuations near a black hole horizon by studying the fluctuations of the event horizon surrounding any geodesic observer in de Sitter spacetime, which exhibits a number of similarities with the event horizon of a black hole in equilibrium [@gibbons77a]. In contrast to the black hole case, it may be possible to obtain exact analytical results for de Sitter space due to its high degree of symmetry.
Furthermore, as explained in Sec. \[sec4\] and shown in detail in Appendices \[appA\] and \[appB\], the noise kernel smeared over the horizon is divergent, and so are the induced metric fluctuations. Hence, one cannot study the fluctuations of the horizon as a three-dimensional hypersurface for each realization of the stochastic source because the amplitude of the fluctuations is infinite, even when restricting one’s attention to the $l=0$ sector. Instead one should regard the horizon as possessing a finite effective width due to quantum fluctuations. In order to characterize its width one must find a sensible way of probing the metric fluctuations near the horizon and extracting physically meaningful information, such as their effect on the Hawking radiation emitted by the black hole. One possibility is to study how metric fluctuations affect the propagation of a bundle of null geodesics [@barrabes99; @barrabes00; @parentani01a; @parentani01b; @parentani02; @ford97]. One expects that this should provide a description of the effects on the propagation of a test field whenever the geometrical optics approximation is valid. However, preliminary analysis of simpler cases with a quantum field theory treatment suggest that when including quantum vacuum metric fluctuations the geometrical optics approximation becomes invalid. Another possibility, which seems to constitute a better probe of the metric fluctuations, is to analyze their effect on the two-point quantum correlation functions of a test field. The two-point functions characterize the response of a particle detector for that field and can be used to obtain the expectation value and the fluctuations of the stress tensor of the test field.
Finally, since the large fluctuations suggested in Refs. [@sorkin95; @sorkin97; @casher97; @marolf03] involve time-scales much shorter than the evaporation time (contrary to those considered in this paper) and high multipoles, one expects that for a sufficiently massive black hole the spacetime near the horizon can be approximated by Rindler spacetime (identifying the black hole horizon and the Rindler horizon) provided that we restrict ourselves to sufficiently small angular scales. Thus, analyzing the effect of including the interaction with the metric fluctuations on the two-point functions of a test field propagating in flat space, which is technically much simpler, could provide useful information for the black hole case.
We thank Paul Anderson, Larry Ford, Valeri Frolov, Ted Jacobson, Don Marolf, Emil Mottola, Don Page, Renaud Parentani and Rafael Sorkin for useful discussions. B. L. H. appreciates the hospitality of Professor Stephen Adler while visiting the Institute for Advanced Study, Princeton in Spring 2007. This work is supported in part by an NSF Grant PHY-0601550. A. R. was also supported by LDRD funds from Los Alamos National Laboratory.
Correlations between outgoing and ingoing fluxes in $1+1$ and $3+1$ dimensions {#appC}
==============================================================================
Any spacetime metric in $1+1$ dimensions is conformally flat (at least locally) and can be written as $$ds^2 = - C(u,v) du dv
\label{metric2}.$$ In terms of these null coordinates, the conservation equation for the stress tensor, $\nabla^a T_{ab} = 0$, reduces to $$\begin{aligned}
\partial_v \hat{T}_{uu} + \partial_u \hat{T}_{vu} - \Gamma^u_{uu} T_{vu} &=& 0
\label{conservation2a}, \\
\partial_u \hat{T}_{vv} + \partial_v \hat{T}_{uv} - \Gamma^v_{vv} T_{uv} &=& 0
\label{conservation2b},\end{aligned}$$ since all the other relevant Christoffel symbols vanish. Taking into account that $\Gamma^u_{uu} = \partial_u (\ln C)$, $\Gamma^v_{vv} = \partial_v (\ln C)$ and combining Eqs. (\[conservation2a\])-(\[conservation2b\]) we get $$\begin{aligned}
\partial_v \hat{T}_{uu} + \partial_u \hat{T}_{vv}
&=& - C \, \partial_t ( \hat{T}_{uv} / C )
\label{conservation3a}, \\
\partial_v \hat{T}_{uu} - \partial_u \hat{T}_{vv}
&=& C \, \partial_x ( \hat{T}_{uv} / C )
\label{conservation3b},\end{aligned}$$ where we introduced the coordinates $t = (u + v)/2$ and $x = (v -
u)/2$.
This result can be applied to the Schwarzschild geometry in $1+1$ dimensions identifying $t$ with the usual Killing time and $x$ with the Regge-Wheeler coordinate $r_*$. In that case we have $C(t,r) =
(1-2M/r)$. If we consider a massless conformally coupled field in $1+1$ dimensions (conformal and minimal coupling are equivalent in that case), the trace of the stress tensor (which is related to the $T_{uv}$ component) vanishes at the classical level and is entirely given by the trace anomaly $\langle \hat{T}^\mu_\mu
\rangle_\mathrm{ren} = R / 24 \pi = M / 6 \pi r^3$. Since both the trace anomaly and the conformal factor are time-independent, the term on the right-hand side of Eqs. (\[conservation3a\]) vanishes. Therefore, it follows from Eq. (\[conservation3a\]) that the generation of left-moving and right-moving null mean fluxes is perfectly anticorrelated, which implies that the positive energy flux of outgoing Hawking radiation equals, in absolute value, the negative energy flux crossing the horizon. Moreover, from Eq. (\[conservation3a\]) and the fact that $\langle \hat{T}_{uv}
\rangle_\mathrm{ren} / C = - \langle \hat{T}^\mu_\mu
\rangle_\mathrm{ren} / 4 = - M / 24 \pi r^3$, which implies $C\, \partial_{r_*}\! (\langle \hat{T}_{uv} \rangle_\mathrm{ren} / C)
= C^2 \, M / 6 \pi r^4$, it is clear that the amount of anticorrelated mean fluxes generated tends to zero for large radii and is largest at $r=3M$.
For energy flux fluctuations, the situation is very different. The trace anomaly does not fluctuate, *i.e.*, the trace of the stress tensor does not fluctuate for conformal fields [@martin99a]. Hence, in $1+1$ dimensions neither correlated nor anticorrelated fluctuations of the left-moving and right-moving null fluxes can be generated in the absence of other interactions: both fluxes are separately conserved.
On the other hand, although in $3+1$ dimensions one can try to use a similar argument when considering sectors with certain symmetries, the final conclusion is different. For instance, if one uses in Minkowski spacetime a coordinate system $\{u,v,y,z\}$ where $u=t-x$ and $v=t+x$ are null coordinates, the components $\hat{T}_{uy}$, $\hat{T}_{uz}$, $\hat{T}_{vy}$, $\hat{T}_{vz}$ and $\hat{T}_{yz}$ vanish in the sector which is rotationally invariant on the $yz$ plane. In that sector, the $u$ and $v$ components of the conservation equation coincide with those in the $1+1$ case, given by Eqs. (\[conservation2a\])-(\[conservation2b\]), with vanishing Christoffel symbols. Therefore, the conditions for the generation of correlated or anticorrelated null fluxes are still given by Eqs. (\[conservation3a\])-(\[conservation3b\]). However, in $3+1$ dimensions the $\hat{T}_{yy}$ and $\hat{T}_{zz}$ components also contribute to the trace of the stress tensor. Hence, although the trace does not fluctuate for conformal fields, $\hat{T}_{uv}$ does fluctuate and its fluctuations coincide with those of $(\hat{T}_{yy}+\hat{T}_{zz})/4$. Correlated and anticorrelated null energy flux fluctuations can thus be generated.
The discussion in the previous paragraph can be extended to a general spherically symmetric spacetime in the region where the gradient of the radial coordinate is spacelike. This can be done by considering angular coordinates for every sphere of constant radius and the coordinates associated with the two radial null directions orthogonal to them. Taking rotational symmetry into account, as done in the flat space case, one can provide an argument similar to that in $1+1$ dimensions. However, as in the flat space case, despite the absence of fluctuations in the trace of the stress tensor, $\hat{T}_{uv}$ will still fluctuate due to the fluctuations of $(\hat{T}_{\theta\theta} +
\hat{T}_{\varphi\varphi} / \sin^2 \theta) / 4r^2$. Moreover, in this case there will be additional contributions due to scattering off the potential barrier (an effect that will be small near the horizon).
Smearing of the noise kernel in flat space {#appA}
==========================================
In this appendix we will consider several kinds of smearings of the noise kernel for the Minkowski vacuum. In Sec. \[sec:null\] we study a product of smearing functions involving two null directions and the two orthogonal spatial directions, and analyze the limit in which the smearing size along one of the null directions vanishes, which is shown to be divergent. On the other hand, in Sec. \[sec:timelike\] a smearing along a timelike direction and the three orthogonal directions is considered and it is shown that a finite smearing size along the timelike direction is sufficient to have a finite result. In contrast, in the limit of a vanishing smearing size along the timelike direction the result is always divergent, even for non-vanishing smearing sizes along all the spatial directions.
The noise kernel should be treated as distribution in spacetime coordinates. It has a divergent coincidence limit and it involves subtle integration prescriptions. However, the Fourier transforms of this kind of distributions are much simpler to deal with. Therefore, it is very convenient to perform our calculations in Fourier space and we will do so throughout this appendix.
The noise kernel $N_{abcd}(x,x') = (1/2)\langle
\{\hat{T}_{ab}(x), \hat{T}_{cd}(x')\} \rangle - \langle
\hat{T}_{ab}(x) \rangle \langle \hat{T}_{cd}(x') \rangle$ for a massless conformally coupled scalar field in the Minkowski vacuum state has been obtained for instance in Ref. [@martin00] and is given in standard inertial coordinates by $$N_{\mu\nu\rho\sigma}(x,x') = \frac{2}{3}
\left( 3 \mathcal{D}_{\mu (\rho} \mathcal{D}_{\sigma) \nu}
- \mathcal{D}_{\mu \nu}\mathcal{D}_{\rho \sigma} \right) N(x-x')
\label{noise1},$$ where $\mathcal{D}_{\mu \nu} \equiv \eta_{\mu \nu} \Box -
\partial_\mu \partial_\nu$ and $$N(x-x') = \frac{1}{(1920 \pi)} \int \frac{d^4p}{(2 \pi)^4} \,
e^{i p (x-x')} \theta(-p^2)
\label{noise2}.$$
Smearing around a null geodesic {#sec:null}
-------------------------------
In this subsection we consider the case in which one smears around a null geodesic. Using the null coordinates $v=t+x$ and $u=t-x$, we define the smeared version of the kernel $N(x-x')$ as $$\mathcal{N} \equiv
\int du\, dv\, d^2 x\, g(u) h (v) f(\vec{x})
\int du' dv' d^2 x' g(u') h (v') f(\vec{x}')
N(v-v',u-u',\vec{x}-\vec{x}')
\label{smearing1},$$ where we integrated the kernel with some smearing functions both for the null coordinates $u$ and $v$, and for the orthogonal spatial directions. If we choose Gaussian smearing functions $$\begin{aligned}
g(u) &=& (2 \pi \sigma_u^2)^{-\frac{1}{2}} \exp(-u^2 / 2 \sigma_u^2)
\label{gaussian1},\\
h(v) &=& (2 \pi \sigma_v^2)^{-\frac{1}{2}} \exp(-v^2 / 2 \sigma_v^2)
\label{gaussian3},\\
f(\vec{x}) &=& (2 \pi \sigma_r^2)^{-1} \exp(-\vec{x}^2 / 2 \sigma_r^2)
\label{gaussian2},\end{aligned}$$ Eq. (\[smearing1\]) can be written as $$\mathcal{N} =
\frac{1}{(2 \pi)^4 \sigma_v^2 \sigma_u^2 \sigma_r^4}
\int dV dU d^2 X e^{-\frac{V^2}{\sigma_v^2}} e^{-\frac{U^2}{\sigma_u^2}}
e^{- \frac{\vec{X}^2}{\sigma_r^2}}
\int d \Delta_v d \Delta_u d^2 \Delta \, e^{-\frac{\Delta_v^2}{4 \sigma_v^2}}
e^{-\frac{\Delta_u^2}{4 \sigma_u^2}} e^{-\frac{\vec{\Delta}^2}{4 \sigma_r^2}}
N(\Delta_v,\Delta_u,\vec{\Delta})
\label{smearing11},$$ where we introduced the semisum and difference variables $U=(u+u')/2$, $V=(v+v')/2$, $\vec{X}=(\vec{x}+\vec{x}')/2$, $\Delta_u=u'-u$, $\Delta_v=v'-v$ and $\vec{\Delta}=\vec{x}'-\vec{x}$. The integrals for $U$, $V$ and $\vec{X}$ can be readily performed and yield the following result: $$\mathcal{N} =
\frac{1}{(4 \pi)^{2} \sigma_v \sigma_u \sigma_r^2}
\int d \Delta_v d \Delta_u d^2 \Delta \, e^{-\frac{\Delta_v^2}{4 \sigma_v^2}}
e^{-\frac{\Delta_u^2}{4 \sigma_u^2}} e^{-\frac{\vec{\Delta}^2}{4 \sigma_r^2}}
N(\Delta_v,\Delta_u,\vec{\Delta})
\label{smearing12}.$$ On the other hand, in order to compute the remaining integrals it is convenient to work in Fourier space. When considering the null coordinates $v$ and $u$, it is useful to introduce the momenta $p_v=(p_x-p_t)/2$ and $p_u=-(p_t+p_x)/2$ so that Eq. (\[noise2\]) becomes $$N(x-x') = \frac{1}{(1920 \pi)} \int \frac{d^4p}{(2 \pi)^4} \,
e^{i [p_v (v-v') + p_u (u-u') + \vec{p} \cdot (\vec{x}-{x}')]}
\theta(p_v p_u - \vec{p}^{\, 2})
\label{noise3},$$ where we used the vector notation for the transverse components associated with the coordinates $y$ and $z$. Eq. (\[smearing12\]) can then be expressed in Fourier space as $$\mathcal{N} =
\frac{1}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int d p_v d p_u d^2 p\,
e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2} e^{-\vec{p}^2 \sigma_r^2}\,
\theta(p_v p_u - \vec{p}^{\, 2})
\label{smearing13}.$$ One can then infer that $\mathcal{N}$ diverges as $\sigma_u
\rightarrow 0$. This can be seen as follows. The integral in Eq. (\[smearing13\]) gets two identical contributions from the quadrants $(p_u,p_v>0)$ and $(p_u,p_v<0)$, whereas the remaining two quadrants give a vanishing contribution. Moreover, since the integrand is positive, the integral is also positive and greater than the same integral restricted to a smaller domain of integration. Taking all that into account, we have $$\begin{aligned}
\mathcal{N} &\geq&
\frac{2}{(1920 \pi)(2 \pi)^{4}}
\int \limits_{\genfrac{}{}{0pt}{}{p_v p_u \geq \sigma_r^{-2}}{p_v,p_u>0}}
d p_u d p_v e^{-p_u^2 \sigma_u^2} e^{-p_v^2 \sigma_v^2}
\int_0^{\sigma_r^{-2}} d |\vec{p}|^2 e^{-1} \nonumber \\
&\geq& \frac{2 e^{-1}}{\sigma_r^2} \frac{1}{(1920 \pi)(2 \pi)^{4}}
\int_{2 \sigma_v/\sigma_r^{2}}^\infty d p_u e^{-p_u^2 \sigma_u^2}
\int_{\sigma_v^{-1}/2}^{\sigma_v^{-1}} d p_v e^{-1} \nonumber \\
&=& \frac{e^{-2}}{\sigma_r^2 \sigma_v} \frac{1}{(1920 \pi)(2 \pi)^{4}}
\int_{2 \sigma_v/\sigma_r^{2}}^\infty d p_u e^{-p_u^2 \sigma_u^2}
\sim \frac{1}{\sigma_u \sigma_v \sigma_r^2}
\label{smearing14}.\end{aligned}$$ The last integral is divergent if one takes $\sigma_u \to 0$ (at least for $\sigma_r \neq 0$ [^12]). Thus, $\mathcal{N}$ diverges unless $\sigma_u \neq 0$.
Using the previous result, it is easy to discuss whether a smeared version of the actual noise kernel (including the differential operators), given by Eq. (\[noise1\]), also diverges when $\sigma_u
\to 0$. Each derivative in Eq. (\[noise1\]) gives rise to an additional factor involving the momentum associated with the corresponding component. Additional factors involving powers of $p_u$, $p_v$, $p_y^2$ and $p_z^2$ (odd powers of $p_y$ or $p_z$ give a vanishing contribution) leave the argument employed in the previous paragraph unchanged and the same conclusions obtained when $\sigma_u
\to 0$ hold for the smeared noise kernel as well (including the differential operators). For example we have $\mathcal{N}_{vvvv} \sim
1/\sigma_u \sigma_v^5 \sigma_r^2$ for the smeared version of the $vvvv$ component of the noise kernel.
On the other hand, when both $\sigma_v \neq 0$ and $\sigma_u \neq 0$ the smeared kernel $\mathcal{N}$ is finite even for $\sigma_r =
0$. This can be seen by taking $\sigma_r = 0$ in Eq. (\[smearing13\]). We then have $$\begin{aligned}
\mathcal{N} &=&
\frac{1}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int d p_v d p_u d^2 p\,
e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2} \,
\theta(p_v p_u - \vec{p}^{\, 2}) \nonumber \\
&=& \frac{1}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int d p_v d p_u
\pi p_v p_u e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2}\,
\theta(p_v p_u) \nonumber \\
&=& \frac{2}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int_0^\infty d p_v
\int_0^\infty d p_u \pi p_v p_u e^{-p_v^2 \sigma_v^2}
e^{-p_u^2 \sigma_u^2}
\sim \frac{1}{\sigma_u^2 \sigma_v^2}
\label{smearing15},\end{aligned}$$ which is finite for $\sigma_u, \sigma_v \neq 0$. It is also clear that the same conclusion applies to the smeared version of the noise kernel. For instance we have $\mathcal{N}_{vvvv} \sim 1/\sigma_u^2
\sigma_v^6$.
Smearing along a timelike curve and on a spacelike hypersurface {#sec:timelike}
---------------------------------------------------------------
In this subsection we consider the smeared kernel $\mathcal{N}$ obtained when working with standard cartesian coordinates in Minkowski spacetime: $$\mathcal{N} \equiv
\int dt\, d^3 x\, g(t) f(\vec{x})
\int dt' d^3 x' g(t') f(\vec{x}')
N(t-t',\vec{x}-\vec{x}')
\label{smearing21},$$ where we used the vector notation for the three spatial components $x$, $y$ and $z$, and the Gaussian smearing functions are now $$\begin{aligned}
g(t) = (2 \pi \sigma_t^2)^{-\frac{1}{2}} \exp(-t^2 / 2 \sigma_t^2),
\label{gaussian4}\\
f(\vec{x}) = (2 \pi \sigma_r^2)^{-\frac{3}{2}} \exp(-\vec{x}^2 / 2 \sigma_r^2).
\label{gaussian5}\end{aligned}$$ Note that $\sigma_r$ corresponds now to the smearing size of the three spatial directions. Introducing the semisum and difference variables $T=(t+t')/2$ and $\Delta_t=t'-t$, Eq. (\[smearing21\]) can be rewritten as $$\mathcal{N} =
\frac{1}{(2 \pi)^4 \sigma_t^2 \sigma_r^6}
\int dT d^3 X e^{-\frac{T^2}{\sigma_t^2}}
e^{- \frac{\vec{X}^2}{\sigma_r^2}}
\int d \Delta_t d^3 \Delta \, e^{-\frac{\Delta_t^2}{4 \sigma_t^2}}
e^{-\frac{\vec{\Delta}^2}{4 \sigma_r^2}} N(\Delta_t,\vec{\Delta})
\label{smearing22}.$$ After integrating over $T$ and $\vec{X}$ we have $$\mathcal{N} =
\frac{1}{(4 \pi)^{2} \sigma_t \sigma_r^3}
\int d \Delta_t d^3 \Delta \, e^{-\frac{\Delta_t^2}{4 \sigma_t^2}}
e^{-\frac{\vec{\Delta}^2}{4 \sigma_r^2}} N(\Delta_t,\vec{\Delta})
\label{smearing23},$$ which can be equivalently rewritten in Fourier space as $$\mathcal{N} =
\frac{1}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int d p_t d^3 p\,
e^{-p_t^2 \sigma_t^2} e^{-\vec{p}^2 \sigma_r^2}\,
\theta(p_t^2 - \vec{p}^{\, 2})
\label{smearing24}.$$ A similar argument to that employed in the previous subsection can be used to show that in this case $\mathcal{N}$ diverges as $\sigma_t
\rightarrow 0$. The integral in Eq. (\[smearing24\]) gets two identical contributions from the intervals $p_t<0$ and $p_t>0$. We will also take into account that since the integrand is positive, the integral is also positive and greater than the same integral restricted to a smaller domain of integration, so that we have $$\begin{aligned}
\mathcal{N} &\geq&
\frac{2}{(1920 \pi)(2 \pi)^{4}}
\int_{\sigma_r^{-1}}^\infty d p_t e^{-p_t^2 \sigma_t^2}
\int_0^{\sigma_r^{-1}} 4\pi |\vec{p}|^2 d |\vec{p}| e^{-1}
\nonumber \\
&=& \frac{8 \pi e^{-1}}{3 \sigma_r^3} \frac{1}{(1920 \pi)(2 \pi)^{4}}
\int_{\sigma_r^{-1}}^\infty d p_t e^{-p_t^2 \sigma_t^2}
\sim \frac{1}{\sigma_t \sigma_r^3}
\label{smearing25}.\end{aligned}$$ The last integral is divergent if one takes $\sigma_t \to 0$ (at least for $\sigma_r \neq 0$ [^13]). Thus, $\mathcal{N}$ diverges unless $\sigma_t \neq 0$.
Using the previous result, it is easy to discuss whether a smeared version of the actual noise kernel, given by Eq. (\[noise1\]), also diverges when $\sigma_t \to 0$. Each derivative in Eq. (\[noise1\]) gives rise to an additional factor involving the momentum associated with the corresponding component. Additional factors involving powers of $p_t^2$, $p_x^2$, $p_y^2$ and $p_z^2$ (odd powers of $p_t$, $p_x$, $p_y$ or $p_z$ give a vanishing contribution) leave the argument employed in the previous paragraph unchanged and the same conclusions obtained when $\sigma_t \to 0$ hold for the smeared noise kernel as well. For example one has $\mathcal{N}_{tttt} \sim 1/\sigma_t
\sigma_r^7$. This result is in agreement with that obtained in Ref. [@ford05]
On the other hand, when $\sigma_t \neq 0$ the smeared noise kernel is finite even for $\sigma_r = 0$. This can be seen by taking $\sigma_r =
0$ in Eq. (\[smearing24\]). We then have $$\mathcal{N} =
\frac{1}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int d p_t d^3 p\,
e^{-p_t^2 \sigma_t^2} \, \theta(p_t^2 - \vec{p}^{\, 2})
= \frac{1}{(1920 \pi)} \frac{1}{(2 \pi)^{4}} \int d p_t
\frac{4 \pi}{3} p_t^3 e^{-p_t^2 \sigma_t^2}
\sim \frac{1}{\sigma_t^4}
\label{smearing26},$$ which is finite for $\sigma_t \neq 0$. It is also clear that the same conclusion applies to the smeared version of the noise kernel, with $\mathcal{N}_{tttt} \sim 1/\sigma_t^8$.
Generalization to curved space, arbitrary Hadamard Gaussian states and general smearing functions {#appB}
=================================================================================================
The results obtained for the Minkowski vacuum in flat space can be generalized to curved space and arbitrary Gaussian Hadamard states. They also apply to more general smearing functions. This will be shown in this appendix.
The key ingredient is the fact that the Wightman function for any Hadamard state has the following form in a sufficiently small normal neighborhood of an arbitrary spacetime:[^14] $$G^+(x,x') = \frac{u(x,x')}{\sigma_+(x,x')}
+ v(x,x') \ln \sigma_+(x,x') + w(x,x')
\label{hadamard1},$$ where $\sigma_+(x,x')$ is the geodetic interval (one half of the geodesic distance) for the geodesic connecting the pair of points $x$ and $x'$ with an additional small imaginary component added to the timelike coordinates (this prescription will be defined more precisely below); $u$, $v$ and $w$ are smooth functions with $v$ and $w$ expandable as $$\begin{aligned}
v(x,x') = \sum_{n=0}^\infty v_n (x,x') \sigma^n (x,x') ,
\label{hadamard2a} \\
w(x,x') = \sum_{n=0}^\infty w_n (x,x') \sigma^n (x,x')
\label{hadamard2b} .\end{aligned}$$ where $u(x,x')$, $v_n(x,x')$ and $w_n(x,x')$ satisfy the Hadamard recursion relations, which uniquely determine $u(x,x')$ and $v(x,x')$. On the other hand, $w_0 (x,x')$ is not uniquely determined and contains the information on the particular state that one is considering, the remaining $w_n (x,x')$ are also determined once a particular choice of $w_0 (x,x')$ has been made. Note that for the Minkowski vacuum in flat space $v(x,x')$ and $w(x,x')$ vanish and $u(x,x')$ is simply given by a constant. Other Hadamard states in flat space have a non-vanishing $w(x,x')$ while $u(x,x')$ and $v(x,x')$ remain unchanged.
The biscalar functions $u(x,x')$, $v_n(x,x')$ and $w_n(x,x')$ can in turn be expanded in the following way: $$A(x,x') = \sum_{m=0}^\infty A_{a_1 \cdots a_m} (x)\, \sigma^{a_1} \cdots
\sigma^{a_m}
\label{hadamard3},$$ where $\sigma^a \equiv g^{ab} \nabla_b \sigma$. Furthermore, it will be convenient to employ Riemann normal coordinates, which can always be introduced in a normal neighborhood. Given a set of normal coordinates $\{y^\mu\}$, the geodetic interval can be simply expressed as $\sigma(y,y') = (1/2)
(y^\mu - y^{\prime \mu})(y^\nu - y^{\prime \nu}) \eta_{\mu\nu}$. The prescription for $\sigma_+$ mentioned above corresponds then to $\sigma_+ (y,y') = (1/2) [ -(y^0 - y^{\prime 0} - i \varepsilon)^2 +
(\vec{y}- \vec{y}')^2 ]$.
Given a smearing function $s(x)$, one can consider the following smearing of the product of two Wightman functions $N(x,x') =
\mathrm{Re} [G^+(x,x') G^+(x,x')]$: $$\mathcal{N} = \int d^4 x\, \sqrt{-g(x)} \, s(x)
\int d^4 x'\, \sqrt{-g(x')} \, s(x') \, N(x,x')
\label{smearing30}.$$ Next, one changes from a set of absolute coordinates for each one of the the two points at which the kernel $N(x,x')$ is evaluated to a set of absolute coordinates for the first point and a set of relative coordinates for the location of the second point with respect to the first one. In particular we will choose Riemann normal coordinates for the relative location of the second point (in order to study the divergences in the coincidence limit it is sufficient, by taking small enough smearing sizes, to consider small enough convex neighborhoods where both normal coordinates can be defined and the series in Eq. (\[hadamard1\]) is convergent). Eq. (\[smearing31\]) then becomes $$\mathcal{N} = \int d^4 x\, \sqrt{-g(x)} \, s(x)
\int d^4 y\, \sqrt{-\tilde{g}_x (y)} \,
\tilde{s}_x (y) \, \tilde{N}(x,y)
\label{smearing31}.$$ For simplicity, in our discussion below we will consider smearing functions $\tilde{s}_x (y)$ which are Gaussian and independent of $x$. However, in Sec. \[sec:general\_smearing\] we will explain how our results can be extended to more general smearing functions. We also note that we will not analyze the integrals in $x$ since those should be finite (we are considering globally Hadamard states and regular smearing functions): only the integrals in $y$ are relevant for the UV divergences associated with the coincidence limit, which corresponds to $y \to 0$. There could still be IR divergences arising from the integrals in $x$, but we will be considering smearing functions which decay sufficiently fast so that this is not the case.
Taking all the previous considerations into account, it becomes clear that when calculating the smeared kernel $\mathcal{N}$, the most divergent terms from $N(x,x') = \mathrm{Re} [G^+(x,x') G^+(x,x')]$ will be of the form $1 / \sigma_+^2(x,x')$, $\ln \sigma_+(x,x') /
\sigma_+(x,x')$ or $1 / \sigma_+(x,x')$. When expressed in normal coordinates, the contribution to $\mathcal{N}$ due to a term of the first kind, which will be denoted by $\mathcal{N}_1$ and corresponds to the product of two $1 / \sigma_+(x,x')$ terms, coincides with the expression for the Minkowski vacuum in flat space. Hence, one can directly apply the results obtained in Appendix A. Furthermore, one expects that the leading divergence to the smeared function $\mathcal{N}$ in the various limits of vanishing smearing sizes will come entirely from that kind of terms and that the other terms will only give rise to subleading divergences. If this is true, the main conclusions in Appendix A will also apply to general Hadamard Gaussian states in curved spacetime. Let us, therefore, study more carefully the contributions to $\mathcal{N}$ from the different kinds of divergent terms and check that this is indeed the case (other possible divergent terms in addition to the three kinds mentioned above correspond to multiplying one of those three by some positive power of $\sigma^a$, and they will be discussed in Sec. \[sec:finite\_terms\]).
In order to analyze the contribution from the other two kinds of divergent terms, denoted by $\mathcal{N}_2$ and $\mathcal{N}_3$, we will proceed analogously to Appendix A and work in Fourier space for the relative normal coordinates. We start by considering a term of the form $\ln \sigma_+(x,x') / \sigma_+(x,x')$. Using the same Gaussian smearing functions as in Appendix A (as explained above, we do not include any dependence on the absolute coordinate for the first point), its contribution to $\mathcal{N}$ when smearing around a null geodesic is given by $$\mathcal{N}_2 = \frac{1}{(2 \pi)^{4}} \int d p_v d p_u d^2 p\,
e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2} e^{-\vec{p}^2 \sigma_r^2} \,
%\frac{1}{2} \left[ L(p_v, p_u, \vec{p}^{\, 2})
%+ \left(L(-p_v, -p_u, \vec{p}^{\, 2})
%\right)^* \right]
\frac{1}{2} \left[ L(p) + \left(L(-p) \right)^* \right]
\label{smearing32},$$ where $L(p_v, p_u, \vec{p}^{\, 2})$ is the Fourier transform of $\ln
(\sigma_+(y,y') /\lambda^2) / \sigma_+(y,y')$ and we need the second term inside the square brackets because we are interested in the real part of the product of two Wightman functions. An explicit expression for $L(p_v, p_u, \vec{p}^{\, 2})$ can be determined using the following two Fourier transforms: $$\frac{1}{\sigma_+(x,x')}
= \frac{1}{\pi} \int d^4 p\, e^{i p (y-y')}\, \theta(-p^0) \delta(-p^2)
\label{fourier1},$$ $$\begin{aligned}
\ln (\sigma_+(y,y') /\lambda^2) = -2 \int d^4 p\, e^{i p (y-y')}
\lim_{m \rightarrow 0} && \!\! \left\{ -\frac{1}{\pi} \theta(-p^0)
\frac{d}{d m^2} \delta(-p^2 - m^2) \right. \nonumber \\
&& \left.
+ \left[ \frac{1}{2} \ln (2 m^2 \lambda^2) + \gamma -1 \right] \delta^4 (p)
\right\}
\label{fourier2}.\end{aligned}$$ where Eq. (\[fourier2\]) was derived in Appendix A.2 of Ref. [@campos95]. We can then write $L(p_v, p_u, \vec{p}^{\, 2})$ as a convolution of those Fourier transforms: $$\begin{aligned}
L(p) &=& - 4 (2 \pi)^3 \int d^4 q\, \theta(-p^0+q^0))
\delta(-(p-q)^2) \nonumber \\
&&\times \lim_{m \rightarrow 0} \left\{ -\frac{1}{\pi} \theta(-q^0)
\frac{d}{d m^2} \delta(-q^2 - m^2)
+ \left[ \frac{1}{2} \ln (2 m^2 \lambda^2) + \gamma -1 \right]
\delta^4 (q) \right\}
\label{fourier3}.\end{aligned}$$ After a certain amount of calculation, one obtains the following result for $L(p)$: $$L(p) = 2 (2 \pi)^3 \left\{ \theta(-p^0) \,
\mathcal{P}\!f \! \left[ \theta(-p^2) \frac{1}{p^2} \right]
- \left[ \ln (2 \lambda^2) + \gamma -1 \right] \theta(-p^0) \delta^4 (-p^2)
\right\}
\label{fourier8},$$ where $\mathcal{P}\!f$ denotes Hadmard’s finite part prescription (a generalization of the principal value prescription) whose precise definition can be found in Refs. [@campos95; @schwartz57; @zemanian87]. We will use this result in the next subsections to compute in Fourier space the contribution $\mathcal{N}_2$ to the smeared kernel $\mathcal{N}$ for different kinds of smearings and analyze under what conditions it is finite.
On the other hand, from Eq. (\[fourier1\]) one can immediately see that the contribution from the term $1 / \sigma_+(x,x')$ is simply given by $$\mathcal{N}_3 = \frac{1}{2 \pi} \int d p_v d p_u d^2 p\,
e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2} e^{-\vec{p}^2 \sigma_r^2} \,
\delta(p_u p_v - \vec{p}^2)
\label{smearing33}.$$
Smearing around null geodesics {#sec:null2}
------------------------------
### Contributions from $\mathcal{N}_2$ and $\mathcal{N}_3$ {#sec:divergent}
As we found in Appendix A, when considering a smearing of the noise kernel for the Minkowski vacuum around a null geodesic, $\mathcal{N}$ diverges as $1 / \sigma_u \sigma_v \sigma_r^2$ in the limit of small $\sigma_u$ (as long as $\sigma_r \neq 0$, otherwise it diverges as $1
/ \sigma_u^2 \sigma_v^2$). Whereas the contribution $\mathcal{N}_1$ from terms of the form $1 / \sigma_+^2(x,x')$ will exhibit the same divergent behavior, we will show in this subsection that $\mathcal{N}_2$ and $\mathcal{N}_3$, the other contributions to $\mathcal{N}$, are arbitrarily smaller than $\mathcal{N}_1$ for sufficiently small $\sigma_u$ or $\sigma_v$.
Let us start with $\mathcal{N}_3$. First, one rewrites Eq. (\[smearing33\]) as follows: $$\begin{aligned}
\mathcal{N}_3 &=& \frac{1}{2 \pi} \int d p_v d p_u d^2 p\,
e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2} e^{-\vec{p}^2 \sigma_r^2} \,
\delta(p_u p_v - \vec{p}^2) \nonumber \\
&=& \int_0^\infty d p_v \int_0^\infty d p_u e^{-p_v^2 \sigma_v^2}
e^{-p_u^2 \sigma_u^2} \int_0^\infty d |\vec{p}| \, e^{-|\vec{p}|^2 \sigma_r^2} \,
\delta(\sqrt{p_u p_v} - |\vec{p}|) \nonumber \\
&=& \int_0^\infty d p_v \int_0^\infty d p_u e^{-p_v^2 \sigma_v^2}
e^{-p_u^2 \sigma_u^2} e^{- p_u p_v \sigma_r^2} \nonumber \\
&=& \int_0^\infty d \xi e^{-\xi \sigma_r^2}
\int_0^\infty \frac{d p_v}{p_v} e^{-p_v^2 \sigma_v^2}
e^{-\frac{\xi^2}{p_v^2} \sigma_u^2}
\label{smearing34},\end{aligned}$$ where we introduced the new variable $\xi = p_v p_u$ in the last equality. Next, using the positivity of the integrand, the fact that the original integral was invariant under interchange of $p_u$ and $p_v$, and the fact that value of the exponentials is always equal or less than one, one can derive the following bound for small $\sigma_u$ and $\sigma_v$: $$\mathcal{N}_3 \leq - \frac{1}{2 \sigma_r^2} (\ln \sigma_u + \ln \sigma_v) + O(1)
\label{smearing36},$$ where the higher-order terms involve positive powers of $\sigma_u$ and $\sigma_v$ (when $\sigma_v$ is not small, one has an expansion only in terms of $\sigma_u$ and the $\ln \sigma_v$ term is absent). On the other hand, for $\sigma_r = 0$ the bound is $\mathcal{N}_3 \leq
(\pi^2/8) (1/\sigma_u \sigma_v)$.
Let us now turn our attention to $\mathcal{N}_2$. Substituting Eq. (\[fourier8\]) into Eq. (\[smearing32\]), we get $$\mathcal{N}_2 = - \int d p_v d p_u d^2 p\,
e^{-p_v^2 \sigma_v^2} e^{-p_u^2 \sigma_u^2} e^{-\vec{p}^2 \sigma_r^2}
\!\! \left\{ \mathcal{P}\!f \! \left[ \theta(p_u p_v - \vec{p}^2) \,
\frac{1}{p_u p_v - \vec{p}^2} \right]
+ \left[ \ln (2 \lambda^2) + \gamma -1 \right]
\delta^4 (p_u p_v - \vec{p}^2) \right\}
\label{smearing37}.$$ The contribution from the second term inside the curly brackets has the same form as $\mathcal{N}_3$. Hence, we need to concentrate on the first term. After a lengthy calculation one can derive the following bound: $$\mathcal{N}_2 \leq L \frac{1}{\sqrt{2 \sigma_u \sigma_v} \, \sigma_r}
+ O (\ln \sigma_u, \ln \sigma_v)
\label{smearing51},$$ where $L$ is a constant of order 1. On the other hand, for $\sigma_r =
0$ the bound is $\mathcal{N}_2 \leq L/\sigma_u \sigma_v$. We can see that $\mathcal{N}_2$ dominates over $\mathcal{N}_3$ in the limit of small $\sigma_u$ or $\sigma_v$. Nevertheless, it is still $\mathcal{N}_1$ that provides the leading contribution in that limit.
### Remaining terms {#sec:finite_terms}
The contribution from the remaining terms in $N(x,x')$ to the smeared function $\mathcal{N}$, which can be seen from terms of the different kinds that we have already analyzed multiplying them by some positive powers of $\sigma$ and $\sigma^a$ can be analyzed as follows. If one is working with Riemann normal coordinates and the corresponding Fourier variables, each $\sigma$ factor will give rise to a momentum d’Alembertian $(1/2) \eta_{\mu\nu} (\partial/\partial p_{\mu})
(\partial/\partial p_{\nu})$ acting on the Fourier transform of that term in $N(x,x')$. Similarly, each $\sigma^\mu$ factor will give rise to a linear differential operator $(\partial/\partial p_\mu)$ acting on the Fourier transform. Integrating by parts in the Fourier space expression for $\mathcal{N}$ so that the d’Alembertian acts on the Fourier transform of the smearing functions, it will produce a factor of the form $(-\sigma_u^2 \sigma_v^2 p_u p_v + 2 \sigma_r^4 \vec{p}^2
-2 \sigma_r^2)$. Proceeding analogously with the linear operator $(\partial/\partial p_\mu)$, one gets the factors $2 \sigma_u^2 \,
p_u$, $2 \sigma_v^2 \, p_v$ and $2 \sigma_r^2 \, p_j$ (where the index $j$ corresponds to one of the two orthogonal spatial components) when $\mu$ equals $u$, $v$ or $j$ respectively.
Next, one needs to see how the results of the integrals in Sec. \[sec:divergent\] change when the integrand is multiplied by positive powers of $p_u$, $p_v$ and $p_j$. Odd powers of $p_j$ for any $j$ give a vanishing result since the integrals and integrands there are symmetric under a sign change of $p_j$, whereas every $(p_j)^2$ factor can be written as $(-1/2) (\partial / \partial
\sigma_r^2)$. Similarly, every factor $p_u^2$ or $p_v^2$ can be written as $(-1/2) (\partial / \partial \sigma_u^2)$ or $(-1/2)
(\partial / \partial \sigma_v^2)$ respectively.[^15] On the other hand, having an odd power of $p_u$ only or $p_v$ only gives a vanishing result because the integrals and integrands in Sec. \[sec:divergent\] are symmetric under a simultaneous sign change of $p_u$ and $p_v$. However, odd powers of $p_u p_v$ do not vanish in general because the integrands are not symmetric under interchange of $p_u$ or $p_v$ only. Therefore, one needs to check how the main results for the integrals in Sec. \[sec:divergent\] change when the integrands are multiplied by a $p_u p_v$ factor. In Eq. (\[smearing34\]) it gives rise to a factor $\xi$, which implies an additional $1 / \sigma_r^2$ factor multiplying the final result for $\mathcal{N}_3$ in Eq. (\[smearing36\]). One can similarly find that an additional $p_u
p_v$ factor in Eq. (\[smearing37\]) also implies a $1 / \sigma_r^2$ factor multiplying the final result for $\mathcal{N}_2$ in Eq. (\[smearing51\]).
We are finally in a position to discuss the effects of $\sigma$ and $\sigma_a$ factors multiplying the contributions to the smeared kernel $\mathcal{N}$. Each power of $p_u^2$ and $p_v^2$ (including their $\sigma_u^4$ and $\sigma_v^4$ accompanying factors) will typically give rise to $\sigma_u^2$ and $\sigma_v^2$ factors respectively. A $p_u p_v$ factor (with its accompanying $\sigma_u^2 \sigma_v^2$ factor) will give rise to a $\sigma_u^2 \sigma_v^2 / \sigma_r^2$ factor. And each $p_j^2$ factor (with its accompanying $\sigma_r^4$ factor) will give rise to a $\sigma_r^2$ factor. Thus, we see that the divergent behavior in the limit of small $\sigma_u$ remains unchanged or even gets improved. (An analogous conclusion would apply in the limit of small $\sigma_r$.) It then follows that the behavior of $\mathcal{N}$ in the small $\sigma_u$ limit is still dominated by the flat space vacuum contribution $\mathcal{N}_1$.
### Smearing of the actual noise kernel {#sec:actual_noise}
The actual noise kernel involves a number of functions multiplying the kernel $N(x,x') = \mathrm{Re} [G^+(x,x') G^+(x,x')]$ and differential operators acting on it. When using relative Riemann normal coordinates for the location of the second point with respect to the first one, the part of these linear operators that depends on the relative coordinates of the second point can be entirely expressed in terms of functions and tensor fields (such as the metric and the curvature tensors) as well as partial derivatives. That is not the case in general for the operators associated with the first spacetime point, but the dependence on the first point does not exhibit a divergent UV behavior.
The functions multiplying $N(x,x')$ can be expanded in terms of the relative Riemann normal coordinates, which involves powers of $\sigma$ and $\sigma^a$ that can be treated as explained in the previous subsubsection. As we saw, they either leave the divergent behavior in the limit of small $\sigma_u$ unchanged or decrease the degree of divergence.
On the other hand, the partial derivatives $(\partial/\partial y^\mu)$ simply correspond to $i p_\mu$ factors in Fourier space, which can also be dealt with as discussed in the previous subsubsection. Even powers of $p_u$ increase the degree of divergence in the limit of small $\sigma_u$: every $p_u^2$ factor gives rise to a $1 /
\sigma_u^2$ factor. All others leave the degree of divergence unchanged. Moreover, since the $p_u^2$ affect both the kind of terms contributing to $\mathcal{N}_1$ and those contributing to $\mathcal{N}_2$ and $\mathcal{N}_3$, the conclusion that the leading divergent behavior when $\sigma_u \to 0$ is given by the Minkowski vacuum result remains unchanged for the actual noise kernel.
Smearing along timelike geodesics and on spacelike hypersurfaces {#sec:timelike2}
----------------------------------------------------------------
The results for smearings of the noise kernel on spacelike hypersurfaces and along timelike geodesics obtained for the vacuum state in Minkowski can be generalized proceeding analogously to what was done in the previous subsection for smearings along null geodesics.
Let us start by considering $\mathcal{N}_3$: $$\begin{aligned}
\mathcal{N}_3 &=& \frac{1}{2 \pi} \int d p_t d^3 p\,
e^{-p_t^2 \sigma_t^2} e^{-\vec{p}^2 \sigma_r^2} \,
\delta(p_t^2 - \vec{p}^2) \nonumber \\
&=& 4 \int_0^\infty d p_t e^{-p_t^2 \sigma_t^2}
\int_0^\infty d |\vec{p}| \, |\vec{p}| e^{-|\vec{p}|^2 \sigma_r^2} \,
\delta(p_t - |\vec{p}|) \nonumber \\
&=& 4 \int_0^\infty d p_t \, p_t e^{-p_t^2 \sigma_t^2}
e^{-p_t^2 \sigma_r^2}
\label{smearing53}.\end{aligned}$$ Hence, we have $$\mathcal{N}_3 = \frac{2}{\sigma_t^2 + \sigma_r^2}
\label{smearing54},$$ which is finite provided that $\sigma_t \neq 0$ or $\sigma_r \neq 0$.
Let us now turn our attention to $\mathcal{N}_2$. Applying the result in Eq. (\[fourier8\]) to this case, we get $$\mathcal{N}_2 = - \int d p_t d^3 p\,
e^{-p_t^2 \sigma_t^2} e^{-\vec{p}^2 \sigma_r^2}
\!\! \left\{ \mathcal{P}\!f \! \left[ \theta(p_t^2 - \vec{p}^2) \,
\frac{1}{p_t^2 - \vec{p}^2} \right]
+ \left[ \ln (2 \lambda^2) + \gamma -1 \right]
\delta^4 (p_t^2 - \vec{p}^2) \right\}
\label{smearing55}.$$ The contribution from the second term inside the curly brackets has the same form as $\mathcal{N}_3$. Hence, we only need to concentrate on the first term. After a slightly lengthy calculation, one gets the following bound for $\mathcal{N}_2$ (when $\sigma_r \neq 0$): $$\mathcal{N}_2 < \frac{C_1}{\sigma_r^2} + \frac{C_2}{\sigma_t^2}
\label{smearing64},$$ where $C_1$ and $C_2$ are positive dimensionless constants which are finite provided that $\sigma_t \neq 0$ (they behave like $- \ln
\sigma_t$ in the limit of small $\sigma_t$). On the other hand, for $\sigma_r = 0$ one has a bound given by $$|\mathcal{N}_2| < \frac{D}{\sigma_t^2}
\label{smearing66},$$ where $D$ is some positive dimensionless constant which is finite provided that $\sigma_t \neq 0$ (it behaves like $- \ln \sigma_t$ in the limit of small $\sigma_t$). This shows that just a temporal smearing is enough to render $\mathcal{N}_2$ finite. Of course $\mathcal{N}_2$ will also be finite if, in addition to the temporal smearing, some but not all of the (orthogonal) spatial directions have a non-vanishing smearing size.
Proceeding similarly to Secs. \[sec:finite\_terms\]-\[sec:actual\_noise\], one can argue that factors involving positive powers of $\sigma$ and $\sigma^a$ as well as the derivative operators in configuration space do not alter the main conclusion in this subsection. Thus, a smearing along the timelike direction is enough to have a finite result for the smeared versions of $N(x,x')$ and the actual noise kernel.
More general smearing functions {#sec:general_smearing}
-------------------------------
In Secs. \[sec:null2\] and \[sec:timelike2\] we considered Gaussian smearing functions for the relative normal coordinates. However, as pointed out at the beginning of this appendix, when transforming from a pair of sets of *absolute* coordinates for the two points where the noise kernel is evaluated to one set of *absolute* coordinates and one set of *relative* ones, the form of the smearing functions will change in general. That will also happen when transforming to Riemann normal coordinates if one had initially chosen a different kind of relative coordinates. Furthermore, even in the flat space case one may be interested in considering other kinds of smearing functions. For instance, one may wish to consider a smearing function adapted to a spherical surface rather than a plane.
The results obtained in these appendices can be extended to more general smearing functions, making it possible to cover the situations described in the previous paragraph. The essential idea is simple: the previous results can be generalized for smearing functions which can be locally approximated by the Gaussian smearing functions in Riemann normal coordinates considered so far. More specifically, by “locally approximated” we mean that in those coordinates the new smearing functions can be expressed as the Gaussian smearing functions times a factor involving an expansion in positive powers of $\sigma$ and $\sigma^a$. The procedure described in Sec. \[sec:finite\_terms\] can then be employed to show that the main results remain unchanged. Moreover, the detailed form of the smearing functions for large values of the Riemann normal coordinates (or even outside the range where they can be defined) is not relevant when concerned with the divergent behavior (when certain smearing sizes tend to zero) as we are are here. Or put in a different way, even though in general the form of the smearing function can be significantly distorted when transforming from the original coordinates to relative Riemann normal coordinates, this effect becomes less and less important when considering sufficiently small smearing sizes (which is in any case the relevant regime to study the UV divergent behavior): for sufficiently small scales the coordinate transformation is characterized by the linear map between tangent spaces (the Jacobian matrix evaluated in the coincidence limit) plus small corrections involving positive powers of $\sigma$ and $\sigma^a$, which can be dealt with following the procedure described in Sec. \[sec:finite\_terms\].
These points can be illustrated with a simple example. Consider a spatial smearing function adapted to a sphere in flat space with the following form (in spherical coordinates): $$f(r,\theta) = \frac{1}{(2 \pi)^{3/2} \sigma_r r_0^2 \sigma_\theta^2 \
b(\sigma_\theta)}
\exp \left[-\frac{(r-r_0)^2}{2 \sigma_r^2} \right]
\exp \left[-\frac{\theta^2}{2 \sigma_\theta^2} \right]
\label{gaussian6},$$ where $b(\sigma_\theta)$ is some dimensionless function which ensures the proper normalization of the smearing function and tends to $1$ when $\sigma_\theta \to 0$ \[note also that for $\theta$ we actually have half a Gaussian since its domain is $(0,\pi)$\]. This can be written in terms of Riemann normal coordinates (cartesian coordinates in this case) adapted to the plane tangent to the sphere at the point $(r=r_0,\ \theta=0)$ as follows: $$\begin{aligned}
f(x,y,z) &=& \frac{1}{(2 \pi)^{3/2} \sigma_r r_0^2 \sigma_\theta^2 \,
b(\sigma_\theta)}
\exp \left[-\frac{(z-r_0)^2}{2 \sigma_r^2} \right]
\exp \left[-\frac{x^2 + y^2}{2 r_0^2 \sigma_\theta^2} \right]
\nonumber \\
&&\times \left[ 1 + O\left( \frac{(z-r_0)^3}{r_0^3 \sigma_\theta^2},
\frac{(x^2 + y^2)(z-r_0)}{r_0^3 \sigma_\theta^2},
\frac{(x^2 + y^2)(z-r_0)}{r_0 \sigma_r^2}
\right) \right]
\label{gaussian7}.\end{aligned}$$ We can see that for any given $r_0 > 0$, if one chooses sufficiently small $\sigma_r$ and $\sigma_\theta$, the higher-order terms become negligible (in the region where there is not a large suppression due to the exponential factors they are very small). Thus, $f(x,y,z)$ corresponds to a Gaussian smearing function in cartesian coordinates with $\sigma_x = \sigma_y = r_0 \sigma_\theta$ and $\sigma_z =
\sigma_r$ plus small corrections.
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[^1]: At least those which provide a relativistic description. The argument in Refs. [@sorkin95; @sorkin97] is based on a non-relativistic description and it is not obvious how to make some of our statements precise in that case. However, a natural generalization to the relativistic case is provided in Ref. [@marolf03], which does fall into this category.
[^2]: Previous attempts on this problem with similar emphasis by Raval, Sinha and one of us have appeared in Refs. [@hu99b; @hu03b]. The apparent difference between the conclusions in Ref. [@hu99b] and what is reported here will be explained below.
[^3]: This approach has been mainly applied to weak field situations, such as the study of quantum corrections to the Newtonian potential for particles in a Minkowski background [@donoghue94; @donoghue97]. However, it is particularly interesting to apply it also to strong field situations involving cosmological [@weinberg05] or black hole spacetimes.
[^4]: See, however, Refs. [@parentani01b; @parentani02], where those correlators were shown to vanish in an effectively two-dimensional model.
[^5]: The natural quantum state for a black hole formed by gravitational collapse is the Unruh vacuum, which corresponds to the absence of incoming radiation far from the horizon. The expectation value of the stress tensor operator for that state is finite on the future horizon of Schwarzschild, which is the relevant one when identifying a region of the Schwarzschild geometry with the spacetime outside the collapsing matter for a black hole formed by gravitational collapse.
[^6]: Throughout the paper we use the notation $\langle \ldots \rangle_\xi$ for stochastic averages over all possible realizations of the source $\xi_{ab}$ to distinguish them from quantum averages, which are denoted by $\langle \ldots \rangle$.
[^7]: This simple relation between the energy flux crossing the horizon and the flux far from it is valid for the expectation value of the stress tensor, which is based on an energy conservation argument for the $T_v^r$ component. In most of the literature this relation is assumed to hold also for fluctuations. However, in the next section we will show that this is an incorrect assumption. Therefore, results derived from this assumption and conclusions drawn are in principle suspect. (This misstep is understandable because most authors have not acquired as much insight into the nature of fluctuations phenomena as now.) Our investigation testifies to the necessity of a complete reexamination of all cases afresh. In fact, an evaluation of the noise kernel near the horizon seems unavoidable for the consideration of fluctuations and back-reaction issues.
[^8]: This means that possible effects on the Hawking radiation due to the fluctuations of the potential barrier for the radial mode functions will be missed by our analysis.
[^9]: A clarification between our results and the claims by Hu, Raval and Sinha in Ref. [@hu99b] is in place here: both use the stochastic gravity framework and perform an analysis based on the Einstein-Langevin equation, so there should be no discrepancy. However, the claim in Ref. [@hu99b] was based on a qualitative argument that focused on the stochastic source, but missed the fact that the perturbations around the mean are unstable for an evaporating black hole. Once this is taken into account, agreement with the result obtained here is recovered.
[^10]: Remember that for large black hole masses this can still correspond to physical distances much larger than the Planck length, as explained in Sec. \[sec3\].
[^11]: Note, however, that in most of these approaches the state of the quantum fields is the Hartle-Hawking vacuum. For an evaporating black hole, one should consider the Unruh vacuum.
[^12]: The divergence of $\mathcal{N}$ as $\sigma_u \rightarrow 0$ can also be proven for $\sigma_r=0$ (*i.e.*, in the absence of smearing along the transverse spatial directions) by taking $\sigma_r=0$ in Eq. (\[smearing13\]) and replacing $\sigma_r^2$ with an arbitrary but fixed positive value in Eq. (\[smearing14\]).
[^13]: The divergence of $\mathcal{N}$ as $\sigma_t \rightarrow 0$ can also be proven for $\sigma_r=0$ (*i.e.*, in the absence of smearing along the spatial directions) by taking $\sigma_r=0$ in Eq. (\[smearing24\]) and replacing $\sigma_r^2$ with an arbitrary but fixed positive value in Eq. (\[smearing25\]).
[^14]: For a general spacetime it may not be guaranteed that the series has a non-vanishing radius of convergence rather than being an asymptotic series [@fulling78]. However, for an analytic spacetime it can be proven that the radius of convergence is non-zero for globally Hadamard states [@fulling78; @garabedian64]. We will restrict ourselves to analytic spacetimes in this appendix, which is anyway the case for the spacetimes considered in the rest of the paper.
[^15]: In general when two functions satisfy a certain inequality that does not imply that their derivatives will satisfy it. This will, however, be the case when considering the bounds derived in the previous sections in order to analyze the leading divergent behavior in the limit $\sigma_u \to 0$.
|
---
abstract: 'We showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative. We show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed $C^1$ manifold with cylindrical singularities.'
address: |
MIT, Dept. of Math.\
77 Massachusetts Avenue, Cambridge, MA 02139-4307.
author:
- Tobias Holck Colding
- 'William P. Minicozzi II'
title: Regularity of the level set flow
---
Introduction
============
The level set method has been used with great success the last thirty years in both pure and applied mathematics to describe evolutions of various physical situations. In mean curvature flow, the evolving hypersurface (front) is thought of as the level set of a function that satisfies a nonlinear degenerate parabolic equation. Solutions are defined weakly in the viscosity sense; in general, they may not even be differentiable (let alone twice differentiable).
For a monotonically advancing front, we showed in [@CM5] that viscosity solutions are in fact twice differentiable and satisfy the equation in the classical sense. Here we characterize when they are $C^2$. As we will see, the situation becomes very rigid when the second derivative is continuous.
When $v: \RR^{n+1} \times \RR \to \RR$ is a function and for each $s$ the level set $t \to \{ x \, | \, v(x,t) = s \}$ evolves by the mean curvature flow, then $v$ satisfies the level set equation $$\begin{aligned}
\label{e:levelsetflow}
\partial_t v=|\nabla v|\,\text{div}\left(\frac{\nabla v}{|\nabla v|}\right)\, .\end{aligned}$$ This equation has been studied extensively. Whereas the work of Osher and Sethian, [@OsSe], was numerical, Evans and Spruck, [@ES], and, independently, Chen, Giga, and Goto, [@ChGG] provided the theoretical justification. This is analytically subtle, principally because the mean curvature evolution equation is nonlinear, degenerate, and indeed defined only weakly at points where $\nabla v = 0$. Moreover, $v$ is a priori not even differentiable, let alone twice differentiable. They resolved these problems by introducing an appropriate definition of a weak solution, inspired by the notion of viscosity solutions, and showed existence and uniqueness.
When the initial hypersurface is mean convex (the mean curvature is non-negative), so are all future ones and the front advances monotonically. In this case, Evans and Spruck, [@ES], showed that $v(x, t) = u(x) - t$, where $u$ is Lipschitz and satisfies (in the viscosity sense) $$\begin{aligned}
\label{e:arrivalu}
-1 = |\nabla u|\,\text{div}\left( \frac{\nabla u}{|\nabla u|}\right)\, .\end{aligned}$$ As the front moves monotonically inwards, it sweeps out the entire domain inside the initial hypersurface. The function $u$ is the [*arrival time*]{} since $u(x)$ is the time when the front passes through $x$. It is defined on the entire compact domain bounded by the initial hypersurface. Singular points for the flow correspond to critical points for $u$: the flow has a singularity at $x$ at time $u(x)$ if and only if $\nabla u (x) = 0$.
When the initial hypersurface is convex, the flow is smooth except at the point it becomes extinct and Huisken showed that the arrival time is $C^2$, [@H1], [@H2]. In [@I1], [@I2], Ilmanen gave an example of a rotationally symmetric mean convex dumbbell in $\RR^3$ for which the arrival time was not $C^2$. There is even more regularity in the plane, where Kohn and Serfaty showed that it is at least $C^3$, [@KS]. For $n > 1$, Sesum, [@S], showed that Huisken’s result is optimal; namely, she gave examples of convex initial hypersurfaces where the arrival time is not three times differentiable.
In the next two theorems and corollary, $u$ is the arrival time of a mean convex flow in $\RR^{n+1}$ starting from a smooth closed connected hypersurface.
\[t:mcvxRn\] $u$ is $C^2$ if and only if both (1) and (2) hold:
1. There is exactly one singular time $T$ (where the flow becomes extinct).
2. The singular set ${{\mathcal{S}}}$ is a $k$-dimensional closed connected embedded $C^1$ submanifold of singularities where the blowup is a cylinder $\SS^{n-k} \times \RR^k$ at each point.
Moreover, ${{\mathcal{S}}}$ is tangent to the $\RR^k$ factor in (2).
In general, even if $u$ is not $C^2$, it follows from [@CM4] that ${{\mathcal{S}}}$ is contained in a union of $C^1$ submanifolds with each submanifold tangent to the axis of the corresponding cylinder at each singular point.[[^1]]{} There are finitely many $(n-1)$-dimensional submanifolds and at most countably many in each lower dimension. Theorem \[t:mcvxRn\] gives a much stronger statement when $u$ is $C^2$: there is only one submanifold, it is closed connected and embedded, it lies in one singular time, and ${{\mathcal{S}}}$ fills out the entire submanifold (rather than being a subset of it).
A convex MCF gives an example where $u$ is $C^2$ and ${{\mathcal{S}}}$ is a point (i.e., $k=0$), while the marriage ring[[^2]]{} gives an example where $u$ is $C^2$ and ${{\mathcal{S}}}$ is a circle of cylindrical singularities. In contrast, any of the examples of rotationally symmetric surfaces studied in [@AAG] has isolated cylindrical singular points and, thus, is not $C^2$.
We can restate the theorem in terms of the function $u$ as follows:
\[c:mcvxRn\] $u$ is $C^2$ if and only if both (1) and (2) hold:
1. There is exactly one critical value $T= \max u$.
2. The critical set ${{\mathcal{S}}}$ is a $k$-dimensional closed connected embedded $C^1$ submanifold. At each critical point, ${{\text {Hess}}}_u$ has a $k$-dimensional kernel tangent to the critical set and is $- \frac{1}{n-k}$ times the identity on the orthogonal complement.
The Hessian is always continuous where the flow is smooth. Thus, discontinuity of ${{\text {Hess}}}_u$ only occurs at critical points of $u$. The next proposition shows that ${{\text {Hess}}}_u$ is still continuous at a critical point if we approach it transversely to the kernel $K$ of ${{\text {Hess}}}_u$ at the critical point: $u$ is $C^2$ where the projection $\Pi_{\text{axis}}$ onto $K$ is bounded by the projection $\Pi $ onto $K^{\perp}$.
\[t:C2trans\] Suppose that $\nabla u (0) = 0$. Given any $C$, there exists $\delta > 0$ so that $u$ is $C^2$ in the region $$\begin{aligned}
\label{e:region}
B_{\delta} \cap \{ x \, | \, \left| \Pi_{\text{axis}} (x) \right| \leq C \, \left| \Pi (x) \right| \} \, .\end{aligned}$$
Thus, any lack of continuity only occurs along paths tangent to the kernel of ${{\text {Hess}}}_u$.
$C^2$ arrival times
====================
In this section, we will prove one direction of the main theorem: If the arrival time is $C^2$, then the flow has the one singular time and the singular set is a closed connected embedded $C^1$ submanifold.
Throughout this section, $u$ is the arrival time of a mean convex flow in $\RR^{n+1}$ starting from a smooth closed connected hypersurface.
The stratification of ${{\mathcal{S}}}$
---------------------------------------
When the initial hypersurface is mean convex, then all singularities are cylindrical; see, [@W1], [@W2], [@H1], [@HS1], [@HS2], [@HaK], [@An]; cf. [@B], [@CM1].
The singular set ${{\mathcal{S}}}$ is stratified into subsets $$\begin{aligned}
{{\mathcal{S}}}_0 \subset {{\mathcal{S}}}_1 \subset \dots \subset {{\mathcal{S}}}_{n-1} = {{\mathcal{S}}}\, , \end{aligned}$$ where ${{\mathcal{S}}}_k$ consists of all singularities where the tangent flow splits off a Euclidean factor of dimension $k$. In particular, ${{\mathcal{S}}}_k \setminus {{\mathcal{S}}}_{k-1}$ is the set where the blow up is $\RR^k \times \SS^{n-k}$. By [@CM5], the Hessian has a special form at a critical point. Namely, if $p \in {{\mathcal{S}}}_k \setminus {{\mathcal{S}}}_{k-1}$, then $$\begin{aligned}
\label{e:1p2}
{{\text {Hess}}}_u (p) = - \frac{1}{n-k} \, \Pi \, , \end{aligned}$$ where $\Pi$ is orthogonal projection onto the orthogonal complement of the $\RR^k$ factor. If $k\geq 1$, let $\Pi_{\text{axis}}$ denote orthogonal projection onto the $k$-plane tangent to the “axis”.
It follows from upper semi-continuity of the density that the top strata ${{\mathcal{S}}}\setminus {{\mathcal{S}}}_{n-2}$ is compact. A priori, it is possible that a sequence of points in one of the lower strata might converge to a point in a higher strata. However, by [(\[e:1p2\])]{}, this is impossible when the arrival time is $C^2$:
\[l:strata\] If $u$ is $C^2$, then each strata ${{\mathcal{S}}}_k \setminus {{\mathcal{S}}}_{k-1}$ is compact.
\[l:normals\] If $u$ is $C^2$ at $p \in {{\mathcal{S}}}_k \setminus {{\mathcal{S}}}_{k-1}$ with $k\geq 1$ and $q_j$ is a sequence of regular points converging to $p$, then $$\begin{aligned}
\Pi_{\text{axis}} ({{\bf{n}}}(q_j)) \to 0 \, .\end{aligned}$$
We will argue by contradiction, so suppose instead that there is a sequence $q_j \to p$ with $|\Pi_{\text{axis}} ({{\bf{n}}}(q_j))| \geq \delta > 0$. Since $\SS^n$ is compact, we can pass to a subsequence so that ${{\bf{n}}}(q_j) \to V \in \SS^n$. In particular, we must have $$\begin{aligned}
\label{e:willcon}
\left|\Pi_{\text{axis}} (V) \right| \geq \delta > 0 \, .\end{aligned}$$ Using the arrival time equation [(\[e:arrivalu\])]{} at the smooth points $q_j$ and then passing to limits since $u$ is $C^2$, we get that $$\begin{aligned}
0 =&\lim_{j \to \infty} \, \left( 1 + \Delta u (q_j) - {{\text {Hess}}}_u (q_j) ({{\bf{n}}}(q_j) , {{\bf{n}}}(q_j) ) \right) = 1 + \Delta u (p) - {{\text {Hess}}}_u (p) (V, V) \notag \\
&= - \frac{1}{n-k} - \left[ - \frac{1}{n-k} \, \langle \Pi (V) , V \rangle \right] = - \frac{1}{n-k} \, \left|\Pi_{\text{axis}} (V) \right|^2 \, .\end{aligned}$$ This contradicts [(\[e:willcon\])]{}, giving the lemma.
The next lemma, which does not assume that $u$ is $C^2$, shows that a plane orthogonal to the axis of a singularity contains a point $q$ where $\Pi (\nabla u (q)) = 0$.
\[l:Piz\] Suppose that $\nabla u (0) = 0$ and ${{\text {Hess}}}_u (0)$ has kernel $K$. There exist $\epsilon > 0$ and $C$ so that if $p \in B_{\epsilon} \cap K$, then there exists $q \in B_{C\, |p|} \cap \left(p+ K^{\perp} \right) $ with $\Pi (\nabla u (q)) = 0$.
By the uniqueness of [@CM2], the flow is cylindrical at time $t = u(0) - \sqrt{\delta}$ in a ball $B_{C' \, \delta}(p)$ for every $\delta \in (0, \epsilon)$ for some $\epsilon > 0$ sufficiently small. Here $C' $ is a large constant.[[^3]]{} Thus, since $p \in B_{\epsilon} \cap K$, the level set $\{ u = u(0) - \sqrt{|p|} \}$ is an approximate cylinder about $K$ in $B_{C'|p|}$. In particular, the intersection $$\begin{aligned}
\{ u = u(0) - \sqrt{|p|} \} \cap \left(p+ K^{\perp} \right)\end{aligned}$$ is close to an $\SS^{n-k+1}$ and, furthermore, $u$ is strictly decreasing at each point of the intersection. Let $q \in \left(p+ K^{\perp} \right)$ be the point where $u$ achieves its maximum inside the subset of $\left(p+ K^{\perp} \right)$ bounded by $\{ u = u(0) - \sqrt{|p|} \} $. It follows that $q$ is in the interior and, thus, $\nabla u(q)$ is orthogonal to $K^{\perp}$ as claimed.
Local lemma
-----------
In this subsection, we assume that $u$ is $C^2$. The key to Theorem \[t:mcvxRn\] is the following local proposition:
\[p:key\] Suppose that $\nabla u (0) = 0$ and ${{\text {Hess}}}_u (0)$ has kernel $K$. Then there exists $\epsilon > 0$ so that $B_{\epsilon} \cap {{\mathcal{S}}}$ is the graph of a $C^1$ map $$\begin{aligned}
f: \Omega \subset K \to K^{\perp} \, ,\end{aligned}$$ where $\Omega$ is a connected open subset of $K$ containing $0$. Furthermore, $u$ is constant on $B_{\epsilon} \cap {{\mathcal{S}}}$.
It follows from theorem $2.5$ and corollary $4.5$ in [@CM4] that there is some $\delta > 0$ so that $B_{\delta} \cap {{\mathcal{S}}}$ is the graph of a $C^1$ map[[^4]]{} $$\begin{aligned}
f: \Omega \subset K \to K^{\perp} \, .\end{aligned}$$ Moreover, $B_{\delta} \cap {{\mathcal{S}}}$ is automatically a (relatively) closed subset of this graph. To prove the the first part of the proposition, we show that we can choose some $\epsilon \in (0 , \delta ]$ so ${{\mathcal{S}}}$ fills out the entire graph in $B_{\epsilon}$. To do this, we must rule out the following possibility:
- There is a sequence $p_j \to 0$ of points $p_j \in K$ so that the plane $P_j$ through $p_j$ and parallel to $K^{\perp}$ misses $B_{\delta} \cap {{\mathcal{S}}}$.
We will show that ($\star$) leads to a contradiction. Namely, for each $j$, Lemma \[l:Piz\] gives a point $q_j \in B_{C|p_j|} \cap P_j$ with $$\begin{aligned}
\label{e:qjp}
\Pi (\nabla u (q_j)) = 0 \, ,\end{aligned}$$ where $\Pi $ is orthogonal projection onto $K^{\perp}$. Since ${{\mathcal{S}}}$ does not intersect $P_j$, we know that $\nabla u ( q_j) \ne 0$. Therefore, [(\[e:qjp\])]{} gives that $$\begin{aligned}
\label{e:qjp2}
\Pi ({{\bf{n}}}(q_j)) = 0 \, .\end{aligned}$$ However, this contradicts Lemma \[l:normals\] since $q_j \to 0$. Thus, we get the desired $\epsilon > 0$. This gives the first part of the proposition.
Next, we must show that this graph is contained in a level set of $u$. This follows immediately from part (B) of theorem $1.2$ in [@CM4] since any two points in the graph can be connected by a $C^1$ curve in ${{\mathcal{S}}}$.
Local extinction after singularities
------------------------------------
In the next lemma, $p \in {{\mathcal{S}}}_k \setminus {{\mathcal{S}}}_{k-1}$ is a singularity of the flow and $K^{\perp}_p$ is the $n+1 - k$ dimensional plane through $p$ orthogonal to the axis of the singularity.
\[l:sepa\] There exists $\epsilon > 0$, depending only on $u$ and not on $p$, so that
- $B_{\epsilon} (p) \cap \{ u > u(p) \}$ does not intersect $K^{\perp}_p$.
By the uniqueness of [@CM2], the flow is cylindrical at time $t = u(p) - \sqrt{\delta}$ in a ball $B_{C \, \delta}(p)$ for every $\delta \in (0, \epsilon)$ for some $\epsilon > 0$ sufficiently small. Here $\epsilon > 0$ depends only on the cylindrical scale and, thus, is uniform in $p$ by theorem $3.1$ in [@CM4] because each strata is compact by Lemma \[l:strata\].
The intersection of the level set $u = u(p) - \sqrt{\delta}$ with $K^{\perp}_p$ is an $(n-k)$ sphere that separates $K^{\perp}_p$ (at least in the ball $B_{\epsilon} (p)$) into an inside containing $p$ and an outside where the flow has recently gone through. Because the flow is monotone, it can never return to this outside region. By assumption, these inside regions shrink to $p$ as $\delta \to 0$.
The next corollary shows that if a critical time can be approached by future regular times, then each critical point at this time is a local maximum.
\[c:extinct\] Suppose that $u$ is $C^2$, $\nabla u (0) = 0$, and there exist $t_i > u(0)$ with $t_i \to u(0)$ and $\nabla u \ne 0 $ on $\{ u = t_i \}$. Then there exists $\delta > 0$ so that $$\begin{aligned}
\sup_{B_{\delta}} u = u (0) \, .\end{aligned}$$
Let $\epsilon > 0$ be from Lemma \[l:sepa\]. We will argue by contradiction, so suppose instead that there is a sequence $p_j \to 0$ with $u(p_j) > u(0)$. By continuity of $u$, $u(p_j) \to u(0)$. Thus, after passing to subsequences for the $p_j$’s and $t_j$’s, we can assume that $$\begin{aligned}
u(p_1) > t_1 > u(p_2) > t_2 > \dots \to u(0) \, .\end{aligned}$$
Suppose that $i$ is large so that $|p_i| < \epsilon$. Since $u$ is continuous and $u(p_i) > t_i > u(0) $, the line segment from $0$ to $p_i$ intersects $\{ u= t_i \}$. Thus, we can choose $q_i$ with $$\begin{aligned}
\label{e:closew}
\left| \Pi_{\text{axis}} (q_i) \right|^2 = \min \left\{ \left| \Pi_{\text{axis}} (q) \right|^2 \, | \, q \in B_{\epsilon} {\text{ and }} u(q) = t_i \right\} \leq |p_i|^2 \, .\end{aligned}$$ This has two consequences: $$\begin{aligned}
\left| q_i \right|^2 & \to 0 \, , \label{e:qit0} \\
\Pi ({{\bf{n}}}(q_i)) &= 0 \, . \label{e:nablaud}\end{aligned}$$ To prove [(\[e:qit0\])]{}, use [(\[e:closew\])]{} to get that $ \left| \Pi_{\text{axis}} (q_i) \right|^2 \to 0$ and then use that the support of the flow for $u> u(0)$ must be close to $K$ near $0$ (by theorem $3.1$ in [@CM4]).
To see [(\[e:nablaud\])]{}, let $h: u^{-1} (t_i) \to \RR$ be given by $h(x) = \left| \Pi_{\text{axis}} (x) \right|^2$, so that $$\begin{aligned}
\frac{1}{2} \, \nabla_x h = \Pi_{\text{axis}} (x)- \langle \Pi_{\text{axis}} (x) , {{\bf{n}}}(x) \rangle \, {{\bf{n}}}(x) \, .\end{aligned}$$ Since $q_i$ is a minimum of $h$, we get that $\nabla_{q_i} h = 0$ and, therefore, $$\begin{aligned}
\Pi_{\text{axis}} (q_i) = \langle \Pi_{\text{axis}} (q_i) , {{\bf{n}}}(q_i) \rangle \, {{\bf{n}}}(q_i) \, .\end{aligned}$$ It follows that $ \Pi_{\text{axis}} (q_i) = \pm \, \left| \Pi_{\text{axis}} (q_i) \right| \, {{\bf{n}}}(q_i)$. This implies that $$\begin{aligned}
\Pi_{\text{axis}} (q_i) = 0 {\text{ or }} \Pi ({{\bf{n}}}(q_i)) = 0 \, .\end{aligned}$$ Lemma \[l:sepa\] rules out the first possibility, so we get [(\[e:nablaud\])]{}.
On the other hand, [(\[e:qit0\])]{} allows us to apply Lemma \[l:normals\] to get that $$\begin{aligned}
\label{e:piax}
\Pi_{\text{axis}} ({{\bf{n}}}(q_i)) \to 0 \, .
\end{aligned}$$ This contradicts [(\[e:nablaud\])]{}, completing the proof.
Proofs of the main results
--------------------------
We will prove one direction of Theorem \[t:mcvxRn\] in the following proposition.
\[p:mcvxRn\] If $u$ is $C^2$, then
1. There is exactly one singular time $T$ (where the flow becomes extinct).
2. The singular set ${{\mathcal{S}}}$ is a $k$-dimensional closed connected embedded $C^1$ submanifold of singularities where the blowup is a cylinder $\SS^{n-k} \times \RR^k$ at each point.
Moreover, ${{\mathcal{S}}}$ is tangent to the $\RR^k$ factor in (2).
Fix a point $p \in {{\mathcal{S}}}$. Let $k$ be the dimension of the kernel of ${{\text {Hess}}}_u (p)$, so $p$ is cylindrical of type $\SS^{n-k} \times \RR^k$. Let ${{\mathcal{S}}}_p$ be the component of ${{\mathcal{S}}}$ containing $p$; note that each point in ${{\mathcal{S}}}_p$ must also be cylindrical of type $\SS^{n-k} \times \RR^k$ by Lemma \[l:strata\]. Given $q \in {{\mathcal{S}}}_p$, let $K^{\perp}_q$ be the $k$-dimensional kernel of ${{\text {Hess}}}_u (q)$.
Proposition \[p:key\] implies that each point $q$ in ${{\mathcal{S}}}_p$ has an $\epsilon_q > 0$ so that
- $B_{\epsilon_q}(q) \cap {{\mathcal{S}}}$ is given as a $C^1$ graph over $K^{\perp}_q$.
- $u$ is constant on this graph.
Since ${{\mathcal{S}}}_p$ is compact and connected, it follows that ${{\mathcal{S}}}_p$ is a closed connected embedded $C^1$ $k$-dimensional submanifold and $u \equiv u(p)$ on ${{\mathcal{S}}}_p$.
Since ${{\mathcal{S}}}$ is compact, we conclude that ${{\mathcal{S}}}$ is given as a finite collection of disjoint embedded $C^1$ closed submanifolds $$\begin{aligned}
{{\mathcal{S}}}= \cup_{j=1}^N {{\mathcal{S}}}_{p_j} {\text{ with }} u ({{\mathcal{S}}}_{p_j}) \equiv u (p_j) \, .\end{aligned}$$
Let $T$ be the first singular time. In the remainder of the proof, we will show that
1. $T$ is also the extinction time and, thus, the only singular time.
2. ${{\mathcal{S}}}$ has only one component.
Let ${{\mathcal{S}}}_T = {{\mathcal{S}}}\cap \{ u = T \}$ be the union of the ${{\mathcal{S}}}_{p_j}$’s where $u(p_j) = T$. Note that ${{\mathcal{S}}}_T$ is compact and there exists $\kappa > 0$ so that $$\begin{aligned}
{{\mathcal{S}}}\cap \{ T < u < T+ \kappa \} = \emptyset \end{aligned}$$ since there are only finitely many singular times. Thus, Corollary \[c:extinct\] gives $\delta > 0$ so that $$\begin{aligned}
\label{e:loce}
\sup_{ T_{\delta} ({{\mathcal{S}}}_T)} \, u = T \, , \end{aligned}$$ where $T_{\delta} ({{\mathcal{S}}}_T)$ is the $\delta$-tubular neighborhood of ${{\mathcal{S}}}_T$.
We can now prove (A) by contradiction. Namely, if (A) does not hold, then [(\[e:loce\])]{} and the monotonicity of the flow imply that $\{ u = t \}$ intersects both inside and outside of $T_{\delta/2} ({{\mathcal{S}}}_T)$ for $t < T$. Since the initial hypersurface is connected and the flow is smooth before $u=T$, we know that $\{ u = t \}$ is connected for each $t< T$. Thus, we get a sequence of points $z_j \in \partial T_{\delta/2} ({{\mathcal{S}}}_T)$ with $u(z_j ) < T$ and $u(z_j) \to T$. By compactness, a subseqence of the $z_j$’s converges to $z \in \partial T_{\delta/2} ({{\mathcal{S}}}_T)$. Continuity of $u$ implies that $u(z) = T$ and, thus, [(\[e:loce\])]{} implies that $z$ is a local maximum for $u$ and $\nabla u (z) = 0$. This contradicts that $z \in \partial T_{\delta/2} ({{\mathcal{S}}}_T)$ is not a critical point, giving (A).
Now that we know that every point in $\{ u = T \}$ is a critical point, the same argument that we used for (A) implies that ${{\mathcal{S}}}= {{\mathcal{S}}}_T$ is connected. This gives (B), completing the proof.
The arrival time is $C^2$ away from the axis
============================================
Throughout this section, $u$ will be the arrival time for a mean convex flow in $\RR^{n+1}$ starting from a smooth closed mean convex hypersurface. By [@CM5], $u$ is twice differentiable everywhere with bounded ${{\text {Hess}}}_u$ and is smooth away from the singular set where $\nabla u = 0$.
It follows from [@CM4] that the region in [(\[e:region\])]{} intersects the singular set only at $0$ for $\delta > 0$ small enough. Thus, by [@CM5], we need only show that any sequence $q_j \to 0$ in [(\[e:region\])]{} must have ${{\text {Hess}}}_u (q_j) \to {{\text {Hess}}}_u (0)$. Furthermore, by lemma $2.11$ of [@CM5], $u(x) \leq u (0)$ in the region [(\[e:region\])]{} with equality only for $x=0$.
If $e_1 ,\dots e_n$ is an orthonormal frame for the level sets of $u$, then $$\begin{aligned}
{{\text {Hess}}}_u (e_i , e_j) &= \frac{A(e_i , e_j)}{H} \, , \\
{{\text {Hess}}}_u ({{\bf{n}}}, {{\bf{n}}}) &= \nabla_{{{\bf{n}}}}|\nabla u| = - \frac{\partial_t H}{H^3} = - \frac{(\Delta + |A|^2) H}{H^3} \, , \\
{{\text {Hess}}}_u (e_i , {{\bf{n}}}) &= \nabla_{e_i} |\nabla u| = - \frac{H_i}{H^2} \, .\end{aligned}$$ In the region [(\[e:region\])]{}, the uniqueness of [@CM2] gives that the rescaled level set flow converges to cylinders with axis $K$. If we let $\rho$ denotes the distance to $K$, then $$\begin{aligned}
\frac{\nabla u}{|\nabla u|} \to \partial_{\rho} {\text{ and }}
\frac{1}{H \rho} = \frac{ |\nabla u|}{\rho} \to 1 \, , \\
-\frac{A}{H} \to \frac{1}{n-k} \, \Pi {\text{ restricted to the tangent space}}, \\
\frac{ \left| \nabla \frac{A}{H} \right|}{H} \, ,
\frac{ | \nabla H |}{H^2} {\text{ and }}
\frac{\left| \Delta H \right|}{H^3} \to 0 \, .\end{aligned}$$ The first three claims are immediate from the uniqueness of the blow up. The last three claims follow from the smooth convergence of the rescaled level sets to the cylinder (where each of these quantities is zero); the powers of $H$ are the appropriate scaling factors.
Combining these facts shows that ${{\text {Hess}}}_u$ is continuous in this conical region.
One direction is given by Proposition \[p:mcvxRn\]. We will suppose therefore that (1) and (2) hold and show that $u$ must be $C^2$. By [@CM5], $u$ is twice differentiable everywhere and smooth away from the singular set ${{\mathcal{S}}}$. Thus, we must show that ${{\text {Hess}}}_u$ is continuous at each point of ${{\mathcal{S}}}$.
Using the form of the Hessian, it follows that if $p , \tilde{p} \in {{\mathcal{S}}}$, then $$\begin{aligned}
\label{e:huclo}
\left| {{\text {Hess}}}_u (p) - {{\text {Hess}}}_u (\tilde{p}) \right| \leq C \, {{\text {dist}}}(T_p {{\mathcal{S}}}, T_{\tilde{p}} {{\mathcal{S}}}) \, .\end{aligned}$$ Fix a point $p \in {{\mathcal{S}}}$ and let $q_j \to p$ be any sequence. We must show that ${{\text {Hess}}}_u (q_j) \to {{\text {Hess}}}_u (p)$. For each $j$, let $p_j$ be a closest point in ${{\mathcal{S}}}$ to $q_j$. It follows that
- $|p_j - q_j| \leq |p- q_j| \to 0$.
- $\langle (p_j - q_j ) , T_{p_j} {{\mathcal{S}}}) \rangle = 0$.
The second property allows us to apply Theorem \[t:C2trans\] to get that $$\begin{aligned}
\left| {{\text {Hess}}}_u (p_j) - {{\text {Hess}}}_u (q_j) \right| \to 0 \, .\end{aligned}$$ Finally, since ${{\mathcal{S}}}$ is $C^1$ and $p_j \to p$, [(\[e:huclo\])]{} implies that $$\begin{aligned}
\left| {{\text {Hess}}}_u (p_j) - {{\text {Hess}}}_u (p) \right| \to 0 \, .\end{aligned}$$
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[^1]: The main theorem of [@CM4] states the submanifolds are Lipschitz, but the proof shows that they are $C^1$.
[^2]: The marriage ring is a thin mean convex torus of revolution in $\RR^3$ where the MCF is smooth until it becomes extinct along a circle.
[^3]: We can make $C'$ as big as we want at the cost of decreasing $\epsilon$.
[^4]: The main theorem of [@CM4] states that the map $f$ below is Lipschitz. However, the regularity of the distribution of $k$-planes implies that it is in fact $C^1$.
|
---
abstract: 'To understand emergent magnetic monopole dynamics in the spin ices [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{}, it is necessary to investigate the mechanisms by which spins flip in these materials. Presently there are thought to be two processes - quantum tunneling at low and intermediate temperatures, and thermally activated at high temperatures. We identify possible couplings between crystal field and optical phonon excitations and construct a strictly constrained model of phonon-mediated spin flipping that quantitatively describes the high temperature processes in both compounds, as measured by quasielastic neutron scattering. We support the model with direct experimental evidence of the coupling between crystal field states and optical phonons in [Ho$_2$Ti$_2$O$_7$]{}.'
author:
- M Ruminy
- S Chi
- S Calder
- T Fennell
title: 'Phonon mediated spin flipping mechanism in the spin ices [Dy$_2$Ti$_2$O$_7$]{} and [Ho$_2$Ti$_2$O$_7$]{}'
---
In rare earth compounds, magnetic responses can be strongly and non-monotonically dependent on the strength or frequency of applied magnetic field, or the temperature. Examples include stepped magnetization curves in single ion magnets [@ishikawa:kx], or the multiply-peaked susceptibility response in LiYF$_4$:Ho$^{3+}$ [@Giraud:2001cp; @Giraud:2003il; @Bertaina:2006gf]. These effects appear because there are competing mechanisms that can contribute to the flipping of large rare earth magnetic moments. Owing to their different origins - conduction electrons [@Becker:1977ge], phonon-mediated (e.g. direct, Raman, Orbach, and phonon bottleneck effects [@ORBACH:1961cd; @Finn:2002fs; @Scott:1962it]), or quantum mechanical (tunneling, thermally assisted tunneling between excited states, resonant tunneling at electronic-nuclear level crossings, and co-tunneling [@Thomas:1996hg; @Bokacheva:2000ey; @Giraud:2001cp; @Giraud:2003il; @Gatteschi:bn]) - these mechanisms have quite different parametric dependencies. Understanding spin flipping (or relaxation) is currently important in rare-earth based single ion magnets [@Blagg:2013bf], especially in the context of applications in quantum information processing [@Leuenberger:2001ft; @Ardavan:2007ci; @Bogani:2008hi] that depend on the stability and control of quantum states [@Prokofev:2000bd; @feynmann; @Ghosh:2002wr; @Stamp:2004kf], and in spin ices, where they determine the mobility of magnetic monopole excitations [@Castelnovo:2008hb].
In a canonical spin ice such as [Dy$_2$Ti$_2$O$_7$]{} or [Ho$_2$Ti$_2$O$_7$]{} [@Bramwell:2001tpa], the magnetization dynamics of the low temperature Coulomb phase [@Fennell:2009ig; @Henley:2010vo] should be described by the cooperative behavior of the thermal population of emergent magnetic monopoles [@Castelnovo:2008hb; @Ryzhkin:2005ko], which form a magnetic Coulomb gas. Indeed, the spin relaxation time, $\tau$, of [Dy$_2$Ti$_2$O$_7$]{}, as extracted from $\chi_{ac}$, has been explained with considerable success by the monopole picture: both a thermally activated regime at $T<1$ K [@Snyder:2001uu; @Matsuhira:2001cp; @Snyder:2003ek; @Snyder:2004it] (which we call low temperature) and temperature independent plateau for $1<T<10$ K (intermediate temperature) are captured well by a theory of monopole hopping in dilute (unscreened) and concentrated (screened) magnetic Coulomb gases respectively [@Jaubert:2009ed]. The reentrant low temperature thermal activation is due to interactions between unscreened monopoles.
For a monopole to hop, a spin must be flipped, and because the plateau of $\tau$ was previously associated with quantum tunneling of the large, Ising-like Dy$^{3+}$ moments between the members of their ground state doublet [@Snyder:2004it], monopoles were assumed to hop by tunneling of the spins with temperature independent attempt frequency [@Jaubert:2009ed]. The resulting picture should describe the Coulomb gas realized in each material by relating the energy for monopole creation and unbinding to the exchange interactions [@BrooksBartlett:2014kf; @Zhou:2011fq]. However, subsequent measurements of [Ho$_2$Ti$_2$O$_7$]{} [@Quilliam:2011df] and [Dy$_2$Ti$_2$O$_7$]{} [@Matsuhira:2011ci; @Yaraskavitch:2012hk; @Kassner:2015jo] have found that in the unscreened regime this relationship is not exactly as expected, while simulations of [Dy$_2$Ti$_2$O$_7$]{} with a temperature-dependent hop rate agree better with the observed relaxation times [@Takatsu:2013bq]. These studies suggest that to understand out-of-equilibrium [@Giblin:2011ft; @Revell:2012hq; @Paulsen:2014cc; @Paulsen:1970de] and quantum dynamics [@Tomasello:2015vg; @Rau:2015jj] in spin ices at low temperature, it is essential to understand all contributions to the monopole hopping dynamics. As in LiYF$_4$:Ho$^{3+}$ [@Bertaina:2006gf], the first requirement is to understand the classical spin flipping mechanism of the spin ices.
Studies of [Dy$_2$Ti$_2$O$_7$]{} [@Snyder:2001uu; @Matsuhira:2001cp; @Snyder:2003ek; @Snyder:2004it] in which the intermediate temperature plateau was ascribed to quantum tunneling of the spins also revealed a second thermally activated regime for $T>10$ K (i.e. high temperature). The response of [Ho$_2$Ti$_2$O$_7$]{} is similar but the relative rate of the low temperature process is much faster than in [Dy$_2$Ti$_2$O$_7$]{} [@Ehlers:2002jz; @sup_mat]. The high temperature process in both spin ices was modeled by an Arrenhius law, with activation energy $\Delta$ attributed to over-barrier hopping via the first crystal field excitation (CFE). However, the best-fitting $\Delta$, although close, is not equal to the energy of any CFE in either material, and this interpretation does not explain how such a process would occur.
![General features of QENS in spin ices, as exemplified by [Ho$_2$Ti$_2$O$_7$]{}. We see a $|\vec{Q}|$-independent $S(|\vec{Q}|,\omega)$ response, with QENS around the elastic line and transitions amongst excited crystal field states (a). In panel b, we show an example of a resolution-convoluted fit of the quasielastic Lorentzian (QENS) and two CFEs (T1 and T2). In panel c we show the general evolution of the QENS and excited state transitions, along with the resolution regimes used in the measurements ($\lambda_{1,2,3}=11,6,4.3$ Å in this case).[]{data-label="fig:f1"}](hto_qens.eps)
We propose that phonon mediated processes involving a higher crystal field state interacting with phonons [@Lovesey:2000jh] provide a quantitative and physical explanation of the high temperature processes. In this mechanism, a rare earth ion is excited from one crystal field state to an intermediate excited state by absorption of a phonon, and then relaxes to a third state by emission of another phonon. Relaxation by a single such process has the characteristic temperature dependence of $n=1/(\exp{(\Delta/k_BT)}-1)$, where $\Delta$ is the energy of the phonon to be absorbed, but more than one process can operate simultaneously, depending which crystal field levels interact with phonons. The time and temperature scales of this type of process mean they can be studied by neutron scattering. Either the width ($\Gamma$) of the quasielastic neutron scattering (QENS) can be understood as lifetime broadening of the ground state doublet and used to give a measure of the spin relaxation time ($\tau$), as was done for rare earth cuprates [@Lovesey:2000jh]; or the width of a CFE can be followed directly, as was done for LiTmF$_4$ [@Babkevich:2015wj]. In the former case the origin of the relaxation was debated [@Boothroyd:2001im], while in the latter full details of the coupled phonons were not established. In the following, we measure $\Gamma$ using QENS, determine the allowed spin-lattice interactions and construct a model of phonon mediated processes in both materials that describes the high temperature processes quantitatively. We provide microscopic evidence of one such coupling.
![The QENS width $\Gamma$ as a function of temperature (a) and relaxation time as a function of inverse temperature (b), measured with different neutron wavelengths shown by the symbols. Solid lines are from the model described in the text, dotted lines in b indicate the resolution limit of the different settings of the spectrometer. The same quantities for [Ho$_2$Ti$_2$O$_7$]{} are shown in panel c and d, incorporating QENS (FOCUS, this study) and neutron spin echo (NSE, [@Ehlers:2002jz]), compared to the full model (FM), the first term of the model ($\Delta=26.3$ meV), and an Arrhenius law (AL) [@Ehlers:2002jz].[]{data-label="fig:f2"}](orbach_model2.eps "fig:")\
We have measured QENS in powder samples of both [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{} [@sup_mat] over a wide range of temperatures using the spectrometer FOCUS [@Juranyi:2003jq] at SINQ. We report results obtained using the $(0,0,2)$ reflection of both the pyrolytic graphite ($\lambda=4.3,5,6$ Å; resolution $\approx100,50,40$ $\mu$eV) and mica ($\lambda=11$ Å; resolution $\approx 20$ $\mu$eV) monochromators, where we selected the wavelength to give appropriate resolution for a range of temperatures. The quasielastic scattering was fitted by a single Lorentzian, adjusted by the Bose factor for detailed balance. The elastic line was removed by fitting with a Gaussian peak, whose parameters were derived from a measurement of the resolution using a vanadium standard. Additional Lorentzians were incorporated in the fit of high temperature data from [Ho$_2$Ti$_2$O$_7$]{} to model excited state CFEs that appear close to the elastic line. Points at the edges of two resolution ranges were measured with both settings to ensure overlap of the fitted peak widths.
An example of the $S(|\vec{Q}|,\omega)$ data obtained for [Ho$_2$Ti$_2$O$_7$]{} is shown in Fig. \[fig:f1\]a, and an example of the $|\vec{Q}|$-integrated data used for fitting is shown in Fig. \[fig:f1\]b. The temperature dependence of the width and intensity of the quasielastic scattering and CFEs can be seen in Fig. \[fig:f1\]c. Below $T\approx50$ K, the spin fluctuation processes are too slow for QENS, and, even with $\lambda=11$ Å, the response is resolution-limited, but as the temperature is further increased, the QENS broadens.
In Fig. \[fig:f2\]a, we show the temperature evolution of $\Gamma$ for both compounds, and also its representation as a relaxation time $\tau$ (panel b). Notably, the QENS spectra of [Dy$_2$Ti$_2$O$_7$]{} are nearly twice as broad as those of [Ho$_2$Ti$_2$O$_7$]{} throughout the sampled temperature range. In Fig. \[fig:f2\]c,d we show our data for [Ho$_2$Ti$_2$O$_7$]{} compared with NSE data from Ref. [@Ehlers:2002jz], which extends to longer times/lower temperatures, in terms of $\Gamma$ and $\tau$ respectively. All the lines in Fig. \[fig:f2\] are derived from models, either the model which we discuss below, or the Arrenhius law used in Ref. [@Ehlers:2004ui]. It can be seen in Ref. [@Ehlers:2004ui] that the relaxation time already departs from the Arrhenius law at the highest temperatures studied there, and this is made plain by the higher temperatures measured in this work (see Fig. \[fig:f2\]c and d).
The phonon-mediated spin relaxation mechanism depends on a magnetoelastic interaction of normal modes of vibration with the single ion crystal field potential [@Lovesey:2000jh]. The contribution to the temperature dependence of $\Gamma$ is given by $$\Gamma(T)=\sum_i\frac{3\pi r n_i}{2M\Delta_i}\zeta_{\mu}^2Z_\mu(\Delta_i)\{|\langle a |Q_\mu |v_i\rangle|^2+|\langle b |Q_\mu|v_i\rangle|^2\},
\label{eq:Orbacheq}$$ where $\zeta_{\mu}$ is the magnetoelastic coupling parameter for a phonon and intermediate crystal field state $|v_i\rangle$ at energy $\Delta_i$, $r$ is the number of ions per unit cell, and $M$ the mass of an oxide ion. The distribution function $n_i=(\exp{(\Delta_i/k_BT)}-1)^{-1}$ provides the temperature dependence of the process, and $Z_{\mu}(\Delta_i)$ is the partial phonon density of states (pPDOS) of the anionic modes of vibration transforming according to the representation $\mu$. The prefactors are absorbed into fitting parameters such that $\Gamma(T)=\sum_{j}B_in_i$ [@Lovesey:2000jh].
For an intermediate state $i$ to enter the summation, we require a finite matrix element for the quadrupolar operator ($Q_\mu$) for the transition between the initial ($\langle a|$) and intermediate ($|v\rangle$) crystal field state and spectral overlap of this state with a phonon ($u_\mu$) of identical symmetry ($\mu$ labels the irreducible representation of the operator or excitation). At the rare earth site, in $D_{3d}$ symmetry, there are three quadrupolar operators with symmetry $(A_1,E,E)$, and the matrix element for a transition $\langle a|Q_\mu|v\rangle$ is finite if the direct product of the representations ($\gamma$) of the two states and the transition operator contain the unit representation, $\gamma_a\times\gamma_v\times\gamma_Q\in A_1$. There are three possible combinations for finite matrix elements of the magnetoelastic interaction operator: $\gamma_{a,Q,v,u}=E$ (1); $\gamma_{a,v}=E,\gamma_{Q,u}=A$ (2); $\gamma_{a,Q,u}=E,\gamma_v=A$ (3). The initial state $|a\rangle$ is a member of the ground state doublet, and $|v\rangle$ is an excited crystal field state, the final state $|b\rangle$ is the other member of the ground state doublet.
The transitions involved in the model are summarized in Fig. \[fig:f3\]. Using the wave functions of crystal field states [@xtal_fields], we evaluated the quadrupolar matrix elements of the crystal field transitions. Quasi-degenerate (at the Brillouin zone center) phonon modes of the correct symmetry were identified from the phonon band structure [@phonons]. The vibrational modes involved are dominated by oxide ions in the $48f$ position, so we approximate $Z_\mu(\Delta_i)$ by the pPDOS of this site.
In [Ho$_2$Ti$_2$O$_7$]{}, we find the two largest matrix elements between the ground state and the doublets at $E=26.3, 60$ meV, and weaker matrix elements between the ground state and the singlet and doublet at $E=21, 22$ meV. Each of these transitions is quasi-degenerate with a phonon of appropriate symmetry, while the remaining transitions have no overlap with a vibrational mode of $E$ or $A$ symmetry. For [Ho$_2$Ti$_2$O$_7$]{}, we construct our model using three intermediate states at $\Delta_i=21.5,26.3,60$ meV, where the first represents the effect of the weak matrix elements. In [Dy$_2$Ti$_2$O$_7$]{}, the intermediate states are those at $E=21,31, 43$ meV. The state at 91 meV also meets the symmetry requirements but is outside the temperature window of this study. Other states have large matrix elements, but no compatible phonon.
For [Ho$_2$Ti$_2$O$_7$]{} we included both QENS and NSE data in the fit, and since the states $B_1$ and $B_2$ have similar energies and nearly identical pPDOS [@phonons], we related their values by the ratio of their quadrupolar matrix elements. The resulting coefficients are $B_{1,2,3}=0.018,0.2,0.79$ meV. For [Dy$_2$Ti$_2$O$_7$]{}, to reduce the number of fitting parameters, the values of the parameters $B_2$ and $B_3$ were linearly related using the energies of their CFEs, unity for the ratio of the oxygen phonon density of states, and their quadrupolar transition matrix elements. We obtained $B_{1,2,3}=0.23, 0.49, 0.38$ meV. As shown in Fig. \[fig:f2\], the model fits the relaxation rates of both compounds very well. For [Ho$_2$Ti$_2$O$_7$]{}, relaxation via the level at 26.3 meV describes the QENS width effectively up to $T\approx250$ K, and the third intermediate state at $E=60$ meV dominates at higher temperatures.
![CFE and phonon interactions in a single crystal of [Ho$_2$Ti$_2$O$_7$]{}. Two CFEs ($E=22,26.3$ meV) and a phonon ($E=31$ meV) can be seen. The CFEs can be measured at the zone center ($l=8$) and boundary ($l=7$) and shift upward between 5 and 200 K. The CFE at $E=22$ meV disperses upward by 0.5 meV at the zone center where it intersects with an optical phonon [@phonons] (panel a). The intensity of the two CFEs follows the magnetic form factor along $(0,0,l)$, except for the CFE at $E=22$ meV which is boosted at zone centers with strong phonon structure factors ($l=4n$) (panel b and c, scan positions of panel a are indicated by colored points).[]{data-label="fig:f5"}](tax_for_resub.eps)
The values for the magnetoelastic coupling constants $\zeta_{\mu}$ extracted from the fitted parameters under these approximations [@sup_mat] suggest that the magnetoelastic coupling is linear (in energy), consistent with physical ingredients of the model. To further verify our model, we sought direct evidence of interactions between CFEs and phonons using a single crystal of [Ho$_2$Ti$_2$O$_7$]{} and the triple axis spectrometer HB3 at HFIR, ORNL. The $(0,0,2)$ reflection of the beryllium monochromator provides access to quite high energy transfers with good energy resolution - using a pyrolytic graphite filter and analyzer ($(0,0,2)$ reflection), the energy resolution was $\Delta E \approx 1.7$ meV in the energy transfer window of $20-30$ meV. With fixed final energies of $E_f=14.7, 30.5$ meV, we measured energy scans in the range $18<E<33$ meV at different $(0,0,l)$ positions that were either Brillouin zone centers $(0,0,l=2,4,6,8,10,12)$ or boundaries $(0,0,l=3,5,7,9)$, at $T=5,200$ K.
Fig. \[fig:f5\]a shows the two CFEs at $E\approx22,26.3$ meV and a phonon at $E\approx 31$ meV, measured at $(0,0,8)$. The intensities of the CFEs decrease as the temperature is raised, and they shift upward in energy, while the intensity of the phonon increases but its energy does not change. The upward shift of the CFEs is also shown by the downward shift of the excited state transitions T1 and T2 in Fig. \[fig:f1\]c. Comparison of the same scan at $l=7,8$ shows a resolution limited sharp peak for both CFEs with a 0.5 meV upward dispersion between zone boundary and center for the first, but at identical energies for the second. The $l$-dependence of the intensity of the CFEs (Fig. \[fig:f5\]b) follows the dipole magnetic form factor at zone boundaries ($l=n$) and at zone centers where the $Fd\bar{3}m$ space group forbids a Bragg reflection ($l=2n$), but the CFE at $E\approx22$ meV has anomalously large intensity at zone centers with strong Bragg reflections ($l=4n$) while the CFE at $E\approx 26.3$ meV also follows the magnetic form factor at these positions.
Phonon calculations [@phonons] show that there is an optical phonon with $E$ symmetry at $E\approx22$ meV at the zone center. The phonon disperses away at the zone boundary, and its structure factor is suppressed at zone centers where the Bragg intensity is not allowed. Hence at all these positions (i.e. $l=n$ and $l=2n$) both CFEs are unaffected and follow the magnetic form factor. At those zone centers with a strong Bragg reflection, the strong phonon structure factor boosts the intensity well above the magnetic form factor, but the observation of a single mode displaced from the energy of the uncoupled zone boundary CFE or phonon shows that the coupling pulls the two excitations into resonance, i.e. they are not just coincident. Conversely, the phonon mode expected to interact with the CFE at 26.3 meV was calculated to have a very weak structure factor along $(0,0,l)$, due to its polarization. Hence we observe no signatures of coupling in this direction, and this CFE also follows the magnetic form factor (Fig. \[fig:f5\]c).
We have shown that the symmetries and wavefunctions CFEs and optical phonons can be used to construct a physically realistic model for phonon mediated spin flipping processes. Modes with the correct symmetry and energy exist in the spin ices [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{}, and we presented direct evidence of one of the couplings in [Ho$_2$Ti$_2$O$_7$]{}. A model based on these spin-lattice interactions describes the high temperature spin relaxation in both compounds very well. We advance this model as the first microscopic description of a spin flipping mechanism in the spin ices [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{}, and also as a quantification of the spin-lattice interactions possible in these materials. Our investigation sets the stage for microscopic investigations of the possible quantum processes at low temperature, and their consequences for collective monopole dynamics.
MR and TF thank ORNL staff for support; P. Santini, B. Tomasello, C. Castelnovo, R. Moessner, J. Quintanilla, and S. Giblin for discussion; and the authors of Refs. [@xtal_fields] and [@phonons] for related collaboration. Neutron scattering experiments were carried out at the continuous spallation neutron source SINQ at the Paul Scherrer Institut at Villigen PSI in Switzerland; and High Flux Isotope reactor (HFIR) of Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA. Work at PSI was partly funded by the Swiss NSF (grants 200021\_140862 and 200020\_162626). Research at Oak Ridge National Laboratory’s HFIR was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U. S. Department of Energy.
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Supplementary Information
=========================
[*Further details of samples and experiments:*]{} The powder samples of [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{} were originally described in Refs. [@phonons; @xtal_fields]. They were prepared from stochiometric ratios of the oxides Ho$_2$O$_3$ or Dy$_2$O$_3$, and TiO$_2$ in a solid state reaction. The oxides, with 99.99% purity, were annealed at 850 $^\mathrm{o}$C for 10 hours, then mixed and ground, and heated at 950-1300 $^\mathrm{o}$C for 140 hours with several intermediate grindings. The structures were verified by combined neutron and x-ray diffraction experiments, which were carried out on HRPT [@Fischer:2000to] at SINQ, PSI and the Materials Science Beamline (MSB) [@Willmott:2013gv] at the SLS, PSI. Rietveld refinement of the structures as implemented in the Fullprof [@fullprof] software proved both samples to be of high quality and single phase.
For the QENS experiment on FOCUS, to overcome the strong absorption of natural isotopic abundance dysprosium we used a flat plate sample holder with very thin layer of sample. The plate was oriented perpendicular to the incident beam to reduce the path length of neutrons in the sample, and we used data from low angle detectors only. To have reasonable statistics requires long counting times. We also confirmed some points by measuring our isotopically enriched crystal of $^{162}$[Dy$_2$Ti$_2$O$_7$]{}, but the crystal is much smaller than the cross section of the beam so intersects relatively few incident neutrons and hence does not greatly improve the situation. [Ho$_2$Ti$_2$O$_7$]{} can be used directly as a packed powder and the count rate is much larger.
[*Magnetoelastic coupling constants:*]{} The determination of an absolute value for the magnetoelastic coupling constant $\zeta$ from the fitted $B$ parameters is difficult. Despite having the full calculated pyrochlore phonon spectrum [@phonons], there is a large uncertainty connected with the evaluation of the partial phonon densities of states $Z_{\mu}(\Delta_i)$, which only includes individual vibrational modes that transform according to the representation $\mu$. We note however, that all vibrational modes involved in the model in both compounds are dominated by the motion of the oxygen O(48$f$) ions, which surround the rare earth ion. Hence, the partial phonon density of states of the O(48$f$) ions gives at least good estimates for the ratios of the $Z_{\mu}(\Delta_i)$ at different energy transfers $\Delta_i$. We therefore estimate all the magnetoelastic coupling constants $\zeta_i$ for the intermediate CEF states in both [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{} relative to the coupling constant $\zeta_3$ of the third intermediate CEF state in [Ho$_2$Ti$_2$O$_7$]{} at $E=60$meV: $$\frac{\zeta_i}{\zeta_3} = \sqrt{\frac{B_i\Delta_i Z_{\text{O}(48f)}(\Delta_3) \left[ \sum_{\mu} \sum_{j=\pm}|\langle j|Q(\mu)|v_3^j\rangle|^2 \right]}{ B_3\Delta_3 Z_{\text{O}(48f)}(\Delta_i) \left[ \sum_{\mu} \sum_{j=\pm}|\langle j|Q(\mu)|v_i^j\rangle|^2 \right] }},$$ where the summation $j$ runs over the two members of the corresponding ground state doublet and all members of the excited intermediate CEF state $v_i$. The ratios of the partial phonon densities of states are taken as: $$\frac{Z_{\text{O}(48f)}(\Delta=21\,\text{meV})}{Z_{\text{O}(48f)}(\Delta=60\,\text{meV})} = \frac{Z_{\text{O}(48f)}(\Delta=26\,\text{meV})}{Z_{\text{O}(48f)}(\Delta=60\,\text{meV})} = 1.5 \nonumber$$ $$\frac{Z_{\text{O}(48f)}(\Delta=31\,\text{meV})}{Z_{\text{O}(48f)}(\Delta=60\,\text{meV})} = 1.$$ In order to compare between two different ions, we calculate the ratios of the bare magnetoelastic coupling constants $\tilde{\zeta_i}$, which follow from $\zeta=\tilde{\zeta}\alpha{J}$ [@Lovesey:2000jh] with the Stevens factors $\alpha(J)$ of the rare earth ions, here $\alpha=-1/450$ and $-2/315$ for Ho$^{3+}$ and Dy$^{3+}$ respectively. The resulting ratios are illustrated in Fig. \[fig:s1\]. We find that the bare magnetoelastic coupling constants of all intermediate CEF states in the two compounds depend linearly on the energy transfer of the intermediate states within errorbars. Given a linear mode coupling such as between the quadrupolar operators and the normal modes of vibration in the magnetoelastic interaction operator of Eq. \[eq:Orbacheq\], it is plausible that the magnetoelastic coupling strength vanishes in the limit of zero energy transfer and is generally highest at high energy transfer. The experimental observations are therefore consistent with the expectation and give strong support for the phonon mediated mechanism with the established intermediate CEF states in both rare earth pyrochlore titanates [Ho$_2$Ti$_2$O$_7$]{} and [Dy$_2$Ti$_2$O$_7$]{}.
![Magneotelastic coupling constants extracted from the fitted parameters $B_j$, using the partial phonon density of states for the O($48f$) for $Z_\mu(\Delta_j)$, normalizing the rare earth ions by the Stevens factors $\alpha(J)$, and scaling to highest energy coupling in [Ho$_2$Ti$_2$O$_7$]{}.[]{data-label="fig:s1"}](coupling_constants.eps)
[*Relation with previously described relaxation processes:*]{} Previously two relaxation processes have been reported in both compounds [@Snyder:2001uu; @Matsuhira:2001cp; @Snyder:2003ek; @Snyder:2004it; @Ehlers:2002jz]. In [Dy$_2$Ti$_2$O$_7$]{} these can be seen in $\chi_{ac}$ experiments because the relative rates of them are favorable for this technique. The low temperature process, ascribed to quantum tunneling due to the plateau in the relaxation time at intermediate temperatures, is sufficiently slow that it does not compete with the high temperature process even as this becomes slow at temperatures of $T\approx20$ K. Hence peaks can be observed in $\chi''$ (as measured by $\chi_{ac}$) as each process dominates the relaxation in the experimental frequency range, first the high temperature process at $T\approx20$ K, then the low temperature process at $T<10$ K. In [Ho$_2$Ti$_2$O$_7$]{}, the low temperature process seems to be relatively much faster, and competes with the high temperature process effectively on the time scales of $\chi_{ac}$. Therefore, one cannot observe a peak in $\chi''$ that corresponds to the high temperature process using $\chi_{ac}$ measurements. To observe the high temperature process one must go to higher temperatures where the low temperature process is no longer competitive and the high temperature process is necessarily much faster. This means that techniques with shorter timescales are required to observe it, which is why it was originally found using neutron spin echo and quasielastic neutron scattering [@Ehlers:2002jz; @Ehlers:2004ui].
The high temprature process in [Dy$_2$Ti$_2$O$_7$]{} was fitted by an Arrhenius law with activation energy $\Delta=17-25$ meV, which was compared to the first CFE. At this time, the exact energy of the CFE was not actually known, but was expected to be of this order. It is now known to be at $21$ meV [@xtal_fields]. In [Ho$_2$Ti$_2$O$_7$]{}, the activation energy was given as $\Delta=25(\pm1)$ meV, but the CFEs are at $E=21.9, 26.3$ meV. The high temperature process was described as a flipping of the spin via the first CFE, but in this case the activation energy should match exactly the energy of the CFE, and the Arrhenius law should describe the relaxation time throughout the temperature range where the process is dominant. Neither of these conditions is actually fulfilled, particularly when the relaxation time is measured to higher temperatures.
However, the previous investigations showed that the presence or absence of a peak in $\chi''_{ac}$ depends on the relative rate of high and low temperature processes. In absence of a competing low temperature process [Ho$_2$Ti$_2$O$_7$]{} would also have a peak in $\chi''_{ac}$ for the high temperature process at a similar temperature to [Dy$_2$Ti$_2$O$_7$]{}. This condition is still satisfied if the temperature dependence of the relaxation rate provided by our model is treated in an analogous fashion. Hence the phonon-mediated model provides a physical explanation and quantitative description of all known observations about the high temperature processes.
|
---
abstract: 'We analyze the discovery potential of the next-to-minimal supergravity motivated model: NmSuGra. This model is an extension of mSuGra by a gauge singlet, and contains only one additional parameter: $\lambda$, the Higgs-singlet-Higgs coupling. NmSuGra solves the $\mu$-problem and reduces the fine tuning of mSuGra. After identifying parameter space regions preferred by present experimental data, we show that these regions of NmSuGra are amenable to detection by the combination of the Large Hadron Collider and an upgraded Cryogenic Dark Matter Search. This conclusion holds strictly provided that the more than three sigma discrepancy in the difference of the experimental and the standard theoretical values of the anomalous magnetic moment of the muon prevails in the future.'
author:
- Csaba Balázs
- Daniel Carter
bibliography:
- 'references.bib'
title: Guaranteed discovery of the NmSuGra model
---
[ address=[School of Physics, Monash University, Melbourne Victoria 3800, Australia]{} ]{}
[ address=[School of Physics, Monash University, Melbourne Victoria 3800, Australia]{} ]{}
Introduction {#sec:Introduction}
============
Supersymmetry is very successful in solving outstanding problems of the standard model (SM) of elementary particles, including the hierarchy of fundamental energy scales, the existence and properties of dark matter and the unification of gauge forces. However, the minimal supersymmetric standard model (MSSM) suffers from several problems. For example, the $\mu$ term is not protected from radiative corrections, and its viable parameter regions are now quite fine tuned [@Giudice:2008bi]. Gauge-singlet extensions of the MSSM offer solutions to these problems. In the next-to-minimal MSSM (NMSSM), the $\mu$ term is dynamically generated and no dimensionful parameters are introduced in the superpotential (other than the vacuum expectation values that are all naturally weak-scale), making the NMSSM a truly natural model (see references in [@Balazs:2008ph]). We define the next-to-minimal supergravity motivated (NmSuGra) model, imposing universality of sparticle masses, gaugino masses, and trilinear couplings at the grand unification theory (GUT) scale.
Using a simple likelihood analysis, we first identify the parameter regions of the NmSuGra model that are preferred by present experimental data. We combine theoretical exclusions with limits from the CERN Large Electron-Positron (LEP) collider, the Fermilab Tevatron, NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) satellite, the Soudan Cryogenic Dark Matter Search (CDMS), the Brookhaven Muon g$-$2 Experiment, and various b-physics measurements including $b \to s \gamma$ and $B_s \to l^+ l^-$. We then show that, assuming recent results on the muon g$-$2 are accurate, the favored parameter space can be detected by the combination of the LHC and an upgraded CDMS. (See Ref.s [@Stockinger:2007pe; @Passera:2008jk] on the theoretical uncertainty of $\Delta a_{\mu}$.)
The NmSuGra model
=================
In this work, we adopt the superpotential $$\begin{aligned}
W = W_{Y} + \lambda \hat{S} \hat{H}_u \cdot \hat{H}_d + \frac{\kappa}{3} \hat{S}^3,
\label{eq:W}\end{aligned}$$ where $W_{Y}$ is the MSSM Yukawa superpotential, $\hat{S}$ ($\hat{H}_{u,d}$) is a standard gauge singlet ($SU(2)_L$ doublet) chiral superfield, $\lambda$ and $\kappa$ are dimensionless couplings, and $\hat{H}_u \cdot \hat{H}_d =
\epsilon_{\alpha\beta} \hat{H}_u^\alpha \hat{H}_d^\beta$ with $\epsilon_{11} =
1$. The corresponding soft supersymmetry breaking terms are $$\begin{aligned}
\mathcal{L}^{soft} = \mathcal{L}^{soft}_{MSSM} + m_S^2 |S|^2 + \nonumber \\
(\lambda A_\lambda S H_u \cdot H_d + \frac{\kappa A_\kappa}{3} S^3 + h.c.),\end{aligned}$$ where $\mathcal{L}^{soft}_{MSSM}$ contains no $B \mu$ term[^1].
We assume that the soft masses of the gauginos unify to $M_{1/2}$, those of the sfermions and Higgses to $M_0$, and all the trilinear couplings (including $A_\kappa$ and $A_\lambda$) to $A_0$ at the GUT scale. Defining $\mu = \lambda
\langle S \rangle$, and $\tan\beta = \langle H_u \rangle/\langle H_d \rangle$ (the ratio of Higgs vevs), our free parameters are [@Balazs:2008ph]: $$\begin{aligned}
M_0, ~ M_{1/2}, ~ A_0, ~ \tan\beta, ~ \lambda, ~ {\rm sign}(\mu).
\label{eq:5Para} \end{aligned}$$ To keep all the attractive features of the CMSSM/ mSuGra, we adhere to universality and use only $\lambda$ to parametrize the singlet sector. This minimal extension alleviates problems of CMSSM/mSuGra rooted in the MSSM. Other constrained versions of the NMSSM have been studied in the recent literature [@Djouadi:2008yj; @Hugonie:2007vd; @Belanger:2005kh; @Cerdeno:2007sn; @Djouadi:2008uw] (see also the contributions by U. Ellwanger in these proceedings).
Our goal is to show that the NmSuGra model can be discovered by nascent experiments in the near future. To this end, for each set of the model parameters, we quantify the experimental preference in terms of: $$\begin{aligned}
\sqrt{\chi^2} = \biggl(\sum_{i=1}^7
\Bigl(\frac{m_i^{experiment}-m_i^{NmSuGra}}{\sigma_i}\Bigr)^2\biggr)^{1/2}
\label{eq:Chi2}\end{aligned}$$ where $m_i$ is the central value of a physical quantity measured by an experiment or calculated in the NmSuGra model, and $\sigma_i$ is the combined experimental and theoretical uncertainty. We include experimental upper limits for $\Omega h^2 = 0.1143 \pm 0.0034$ [@Komatsu:2008hk], $Br(B_s \to \mu^+\mu^-) = 5.8 \times 10^{-8}$ (95 % CL) [@Barberio:2006bi] and $\sigma_{SI}$ by CDMS [@Ahmed:2008eu]. We also include the LEP lower limits of the lightest scalar Higgs and chargino masses (which can be approximately stated as) [@Abbiendi:2003sc]: $m_h > 114.4$ GeV for $\tan\beta {\scriptstyle \stackrel{<}{\sim}} 10$, $m_h > 91$ GeV for $\tan\beta {\scriptstyle \stackrel{>}{\sim}} 10$, $m_{\tilde{W}_1} > 104$ GeV. Finally, we consider the central values of $\Delta a_\mu = 29.5 \pm 8.8 \times 10^{-10}$ [@Stockinger:2007pe], and $Br(b \to s \gamma) = 3.55 \pm 0.26 \times 10^{-4}$ [@Barberio:2006bi]. The related uncertainties are given above at 68 % CL, unless stated otherwise. Theoretical uncertainties are calculated using NMSSMTools [@Ellwanger:2006rn] for the b-physics related quantities.
![\[fig:Chi2VsInput\] The log-likelihood vs. four of the NmSuGra input parameters for a random sample of models. The combination of the experimental quantities included in $\chi^2$ favor low values of $M_0$, $M_{1/2}$ and $|A_0|$. ](chi2_vs_inputs_b.eps){width="48.00000%" height="43.00000%"}
A glance at the likelihood reveals a significant statistical preference for relatively narrow intervals of $M_0$, $M_{1/2}$ and $A_0$, as shown in figure \[fig:Chi2VsInput\] for a randomly selected set of models with ${\rm
sign}(\mu) > 0$. At high values of $M_0$, $M_{1/2}$ and $|A_0|$, $\chi^2$ is dominated by $\Delta a_\mu$, similar to the CMSSM. Based on this, we limit our study to the following ranges of the continuous parameters: $0 < M_0 < 4 ~{\rm
TeV}, 0 < M_{1/2} < 2 ~{\rm TeV}, 0 < |A_0| < 5 ~{\rm TeV}, 1 < \tan\beta < 60, and
0.01 < \lambda < 0.7$.
Detectability of NmSuGra
========================
Having defined the NmSuGra model and the range of its parameters, we set out to show that this parameter region will be detectable by the LHC and an upgraded CDMS detector. Two million theoretically allowed representative model points (from an initial sample of 20 million) are projected in figure \[fig:Oh2vsAdMix\] to the plane of $\Omega h^2$ vs. the gaugino admixture of the lightest neutralino. From figure \[fig:Oh2vsAdMix\] it is evident that the WMAP upper limit (green horizontal line) favors models with mostly bino- (red circles) and higgsino-like (magenta squares) lightest neutralino, while the fraction of allowed models with singlino-like (blue pluses) dark matter is negligible. By checking mass relations and couplings, we can easily establish that branch 4 contains only models with dominant neutralino-stop coannihilation, while branch 3 corresponds to neutralino-stau coannihilation. Branch 2 represents the Higgs resonance corridors, and branch 1 is the equivalent of the CMSSM/mSuGra focus point region.
![\[fig:Oh2vsAdMix\] Relic abundance of the lightest neutralino as the function of its gaugino admixture. Right of the vertical line the neutralino is mostly gaugino. The horizontal lines shows the WMAP upper limit (95 % CL). ](omegah2_vs_binoadmix_random_b.eps){width="48.00000%" height="43.00000%"}
To gauge the detectability of the NmSuGra model, first we identify model points that could have been seen at LEP [@Balazs:2008ph]. The top left frame of figure \[fig:Oh2vsAdMix5\] shows model points from figure \[fig:Oh2vsAdMix\] colored either green (plus) if the model would be accessible to LEP or red (cross) if it passes the above LEP constraints and is therefore allowed. Just as in the CMSSM/mSuGra the neutralino-stop coannihilation region is mostly covered by LEP. For the LHC reach we use a conservative approximation relying on the similarity between the mSuGra and NmSuGra models. According to Ref. [@Baer:2003wx] the reach of the LHC for mSuGra can be well approximated by the combined reach for gluinos and squarks. Based on this, if either the gluino mass is below 1.75 TeV, or the geometric mean of the stop masses is below 2 TeV for a given model point, we consider it discoverable at the LHC.
![\[fig:Oh2vsAdMix5\] Same as figure \[fig:Oh2vsAdMix\], but green (plus) models can be reached by the combination of LEP, the LHC and CDMS1T, while red ones (crosses) cannot. The last frame dismisses experimentally inaccessible points which have $\sqrt{\chi^2}>3$. ](omegah2_vs_binoadmix_random_5_b.eps){width="48.00000%" height="43.00000%"}
The top right frame of figure \[fig:Oh2vsAdMix5\] shows the model points that can be reached by LEP and the LHC using the above criteria. As in the CMSSM/mSuGra, most of the slepton coannihilation and the bulk of the Higgs resonance branches are covered by the LHC. A good part of the focus point is also within reach of the LHC, with the exception of models with high $M_0$ and/or $M_{1/2}$. The bottom left frame of figure \[fig:Oh2vsAdMix5\] shows the reach of a one ton equivalent of CDMS (CDMS1T). As expected from the CMSSM/mSuGra, the rest of the focus point and most of the remaining Higgs resonances are in the reach of CDMS1T. The small number of models that remain inaccessible are all located in regions that have relatively low $M_0$ and high $M_{1/2}$ with dominant neutralino annihilation via s-channel Higgs resonances. The NmSuGra contribution to $\Delta a_\mu$ in these model points is outside the preferred 99 % CL region as shown by the last frame.
Assuming that the NmSuGra contribution to the anomalous magnetic moment of the muon is larger than $3.1 \times 10^{-10}$ constrains slepton and chargino masses below 3 and 2.5 TeV, respectively. Since universality restricts the mass hierarchy within NmSuGra, the resulting mass spectrum is typically mSuGra-like. Thus, the cascade decays and their signatures at LHC are not expected to deviate significantly from that of the mSuGra case.
Conclusions
===========
Analyzing the next-to-minimal supergravity motivated (NmSuGra) model, we found that the LHC and an upgraded CDMS experiment will be able to discover the experimentally favored regions of this model provided that the present deviation between the experimental and standard theoretical values of the muon anomalous magnetic moment prevails.
We thank M. Carena, U. Ellwanger, A. Menon, D. Morrissey, C. Munoz and C. Wagner for invaluable discussions on various aspects of the NMSSM. This research was funded in part by the Australian Research Council under Project ID DP0877916.
[^1]: Radiative breaking the $Z_3$ symmetry may destabilize the hierarchy of vevs in the NMSSM, however by imposing a $Z_2$ R-symmetry these problems can be alleviated without affecting the phenomenology [@Panagiotakopoulos:1998yw].
|
---
abstract: 'The SNe type Ia data admit that the Universe today may be dominated by some exotic matter with negative pressure violating all energy conditions. Such exotic matter is called [*phantom matter*]{} due to the anomalies connected with violation of the energy conditions. If a phantom matter dominates the matter content of the universe, it can develop a singularity in a finite future proper time. Here we show that, under certain conditions, the evolution of perturbations of this matter may lead to avoidance of this future singularity (the Big Rip). At the same time, we show that local concentrations of a phantom field may form, among other regular configurations, black holes with asymptotically flat static regions, separated by an event horizon from an expanding, singularity-free, asymptotically de Sitter universe.'
address: 'Departamento de Física, Universidade Federal do Espírito Santo, Vitória, ES, Brazil'
author:
- 'K.A. Bronnikov[^1], J.C. Fabris[^2] and S.V.B. Gonçalves[^3]'
title: Different faces of the phantom
---
Introduction
============
The evidence for an accelerating expanding phase of the universe seems to be robust [@SN]. If this is the case, the deceleration parameter $q = -
\ddot a\,a/\dot a^2$, must be negative, implying that the matter dominating the matter content of the universe, if described in terms of a perfect fluid, must have an equation of state $p = w\rho$, with $w = - 1/3$. Hence, the strong energy condition must be violated. More recently, there has been claims that the observational data favour an equation of state with $w < -
1$ [@PHANTOM]. Matter with such an equation of state violates all the energy conditions. This kind of matter can be represented, in a more fundamental way, by a self-interacting scalar field, whose kinetic energy appears with the “wrong sign”. For this reason, it is more popularly called a [*phantom field*]{}. In the usual hydrodynamics representation, a matter with this equation of state is unstable. However, such an instability may disappear when a fundamental description is employed, for example, using a self-interacting scalar field, as stated before.
If phantom matter dominates the matter content of the Universe, a future singularity may develop in a finite proper time since its density grows as the universe expands. Such a singularity inevitably appears in an isotrpic Universe if $w=\const < -1$). This possible future singularity has been named a [*Big Rip*]{}, leading to the notion of a [*phantom menace*]{} [@caldwell]. However, it must be stressed that the possibility that phantom matter dominates the universe today is yet a matter of debate: the observational data, mainly those coming from the supernovae type Ia of high redshift, lead to different conclusions depending on how the sample is selected, and even how the statistical analysis is performed [@padmana1]. In any case, the possibility that a phantom matter has something to do with the actual universe must be taken seriously.
The goal of the present work is twofold. First, we intend to analyse the evolution of scalar perturbations for phantom matter. This is an important point, since we are dealing with matter with negative pressure, and instabilities may develop, mainly at small scales [@jerome]. It is possible to get rid of these instabilities if a scalar field representation is used: the behaviour of the perturbations on small scales are quite sensitive to the description used for the matter [@nazira]. On the other hand, the behaviour of the perturbations at large scales are quite insensitive to the description employed. We will show, using a self-interacting scalar field [simulating a perfect fluid with $w
=\const$]{}, that phantom matter may lead to growing perturbations at large scales if the pressure is negative enough. This may lead to a very inhomogenous universe deep in the phantom era, and such inhomogeneties may lead to avoidance of a Big Rip.
Besides this perturbative analysis of a cosmological scenario where the phantom matter dominates the matter content of the universe, we will study local configurations with spherical symmetry. In this case, the results are still more unexpected. The fact that the kinetic term has a “wrong” sign, may lead to a minimum of the radius of coordinate 2-spheres, so that a central singularity is avoided by having no center at all, as is the case with wormholes. However, such configurations may contain one or two Killing horizons, and, among others, it is possible to have configurations where there is a static, asymptotically flat region which is separated by an event horizon from an expanding singularity-free, Kantowski-Sachs type universe. It is thus a black hole in which an explorer may survive after crossing the horizon.
Evolution of scalar perturbations
=================================
When a barotropic fluid with the equation of state $p = w\rho$ is introduced in the Einstein’s equations, with a Friedmann-Robertson-Walker flat line element $ds^2 = a^2(d\eta^2 - dx^2 - dy^2 - dz^2)$, $\eta$ being the conformal time, the conservation equation $$\dot\rho + 3\frac{\dot a}{a}(\rho + p) = 0 \quad ,$$ leads, in case $w = \const$, to $\rho \propto a^{-3(1 + w)}$. Inserting this into the Friedmann equation $$\biggr(\frac{a'}{a}\biggl) = \frac{8\pi G}{3}\rho\,a^2 \quad ,$$ we find that the scale factor behaves as $a \propto \eta^\frac{2}{1 + 3w}$. One important feature of this solution that must be stressed in order to understand the behaviour at perturbative level, is the character of the “future”. If $w > - 1/3$, when all energy conditions hold, $\eta \rightarrow \infty$ means $a
\rightarrow \infty$; when $w < - 1/3$, on the other hand, the universe is expanding as $\eta \rightarrow 0_-$.
In general, fluids with negative pressure contain, at a perturbative level, decreasing modes at large scales and unstable models at small scales [@jerome]. The instabilities at small scales must not be taken so seriously, since it must be due mainly to the hydrodynamical approximation [@nazira]. It is possible to use a more fundamental representation for such exotic matter by considering a self-interacting scalar field, which reproduces, from the point of view of the background behaviour, the hydrodynamical approach employed until now. A scale factor which evolves as $a \propto \eta^\frac{2}{1 + 3w}$, with $w < - 1$, can be achieved by considering a self-interacting minimally coupled scalar field, such that, $$\label{potential}
V(\phi) = V_0\exp\biggr(\pm\sqrt{-3(1 + w)}\phi\biggl) \quad ,
\quad \phi = \pm 2\frac{\sqrt{-3(1 + w)}}{1 + 3w}\ln\eta \quad .$$ A similar model can be constructed when $w > - 1$ by just changing the sign of the term inside the square roots.
For gravity minimally coupled to a (self-interacting) scalar field, the equations for the perturbed quantities reduce to a single equation for the metric perturbed function $\Phi$, called Bardeen’s potential, which is [@brand] $$\Phi'' + 2\biggr\{H - \frac{\phi''}{\phi'}\biggl\}\Phi'
+ \biggr\{k^2 + 2\biggr[H'- H\frac{\phi''}{\phi'}\biggl]\biggl\}
\Phi = 0 \quad .$$ Using the background expressions for $H$ and $\phi$, this equation becomes, $$\Phi'' + 2 \frac{3(1 + w)}{1 + 3w}\frac{\Phi'}{\eta} + k^2\Phi = 0 \quad ,$$ with the solutions $$\Phi = (k\eta)^{-\nu}\biggr\{c_1(k)J_\nu(k\eta)
+ c_2(k)J_{-\nu}(k\eta)\biggl\} \quad , \quad
\forall\, w \quad ,$$ where $\nu = (5 + 3w)/[2(1 + 3w)]$ and $k$ is the wavenumber of the perturbations, resulting from a plane wave expansion of the spatial part of the perturbed quantities.
In the small-scale asymptotic limit defined by $k\eta \ll 1$, the solutions behave as in the hydrodynamical representation: $$\Phi \propto c_1 + c_2(k\eta)^{-2\nu} \quad .$$ In all cases, there is a constant mode. However, if $\omega > - 5/3$ the second mode decreases as the universe expand; but there is a growing mode when $w \leq - 5/3$, what can lead to formation of large inhomogeneties. There is an asymptotic logarithmic divergence for $w = - 5/3$. Using, on the other hand, the asymptotic expression for the Bessel functions for large values of the argument $k\eta \gg 1$, the potential can be expressed as $$\Phi \sim (k\eta)^{-\frac{1 + w}{1 + 3w}}\cos(k\eta + \delta) \quad ,$$ $\delta$ being a phase. It is easy to verify that for $w > - 1$, the potential oscillates with decreasing amplitude, while for $w < - 1$, the potential oscillates with increasing amplitude. Hence, for $w < - 5/3$ the phantom field may exhibit instability at large and small scales. However, it must be stressed that the behaviour at small scale is quite model-dependent, and another field representation of the phantom field can modify the conclusions at small scales, like considering the phantom field as a ghost condensation [@piazza] or a tachyon [@padmana2]. But, at large scales, it seems that there is always a growing mode, for $w \leq - 5/3$, irrespective of the representation chosen.
It is fundamental now to understand the meaning of small and large asymptotic limits, $k\eta \ll 1$ or $k\eta \gg 1$ respectively. We normalize the scale factor by fixing $a_0 = 1$ at the present time. This implies that the Hubble parameter, expressed in terms of the cosmic time, is given by ${\cal H}_0 = \frac{\dot a}{a}|_{t=t_0} = \frac{a'}{a^2}|_{\eta=\eta_0} =
\frac{2}{|1 + 3w|}\frac{1}{|\eta_0|}$, where $\eta_0$ is the conformal time today. Hence, $\eta_0 \sim {\cal H}_0^{-1} = l_H$ (with $c = 1$). This implies that the separation point between the large and small scale regimes is given by the Hubble length $l_H$. When $w > - 1/3$, the conformal time increases as the Universe expands, implying that, as time goes on, more and more modes satisfy the condition $k\eta \gg 1$, which can be re-expressed by saying that the modes enter the Hubble horizon as time goes on. The opposite occurs when $w < - 1/3$, and as time goes on, more and more physical modes satisfy the condition $k\eta \ll 1$, the usual situation of an accelerated expansion phase. In the interval $- 5/3 < w < -
1/3$, these modes that are stretched outside the Hubble horizon are frozen or decay, and there is no danger for homogeneity. But, for $w < - 5/3$, these modes begin to become strongly unstable and homogeneity can be destroyed. The above results can be re-expressed using, as a reference parameter, the Hubble horizon as a function of time. In fact, using the expression for the Hubble length at any time, we have $l_H(\eta) = \frac{|1
+ 3w|}{2}|\eta|^\frac{3(1 + w)}{1 + 3w}|\eta_0|^\frac{-2}{1 + 3w}$ and the argument of the Bessel functions written above can be expressed as $k\eta
\sim k\,[l_H(\eta)]^\frac{1 + 3w}{3(1 + w)}$. For the phantom field, $l_H(\eta)$ decreases as the Universe expands. Hence, for $w < - 5/3$ more and more modes go out of the Hubble horizon and begins to grow, enchancing the inhomogeneity. Notice that for $w = - 1$ (cosmological constant case), the Hubble horizon remains constant.
Even if the calculations exposed here were done for the flat case, they also apply to the open and closed cases [@deborah]. These results have been obtained with the potential (\[potential\]) that reproduces the equation of state $w = \const$ in an isotropic universe. It should be mentioned that other froms of the potential may not lead to a Big Rip even in the isotropic case: e.g., if $V$ is bounded above, a phantom-dominated universe evolves, in general, toward a de Sitter attractor solution [@fara05].
Local configurations
====================
Let us consider now the Hilbert-Einstein Lagrangian coupled to a self-interacting scalar field with an unspecified sign for the kinetic term. For the moment, we do not specify the potential. Considering the spherically symmetric metric written in the form $$ds^2 = Adt^2 - A^{-1}d\rho^2 - r(\rho)^2d\Omega^2 \quad,$$ $\rho$ being the radial variable, we find the following set of coupled differential equations: $$\begin{aligned}
\label{ne1}
(A\,r^2\,\phi')' = \epsilon r^2\,V_\phi \quad ,
\\ \label{ne2}
(A'\,r^2)' = - 2V\,r^2 \quad ,
\\ \label{ne3}
2r''/r = - \epsilon\,\phi'^2\quad ,
\\ \label{ne4}
(r^2)''\,A - A''\,r^2 = 2\quad .\end{aligned}$$ If $\epsilon = 1$ we have a “normal” scalar field, while if $\epsilon = - 1$ we have a phantom scalar field. Equation (\[ne4\]) is once integrated giving $$\label{ne4i}
(A/r^2)' = 2 (\rho_0 - \rho)/r^4 \quad , \quad \rho_0 = \mbox{const} \quad .$$ We will summarize the possible configurations in what follows without solving explicitly the equations (\[ne1\]-\[ne4\]) [@kirill].
Let us indicate the possible kinds of nonsingular solutions without restricting the shape of $V(\phi)$. Assuming no pathology at intermediate $\rho$, regularity is determined by the system behavior at the ends of the $\rho$ range. The latter may be classified as a regular infinity ($r \rightarrow \infty$), which may be flat, de Sitter or AdS, a regular center $r \rightarrow 0$, and the intermediate case $r \rightarrow r_0 > 0$. Suppose we have a regular infinity as $\rho \rightarrow \infty$, so that $V \rightarrow V_+ = \mbox{const}$ while the metric becomes Minkowski (M), de Sitter (dS) or AdS according to the sign of $V_+$. In all cases $r \approx
\rho$ at large $\rho$.
For $\epsilon = +1$, due to $r'' \leq 0$, $r$ necessarily vanishes at some $\rho=\rho_c$, which means a center, and the only possible regular solutions interpolate between a regular center and an AdS, flat or dS asymptotic; in the latter case the causal structure coincides with that of de Sitter space-time.
For $\epsilon = -1$, there are similar solutions with a regular center, but due to $r''\geq 0$ one may obtain either $r \rightarrow r_0 = \mbox{const} > 0$ or $r \rightarrow \infty$ as $\rho \rightarrow -\infty$. In other words, all kinds of regular behavior are possible at the other end. In particular, if $r \rightarrow r_0$, we get $A \approx -\rho^2/r_0^2$, i.e., a cosmological region comprising a highly anisotropic Kantowski-Sachs cosmology (KS) with one scale factor ($r$) tending to a constant while the other ($A$) inflates. The scalar field tends to a constant, while $V(\phi) \rightarrow 1/r_0^2$.
Thus there are three kinds of regular asymptotics at one end, $\rho\rightarrow \infty$ (M, dS, AdS), and four at the other, $\rho \rightarrow -\infty$: the same three plus $r \rightarrow r_0$, simply $r_0$ for short. (The asymmetry has appeared since we did not allow $r \rightarrow \mbox{const}$ as $\rho \rightarrow \infty$. The inequality $r''> 0$ forbids nontrivial solutions with two such $r_0$-asymptotics.) This makes nine combinations shown in Table 1. Moreover, each of the two cases labelled KS\* actually comprises three types of solutions according to the properties of $A(\rho)$: there can be two simple horizons, one double horizon or no horizons between two dS asymptotics. Recalling 3 kinds of solutions with a regular center, we obtain as many as 16 qualitatively different classes of globally regular configurations of phantom scalar fields.
AdS M dS $r_0$
----- ---------- ---------- ------------ ------------
AdS wormhole wormhole black hole black hole
M sym wormhole black hole black hole
dS sym sym KS\* KS\*
: Regular solutions with for $\epsilon = -1$. Each row corresponds to a certain asymptotic behavior as $\rho\rightarrow +\infty$, each column — to $\rho\rightarrow -\infty$. The mark “sym” refers to combinations obtained from others by symmetry $\rho \leftrightarrow -\rho$.
Conclusions
===========
Phantom matter implies violation of all energy conditions. In a hydrodynamical representation, phantom matter is described by $p = w\rho$ with $w < - 1$. Phantom matter can be described by a self-interacting scalar field with a “wrong” sign in the kinetic term. There is some evidence that the exotic matter responsable for the actual phase of accelerated expansion of the universe may be a kind of phantom field. In this work, we have studied some properties of a phantom field, specifically with respect to the evolution of scalar perturbations and with respect to local configurations.
We found that, under certain conditions, a universe dominated by a phantom matter may develop high inhomogeneities even at large scales. Hence, after a certain stage of its evolution, the hypothesis of homogeneity and isotropy becomes no more valid. As the big rip scenario depends on these hypothesis, it is possible that a phantom universe brings in itself a mechanism of avoiding a future singularity even in the case when $w = \const$ in homogeneos and isotropic space-time.
In what concerns local configurations, we find a wealth of nonsingular models among which of particular interest are asymptotically flat black holes with an expanding universe beyond the event horizon. This provides an interesting singularity-free cosmological scenario: one may speculate that our Universe could appear from collapse to such a phantom black hole in another, “mother” universe and undergo isotropization (e.g., due to particle creation) soon after crossing the horizon.
There are no similar configurations with a “normal” scalar field. In any case, violation of all energy condition inevitably leads to completely new configurations.
[**Acknowledgments.**]{} This work was supported by CNPq (Brazil); KB was also supported by DFG Project 436/RUS 113/807/0-1(R).
References {#references .unnumbered}
==========
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[^1]: E-mail: kb20@yandex.ru. Permanent address: Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St., Moscow, Russia; Institute of Gravitation and Cosmology, PFUR, 6 Miklukho-Maklaya St., Moscow 117198, Russia.
[^2]: E-mail: fabris@cce.ufes.br
[^3]: E-mail: sergio@cce.ufes.br
|
---
abstract: 'We compute the Dirac spectrum of SU(3) for a one parameter family of Dirac operators, including the Levi-Civita, cubic, and trivial Dirac operators. We then proceed to compute the spectral action for the entire family.'
address: |
Department of Mathematics\
California Institute of Technology\
Pasadena, CA 91125, USA
author:
- 'Alan Lai, Kevin Teh'
title: 'Dirac spectrum and spectral action of SU(3)'
---
Dirac Laplacian of a Lie group
==============================
In this section, we define the one-parameter family of Dirac operators of a Lie group equipped with a positive definite, symmetric bilinear form on its Lie algebra. This family was studied in [@agricola]. The definition of the Dirac Laplacian is the same as in [@SU2], but we reproduce it here for convenience.
In this paper, we consider the one-parameter family of Dirac operators on $SU(3)$ corresponding to the one-parameter family of connections,
$$\nabla _X^t := \nabla _X^0 + t [X,\cdot] ,$$
with $\nabla_X^0$ being the trivial connection of the Lie group coming from the left-multiplication trivialization of the tangent bundle. $\nabla_X^1$ is the trivial connection of the right-multiplication trivialization.
The connections so defined are metric connections, with torsion given by $$T(X,Y) =(2t-1)[X,Y],$$ and hence $\nabla_X^{1/2}$ is the Levi-Civita connection. The operator corresponding to $t=1/3$, is the cubic Dirac operator studied in [@cubic]. Let $\langle \cdot, \cdot \rangle$ be the metric on $G$ and let $\{X_k \}$ be an orthonormal basis with respect to the metric. The $\mathfrak{so}(\fg)$ connection $\nabla_X^t$ lifts to a metric $\mathfrak{spin}(\fg)$ connection $\widehat{\nabla_X^t}$ given by
$$\label{connect}
\widehat{\nabla_X^t} = \nabla_X^0 + t \frac{1}{4}\sum_{k,l}\langle X, [X_k, X_l] \rangle X_k X_l,$$
see [@par].
Let $\C l (\fg)$ be the Clifford algebra generated by $\fg$ and the relation $$XY +YX = -2\langle X, Y \rangle,$$ and let $\cU (\fg)$ be the universal enveloping algebra. Then the Dirac operator, $\cD_t$ of the connection \[connect\], as an element of $\C l (\fg) \otimes \cU(\fg)$ is given by
$$\cD_t = \sum_i X_i \otimes X_i + t H,$$
where
$$H = \frac{1}{4} \sum_{j,k,l}X_j X_k X_l \otimes \langle X_j, [X_k,X_l] \rangle.$$
Spectrum of Dirac Laplacian of $SU(3)$
======================================
We claim that the spectrum of the Dirac Laplacian of $SU(3)$ is as follows.
\[spec\] In each row of the table below, for each pair $(p,q)$ with $p,q$ in the set of parameter values displayed, the Dirac Laplacian $\cD_t^2$ of $SU(3)$ has an eigenvalue in the first column of the multiplicity listed in the center column.
Let $$\lambda(u,v) = u^2 + v^2 +u v.$$ Let $$m(a,b) = \frac{(p+1)(q+1)(p+q+2)(p+1+a)(q+1+b)(p+q+2+a+b)}{4}$$ We denote by $\N^{\geq a}$, the set $\{n \in \N : n \geq a\}$, and we take $\N$ to be the set of integers greater than or equal to zero.
$$\begin{array}{|c|c|c|}
\hline
\mathrm{Eigenvalue} & \mathrm{Multiplicity} & \mathrm{Parameter~Values}\\
\hline
\lambda(p+3t,q+3t)& m(1,1) & p \in \N, q \in \N \\
\hline
\lambda(p+2-3t,q-1+6t)& m(-1,2) & p \in \N, q \in \N \\
\hline
\lambda(p+1,q+1) + 3(3t-1)(3t-2)& m(0,0) & p \in \N, q \in \N, (p,q)\neq (0,0) \\
\hline
\lambda(p+3-6t,q+3t)& m(-2,1) & p \in \N^{\geq 1}, q \in \N \\
\hline
\lambda(p-1+6t,q+2-3t)& m(2,-1) & p \in \N, q \in \N \\
\hline
\lambda(p+1,q+1) + 3(3t-1)(3t-2)& m(0,0) & p \in \N^{\geq 1}, q \in \N^{\geq 1} \\
\hline
\lambda(p+3t,q+3-6t)& m(1,-2) & p \in \N, q \in \N^{\geq 1} \\
\hline
\lambda(p+2-3t,q+2-3t) + 3(3t-1)(3t-2)& m(0,0) & p \in \N, q \in \N \\
\hline
\end{array}$$
There is some flexibility in the set of parameter values. For instance in the second line, we could have used instead $p \in \N^{\geq 1}$ since for that line the multiplicity is zero whenever $p=0$.
Spectrum for $t = 1/3$
----------------------
In this case, the expression of the spectrum becomes much simpler
\[specThird\] The spectrum for the Dirac Laplacian $\cD_{1/3}^2$ of SU(3) is given in the following table.
$$\begin{array}{|c|c|c|}
\hline
\mathrm{Eigenvalue} & \mathrm{Multiplicity} & \mathrm{Parameter~Values}\\
\hline
p^2 + q^2 + pq & 2 p^2 q^2(p+q)^2 & p \in \N, q \in \N \\
\hline
\end{array}$$
Derivation of the spectrum
--------------------------
We compute the spectrum of the Dirac Laplacian of $SU(3)$ using a representation-theoretic approach. First, we recall the fact that the Dirac Laplacian of a Lie group may be represented in terms of the Casimir element.
[@SU2] Theorem 2.3, \[Dt2\] Let $\pi$ be the natural homomorphism $\cU(\fg) \rightarrow \C l(\fg)$. Let $\rho$ be the Weyl vector, the half sum of all positive roots of $\fg$. Then, $\cD _t ^2 = (\pi \otimes 1)T_t$ where $T_t \in \cU (\fg) \otimes \cU(\fg)$ is given by $$T_t = (1-3t)(1 \otimes \Cas + \Cas \otimes 1 - \Delta \Cas) + 1 \otimes \Cas + 9t^2 |\rho|^2.$$
Next we use the Peter-Weyl theorem to reduce the computation to understanding the irreducible representations of $SU(3)$. On irreducible representations, the action of the Casimir is well understood. For any irreducible representation $V_{\mu}$, of highest weight $\mu$, the Casimir acts as scalar multiplication by the scalar $$\label{cas}
\pi_{\mu}(\mathrm{Cas})=(\mu + \rho, \mu + \rho) - (\rho,\rho),$$ where $\rho$ is the Weyl vector, the half-sum of all positive simple roots. The pairing $(\cdot,\cdot)$ is the dual pairing, on the weight space of a nondegenerate symmetric bilinear form on the Cartan subalgebra of $\mathfrak{g}$. Such a pairing is necessarily a constant multiple of the Killing form, which for $SU(3)$ is given by $$\label{killing}
\kappa(X,Y) = 6 \Tr(XY),$$ where the trace and multiplication are taken in the natural representation of $X,Y$ as $3\times3$ matrices. One may identify $\fg ^*$ with $\fg$ by identifying $\lambda \in \fg^*$ with the unique $X_\lambda$ such that $(X_\lambda,Y) = \lambda(Y)$, for all $Y \in \fg$. This is possible due to the nondegeneracy of the pairing on $\fg$. In this way, one defines the dual pairing on $\fg^*$. The particular pairing which occurs depends on the normalization of the Riemannian metric. More specifically, the pairing is related to the Casimir operator by the following relation (the Casimir in turn being determined by the metric on $\mathfrak{g}$).
[@kostant] Given a symmetric nondegenerate bilinear form on $\mathfrak{g}$, the corresponding Casimir element, and bilinear form on weights are related by $$\frac{1}{24}\Tr(\mathrm{ad} (\mathrm{Cas})) = (\rho,\rho).$$
Henceforth, we assume that the metric is normalized so that $(\rho,\rho)=3$. This leads to the simplest expressions for the spectrum.
Therefore in order to derive the spectrum of the Dirac Laplacian, we must first analyze the pairing of weights. We take for our basis of the Cartan subalgebra to be $\{H_1, H_2 \}$, $$H_1 =
\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{array}
\right),
~H_2 =
\left(
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -1
\end{array}
\right).$$ We identify weights concretely using the basis. I.e. for $\lambda \in \mathfrak{h}^*$, we identify $\lambda$ with $(\lambda(H_1),\lambda(H_2))$. Then, the Weyl vector is given by $\rho = (1,1)$, and the weights $\lambda_1 = (1,0)$ and $\lambda_2 = (0,1)$ form an $\N$-basis of the highest weights of irreducible representations of $SU(3)$. The pairing of weights can be determined up to normalization, using duality, and the Killing form \[killing\], from which one deduces the relations $$(\lambda_1,\lambda_1) = (\lambda_2,\lambda_2) = 2 (\lambda_1,\lambda_2).$$
\[caspq\] On the irreducible representation of highest weight $(p,q)$, $p,q \in \N$, the Casimir element acts by the scalar $$\pi_{p,q}(\mathrm{Cas}) = (p^2 + q^2 + 3 p + 3 q + p q)(\lambda_1,\lambda_1).$$
For the normalization that we are considering, we have $(\lambda_1,\lambda_1)=1$.
We have listed the irreducible representations of $SU(3)$ as well as the action of the Casimir operator on them. To write down the spectrum of the Dirac Laplacian the only obstacle now is to understand the term $\Delta \Cas$ in Theorem \[Dt2\]; i.e. we need to know the Clebsch-Gordan coefficients of the tensor products $V_{\rho} \otimes V_{(p,q)}$. These were computed in [@CG]. We recall the Clebsch-Gordan coefficients that we will need below.
\[ClebGord\] The decomposition of $V_{\rho} \otimes V_{(p,q)}$ into irreducible representations is $$V_{\rho} \otimes V_{(p,q)} = \oplus_{\mu} V_{\mu},$$ where the summands $V_{\mu}$ appearing in the direct sum are given by the following table: $$\begin{array}{|c|c|}
\hline
\mathrm{Summand} & \mathrm{Parameter~Values}\\
\hline
V_{(p+1,q+1)} & p \in \N, q \in \N \\
\hline
V_{(p-1,q+2)} & p \in \N^{\geq 1}, q \in \N \\
\hline
V_{(p,q)} & p \in \N, q \in \N, (p,q)\neq (0,0) \\
\hline
V_{(p-2,q+1)} & p \in \N^{\geq 2}, q \in \N \\
\hline
V_{(p+2,q-1)} & p \in \N, q \in \N^{\geq 1} \\
\hline
V_{(p,q)} & p \in \N^{\geq 1}, q \in \N^{\geq 1} \\
\hline
V_{(p+1,q-2)} & p \in \N, q \in \N^{\geq 2} \\
\hline
V_{(p-1,q-1)} & p \in \N^{\geq 1}, q \in \N^{\geq 1} \\
\hline
\end{array}$$
Each summand in the left column appears once if $(p,q)$ lies in the set of parameter values listed on the right column. For instance for $(p,q) =(1,1)$, the summand $V_{(p,q)} = V_{(1,1)}$ appears twice in the direct sum decomposition, since $V_{(p,q)}$ appears twice in the left column, and $(p,q)$ is in the set of parameter values in each of the two rows.
By combining Theorem \[Dt2\], Lemma \[caspq\], and Lemma \[ClebGord\], we obtain Theorem \[spec\]. The multiplicities are obtained using the Weyl dimension formula $$\dim V_{(p,q)} = \frac{1}{2}(p+1)(q+1)(p+q+2).$$
When $t=1/3$, the formula for the Dirac Laplacian in Theorem \[Dt2\] simplifies to $$\cD_{1/3}^2 = 1 \otimes \mathrm{Cas} + 3.$$ Therefore, we no longer need to decompose any tensor products into irreducible components, and using just \[caspq\], one obtains Theorem \[specThird\].
Spectral action of $SU(3)$
==========================
In this section, we compute the spectral action, $\Tr f(\cD_t^2/\Lambda^2)$. In the case $t=1/3$, one may apply the Poisson summation formula as in [@uncanny] to quickly obtain the full asymptotic expansion for the spectral action. For general $t$ however, this approach no longer works. An expansion can however still be generated using a two variable generalization of the Euler-Maclaurin formula [@karshon]. However, this requires more work to produce, and produces the full expansion of the spectral action only if one assumes in addition that the test function $f$ has all derivatives equal to zero at the origin. Here, we compute the spectral action to order $\Lambda^0$.
t = 1/3
-------
Let $f \in \cS (\R)$ be a Schwarz function. By Theorem \[specThird\], the spectral action of $SU(3)$, for $t = 1/3$ is given by $$\Tr f(\cD_{1/3}^2/\Lambda^2) = \sum_{p=0}^{\infty} \sum_{q=0}^{\infty} 2 p^2 q^2(p+q)^2f(\frac{p^2+q^2+pq}{\Lambda^2}).$$ In order to apply the Poisson summation formula, one needs to turn this sum into a sum over $\Z ^2$. For this purpose, one takes advantage of the fact that the expressions for the eigenvalues and multiplicities are both invariant under a set of transformations of $\N^2$ which together cover $\Z^2$. The linear transformations of $\N^2$ which together cover $\Z^2$ are $$\begin{aligned}
T_1 (p,q) &= (p,q), \\
T_2(p,q) &= (-p,p+q), \\
T_3(p,q) &= (-p-q,p), \\
T_4(p,q) &= (-p,-q), \\
T_5(p,q) &= (p,-p-q), \\
T_6(p,q) &= (p+q,-p). \\\end{aligned}$$ Each of the transformations is injective on $\N^2$. The union of the images is all of $\Z^2$. The six images of $\N ^2$ overlap on the sets $\{(p,q):p=0\}$ and $\{(p,q):q=0\}$. However, the multiplicity is equal to zero at these points, and so this overlap is of no consequence. Therefore, we may now write the spectral action as a sum over $\Z ^2$ $$\label{spactThird}
\Tr f(\cD_{1/3}^2/\Lambda^2) = \frac{1}{6}\sum_{p=-\infty}^{\infty} \sum_{q=-\infty}^{\infty} 2 p^2 q^2(p+q)^2f(\frac{p^2+q^2+pq}{\Lambda^2})$$
For a sufficiently regular function, the Poisson summation formula (in two variables) is $$\label{psf}
\sum_{\Z^2} g(p,q) = \sum_{\Z^2}\widehat{g}(x,y).$$
Applying Equation \[psf\] to \[spactThird\], and applying the argument used in [@uncanny] we get the following result.
\[spactThird\] Let $f \in \cS (\R)$ be a Schwarz function. For $t=1/3$, the spectral action of SU(3) is $$\Tr f(\cD_{1/3}^2/\Lambda^2) = \frac{1}{3}\iint_{\R^2}x^2y^2(x+y)^2f(x^2+y^2+xy)dxdy ~\Lambda^8 + O(\Lambda^{-k}),$$ for any integer $k$.
General $t$ and the Euler-Maclaurin formula
-------------------------------------------
The one-variable Euler-Maclaurin formula was used in [@CCRobWalk] to compute the spectral action of $SU(2)$ equipped with the Robertson-Walker metric. A two-variable Euler-Maclaurin formula may be applied here to compute the spectral action on $SU(3)$ for all values of $t$.
Let $m$ be a positive integer. Let $g$ be a function on $\R ^2$ with compact support. One instance of the two-variable Euler-Maclaurin formula is [@karshon] $$\label{eulmac}
\sum_{p=0}^{\infty}\sum_{q=0}^{\infty}\mathrm{'} g(p,q) = L^{2k}(\frac{\partial}{\partial h_1})L^{2k}(\frac{\partial}{\partial h_2}) \int_{h_1}^{\infty}\int_{h_2}^{\infty}g(p,q)dp dq \vert_{h_1=0,h_2=0} +R_m^{st}(g).$$
The notation $\sum\sum'$ indicates that terms of the form $g(0,q)$, $q\neq 0$, and $g(p,0)$, $p\neq0$ have a coefficient of 1/2, $g(0,0)$ has a coefficient of 1/4 and the rest of the terms are given the usual coefficient of 1. The operator $L^{2k}(S)$ is defined to be $$\label{em}
L^{2k}(S) = 1 + \frac{1}{2!}b_2 S^2 + \ldots + \frac{1}{(2k)!}b_{2k} S^{2k},$$ where $b_j$ is the $j$th Bernoulli number. The number $k$ is defined by $k = \lfloor m/2 \rfloor$ The remainder $R_m^{st}(g)$ is
$$\label{rem}
\begin{array}{l}
R_m^{st}(g) = \sum_{I \subsetneq \{1,2\}} (-1)^{(m-1)(2-|I|)} \times \\
\times \prod_{i \in I} L^{2k}(\frac{\partial}{\partial h_i})\int_{h_1}^{\infty} \int_{h_2}^{\infty} \prod_{i \notin I}P_m(x_i)\prod_{i\notin I} \left( \frac{\partial}{\partial x_i}\right)^m g(x_1,x_2)dx_1 dx_2 \vert_{h=0}.
\end{array}$$
Equation \[eulmac\] is proved in an elementary way in [@karshon], by casting the one-variable Euler-Maclaurin formula in a suitable form, and then iterating it two times.
Using Theorem \[spec\], one may write the spectral action in terms of eight summations of the form
$$\sum_{(p,q) \in \N ^2} g_i(p,q),$$ where $$g_i(p,q) = f\left( \frac{\lambda_i(p,q)}{\Lambda^2} \right)m_i(p,q), \quad i = 1,\ldots,8.$$
The notations $\lambda_i(p,q)$ and $m_i(p,q)$ denote the eigenvalues and multiplicities of the spectrum in Theorem \[spec\].
One then applies the two-variable Euler-Maclaurin formula to each of the eight summations to replace the sums with integrals. Then to obtain an asymptotic expression in $\Lambda$, one controls the remainder, $R_m^{st}(g)$, to arbitrary order in $\Lambda$ by taking $m$ to be sufficiently large, and computes the big-$O$ behavior of the other integrals to arbitrary order in $\Lambda$ by applying the multivariate Taylor’s theorem to a large enough degree. The terms in the Taylor expansions of the integrals yield the asymptotic expansion of the spectral action.
Analysis of remainders
----------------------
Let us consider in detail the case $I = \{ \}$, of the remainder, \[rem\]. The functions $P_m(x_i)$ are periodic, and hence bounded. Furthermore, they are independent of $\Lambda$. Therefore to study the big-$O$ behavior with respect to $\Lambda$ of the remainder, \[rem\], we only need to estimate the integral $$\iint \left| \frac{\partial ^m}{\partial p ^m} \frac{\partial ^m}{\partial q ^m} f(s \lambda(p,q)) m(p,q)\right|.$$ The integration happens over $(\R^+)^2 = [0,\infty)\times[0,\infty)$, and $m(p,q)$ is the multiplicity polynomial. The differentiated function is a sum of terms, whose general term is given by $$C s^if^{(i)}(t \lambda)\lambda^{(a_1,b_1)(p,q)} \ldots \lambda^{(a_i,b_i)}(p,q)m^{(j,k)}(p,q),$$ where $C$ is a combinatorial constant, $s = \Lambda ^{-2}$, and where $j$ and $k$ are less than or equal to $m$ and $0\leq i \leq 2m-j-k$ and $$\sum (a_i,b_i) = (m-j,m-k).$$
Since $m$ is degree 4 in both $p$ and $q$, we know that $j \leq 4$ and $k \leq 4$. Since $\lambda$ is degree 2 in both $p$ and $q$ we know that each of the coefficients $a_k$, $b_k$ is less than or equal to 2. Therefore, one has the estimate $$2 i \geq \sum a_i = m-j \geq m-4,$$ and so $$i \geq \frac{m-4}{2}.$$ It is not too hard to see that $$\iint f^{(i)}(s \lambda)\lambda^{(a_1,b_1)(p,q)} \ldots \lambda^{(a_i,b_i)}(p,q)m^{(j,k)}(p,q)dpdq$$ is uniformly bounded as $s$ approaches zero, and therefore we have that the integral has a big-$O$ behavior of $O(s^{\frac{m-4}{2}})$ as $s$ goes to zero.
The same argument gives the same estimate for the terms in the cases $I =\{1\}$ and $I=\{2\}$. Therefore we have shown
The remainder $R^{st}_m(g)$ behaves like $O(\Lambda^{-(m-4)})$ as $\Lambda$ approaches infinity.
Since the sum in the Euler-Maclaurin formula, \[eulmac\] gives only a partial weight to terms on the boundary, and since the functions $g_i(p,q)$, are at times nonzero on the boundary, $\{p=0\} \cup \{ q=0\}$ even when there are no eigenvalues there, we must compensate at the boundary in order to obtain an accurate expression for the spectral action.
In doing so, one considers sums of the form $$\begin{aligned}
& \sum_{p=0}^{\infty}g_i(p,0), \\
& \sum_{q=0}^{\infty}g_i(0,q).\end{aligned}$$ One treats these sums using the usual one-variable Euler-Maclaurin formula, which for a function, $h$, with compact support is $$\sum_{p=0}^{\infty}h(p) = \int_0^{\infty}h(x) dx + \frac{1}{2}h(0) -\sum_{j=1}^{m} \frac{b_{2j}}{(2j)!}h^{(2j-1)}(0) + R_m(h),$$ where the remainder is given by $$\label{onerem}
R_m(h) = \int_0^{\infty} P_m(x) \left( \frac{\partial}{\partial x}\right)^{m} h(x)dx.$$
The necessary estimate for the remainder \[onerem\] is as follows.
$R_m(g(p,\cdot))$ and $R_m(g(\cdot,q))$ behave as $O(\Lambda^{-m+4})$ as $\Lambda$ approaches infinity.
To prove this, we observe that since the polynomial $P_m(x)$ is bounded and independent of $x$, we only need to estimate for instance $$\left| \int_0^{\infty} \left( \frac{\partial}{\partial x}\right)^{m} g_i(x,0)dx\right|.$$ The function $g_i(x,0)$ is of the form $$\label{diffand}
f(\frac{a x^2 + b x + c}{\Lambda ^2} + d)m(x,0),$$ where $a,b,c$ are independent of $\Lambda$ and $x$, and $d$ is independent of $x$. The polynomial $m(x,0)$ is of degree 4 in $x$. Therefore, when one expands the derivative of \[diffand\] using the product rule, the derivatives of $f(\frac{a x^2 + b x + c}{\Lambda ^2} + d)$ are all of order $j \geq m-4$. A simple inductive argument shows that the expansion of $(\partial / \partial x)^j f(\frac{a x^2 + b x + c}{\Lambda ^2} + d)$ under the chain rule the terms are all of the form $$\frac{1}{\Lambda^k} f^{(i)}(\frac{a x^2 + b x + c}{\Lambda ^2} + d) \alpha(x),$$ where $k \geq j$, and $\alpha(x)$ is a polynomial. Finally we conclude the proof of the lemma by observing that $$\int_0^{\infty} \left( \frac{\partial}{ \partial x}\right)^j f(\frac{a x^2 + b x + c}{\Lambda ^2} + d)\alpha(x)dx$$ is uniformly bounded as $\Lambda$ goes to infinity.
Analysis of main terms
----------------------
With the remainders taken care of, one still needs to work out the big-$O$ behavior of the spectral action with respect to $\Lambda$ of the remaining terms coming from the two-variable and one-variable Euler-Maclaurin formulas.
The calculation required is lengthy, but the technique is elementary. One changes variables to remove (most of) the $\Lambda$ dependence from the argument of the test function $f$. Then, one uses Taylor’s theorem to remove the $\Lambda$ dependence from the limits of integration, and whatever $\Lambda$ dependence remains in the argument of $f$. In this way, one can obtain the big-$O$ behavior of the spectral action with respect to $\Lambda$ to any desired order. Here, we do the computation up to constant order in $\Lambda$. If one assumes that the test function $f$ has all derivatives equal to zero at the origin, then one obtains the asymptotic expansion to all orders in $\Lambda$.
To give a better idea of how the calculation proceeds, let us consider in detail a couple of terms coming from the Euler-Maclaurin formulas.
One term that appears upon application of the Euler-Maclaurin formula is $$\int_{0}^{\infty}\int_0^{\infty}g_1(p,q)dpdq.$$ where $$\begin{aligned}
g_1(p,q) &= f(\frac{(p+3t)^2 + (q+3t)^2+(p+3t)(q+3t)}{\Lambda^2})\times \\
&\times \frac{1}{4}(p+1)(q+1)(p+q+2)(p+2)(q+2)(p+q+4)\end{aligned}$$
First, one performs the change of variables, $$x = \frac{p+3t}{\Lambda}, y = \frac{q+3t}{\Lambda},$$ whereby one obtains $$\begin{aligned}
&\frac{1}{4}\int_{3t/\Lambda}^{\infty}\int_{3t/\Lambda}^{\infty} f(x^2 + y^2 +x y) (1-3t+x\Lambda)(2-3t+x\Lambda)\times \\
&\times(1-3t+y\Lambda)(2-3t+y\Lambda)(2-6t+x\Lambda+y\Lambda)(4-6t+x\Lambda+y\Lambda)\Lambda^2 dx dy\end{aligned}$$ Next, one does a Taylor expansion on the two lower limits of integration about 0. The first term in this Taylor series is obtained by setting the limits of integration to zero.
$$\label{zeroTerm}
\begin{array}{l}
\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty} f(x^2 + y^2 +x y) (1-3t+x\Lambda)(2-3t+x\Lambda)\times\\
\times(1-3t+y\Lambda)(2-3t+y\Lambda)(2-6t+x\Lambda+y\Lambda)(4-6t+x\Lambda+y\Lambda)\Lambda^2 dx dy
\end{array}$$
Remarkably, if one sums the analog of \[zeroTerm\] for $g_1, \ldots, g_8$ one obtains the complete spectral action to constant order. All of the other terms which appear in the computation (of which there are many) cancel out, to constant order in $\Lambda$, in an intricate manner.
The end result of the calculation is the following.
\[spact\] Let $f$ be a real-valued function on the real line with compact support. To constant order, the spectral action, $\Tr(f(\cD_t ^2 /\Lambda^2))$ of $SU(3)$ is equal to $$\begin{aligned}
&2\iint_{(\R^+)^2} f(x^2+y^2+xy)x^2y^2(x+y)^2dx dy ~\Lambda^8 \\
&+ 3(3t-1)(3t-2)\iint_{(\R^+)^2} f(x^2+y^2+xy)(x^4+2x^3y+3x^2y^2+2xy^3+y^4)dx dy~\Lambda^6 \\
&+ 9(3t-1)^2(3t-2)^2 \iint_{(\R^+)^2} f(x^2+y^2+xy)(x^2 + xy +y^2)dx dy~ \Lambda^4\\
&+ 6 (3t-1)^3(3t-2)^3\iint_{(\R^+)^2} f(x^2+y^2+xy)dx dy ~\Lambda^2\\
&+ O(\Lambda^{-1}).\end{aligned}$$ Here, the integrals are taken over the set $(\R^+)^2 = [0,\infty) \times [0,\infty)$. If $f$, in addition, has all derivatives equal to zero at the origin, then this expression gives the full asymptotic expansion of the spectral action.
The linear transformations, $T_1 \ldots T_6$, are all unimodular, and the images of $(\R^+)^2$ cover $\R ^2$, up to a set of measure zero. Therefore, in the case of $t=1/3$, integrating over $\R^2$ multiplies the result by a factor of 6, and we see that Theorem \[spact\] agrees with Theorem \[spactThird\]
When computing the asymptotic expansion of the spectral action using the Euler-Maclaurin formula, as a result of the chain rule, the negative powers, $\Lambda^{-j}$ appear only with derivatives $f^{(k)}(0)$, $k \geq j$. This is why the terms of the asymptotic expansion vanish for negative powers of $\Lambda$, when the derivatives of $f$ vanish at zero.
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|
---
author:
- 'Kazuhiro [Sano]{} and Ken’ichi [Takano]{}$^{1}$'
title: |
Spin Gap of $S=1/2$ Heisenberg Model\
on Distorted Diamond Chain
---
Recently, Ishii et al.[@Ishii] measured the magnetic susceptibility $\chi$ for $\rm Cu_3 Cl_6 (H_2 O)_2 \cdot 2H_8 C_4 SO_2$, which is considered to be a quasi-one-dimensional material consisting of $S=\frac{1}{2}$ trimer spin chains. The result indicates that $\chi$ vanishes in the low temperature limit. They also measured the magnetization process for this material, and showed that there is a plateau of zero magnetization below the critical field $H_c \simeq 3.9$ T. From these experimental results, they concluded that the ground state is a singlet state with spin gap. The spin gap $\Delta$ is estimated as $\Delta/k_{\rm B} \simeq 5.2$ K from the value of $H_c$.
The proposed Hamiltonian[@Ishii] representing a spin chain in this material is given by $$\begin{aligned}
H&=&J_1 \sum_j \left( {\mib S}_{3j-1} \cdot {\mib S}_{3j}
+ {\mib S}_{3j} \cdot {\mib S}_{3j+1} \right) \nonumber \\
&+&J_2 \sum_j {\mib S}_{3j+1} \cdot {\mib S}_{3j+2} \nonumber \\
&+&J_3 \sum_j \left( {\mib S}_{3j-2} \cdot {\mib S}_{3j}
+ {\mib S}_{3j} \cdot {\mib S}_{3j+2} \right),
\label{eq1}\end{aligned}$$ where ${\mib S}_{j}$ is the $S=\frac{1}{2}$ spin on site $j$. Three spins ${\mib S}_{3j-1}$, ${\mib S}_{3j}$ and ${\mib S}_{3j+1}$ form a trimer. The lattice structure is shown in Fig. \[fig1\]. Three kinds of exchange constants $J_1$, $J_2$ and $J_3$ are inferred to be positive and to satisfy the relation $J_1 > J_2, J_3$ from the lattice parameters of the material [@Swank]. Hereafter, we use the unit of $J_1 = 1$.
The symmetric case of $J_3 = J_1 (=1)$ has been studied by Takano et al.[@Takano] and the system has been called the [*diamond chain*]{}. They almost exactly showed that there exist three phases in the parameter space; the ferrimagnetic phase for $J_2 < 0.909$, the tetramer-dimer (TD) singlet phase for $0.909 < J_2 < 2$ and the dimer-monomer (DM) singlet phase for $ J_2 > 2$. The TD phase is a disordered phase with spin gap which originates from frustration among exchange interactions, while the DM phase is a spin fluid phase without spin gap. Okamoto et al.[@Okamoto] studied the general case of $J_3 \neq 1$; i. e. the [*distorted diamond chain*]{}. The three phases develop in the $J_2$-$J_3$ plane. They numerically determined the phase boundaries. Also Tonegawa et al.[@Tonegawa] numerically studied the magnetization process and showed plateaux for $\frac{1}{3}$ and $\frac{2}{3}$ of the saturation field.
In this article, we estimate the values of the spin gap by the numerical diagonalization. Then we produce a contour map in the $J_2$-$J_3$ parameter space. The contour map represents an overall feature of the gapped phase of the $S=\frac{1}{2}$ Heisenberg model on the distorted diamond chain. When further experimental information on $\rm Cu_3 Cl_6 (H_2 O)_2 \cdot 2H_8 C_4 SO_2$ is given, the contour map will be useful to determine the values of the exchange constants for the real material.
We first calculate the spin gap $\Delta_L$ for finite chains with system size $L$. The spin gap $\Delta_{\infty}$ in the thermodynamic limit is evaluated by extrapolation. We assume the size dependence of $\Delta_L$ as $$\Delta_L= \Delta_{\infty}+ \frac{c_1}{L} + \frac{c_2}{L^2}
\label{eq2}$$ with constants $c_1$ and $c_2$. The numerical diagonalization has been done for $L=$12, 18 and 24 under the periodic boundary condition. We determine $c_1$, $c_2$ and $\Delta_{\infty}$ by fitting. In Fig. \[fig2\], we show $\Delta_L$ as a function of $L$ for several values of $J_3$ at $J_2=1$. For $J_3=0$, the estimated value of $\Delta_{\infty}$ is about 0.002 and is close to zero; the nonzero value is interpreted as an extrapolation error [@exact]. For $0 < J_3 {\stackrel{<}{_\sim}}0.4$, the true value of the spin gap is very small or may be regarded as zero, since the estimated values are less than 0.002 and are within the extrapolation error. For $J_3 {\stackrel{>}{_\sim}}0.6$, the figure shows that the system has a finite spin gap. For $J_3=0.5$, $\Delta_{\infty}$ is 0.0034, which is small but seems to be finite. This is consistent with the result of Okamoto et al. that the spin gap opens at the critical value $J_3^c \simeq 0.35$ for $J_2=1$ [@Okamoto].
In general, the spin gap in a dimer phase is exponentially small near the phase boundary to a spin fluid phase. Hence it is difficult to estimate $\Delta_{\infty}$ near the boundary in the present case. To overcome this difficulty, we assume that $J_3$ dependence of the spin gap is given by $$D(J_3) = a_1 \sqrt{J_3-J_3^c}
\exp\left( -\frac{a_2}{J_3-J_3^c} \right)
\label{eq3}$$ for $J_3 \sim J_3^c$[@Haldane], where $a_1$ and $a_2$ are constants. We have the values of $J_3^c$ by inspecting the phase diagram of Okamoto et al. [@Okamoto]; e. g. $J_3^c$ = 0.374, 0.354 and 0.460 for $J_2$ = 0.7, 1.0 and 1.4, respectively. We carry out the fitting of the extrapolation data $\Delta_{\infty}$ to eq. (\[eq3\]) and determine $a_1$ and $a_2$. Figure \[fig3\] represents the fitting function $D(J_3)$ and the extrapolation data. We find that the extrapolation data are well reproduced by eq. (\[eq3\]) for $\Delta_{\infty} {\stackrel{>}{_\sim}}0.02$. Hence the function form in eq. (\[eq3\]) is reliable. We use eq. (\[eq3\]) to estimate the spin gap for $\Delta_{\infty} {\stackrel{<}{_\sim}}0.02$ near the critical value $J_3^c$. For example, the spin gap is estimated as $1.0 \times 10^{-2}$ at $J_3 \simeq 0.57$, $1.0 \times 10^{-3}$ at $J_3 \simeq 0.50$, $1.0 \times 10^{-4}$ at $J_3 \simeq 0.47$ and $1.0 \times 10^{-5}$ at $J_3 \simeq 0.45$ for $J_2=1.0$.
Using these results, we draw contour lines of the spin gap in the $J_2$-$J_3$ plain. The resultant contour map is shown in Fig. \[fig4\]. We have calculated $\Delta_{\infty}$ at the discrete positions ($J_2$, $J_3$) with $J_2$ = 0.7, 0.8, ..., 2.0 and $J_3$ = 0.5, 0.55, ..., 1.0. For $\Delta_{\infty} > 0.02$, the positions of solid circles are determined by the linear interpolation among the spin gaps $\Delta_{\infty}$ at the discrete positions. For $\Delta_{\infty} < 0.02$, the positions of open circles are determined by using $D(J_3)$ (eq. (\[eq3\])) instead of $\Delta_{\infty}$.
The temperature dependence of the experimental magnetic susceptibility has a broad peak at $\sim$70 K. It suggests that the energy scale of the characteristic exchange constant $J_1$ is larger than 70 K. Here we consider a case of $J_1$ being 100 K as an example.[@J1] In this case, we have $\Delta_{\infty} \sim 0.05$ according to the observed spin gap $\sim$5 K. Then $J_2$ and $J_3$ are limited to values close to the contour line of $\Delta_{\infty}=0.05$ and of $J_2 < 1$.
One of the authors (K. T.) would like to thank H. Tanaka and M. Ishii for explaining their experimental results, and K. Okamoto for discussion. This work is partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan.
[99]{} M. Ishii, H. Tanaka, M. Mori, H. Uekusa, Y. Ohashi, K. Tatani, Y. Narumi and K. Kindo: J. Phys. Soc. Jpn. [**69**]{} (1999) 340. D. D. Swank and R. D. Willett: Inorg. Chimica Acta [**8**]{} (1974) 143. K. Takano, K. Kubo and H. Sakamoto: J. Phys. :Condens. Matter [**8**]{} (1996) 6405. K. Okamoto, T. Tonegawa, Y. Takahashi and M. Kaburagi: J. Phys.: Condens. Matter [**11**]{} (1999) 10485. T. Tonegawa, K. Okamoto, T. Hikihara, Y. Takahashi and M. Kaburagi: cond-mat/9912482 (1999). In this case, the Bethe Ansatz solution exactly shows that the system has a gapless excitation. F. D. M. Haldane: Phys. Rev. [**B25**]{} (1982) 4925. For example, the peak of the magnetic susceptibility is located at $\sim$0.6$J$ for the spin chain with uniform nearest-neighbor interactions (J. C. Bonner and M. E. Fisher: Phys. Rev. [**135**]{} (1964) A640), and at $\sim$0.7$J$ for the two-leg spin ladder and the spin chain with bond alternation. (T. Barnes and J. Riera: Phys. Rev. [**B50**]{} (1994) 6816).
|
---
abstract: 'We prove that the interval $(5/6,\, 1)$ contains no $3$-dimensional canonical thresholds.'
address: ' Yuri Prokhorov, Department of Higher Algebra, Faculty of Mathematics and Mechanics, Moscow State Lomonosov University, Vorobievy Gory, Moscow, 119 899, RUSSIA'
author:
- Yuri Prokhorov
title: 'Gap conjecture for $3$-dimensional canonical thresholds'
---
Introduction
============
Let $(X\ni P)$ be a three-dimensional canonical singularity and let $S\subset X$ be a ${{\mathbb Q}}$-Cartier divisor. The *canonical threshold* of the pair $(X,S)$ is $${\operatorname{ct}}(X,S):=\sup \{ c \mid \text{the pair $(X,cS)$ is canonical}\}.$$ It is easy to see that ${\operatorname{ct}}(X,S)$ is rational and non-negative. Moreover, if $S$ is effective and integral, then ${\operatorname{ct}}(X,S)\in [0,\, 1]$. Define the subset ${{\mathcal T}^{\mathrm{can}}}_n\subset [0,\, 1]$ as follows $${{\mathcal T}^{\mathrm{can}}}_n:=\{{\operatorname{ct}}(X,S) \mid \text{$\dim X=n$, $S$ is
integral and effective}
\}.$$
The following conjecture is an analog of corresponding conjectures for log canonical thresholds and minimal discrepancies, see [@Shokurov1988p], [@Utah], [@Kollar-1995-pairs], [@Prokhorov-McKernan-2004], [@kollar-2008].
\[conj-acc\] The set ${{\mathcal T}^{\mathrm{can}}}_n$ satisfies the ascending chain condition.
The conjecture is interesting for applications to birational geometry, see, e.g., [@Corti1995]. It was shown in [@Birkar-Shokurov] that much more general form of \[conj-acc\] follows from ACC for minimal log discrepancies and weak Borisov-Alexeev conjecture. The important particular case of \[conj-acc\] is the following
\[conj-gap\] $\epsilon_n^{\mathrm{can}}:=1-\sup ({{\mathcal T}^{\mathrm{can}}}_n\setminus \{1\})>0$.
The aim of this note is to prove Conjecture \[conj-gap\] for $n=3$ in a precise form:
\[main-1\] $\epsilon^{\mathrm{can}}_3=1/6$.
An analog of this theorem for log canonical thresholds was proved by J. Kollár [@Kollar1994]: $\epsilon^{\mathrm{lc}}_3=1/42$.
Note that replacing $(X\ni P)$ with its terminal ${{\mathbb Q}}$-factorial modification we may assume that $(X\ni P)$ is terminal. Thus the following is a stronger form of Theorem \[main-1\]:
\[main-2\] Let $(X\ni P)$ be a three-dimensional terminal singularity and let $S\subset X$ be an (integral) effective Weil ${{\mathbb Q}}$-Cartier divisor such that the pair $(X,S)$ is not canonical. Then ${\operatorname{ct}}(X,S)\le 5/6$. Moreover, if $(X\ni P)$ is singular, then ${\operatorname{ct}}(X,S)\le 4/5$.
The proof is rather standard. We use the classification of terminal singularities and weighted blowups techniques, cf. [@Kawamata-1992-discr], [@Kollar1994], [@Markushevich-1996-discr].
Preliminaries
=============
**Notation.** For a polynomial $\phi$, ${\operatorname{ord}}_0 \phi$ denotes the order of vanishing of $\phi$ at $0$ and $\phi_d$ is the homogeneous component of degree $d$.
Throughout this paper we let $(X\ni P)$ be the germ of a three-dimensional terminal singularity and let $S\subset X$ be an effective Weil ${{\mathbb Q}}$-Cartier divisor such that the pair $(X,S)$ is not canonical. Put $c:={\operatorname{ct}}(X,S) >0$. Since $(X,S)$ is not canonical, $c<1$.
We work over the complex number field ${{\mathbb C}}$.
In the above notation the singularity $(S\ni P)$ is not Du Val.
This is well-known, see e.g. [@Reid-1980can Th. 2.6].
We use the techniques of weighted blowups. For definitions and basic properties we refer, for example, to [@Markushevich-1996-discr], [@Reid-YPG1987]. By fixing coordinates $x_1,\dots,x_n$ we regard the affine space ${{\mathbb C}}^n$ as a toric variety. Let ${{\boldsymbol{\alpha}}}=(\alpha_1,\dots,\alpha_n)$ be a weight (a primitive lattice vector in the positive octant) and let $\sigma_{\alpha}\colon {{\mathbb C}}^n_{{{\boldsymbol{\alpha}}}}\to {{\mathbb C}}^n$ be the weighted blowup with weight ${{\boldsymbol{\alpha}}}$ (${{\boldsymbol{\alpha}}}$-blowup). The exceptional divisor $E_{{{\boldsymbol{\alpha}}}}$ is irreducible and determines a discrete valuation $v_{{{\boldsymbol{\alpha}}}}$ of the function field ${{\mathbb C}}({{\mathbb C}}^n)$ such that $v_{{{\boldsymbol{\alpha}}}}(x_i)=\alpha_i$.
Now let $X\subset {{\mathbb C}}^n$ be a hypersurface given by the equation $\phi=0$ and let $X_{{{\boldsymbol{\alpha}}}}\subset {{\mathbb C}}^n_{{{\boldsymbol{\alpha}}}}$ be its proper transform. Fix an irreducible component $G$ of $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ such that $X_{{{\boldsymbol{\alpha}}}}$ is smooth at the generic point of $G$. Let $v_G$ be the corresponding discrete valuation of ${{\mathbb C}}(X)$. Write $$E_{{{\boldsymbol{\alpha}}}}\mid_{ X_{{{\boldsymbol{\alpha}}}}}=m_G G+(\text{other components}).$$ Assume that $d_G=1$ and $G$ is not a toric subvariety in ${{\mathbb C}}^n_{{{\boldsymbol{\alpha}}}}$. Then the discrepancy of $G$ with respect to $K_X$ is computed by the formula $$a(G,K_X)=|{{\boldsymbol{\alpha}}}|-1-v_{{{\boldsymbol{\alpha}}}}(\phi), \quad |{{\boldsymbol{\alpha}}}|=\sum \alpha_i,$$ see [@Markushevich-1996-discr]. Let $S\subset X$ be a Cartier divisor and let $\psi$ be a local defining equation of $S$ in ${{\mathscr{O}}}_{0,X}$. Then $v_G(\psi)=v_{{{\boldsymbol{\alpha}}}}(\psi)$ and the discrepancy of $G$ with respect to $K_X+cS$ is computed by the formula $$a(G,K_X+cS)=a(G,K_X)-cv_{G}(\psi)=|{{\boldsymbol{\alpha}}}|-1-v_{{{\boldsymbol{\alpha}}}}(\phi)-cv_{{{\boldsymbol{\alpha}}}}(\psi).$$ Therefore, $$c \le a(G,K_X)/v_{{{\boldsymbol{\alpha}}}}(\psi)= (|{{\boldsymbol{\alpha}}}|-1-v_{{{\boldsymbol{\alpha}}}}(\phi))/v_{{{\boldsymbol{\alpha}}}}(\psi).$$
A weight ${{\boldsymbol{\alpha}}}$ is said to be *admissible* if $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ contains at least one reduced non-toric component.
Gorenstein case
===============
In this section we consider the case where $(X\ni P)$ is either smooth or an index one singularity.
\[l-G-1\] If $(X\ni P)$ is smooth, then $c\le 5/6$.
Let $c>5/6$. We may assume that $X={{\mathbb C}}^3$. Let $\psi(x,y,z)=0$ be an equation of $S$. Consider a weighted blowup $\sigma_{{{\boldsymbol{\alpha}}}}\colon {{\mathbb C}}^3_{{{\boldsymbol{\alpha}}}}\to {{\mathbb C}}^3$ with a suitable weight ${{\boldsymbol{\alpha}}}$. Let $E_{{\boldsymbol{\alpha}}}$ be the exceptional divisor. Recall that $(S\ni P)$ is not Du Val. Up to analytic coordinate change there are the following cases (cf. [@Kollar-Mori-19988 4.25]):
**Case ${\operatorname{ord}}_0 \psi\ge 3$.** Take ${{\boldsymbol{\alpha}}}=(1,1,1)$ (usual blowup of $0$). Then $a(E_{{{\boldsymbol{\alpha}}}}, K_X) =2$, $v_{{{\boldsymbol{\alpha}}}}(\psi)={\operatorname{ord}}_0 \psi\ge 3$. Hence $c\le a(E_{{{\boldsymbol{\alpha}}}}, K_X)/v_{{{\boldsymbol{\alpha}}}}(\psi) \le 2/3$, a contradiction.
**Case $\psi=x^2+\eta(y,z)$, where ${\operatorname{ord}}_0 \eta\ge 4$.** Take ${{\boldsymbol{\alpha}}}=(2,1,1)$. Then $a(E_{{{\boldsymbol{\alpha}}}}, K_X) =3$, $v_{{{\boldsymbol{\alpha}}}}(\psi)=4$. Hence $c\le a(E_{{{\boldsymbol{\alpha}}}}, K_X) /v_{{{\boldsymbol{\alpha}}}}(\psi)\le 3/4$, a contradiction.
**Case $\psi=x^2+y^3+\eta(y,z)$, where ${\operatorname{ord}}_0 \eta\ge 4$.** Here $\eta$ contains no terms $yz^l$, $l\le 3$ and $z^l$, $l\le 5$ (see, e.g., [@Kollar-Mori-19988 4.25]). Take ${{\boldsymbol{\alpha}}}=(3,2,1)$. Then $a(E_{{{\boldsymbol{\alpha}}}}, K_X) =5$, $v_{{{\boldsymbol{\alpha}}}}(\psi)=6$. Hence $c\le a(E_{{{\boldsymbol{\alpha}}}}, K_X) /v_{{{\boldsymbol{\alpha}}}}(\psi)= 5/6$, a contradiction.
\[l-G-2\] Assume that $(X\ni P)$ is a Gorenstein terminal singularity and $(X\ni P)$ is not smooth. Then $c\le 4/5$.
Let $c>4/5$. We may assume that $X$ is a hypersurface in ${{\mathbb C}}^4$ (it is an isolated cDV-singularity [@Reid-1980can]). Let $\phi(x,y,z,t)=0$ be the equation of $X$. Since $(X\ni P)$ is a cDV-singularity, ${\operatorname{ord}}_0 \phi=2$. According to [@Markushevich-1996-discr], in a suitable coordinate system $(x,y,z,t)$, there is an admissible weighted blowup $\sigma_{{{\boldsymbol{\alpha}}}} \colon {{\mathbb C}}^4_{{{\boldsymbol{\alpha}}}}\to {{\mathbb C}}^4$ such that at least for one component $G$ of $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ we have $a(G,K_X)=1$. Then $c\le 1/v_{{{\boldsymbol{\alpha}}}}(\psi)$, so $v_{{{\boldsymbol{\alpha}}}}(\psi)=1$. This means, in particular, that ${\operatorname{ord}}_0 \psi=1$. Up to coordinate change we may assume that $\psi=t$. Write $$\phi=\eta(x,y,z)+t \zeta (x,y,z,t).$$ Then $S$ is a hypersurface in ${{\mathbb C}}^3_{x,y,z}$ given by $\eta(x,y,z)=0$. As in the proof of Lemma \[l-G-1\], using Morse Lemma we get the following cases:
**Case ${\operatorname{ord}}_0 \eta\ge 3$.** Take ${{\boldsymbol{\alpha}}}=(1,1,1,2)$. By the terminality condition [@Reid-YPG1987 Th. 4.6], we have $4=v_{{{\boldsymbol{\alpha}}}}(xyzt)-1>v_{{{\boldsymbol{\alpha}}}}(\phi)$. Hence, $v_{{{\boldsymbol{\alpha}}}}(\eta)\le 3$ and $\eta_3\neq 0$. We claim that ${{\boldsymbol{\alpha}}}$ is admissible whenever $\eta_3$ is not a cube of a linear form. Indeed, in the affine chart $U_x:=\{x\neq 0\}$ the map $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}$ is given by $$\label{eq-odd3}
x \mapsto x',\quad y \mapsto y'x',\quad z\mapsto z'x',\quad
t \mapsto t'x'^2.$$ First we assume that $\zeta$ contains the term $x$. After the coordinate change $x\mapsfrom \zeta (x,y,z,t)$ we obtain $$\phi=\eta(x,y,z)+tx.$$ Using we see that $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ is given in $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_x)\simeq
{{\mathbb C}}^4$ by $$x'=\eta_3(1,y',z')+t'=0.$$ Hence ${{\boldsymbol{\alpha}}}$ is admissible, i.e., $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ has a reduced non-toric component $G$. Then $a(G, K_X) =1$, $v_{G}(\psi)=2$ and $c\le a(G, K_X) /v_{G}(\psi)=1/2$, a contradiction.
Thus by symmetry we may assume that $\zeta$ contains no terms $x$, $y$, $z$. Since ${\operatorname{ord}}_0 \phi=2$, $\zeta$ contains $t$. So, $$\phi=\eta(x,y,z)+t^2+ t\xi (x,y,z,t),\quad {\operatorname{ord}}_0 \xi\ge 2.$$ As above, $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ is given in ${{\mathbb C}}^4$ by $x'=\eta_3(1,y',z')=0$. If $\eta_3$ is not a cube of a linear form, then $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ has a reduced non-toric component $G$. Then, as above, $c\le 1/2$, a contradiction.
Consider the case where $\eta_3$ is a cube of a linear form. Then we may assume that $\eta_3(x,y,z)=y^3$, so $$\phi=y^3+\eta^{\bullet}(x,y,z)+t^2+ t\xi (x,y,z,t),\quad {\operatorname{ord}}_0 \xi\ge 2, \quad
{\operatorname{ord}}_0\eta^{\bullet} \ge 4.$$ Put ${{\boldsymbol{\alpha}}}'=(2,2,2,3)$. Again, in the affine chart $U_x:=\{x\neq 0\}$ the map $\sigma_{{{\boldsymbol{\alpha}}}'}^{-1}$ is given by $x \mapsto x'^2$, $y \mapsto y'x'^2$, $z\mapsto z'x'^2$, $t \mapsto t'x'^3$, where $\sigma_{{{\boldsymbol{\alpha}}}'}^{-1}(U_x)\simeq {{\mathbb C}}^4/{{\boldsymbol{\mu}}}_2(1,0,0,1)$ and $$E_{{{\boldsymbol{\alpha}}}'}\cap X_{{{\boldsymbol{\alpha}}}'}\cap \sigma_{{{\boldsymbol{\alpha}}}'}^{-1}(U_x)=
\{
x'=0,\ y'^3+t'^2 =0\}.$$ Thus ${{\boldsymbol{\alpha}}}'$ is admissible and for some component $G'$ of $X_{{{\boldsymbol{\alpha}}}'}\cap E_{{{\boldsymbol{\alpha}}}'}$ we have $a(G', K_X)=2$, $v_{G'}(\psi)=3$, $c\le 2/3$, a contradiction.
**Case $\eta=x^2+\xi(y,z)$, where ${\operatorname{ord}}_0 \xi\ge 4$.** By Morse Lemma we may assume that $\zeta$ does not depend on $x$. Write $\zeta_1=\delta_1y+\delta_2z+\delta_3t$, $\delta_i\in {{\mathbb C}}$. Take ${{\boldsymbol{\alpha}}}=(2,1,1,3)$. In the affine chart $U_y:=\{y\neq 0\}$ the map $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}$ is given by $x \mapsto x'y'^2$, $y \mapsto y'$, $z\mapsto z'y'$, $t \mapsto t'y'^3$ and $$E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}\cap \sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_y)=
\{ y'=0,\ x'^2+\xi_4(1,z')+\delta_1t' +\delta_2t'z'=0\}.$$ If either $\delta_1\neq 0$ or $\delta_2\neq 0$ or $\xi_4\neq 0$, then $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ is reduced (at least over $U_y$). Hence, ${{\boldsymbol{\alpha}}}$ is admissible and for some component $G$ of $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ we have $c\le a(G, K_X) /v_{G}(\psi)= 2/3$, a contradiction. Thus $\delta_1=\delta_2= 0$ and $\xi_4=0$. Then we can write $$\phi=x^2+\xi(y,z)+\delta_3t^2+t\zeta^{\bullet}(y,z,t), \quad {\operatorname{ord}}_0\xi\ge 5,
\quad {\operatorname{ord}}_0\zeta^{\bullet}\ge 2.$$ Take ${{\boldsymbol{\alpha}}}'=(2,1,1,2)$. In the affine chart $U_y:=\{y\neq 0\}$ the map $\sigma_{{{\boldsymbol{\alpha}}}'}^{-1}$ is given by $x \mapsto x'y'^2$, $y \mapsto y'$, $z\mapsto z'y'$, $t \mapsto t'y'^2$ and $$E_{{{\boldsymbol{\alpha}}}'}\cap X_{{{\boldsymbol{\alpha}}}'}\cap \sigma_{{{\boldsymbol{\alpha}}}'}^{-1}(U_y)=
\{ y'=0,\ x'^2+\delta_3t'^2+t\lambda(1,z')=0\},$$ where $\lambda$ is the degree $2$ homogeneous part of $\zeta(y,z,0)$. If $\delta_3\neq 0$ or $\lambda\neq 0$, as above, ${{\boldsymbol{\alpha}}}'$ is admissible and $c\le 1/2$, a contradiction. Thus $\delta_3=0$, $\lambda= 0$, and $$\phi=x^2+\xi(y,z)+\delta t^3+t\zeta^{\circ}(y,z,t), \quad \delta\in {{\mathbb C}}, \quad{\operatorname{ord}}_0\xi\ge 5,
\quad {\operatorname{ord}}_0\zeta^{\circ}\ge 3.$$ Applying the terminality condition [@Reid-YPG1987 Th. 4.6] with weight $(2,1,1,1)$ we get that $\delta\neq 0$.
Take ${{\boldsymbol{\alpha}}}''= (3,1,1,2)$. As above we get that ${{\boldsymbol{\alpha}}}''$ is admissible and then $c\le 1/2$, a contradiction.
**Case $\eta=x^2+y^3+\xi(y,z)$, where ${\operatorname{ord}}_0 \xi\ge 4$.** Here $\xi$ contains no terms $yz^l$, $l\le 3$ and $z^l$, $l\le 5$ (see, e.g., [@Kollar-Mori-19988 4.25]). Write $\zeta_1=cz+\ell(x,y,t)$ and $\xi=\xi_{(6)}+\xi_{(7)}+\cdots$, where $\xi_{(d)}$ is the degree $d$ weighted homogeneous part of $\xi$ with respect to ${\operatorname{wt}}(y,z)=(2,1)$. Here $\xi_{(6)}$ is a linear combination of $z^6$, $yz^4$, $y^2z^2$. Take ${{\boldsymbol{\alpha}}}=(3,2,1,5)$. In the affine chart $U_z:=\{z\neq 0\}$ the map $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}$ is given by $x \mapsto x'z'^3$, $y \mapsto y'z'^2$, $z\mapsto z'$, $t \mapsto t'z'^5$ and $$E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}\cap \sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_z)=
\{z'=0,\ x'^2+y'^3+\xi_{(6)}(y',1)+\delta t'=0\},$$ where $\delta$ is a constant and $\xi_{(6)}(y',1)$ contains no $y'^3$. Hence ${{\boldsymbol{\alpha}}}$ is admissible, i.e., $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ has a reduced non-toric component $G$. Then $a(G, K_X) =4$, $v_{G}(\psi)=5$, and $c\le a(G, K_X) /v_{G}(\psi)\le 4/5$, a contradiction.
The following examples show that bounds ${\operatorname{ct}}(X,S)\le 5/6$ and $\le 4/5$ in Theorem \[main-2\] are sharp.
Let $X={{\mathbb C}}^3$ and let $S=S^d$ is given by $x^2+y^3+z^d$, $d\ge 6$. Then ${\operatorname{ct}}({{\mathbb C}}^3,S^d)=5/6$. We prove this by descending induction on ${\left\lfloor d/6\right\rfloor}$. Take ${{\boldsymbol{\alpha}}}=(3,2,1)$ and consider the ${{\boldsymbol{\alpha}}}$-blowup $\sigma_{{{\boldsymbol{\alpha}}}}\colon {{\mathbb C}}^3_{{{\boldsymbol{\alpha}}}}\to {{\mathbb C}}^3$. Let $S_{{{\boldsymbol{\alpha}}}}\subset X_{{{\boldsymbol{\alpha}}}}$ be the proper transform of $S$. We have $a(E_{{{\boldsymbol{\alpha}}}}, K_X) =5$ and $v_{{{\boldsymbol{\alpha}}}}(\psi)=6$. Hence, ${\operatorname{ct}}({{\mathbb C}}^3,S^d)\le 5/6$. Further, $$\textstyle
K_{{{\mathbb C}}^3_{{{\boldsymbol{\alpha}}}}}+\frac56S_{{{\boldsymbol{\alpha}}}}=\sigma_{{{\boldsymbol{\alpha}}}}^*(K_{{{\mathbb C}}^3}+\frac56S).$$ Thus it is sufficient to show that ${\operatorname{ct}}(X_{{{\boldsymbol{\alpha}}}},\frac 56 S_{{{\boldsymbol{\alpha}}}})$ is canonical. We have three affine charts:
- $U_x:=\{x\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon$ $x \mapsto x'^3$, $y \mapsto y'x'^2$, $z\mapsto z'x'$, $S_{{{\boldsymbol{\alpha}}}}$ is given in $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_x)\simeq{{\mathbb C}}^3/{{\boldsymbol{\mu}}}_3(-1,2,1)$ by the equation $1+y'^3+z'^dx'^{d-6}=0$. Hence, in this chart, $S_{{{\boldsymbol{\alpha}}}}$ is smooth and does not pass through a (unique) singular point of $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_x)$.
- $U_y:=\{y\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon$ $x \mapsto x'y'^3$, $y \mapsto y'^2$, $z\mapsto z'y'$, $S_{{{\boldsymbol{\alpha}}}}$ is given in $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_y)\simeq {{\mathbb C}}^3/{{\boldsymbol{\mu}}}_2(3,-1,1)$ by the equation $x'^2+1+z'^dy'^{d-6}=0$. Again, in this chart, $S_{{{\boldsymbol{\alpha}}}}$ is smooth and does not pass through a (unique) singular point of $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_y)$.
- $U_z:=\{z\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon$ $x \mapsto x'z'^3$, $y \mapsto y'z'^2$, $z\mapsto z'$, $S_{{{\boldsymbol{\alpha}}}}$ is given in $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_z)\simeq {{\mathbb C}}^3$ by the equation $x'^2+y'^3+z'^{d-6}=0$. In this chart, $(X_{{{\boldsymbol{\alpha}}}},S_{{{\boldsymbol{\alpha}}}})\simeq ({{\mathbb C}}^3, S^{d-6})$.
Thus $X_{{{\boldsymbol{\alpha}}}}$ has only terminal singularities, $S_{{{\boldsymbol{\alpha}}}}$ does not pass through any singular point of $X_{{{\boldsymbol{\alpha}}}}$, and the pair $(X_{{{\boldsymbol{\alpha}}}}, S_{{{\boldsymbol{\alpha}}}})$ is terminal in charts $U_x$ and $U_y$. In the chart $U_z$ the pair by induction $(X_{{{\boldsymbol{\alpha}}}},\frac 56 S_{{{\boldsymbol{\alpha}}}})$ is canonical (moreover, $(X_{{{\boldsymbol{\alpha}}}}, S_{{{\boldsymbol{\alpha}}}})$ is canonical if $d\le 11$). Therefore, ${\operatorname{ct}}(X,S)=5/6$.
Let $X\subset {{\mathbb C}}^4$ is given by $x^2+y^3+z^d+tz=0$, $d\ge 7$ and let $S$ cut out by $t=0$. Take ${{\boldsymbol{\alpha}}}=(3,2,1,5)$ and consider the ${{\boldsymbol{\alpha}}}$-blowup $\sigma_{{{\boldsymbol{\alpha}}}} \colon X_{{{\boldsymbol{\alpha}}}}\to X$. Let $S_{{{\boldsymbol{\alpha}}}}\subset X_{{{\boldsymbol{\alpha}}}}$ be the proper transform of $S$. We see below that ${{\boldsymbol{\alpha}}}$ is admissible. Moreover, the exceptional divisor $G:=E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ is reduced and irreducible. We have four charts:
- $U_x:=\{x\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon
x \mapsto x^3$, $y \mapsto yx^2$, $z\mapsto zx$, $t\mapsto tx^5$, $X_{{{\boldsymbol{\alpha}}}}$ is given in $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_x)\simeq{{\mathbb C}}^4/{{\boldsymbol{\mu}}}_3(-1,2,1,5)$ by the equation $1+y^3+z^dx^{d-6}+tz=0$ and $S_{{{\boldsymbol{\alpha}}}}$ by two equations $x=1+y^3+tz=0$. Hence, in this chart, both $X_{{{\boldsymbol{\alpha}}}}$ and $S_{{{\boldsymbol{\alpha}}}}$ are smooth.
- $U_y:=\{y\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon
x \mapsto xy^3$, $y \mapsto y^2$, $z\mapsto zy$, $t\mapsto ty^5$, $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_y)\simeq {{\mathbb C}}^4/{{\boldsymbol{\mu}}}_2(3,-1,1,5)$, $X_{{{\boldsymbol{\alpha}}}}=\{x^2+1+z^dy^{d-6}+tz=0\}$, and $S_{{{\boldsymbol{\alpha}}}}=\{y=x^2+1+tz=0\}$. As above, both $X_{{{\boldsymbol{\alpha}}}}$ and $S_{{{\boldsymbol{\alpha}}}}$ are smooth in this chart.
- $U_z:=\{z\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon
x \mapsto xz^3$, $y \mapsto yz^2$, $z\mapsto z$, $t\mapsto tz^5$, $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_z)\simeq {{\mathbb C}}^4$, $X_{{{\boldsymbol{\alpha}}}}=\{x^2+y^3+z^{d-6}+t=0\}$, and $S_{{{\boldsymbol{\alpha}}}}=\{z=x^2+y^3+t=0\}$. As above, both $X_{{{\boldsymbol{\alpha}}}}$ and $S_{{{\boldsymbol{\alpha}}}}$ are smooth in this chart.
- $U_t:=\{t\neq 0\}$. Here $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}\colon
x \mapsto xt^3$, $y \mapsto yt^2$, $z\mapsto zt$, $t\mapsto t^5$, $\sigma_{{{\boldsymbol{\alpha}}}}^{-1}(U_t)\simeq {{\mathbb C}}^4/{{\boldsymbol{\mu}}}_5(3,2,1,-1)$, $X_{{{\boldsymbol{\alpha}}}}=\{x^2+y^3+z^dt^{d-6}+z=0\}$, and $S_{{{\boldsymbol{\alpha}}}}=\{t=x^2+y^3+z=0\}$. The variety $X_{{{\boldsymbol{\alpha}}}}$ has a unique singular point $Q$ at the origin and this point is terminal of type $\frac15(3,2,-1)$ the surface $S_{{{\boldsymbol{\alpha}}}}$ is smooth and does not pass through $Q$.
Thus we have $a(G, K_X) =4$, $v_{{{\boldsymbol{\alpha}}}}(\psi)=5$, and $a(G, K_X+ \frac45 S) =0$. Therefore, $$\textstyle
K_{X_{{{\boldsymbol{\alpha}}}}}+\frac45S_{{{\boldsymbol{\alpha}}}}=\sigma_{{{\boldsymbol{\alpha}}}}^*(K_{X}+\frac45S).$$ Since the pair $K_{X_{{{\boldsymbol{\alpha}}}}}+\frac45S_{{{\boldsymbol{\alpha}}}}$ is canonical, ${\operatorname{ct}}(X,S)=4/5$.
Non-Gorenstein case
===================
Now we assume that $(X\ni P)$ is a (terminal) point of index $r>1$. Let $\pi \colon (X^\sharp\ni P^\sharp)\to (X\ni P)$ be the index-one cover and let $S^\sharp:=\pi^{-1}(S)$.
\[l-nG-1\] If $(X\ni P)$ is a cyclic quotient singularity, then ${\operatorname{ct}}(X,S)\le 1/2$.
By our assumption we have $X\simeq {{\mathbb C}}^3/{{\boldsymbol{\mu}}}_r(a,-a,1)$ for some $r\ge2$, $1\le a <r$, $\gcd(a,r)=1$. Assume that $c={\operatorname{ct}}(X,S)> 1/2$. Let $\psi=0$ be a defining equation of $S^\sharp$. Consider the weighted blowup $\sigma_{{{\boldsymbol{\alpha}}}}\colon X_{{{\boldsymbol{\alpha}}}}\to X$ with weights ${{\boldsymbol{\alpha}}}=\frac1r (a,r-a,1)$. Then $a(E_{{{\boldsymbol{\alpha}}}}, K_X)=1/r$. Since $a(E_{{{\boldsymbol{\alpha}}}}, K_X)-c v_{{\boldsymbol{\alpha}}}(\psi)\ge 0$, we have $v_{{\boldsymbol{\alpha}}}(\psi)\le a(E_{{{\boldsymbol{\alpha}}}}, K_X)/c< 2a(E_{{{\boldsymbol{\alpha}}}}, K_X)= 2/r$ and so $v_{{\boldsymbol{\alpha}}}(\psi)=1/r$. Thus we may assume that $\psi$ contains $x_3$ (if $a\equiv \pm 1$ we possibly have to permute coordinates). Then $S^\sharp\simeq {{\mathbb C}}^2$ is smooth and $S\simeq {{\mathbb C}}^2/{{\boldsymbol{\mu}}}_r(a,-a)$, i.e., $S$ is Du Val of type $A_{r-1}$.
\[l-nG-2\] If $(X\ni P)$ is a terminal singularity of index $r>1$ and ${\operatorname{ct}}(X,S)>1/2$, then $K_X+S\sim 0$.
By Lemma \[l-nG-1\] $(X\ni P)$ is not a cyclic quotient singularity. There is an analytic ${{\boldsymbol{\mu}}}_r$-equivariant embedding $(X^\sharp, P^\sharp) \subset ({{\mathbb C}}^4,0)$. Let $(x_1,x_2,x_3,x_4)$ be coordinates in ${{\mathbb C}}^4$, let $\phi=0$ be an equation of $X^\sharp$, and let $\psi=0$ be an equation of $S^{\sharp}$. We can take $(x_1,x_2,x_3,x_4)$ and $\phi$ to be semi-invariants such that one of the following holds [@Reid-YPG1987]:
1. **Main series.** ${\operatorname{wt}}(x_1,x_2,x_3,x_4; \phi)\equiv (a,-a, 1, 0;0)\mod r$, where $\gcd (a,r)=1$.
2. **Case $cAx/4$.** $r=4$, ${\operatorname{wt}}(x_1,x_2,x_3,x_4; \phi)\equiv (1,3, 1, 2;2)\mod 4$.
In both cases ${\operatorname{wt}}(x_1x_2x_3x_4)- {\operatorname{wt}}\phi\equiv {\operatorname{wt}}x_3\mod r$. According to [@Kawamata-1992-discr] there is a weight ${{\boldsymbol{\alpha}}}$ such that for the corresponding ${{\boldsymbol{\alpha}}}$-blowup $\sigma_{{{\boldsymbol{\alpha}}}}\colon X_{{{\boldsymbol{\alpha}}}}\subset W\to X\subset {{\mathbb C}}^4/{{\boldsymbol{\mu}}}_r$ the exceptional divisor $E_{{{\boldsymbol{\alpha}}}}\cap X_{{{\boldsymbol{\alpha}}}}$ has a reduced component $G$ of discrepancy $a(G, K_X)=1/r$. Moreover, $r\alpha_i\equiv {\operatorname{wt}}x_i \mod r$, $i=1,2,3,4$. Since $c>1/2$, we have $1/r-cv_{{\boldsymbol{\alpha}}}(\psi)\ge 0$, i.e., $rv_{{\boldsymbol{\alpha}}}(\psi)< 2$, so $rv_{{\boldsymbol{\alpha}}}(\psi)=1$. In particular, ${\operatorname{wt}}\psi\equiv 1 \mod r$.
Let $\omega$ be a section of ${{\mathscr{O}}}_X(-K_X)$. Then $\omega$ can be written as $$\omega=\lambda (\partial \phi/\partial x_4)(dx_1 \wedge dx_2\wedge dx_3)^{-1},$$ where $\lambda$ is a semi-invariant function with $${\operatorname{wt}}\lambda-{\operatorname{wt}}(x_1x_2x_3x_4)+{\operatorname{wt}}\phi\equiv {\operatorname{wt}}\omega \equiv 0\mod r.$$ Thus, ${\operatorname{wt}}\psi \equiv {\operatorname{wt}}\lambda\mod r$. Hence, $S\sim -K_X$.
\[l-nG-3\] If $(X\ni P)$ is a terminal singularity of index $r>1$, then $c\le 4/5$.
Since $\pi$ is étale in codimension one, we have $K_{X^\sharp }+cS^\sharp=\pi^*(K_X+cS)$. Hence the pair $(X^\sharp,\, cS^\sharp)$ is canonical (see, e.g., [@Kollar-1995-pairs 3.16.1]). Assume that $c> 4/5$. By Lemma \[l-nG-1\] the point $(X^\sharp\ni P^\sharp)$ is singular. Then by Lemma \[l-G-2\] the pair $(X^\sharp,\, S^\sharp)$ is canonical. Therefore, $(S^\sharp\ni P^\sharp)$ is a Du Val singularity. Then the singularity $(S\ni P)=(S^\sharp\ni P^\sharp)/{{\boldsymbol{\mu}}}_r$ is log terminal. On the other hand, by Lemma \[l-nG-2\] the divisor $K_S$ is Cartier. Hence, $(S\ni P)$ is Du Val, a contradiction.
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abstract: |
The adsorption effects of small molecules (H$_{2}$O, CO, NH$_{3}$, NO$_{2}$) and large molecules (Tetracyanoquinodimethane (TCNQ) and Tetrafluoro-tetracyanoquinodimethane (F4TCNQ)) on electronic and magnetic properties of two triangular graphene antidot lattices (GALs), $[10,3,6]_{RTA}$ and $[10, 5]_{ETA}$, are investigated by means of first-principles calculations. We find that CO, NO$_{2}$, TCNQ, and F4TCNQ molecules are chemisorbed by both antidots, whereas NH$_{3}$ is physisorbed (chemisorbed) by $[10, 5]_{ETA}$ ($[10,3,6]_{RTA}$) structure. H$_{2}$O, CO, NH$_{3}$ molecules reveal no significant effect on electronic and magnetic properties of these antidot structures. The adsorbed NO$_{2}$ molecules affect the energy gap of GALs by changing their electronic structure from semiconducting to half-metal nature. This suggests that both GALs can act as efficient NO$_{2}$ sensors. The adsorption of TCNQ and F4TCNQ molecules on GALs induces flat bands in the vicinity of the Fermi energy and also turn the electronic structure of antidot lattices to half-metallicity. Among the small and large molecules, NO$_{2}$ molecules induce the most total magnetic moment, paving the way to make magnetic GAL-based devices.
Keywords: gas adsorption, graphene antidots, electronic properties, magnetic moments
author:
- Zahra Talebi Esfahani
- Alireza Saffarzadeh
- Ahmad Akhound
- Amir Abbas Sabouri Dodaran
title: Gas adsorption effects on electronic and magnetic properties of triangular graphene antidot lattices
---
Introduction
============
Graphene, as a single atomic layer of graphite, has been the subject of a large amount of theoretical and experimental studies during the last decades, because of its rich and fascinating physical properties [@nov1; @cas2; @nov3; @bol4; @nai5] which can lead to many novel applications from more efficient solar cells to medicinal technologies. For instance, due to the Dirac-like spectrum of charge carriers in its gapless band structure, the electronic properties of graphene have attracted a great deal of research interest. Nevertheless, the zero band gap nature of graphene limits its practical applications in optoelectronics and photonics. To open an energy gap at Dirac point one can cut graphene into nanoribbons, create defects or add gas atoms [@lee7; @wan8]. One way to create defects is to remove some atoms and make a periodic array of holes, known as graphene antidot lattice (GAL) [@fur9; @van10; @ped11; @ouy12; @pet13; @ped14; @pow15].
The electric and magnetic properties of some GALs have been extensively studied in recent years [@ped11; @ouy12; @pet13; @ped14; @pow15; @fur16; @van17; @our18; @sch23]. Moreover, the investigation of sensing properties of GALs in the presence of small and large gas molecules may promote our fundamental understanding of the underlying physics and their possible technological applications. In this regard, Brun et al. [@sor34] studied the adsorption of boron and nitrogen doping on the hexagonal antidots and that how the size of antidot supercell and the number of passivated atoms affect the GAL properties.
The small molecules, such as CO, H$_{2}$O, NH$_{3}$ and NO$_{2}$ are common adsorbates on graphene substrate [@Schedin2007; @lee24; @you25; @lin26]. It has been shown experimentally [@Schedin2007] that to make highly sensitive graphene-based sensors, one can increase the graphene charge carrier concentration by adsorption of these small gas molecules. Moreover, using the density functional theory (DFT), it has been reported that H$_{2}$O and NO$_{2}$ molecules on graphene act as acceptors, while NH$_{3}$ and CO molecules exhibit electron donors [@lee24]. Among these adsorbates, the paramagnetic molecule NO$_{2}$ is a strong dopants [@lee24]. The charge transfer between the gas molecules and graphene surface depends strongly on the orientation of the adsorbate with respect to the surface but it is almost independent of the adsorption site. On the other hand, F4TCNQ and TCNQ are large molecules with high electron affinity [@ger; @men; @hsi; @bal27; @pin28; @xia29]. TCNQ is an organic compound with chemical formula C$_{12}$N$_{4}$H$_{4}$ acting as an electron acceptor. This molecule which forms charge-transfer salts has received a great deal of attention due to its high electrical conductivity at room temperature [@Ferraris1973]. F4TCNQ molecule with chemical formula C$_{12}$N$_{4}$F$_{4}$ has the same TCNQ structure and only the four hydrogen atoms are replaced by fluorine (see Fig. 1(c) and (d)). Since fluorine has higher electronegativity than hydrogen, F4TCNQ is more electronegative than TCNQ. Both of these adsorbates are strong electron acceptors. As a result, the adsorption of these large molecules on graphene surface modifies strongly the electric and magnetic properties of the surface [@ger; @men; @hsi; @bal27; @pin28; @xia29]. Nevertheless, the interplay between these molecules and GALs has, to our knowledge, not been reported previously.
![Optimized geometry of (a) $[10,5]_{ETA}$, (b) $[10, 3,6]_{RTA}$ supercells and (c) TCNQ and (d) F4TCNQ molecules. The yellow, blue, purple, and green balls represent carbon, hydrogen, nitrogen, and fluorine atoms, respectively.[]{data-label="image1"}](Fig1.pdf){width="0.95\linewidth"}
In this paper, we explore the adsorption of small and large molecules on the surface of two different shapes of triangular GALs, namely, $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$. The type of gas adsorption, semiconducting properties and magnetic moments of the antidot lattices in the presence of H$_{2}$O, CO, NH$_{3}$, NO$_{2}$, TCNQ, and F4TCNQ molecules are studied by means of density functional theory (DFT) calculations. To do this we present our computational method in section II. The numerical results of band structures and induced magnetic moments for the adsorption of gas molecules on the antidot lattices are given in section III. Finally, we present a brief conclusion in section IV.
![ The front view and side view (on top) of GALs in the presence of small and large molecules. (a1) pristine $[10,5]_{ETA}$, after adsorbing (a2) H$_{2}$O, (a3) CO, (a4) NH$_{3}$, (a5) 2NH$_{3}$, (a6) NO$_{2}$, (a7) TCNQ, (a8) F4TCNQ, (a9) F4TCNQ$_{90^{\circ}}$ molecules, and (b1) pristine $[10,3,6]_{RTA}$, after adsorbing (b2) H$_{2}$O, (b3) CO, (b4) NH$_{3}$, (b5) NO$_{2}$, (b6) TCNQ, and (b7) F4TCNQ molecules. The blue and red regions around each atom represent the charge densities corresponding to spin-up and spin-down electrons, respectively, with isosurface value 0.003 e/[Å]{}$^{3}$.[]{data-label="image2"}](Fig2.pdf){width="0.95\linewidth"}
Computational methods
=====================
To investigate the electronic and magnetic properties of triangular GALs in the presence of gas molecules, we use two notations, $[L,D]_{ETA}$ for equilateral triangular antidot ( ETA) and $[L,Z,A]_{RTA}$ for right triangular antidot (RTA) patterns, created in the surface of graphene layers. We choose triangular antidot lattices, because the edge states of these structures are highly sensitive to the spatial arrangement of the atoms [@pow15; @faragh2014] and induce magnetic moments on the zigzag-shaped edges in graphene nanoribbons and nanorings [@faragh2013; @Fujita1996; @Grujic2013]. Therefore, by examining the triangular antidots with pure zigzag edges (i.e., ETAs) and also the antidots with mixed armchair and zigzag edges (i.e., RTAs) in graphene lattices and comparing their spin-dependent electronic structures, we may provide a guideline for the future magneto-optical experiments in graphene-based nanodevices. In this regard, the triangular antidots in graphene have recently been fabricated [@Autin2016], and experiments suggest the possibility of nanostructured zigzag-edged devices [@Bai2010; @Shi2011; @Oberhuber2013; @Stehle2017]. Here $L$ represents the number of carbon atoms along the supercell edges, $D$ denotes the number of passivated carbon atoms along the ETA sides, and $Z$ and $A$ express the number of passivated carbon atoms along the two perpendicular sides of RTA with zigzag and armchair edges, respectively [@our18]. In this study, we consider superlattices with $L=10$, $D=5$, and $L=10$, $Z=3$, $A=6$, as shown in Fig. 1(a) and (b).
The DFT calculations are performed using the SIESTA code [@sol35; @ord36]. The exchange-correlation energy is calculated within the generalized gradient approximation (GGA) [@per37]. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, norm-conserving Troullier-Martins pseudo potentials, and a double-polarized basis (DZP) are used for this calculation [@art38]. Here, we should point out that although the PBE functional underestimates the electronic band gap [@Cohen2008; @Mori2008], it accurately describes the ground state properties. We use a $10 \times10$ ($L = 10$) supercell geometry with basis vectors $a$(1, 0, 0) and $a$($\frac{1}{2},-\frac{\sqrt{3}}{2},0$) in the $xy$ plane where $a = 24.63$[Å]{}. In addition, a vacuum space of size $a$ is applied along the $z$-axis to avoid the interactions between periodic images in the lattice. All structures are fully relaxed until the forces on each atom become smaller than 0.06 eV [Å]{}$^{-1}$. The cut-off energy is set to 300 Ry and the Brillouin zone sampling is performed by the Monkhorst-pack mesh of $\bf k$-points. A mesh of $(10 \times 10 \times 2)$ has been adopted for discretization of ${\bf k}$-points. The spin-polarized calculations are performed to obtain the total magnetic moment. Also, the Mulliken population analysis is used in the calculation of local magnetic moment of each carbon atom.
The adsorption energy, $E_{ads}$, of a single gas molecule is calculated using the following formula [@dam39] $$\label{eq1}
E_{ads} = E_{\mathrm{GAL+molecule}}- (E_{\mathrm{GAL}}- E_{\mathrm{molecule}})\ ,$$ where $E_{\mathrm{GAL+molecule}}$ is the optimized energy of the GAL in the presence of adsorbed molecule, $E_{\mathrm{GAL}}$ and $E_{\mathrm{molecule}}$ are the optimized energies of the pristine GAL and the isolated molecule, respectively [@dam39; @tay40]. It is worth mentioning that our calculations are limited to zero temperature. However, we should take the entropy corrections into account, when the gas adsorption is investigated at finite temperature.
Note that the SIESTA code calculates the Mulliken charge population for every single atom in the gas molecule before and after the adsorption. Therefore, the charge population determines whether the adsorbed molecule turns the system as a $p$-type (acceptor) or $n$-type (donor) semiconductor [@oli41]. The adsorption can be classified into physisorption (week van der Waals interaction) or chemisorption (covalent bonding). In general, the chemisorption is defined when the absolute value of adsorption energy is greater than 0.2 eV and the distance between the adsorbate and adsorbent surface is not large, otherwise the process is called physisorption [@XChen2017; @tin30]. Here, we use the same definition for both small and large adsorbates.
Results
=======
We start with the small adsorbates H$_{2}$O, CO, NH$_{3}$ and NO$_{2}$ on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices. Each of these molecules are placed in the middle of the respective triangular hole created in the $10 \times10$ superlattices. Then the structures are optimized by means of the procedure explained above. From the optimized structures we found that the $[10,5]_{ETA}$ lattice does not exhibit any structural distortion in the presence of gas molecules. In other words, the small gas molecules are not able to bend or distort the pristine ETA lattices, and as a result, the corresponding lattice remains flat. The optimized superlattices before and after introduction of a small gas adsorbate are shown in Figs. 2(a1)-(a4) and 2(a6). The same procedure happens for the $[10,3,6]_{RTA}$ lattice before and after introduction of H$_{2}$O molecules as shown in Fig. 2(b1) and (b2), while the structure exhibits a small distortion in the presence of CO molecule, as depicted in Fig. 2(b3). On the other hand, the optimized structures with other adsorbed molecules show a considerable out of plane distortion as can be seen in Fig. 2(a7)-(a9) and also Fig. 2(b4)-(b7). Such distortions in RTA lattices may happen due to the lack of structural symmetry, and also, as a result of the introduction of large molecules in both RTA and ETA lattices. The large gas molecules are placed over the hole at the distance of 3[Å]{} from the plane of GAL. The ETA lattices possess zigzag edges only, while the RTA structures have the mixed armchair and zigzag edges. Therefore, the adsorbates in $[10,3,6]_{RTA}$ lattice move closer to the atoms of the hole edges compared to the $[10,5]_{ETA}$ lattice. Note that in Fig. 2(a5), we have also shown the optimized structure of $[10,5]_{ETA}$ lattice with two adsorbed NH$_{3}$ molecules, causing a considerable out of plane distortion.
In Table \[Table1\], the values of magnetic moment, adsorption energy, energy gaps, and the type of adsorbate are listed for both RTA and ETA lattices in the presence of small and large gas molecules. The adsorption energy of all structures is negative indicating that all optimized structures are stable. We discuss the effect of each adsorbate over GAL in details below.
Structure $~M_{t}(\mu_B)~$ $~~~E_{ads}$(eV) $E_{gap}^{\uparrow}$(eV) $E_{gap}^{\downarrow}$(eV) $M_{a}(\mu_{B})$ Adsorbate type
---------------------------------------- ------------------ -------------------- -------------------------- ---------------------------- ------------------ ----------------
$[10,3,6]_{RTA}$ 2.954 0.517 0.550
$[10,3,6]_{RTA}$+H$_{2}$O 2.943 -0.523 0.512 0.543 0 Donor
$[10,3,6]_{RTA}$ + CO 2.965 -1.730 0.532 0.555 0 Donor
$[10,3,6]_{RTA}$ + NH$_{3}$ 2.974 -2.745 0.529 0.542 0 Donor
$[10,3,6]_{RTA}$ + NO$_{2}$ 2.723 -3.855 0.168 1 Acceptor
$[10,3,6]_{RTA}$ + TCNQ 1.012 -5.149 0.234 0 Acceptor
$[10,3,6]_{RTA}$ + F4TCNQ 3.003 -5.313 0.485 0 Acceptor
$[10,5]_{ETA}$ 4.984 0.793 0.802
$[10,5]_{ETA}$ + H$_{2}$O 4.985 -0.151 0.790 0.798 0
$[10,5]_{ETA}$ + CO 4.986 -0.428 0.794 0.815 0 Donor
$[10,5]_{ETA}$ + NH$_{3}$ 4.985 -0.101 0.789 0.799 0
$[10,5]_{ETA}$ + 2NH$_{3}$ 4.993 -4.880 0.804 0.823 0 Donor
$[10,5]_{ETA}$ + NO$_{2}$ 5.291 -0.931 0.572 1 Acceptor
$[10,5]_{ETA}$ + TCNQ 3.496 -5.105 0.245 0 Acceptor
$[10,5]_{ETA}$ + F4TCNQ 4.985 -5.449 0.680 0 Acceptor
$[10,5]_{ETA}$ + F4TCNQ$_{90^{\circ}}$ 3.040 -5.614 0.281 0 Acceptor
\[Table1\]
H$_{2}$O on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices
---------------------------------------------------------
The optimized structure of $[10,3,6]_{RTA}$ supercell in the presence of a single water molecule revealed that the molecule is located inside the hole at the distance of 1.97[Å]{} from the edge atoms and that the calculated adsorption energy is -0.523 eV. On the contrary, the adsorption energy of a single H$_{2}$O on $[10,5]_{ETA}$ structure is -0.151 eV and the molecule is located at the distance of 2.72 [Å]{} from the edge atoms (see Table 1). This shows that the binding interaction between water molecule and $[10,3,6]_{RTA}$ structure is much stronger than that between H$_{2}$O and $[10,5]_{ETA}$ lattice. Using the Mulliken charge analysis, one can conclude that H$_{2}$O molecule in $[10,3,6]_{RTA}$ lattice acts as a donor, while the amount of charge transfer from $[10,5]_{ETA}$ lattice is very small. As shown in Figs. 3(b) and 4(b), the band structure of both lattices is not affected by adsorption of water molecules due to the weak bonding between H$_{2}$O and carbon atoms of the GAL, in agreement with the intrinsic hydrophobicity of graphene and graphitic surfaces [@Hong2016; @LiuLi2017]. The water molecule adsorption on the antidot lattices does not change the total magnetic moment of the respective pristine lattices, as given in Table. 1. Moreover, the distribution of charge densities remains unchanged if one compares Fig. 2(a2) with (a1) and Fig. 2(b2) with (b1).
It is worth mentioning that H$_{2}$O molecule over graphene surface acting as an acceptor results an adsorption energy of -0.047 eV with the bond length of 3.5[Å]{} [@lee24]. The water adsorption on antimonene (InS) which also behaves as an acceptor (acceptor) gives the adsorption energy of -0.20 eV (-0.17 eV) with the bond length of 2.88 [Å]{} (2.37 [Å]{}) [@oli41; @ant-and]. On the contrary, the binding energy of H$_{2}$O over phosphorene is -0.14 eV with the bond length of 2.71[Å]{} and for this case the molecule acts as a donor [@p-cai].
CO on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices
---------------------------------------------------
The adsorption of carbon monoxide molecule, CO, on the GALs results the adsorption energies of -1.730 eV in $[10,3,6]_{RTA}$ and -0.428 eV in $[10,5]_{ETA}$ lattices, suggesting that CO molecule is chemisorbed by both lattices (see Table I). Also, the molecule is located at the distance of 2.25[Å]{} and 2.67[Å]{} from the antidot edges of $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices, respectively. Figures 3(c) and 4(c) show that the adsorption of CO molecule has a negligible effect on the gap and energy bands of the GALs.
Our Mulliken charge analysis showed that CO molecule acts as a donor in both GAL structures, that is the same as graphene and phosphorene [@lee24; @p-cai]. Nevertheless, the binding energy of the molecule at the distance of 3.74[Å]{} (3.06[Å]{}) above pristine graphene (phosphorene) is -0.014 eV (-0.31 eV) [@lee24; @p-cai]. For comparison, its adsorption energy above InS (antimonene) at the distance of 3.08[Å]{} (3.72[Å]{}) is -0.13 eV (-0.12 eV) [@in-cai; @ant-and]. However, the charge transfer between the molecule and both InS and antimonene surfaces is very small [@in-cai; @ant-and]. The distribution of charge densities is not affected by the adsorption of CO molecule on both GAL structures. This can be confirmed by comparing the blue and red regions of Fig. 2(a3) with 2(a1), and that of 2(b3) with 2(b1).
![ Spin-dependent band-structure of (a) pristine $[10,3,6]_{RTA}$ lattice and the same structure after adsorption of the small gas molecules (b) H$_{2}$O, (c) CO, (d) NH$_{3}$, and (e) NO$_{2}$.[]{data-label="image3"}](Fig3.pdf){width="0.98\linewidth"}
![Spin-dependent band-structure of (a) pristine $[10,5]_{ETA}$ lattice and the same structure after adsorption of the small gas molecules (b) H$_{2}$O, (c) CO, (d) NH$_{3}$, and (e) NO$_{2}$.[]{data-label="image4"}](Fig4.pdf){width="0.98\linewidth"}
NH$_{3}$ on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices
---------------------------------------------------------
Our ab initio calculations showed that the adsorption energy and the position of NH$_{3}$ molecule on $[10,3,6]_{RTA}$ are -2.75 eV and 2.192[Å]{}, while these values are -0.1 eV and 2.05[Å]{} for $[10,5]_{ETA}$ lattice (see Table I). This means that NH$_{3}$ molecule is chemisorbed by $[10,3,6]_{RTA}$ structure, while it is physisorbed by $[10,5]_{ETA}$ lattice. Again, the adsorption of NH$_{3}$ molecule does not change the energy bands of the GALs, as can be seen in Figs. 3(d) and 4(d).
The Mulliken charge analysis showed that the amount of charge transfer between NH$_{3}$ molecule and $[10,5]_{ETA}$ is very small but it is considerable in the case of $[10,3,6]_{RTA}$ lattice where the molecule acts as a donor. This situation is the same as NH$_{3}$ adsorption over graphene and phosphorene [@lee24; @p-cai]. It has been reported that the adsorption energy of NH$_{3}$ molecule at the distance of 3.81[Å]{} (2.59[Å]{}) above graphene (phosphorene) surface is -0.031 eV (-0.18 eV) and the molecule acts as a donor [@lee24; @p-cai]. On the other hand, the adsorption energy of the molecule at 2.49[Å]{} (3.41[Å]{}) above InS (antimonene) is -0.20 eV (-0.12 eV) and the molecule acts as an acceptor [@in-cai; @ant-and].
In the case of small adsorbates, we have also examined the adsorption effect of two NH$_{3}$ molecules on $[10,5]_{ETA}$ holes by optimizing the total structure (see Fig. 2(a5)). The Mulliken charge analysis showed that the tendency of the molecules for charge transfer becomes smaller, while the adsorption energy increases significantly. Note that the gap value and the energy bands (not shown here) are not considerably affected by the presence of two NH$_{3}$ molecules.
NO$_{2}$ on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices
---------------------------------------------------------
Among the various small adsorbates that we consider here, NO$_{2}$ molecule exhibits a nonzero magnetic moment in both GALs (see table \[Table1\]). The distributions of charge densities in $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices and also in the adsorbed molecule are shown in Fig. 2(a6) and (b5). The total magnetic moment is maximum in $[10,5]_{ETA}$ lattice while the molecule causes a reduction in the total magnetic moment of $[10,3,6]_{RTA}$ lattice. The adsorption energy and the position of NO$_{2}$ molecule on $[10,3,6]_{RTA}$ lattice are -3.855 eV and 2.03[Å]{}, while these values are -0.931 eV and 2.66[Å]{} for $[10,5]_{ETA}$ lattice. The energy values predict that the molecule is chemisorbed by both GALs.
As shown in Figs. 3(d) and 4(d), the Fermi energy crosses the spin-up energy bands of both GALs, whereas it falls in the band gap of spin-down band-structure with the energy gap value $\sim 0.168$ eV for $[10,3,6]_{RTA}$ and $\sim 0.572$ eV for $[10,5]_{ETA}$. This suggests that adsorbing NO$_{2}$ molecules by the GALs may turn the semiconductor characteristics of these systems into a half-metallicity behavior. The energy gap of $[10,5]_{ETA}$ lattice is more than three times greater than that of $[10,3,6]_{RTA}$ lattice. Therefore, in the presence of adsorbed NO$_{2}$ molecule, it will be more easier for electrons to make a transition from valence band of the $[10,3,6]_{RTA}$ lattice compared to $[10,5]_{ETA}$ structure.
In the context of half-metallicity, the energy gap and the electronic structure which are related to each other, play an important role. Therefore, to ensure that the half-metallicity behavior, seen in the GALs in the presence of NO$_{2}$ molecules and obtained by PBE (GGA) functional is reliable enough, we have also performed DFT+$U$ calculations for NO$_{2}$ adsorption over $[10,5]_{ETA}$ structure with several $U$ values for oxygen atoms. Our results showed that although some energy bands are shifted in energy as $U$ parameter changes, the energy gap value and also the energy bands around the Fermi energy are not affected by on-site Coulomb correction. Note that the GGA+$U$ method can be applied to any states which are too high (or shallow) in energy, too delocalized (due to the GGA-inherent self-interaction error), and hence, too metallic (the gap closes or becomes negative) [@Keating2012].
In addition, the shift of Fermi level into the valence band of spin-up electrons indicates that NO$_{2}$ may act as a $p$-type impurity in GALs, similar to the case of graphene [@tay40]. However, the spin-down band-structures demonstrate a mid-gap state (flat band), induced by NO$_{2}$ molecule. The mid-gap state is an unoccupied state at 0.014 eV above the Fermi level in the $[10,5]_{ETA}$ lattice, whereas it is an occupied state at 0.014eV below the Fermi level in the $[10,3,6]_{RTA}$ lattice. Such a flat band has also been reported for the case of NO$_{2}$ molecule on monolayer of MoS$_{2}$ [@yue] and tetragonal GaN [@yon]. Therefore, the $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices can act as viable NO$_{2}$ detectors. Note that the acceptor action of NO$_{2}$ molecules on GALs, has also been predicted for this adsorbed molecule on graphene ($d=3.38$[Å]{} and $E_{ads}=-0.067$ eV) [@lee24], antimonene ($d= 2.44$[Å]{}, $E_{ads}=-0.81$ eV) [@ant-and], phosphorene ($d=2.27$[Å]{} and $E_{ads}=-0.51$ eV) [@p-cai], and InS ($d=2.71$[Å]{} and $E_{ads}=-0.24$ eV) [@in-cai].
Note that to see to what extent the van der Waals (vdW) forces can affect the electronic structure of GALs in the presence of gas molecules, we have also included the vdW corrections in our band structure calculations of NO$_{2}$ molecule on the GALs. Our results showed that the energy bands (not given here) around the Fermi energy and the band gap values obtained from GGA calculations are not affected by the inclusion of vdW interactions, originated from quantum fluctuations of electric charge.
![Spin-dependent band-structure of $[10,3,6]_{RTA}$ lattice before (grey curves) and after (blue and red curves) adsorbing the large molecules (a) TCNQ and (b) F4TCNQ. Blue (red) curves represent the energy bands of spin-up (spin-down) electrons.[]{data-label="image5"}](Fig5.pdf){width="0.68\linewidth"}
![ Spin-dependent band-structure of $[10,5]_{ETA}$ lattice before (grey curves) and after (blue and red curves) adsorbing the large molecules (a) TCNQ, (b) F4TCNQ, and (c) F4TCNQ$_{+90^{\circ}}$. Blue (red) curves represent the energy bands of spin-up (spin-down) electrons.[]{data-label="image6"}](Fig6.pdf){width="0.95\linewidth"}
TCNQ on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices
-----------------------------------------------------
The TCNQ molecule is a large organic adsorbate that is known as a strong electron acceptor. We obtained the energy adsorption of the molecule on the $[10,3,6]_{RTA}$ lattice $E_{ads}= -5.149$ eV and the shortest distance between the molecule and the GAL $d=2.17$[Å]{}, while the values are -5.105 eV and 2.07[Å]{} for the $[10,5]_{ETA}$ structure, as given in Table I. We have depicted in Figs. 5(a) and 6(a) the band-structures of GALs after adsorption of TCNQ. As shown in the Table \[Table1\], the adsorbed molecule reduces the total magnetic moment of both structures, compared to the relative pristine lattices. The reduction in the magnetic moment of the $[10,5]_{ETA}$ structure is less than that of the $[10,3,6]_{RTA}$ lattice. The adsorbed molecule induces several flat bands in the energy gap of both systems with a shift towards their valence band. This suggests the TCNQ molecule as a $p$-type impurity which can act as an acceptor in the GALs. The spin-up energy bands cross the Fermi level, while the spin-down band structures exhibit the energy gaps of 0.234 eV and 0.245 eV in $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices, respectively. This behavior shows that the adsorption of TCNQ molecule by GALs may transform these semiconductors into half metals.
In this regard, Yang et al. [@yan] showed that the adsorption energy of TCNQ molecule on graphene varies from -1.19 eV to -1.53 eV by changing the calculation method and the orientation of adsorbate. Also, its adsorption energy over phosphorene and Al(100) reported about -0.97 eV [@yu] and -3.66 eV [@vin], respectively. In these cases, the TCNQ molecule also acts as a strong acceptor.
F4TCNQ on $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices
------------------------------------------------------
The F4TCNQ molecule is considered as an excellent $p$-type dopant on graphene surface [@wee] with adsorption energy ranging from -1.42 eV to -1.81 eV, depending on the calculation methods and the change in orientation of the molecule [@yan]. From optimized structures, we obtained $E_{ads}=-5.313$ eV and $d=2.13$[Å]{} in $[10,3,6]_{RTA}$, while $E_{ads}=-5.449$ eV and $d=1.99$[Å]{} in $[10,5]_{ETA}$ lattice. This means that F4TCNQ molecule can be considered as a stronger adsorbate in GALs, compared to TCNQ molecule. The band structures of both GALs in the presence of F4TCNQ are shown in Figs. 5(b) and 6(b). The Fermi energy of both lattices crosses the spin-up bands, while it falls in the gap of spin-down band structures with energy of 0.485 eV in $[10,3,6]_{RTA}$ lattice and 0.680 eV in $[10,5]_{ETA}$ lattice. Note that in both lattices the total magnetic moment is strong (see Table \[Table1\] and also distribution of charge densities in Fig. 2(a8) and (b7)).
In the case of $[10,5]_{ETA}$ lattice, we have also considered the effect of 90$^{\circ}$ rotation of the molecule above the antidot, as shown in Fig. 2(a9). This rotation which is parallel to the GAL plane results $E_{ads}=-5.614$ eV and $d=2.36$[Å]{}. Comparing $E_{ads}$ values before and after 90$^{\circ}$ rotation of the molecule, we find that such a rotation can slightly increase the adsorption energy while it decreases the total magnetic moment considerably, as can be predicted from spin-dependent charge densities of Fig. 2(a9). The degeneracy of energy bands is broken slightly in the vicinity of the Fermi energy and a flat band is shifted above (below) the Fermi energy in the spin-up (spin-down) band structure, as depicted in Fig. 6(c). Nevertheless, the electronic structure of the system is not considerably affected by rotation of the molecule. The system is conductive for spin-up electrons, whereas it acts as a semiconductor with the energy gap of 0.281 eV for spin-down electrons.
Therefore, the above features of GALs in the presence of large adsorbed molecules suggest a half-metallic nature with metallic behavior in one spin channel and insulating in the other. The energy bands of GALs are shifted towards higher energies by these molecules. In addition, the electronic and magnetic properties of these structures can be controlled by the type and orientation of the large molecules. The appropriate molecule can be selected by considering the required energy gap and the total magnetic moment.
conclusion
==========
In this paper we have studied the electric and magnetic properties of $[10,3,6]_{RTA}$ and $[10,5]_{ETA}$ lattices before and after adsorbing small and large gas molecules using DFT calculations, implemented in the SIESTA package with periodic boundary conditions. Our findings show that NH$_{3}$ is physisorbed (chemisorbed) by $[10, 5]_{ETA}$ ($[10,3,6]_{RTA}$) lattice, while CO, NO$_{2}$, TCNQ, and F4TCNQ molecules are chemisorbed by both lattices. The binding interaction between water molecule and $[10,3,6]_{RTA}$ lattice is stronger than that between H$_{2}$O and $[10,5]_{ETA}$ structure. Although the electronic and magnetic properties of these GALs are not considerably affected by H$_{2}$O, CO, NH$_{3}$ molecules, the NO$_{2}$ adsorbate modulates the energy gap by transforming the electronic structure from semiconducting to half-metal nature. Also, the adsorption of TCNQ and F4TCNQ molecules on these lattices can turn the system into a half-metallicity as a result of shifting the energy bands around the Fermi energy. Our calculations suggest that the adsorption of NO$_{2}$ molecules with maximum induced magnetic moment, can be a promising candidate to design nanoscale spintronic devices and gas sensors, based on triangular graphene antidot lattices.
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abstract: 'We have theoretically investigated two-band models of graded-gap superlattices within the envelope-function approximation. Assuming that the gap varies linearly with spatial coordinate, we are able to find exact solutions of the corresponding Dirac-like equation describing the conduction- and valence-band envelope-functions. The dispersion relation inside allowed miniband of the superlattice may be expressed in terms of confluent hypergeometric functions in a closed form.'
address: 'Departamento de Física de Materiales, Universidad Complutense, 28040 Madrid, Spain'
author:
- 'B. Méndez and F. Domínguez-Adame'
title: 'Exact solutions of two-band models of graded-gap superlattices'
---
= 10000
During the last years, graded-gap superlattices have been the subject of very detailed investigations as interesting materials for device applications [@Capasso; @Brum]. The graded doping creates a modulation of both conduction- and valence-bands, which may be approximated by a sawtooth potential. The miniband structure can be obtained within the envelope-function approximation [@Bastard], the system being usually described by a scalar Hamiltonian (Schrödinger-like) corresponding to decoupled bands. However, this approach cannot adequately describe those graded-gap superlattices whose band modulation is comparable to the magnitude of the gap, and a more realistic band structure is essential to properly describe the electronic structure. In this paper we calculate the miniband structure of graded-gap superlattices within a two-band model, which is known to be valid in a large variety of semiconductor superlattices where the coupling of bands is not negligible, as occurs in some narrow-gap III-V compounds (InAs, InSb, GaSb).
We obtain the miniband structure in the superlattice by means of the effective-mass $\bf k\cdot p$ approximation. There are two coupled envelope-functions describing the conduction-band and valence-band states of the semiconductor, subject to an effective $2\times 2$ Dirac-like equation along the growth direction $z$ $$\left( -i\hbar v \sigma_x\partial +{1\over 2}E_g(z)\sigma_z-E \right)
\left( \begin{array}{c} f_c (z)\\ f_v(z)\end{array} \right) =0,
\label{Dirac}$$ where $\partial =d/dz$, $\sigma_x$ and $\sigma_z$ are Pauli matrices, and $E_g(z)$ stands for the position-dependent gap in the two-band semiconductor superlattice. The spatial periodicity of the lattice implies that $E_g(z+L)=E_g(z)$, $L$ being the period of the superlattice. We assume that the centre of the gap remains unchanged when doping; this simplifies calculations and is a good approximation in several cases (for instance, in GaAs-Ga$_{1-x}$Al$_x$As the centre of the gap varies only 10% of the gap difference in both materials). The velocity $v=(E_g/2m^*)^{1/2}$ is almost constant in direct gap III-V semiconductors, and we will assume this constancy hereafter. In graded-gap structures the gap varies linearly with position so that we can write $$E_g(z)=E_{g0}+(E_{gL}-E_{g0})\left( \frac{z}{L} \right)
\equiv E_{g0}+\Delta E_g \left( \frac{z}{L} \right),
\label{gap}$$ where $E_{g0}=E_g(0)$ and $E_{gL}=E_g(L)$ for $0<z<L$. Note that $E_g(z)$ is equivalent to a relativistic scalar-like potential in the Dirac theory, that is to say, Eq. (\[Dirac\]) is analogous to the Dirac equation for a relativistic particle with a position-dependent mass. We exploit this analogy to find the exact solutions of Eq. (\[Dirac\]) for the graded-gap (\[gap\]). It is well-known that the Dirac equation for linear scalar-like potentials admits exact solutions (see Refs. and references therein) and we can use a similar method to solve Eq. (\[Dirac\]). Therefore we define $$\left( \begin{array}{c} f_c (z)\\ f_v(z)\end{array} \right) =
\left( -i\hbar v \sigma_x\partial +{1\over 2}E_g(z)\sigma_z+E \right)
\left( \begin{array}{c} i\phi (z)\\ \phi(z)\end{array} \right),
\label{phi}$$ and inserting (\[phi\]) in (\[Dirac\]) one obtains the equation for the function $\phi (z)$ as $$\left[ -\,\hbar^2 v^2 \partial^2 + {1\over 4} E^2_g(z) -E^2
-\, \left( {\hbar v \Delta E_g \over 2L} \right)\right] \phi(z)=0.
\label{kg}$$
The equation (\[kg\]) may be reduced to a standard form, the equation of the parabolic cylinder, by making the change of parameters according to
\[change\] $$\begin{aligned}
x&=&\sqrt{L\over \hbar v \Delta E_g}\, \left( E_{g0}+
\Delta E_g {z\over L} \right), \label{changea}\\
\eta &=& \frac{E^2L}{\hbar v \Delta E_g}. \label{changeb}\end{aligned}$$
On making these substitutions one gets $$\frac{d^2\phi (x)}{d\,x^2} + \left( -\,{x^2\over 4}+\eta+{1\over 2}
\right) \phi (x)=0,
\label{parabol}$$ whose two independent solutions are parabolic cylinder functions $D_\eta
(x)$ and $D_\eta(-x)$. Using Eq. (\[phi\]) we find that the envelope-functions in the conduction- and valence-bands can be cast in the matrix form $$\left( \begin{array}{c} f_c (z)\\ f_v(z)\end{array} \right) =
{\bf D}[x(z)] \left( \begin{array}{c} A\\ B\end{array} \right),
\hspace{1cm} 0<z<L,
\label{result}$$ where $A$ and $B$ are arbitrary constants and the $2\times 2$ matrix ${\bf D}[x(z)]$ is written out explicitly as $${\bf D}[x(z)]= \left[ \begin{array}{cc}
-i\{ D_\eta(x)-\sqrt{\eta} D_{\eta-1}(x) \}&
-i\{ D_\eta(-x)+\sqrt{\eta} D_{\eta-1}(-x) \}\\
D_\eta(x)+\sqrt{\eta} D_{\eta-1}(x) &
D_\eta(-x)-\sqrt{\eta} D_{\eta-1}(-x) \end{array} \right].
\label{matrix}$$
Once the general solution of the Dirac equation (\[Dirac\]) is obtained, appropriate boundary conditions should be used to find eigenenergies. We assume the continuity of the envelope-functions at the interface $z=L$, namely, $$\left( \begin{array}{c} f_c (L^-)\\ f_v(L^-)\end{array} \right) =
\left( \begin{array}{c} f_c (L^+)\\ f_v(L^+)\end{array} \right),
\label{conti}$$ along with the Bloch condition in the growth direction $$\left( \begin{array}{c} f_c (L)\\ f_v(L)\end{array} \right) =
\exp(ikL)\, \left( \begin{array}{c} f_c(0)\\ f_v(0)\end{array} \right),
\label{Bloch}$$ where $k$ denotes the component of the momentum along the growth direction $z$. By means of the general solution (\[result\]) we can find the dispersion relation as $$\cos kL = {1\over 2} \mbox{Tr} \left( {\bf D}^{-1}(x_L) {\bf D}(x_0)
\right) \label{tr}$$ where for brevity we have defined $$x_0 = \sqrt{E^2_{g0}L \over \hbar v \Delta E_g}, \hspace{1cm}
x_L = \sqrt{E^2_{gL}L \over \hbar v \Delta E_g}.
\label{x}$$ Finally, using the relationship between parabolic cylinder functions and the confluent hypergeometric functions $M(\alpha,\beta;t)$ [@Abra], it is straightforward although somewhat tedious to demonstrate that the dispersion relation can be expressed as $$\begin{aligned}
\cos kL&=& {e^{-(x_0^2+x_L^2)/4}\over 2}
\left\{
M(-\, {\eta \over 2}, {1\over 2}; {x_0^2\over 2})
M({1-\eta \over 2}, {1\over 2}; {x_L^2\over 2})
+ M(-\, {\eta \over 2}, {1\over 2}; {x_L^2\over 2})
M({1-\eta \over 2}, {1\over 2}; {x_0^2\over 2}) \right. \nonumber \\
+\eta x_0 x_L &&\hspace{-5mm} \left. \left[
M({1-\eta \over 2}, {3\over 2}; {x_0^2\over 2})
M(1-{\eta \over 2}, {3\over 2}; {x_L^2\over 2})+
M({1-\eta \over 2}, {3\over 2}; {x_L^2\over 2})
M(1-{\eta \over 2}, {3\over 2}; {x_0^2\over 2})\right] \right\}.
\label{porfin}\end{aligned}$$ Whenever the absolute value of the right-hand-side of this equation is less than unity, a real value of $k$ is found and hence the dispersion relation inside allowed minibands is obtained. Conversely, if the absolute value is larger than unity, the energy corresponds to a minigap of the superlattice.
As a specific example we have considered graded structures with $E_{g0}=0.18\,$eV (corresponding to InSb), $E_{gL}=0.27\,$eV, $\hbar
v=0.70\,$eVnm and superlattice periods $L$ ranging from $5\,$nm up to $40\,$nm. Results of the allowed minibands and minigaps as a function of the lattice period are shown in Fig. \[fig1\]. Note that allowed minibands shrink on increasing superlattice period due to the reduction of the overlap of neighbouring cells.
In conclusion, we have described theoretically the miniband structure in graded-gap superlattices within a two-band semiconductor model, that is, we have taking into account the coupling between the conduction- and valence-bands in the host semiconductor. Assuming that the gap increases linearly with position, we are able to solve exactly the $2\times 2$ Dirac-like equation of the model. The dispersion relation inside allowed minibands may be expressed in a closed form in terms of the confluent hypergeometric functions.
F. Capasso, S. Lurti, W. T. Tsang, C. G. Bethea, and B. F. Levine, Phys. Rev. Lett. [**51**]{}, 2318 (1983).
J. A. Brum, P. Voisin, and G. Bastard, Phys. Rev. B [**33**]{}, 1063 (1986).
G. Bastard, Phys. Rev. B [**24**]{}, 5693 (1981). Rev. B [**40**]{}, 6420 (1989).
F. Domínguez-Adame and M. A. González, Europhys. Lett. [**13**]{}, 193 (1990).
F. Domínguez-Adame and B. Méndez, Il Nuovo Cim. B [**107**]{}, 489 (1992).
M. Abramowitz and I. Stegun (ed), [*Handbook of Mathematical functions*]{} (Dover, New York, 1972).
|
---
abstract: 'To support the validity of a factorizable metric ansatz used in string cosmology, we investigate a toy problem in RSI model. For this purpose, we revise the gradient expansion method to conform to the factorizable metric ansatz. By solving the 5-dimensional equations of motion and substituting the results into the action, we obtain the 4-dimensional effective action. It turns out that the resultant action is equivalent to that obtained by assuming the factorizable metric ansatz. Our analysis gives the support of the validity of the factorizable metric ansatz.'
author:
- Sugumi Kanno
- Jiro Soda
title: On the Validity of a Factorizable Metric Ansatz in String Cosmology
---
Introduction
============
Results from WMAP strongly support the idea of the inflationary universe. Hence, it is an urgent matter to construct an inflaton potential that agrees with observations based on the fundamental theory such as the string theory. Recent intense researches on inflationary models in string theory have stemmed from the success to construct brane inflation with moduli stabilization [@Kachru:2003aw]. However, almost all studies computing potentials for moduli in type IIB string theory suppose a factorizable ansatz for the ten dimensional metric: $$\begin{aligned}
ds^2 = e^{2\omega (y) } g_{\mu\nu} (x) dx^\mu dx^\nu
+ e^{-2\omega (y) + 2u(x) }{\tilde g}_{ab} (y) dy^a dy^b\end{aligned}$$ where ${\tilde g}_{ab}(y)$ is the metric on the internal Calabi-Yau manifold, $\omega (y)$ is the warp factor, and $u(x)$ represents the volume modulus. This ansatz is based on the static solution [@Giddings:2001yu] $$\begin{aligned}
ds^2=e^{2\omega (y)}\eta_{\mu\nu}dx^\mu dx^\nu
+e^{-2\omega (y)}{\tilde g}_{ab}(y)dy^ady^b \ .\end{aligned}$$ It should be stressed that, in the presence of the moving branes, no proof for the factorizable ansatz exists even at low energy. If this ansatz is not correct, any conclusion derived using it is not reliable. In fact, recently, this ansatz is challenged by de Alwis [@deAlwis:2004qh]. As this factorizable ansatz for the metric is crucial in the discussion of the D-brane inflation, it is important to examine its validity.
In order to investigate the validity of this ansatz, as a modest step, we focus on the Randall-Sundrum (RS) I model [@RS1]. Here, we also have a similar problem. In considering the cosmology, if we follow the factorizable metric ansatz, what we should do is to replace the Minkowski metric in a static solution $$\begin{aligned}
ds^2 = a^2 (y) \eta_{\mu\nu} dx^\mu dx^\nu +dy^2
\ , \quad a(y)=e^{-y/\ell} \end{aligned}$$ with a spacetime dependent metric $g_{\mu\nu}(x)$ as $$\begin{aligned}
ds^2 = a^2 (y) g_{\mu\nu} (x) dx^\mu dx^\nu
+ {\cal G}_{yy}(x) dy^2 \ ,\end{aligned}$$ where we also included the modulus field ${\cal G}_{yy}$. The question is the validity of this assumption in the context of the brane cosmology. In this paper, we derive the four dimensional low energy effective action on the brane without using this ansatz. Then, we compare the result with the effective action derived using the factorizable ansatz to examine its validity.
Organization of this paper is as follows. In Sec.II, we present our strategy to attack the issue. In Sec.III, we solve the bulk equations of motion using the revised gradient expansion method. In the Sec.IV, we derive the 4-dimensional effective action and discuss the validity of the factorizable metric ansatz. The final section is devoted to the conclusion.
How to justify the metric ansatz?
=================================
We consider an $S_1/Z_2$ orbifold spacetime with the two branes as the fixed points. In the RSI model, the two flat 3-branes are embedded in the 5-dimensional asymptotically anti-deSitter (AdS) bulk with the curvature radius $\ell$ with brane tensions given by $\sigma_+=6/(\kappa^2\ell)$ and $\sigma_-=-6/(\kappa^2\ell)$. The model is described by the action $$\begin{aligned}
S&=&{1\over 2\kappa^2}\int d^5x\sqrt{-\cal G}
\left[{\cal R}+{12\over\ell^2}\right]\nonumber\\
&& -{6\over\kappa^2\ell}\int d^4x\sqrt{-g_+}
+{6\over\kappa^2\ell}\int d^4x\sqrt{-g_-}\nonumber\\
&& +{2\over\kappa^2}\int d^4x\sqrt{-g_+}{\cal K}_+
-{2\over\kappa^2}\int d^4x\sqrt{-g_-}{\cal K}_-\ ,
\label{action:5-dim}\end{aligned}$$ where $\kappa^2$ is the five-dimensional gravitational coupling constant and ${\cal R}$ is the curvature scalar. We denoted the induced metric on the positive and negative tension branes by $g_{\mu\nu}^+$ and $g_{\mu\nu}^-$, respectively. In the last line, we have taken into account the Gibbons-Hawking boundary terms instead of introducing delta-function singularities in the curvature. The factor 2 in the Gibbons-Hawking term comes from the $Z_2$ symmetry of this spacetime. ${\cal K}_\pm$ is the trace part of the extrinsic curvature of the boundary near each one of the branes.
Here, the question is how to obtain the effective action for discussing the cosmology. One often takes the metric ansatz and substitutes the ansatz into the action to get the 4-dimensional effective action. Let us assume that the metric is factorizable $$\begin{aligned}
ds^2= a^2 (y) g_{\mu\nu}(x) dx^\mu dx^\nu + {\cal G}_{yy}(x) dy^2
\label{factor1}\end{aligned}$$ and the branes are located at the fixed coordinate points. Substituting this metric into the action and integrating out the result with respect to $y$, we obtain the 4-dimensional action. However, an inadequate restriction of the functional space in the variational problem yields the wrong result. The correct procedure to obtain the 4-dimensional effective action is first to solve the bulk equations of motion and substitute the results into the original action. To solve the bulk equations, we can employ the gradient expansion method [@KS2; @KS; @wiseman; @sugumi]. Our analysis using the gradient expansion method shows that the correct metric takes the form [@KS2]: $$\begin{aligned}
ds^2= a^2 (y \sqrt{{\cal G}_{yy}(x)}) g_{\mu\nu}(x) dx^\mu dx^\nu
+ {\cal G}_{yy}(x) dy^2\end{aligned}$$ which clearly reject the factorizable ansatz (\[factor1\]).
However, there is another possibility. One can assume the following factorizable metric $$\begin{aligned}
ds^2 = a^2 (y) g_{\mu\nu} (x) dx^\mu dx^\nu + dy^2 \ ,\end{aligned}$$ and the branes are moving in the above coordinates. Namely, the positive and negative tension branes are respectively placed at $$\begin{aligned}
y=\phi_+(x),\qquad y= \phi_-(x) \ ,\end{aligned}$$ which are often referred to as the moduli fields. This is another description of the RSI cosmology and often called as moduli approximation in the literature [@Khoury:2001wf]. Although two scalar fields are introduced, one of which is the extra degree of freedom as we will see later in Eq. (\[ST\]). The physical quantity is the difference of these moduli fields which corresponds to the radion in our previous work [@KS2]. This ansatz leads to the action [@Khoury:2001wf] $$\begin{aligned}
S&=& \frac{\ell}{2\kappa^2}\int d^4x\sqrt{-g}
\left[\left\{
a^2(\phi_+)-a^2(\phi_-)\right\}R(g) \right.\nonumber\\
&&\quad
\left.
+\frac{6}{\ell^2}\left\{
a^2(\phi_+)(\partial\phi_+)^2
-a^2(\phi_-)(\partial\phi_-)^2
\right\}\right] \ .
\label{moduliapp}\end{aligned}$$ In this case, we do not have the result to be compared, because the gradient expansion method is not prepared for this parameterization of the model. Our aim is to examine the validity of this ansatz by conforming the gradient expansion method to the factorizable metric ansatz.
Revised Gradient Expansion Method
=================================
The metric we take in solving the bulk equations of motion is the one in the Gaussian normal coordinate system $$\begin{aligned}
ds^2=\gamma_{\mu\nu}(y,x)dx^\mu dx^\nu+dy^2 \ ,\end{aligned}$$ where the factorized metric is not assumed.
Now we give the basic equations in the bulk. When solving the bulk equations of motion, it is convenient to define the extrinsic curvature on the $y = {\rm constant} $ slicing as $K_{\mu\nu}
= - {1\over 2} {\partial \over \partial y}
\gamma_{\mu\nu} $. Decomposing this extrinsic curvature into the traceless part and the trace part $$K_{\mu\nu}=\Sigma_{\mu\nu}+{1\over 4}\gamma_{\mu\nu}K \ , \quad
K = - {\partial \over \partial y}\log \sqrt{-\gamma} \ ,
\label{K}$$ we obtain the basic equations which hold in the bulk; $$\begin{aligned}
&&\Sigma^\mu{}_{\nu ,y}-K\Sigma^\mu{}_{\nu}
=-\left[
R^\mu{}_\nu(\gamma)
-{1\over 4} \delta^\mu_\nu R(\gamma)
\right]
\label{munu-trc} \ , \\
&&{3\over 4}K^2-\Sigma^\alpha{}_{\beta}\Sigma^\beta{}_{\alpha}
=R(\gamma)
+{12\over\ell^2}
\label{munu-trclss} \ , \\
&&\nabla_\lambda\Sigma_{\mu}{}^{\lambda}
-{3\over 4}\nabla_\mu K = 0
\label{ymu} \ ,\end{aligned}$$ where $\nabla_\mu $ denotes the covariant derivative with respect to the metric $\gamma_{\mu\nu}$ and $R^\mu{}_\nu(\gamma)$ is the corresponding curvature.
The effective action have to be derived by substituting the solution of Eqs. (\[munu-trc\])$\sim$(\[ymu\]) into the action (\[action:5-dim\]) and integrating out the result over the bulk coordinate $y$. In reality, it is difficult to perform this general procedure. However, what we need is the low energy effective theory. At low energy, the energy density of the matter, $\rho$, on a brane is smaller than the brane tension, i.e., $\rho /|\sigma| \ll 1$. In this regime, the 4-dimensional curvature can be neglected compared with the extrinsic curvature. Thus, the Anti-Newtonian or gradient expansion method used in the cosmological context [@tomita] is applicable to our problem.
Zeroth Order
------------
At zeroth order, we can neglect the curvature term in Eqs. (\[munu-trc\])$\sim$(\[ymu\]). Moreover, the tension term only induces the isotropic bending of the brane. Thus, an anisotropic term vanishes at this order, $\overset{(0)}{\Sigma}{}^\mu{}_\nu=0$. As the result, we obtain $$\overset{(0)}{K} = {4\over \ell} \quad {\rm or} \quad
\overset{(0)}{K}{}^\mu{}_{\nu} = {1\over\ell}
\delta^{\mu}_{\nu} \ .$$ Using the definition of the extrinsic curvature $$\overset{(0)}{K}{}_{\mu\nu}
= - {1\over 2} {\partial \over \partial y}
\overset{(0)}{\gamma}{}_{\mu\nu} \ ,$$ we get the zeroth order metric as $$ds^2 = dy^2 + a^2 (y) g_{\mu\nu}(x) dx^\mu dx^\nu\ ,
\quad
a(y)=e^{-y/\ell} \ ,$$ where the tensor $g_{\mu\nu}$ is the constant of integration which weakly depends on the brane coordinates $x^\mu$.
First Order
-----------
Our iteration scheme is to write the metric $\gamma_{\mu\nu}$ as a sum of local tensors built out of $g_{\mu\nu}$, with the number of derivatives increasing with the order of iteration, that is, $ O((\ell/L)^{2n})$, $n=0,1,2,\cdots$. Here, $L$ represents the characteristic length scale of the 4-dimensional curvature. Hence, we seek the metric as a perturbative series $$\begin{aligned}
&&\gamma_{\mu\nu} (y,x) =
a^2(y)\left[ g_{\mu\nu} (x)
+f_{\mu\nu}(y,x)
+ \cdots \right] \ .
\label{expansion:metric}\end{aligned}$$ The effective action can be constructed with the knowledge of the leading order metric $f_{\mu\nu}(y,x)$. Other quantities can be also expanded as $$\begin{aligned}
K^\mu{}_{\nu}&=&
{1\over\ell}
\delta^{\mu}_{\nu}
+\overset{(1)}{K}{}^{\mu}{}_{\nu}
+\overset{(2)}{K}{}^{\mu}{}_{\nu}+\cdots \ , \nonumber\\
\Sigma^\mu{}_{\nu}
&=& \ \ 0 \ \
+\overset{(1)}{\Sigma}{}^{\mu}{}_{\nu}
+\overset{(2)}{\Sigma}{}^{\mu}{}_{\nu} + \cdots \ .\end{aligned}$$
The first order solutions are obtained by taking into account the terms neglected at zeroth order. At first order, Eqs. (\[munu-trc\])$\sim$(\[ymu\]) become $$\begin{aligned}
&&\overset{(1)}{\Sigma}{}^{\mu}{}_{\nu , y}
-{4\over\ell} \overset{(1)}{\Sigma}{}^{\mu}{}_{\nu}
=-\left[R^\mu{}_\nu(\gamma)
-{1\over 4} \delta^\mu_\nu R(\gamma)\right]^{(1)}
\label{1:munu} \ , \\
&&{6 \over\ell} \overset{(1)}{K} = \left[~R(\gamma)
~\right]^{(1)}
\label{1:trace} \ ,\\
&&\overset{(1)}{\Sigma}{}_{\mu}{}^{\lambda}{}_{|\lambda}
-{3\over 4}\overset{(1)}{K}{}_{|\mu} = 0
\label{1:ymu}\ .\end{aligned}$$ where the superscript $(1)$ represents the order of the derivative expansion and $|$ denotes the covariant derivative with respect to the metric $g_{\mu\nu}$. Here, $[R^\mu{}_\nu(\gamma)]^{(1)} $ means that the curvature is approximated by taking the Ricci tensor of $a^2(y)g_{\mu\nu}(x)$ in place of $R^{\mu}{}_{\nu}(\gamma)$. It is also convenient to write it in terms of the Ricci tensor of $g_{\mu\nu}$, denoted by $R^\mu{}_\nu (g)$.
Substituting the zeroth order metric into $R(\gamma)$, we can write Eq. (\[1:trace\]) as $$\overset{(1)}{K} = {\ell\over 6a^2} R(g)
\label{1:trc} \ .$$ Hereafter, we omit the argument of the curvature for simplicity. Simple integration of Eq. (\[1:munu\]) also gives the traceless part of the extrinsic curvature as $$\overset{(1)}{\Sigma}{}^{\mu}{}_{\nu}={\ell\over 2a^2}
(R^\mu{}_{\nu}-{1\over 4}\delta^\mu_\nu R)
+{\chi^{\mu}{}_{\nu}(x)\over a^4}
\label{1:trclss} \ ,$$ where $\chi^\mu{}_\nu$ is the constant of integration which satisfies $$\begin{aligned}
\chi^{\mu}{}_{\mu}=0 \ , \quad\chi^{\mu}{}_{\nu|\mu}=0 \ .
\label{TT}\end{aligned}$$ Here, the latter condition came from Eq. (\[1:ymu\]). In our previous work [@KS2] [@KS], we find this term corresponds to the dark radiation at this order. Due to the traceless property, $\chi^\mu{}_{\nu}$ is not relevant to the derivation of the effective action.
From Eqs.(\[1:trc\]) and (\[1:trclss\]), the correction to the metric $g_{\mu\nu}$ at this order can be obtained as $$\begin{aligned}
f_{\mu\nu}(y,x^\mu)&=&-{\ell^2\over 2a^2}
\left( R_{\mu\nu}-{1\over 6} g_{\mu\nu} R \right)
\nonumber\\
&&\qquad\qquad
-{\ell \over 2a^4}\chi_{\mu\nu}+C_{\mu\nu}(x) \ ,\end{aligned}$$ where $C_{\mu\nu}$ is the constant of integration which will be fixed later in Eq. (\[C\]).
Effective Action
================
Now, up to the first order, we have $$\begin{aligned}
g_{\mu\nu}(y,x)=a^2(y)
\left[
g_{\mu\nu}(x)+f_{\mu\nu}(y,x)
\right] \ . \end{aligned}$$ In the following, we will calculate the bulk action, $S_{\rm bulk}$, the actions for each brane, $S_\pm$ and the Gibbons-Hawking term, $S_{\rm GH}$, separately. After that, we collect all of them and obtain the 4-dimensional effective action.
In order to calculate the bulk action, we need the determinant of the bulk metric $$\begin{aligned}
\sqrt{\cal -G}&=&a^4(y)\sqrt{-g}\sqrt{1-\frac{\ell^2}{6a^2}R+C^\mu{}_\mu}
\nonumber\\
&\approx&
a^4(y)\sqrt{-g}\left(
1-\frac{\ell^2}{12a^2}R\right)
\left(1+\frac{C^\mu{}_\mu}{2}\right) \ , \end{aligned}$$ where we neglected the second order quantities. Then the bulk action becomes $$\begin{aligned}
S_{\rm bulk}&\equiv&{1\over 2\kappa^2}\int d^5x\sqrt{\cal -G}
\left[{\cal R}+{12\over\ell^2}\right]
\nonumber\\
&=&-\frac{8}{\kappa^2\ell^2}\int d^4x\sqrt{-g}
\left[\frac{\ell}{4}\left\{
a^4(\phi_+)-a^4(\phi_-)\right\}
\right.\nonumber\\
&& \left.
-\frac{\ell^3}{24}\left\{
a^2(\phi_+)-a^2(\phi_-)\right\}R~
\right]\left[1+\frac{C^\mu{}_\mu}{2}\right] \ ,
\label{bulk}\end{aligned}$$ where we have used the the equation ${\cal R} = -20/\ell^2$ which holds in the bulk. Notice that the Ricci scalar came from ${\rm tr}f_{\mu\nu}$ in $\sqrt{\cal -G}$.
Next, let us calculate the action for the brane tension. The induced metric on each brane is written by $$\begin{aligned}
g^\pm_{\mu\nu}(\phi_\pm,x)=
a^2g_{\mu\nu}(x)+a^2f_{\mu\nu}(\phi_\pm,x)
+\partial_\mu\phi_\pm\partial_\nu\phi_\pm \ .\end{aligned}$$ The determinant of the induced metric can be calculated as $$\begin{aligned}
\sqrt{-g_\pm}&=&a^4(\phi_+)\sqrt{-g}
\sqrt{1+\frac{1}{a^2}(\partial\phi_\pm)^2
-\frac{\ell}{6a^2}R+C^\mu{}_\mu}
\nonumber\\
&\approx&
a^4(\phi_+)\sqrt{-g}\left(
1+\frac{1}{2a^2}(\partial\phi_\pm)^2
-\frac{\ell}{12a^2}R\right)
\nonumber\\
&&\times\left(1+\frac{C^\mu{}_\mu}{2}\right) \ ,\end{aligned}$$ where $(\partial\phi_\pm)^2$ means $\partial^\alpha\phi_\pm\partial_\alpha\phi_\pm$. Thus, the action for each brane becomes $$\begin{aligned}
S_\pm &\equiv& \mp {6\over\kappa^2\ell}\int d^4x\sqrt{-g_\pm}
\nonumber\\
&=& \mp {6\over\kappa^2\ell}\int d^4x\sqrt{-g}
\left[a^4(\phi_\pm)
+\frac{a^2(\phi_\pm)}{2}(\partial\phi_\pm )^2
\right.\nonumber\\
&& \left.
-\frac{\ell^2}{12}a^2(\phi_\pm)~R~\right]\left[
1+\frac{C^\mu{}_\mu}{2}\right] \ .
\label{tension}\end{aligned}$$ Note that the Ricci scalar came from ${\rm tr}f_{\mu\nu}$ in $\sqrt{-g_\pm}$.
In order to calculate the Gibbons-Hawking term, we need the extrinsic curvature defined by $$\begin{aligned}
{\cal K}_{\mu\nu}\equiv n_A\left(
\frac{\partial^2x^A}{\partial\xi^\mu\partial\xi^\nu}
\right)
+\Gamma^A_{BD}
\frac{\partial x^B}{\partial\xi^\mu}
\frac{\partial x^D}{\partial\xi^\nu} \ ,\end{aligned}$$ where $x^A$ is the coordinate of the brane, $\xi^\mu = x^\mu $ is the one on the brane and $n_A$ is the normal vector to the brane. Note that ${\cal K}_{\mu\nu}$ is different from $K_{\mu\nu}$ in Eq. (\[K\]). The Christoffel symbols we need are $$\begin{aligned}
\Gamma^y_{\mu\nu}&=&\frac{1}{\ell}a^2\left(
g_{\mu\nu}+f_{\mu\nu}\right)
-\frac{1}{2}a^2f_{\mu\nu,y}\ \ , \\
\Gamma^\alpha_{y\mu}&=&-\frac{1}{\ell}\delta^\alpha_\mu
+\frac{1}{2}g^{\alpha\beta}f_{\beta\mu,y} \ .\end{aligned}$$ The tangent basis on the brane are given by $$\begin{aligned}
\frac{\partial x^A}{\partial\xi^\mu}
=(\delta^\alpha_\mu,\partial_\mu\phi_\pm) \ .\end{aligned}$$ Thus, the normal vector takes the form $$\begin{aligned}
n_A=(-n_y\partial_\alpha\phi_\pm,n_y) \ .\end{aligned}$$ From the normalization condition $n_A n^A =1$, we have $$\begin{aligned}
n_y=\frac{1}{\sqrt{1
+\frac{1}{a^2}(\partial\phi_\pm)^2}} \ .\end{aligned}$$ Then the extrinsic curvature is calculated as $$\begin{aligned}
{\cal K}^\pm_{\mu\nu}&=&n_y\left[
\nabla_\mu\nabla_\nu\phi_\pm
+\frac{a^2}{\ell}\left(
g_{\mu\nu}+f_{\mu\nu}-\frac{\ell}{2}f_{\mu\nu,y}
\right)\right.\nonumber \\
&& \left.\qquad
+\frac{2}{\ell}\partial_\mu\phi_\pm\partial_\nu\phi_\pm
\right] \ .\end{aligned}$$ The trace part of extrinsic curvature on each brane is $$\begin{aligned}
{\cal K}_\pm&=&g^{\mu\nu}_\pm{\cal K}^\pm_{\mu\nu}\nonumber\\
&=&n_y\left[
\frac{4}{\ell}+\frac{1}{a^2}\square\phi_\pm
+\frac{1}{\ell a^2}(\partial\phi_\pm)^2
+\frac{\ell}{6a^2}R
\right] \ .\end{aligned}$$ Therefore, the Gibbons-Hawking term is obtained as $$\begin{aligned}
S_{\rm GH}&\equiv&{2\over\kappa^2}\int d^4x\sqrt{-g_+}{\cal K}_+
-{2\over\kappa^2}\int d^4x\sqrt{-g_-}{\cal K}_-
\nonumber\\
&=& {2\over\kappa^2}\int d^4x\sqrt{-g}\left[
\frac{4}{\ell}a^4(\phi_+)+\frac{3}{\ell}a^2(\phi_+)
(\partial\phi_+)^2
\right.\nonumber\\
&& \left.
-\frac{\ell}{6}a^2(\phi_+)~R~\right]\left[
1+\frac{C^\mu{}_\mu}{2}\right]
-\left(\phi_+\rightarrow\phi_-\right) \ .
\label{GH}\end{aligned}$$ Note that the Ricci scalar came from ${\rm tr}f_{\mu\nu}$ in $\sqrt{-g_\pm}$ and ${\rm tr}f_{\mu\nu,y}$ in ${\cal K}_\pm$.
Substituting the results Eqs. (\[bulk\]), (\[tension\]) and (\[GH\]) into the 5-dimensional action Eq. (\[action:5-dim\]), we get the 4-dimensional effective action $$\begin{aligned}
S&=&S_{\rm bulk}+S_++S_-+S_{\rm GH}\nonumber\\
&=& \frac{\ell}{2\kappa^2}\int d^4x\sqrt{-g}
\left[\left\{
a^2(\phi_+)-a^2(\phi_-)\right\}R \right.\nonumber\\
&&\quad
\left.
+\frac{6}{\ell^2}\left\{
a^2(\phi_+)(\partial\phi_+)^2
-a^2(\phi_-)(\partial\phi_-)^2
\right\}\right]\nonumber\\
&&\quad
\times\left[
1+\frac{C^\mu{}_\mu}{2}\right] \ .
\label{4d}\end{aligned}$$ Here, $C^\mu{}_\mu$ is the first order quantity, so we can ignore this term at leading order. We see that this effective action (\[4d\]) is indistinguishable from Eq. (\[moduliapp\]) obtained by assuming the factorizable metric. Thus, we have shown that the action obtained from the factorizable ansatz is correct at the leading order.
Here, it should be stressed that the Einstein-Hilbert term is originated from the contributions of $f_{\mu\nu}$ in each $S_{\rm bulk},
S_\pm$ and $S_{\rm GH}$, so the correction $f_{\mu\nu}$ to the metric $g_{\mu\nu}$ plays an important role.
Note that the induced metric on the positive tension brane is $$\begin{aligned}
g^+_{\mu\nu}(\phi_+,x)&=&a^2(\phi_+)g_{\mu\nu}(x)\ ,\end{aligned}$$ where we have chosen the constant of integration $C_{\mu\nu}$ to be $$\begin{aligned}
f_{\mu\nu}(\phi_+,x)=-\frac{1}{a^2(\phi_+)}\partial_\mu\phi_+\partial_\nu\phi_+
\label{C}
\ .\end{aligned}$$ We see that the induced metric on the positive tension brane is different from the factorized metric $g_{\mu\nu}$. Using a conformal transformation: $g_{\mu\nu}=(1/a^2(\phi_+))g^+_{\mu\nu}$ to rewrite the effective action in terms of the induced metric, we finally get $$\begin{aligned}
S&=&\frac{\ell}{2\kappa^2}\int d^4x\sqrt{-g_+}
\left[\left\{1-\left(\frac{a(\phi_-)}{a(\phi_+)}\right)^2\right\}
R(g_+)\right.\nonumber\\
&&\qquad\quad\left.
-6~\partial_\mu\left(\frac{a(\phi_-)}{a(\phi_+)}\right)
\partial^\mu\left(\frac{a(\phi_-)}{a(\phi_+)}\right)
\right] \ ,
\label{ST}\end{aligned}$$ where $$\begin{aligned}
\frac{a(\phi_-)}{a(\phi_+)}={\rm exp}\left[-\frac{1}{\ell}
\left(\phi_--\phi_+\right)\right]\ .\end{aligned}$$ Two moduli fields appear only in the form of the difference, which corresponds to the radion field. In physical frame, the extra degree of freedom disappears.
Conclusion
==========
To support the validity of the factorizable metric ansatz used in string cosmology, we investigated a toy problem in RSI model. For this purpose, we have revised the gradient expansion method to conform to the the factorizable metric ansatz. We have solved the 5-dimensional equations of motion and substituted the results into the action. Consequently, we have obtained the 4-dimensional effective action which is equivalent to that obtained by assuming the factorizable metric ansatz. Hence, our calculation supports the factorizable metric ansatz.
In the higher order analysis including Kaluza-Klein corrections, the factorizable metric ansatz cannot be correct anymore [@sugumi]. However, in string cosmology, what we want is the leading order action. Then the factorizable metric ansatz becomes useful method when discussing the cosmology without solving the bulk equations of motion at least at the leading order.
Finally, we should mention the limitation of our approach. Although we have shown the validity of the factorizable ansatz in 5-dimensions, the issue is still unclear in the case of higher codimension. This is because the higher codimension objects is difficult to treat in a relativistic manner. To give a more strong support of the factorizable metric ansatz, we need to settle this issue.
This work was supported in part by Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science and Culture of Japan No. 155476 (SK) and No.14540258 (JS) and also by a Grant-in-Aid for the 21st Century COE “Center for Diversity and Universality in Physics".
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---
abstract: 'The search for the Standard Model (SM) Higgs boson (H) produced through the Vector Boson Fusion (VBF) mechanism and decaying to a pair of bottom quarks is reported. The used data have been collected with the CMS detector and correspond to an integrated luminosity of 19.8 fb$^{-1}$ of proton-proton collisions at €‰$\sqrt{s}=8$ TeV at the CERN LHC. Parked data have been exploited as well. This search resulted in an observed (expected) significance in these data samples for a H $\rightarrow$ b$\bar{\textnormal{b}}$ signal at a mass of 125 GeV of 2.2 (0.8) standard deviations. The cited signal strength, $\mu=\sigma/\sigma_{\textrm{SM}}$, was measured to be 2.8$^{+1.6}_{-1.4}$. This result has been combined with other CMS searches for the SM Higgs boson decaying in a pair of bottom quarks exploiting other Higgs production mechanisms. The obtained combined signal strength is 1.0 $\pm$ 0.4, corresponding to an observed signal significance of 2.6 standard deviations for a Higgs boson mass of 125 GeV.'
author:
- 'Giorgia Rauco [^1]'
date: 'Dated: '
title: Search for the Standard Model Higgs boson produced by vector boson fusion and decaying to bottom quarks
---
Introduction
============
In the Standard Model theory [@Glashow1961; @Salam:1961en; @Weinberg1967] the electroweak symmetry breaking is explained by the Brout-Englert-Higgs mechanism [@PhysRevLett.13.321; @Unnamed; @PhysRevLett.13.585] which is responsible for the electroweak gauge bosons to acquire mass. This mechanism predicts the existence of a Higgs scalar boson. The observation of a new particle in the mass region around $125$ GeV, consistent with the Higgs boson, was announced by the ATLAS [@Aad2012] and CMS [@Chatrchyan12] experiments at CERN’s [Large Hadron Collider]{} (LHC) on $4$ July 2012.
After the Higgs boson discovery, the main goal is now to precisely measure and study the properties of the recently discovered particle and to compare them with the ones expected for the predicted boson. One of the most interesting aspects to analyze is the coupling between the particle and the fermions and, in the SM theory, the most probable decay of the Higgs boson of m$_{\textnormal{H}} = 125$ GeV is in a pair of b-quarks. This process is particularly difficult to be observed at the LHC in the inclusive production (dominated by [Gluon Fusion]{}) since the QCD background is overwhelming. It is therefore searched in other production channels: [Vector Boson Fusion]{} (VBF), [Associated Vector Boson Production]{} (VH) and [Associated Top-Pair Production]{} (ttH), where the Higgs boson is produced in association with other particles, resulting in a more distinguishing signal topology. The VBF production channel is the one exploited in the analysis here presented.
In the VBF process a quark of each one of the colliding protons radiates a W or Z boson that subsequently interact or fuse.
The two valence quarks are typically scattered away from the beam line and inside the detector acceptance, where they can be revealed as hadronic jets. The prominent signature of VBF is therefore the presence of two energetic hadronic jets, roughly in the forward and backward direction with respect to the proton beam line. As a result, the signal final state features are a central b-quark pair (from the Higgs decay) and a light- quark pair (u,d-type) from each of the colliding protons, in the forward and backward regions.
The overwhelmingly most relevant and irreducible background to the signal search comes from the QCD production of four jets events with true or mistagged b-jets. Other backgrounds arise from: (i) hadronic decays of Z or W bosons produced in association with additional jets, (ii) hadronic decays of top quark pairs, and (iii) hadronic decays of singly produced top quarks. The final expected signal yield includes also the contribution of the Higgs bosons produced in Gluon Fusion processes with at least two associated jets.
Trigger
=======
The data used for this analysis were collected with two different trigger strategies:
1. [**Dedicated VBF qqH $\mathbf{\rightarrow}$ qqb$\mathbf{\bar{\textnormal{{\bf b}}}}$ trigger**]{}. A set of dedicated trigger paths was specifically designed and deployed for the VBF qqH $\rightarrow$ qqbb signal search, both for the L1 and HLT levels, and operated during the full 2012 data-taking period. This set of triggers, called nominal, collected the largest fraction of the signal event.
The L1 paths require the presence of three jets with p$_{\textrm{T}}$ above optimized thresholds X, Y, Z (X = 64 - 68 GeV, Y = 44 - 48 GeV, Z = 24 - 32 GeV) according to instantaneous luminosity. Among the three jets, at most one among the two p$_{\textrm{T}}$ leading jets can be in the forward pseudorapidity region, while the remaining two have to be central. The HLT paths are seeded by the L1 paths described above, and require the presence of four jets with p$_{\textrm{T}}$ above thresholds that are again adjusted to the data-taking luminosity, p$_{\textrm{T}} >$ 75 - 82, 55 - 65, 35 - 48, and 20 - 35 GeV, respectively. At least one of the selected four jets must further fulfill minimum b-tagging requirements. To identify the two VBF-tagging jets two criteria have been exploited: (i) the pair with the smallest HLT b-tagging values; (ii) the pair with the maximum pseudorapidity opening. Both pairs are required to exceed variable minimum thresholds on $|\Delta\eta_{jj}|$ of 2.2-2.5, and of 200-240 GeV on the dijet invariant mass $m_{jj}$, depending on the instantaneous luminosity.
2. [**General-purpose VBF trigger.**]{} The L1 paths for the general-purpose VBF trigger require minimum hadronic activity in the event with a scalar p$_{\textrm{T}}$ sum of 175 or 200 GeV, depending on the instantaneous luminosity. The HLT path is seeded by the L1 path described above, and requires the presence of at least two CaloJets with p$_{\textrm{T}} >$ 35 GeV. Out of all the possible jet pairs in the event the pair with the highest invariant mass is selected as the most probable VBF tagging pair. The corresponding invariant mass m$_{jj}$ and absolute pseudorapidity difference $|\Delta\eta_{jj}|$ are required to be larger than 700 GeV and 3.5.
The integrated luminosity collected with the first set of triggers was 19.8 fb$^{-1}$, while for the second trigger it was 18.2 fb$^{-1}$ .
Event reconstruction and selection
==================================
The offline analysis uses reconstructed charged-particle tracks and candidates of the Particle-Flow (PF) algorithm [@CMS-PAS-PFT-10-001; @CMS-PAS-PFT-1002; @CMS-PAS-PFT-09-001]. Jets are reconstructed by clustering the PF candidates with the anti-k$_{\textnormal{T}}$ algorithm with distance parameter 0.5 and jets that are likely to be originated from the hadronization of b quarks are identified with the CSV b-tagger [@Chatrchyan:2012jua].
The events used in the offline analysis are required to have at least four reconstructed jets and the four p$_{\textnormal{T}}$-leading ones are considered as the most probable b-jet and VBF jet candidates. A multivariate discriminant taking into account the b-tag value, the b-tag ordering, the $\eta$ value, and the $\eta$ ordering is exploited to distinguish between the two jet types.
The offline event selection is based upon the kinematic properties of the b-jet and VBF jets. Selected events are divided into two sets: [set A]{} and [set B]{}, whereof the selection requirements are shown in Table \[tab:sel\].
[set A]{} [set B]{}
------------------------------ ------------------------------------------------------------- -------------------------------------------------------------------------------------
[trigger]{} [dedicated VBF qqH ${\rightarrow}$ qqb$\rm{\bar{{{b}}}}$]{} [general-purpose VBF trigger]{}
p$_\textnormal{T,1,2,3,4}>30$GeV
jets p$_{\textnormal{T}}$ p$_\textnormal{T,1,2,3,4}>80,70,50,40$GeV
p$_\textnormal{T,1}+p_\textnormal{T,2}>160$GeV
jets $|\eta|$ $<4.5$ $<4.5$
b-tag at least 2 CSVL jets at least 1 CSVM and 1 CSVL jets
$\Delta\phi_\textnormal{bb}$ $<2.0$ $<2.0$
$m_\textnormal{jj}>250$GeV $m_\textnormal{jj},\,m_\textnormal{jj}^\textnormal{trig}>700$GeV
VBF topology
$|\Delta\eta_\textnormal{jj}| > 2.5$ $|\Delta\eta_\textnormal{jj}|,\,|\Delta\eta_\textnormal{jj}^\textnormal{trig}|>3.5$
veto none events that belong to set A
: Summary of selection requirements for the two analysis sets. Reprinted from [@PhysRevD.92.032008].[]{data-label="tab:sel"}
After all the selection requirements, $2.3\%$ of the VBF simulated signal events end up in set A and $0.8\%$ end up in set B. In set B $39\%$ of the signal events would also satisfy the requirements to enter set A. Such events are taken into set A and vetoed from set B, as noted in the last line of Table \[tab:sel\].
Signal properties
=================
Jet transverse-momentum regression
----------------------------------
In order to improve the b$\bar{\textnormal{b}}$ mass resolution a regression technique is applied. It is essentially a refined calibration for individual b-jets which takes into account the jet composition properties beyond the default jet-energy corrections. This regression technique mainly targets the b decays in a neutrino that lead to a substantial mismeasurement of the jet p$_{\textnormal{T}}$.
For this purpose a regression Boost Decision Tree (BDT), trained on simulated signal events, is applied. Its inputs include: (i) the jet p$_{\textnormal{T}}$, $\eta$ and mass; (ii) the jet-energy fractions carried by neutral hadrons and photons; (iii) the mass and the uncertainty on the decay length of the secondary vertex, when present; (iv) the event missing transverse energy and its azimuthal direction relative to the jet; (v) the total number of jet constituents; (vi) the p$_{\textnormal{T}}$ of the soft-lepton candidate inside the jet, when present, and its p$_{\textrm{T}}$ component perpendicular to the jet axis; (vii) the p$_{\textnormal{T}}$ of the leading track in the jet; (viii) the event’s average p$_{\textrm{T}}$ density in the $y-\phi$ space.
The improvement on the jet p$_{\textnormal{T}}$ leads to an improvement on the dijet invariant mass resolution by approximately 17$\%$.
Discrimination between quark-and gluon-originated jets
------------------------------------------------------
The VBF-tagging jets originate from the hadronization of a light (u,d-type) quark, while the jets produced in QCD processes are more likely to come from gluons. As a consequence, in order to further identify if the jet pair with the smallest b-tagging values among the four selected jets is a signal event or a background event, a quark-gluon discriminator [@Chatrchyan:2013jya; @Charticyian2015; @CMS-PAS-JME-13-002] is applied to the b-tag sorted jj candidate jets.\
The discriminator exploits the differences in the showering and the fragmentation of gluons and quarks and it uses, as an input to a likelihood trained on gluon and quark jets from simulated QCD events, the following variables: (i) the jet constituents’ major quadratic mean (RMS) in the $\eta-\phi$ plane; (ii) the jet constituents’ minor quadratic mean (RMS) in the $\eta-\phi$ plane; (iii) the jet asymmetry pull (essentially a p$_{\mathrm{T}}$-weighted vector); (iv) the jet particle multiplicity; (v) the maximum energy fraction carried by a jet constituent.
Soft QCD activity
-----------------
In the region between the two VBF-tagging jets (with the exception of the more centrally produced Higgs decay products), the QCD color flow is suppressed. In order to measure the additional hadronic activity associated with the main primary vertex, only charged tracks are used.
A collection of *additional tracks* is built, selecting reconstructed tracks that: (i) have a [*high purity*]{} quality flag; (ii) have p$_{\textnormal{T}}>$300 MeV; (iii) are not associated to any of the four leading jets; (iv) have a minimum longitudinal impact parameter, $|d_{z}(PV)|$ with respect to the event’s main primary vertex; (v) satisfy $|d_{z}(PV)|<2$ mm and $|d_{z}(PV)|<3\sigma_{z}(PV)$ where $\sigma_{z}(PV)$ is the uncertainty on $d_{z}(PV)$; (vi) are not in the region between the most b-tagged jets. This region is defined as an ellipse in the $\eta-\phi$ plane around the b-jets with axis $(a,b) = (\Delta R(bb)+1,1)$ where $\Delta R = \sqrt{(\Delta\eta_{bb})^2+(\Delta\phi_{bb})^2}$.
The additional tracks are then clustered in *soft TrackJets* within the anti-$k_{\textnormal{T}}$ algorithm [@Cacciari:2008gp] (with R = 0.5).
In order to discriminate between the signal and the QCD background, a discriminating variable $H_{T}^{soft}$ is used and it is defined as the scalar p$_{\textnormal{T}}$ sum of the soft TrackJets with p$_{\textnormal{T}}>$1 GeV.
Search for a Higgs boson
========================
In order to separate the overwhelmingly large QCD background from the Higgs boson signal, all the discriminating features have to be used in an optimal way. This is achieved by using a BDT multivariate discriminant, which exploits as input, variables very weakly correlated to the dynamics of the $\textnormal{b}\bar{\textnormal{b}}$ system, in particular to m$_{{\textnormal b}\bar{\textnormal{b}}}$. These variables are conceptually grouped into five groups: (i) the dynamics of the VBF jet system, expressed by $\Delta\eta_{\rm jj}$, $\Delta\phi_{\rm jj}$, and $ m_{\rm jj}$; (ii) the b jet content of the event, expressed by the CSV output for the two most b-tagged jets; (iii) the jet flavor of the event: quark-gluon likelihood (QGL) for all four jets; (iv) the soft activity, quantified by the scalar $p_{\textnormal{T}}$ sum $H_T^{\rm soft}$ of the additional “soft” TrackJets with $p_{\textnormal{T}}>1$ GeV, and the number $N^{\rm soft}$ of “soft” TrackJets with $p_{\textnormal{T}}>2$ GeV; (v) the angular dynamics of the production mechanism, expressed by the cosine of the angle between the $\rm jj$ and $\textnormal{b}\bar{\textnormal{b}}$ vectors in the center-of-mass frame of the four leading jets $\cos\theta_{\rm jj,bb}$.
Since the properties of the selected events are significantly different between the two selections (set A and set B) and two BDT’s are trained.
According to the BDT outputs, seven categories are defined: four for set A and three for set B.
The QCD m$_{\textnormal{b}\bar{\textnormal{b}}}$ spectrum shape is assumed to be the same in all BDT categories of the same set of events. In reality small differences between the categories are present and to take into account this effect transfer functions are exploited (linear function in set A and quadratic in set B). With the introduction of the transfer functions, the fit model for the Higgs boson signal is $$\label{eq:fitH}
\begin{split}
f_i(m_{\textnormal{b}\bar{\textnormal{b}}})=\mu_{\rm H}\cdot N_{i,\rm H}\cdot H_i(m_{\textnormal{b}\bar{\textnormal{b}}};k_{\rm JES},k_{\rm JER})+N_{i,\rm Z}\cdot Z_i(m_{\textnormal{b}\bar{\textnormal{b}}};k_{\rm JES},k_{\rm JER})+ \\
N_{i,\rm Top}\cdot T_i(m_{\textnormal{b}\bar{\textnormal{b}}};k_{\rm JES},k_{\rm JER})+N_{i,\rm QCD}\cdot K_i(m_{\textnormal{b}\bar{\textnormal{b}}})\cdot B(m_{\textnormal{b}\bar{\textnormal{b}}};\vec{p}_{\rm set}),
\end{split}$$ where the subscript $i$ denotes the category and $\mu_{\rm H},\,N_{i,\rm QCD}$ are free parameters for the signal strength and the QCD event yield. $N_{i,\rm H}$, $N_{i,\rm Z}$, and $N_{i,\rm Top}$ are the expected yields for the Higgs boson signal, the Z+jets, and the top background respectively. The shape of the top background $T_i(m_{\textnormal{b}\bar{\textnormal{b}}};k_{\rm JES},k_{\rm JER})$ is taken from the simulation (sum of the $\textnormal{t}\bar{\textnormal{t}}$ and single-top contributions) and is described by a broad gaussian. The Z/W+jets background $Z_i(m_{\textnormal{b}\bar{\textnormal{b}}};k_{\rm JES},k_{\rm JER})$ and the Higgs boson signal $H_i(m_{\textnormal{b}\bar{\textnormal{b}}};k_{\rm JES},k_{\rm JER})$ shapes are taken from the simulation and are parameterized as a crystal-ball function on top of a polynomial background. The position and the width of the gaussian core of the MC templates (signal and background) are allowed to vary by the free factors $k_{\rm JES}$ and $k_{\rm JER}$, respectively, which quantify any mismatch of the jet energy scale and resolution between data and simulation. Finally, the QCD shape is described by a Bernstein polynomial $B(m_{\textnormal{b}\bar{\textnormal{b}}};\vec{p}_{\rm set})$, common within the categories of each set, and whose parameters $\vec{p}_{\rm set}$ are determined by the fit, and a multiplicative transfer function $K_i(m_{\textnormal{b}\bar{\textnormal{b}}})$ that accounts for the shape differences between the categories. For set A, the Bernstein polynomial is of 5th order, while for set B it is of 4th order. Figure \[fig:fitHiggs\] show the simultaneously fitted m$_{\textnormal{b}\bar{\textnormal{b}}}$ distributions in the signal enriched categories for set A and set B, respectively.
![Fit for the Higgs boson signal (m$_\textnormal{H}=125$ GeV) on the invariant mass of the two b-jet candidates in the signal enriched event category of set A (left) and set B (right). Data is shown with markers. The solid line is the sum of the post-fit background and signal shapes, the dashed line is the background component, and the dashed-dotted line is the QCD component alone. The bottom panel shows the background-subtracted distribution, overlaid with the fitted signal, and with the 1-$\sigma$ and 2-$\sigma$ background uncertainty bands. Reprinted from [@PhysRevD.92.032008]. []{data-label="fig:fitHiggs"}](Fit_mH125_CAT3.pdf "fig:"){width="45.00000%"} ![Fit for the Higgs boson signal (m$_\textnormal{H}=125$ GeV) on the invariant mass of the two b-jet candidates in the signal enriched event category of set A (left) and set B (right). Data is shown with markers. The solid line is the sum of the post-fit background and signal shapes, the dashed line is the background component, and the dashed-dotted line is the QCD component alone. The bottom panel shows the background-subtracted distribution, overlaid with the fitted signal, and with the 1-$\sigma$ and 2-$\sigma$ background uncertainty bands. Reprinted from [@PhysRevD.92.032008]. []{data-label="fig:fitHiggs"}](Fit_mH125_CAT6.pdf "fig:"){width="45.00000%"}
Results
=======
The models representing the two hypotheses, of background only, and of background+signal are fitted to the data, simultaneously in all the categories. The limits on the signal strength are computed with the Asymptotic CLs method [@CLs]. Figure \[fig:limits\] shows the observed (expected) $95\%$ C.L. limit on the signal strength, as a function of the Higgs boson mass, which ranges from $5.1$ ($2.2$) at m$_\textnormal{H}=115$ GeV to $5.9$ ($3.8$) at m$_\textnormal{H}=135$ GeV, together with the expected limits in the presence of a SM Higgs boson with mass 125 GeV. For a 125 GeV Higgs boson signal the observed (expected) significance is 2.2 (0.8) standard deviations, and the fitted signal strength is $\mu=\sigma/\sigma_{\rm SM}=2.8^{+1.6}_{-1.4}$. The measured signal strength is compatible with the SM Higgs boson prediction $\mu=1$ at the $8\%$ level.
![Expected and observed $95\%$ confidence level limits on the signal cross section in units of the SM expected cross section, as a function of the Higgs boson mass, including all event categories. The limits expected in the presence of a SM Higgs boson with mass 125 GeV are indicated by the dotted curve. Reprinted from [@PhysRevD.92.032008].[]{data-label="fig:limits"}](limit.pdf){width="60.00000%"}
The search for the Standard Model Higgs boson decaying in a pair of bottom quarks, as described in the previous Sections, is performed by the CMS Collaboration in the VH [@vh], VBF and ttH [@tth] production channels. The results of these searches are summarized in Table \[tab:summary\], along with the resulting combined results.
------------------------------------------------- ---------------------------- -------------- -------------- -------------- --------------
[H $\rightarrow$ b${\bar{\textnormal{{b}}}}$]{} [Best-fit (${68\%}$ CL)]{}
[channel]{} [Observed]{} [Observed]{} [Expected]{} [Observed]{} [Expected]{}
VH 0.89 $\pm$ 0.43 1.68 0.85 2.08 2.52
ttH 0.7 $\pm$ 1.8 4.1 3.5 0.37 0.58
VBF 2.8$^{+1.6}_{-1.4}$ 5.5 2.5 2.20 0.83
combined 1.03$^{+0.44}_{-0.42}$ 1.77 0.78 2.56 2.70
------------------------------------------------- ---------------------------- -------------- -------------- -------------- --------------
: Observed and expected 95$\%$CL limits, best fit values and significance on the signal strength parameter $\mu = \sigma/\sigma_{\textnormal{SM}}$ at m$_{\textnormal{H}}$ = 125 GeV, for each H $\rightarrow$ b$\bar{\textnormal{b}}$ channel and combined. Reprinted from [[@PhysRevD.92.032008]]{}.[]{data-label="tab:summary"}
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[^1]: Electronic address: `giorgia.rauco@cern.ch`
|
---
abstract: 'High pressure nuclear magnetic resonance is among the most challenging fields of research for every NMR spectroscopist due to inherently low signal intensities, inaccessible and ultra-small samples, and overall extremely harsh conditions in the sample cavity of modern high pressure vessels. This review aims to provide a comprehensive overview of the topic of high pressure research and its fairly young and brief relationship with NMR.'
address: 'Bayerisches Geoinstitut, Universität Bayreuth,Universitätsstraße 30, D-95447 Bayreuth, Germany'
author:
- Thomas Meier
title: 'Journey to the Centre of the Earth: Jule Vernes’ dream in the laboratory from an NMR perspective'
---
NMR, High Pressure Research, Diamond Anvil Cells, Micro-resonators, Sensitivity enhancement
An unexpected journey for an NMR spectroscopist, a motivation {#intro}
=============================================================
“Pressure. To most people the word brings to mind the stress of our lives in a time of economic crisis. Yet to many scientists, pressure means something very different; it is an idea filled with wonder and power – a phenomenon unlike anything else we know. Pressure shapes the stars and planets, forges the continents and oceans, and influences our lives every moment of every day.”
These are the opening lines of Robert Hazen’s intriguing monograph “The New Alchemists”, in which the author describes the early days of modern high pressure science, and its evolution towards one of the most challenging and fascinating research branches today[@Hazen1993]. It is precisely because of this challenge and uniqueness of high pressure research that we need to take a closer look at the methods available today, which allow us to recreate extreme conditions in the tiniest of spaces inside a laboratory. Thus, in this review, we will metaphorically retrace the footsteps of Jules Vernes’ Professor Lidenbrock as he descends beneath the surface of the Earth, towards a strange and unknown and yet utterly enticing world - which we now know to be under extreme pressure.\
But why investigate pressure with NMR?\
While this simple question is frequently raised in an NMR environment, it is not that easy to answer.
![Even on our planet Earth, pressure ranges over about seven orders of magnitude from its surface, at one bar of atmospherical pressure, to the gravitational centre with about 360 GPa. The schematic picture shows the inner structure of the Earth together with a pressure-temperature scale and corresponding depths. Up to now, NMR techniques are able to mimic the extreme conditions of the inner mantle – disregarding elevated temperatures for now– corresponding to a depth of about 2000 km. \[fig1\]](Front)
As far as NMR is concerned, the most typical ways to alter a given system are by altering its chemical composition, e.g. through doping, increasing or decreasing temperature, or using high or low magnetic fields. Pressure modifying vessels, on the other hand, are only rarely employed in NMR research, owing to substantial technical difficulties associated with the design of such apparatuses; their demand for radio-frequency (RF) resonators, which require most NMR spectroscopists to look over the rim of their tea cup; as well as their inherent inability to use modern high sensitivity and line narrowing methods like MAS or DNP.\
Nevertheless, in condensed matter systems, where the interatomic bond strength easily exceeds several eV, utilisation of immense pressures is needed to induce significant structural or electronic changes. Other ways of increasing energy density would require tremendous resources, or else are simply extremely impractical. For example, the application of an external magnetic field of 50 T would only correspond to the application of about 1 GPa (10.000 atmospheres)[^1]; a pressure which is easily reached in modern diamond anvil cells.\
If we would start to compare these pressures with the geothermal gradient of our planet Earth, we would have started our hypothetical journey at the bottom of the Mariana’s trench, corresponding to about 1 kbar or 100 MPa of pressure exerted by the 11 km high water pillar above us. Reaching deeper into Earth’s interior, say to a depth of about 200 km in the middle of the upper mantle, the experienced pressure would be similar to the contact pressure of the Eiffel tower turned up side down and balanced on its tip, which would only be 10 GPa (about 100.000 bar). At one mega-bar, or 100 GPa, we would already have “dived” down half way through the lower mantle, and at a crushing 350 GPa our journey would come to a violent end upon reaching Earths’ gravitational centre, a solid ball primarily comprised of iron and nickel. Clearly, Professor Lidenbrock and his companions would have come to a literally crushing end as well, had they succeeded in their quest.\
In modern high pressure laboratories, of course, Jules Vernes’ fictional journey takes a somewhat different path. Here, pressures are generated by the application of pressure sustaining vessels, which are often composed of a movable piston and an enclosed pressure chamber[@Bridgman1952a]. Especially with the invention of the diamond anvil cell by Charles Weir[@Weir1959] and Alvin van Valkenburg[@Piermarini1993] in 1959, high pressure science took up momentum and became an integral part of contemporary chemistry, biochemistry, physics, and geophysics[@Grochala2007; @Bassett2009; @Hemley2010]. As the technique evolved with higher quality diamonds of increasingly complex geometries, and harder materials used for the pressure vessels, the range of applicable pressures rapidly increased well into the megabar regime (1 Mbar = 1 million atmospheres)[@Mao1985; @Hazen1993], mimicking the extreme conditions at the centre of the Earth. Recently, record pressures of up to 1 TPa were achieved in double-stage diamond anvil cells[@Dubrovinskaia2016].\
This review’s aim is threefold. Firstly, readers not familiar with this rather exotic application of magnetic resonance[^2] should gain a general impression of the “nuts and bolts” approach associated with these experimental set-ups. Secondly, I will review NMR experiments obtained using micro-coil and magnetic flux tailoring techniques, which are capable of reaching pressures from between 1 GPa and close to 1 Mbar. Closely related to this is a critical overview of some methodological difficulties arising within certain experiments, which can complicate data analysis, or might even lead to false interpretations.\
From Psi over Torr to kbar. The Low pressure regime as the playground for bio-chemistry and life-sciences. {#lowp}
==========================================================================================================
Let us begin our journey with a pressure regime ranging over almost four orders of magnitude from ambient conditions to about 1 GPa. This is the realm of high pressure Bio-NMR, were one of the more commonly used and better known high pressure NMR set-ups is used. This approach uses so-called clamp cells, which are basically comprised of a movable piston exerting the pressure on a sample volume often as big as 100 $\mu l$ .\
As this pressure range is still below solidification transitions of most liquid buffer media, it has been proven to be an ideal tool investigating pressure driven protein folding and unfolding dynamics in liquids[@Kitahara2003; @Li2006] under increasing compressional stages. Figure \[fig2\] schematically demonstrates the famous protein volume theorem first proposed by Kitahara et al.[@Kitahara2003] and Li et al.[@Li2006].
As this field of high pressure NMR research is extremely extensive and already very well studied, I would like to draw the attention of the reader to the comprehensive review articles from Jonas[@Jonas1994], Ballard[@Ballard1997], and Roche et al.[@Roche2017].
![Schematic representation of the protein-volume-theorem of an arbitrary protein in an arbitrary energy landscape. The native folded molecules often occupy the highest molar volume, whereas a decrease in the overall volume leads to a destabilisation of the protein structure. \[fig2\]](proteinfolding)
Within the Upper Mantle: First Endeavours with DACs {#midp}
===================================================
Up to this point in the pressure-temperature landscape, experimental conditions mimicking the pressure at the bottom at the Marianas trench, and some kilometers deeper in the Earth’s crust, could easily be achieved without the application of diamond anvil cells. However, conditions beyond 1 GPa demand a much more powerful device.\
In this sense, the DAC turned out to be one of the most versatile pressure generating vessels, as it enables the experimenter to not only reach very high static pressures, but also provides him with an astonishing variability of set-ups which can be adopted to a plethora of different experimental environments.\
Nevertheless, performing NMR experiments in a DAC can at best be considered problematic due to the following reasons.\
1) The available sample volume in a DAC is often several orders of magnitude smaller than in a standard NMR experiment. The reason behind this is obvious: We can either generate high pressures by applying a huge force on a sample of some dozens of mm$^3$, which becomes exceedingly unpractical as we reach pressures above, say, 2 or 3 GPa. Also, these such so-called “large volume” presses are typically fairly large [@Liebermann2011] (several meters in height, for example), thus an application in a superconducting NMR magnet would be out of the question. The other possibility is, of course, to reduce the size of the pressurising area. In a DAC, typical culet sizes[^3] are between 1.2 and 1 mm. As the sample cavity should be a bit smaller than the diameter and rather flat, we have to work with a sample of roughly 500 $\mu$m in diameter and 120 $\mu$m in height, which some NMR spectroscopists might already consider impossible to work with.\
2) The cavity is tightly enclosed by diamond from two sides, and by a very hard and often metallic gasket which seals the cavity, and provides additional “massive support” of the diamond anvils[@Yousuf1982]. Thus, any available free space is located far off the actual sample.\
3) If we are talking about pressures exceeding 1 GPa, we have to start thinking about hydrostaticity, that is in NMR we need a more or less uniform pressure distribution in the sample cavity, as we are detecting NMR signals from the bulk of the sample. Thus, non-hydrostatic conditions which arise if pressure media turn solid, either at cryogenic temperatures or at high pressures, can lead to ambiguous and distorted NMR spectra.\
The first NMR experiments in DACs emerged in the late 1980s. The main idea of these pioneering groups was to place a small RF coil operating at predominantly hydrogen frequencies as close as possible to the sample cavity without distorting the diamonds or the metallic gasket. These set-ups include resonators which comprised, for example, a pair of coils placed on the diamonds pavilion[@Yarger1995; @Lee1989a], a gradient-field Maxwell coil[@Lee1992], or a single loop cover inductor coupled with a split rhenium gasket[@Pravica1998]. A more detailed overview of the development of these high pressure NMR techniques is given elsewhere[@Meier2017b].
![Two possible arrangements of NMR resonators in a DAC. A) The “Hairpin” resonator could be placed on top and bottom of the rhenium gaskets, thus forming a gradient-field Maxwell coil. After Lee et al.[@Lee1992] B) The “Key Hole” gasket resonator basically consists of a copper cover inductor, which is directly connected to the spectrometer and is in electrical contact to a split rhenium gasket, leading to a focusing effect of the RF B$_1$ field at the sample cavity. After Pravica and Silvera [@Pravica1998] \[fig3\] ](olddesigns)
The hairpin and gasket resonator approaches shown in figure \[fig3\] demonstrate a certain amount of ingenuity needed to overcome the obstacles described above. With these set-ups, pressures as high as 13 GPa could be reached[@Pravica1998a]. Nonetheless, the set-ups are far from ideal.\
As the hairpin resonator is far off the actual samples, the filling factors in this approach are in the order of a fraction of a percent, leading to a spin sensitivity of about $10^{19} \textrm{spin}/\sqrt{\textrm{Hz}}$, which is roughly a factor of 100 lower compared to the standard NMR sensitivity of a static non-DNP experiment. While this improvement sounds very promising, we have to keep in mind that the sample dimensions are already decreased by a factor of about $10^{6}$ in a DAC such as was used in these experiments. Thus, very long data acquisition times hampered a further application of this technique beyond its use for hydrogen NMR in liquid samples, where NMR signals are typically sharp enough to be detected after a couple of thousand scans[@Lee1987; @Lee1989; @Lee2008].\
Experience has shown that the problem can only be solved if the RF resonator’s filling factor could be significantly improved. At the end of the 1990s, Pravica and Silvera came up with one of the most interesting ideas so far: the electrically conductive rhenium metal gasket was cut open from the sample hole radially outwards, resembling a key hole[^4]. The slit was filled with a mixture of diamond powder and NaCl, which, after careful melting of the NaCl powder, formed a homogeneous filling of the slit. Afterwards, a copper cover inductor, connected directly to the NMR spectrometer, was placed in contact with the slitted gasket. Therewith, NMR experiments could be performed using the electric coupling of both gasket and cover inductor, leading to a locally enhanced B$_1$ in the sample chamber, and an increase of spin sensitivity by one order of magnitude compared to hairpin resonators.\
However, the downfall of this approach is a bit more subtle. First, as the conductivity of rhenium is one order of magnitude less than that of copper, the quality factor of the key hole resonator gasket is rather low. Coupling of both resonators also turns the copper cover inductor into a lossy resonator. Secondly, the slit in the gasket forms a capacitor with the NaCl grains due to its conductivitiy, breaking down the electric field into small steps of floating potential in the capacitor. Now, since this capacitor is rather inaccurate, the self resonance of the slit gasket will be far off the desired resonance frequency of the nuclei in the sample, and the slit gasket basically forms a lossy inductance. Furthermore, both sodium and chlorine ions show increasing mobility when under pressure or stress, thus they move due to applied magnetic fields, warming up the gasket through thermal dissipation, and reducing the Q even further. Therefore, all proximity advantages due to the geometry are negated.\
These developments mark the apex of the first evolution period in DAC-NMR research. Unfortunately, only a handful of groups tackled this demanding task. Also, as sensitivities were very low and actual acquisition times exceedingly long, it was widely believed that this method could only be applied for proton NMR.\
Nonetheless, while being able to perform NMR experiments at pressures of around 5 GPa seems to be a great accomplishment indeed, we still need to dig deeper into our metaphorical hole in the ground. Far deeper.
Reaching the Lower Mantle: where things become tricky {#micro-coil}
=====================================================
Up to this point, we were merely able to literally scratch the surface of our planet. The analogy of an apple seems fitting: We would have penetrated the apple to just below its peel. So, most of the interesting things are still deeper below, awaiting their discovery.\
This demand for higher pressures can easily be understood then we think about chemical bonding and crystal structures. To investigate transitions in the electronic or atomic environment of a solid, we need to be able to increase its energy density, i.e. the pressure, up to a point where atomic distances in a system are below a certain threshold, triggering phase transitions. Here, we are typically not only talking about structural phase transitions but also about higher order phase transitions like electronic or magnetic transitions[^5].\
Of course, the pressure needed to trigger these transitions changes from system to system. For example, the cuprate high-temperature super-conductors exhibit a layered structure with copper-oxygen layers separating their charge reservoirs[@Hazen1987]. These systems are prone to react rather sensitively to the application of pressure. For example, it was reported that the super-conducting transition temperature T$_c$ in Hg$_{1-x}$Pb$_x$Ba$_2$Ca$_2$Cu$_3$O$_{8+\delta}$ rises significantly under an application of about 30 GPa with a maximum in T$_c$ of 164 K [@Gao1994]. Inducing structural phase changes in these systems can potentially occur at pressures far below 10 GPa[^6], as it was reported in YBa$_2$Cu$_4$O$_8$[@Souliou2014], followed by a complete collapse of super-conductivity[@Tissen1991; @Mito2014]. More robust systems like atomic metals, e.g. sodium or lithium, require a much higher energy density before any electronic or structural changes can occur[@Ma2009; @Han2000].\
At this point, DAC-NMR appeared to be stuck in a crisis until into the late 2000s. It was quickly realised that two major problems should be solved, the first being to achieve stable pressures above 10 GPa with good sensitivities, allowing for realistic and time-saving experiments on nuclei other than $^1$H or $^{19}$F, with sample dimensions rapidly decreasing due to the demand for higher pressures. The second issue was a minimisation of the diamond anvils to a point below what was possible at that time in NMR spectroscopy.\
Apparently, the most promising solution was to use RF micro-coils as close as possible to the sample, even if that would mean to place them directly in the sample chamber. From an NMR perspective, the use of micro-coils is preferable to other methods, as they were shown to exhibit excellent mass sensitivities and large bandwidths due to their small size.[@Lacey1999; @Olson1995]. Placing such minuscule coils in the pressure chamber of a DAC, however, turned out to be a demanding task. To begin with, the micro-coils would have to be about a factor of 4 to 5 smaller, compared to micro-coils pioneered and characterised before [@Stocker1997; @Webb1997]. Furthermore, the issue of safely guiding the coil’s leads out of the chamber requires either the use of gold liners, which are prone to rupture under stress, or the carving of channels into the metallic gaskets, which is greatly compromising the overall stability of the DAC under load.\
In 2009 Suzuki et al.[@Suzuki2009] presented an intriguing study on the $^{51}$V-NMR of the one dimensional conductor $\beta$-Na$_{0.33}$V$_2$O$_5$ up to 8.8 GPa at cryogenic temperatures. In their figure 1b, a microcoil can be seen placed in the cavity of a Bridgman-type pressure cell[^7]. Unfortunately, the authors did not celebrate this ground-breaking advancement of the field with a separate publication, introducing this approach to a wider NMR community. That was done a short time later in the same year by another group using a strikingly similar set-up[@Haase2009].\
Both these set-ups were predominantly used by solid-state physicists investigating highly correlated electron systems at low temperatures. In 2011, about the time when I started to work in this field at the University of Leipzig, Meissner et al.[@Meissner2011] reported the pressure induced closing of the spin pseudo-gap in YBa$_2$Cu$_4$O$_8$ at pressures up to 6 GPa and temperatures of about 100 K.
![Left: Miniature non-magnetic DAC made from Titanium-6%wt Aluminium-4%wt Vanadium Right: Schematic diagram of all parts of the DAC. Figure from [@Meier2014] \[fig4\]](Figure1-eps-converted-to)
The pressure cells were manufactured to be fairly small, only 22 mm in length and 18 mm in diameter, and thus could be used in a small bore super-conducting NMR magnet, see figure \[fig4\]. Similar to the work of Suzuki et al. an average pressure of about 4 to 7 GPa could be realised easily[@Meissner2012; @Meier2016].\
Even more important than the achieved pressures, which were comparable to the set-ups discussed in the last chapter, was the finding that the microcoil set-up yielded very high signal-to-noise ratios, see figure \[fig5\], which could be translated to a spin sensitivity of about $10^{13} \textrm{spin}/\sqrt{\textrm{Hz}}$ which is almost four orders of magnitude lower, and thus much more sensitive compared to the set-ups shown in chapter \[midp\][@Meier2014a].\
![$^1$H-NMR spectrum after a single shot on water at ambient conditions at a magnetic field of 7 T. Left inset: proton background of the empty cell. Right inset: recorded proton nutation data. Figure from [@Meier2014] \[fig5\]](JMRFig3)
Working in an environment dominated by physicists, we focused our research not on structural determination, or more chemically motivated questions under pressure[^8]. Our main focus was the change of electronic properties of solids, i.e. pressure induced changes in a solids’ band structure, or changes in the occupancy of energy levels of a metals’ conduction electrons.\
If we think about a solid, be it a metal, semiconductor, or insulator, we realise immediately that reducing the inter-atomic distances will inevitably have a large effect on the solids band structure. The most pronounced of these effects is the transition from an insulator to a metal, i.e. a pressure triggered electron delocalisation. Decreasing the distances between atoms often leads to a broadening of valence and conduction bands, culminating in an overlap of both bands triggering electronic conductivity.\
Of course, such a drastic effect will also significantly influence observable NMR parameters, like spin relaxation, or resonance frequencies.\
Typically, we can define two distinctively different regimes in NMR. On the one hand, there are insulators, e.g. most organic material is insulating, where the shift of resonance frequency is mainly governed by the diamagnetic shielding of the nuclei by low energy paired-up electrons. Thus, the shift is often small, in the range of some ppm. Spin lattice relaxation is predominantly given by dipole-dipole coupling for $I=\frac{1}{2}$ nuclei or quadrupolar interactions for $I>\frac{1}{2}$ nuclei, and is often in the range from 1 ms up to hours[@Bloembergen1948].\
On the other hand, as was realised already in the early days in NMR, resonance frequencies in a metal are profoundly higher compared to a non-conducting salt containing the same nucleus[@Knight1949]. This so-called Knight shift, named after Walter D. Knight in 1949, is a direct consequence of Pauli-paramagnetism, i.e. the hyperfine interaction of the s-like conduction electrons with the nucleus. This electron-nuclear coupling in fact proved to be so dominant that observable Knight shifts are often two to four orders of magnitude higher than the chemical shifts of the insulating compounds of a given metal. Furthermore, Korringa found[@Korringa1950], based on nuclear relaxation theory from Heitler and Teller[@Heitler1936], that the spin lattice relaxation times in a metal must also be directly correlated to the hyperfine interaction felt by the nucleus. The famous Korringa relation combines both Knight shift $K$ and T$_1$, and shows that the ratio of $K^2$ and T$_1$ at constant temperatures only depend on natural constants, and the gyromagnetic ratios of the electron and the nuclei. As the Korringa relation should also be independent of volume, it is a perfect tool to probe and identify metallisation processes.\
Coincidentally, we were given a sample of nano-crystalline AgInTe$_2$ powder at the time, synthesized by our chemistry department in Leipzig University [@Schroder2013; @Welzmiller2014]. This compound, which is semiconducting at ambient conditions, was believed to become fully conducting at the chalcopyrite to rocksalt structural transition[@Range1971], occurring in a pressure range between 4 and 6 GPa.\
First experiments on AgInTe$_2$ powdered samples, which has not been characterised by NMR so far, showed that the $^{115}$In spectra displayed a first order quadrupole interaction – indium is nuclear spin 9/2 – with a quadrupole frequency $\nu_q$ of about 45 kHz, thus the 8 satellite transitions were found to be heavily broadened and merged into a broad symmetric background around the sharp central transition, see figure \[AgInTe\]. The spectra were found to be relatively strong shielded, having chemical shifts of about -400 ppm relative to an aqueous solution of an indium salt. Furthermore, T$_1$ relaxation times were found to be in the range of some 10 ms, indicating relaxtion mechanisms governed by quadrupole interaction.\
![Top panel: Recorded $^{115}$In-NMR powder spectra recorded using a quadrupole echo sequence at ambient pressure (red) and at 20 GPa (blue). The vertical shift was introduced for better comparison. Inset: Obtained frequency shift values recorded over the full pressure range in these experiments. Bottom panel: Magnetisation recovery curves obtained during inversion recovery experiments as a function of the separation pulse $\Delta$ between the 180$^{\circ}$ inversion pulse and the detection pulses. Inset: Korringa ratio as a function of pressure. The dotted line depicts the expected values from a free electron metal of Indium atoms. \[AgInTe\]](zusammenfassung.pdf)
Up to about a pressure of 4 to 5 GPa, these parameters were not found to change significantly. Above 5 GPa, however, both the resonance shift as well as T$_1$ changed rather drastically by about 9000 ppm higher in frequency, and two orders of magnitude faster relaxation times. Combining both effects, the Korringa relation was found to suddenly become volume independent, and not change in a pressure range from 8 to 20 GPa, indicationg electron delocalisation in AgInTe$_2$.\
Up to this point, we had not payed much attention to the problem of hydrostaticity for NMR experiments at pressures above 7 GPa. At these compressions, most of the commonly used pressure transmitting media, like glycerol or Daphne, are solidified. This leads to so-called dry contact of the diamond faces with the gasket and sample, and will result in pronounced pressure gradients. Unfortunately, the influence of non-hydrostatic pressure conditions on NMR spectra or relaxation mechanisms has not been investigated in detail so far. Nevertheless, there is mounting evidence of the importance of this issue for some systems.\
A good illustrative example is the behaviour of metallic aluminium under pressures up to 10 GPa. In 2014, Meissner et al.[@Meissner2014] presented experiments on the $^{27}$Al-NMR spectra of metallic Al powder. There, the authors claimed that the observed deviation of the Knight shift and the sudden increase in linewidth at about 4 GPa must be due to a so-called Lifshitz transition, which occurs if a van-Hove singularity of a given energy band in the solid’s band structure crosses the Fermi energy E$_F$, and becomes partly filled or unfilled. One might argue that such transitions should be ubiquitous in solids under pressure, as the band structure typically changes quite significantly under compression. However, direct experimental observation of such an effect has been scarce, because these effects are typically smeared out by thermal excitations close to E$_F$. Thus, they should only be observable at low temperatures of about 10 K or below. Unfortunately, no low temperature experiments could be published confirming these findings at 300 K.\
Careful re-examination of the experimental conditions, however, led to a slightly different, and much more simple, interpretation of these findings. In fact, the ’smoking-gun’ evidence correlating the transitions found with the experiment itself was that 4 GPa, the pressure where both observed effects on the $^{27}$Al spectra became dominant, coincides with the reported crystallisation point of the glycerol pressure medium used. Thus, new sets of experiments using paraffin oil as a pressure medium, which solidifies at much higher pressures of about 12 to 13 GPa, showed that both the deviation of the Knight shift as well as the increase in linewidth strongly followed the onset of non-hydrostatic pressure conditions. A more detailed account for these effects are given in [@Meier2017b]. Thus, by now, every NMR spectroscopist should be aware of the deceptional effects occuring when at sufficiently high pressures, their pressure media begin to cristallise.\
![Height of the sample cavity for different pressures for various gasket materials. Taken from [@Meier2015a] \[heights\]](height)
Within the five odd years of my working with micro-coils, some serious limitations became obvious. Due to the limited space available in the sample chambers in a DAC, very thin insulated wires had to be used to prepare the coils. Of course, companies providing thin insulation wires made from copper or gold are sparse, and acquiring larger amounts often rather expensive. Furthermore, only organic insulating materials were possible to deposit on the wires, thus limiting their application to low $\gamma_n$ NMR nuclei, because hydrogen backgrounds excessively overlapped with $^1$H-NMR spectra. Finally, copper wires sold by *Polyfil* could be acquired. With these 18 $\mu$m thick wires, insulated with Polyurethane, coils of 3 to 6 windings could be manufactured. Thus, the coils had approximate dimensions of 200 - 500 $\mu$m in diameter and 80 to 160 $\mu$m in height. To reach higher pressures, smaller culet faces must be applied, which reduces the initial sample cavity quite significantly. This leads to certain boundaries of the applicability of micro-coils in DACs. To give one example, using a pair of two 500 $\mu$m culeted anvils, reaching about 40 GPa on average, requires a sample volume of 160 $\mu$m in diameter and 40 to - 50 $\mu$m in height for best stability; but this would require micro-coils to be made having only two turns, which is almost impossible to manufacture.\
![Photograph of the composite gasket assembly. The diamonds used in this photograph were culeted to 600 $\mu$m. Taken from [@Meier2015a] \[gasket\]](gasket)
Another serious problem originated in the use of the gasket material. Due to its low magnetic susceptibility, Cu-Be chips were used as gaskets blanks. Unfortunately, this alloy turned out to be quite soft under compression, leading to sample height reductions of almost a factor of four within some GPa, see figure \[heights\]. Such a pronounced cavity collapse would lead to significant deformations of the RF micro-coils, leading to B$_1$ field inhomogeneities and, thus, reduced sensitivity within a single pressure run.\
Therewith, the quest for a method for gasket stabilisation was on. A possible solution was found by using so-called composite gaskets, which are made by replacing the pre-indented part of the gasket with a rigid matrix of a nano-crystalline ultra hard material, like diamond or c-BN. A photograph of such a gasket design is shown in figure \[gasket\]. Here, an amorphous mixture of $\alpha-Al_2O_3$ and epoxy was used within a DAC, using a pair of 600 $\mu$m anvils.\
It could be shown that these gaskets allow for significantly stabilised cavities as illustrated by the much bigger recovered gasket heights in figure \[heights\]. Figure \[FID\] shows time domain solid echoes, as well as their Fourier transform, at pressures up to 30 GPa.
![Upper part: $^{27}$Al solid echoes of Cr:Al$_2$O$_3$ up to 40 GPa. The spectrometer was blanked off for 5 $\mu$s. Lower panel: Corresponding NMR spectra after Fourier transform. Both spectra and time domain signals were offset in the y axis for better comparison. \[FID\]](FID)
As can be seen, even at 30 GPa, features in both time and frequency domain remain sharp, while at the same time allowing for rather high excitation bandwidths.\
These experiments mark the high point in micro-coil high pressure research. The difficulty in preparing the set-ups, however, greatly limited the applicability of this approach, since they seem to resemble an art form rather than a reliable and reproducible method of science. Also, the results shown in figure \[AgInTe\] and \[FID\] can actually be considered singular events, and an average pressure limit of about 7 to 8 GPa using micro-coils in DACs would be reasonable. Unfortunately, most prominent questions in contemporary physics, chemistry and the geosciences appear to happen at considerably higher pressures. Thus, in order to reach even deeper into the Earth, completely new resonator structures needed to be developed.\
The Mega-Bar regime awaits!
Think Mega-bars! {#mbar}
================
Our journey is almost at an end. Despite what the title of this chapter suggests, NMR experiments above 1 Mbar, or 100 GPa, have not been realised so far. But it is a close call.\
In the course of the last year, high pressure NMR gained momentum, and routinely reaching pressures above 30 to 40 GPa could be feasible even for a broader NMR community. But let us start at the beginning.\
At the end of the last chapter I have summarised attempts to conduct magnetic resonance at pressures well above 10 GPa. This research also coincided with me obtaining my PhD, and moving on to a purely high pressure oriented institute. Here, it instantly became clear that pressures well above the state-of-the-art must be realised. Having gathered all the experience from implementing micro-coils in DACs, we began looking for a completely different approach.\
Experimenting with planar micro-coils or even micro-striplines, it quickly became apparent that something much more robust must be used, as both micro coils and striplines did not sustain the exceedingly high deviatoric stresses in a DAC made for Mbar measurements. The revelation came in form of magnetic flux tailoring Lenz lenses (LL), pioneered by the group of Prof. Korvink in the KIT[@Spengler2017; @Korvink2017].\
These passive electro-magnetic devices are governed by Lenz’ law of induction, and could be shown to locally amplify the RF B$_1$ field at the sample chamber. This very handy focussing effect is predominantly given by the geometry of these lenses. In a two dimensional plane, an outer winding builds up current, which is induced within a small region along the rim of the lens by an excitation coil. The current is fed into the inner part of the lens, where an anti-winding deposits the magnetic field within the LL’s inner diameter[@Jouda2017]. Any sample within this inner hole would consequently feel a much higher B$_1$ than without a lens. In this sense, these LLs work as a flux transformer, and display an astonishing degree of flexibility in terms of their field of application.\
Of course, we immediately tried to implement these fascinating devices into one of our pressure cells. We used not a symmetric arrangement of diamond anvils at first, but two anvils of much different culet sizes. These so-called indenter cells are often used for feeding small wires into the sample cavities, as the gasket only deforms in the direction of the sharper anvil, leaving a lot of space under the gasket open for further manipulations. Thus, we decided to use a 800 $\mu$m base and a 250 $\mu$m primary anvil. The LLs were cut out of 5 $\mu$m thin gold foil – 600 $\mu$m outer diameter and 100 $\mu$m inner diameter – carefully placed at the centre of the base anvil, and aligned with the 100 $\mu$m sample hole of a rhenium gasket. The gasket was covered with a thin layer of korundum in order to electrically insulate it from the LL. The excitation coil was placed directly on the pavilion of the base anvil to achieve sufficient inductive coupling into the LL. Figure \[SciAdvfig1\] shows this set-up.
![Schematic explosion diagram of the resonator setup and the anvil/gasket arrangement. The blue and red arrows denote the directions of the external magnetic field B$_0$ and the RF magnetic field B$_1$, respectively, generated by the excitation coil and the lens, which is compressed between the rhenium gasket and the 800 $\mu$m culeted diamond anvil. The enlarged picture shows the RF arrangement of the excitation coil with the Lenz lens. Black arrows denote the directions of the high-frequency current. Taken from [@Meier2017] \[SciAdvfig1\]](SciAdvfig1)
This set-up proved not only to be exceedingly stable under compression, but also to yield much higher sensitivities compared to all other methods so far. Figure \[SciAdvfig567\]a shows one of the first test scans using paraffin oil as a sample. As can be seen, in the case when no LL is used (i.e. NMR experiments only performed with the bigger excitation coil residing at the base anvils pavilion), SNR is very poor, in the order of $10^{-3}$ after a single scan. Using the LL, however, leads to a significant increase in SNR by almost four orders of magnitude, corresponding to detection limits of only $10^{12}$ spins/$\sqrt{Hz}$.
The lenses were found to be mechanically stable under compression, see \[SciAdvfig567\]b. The overall shape of the LL was kept more or less intact up to a pressure of 72 GPa. Further increase in pressure led to the destruction of the anvils. In \[SciAdvfig567\]c, $^1$H-NMR spectra of paraffin at increasing pressures of up to 72 GPa are shown. These spectra are all acquired after a single scan and demonstrate impressively how the spin sensitivity basically remains constant under compression; this is in drastic contrast to what was observed with the micro-coil set-up.\
The attentive reader might have recognised that the spin sensitivities realised with the LL resonators are about 2 up to 4 orders of magnitude lower, thus more sensitive, compared to the micro-coil set-ups. The reason for this might strike some as trivial, but I would like to explain it nonetheless. Consider using a micro-coil of 150–200 $\mu$m in height and about 300-400 $\mu$m in diameter in a DAC equipped with two 800 $\mu$m anvils. The sample cavity in such a DAC is most often much bigger than what is considered a ’safe’ pressure cell[^9].\
Under compression, of course, such a DAC can be at best considered rather unstable, and the sample cavity is prone to collapse due to significant flow of the Cu-Be metal. Thus, every micro-coil will be subject to immense deformations, leading to a decrease in the ’effective’ sample cavity. To underline this thought, numerical simulations have been conducted of the RF B$_1$ fields in a micro coil set-up as well as for a LL-resonator, see figure \[fig9\].\
In accordance with similar calculations from van Bentum et al.[@VanBentum2007], the B$_1$ field map in the x-z plane of a flat micro-coil, with a length-to-diameter ratio of less than unity, the magnetic field is fairly inhomogeneously distributed, with the highest magnetic fields close to the respective windings. The effective observable sample volume, V$_{eff}$, with a B$_1$ homogeneity within 20% of the central field, is about 1.7 nl, which is about 14% of the total available sample space, and stores only 6% of the total magnetic field energy of the micro-coil. Moreover, as indicated by the “deformed resonator”, the B$_1$ field homogeneity greatly suffers from an irregular arrangement of the current-carrying wire segments of the micro-coil. This deformed state typically arises already at relatively low pressures. Depending on the choice of gasket materials and the geometry of the sample cavity, a collapse can occur at relatively low pressures of some GPa, which will lead to significant deformations of the interior of the cavity, including the placed micro-coils. In this particular case, V$_{eff}$ drops to a 1/20th of a percent because of significant B$_1$ field inhomogeneities, while at the same time storing only about 0.003% of the total magnetic field energy. In addition, at these compressions, the risk for coil gasket or inter turn short circuits increases rapidly[@Meier2016], rendering the application of micro-coils in DACs increasingly unreliable above 10 GPa.\
In the case of the Lenz lenses, on the other hand, the RF B$_1$ field appears to be homogeneous over more than at least 40 to 50% of the total sample cavity, storing about 30% of the magnetic field energy. Strikingly, under compression, the situation does not deteriorate significantly, and both the stored energy (≈35 %) and V$_{eff}$(≈ 47 %) remain almost constant. The B$_1$ field strengths of the LL resonators found in the simulations compare well with the actual field strengths found via nutation experiments, which is further evidence of the applicability of this approach.\
Application of LLs in DACs proved to be a real game changer. From now on, diamond anvil cells of standard design could be used, closely following well established preparation guide lines[@Eremets1996]. This opens, at least in principle, the door for NMR at 100 GPa!\
Let us proceed to the most recent technical advancement. Using a diamond indenter cell, as in the aforementioned case, poses some limitation on the maximal reachable pressure ranges (if two diamonds of different culets typically have a somewhat lower maximal pressure than two identical diamonds). For example, two diamonds of 500 $\mu$m culets might reach pressures of about 40 GPa. But if we were to substitute one anvil with a 700 $\mu$m culeted anvil, the maximal pressure could drop to below 30 GPa. Thus, if we want to penetrate the Mbar regime, we need to develop a method for using the LLs in a symmetrical arrangement of anvils.\
With the introduction of the so-called Double Stage Lenz Lens (DSLL) resonator, this problem could be solved[@Meier2018a]. The set-up uses a thin layer of deposited copper on the complete surface of the anvils. A structure of two lenses was cut into this layer using a focused ion beam. The first stage LL is situated along the pavilion of the anvil, with its inner diameter slightly below the diamond’s culet face. The second stage LL is the main driving LL and lies directly on the culet. It has an outer diameter of 250 $\mu$m and an inner diameter of 80 $\mu$m, closely following the sample hole diameter. The DSLL resonator is driven by a multi-turn copper coil of 4 mm in diameter placed around the diamond anvil. This resonator structure was also realised on the other anvil. After loading and pressurising, both driving coils were connected to form a Helmholtz coil arrangement. Figure \[cell\] shows photographs into the pressure cell equipped with such a DSLL resonator, and a BX90 pressure cell on a wide-bore NMR probe.\
The working principle of these DSLL-resonators is similar to the single LL resonators with an additional LL to further focus the B$_1$ field at the sample chamber. One could ask why an application of a single LL would not work as well; the reason is that a single LL running over the sharp edge of the culet would have been cut off under compressional strain. Furthermore, as this ripping off would most likely not occur in an evenly matter, the complete resonator would inevitably turn into a lossy inductor.
![Upper photo: completely prepared DSLL-resonator for a DAC equipped with two 250 $\mu$m culeted anvils. Lower photo: mounted pressure cell on a home-built NMR probe \[cell\]](cell2)
First experiments have been performed recently on an initial sample of 100 pl of water. Figure \[echo\] shows an obtained solid echo train at 90 GPa. For these experiments, only 1000 scans were accumulated, resulting in a SNR per shot of about 39, and a time domain limit of detection of $5\cdot10^{11}$spins/$\sqrt{Hz}$, which is about a factor of two lower than in the case of a single LL.
![Recorded solid echo train of high pressure ice X at 90 GPa. The spectrometer was blanked off for 4.6 µs after the second 90° pulse (grey area). Taken from [@Meier2018a]. \[echo\]](echotrain)
This last development is particularly easy to prepare, as no time consuming alignment processes, or meddling with micro-coils, is necessary. Furthermore, these are the first experiments with a standard DAC allowing to access the complete available pressure range of the chosen diamond. In this case, 90 GPa is widely considered to be the limit for standard Drukker-type diamonds of 250 $\mu$m without a bevel. In fact, X-ray absorption measurements on these cells at 90 GPa have shown the diamonds to be heavily stressed, leading to a cupping of the culet faces to a degree that both diamond rims are almost touching one another. Thus, a further increase in pressure will certainly result in complete destruction of the anvils.\
As already stated at the beginning of this chapter, the border to the Mbar regime has not been passed yet, but the recent developments leaves one to argue that this might only be a matter of time. Of course, application of smaller culet sizes, and thus higher pressures, will inevitably reduce the available initial sample volume; but on the other hand, the sensitivity of the DSLL-resonator will benefit from the increased proximity due to smaller inner diameters of the second stage LL, as well as a reduced distance between both pairs of LL on the advancing diamonds.\
A Few Last Words {#end}
================
Within this admittedly rather short overview about the technical evolution of high pressure NMR devices, we came across an extraordinary degree of ingenuity. Every little step towards the final installments has certainly been difficult, riddled with mishap and frustration. Nonetheless, the pressure possible nowadays surpasses what was thought possible some years ago almost a hundred fold.\
It is often said that progress in diamond anvil cell research is by no means revolutionary but evolutionary, every advancement in the field comes together with a new problem longing for a solution. This thought seems to be particularly accurate for high pressure NMR.\
At this point, DAC-NMR advanced both in terms of NMR sensitivity and in achievable pressures to an amount which was widely believed impossible. In fact, over the course of my relatively short academic career, I was met with both disbelieve and ridicule about my vision to perform NMR above 100 GPa. But, as years went by and the field went forward, the voices of the skeptics became increasingly quieter. And as we face the world of *real* high pressure, both NMR and high pressure communities slowly begin to show interest in each other.
Acknowledgements
================
I am particularly thankful to Leonid Dubrovinsky, who offered me a safe haven to pursue my research. Without his encouragement and enthusiasm, we would not stand were we are today. The same holds true for Jan Korvink, who showed me that it is always worthwhile to look beyond the own comfort zone and find solutions where you would not expect them. Also, I want to thank my wife who, despite endless proof readings, always encouraged me, even in times of upmost discord. Finally, I want to thank the editorial board of PNMR for the chance to write about my story and my contribution to, what I think, is one of the most intriguing new research brands for the magnetic resonance community.
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[^1]: The energy density of a magnetic field is given by $\rho_{B}=B^2/2\mu_0$, yielding $9.94\cdot 10^8 Jm^{-3}$ for 50 T which is roughly 1 GPa (pressure has the same dimension as energy density)
[^2]: I am omitting the specification of “nuclear” here on purpose, because the techniques presented are also applicable for pulsed ESR methods.
[^3]: We often refer to the culet as the flattened area on the tip of the diamond anvil.
[^4]: In fact, these resonators are often referred to as “key hole resonators”.
[^5]: which do not necessarily coincide with first order phase transitions.
[^6]: If the experiments were conducted carefully, and special care has been taken to ensure hydrostatic pressure conditions.
[^7]: which has the same working principle as a DAC, but uses metallic anvils often made from non-magnetic WC.
[^8]: Which will, without a doubt, yield an amazing amount of new phenomena even in the lower pressure range in the near future.
[^9]: In fact, we often prepared DACs like this to use as much sample as possible due to limited sensitivity.
|
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abstract: 'We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $\Gamma\wr\SG_n$ and the corresponding generalizations of quasi-symmetric functions. Finally, we obtain Hopf algebras of colored parking functions, colored non-crossing partitions and parking functions of type $B$.'
address: |
Institut Gaspard Monge, Université de Marne-la-Vallée\
5 Boulevard Descartes\
Champs-sur-Marne\
77454 Marne-la-Vallée cedex 2\
FRANCE
title: 'Free quasi-symmetric functions of arbitrary level'
---
Introduction
============
The Hopf algebra of Free quasi-symmetric functions $\FQSym$ [@NCSF6] is a certain algebra of noncommutative polynomials associated with the sequence $(\SG_n)_{n\geq0}$ of all symmetric groups. It is connected by Hopf homomorphisms to several other important algebras associated with the same sequence of groups : Free symmetric functions (or coplactic algebra) $\FSym$ [@PR; @NCSF6], Non-commutative symmetric functions (or descent algebras) $\NCSF$ [@NCSF1], Quasi-Symmetric functions $\QSym$ [@Ge84], Symmetric functions $\sym$, and also, Planar binary trees $\PBT$ [@LR1; @HNT], Matrix quasi-symmetric functions $\MQSym$ [@NCSF6; @Hiv], Parking functions $\PQSym$ [@KW; @NT], and so on.
Among the many possible interpretations of $\sym$, we can mention the identification as the representation ring of the tower of algebras $$\C \to \C\SG_1 \to \C\SG_2 \to \cdots \to \C\SG_n \to \cdots,$$ that is $$\sym \simeq \oplus_{n\geq0} R(\C\SG_n),$$ where $R(\C\SG_n)$ is the vector space spanned by isomorphism classes of irreducible representations of $\SG_n$, the ring operations being induced by direct sum and outer tensor product of representations [@Mcd].
Another important interpretation of $\sym$ is as the support of Fock space representations of various infinite dimensional Lie algebras, in particular as the level $1$ irreducible highest weight representations of $\glchap_\infty$ (the infinite rank Kac-Moody algebra of type $A_\infty$, with Dynkin diagram $\Z$, see [@Kac]).
The analogous level $l$ representations of this algebra can also be naturally realized with products of $l$ copies of $\sym$, or as symmetric functions in $l$ independent sets of variables $$(\sym)^{\otimes l} \simeq \sym(X_0 ; \ldots ; X_{l-1}) =: \sym^{(l)},$$ and these algebras are themselves the representation rings of wreath product towers $(\Gamma\wr\SG_n)_{n\geq0}$, $\Gamma$ being a group with $l$ conjugacy classes [@Mcd] (see also [@Zel; @Wang]).
We shall therefore call for short $\sym(X_0 ; \ldots ; X_{l-1})$ the algebra of symmetric functions of level $l$. Our aim is to associate with $\sym^{(l)}$ analogues of the various Hopf algebras mentionned at the beginning of this introduction.
We shall start with a level $l$ analogue of $\FQSym$, whose bases are labelled by $l$-colored permutations. Imitating the embedding of $\NCSF$ in $\FQSym$, we obtain a Hopf subalgebra of level $l$ called $\NCSF^{(l)}$, which turns out to be dual to Poirier’s quasi-symmetric functions, and whose homogenous components can be endowed with an internal product, providing an analogue of Solomon’s descent algebras for wreath products.
The Mantaci-Reutenauer descent algebra arises as a natural Hopf subalgebra of $\NCSF^{(l)}$ and its dual is computed in a straightforward way by means of an appropriate Cauchy formula.
Finally, we introduce a Hopf algebra of colored parking functions, and use it to define Hopf algebras structures on parking functions and non-crossing partitions of type $B$.
[*Acknowledgements*]{} This research has been partially supported by EC’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe".
Free quasi-symmetric functions of level $l$
===========================================
$l$-colored standardization
---------------------------
We shall start with an $l$-colored alphabet $$A = A^0 \sqcup A^1 \sqcup \cdots \sqcup A^{l-1},$$ such that all $A^i$ are of the same cardinality $N$, which will be assumed to be infinite in the sequel. Let $C$ be the alphabet $\{c_0,\ldots,c_{l-1}\}$ and $B$ be the auxiliary ordered alphabet $\{1,2,\ldots,N\}$ (the letter $C$ stands for *colors* and $B$ for *basic*) so that $A$ can be identified to the cartesian product $B\times C$: $$A \simeq B \times C = \{ (b,c), b\in B,\ c\in C \}.$$ Let $w$ be a word in $A$, represented as $(v,u)$ with $v\in B^*$ and $u\in C^*$. Then the *colored standardized word* $\cstd(w)$ of $w$ is $$\cstd(w) := (\Std(v),u), $$ where $\Std(v)$ is the usual standardization on words.
Recall that the standardization process sends a word $w$ of length $n$ to a permutation $\Std(w)\in\SG_n$ called the *standardized* of $w$ defined as the permutation obtained by iteratively scanning $w$ from left to right, and labelling $1,2,\ldots$ the occurrences of its smallest letter, then numbering the occurrences of the next one, and so on. Alternatively, $\Std(w)$ is the permutation having the same inversions as $w$.
$\FQSym^{(l)}$ and $\FQSym^{(\Gamma)}$
--------------------------------------
A *colored permutation* is a pair $(\sigma,u)$, with $\sigma\in\SG_n$ and $u\in C^n$, the integer $n$ being the *size* of this permutation.
The *dual free $l$-quasi-ribbon* $\G_{\sigma,u}$ labelled by a colored permutation $(\sigma,u)$ of size $n$ is the noncommutative polynomial $$\G_{\sigma,u} := \sum_{w\in A^n ; \cstd(w)=(\sigma,u)} w \quad\in\Z{\langleA\rangle}.$$
Recall that the *convolution* of two permutations $\sigma$ and $\mu$ is the set $\sigma\convol\mu$ (identified with the formal sum of its elements) of permutations $\tau$ such that the standardized word of the $|\sigma|$ first letters of $\tau$ is $\sigma$ and the standardized word of the remaining letters of $\tau$ is $\mu$ (see [@Reu]).
\[prodG\] Let $(\sigma',u')$ and $(\sigma'',u'')$ be colored permutations. Then $$\G_{\sigma',u'}\,\,\G_{\sigma'',u''} = \sum_{\sigma\in \sigma'\convol\sigma''}
\G_{\sigma,u'\cdot u''},$$ where $w_1\cdot w_2$ is the word obtained by concatenating $w_1$ and $w_2$. Therefore, the dual free $l$-quasi-ribbons span a $\Z$-subalgebra of the free associative algebra.
Moreover, one defines a coproduct on the $\G$ functions by
$$\label{deltaG}
\Delta \G_{\sigma,u} := \sum_{i=0}^n
\G_{(\sigma,u)_{[1,i]}} \otimes \G_{(\sigma,u)_{[i+1,n]}},$$
where $n$ is the size of $\sigma$ and $(\sigma,u)_{[a,b]}$ is the standardized colored permutation of the pair $(\sigma',u')$ where $\sigma'$ is the subword of $\sigma$ containing the letters of the interval $[a,b]$, and $u'$ the corresponding subword of $u$.
For example, $$\begin{split}
\Delta \G_{3142,2412} =
&\ 1\otimes \G_{3142,2412} + \G_{1,4}\otimes \G_{231,212} +
\G_{12,42}\otimes \G_{12,21} \\
& + \G_{312,242}\otimes \G_{1,1} + \G_{3142,2412}\otimes 1.
\end{split}$$
The coproduct is an algebra homomorphism, so that $\FQSym^{(l)}$ is a graded bialgebra. Moreover, it is a Hopf algebra.
The *free $l$-quasi-ribbon* $\F_{\sigma,u}$ labelled by a colored permutation $(\sigma,u)$ is the noncommutative polynomial $$\F_{\sigma,u} := \G_{\sigma^{-1},u\cdot\sigma^{-1}},$$ where the action of a permutation on the right of a word permutes the positions of the letters of the word.
For example, $$\F_{3142,2142} = \G_{2413,1422}\,.$$
The product and coproduct of the $\F_{\sigma,u}$ can be easily described in terms of shifted shuffle and deconcatenation of colored permutations.
Let us define a scalar product on $\FQSym^{(l)}$ by $$\langle \F_{\sigma,u} , \G_{\sigma',u'} \rangle :=
\delta_{ \sigma,\sigma'} \delta_{u,u'},$$ where $\delta$ is the Kronecker symbol.
For any $U,V,W\in\FQSym^{(l)}$, $$\langle \Delta U, V\otimes W \rangle =
\langle U, V W \rangle,$$ so that, $\FQSym^{(l)}$ is self-dual: the map $\F_{\sigma,u} \mapsto {\G_{\sigma,u}}^*$ is an isomorphism from $\FQSym^{(l)}$ to its graded dual.
Let $\bij$ be any bijection from $C$ to $C$, extended to words by concatenation. Then if one defines the free $l$-quasi-ribbon as $$\F_{\sigma,u} := \G_{\sigma^{-1},\bij(u)\cdot\sigma^{-1}},$$ the previous theorems remain valid since one only permutes the labels of the basis $(\F_{\sigma,u})$.
Moreover, if $C$ has a group structure, the colored permutations $(\sigma,u)\in\SG_n\times C^n$ can be interpreted as elements of the semi-direct product $H_n := \SG_n\ltimes C^n$ with multiplication rule $$(\sigma ; c_1,\ldots,c_n) \cdot (\tau ; d_1,\ldots,d_n) :=
(\sigma\tau ; c_{\tau(1)}d_1, \ldots, c_{\tau(n)}d_n).$$ In this case, one can choose $\bij(\gamma):=\gamma^{-1}$ and define the scalar product as before, so that the adjoint basis of the $(\G_{h})$ becomes $\F_h := \G_{h^{-1}}$. In the sequel, we will be mainly interested in the case $C:=\Z/l\Z$, and we will indeed make that choice for $\bij$.
Algebraic structure
-------------------
Recall that a permutation $\sigma$ of size $n$ is *connected* [@MR; @NCSF6] if, for any $i<n$, the set $\{\sigma(1),\ldots,\sigma(i)\}$ is different from $\{1,\ldots,i\}$.
We denote by $\conn$ the set of connected permutations, and by $c_n:=|\conn_n|$ the number of such permutations in $\SG_n$. For later reference, we recall that the generating series of $c_n$ is $$c(t) := \sum_{n\ge 1} c_n t^n
= 1 - \left(\sum_{n\ge 0} n! t^n\right)^{-1}\\
= t+{t}^{2}+3\,{t}^{3}+13\,{t}^{4}+71\,{t}^{5}+461\,{t}^{6} +O(t^7)\,.$$
Let the *connected colored permutations* be the $(\sigma,u)$ with $\sigma$ connected and $u$ arbitrary. Their generating series is given by $c(lt)$.
It follows from [@NCSF6] that $\FQSym^{(l)}$ is free over the set $\G_{\sigma,u}$ (or $\F_{\sigma,u}$), where $(\sigma,u)$ is connected.
Since $\FQSym^{(l)}$ is self-dual, it is also cofree.
Primitive elements
------------------
Let $\L^{(l)}$ be the primitive Lie algebra of $\FQSym^{(l)}$. Since $\Delta$ is not cocommutative, $\FQSym^{(l)}$ cannot be the universal enveloping algebra of $\L^{(l)}$. But since it is cofree, it is, according to [@LRdip], the universal enveloping dipterous algebra of its primitive part $\L^{(l)}$. Let $d_n = \dim\, \L^{(l)}_n$.
Recall that the *shifted concatenation* $w\bullet w'$ of two elements $w$ and $w'$ of $\N^*$, is the word obtained by concatenating to $w$ the word obtained by shifting all letters of $w'$ by the length of $w$. We extend it to colored permutations by simply concatenating the colors and concatenating *with shift* the permutations. Let $\G^{\sigma,u}$ be the multiplicative basis defined by $\G^{\sigma,u}=\G_{\sigma_1,u_1}\cdots\G_{\sigma_r,u_r}$ where $(\sigma,u)=(\sigma_1,u_1)\bullet\cdots\bullet(\sigma_r,u_r)$ is the unique maximal factorization of $(\sigma,u)\in\SG_n\times C^n$ into connected colored permutations.
Let $\V_{\sigma,u}$ be the adjoint basis of $\G^{\sigma,u}$. Then, the family $(\V_{\alpha,u})_{\alpha\in\conn}$ is a basis of $\L^{(l)}$. In particular, we have $d_n=l^n c_n$.
As in [@NCSF6], we conjecture that $\L^{(l)}$ is free.
Non-commutative symmetric functions of level $l$
================================================
Following McMahon [@McM], we define an *$l$-partite number* $\npn$ as a column vector in $\N^l$, and a *vector composition of $\npn$* of weight $|\npn|:=\sum_{1}^l{n_i}$ and length $m$ as a $l\times m$ matrix $\bf I$ of nonnegative integers, with row sums vector $\npn$ and no zero column.
For example, $$\label{exM}
{\bf I} =
\begin{pmatrix}
1 & 0 & 2 & 1 \\
0 & 3 & 1 & 1 \\
4 & 2 & 1 & 3 \\
\end{pmatrix}$$ is a vector composition (or a $3$-composition, for short) of the $3$-partite number $\begin{pmatrix} 4\\ 5\\ 10\end{pmatrix}$ of weight $19$ and length $4$.
For each $\npn\in\N^l$ of weight $|\npn|=n$, we define a level $l$ *complete homogeneous symmetric function* as $$S_{\npn} := \sum_{u ; |u|_i=n_i} \G_{1\cdots n, u}.$$ It is the sum of all possible colorings of the identity permutation with $n_i$ occurrences of color $i$ for each $i$.
Let $\NCSF^{(l)}$ be the subalgebra of $\FQSym^{(l)}$ generated by the $S_{\npn}$ (with the convention $S_{\bf 0}=1$). The Hilbert series of $\NCSF^{(l)}$ is easily found to be $$S_l(t) := \sum_{n} {\dim\ \NCSF_n^{(l)}t^n} = \frac{(1-t)^l}{2(1-t)^l-1}.$$
$\NCSF^{(l)}$ is free over the set $\{S_{\npn}, |\npn|>0 \}$. Moreover, $\NCSF^{(l)}$ is a Hopf subalgebra of $\FQSym^{(l)}$.
The coproduct of the generators is given by $$\Delta S_\npn = \sum_{{\bf i}+{\bf j}= {\bf n}} S_{\bf i}\otimes S_{\bf j},$$ where the sum ${\bf i}+{\bf j}$ is taken in the space $\N^l$. In particular, $\NCSF^{(l)}$ is cocommutative.
We can therefore introduce the basis of products of level $l$ complete function, labelled by $l$-compositions $$S^{\bf I} = S_{{\bf i}_1} \cdots S_{{\bf i}_m},$$ where ${\bf i}_1,\cdots,{\bf i}_m$ are the columns of $\bf I$.
If $C$ has a group structure, $\NCSF_n^{(l)}$ becomes a subalgebra of $\C[C\wr\SG_n]$ under the identification $\G_h \mapsto h$.
This provides an analogue of Solomon’s descent algebra for the wreath product $C\wr\SG_n$. The proof amounts to check that the Patras descent algebra of a graded bialgebra [@Patras] can be adapted to $\N^l$-graded bialgebras.
As in the case $l=1$, we define the *internal product* $*$ as being opposite to the law induced by the group algebra. It can be computed by the following splitting formula, which is a straightforward generalization of the level 1 version.
Let $\mu_r: (\NCSF^{(l)})^{\otimes r} \to \NCSF^{(l)}$ be the product map. Let $\Delta^{(r)} : (\NCSF^{(l)}) \to (\NCSF^{(l)})^{\otimes r}$ be the $r$-fold coproduct, and $*_r$ be the extension of the internal product to $(\NCSF^{(l)})^{\otimes r}$. Then, for $F_1,\ldots,F_r$, and $G\in\NCSF^{(l)}$,
$$(F_1\cdots F_r) * G = \mu_r [ (F_1\otimes\cdots\otimes F_r) *_r
\Delta^{(r)}G ].$$
The group law of $C$ is needed only for the evaluation of the product of one-part complete functions $S_{\bf m}*S_{\bf n}$.
With $l=2$ and $C=\Z/2\Z$,
$$\begin{split}
S^{\hbox{\scriptsize$\begin{pmatrix}1&0\\1&1\end{pmatrix}$}} *
S^{\hbox{\scriptsize$\begin{pmatrix}0&2\\1&0\end{pmatrix}$}} &=
\mu_2 \left[ \left(
S^{\hbox{\scriptsize$\begin{pmatrix}1\\1\end{pmatrix}$}}\otimes
S^{\hbox{\scriptsize$\begin{pmatrix}0\\1\end{pmatrix}$}} \right)
*_2
\Delta S^{\hbox{\scriptsize$\begin{pmatrix}0&2\\1&0\end{pmatrix}$}}
\right]\\
& = \left(S^{\hbox{\scriptsize$\begin{pmatrix}1\\1\end{pmatrix}$}}*
S^{\hbox{\scriptsize$\begin{pmatrix}2\\0\end{pmatrix}$}}\right)
\left(S^{\hbox{\scriptsize$\begin{pmatrix}0\\1\end{pmatrix}$}}*
S^{\hbox{\scriptsize$\begin{pmatrix}1\\0\end{pmatrix}$}}\right)
+ \left(S^{\hbox{\scriptsize$\begin{pmatrix}1\\1\end{pmatrix}$}}*
S^{\hbox{\scriptsize$\begin{pmatrix}0&1\\1&0\end{pmatrix}$}}\right)
\left(S^{\hbox{\scriptsize$\begin{pmatrix}0\\1\end{pmatrix}$}}*
S^{\hbox{\scriptsize$\begin{pmatrix}1\\0\end{pmatrix}$}}\right) \\
& = S^{\hbox{\scriptsize$\begin{pmatrix}1&1\\1&0\end{pmatrix}$}}
+ S^{\hbox{\scriptsize$\begin{pmatrix}1&1&0\\0&0&1\end{pmatrix}$}}
+ S^{\hbox{\scriptsize$\begin{pmatrix}0&0&0\\1&1&1\end{pmatrix}$}}.
\end{split}$$
Recall that the underlying colored alphabet $A$ can be seen as $A^0 \sqcup \cdots \sqcup A^{l-1}$, with $A^i = \{ a^{(i)}_j, j\geq1 \}$. Let ${\bf x} = (x^{(0)}, \ldots, x^{(l-1)})$, where the $x^{(i)}$ are $l$ commuting variables. In terms of $A$, the generating function of the complete functions can be written as $$\sigma_{\bf x}(A) = \prod_{i\geq0}^{\rightarrow}
\left(1-\sum_{0\leq j\leq l-1} x^{(j)} a_{i}^{(j)} \right)^{-1}
= \sum_\npn {S_{\bf n}(A) {\bf x}^{\bf n}},$$ where ${\bf x}^{\bf n} = (x^{(0)})^{n_0} \cdots (x^{(l-1)})^{n_{l-1}}$.
This realization gives rise to a Cauchy formula (see [@NCSF2] for the $l=1$ case), which in turn allows one to identify the dual of $\NCSF^{(l)}$ with an algebra introduced by S. Poirier in [@Poi].
Quasi-symmetric functions of level $l$
======================================
Cauchy formula of level $l$
---------------------------
Let $X = X^0 \sqcup \cdots \sqcup X^{l-1}$, where $X^i=\{ x_j^{(i)},j\geq1\}$ be an $l$-colored alphabet of commutative variables, also commuting with $A$. Imitating the level $1$ case (see [@NCSF6]), we define the Cauchy kernel
$$K(X,A) = \prod_{j\geq1}^{\rightarrow}
\sigma_{\left(x_j^{(0)}, \ldots, x_j^{(l-1)}\right)} (A).$$
Expanding on the basis $S^{\bf I}$ of $\NCSF^{(l)}$, we get as coefficients what can be called the *level $l$ monomial quasi-symmetric functions* $M_{\bf I}(X)$
$$K(X,A) = \sum_{\bf I} M_{\bf I}(X) S^{\bf I}(A),$$
defined by $$M_{\bf I}(X) = \sum_{j_1<\cdots<j_m}
{\bf x}^{{\bf i}_1}_{j_1} \cdots
{\bf x}^{{\bf i}_m}_{j_m},$$ with ${\bf I}=({\bf i}_1,\ldots,{\bf i}_m)$.
These last functions form a basis of a subalgebra $\QSym^{(l)}$ of $\K[X]$, which we shall call the *algebra of quasi-symmetric functions of level $l$*.
Poirier’s Quasi-symmetric functions
-----------------------------------
The functions $M_{\bf I}(X)$ can be recognized as a basis of one of the algebras introduced by Poirier: the $M_{\bf I}$ coincide with the $M_{(C,v)}$ defined in [@Poi], p. 324, formula (1), up to indexation.
Following Poirier, we introduce the level $l$ quasi-ribbon functions by summing over an order on $l$-compositions: an $l$-composition $C$ is finer than $C'$, and we write $C\finer C'$, if $C'$ can be obtained by repeatedly summing up two consecutive columns of $C$ such that no non-zero element of the left one is strictly below a non-zero element of the right one. This order can be described in a much easier and natural way if one recodes an $l$-composition ${\bf I}$ as a pair of words, the first one $d({\bf I})$ being the set of sums of the elements of the first $k$ columns of $\bf I$, the second one $c({\bf I})$ being obtained by concatenating the words $i^{{\bf I}_{i,j}}$ while reading of $\bf I$ by columns, from top to bottom and from left to right. For example, the $3$-composition of Equation (\[exM\]) satisfies $$d({\bf I}) = \{5, 10, 14, 19\} \text{\quad and\quad}
c({\bf I}) = 13333\, 22233\, 1123\, 12333\,.$$ Moreover, this recoding is a bijection if the two words $d({\bf I})$ and $c({\bf I})$ are such that the descent set of $c({\bf I})$ is a subset of $d({\bf I})$. The order previously defined on $l$-compositions is in this context the inclusion order on sets $d$: $(d',c)\finer (d,c)$ iff $d'\subseteq d$.
It allows us to define the *level $l$ quasi-ribbon functions* $F_{\bf I}$ by $$F_{\bf I} = \sum_{{\bf I'}\finer {\bf I}} M_{\bf I'}.$$ Notice that this last description of the order $\finer$ is reminiscent of the order $\finer'$ on descent sets used in the context of quasi-symmetric functions and non-commutative symmetric functions: more precisely, since it does not modify the word $c({\bf I})$, the order $\finer$ restricted to $l$-compositions of weight $n$ amounts for $l^n$ copies of the order $\finer'$. The computation of its Möbius function is therefore straightforward.
Moreover, one can directly obtain the $F_{\bf I}$ as the commutative image of certain $\F_{\sigma,u}$: any pair $(\sigma,u)$ such that $\sigma$ has descent set $d({\bf I})$ and $u=c({\bf I})$ will do.
The Mantaci-Reutenauer algebra
==============================
Let ${\bf e}_i$ be the canonical basis of $\N^l$. For $n\geq1$, let $$S_n^{(i)} = S_{n\cdot {\bf e}_i} \in\NCSF^{(l)},$$ be the *monochromatic complete symmetric functions*.
The $S_n^{(i)}$ generate a Hopf-subalgebra $\MR^{(l)}$ of $\NCSF^{(l)}$, which is isomorphic to the Mantaci-Reutenauer descent algebra of the wreath products $\SG_n^{(l)} = (\Z/l\Z) \wr\SG_n$.
It follows in particular that $\MR^{(l)}$ is stable under the composition product induced by the group structure of $\SG_n^{(l)}$. The bases of $\MR^{(l)}$ are labelled by colored compositions (see below).
The duality is easily worked out by means of the appropriate Cauchy kernel. The generating function of the complete functions is $$\sigma^{\MR}_{\bf x}(A) := 1 + \sum_{j=0}^{l-1} \sum_{n\geq1}
S_n^{(j)}.(x^{(j)})^n,$$ and the Cauchy kernel is as usual $$K^{\MR}(X,A) = \prod_{i\geq1}^\rightarrow \sigma^{\MR}_{{\bf x}_i}(A)
= \sum_{(I,u)} M_{(I,u)}(X) S^{(I,u)}(A),$$ where $(I,u)$ runs over colored compositions $(I,u) = ((i_1,\ldots,i_m),(u_1,\ldots,u_m))$ that is, pairs formed with a composition and a color vector of the same length. The $M_{I,u}$ are called the *monochromatic monomial quasi-symmetric functions* and satisfy $$M_{(I,u)}(X) = \sum_{j_1<\cdots<j_m}
(x_{j_1}^{(u_1)})^{i_1} \cdots (x_{j_m}^{(u_m)})^{i_m}.$$
The $M_{(I,u)}$ span a subalgebra of $\C[X]$ which can be identified with the graded dual of $\MR^{(l)}$ through the pairing $$\langle M_{(I,u)}, S^{(J,v)} \rangle = \delta_{I,J} \delta_{u,v},$$ where $\delta$ is the Kronecker symbol.
Note that this algebra can also be obtained by assuming the relations $$x_i^{(p)} x_i^{(q)} = 0, \text{\ for $p\not=q$}$$ on the variables of $\QSym^{(l)}$.
Baumann and Hohlweg have another construction of the dual of $\MR^{(l)}$ [@BH] (implicitly defined in [@Poi], Lemma 11).
Level $l$ parking quasi-symmetric functions
===========================================
Usual parking functions
-----------------------
Recall that a *parking function* on $[n]=\{1,2,\ldots,n\}$ is a word $\park=a_1a_2\cdots a_n$ of length $n$ on $[n]$ whose nondecreasing rearrangement $\park^\uparrow=a'_1a'_2\cdots a'_n$ satisfies $a'_i\le i$ for all $i$. Let $\PF_n$ be the set of such words. It is well-known that $|\PF_n|=(n+1)^{n-1}$.
Gessel introduced in 1997 (see [@Stan2]) the notion of *prime parking function*. One says that $\park$ has a *breakpoint* at $b$ if $|\{\park_i\le b\}|=b$. The set of all breakpoints of $\park$ is denoted by $\bp(\park)$. Then, $\park\in \PF_n$ is prime if $\bp(\park)=\{n\}$.
Let $\PPF_n\subset\PF_n$ be the set of prime parking functions on $[n]$. It can easily be shown that $|\PPF_n|=(n-1)^{n-1}$ (see [@Stan2]).
We will finally need one last notion: $\park$ has a *match* at $b$ if $|\{\park_i< b\}|=b-1$ and $|\{\park_i\leq b\}|\geq b$. The set of all matches of $\park$ is denoted by $\match(\park)$.
We will now define generalizations of the usual parking functions to any level in such a way that they build up a Hopf algebra in the same way as in [@NT].
Colored parking functions
-------------------------
Let $l$ be an integer, representing the number of allowed colors. A *colored parking function* of level $l$ and size $n$ is a pair composed of a parking function of length $n$ and a word on $[l]$ of length $l$.
Since there is no restriction on the coloring, it is obvious that there are $l^n (n+1)^{n-1}$ colored parking functions of level $l$ and size $n$.
It is known that the convolution of two parking functions contains only parking functions, so one easily builds as in [@NT] an algebra $\PQSym^{(l)}$ indexed by colored parking functions:
$$\G_{(\park',u')} \G_{(\park'',u'')} = \sum_{\park\in \park'\convol\park''}
\G_{(\park,u'\cdot u'')}.$$
Moreover, one defines a coproduct on the $\G$ functions by $$\Delta\G_{(\park,u)} = \sum_{i\in \bp(\park)}
\G_{(\park,u)_{[1,i]}} \otimes \G_{(\park,u)_{[i+1,n]}}$$ where $n$ is the size of $\park$ and $(\park,u)_{[a,b]}$ is the parkized colored parking function of the pair $(\park',u')$ where $\park'$ is the subword of $\park$ containing the letters of the interval $[a,b]$, and $u'$ the corresponding subword of $u$.
The coproduct is an algebra homomorphism, so that $\PQSym^{(l)}$ is a graded bialgebra. Moreover, it is a Hopf algebra.
Parking functions of type $B$
-----------------------------
In [@Rei], Reiner defined non-crossing partitions of type $B$ by analogy to the classical case. In our context, he defined the level $2$ case. It allowed him to derive, by analogy with a simple representation theoretical result, a definition of parking functions of type $B$ as the words on $[n]$ of size $n$.
We shall build another set of words, also enumerated by $n^n$ that sheds light on the relation between type-$A$ and type-$B$ parking functions and provides a natural Hopf algebra structure on the latter.
First, fix two colors $0<1$. We say that a pair of words $(\park,u)$ composed of a parking function and a binary colored word is a *level $2$ parking function* if
- the only elements of $\park$ that can have color $1$ are the matches of $\park$.
- for all element of $\park$ of color $1$, all letters equal to it and to its left also have color $1$,
- all elements of $\park$ have at least once the color $0$.
For example, there are $27$ level $2$ parking functions of size $3$: there are the $16$ usual ones all with full color $0$, and the eleven new elements $$\begin{split}
& (111,100), (111,110), (112,100), (121,100), (211,010),\\
& (113,100), (131,100), (311,010),
(122,010), (212,100), (221,100). \\
\end{split}$$
The first time the first rule applies is with $n=4$, where one has to discard the words $(1122,0010)$ and $(1122,1010)$ since $2$ is not a match of $1122$. On the other hand, both words $(1133,0010)$ and $(1133,1010)$ are $B_4$-parking functions since $1$ and $3$ are matches of $1133$.
The restriction of $\PQSym^{(2)}$ to the $\G$ elements indexed by level $2$ parking functions is a Hopf subalgebra of $\PQSym^{(2)}$.
Non-crossing partitions of type $B$
-----------------------------------
Remark that in the level $1$ case, non-crossing partitions are in bijection with non-decreasing parking functions. To extend this correspondence to type $B$, let us start with a non-decreasing parking function ${\bf b}$ (with no color). We factor it into the maximal shifted concatenation of prime non-decreasing parking functions, and we choose a color, here 0 or 1, for each factor. We obtain in this way $\binom{2n}{n}$ words $\pi$, which can be identified with *type $B$ non-crossing partitions*.
Let $${\bf P}^{\pi}=\sum_{{\bf a}\sim{\pi}}\F_{\bf a}\,$$ where $\sim$ denotes equality up to rearrangement of the letters. Then,
The ${\bf P}^{\pi}$, where $\pi$ runs over the above set of non-decreasing signed parking functions, form the basis of a cocommutative Hopf subalgebra of $\PQSym^{(2)}$.
All this can be extended to higher levels in a straightforward way: allow each prime non-decreasing parking function to choose any color among $l$ and use the factorization as above. Since non-decreasing parking functions are in bijection with Dyck words, the choice can be described as: each block of a Dyck word with no return-to-zero, chooses one color among $l$. In this version, the generating series is obviously given by $$\frac{1}{1- l\frac{1-\sqrt{1-4t}}{2}}.$$ For $l=3$, we obtain the sequence A007854 of [@Slo].
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|
---
abstract: |
Background
: The dense neutron-rich matter found in supernovae and inside neutron stars is expected to form complex nonuniform phases, often referred to as *nuclear pasta*. The pasta shapes depend on density, temperature and proton fraction and determine many transport properties in supernovae and neutron star crusts.
Purpose
: To characterize the topology and compute two observables, the radial distribution function (RDF) $g(r)$ and the structure factor $S(q)$, for systems with proton fractions $Y_p=0.10,\,0.20,\,0.30$ and $0.40$ at about one third of nuclear saturation density, $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$, and temperatures near $kT=1{\ensuremath{\, \mathrm{MeV}}}$.
Methods
: We use two recently developed hybrid CPU/GPU codes to perform large scale molecular dynamics (MD) simulations with 51200 and 409600 nucleons. From the output of the MD simulations we obtain the two desired observables.
Results
: We compute and discuss the differences in topology and observables for each simulation. We observe that the two lowest proton fraction systems simulated, $Y_p=0.10$ and $0.20$, equilibrate quickly and form liquid-like structures. Meanwhile, the two higher proton fraction systems, $Y_p=0.30$ and $0.40$, take a longer time to equilibrate and organize themselves in solid-like periodic structures. Furthermore, the $Y_p=0.40$ system is made up of slabs, *lasagna phase*, interconnected by defects while the $Y_p=0.30$ systems consist of a stack of perforated plates, the *nuclear waffle* phase.
Conclusions
: The periodic configurations observed in our MD simulations for proton fractions $Y_p\ge0.30$ have important consequences for the structure factors $S(q)$ of protons and neutrons, which relate to many transport properties of supernovae and neutron star crust. A detailed study of the waffle phase and how its structure depends on temperature, size of the simulation and the screening length showed that finite-size effects appear to be under control and, also, that the plates in the waffle phase merge at temperatures slightly above 1.0[$\, \mathrm{MeV}$]{} and the holes in the plates form an hexagonal lattice at temperatures slightly lower than 1.0[$\, \mathrm{MeV}$]{}.
author:
- 'A. S. Schneider'
- 'D. K. Berry'
- 'C. M. Briggs'
- 'M. E. Caplan'
- 'C. J. Horowitz'
bibliography:
- 'Waffle.bib'
title: Nuclear Waffles
---
Introduction {#sec:Intro}
============
It is widely accepted that dense neutron-rich matter forms during a core-collapse supernova and exists between the crust and the core of a neutron star. A combination of theoretical arguments and numerical simulations suggests that this type of matter forms complex nonuniform structures, nowadays referred to as *nuclear pasta*. These complex nonuniform structures form because the system is unable to minimize all its fundamental interactions [@Pethick19987]. Here the interactions are the attractive short-range nuclear force, $\mathcal{O}\sim 1{\ensuremath{\, \mathrm{fm}}}$, and the repulsive long-range Coulomb force, $\mathcal{O}\sim 10-100{\ensuremath{\, \mathrm{fm}}}$. There is an ongoing effort aiming to determine the possible shapes of the pasta, its phase-transitions and their properties as these are relevant to the equation of state of nuclear matter [@PhysRevLett.41.1623], neutrino opacities in supernovae [@PhysRevC.69.045804; @PhysRevC.83.035803] and electric transport in the neutron star crust [@pons2013highly].
Often, studies of nuclear pasta make use of symmetry arguments to determine what is the most favored structure at a given density, temperature and proton fraction. For example, some mean-field calculations solve the equations of motion of dense matter in a Wigner-Seitz approximation in one, two and three dimensions and choose the favored geometry as the one that minimizes the energy density of the system for a given density and proton fraction [@PhysRevC.72.015802; @PhysRevC.78.015802; @PhysRevC.79.035804; @PhysRevC.87.028801; @PhysRevC.89.045807]. Other works based on liquid-drop models and Thomas-Fermi approximation also have explicit assumptions about the geometrical shapes of nuclear pasta [@PhysRevLett.50.2066; @PTP.71.320; @1984PThPh..72..373O; @Watanabe2000455]. As noted by Williams and Koonin in Reference [@Williams1985844] these symmetry arguments limit the possible structures to uniform and five geometries: spheres (3D), cylinders (2D), plates (1D), tubes (2D) and bubbles (3D). Thus, they performed simulations without any *a priori* assumption on the pasta geometry and were able to show that the five assumed geometries are good descriptions of nuclear matter in certain density ranges.
The way the primitive cells are stacked in small volume simulations limits the configurations of the 3D phases to simple cubic lattices while the 2D phases can only form a square lattice. With this in mind Oyamatsu [*et al.*]{} added extra configurations in which the primitive cells could be stacked to include bcc and fcc lattice configurations to the 3D phases as well as an hexagonal lattice configuration to the two-dimensional phases [@1984PThPh..72..373O]. Though their calculations still had assumptions on the nuclear structures formed within their unit cells they concluded that for symmetric nuclear matter the 3D phases prefer to form bcc lattices while the 2D phases form hexagonal lattices. Recently Okamoto [*et al.*]{} used Thomas-Fermi approximation to determine pasta structures within a volume large enough to include more than a single periodic image of its structure [@Okamoto2012284]. Their computations, which had no assumptions about the pasta geometries or lattice structures formed, indicated that symmetric nuclear matter can transition from bcc to fcc structures in the three-dimensional phases and from a honeycomb to a square lattice in the two-dimensional phases. For lower proton fractions, $Y_p=0.10$ and $0.30$, they demonstrated that the fcc and simple cubic structures are favoured. Furthermore, in their search for the ground-state they found more exotic pasta shapes, albeit in a metastable state. Amongst the geometries obtained were dumbbell-like and diamond-like structures, as well as coexistence of phases of different dimensionalities, for instance mixtures of droplets and rods appeared at low densities and slabs and tubes at intermediate densities. Other works explored structures such as gyroid and double-diamond morphologies [@PhysRevLett.103.132501; @PhysRevC.83.065811] as the existence of these exotic shapes may have important implications to the pasta properties.
Advances in computational power in the past decade have allowed for sophisticated calculations beyond mean-field, Thomas-Fermi and liquid drop model approximations. These include fully self-consistent calculations using a Skyrme-Hartree-Fock+BCS calculation at finite temperature [@PhysRevC.79.055801; @PhysRevLett.109.151101] and time-dependent Hartree-Fock simulations [@PhysRevC.87.055805; @Schuetrumpf:2014aea; @Schutrumpf:2014vqa]. These computations showed a richer variety of pasta shapes than the five geometries typically reproduced. However, due to their complexity, these calculations are often limited to a single periodic structure so that the pasta shapes obtained may exhibit significant dependence on the finite-size of the simulation. In fact, it was recently showed by Molinelli [*et al.*]{} for molecular dynamics simulations of about less than 10000 nucleons that the pasta shapes may differ based on the geometry chosen for the simulation volume [@Molinelli:2014uta]. Therefore, it is necessary to perform simulations with a much larger number of nucleons so that finite-size effects can be overcome.
Because of limitations in computational power calculations with more than a few thousand nucleons are only manageable by considerably simplifying nucleon interactions. That can be attained by replacing the nucleon interactions by schematic forces that reproduce some of the properties of finite nuclei and nuclear matter, even if that implies ignoring shell effects and other important physics. This is what is done in works that study nuclear pasta using semi-classical molecular dynamics (MD) [@PhysRevC.69.045804; @PhysRevC.70.065806; @PhysRevC.72.035801; @PhysRevC.78.035806; @PhysRevC.86.055805; @GiménezMolinelli201431; @PhysRevC.88.065807; @PhysRevC.89.055801], quantum molecular dynamics (QMD) [@PhysRevC.57.655; @PhysRevC.66.012801; @PhysRevC.68.035806; @PhysRevC.69.055805; @PhysRevC.77.035806; @PhysRevLett.94.031101; @PhysRevLett.103.121101] and Monte-Carlo methods [@PhysRevC.88.025807; @PhysRevC.85.015807]. So far, the largest simulations reported in literature were performed by Horowitz [*et al.*]{} and included up to 100000 nucleons though it is not clear if those simulations were run for long enough for the system to equilibrate [@PhysRevC.70.065806; @PhysRevC.72.035801].
In a previous paper we studied nuclear pasta formation using MD [@PhysRevC.88.065807]. In that work we evolved dense matter with a proton fraction of $Y_p=0.40$ at a temperature of 1[$\, \mathrm{MeV}$]{} from high to low densities, $n=0.10{\ensuremath{\, \mathrm{fm}}}^{-3}$ to $n\sim0.01{\ensuremath{\, \mathrm{fm}}}^{-3}$, by expanding the simulation volume at different rates. We explicitly observed the nucleation mechanism as the pasta transitioned from one phase to the next and quantified the topologycal transitions by calculating Minkowski functionals on a suitable isosurface of the structures formed. Specifically, we noted that once the density reached approximately $n=0.04{\ensuremath{\, \mathrm{fm}}}^{-3}$ the system transitioned from plates (“lasagna” phase) to cylinders (“spaghetti phase”). During the transition, holes appeared in the lasagna plates and a phase similar to perforated plates or cross linked network of spaghetti formed. As density was decreased further the cross links disappeared to produce isolated nearly straight spaghetti strands. Study of finite size effects of this phase of perforated plates is the main focus of this work. This is done running MD simulations of 51200 and 409600 nucleons at a density of $n=0.05{\ensuremath{\, \mathrm{fm}}}^{-3}$ and proton fraction of $Y_p=0.30$ for up to $15\times10^6$ MD time steps. This proton fraction is slightly lower than in our previous work and was chosen as the cross-linked phase was more stable. Besides that it also allows us to compare the topology of our results to the work of Pais and Stone [@PhysRevLett.109.151101] and Schuetrumpf [*et al.*]{} [@PhysRevC.87.055805; @Schutrumpf:2014vqa] as they obtained a similar phase at similar proton fractions and densities. We also note here that Sebille [*et al.*]{} using a dynamic self-consistent mean-field model also obtained a similar phase of stacked perforated plates at this same density for symmetric nuclear matter [@PhysRevC.84.055801].
One of the purposes of performing large nuclear pasta simulations is to determine how the pasta phases affect observable quantities present in supernovae and neutron stars. While nucleon clustering and long range order of the pasta structures are relevant for neutrino pasta-scattering [@PhysRevC.69.045804], impurities and/or defects may be important for heat and electrical conductivity and pulsar spin-down [@pons2013highly; @PhysRevLett.93.221101]. Because MD allows for much larger simulations than possible with quantum calculations and it is straightforward to track the time evolution of the system, we can directly calculate observables like radial distribution function (RDF), $g(r)$, and its Fourier transform, the static-structure factor $S(q)$. Therefore, besides the study of the perforated plates phase, we also calculate the topology and observables of structures formed at a density of $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$ for four proton fractions, $Y_p=0.10,\,0.20\,0.30$ and $0.40$ and compare their properties. Time and frequency dependent observables may also be computed and will be the topic of a future work.
This manuscript is arranged as follows. In Section \[ssec:Formalism\] we review our MD formalism while Section \[ssec:codes\] is devoted to the CPU/GPU codes used in our simulations. Afterwards, in Section \[sec:Results\] we present our results. The section starts with a discussion of four 51200 nucleon simulations with different proton fractions, Section \[ssec:Yp\]. We then move our focus to simulations with proton fraction $Y_p=0.30$ of different sizes and screening lengths, Section \[ssec:30\], and finish with a discussion of some observables that can be obtained from the MD simulations, Section \[ssec:Obs\]. Finally, we conclude in Sec. \[sec:Conclusions\].
MD Code {#sec:Formalism}
=======
We start this section discussing the formalism used in our codes, Sec. \[ssec:Formalism\], a brief description of how we obtain the relevant Minkowski functionals, Sec. \[ssec:minkowski\] and then describe how the CPU/GPU codes work in Sec. \[ssec:codes\].
Formalism {#ssec:Formalism}
---------
Following is a review of our MD formalism, as it is the same as the one used by Horowitz [*et al.*]{} and others in previous works [@PhysRevC.69.045804; @PhysRevC.70.065806; @PhysRevC.72.035801; @PhysRevC.78.035806; @PhysRevC.86.055805; @GiménezMolinelli201431; @PhysRevC.88.065807; @PhysRevC.89.055801]. We use a cubic box with periodic boundary conditions to simulate systems of neutrons and protons immersed in a degenerate relativistic free Fermi electron gas. The nucleons are mass $M=939{\ensuremath{\, \mathrm{MeV}}}$ point-like particles that interact via two-body potentials of the form
$$\begin{aligned}
V_{np}(r)&=a e^{-r^2/\Lambda}+[b-c]e^{-r^2/2\Lambda}\\
V_{nn}(r)&=a e^{-r^2/\Lambda}+[b+c]e^{-r^2/2\Lambda}\\
V_{pp}(r)&=a e^{-r^2/\Lambda}+[b+c]e^{-r^2/2\Lambda}+\frac{\alpha}{r}e^{-r/\lambda}.\end{aligned}$$
The $n$ and $p$ indexes denote whether the potential is for a neutron-proton, neutron-neutron or proton-proton interaction. In the equations above, $r$ is the distance between the two nucleons and $a$, $b$, $c$ and $\Lambda$ are constants adjusted to approximately reproduce some bulk properties of pure neutron matter and symmetric nuclear matter as well as the binding energies of selected nuclei [@PhysRevC.69.045804]. Their values are given in Table \[Tab:parameters\]. As there have been studies on the dependence of the pasta phases on the density dependence of the nuclear symmetry energy, for an example see Reference [@PhysRevC.89.045807], we quote our value for this quantity: $S=40.7{\ensuremath{\, \mathrm{MeV}}}$. We also obtain a value of $K=372{\ensuremath{\, \mathrm{MeV}}}$ for the nuclear compressibility, although we do not expect our results to be very sensitive to this somewhat high value.
The proton-proton interaction $V_{pp}$ also has a term proportional to the fine-structure constant $\alpha$. This is the Coulomb repulsion between protons screened by the background electron gas. The screening has a characteristic length $\lambda$ that depends on the electron Fermi momentum $k_F=(3\pi^2n_e)^{1/3}$, where $n_e$ is the electron density and the electron mass $m_e$. Its value is $$\lambda=\frac{\pi^{1/2}}{2\alpha^{1/2}}\left(k_F\sqrt{k_F^2+m_e^2}\right)^{-1/2}
\label{eq:lambda}$$ In most previous works $\lambda$ was fixed to an arbitrary value $\lambda=10{\ensuremath{\, \mathrm{fm}}}$. Though we do that in Section \[ssec:Yp\], in Section \[ssec:30\] we compare our results for runs with both $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ and $\lambda=\lambda_{TF}$ given by Eq. , [*i.e.*]{} $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$.
[\*[4]{}[c]{}]{} $a$ (MeV) &$b$ (MeV)&$c$ (MeV)&$\Lambda$ (fm$^{2}$)\
110 & $-$26 & 24 & 1.25\
Minkowski functionals {#ssec:minkowski}
---------------------
To quantify the shapes of the structures formed in our simulations we use Minkowski functionals. In three dimensions any shape may be classified in terms of four Minkowski functionals: volume $V$, area $A$, mean breath $B$ and Euler characteristic $\chi$. In our simulations the occupied volume $V$ is defined by the region enclosed by a nuclear surface of total area $A$. Meanwhile, the mean breadth $B$ and Euler characteristic $\chi$ are, respectively, proportional to the surface integrals of the mean curvature $\tfrac{1}{2}(\kappa_\text{min}+\kappa_\text{max})$ and the Gaussian curvature $(\kappa_\text{min}\kappa_\text{max})$. Here $\kappa_\text{min}$ and $\kappa_\text{max}$ are the minimum and maximum values for the curvature on each point of the surface. Furthermore, the Gaussian curvature may be related to the number of structures or the connectivity of the shapes formed [@Michielsen2001461].
As in our previous work, Reference [@PhysRevC.88.065807], the nuclear surface is defined as isosurfaces of charge density $n_{\rm{ch}}=0.03{\ensuremath{\, \mathrm{fm}}}^{-3}$ obtained by folding a three-dimensional unitary Gaussian with standard deviation of $\sigma=1.5{\ensuremath{\, \mathrm{fm}}}$ around each proton of the system. The surface integrals were performed using the prespcription of Lang [*et al.*]{}[@lang01]. To track the evolution of a system and to directly compare the topology of simulations of different sizes we calculate the average mean curvature, $B/A$, and the average Gaussian curvature, $\chi/A$.
GPU codes {#ssec:codes}
---------
The most time consuming task when solving the equations of motion of the system described above is the computation of the forces acting on each nucleon. In this work we use an upgraded version of the Indiana University Molecular Dynamics (IUMD) Fortran code used in our previous paper, Reference [@PhysRevC.88.065807]. Amongst the upgrades are a neighbor-list scheme to calculate the nuclear forces using CPUs and the use of GPUs, whenever available, to calculate the long-range Coulomb interaction between protons. The details of the code are described in Sec. \[sssec:IUMD\]. We also describe another newly developed Fortran code, <span style="font-variant:small-caps;">CubeMD</span>, which also makes use of CPUs and GPUs. This code is discussed in Sec. \[sssec:CubeMD\]. In a forthcoming paper we will discuss the performance of each code as it depends on density, temperature and proton fraction of the simulation.
### The IUMD code {#sssec:IUMD}
The IUMD code has been developed for the past decade and has recently undergone a major reformulation to take full advantage of the <span style="font-variant:small-caps;">Big Red II</span> supercomputer acquired by Indiana University last year. <span style="font-variant:small-caps;">Big Red II</span> is a Cray XE6/XK7. The XE6 part of the machine consists of 344 dual CPU compute nodes, where each CPU is an Advanced Micro Devices 16-core Abu Dhabi Opteron. Each of these nodes has 64 GB of RAM. The XK7 part consists of 676 CPU/GPU compute nodes, each containing one 16-core AMD Interlagos Opteron CPU, one Nvidia Kepler K20 GPU, and 32 GB of RAM [@BigRed2]. IUMD is a parallel code that can run on either the dual CPU nodes, or the CPU/GPU nodes, using MPI (Message Passing Interface) to communicate between nodes, OpenMP threads on the 16 cores of each CPU, and Portland Group CUDA Fortran on each GPU. IUMD takes full advantage of the compute power of CPU/GPU nodes by calculating nuclear forces on the CPUs while computing the Coulomb forces on the GPUs via a straightforward particle-particle algorithm. On CPU-only nodes of <span style="font-variant:small-caps;">Big Red II</span>, and any other machines that do not have hybrid CPU/GPU architecture it is also possible to run the code using only CPUs.
Decomposition of the force calculation among compute nodes is best understood by thinking of all the two-particle interactions as making up a *force matrix*. Element $ij$ of the matrix corresponds to the force $\boldsymbol{f}_{ij}$ that *source* particle $j$ exerts on *target* particle $i$. Of course, sources and targets are the same $N$ particles overall, but thinking of them as sources acting on targets simplifies explanation. In the parallel code, the force matrix is decomposed into $P$ block rows and $Q$ block colums, where $PQ$ is the total number of MPI processes (one process per compute node). Each process is assigned one block, and is responsible for calculating the action of its $N/Q$ sources on its $N/P$ targets. In order to simplify communication between processes, as well as the Coulomb calculation on the GPUs, IUMD does not use Newton’s third law to calculate the reaction of targets on sources. This decomposition resembles a customary cell algorithm, except that the cells are abstract rather than a geometrical division of real space. Once assigned to a process, particles stay there; they do not need to be moved from process to process as they would in a spatially based cell algorithm.
After all processes have calculated the forces their sources exert on their targets, forces are summed along the $Q$ processes of each row to get the total force on each target. This is done by an MPI *allreduce* which leaves each process with the total force on each of its targets. Note that these are row-wise allreduces, so that in principle the $P$ allreduces can be done concurrently. Thus the code should scale to very large node counts on machines that can actually do them concurrently. A time step is finished by each process applying a velocity Verlet update to its targets, followed by another set of allreduces, this time along each block column, to copy the new target positions to sources belonging to that column. Since each column has the complete set of new target positions, these allreduces can also be done concurrently. The only time an allreduce over all processes is required, is when calculating total potential energy, or virial for the pressure. However, these calculations are needed relatively infrequently.
Each MPI process calculates Coulomb forces by a simple particle-particle algorithm. All source and target proton positions are sent to the GPU, which sums the force of all sources on each target, and returns the forces to the CPU. The GPU version of the code does not use a cut-off or other work reducing measure, so the Coulomb calculation has computational complexity $\mathcal{O}((Y_p N)^2/(PQ))$. In the CPU only version the performance of the Coulomb force calculations can be improved by setting a cut-off to the Coulomb interaction, though this cut-off still has to be large enough to allow distant protons to interact with each other. More details of the Coulomb force calculations using GPUs or CPUs were described in Reference [@berry2013experiences]. While the GPU calculates the Coulomb interaction, the CPU calculates nuclear forces via a cell and neighbor list algorithm. Since the nuclear force has a range of only few fermi, at most a few thousand source nucleons will be within range of each target nucleon, even at saturation density. Because sources are randomly distributed among $Q$ nodes of each block row of the force matrix, this is reduced to perhaps hundreds per target per node, making a neighbor list algorithm very efficient.
In detail, the code builds a neighbor list $L_i$ of all sources within a distance $r_{nuc}+\delta r_{nuc}$ of target $i$. The force on $i$ is calculated only from its interaction with sources in $L_i$ that are within distance $r_{nuc}$. We set $r_{nuc}=11.5{\ensuremath{\, \mathrm{fm}}}$ in all runs reported in this paper, as the nuclear force drops well below machine precision by this distance, even for IEEE 64-bit arithmetic. We could probably reduce $r_{nuc}$ to 9 or 8 fm, but took a conservative approach for these runs. Sources are included in $L_i$ from the buffer zone of thickness $\delta r_{nuc}$ about the interaction sphere so lists do not have to be rebuilt as nucleons move in and out of interaction range. Rather, list $L_i$ needs to be rebuilt only when the distance $i$ has moved, plus the maximum distance any source on a node has moved since the last build is greater than $\delta r_{nuc}$. We have found it more efficient to rebuild all lists on all nodes when any one of them needs rebuilding, as the list-building procedure takes some time, and interrupts flow of the simulation. This requires an MPI allreduce of a logical variable from each process telling whether it needs to do a rebuild. For the size of runs we have done, this all-process allreduce is not too time-consuming, and results in more efficient runs. However in principle, the decision to rebuild lists only needs to be done on a process-by-process basis.
As just described neighbor list builds would be of $\mathcal{O}(N^2/PQ)$ computational complexity, as distances between all targets and all sources on a node must be checked. This complexity is reduced considerably by coupling the algorithm with a cell algorithm. Each process divides the whole simulation volume into cells of width $r_{nuc}+ \delta r_{nuc}$, and figures out which cell each source is in. This is an order $\mathcal{O}(N/Q)$ operation. Then for each target $i$, only $i$’s cell and its 26 neighboring cells must be checked in order to build $L_i$. Note that this requires no communication between processes. Even though this reduces work required to build the $L_i$, builds should still be done as infrequently as possible, implying $\delta r_{nuc}$ should be large. However the number of sources in each list grows as $(r_{nuc}+ \delta r_{nuc})^3$, so $r_{nuc}+ \delta r_{nuc}$ should be kept small. We have chosen $\delta r_{nuc}=4.0{\ensuremath{\, \mathrm{fm}}}$, as a good trade-off of list size vs. frequency of builds. For $Q=1$ this would result in about $3\,200$ sources in each list for density $n=0.20 {\ensuremath{\, \mathrm{fm}}}^{-3}$, well above saturation density. Of these, only about $1\,350$ would be within interaction range $r_{nuc}$. Note that for parallel runs these numbers are reduced by the number $Q$ of MPI processes in each row. For the densities and temperatures we usually consider in our works the neighbor lists are rebuilt every dozen or so time steps depending on how close to equilibrium the system is.
The algorithm just described is a significant improvement over the one used in our previous paper where the distance over every pair of particles had to be calculated and the code would scale with $\mathcal{O}(N^2)$.
### The <span style="font-variant:small-caps;">CubeMD</span> code {#sssec:CubeMD}
The <span style="font-variant:small-caps;">CubeMD</span> code is also a hybrid CPU/GPU code that works similarly to the IUMD code. It calculates the nuclear forces on the CPUs while the GPUs take care of the Coulomb interactions amongst the protons. The difference is in how the nuclear forces are calculated; while the IUMD code builds neighbor lists for each nucleon the <span style="font-variant:small-caps;">CubeMD</span> code divides the simulations volume into cubes of sides of approximately 4[$\, \mathrm{fm}$]{}. Each nucleon is then tagged with a number that specifies which of the smaller cubes it belongs to. The force on a target nucleon is computed only for the potential due to source nucleons in the same cube as the target or in adjacent ones. The adjacent cubes are determined in such way as to preserve periodic boundary conditions. We note that as of now the <span style="font-variant:small-caps;">CubeMD</span> code uses only a single CPU/GPU compute node. Its performance is slightly better than the IUMD code running on a single compute node.
Results {#sec:Results}
=======
In this section we describe our simulations and what we have learned from them. Though all the results presented here are from simulations performed with the <span style="font-variant:small-caps;">IUMD</span> code we did obtain very similar results with the <span style="font-variant:small-caps;">CubeMD</span> code. However, we decided to omit those results from this work to make the presentation of our results clearer.
We start in Sec. \[ssec:Yp\] with a comparison of the topologies of systems evolved at a constant density of $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$ and temperature $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ for different proton fractions. The topology is characterized by the average mean $B/A$ and Gaussian $\chi/A$ curvatures [@Michielsen2001461]. In Sec. \[ssec:30\] we focus on systems with proton fraction $Y_p=0.30$. We compare how the average mean and Gaussian curvatures evolve for simulations of 51200 and 409600 nucleons from different initial conditions, sizes and screening length and discuss their topological structures. Finally we finish Sec. \[ssec:Obs\] discussing how we obtain the radial distribution functions (RDFs) $g(r)$ and the structure factors $S(q)$ from MD simulations.
Systems of different proton fractions {#ssec:Yp}
-------------------------------------
We start this section discussing the topologies of systems of different proton fractions. Using the IUMD code we simulated systems with 51200 nucleons in a cubic box with nucleon number density $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$, temperature $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ and proton fractions $Y_p=0.10,\,0.20,\,0.30$ and $0.40$. For the simulations discussed in this section we fixed the screening length to $10{\ensuremath{\, \mathrm{fm}}}$. Since a constant density of $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$ implies a box with length size $100.8{\ensuremath{\, \mathrm{fm}}}$ the ratio of box length $L$ to screening length $\lambda$ is approximately 10. Had we used the screening length $\lambda_{TF}$ obtained in the relativistic Thomas-Fermi approximation the ratio of box length $L$ to screening length $\lambda_{TF}$ would be somewhat smaller, see Table \[tab:comp\], and increase with lower proton fractions.
[D[.]{}[.]{}[1.2]{} D[.]{}[.]{}[2.2]{} D[.]{}[.]{}[1.2]{} D[.]{}[.]{}[1.2]{}]{} & & &\
0.10 & 19.610 & 5.14 & 10.08\
0.20 & 15.565 & 6.48 & 10.08\
0.30 & 13.597 & 7.41 & 10.08\
0.40 & 12.354 & 8.16 & 10.08\
From Equation and the results in Table \[tab:comp\] we note that, for electrically neutral ultra-relativistic systems ($k_F\gg m_e$) such as the ones where nuclear pasta forms, the Thomas-Fermi screening lengths is proportional to $Y_p^{-1/3}$, [*i.e.*]{} $\lambda_{TF}\propto Y_p^{-1/3}$. We also note that in the worst case scenario presented above, $Y_p=0.10$, the value of $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ is within a factor of two of the screening predicted by the Thomas-Fermi approximation. These values for the screening $\lambda_{TF}$ are much smaller than the ones estimated by Alcain [*et al.*]{} in Reference [@PhysRevC.89.045807] using a non-relativistic approximation, $m_e>>k_F$ in Eq. \[eq:lambda\]. In their work they simulated isospin symmetric nuclear matter which, in a relativistic approximation ($m_e<<k_F$), implies a screening length $\lambda_{TF}\sim11.5{\ensuremath{\, \mathrm{fm}}}$ at the density used in this work, $n=0.050{\ensuremath{\, \mathrm{fm}}}^3$. Thus, following their conclusions we expect that for large simulations such as the ones presented here, a screening of $10{\ensuremath{\, \mathrm{fm}}}$ should be sufficient to at least correctly predict the signs for the average mean and Gaussian curvatures of the systems with higher proton fractions, $Y_p\gtrsim0.30$. The differences between the predictions of fixing $\lambda$ for $Y_p=0.30$ and using the Thomas-Fermi approximation will be explored in Section \[ssec:30\].
![\[fig:Yp\] (Color on line) Charge density isosurfaces of runs with 51200 nucleons, mean density $n=0.05{\ensuremath{\, \mathrm{fm}}}^{-3}$, temperature $kT=1.00$ MeV, and proton fractions $Y_p = 0.10, 0.20, 0.30$ and 0.40 after $10^7$ fm/c evolution time. In this figure, and all similar ones throughout this paper, the golden surfaces represent isosurfaces of charge density $n_{\rm{ch}}=0.03{\ensuremath{\, \mathrm{fm}}}^{-3}$, while the cream color shows regions such that $n_{\rm{ch}}>0.03
{\ensuremath{\, \mathrm{fm}}}^{-3}$. All such figures were generated using ParaView [@Paraview].](figures/sidebyside1b.pdf){width="50.00000%"}
Each simulation described in this section was evolved for $10^7{\ensuremath{\, \mathrm{fm/c}}}$ in time steps of $2{\ensuremath{\, \mathrm{fm/c}}}$. The final configuration of each simulation is shown in Figure \[fig:Yp\]. To generate Figure 1, a gaussian of unit volume was folded about each proton. These gaussians were then summed at each point of a fine 3D grid overlaying the simulation volume, and an isosurface corresponding to charge density $n_{\rm{ch}}=0.03{\ensuremath{\, \mathrm{fm}}}^{-3}$ constructed. Details of this construction are given in Reference [@PhysRevC.88.065807]. We see that the lowest proton fraction, $Y_p=0.10$, formed a phase that consist of small deformed nuclei while the $Y_p=0.20$ system is mostly formed of deformed elongated nuclei that resemble the spaghetti phase. The two larger proton fractions, $Y_p=0.30$ and $0.40$, formed structures that spread along the whole length of the simulation volume; the $Y_p=0.40$ proton fraction formed flat sheets interconnected by defects, while the $Y_p=0.30$ proton fraction formed perforated plates we named *nuclear waffles*. The waffle phase is the subject of the following section while the defects in pasta structure will be the subject of a forthcoming paper.
We quantify the shapes formed by calculating the Minkowski functionals (area, mean curvature and Gaussian curvature) of the charge isosurface of density $n_{\rm{ch}}=0.030{\ensuremath{\, \mathrm{fm}}}^{-3}$. We calculate these quantities the same way as in Reference [@PhysRevC.88.065807], and refer the reader to that paper for details. The reason we evolved our simulations for $10^7$ fm/c was that that was the time the slowest converging run took to appear to equilibrate. While the Minkowski functionals of the $Y_p=0.10$ and $0.20$ runs stopped evolving after about $2\times10^5{\ensuremath{\, \mathrm{fm/c}}}$, the $Y_p=0.40$ run took about $2\times10^6{\ensuremath{\, \mathrm{fm/c}}}$ to reach equilibrium. The slowest converging run was the $Y_p=0.30$. The Minkowski functionals took about $5\times10^6{\ensuremath{\, \mathrm{fm/c}}}$ to reach an apparent asymptotic value. In Table \[tab:mf1\] we show the mean and Gaussian curvatures per unit area averaged over the last $10^6{\ensuremath{\, \mathrm{fm/c}}}$ of the run. We note that for the lowest proton fraction, $Y_p=0.10$ both values are positive, which means several separated convex structures. For $Y_p=0.20$ the average Gaussian curvature is very close to zero while the average mean curvature is positive. This is characteristic of convex structures that are on average flat along one direction, such as cylinders. Meanwhile, both the $Y_p=0.30$ and $Y_p=0.40$ systems have positive average mean curvature and negative Gaussian curvatures characteristic of network-like structures [@PhysRevLett.111.138301; @Schuetrumpf:2014aea]. We note that for the $Y_p=0.40$ system both curvatures are close to zero, as the system consists mostly of flat plates. As we shall see in our discussion of observables, Sec. \[ssec:Obs\], the $Y_p=0.10$ and $Y_p=0.20$ proton fraction simulations exhibit structure factors that resemble those of a liquid phase. Meanwhile, the $Y_p=0.30$ and $Y_p=0.40$ simulations display Bragg peaks in their structure factor characteristic of a phase with periodic structures.
[D[.]{}[.]{}[1.2]{} D[.]{}[.]{}[1.5]{} D[,]{}[5.5]{}]{} & &\
0.10 & 0.415(5) & 1.23(4) ,10\^[-2]{}\
0.20 & 0.170(1) & 6.(12.),10\^[-5]{}\
0.30 & 0.0718(9) & -1.15(3) ,10\^[-3]{}\
0.40 & 0.0127(3) & -3.51(3) ,10\^[-4]{}\
The *waffle* phase {#ssec:30}
------------------
In this section we focus on systems with proton fractions of $Y_p=0.30$ at a density of $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$. As seen in the previous section this system has an interesting topology formed of perforated plates we call the “waffle” phase. This phase lies in the transition between a phase formed of flat plates, “lasagna” phase, and one made up of elongated cylindrical nuclei, “spaghetti” phase.
We first discuss simulations performed at a temperature of $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ started from a random configuration. To study finite size effects we simulated systems of 51200 and 409600 nucleons and compared their topologies. We also compare the results obtained from systems that use the artificially decreased screening length $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ and those obtained from the relativistic Thomas-Fermi approximation, $\lambda_{TF}=13.6{\ensuremath{\, \mathrm{fm}}}$. All systems were evolved for about $3\times10^7{\ensuremath{\, \mathrm{fm/c}}}$ in time steps of $2{\ensuremath{\, \mathrm{fm/c}}}$.
Comparisons of their topologies can be seen in Figure \[fig:bcplot\]. The top plot, Figure , shows the mean curvature per unit area while the bottom one, Figure , shows the Gaussian curvature per unit area of the system as a function of simulation time. In Figure we see that all systems have initially a mean curvature $B/A\gtrsim0.08{\ensuremath{\, \mathrm{fm}}}^{-1}$ that decreases to $B/A\sim0.07{\ensuremath{\, \mathrm{fm}}}^{-1}$ as the system evolves. As expected the 51200 nucleon systems equilibrate faster than their 409600 counterparts. Note that here we define equilibrium state as the point where the average mean curvature of the system stops evolving significantly. In fact, the small systems with screening lengths $\lambda_{TF}=13.6{\ensuremath{\, \mathrm{fm}}}$ and $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ seem to have reached some sort of equilibrium state in about $2\times10^6{\ensuremath{\, \mathrm{fm/c}}}$ and $10^7{\ensuremath{\, \mathrm{fm/c}}}$, respectively. Meanwhile, the larger systems with screening lengths $\lambda_{TF}=13.6{\ensuremath{\, \mathrm{fm}}}$ and $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ take somewhat longer to equilibrate. While the first reaches equilibrium in $2\times10^7{\ensuremath{\, \mathrm{fm/c}}}$ it is not clear whether the second has reached equilibrium after $3\times10^7{\ensuremath{\, \mathrm{fm/c}}}$.
[figures/bplot.pdf]{} (25,62) [(a)]{}
[figures/cplot.pdf]{} (25,62) [(b)]{}
While we can infer a time scale for equilibration of the system from the average mean curvature of each simulation the average Gaussian curvatures only oscillate around an average value soon after the start of the simulation. As expected the average curvature values depend mostly on the screening length while the deviations from average depend on the number of nucleons in each simulation. In Table \[tab:30\] we show the average mean and Gaussian curvatures over the last one tenth of each run. We see that, even though its not clear whether the larger systems have equilibrated, all values agree well within their standard deviations.
[D[6.0]{} D[.]{}[.]{}[2.1]{} D[.]{}[.]{}[1.7]{} D[.]{}[.]{}[1.7]{}]{} & & &\
51200 & 10.0 & 0.714(14) & -0.113(3)\
51200 & 13.6 & 0.735(13) & -0.123(4)\
409600 & 10.0 & 0.700(05) & -0.113(1)\
409600 & 13.6 & 0.731(03) & -0.120(1)\
{width="100.00000%"}
In Figure \[fig:30\] we show the last configuration of each run from two different points of view. We observe that in every run the final state was formed of perforated plates parallel to each other. Furthermore, in the run with 51200 nucleons with $\lambda_{TF}=13.6{\ensuremath{\, \mathrm{fm}}}$, the plates are also parallel to one of the sides of the box. We also note that even after the long simulation time the larger run with $\lambda_{TF}=13.6{\ensuremath{\, \mathrm{fm}}}$ exhibited several defects that connected perforated plates aligned along two different directions. This will become clearer in the following discussion of observables, specifically the structure factor $S(q)$.
In order to test the stability of these phases we selected the last configuration of the two smaller simulations and slowly increased (decreased) their temperature from $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ to $kT=1.5{\ensuremath{\, \mathrm{MeV}}}$ ($kT=0.5{\ensuremath{\, \mathrm{MeV}}}$) at a rate of $d(kT)/dt=10^{-7}·{\ensuremath{\, \mathrm{MeV/(fm/c)}}}$. We then measured the topological characteristics as the system evolved. We noticed that when the temperature was increased some connections between adjacent plates appeared and at high enough temperatures the pattern of perforated parallel plates merged as the temperature reached $kT=1.3{\ensuremath{\, \mathrm{MeV}}}$, for an example see Figure \[fig:hot1\]. This transition is characterized by a sudden increase (decrease) in the average mean (Gaussian) curvatures away from their values at $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$, see Figure \[fig:plot\_kT\]. Meanwhile, when the temperature is decreased the holes in the perforated plates form a structure close to an hexagonal lattice. Note also that this 2D hexagonal lattice of holes is displaced by about half of a lattice spacing in nearest neighbor plates and, thus, is aligned to the holes in next-nearest neighbor plates. Though this happens in both the $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ and $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$ simulations it is easier to see what happens in the latter as the plates are parallel to one of the sides of the box. Therefore, we chose to show only the $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$ plates in Figure \[fig:cold1\]. It should be clear comparing the two figures that neighboring plates have holes displaced by half of a lattice spacing so that next-nearest neighbor plates have their holes aligned.
{width="100.00000%"}
[figures/bplot1.pdf]{} (25,62) [(a)]{}
[figures/cplot1.pdf]{} (25,62) [(b)]{}
[figures/plates\_1\_5.png]{} (0,90) [(a)]{}
[figures/plates\_4\_5.png]{} (0,90) [(b)]{}
Besides the simulations described above we also performed two simulations starting at temperature $kT=2.5$ MeV, and cooled at rate $d(kT)/dt = -10^{-7}$ MeV/(fm/c), in order to test the stability of the phase of perforated plates. We used 51200 nucleons for both simulations, with $\lambda=10.0$ fm for one and $\lambda=13.6$ fm for the other. We expected that cooling the system slowly enough would allow it to reach an equilibrium state similar to the one found in the constant temperature simulations when it reached $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$. We expected this since at higher temperatures it is easier for the system to jump the potential barrier that separates states with similar energies. Therefore, once the simulations reached a temperature of slightly below the plate melting temperatures of $kT=1.3{\ensuremath{\, \mathrm{MeV}}}$ we expected plates to form. However, this only happened for the simulation with $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$. In this case the topological characteristics of the system at $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ are very similar to those obtained by evolving a random configuration for a long time at $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$. The systems also look very similar: six parallel perforated plates though their potential energies are slightly different, see Table \[tab:comparison\]. This may be due to small differences in the number of nucleons on each plate. The time evolution of this system can be seen in Figure \[fig:kT2\].
{width="100.00000%"}
Meanwhile, when the run with screening length $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ reached a temperature of $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$, down from $kT=2.5{\ensuremath{\, \mathrm{MeV}}}$, it formed a phase that resembles more several interconnected spaghetti than the perforated plates obtained from evolving an initial random configuration for a long time at a constant $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ temperature. The difference in potential energy between the systems at 1.0[$\, \mathrm{MeV}$]{} is of the same order of magnitude as the systems run with $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$, see Table \[tab:comparison\]. This stresses the fact that the difference in energy of systems with significantly different topological characteristics is indeed small. The cooled system may not have reached the waffle phase due to a possible energy barrier once it formed interconnected spaghetti. The evolution of this system can be seen in Figure \[fig:kT1\]. Besides that we also plot the evolution of the topological characteristics of the cooled down systems in Figure \[fig:plot\_kT2\]. We see that at a temperature of 1.0[$\, \mathrm{MeV}$]{} the average curvatures of the system cooled down from 2.5[$\, \mathrm{MeV}$]{} are close to the ones obtained from the constant temperature runs for the simulation with screening $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$. On the other hand, there are significant differences for the average mean curvature of the two simulations that used a screening length of $\lambda=10{\ensuremath{\, \mathrm{fm}}}$. These values are also shown in Table \[tab:comparison\].
{width="100.00000%"}
[figures/bplot2.pdf]{} (25,62) [(a)]{}
[figures/cplot2.pdf]{} (25,62) [(b)]{}
[l D[.]{}[.]{}[2.1]{} D[.]{}[.]{}[1.7]{} D[.]{}[.]{}[1.7]{} D[.]{}[.]{}[1.7]{}]{} & & & &\
constant & 10.0 & 0.714(14) & -0.113(3) & -5.6304(1)\
cooled & 10.0 & 0.790(10) & -0.116(3) & -5.6246(6)\
constant & 13.6 & 0.735(13) & -0.123(4) & -2.5807(1)\
cooled & 13.6 & 0.736(12) & -0.124(3) & -2.5763(6)\
Observables {#ssec:Obs}
-----------
In this section we discuss two observables that can also help us quantify the different pasta structures. We start with the pair correlation function or radial distribution function (RDF) $g(r)$ and then discuss the structure factor of the pasta shapes $S(q)$.
The RDF $g(r)$ defines the normalized probability of finding a particle of type $a$ at a distance $r$ from a particle of type $b$, [*i.e.*]{}, $$g_{ab}(r)=\frac{1}{4\pi r^2}\frac{1}{N_aN_b}\sum_{i=1}^{N_a}\sum_{j=1}^{N_b}\langle\delta(\vert\boldsymbol{r}_i-\boldsymbol{r}_j\vert-r)\rangle.$$ If $a$ and $b$ are the same type then the sum runs over $i\ne j$ and $N_b=N_a-1$. In Figures , and , we compare, respectively, $g(r)$ for proton-proton, proton-neutron and neutron-neutron pairs for systems simulated with different proton fractions. In order to obtain the RDFs we analyzed the positions of all nucleons every 100 time steps over the last $10^6$ time steps of the run.
[figures/gppr.pdf]{} (20,50) [(a)]{}
[figures/gnpr.pdf]{} (20,50) [(b)]{}
[figures/gnnr.pdf]{} (20,50) [(c)]{}
First we compare the short range behavior of the RDFs. Note that the three systems with higher proton fractions, $Y_p=0.20$, $0.30$ and $0.40$, exhibit similar qualitative behaviors for short range correlations, $r\lesssim10{\ensuremath{\, \mathrm{fm}}}$; all of their maxima and minima in this region are approximately in the same places regardless of proton fraction, although the heights of these peaks and valleys changes significantly from one system to the next.
The behavior of the RDFs of the low proton fraction system, $Y_p=0.10$, is somewhat different to those of higher proton fractions. For instance, the positions of the first maxima and minima of the low $Y_p$ system of the proton-proton correlations do not match that of the other systems. Also, the neutron-neutron correlations of this system have only two maxima in the $r<10{\ensuremath{\, \mathrm{fm}}}$ range while the others have four. We also note that the neutron-neutron and neutron-proton RDF of the $Y_p=0.10$ reach their asymptotic limit of one ($g(r)\rightarrow1$) at $r\sim7{\ensuremath{\, \mathrm{fm}}}$, while the proton-proton correlations reach this limit at about twice that value. These features may be explained by the fact that the $Y_p=0.10$ system only forms small clusters that are not organized in any particular way and have a large amount of free neutrons in their proximity.
As for the long range behavior, the larger the proton fraction the larger are the oscillations around the asymptotic limit of $g(r)$. This is because the larger proton fraction systems, $Y_p=0.30$ and $0.40$, formed somewhat periodic structures within the simulation volume while the lower proton fraction systems, $Y_p=0.10$ and $0.20$, did not. Also, the long range correlations between proton-proton pairs are stronger than between neutron-neutron and neutron-proton pairs. This is due to two facts. First, there are free neutrons roaming the simulation volume not bound to any cluster and their numbers are larger for the lower the proton fraction. Second, only proton pairs have long range interactions and, therefore, long range correlations that involve a neutron depend on those being bound to nucleon clusters.
Besides the RDFs $g(r)$ we may also obtain the static structure factor $S_a(\boldsymbol{q})$ for nucleons of species $a=n,p$ of the system. This quantity is related to the Fourier transform of the pair correlation function $g_{aa}(r)$ [@PhysRevC.70.065806] $$S_a(\boldsymbol{q})=1+\rho_a\int_V(g_{aa}(r)-1)e^{i\boldsymbol{q}\cdot\boldsymbol{r}}d^3r.$$ The structure factor $S_n(\boldsymbol{q})$ of neutrons ($S_p(\boldsymbol{q})$ of protons) can be used to determine the scattering cross section of neutrinos (electrons) by the pasta shapes. While the neutron structure factor $S_n(\boldsymbol{q})$ may be used to compute neutrino mean-free paths in supernovae and how they are initially trapped, see Reference [@PhysRevC.70.065806], the proton structure factor $S_p(\boldsymbol{q})$ is used to compute thermal conductivity, shear viscosity and electrical conductivity of the pasta, see Reference [@PhysRevC.78.035806]. To first order, the cross section per neutron of a neutrino of energy $E$ scattered by the pasta is [@PhysRevC.70.065806] $$\frac{1}{N}\frac{d\sigma}{d\Omega}=S_n(\boldsymbol{q})\frac{G_F^2E^2}{4\pi^2}\frac{1}{4}(1+\cos\theta).$$ Here $G_F$ is the Fermi coupling constant, $\theta$ the scattering angle and $\boldsymbol{q}$ the momentum transferred to the system by the incident particle. The transferred momentum $\boldsymbol{q}$, the scattering angle $\theta$ and the incident energy are related by $$q^2=2E^2(1-\cos\theta).$$ Thus, a large structure factor at some transferred momentum $\boldsymbol{q}$ means a large probability that a scattered particle will transfer that momentum to the system. This occurs whenever the system has a (quasi) periodicity along a direction $\boldsymbol{r}$ such that $\boldsymbol{q}\cdot\boldsymbol{r}\simeq\pm2\pi$.
When calculating the structure factor directly from the Fourier transform of the RDFs obtained from the MD simulations one has to deal with significant finite-size effects as it is difficult to obtain $g(r)$ for $r>L/2$, where $L$ is the size of the simulations cube. This becomes even more troublesome for simulations with higher proton fractions where significant oscillations around the asymptotic limit continue for a distance $r$ much larger than the size of the box. Horowitz [*et al.*]{} in Reference [@PhysRevC.69.045804] tried to circumvent that by fitting an exponentially decaying sine function to the tail of $g(r)$. However, this was not helpful in our simulations with $Y_p\geq0.30$. In these cases we noticed that we missed important information about the Bragg peaks in the structure factors that were obtained from the method described next.
As in Horowitz [*et al.*]{}, Reference [@PhysRevC.78.035806], we calculate the neutron and proton structure factors $S_a(\boldsymbol{q})$ from the density-density correlation function $$\label{eq:Sq}
S_a(\boldsymbol{q})=\langle\rho^*_a(\boldsymbol{q})\rho_a(\boldsymbol{q})\rangle
-\langle\rho^*_a(\boldsymbol{q})\rangle\langle\rho_a(\boldsymbol{q})\rangle.$$ The equation above determines the density-density correlations of the neutron and proton densities in momentum space of the system, $$\rho_a(\boldsymbol{q})=\frac{1}{\sqrt{N_a}}\sum_{i=1}^{N_a}e^{i\boldsymbol{q}\cdot\boldsymbol{r}_i}.$$ In order to avoid finite size effects due to the finite simulation volumes we only take into account transferred momenta $\boldsymbol{q}$ such that $$\boldsymbol{q}=2\pi\left(\frac{n_x}{L_x},\frac{n_y}{L_y},\frac{n_z}{L_z}\right)$$ where the $n_i\in\mathbb{Z}$ and $L_i$ is the side of the box along the $i$ direction. This choice should be clear since $e^{i\boldsymbol{q}\cdot\boldsymbol{r}}=e^{i\boldsymbol{q}\cdot(\boldsymbol{r+\boldsymbol{L}})}$ for all $\boldsymbol{L}=(m_xL_x,m_yL_y,m_zL_z)$ with $m_i\in\mathbb{Z}$. Note that since our simulation volumes are cubic all $L_i=L$. In order to obtain the structure factors we saved the configurations of the 51200 nucleon runs every 10 time steps over the last $10^6$ time steps of each run. For the larger 409600 nucleon runs we saved $10^4$ configurations over the last $10^6$ time steps of each run.
In Figure \[fig:sq\] we plot the angle averaged structure factor $S(q)=\langle{S(\boldsymbol{q})}\rangle$ for protons, Figure , and neutrons, Figure , for the four simulations discussed in Section \[ssec:Yp\]. First we observe that the two simulations with lower proton fractions, $Y_p=0.10$ and $0.20$, have smooth structure factor curves that are characteristic of liquid-like systems. As seen in Figure \[fig:Yp\] neither of these two simulations formed periodic structures within the simulation volume. The peaks near $q=0.36{\ensuremath{\, \mathrm{fm}}}^{-1}$ arise from the average distance between the clusters formed, approximately $L/6$. The height of the peaks is proportional to the contrast in the proton and neutron densities. Therefore, since the $Y_p=0.20$ simulations formed larger clusters than the $Y_p=0.10$ system and the free neutrons gas between its clusters is less dense its peaks are larger.
Meanwhile, the other two simulations, $Y_p=0.30$ and $0.40$, have diffraction peaks characteristic of periodic or solid-like systems. These diffraction peaks come from the values of $\boldsymbol{q}$ perpendicular to the plates formed in the simulation volume. For example, in the $Y_p=0.30$ simulation at $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ the transferred momentum that contributes the most to the Bragg peak is the $\boldsymbol{q}=\pm\tfrac{2\pi}{L}(4,3,3)$. This can be checked by looking at the $Y_p=0.30$ configuration in Figure \[fig:Yp\]. Note that starting from one of the plates and moving up along the box one reaches another plate every $L/3$. If one moves along one of the horizontal axis we see plates separated by $L/3$ (left side of the figure) and $L/4$ (right side of the figure). Thus, $\boldsymbol{q}=\pm\tfrac{2\pi}{L}(4,3,3)$ produces the strongest Bragg peak. Its absolute value, $q=0.363{\ensuremath{\, \mathrm{fm}}}^{-1}$, can be used to estimate the distance $d=2\pi/q\simeq17.3{\ensuremath{\, \mathrm{fm}}}^{-1}$ between the plates. Also, one expects that for a transferred momentum $\boldsymbol{q}$ that is double of the first peaks, $\boldsymbol{q}=\pm\tfrac{2\pi}{L}(8,6,6)$, there would be another diffraction peak. Though this happens for the proton structure factor $S_p(q)$, it does not for the neutron structure factor $S_n(q)$. This may be due to how the bound neutrons move in the plates or the free neutrons move between them. Another important point is that there does not seem to be any significant diffraction peaks related to the holes in the plates. This is because at $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$ the shape and position of the holes is constantly changing. This might not be true for that system at lower temperature where the holes in the plates form a two dimensional lattice. Also, it is likely that at slightly higher temperatures than 1.0[$\, \mathrm{MeV}$]{} the diffraction peaks disappear altogether as the systems does not have any visible periodic structures within the simulation volume.
The structure factor of the $Y_p=0.40$ run exhibits several prominent peaks, the largest one being near $q=0.34{\ensuremath{\, \mathrm{fm}}}^{-1}$. This peak has significant contribution from four different orientations of $\boldsymbol{q}$: $\boldsymbol{q}_1=\pm\tfrac{2\pi}{L}(5,-2,-1)$, $\boldsymbol{q}_2=\pm\tfrac{2\pi}{L}(5,-2,1)$, $\boldsymbol{q}_3=\pm\tfrac{2\pi}{L}(5,2,-1)$ and $\boldsymbol{q}_4=\pm\tfrac{2\pi}{L}(5,2,1)$. The main contribution is from $\boldsymbol{q}_1$ while the other large contributions likely arise from the defects on the pasta structure. In this case, due to the very low number of free neutrons the diffraction peaks appear even in the neutron structure factor at twice and thrice (not shown) the value of $\boldsymbol{q}=0.34{\ensuremath{\, \mathrm{fm}}}^{-1}$.
[figures/spq.pdf]{} (20,50) [(a)]{}
[figures/snq.pdf]{} (20,50) [(b)]{}
Another comparison we make is between the structure factors obtained for all of the $Y_p=0.30$ simulations discussed in Section \[ssec:30\]. In our comparisons, see Figure \[fig:sq\_30\], we first note that finite size effects for the long wavelength limit, $q\lesssim0.30$, of both proton, Figure , and neutron, Figure , structure factors seem to be well constrained by our simulations. In this region the values for the structure factors only depend on our choice of screening length. On the other hand, for $q\gtrsim0.40$ all the curves are very close to each other provided we ignore the eventual diffraction peaks in the proton structure factors. As discussed above there are no diffraction peaks for the neutron structure factor for $q\gtrsim0.40$ for the $Y_p=0.30$ runs. Though we expect some differences in the structure factors of different runs with different screening lengths we also noted that the number of diffraction peaks and their height and position still depend on the size of the simulation. This implies that, as far as structure factors go, we may need even larger simulations in order to accurately quantify the pattern of diffraction peaks.
[figures/spq1.pdf]{} (20,50) [(a)]{}
[figures/snq1.pdf]{} (20,50) [(b)]{}
[D[.]{}[3.3]{} D[.]{}[.]{}[2.1]{} D[.]{}[.]{}[1.3]{} D[.]{}[3.0]{} D[.]{}[.]{}[2.4]{} D[.]{}[.]{}[2.4]{} c D[.]{}[.]{}[3.2]{} D[.]{}[.]{}[3.2]{} c D[.]{}[.]{}[3.2]{} D[.]{}[.]{}[3.2]{}]{} &&&&& &&& &&&\
&&&&&&&&&&&\
51.200 &10.0 & 0.358 & 48.& 31.3(4.0) & 34.8 (5.6) & $\pm\tfrac{2\pi}{L}(4,4,1)$ & 440.6 & 476.0 & $\pm\tfrac{2\pi}{L}(4,-4,-1)$ & 3.8 & 4.1\
51.200 &13.6 & 0.374 & 30.& 377.5(34.5) & 411.1(48.1) & $\pm\tfrac{2\pi}{L}(0,6,0)$ & 5566.5 & 6053.0 & $\pm\tfrac{2\pi}{L}(6,0,0)$ & 4.4 & 4.7\
409.600 &10.0 & 0.354 & 144.& 570.6(3.8) & 615.4(3.9) &$\pm\tfrac{2\pi}{L}(2,2,11)$ &40175. &43266. & $\pm\tfrac{2\pi}{L}(11,-2,-2)$& 4.7 & 5.1\
409.600 &13.6 & 0.363 & 48.& 403.8(5.6) & 437.2(14.8) &$\pm\tfrac{2\pi}{L}(0,10,-6)$& 9557.3 &10162. & $\pm\tfrac{2\pi}{L}(6,-6,-8)$ & 4.9 & 5.4\
In Table \[tab:peaks\] we tabulate properties of the diffraction peaks such as their height, position and which orientations of $\boldsymbol{q}$ contribute the most and the least to the peak. First we observe that in all runs the height of $S_p(q_{\text{max}})$ is about $10\pm2\%$ larger than $S_n(q_{\text{max}})$. Also, the main peak positions are within 1% of each other for the $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ runs and 3% for the $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$ runs. However, while the peak heights agree within 10% for the $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$ runs there is a factor of 18 in height difference between the 51200 and 409600 $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ runs. The short peak in the small run with $\lambda=10{\ensuremath{\, \mathrm{fm}}}$ screening length comes from a strong cancellation between the first and second terms in the right hand side of Equation for the orientation of $\boldsymbol{q}$ that contributes the most to the peak; see the $\boldsymbol{q}_{\text{max}}$ column in Table \[tab:peaks\]. In fact, while in this run both terms are within 10% of each other for both protons and neutrons, in the three other runs the first term is a factor of 10 to 60 larger than the second (not explicitly shown). Finally, we note that the smallest contribution to the peaks are often from orientations $\boldsymbol{q}_{\text{min}}$ such that $\boldsymbol{q}_{\text{min}}\cdot\boldsymbol{q}_{\text{max}}\simeq0$. The only run where this is not the case is the 409600 nucleon run with screening length $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$. However, even in this case the contribution to $S_n(q)$ and $S_p(q)$ from the orientation orthogonal to $\boldsymbol{q}_{\text{max}}$, $\boldsymbol{q}'=\pm\tfrac{2\pi}{L}(0,10,6)$, is of the same order of magnitude as the contribution from $\boldsymbol{q}_{\text{min}}$: $S_n(\boldsymbol{q}')=5.7{\ensuremath{\, \mathrm{fm}}}$ and $S_p(\boldsymbol{q}')=6.2{\ensuremath{\, \mathrm{fm}}}$.
There is a second range in momentum transfer $q$ where diffraction peaks appear for the proton structure factors. These are located at approximately twice in momentum transfer value as $q$ of the largest peak. As before there are also small differences in the number of peaks around the largest peak and in their positions and magnitudes.
Conclusions {#sec:Conclusions}
===========
Using the recently upgraded <span style="font-variant:small-caps;">IUMD</span> code and the newly developed <span style="font-variant:small-caps;">CubeMD</span> we studied nuclear systems at a density of $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$. First we discussed the differences in topologies (Minkowski functionals) of four 51200 nucleon simulations with different proton fractions at a temperature of $kT=1.0{\ensuremath{\, \mathrm{MeV}}}$. We observed that the system with a proton fraction of $Y_p=0.10$ formed several small deformed nuclei while the $Y_p=0.20$ system formed elongated nuclei that resembled spaghetti. Meanwhile, both the $Y_p=0.30$ and $0.40$ systems formed network-like structures that spread along the whole length of the simulation volume. By calculating the radial distribution function $g(r)$ we observed that the lower the proton fraction of the system the smaller were the long-range correlations. We also noted that proton-proton correlations $g_{pp}(r)$ exhibited oscillations around the asymptotic value of $g_{pp}(r)$ much larger than the neutron-neutron $g_{nn}(r)$ and neutron-proton $g_{np}(r)$ correlations. Also, except for the lowest proton fraction run, $Y_p=0.10$, all runs had a similar qualitative behavior for the short-range correlations. When we examined the structure factor $S(q)$ of the four systems it became evident that the two systems with lower proton fractions exhibited a liquid-like behavior while the two higher proton fraction systems showed diffraction peaks characteristic of periodic structures inside the simulations volume.
For systems of proton fraction $Y_p=0.30$ we first noticed that the time it takes for the system to equilibrate from a random initial configuration at a temperature of $1.0{\ensuremath{\, \mathrm{MeV}}}$ depended on system size (51200 or 409600) and screening length ($\lambda=10.0{\ensuremath{\, \mathrm{fm}}}$ or $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$) used. The system that reached equilibrium fastest was the 51200 nucleon run with $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$ screening length. It did that in about $2\times10^6{\ensuremath{\, \mathrm{fm/c}}}$. On the other hand, it was not clear whether the 409600 system with $\lambda=10.0{\ensuremath{\, \mathrm{fm}}}$ reached equilibrium after a $3\times10^7{\ensuremath{\, \mathrm{fm/c}}}$ simulation time. However, it was obvious that all $Y_p=0.30$ systems were converging to the same phase, a stack of perforated parallel plates. Though the plates formed were stable if the system was kept at a constant temperatures $kT\lesssim1.0{\ensuremath{\, \mathrm{MeV}}}$ they quickly merged at slightly higher temperatures, $kT\gtrsim1.30{\ensuremath{\, \mathrm{MeV}}}$. Also, while at temperatures of $kT\simeq1.0{\ensuremath{\, \mathrm{MeV}}}$ the number, position and shape of the holes were constantly changing. Once the system was cooled to slightly lower temperatures, $kT\lesssim0.75{\ensuremath{\, \mathrm{MeV}}}$, their positions became approximately fixed and their sizes and shapes were uniform, forming a two dimensional hexagonal lattice. Similar phases have been reported elsewhere in the literature for similar densities and proton fractions, see for example the cross-rods in Reference [@PhysRevLett.109.151101] and rod-2 phase in References [@PhysRevC.87.055805; @Schutrumpf:2014vqa]. However, those simulations had much smaller simulation volumes and, therefore, the two dimensional lattice structure formed by the perforations in the lattice may suffer from significant finite-size effects.
Finally we obtained the structure factor for the $Y_p=0.30$ systems of different sizes and screening lengths. We observed that the qualitative behavior of all structure factors were about the same. At low momentum transferred $q\lesssim0.3{\ensuremath{\, \mathrm{fm}}}^{-1}$ the structure factors for both neutrons, $S_n(q)$, and protons, $S_p(q)$, depended mostly on the screening length used and was almost independent on the system sizes for our runs. This should be clear since in our simulations the long range periodicity comes from the long range repulsive Coulomb forces and, thus, the distance between structures is highly dependent on the strength of the repulsion. At intermediate momentum transfer, $0.30{\ensuremath{\, \mathrm{fm}}}^{-1}\lesssim{q}\lesssim0.40{\ensuremath{\, \mathrm{fm}}}^{-1}$ the structure factors had large Bragg peaks caused by coherent scattering from the periodic structures. Their positions and magnitudes, as well as the vector $\boldsymbol{q}$ that contributed most to the peak, was different for one run to the next. While for the $\lambda=13.6{\ensuremath{\, \mathrm{fm}}}$ runs the magnitude of the peaks were within 10% of each other and their positions differed by about 3%, for the $\lambda=10.0{\ensuremath{\, \mathrm{fm}}}$ runs the peak positions were within 1% of each other and their heights differed by a factor of 18. At large momentum transfer, $q\gtrsim0.4{\ensuremath{\, \mathrm{fm}}}^{-1}$ the curves of $S_n(q)$ and $S_p(q)$ had the same qualitative behavior which was independent of the screening length used in the simulation, though differences would probably appear had the screening lengths been different enough. The proton structure factor showed a second range of peaks at about twice the value of the first diffraction peaks.
Thus, we conclude that we are able to simulate nucleon systems large enough and for enough time for them to appear to equilibrate. We were able to demonstrate with an independent method from others that there is a stable phase of perforated plates for proton fraction of $Y_p=0.30$ at density $n=0.050{\ensuremath{\, \mathrm{fm}}}^{-3}$ at low temperatures $kT\lesssim1.0{\ensuremath{\, \mathrm{MeV}}}$. We also showed how to predict qualitatively and quantitatively the diffraction peaks in the structure factor that should affect heat and thermal conductivities in a neutron star crust and the neutrino opacities of supernovae.
We would like to thank Indiana University for time to run our simulations on the <span style="font-variant:small-caps;">Big Red II</span> supercomputer and acknowledge that figures showing isosurfaces were generated using the software [@Paraview].
This research was supported in part by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute, and in part by the Indiana METACyt Initiative. The Indiana METACyt Initiative at IU is also supported in part by Lilly Endowment, Inc. This research was also supported by DOE grants DE-FG02-87ER40365 (Indiana University) and DE-SC0008808 (NUCLEI SciDAC Collaboration).
|
---
abstract: 'Nearly all statistical inference methods were developed for the regime where the number $N$ of data samples is much larger than the data dimension $p$. Inference protocols such as maximum likelihood (ML) or maximum a posteriori probability (MAP) are unreliable if $p=\order(N)$, due to overfitting. This limitation has for many disciplines with increasingly high-dimensional data become a serious bottleneck. We recently showed that in Cox regression for time-to-event data the overfitting errors are not just noise but take mostly the form of a bias, and how with the replica method from statistical physics one can model and predict this bias and the noise statistics. Here we extend our approach to arbitrary generalized linear regression models (GLM), with possibly correlated covariates. We analyse overfitting in ML/MAP inference without having to specify data types or regression models, relying only on the GLM form, and derive generic order parameter equations for the case of $L2$ priors. Second, we derive the probabilistic relationship between true and inferred regression coefficients in GLMs, and show that, for the relevant hyperparameter scaling and correlated covariates, the $L2$ regularization causes a predictable direction change of the coefficient vector. Our results, illustrated by application to linear, logistic, and Cox regression, enable one to correct ML and MAP inferences in GLMs systematically for overfitting bias, and thus extend their applicability into the hitherto forbidden regime $p\!=\!\order(N)$.'
address: |
$\dag$ Dept of Biophysics, Radboud University, 6525AJ Nijmegen, The Netherlands\
$\S$ Saddle Point Science Ltd, 35A South St, London W1K2XF, UK\
$\P$ London Inst for Mathematical Sciences, 35A South St, London W1K2XF, UK\
$\ddag$ Dept of Mathematics, King’s College London, London WC2R 2LS, UK
author:
- |
ACC Coolen$^{\dag\S\P}$, M Sheikh$^{\ddag}$, A Mozeika$^{\P}$,\
F Aguirre-Lopez$^{\ddag}$ and F Antenucci$^\S$\
title: |
Replica analysis of overfitting in\
generalized linear regression models
---
[*Keywords*]{}: Generalized linear models, overfitting, regression, replica method
Introduction
============
Extensive quantities of data are now available in many commercial, scientific and medical settings, due to the decreasing cost of high throughput measurement devices and data storage, and rapidly increased computing power. Here we will be concerned with data where each sample is a pair $(\bz,s)$, with $\bz\in\R^p$ (the input, or covariate vector) and with an output variable $s$. The latter can be real-valued, discrete, or even composite. The aim is to determine from a given set ${\mathscr{D}}=\{(\bz_1,s_1),\ldots, (\bz_N,s_N)\}$ of randomly drawn historic samples whether there is information in $\bz$ about $s$, and to predict the value of $s$ associated with any vector $\bz$. In parametric statistical inference one approaches this question by postulating a parametrized probabilistic model $p(s|\bz,\btheta)$ for the dependence of $s$ on $\bz$, followed by defining a function $\Omega(\btheta|{\mathscr{D}})$ whose minimization gives a sensible estimate for the parameters $\btheta$. In this work we will focus on generalized linear models (GLMs, [@MccullaghNelder]), which are regression models $p(s|\bz,\btheta)$ in which the covariates $\bz$ enter strictly via a linear combination $\bbeta\!\cdot\!\bz\!=\!\sum_{\mu=1}^p \beta_\mu z_\mu $, with coefficients $\bbeta\in\R^p$. For the $\Omega(\btheta|{\mathscr{D}})$ function, common choices are $\Omega(\btheta|{\mathscr{D}})=-\log p({\mathscr{D}}|\btheta)$ (ML regression) and $\Omega(\btheta|{\mathscr{D}})=-\log p(\btheta|{\mathscr{D}})$ (MAP regression). Here $p(\btheta|{\mathscr{D}})\propto p({\mathscr{D}}|\btheta)p(\btheta)$, and $p({\mathscr{D}}|\btheta)=\prod_i p(s_i|\bz_i,\btheta)$. MAP requires the specification of a prior parameter distribution $p(\btheta)$. ML and MAP can be seen as approximations of a computationally often intractable Bayesian approach, where one works with the full posterior distribution $p(\btheta|{\mathscr{D}})$. MAP replaces this posterior by a delta peak at the most probable point, and ML follows from MAP by choosing a flat prior $p(\btheta)$. ML performs well when $p\!\ll \!N$, but its estimators become increasingly incorrect in the high-dimensional regime, which for GLMs involves both the number $N$ of samples and the number $p$ of covariates diverging, with finite ratio $\zeta=p/N$. Remedial attempts can be categorized into corrective [@bartlett1953approximateI; @bartlett1953approximateII; @haldane1956sampling; @anderson1979logistic; @CoxSnell; @shenton1963higher; @shenton1969maximum; @gauss1991] or preventative ones [@firth1993bias]. The former route seeks to construct better estimators via a power series in $N^{-1}$ (for fixed $p$), with the ML estimator as zeroth term, but becomes unwieldy beyond the linear term [@BowmanShenton]. Also various computational resampling recipes were proposed [@Efron; @Quenouille], and a wider family of estimators beyond ML/MAP [@Kosmidis]. All these remedial and corrective approaches tend to work only when the data dimension $p$ is small and fixed. The most popular remedial approaches to overfitting in the statistics and machine learning communities are regularization, i.e. MAP inference with optimized priors, and variable selection, i.e. regression with low-dimensional representations of the vectors $\bz$.
We tackle the high-dimensional statistics regime adopting a statistical physics perspective. Statistical physics provided many contributions to our understanding of parametric statistical inference in this regime, that is not captured in traditional statistics. Statistical physics tends to deal with ‘typical-case’ scenarios, unlike the ‘worst-case’ analysis more prevalent in statistics and computer science (see e.g. [@FML18]). The two approaches are complementary, with ‘typical-case’ behaviour becoming relevant if the ‘worst-case’ scenario is very rare as $N\! \to\! \infty$. Within statistical physics, the techniques from the field of spin glasses have been particularly effective, especially replica theory and the cavity method (and related message passing algorithms).
The replica method [@MPV87] gives relatively simple prescriptions for computing asymptotic joint distributions of model parameters, and allows one to predict asymptotic values of statistical estimators. It led to valuable progress in various areas of computer science [@mezard2009information], in particular in machine learning [@seung1992statistical; @watkin1993statistical; @opper1996statistical; @dietrich1999statistical; @gardner1988space]. Although it is adaptable to many machine learning problems, the replica method is not always provably exact. Hence considerable effort has been dedicated to proving rigorously the replica predictions in specific settings, for problems originating from statistical physics [@talagrand2003spin], and in machine learning (e.g. low-rank matrix factorisation [@dia2016mutual]). Alternative methods were also proposed to derive the replica results, based on the elegant interpolation technique [@guerra2003broken], and later extended to Bayesian inference [@barbier2019adaptive].
Inference problems are intrinsically algorithmic. One ideally wants computationally efficient methods for finding the answer to any problem instance. In this aspect, the insights from statistical physics originate from the iterative procedure for computing marginals in Ising spin models with pair interactions, known as the Thouless-Anderson-Palmer (TAP) equations [@ThoulessAnderson77]. When implemented correctly [@bolthausen2014iterative], it is equivalent to the belief propagation (BP) approach in computer science [@gallager1968information; @pearl1982reverend], as was realised in [@kabashima1998belief]. For continuous variables and multibody interactions, the TAP approach to inference is now commonly referred to as Approximate Message Passing (AMP) [@donoho2009message]. AMP is computational simple, usually competitive with the fastest algorithms, rigorously characterized in the large system size limit by the cavity method (or state evolution), and used to predict accurately performance metrics such as the mean-squared error (MSE) or the detection accuracy [@bayati2015universality; @bayati2011dynamics; @deshpande2014information; @matsushita2013low; @lesieur2017statistical]. A generalization of AMP to arbitrary priors and component-wise output functions is found in [@rangan2011generalized], which coined the name generalized approximate message passing (GAMP) for the generalized linear model. Message passing tools were also used to study logistic regression in the high-dimensional regime, for ML and MAP [@sur2019modern; @salehi2019impact]. Implications of overfitting for likelihood ratio tests in the $p=\order(N)$ regime were explored in [@sur2019likelihood].
Despite its successes, the application of AMP to real-world problems has been limited by its dependence on a Bayes-optimal setting and specific structural features of the data matrix. When there is model mismatch, replica symmetry may be broken and AMP may stop converging [@antenucci2019approximate; @takahashi2020macroscopic]. When the distributions are unknown, one could try to find a minimax estimator over a class of distributions [@donoho2011compressed], or combine GAMP with expectation-maximization (EM) estimation [@krzakala2012statistical; @krzakala2012probabilistic; @kamilov2012approximate; @vila2011expectation; @vila2013expectation]. Alternatively, AMP can be modified to a replica symmetric broken (RSB) structure, but this algorithm becomes computationally more demanding with each RSB step, and requires introducing additional parameters whose values are not easily determined [@antenucci2019approximate]. Nevertheless, for linear models the main limitation of AMP is often the structure of the data matrix. AMP’s original version holds only for i.i.d. sub-Gaussian random data matrices [@bayati2015universality; @bayati2011dynamics; @rangan2011generalized], and AMP is fragile with regard to alternative choices. For example, it diverges for even mildly ill-conditioned or non-zero-mean data matrices [@rangan2019convergence; @caltagirone2014convergence; @vila2015adaptive]. Several heuristic strategies have been proposed for inducing AMP to converge [@rangan2019convergence; @vila2015adaptive; @vila2015adaptive; @manoel2015swept; @rangan2016inference] but their effectiveness is limited. Other algorithms for linear regression have been designed using approximations of belief propagation (BP) and/or free-energy minimization, such as Adaptive TAP [@opper2001adaptive], Expectation Consistent Approximation (EC) [@opper2005expectation; @kabashima2014signal; @fletcher2016expectation], (S-transform AMP) S-AMP [@cakmak2014s], and (Orthogonal AMP) OAMP [@ma2017orthogonal]. Numerical experiments suggest that some are indeed more robust than AMP, but their convergence has not been rigorously determined. Currently, the AMP-like algorithm with a rigorous underpinning that is able to handle the broadest class of data matrices is Vector Approximate Message Passing (VAMP) [@rangan2019vector], which converges correctly for all large random data matrices that are right-orthogonally invariant. While this class of matrices relax the need for fully independent matrix elements, it is still an excessively specific requirement for most practical applications.
In this work we consider GLMs and focus on generalising the structure of the data matrix, employing the replica method – the most versatile of our theoretical tools. We build on recent studies [@coolen2017replica; @SheikhCoolen2019] which gave an accurate quantitative analysis of overfitting in (regularized) Cox models [@Cox; @Cox_book] for time-to-event data. We show how the calculations of [@coolen2017replica; @SheikhCoolen2019] can be generalized to ML or MAP regression with arbitrary GLMs. Here we consider only models with a single linear combination of covariates (which includes logistic and ordinal class regression, perceptrons [@Coolenbook], and other survival analysis models such as frailty and random effects models [@survival_analysis]). Generalization to models with multiple linear combinations (e.g. neural networks with hidden layers, or survival analysis with competing risks [@Hougaard]) is straightforward. We analyse overfitting in ML or MAP inference with GLMs without requiring the entries of the data matrix to be uncorrelated. We only assume that there is no model mismatch, and that the $N$ covariate vectors $\{\bz^i\}$ are drawn independently from some distribution $p(\bz)$. This distribution may describe correlated entries, provided some weak conditions on the spectrum of the correlation matrix are met. We refer to this setting as row-independent data matrix. Using only the generalized linear form of the models, we derive generic replica order parameter equations within the replica-symmetric ansatz (RS), for the case of Gaussian priors[^1]. Only at the stage of solving order parameter equations will one have to specify model details. We also calculate the probabilistic relationship between true and inferred association coefficients, and show that, when covariates are correlated and $L2$ regularizers are sufficiently strong to have an effect, the latter induce a predictable direction change of the coefficient vector. For linear regression problems, similar studies of MAP estimators are present in literature. Again these assume i.i.d. elements or some form of rotational invariance for the data matrix, either within an RS [@rangan2009asymptotic; @vehkapera2016analysis] or an RSB ansatz [@bereyhi2019statistical]. Within the setting of rotationally invariant data matrices, the authors of [@gerbelot2020asymptotic] define an oracle version of VAMP and show rigorously that the corresponding state evolution converges to the MSE state evaluated via replica theory in [@kabashima2014signal; @rangan2009asymptotic; @kabashima2012typical]. Similar proofs of replica results can be found also for MMSE estimators in linear regression, see e.g. [@barbier2016mutuallinear; @reeves2016replica; @barbier2019optimal] for Gaussian data matrices and [@barbier2018mutualbeyond] for rotationally invariant data matrices. They cannot immediately be extended to the row-independent data matrices we consider in this work. In [@krzakala2012statistical] the replica method is used to analyse properties of the MAP estimator in compressed sensing. Our present setting differs in two key aspects: we consider generalized linear models and we remove the need for i.i.d. entries of the data matrix. There is presently no AMP-like algorithm that provably works for independently drawn covariate vectors with correlated entries. The present RS replica calculation, however, is able to deal with such more realistic data, and the mathematical physics literature provides evidence that any failures of the replica solution in practical applications are most likely to reflect model mismatch, i.e. a breaking of the replica symmetry, as opposed to fundamental features of the analytical continuation. Our limitation to ML and MAP estimators is also not crucial, and the results could be easily extended to e.g. the minimum mean square error (MMSE) estimator. We concentrate on ML and MAP because they are less computationally demanding, in the absence of an AMP-like algorithm for row-independent data matrices, and their practical evaluation is usually feasible using standard numerical methods.
This paper is organized as follows. We first generalize in section 2 the replica analysis of [@coolen2017replica; @SheikhCoolen2019] to arbitrary GLMs. In section 3 we calculate the quantitative relation between true and inferred association parameters in the overfitting regime, for potentially correlated covariates, and show how our results can be used to compute new estimators that are decontaminated for overfitting distortions (via bias removal, or MSE minimization). In section 4 we test our theoretical predictions via application to linear, logistic and Cox regression, recovering some known results as a test, and deriving several new ones. We close with a discussion of present and future work. Most of the more technical calculations are relegated to appendices, to focus the main text on the key ideas and outcomes. In contrast to most analytical studies on overfitting in literature, our theory is not limited to linear models, to uncorrelated covariates, to scalar outputs, or to models with output noise. Our present results enable one to correct ML and MAP inferences in generalized linear regression models for overfitting distortions, and thereby extend the applicability of these popular regression methods into the hitherto forbidden regime $p\!=\! \order(N)$.
General theory of GLM regression
================================
Definitions and notation
------------------------
In generalized linear regression models, the probability (density) of observing an outcome $s\in\Omega$ depends on the values of covariates $\bz\in\R^p$ via an expression of the form ${\rm Prob}(s|\bz)=p(s|\bbeta\cdot\!\bz/\!\sqrt{p},\theta)$, with $\theta$ representing any auxiliary parameters that are not coupled to covariates. The covariates appear only in an inner product with a vector $\bbeta$ of so-called association parameters. The outcome set $\Omega$ can be continuous, discrete, or a combination of both (for multi-valued outcomes), and the auxiliary parameters $\theta$ can even be a function, as in the Cox model [@Cox]. We consider MAP inference without model mismatch, where upon observing a data set ${\mathscr{D}}=\{(\bz_1,s_1),\ldots,(\bz_N,s_N)\}$ in which all samples $(\bz_i,s_i)$ are assumed to have been drawn randomly and independently from a distribution of the form $p(\bz,s)= p(\bz) p(s|\bbeta^\star\!\cdot\bz/\!\sqrt{p},\theta^\star)$, the inferred parameters $(\hat{\bbeta},\hat{\theta})$ are those that maximize the Bayesian posterior parameter probability $p(\bbeta,\theta|{\mathscr{D}})$: $$\begin{aligned}
p(\bbeta,\theta|{\mathscr{D}})&=& \frac{p({\mathscr{D}}|\bbeta,\theta) p(\bbeta,\theta)}{\int\!\rmd\bbeta^\prime \rmd\theta^\prime~p({\mathscr{D}}|\bbeta^\prime,\theta^\prime) p(\bbeta^\prime,\theta^\prime)}
\nonumber
\\
&=& \frac{p(\bbeta,\theta)\prod_{i=1}^N p(s_i|\bbeta\!\cdot\!\bz_i/\sqrt{p},\theta)}{\int\!\rmd\bbeta^\prime \rmd\theta^\prime~ p(\bbeta^\prime,\theta^\prime)\prod_{i=1}^N p(s_i|\bbeta^\prime\!\cdot\!\bz_i/\sqrt{p},\theta^\prime)}.
\end{aligned}$$ Hence, upon taking a logarithm and discarding an irrelevant constant, $$\begin{aligned}
(\hat{\bbeta},\hat{\theta})_{\rm MAP}&=& {\rm argmax}_{\bbeta,\theta}\Big\{\sum_{i=1}^N \log p\Big(s_i|\frac{\bbeta\!\cdot\!\bz_i}{\sqrt{p}},\theta\Big)+\log p(\bbeta,\theta)\Big\}.
\label{eq:MAPdefn}\end{aligned}$$ Choosing a regression model implies choosing a parametrization $p(s|\xi,\theta)$ and a prior $p(\bbeta,\theta)$. We recover ML from MAP regression by choosing the prior to be constant. Our convention to define GLMs in terms of $\bbeta\!\cdot\!\bz/\sqrt{p}$ as opposed to $\bbeta\!\cdot\!\bz$ ensures that even for $p\to\infty$ the components of $\bbeta$ will typically scale as $\beta_\mu=\order(1)$. Following mainstream literature, we will for simplicity choose nontrivial priors only for the coefficients $\bbeta$, where their inclusion is indeed most critical, so $p(\bbeta,\theta)\propto p(\bbeta)$.
For instance, the simplest GLM is linear regression, where one has outcomes from $\Omega=\R$, two auxiliary parameters $\theta=(\beta_0,\sigma)$ with $\beta_0\in\R$ and $\sigma>0$, and $$\begin{aligned}
p(s|\xi,\beta_0,\sigma)&=&(2\pi\sigma^2)^{-\frac{1}{2}}\rme^{-\frac{1}{2}(s-\xi-\beta_0)^2/\sigma^2}\!.\end{aligned}$$ In logistic regression, which can be seen as a stochastic generalization of the binary perceptron [@Coolenbook], we have $\Omega=\{-1,1\}$, one auxiliary parameter $\theta=\beta_0\in\R$, and $$\begin{aligned}
p(s|\xi,\beta_0)&=& \frac{1}{2}+\frac{1}{2}s\tanh(\xi\!+\!\beta_0)
=\frac{\rme^{s(\xi+\beta_0)}}{2\cosh(\xi\!+\!\beta_0)}.\end{aligned}$$ In Cox regression [@Cox] without censoring we have $\Omega=[0,\infty)$ and a functional auxiliary parameter $\theta=\{\lambda(t)\}$, the base hazard rate, with $$\begin{aligned}
p(s|\xi, \lambda)&=& \lambda(s)\rme^{\xi-\exp(\xi)\int_0^s\!\rmd s^\prime~\lambda(s^\prime)}
=-\frac{\rmd}{\rmd s}\rme^{-\exp(\xi)\int_0^s\!\rmd s^\prime~\lambda(s^\prime)}.\end{aligned}$$ For Cox regression with censoring, $\Omega=[0,\infty)\times \{0,1\}$ (the outcome is a pair $(t,r)$ of an event time $t$ and a binary label $r$ indicating whether the event was a primary one or censoring), with two functional auxiliary parameters $\theta=\{\lambda_0(t),\lambda_1(t)\}$ (the base rates of the primary and the censoring events), and $$\begin{aligned}
p(t,r|\xi, \lambda_0,\lambda_1)&=& \lambda_r(t)\rme^{\xi\delta_{r 1}-\int_0^t\!\rmd t^\prime~\lambda_0(t^\prime)-\exp(\xi)\int_0^t\!\rmd t^\prime~\lambda_1(t^\prime)}.\end{aligned}$$ In proportional hazards ordinal class regression with $C$ discrete possible outcomes we have $\Omega=\{1,2,\ldots,C\}$ and $\theta=(\lambda_2,\ldots,\lambda_C)\in\R^{C-1}$, with $$\begin{aligned}
\hspace*{-5mm}
&& p(c|\xi,\lambda_2,\ldots,\lambda_C)= \Big(1\!-\!\tilde{p}(c|\xi,\lambda_c)\Big)\prod_{c^\prime>c}^{C+1} \tilde{p}(c^\prime|\xi,\lambda_{c^\prime})
\\
\hspace*{-5mm}
&& \tilde{p}(1|\xi,\lambda)=0,~~~\tilde{p}(C\!+\!1|\xi,\lambda)=1,~~~~~~
1\!<c\leq C\!:~~ \tilde{p}(c|\xi,\lambda)= \rme^{- \lambda\exp(\xi)}.
\nonumber\end{aligned}$$
The information-theoretic overfitting measure
---------------------------------------------
We follow closely the procedure in [@coolen2017replica; @SheikhCoolen2019], which can be adapted to arbitrary GLMs with only minimal change. We start from the observation that MAP regression for any model of the type $p(s|\bz,\btheta)$ (whether or not of the GLM form) is equivalent to minimization over the model parameters $\btheta$ of the quantity $$\begin{aligned}
\Omega(\btheta|{\mathscr{D}})&=& D(\hat{p}_{{\mathscr{D}}}|| p_{\btheta})-N^{-1}\log p(\btheta).\end{aligned}$$ Here $\hat{p}_{{\mathscr{D}}}$ is the empirical distribution of covariates and outcomes in the data, $\hat{p}(s,\bz|{\mathscr{D}})=N^{-1}\sum_{i\leq N}\delta(s\!-\!s_i)\delta(\bz\!-\!\bz_i)$, $D(\hat{p}_{{\mathscr{D}}}|| p_{\btheta})$ is the Kullback-Leibler distance $$\begin{aligned}
D(\hat{p}_{{\mathscr{D}}}||p_{\btheta})=\int\!\rmd\bz \rmd s ~\hat{p}(s,\bz|{\mathscr{D}})\log\Big(\frac{\hat{p}(s|\bz,{\mathscr{D}})}{p(s|\bz,\btheta)}
\Big),\end{aligned}$$ and $p_{\btheta}$ is the assumed parametrized regression model, with covariate-conditioned outcome probabilities $p(s|\bz,\btheta)$. For discrete variables, delta functions are replaced by Kronecker delta symbols. Assuming that our data were indeed generated from a model of the assumed form, with (unknown) parameters $\btheta^\star\!$, a transparent overfitting measure can be defined as $E(\btheta^\star\!, {\mathscr{D}})={\rm min}_{\btheta}~ \Omega(\btheta|{\mathscr{D}})-\Omega(\btheta^\star|{\mathscr{D}})$, giving $$\begin{aligned}
E(\btheta^\star\!,{\mathscr{D}})&=& \min_{\btheta}\Bigg\{
\!\frac{1}{N}\sum_{i=1}^N \log\Big(\frac{p(s_i|\bz_i,\btheta^\star)}{p(s_i|\bz_i,\btheta)}\Big)
+\frac{1}{N}\log\Big(\frac{p(\btheta^\star)}{p(\btheta)}\Big)\Bigg\}.
\label{eq:Hamiltonian}\end{aligned}$$ Perfect regression would give $E(\btheta^\star\!,{\mathscr{D}})=0$, finding $E(\btheta^\star\!,{\mathscr{D}})<0$ implies overfitting, and finding $E(\btheta^\star\!,{\mathscr{D}})>0$ implies underfitting. In GLMs with the MAP regression protocol (\[eq:MAPdefn\]) the parameters would be $\btheta=(\bbeta,\theta)$. Our calculations focus on evaluating the average of (\[eq:Hamiltonian\]) over the possible realizations of the data set ${\mathscr{D}}$, whose samples are drawn randomly from $p(s,\bz)=p(s|\bz,\btheta^\star)p(\bz)$ for some $p(\bz)$. The average is handled using the replica identity $\bra \log Z\ket=\lim_{n\to 0}n^{-1}\log\bra Z^n\ket$, and we write the minimization as the computation of the ground state energy density of a statistical mechanical system with degrees of freedom $\btheta\in\R^p$ and Hamiltonian $$\begin{aligned}
H(\btheta|\btheta^\star\!,{\mathscr{D}})&=& \sum_{i=1}^N \log\Big(\frac{p(s_i|\bz_i,\btheta^\star)}{p(s_i|\bz_i,\btheta)}\Big)+\log\Big(\frac{p(\btheta^\star)}{p(\btheta)}\Big).\end{aligned}$$ We can thus model MAP regression as the zero noise limit of a stochastic minimization of $H(\btheta|\btheta^\star\!,{\mathscr{D}})$ at inverse noise level $\gamma$, giving $$\begin{aligned}
\hspace*{-10mm}
\bra E(\btheta^\star\!,{\mathscr{D}})\ket_{{\mathscr{D}}}&=& \lim_{\gamma\to \infty}E_\gamma(\btheta^\star),
\\
\hspace*{-10mm}
E_\gamma(\btheta^\star)
&=&- \frac{\partial}{\partial\gamma}\frac{1}{N}\Big\bra \log \int\!\rmd\btheta~\rme^{-\gamma H(\btheta|\btheta^\star\!,{\mathscr{D}})}\Big\ket_{\!{\mathscr{D}}}
\nonumber
\\
\hspace*{-10mm}
&=& - \lim_{n\to 0}\frac{\partial}{\partial\gamma}\frac{1}{Nn}\log \int\!\rmd\btheta^1\!\ldots\rmd\btheta^n \Big\bra\rme^{-\gamma \sum_{\alpha=1}^n H(\btheta^\alpha|\btheta^\star\!,{\mathscr{D}})}\Big\ket_{\!{\mathscr{D}}}
\nonumber
\\
\hspace*{-10mm}
&=& - \lim_{n\to 0}\frac{\partial}{\partial\gamma}\frac{1}{Nn}\log \int\!\rmd\btheta^1\!\ldots\rmd\btheta^n
\prod_{\alpha=1}^n
\Big[\frac{p(\btheta^\alpha)}{p(\btheta^\star)}\Big]^\gamma
\nonumber
\\
\hspace*{-10mm}
&&\hspace*{5mm} \times
\Big\{
\int\!\rmd\bz \rmd s~p(\bz)p(s|\bz,\btheta^\star)
\prod_{\alpha=1}^n
\Big[\frac{p(s|\bz,\btheta^\alpha)}{p(s|\bz,\btheta^\star)}\Big]^\gamma\Big\}^N.
\label{eq:E_gamma}\end{aligned}$$ Integrals over outcome variables become summations when these variables are discrete, and integrals over functional parameters are interpreted as path integrals. In the alternative limit $\gamma\to 1$ the quantity $E_\gamma(\btheta^\star)$ would involve the average over all data realizations of the Bayesian estimator, $E_1(\btheta^\star)=\bra \int\!\rmd\btheta~p(\btheta|{\mathscr{D}})\Omega(\btheta|{\mathscr{D}})-\Omega(\btheta^\star|{\mathscr{D}})\ket_{{\mathscr{D}}}$.
Application of expression (\[eq:E\_gamma\]) to generalized linear regression models implies setting $\btheta\to (\bbeta,\theta)$ and $p(s|\bz,\btheta)\to
p(s|\bbeta\!\cdot\!\bz/\sqrt{p},\theta)$, so that we obtain $$\begin{aligned}
\bra E(\bbeta^\star\!,\theta^\star\!,{\mathscr{D}})\ket_{{\mathscr{D}}}&=& \lim_{\gamma\to \infty}E_\gamma(\bbeta^\star\!,\theta^\star),\end{aligned}$$ with $$\begin{aligned}
\hspace*{-10mm}
E_\gamma(\bbeta^\star,\theta^\star)
&=& - \lim_{n\to 0}\frac{\partial}{\partial\gamma}\frac{1}{Nn}\log \int\!\rmd\theta^1\!\ldots\rmd\theta^n
\int\!\rmd\bbeta^1\!\ldots\rmd\bbeta^n
\prod_{\alpha=1}^n
\Big[\frac{p(\bbeta^\alpha)}{p(\bbeta^\star)}\Big]^\gamma
\nonumber
\\
\hspace*{-10mm}
&&\hspace*{0mm} \times
\Big\{
\int\!\rmd\bz \rmd s~p(\bz)p(s|\bz,\bbeta^\star,\theta^\star)
\prod_{\alpha=1}^n
\Big[\frac{p(s|\bz,\bbeta^\alpha,\theta^\alpha)}{p(s|\bz,\bbeta^\star,\theta^\star)}\Big]^\gamma\Big\}^N\!.~~
\label{eq:starting_point_of_replicas}\end{aligned}$$
Replica analysis in the regime $p=\order(N)$
--------------------------------------------
In \[app:SheikhCoolen\] we compute (\[eq:starting\_point\_of\_replicas\]) for $N,p\to\infty$ with fixed ratio $\zeta=p/N$, assuming $p(\bz)$ to be a zero-average distribution on $\R^p$, and for $L2$ priors $p(\bbeta)\propto\exp(-\eta\bbeta^2)$[^2]. We include this derivation, which follows [@SheikhCoolen2019], for completeness. The outcome of the regression process is characterized by the values of a finite number of order parameters[^3], in terms of which one can quantify the relation between inferred and true regression coefficients. The result of \[app:SheikhCoolen\] is: $$\begin{aligned}
\hspace*{-20mm}
\lim\limits_{N \to \infty} E_{\gamma}({\bbeta}^\star, \theta^\star) &=&
\int\! {\rm D}y_0\!\int\!\rmd s~ p(s |S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
\log p(s | S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
-\zeta\eta S^2
\nonumber\\
\hspace*{-20mm}
&&
\hspace*{-20mm}
+~\eta\zeta\Bigg\{
w^2\bra a\ket
\Big\langle \frac{a^2}{2 \eta \gamma\! +\! ga}\Big\rangle^{\!\!-2}
\Big\langle \frac{a^2}{(2 \eta \gamma\! +\! ga)^2}\Big\rangle
+
\Big\langle\! \frac{1}{2 \eta \gamma\!+\!ga} \Big\rangle- f\Big\langle\frac{a}{(2 \eta \gamma\! +\! ga)^2} \Big\rangle
\Bigg\}
\nonumber \\
\hspace*{-20mm}
&&
\hspace*{-31mm}
-\! \int\!{\rm D}z {\rm D}y_0\!\int\!\!\rmd s~ p(s |S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
\frac{ \int \!{\rm D} y~
p^\gamma(s | uy \!+\! wy_0\!+\!vz, \theta)\log p(s | uy \!+\! wy_0\!+\!vz, \theta) }
{ \int \!{\rm D} y~
p^\gamma(s| uy \!+\! wy_0\!+\!vz, \theta) },
\nonumber
\\[-1mm]
\hspace*{-20mm}
&&\end{aligned}$$ with the shorthand ${\rm D}z=(2\pi)^{-\frac{1}{2}}\rme^{-\frac{1}{2}z^2}\rmd z$. Brackets denote averages over the limit $p\to\infty$ of the eigenvalue spectrum $\varrho(a) $ of the $p\times p$ covariate correlation matrix $\bA$, with entries $A_{\mu\nu}=\int\!\rmd\bz~p(\bz)z_\mu z_\nu$. This result depends on the true association parameter vector $\bbeta^\star$ only via the asymptotic rescaled amplitude $S=\lim_{p\to\infty}p^{-1}\bbeta^{\star 2}$, assuming the components of $\bbeta^\star$ to have been drawn randomly from a symmetric distribution with finite second and fourth moments. Of the covariate covariance matrix $\bA$ we only require that its eigenvalue spectrum obeys $\lim_{p\to\infty} \int\!\rmd a~\varrho(a)a \in \R$ and $\lim_{p\to\infty}p^{-1}\int\!\rmd a~\varrho(a)a^2 =0$. The order parameters $(u,v,w,f,g,\theta)$ are determined by extremization of the following quantity, which acts as a free energy density: $$\begin{aligned}
\Psi_{\rm RS}(u,v,w,f,g,\theta)&=& \Psi^A_{\rm RS}(u,v,w,f,g) - \Psi^B_{\rm RS}(u,v,w,\theta),\end{aligned}$$ with $$\begin{aligned}
\hspace*{-20mm}
\Psi^A_{\rm RS}(\ldots)&=&
-\frac{1}{2}\zeta (g\!+\!f) u^2-\frac{1}{2}\zeta g(v^2\!+\! w^2)
\nonumber
\\
\hspace*{-20mm}
&&
\hspace*{0mm}
+
\frac{1}{2}\zeta\Big\{
w^2\bra a\ket
\Big\bra \frac{a^2}{
2\gamma\eta\!+\!ga}\Big\ket^{\!-1}\!\!\!
+\Big\bra\log\Big(2\gamma\eta\!+\!ga\Big)\Big\ket
+
f\Big\bra
\frac{a}{2\gamma\eta\!+\!ga}\Big\ket
\Big\},
\\
\hspace*{-20mm}
\Psi^B_{\rm RS}(\ldots)&=&
\int\!{\rm D}y_0{\rm D}z\!\int\!\rmd s~ p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
\log \!
\int\!{{\rm D}}y~
p^\gamma (s|uy\!+\!
wy_0\!+\!vz,\theta).~
\label{eq:Psi_before_gamma_limit}\end{aligned}$$ Here $\theta^\star$ are the true (unknown) auxiliary model parameters assumed to have been used to generate the observed data. The physical meaning of the two main order parameters, expressed in terms of the MAP-inferred parameters $\hat{\bbeta}$ and the true parameters $\bbeta^\star$ responsible for the data, is $$\begin{aligned}
v&=& \lim_{p\to\infty}
\Bigg\bra \frac{1}{\sqrt{p}}\Big\{\hat{\bbeta}\cdot\bA\hat{\bbeta}-\frac{\big(\bbeta^\star\!\cdot\bA\hat{\bbeta}\big)^2}{\bbeta^\star\!\cdot\bA\bbeta^\star}\Big\}^{\frac{1}{2}}\Bigg\ket_{\!\!{\mathscr{D}}}
\label{eq:meaning_v}
\\
w&=& \lim_{p\to\infty}\Bigg\bra \frac{1}{\sqrt{p}}\frac{\bbeta^\star\!\cdot\bA\hat{\bbeta}}{\sqrt{\bbeta^\star\!\cdot\bA\bbeta^\star}}
\Bigg\ket_{\!\!{\mathscr{D}}}
\label{eq:meaning_w}\end{aligned}$$ Perfect regression, i.e. $\hat{\bbeta}=\bbeta^\star$, would give $v=0$ and $w=\lim_{p\to\infty}\bra \sqrt{\bbeta^\star\!\cdot\bA\bbeta^\star/p}\ket_{{\mathscr{D}}}$.
In the limit $\gamma\to\infty$ the maximization of the posterior becomes deterministic, and we recover the formulae describing MAP inference. In the precursor studies [@coolen2017replica; @SheikhCoolen2019] it was found that the canonical scaling of the RS order parameters for large $\gamma$ is $$\begin{aligned}
u=\tilde{u}/\sqrt{\gamma},~~~~~~v,w,\theta=\order(1),~~~~~~g=\tilde{g}\gamma,~~~~~~f=\tilde{f}\gamma^2.\end{aligned}$$ Assuming this scaling to hold more generally gives $$\begin{aligned}
\hspace*{-22mm}
\lim_{\gamma\to\infty}\frac{1}{\gamma}\Psi^A_{\rm RS}(\ldots)&=&
\frac{1}{2}\zeta\Big\{\!
w^2\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-1}\!\!
+
\tilde{f}\Big[\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
\!-\! \tilde{u}^2\Big]- \tilde{g}(v^2\!+\! w^2)
\Big\},
\label{eq:Psi_A_large_gamma}
\\
\hspace*{-22mm}
\lim_{\gamma\to\infty}\frac{1}{\gamma}\Psi^B_{\rm RS}(\ldots)&=&
\int\!{\rm D}y_0{\rm D}z\int\!\rmd s~ p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
\nonumber
\\
\hspace*{-22mm}
&&\times
\lim_{\gamma\to\infty}\frac{1}{\gamma}
\log
\int\!\rmd y~\rme^{-\frac{1}{2}y^2}
p^\gamma (s|\tilde{u}y/\sqrt{\gamma}\!+\!
wy_0\!+\!vz,\theta)
\nonumber
\\
\hspace*{-22mm}
&&\hspace*{-15mm} =
\int\!{\rm D}y_0{\rm D}z\!\int\!\rmd s~ p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
{\rm max}_{y\in\R}\Big[
\log p(s|\tilde{u}y\!+\!
wy_0\!+\!vz,\theta)
-\frac{1}{2} y^2\Big]
\nonumber
\\
\hspace*{-22mm}
&&\hspace*{-15mm} =
\int\!{\rm D}y_0{\rm D}z\!\int\!\rmd s~ p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
{\rm max}_{\xi\in\R}\Big[
\log p(s|\xi,\theta)
\!-\!\frac{(\xi\!-\!wy_0\!-\!vz)^2}{2\tilde{u}^2}\Big].
\nonumber
\\[-0mm]\hspace*{-22mm}&&
\label{eq:Psi_B_large_gamma}\end{aligned}$$ We next abbreviate $\Xi_A(\tilde{f},\tilde{g},\tilde{u},v,w)=\lim_{\gamma\to\infty}\gamma^{-1}\Psi^A_{\rm RS}(\ldots)$ and $\Xi_B(\tilde{u},v,w,\theta)=\lim_{\gamma\to\infty}\gamma^{-1}\Psi^B_{\rm RS}(\ldots)$, and write the various averages as $\bra\!\bra \ldots \ket\!\ket=\int\!{\rm D}y_0{\rm D}z \ldots$ and $\bra f(s)\ket_s
=\int\!\rmd s~ p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)f(s)$ (with the integral over $s$ replaced by a sum if $s$ is discrete). We also define $$\begin{aligned}
\xi(\mu,\sigma,s,\theta)&=& {\rm argmax}_{\xi\in\R}\Big[
\log p(s|\xi,\theta)
\!-\!\frac{1}{2} (\xi\!-\!\mu)^2/\sigma^2\Big].\end{aligned}$$ A sufficient condition for $\xi(\mu,\sigma,s,\theta)$ to exists is that ${\rm argmax}_{\xi\in\R} p(s|\xi,\theta)$ exists for all $(s,\theta)$, which we have found to be true in all GLM models considered so far. Note that $ \xi(\mu,\sigma,s,\theta)$ is the solution of $$\begin{aligned}
\frac{\partial}{\partial\xi}\log p(s|\xi,\theta)=
(\xi\!-\!\mu)/\sigma^2.
\label{eq:xi_equation}\end{aligned}$$ Hence we may write the model-independent part of the quantity to be extremized as $$\begin{aligned}
\hspace*{-20mm}
\Xi_A(\tilde{f}\!,\tilde{g},\tilde{u},v,w)&=&
\frac{1}{2}\zeta\Big\{
w^2\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!\!-1}\!\!
\!+
\tilde{f}\Big[\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
\!-\! \tilde{u}^2\Big]\!- \tilde{g}(v^2\!+\! w^2)
\Big\},~\end{aligned}$$ and the model-dependent part as $$\begin{aligned}
\hspace*{-15mm}
\Xi_B(\tilde{u},v,w,\theta)&=&
\Big\bra\!\Big\bra\! \Big\bra
{\rm max}_{\xi\in\R}\Big[
\log p(s|\xi,\theta)
\!-\!\frac{1}{2} (\xi\!-\!wy_0\!-\!vz)^2/\tilde{u}^2\Big]
\Big\ket_{\!s}\Big\ket\!\Big\ket
\nonumber
\\
\hspace*{-15mm}
&=&
\Big\bra\!\Big\bra \!\Big\bra
\Big[
\log p(s|\xi,\theta)
\!-\!\frac{(\xi\!-\!wy_0\!-\!vz)^2}{2\tilde{u}^2}\Big]\Big|_{\xi=\xi(wy_0+vz,\tilde{u},s,\theta)}\Big\ket_{\!s}\Big\ket\!\Big\ket.
\label{eq:Psi_B_large_gamma_compact}\end{aligned}$$ The RS order parameter equations can then be written as $$\begin{aligned}
\hspace*{-10mm}
&& \frac{\partial\Xi_A}{\partial\tilde{f}}=\frac{\partial\Xi_A}{\partial\tilde{g}}= \frac{\partial\Xi_B}{\partial\theta}=0,
\label{eq:RS_eqns_uncoupled}
\\
&&
\frac{\partial\Xi_A}{\partial\tilde{u}}=\frac{\partial\Xi_B}{\partial\tilde{u}},~~~~~~
\frac{\partial\Xi_A}{\partial v}=\frac{\partial\Xi_B}{\partial v},~~~~~~
\frac{\partial\Xi_A}{\partial w}=\frac{\partial\Xi_B}{\partial w}.
\label{eq:RS_eqns_coupled}\end{aligned}$$ In \[app:further\] we analyse and simplify these RS order parameter equations further, and find that we can rewrite our closed MAP order parameter equation set as: $$\begin{aligned}
\hspace*{-20mm}
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
&=& \tilde{u}^2,
\label{eq:ddf2=0}
\\
\hspace*{-20mm}
w^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!\!-2}
\!
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
\!-\!1\Big]
\!-\!
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket
&=&v^2,
\label{eq:ddg2=0}
\\
\hspace*{-20mm}
\Big\bra\!\Big\bra\!\Big\bra
[\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\!-\!wy_0\!-\!vz]^2\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& - \zeta
\tilde{f} \tilde{u}^4,
\label{eq:ddu2=0}
\\[2mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra \!\Big\bra
(\partial_1\xi)(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& 1- \zeta \tilde{g}\tilde{u}^2,
\label{eq:ddv2=0}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra\!\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta) \frac{\partial \log p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)}{\partial y_0}\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& \zeta w \tilde{u}^2
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-1}\!\!,
\label{eq:ddw2=0}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra\! \Big\bra \frac{\partial \log p(s|\xi,\theta)}{\partial\theta}\Big|_{\xi=\xi(wy_0+vz,\tilde{u},s,\theta)}
\Big\ket_{\!s}\Big\ket\!\Big\ket&=& 0.
\label{eq:ddtheta2=0}\end{aligned}$$ The function $\xi(\mu,\sigma,s,\theta)$, defined as the solution of (\[eq:xi\_equation\]), obeys $\lim_{\sigma\to 0} \xi(\mu,\sigma,s,\theta)=\mu$. Its partial derivative $(\partial_1\xi)(\mu,\sigma,s,\theta)$ follows upon working out the partial derivative with respect to $\mu$ of (\[eq:xi\_equation\]), $$\begin{aligned}
0&=&\frac{1}{\sigma^2}+ \frac{\partial\xi}{\partial\mu}\frac{\partial}{\partial\xi}\Big[
\frac{\partial}{\partial\xi}\log p(s|\xi,\theta)+
\frac{\mu\!-\!\xi}{\sigma^2}\Big].\end{aligned}$$ Hence $$\begin{aligned}
\hspace*{-5mm}
(\partial_1\xi)(\mu,\sigma,s,\theta)&=&\Bigg[1-
\sigma^2\frac{\partial^2\log p(s|\xi,\theta)}{\partial\xi^2}\Bigg]^{-1}_{\xi=\xi(\mu,\sigma,s,\theta)}\end{aligned}$$ and $$\begin{aligned}
\hspace*{-5mm}
(\partial_1\xi)(wy_0\!+\!vz,\tilde{u},s,\theta)&=&\Bigg[1-
\tilde{u}^2\frac{\partial^2\log p(s|\xi,\theta)}{\partial\xi^2}\Bigg]^{-1}_{\xi=\xi(wy_0+vz,\tilde{u},s,\theta)}.\end{aligned}$$
The limit $\zeta\to 0$ for ML regression
----------------------------------------
For $\eta=0$ we revert back to ML regression. Here equations (\[eq:ddf2=0\],\[eq:ddg2=0\]) simplify to $\tilde{g}=1/\tilde{u}^2$ and $\tilde{f}=-v^2/\tilde{u}^4$, equation (\[eq:ddtheta2=0\]) remains unaltered, and the three equations (\[eq:ddu2=0\], \[eq:ddv2=0\], \[eq:ddw2=0\]) referring to extremization over $(\tilde{u},v,w)$ simplify to $$\begin{aligned}
\Big\bra\!\Big\bra \!\Big\bra
[\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\!-\!wy_0\!-\!vz]^2\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& \zeta v^2,
\label{eq:ddu_ML=0}
\\[2mm]
\Big\bra\!\Big\bra \!\Big\bra
(\partial_1\xi)(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{s}\Big\ket\!\Big\ket
&=& 1- \zeta,
\label{eq:ddv_ML=0}
\\[1mm]
\Big\bra\!\Big\bra\!\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta) \frac{\partial \log p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)}{\partial y_0}\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& \zeta w.
\label{eq:ddw_ML=0}\end{aligned}$$ As a test, let us consider the classical regime $\zeta\to 0$ where the number of samples is much larger than the number of covariates. We can show relatively easily, for any model $p(s|\xi,\theta)$, that in this limit the remaining ML equations are solved by the correct solution $(\tilde{u},v,w,\theta)=(0,0,S\bra a\ket^{\frac{1}{2}},\theta^\star)$, as one should expect. To see this we use $$\begin{aligned}
\lim_{\tilde{u}\to 0}\xi(wy_0\!+\!vz,\tilde{u},s,\theta)&=& wy_0\!+\!vz,
\\
\lim_{\tilde{u}\to 0}(\partial_1\xi)(wy_0\!+\!vz,\tilde{u},s,\theta)&=&1.\end{aligned}$$ Upon inserting $(\tilde{u},v,w,\theta)=(0,0,S\bra a\ket^{\frac{1}{2}},\theta^\star)$ we find that (\[eq:ddu\_ML=0\]) and (\[eq:ddv\_ML=0\]) are trivially satisfied, whereas (\[eq:ddtheta2=0\]) and (\[eq:ddw\_ML=0\]) reduce, respectively, to the trivial statements $$\begin{aligned}
0&=& \Big\bra\!\Big\bra \frac{\partial \log p(s| wy_0,\theta^\star)}{\partial \theta^\star}\Big\ket_{\!s}\Big\ket
\nonumber
\\&=&
\int\!{\rm D}y_0
\int\!\rmd s ~\frac{\partial }{\partial\theta^\star}p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)=0\end{aligned}$$ and $$\begin{aligned}
0&=&
\Big\bra y_0\Big\bra \frac{\partial \log p(s|wy_0,\theta^\star)}{\partial y_0}\Big\ket_{\!s}\Big\ket
\nonumber
\\&=&
S\bra a\ket^{\frac{1}{2}} \int\!{\rm D}y_0~y_0\int\!\rmd s~\frac{\partial}{\partial y_0} p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
=0.\end{aligned}$$
Link between true and inferred association parameters {#app:the_link}
=====================================================
Replica evaluation of the joint distribution
--------------------------------------------
We can also calculate the (probabilistic) relation[^4] between the true parameters $\beta_\mu^\star$ and the MAP-inferred values $\hat{\beta}_\mu$ in regression models of the GLM form. The relevant object to be computed in the case of MAP regression with prior $p(\bbeta)$ and data ${\mathscr{D}}=\{(\bz_1,s_1),\ldots,(\bz_N,s_N)\}$ is the joint distribution $$\begin{aligned}
\hspace*{-10mm}
{\mathscr{P}}(\beta,\beta^\star|{\mathscr{D}})&=& \lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \frac{\int\!\rmd\theta\rmd\bbeta~\rme^{\gamma \log p(\theta,\bbeta|{\mathscr{D}})}\delta(\beta-\beta_\mu)}
{\int\!\rmd\theta\rmd\bbeta~\rme^{\gamma \log p(\theta,\bbeta|{\mathscr{D}})}}
\delta(\beta^\star-\beta^\star_\mu),
\label{eq:link_between_betas_start}
\end{aligned}$$ with the posterior parameter likelihood $$\begin{aligned}
p(\theta,\bbeta|{\mathscr{D}})&=& \frac{p(\bbeta,\theta)\prod_{i=1}^N p(s_i|\bbeta\cdot\bz_i/\sqrt{p},\theta)}
{\int\!\rmd\bbeta^\prime \rmd\theta^\prime~p(\bbeta^\prime,\theta^\prime)\prod_{i=1}^N p(s_i|\bbeta^\prime\!\cdot\bz_i/\sqrt{p},\theta^\prime)}.
\label{eq:posterior}
\end{aligned}$$ The limit $\gamma\to\infty$ ensures that the integrations in (\[eq:link\_between\_betas\_start\]) are dominated by the parameter values where $p(\theta,\bbeta|{\mathscr{D}})$ is maximized. Hence the fraction in (\[eq:link\_between\_betas\_start\]) reduces to $\delta(\beta-\hat{\beta}_\mu)$, where $\hat{\beta}_\mu$ is the MAP estimator of the parameter $\beta_\mu$, given dataset ${\mathscr{D}}$. Note that for $\gamma\to 1$ expression (\[eq:link\_between\_betas\_start\]) would have reduced to the joint distribution of true coefficients and their Bayesian estimators, viz. $$\begin{aligned}
{\mathscr{P}}(\beta,\beta^\star|{\mathscr{D}})&=&
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta^\star_\mu)\int\!\rmd\theta\rmd\bbeta~p(\theta,\bbeta|D)\delta(\beta-\beta_\mu).
\end{aligned}$$
As always we are interested mainly in the typical form of ${\mathscr{P}}(\beta,\beta^\star|{\mathscr{D}})$, so we average over the possible realizations of the data set ${\mathscr{D}}$, assuming all samples $(\bz_i,s_i)$ to be drawn randomly and independently from $p(\bz)p(s|\bbeta^\star\!\!\cdot\bz/\!\sqrt{p},\theta^\star)$: $$\begin{aligned}
\hspace*{-10mm}
{\mathscr{P}}(\beta,\beta^\star)&=& \bra {\mathscr{P}}(\beta,\beta^\star|{\mathscr{D}})\ket_{{\mathscr{D}}}
\nonumber
\\
\hspace*{-10mm}
&=&
\lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta^\star_\mu)~\Bigg\bra\frac{\int\!\rmd\theta\rmd\bbeta~\rme^{\gamma \log p(\theta,\bbeta|{\mathscr{D}})}\delta(\beta-\beta_\mu)}
{\int\!\rmd\theta\rmd\bbeta~\rme^{\gamma \log p(\theta,\bbeta|{\mathscr{D}})}}
\Bigg\ket_{\!\!{\mathscr{D}}}.
\label{eq:joint_beta_stats_nearly}
\end{aligned}$$ Upon inserting (\[eq:posterior\]) into (\[eq:joint\_beta\_stats\_nearly\]) we obtain for this ${{\mathscr{D}}}$-independent joint distribution: $$\begin{aligned}
\hspace*{-10mm}
{\mathscr{P}}(\beta,\beta^\star)&=&
\lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p
\delta(\beta^\star\!-\beta^\star_\mu)~
\nonumber
\\[-1mm]
\hspace*{-10mm}
&&
\hspace*{-5mm} \times\Bigg\bra\frac{\int\!\rmd\theta\rmd\bbeta~\Big[p(\bbeta)p(\theta)\prod_{i=1}^N p(s_i|\bbeta\cdot\bz_i/\sqrt{p},\theta)\Big]^\gamma \delta(\beta\!-\!\beta_\mu)}
{\int\!\rmd\theta\rmd\bbeta~\Big[p(\bbeta)p(\theta)\prod_{i=1}^N p(s_i|\bbeta\cdot\bz_i/\sqrt{p},\theta)\Big]^\gamma}
\Bigg\ket_{\!\!{\mathscr{D}}}\!.
\label{eq:joint_beta_stats}
\end{aligned}$$ We can evaluate (\[eq:joint\_beta\_stats\]) using the following alternative form of the replica identity, which can be shown to be equivalent to the previous version $\bra \log Z\ket=\lim_{n\to 0}n^{-1}\log \bra Z^n\ket$, $$\begin{aligned}
\hspace*{-15mm}
\Bigg\bra\frac{\int\!\rmd x~w(x,y)f(x)}{\int\!\rmd x~w(x,y)}\Bigg\ket_{\!\!y}&=& \lim_{n\to 0}
\Big\bra\Big[ \int\!\rmd x~w(x,y)f(x)\Big]\Big[ \int\!\rmd x~w(x,y)\Big]^{n-1} \Big\ket_{\!y}
\nonumber
\\[-1mm]
&=& \lim_{n\to 0}
\int\!\Big[\prod_{\alpha=1}^n \rmd x^\alpha\Big] f(x^1) \Big\bra \prod_{\alpha=1}^n w(x^\alpha,y)\Big\ket_{\!y}.
\end{aligned}$$ Application of this identity to (\[eq:joint\_beta\_stats\]), with the choices $x\to (\theta,\bbeta)$, $y\to {\mathscr{D}}$, $w(x,y)\to
[p(\bbeta,\theta)\prod_{i=1}^N p(t_i|\bbeta\cdot\bz_i/\sqrt{p},\theta)]^\gamma$, and $f(x)\to \delta(\beta\!-\!\beta_\mu)$, followed by working out the definition of the average over the data sets ${\mathscr{D}}$, gives $$\begin{aligned}
\hspace*{-15mm}
{\mathscr{P}}(\beta,\beta^\star) &=&
\lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^\star)
\int\!\Big\{\prod_{\alpha=1}^n \rmd \theta^\alpha\rmd\bbeta^\alpha [p(\bbeta^\alpha)p(\theta^\alpha)]^\gamma \Big\}
~\delta(\beta\!-\!\beta^1_\mu)\nonumber
\\
\hspace*{-15mm}
&&\hspace*{-3mm}\times
\Big\{\int\!\rmd\bz \rmd t~p(\bz)p(t|\bbeta^\star\!\cdot\bz/\sqrt{p},\theta^\star)
\prod_{\alpha=1}^n
\Big[p(t|\bbeta^\alpha\!\cdot\bz/\sqrt{p},\theta^\alpha)\Big]^\gamma
\Big\}^N.
\label{eq:betas_starting_point}
\end{aligned}$$ In \[app:beta\_stats\] w ecalculate the limit $p,N\to\infty$ of (\[eq:betas\_starting\_point\]), for finite ratio $\zeta=p/N$. This builds on the replica calculation in \[app:SheikhCoolen\]. For the simplest case of uncorrelated and normalized covariates, i.e. $\bA=\one$, we find that $$\begin{aligned}
\lim_{N\to\infty}{\mathscr{P}}(\beta|\beta^\star) &=&
\frac{1}{v\sqrt{2\pi}}\rme^{-\frac{1}{2}(\beta-w\beta^\star/S)^2/v^2}\end{aligned}$$ This confirms what was suggested by simulation data and exploited in [@coolen2017replica]: if we plot inferred versus true association parameters in a plane, we will find for $L2$ priors and uncorrelated covariates a linear cloud with slope $w/S$ and zero-average Gaussian noise of width $v$. We have now proved this analytically, for [*any*]{} generalized linear model. For the more tricky case of correlated covariates, i.e. arbitrary covariance matrices $\bA$ subject only to the spectral conditions $\lim_{p\to\infty}\bra a\ket\in \R$ and $\lim_{p\to\infty}\bra a^2\ket\in \R$ of \[app:self\_averaging\], we find $$\begin{aligned}
\hspace*{-20mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=&
\lim_{p\to\infty}
\frac{1}{p}\sum_{\mu=1}^p
\frac{\delta(\beta^\star\!-\beta_\mu^0)}{\sqrt{2\pi |\tilde{f}|[
(\tilde{g} \bA+2\eta\one)^{-1}\bA(\tilde{g} \bA+2\eta\one)^{-1}]_{\mu\mu}}}
\label{eq:beta_relation_derived}
\\
\hspace*{-20mm}
&&
\times
\rme^{-\frac{1}{2}\Big[\beta+ \tilde{d}_0
[ (\tilde{g}\one+2\eta\bA^{-1})^{-1}
\bbeta^0]_\mu\Big]^2/|\tilde{f}|[
(\tilde{g} \bA+2\eta\one)^{-1}\bA(\tilde{g} \bA+2\eta\one)^{-1}]_{\mu\mu}
}.
\nonumber\end{aligned}$$ Once more the inferred vector $\hat{\bbeta}$ depends linearly on the true vector $\bbeta^\star$, supplemented with Gaussian noise. However, in the presence of covariate correlations, we obtain a scalar relation $\hat{\bbeta}=\kappa\bbeta^\star+{\it noise}$ typically only when $\eta=0$ (i.e. no regularizer, giving ML regression).
Expression (\[eq:beta\_relation\_derived\]) is consistent with the more general propositions $$\begin{aligned}
\bra \hat{\bbeta}\ket&=&-\tilde{d}_0 [\tilde{g}\one+2\eta\bA^{-1}]^{-1}
\bbeta^\star,
\label{eq:average_beta}
\\[1mm]
\bra \hat{\beta}_\mu\hat{\beta}_\nu\ket-\bra \hat{\beta}_\mu\ket \bra \hat{\beta}_\nu\ket &=&|\tilde{f}|[
(\tilde{g} \bA+2\eta\one)^{-1}\bA(\tilde{g} \bA+2\eta\one)^{-1}]_{\mu\nu}.
\label{eq:covariance_beta}
\end{aligned}$$ Using expression (\[eq:found\_d0\]) for $d_0$, and $c_0=S\bra a\ket^{\frac{1}{2}}w$, we can write (\[eq:average\_beta\]) also as $$\begin{aligned}
\bra \hat{\bbeta}\ket&=& \frac{w\bra a\ket^{\frac{1}{2}}}{S} \Big\bra\frac{a^2}{2\eta\!+\!a\tilde{g}}\Big\ket^{\!-1} [\tilde{g}\one+2\eta\bA^{-1}]^{-1}
\bbeta^\star.
\label{eq:average_beta_again}
\end{aligned}$$ When covariates are correlated, MAP regression with Gaussian priors sufficiently strong to have an impact will for finite $\zeta>0$ not just rescale the length of the inferred association vectors but also change its direction. Only for small $\eta$ or weak correlations (or if by accident $\bbeta^\star$ happens to be an eigenvector of $\bA$) will the relation between $\bra \hat{\beta}\ket$ and $\bbeta^\star$ reduce to scalar multiplication. This is reminiscent of what happens for linear regression, and we will explore the connection in more detail in a subsequent section. It is interesting that the form of the above expressions is universal; GLM model dependencies enter only via the order parameters $(\tilde{d}_0,\tilde{f},\tilde{g})$.
Since the vectors and matrices in (\[eq:average\_beta\],\[eq:covariance\_beta\]) have diverging dimensionality as $p\to\infty$, it is not possible to derive these propositions directly using the steepest descent route followed in deriving (\[eq:beta\_relation\_derived\]). Only for linear ML regression will we be able to confirm (\[eq:average\_beta\],\[eq:covariance\_beta\]) rigorously. However, in addition to simulation experiments with different models (described in a subsequent section), one can envisage several indirect mathematical tests of expressions (\[eq:average\_beta\],\[eq:covariance\_beta\]). First, they can be used to compute the two order parameters $c_0$ and $C$, testing their consistency with our RS order parameter equations derived earlier. This gives, using \[app:self\_averaging\], $$\begin{aligned}
\hspace*{-10mm}
c_0&=& \lim_{p\to\infty}\frac{1}{p}\sum_{\mu\nu=1}^p \bra \hat{\beta}_\mu\ket A_{\mu\nu}\beta_\nu^\star
\label{eq:check_c0}
\\
\hspace*{-10mm}
&=& \frac{w\bra a\ket^{\frac{1}{2}}}{S} \Big\bra\frac{a^2}{2\eta\!+\!a\tilde{g}}\Big\ket^{\!-1}\! \!\lim_{p\to\infty}\frac{1}{p} \sum_{\mu\nu=1}^p
\beta^\star_\mu [(\tilde{g}\bA\!+\!2\eta\one)^{-1}\bA^2]_{\mu\nu}\beta_\nu^\star
~=~ w\bra a\ket^{\frac{1}{2}}S,
\nonumber
\\
\hspace*{-10mm}
C&=& \lim_{p\to\infty}\frac{1}{p}\sum_{\mu\nu=1}^p \bra \hat{\beta}_\mu \hat{\beta}_\nu\ket A_{\mu\nu}
\nonumber
\\
\hspace*{-10mm}
&=& \lim_{p\to\infty}\frac{1}{p}{\rm Tr}
\Big[|\tilde{f}|[
(\tilde{g} \bA\!+\!2\eta\one)^{-1}\!\bA^2(\tilde{g} \bA\!+\!2\eta\one)^{-1}\Big]
\nonumber
\\
\hspace*{-10mm}
&&+ \frac{w^2\bra a\ket}{S^2} \Big\bra\frac{a^2}{2\eta\!+\!a\tilde{g}}\Big\ket^{\!-2} \!
\lim_{p\to\infty}\frac{1}{p}\!\sum_{\mu\nu=1}^p
[ (\tilde{g}\bA\!+\!2\eta\one)^{-1}\!\bA^3
(\tilde{g}\bA\!+\!2\eta\one)^{-1}]_{\mu\nu}
\beta^\star_\mu\beta^\star_\nu
\nonumber
\\
\hspace*{-10mm}
&=&|\tilde{f}|\Big\bra\frac{a^2}{(2\eta\!+\!a\tilde{g})^2}\Big\ket
+ w^2\bra a\ket\Big\bra\frac{a^2}{2\eta\!+\!a\tilde{g}}\Big\ket^{\!-2}
\Big\bra\frac{a^3}{(2\eta\!+\!a\tilde{g})^2}\Big\ket.
\label{eq:check_C}
\end{aligned}$$ Clearly, (\[eq:check\_c0\]) is identical to the result of combining the first identity of (\[eq:inverse\_transformation\]) with the expression for $\tilde{S}$ in (\[eq:Stilde\_to\_S\]). Similarly, upon combining the third identity of (\[eq:inverse\_transformation\]) with $\lim_{\gamma\to\infty}u=\lim_{\gamma\to\infty}\tilde{u}/\sqrt{\gamma}=0$, we find that (\[eq:check\_C\]) gives in the limit $\gamma\to\infty$: $$\begin{aligned}
v^2+w^2=|\tilde{f}|\Big\bra\frac{a^2}{(2\eta\!+\!a\tilde{g})^2}\Big\ket
+ w^2\bra a\ket\Big\bra\frac{a^2}{2\eta\!+\!a\tilde{g}}\Big\ket^{\!-2}
\Big\bra\frac{a^3}{(2\eta\!+\!a\tilde{g})^2}\Big\ket,\end{aligned}$$ which, in combination with $\tilde{f}<0$, reproduces equation (\[eq:ddg2=0\]). Hence one can compute the correct RS order parameter equations from (\[eq:covariance\_beta\],\[eq:average\_beta\_again\]). Secondly, in the ML limit $\eta\to 0$, where we know that $|\tilde{f}|=v^2\tilde{g}^2$, the formulae are seen to simplify as follows, confirming [*en passant*]{} an ansatz made in [@SheikhCoolen2019]: $$\begin{aligned}
\bra \hat{\bbeta}\ket= (w\bra a\ket^{\frac{1}{2}}\!/S) \bbeta^\star,~~~~~~
\bra \hat{\beta}_\mu^2\ket\!-\!\bra \hat{\beta}_\mu\ket^2=v^2
(\bA^{-1})_{\mu\mu}.
\end{aligned}$$ As a third test we can also verify from (\[eq:average\_beta\],\[eq:covariance\_beta\]) our earlier results for uncorrelated and normalized covariates. Substitution of the appropriate values $\bA=\one$, $\tilde{f}=-v^2/\tilde{u}^4$, $2\eta+\tilde{g}=\tilde{u}^{-2}$, and $\tilde{d}_0=-w/S\tilde{u}^2$ into (\[eq:average\_beta\],\[eq:covariance\_beta\]) gives indeed the correct expressions $$\begin{aligned}
\bra \hat{\bbeta}\ket= (w/S) \bbeta^\star,~~~~~~
\bra \hat{\beta}_\mu^2\ket-\bra \hat{\beta}_\mu\ket^2=
v^2.
\end{aligned}$$
Correction of association parameters for overfitting effects
------------------------------------------------------------
To work out the replica order parameters and all associated theoretical predictions in practice, we first need to estimate the true covariate correlation matrix $\bA$ from the available covariate samples $\{\bz_1,\ldots,\bz_N\}$ (which is a standard statistical problem in portfolio theory), and the value of $S$ (which controls the amplitude of the unknown vector $\bbeta^\star$). The latter can be found for sufficiently large $p$ by evaluation of $p^{-1}\hat{\bbeta}\cdot\bA\hat{\bbeta}$, using the outcome $\hat{\bbeta}$ of MAP/ML inference on the given data, in combination with equation (\[eq:check\_C\]). One similarly uses the MAP/ML inferred auxiliary parameters $\hat{\theta}$ together with their associated order parameter equations that express the link between $\hat{\theta}$ and $\theta^\star$ to eliminate the need to know $\theta^\star$. Once $\bA$, $S$, $\theta^\star$, and the solution of our RS equations are known, expressions (\[eq:average\_beta\],\[eq:covariance\_beta\]) allow us to construct alternative estimators from the MAP estimator $\hat{\bbeta}$ of the association parameters, decontaminated from the distorting effects of overfitting. To compactify notation we first define two $p\times p$ matrices $\bG$ and $\bXi$: $$\begin{aligned}
\bG&=& |\tilde{d}_0|[\tilde{g}\one\!+\!2\eta \bA^{-1}]^{-1},
\\
\bXi&=& |\tilde{f}|(\tilde{g}\bA\!+\!2\eta\one)^{-1}\bA(\tilde{g}\bA\!+\!2\eta\one)^{-1},\end{aligned}$$ with which (\[eq:average\_beta\],\[eq:covariance\_beta\]) become $$\begin{aligned}
\bra \hat{\bbeta}\ket=\bG\bbeta^\star,~~~~~~
\bra \hat{\beta}_\mu\hat{\beta}_\nu\ket-\bra \hat{\beta}_\mu\ket\bra \hat{\beta}_\nu\ket=\Xi_{\mu\nu}.\end{aligned}$$ Both $\bG$ and $\bXi$ are symmetric matrices, which commute, and $\bXi=|\tilde{f}|\tilde{d}_0^{-2}\bG\bA^{-1}\bG$. We will limit ourselves to linear correction protocols of the form $\hat{\bbeta}\to \hat{\bbeta}^\star\!=\bF\hat{\bbeta}$, where the correction matrix $\bF$ is restricted to be non-stochastic. One could a priori envisage several natural criteria for determining $\bF$, dependent upon the desired properties of the new estimator $\hat{\bbeta}^\star$, such as:
1. Removal of the inference bias, i.e. $\bra \hat{\bbeta}^\star\ket=\bbeta^\star $.
2. Minimization of the expected MSE (mean squared error) $\sum_{\mu=1}^p \bra (\hat{\beta}_\mu^\star\!-\beta_\mu^\star)^2\ket$.
3. Minimization of the expected generalization error.
In \[app:generalization\_error\] we show that, somewhat counterintuitively, minimization of the generalization error can lead to nonsensical results (an excessive bias and a hyperconfident outcome prediction model), and should therefore not be used. We will next compute the correction matrices and corresponding new estimators for the more reliable criteria (i) and (ii) in explicit form.
Criterion (i), removal of inference bias, is immediately seen to require choosing $\bF_{\rm opt}=\bG^{-1}$, giving the new and unbiased estimator $$\begin{aligned}
\hat{\bbeta}^\star
&=& |\tilde{d}_0|^{-1} [\tilde{g}\one+2\eta\bA^{-1}]\hat{\bbeta}.
\label{eq:Correction_debias}
\end{aligned}$$ Its variance is $$\begin{aligned}
\bra \hat{\beta}^{\star 2}_\mu \ket-\bra \hat{\beta}^\star_\mu\ket^2&=&
( |\tilde{f}| /\tilde{d}^2_0)
(\bA^{-1})_{\mu\mu}.\end{aligned}$$
Next we work out criterion (ii) for large $p$, assuming the various matrices to obey the conditions of \[app:self\_averaging\], so that we may use expressions such as $p^{-1}\sum_{\mu\nu\leq p}\beta_\mu^\star M_{\mu\nu}\beta_\nu^\star=(S^2/p){\rm Tr}\bM+{\it o}(1)$. The objective function to be minimized over $\bF$ then becomes, after a rescaling by $p$ to ensure that it is $\order(1)$ as $p\to\infty$: $$\begin{aligned}
\hspace*{-15mm}
\Omega(\bF)&=& \frac{1}{p} \bra (\bF\hat{\bbeta}-\bbeta^\star)^2\ket
\nonumber
\\
\hspace*{-15mm}
&=& \frac{1}{p}\sum_{\mu} \Big(\sum_{\nu\rho} F_{\mu\nu}F_{\mu\rho}\bra \hat{\beta}_\nu\hat{\beta}_\rho\ket+(\beta^\star_\mu)^2-2\beta^\star_\mu
\sum_\nu F_{\mu\nu}\bra \hat{\beta}_\nu\ket\Big)
\nonumber
\\
\hspace*{-15mm}
&=&
\frac{1}{p} \sum_{\mu\nu\rho} F_{\mu\nu}F_{\mu\rho}
[\Xi_{\nu\rho}+(\bG\bbeta^\star)_\nu(\bG\bbeta^\star)_{\rho}]+S^2 - \frac{2}{p}\sum_{\mu\nu\rho}\beta^\star_\mu
F_{\mu\nu}G_{\nu\rho}\beta_\rho^\star+{\it o}(1)
\nonumber
\\
\hspace*{-15mm}
&=&
\frac{1}{p}{\rm Tr}(\bF\bXi\bF^\dag)+ \frac{S^2}{p}
\sum_{\mu\nu\rho\lambda} F_{\mu\nu}F_{\mu\rho}G_{\nu\lambda} G_{\rho\lambda}
+S^2 - \frac{2S^2}{p}\sum_{\mu\nu}
F_{\mu\nu}G_{\nu\mu}+{\it o}(1)
\nonumber
\\
\hspace*{-15mm}
&=&
\Omega_1(\bF)+\Omega_2(\bF)+{\it o}(1).
\label{eq:Omega}\end{aligned}$$ with $$\begin{aligned}
\hspace*{-10mm}
\Omega_1(\bF)= \frac{1}{p}{\rm Tr}(\bF\bXi\bF^\dag),~~~~~~
\Omega_2(\bF)=\frac{S^2}{p}{\rm Tr}[(\bF\bG\!-\!\one)^\dag(\bF\bG\!-\!\one)].
\label{eq:Omega_terms}\end{aligned}$$ Removal of the inference bias gives $\Omega_2(\bF)=0$ (achieved for $\bF=\bG^{-1}$, following the previous criterion (i)), and removal of the inference noise gives $\Omega_1(\bF)=0$ (achieved for $\bF=\bnull$, or for any $\bF$ if $\bXi=\bnull$, i.e. if the MAP inference is already noise-free). Hence we can interpret $\Omega_1(\bF)$ as the error contribution from the noise, and $\Omega_2(\bF)$ as the error contribution from the bias. In criterion (i) we minimized $\Omega_2(\bF)$ and this would generally increase $\Omega_1(\bF)$. Minimizing (\[eq:Omega\]) requires balancing the two error sources. This is the bias-variance trade-off in inference [@Bishop; @CasellaBerger]. However, since $\Omega(\bF)$ is quadratic in $\bF$ we can find the location of the overall minimum in explicit form: $$\begin{aligned}
\bF_{\rm opt}&=&(\bXi/S^2\!+\!\bG^2)^{-1}\bG,
\label{eq:Correction_mse}
\\
\Omega_{\rm min}&=& \frac{1}{p}{\rm Tr}(S^{-2}\one\!+\!\bXi^{-1}\bG^{2})^{-1}.\end{aligned}$$
Applications to specific regression models
==========================================
We now apply the generic replica symmetric MAP order parameter equations (\[eq:ddf2=0\])–(\[eq:ddtheta2=0\]), where $\xi(\mu,\sigma,s,\theta)$ represents the solution of (\[eq:xi\_equation\]), to different regression models of the GLM family. We test the predictions of the theory for MAP and ML regression against measurements of simulations with different outcome types and models, and with synthetic data. In all cases we will for simplicity choose the covariate distribution $p(\bz)$ to be Gaussian, with zero average but potentially correlated components $\{z_\mu\}$. For the $p\times p$ covariance matrix $\bA$ with entries $A_{\mu\nu}=\bra z_\mu z_\nu\ket$ we will make the following choice, with $\epsilon\in[0,1]$: $$\begin{aligned}
\begin{array}{lll}
A_{\mu\mu}&\!\!\!=\!\!\!&1
\\[1mm]
A_{\mu,\mu+1}&\!\!\!=\!\!\!&A_{\mu+1,\mu}~=~\epsilon
\\[1mm]
A_{\mu\nu}&\!\!\!=\!\!\!&0~~~{\rm for~all~other~entries}.
\end{array}
\label{eq:chosen_A}\end{aligned}$$ This describes pairwise correlated covariates. The matrix (\[eq:chosen\_A\]) obeys the conditions in \[app:self\_averaging\], and is trivially diagonalised to give $\varrho(a)=\frac{1}{2}\delta(a\!-\!1\!+\!\epsilon)+\frac{1}{2}\delta(a\!-\!1\!-\!\epsilon)$, enabling precise tests of the predictions of the theory.
Linear regression
-----------------
[*Replica equations for MAP linear regression.*]{} The simplest case of a GLM corresponds to linear regression, where the outcomes of ML and MAP regression can in principle be computed in explicit form. It therefore serves as the simplest test for our general equations. In linear regression we have $\theta=(\beta_0,\Sigma)\in\R^2$ and $$\begin{aligned}
p(s|
\xi,\theta)=(2\pi\Sigma^2)^{-\frac{1}{2}}\rme^{-\frac{1}{2}(s-\xi-\beta_0)^2/\Sigma^2}.
\label{eq:linear_case}\end{aligned}$$ Here we find that $$\begin{aligned}
\xi(\mu,\sigma,s,\theta)&=& \frac{\mu\Sigma^2+
\sigma^2 (s-\beta_0)}{\Sigma^2+\sigma^2}.\end{aligned}$$ Hence $(\partial_1\xi)(\mu,\sigma,s,\theta)=\Sigma^2/(\sigma^2\!+\!\Sigma^2)$, and upon working out the relevant derivatives of $p(s|
\xi,\theta)$, we find the following closed set of MAP order equations: $$\begin{aligned}
\hspace*{-15mm}
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
&=& \tilde{u}^2,
\label{eq:ddf_lin=0}
\\
\hspace*{-15mm}
w^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-2}
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
\!-\!1\Big]
-
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket
&=&v^2,
\label{eq:ddg_lin=0}
\\
\hspace*{-15mm}
\int\!{\rm D}t ~\Big\bra\!\Big\bra \bra
(\beta_0^\star\!-\!\beta_0\! +\!(S\bra a\ket^{\frac{1}{2}}\!-\!w)y_0\!+\!\Sigma^\star t\!-\!vz)^2\ket_{s}\Big\ket\!\Big\ket
&=& - \!\zeta
\tilde{f}(\Sigma^2\!+\! \tilde{u}^2)^2,
\label{eq:ddu_lin=0}
\\[1mm]
\hspace*{-15mm}
\frac{1}{\tilde{u}^2+\Sigma^2}
&=&\zeta \tilde{g},
\label{eq:ddv_lin=0}
\\[1mm]
\hspace*{-15mm}
\frac{S}{\Sigma^2+\tilde{u}^2}
&=& \frac{\zeta w
\bra a\ket^{\frac{1}{2}}}
{ \Big\bra \frac{a^2}{
2\eta+\tilde{g}a}\Big\ket},
\label{eq:ddw_lin=0}
\\[1mm]
\hspace*{-15mm}
\beta_0^\star-\beta_0&=& 0,
\label{eq:ddbeta0_lin=0}
\\[1mm]
\hspace*{-15mm}
\int\!{\rm D}t~ \Big\bra\!\Big\bra \bra
[\beta_0^\star\!-\!\beta_0\!+\!(S\bra a\ket^{\frac{1}{2}}\!-\!w)y_0\!+\!\Sigma^\star t\!-\!vz]^2\Big\ket_{\!s}\Big\ket\!\Big\ket
&=&\frac{(\Sigma^2+\tilde{u}^2)^2}{\Sigma^2}.
\label{eq:ddSigma_lin=0}\end{aligned}$$ Thus one always has $\beta_0=\beta_0^\star$, and the other equations can be compactified to $$\begin{aligned}
&&
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
= \tilde{u}^2, ~~~~~~
1/\Sigma^2
= - \zeta
\tilde{f}, ~~~~~~
\frac{1}{\tilde{u}^2+\Sigma^2}
=\zeta \tilde{g},
\\
&&
w^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-2}
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
-1\Big]
-
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket
=v^2,
\\[2mm]
&&
\frac{S}{\Sigma^2+\tilde{u}^2}
= \zeta w
\bra a\ket^{\frac{1}{2}}
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{-1},
\\[1mm]
&&
(S\bra a\ket^{\frac{1}{2}}\!-\!w)^2 \!+\!v^2\!+\!\Sigma^{\star 2}
= (\Sigma^2+\tilde{u}^2)^2/\Sigma^2.\end{aligned}$$ Via substitutions one can reduce these coupled equations to a single nonlinear equation for $\tilde{g}$, the numerical solution of which then generates the other order parameters $(v,w,\tilde{f},\tilde{u},\Sigma)$. With the short-hand $\alpha_{k\ell}=\bra a^k/(2\eta\!+\!\tilde{g}a)^\ell\ket$ this equation takes the form $$\begin{aligned}
\tilde{g}^{-1}&=& \zeta(1\!-\!\zeta\tilde{g}\alpha_{11})\big[S^2(\alpha_{10}\!-\!2\tilde{g}\alpha_{21}\!+\!\tilde{g}^2 \alpha_{32})\!+\!\Sigma^{\star 2}\big]+\zeta\tilde{g}\alpha_{22}.\end{aligned}$$ Similarly, using the above formulae we can also simplify the predictions (\[eq:average\_beta\],\[eq:covariance\_beta\]) to $$\begin{aligned}
\bra \hat{\bbeta}\ket &=& (\one+\frac{2\eta}{\tilde{g}}\bA^{-1})^{-1}\bbeta^\star,
\\
\bra\hat{\beta}^2_\mu\ket-\bra \hat{\beta}_\mu\ket^2&=& \frac{1}{\zeta\Sigma^2\tilde{g}^2}[(\one\!+\!\frac{2\eta}{\tilde{g}}\bA^{-1})\bA^{-1}(\one\!+\!\frac{2\eta}{\tilde{g}}\bA^{-1})]_{\mu\mu}.\end{aligned}$$ For uncorrelated covariates, i.e. $\bA=\one$, these results are consistent with the well-known asymptotic behaviour of linear estimators with large random measurement matrices [@FA1; @FA2; @FA3; @FA4]. Setting $\eta=0$ brings us from MAP regression to ML regression. Here we find that the above equations reduce after some simple manipulations to $$\begin{aligned}
&&
\hspace*{-10mm}
w=S\bra a\ket^{\frac{1}{2}},~~~~~~\Sigma=\Sigma^\star\sqrt{1\!-\!\zeta},~~~~~~\tilde{u}=\Sigma^\star\sqrt{\zeta},~~~~~~v=\Sigma^\star\sqrt{\frac{\zeta}{1\!-\!\zeta}},
\label{eq:linear_ML_replica}
\\[-1mm]
&&
\hspace*{-10mm}
\bra \hat{\bbeta}\ket = \bbeta^\star,~~~~~~~~
\bra\hat{\beta}^2_\mu\ket-\bra \hat{\beta}_\mu\ket^2= \frac{\zeta}{1\!-\!\zeta}(\Sigma^{\star})^2 (\bA^{-1})_{\mu\mu}.\end{aligned}$$ Thus also the association parameters will on average be inferred correctly in ML, but there will be increasing overfitting induced noise (diverging at the transition point $\zeta\!=\!1$), and under-estimation of the true uncertainty $\Sigma^\star$ in the outcome predictions.
[*Direct solution.*]{} For linear regression we can go beyond testing the replica predictions against numerical simulations, since the regression problem allows for exact solution. The parameter to be inferred are $\bbeta$ and $\Sigma$, whose MAP estimators are $$\begin{aligned}
(\hat{\bbeta},\hat{\Sigma})&=& {\rm argmin}_{\bbeta,\Sigma}\Big\{
\frac{1}{2\Sigma^2}\sum_{i=1}^N (s^i\!-\!\frac{\bbeta\cdot\bz^i}{\sqrt{p}})^2+N\log\Sigma
+\eta\bbeta^2\Big\}.\end{aligned}$$ This minimization results in the following coupled equations, with the empirical $p\times p$ covariance matrix $\hat{\bA}$ with entries $\hat{A}_{\mu\nu}=N^{-1}\sum_{i\leq N} z_{i\mu}z_{i\nu}$: $$\begin{aligned}
\hat{\Sigma}^2&=& \frac{1}{N}\sum_{i=1}^N (s_i-\frac{\hat{\bbeta}\cdot\bz_i}{\sqrt{p}})^2,
\\[-1mm]
\hat{\beta}_\mu&=&\sum_{\nu}\Big(2\eta\zeta\hat{\Sigma}^2\one+\hat{\bA}\Big)^{\!-1}_{\!\mu\nu}
\frac{\zeta}
{\sqrt{p}}\sum_{i=1}^N z_{i\nu} s_i.\end{aligned}$$ The direct solution is formulated in terms of the empirical covariate covariance matrix $\hat{\bA}$, whereas the replica analysis involves the true population covariance matrix $\bA$. To understand the connection between the two descriptions, we need to compute disorder-averaged quantities from the above equations. To do this, we assume, as in the replica analysis, that the data are generated by a linear model of the type (\[eq:linear\_case\]), with unknown parameters $(\bbeta^\star,\Sigma^\star)$. Hence $s_i=\bbeta^\star\!\cdot\bz_i/\sqrt{p}+\Sigma^\star\xi_i$, in which all $\xi$ are i.i.d. random variables, drawn from $p(\xi)=(2\pi)^{-\frac{1}{2}}\rme^{-\frac{1}{2}\xi^2}$. We will show below that in ML regression $\hat{\Sigma}$ is self-averaging for $p\to\infty$, so that for large $p$ we can evaluate the distribution of inferred association parameters, averaged over all possible realizations of the data, i.e. over all $\{\xi_i,\bz_i\}$: $$\begin{aligned}
\hspace*{-0mm}
P(\hat{\bbeta})&=&
\Big\bra\!\Big\bra \delta\Big[\hat{\bbeta}-(2\eta\zeta\hat{\Sigma^2}\one+\hat{\bA})^{-1}\Big(\hat{\bA}\bbeta^\star+\frac{\zeta \Sigma^\star}
{\sqrt{p}}\sum_{i=1}^N \bz_i \xi_i\Big)\Big]\Big\ket\!\Big\ket_{\{\xi,\bz\}}
\nonumber
\\
\hspace*{-0mm}
&=&\Big\bra \int\!\frac{\rmd\bx}{(2\pi)^d}~\rme^{\rmi\bx\cdot(\bbeta-\hat{\bG}\bbeta^\star)
-\frac{1}{2} \zeta (\Sigma^\star)^2\bx\cdot \hat{\bG}\!\hat{\bA}^{-1}\hat{\bG}\bx}
\Big\ket_{\{\bz\}}
\nonumber
\\
\hspace*{-0mm}
&=&\int\!\rmd\hat{\bA} ~W(\hat{\bA})~
{\mathcal N}(\hat{\bbeta}|\hat{\bG}\bbeta^\star,2\zeta\Sigma^{\star 2}\bG
\hat{\bA}^{-1}\bG),
\label{eq:linear_betastats_1}\end{aligned}$$ with $\hat{\bG}=(\one\!+\!2\eta\zeta\hat{\Sigma^2}\hat{\bA}^{-1})^{-1}$, and $$\begin{aligned}
{\mathcal N}(\bbeta|\bmu,\bSigma)&=&
\frac{\rme^{-\frac{1}{2}(\bbeta-\bmu)\cdot\bSigma^{-1}
(\bbeta-\bmu)}}
{\sqrt{(2\pi)^d{\rm Det}\bSigma}},
\\
P(\hat{\bA})&=& \Big\bra \prod_{\mu\nu=1}^p \delta\Big[\hat{A}_{\mu\nu}-\frac{1}{N}\sum_{i=1}^N z_{i\mu} z_{i\nu} \Big]\Big\ket_{\{\bz\}}.
\label{eq:Ahat_measure}\end{aligned}$$ Thus $P(\hat{\bbeta})$ is an average of Gaussian distributions, each weighted by the measure $P(\hat{\bA})$ of empirical covariate covariance matrices. The integral in (\[eq:linear\_betastats\_1\]) is still defined over all $p\times p$ matrices $\hat{\bA}$.
In \[app:towards\_Wishart\] we evaluate (\[eq:Ahat\_measure\]) and (\[eq:linear\_betastats\_1\]) further for the choice $p(\bz)=[(2\pi)^{-p}{\rm Det}\bA]^{\frac{1}{2}}\rme^{-\frac{1}{2}\bz\cdot\bA\bz}$, and show that here $P(\hat{\bbeta})$ can be written as the following integral over the space $\Omega_p$ of symmetric positive definite matrices, involving the Wishart distribution $W(\hat{\bA})$ [@Wishart] with $N$ degrees of freedom: $$\begin{aligned}
P(\hat{\bbeta})&=&
\int_{\Omega_p}\!\rmd\hat{\bA} ~W(\hat{\bA})
~{\cal N}(\hat{\bbeta}|\hat{\bG}\bbeta^\star,2\zeta\Sigma^{\star 2}\hat{\bG}
\hat{\bA}^{-1}\hat{\bG}),
\label{eq:linear_betastats_2}\end{aligned}$$ with $$\begin{aligned}
W(\hat{\bA})&=& \frac{\rme^{-\frac{1}{2}N{\rm Tr}(\hat{\bA}\bA^{-1})}({\rm Det}\hat{\bA})^{\frac{1}{2}(N-p-1)}}{{\cal Z}(\bA)}
\\[1mm]
{\cal Z}(\bA) &=&
\int_{\Omega_p}\!\!\rmd\hat{\bA} ~
\rme^{-\frac{1}{2}N{\rm Tr}(\hat{\bA}\bA^{-1})}({\rm Det}\hat{\bA})^{\frac{1}{2}(N-p-1)}
\nonumber
\\[-1mm]
&=& \Big(\frac{2}{N}\Big)^{\!Np/2}
\pi^{\frac{1}{4}p(p-1)}({\rm Det}\bA)^{\frac{1}{2}N} \prod_{j=\frac{1}{2}(N-p+1)}^{\frac{1}{2}N} \Gamma(j).\end{aligned}$$ From the properties of the Wishart distribution follow average and variance of the entries of $\hat{\bA}$, which confirm, as expected, that $\hat{A}_{\mu\nu}=A_{\mu\nu}+\order(N^{-\frac{1}{2}})$: $$\begin{aligned}
\bra \hat{A}_{\mu\nu}\ket = A_{\mu\nu},~~~~~~
\bra \hat{A}^2_{\mu\nu}\ket-\bra \hat{A}_{\mu\nu}\ket^2= N^{-1}A_{\mu\nu}\!+\!N^{-2}A_{\mu\mu}A_{\nu\nu}.\end{aligned}$$
The integral in (\[eq:linear\_betastats\_2\]) is still nontrivial, so we now focus on the case of linear ML regression and take the limit $\eta\to 0$, where our previous result simplifies considerably. We define the $p\times p$ matrix with entries $C_{\mu\nu}=p^{-1}(\hat{\beta}_\mu\!-\!\beta^\star_\mu)(\hat{\beta}_\nu\!-\!\beta^\star_\nu)$. This leads to $$\begin{aligned}
\hspace*{-15mm}
P(\hat{\bbeta})&=&
\int_{\Omega_p}\!\!\frac{\rmd\hat{\bA} ~
\rme^{-\frac{1}{2}N{\rm Tr}(\hat{\bA}\bA^{-1})}({\rm Det}\hat{\bA})^{\frac{1}{2}(N-p-1)}}{{\cal Z}(\bA)}
~{\cal N}(\bbeta|\bbeta^\star\!,\zeta (\Sigma^\star)^2\hat{\bA}^{-1})
\nonumber
\\
\hspace*{-15mm}
&=& \frac{[2\pi \zeta (\Sigma^\star)^2]^{-p/2}}{{\cal Z}(\bA)}
\int_{\Omega_p}\!\rmd\hat{\bA} ~
\rme^{-\frac{1}{2}N{\rm Tr}[\hat{\bA}(\bA^{-1}\!+\bC/(\Sigma^\star)^2)]}({\rm Det}\hat{\bA})^{\frac{1}{2}(N-p)}.
~~~~\end{aligned}$$ This is again an integral of the Wishart form that can be evaluated analytically, now with $N+1$ degrees of freedom. Thus we get $$\begin{aligned}
\hspace*{-15mm}
P(\hat{\bbeta})
&=& \frac{N^{Np/2-\frac{1}{2}p(N+1)} \Gamma(\frac{1}{2}(N\!+\!1))\sqrt{{\rm Det}\bA}}{[\pi \zeta (\Sigma^\star)^2]^{p/2} \Gamma(\frac{1}{2}(N\!-\!p\!+\!1))}
[{\rm Det}(\one\!+\!\bA\bC/(\Sigma^\star)^2)]^{-\frac{N+1}{2}}.\end{aligned}$$ Finally we use the identity $$\begin{aligned}
{\rm Det}(\one\!+\!\bA\bC/(\Sigma^\star)^2)&=& 1+\frac{(\hat{\bbeta}\!-\!\bbeta^\star)\cdot\bA(\hat{\bbeta}\!-\!\bbeta^\star)}{p (\Sigma^\star)^{2}}\end{aligned}$$ to show that $P(\hat{\bbeta})$ is for any $(p,N)$ a multivariate student’s $t$-distribution with $N\!-\!p\!+\!1$ degrees of freedom: $$\begin{aligned}
\hspace*{-15mm}
P(\hat{\bbeta})
&=&\pi^{-p/2} \frac{\Gamma(\frac{1}{2}(N\!+\!1))}{\Gamma(\frac{1}{2}(N\!-\!p\!+\!1))}
\frac{\sqrt{{\rm Det}\bA}}{[p (\Sigma^\star)^2]^{p/2} }
\Big[ 1\!+\!\frac{(\hat{\bbeta}\!-\!\bbeta^\star)\cdot\bA(\hat{\bbeta}\!-\!\bbeta^\star)}{p (\Sigma^\star)^{2}}\Big]^{-\frac{N+1}{2}}.
\label{eq:Exact_ML}\end{aligned}$$ Equivalently we can write $\hat{\bbeta}=\bbeta^\star+\bA^{\frac{1}{2}}\bx$, where $$\begin{aligned}
P(\bx)
&=&\frac{\Gamma(\frac{1}{2}(N\!+\!1))}{\Gamma(\frac{1}{2}(N\!-\!p\!+\!1))}
\frac{\pi^{-p/2} }{[p (\Sigma^\star)^2]^{p/2} }
\Big[ 1\!+\!\frac{\bx^2}{p (\Sigma^\star)^{2}}\Big]^{-\frac{N+1}{2}}
\label{eq:Px}\end{aligned}$$ Mean and covariance matrix of(\[eq:Exact\_ML\]) are in the limit $N,p\to\infty$, with $p/N=\zeta$ fixed, exactly as predicted by the replica theory, since $$\begin{aligned}
\bra \hat{\beta}_\mu\ket&=& \beta^\star_\mu,
\label{eq:t_moment_1}
\\
\bra \hat{\beta}_\mu\hat{\beta}_\nu\ket- \bra \hat{\beta}_\mu\ket\bra \hat{\beta}_\nu\ket&=&
\frac{\zeta (\Sigma^\star)^2}{1\!-\!\zeta}(\bA^{-1})_{\mu\nu}+\order(\frac{1}{N}).
\label{eq:t_moment_2}\end{aligned}$$ Along the same lines one can also compute higher order moments of $P(\hat{\bbeta})$, giving results such as $$\begin{aligned}
\Big\bra \Big[\frac{1}{p}(\hat{\bbeta}\!-\!\bbeta^\star)^2\Big]^2\Big\ket-\Big\bra\frac{1}{p}(\hat{\bbeta}\!-\!\bbeta^\star)^2\Big\ket^2=\order(p^{-1})\end{aligned}$$ Although (\[eq:Exact\_ML\]) is itself not a Gaussian distribution, for the marginal distribution of any finite set of components of $\hat{\bbeta}$ it predicts Gaussian statistics in the limit $p,N\to\infty$ with fixed ratio $\zeta=p/N$. It is a general property of the multivariate student’s $t$-distribution that all its marginals also obey multivariate student’s $t$-distributions [@Nadarajah]. Let us define the set of indices corresponding to non-marginalized components of $\hat{\bbeta}$ as ${\cal S}\subset\{1,\ldots,p\}$, and write these components as $\tilde{\bbeta}=\{\beta_\mu,\mu\!\in\! {\cal S}\}$. We also define an $|{\cal S}|\!\times\!|{\cal S}|$ matrix $\tilde{\bA}$, defined by the property that $(\tilde{\bA}^{-1})_{\mu\nu}=(\bA^{-1})_{\mu\nu}$ for all $(\mu,\nu)\in|{\cal S}|^2$. Then the marginal distribution for $\tilde{\bbeta}$ is [@Nadarajah]: $$\begin{aligned}
P(\tilde{\bbeta}) &\propto &
\Big[1\!+\!\frac{(\tilde{\bbeta}-\tilde{\bbeta}^\star)\cdot\tilde{\bA}(\tilde{\bbeta}-\tilde{\bbeta}^\star)}{p(\Sigma^\star)^2}\Big]^{-\frac{1}{2}(N-p+1+|S|)}.\end{aligned}$$ For $|{\cal S}|$ is finite and $p,N\!\to\!\infty$ with $p/N=\zeta$ fixed, we can expand this and find $$\begin{aligned}
P(\tilde{\bbeta}) &\propto & \rme^{-\frac{1}{2}\frac{1-\zeta}{\zeta(\Sigma^\star)^2}
(\tilde{\bbeta}-\tilde{\bbeta}^\star)\cdot\tilde{\bA}(\tilde{\bbeta}-\tilde{\bbeta}^\star)+\order(p^{-1})}.\end{aligned}$$ So in the relevant limit, exact evaluation of $P(\hat{\bbeta})$ gives for linear ML regression a Gaussian distribution for the marginals (if $|{\cal S}|$ is finite), with, upon using $(\tilde{\bA}^{-1})_{\mu\nu}=({\bA}^{-1})_{\mu\nu}$ and in accordance with (\[eq:t\_moment\_1\], \[eq:t\_moment\_2\]), $$\begin{aligned}
&&\hspace*{-10mm}
\mu,\nu\in {\cal S}:~~~~
\bra \hat{\beta}_\mu\ket=\beta_\mu^\star,~~~~ \bra \hat{\beta}_\mu\hat{\beta}_\nu\ket-\bra \hat{\beta}_\mu\ket\bra\hat{\beta}_\nu\ket=\frac{\zeta(\Sigma^\star)^2}{1\!-\!\zeta}(\bA^{-1})_{\mu\nu}.
\end{aligned}$$ We conclude from the above analysis that, in those cases where exact evaluation enables direct comparison with the predictions of the replica theory (i.e. for linear ML regression), there is full agreement between the two, and that the two propositions (\[eq:average\_beta\],\[eq:covariance\_beta\]) hold. Going beyond ML to do the same test for linear MAP regression requires evaluation of the integral in (\[eq:linear\_betastats\_2\]), which we have so far been unable to do.
The direct calculation of the statistics of the inferred noise parameter $\hat{\Sigma}$ in ML linear regression also confirms the replica prediction. After some simple manipulations one finds that $\hat{\Sigma}^2$ can be written in terms of the data as $$\begin{aligned}
\hat{\Sigma}^2&=&
\frac{\Sigma^{\star 2}}{N}\sum_{ij=1}^N \xi_i\Big\{
\delta_{ij}
- \frac{1}{N}
\bz^i\cdot
\hat{\bA}^{-1}\bz^j
\Big\}\xi_j.\end{aligned}$$ We define the $p\!\times\! N$ matrix $\bZ$ with entries $Z_{\mu i}=z_\mu^i/\sqrt{N}$. The characteriztic function of the distribution $P(\hat{\Sigma}^2)$ of $\hat{\Sigma}^2$ over the realizations of the outcome noise $\bxi$ is $$\begin{aligned}
\phi(k)&=&\int\!{\rm D}\bxi~ \rme^{\rmi k \frac{\Sigma^{\star 2}}{N}\bxi\cdot[
\one
- \bZ^\dag
\hat{\bA}^{-1}\bZ
]\bxi}
\nonumber
\\
&=&
\Big[
{\rm Det}
\Big(
(1\! -\frac{2\rmi k}{N} \Sigma^{\star 2})\one +\frac{2\rmi k}{N} \Sigma^{\star 2} \bZ^\dag\hat{\bA}^{-1}\bZ
\Big)\Big]^{-\frac{1}{2}}.\end{aligned}$$ We note that $\bZ\bZ^\dag\!=\!\hat{\bA}$, from which it follows in turn that $(\bZ^\dag\hat{\bA}^{-1}\bZ )^2=
\bZ^\dag\hat{\bA}^{-1}\bZ $. Hence $\bZ^\dag\hat{\bA}^{-1}\bZ $ is a projection matrix, with eigenvalues 0 and 1. Moreover, since ${\rm Tr}(\bZ^\dag\hat{\bA}^{-1}\bZ )=p$, we know in fact that it has precisely $\zeta N$ eigenvalues 1 and $(1\!-\!\zeta)N$ eigenvalues 0. Hence, for any realization of the covariates we have $$\begin{aligned}
{\rm Det}
\Big[(1\! -\!\frac{2\rmi k \Sigma^{\star 2}}{N})\one \!+\!\frac{2\rmi k \Sigma^{\star 2} }{N}\bZ^\dag\hat{\bA}^{-1}\bZ
\Big)\Big]=(1\! -\!\frac{2\rmi k \Sigma^{\star 2}}{N})^{(1-\zeta)N}\!,~~\end{aligned}$$ so that $$\begin{aligned}
\phi(k)=(1\! -\frac{2\rmi k \Sigma^{\star 2}}{N})^{-\frac{1}{2}(1-\zeta)N}.\end{aligned}$$ We recognize that this is the characteriztic function of the gamma distribution, with average $(1\!-\!\zeta)\Sigma^{\star 2}$ and width $\Sigma^{\star 2}\sqrt{2(1\!-\!\zeta)/N}$. Hence $\hat{\Sigma}$ obeys the gamma distribution and is self-averaging with respect to the realization of the data for $N\to\infty$, and $\lim_{N\to\infty}\hat{\Sigma}=\Sigma^{\star}\sqrt{1\!-\!\zeta}$, confirming the prediction in (\[eq:linear\_ML\_replica\]) of the replica theory.
=0.337mm
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(25,305)[$\eta=0.01$, $\epsilon\!=\!0$, ]{}(25,291)[$\Sigma^\star\!=\!0.1$]{} (125,305)[$\eta=0.01$, $\epsilon\!=\!0.75$,]{}(125,291)[$\Sigma^\star\!=\!0.1$]{} (225,305)[$\eta=0.01$, $\epsilon\!=\!0$, ]{}(225,291)[$\Sigma^\star\!=\!0.5$]{} (325,305)[$\eta=0.1$, $\epsilon\!=\!0.75$, ]{}(325,291)[$\Sigma^\star\!=\!0.5$]{}
(3,233)[$\hat{\beta}_\mu$]{} (60,175)[$\beta^\star_\mu$]{}(160,175)[$\beta^\star_\mu$]{} (260,175)[$\beta^\star_\mu$]{}(360,175)[$\beta^\star_\mu$]{} (28,265)[$\zeta\!=\!0.5$]{}(128,265)[$\zeta\!=\!0.5$]{} (228,265)[$\zeta\!=\!0.5$]{}(328,265)[$\zeta\!=\!0.5$]{} (0,185)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_eta001_sig01_eps000_S2_betas_zeta050.eps "fig:"){width="135\unitlength"}]{} (100,185)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_eta001_sig01_eps075_S2_betas_zeta050.eps "fig:"){width="135\unitlength"}]{} (200,185)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_eta001_sig05_eps000_S2_betas_zeta050.eps "fig:"){width="135\unitlength"}]{} (300,185)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_eta01_sig05_eps075_S2_betas_zeta050.eps "fig:"){width="135\unitlength"}]{}
(2,120)[$w$]{} (0,70)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_w_eta001_sig01_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (100,70)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_w_eta001_sig01_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (200,70)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_w_eta001_sig05_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (300,70)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_w_eta010_sig05_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (2,20)[$v$]{} (0,-30)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_v_eta001_sig01_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (100,-30)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_v_eta001_sig01_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (200,-30)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_v_eta001_sig05_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (300,-30)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_v_eta010_sig05_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (2,-80)[$\Sigma$]{} (0,-130)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_sigma_eta001_sig01_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (63,-140)[$\zeta$]{} (100,-130)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_sigma_eta001_sig01_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (163,-140)[$\zeta$]{} (200,-130)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_sigma_eta001_sig05_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (263,-140)[$\zeta$]{} (300,-130)[![Results of linear MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer), $\epsilon$ (covariate correlations), and $\Sigma^\star$ (true noise strength). In all cases $S=2$. Top row: inferred versus true association parameters for $\zeta=0.5$. Second and third row: order parameters $w$ and $v$ plotted versus $\zeta$. Bottom row: inferred noise strength $\Sigma$ versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves: theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_linear_vw"}](linear_MAP_sigma_eta010_sig05_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (363,-140)[$\zeta$]{}
=0.337mm
(300,233)
(25,219)[$\eta=0.01$, $\epsilon\!=\!0$, ]{}(25,205)[$\Sigma^\star\!=\!0.1$]{} (125,219)[$\eta=0.01$, $\epsilon\!=\!0.75$,]{}(125,205)[$\Sigma^\star\!=\!0.1$]{} (225,219)[$\eta=0.01$, $\epsilon\!=\!0$, ]{}(225,205)[$\Sigma^\star\!=\!0.5$]{} (325,219)[$\eta=0.1$, $\epsilon\!=\!0.75$, ]{}(325,205)[$\Sigma^\star\!=\!0.5$]{}
(-12,150)[*slope*]{} (0,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_slopes_eta001_sig01_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (100,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_slopes_eta001_sig01_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (200,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_slopes_eta001_sig05_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (300,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_slopes_eta010_sig05_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (-20,50)[*$\sqrt{\it MSE}$*]{} (0,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_mse_eta001_sig01_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (63,-10)[$\zeta$]{} (100,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_mse_eta001_sig01_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (163,-10)[$\zeta$]{} (200,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_mse_eta001_sig05_eps000_S2.eps "fig:"){width="135\unitlength"}]{} (263,-10)[$\zeta$]{} (300,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias. Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computed via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_linear_correction"}](linear_MAP_mse_eta010_sig05_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (363,-10)[$\zeta$]{}
[*Numerical simulations of MAP linear regression with correlated covariates.*]{} The result of solving numerically the RS order parameter equations in the presence of covariate correlations of the type (\[eq:chosen\_A\]) is shown in Figure \[fig:MAP\_linear\_vw\], where we plot the resulting values of the order parameters $v$ and $w$ and the inferred noise strength $\Sigma$ together with regression simulation data (for synthetic Gaussian covariates), as functions of the ratio $\zeta=p/N$. In these experiments we chose $\beta_0=\beta_0^\star=0$, for simplicity. Once more we observe excellent agreement between theory and simulation. In the top row we also plot for each parameter combination the MAP-inferred parameters $\hat{\beta}_\mu$ versus the corresponding true association strengths $\beta_\mu^\star$, for pooled data from 20 regressions. In the two columns on the right we see that the width of the data cloud (top picture) reflects inference noise in the case where $\epsilon=0$ (no covariate correlations), with a larger $v$, whereas for $\epsilon=0.75$ the inference noise is reduced, so that there the wider cloud reflects the correlation-induced bias.
These data also enable us to tests our two protocols (\[eq:Correction\_debias\],\[eq:Correction\_mse\]) for correcting the MAP estimator of the association parameters for the distortions caused by overfitting. See Figure \[fig:MAP\_linear\_correction\]. We plot the slopes of the data clouds of estimators versus true parameter values, as shown for $\zeta=0.5$ in the top row of Figure \[fig:MAP\_linear\_vw\], including the MAP estimator (red circles), the minimum bias estimator (\[eq:Correction\_debias\]) (blue squares), and the minimum MSE estimator (\[eq:Correction\_mse\]) (green crosses). For linear regression with small regularizers (i.e. small $\eta$) the slopes of the data clouds for the MAP estimator are very close to unity, the inference bias is weak, and hence the red and blue data points coincide (debiasing makes no difference). For stronger correlations and stronger regularization (column on the right), this si no longer true. Similarly, the MSE values of the minimum MSE estimator (\[eq:Correction\_mse\]) (green) are as predicted indeed always identical to or below those of the other two estimators.
Logistic regression
-------------------
[*Equations for MAP logistic regression.*]{} In logistic regression we have $s\in\{-1,1\}$ (alternatively one could define $s\in\{0,1\}$; with our present choice the equations will be somewhat more compact), with $\theta=\beta_0\in\R$, and $$\begin{aligned}
\label{eq:logistic_regression_model}
p(s|\xi,\beta_0)&=&\frac{\rme^{s(\xi+\beta_0)}}{2\cosh(\xi\!+\!\beta_0)}.\end{aligned}$$ Hence $\partial_\xi\log p(s|\xi,\beta_0)= s-\tanh(\xi\!+\!\beta_0)$ and $\partial_\xi^2\log p(s|\xi,\beta_0)=\tanh^2(\xi\!+\!\beta_0)-1$. We will now compute the various model-dependent building blocks of our general order parameter equations (\[eq:ddf2=0\]–\[eq:ddtheta2=0\]). The function $\xi(\mu,\sigma,s,\beta_0)$ is the solution of $$\begin{aligned}
s-\tanh(\xi\!+\!\beta_0)=(\xi-\mu)/\sigma^2.\end{aligned}$$ We switch from $\xi$ to the new variable $x=s(\xi+\beta_0)$, so $sx-\beta_0=\xi$. Now $$\begin{aligned}
1-[x-s(\beta_0+\mu)]/\sigma^2=\tanh(x).\end{aligned}$$ We next define $\tilde{x}(\mu,\sigma)$ as the solution of the following transcendental equation, whose solution is unique since the right-hand side increases monotonically from $-1$ to $1$, and the left-hand side decreases monotonically from $+\infty$ to $-\infty$: $$\begin{aligned}
\tilde{x}(\mu,\sigma):&~~~& {\rm solution~of}~~\tanh(x)=1-(x\!-\!\mu)/\sigma^2.\end{aligned}$$ Graphical inspection shows that $\tilde{x}(\mu,\sigma)$ increases monotonically with both $\mu$ and $\sigma\geq 0$, and that $\mu=\tilde{x}(\mu,0)\leq \tilde{x}(\mu,\sigma)\leq \tilde{x}(\mu,\infty)=\infty$. In terms of $\tilde{x}$ we may write $$\begin{aligned}
\xi(\mu,\sigma,s,\beta_0)=s\tilde{x}(s(\beta_0\!+\!\mu),\sigma)-\beta_0.\end{aligned}$$ We can now work out the relevant derivatives required in our equations: $$\begin{aligned}
(\partial_1\xi)(\mu,\sigma,s,\beta_0)&=& \Big[1+\sigma^2[1\!-\!\tanh^2(\xi\!+\!\beta_0)]\Big]^{-1},
\\
\frac{\partial}{\partial\beta_0}\log p(s|\xi,\beta_0)&=&s-\tanh(\xi\!+\!\beta_0),
\\
\frac{\partial}{\partial y_0}\log p(s|S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)&=&
S\bra a\ket^{\frac{1}{2}}\Big[s-\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta^\star_0)\Big].\end{aligned}$$ Upon substituting the above model-specific expressions for logistic regression into our general RS order parameter equations (\[eq:ddf2=0\]–\[eq:ddtheta2=0\]), we obtain $$\begin{aligned}
\hspace*{-20mm}
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
&=& \tilde{u}^2,
\label{eq:logistic_MAP_1}
\\
\hspace*{-20mm}
w^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{-2}
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
-1\Big]
-
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket
&=&v^2,
\label{eq:logistic_MAP_2}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra \bra
[s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})\!-\!\beta_0\!-\!wy_0\!-\!vz]^2\ket_{s}\Big\ket\!\Big\ket
&=& - \zeta
\tilde{f} \tilde{u}^4,
\label{eq:logistic_MAP_3}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra \bra
\Big[1+\tilde{u}^2[1\!-\!\tanh^2(s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))]\Big]^{-1}
\ket_{s}\ket\!\Big\ket
&=& 1- \zeta \tilde{g}\tilde{u}^2,
\label{eq:logistic_MAP_4}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra\Big\bra
\Big[
s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})\!-\!\beta_0
\Big]
\Big[s\!-\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta^\star_0)\Big]
\Big\ket_{\!s}\big\ket\!\Big\ket
&=& \frac{\zeta w \tilde{u}^2
\bra a\ket^{\frac{1}{2}}}{S \Big\bra \frac{a^2}{
2\eta+\tilde{g}a}\Big\ket},
\label{eq:logistic_MAP_5}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra \Big\bra
s-\tanh(s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))
\Big\ket_{\!s}\Big\ket\!\Big\ket&=& 0.
\label{eq:logistic_MAP_6}\end{aligned}$$
[*Equations for ML logistic regression.*]{} For $\eta=0$ we revert from MAP to ML regression. Here we find the usual model-independent simplifications $\tilde{g}=1/\tilde{u}^2$ and $\tilde{f}=-v^2/\tilde{u}^4$, the covariate correlations (if present) drop out of the theory, and the remaining equations simplify to $$\begin{aligned}
\hspace*{-15mm}
\Big\bra\!\Big\bra \bra
[s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})\!-\!\beta_0\!-\!wy_0\!-\!vz]^2\ket_{s}\Big\ket\!\Big\ket
&=& \zeta
v^2,
\label{eq:ML_logistic_1}
\\[1mm]
\hspace*{-15mm}
\Big\bra\!\Big\bra \bra
\Big[1+\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))]\Big]^{-1}
\ket_{s}\ket\!\Big\ket
&=& 1- \zeta,
\label{eq:ML_logistic_2}
\\[1mm]
\hspace*{-15mm}
\Big\bra\!\Big\bra\Big\bra
\Big[
s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})\!-\!\beta_0
\Big]
\Big[s\!-\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta^\star_0)\Big]
\Big\ket_{\!s}\big\ket\!\Big\ket
&=& \frac{\zeta w}{
S \bra a\ket^{\frac{1}{2}}},
\label{eq:ML_logistic_3}
\\[1mm]
\hspace*{-15mm}
\Big\bra\!\Big\bra \Big\bra
s-\tanh(s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))
\Big\ket_{\!s}\Big\ket\!\Big\ket&=& 0.
\label{eq:ML_logistic_4}\end{aligned}$$ For numerical evaluation it is helpful to write (\[eq:ML\_logistic\_4\]) in an alternative form, exploiting the equation that defines $\tilde{x}(\mu,\sigma)$: $\tanh(\tilde{x}(\mu,\sigma))=1-[\tilde{x}(\mu,\sigma)\!-\!\mu]/\sigma^2$. We see that $$\begin{aligned}
\hspace*{-10mm}
\tanh(s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))&=& s\tanh(\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))
\nonumber
\\
\hspace*{-10mm}
&&\hspace*{-25mm} =~ s-\frac{1}{\tilde{u}^2}\Big[s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})\!-\!(\beta_0\!+\!wy_0\!+\!vz)\Big].
\label{eq:rewrite_beta0_logistic}\end{aligned}$$ This enables us to write (\[eq:ML\_logistic\_4\]) as $$\begin{aligned}
\beta_0&=&
\Big\bra\!\Big\bra \Big\bra
s\tilde{x}(s(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})
\Big\ket_{\!s}\Big\ket\!\Big\ket.\end{aligned}$$ Upon finally writing in explicit form all the averages, and after some simple rewriting, we obtain four equations that can be solved numerically via fixed-point iteration: $$\begin{aligned}
\hspace*{-20mm}
\zeta v^2\!
&=&\! \int\!{\rm D}y_0{\rm D}z \Bigg\{
\frac{1}{2}\Big[1\!+\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\Big[\tilde{x}(\beta_0\!+\!wy_0\!+\!vz,\tilde{u})\!-\!(\beta_0\!+\!wy_0\!+\!vz)\Big]^2
\nonumber
\\
\hspace*{-20mm}
&&
\hspace*{0mm} +
\frac{1}{2}\Big[1\!-\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\Big[\tilde{x}(-(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})\!+\!(\beta_0\!+\!wy_0\!+\!vz)\Big]^2
\Bigg\},
\nonumber
\\[-0mm]
\hspace*{-20mm}
&&
\label{eq:ML_logistic_final_1}
\\[1mm]
\hspace*{-20mm}
\zeta &=&\!
\int\!{\rm D}y_0{\rm D}z
\Bigg\{
\frac{1}{2}\Big[1\!+\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\frac{\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(\beta_0\!+\!wy_0\!+\!vz,\tilde{u}))]}
{1\!+\!\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(\beta_0\!+\!wy_0\!+\!vz,\tilde{u}))]}
\nonumber
\\
\hspace*{-20mm}
&& \hspace*{0mm}
+\frac{1}{2}\Big[1\!-\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\frac{\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(-(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))]}
{1\!+\!\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(-(\beta_0\!+\!wy_0\!+\!vz),\tilde{u}))]}
\Bigg\},
\nonumber
\\[-0mm]
\hspace*{-20mm}
&&
\label{eq:ML_logistic_final_2}
\\[1mm]
\hspace*{-20mm}
w
&=&
\frac{S \bra a\ket^{\frac{1}{2}}}{2\zeta}\int\!{\rm D}y_0{\rm D}z
\Big[1\!-\!\tanh^2(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\nonumber
\\
\hspace*{-20mm}&&\hspace*{20mm}\times
\Big[
\tilde{x}(\beta_0\!+\!wy_0\!+\!vz,\tilde{u})+
\tilde{x}(-(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})
\Big],
\label{eq:ML_logistic_final_3}
\\[1mm]
\hspace*{-20mm}
\beta_0&=&\int\!{\rm D}y_0{\rm D}z
\Bigg\{
\frac{1}{2}\Big[1\!+\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\tilde{x}(\beta_0\!+\!wy_0\!+\!vz,\tilde{u})
\nonumber
\\[-1mm]
\hspace*{-20mm}
&& \hspace*{20mm}
-
\frac{1}{2}\Big[1\!-\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0\!+\!\beta_0^\star)\Big]
\tilde{x}(-(\beta_0\!+\!wy_0\!+\!vz),\tilde{u})
\Bigg\}.
\label{eq:ML_logistic_final_4}\end{aligned}$$
=0.32mm
(400,175) (0,0)[![Regression simulations (markers) versus theoretical predictions (lines) for logistic ML regression with $Np=400,000$, $\bA=\one$, and $\beta_0^\star=0$. Left: order parameter $w$ versus $\zeta$, for different values of $S$. Right: order parameter $v$ versus $\zeta$, for different values of $S$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. []{data-label="fig:ML_logistic_vw"}](LR_varyS_w.eps "fig:"){width="240\unitlength"}]{} (220,0)[![Regression simulations (markers) versus theoretical predictions (lines) for logistic ML regression with $Np=400,000$, $\bA=\one$, and $\beta_0^\star=0$. Left: order parameter $w$ versus $\zeta$, for different values of $S$. Right: order parameter $v$ versus $\zeta$, for different values of $S$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. []{data-label="fig:ML_logistic_vw"}](LR_varyS_v.eps "fig:"){width="240\unitlength"}]{}
=0.337mm
(300,318)
(25,300)[$\eta\!=\!\frac{1}{10}$, $\epsilon\!=\!\frac{1}{2}$, $S\!=\!1$]{} (125,300)[$\eta\!=\!\frac{1}{20}$, $\epsilon\!=\!0$, $S\!=\!1$]{} (225,300)[$\eta\!=\!\frac{1}{20}$, $\epsilon\!=\!\frac{3}{4}$, $S\!=\!1$]{} (325,300)[$\eta\!=\!\frac{1}{20}$, $\epsilon\!=\!\frac{3}{4}$, $S\!=\!2$]{}
(3,233)[$\hat{\beta}_\mu$]{} (60,175)[$\beta^\star_\mu$]{}(160,175)[$\beta^\star_\mu$]{} (260,175)[$\beta^\star_\mu$]{}(360,175)[$\beta^\star_\mu$]{} (28,265)[$\zeta\!=\!0.5$]{}(128,265)[$\zeta\!=\!0.5$]{} (228,265)[$\zeta\!=\!0.5$]{}(328,265)[$\zeta\!=\!0.5$]{} (0,185)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_eta010_eps050_S1_betas_zeta050.eps "fig:"){width="135\unitlength"}]{} (100,185)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_eta005_eps000_S1_betas_zeta050.eps "fig:"){width="135\unitlength"}]{} (200,185)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_eta005_eps075_S1_betas_zeta050.eps "fig:"){width="135\unitlength"}]{} (300,185)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_eta005_eps075_S2_betas_zeta050.eps "fig:"){width="135\unitlength"}]{}
(2,120)[$w$]{} (0,70)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_w_eta010_eps050_S1.eps "fig:"){width="135\unitlength"}]{} (100,70)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_w_eta005_eps000_S1.eps "fig:"){width="135\unitlength"}]{} (200,70)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_w_eta005_eps075_S1.eps "fig:"){width="135\unitlength"}]{} (300,70)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_w_eta005_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (2,20)[$v$]{} (0,-30)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_v_eta010_eps050_S1.eps "fig:"){width="135\unitlength"}]{} (63,-40)[$\zeta$]{} (100,-30)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_v_eta005_eps000_S1.eps "fig:"){width="135\unitlength"}]{} (163,-40)[$\zeta$]{} (200,-30)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_v_eta005_eps075_S1.eps "fig:"){width="135\unitlength"}]{} (263,-40)[$\zeta$]{} (300,-30)[![Results of logistic MAP regression simulations with $Np=400,000$ and $\beta_0^\star=0$, for different combinations of $\eta$ (regularizer strength), $\epsilon$ (covariate correlations), and $S$ (true association strengths). Top row: inferred versus true association parameters for the ratio $\zeta=p/N=0.5$. Middle and lower row: order parameters $w$ and $v$ plotted versus $\zeta$. Each simulation data point represents average and standard deviation computed over 400 synthetic data sets and regressions. Solid curves give the theoretical predictions obtained by solving the RS equations. []{data-label="fig:MAP_logistic_vw"}](logistic_MAP_v_eta005_eps075_S2.eps "fig:"){width="135\unitlength"}]{} (363,-40)[$\zeta$]{}
=0.337mm
(300,220)
(25,208)[$\eta\!=\!\frac{1}{10}$, $\epsilon\!=\!\frac{1}{2}$, $S\!=\!1$]{} (125,208)[$\eta\!=\!\frac{1}{20}$, $\epsilon\!=\!0$, $S\!=\!1$]{} (225,208)[$\eta\!=\!\frac{1}{20}$, $\epsilon\!=\!\frac{3}{4}$, $S\!=\!1$]{} (325,208)[$\eta\!=\!\frac{1}{20}$, $\epsilon\!=\!\frac{3}{4}$, $S\!=\!2$]{}
(-12,150)[*slope*]{} (0,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta010_eps050_S1_slopes.eps "fig:"){width="135\unitlength"}]{} (100,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta005_eps000_S1_slopes.eps "fig:"){width="135\unitlength"}]{} (200,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta005_eps075_S1_slopes.eps "fig:"){width="135\unitlength"}]{} (300,100)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta005_eps075_S2_slopes.eps "fig:"){width="135\unitlength"}]{} (-20,50)[*$\sqrt{\it MSE}$*]{} (0,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta010_eps050_S1_mse.eps "fig:"){width="135\unitlength"}]{} (63,-10)[$\zeta$]{} (100,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta005_eps000_S1_mse.eps "fig:"){width="135\unitlength"}]{} (163,-10)[$\zeta$]{} (200,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta005_eps075_S1_mse.eps "fig:"){width="135\unitlength"}]{} (263,-10)[$\zeta$]{} (300,0)[![Tests of the correction protocols for MAP estimators, applied to the data of the previous figure. Red circles: the MAP estimator $\hat{\beta}_\mu$. Blue squares: the corrected estimator (\[eq:Correction\_debias\]), aimed at removing inference bias . Green crosses: the corrected estimator (\[eq:Correction\_mse\]), aimed at minimizing the MSE. The top row show as a function of $\zeta$ the slopes of the relation between the three estimators and the true values $\beta_\mu^\star$ of association parameters (the slopes of the data clouds in the top row of the previous figure, computes via least squares analysis); this slope would be unity for unbiased estimators. The bottom row shows the values of $\sqrt{\rm MSE}$, where ${\rm MSE}=p^{-1}\sum_{\mu\leq p}(\hat{\beta}_\mu\!-\!\beta_\mu)^2$. []{data-label="fig:MAP_logistic_correction"}](logistic_MAP_eta005_eps075_S2_mse.eps "fig:"){width="135\unitlength"}]{} (363,-10)[$\zeta$]{}
=0.42mm
(400,130)
(70,110)[$\eta\!=\!0$]{} (105,85)[$\eta\!=\!0.025$]{} (107,33)[$\eta\!=\!0.05$]{}
(9,65)[$\hat{\beta}_0$]{} (0,0)[![Histograms of rescaled values of the MAP estimators, each defined as $\hat{\beta}^\prime_\mu=
\big[\hat{\beta}_\mu+ \tilde{d}_0
[ (\tilde{g}\one+2\eta\bA^{-1})^{-1}
\bbeta^\star]_\mu\big]/\sqrt{|\tilde{f}|[
(\tilde{g} \bA+2\eta\one)^{-1}\bA(\tilde{g} \bA+2\eta\one)^{-1}]_{\mu\mu}}$, computed for $10^6$ independent regularized logistic regression experiments with correlated covariates ($\epsilon=0.75$, $\eta=0.01$, $S=1$, $p=500$ and $N=1000$). The resulting 500 histograms are plotted together in the present figure, one for each value of $\mu$. According to the theoretical prediction (\[eq:beta\_relation\_derived\]), these histograms should all asymptotically become zero average and unit average distributions. This is indeed seen to be the case.[]{data-label="fig:logistic_gaussian"}](Gaussian.eps "fig:"){width="180\unitlength"}]{} (91,-4)[$\hat{\beta}^\prime$]{} (-12,70)[$P(\hat{\beta}^\prime)$]{}
[*ML logistic regression for data with zero offset.*]{} Further simplifications arise when we have $\beta_0^\star=0$. By symmetry of the Gaussian averages we now immediately obtain $\beta_0=0$ in (\[eq:ML\_logistic\_final\_4\]), which leaves us with just three coupled equations to be solved numerically. Upon using wherever possible the symmetry of the Gaussian averages these final equations take the relatively simple form: $$\begin{aligned}
\hspace*{-15mm}
\zeta v^2
&=&\int\!{\rm D}y_0{\rm D}z ~
\Big[1\!+\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0)\Big]
\Big[\tilde{x}(wy_0\!+\!vz,\tilde{u})\!-\!(wy_0\!+\!vz)\Big]^2,
\label{eq:ML_logistic_final_betazero_1}
\\[1mm]
\hspace*{-15mm}
\zeta &=&
\int\!{\rm D}y_0{\rm D}z~
\Big[1\!+\!\tanh(S\bra a\ket^{\frac{1}{2}}y_0)\Big]
\frac{\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(wy_0\!+\!vz,\tilde{u}))]}
{1\!+\!\tilde{u}^2[1\!-\!\tanh^2(\tilde{x}(wy_0\!+\!vz,\tilde{u}))]},
\label{eq:ML_logistic_final_betazero_2}
\\[1mm]
\hspace*{-15mm}
\zeta w
&=& S \bra a\ket^{\frac{1}{2}}
\int\!{\rm D}y_0{\rm D}z~
\Big[1\!-\!\tanh^2(S\bra a\ket^{\frac{1}{2}}y_0)\Big]
\tilde{x}(wy_0\!+\!vz,\tilde{u}).
\label{eq:ML_logistic_final_betazero_3}\end{aligned}$$ In Figure \[fig:ML\_logistic\_vw\] we plot the resulting values of the order parameters $v$ and $w$, whose physical meaning is given in (\[eq:meaning\_v\],\[eq:meaning\_w\]), as functions of $\zeta=p/N$, together with the corresponding results of regression simulations on synthetic data with Gaussian covariates. The agreement between theory and simulations is very good.
[*MAP logistic regression with correlated covariates.*]{} The result of solving numerically the MAP equations (\[eq:logistic\_MAP\_1\]-\[eq:logistic\_MAP\_6\]) in the presence of covariate correlations of the type (\[eq:chosen\_A\]) is shown in Figure \[fig:MAP\_logistic\_vw\], where we plot the resulting values of the order parameters $v$ and $w$ together with regression simulation data (for synthetic Gaussian covariates) as functions of the ratio $\zeta=p/N$. In these experiments we chose $\beta_0=\beta_0^\star=0$; we will address the intercept parameter below. Once more we observe excellent agreement between theory and simulation. In the top row we also plot for each parameter combination the MAP-inferred parameters $\hat{\beta}_\mu$ versus the corresponding true association strengths $\beta_\mu^\star$, for pooled data from 20 regressions and $\zeta=0.5$.
Again we can also for logistic regression test our two protocols (\[eq:Correction\_debias\],\[eq:Correction\_mse\]) for correcting the MAP estimator of the association parameters for the distortions caused by overfitting. See Figure \[fig:MAP\_logistic\_correction\]. The slopes of the data clouds of estimators versus true parameter values, as shown for $\zeta=0.5$ in the top row of Figure \[fig:MAP\_logistic\_vw\], are indeed typically away from unity (implying inference bias), both for the MAP estimator (red circles) and the minimum MSE estimator (\[eq:Correction\_mse\]) (green crosses). For the debiased estimator (\[eq:Correction\_debias\]) (blue squares), in contrast, the slope is indeed unity, indicating that bias has been removed successfully. Similarly, the MSE values of the minimum MSE estimator (\[eq:Correction\_mse\]) (green) are as predicted indeed always below those of the other two estimators.
In order to test prediction (\[eq:beta\_relation\_derived\]) for the distribution of inferred regression parameters we next generated $10^6$ data sets, each with $p=500$ and $N=1000$ (so $\zeta=0.5$), with Gaussian covariates that are pairwise correlated according to (\[eq:chosen\_A\]) and $\epsilon=0.75$. The true association parameters were drawn as i.i.d. Gaussian random variables with amplitude $S=1$. After carrying our regularized logistic regression with $\eta=0.05$, we carried out on each of the resulting MAP estimators $\{\hat{\beta}_\mu\}$ of each dataset the specific linear transformation that according to (\[eq:beta\_relation\_derived\]) should transform these into zero-average and unit variance Gaussian random variables (using the order parameters computed from the theory). Upon creating for each value of $\mu$ a histogram of the rescaled estimators $\hat{\beta}_\mu^\prime$, we obtain 500 histograms which according to theory should all collapse asymptotically to a zero average unit variance Gaussian. The result is shown in Figure \[fig:logistic\_gaussian\]. This figure confirms that, even for the modest values of $p$ and $N$ used, the predicted Gaussian statistics of the estimators with the predicted values of average and width given in (\[eq:beta\_relation\_derived\]) are indeed correct.
[*Intercept parameter for imbalanced class sizes.*]{} Having training data with vastly different outcome class sizes leads to the minority outcome rarely being predicted [@wallace2011class] in logistic regression. As this imbalance increases, especially in the overfitting regime the intercept term $\beta_0$ in parametrized models diverges [@owen2007infinitely], and all new samples are assigned the majority outcome. Medical data often exhibit large imbalances between numbers of diseased and healthy samples, with the clinically important decision relying on identifying the rare cases correctly. Similarly, in financial fraud detection there may be millions of legitimate transactions against a handful of fraudulent ones, and we seek to identify the minority class. Existing methods to mitigate the effect of class imbalance have focused on data pre-processing [@chawla2002smote; @drummond2003c4] or incorporating a cost function into the classification algorithm [@wallace2011class]. While these methods are useful to the practitioner, theoretical explanations are limited [@owen2007infinitely; @sei2014infinitely]. Our present theory enables us to investigate class imbalance effects analytically.
=0.42mm
(400,130)
(70,103)[$\eta\!=\!0$]{} (135,80)[$\eta\!=\!0.025$]{} (137,30)[$\eta\!=\!0.05$]{}
(15,65)[$\hat{\beta}_0$]{} (0,0)[![Predicted values of the offset parameter $\beta_0$ are drawn as solid curves, for ${\bA}= \one$ (uncorrelated covariates), $S=1$, and three regularization strengths ($\eta=0$: blue circles; $\eta=0.025$: red squares; $\eta=0.05$: green crosses). Full circles give the average values of $\hat{\beta}_0$ found in MAP regression, for 400 simulations with Gaussian covariates and $NP=400,000$. Standard deviations are not shown in order to reduce visual clutter, but range between 0.003 for small $\zeta$ and 0.1 for large $\zeta$). The true offset used in generating the data was $\beta_0^\star=0.25$, representing an average class imbalance of $42:58$ according to (\[eq:imbalance\]). []{data-label="fig:intercept"}](beta0_MAP_400000_noerrorbars_new.eps "fig:"){width="221\unitlength" height="120\unitlength"}]{} (105,-9)[$\zeta$]{}
The outcome class imbalance in logistic regression data is measured by $m= N^{-1} \sum_{i=1}^N s_i \in [-1,1]$. Averaging over the data in (\[eq:logistic\_regression\_model\]) gives the expectation value $$\begin{aligned}
\label{eq:imbalance}
\bra m \ket &= &\int \!\rmd {\bz}~ p({\bz}) \tanh\big(\beta_0^\star \!+\!{\bbeta}^\star\!\!\cdot {\bz}/\sqrt{p} \big).\end{aligned}$$ Figure \[fig:intercept\] shows our theoretical predictions for the inferred order parameter $\beta_0$ in ML and MAP logistic regression, together with the result of regression simulations. For any given level of outcome imbalance, controlled by $\beta_0^\star$, the bias in the inferred intercept increases with ${\zeta}$. Regularization mitigates this effect, leading to the possibility of correcting inferred class imbalances, in spite of the regularization being applied only to the coefficients $\{\beta_{\mu}\}_{\mu=1}^p$, not to $\beta_0$ itself. This situation is reminiscent of Cox’s survival analysis model [@Cox], where the inferred hazard rate (given by Breslow’s estimator [@breslow1972contribution]) can be expressed in terms of the inferred regression coefficients, and thereby inherits their inference bias. In the $\eta>0$ case, we find the intercept inflation differs from that of the association parameters, due to their rescaling with $\sqrt{p}$. Again the agreement between theory and experiment is very satisfactory.
Regularized Cox regression
--------------------------
[*Equations for MAP Cox regression.*]{} In Cox regression without censoring[^5] we have $s=t\in[0,\infty)$, $\theta=\{\lambda(t)\}$ (the so-called base hazard rate, a nonnegative function on the time interval $[0,\infty)$ such that $\Lambda(t)=\int_0^t\!\rmd t^\prime~\lambda(t^\prime)$ diverges for $t\to\infty$), and $$\begin{aligned}
p(t|\xi,\lambda)&=& \lambda(t)\rme^{\xi-\exp(\xi)\Lambda(t)},\\
\log p(t|\xi,\lambda)&=& \log\lambda(t)+\xi-\rme^\xi\Lambda(t).\end{aligned}$$ Hence $$\begin{aligned}
\frac{\partial}{\partial\xi}\log p(t|\xi,\lambda)= 1-\rme^\xi\Lambda(t),
~~~~~~
\frac{\partial^2}{\partial\xi^2}\log p(t|\xi,\lambda)=-\rme^\xi\Lambda(t).\end{aligned}$$ The analysis of overfitting in MAP Cox regression with arbitrary covariate covariance matrices $\bA$ was first carried out in [@SheikhCoolen2019]. We will now show that from the general equations (\[eq:ddf2=0\],\[eq:ddg2=0\],\[eq:ddu2=0\],\[eq:ddv2=0\],\[eq:ddw2=0\],\[eq:ddtheta2=0\]) one indeed recovers the results of [@SheikhCoolen2019], and more (e.g. the explicit link between true and inferred association parameters).
We will first compute the various model-dependent building blocks of the RS equations. The relevant (functional and partial) derivatives of the logarithm of the outcome probability density are $$\begin{aligned}
\frac{\delta}{\delta\lambda(t)}\log p(s|\xi,\lambda)&=&\frac{\delta(s\!-\!t)}{\lambda(t)}-\rme^\xi \theta(s\!-\!t),
\\
\frac{\partial}{\partial y_0}\log p(s|S\bra a\ket^{\frac{1}{2}}y_0,\lambda^\star)&=& S\bra a\ket^{\frac{1}{2}}\Big[1-\rme^{S\bra a\ket^{\frac{1}{2}}y_0}\Lambda^\star(s)\Big].\end{aligned}$$ The function $\xi(\mu,\sigma,s,\beta_0)$ is here the solution of $$\begin{aligned}
1-\rme^\xi\Lambda(s)=(\xi-\mu)/\sigma^2.\end{aligned}$$ Upon switching from $\xi$ to the variable $x=\mu\!-\!\xi\!+\!\sigma^2$, we can solve $x$ in explicit form: $$\begin{aligned}
x=W(\sigma^2\rme^{\mu+\sigma^2}\Lambda(s)).\end{aligned}$$ Here $W(x)$ denotes the Lambert $W$-function. i.e. the inverse of $f(x)=x\exp(x)$, with derivative $W^\prime(x)=W(x)/x[1\!+\!W(x)]$. It then follows that $$\begin{aligned}
\xi(\mu,\sigma,s,\lambda)&=&\mu+\sigma^2-W(\sigma^2\rme^{\mu+\sigma^2}\Lambda(s)),
\\
\frac{\partial}{\partial_\mu}\xi(\mu,\sigma,s,\lambda)&=& \frac{1}{1\!+\!W(\sigma^2\rme^{\mu+\sigma^2}\Lambda(s))}.\end{aligned}$$
[ *Order parameter equations.*]{} We insert the above formulae into our RS order parameter equations (\[eq:ddf2=0\]–\[eq:ddtheta2=0\]), and use identities such as $\bra \exp(S\bra a\ket^{\frac{1}{2}}y_0)\Lambda^\star(s)\ket_{s}=1$, to simplify our equations to the following set: $$\begin{aligned}
\hspace*{-15mm}
\tilde{u}^2
&=&
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket,
\label{eq:Cox1}
\\
\hspace*{-15mm}
v^2 &=&
w^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-2}
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
-1\Big]
-
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket,
\label{eq:Cox2}
\\
\hspace*{-15mm}
- \zeta
\tilde{f} \tilde{u}^4
&=&
\Big\bra\!\Big\bra \Big\bra
\Big[
\tilde{u}^2-W(\sigma^2\rme^{wy_0+vz+\tilde{u}^2}\Lambda(s))
\Big]^2 \Big\ket_{\!s}\Big\ket\!\Big\ket,
\label{eq:Cox3}
\\[2mm]
\hspace*{-15mm}
\zeta \tilde{g}\tilde{u}^2&=&
\Big\bra\!\Big\bra \Big\bra
\frac{W(\sigma^2\rme^{wy_0+vz+\tilde{u}^2}\Lambda(s))}{1\!+\!W(\sigma^2\rme^{wy_0+vz+\tilde{u}^2}\Lambda(s))}
\Big\ket_{s}\ket\!\Big\ket,
\label{eq:Cox4}
\\[1mm]
\hspace*{-15mm}
\frac{ \zeta w \tilde{u}^2 \bra a\ket^{\frac{1}{2}}}{S
\big\bra \frac{a^2}{
2\eta+\tilde{g}a}\big\ket}
&=&
- \Big\bra\!\Big\bra\Big\bra \Big[
W(\tilde{u}^2\rme^{wy_0+vz+\tilde{u}^2}\Lambda(s))
\Big]
\Big[1-\rme^{S\bra a\ket^{\frac{1}{2}}y_0}\Lambda^\star(s)\Big]
\Big\ket_{\!s}\Big\ket\!\Big\ket,
\label{eq:Cox5}
\\[1mm]
\hspace*{-15mm}
\frac{\big\bra\!\big\bra \bra
\delta(s\!-\!t)\ket_s\big\ket\!\big\ket}{\lambda(t)}
&=&\Big\bra\!\Big\bra \Big\bra \rme^{wy_0+vz+\tilde{u}^2-W(\tilde{u}^2\rme^{wy_0+vz+\tilde{u}^2}\Lambda(s))}\theta(s\!-\!t)
\Big\ket_{\!s}\Big\ket\!\Big\ket.
\label{eq:Cox6}\end{aligned}$$ The first three are immediately recognised from [@SheikhCoolen2019]. With the identity $\exp(-W(x))=W(x)/x$, we find also that (\[eq:Cox4\]) reduces to the corresponding equation in [@SheikhCoolen2019]. This leaves only the identification of equation (\[eq:Cox5\]). Let us start from the corresponding equation in [@SheikhCoolen2019], which reads: $$\begin{aligned}
\hspace*{-20mm}
\zeta w\Big[\tilde{g}-\bra a\ket\Big\bra \frac{a^2}{2\eta\!+\!\tilde{g}a}\Big\ket^{-1}\Big]&=&\frac{1}{\tilde{u}^2}\Big\bra\!\Big\bra y_0\Big\bra W(\tilde{u}^2\rme^{\tilde{u}^2+wy_0+vz}\Lambda(s))\Big\ket_{\!s}\Big\ket\!\Big\ket
\nonumber
\\
\hspace*{-20mm}
&=&\frac{1}{\tilde{u}^2}\Big\bra\!\Big\bra \frac{\partial} {\partial y_0}\Big\bra W(\tilde{u}^2\rme^{\tilde{u}^2+wy_0+vz}\Lambda(s))\Big\ket_{\!s}\Big\ket\!\Big\ket
\nonumber
\\[1mm]
\hspace*{-20mm}
&& \hspace*{-30mm} =\frac{1}{\tilde{u}^2}\Big\bra\!\Big\bra \Big\bra
\frac{\partial W(\tilde{u}^2\rme^{\tilde{u}^2+wy_0+vz}\Lambda(s))}{\partial y_0}
\nonumber
\\[-1mm]
\hspace*{-20mm}
&&
+W(\tilde{u}^2\rme^{\tilde{u}^2+wy_0+vz}\Lambda(s))
\frac{\partial \log p(s|S\bra a\ket^{\frac{1}{2}}y_0,\lambda_0^\star)}{\partial y_0}
\Big\ket_{\!s}\Big\ket\!\Big\ket
\nonumber
\nonumber
\\
\hspace*{-20mm}
&& \hspace*{-30mm} = w\zeta\tilde{g}
+
\frac{S\bra a\ket^{\frac{1}{2}}}{\tilde{u}^2}\Big\bra\!\Big\bra \Big\bra W(\tilde{u}^2\rme^{\tilde{u}^2+wy_0+vz}\Lambda(s))
\Big[1\!-\!\rme^{S\bra a\ket^{\frac{1}{2}}y_0}\Lambda^\star(s)\Big]
\Big\ket_{\!s}\Big\ket\!\Big\ket.~\end{aligned}$$ From this we directly recover (\[eq:Cox5\]), as required. This confirms that from our general theory for GLMs we can indeed recover also for the example choice of regularized Cox regression the complete results of [@SheikhCoolen2019]. Moreover, we now have the additional identities (\[eq:average\_beta\],\[eq:covariance\_beta\]), which were not available in that earlier study, and reveal the nontrivial impact of regularization in the more realistic scenario of correlated covariates (which cannot be extracted from overlap order parameters alone).
Discussion
==========
In this paper we have extended to arbitrary generalized linear regression models (GLM) the replica analysis of overfitting in MAP and ML inference that was developed initially in [@coolen2017replica; @SheikhCoolen2019] for Cox regression [@Cox] with time-to-event data. Parameter inference methods such as MAP and ML were derived and work well for the regime $p\ll N$, where $p$ is the dimensionality and $N$ is the number of samples. But they can produce large inference errors when $p=\order(N)$. This seriously hampers statistical inference in high dimensions, and thereby limits progress in many data-driven scientific disciplines.
In all GLMs, the ML/MAP overfitting-induced parameter inference errors consist of a combination of a reproducible bias and excess noise, both of which disappear when $p/N\to 0$ but become more prominent as the ratio $p/N$ increases. In the regime $p,N\to \infty$ with fixed $\zeta=p/N$, the replica method enables us to predict analytically both this inference bias and the distribution of the excess noise, expressed in terms of the true parameters of the model that generated the data, and the distribution of covariates from which the samples were drawn. In contrast to some recent alternative approaches, such as [@sur2019modern; @salehi2019impact; @barbier2019optimal], by using the replica method we are not restricted to uncorrelated covariates or to models with output noise only, and we can calculate in explicit form the relation between MAP/ML estimators and the true (but unknown) model parameters responsible for the data. Covariate correlations are in fact found to play an important role in this relation. Our results pave the way for correcting ML and MAP inferences in GLMs systematically for overfitting bias, and thereby extend the applicability of such models into the hitherto forbidden regime $p\!=\!\order(N)$.
We found that in our analysis, the choice of outcome data types and regression models can be left until after the derivation of replica symmetric order parameter equations (there is no evidence for replica symmetry breaking, which is reasonable since we have assumed absence of model mismatch). Our derivation relies only on the generalized linear form of GLMs and on choosing $L2$ priors. Hence the replica calculation need not be repeated for every new GLM model instance; as always with the replica method, it served as a relatively painless and elegant but powerful vehicle for arriving at a closed set of order parameter equations, together with formulae expressing the relation between the ML/MAP parameter estimators and the true values of these parameters. The latter equations can serve as a natural and convenient starting point for practical applications, even for those with no interest in their derivation. We illustrate our results in this paper by applying the general theory to linear, logistic, and Cox regression, and find excellent agreement with simulations and earlier results. We have limited our experiments to $\zeta\leq 1$. In ML regression this marks the point by which a phase transition will have occurred (even earlier in logistic regression), whereas in MAP regression with $\eta>0$ once can in fact continue regression and find agreement between theory and simulations into the $\zeta>1$ regime (data not shown here).
This work can be extended in both practical and theoretical ways, several of which are presently being explored. Our theory was built upon the idealized scenario of knowledge of the underlying data-generating model. To put it into practice, the variance $S$ of the true regression parameters and the population covariance matrix $\mathbf{A}$ need to be estimated. The former is available through the inferred MAP estimators and the existing order parameters. The latter can be estimated from the empirical covariate statistics, building on methods such as [@A1; @A2; @burda2005spectral; @el2008spectrum]. For time-to-event models, the next obvious step would be to include censored data. More general extensions of the present theory include working with alternative non-Gaussian priors, inspecting more pathological models or data where some of our mathematical assumptions no longer hold, or generalizing the theory to regression models with multiple linear predictors, such as multinomial regression [@obuchi2018accelerating], multiple risks and latent classes in survival analysis [@Cox_book; @multiple_risks], and multilayer neural networks [@Coolenbook; @MacKay; @li2018exploring].
[**Acknowledgements**]{}\
The authors gratefully acknowledge valuable and stimulating discussions with Sir David Cox and Dr Heather Battey. MS is supported by the Biotechnology and Biological Sciences Research Council (award 1668568) and GSK Ltd. AM is supported by Cancer Research UK (award C45074/A26553) and the UK’s Medical Research Council (award MR/R014043/1). FAL is supported through a scholarship from Conacyt (Mexico).
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Derivation of the generic saddle point form {#app:SheikhCoolen}
===========================================
Preparation
-----------
We start with expression (\[eq:starting\_point\_of\_replicas\]), with the $L_2$ prior $p({\bbeta})\propto \exp(-p\eta{\bbeta}^2)$: $$\begin{aligned}
\hspace*{-10mm}
E_\gamma(\bbeta^\star,\theta^\star)
&=& - \lim_{n\to 0}\frac{\partial}{\partial\gamma}\frac{1}{Nn}\log \int\!\rmd\theta^1\!\ldots\rmd\theta^n
\int\!\rmd\bbeta^1\!\ldots\rmd\bbeta^n
\prod_{\alpha=1}^n
\Big[\frac{p(\bbeta^\alpha)}{p(\bbeta^\star)}\Big]^\gamma
\nonumber
\\
\hspace*{-10mm}
&&\hspace*{0mm} \times
\Big\{
\int\!\rmd\bz \rmd s~p(\bz)p(s|\bz,\bbeta^\star,\theta^\star)
\prod_{\alpha=1}^n
\Big[\frac{p(s|\bz,\bbeta^\alpha,\theta^\alpha)}{p(s|\bz,\bbeta^\star,\theta^\star)}\Big]^\gamma\Big\}^N.\end{aligned}$$ The covariate distribution $p(\bz)$ is assumed to have have zero mean and covariance matrix ${\bA}$, with entries $A_{\mu \nu} = \int\!\rmd{\bz}~p({\bz}) z_{\mu} z_{\nu}$. We consider the regime where $N,p\rightarrow\infty$ with fixed ratio ${\zeta}= p/N$. Following [@coolen2017replica; @SheikhCoolen2019] we next introduce $$\begin{aligned}
p({\by}| {\bbeta}^0\!, \ldots, {\bbeta}^n) = \int\!\rmd {\bz}~ p({\bz}) \prod_{\alpha=0}^n \delta \Big[ y^{\alpha} - \frac{{\bbeta}^{\alpha}\! \cdot {\bz}}{\sqrt{p}} \Big],
\label{eq:switch_to_y}\end{aligned}$$ where ${\by}= \{ y^0 , y^1, \ldots, y^n \}\! \in\! \R^{n+1}$ (which in survival analysis would be interpreted as risk scores) and $\bbeta^0\equiv\bbeta^\star$. Now $$\begin{aligned}
\hspace*{-15mm}
\label{eq:energy2}
E_{\gamma}({\bbeta}^\star, \theta^\star) &=& - \frac{\partial }{\partial \gamma} \lim\limits_{n \to 0} \frac{1}{Nn}
\log \int \!\rmd\theta^1\! \ldots \rmd\theta^n \int \!\rmd{\bbeta}^1\! \ldots \rmd{\bbeta}^n \prod_{\alpha=1}^n \Big[\frac{ p({\bbeta}^{\alpha}) }{p({\bbeta}^0)} \bigg]^{\gamma}
\nonumber \\
\hspace*{-15mm}
&&\hspace*{-8mm} \times \Big\{
\int\! \rmd {\by}~ p({\by}| {\bbeta}^0\!, \ldots, {\bbeta}^n) \int \!\rmd s~ p(s |y^0\!, \theta^\star)
\prod_{\alpha=1}^n \Big[ \frac{p(s |y^{\alpha}\!, \theta^{\alpha}) }{p(s |y^0\!, \theta^\star) } \Big]^{\gamma}
\Big\}^N\!.~~ \end{aligned}$$ To proceed we assume that $p({\by}| {\bbeta}^0, \ldots, {\bbeta}^n)$ is Gaussian, via the Central Limit Theorem. Since $\int\!\rmd{\bz}~p({\bz}){\bz}=\bnull$, the distribution $p({\by}| {\bbeta}^0, \ldots, {\bbeta}^n) $ is now given by $$\begin{aligned}
p({\by}| {\bbeta}^0, \ldots, {\bbeta}^n) = \frac{\rm\rme^{-{{\frac{1}{2}}}{\by}\cdot {\bC}^{-1}[\{{\bbeta}\}] {\by}}}{\sqrt{(2 \pi)^{n+1} \det {\bC}[\{{\bbeta}\}]}}.
\label{eq:y_mvn}\end{aligned}$$ It is determined in full by the $(n\!+\!1)\! \times\! (n\!+\!1)$ covariance matrix ${\bC}[\{{\bbeta}\}]$, with entries $$\begin{aligned}
C_{\alpha \rho}[\{{\bbeta}\}] &=& \int\! \rmd{\bz}~ p({\bz}) \Big(\frac{{\bbeta}^{\alpha}\!\cdot{\bz}}{\sqrt p}\Big)\Big( \frac{{\bbeta}^{\rho}\!\cdot{\bz}}{\sqrt p} \Big)
~= \frac{1}{p} {\bbeta}^{\alpha}\!\cdot {\bA}{\bbeta}^{\rho}.
\label{eq:correlated}\end{aligned}$$ For each replica pair $(\alpha,\rho)$ we use the integral representation of the Dirac delta function, and rescale the conjugate integration parameter by $p$, substituting $$\begin{aligned}
\hspace*{-10mm}
1 = \int\! \rmd C_{\alpha \rho}~\delta \big[ C_{\alpha \rho} \!-\! \frac{1}{p} {\bbeta}^{\alpha}\!\cdot {\bA}{\bbeta}^{\rho} \big] =
\int \!\frac{\rmd C_{\alpha \rho} \rmd {\hat{C}}_{\alpha \rho}}{{2 \pi}/p} \rm\rme^{\rmi p {\hat{C}}_{\alpha \rho} (C_{\alpha \rho} - \frac{1}{p} {\bbeta}^{\alpha}\!\cdot {\bA}{\bbeta}^{\rho} )},\end{aligned}$$ in order to simplify expression (\[eq:energy2\]) to $$\begin{aligned}
\hspace*{-20mm}
E_{\gamma}({\bbeta}^\star, \theta^\star) &=& - \frac{\partial }{\partial \gamma} \lim\limits_{n \to 0} \frac{1}{Nn}
\log \int \!\{\rmd \theta^1\! \ldots \rmd \theta^n\} \int\! \rmd {\bC}\, \rmd \hat{{\bC}}~\frac{\rm\rme^{\rmi p \, \sum_{\alpha ,\rho=0}^n {\hat{C}}_{\alpha \rho} \, C_{\alpha \rho} }}{(2 \pi / p )^{(n+1)^2}} \nonumber \\
\hspace*{-20mm}
&& \times \Bigg[
\int \! \frac{ \rmd {\by}~\rm\rme^{-{{\frac{1}{2}}}{\by}\cdot {\bC}^{-1} {\by}}}{\sqrt{(2 \pi)^{n+1} \det {\bC}}} \int \! \rmd s~ p(s | y^0\!, \theta^\star)
\prod_{\alpha=1}^n \Big[ \frac{p(s | y^{\alpha}\!, \theta^{\alpha}) }{p(s | y^0\!, \theta^\star) } \Big]^{\gamma}
\Bigg]^N \nonumber \\
\hspace*{-20mm}
&& \times \int\! \rmd{\bbeta}^1\! \ldots d{\bbeta}^n~ \rm\rme^{-\eta \gamma \sum_{\alpha=1}^n [ ({\bbeta}^{\alpha})^2 - ({\bbeta}^{0})^2 ] -\rmi \sum_{\alpha, \rho=0}^n {\hat{C}}_{\alpha \rho} {\bbeta}^{\alpha} \cdot{\bA}{\bbeta}^{\rho }}.
\label{eq:energy3a}\end{aligned}$$
Conversion into a saddle point problem
--------------------------------------
We next transform $\hat{{\bC}}=-\frac{1}{2}\rmi{\bD}$, define ${\tilde{\bbeta}}\equiv {\bA}^{{{\frac{1}{2}}}} {\bbeta}$ and introduce the $np\times np$ matrix $\bXi $ and the $np$-dimensional vector $\bxi$, with entries $$\begin{aligned}
\Xi_{\alpha\mu;\beta\nu}= 2 \eta \gamma \delta_{\alpha \beta} ({\bA}^{-1})_{\mu \nu} + \delta_{\mu \nu}D_{\alpha \beta},~~~~~~ \xi_{\mu}^{\alpha}= -D_{0 \alpha} {\tilde{\beta}}_{\mu }^{0}
\label{eq:define_xi}\end{aligned}$$ The Gaussian integral in (\[eq:energy3a\]) then becomes $$\begin{aligned}
&&\hspace*{-15mm}
\int \Big( \prod_{\alpha=1}^n \rmd {\tilde{\bbeta}}^{\alpha} \rme^{-\eta \gamma {\tilde{\bbeta}}^{\alpha}\cdot {\bA}^{-1} {\tilde{\bbeta}}^{\alpha}}\Big)
\rme^{ -\frac{1}{2} \sum_{\alpha,\rho=1}^n D_{\alpha \rho} {\tilde{\bbeta}}^{\alpha} \cdot {\tilde{\bbeta}}^{\rho } - \sum_{\rho=1}^n D_{0 \rho} {\tilde{\bbeta}}^{0} \cdot {\tilde{\bbeta}}^{\rho } }
\nonumber
\\
&=&
\frac{(2\pi)^{\frac{np}{2}}}{\sqrt{\det\bXi}} \rme^{{{\frac{1}{2}}}\bxi\cdot \bXi^{-1} \bxi}.
\label{eq:gaussian}\end{aligned}$$ Let $\{a_\mu\}$ and $\{b_\alpha\}$ denote the eigenvalues of ${\bA}$ and ${\bD}$. The two terms ${\bP}$ and ${\bQ}$ of $\bXi$, with components $P_{\alpha\mu,\beta\nu}=2 \eta \gamma \delta_{\alpha \beta} ({\bA}^{-1})_{\mu \nu} $ and $Q_{\alpha\mu,\beta\nu}= \delta_{\mu \nu} D_{\alpha \beta}$, commute. The eigenvectors of $\bXi$ can therefore be written as $\{\hat{{\bu}}^{\mu\alpha}\}$, with components $\hat{u}^{\mu\alpha}_{\nu\rho}=u^\alpha_\rho v^\mu_{\nu}$, and where $\sum_{\rho\leq n} D_{\lambda\rho}u_\rho^\alpha=b_\alpha u_\rho^\lambda$ and $\sum_{\nu\leq p}A_{\lambda\nu} v^{\mu}_{\nu}=a_\mu v^{\mu}_{\lambda}$, and where both are normalised according to $\sum_{\rho\leq n}(u^\alpha_\rho)^2=\sum_{\nu\leq p} (v^\mu_{\nu})^2=1$. The eigenvalues of $\bXi$ are then $\xi_{\mu\alpha} =2 \eta \gamma/a_\mu + b_\alpha$, and $$\begin{aligned}
\hspace*{-15mm}
\det \bXi = \prod_{\mu=1}^p \prod_{\alpha=1}^n \big( \frac{2 \eta
\gamma}{a_\mu} \!+\! b_\alpha \big),~~~~~~
(\bXi^{-1})_{\alpha\mu,\alpha^\prime\mu^\prime}= \sum_{\beta=1}^n \sum_{\nu=1}^p
\frac{u^\beta_\alpha v^\nu_{\mu} u^\beta_{\alpha^\prime} v^\nu_{\mu^\prime}}{2 \eta \gamma/a_\nu + b_\beta}.\end{aligned}$$ Hence the integral (\[eq:gaussian\]) can be written as $$\begin{aligned}
\hspace*{-18mm}
\frac{(2\pi)^{\frac{np}{2}} \rme^{{{\frac{1}{2}}}\bxi\cdot \bXi^{-1} \bxi} }{\sqrt{{\rm det}\bXi}}&=&
\rme^{\frac{1}{2}np\log(2\pi)-\frac{1}{2}np \big\langle\!\log ( 2 \eta
\gamma/a +b )\big\rangle+
\frac{1}{2} np\big\langle (\bxi\cdot\hat{{\bu}})^2
(2 \eta \gamma/a + b)^{-1}\big\rangle},~~
\label{eq:gaussian_done}\end{aligned}$$ where the averages are over the eigenvalues and orthonormal eigenvectors of $\bXi$, i.e. $\langle f(a,b,\hat{{\bu}})\rangle\!=\!(np)^{-1}\sum_{\mu=1}^p\sum_{\alpha=1}^n f(a_\mu,b_\alpha,\hat{{\bu}}^{\mu\alpha})$. Since $p=\zeta N$ with $\zeta\!>\!0$, the integrals over ${\bC}$, $\hat{{\bC}}$ and the base hazard rates in (\[eq:energy3a\]) can for $N\to\infty$ be evaluated by steepest descent, provided the limits $n\!\to\! 0$ and $N\!\to\! \infty$ commute. Expression (\[eq:gaussian\_done\]) then enables us to write the result as $$\begin{aligned}
\lim\limits_{N \to \infty} E_{\gamma}({\bbeta}^\star, \theta^\star) &=& \frac{\partial }{\partial \gamma}\lim\limits_{n \to 0} \frac{1}{n} \mbox{extr}\, \Psi({\bC}, {\bD}, \theta^1 \ldots \theta^n),
\label{eq:energy3d}\end{aligned}$$ with $$\begin{aligned}
\hspace*{-20mm}
\Psi ({\bC}, {\bD}, \theta^1\! \ldots \theta^n) &=& -\frac{1}{2} \zeta \, \bigg[\sum_{\alpha, \rho=0}^n D_{\alpha \rho} C_{\alpha \rho} -
\frac{1}{p}D_{00} ({\tilde{\bbeta}}^0)^2 \bigg]
+{{\frac{1}{2}}}(n\!+\!1\!-\!n\zeta) \log(2 \pi)
\nonumber\\
\hspace*{-20mm}
&&\hspace*{-10mm}
+ {{\frac{1}{2}}}\log \det {\bC}- n \eta {\zeta}\gamma S^2
+ {{\frac{1}{2}}}n{\zeta}\Big\langle\! \log \Big( \frac{2 \eta \gamma}{a}\! +\! b \Big) \Big\rangle
- {{\frac{1}{2}}}n{\zeta}\Big\langle \frac{(\bxi\cdot\hat{{\bu}})^2}{2 \eta \gamma/a\! +\! b} \Big\rangle
\nonumber \\
\hspace*{-20mm}
&& \hspace*{-10mm}
- \log \int \!\rmd {\by}~\rme^{-{{\frac{1}{2}}}{\by}\cdot {\bC}^{-1} {\by}} \! \int\!\rmd s~ p(s |y^0\!, \theta^\star)
\prod_{\alpha=1}^n \Big[ \frac{p(s | y^{\alpha}\!, \theta^{\alpha}) }{p(s | y^0\!, \theta^\star) } \Big]^{\gamma}.
\label{eq:energy3c}\end{aligned}$$ where $S^2=\lim_{p\to\infty}p^{-1}({\bbeta}^0)^2$. Differentiating $\Psi(\ldots)$ with respect to $D_{00}$ immediately gives $C_{00} = p^{-1}{\bbeta}^0\cdot {\bA}{\bbeta}^0 \equiv {\tilde{S}}^2 $.
Replica symmetric saddle points
-------------------------------
Replica symmetric (RS) saddle points are fully invariant under all permutations of the replica labels $\{1,\ldots,n\}$. For the present model the RS ansatz takes the form $$\begin{aligned}
\hspace*{-10mm}
\theta^{\alpha} = \theta,~~~~~~
\begin{array}{l}
C_{0 \alpha} = c_0
\\[1mm]
D_{0\alpha}=d_0
\end{array},~~~~~~
\begin{array}{ll}
C_{\alpha \rho} & = C \delta_{\alpha \rho} + c (1 - \delta_{\alpha \rho} ),
\\[1mm]
D_{\alpha \rho} & = D \delta_{\alpha \rho} + d (1 - \delta_{\alpha \rho} ).
\end{array}\end{aligned}$$ Both ${\bC}$ and ${\bD}$ are positive definite, so $C>c$ and $D>d$. We may now write $$\begin{aligned}
&&
\hspace*{-10mm}
{\bC}= \pmatrix{C_{00} & c_0 &\ldots & \ldots & c_0 \cr
c_0 & C & c & \ldots & c \cr
\vdots & c & C & \ldots & c \cr
\vdots & \vdots & \vdots & \ddots & \vdots \cr
c_0 & c & c & \ldots & C \cr
},~~~~~~
{\bC}^{-1} = \pmatrix{B_{00} & b_0 &\ldots & \ldots & b_0 \cr
b_0 & B & b & \ldots & b \cr
\vdots & b & B & \ldots & b \cr \nonumber
\vdots & \vdots & \vdots & \ddots & \vdots \cr
b_0 & b & b & \ldots & B \cr
}.
\label{eq:Cmatrix}\end{aligned}$$ ${\bC}$ has two nondegenerate eigenvalues $\lambda_{\pm}$ with $\lambda_+ \lambda_- = [C+(n\!-\!1)c ] C_{00} - nc_0^2$, and a further $n\!-\!1$ fold degenerate eigenvalue $\lambda_0=C-c$. Hence $$\begin{aligned}
\label{eq:logdetC}
\log \det {\bC}&=& \log C_{00} + n \log (C\!-\!c) + \frac{n\big(c\! -\! c_0^2 / C_{00} \big)}{C-c} + \mathcal{O}(n^2). \end{aligned}$$ The entries of ${\bC}^{-1}$ are found to be $$\begin{aligned}
&&
\hspace*{-15mm} B_{00} = \frac{C + (n-1)c}{C_{00} [C + (n-1)c] - nc_0^2},~~~~~~
b_0= - \frac{c_0}{C_{00} [C + (n-1)c] - nc_0^2},
\\
&&
\hspace*{-15mm} B = b + \frac{1}{C-c},~~~~~~\hspace*{15mm}
b = \frac{c_0^2 - c C_{00}}{(C_{00} [C + (n-1)c] - nc_0^2)(C-c)}.
\label{eq:Cinv}\end{aligned}$$ Hence $$\begin{aligned}
\label{eq:quadratic}
\hspace*{-15mm}
{\by}\cdot {\bC}^{-1} {\by}&=& B_{00} (y^0)^2 + (B\!-\!b) \sum\limits_{\alpha=1}^n (y^{\alpha})^2 + b \Big( \sum\limits_{\alpha=1}^n y^{\alpha} \Big)^2 + 2 b_0 y^0 \sum\limits_{\alpha=1}^n y^{\alpha}.\end{aligned}$$ The matrix ${\bD}$ has one eigenvalue $D\!+\!(n\!-\!1)d$ with eigenvector ${\bv}= (1,\ldots, 1)$, and the $n\!-\!1$ fold degenerate eigenvalue $D-d$ with eigenspace $ (1,\ldots, 1)^{\perp}$. Hence $$\begin{aligned}
\hspace*{-15mm}
\Big\langle\! \log \Big( \frac{2 \eta \gamma}{a}\! +\! b \Big) \Big\rangle
&=&\Big\langle \log \Big( \frac{2 \eta \gamma}{a}\! +\! D\!-\!d \Big) \Big\rangle
+ \Big\langle\frac{da}{2 \eta \gamma\! +\! (D\!-\!d)a} \Big\rangle +O(n).
\label{eq:RSterm1}
\end{aligned}$$ Similarly, using the RS form of $\xi_\mu^\alpha=-d_0({\bA}^{\frac{1}{2}}{\bbeta}^0)_\mu$, we may write $$\begin{aligned}
\hspace*{-5mm}
\Big\langle \frac{(\bxi\cdot\hat{{\bu}})^2}{2 \eta \gamma/a\! +\! b} \Big\rangle
&=&
d_0^2 ~
\Big\langle
\frac{a^2 ({\bbeta}^0\!\cdot\! {\bv})^2}{2 \eta \gamma\! +\! (D\!-\!d)a}\Big\rangle+O(n).
\label{eq:RSterm2}\end{aligned}$$ Inserting the above RS expressions into (\[eq:energy3c\]), and using $C_{00}=\tilde{S}^2$, then gives us $$\begin{aligned}
\hspace*{-20mm}
\frac{1}{n} \Psi_{\rm RS} (\ldots) &=&
-\frac{1}{2} \zeta (
2d_0c_0+DC-dc)
+{{\frac{1}{2}}}(1\!-\!\zeta) \log(2 \pi) - \eta {\zeta}\gamma S^2
+O(n)
\nonumber\\
\hspace*{-20mm}
&&
+ {{\frac{1}{2}}}\Big[
\log (C\!-\!c) + \frac{c\! -\! c_0^2 /\tilde{S}^2}{C\!-\!c}
\Big]
- {{\frac{1}{2}}}{\zeta}d_0^2 ~
\Big\langle
\frac{a^2 ({\bbeta}^0\!\cdot\! {\bv})^2}{2 \eta \gamma\! +\! (D\!-\!d)a}\Big\rangle
\nonumber
\\
\hspace*{-20mm}
&&
+ {{\frac{1}{2}}}{\zeta}\Big\langle \log \Big( \frac{2 \eta \gamma}{a}\! +\! D\!-\!d \Big) \Big\rangle
+ {{\frac{1}{2}}}{\zeta}\Big\langle\frac{da}{2 \eta \gamma\! +\! (D\!-\!d)a} \Big\rangle
+\frac{1}{2n}\log(\tilde{S}^2 B_{00})
\nonumber \\
\hspace*{-20mm}
&&
-\frac{1}{n}
\log \int\!{\rm D}z {\rm D}y_0\int\!\rmd s~ p(s |y_0/\sqrt{B_{00}}, \theta^\star)
\nonumber
\\
\hspace*{-20mm}
&&~~~\times
\Big[
\int \!\rmd y~
\rme^{-{{\frac{1}{2}}}(B-b) y^2 + y(\rmi \sqrt{b} -b_0 y_0/\sqrt{B_{00}})}
\frac{p^\gamma(s | y, \theta) }{p^\gamma(s | y_0/\sqrt{B_{00}}, \theta^\star) } \Big]^n.\end{aligned}$$ We note that $$\begin{aligned}
&&\hspace*{-5mm}
B_{00}^{-1}=\tilde{S}^2-nc_0^2/(C\!-\!c)+O(n^2),~~~~~~
B-b= 1/(C\!-\!c),
\\
&&\hspace*{-5mm}
b_0=-c_0/\tilde{S}^2(C\!-\!c)+O(n),~~~~~~~~~~~~~~
b= \frac{c_0^2 - c\tilde{S}^2}{\tilde{S}^2 (C\!-\!c)^2}+O(n).\end{aligned}$$ This enable us to write the limit $\Psi_{\rm RS}(\ldots)=\lim_{n\to 0}n^{-1}\Psi_{\rm RS}(\ldots)$ in the simpler form $$\begin{aligned}
\hspace*{-20mm}
\Psi_{\rm RS}(\ldots)&=&
-\frac{1}{2} \zeta\Bigg\{
2d_0c_0+DC-dc
+ \log(2 \pi) +2\eta \gamma S^2
\nonumber\\[-1mm]
\hspace*{-20mm}
&&~~
+
d_0^2 ~
\Big\langle
\frac{a^2 ({\bbeta}^0\!\cdot\! {\bv})^2}{2 \eta \gamma\! +\! (D\!-\!d)a}\Big\rangle
-
\Big\langle \log \Big( \frac{2 \eta \gamma}{a}\! +\! D\!-\!d \Big) \Big\rangle
- \Big\langle\frac{da}{2 \eta \gamma\! +\! (D\!-\!d)a} \Big\rangle
\Bigg\}
\nonumber \\
\hspace*{-20mm}
&&
\hspace*{-14mm}
-\! \int\!{\rm D}z {\rm D}y_0\!\int\!\rmd s~ p(s |\tilde{S}y_0, \theta^\star)
\log\!
\int \!{\rm D} y~
\frac{p^\gamma(s | y\sqrt{C\!-\!c}+\!z(c\!-\!c_0^2/\tilde{S}^2)^{\frac{1}{2}} \!+\! y_0c_0/\tilde{S}, \theta) }{p^\gamma(s | \tilde{S}y_0, \theta^\star) }.
\nonumber
\\[-1mm]
\hspace*{-20mm}&&
\label{eq:Psi_RS_before_uvw}\end{aligned}$$ Here the brackets denote averages over eigenvectors and eigenvalues of the covariate correlation matrix $\bA$: $\bra f(\bv,a)\ket=\lim_{p\to\infty}p^{-1}\sum_{\mu=1}^p f(\bv_\mu,a_\mu)$, with $\bA\bv_\mu=a_\mu\bv_\mu$ for all $\mu=1\ldots p$.
Simplification of the theory
----------------------------
We extremize (\[eq:Psi\_RS\_before\_uvw\]) over $d_0$, which removes an order parameter, and we transform $$\begin{aligned}
\hspace*{-15mm}
&& u= \sqrt{C\!-\!c}, ~~~~ v =\sqrt{c\! -\! (c_0/{\tilde{S}})^2},~~~~ w = c_0/{\tilde{S}}, ~~~~f = d,~~~~ g = D\!-\!d,
\label{eq:transform}\end{aligned}$$ with $u,v,w\in[0,\infty)$ and with the inverse transformations $$c_0 = {\tilde{S}}w, \hsp \hsp c = v^2 \!+\! w^2, \hsp \hsp C = u^2 \!+\!v^2\! +\! w^2.
\label{eq:inverse_transformation}$$ These steps result in $$\begin{aligned}
\lim\limits_{N \to \infty} E_{\gamma}({\bbeta}^\star, \theta^\star) &=& \frac{\partial }{\partial \gamma}\mbox{extr}_{u,v,w,f,g,\lambda} \Psi_{\rm RS}(u,v,w,f,g, \theta),\end{aligned}$$ in which $$\begin{aligned}
\hspace*{-20mm}
\Psi_{\rm RS}(\ldots) &=&
-\frac{1}{2} \zeta
(g\!+\!f)u^2 -\frac{1}{2} \zeta
g(v^2\!+\!w^2)
-\zeta\eta \gamma S^2
\nonumber\\
\hspace*{-20mm}
&&
\hspace*{0mm}
+\frac{1}{2}\zeta\Bigg\{
\tilde{S}^2w^2~ \Big\langle
\frac{a^2 ({\bbeta}^0\!\cdot\! {\bv})^2}{2 \eta \gamma\! +\! ga}\Big\rangle^{\!-1}\!
+
\Big\langle\! \log \Big( \frac{2 \eta \gamma\!+\!ga}{a} \Big) \Big\rangle
+ f~\Big\langle\frac{a}{2 \eta \gamma\! +\! ga} \Big\rangle
\Bigg\}
\nonumber \\
\hspace*{-20mm}
&&
-\! \int\!{\rm D}z {\rm D}y_0\!\int\!\rmd s~ p(s |\tilde{S}y_0, \theta^0)
\log\!
\int \!{\rm D} y~
\frac{p^\gamma(s | uy \!+\! wy_0\!+\!vz, \theta) }{p^\gamma(s | \tilde{S}y_0, \theta^0) }.\end{aligned}$$ We could also extremize over $f$, leading to a simple expression with which to remove $f$ and either $u$ or $g$. The true association parameters ${\bbeta}^0$ are seen to enter the asymptotic theory only in quadratic functions of ${\bbeta}^0$. In \[app:self\_averaging\] we show that, if the true associations $\{\beta_\mu^0\}$ are drawn randomly and independently from a zero-average distribution, and under mild conditions on the spectrum $\varrho(a)$ of the covariate correlation matrix ${\bA}$, both terms will be self-averaging with respect to the realization of ${\bbeta}^0$. Consequently, with $S^2=\lim_{p\to\infty}p^{-1}({\bbeta}^0)^2$ we may then write $$\begin{aligned}
\tilde{S}^2= S^2\bra a\ket,~~~~~~
\Big\bra \frac{a^2({\bbeta}^0\!\cdot{\bv})^2}{2\eta\gamma\!+\!ga}\Big\ket=
\bra \frac{S^2 a^2}{2\eta\gamma\!+\!ga}\ket,
\label{eq:Stilde_to_S}\end{aligned}$$ (where we used the fact that the eigenvectors ${\bv}$ of ${\bA}$ were normalized). Hence $$\begin{aligned}
\hspace*{-20mm}
\lim\limits_{N \to \infty} E_{\gamma}({\bbeta}^0, \theta^\star) &=&
\int\! {\rm D}y_0\!\int\!\rmd s~ p(s |S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
\log p(s | S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
-\zeta\eta S^2
\nonumber\\
\hspace*{-20mm}
&&
\hspace*{-20mm}
+~\eta\zeta\Bigg\{
w^2\bra a\ket
\Big\langle \frac{a^2}{2 \eta \gamma\! +\! ga}\Big\rangle^{\!\!-2}
\Big\langle \frac{a^2}{(2 \eta \gamma\! +\! ga)^2}\Big\rangle
+
\Big\langle\! \frac{1}{2 \eta \gamma\!+\!ga} \Big\rangle- f\Big\langle\frac{a}{(2 \eta \gamma\! +\! ga)^2} \Big\rangle
\Bigg\}
\nonumber \\
\hspace*{-20mm}
&&
\hspace*{-31mm}
-\! \int\!{\rm D}z {\rm D}y_0\!\int\!\!\rmd s~ p(s |S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
\frac{ \int \!{\rm D} y~
p^\gamma(s | uy \!+\! wy_0\!+\!vz, \theta)\log p(s | uy \!+\! wy_0\!+\!vz, \theta) }
{ \int \!{\rm D} y~
p^\gamma(s| uy \!+\! wy_0\!+\!vz, \theta) }.
\nonumber
\\[-1mm]
\hspace*{-20mm}
&&
\label{eq:Evalue_before_scaling}\end{aligned}$$ The order parameters $(u,v,w,f,g,\theta\}$ are computed by extremization of the following function, from which we removed any constant terms: $$\begin{aligned}
\hspace*{-20mm}
\Psi_{\rm RS}(\ldots) &=&
-\frac{1}{2} \zeta
(g\!+\!f)u^2 -\frac{1}{2} \zeta
g(v^2\!+\!w^2)
\nonumber\\
\hspace*{-20mm}
&&
\hspace*{0mm}
+\frac{1}{2}\zeta\Bigg\{
w^2\bra a\ket \Big\langle
\frac{a^2}{2 \eta \gamma\! +\! ga}\Big\rangle^{\!-1}\!
+
\Big\langle\! \log (2 \eta \gamma\!+\!ga) \Big\rangle
+ f~\Big\langle\frac{a}{2 \eta \gamma\! +\! ga} \Big\rangle
\Bigg\}
\nonumber \\
\hspace*{-20mm}
&& \hspace*{-1mm}
-\! \int\!{\rm D}z {\rm D}y_0\!\int\!\rmd s~ p(s |S\bra a\ket^{\frac{1}{2}}y_0, \theta^\star)
\log\!
\int \!{\rm D} y~
p^\gamma(s | uy \!+\! wy_0\!+\!vz, \theta).
\label{eq:RS_Psi_before_scaling}\end{aligned}$$
Self-averaging with respect to true associations {#app:self_averaging}
================================================
The results of this Appendix were derived in [@SheikhCoolen2019], but will be briefly recapitulated, for completeness and because they are also needed in deriving (\[eq:average\_beta\],\[eq:covariance\_beta\]). We investigate random variables of the form $\mathcal{R}=p^{-1}{\bbeta}^0\cdot {\bf P}{\bbeta}^0$, where the true association vectors ${\bbeta}^0=\{\beta_\mu^0\}$ are drawn randomly from some distribution $p({\bbeta}^0)$, and ${\bf P}$ is a fixed symmetric positive definite $p\times p$ matrix, which is independent of ${\bbeta}^0$. We wish to know the conditions under which $\mathcal{R}$ will be self-averaging, i.e. $\lim_{p\to\infty}\bra \mathcal{R}\ket>0$ exists, and $\lim_{p\to\infty}[\bra \mathcal{R}^2\ket-\bra \mathcal{R}\ket^2]=0$ (brackets denote averaging over $p({\bbeta}^0)$). We assume:
1. The $\{\beta_\mu^0\}$ are independent and identically distributed, i.e. $p({\bbeta}^0) = \prod_{\mu=1}^p p(\beta_{\mu}^0)$.
2. $p(\beta_\mu^0)$ is symmetric in $\beta_\mu^0$, with finite second and fourth order moments.
3. $\lim_{p\to\infty} p^{-1}\sum_{\mu=1}^p P_{\mu\mu}\in \R$.
4. $\lim_{p\to\infty}p^{-2}\sum_{\mu\nu=1}^p P^2_{\mu\nu}=0$.
Given that $S^2=\lim_{p\to\infty}p^{-1}({\bbeta}^0)^2\!$, we must identify $\bra (\beta_\mu^0)^2\ket=S^2$. It was shown in [@SheikhCoolen2019] that the above conditions are sufficient for $\mathcal{R} $ to be self-averaging. This enabled us to infer that the following identities hold (for $g>0$), as soon as average and width of the eigenvalue distribution $\varrho(a)$ of ${\bA}$ remain finite in the limit $p\to\infty$: $$\begin{aligned}
\lim_{p\to \infty}\frac{1}{p}{\bbeta}^0\cdot{\bA}{\bbeta}^0&=& S^2\int\!\rmd a~\varrho(a) a,
\\
\lim_{p\to\infty} \frac{1}{p} \sum_{\rho=1}^p \frac{a_\rho^2 ({\bbeta}^0\!\cdot {\bv}^\rho)^2 }{2\eta\gamma+ga_\rho}&=&
\int\!\rmd a~\varrho(a) \frac{S^2a^2}{2\eta\gamma\!+\!ga}.
\label{eq:selfav_tricky}\end{aligned}$$
Further evaluation of the RS order parameter equations {#app:further}
======================================================
One can take further steps in evaluating the RS order parameter equations (\[eq:RS\_eqns\_uncoupled\],\[eq:RS\_eqns\_coupled\]), without specifying any specific GLM model, exploiting the structural features of the theory only. For instance, the order parameter equations for $(\tilde{f},\tilde{g})$ are not model dependent, and give: $$\begin{aligned}
&&\hspace*{-10mm}
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
= \tilde{u}^2,
\label{eq:ddf=0}
\\[0.5mm]
&&\hspace*{-10mm}
w^2\Bigg[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{-2}
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
-1\Bigg]
-
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket
=v^2.
\label{eq:ddg=0}\end{aligned}$$ For uncorrelated and normalized data, where $\varrho(a)=\delta(a\!-\!1)$, this reduces to $$\begin{aligned}
2\eta\!+\!\tilde{g}
= \tilde{u}^{-2},
~~~~~~
\tilde{f}
=-v^2/\tilde{u}^4.\end{aligned}$$ Alternatively, for $\eta\to 0$ (ML regression) equations (\[eq:ddf=0\],\[eq:ddg=0\]) become $$\begin{aligned}
\tilde{g}=1/ \tilde{u}^2, ~~~~~~
\tilde{f}=-
v^2/\tilde{u}^4.\end{aligned}$$ In addition to the two derivatives of $\Xi_A$ required in the theory, we will also require derivatives with respect to $(\tilde{u},v,w)$. These are $$\begin{aligned}
&&\hspace*{-15mm}
\frac{\partial \Xi_A}{\partial\tilde{u}}=
- \zeta
\tilde{f} \tilde{u},~~~~~~
\frac{\partial \Xi_A}{\partial v}=
- \zeta \tilde{g}v,~~~~~~
\frac{\partial \Xi_A}{\partial w}=
\zeta w\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-1}\!\!
\!- \tilde{g}
\Big].~\end{aligned}$$ We also need to compute the derivatives of (\[eq:Psi\_B\_large\_gamma\_compact\]). We note that all partial derivatives of the argument of (\[eq:Psi\_B\_large\_gamma\_compact\]) that are channelled indirectly via the variable $\xi$ vanish at the point $\xi=\xi(wy_0+vz,\tilde{u},s,\theta)$, by definition. Hence $$\begin{aligned}
\frac{\partial \Xi_B}{\partial \tilde{u}}&=& \frac{1}{\tilde{u}^3}
\Big\bra\!\Big\bra\! \Big\bra
[\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\!-\!wy_0\!-\!vz]^2\Big\ket_{\!s}\Big\ket\!\Big\ket,
\label{eq:ddu}
\\[1mm]
\frac{\partial \Xi_B}{\partial v}&=& \frac{1}{\tilde{u}^2}\Big\{
\Big\bra\!\Big\bra z\Big\bra
\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket-v\Big\},
\label{eq:ddv}
\\[1mm]
\frac{\partial \Xi_B}{\partial w}&=& \frac{1}{\tilde{u}^2}\Big\{
\Big\bra\!\Big\bra y_0
\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket
-w\Big\},
\label{eq:ddw}
\\[1mm]
\frac{\partial \Xi_B}{\partial \theta}&=&
\Big\bra\!\Big\bra \!\Big\bra \frac{\partial \log p(s|\xi,\theta)}{\partial\theta}\Big|_{\xi=\xi(wy_0+vz,\tilde{u},s,\theta)}
\Big\ket_{\!s}\Big\ket\!\Big\ket.
\label{eq:ddtheta}\end{aligned}$$ The remaining four order parameter equations, in addition to the previously derived pair (\[eq:ddf=0\],\[eq:ddg=0\]), then become $$\begin{aligned}
\hspace*{-15mm}
\Big\bra\!\Big\bra \!\Big\bra
[\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\!-\!wy_0\!-\!vz]^2\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& - \zeta
\tilde{f} \tilde{u}^4,
\label{eq:ddu=0}
\\[1mm]
\hspace*{-15mm}
\Big\bra\!\Big\bra z\Big\bra
\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& v\Big(1- \zeta \tilde{g}\tilde{u}^2\Big),
\label{eq:ddv=0}
\\[1mm]
\hspace*{-15mm}
\Big\bra\!\Big\bra y_0
\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& w+ \zeta w \tilde{u}^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!\!-1}\!\!
- \tilde{g}
\Big],
\label{eq:ddw=0}
\\[1mm]
\hspace*{-15mm}
\Big\bra\!\Big\bra \!\Big\bra \frac{\partial \log p(s|\xi,\theta)}{\partial\theta}\Big|_{\xi=\xi(wy_0+vz,\tilde{u},s,\theta)}
\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& 0.
\label{eq:ddtheta=0}\end{aligned}$$ Equations (\[eq:ddv=0\],\[eq:ddw=0\]) can be simplified further upon integrating by parts over $z$ and $y_0$. We need to take care that $y_0$ appears also in the distribution $p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)$ used to define the measure $\bra \ldots\ket_s$. We first turn to the average in (\[eq:ddv=0\]): $$\begin{aligned}
\Big\bra\!\Big\bra z\Big\bra
\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\ket\!\Big\ket &=& \Big\bra\!\Big\bra \!\Big\bra
\frac{\partial}{\partial z}\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{s}\Big\ket\!\Big\ket
\nonumber
\\
&=& v\Big\bra\!\Big\bra \!\Big\bra
(\partial_1\xi)(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket.
\label{eq:partial_z}\end{aligned}$$ Next we work on the average in (\[eq:ddw=0\]): $$\begin{aligned}
\hspace*{-20mm}
\Big\bra\!\Big\bra y_0
\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket&=&
\Big\bra\!\Big\bra \frac{\partial}{\partial y_0}\int\!\rmd s~p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket\!\Big\ket
\nonumber
\\
\hspace*{-20mm}
&=&
\Big\bra\!\Big\bra\int\!\rmd s\Big\{
\xi(wy_0\!+\!vz,\tilde{u},s,\theta)
\frac{\partial}{\partial y_0}p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)
\nonumber
\\
\hspace*{-20mm}
&&
\hspace*{6mm}+ w p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)(\partial_1 \xi)(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\}
\Big\ket\!\Big\ket
\nonumber
\\
\hspace*{-20mm}
&&\hspace*{-20mm} =
\Big\bra\!\Big\bra\!\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta) \frac{\partial \log p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)}{\partial y_0}\Big\ket_{\!s}\Big\ket\!\Big\ket
\nonumber
\\
\hspace*{-20mm}&&\hspace*{6mm}
+w \Big\bra\!\Big\bra \!\Big\bra (\partial_1 \xi)(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s} \Big\ket\!\Big\ket.\end{aligned}$$ With the above results, and upon discarding the trivial solution $v=0$ and using (\[eq:ddv=0\]) to simplify (\[eq:ddw=0\]), we can rewrite our closed MAP order parameter equation set as: $$\begin{aligned}
\hspace*{-20mm}
\Big\bra
\frac{a}{2\eta\!+\!\tilde{g}a}\Big\ket
&=& \tilde{u}^2,
\label{eq:ddf2_app=0}
\\
\hspace*{-20mm}
w^2\Big[
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!\!-2}
\!
\Big\bra \frac{a^3}{
(2\eta\!+\!\tilde{g}a)^2}\Big\ket
\!-\!1\Big]
\!-\!
\tilde{f}\Big\bra
\frac{a^2}{(2\eta\!+\!\tilde{g}a)^2}\Big\ket
&=&v^2,
\label{eq:ddg2_app=0}
\\
\hspace*{-20mm}
\Big\bra\!\Big\bra\!\Big\bra
[\xi(wy_0\!+\!vz,\tilde{u},s,\theta)\!-\!wy_0\!-\!vz]^2\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& - \zeta
\tilde{f} \tilde{u}^4,
\label{eq:ddu2_app=0}
\\[2mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra \!\Big\bra
(\partial_1\xi)(wy_0\!+\!vz,\tilde{u},s,\theta)\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& 1- \zeta \tilde{g}\tilde{u}^2,
\label{eq:ddv2_app=0}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra\!\Big\bra \xi(wy_0\!+\!vz,\tilde{u},s,\theta) \frac{\partial \log p(s|S\bra a\ket^{\frac{1}{2}}y_0,\theta^\star)}{\partial y_0}\Big\ket_{\!s}\Big\ket\!\Big\ket
&=& \zeta w \tilde{u}^2
\bra a\ket
\Big\bra \frac{a^2}{
2\eta\!+\!\tilde{g}a}\Big\ket^{\!-1}\!\!,
\label{eq:ddw2_app=0}
\\[1mm]
\hspace*{-20mm}
\Big\bra\!\Big\bra\! \Big\bra \frac{\partial \log p(s|\xi,\theta)}{\partial\theta}\Big|_{\xi=\xi(wy_0+vz,\tilde{u},s,\theta)}
\Big\ket_{\!s}\Big\ket\!\Big\ket&=& 0.
\label{eq:ddtheta2_app=0}\end{aligned}$$
Statistics of inferred association parameters {#app:beta_stats}
=============================================
Asymptotic form
---------------
In this Appendix we give the details of the evaluation of the distribution (\[eq:betas\_starting\_point\]), in the limit $p,N\to\infty$ with $\zeta=p/N$, first for uncorrelated and then for correlated covariates. We write $\bbeta^0=\bbeta^\star$, assume flat priors for the non-association parameters $\theta$, and use the definition (\[eq:switch\_to\_y\]). Due to the limit $n\to 0$, we may also insert into the above expression without consequence quantities such as $p^{-\gamma n}(\bbeta^0)$ and $p^{-\gamma n}(s|y^0\!,\theta^\star)$, in order to bring it closer to the integrals found in \[app:SheikhCoolen\]. The result is $$\begin{aligned}
\hspace*{-15mm}
{\mathscr{P}}(\beta,\beta^\star) &=&
\lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)
\nonumber
\\[-1mm]
\hspace*{-15mm}
&&\times
\int\{\rmd\theta^1\ldots\rmd\theta^n\}\int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \delta(\beta\!-\!\beta^1_\mu)
\prod_{\alpha=1}^n\Big[\frac{p(\bbeta^\alpha)}{p(\bbeta^0)}\Big]^\gamma
\nonumber
\\
\hspace*{-15mm}
&&\hspace*{-5mm}\times
\Big\{
\int\!\rmd\by~p(\by|\bbeta^0\!,\ldots,\bbeta^n)
\int\!\rmd s~p(s|y^0\!,\theta^\star)
\prod_{\alpha=1}^n
\Big[\frac{p(s|y^\alpha,\theta^\alpha)}{p(s|y^0,\theta^\star)}\Big]^\gamma
\Big\}^N.
\end{aligned}$$ We can now repeat the manipulations of \[app:SheikhCoolen\], with slight modifications. It will in fact be useful to work with the more general family of factorizing priors $p(\bbeta)\propto \prod_{\mu\leq p}p(\beta_\mu)$, of which the Gaussian one is a special case, but which also allows us to inspect e.g. $L1$ priors. Our expression for ${\mathscr{P}}(\beta,\beta^\star)$ then becomes $$\begin{aligned}
\hspace*{-20mm}
{\mathscr{P}}(\beta,\beta^\star) &=&
\lim_{\gamma\to\infty} \lim_{n\to 0}\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)
\int\{\rmd\theta^1\ldots\rmd\theta^n\}
\int\!\rmd\bC\rmd\hat{\bC}~\frac{\rme^{\rmi p\sum_{\alpha,\rho=0}^n \hat{C}_{\alpha\rho}C_{\alpha\rho}}}{(2\pi/p)^{(n+1)^2}}
\nonumber
\\
\hspace*{-20mm}
&&\times
\Bigg[
\int\!\frac{\rmd\by~\rme^{-\frac{1}{2}\by\cdot\bC^{-1}\by}}{\sqrt{(2\pi)^{n+1}{\rm det}\bC}}
\int\!\rmd s~p(s|y^0\!,\theta^\star)
\prod_{\alpha=1}^n
\Big[\frac{p(s|y^\alpha,\theta^\alpha)}{p(s|y^0,\theta^\star)}\Big]^\gamma
\Bigg]^N
\nonumber
\\
\hspace*{-20mm}
&&\hspace*{-5mm} \times \int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \delta(\beta\!-\!\beta^1_\mu)\rme^{-\rmi\sum_{\alpha,\rho=0}^n \hat{C}_{\alpha\rho}\bbeta^\alpha\cdot\bA\bbeta^\rho}
\prod_{\alpha=1}^n\prod_{\nu=1}^p \Big[\frac{p(\beta_\nu^\alpha)}{p(\beta_\nu^0)}\Big]^\gamma.
\label{eq:beta_beta_intermediate}
\end{aligned}$$ We proceed to the limit $p,N\to\infty$ with fixed $\zeta=p/N$. In view of our previous calculations we define the following quantity: $$\begin{aligned}
\hspace*{-20mm}
\Psi(\bC,\hat{\bC},\theta^1,\ldots,\theta^n)&=& -\rmi\zeta \sum_{\alpha,\rho=0}^n\hat{C}_{\alpha\rho}C_{\alpha\rho}
+ \frac{1}{2}(n\!+\!1)\log (2\pi)
+ \frac{1}{2}\log {\rm det}\bC
\nonumber
\\
\hspace*{-20mm}
&& \hspace*{-18mm}
- \log \int\!\rmd\by~\rme^{-\frac{1}{2}\by\cdot\bC^{-1}\by}
\int\!\rmd s~p(s|y^0\!,\theta^\star)
\prod_{\alpha=1}^n
\Big[\frac{p(s|y^\alpha,\theta^\alpha)}{p(s|y^0,\theta^\star)}\Big]^\gamma
\nonumber
\\
\hspace*{-20mm}
&& \hspace*{-18mm} -\frac{1}{N}\log\int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \rme^{-\rmi\sum_{\alpha,\rho=0}^n \hat{C}_{\alpha\rho}\bbeta^\alpha\cdot\bA\bbeta^\rho}
\prod_{\alpha=1}^n\prod_{\nu=1}^p \Big[\frac{p(\beta_\nu^\alpha)}{p(\beta_\nu^0)}\Big]^\gamma.
\end{aligned}$$ This enables us to write (\[eq:beta\_beta\_intermediate\]) as $$\begin{aligned}
\hspace*{-23mm}
{\mathscr{P}}(\beta,\beta^\star) &=&
\lim_{\gamma\to\infty} \lim_{n\to 0}
\int\{\rmd\theta^1\ldots\rmd\theta^n\}
\int\!\rmd\bC\rmd\hat{\bC}~
\rme^{-N \Psi(\bC,\hat{\bC},\theta^1,\ldots,\theta^n)}
\label{eq:beta_relation_nearly}
\\
\hspace*{-23mm}
&&
\hspace*{-17mm} \times \frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\frac{ \int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \delta(\beta\!-\!\beta^1_\mu)\rme^{-\rmi\sum_{\alpha,\rho=0}^n \hat{C}_{\alpha\rho}\bbeta^\alpha\cdot\bA\bbeta^\rho}
\! \prod_{\alpha=1}^n\prod_{\nu=1}^p p^\gamma(\beta_\nu^\alpha)}
{ \int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \rme^{-\rmi\sum_{\alpha,\rho=0}^n \hat{C}_{\alpha\rho}\bbeta^\alpha\cdot\bA\bbeta^\rho}
\!\prod_{\alpha=1}^n\prod_{\nu=1}^p p^\gamma (\beta_\nu^\alpha)}.
\nonumber
\\[-2mm]
\hspace*{-20mm}&&\nonumber\end{aligned}$$ Integrating both sides over $(\beta,\beta^\star)$ shows that the first line of (\[eq:beta\_relation\_nearly\]) on its own would equal one. Hence for $N\to\infty$ we will be left simply with the limit $p\to\infty$ of the second line, which is an $\order(1)$ object, evaluated at the the saddle point of $\Psi(\bC,\hat{\bC},\theta^1,\ldots,\theta^n)$. For Gaussian priors, the saddle point is the one computed in \[app:SheikhCoolen\]. Hence, upon transforming as before $\hat{\bC}=-\frac{1}{2}\rmi \bD$, and choosing the saddle point values, $$\begin{aligned}
\hspace*{-12mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=& \lim_{p\to\infty} \lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)
\label{eq:link_arbitrary_A}
\\
\hspace*{-12mm}
&&\hspace*{-15mm}\times \frac{ \int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \delta(\beta\!-\!\beta^1_\mu)\rme^{-\frac{1}{2}\sum_{\alpha,\rho=0}^n D_{\alpha\rho}\bbeta^\alpha\cdot\bA\bbeta^\rho}
\prod_{\alpha=1}^n\prod_{\nu=1}^p p^\gamma(\beta_\nu^\alpha)}
{ \int\!\rmd\bbeta^1\ldots\rmd\bbeta^n~ \rme^{-\frac{1}{2}\sum_{\alpha,\rho=0}^n D_{\alpha\rho}\bbeta^\alpha\cdot\bA\bbeta^\rho}
\prod_{\alpha=1}^n\prod_{\nu=1}^p p^\gamma (\beta_\nu^\alpha)}.
\nonumber\end{aligned}$$ This expression depends on the choice of $p(s|\xi,\theta)$ only indirectly, via the values of the order parameters $\{D_{\alpha\rho}\}$. We will now work out (\[eq:link\_arbitrary\_A\]) first for uncorrelated and normalized covariates, followed by evaluation for arbitrary covariate correlations.
Uncorrelated and normalized covariates
--------------------------------------
This is the simplest case, where $A_{\mu\nu}=\delta_{\mu\nu}$. The integrations in the above formula now factorize over all components of $\bbeta$, giving $$\begin{aligned}
\hspace*{-22mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=& \Big(\lim_{p\to\infty} \frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\Big)\times
\\
\hspace*{-22mm}
&&\hspace*{-23mm}\lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{ \int\!\rmd\beta^1\ldots\rmd\beta^n~ \delta(\beta\!-\!\beta^1)\rme^{-\frac{1}{2}\sum_{\alpha,\rho=1}^n D_{\alpha\rho}\beta^\alpha \beta^\rho
-\beta^\star\sum_{\alpha=1}^n D_{\alpha 0}\beta^\alpha}
\prod_{\alpha=1}^n p^\gamma(\beta^\alpha)}
{ \int\!\rmd\beta^1\ldots\rmd\beta^n~ \rme^{-\frac{1}{2} \sum_{\alpha,\rho=1}^n D_{\alpha\rho}\beta^\alpha \beta^\rho
-\beta^\star\sum_{\alpha=1}^n D_{\alpha 0}\beta^\alpha}
\prod_{\alpha=1}^n p^\gamma (\beta^\alpha)}.
\nonumber\end{aligned}$$ Hence $$\begin{aligned}
&&
\hspace*{-22mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta|\beta^\star)~ =
\\[-1mm]
\hspace*{-22mm}
&& \hspace*{-18mm}
\lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{ \int\!\rmd\beta^1\ldots\rmd\beta^n~ \delta(\beta\!-\!\beta^1)\rme^{-\frac{1}{2}\sum_{\alpha,\rho=1}^n D_{\alpha\rho}\beta^\alpha \beta^\rho
-\beta^\star\sum_{\alpha=1}^n D_{\alpha 0}\beta^\alpha}
\prod_{\alpha=1}^n p^\gamma(\beta^\alpha)}
{ \int\!\rmd\beta^1\ldots\rmd\beta^n~ \rme^{-\frac{1}{2} \sum_{\alpha,\rho=1}^n D_{\alpha\rho}\beta^\alpha \beta^\rho
-\beta^\star\sum_{\alpha=1}^n D_{\alpha 0}\beta^\alpha}
\prod_{\alpha=1}^n p^\gamma (\beta^\alpha)}.
\nonumber\end{aligned}$$ We next use the replica symmetric form of the matrix $\bD$, i.e. $D_{\alpha\rho}=D\delta_{\alpha\rho}+d(1\!-\!\delta_{\alpha\rho})$ and $D_{\alpha 0}=d_0$ and for $\alpha,\rho=1\ldots n$, and carry out a Gaussian linearization: $$\begin{aligned}
\hspace*{-20mm} \lim_{N\to\infty}{\mathscr{P}}(\beta|\beta^\star) &=&
\nonumber
\\
\hspace*{-20mm} &&\hspace*{-31mm}
\lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{ \int\!{\rm D}z
\Big[ \int\!\rmd\beta^\prime~ \rme^{
\rmi z\sqrt{d}\beta^\prime
-\frac{1}{2}(D-d) (\beta^\prime)^2
-d_0\beta^\star \beta^\prime}
p^\gamma (\beta^\prime)\Big]^{n-1}
\!
\rme^{
\rmi z\sqrt{d} \beta
-\frac{1}{2}(D-d)\beta^2
-d_0\beta^\star\beta}
p^\gamma(\beta)}
{\int\!{\rm D}z\Big[ \int\!\rmd\beta^\prime~ \rme^{
\rmi z\sqrt{d}\beta^\prime
-\frac{1}{2}(D-d) (\beta^\prime)^2
-d_0\beta^\star \beta^\prime}
p^\gamma (\beta^\prime)\Big]^n}
\nonumber\hspace*{-1mm}
\\
\hspace*{-20mm} &=&
\lim_{\gamma\to\infty}
\int\!{\rm D}z\left[ \frac{
\rme^{
-\frac{1}{2}(D-d)\beta^2
+\beta(\rmi z\sqrt{d}-d_0\beta^\star)}
p^\gamma(\beta)}
{ \int\!\rmd\beta^\prime~ \rme^{
-\frac{1}{2}(D-d) (\beta^\prime)^2
+\beta^\prime( \rmi z\sqrt{d}
-d_0\beta^\star )}
p^\gamma (\beta^\prime)}
\right].\end{aligned}$$ In terms of the transformed order parameters $f=d$ and $g=D-d$ this becomes $$\begin{aligned}
\hspace*{-10mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta|\beta^\star) &=&
\lim_{\gamma\to\infty}
\int\!{\rm D}z\left[ \frac{
\rme^{
-\frac{1}{2}g\beta^2
+\beta(\rmi z\sqrt{f}-d_0\beta^\star)}
p^\gamma(\beta)}
{ \int\!\rmd\beta^\prime~ \rme^{
-\frac{1}{2}g(\beta^\prime)^2
+\beta^\prime( \rmi z\sqrt{f}
-d_0\beta^\star )}
p^\gamma (\beta^\prime)}
\right].\end{aligned}$$ The order parameter $d_0$, which we could remove from the general theory, here needs to be computed after all. For the $L2$ (i.e. Gaussian) prior $p(\beta)\propto\exp(-\eta \beta^2)$ we can find $d_0$ via differentiation of (\[eq:Psi\_RS\_before\_uvw\]) and subsequently use (\[eq:selfav\_tricky\]), giving $$\begin{aligned}
d_0&=& -c_0\Big\langle
\frac{a^2 ({\bbeta}^0\!\cdot\! {\bv})^2}{2 \eta \gamma\! +\! (D\!-\!d)a}\Big\rangle^{-1}
= -\frac{c_0}{S^2}
\Big\bra \frac{a^2}{2\eta\gamma\!+\!ga}\Big\ket^{-1}.
\label{eq:found_d0}
\end{aligned}$$ For uncorrelated and normalized covariates we have $\varrho(a)=\delta(a\!-\!1)$, so $c_0=Sw$ and $$\begin{aligned}
d_0&=& -\frac{c_0}{S^2}
(2\eta\gamma\!+\!g)
= -\frac{\gamma w}{S}
(2\eta\!+\!\tilde{g}).
\end{aligned}$$ We thus find with $f=\tilde{f}\gamma^2$, $$\begin{aligned}
\hspace*{-10mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta|\beta^\star) &=&
\lim_{\gamma\to\infty}
\int\!{\rm D}z\left[ \frac{
\rme^{\gamma\big[
-\frac{1}{2}(2\eta+\tilde{g})\beta^2
+\beta(\rmi z\sqrt{\tilde{f}}+w
(2\eta+\tilde{g})\beta^\star/S)\big]}}
{ \int\!\rmd\beta^\prime~
\rme^{\gamma\big[
-\frac{1}{2}(2\eta+\tilde{g})(\beta^\prime)^2
+\beta^\prime(\rmi z\sqrt{\tilde{f}}+w
(2\eta+\tilde{g})\beta^\star/S)\big]}
}
\right]
\nonumber
\hspace*{-10mm}
\\[0.5mm]
&=& \lim_{\gamma\to\infty}
\frac{\sqrt{\gamma(2\eta\!+\!\tilde{g})}}{\sqrt{2\pi}} \int\!{\rm D}z~
\rme^{-\frac{1}{2}\gamma(2\eta+\tilde{g})\big[
\beta
-\rmi z\sqrt{\tilde{f}}/(2\eta+\tilde{g})-w
\beta^\star/S\big]^2}.
\nonumber
\\[-1mm]
\hspace*{-10mm}&&\end{aligned}$$ For $\varrho(a)=\delta(a\!-\!1)$ we also know that $2\eta+\tilde{g}=\tilde{u}^{-2}$ and $\tilde{f}=-v^2/\tilde{u}^4$. Hence the above integral reduces to $$\begin{aligned}
\lim_{N\to\infty}{\mathscr{P}}(\beta|\beta^\star) &=&
\frac{1}{v\sqrt{2\pi}}\rme^{-\frac{1}{2}(\beta-w\beta^\star/S)^2/v^2}\end{aligned}$$ This confirms what was suggested by simulation data and exploited in [@coolen2017replica]: if we plot inferred versus true association parameters in a plane, we will find for $L2$ priors and uncorrelated covariates a linear cloud with slope $w/S$ and zero-average Gaussian noise of width $v$. We have now proved this analytically, for [*any*]{} generalized linear model.
Correlated covariates
---------------------
This is the more tricky case. We return to (\[eq:link\_arbitrary\_A\]) and implement first the replica symmetry ansatz, i.e. $D_{\alpha 0}=d_0$ and $D_{\alpha\rho}=D\delta_{\alpha\rho}+d(1\!-\!\delta_{\alpha\rho})$ for $\alpha,\rho=1\ldots n$, so that we can proceed with our calculation: $$\begin{aligned}
\hspace*{-25mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=& \lim_{p\to\infty} \lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\times
\nonumber
\\
\hspace*{-25mm}
&&\hspace*{-25mm}\frac{\int\!\rmd\tilde{\bbeta}
\rme^{-\frac{1}{2}d\tilde{\bbeta}\cdot\bA\tilde{\bbeta} -d_0\tilde{\bbeta}\cdot\bA\bbeta^0}
\int\!
\prod_{\alpha=1}^n\!\Big[ \rmd\bbeta^\alpha \rme^{
-\frac{1}{2}(D-d) \bbeta^\alpha\cdot\bA\bbeta^\alpha
}\!
\prod_{\nu=1}^p p^\gamma(\beta_\nu^\alpha)\Big] \delta(\tilde{\bbeta}\!-\!\sum_{\alpha=1}^n\bbeta^\alpha) \delta(\beta\!-\!\beta^1_\mu)
}
{ \int\!\rmd\tilde{\bbeta}\rme^{-\frac{1}{2}d\tilde{\bbeta}\cdot\bA\tilde{\bbeta} -d_0\tilde{\bbeta}\cdot\bA\bbeta^0}
\int\!\prod_{\alpha=1}^n\!\Big[ \rmd\bbeta^\alpha \rme^{
-\frac{1}{2}(D-d) \bbeta^\alpha\cdot\bA\bbeta^\alpha
}\!
\prod_{\nu=1}^p p^\gamma(\beta_\nu^\alpha)\Big] \delta(\tilde{\bbeta}\!-\!\sum_{\alpha=1}^n\bbeta^\alpha)
}
\hspace*{-25mm}
\nonumber
\\
\hspace*{-25mm}
&=& \lim_{p\to\infty} \lim_{\gamma\to\infty} \lim_{n\to 0}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\times
\nonumber
\\
\hspace*{-25mm}
&&\hspace*{-15mm}\frac{\int\!\rmd\tilde{\bbeta}\rmd\hat{\bbeta}~W(\hat{\bbeta},\tilde{\bbeta}) H^{n-1}\!(\hat{\bbeta})
\int\!\rmd\bbeta^1 \rme^{-\rmi\hat{\bbeta}\cdot\bbeta^1\!
-\frac{1}{2}(D-d) \bbeta^1\cdot\bA\bbeta^1
}\!
\delta(\beta\!-\!\beta^1_\mu)\prod_{\nu=1}^p p^\gamma(\beta_\nu^1)
}
{ \int\!\rmd\tilde{\bbeta}\rmd\hat{\bbeta}~ W(\hat{\bbeta},\tilde{\bbeta})
H^{n}(\hat{\bbeta})
},
\nonumber
\\[-1mm]
\hspace*{-25mm}&&\label{eq:beta_beta_corr_1}\end{aligned}$$ with $$\begin{aligned}
W(\hat{\bbeta},\tilde{\bbeta})&=&\rme^{\rmi\hat{\bbeta}\cdot\tilde{\bbeta}-\frac{1}{2}d\tilde{\bbeta}\cdot\bA\tilde{\bbeta} -d_0\tilde{\bbeta}\cdot\bA\bbeta^0},
\\
H(\hat{\bbeta})&=&\int\! \rmd\bbeta^\prime \rme^{-\rmi\hat{\bbeta}\cdot\bbeta^\prime
-\frac{1}{2}(D-d) \bbeta^\prime\cdot\bA\bbeta^\prime
}\!
\prod_{\nu=1}^p p^\gamma(\beta_\nu^\prime).\end{aligned}$$ Note that $\hat{\bbeta},\tilde{\bbeta}\in\R^p$. For $n\to 0$ the denominator evaluates to $(2\pi)^p$. Hence expression (\[eq:beta\_beta\_corr\_1\]) can be simplified to $$\begin{aligned}
\hspace*{-10mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=& \lim_{p\to\infty} \lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\times
\\
\hspace*{-10mm}
&&\hspace*{-21mm}\int\!\frac{\rmd\tilde{\bbeta}\rmd\hat{\bbeta}}{(2\pi)^p}~
W(\hat{\bbeta},\tilde{\bbeta})
\left\{\!
\frac{
\int\!\rmd\bbeta^\prime \rme^{-\rmi\hat{\bbeta}\cdot\bbeta^\prime\!
-\frac{1}{2}(D-d) \bbeta^\prime\cdot\bA\bbeta^\prime
}\!
\delta(\beta\!-\!\beta^\prime_\mu)\prod_{\nu=1}^p p^\gamma(\beta_\nu^\prime) }
{
\int\!\rmd\bbeta^\prime \rme^{-\rmi\hat{\bbeta}\cdot\bbeta^\prime\!
-\frac{1}{2}(D-d) \bbeta^\prime\cdot\bA\bbeta^\prime
}\!
\prod_{\nu=1}^p p^\gamma(\beta_\nu^\prime) }
\right\}.
\nonumber\end{aligned}$$ We choose the Gaussian prior $p(\beta)\propto \exp(-\eta\beta^2)$, we write $\delta(\beta-\!\beta^\prime_\mu) $ in integral form, we introduce the unit vector $\hat{{\bf e}}^\mu$ with components $\hat{e}^\mu_\nu=\delta_{\mu\nu}$, we use $f=d$ and $g=D-d$, and we do the Gaussian integrals where possible. This gives $$\begin{aligned}
\hspace*{-25mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=&\lim_{p\to\infty} \lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\int\!\frac{\rmd k}{2\pi}\rme^{\rmi k\beta}
\\
&&\times\int\!\frac{\rmd\tilde{\bbeta}\rmd\hat{\bbeta}}{(2\pi)^p}~
W(\hat{\bbeta},\tilde{\bbeta})
\left\{
\frac{
\int\!\rmd\bbeta^\prime \rme^{-\rmi(\hat{\bbeta}+k\hat{\bf e}^\mu)\cdot\bbeta^\prime\!
-\frac{1}{2}\bbeta^\prime\cdot[(D-d) \bA+2\gamma\eta\one]\bbeta^\prime
}
}
{
\int\!\rmd\bbeta^\prime \rme^{-\rmi\hat{\bbeta}\cdot\bbeta^\prime\!
-\frac{1}{2}\bbeta^\prime\cdot[(D-d) \bA+2\gamma\eta\one]\bbeta^\prime}
}
\right\}
\nonumber
\\
\hspace*{-25mm}
&=&
\lim_{p\to\infty} \lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\int\!\frac{\rmd k}{2\pi}\rme^{\rmi k\beta-\frac{1}{2}k^2[(D-d) \bA+2\gamma\eta\one]^{-1}_{\mu\mu}}
\nonumber
\\
\hspace*{-25mm}
&&\times\! \int\!\frac{\rmd\hat{\bbeta}}{(2\pi)^p}\rme^{-k\hat{\bf e}^\mu\cdot[(D-d) \bA+2\gamma\eta\one]^{-1}\hat{\bbeta}
}\!
\int\!\rmd\tilde{\bbeta}~\rme^{-\frac{1}{2}d\tilde{\bbeta}\cdot\bA\tilde{\bbeta} -\tilde{\bbeta}\cdot (d_0\bA\bbeta^0-\rmi\hat{\bbeta})}
\nonumber
\\
\hspace*{-25mm}
&=&
\lim_{p\to\infty} \lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\int\!\frac{\rmd k}{2\pi}
\rme^{\rmi k\big[\beta
+ d_0
\hat{\bf e}^\mu\cdot [(D-d) \bA+2\gamma\eta\one]^{-1}
\bA
\bbeta^0\big]
}
\nonumber
\\
\hspace*{-25mm}
&&\hspace*{0mm} \times
\rme^{-\frac{1}{2}k^2\Big[ [(D-d) \bA+2\gamma\eta\one]^{-1}
-d
[(D-d) \bA+2\gamma\eta\one]^{-1}
\bA
[(D-d) \bA+2\gamma\eta\one]^{-1}\Big]_{\mu\mu}
}
\nonumber
\\
\hspace*{-25mm}
&=&
\lim_{p\to\infty} \lim_{\gamma\to\infty}
\frac{1}{p}\sum_{\mu=1}^p \delta(\beta^\star\!-\beta_\mu^0)\int\!\frac{\rmd k}{2\pi}
\rme^{\rmi k\Big[\beta+ d_0
[ (g\bA+2\gamma\eta\one)^{-1}
\bA
\bbeta^0]_\mu\Big]
}
\nonumber
\\
\hspace*{-25mm}
&&\hspace*{5mm} \times
\rme^{-\frac{1}{2}k^2\Big[ (g\bA+2\gamma\eta\one)^{-1}
-f (g \bA+2\gamma\eta\one)^{-1}\bA(g \bA+2\gamma\eta\one)^{-1}\Big]_{\mu\mu}
}.\end{aligned}$$ Next we use the scaling with $\gamma$ of the order parameters, $f=\tilde{f}\gamma^2$, $g=\tilde{g}\gamma$ and $d_0=\gamma\tilde{d}_0$. For the integrals to converge we must have $\tilde{f}<0$ (which follows from solving the order parameter equations). We can then take $\gamma\to\infty$ and do the integral over $k$, giving $$\begin{aligned}
\hspace*{-20mm}
\lim_{N\to\infty}{\mathscr{P}}(\beta,\beta^\star) &=&
\lim_{p\to\infty}
\frac{1}{p}\sum_{\mu=1}^p
\frac{\delta(\beta^\star\!-\beta_\mu^0)}{\sqrt{2\pi |\tilde{f}|[
(\tilde{g} \bA+2\eta\one)^{-1}\bA(\tilde{g} \bA+2\eta\one)^{-1}]_{\mu\mu}}}
\label{eq:beta_relation_derived_app}
\\
\hspace*{-20mm}
&&
\times
\rme^{-\frac{1}{2}\Big[\beta+ \tilde{d}_0
[ (\tilde{g}\one+2\eta\bA^{-1})^{-1}
\bbeta^0]_\mu\Big]^2/|\tilde{f}|[
(\tilde{g} \bA+2\eta\one)^{-1}\bA(\tilde{g} \bA+2\eta\one)^{-1}]_{\mu\mu}
}.
\nonumber\end{aligned}$$ This is expression (\[eq:beta\_relation\_derived\]) in the main text.
Pathologies of generalization error minimization {#app:generalization_error}
================================================
Here we illustrate the dangers of using the generalization error as an objective function to be minimized, by using logistic regression as an example. The generalization error $E_g\in[0,1]$ is the expected fraction of samples for which the true and the inferred model disagree on the outcome values, for samples drawn randomly from the population (as opposed to from the training set). In logistic regression we have $s=\pm 1$ and write $p(s|\bz,\bbeta)=\frac{1}{2}+\frac{1}{2}s\tanh(\bbeta\cdot\bz)$ (rescaling by $\sqrt{p}$ is not relevant here). If the true and inferred parameters are $\bbeta^\star$ and $\bbeta$, the generalization error is $$\begin{aligned}
E_g&=& \int\!\rmd\bz~p(\bz)\sum_{s,s^\prime=\pm 1}\frac{1}{2}(1\!-\!ss^\prime)p(s|\bz,\bbeta^\star)p(s^\prime|\bz,\bbeta)
\nonumber
\\
&=& \frac{1}{2}-\frac{1}{2}\int\!\rmd\bz~p(\bz)\tanh(\bbeta^\star\!\cdot\bz)\tanh(\bbeta\cdot\bz).\end{aligned}$$ If we were to use $E_g$ to optimize the inferred parameters (assuming it could be estimated without explicit knowledge of the true parameters $\bbeta^\star$), we would seek to minimize $E_g$ over $\bbeta$. We note the lower bound $$\begin{aligned}
E_g & \geq & \frac{1}{2}-
\frac{1}{2}\int\!\rmd\bz~p(\bz)\Big|\tanh(\bbeta^\star\!\cdot\bz)\Big|.
\label{eq:Eg_bound}\end{aligned}$$ While $E_g$ indeed computes the fraction of samples for which the two models disagree on the outcome, it does [*not*]{} measure whether the two models also use the same outcome probabilities. To see this, imagine choosing $\bbeta=\kappa\bbeta^\star$. Here one would find $$\begin{aligned}
\hspace*{-10mm}
\frac{\rmd}{\rmd\kappa}E_g&=& -\int\!\rmd\bz~p(\bz)\Big(\frac{\bbeta^\star\!\cdot\bz}{2\sqrt{p}}\Big)
\tanh(\frac{\bbeta^\star\!\cdot\bz}{\sqrt{p}})\Big[1\!-\!\tanh^2(\kappa\frac{\bbeta^\star\!\cdot\bz}{\sqrt{p}})\Big]
<0.\end{aligned}$$ Hence the of $E_g$ minimum is found for $\kappa\to\infty$, where the (diverging) estimator $\bbeta=\kappa\bbeta^\star$ satisfies the lower bound (\[eq:Eg\_bound\]). The inferred model $p(s|\bz,\bbeta)=\frac{1}{2}+\frac{1}{2}s~ {\rm sgn}(\bbeta^\star\cdot\bz)$ would indeed get the maximum achievable fraction of binary outcomes predicted correctly, but it would believe erroneously that it has 100% prediction accuracy.
Distribution of empirical covariance matrices {#app:towards_Wishart}
=============================================
Here we evaluate expression (\[eq:Ahat\_measure\]) further, to convert the integral in (\[eq:linear\_betastats\_1\]) over all $p\times p$ matrices into an integral over positive definite and symmetric ones. We write symmetric and antisymmetric parts of matrices $\bM$ as $\bM^s$ and $\bM^a$, and transform integrations over all $\hat{\bA}$ into integrations over symmetric and antisymmetric parts. The (anti)symmetrization transformations involved induce identities such as $\rmd\bM=2^{\frac{1}{2}p(p-1)}\rmd\bM^s\rmd\bM^a$ and $\delta(\bM)=2^{-\frac{1}{2}p(p-1)} \delta(\bM^s)\delta(\bM^a)$, where $\rmd\bM^s=\prod_{\mu\leq \nu}\rmd M^s_{\mu\nu}$ and $\rmd\bM^a=\prod_{\mu< \nu}\rmd M^a_{\mu\nu}$. Moreover, $$\begin{aligned}
\int\!\rmd \bM^a ~\rme^{\rmi {\rm Tr}(\bA^a\bM^a)}&=& \pi^{\frac{1}{2}p(p-1)}\prod_{\mu<\nu}\delta(A^a_{\mu\nu}),
\\
\int\!\rmd \bM^s ~\rme^{\rmi {\rm Tr}(\bA^s\bM^s)}&=& 2^p\pi^{\frac{1}{2}p(p+1)}\prod_{\mu\leq \nu}\delta(A^s_{\mu\nu}).\end{aligned}$$ We can now compute $P(\hat{\bA})$ for the case where $p(\bz)=[(2\pi)^{-p}{\rm Det}\bA]^{\frac{1}{2}}\rme^{-\frac{1}{2}\bz\cdot\bA\bz}$, giving $$\begin{aligned}
P(\hat{\bA})&=&
\int\!\frac{\rmd\bQ}{(2\pi)^{p^2}}~\rme^{\rmi {\rm Tr}(\bQ^s\hat{\bA}^s)+\rmi {\rm Tr}(\bQ^a\hat{\bA}^a)}
[{\rm Det}(\one\!+\!\frac{2\rmi}{N}\bA\bQ^s)]^{-N/2}
\nonumber
\\
&=&\delta(\hat{\bA}^a)\Big(\frac{2\rmi}{N}\Big)^{-Np/2}\!({\rm Det}\bA)^{-N/2}
\nonumber
\\
&&\times
\int\!\frac{\rmd\bQ^s}{(2\pi)^{\frac{1}{2}p(p+1)}}~\rme^{\rmi {\rm Tr}(\bQ^s\hat{\bA}^s)}
[{\rm Det}(\bQ^s\!-\!\frac{1}{2}N\rmi\bA^{-1})]^{-N/2}.\end{aligned}$$ Thus $P(\hat{\bA})\rmd\hat{\bA}=2^{-\frac{1}{2}p(p-1)}[\delta(\hat{\bA}^a) \rmd\hat{\bA}^a][P(\hat{\bA}^s)\rmd\hat{\bA}^s]$, where $$\begin{aligned}
P(\hat{\bA}^s)&=& 2^{\frac{1}{2}p(p-1)}\Big(\frac{2}{N}\Big)^{\!-Np/2}\!({\rm Det}\bA)^{-N/2}
\nonumber
\\
&&\hspace*{-2mm} \times\!
\int\!\frac{\rmd\bQ^s}{(2\pi)^{\frac{1}{2}p(p+1)}}~\rme^{\rmi {\rm Tr}(\bQ^s\hat{\bA}^s)}
[{\rm Det}(\rmi\bQ^s\!+\!\frac{1}{2}N\bA^{-1})]^{-N/2}.~~~
\label{eq:PhatA_intermediate}\end{aligned}$$ We can now forget about the antisymmetric parts of $\hat{\bA}$, and average only over all symmetric matrices. The nontrivial integral in (\[eq:PhatA\_intermediate\]) is found in [@Ingham], giving $$\begin{aligned}
P(\hat{\bA}^s)&=& \Big(\frac{2}{N}\Big)^{\!-Np/2}\!
\frac{\rme^{-\frac{1}{2}N{\rm Tr}(\hat{\bA}^s\! \bA^{-1})}({\rm Det}\hat{\bA}^s)^{\frac{1}{2}(N-p-1)}}
{\pi^{\frac{1}{4}p(p-1)}({\rm Det}\bA)^{N/2}\prod_{j=\frac{1}{2}(N-p+1)}^{N/2} \Gamma(j)}.
\label{eq:Wishart}\end{aligned}$$ Hence $P(\hat{\bA}^s)$ is a Wishart distribution with $N$ degrees of freedom. With $\Omega_p$ denoting the space of positive definite symmetric $p\times p$ matrices, and dropping the superscript $s$, we may then summarize our result for (\[eq:linear\_betastats\_1\]) as: $$\begin{aligned}
P(\hat{\bbeta})&=&
\int_{\Omega_p}\!\rmd\hat{\bA} ~\Big(\frac{2}{N}\Big)^{\!-Np/2}\!
\frac{\rme^{-\frac{1}{2}N{\rm Tr}(\hat{\bA}^s\! \bA^{-1})}({\rm Det}\hat{\bA}^s)^{\frac{1}{2}(N-p-1)}}
{\pi^{\frac{1}{4}p(p-1)}({\rm Det}\bA)^{N/2}\prod_{j=\frac{1}{2}(N-p+1)}^{N/2} \Gamma(j)}
\nonumber
\\&&\hspace*{20mm}
\times {\cal N}(\hat{\bbeta}|\hat{\bG}\bbeta^\star\!,\zeta (\Sigma^\star)^2\hat{\bG}\hat{\bA}^{-1}\hat{\bG}).\end{aligned}$$
[^1]: The present limitation to $L2$ (i.e. Gaussian) priors is not critical, alternative choices such as $L1$ (or lasso) priors simply lead to more complicated integrals.
[^2]: Note that this latter choice would become $p(\bbeta)\propto\exp(-\eta p\bbeta^2)$ for the alternative convention where the association coefficients are not rescaled by $\sqrt{p}$, i.e. for GLMs written as $p(s|\bbeta\cdot\bz,\theta)$.
[^3]: The order parameters have been determined within the so-called replica-symmetric (RS) ansatz, which implies the assumption that the stochastic optimization process at finite $\gamma$ is ergodic.
[^4]: Note that, although anticipated at the time, this was not yet done in the previous studies [@coolen2017replica; @SheikhCoolen2019].
[^5]: Also the more complicated case of Cox regression with censoring falls within the scope of our present GLM equations, provided the censoring events are non-informative (as with end-of-trial censoring). For informative censoring, such as censoring caused by nontrivial competing risks, we first need to generalize the theory further to models in which outcome probabilities involve multiple linear combinations of covariates. This should be straightforward, but will be the subject of a future study.
|
---
abstract: 'We describe an algebraic proof of the well-known topological fact that $\pi_1(SO(n)) \cong \bBZ/2\bBZ$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional relation and turns out to be a nontrivial central extension by $\bBZ/2\bBZ$ of the corresponding group of rotational symmetries of the hyperoctahedron in dimension $n$.'
author:
- 'Ina Hajdini[^1] and Orlin Stoytchev[^2]'
title: 'The Fundamental Group of $SO(n)$ Via Quotients of Braid Groups'
---
Introduction.
=============
The set of all rotations in $\Rn$ forms a group denoted by $SO(n)$. We may think of it as the group of $n\times n$ orthogonal matrices with unit determinant. As a topological space it has the structure of a smooth $(n(n-1)/2)$-dimensional submanifold of $\bBR^{n^2}$. The group structure is compatible with the smooth one in the sense that the group operations are smooth maps, so it is a Lie group. The space $SO(n)$ when $n\ge 3$ has a fascinating topological property—there exist closed paths in it (starting and ending at the identity) that cannot be continuously deformed to the trivial (constant) path, but going twice along such a path gives another path, which is deformable to the trivial one. For example, if you rotate an object in $\R3$ by $2\pi$ along some axis, you get a motion that is not deformable to the trivial motion (i.e., no motion at all), but a rotation by $4\pi$ is deformable to the trivial motion. Further, a rotation by $2\pi$ along any axis can be deformed to a rotation by $2\pi$ along any other axis. We shall call a [*full rotation*]{} in $\Rn$ any motion that corresponds to a closed path in $SO(n)$, starting and ending at the identity. Thus, it turns out that there are two classes of full rotations: topologically trivial, i.e., deformable to the trivial motion, and topologically nontrivial. Every nontrivial full rotation can be deformed to any other nontrivial full rotation. Two consecutive nontrivial full rotations produce a trivial one.
For any topological space, one can consider the set of closed paths starting and ending at some fixed point, called [*base-point*]{}. Two closed paths that can be continuously deformed to each other, keeping the base-point fixed, are called *homotopic*. One can multiply closed paths by [*concatenation*]{}, i.e., take the path obtained (after appropriate reparametrization) by traveling along the first and then along the second. There is also an inverse for each path–the path traveled in reverse direction. These operations turn the set of homotopy classes of closed paths (with a given base-point) into a group and it is an important topological invariant of any topological space. It was introduced by Poincaré and is called the [*first homotopy group*]{} or the *fundamental group* of the space, denoted by $\pi_1$. Thus, the property of $SO(n)$ stated above is written concisely as $\pi_1(SO(n)) \cong \bBZ/2\bBZ\equiv \Zz$.
This specific topological property in the case $n=3$ plays a fundamental role in our physical world. To the two homotopy classes of closed paths in $SO(3)$ correspond precisely two principally different types of elementary particles: bosons, with integer spin, and fermions, with half-integer spin, having very distinct physical properties. The difference can be traced to the fact that the complex (possibly multicomponent) wave function determining the quantum state of a boson is left unchanged by a rotation by $2\pi$ of the coordinate system while the same transformation multiplies the wave function of a fermion by $-1$. This is possible since only the modulus of the wave function has a direct physical meaning, so measurable quantities are left invariant under a full rotation by $2\pi$. However, as discovered by Pauli and Dirac, one needs to use wave functions having this (unexpected) transformation property for the correct description of particles with half-integer spin, such as the electron. The careful analysis showed ([@Wig; @Barg]) that the wave function has to transform properly only under transformations which are in a small neighborhood of the identity. A “large” transformation such as a rotation by $2\pi$ can be obtained as a product of small transformations, but the transformed wave function need not come back to itself–there may be a complex phase multiplying it. From continuity requirements it follows that if one takes a closed path in $SO(3)$ which is contractible, the end-point wave function must coincide with the initial one. Therefore, a rotation by $4\pi$ should bring back the wave function to its initial value and so the phase factor corresponding to a $2\pi$-rotation can only be $-1$. What we have just described is the idea of the so-called [*projective*]{} representations of a Lie group, which we have to use in quantum physics. As we see on the example of $SO(3)$, they exist because the latter is not simply-connected, i.e., $\pi_1(SO(3))$ is not trivial. Projective representations of a non-simply-connected Lie group are in fact representations of its covering group. In the case of $SO(3)$ this is the group $SU(2)$ of $2\times 2$ unitary matrices with unit determinant. Topologically, this is the three-dimensional sphere $S^3$; it is a double cover of $SO(3)$ and the two groups are locally isomorphic and have the same Lie algebra.
The standard proof that $\pi_1(SO(n)) \cong \bBZ/2\bBZ$ when $n=3$ uses substantially Lie theory. A $2-1$ homomorphism $SU(2)\rightarrow SO(3)$ is exhibited, which is a local isomorphism of Lie groups. This is the double covering map in question, sending any two antipodal points of $SU(2)$ (i.e., $S^3$) to one point in $SO(3)$. The case $n>3$ reduces to the above result by applying powerful techniques from homotopy theory.
There are several more or less easy geometric methods to unveil the nontrivial topology of $SO(3)$. Among them, a well-known demonstration is the so-called “Dirac’s belt trick” in which one end of a belt is fastened, the other (the buckle) is rotated by $4\pi$. Then without changing the orientation of the buckle, the belt is untwisted by passing it around the buckle (see, e.g., [@Egan; @Palais] for nice Java applets and explanations). A refinement of “Dirac’s belt trick” was proposed in [@Sto] where an isomorphism was constructed between homotopy classes of closed paths in $SO(3)$ and a certain factor group of the group $P_3$ of pure braids with three strands. This factor group turns out to be isomorphic to $\bBZ/2\bBZ$. The idea is fairly simple and is based on the following experiment: attach the ends of three strands to a ball, attach their other ends to the desk, perform an arbitrary number of full rotations of the ball to obtain a plaited braid. Then try to unplait it without further rotating the ball. As expected, braids that correspond to contractible paths in $SO(3)$ are trivial, while those corresponding to noncontractible paths form a single nontrivial class.
While the method of [@Sto] is simple and easy to visualize, it has the disadvantage that it does not lend itself to a generalization to higher dimensions. (A geometric braid in $\Rn$ is always trivial when $n>3$.) The present paper takes a different, more algebraic approach. We study a certain discrete (in fact finite) group of homotopy classes of paths in $SO(n)$, starting at the identity and ending at points which are elements of some fixed finite subgroup of $SO(n)$. It turns out that it is convenient to use the finite group of rotational symmetries of the hyperoctahedron (the polytope in dimension $n$ with vertices $\{(\pm 1,0,...0), (0,\pm 1,...,0),...,(0,0,...,\pm 1)\}$). Each homotopy class contains an element consisting of a chain of rotations by $\pi/2$ in different coordinate planes. These simple motions play the role of generators of our group. Certain closed paths obtained in this way remain in a small neighborhood of the identity (in an appropriate sense, explained later) and can be shown explicitly to be contractible. Thus, certain products among the generators must be set to the identity and we get a group defined by a set of generators and relations. Interestingly, the number of (independent) generators is $n-1$ and the relations, apart from one of them, are exactly Artin’s relations for the braid group $B_n$. In this way we obtain for each $n$ a finite group, which is the quotient of $B_n$ by the normal closure of the group generated by the additional relation. When $n=3$ the order of the group turns out to be 48 and it is the so-called [*binary octahedral group*]{} which is a nontrivial extension by $\bBZ/2\bBZ$ of the group of rotational symmetries of the octahedron. We may think of the former as a double cover of the latter and this is a finite version of the double cover $SU(2)\rightarrow SO(3)$. The next groups in the series have orders 384, 3840, 46080, etc.; in fact the order is given by $2^nn!$. Note that these groups have the same orders as the Coxeter groups of all symmetries (including reflections as well as rotations) of the respective hyperoctahedra, but they are different. This is analogous to the relationship between $O(n)$ and $SO(n)$ on the one hand, and $Spin(n)$ and $SO(n)$ on the other.
It turns out that the subgroup of homotopy classes of closed paths, i.e. $\pi_1(SO(n))$, in each case either coincides or lies in the center of the respective group and is isomorphic to $\bBZ/2\bBZ$. Factoring by it, the respective rotational hyperoctahedral groups are obtained. We believe that these are new presentations of all rotational hyperoctahedral groups and their double covers.\
Groups of Homotopy Classes of Paths.
====================================
We will consider paths in $SO(n)$ starting at the identity. In other words, we have continuous functions $R:[0,1]\rightarrow SO(n)$, subject to the restriction $R(0)=Id$. Because the target space is a group, there is a natural product of such paths, i.e. if $R_1$ and $R_2$ are paths of this type, we first translate $R_1$ by the constant element $R_2(1)$ to the path $R_1R_2(1)$. Then we concatenate $R_2$ with $R_1R_2(1)$. Thus, by $R_1R_2$ we mean the path:\
$$\text{$(R_1R_2)(t)$}=\begin{cases}
\text{$R_2(2t)$}, & \text{if $t<{1\over 2}$} \\
\text{$R_1(2t-1)R_2(1)$}, & \text{if $t\ge{1\over 2}$} \\
\end{cases}\nonumber$$ For each $R$ we denote by $R^{-1}$ the path given by $$R^{-1}(t)=(R(t))^{-1}\ .$$ The set formed by continuous paths in $SO(n)$ obviously contains an identity, which is the constant path. The product in this set is not associative since $(R_1R_2)R_3$ and $R_1(R_2R_3)$ are different paths (due to the way we parametrize them), even though they trace the same curve. Also, $R^{-1}$ is by no means the inverse of $R$. In fact, there are no inverses in this set. However, the set of homotopy classes of such paths (with fixed ends) is a group with respect to the induced operations. Further, we will use $R$ to denote the homotopy class of $R$. Now $R^{-1}$ is the inverse of $R$. Recall that in algebraic topology one constructs the universal covering space of a non-simply connected space by taking the homotopy classes of paths starting at some fixed point (base-point). Within a given homotopy class, what is important is just the end-point and so the covering space is locally homeomorphic to the initial one. However, paths with the same end-point belonging to different classes are different points in the covering space. Thus, one effectively “unwraps” the initial space. We see that the group we have defined is none other than the universal covering group of $SO(n)$. In the case $n=3$ this is the group $SU(2)$; in general it is the group denoted by $Spin(n)$.
Our aim is of course to show that the subgroup corresponding to closed paths is $\bBZ_2$ or equivalently that the covering map, which is obtained by taking the end-point of a representative path, is $2\rightarrow 1$. Since the full group of homotopy classes of paths starting at the identity is uncountable and difficult to handle, the main idea of this paper is to study a suitable discrete, in fact finite, subgroup by limiting the end-points to be elements of the group of rotational symmetries of the hyperoctahedron in dimension $n$. In what follows we denote this group by $G$. (One could perhaps take any large enough finite subgroup of $SO(n)$, but using the hyperoctahedral group seems the simplest.)
The further study of $G$ requires algebra and just a couple of easy results from analysis and geometry. We state these here and leave the proofs for the appendix.
By a [*generating path*]{} $R_{ij}$ we will mean a rotation from $0$ to $\pi/2$ in the coordinate plane $ij$.
Every homotopy class of paths in $SO(n)$ starting at the identity and ending at an element of the rotational hyperoctahedral group contains a representative which is a product of generating paths.
Each vertex of the hyperoctahedron lies on a coordinate axis—either in the positive or negative direction — and determines a closed half-space (of all points having the respective coordinate nonnegative or nonpositive) to which it belongs. Let us call [*local closed paths*]{} those closed paths in $SO(n)$ for which no vertex of the hyperoctahedron leaves the closed half-space to which it belongs initially.
Local closed paths are contractible. A closed path consisting of generating paths is contractible if and only if the word representing it can be reduced to the identity by inserting expressions describing local closed paths.
The case $\boldsymbol{n=3}$
===========================
We consider three-dimensional rotations separately since the essential algebraic properties of $G$ are present already here and in a sense the higher-dimensional cases are a straightforward generalization. It is worth recalling some facts about the group of rotational symmetries of the octahedron in three-dimensions. The octahedron is one of the five regular convex polyhedra, known as Platonic solids. It has six vertices and eight faces which are identical equilateral triangles. Each vertex is connected by an edge to all other vertices except the opposite one. We may assume that the vertices lie two by two on the three coordinate axes. We will enumerate the vertices from 1 to 6 as follows: $1=(1, 0, 0)$, $2=(0, 1, 0)$, $3=(0, 0, 1)$, $4=(0,0,-1)$, $5=(0,-1,0)$ and $6=(-1,0,0)$. The octahedron is the dual polyhedron of the cube and so they have the same symmetry group. The analogous statement is true in any dimension. We find it convenient to think of a spherical model of the octahedron — the edges connecting the vertices are parts of large circles on the unit sphere (Figure \[octah\]).\
-10mm ![A spherical octahedron with its three axes of rotational symmetries of order 4[]{data-label="octah"}](Fig2.pdf "fig:"){width="60mm"}
The octahedron has three types of rotational symmetries belonging to cyclic subgroups of different orders:\
*Type 1*: Three cyclic subgroups of order $4$ generated by rotations by $\pi/2$ around the three coordinate axes. Each such symmetry preserves the two vertices lying on that axis while permuting the rest.\
*Type 2*: Six cyclic subgroups of order $2$, generated by rotations by $\pi$ around axes connecting the centers of six pairs of opposite edges.\
*Type 3*: Four cyclic subgroups of order $3$ generated by rotations by $2\pi/3$ around axes connecting the centers of four pairs of opposite faces.\
Counting the number of nontrivial elements in each subgroup and adding $1$ shows that the group of rotational symmetries of the octahedron has order $24$. As each symmetry permutes the six vertices, it is obviously a subgroup of the symmetric group $S_6$. It is well-known and easy to see experimentally that the group of rotational symmetries of the octahedron is faithfully represented as the group of permutations of the four pairs of opposite faces. Therefore, it is isomorphic to $S_4$. All symmetric groups can be realized as finite reflection groups and thus finite Coxeter groups of type $A_n$ [@Humph]. We have $S_n\cong A_{n-1}$ and in the case at hand the group is $A_3$ — the full symmetry group (including reflections) of the tetrahedron. The case $n=3$ is an exception. When $n>3$ the respective rotational hyperoctahedral group is not a Coxeter group. For any $n$ however, it is a normal subgroup of index $2$ of the respective full hyperoctahedral group which is a Coxeter group of type $B_n$ (not to be confused with the braid group on $n$ strands, for which the same notation is used).
-10mm ![Three simple roots for $B_3$[]{data-label="roots"}](Fig4.pdf "fig:"){width="100mm"} -10mm
For pedagogical purposes we choose to make a short description of the full hyperoctahedral group as a symmetry group of the hyperoctahedron, which is generated by reflections. Starting with the octahedron in dimension 3, we see that it is preserved by reflections with respect to planes perpendicular to the vectors $(\pm1,0,0),\,(0,\pm1,0),\,(0,0,\pm1)$ (the six vertices) and $(\pm1,\pm1,0),\\\,(\pm1,0,\pm1),\,(0,\pm1,\pm1)$ (the middles of the twelve edges). The vectors just listed are called [*roots*]{} and they satisfy two essential properties which are axioms for a [*root system*]{}: 1) Each reflection determined by a root maps the root system onto itself, and 2) The root system contains together with each root its negative but no other multiples of that root. Among the 18 roots, we can choose (not uniquely) 3 roots, called [*simple roots*]{} with the following properties: 3) They span the whole space in which the root system lives, and 4) Each root is either entirely positive linear combination of the simple ones or is entirely negative linear combination of those. One possible choice of simple roots is (see Figure \[roots\]) $$\boldsymbol{\alpha}_1=(1,0,0),\quad\boldsymbol{\alpha}_2=(-1,1,0),\quad\boldsymbol{\alpha}_3=(0,-1,1)\,.$$ One can check that indeed every root is a linear combination of these three with either entirely positive or entirely negative coefficients. For example $(0,1,1)=2\boldsymbol{\alpha}_1+2\boldsymbol{\alpha}_2+\boldsymbol{\alpha}_3$. Notice that the angle between $\boldsymbol{\alpha}_1$ and $\boldsymbol{\alpha}_2$ is $135^{\circ}$ or $3\pi/4$, the angle between $\boldsymbol{\alpha}_2$ and $\boldsymbol{\alpha}_3$ is $120^{\circ}$ or $2\pi/3$, while the angle between $\boldsymbol{\alpha}_1$ and $\boldsymbol{\alpha}_3$ is $90^{\circ}$ or $\pi/2$. This is always the case — any two simple roots form either obtuse angle or are orthogonal. The angle between two roots is further restricted by the fact that the product of two reflections corresponding to two different roots is a rotation at an angle linked to the angle between the roots. More precisely, if the angle between $\boldsymbol{\alpha}_i$ and $\boldsymbol{\alpha}_j$ is $\pi-\theta$ and the corresponding reflections are denoted as $s_{\alpha_i}$ and $s_{\alpha_j}$, then $s_{\alpha_i}s_{\alpha_j}$ is a rotation by $2\theta$. Since this rotation must have some finite order $m$, we have $\theta={\pi\over m}$. The whole finite reflection group is determined by the positive integers $m(\boldsymbol{\alpha}_i,\boldsymbol{\alpha}_j)$ for each pair of simple roots. Note that because all $\boldsymbol{\alpha}_i$ are reflections and thus have order two, the diagonal entries $m(\boldsymbol{\alpha}_i,\boldsymbol{\alpha}_i)=1$ always. For the octahedron from the angles between simple roots we get $m(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2)=4$, $m(\boldsymbol{\alpha}_2,\boldsymbol{\alpha}_3)=3$, and $m(\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_3)=2$. The information is traditionally encoded in the so-called [*Coxeter – Dynkin diagram*]{} where each simple root is represented by a node, two nodes are connected by an edge if $m>2$ and $m$ is written as a label below the corresponding edge if $m>3$. The Coxeter – Dynkin diagram for the octahedral group is shown on Figure \[Dynkin\].
![The Coxeter-Dynkin diagrams for $B_3$ and $B_n$[]{data-label="Dynkin"}](B3.pdf "fig:"){width="30mm"}![The Coxeter-Dynkin diagrams for $B_3$ and $B_n$[]{data-label="Dynkin"}](Bn.pdf "fig:"){width="30mm"}
The generalization to higher $n$ is pretty straightforward. The roots will be all $2n$ vertices and the middles of all $2n(n-1)$ edges of the respective hyperoctahedron. It is easy to figure out that one can choose the following $n$ simple roots: $(1,0,...,0),\,(-1,1,0,...,0),...,(0,...,0,-1,1)$, and calculating the integers $m$, we see that the group has a Coxeter – Dynkin diagram as in Figure \[Dynkin\]. The groups of this type are denoted by $B_n$, where $n$ is the number of simple roots and is called the [*rank*]{}.
Alternatively, the full hyperoctahedral group can be thought of as the [*wreath product*]{} $S_2\wr S_n$. The wreath product in this special case is the semidirect product of the product of $n$ copies of $S_2$ with $S_n$, where $S_n$ acts on the first factor by permuting its components. More precisely, let $\Sigma=S_2\times S_2\times \cdots\times S_2$ and let $\sigma=(\sigma_1,\sigma_2,...,\sigma_n)\in \Sigma$. The symmetric group $S_n$, considered as a permutation group of $(1,2,...,n)$, acts naturally by automorphisms on $\Sigma$, namely, if $h\in S_n$ $$(h\sigma)_i=\sigma_{h^{-1}(i)}\,.$$ Now, the wreath product $S_2\wr S_n$ is just the semidirect product $\Sigma\rtimes S_n$. The $n$ pairs of opposite vertices determine $n$ mutually orthogonal (non-oriented) lines in $\Rn$. A rotation in the $ij$th plane by $\pi/2$ permutes the $i$th and $j$th lines. An arbitrary symmetry can be realized by an arbitrary permutation of the lines plus possible reflections with respect to the hyperplanes perpendicular to the $n$ lines. This explains the structure of the full hyperoctahedral group as a wreath product. One advantage is that it is easy to see that the order of the group is $2^nn!$.
The rotational hyperoctahedral group forms a normal subgroup of index 2 in the full hyperoctahedral group. In terms of orthogonal matrices, this is the subgroup of matrices with unit determinant. Note that each rotation is a product of two reflections. Coming back to the rotational octahedral group (i.e. the case $n=3$), we observe that it is generated by three elements, denoted by $r_1$, $r_2$ and $r_3$, of *Type 1* — rotations by $\pi/2$ around the three coordinate axes. They permute the vertices of the octahedron and using standard notations for permutations in terms of cycles (omitting the trivial 1-cycles) we can write: $$r_1= (2354),\quad r_2= (1463),\quad r_3= (1265).$$ Obviously, these three elements have order 4, and generate the respective cyclic groups of *Type 1* but they also generate all symmetries of *Type 2* and *Type 3*:
The six rotations by $\pi$ around the respective rotational axis (as described above) in terms of $r_1$, $r_2$ and $r_3$ are $r_2r_3r_1r_3^{\,2}$, $r_2r_3r_1$, $r_3r_1r_2r_1^{\,2}$, $r_3r_1r_2$, $r_1r_2r_3r_2^{\,2}$, and $r_1r_2r_3$.
The eight rotations by $\pm2\pi/3$ around the respective rotational axis (as described above) in terms of $r_1$, $r_2$ and $r_3$ are $r_1r_2$, $r_1r_3$, $r_1r_2^{\,3}$, $r_1r_3^{\,3}$, and their inverses.
We consider the discrete group $G$ generated by the generating paths $R_1:=R_{23}$, $R_2:=R_{31}$, and $R_3:=R_{12}$, treated as homotopy classes. (We have $R_i(1)=r_i,\ i=1,2,3$.) Local closed paths built out of the generators $R_i$ and their inverses must be set to identity. Apart from trivial cases where an $R_i$ is followed by its inverse, we have a family of paths for which each vertex either goes around the edges of a single triangular face of the octahedron or moves along an edge and comes back. Inspecting all possible ways in which such “triangular” closed paths can be built, we see that each one is represented by a word of four letters. Each word contains either $R_i$ or $R_i^{-1}$ for each $i$. No word contains twice a given letter but it may contain a letter together with its inverse. In that case this letter and its inverse conjugate one of the other two letters. Here is a list of all identities that follow: $$\begin{aligned}
\label{Id}
1&=&R_3^{-1}R_1^{-1}R_2^{-1}R_1=
R_3R_1^{-1}R_2R_1=
R_2R_3^{-1}R_2^{-1}R_1=
R_2^{-1}R_3R_2R_1\\\nonumber
&=&R_3^{-1}R_2^{-1}R_3R_1=
R_3R_2R_3^{-1}R_1=
R_2^{-1}R_1^{-1}R_3R_1=
R_2R_1^{-1}R_3^{-1}R_1\\\nonumber
&=&R_1R_3R_1^{-1}R_2=
R_1^{-1}R_3^{-1}R_1R_2=
R_3R_1^{-1}R_3^{-1}R_2=
R_3^{-1}R_1R_3R_2 \\\nonumber
&=&R_3^{-1}R_2^{-1}R_1R_2=
R_3R_2^{-1}R_1^{-1}R_2=
R_1R_2^{-1}R_3R_2=
R_1^{-1}R_2^{-1}R_3^{-1}R_2\\\nonumber
&=&R_1^{-1}R_2R_1R_3=
R_1R_2^{-1}R_1^{-1}R_3=
R_2R_3^{-1}R_1R_3=
R_2^{-1}R_3^{-1}R_1^{-1}R_3 \\\nonumber
&=&R_2^{-1}R_1^{-1}R_2R_3=
R_2R_1R_2^{-1}R_3=
R_1^{-1}R_3^{-1}R_2R_3=
R_1R_3^{-1}R_2^{-1}R_3. \\\nonumber\end{aligned}$$
![Visualization of the triangular closed path $R_3^{-1}R_1^{-1}R_2^{-1}R_1$ (the path traced by vertex $1$)[]{data-label="triang"}](Fig5.pdf){width="80mm"}
Very few of these 24 identities are actually independent. First, we notice that one of the generators, e.g. $R_3$ can be expressed as a combination of the other two and their inverses, in several different ways. For example, if we use the first identities in the last two rows we get $R_3=R_2^{-1}R_1R_2=R_1^{-1}R_2^{-1}R_1$, from which follows $$\label{Artin}
R_2R_1R_2=R_1R_2R_1,$$
while if we use $R_3=R_1^{-1}R_2^{-1}R_1=R_1R_2R_1^{-1}$, and also $R_3=R_2^{-1}R_1R_2=R_2R_1^{-1}R_2^{-1}$, we obtain: $$\label{add}
R_1^2=R_2R_1^2R_2,\quad\quad R_2^2=R_1R_2^2R_1.$$ All other identities are consequences of these three. Therefore, $G$ is presented as a group generated by two generators and a set of relations, one of which (Equation \[Artin\]) is precisely Artin’s braid relation for the braid group with three strands $B_3$. In other words $G$ is the quotient of $B_3$ by the normal closure of the subgroup generated by the additional relations (\[add\]).
Actually, only one of the identities (\[add\]) is independent:
\[Second\][The second identity in (\[add\]) follows from the first one and Artin’s braid relation (\[Artin\]).]{}
Using Artin’s relation twice it is almost immediate that $$R_2^{\, 2}=R_1R_2R_1^{\, 2}R_2^{-1}R_1^{-1}.$$ Now using the first of the identities (\[add\]) inside the expression above we get $$R_2^{\, 2}=R_1R_2R_1^{\, 2}R_2^{-1}R_1^{-1}=R_1R_2R_2R_1^{\, 2}R_2R_2^{-1}R_1^{-1}=R_1R_2^{\, 2}R_1.$$
-10mm ![Geometric Proof of Lemma \[Second\] that $R_2^2=R_1R_2^2R_1$ follows from $R_1^2=R_2R_1^2R_2$[]{data-label="braiding"}](Fig7.pdf "fig:"){width="80mm"}
[**Note:**]{} As shown by Artin [@Artin], the braid group $B_n$ can be thought as a group of isotopy classes of geometric braids with $n$ strands or as a group generated by $n-1$ generators satisfying what came to be called Artin’s braid relations. When $n=3$ there is just one relation (\[Artin\]) between the two generators. The geometric picture has the advantage of being more intuitive and providing us with ways to see identities, which can then be shown algebraically. Thus, for example, the proof above has a geometric version which is easy to visualize (Figure \[braiding\]). The same situation will be in place when we consider $n>3$. Our group $G$ will be generated by $n-1$ generators satisfying the standard braid relations plus some additional ones. Using the geometric picture will allow us to arrive at conclusions which are difficult to see algebraically.
[**Corollary:**]{} [*The group $G$ has a presentation*]{} $$\label{Biocta}
G=\left<R_1, R_2\,|\,R_1R_2R_1=R_2R_1R_2, R_1^{\, 2}=R_2R_1^{\, 2}R_2\right>\,.$$
\[R4\] The order of $R_1$ and $R_2$ in $G$ is eight.
Using relations (\[add\]) we conclude that $R_1^{\, 2}=R_2R_1^{\, 2}R_2=R_2^{\, 2}R_1^{\, 2}R_2^{\, 2}$ and similarly $R_2^{\, 2}=R_1^{\, 2}R_2^{\, 2}R_1^{\, 2}$. Therefore $$R_1^{\, 2}=R_2^{\, 2}R_1^{\, 2}(R_1^{\, 2}R_2^{\, 2}R_1^{\, 2}) \Longrightarrow
Id=R_2^{\, 2}R_1^{\, 4}R_2^{\, 2}\Longrightarrow
R_1^{4}=R_2^{-4}.$$ Next, one can write $$R_2^{\, 4}=R_2R_2^{\, 2}R_2=R_2R_1R_2^{\, 2}R_1R_2=R_2R_1R_2R_1R_2R_1=R_1R_2R_1^{\, 2}R_2R_1=R_1^{\, 4}.$$ Putting together the results in the last two equations we see that $R_1^{\, 4}=R_1^{-4}$ and similarly $R_2^{\, 4}=R_2^{-4}$, which imply $R_1^{\, 8}=R_2^{\, 8}=Id$.
From this we can tell that the order of $R_1$ and $R_2$ is at most 8. To conclude that it is exactly 8 (and not for example 4) is not a trivial problem. One way to do this is to perform the Todd – Coxeter algorithm (see e.g. Ken Brown’s short description [@Brown]). If the group defined by a finite set of generators and relations is finite, the algorithm will (in theory) close and stop and will produce the order of the group and a table for the action of the generators on all elements. We used the simple computer program graciously made available by Ken Brown on his site. The resulting table showed that the order of $G$ is 48 and the central element $R_1^{\, 4}=R_2^{\, 4}$ is not trivial. The results are represented graphically in the so-called Cayley graph (Figure \[Cayley\]) where dotted lines represent multiplication by $R_1$ and solid lines — by $R_2$ (The dotted lines wrap around horizontally).
![The Cayley graph of $G$[]{data-label="Cayley"}](Cayley48n.pdf){width="100mm"}
We know that taking the end-point of any element $R\in G$ gives a homomorphism from $G$ onto the group of rotational symmetries of the octahedron and the latter can be considered as a subgroup of the symmetric group $S_6$. So we have a homomorphism $\theta: G \rightarrow S_6$. In particular, $\theta(R_1)=(2354)$ and $\theta(R_2)=(1463)$. To prove that the fundamental group of $SO(3)$ is $\Zz$, we need to show that the kernel of the homomorphism described above is $\Zz$, because the kernel consists of those elements of $G$ which are classes of closed paths in $SO(3)$. In other words, these are the motions that bring the octahedron back to its original position. It will be helpful to come up with a way to list all elements of $G$ as words in the generators $R_1$, $R_2$ and their inverses, i.e., find a [*canonical form*]{} for the elements of $G$.
\[elements\] Any $x\in G$ can be written uniquely in one of the three forms:\
1. $R_1^{\, m}R_2^{\, n}$ (32 elements),\
2. $R_1^{\, m}R_2R_1$ (8 elements),\
3. $R_1^{\, m}R_2^{\, 3}R_1$ (8 elements),\
where $m\in \{0, 1, 2, 3, . . ., 7\}$ and $n\in \{0, 1 ,2, 3\}$.
Any element of $G$ can be obtained by multiplying the identity by a sequence of the generators and their inverses either on the right or on the left. We are using right multiplication. The idea is to show that when multiplying an element $x\in G$, which is written in the form 1, 2 or 3 by any of the two generators of $G$ or their inverses, we get again an expression of these three types. Since $R_i^{-1}=R_i^7$ it is enough to check the above for positive powers.
The proof is a direct verification using Artin’s braid relation and the identity $R_2^{\, 2}R_1=R_1^{-1}R_2^{\, 2}$ (and the symmetric one with $R_1$ and $R_2$ interchanged). Thus, e.g., multiplying the first expression by $R_1$ we get $R_1^{\, m\pm 1}R_2^{\, n}$, if $n$ is even and either expression 2 or 3, if $n$ is odd.
Similarly, multiplying expression 3 by $R_1$ we get
$$R_1^{\, m}R_2^{\, 3}R_1^{\, 2}=R_1^{\, m+2}R_2^{-3}=R_1^{\, m+2}R_2^{\, 5}=R_1^{\, m+6}R_2$$ while multiplying it by $R_2$ gives $$R_1^{\, m}R_2^{\, 3}R_1R_2=R_1^{\, m}R_2^{\, 2}R_2R_1R_2=R_1^{\, m}R_2^{\, 2}R_1R_2R_1
=R_1^{\, m-1}R_2^{\, 3}R_1\ .$$
Uniqueness is proven by inspection. Let us show as an example that $R_1^{\, m}R_2^{\, n}\ne R_1^{\, k}R_2R_1$. Indeed, the assumption that the two are equal leads to the following sequence of equivalent statements: $$\begin{aligned}
\nonumber
&R_1^{\, m-k}R_2^{\,n}=R_2R_1\Rightarrow R_1^{\, m-k+1}R_2^{\, n}=R_1R_2R_1=R_2R_1R_2\Rightarrow\\
& R_1^{\, m-k+1}R_2^{\, n-1}=R_2R_1\Rightarrow\dots\Rightarrow
R_1^{\, m-k+n}=R_2R_1\Rightarrow R_1^{\, m-k+n-1}=R_2.\\\end{aligned}$$ The last identity is apparently wrong since no power of $R_1$ can be equal to $R_2$.
The kernel of the homomorphism $\theta:G\rightarrow S_6$ is isomorphic to $\Zz$.
We study how the images under $\theta$ of the elements of $G$ permute the numbers $\{1,2,3,4,5,6\}$. First, $\theta(R_1)$ leaves $1$ in place, then $\theta(R_2)$ sends $1$ to $4$, while $\theta(R_2^{\, 3})$ sends $1$ to $3$ . Finally, recalling that $\theta(R_1)=(2354)$ we see that $1$ is not in the orbit of $4$ or $3$ under the action of $\theta(R_1)$, so there is no way that $\theta(R_1^{\, m}R_2R_1)$ or $\theta(R_1^{\, m}R_2^{\, 3}R_1)$ can bring $1$ back to itself. Next we look at $\theta(R_1^{\, m}R_2^{\, n})$. Recall that because $R_2^{\, 4}=R_1^{\, 4}$ and $R_1^{\, 8}=Id$ we have $n\in\{0,1,2 ,3\}$ and $m\in \{0,\dots,7\}$.Unless $\theta(R_2^{\, n})=Id$, which happens if and only if $n=0$, one of the numbers $\{3,4,6\}$ will be sent to $1$. Then, since $1$ is fixed by $\theta(R_1^{\, m})$, we see that $\theta(R_1^{\, m}R_2^{\, n})$ cannot be trivial unless $n= 0$. In that case it is obvious that $\theta(R_1^{\, m})$ can be the identity only when $m=0\,(\text{mod}\,4)$. Thus, $$\pi_1(SO(3))\cong \text{ker}(\theta)=\{Id, R_1^{\, 4}\}\cong \Zz.$$
Since $R_1^{\, 4}$ is a central element of order two, the group generated by it, $\{Id, R_1^{\, 4}\}$, belongs to the center $Z(G)$. It is not obvious a priori that there are no other central elements. We make the following observation: the center of the braid group $B_n$ is known to be isomorphic to $\Z$. It is generated by a full twist of all $n$ strands — $(\sigma_1\sigma_2\cdots\sigma_{n-1})^n$ (using $\sigma_i$ for the generators of $B_n$). Since $G$ is obtained from $B_n$ by imposing one more relation, any central element in $B_n$ will be central in $G$, although there is no guarantee that we will obtain a nontrivial element in $G$ in this way. In addition, there may be central elements in $G$ which come from non-central elements in $B_n$. When $n=3$ the above argument ensures that $(R_1R_2)^3$ is a central element in $G$. The calculation in Proposition \[R4\] shows that $$(R_1R_2)^3=R_1R_2R_1R_2R_1R_2=R_1^{\, 4}=R_2R_1R_2R_1R_2R_1=R_2^{\, 4},$$ so we do not get any new central element, different from the one we have already found. When $n=4$, however the analogous calculation, using the braid relations plus the additional relations as in equation \[add\], gives $$(R_1R_2R_3)^{\, 4}=R_1^{\, 2}R_3^{-2}.$$ This is another central element, different from $R_1^{\, 4}=R_2^{\, 4}=R_3^{\, 4}$. Taking the product of the two we obtain a third central element $R_1^{\, 2}R_3^{\, 2}$. Therefore when $n=4$, $Z(G)$ contains (in fact coincides with) the product of two copies of $\Zz$. This obviously generalizes to any even $n$ — the element $R_1^{\, 2}R_3^{\, 2}\cdots R_{n-1}^{\, 2}$ is central, as can be checked explicitly.
The difference between even and odd dimensions can be traced back to the difference between rotation groups in even and odd dimensions. As we shall see shortly, when we factor $G$ by $\text{ker}\,\theta$ we obtain the rotational hyperoctahedral group. In odd dimensions this has trivial center, which follows for example from Schur’s lemma and the fact that the matrix $-\mathbf{1}$ is not a rotation. However, in even dimensions reflection of all axes, given by $-\bold{1}$, is a rotation and therefore the corresponding rotational hyperoctahedral group has a nontrivial center.
According to the general construction, we expect that when we factorize $G$ by the kernel of the covering map, which is nothing but $\text{ker}\, \theta$, we should obtain the rotational octahedral group. This can also be established directly. Denoting by $G_1$ the quotient, we have a presentation for it: $$\label{S4}
G_1=\left<R_1, R_2 \,|\, R_1R_2R_1=R_2R_1R_2, R_1^{\, 2}=R_2R_1^{\, 2}R_2, R_1^{\, 4}=Id\right>.$$ The order of $G_1$ is 24. We can list its elements using the same expressions as in Proposition \[elements\], except that now the two integers $n$ and $m$ run from $0$ to $3$. An easy calculation shows that $G_1$ contains nine elements of order 2, eight elements of order 3, and six elements of order 4. Among all 15 classified groups of order 24, the only one that has this structure is the symmetric group $S_4$, which on the other hand is the group of the octahedral rotational symmetries. Given the structure of $G$ that can be visualized through its Cayley graph, we conclude that it is a nontrivial extension by $\Zz$ of $S_4$ described by a non-split short exact sequence $$1\longrightarrow \Zz\longrightarrow G\longrightarrow S_4\longrightarrow 1\ .$$ The non-isomorphic central extensions by $\Zz$ of $S_4$ are in one-to-one correspondence with the elements of the second cohomology group (for trivial group action) of $S_4$ with coefficients in $\Zz$ and the latter is known to be isomorphic to the Klein four-group $\Zz\times \Zz$. The identity in cohomology corresponds to the trivial extension $\Zz\times S_4$ (more generally, a semidirect product is also considered trivial). The other three elements of the cohomology group classify the three non-isomorphic nontrivial extensions, namely the binary octahedral group $2O$, the group $GL(2,3)$ of nonsingular $2\times 2$ matrices over the field with three elements, and the group $SL(2,4)$ of $2\times 2$ matrices with unit determinant over the ring of integers modulo 4.
\[2O\] $G$ is isomorphic to the binary octahedral group $2O$.
[**Note:**]{} By construction, we expect the group $G$ to be a subgroup of $SU(2)$ and the covering map $G\rightarrow S_4$ to be a restriction of the covering map $SU(2)\rightarrow SO(3)$. The fact that $G$ with the presentation (\[Biocta\]) is isomorphic to $2O$ is mentioned in Section 6.5 of [@Cox]. The GAP ID of $G$ is \[48, 28\].
The binary octahedral group, being a subgroup of $SU(2)$, can be realized as a group of unit quaternions. In fact it consists of the 24 Hurwitz units $$\{\pm1, \pm i, \pm j, \pm k, \frac{1}{2}(\pm 1 \pm i \pm j \pm k)\}$$ and the following 24 additional elements: $$\{\frac{1}{\sqrt{2}}(\pm 1 \pm i),\frac{1}{\sqrt{2}}(\pm 1 \pm j),\frac{1}{\sqrt{2}}(\pm 1\pm k), \frac{1}{\sqrt{2}}(\pm i \pm j),\frac{1}{\sqrt{2}}(\pm i\pm k),
\frac{1}{\sqrt{2}}(\pm j\pm k)\}$$ To prove the isomorphism it is enough to find two elements in $2O$ which generate the whole group and which satisfy the same relations as the defining relations of $G$. Obviously we are looking for elements of order 8. Let, e.g., $u_1=\frac{1}{\sqrt{2}}(1-k), u_2=\frac{1}{\sqrt{2}}(1-j)\in 2O$. It is a simple exercise in quaternion algebra to prove that $u_2u_1u_2=u_1u_2u_1$. Similarly, we check that $u_1^{\, 2}=u_2u_1^{\, 2}u_2$, or equivalently that $u_1^{\, 2}u_2=u_2^{-1}u_1^{\, 2}$. Indeed, since $u_1^{\, 2}=-k$ and since $k$ and $j$ anticommute, $$u_1^{\, 2}u_2=-k\frac{1}{\sqrt{2}}(1-j)=\frac{1}{\sqrt{2}}(1+j)(-k)=u_2^{-1}u_1^{\, 2} \ .$$ A direct verification further shows that the whole $2O$ is generated by $u_1$ and $u_2$.
The 24 Hurwitz units, when considered as points in $\bBR ^4$, lie on the unit sphere $S^3$ and are the vertices of a regular 4-polytope — the 24-cell, one of the exceptional regular polytopes with symmetry — the Coxeter group $F_4$. The set is also a subgroup of $2O$ — the binary tetrahedral group, denoted as $2T$. The second set of 24 unit quaternions can be thought of as the vertices of a second 24-cell, obtained from the first one by a rotation, given by multiplication of all Hurwitz units by a fixed element, e.g. ${1\over \sqrt{2}}(1+i)$. The convex hull of all 48 vertices is a 4-polytope, called disphenoidal 288-cell.
It may be instructive to consider the symmetric group $S_4$ with its presentation given by equation (\[S4\]) and try to construct explicitly all non-isomorphic central extensions by $\Zz$. This means that the three relations in the presentation of $S_4$ must now be satisfied up to a central element, belonging to the (multiplicative) cyclic group with two elements $\{1, -1\}$: $$R_1R_2R_1=aR_2R_1R_2,\quad R_1^{\, 2}=bR_2R_1^{\, 2}R_2,\quad R_1^{\, 4}=c,\quad a,b,c\in \Zz.$$ The element $a$ in the first relation can be absorbed by replacing $R_1$ by $aR_1$, so Artin’s braid relation remains unchanged. We are left with four choices for $b$ and $c$:\
1. The choice $b=c=1$ leads to the trivial extension as a direct product $\Zz\times S_4$.\
2. The choice $b=1, c=-1$ leads to the already familiar group $$G\cong 2O=\left<R_1, R_2\,|\,R_1R_2R_1=R_2R_1R_2, R_1^{\, 2}=R_2R_1^{\, 2}R_2\right>.$$ 3. The choice $b=c=-1$ leads to the group $GL(2,3)$ with presentation $$GL(2,3)=\left<R_1, R_2\,|\,R_1R_2R_1=R_2R_1R_2, R_1^{\, 2}=R_2R_1^{\, 6}R_2, R_2R_1^{\, 4}=R_1^{\, 4}R_2\right>.$$ 4. The choice $b=-1, c=1$ leads to the group $SL(2,4)$ with presentation $$SL(2,4)=\left<R_1, R_2,b\,|\,R_1R_2R_1=R_2R_1R_2, R_1^{\, 2}=bR_2R_1^{\, 2}R_2,R_1^{\, 4}=b^2=1\right>.$$ or, after simplifications, to $$SL(2,4)=\left<R_1, R_2\,|\,R_1R_2R_1=R_2R_1R_2, R_1^{\, 4}=(R_1R_2)^6=1\right>.$$ The proofs of points 3 and 4 above repeat the logic of the proof of Proposition \[2O\]. For the group $GL(2,3)$ we can make the following identifications $$R_1=\begin{pmatrix}
1&1\\
1&0
\end{pmatrix},
R_2=\begin{pmatrix}
1&2\\
2&0
\end{pmatrix}\in GL(2,3)$$ and then check that they generate the whole $GL(2,3)$ and satisfy the respective relations. Similarly, for $SL(2,4)$ we can set $$R_1=\begin{pmatrix}
1&0\\
1&1
\end{pmatrix},
R_2=\begin{pmatrix}
3&3\\
0&3
\end{pmatrix}
\in SL(2,4).$$ Notice that there is an essential difference between the extensions described in 2 and 3, and the extension in 4. The extensions in 2 and 3 have presentations as the presentation of $S_4$ (equation \[S4\]) with the same set of generators and some relation removed. This is not the case with $SL(2,4)$ where, if we want to keep the form of the relations, we need three generators. Notice also that for $2O$ and $GL(2,3)$ the order of $R_1$ and $R_2$ becomes 8 (it is 4 in $S_4$), while in $SL(2,4)$ the order of $R_1$ and $R_2$ remains 4. (It may be worth pointing out that the properties of the braid group $B_n$ ensure that in any of its factors the order of any two (standard) generators has to be the same.)
![The Cayley graph of $S_4$[]{data-label="Cayley24"}](Cayley24new.pdf){width="50mm"}
The groups $2O$ and $GL(2,3)$ are so-called [*stem extensions*]{} of $S_4$, i.e., the abelian group by which we extend is not only contained in the center of the extended group, but also in its commutator subgroup. One can check explicitly that for $2O$ and $GL(2,3)$ the corresponding central element $R_1^{\, 4}=R_2^{\, 4}$ can be written as a commutator, while in the case $SL(2,4)$ the center is generated by $(R_1R_2)^3$ and it is not in the commutator subgroup. If one looks at the Cayley graph of $2O$ (Figure \[Cayley\]), one can see that it really looks like a (topological) double cover of the corresponding Cayley graph of $S_4$ (Figure \[Cayley24\] ). For this reason stem extensions are called covering groups and are discrete versions of covering groups of Lie groups. It is intriguing that while $SO(3)$ has $SU(2)$ as its unique (double) cover, the finite subgroup $S_4$ has one additional double cover, namely $GL(2,3)$, which does not come from lifting $SO(3)$ to $SU(2)$.
Finally, we may notice that in the presentations of $2O$ and $GL(2,3)$ there is no condition imposed on the order of the central element. Its order comes out to be 2 automatically, which means that $S_4$ does not admit bigger stem extensions and the two groups considered are maximal stem extensions, i.e., [*Schur extensions*]{}.
Generalization to arbitrary $\boldsymbol{n}$
============================================
The $n$-dimensional case is an easy generalization of the three-dimensional one. We consider products of generating paths $R_{ij}$ in $SO(n)$ as defined in Section 2. If we take a closed path of the form $R_{ij}R_{kl}R_{ij}^{-1}R_{kl}^{-1}$, where all four indices are different, it is clear that this is contractible as these are rotations in two separate planes and at the level of homotopy classes the generators $R_{ij}$ and $R_{kl}$ will commute. When one of the indices coincides, we have a motion that takes place in a 3-dimensional subspace of $\Rn$ and we may use the algebraic relations we had in the previous section. In particular, if we consider the elements $R_{ij}$, $R_{jk}$ and $R_{ki}$, we can express one, e.g. the third, in terms of the other two, just as we expressed $R_3=R_{12}$ as the conjugation of $R_2=R_{31}$ by $R_1=R_{23}$. At this point, it seems convenient to choose a different notation, where $R_1:=R_{12},
R_2:=R_{23},\dots, R_{n-1}:=R_{n-1\,n}$. With this notation we have $R_{31}=R_{13}^{-1}=R_1R_2R_1^{-1}$, $R_{41}=R_{13}R_{34}R_{13}^{-1}=R_1R_2^{-1}R_1^{-1}R_3R_1R_2R_1^{-1}$, etc. In this way, all $R_{ij}$ are products of the $n-1$ generators $R_i$ and their inverses. We have $R_iR_j=R_jR_i$ when $|i-j|\ge 2$. Further, since local closed paths that correspond to rotations in any 3-dimensional subspace must be set to identity when passing to homotopy classes, we have identities analogous to the ones in equation \[Id\] for any two generators $R_i$ and $R_{i+1}$. In particular, Artin’s braid relation is satisfied for any $i=1,\dots , n-2$: $$R_iR_{i+1}R_i=R_{i+1}R_iR_{i+1}.$$ The other relation must also hold: $$R_iR_{i+1}^{\, 2}R_i=R_{i+1}^{\, 2}.$$ The fact that there are no additional relations follows from the observation that any contractible closed path in $SO(n)$ which is a product of generating paths can be written as a product of “triangular” local closed paths of the type described in Section 3 (see Appendix).
The properties of the braid group lead to some interesting restrictions on the type of additional relations that can be imposed on the generators. In particular, if $P(R_1,R_2)=1$ is some relation involving the first two generators and their inverses, then it follows that $P(R_i,R_{i+1})=1$ (translation) and $P(R_{i+1},R_i)=1$ (symmetry) will be satisfied automatically. These have a simple geometric explanation. For example, the first relation involves the first three strands of the braid, so if we want to prove the relation for $R_2$ and $R_3$, we flip the first strand over the next three, apply the relation and then flip back the former first strand to the first place. The symmetry property follows from the first and the fact that the braid group has an outer automorphism $R_i\rightarrow R_{n-i}$ which correspond to “looking at the same braid from behind.” It is also clear that the order of all generators in the factor group will be the same. Therefore, for any $n$ the group $G$ generated by the generating paths, up to homotopy, has presentation
$$\label{n-dim}
G=\left< R_1,\dots,R_{n-1}\,|R_iR_{i+1}R_i=R_{i+1}R_iR_{i+1}; \\
R_iR_j=R_jR_i, |i-j|\ge 2; R_1^{\, 2}=
R_2R_1^{\, 2}R_2\right>.$$
The group $G$ for arbitrary $n$ has many features in common with the case $n=3$. In particular, all $R_i$ have order $8$ and $R_1^{\, 4}=R_2^{\, 4}=\cdots=R_{n-1}^{\, 4}$ is central. The Todd-Coxeter algorithm when run on the computer gives the following results for the order of $G$ when $n=3, 4, 5, 6$ respectively — 48, 384, 3840, 46080. In fact, we have $|G|=2^nn!$, which will be shown next. As mentioned already in Section 3, the full hyperoctahedral group in dimension $n$, which is the Coxeter group $B_n$, has the same order, but $G$ is a different group — it is a non-trivial double cover of the orientation-preserving subgroup of the full hyperoctahedral group.
A canonical form of the elements of $G$ can be defined inductively as follows:\
Let $x_{(i)}\in G$ denote a word which contains no $R_j$ with $j>i$. Then we will say that $x_{(i)}$ is in canonical form if it is written as $x_{(i)}=x_{(i-1)}y_{(i)}$ with $x_{(i-1)}$ being in canonical form and $y_{(i)}$ being an expression of one of the types:\
1. $R_i^{\, k},\ k\in\{0,1,2,3\}$,\
2. $R_iR_{i-1}\cdots R_{i-j},\ j\in\{1,\dots,i-1\}$,\
3. $R_i^{\, 3}R_{i-1}\cdots R_{i-j},\ j\in\{0,\dots,i-1\}$,\
As an example, let us list all elements of $G$ in the case $n=4$, by multiplying all canonical expressions containing $R_1$ and $R_2$ (see Proposition \[elements\]) with all the expressions as above, with $i=3$. We have $R_1^{\, m}R_2^{\, n}R_3^{\, k}$ (128 elements), $R_1^{\, m}R_2^{\, n}R_3R_2$ (32 elements), $R_1^{\, m}R_2^{\, n}R_3R_2R_1$ (32 elements), $R_1^{\, m}R_2^{\, n}R_3^{\, 3}R_2$ (32 elements), $R_1^{\, m}R_2^{\, n}R_3^{\, 3}R_2R_1$ (32 elements), $R_1^{\, m}R_2R_1R_3^{\, k}$ (32 elements), $R_1^{\, m}R_2R_1R_3R_2$ (8 elements), $R_1^{\, m}R_2R_1R_3R_2R_1$ (8 elements), $R_1^{\, m}R_2R_1R_3^{\, 3}R_2$ (8 elements), $R_1^{\, m}R_2R_1R_3^{\, 3}R_2R_1$ (8 elements),\
$R_1^{\, m}R_2^{\, 3}R_1R_3^{\, k}$ (32 elements), $R_1^{\, m}R_2^{\, 3}R_1R_3R_2$ (8 elements), $R_1^{\, m}R_2^{\, 3}R_1R_3R_2R_1$ (8 elements), $R_1^{\, m}R_2^{\, 3}R_1R_3^{\, 3}R_2$ (8 elements), $R_1^{\, m}R_2^{\, 3}R_1R_3^{\, 3}R_2R_1$ (8 elements). The following is a straightforward generalization of Proposition \[elements\].
Any element of $G$ can be written uniquely in the canonical form defined above. The number of elements is $2^nn!$.
The idea is to show that by multiplying an element in canonical form on the right by any $R_j$, one gets another element that can be brought to a canonical form as well. There is nothing conceptually different from the proof of Proposition \[elements\] and we skip the details. In order to calculate the order of $G$, we notice that the elements in canonical form of type $x_{(i)}$ are obtained by all possible products of all elements of type $x_{(i-1)}$ with the $4+2(i-1) =2(i+1)$ different expressions of type $y_{(i)}$. Starting with $i=2$ where we have $48=2^33!$ and remembering that $i$ runs from $1$ to $n-1$ we get $$|G|=2^3\cdot 3!\cdot 2\cdot 4\cdot 2 \cdot 5\cdots 2\cdot n=2^n n!.$$
The group $G$ consists of homotopy classes of paths in $SO(n)$ starting at the identity and ending at an element of $SO(n)$ which is a rotational symmetry of the hyperoctahedron in $n$ dimensions (also called $n$-orthoplex or $n$-cross polytope). In particular we have a homomorphism $\theta :G\rightarrow S_{2n}$ because we have a permutation of the $2n$ vertices of the hyperoctahedron (we take left action of the group on the set). Since the fundamental group of $SO(n)$ consists of the homotopy classes of closed paths, we investigate the kernel of $\theta$.
$$\pi_1(SO(n))\cong \text{ker}\theta=\{Id, R_1^{\, 4}\}\cong \Zz.$$
We choose to enumerate the $2n$ vertices of the hyperoctahedron as $\{1,-1,2,$\
$-2,\dots n,-n\}$ taking $\pm 1$ to denote the two opposite vertices on the first axis, $\pm 2$ on the second axis, etc. Then the element $\theta(R_i)$ permutes cyclically only the elements $\{i, i+1,-i,-(i+1)\}$ leaving the rest in place. Consider an element $x_{(n-1)}=x_{(n-2)}y_{(n-1)}\in G$ in canonical form. By looking at the three possible expressions for $y_{(n-1)}$ we see that the leftmost letter is $R_{n-1}$ to some power, possibly preceded (on the right) by a sequence of $R_i$s with decreasing $i$. Since in the word $y_{(n-1)}$ only $\theta(R_{n-1})$ moves $n$ and $-n$, we have $\theta(y_{(n-1)})(n)\ne n$, unless $y_{(n-1)}=1$. Now, because $x_{(n-2)}$ does not contain $R_{n-1}$, the number $n$ is not the image of any number $k\ne n$ under the action of $\theta(x_{(n-2)})$. Therefore we have $\theta(x_{(n-1)})(n)\ne n$, unless $x_{(n-1)}=x_{(n-2)}$. Proceeding in this way we see that $x\in ker \,\theta$ if and only if $x=R_1^{\, m}$. Finally, as $\theta(R_1)=(1\ 2-1-2)$ (cyclic permutation of $\{1,2,-1,-2\}$; the rest fixed), the only possible cases are when $m=0, 4$. Therefore, $R_1^{\, 4}$ is the only nontrivial element of $ker\,\theta$ and it has order $2$.
We obtained a series of finite groups $G$ from the braid groups $B_n$ by imposing one additional relation, namely $R_1^{\, 2}=R_2R_1^{\, 2}R_2$. These groups are nontrivial double covers of the corresponding rotational hyperoctahedral groups and have order $2^nn!$. It is quite obvious that we obtain a second, nonisomorphic series of double covers if we impose the relations $R_1^{\, 2}=
R_2R_1^{\, 6}R_2$ and $R_1^{\, 4}R_2=R_2R_1^{\, 4}$, instead. Note that for the second series we need the additional condition, which then implies that $R_1^{\, 4}=R_2^{\, 4}=\cdots=R_{n-1}^{\, 4}$ is central. When $n=3$ the two groups are the two Schur extensions of the base group. This is perhaps the case also for arbitrary $n$.
Appendix
========
[*Proof of Lemmas 1 and 2.*]{} It is helpful to introduce a function, measuring the (square of a) “distance” between two points on $SO(n)$. Let $X,\ Y\in SO(n)$ be written as $n\times n$ orthogonal matrices and define $$D(X,Y){\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\operatorname{Tr}({\mathbf 1}- X^TY)\ .$$ It is a simple exercise to show that this is a positive-definite symmetric function on $SO(n)\times SO(n)$. Notice that $(X^TY)_{ii}$ is the cosine of the angle between the image of the standard coordinate basis vector $\mathbf e_i$ under the action of $X$ and its image under the action of $Y$. The function $D$ does not satisfy the triangle inequality and is not a true distance, but this causes no difficulties in our considerations.
Further we write the proof for $SO(3)$ for brevity. It is obvious that the same method works in general. Let $X\in SO(3)$ be written as a $3\times 3$ orthogonal matrix: $$X=\begin{pmatrix}
x_{11}&x_{12}&x_{13}\\
x_{21}&x_{22}&x_{23}\\
x_{31}&x_{32}&x_{33}
\end{pmatrix}
\equiv\begin{pmatrix}
{\mathbf x_1}&{\mathbf x_2}&{\mathbf x_3}
\end{pmatrix}.$$ The “distance” of $X$ to the identity (writing just one argument) is : $$D(X)=3-x_{11}-x_{22}-x_{33}.$$
Thus $D(X)\ge 0$ and $D(X)=0$ implies $X=\mathbf 1$. The gradient of $D(X)$ is a 9-dimensional vector field, which we choose to write as a $3\times 3$ matrix. We have $${\rm grad}\, D(X)=
\begin{pmatrix}
{\mathbf e_1}&{\mathbf e_2}&{\mathbf e_3}
\end{pmatrix}.$$ $SO(3)$ is a 3-dimensional submanifold of $\bBR^9$ consisting of the points satisfying six algebraic equations, ensuring that the $3\times 3$ matrix $X$ is orthogonal. In vector form these are equivalent to the statement that the (column) vectors $\{{\mathbf x_1},{\mathbf x_2},{\mathbf x_3}\}$ form an orthonormal basis:
$${\mathbf x_1}\cdot {\mathbf x_1}=1,\quad {\mathbf x_2}\cdot {\mathbf x_2}=1,\quad
{\mathbf x_3}\cdot {\mathbf x_3}=1,\quad {\mathbf x_1}\cdot {\mathbf x_2}=0,\quad
{\mathbf x_1}\cdot {\mathbf x_3}=0,\quad {\mathbf x_2}\cdot {\mathbf x_3}=0.$$
The respective gradients of these six functions are $$\begin{pmatrix}
{\mathbf x_1}&{\mathbf 0}&{\mathbf 0}
\end{pmatrix}, \quad
\begin{pmatrix}
{\mathbf 0}&{\mathbf x_2}&{\mathbf 0}
\end{pmatrix}, \quad
\begin{pmatrix}
{\mathbf 0}&{\mathbf 0}&{\mathbf x_3}
\end{pmatrix}, \quad$$ $$\begin{pmatrix}
{\mathbf x_2}&{\mathbf x_1}&{\mathbf 0}
\end{pmatrix},\quad
\begin{pmatrix}
{\mathbf x_3}&{\mathbf 0}&{\mathbf x_1}
\end{pmatrix},\quad
\begin{pmatrix}
{\mathbf 0}&{\mathbf x_3}&{\mathbf x_2}
\end{pmatrix}.$$ It is well known and an easy exercise that for each $X\in SO(3)$ these six vectors are linearly independent and their orthogonal complement is precisely the tangent space of $SO(3)$. (This is in fact how it is shown that these six equations in $\bBR^9$ define indeed a three-dimensional submanifold.) We want to show that under our assumptions ${\rm grad}\, D(X)$ has a nonzero tangential component. Indeed, supposing that ${\rm grad}\, D(X)$ is in the span of the six vectors above leads to $$\begin{aligned}
\nonumber
\begin{pmatrix}
{\mathbf e_1}&{\mathbf e_2}&{\mathbf e_3}
\end{pmatrix}&=&
a\begin{pmatrix}
{\mathbf x_1}&{\mathbf 0}&{\mathbf 0}
\end{pmatrix}+
b\begin{pmatrix}
{\mathbf 0}&{\mathbf x_2}&{\mathbf 0}
\end{pmatrix}+
c\begin{pmatrix}
{\mathbf 0}&{\mathbf 0}&{\mathbf x_3}
\end{pmatrix}\\\nonumber
&+&
d\begin{pmatrix}
{\mathbf x_2}&{\mathbf x_1}&{\mathbf 0}
\end{pmatrix}
+
e\begin{pmatrix}
{\mathbf x_3}&{\mathbf 0}&{\mathbf x_1}
\end{pmatrix}
+
f\begin{pmatrix}
{\mathbf 0}&{\mathbf x_3}&{\mathbf x_2}
\end{pmatrix},\\\nonumber\end{aligned}$$ which in turn is equivalent to the three vector equations $${\mathbf e_1}=a{\mathbf x_1}+d{\mathbf x_2}+e{\mathbf x_3},\quad
{\mathbf e_2}=b{\mathbf x_2}+d{\mathbf x_1}+f{\mathbf x_3},\quad
{\mathbf e_3}=c{\mathbf x_3}+e{\mathbf x_1}+f{\mathbf x_2}.$$ Taking the scalar product of the first equation with ${\mathbf x_2}$ yields $d=x_{12}$, while taking the scalar product of the second equation with ${\mathbf x_1}$ yields $d=x_{21}$. In a similar way we see that $x_{ij}=x_{ji}$ for any $i,j$. Therefore the matrix $X$ must be symmetric and being also orthogonal its square is the identity. The eigenvalues can only be $1$ and $-1$ but the latter is excluded by the assumption that under the transformation corresponding to $X$ the coordinate axes do not leave the closed half-space they belong to initially.
Thus that the tangential component of $-{\rm grad}\, D(X)$ defines a vector field on $SO(3)$ which will be nonzero for any $X=R(t)$ where $R:[0,1]\rightarrow SO(3)$ is a local closed path. This means that the flow along this vector field defines a homotopy from $R(t)$ to the identity of $SO(3)$. This proves the first part of Lemma 2.
Let us denote by $G'\subset SO(n)$ the respective rotational hyperoctahedral group in dimension $n$. Taking an arbitrary path $R: [0,1]\rightarrow SO(n)$ with $R(0)=Id$ and $R(1)=r\in G'$, we want to construct another path, homotopic to the first one, which is a product of generating paths $R_i$. We can proceed as follows: If $t_1\in [0,1]$ the smallest $t$ for which the “distance” from $R(t)$ to some $r_1\in G'$ becomes equal to the “distance” to $Id$, we take a product of generating paths $R_{k_1}^{(1)}\cdots R_1^{(1)}$ with $(R_{k_1}^{(1)}\cdots R_1^{(1)})(0)=Id$ and $(R_{k_1}^{(1)}\cdots R_1^{(1)})(t_1)=r_1$. Note that the transformation $(R_{k_1}^{(1)}\cdots R_1^{(1)})(t)$ leaves every vertex of the hyperoctahedron in the closed half-space determined by it (as defined in Section 2) or, equivalently, the angle between the standard basis vector ${\mathbf e_i}$ and its image under $(R_{k_1}^{(1)}\cdots R_1^{(1)})(t)$ does not exceed $\pi/2$ for any $i$ and any $t$. Indeed, as we consider a continuous path in $SO(n)$ starting at the identity, we may think of the motion of the $n$ points on the $(n-1)$–dimensional sphere (the images of the vectors ${\mathbf e_i}$ under $R(t)$). Initially all points remain in some respective adjacent $(n-1)$-cells (these are $(n-1)$–simplices (spherical)) forming the spherical hyperoctahedron. If the point $i$ is to leave the closed half-space to which it initially belonged, it must reach the $(n-2)$-dimensional boundary opposite to ${\mathbf e_i}$ for some $t$. (At the same time there will be at least one more point $i'$ belonging to a boundary of an $(n-1)$-cell since two points cannot belong to the interior of the same $(n-1)$–cell). But then it follows that there will be a vertex $j$ the angular distance to which, from $i$ is less than or equal to $\pi/2$. This implies that the element $r_{ij}\in G'$ giving rotation by $\pi/2$ in the $ij$th plane is not further to $R(t)$ than the “distance” between $R(t)$ and the identity. The above argument shows that even though the element $r_1$ above does not determine the product $R_{k_1}^{(1)}\cdots R_1^{(1)}$ uniquely, it is unique up to homotopy.
Next, we take $t_2$ as the smallest $t\ge t_1$ for which the “distance” from $R(t)$ to some $r_2$ becomes equal to the “distance” to $r_1$. There is an element $r_2'$ in $G'$, such that $r_2=r_2'r_1$ and $r_2'$ satisfies the same property as $r_1$ above. We take a product of generating paths $R_{k_2}^{(2)}\cdots R_1^{(2)}$ with $(R_{k_2}^{(2)}\cdots R_1^{(2)})(t_1)=Id$ and $(R_{k_2}^{(2)}\cdots R_1^{(2)})(t_2)=r_2'$. Proceeding in this way and taking the product of products of generating paths we produce a path (after renumbering) $R'{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}R_k\cdots R_1$ with $(R_k\cdots R_1)(0)=Id$ and $(R_k\cdots R_1)(1)=r$. The elements $r_i$ which we have to use at each step may not be unique and we will have to make a choice but the end result will lead to homotopic paths. To show that the path we have constructed is homotopic to the original path $R$ we can use the following trick — for each fixed $t$ take $R''(t){\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}R'(t)R^{-1}(t)$. The path $R''$ is closed and is local by the construction of $R'$. Therefore it is homotopic to the identity by the first part of Lemma 2. With this Lemma 1 is proven.
Finally, we need to show that the expression corresponding to any contractible closed path consisting of generating paths can be reduced to the identity by inserting in it words giving local closed paths. More precisely, the words that must be inserted give triangular closed paths as described in Section 3. This is essential as we must be sure that there are no additional relations other than the ones as in Equations \[Artin\] and \[add\]. First we carry out the proof for local closed paths consisting of generating paths (these are contractible because of locality) by induction on the length of the word representing the path. A single-letter path $R_{ij}$ cannot be closed. A two-letter path can only be closed if the word is $R_{ji}R_{ij}$. A three-letter path cannot be closed as can easily be seen considering all possibilities. For example, if we want to try to close the path starting with $R_{23}R_{12}$, we have to add $R_{31}$ on the right so that vertex 1 goes back to 1 but $R_{31}R_{23}R_{12}$ is not closed since 2 goes to -3 and 3 goes to -2. (We adopt enumeration of the $2n$ vertices with $\{1,2,\dots,n,-1,-2,\dots,-n\}$ where $-i$ denotes the vertex opposite to $i$. The element $R_{ij}$ moves vertex $i$ to vertex $j$, vertex $j$ to vertex $-i$ and leaves all other vertices in place.) Considering four-letter paths, it is easy to check that they can be closed either if they consist of a product of two closed two-letter paths or if they involve only rotations in a three-dimensional subspace spanned by some three axes $i$, $j$ and $k$. These are precisely what we called triangular closed paths in Section 3. In general notations it turns out that all such closed paths are words that are cyclic permutations of the following four expressions: $$\label{triang}
R_{kj}R_{ki}R_{jk}R_{ij},\quad R_{jk}R_{ik}R_{kj}R_{ij},\quad R_{ki}R_{jk}R_{ik}R_{ij},\quad
R_{ik}R_{kj}R_{ki}R_{ij}.$$ Notice that setting these expressions to one gives conjugation identities, e.g. $R_{kj}R_{ik}R_{jk}$\
$=R_{ij}$, etc. Suppose now that the statement we want to prove is valid for all words with length $2n$ and consider a word of $2n+2$ letters. Suppose that the first letter on the right is $R_{ij}$. We can move $R_{ij}$ to the left across any $R_{kl}$ with $k$, $l$ different from $i$ and $j$, as rotations in two such planes commute. If $R_{ij}$ gets next to a word $R_{ji}$ we are allowed to cancel the two and obtain a word of length $2n$ and we are done. The possibility to obtain $R_{ij}R_{ij}$ is excluded by locality since anything to the right leaves vertex $i$ invariant and $R_{ij}^2$ sends $i$ to $-i$. There are four additional possibilities in which $R_{ij}$ gets next to a letter with which it does not commute. These are $R_{jk}R_{ij}$, $R_{kj}R_{ij}$, $R_{ik}R_{ij}$ and $R_{ki}R_{ij}$. Using the identities following from setting the expressions in Equation (\[triang\]) to one, we replace the above combinations by products of two letters in which the index $i$ appears only in the letter on the left. Here are the actual identities that can be used: $$R_{jk}R_{ij}=R_{ik}R_{jk},\quad R_{kj}R_{ij}=R_{ki}R_{kj},\quad R_{ik}R_{ij}=R_{ij}R_{kj},\quad
R_{ki}R_{ij}=R_{ij}R_{jk}.$$ Next, we continue moving the letter involving $i$ to the left. Notice that the index $i$ may now appear as the second index, so we may need to use the additional identities $$R_{jk}R_{ji}=R_{ki}R_{jk},\quad R_{kj}R_{ji}=R_{ik}R_{kj},\quad R_{ik}R_{ji}=R_{ji}R_{jk},\quad
R_{ki}R_{ji}=R_{ji}R_{kj}.$$ Eventually the letter involving the index $i$ will reach the left end of the word and there will be no letters involving $i$ to its right. Such a path cannot be closed as it does not leave vertex $i$ in place. Therefore, the only possibility is that the letter involving $i$ gets canceled in the process and the length of the word becomes $2n$.
Now consider a general closed path $R$ consisting of generating paths and homotopic to the identity. In general the homotopy from the identity to $R$ need not pass through paths consisting of generating paths but we can modify it so that it does. Indeed, if $R(s)$ is the homotopy from the identity to $R$, i.e. $R(0)\equiv Id$, $R(1)=R$, for each $s$ we can homotope $R(s)$ to the nearest closed path consisting of generating paths, as described earlier in this section. For $s$ small enough the nearest will be $Id$. For some $s_1$ the construction will yield a path $R'$ consisting of generating paths, such that $R'$ is homotopic to $R(s_1)$. Furthermore, the construction is such that $R'$ is local closed path as any vertex remains in some $(n-1)$-cell without leaving it. Then for some $s_2>s_1$ we will get $R''$ which is nearest to $R(s_2)$. The path $R''$ is not local but $R'^{-1}R''$ (usual product of paths by concatenation) is local. Since the words giving $R'$ and $R'^{-1}R''$ can be reduced to the identity by inserting words giving triangular closed paths, the same will be true for $R''$. Proceeding in this way, after a finite number of steps we show that the word for the closed path $R$ has the same property.
Acknowledgement\
We would like to thank Tatiana Gateva-Ivanova for some helpful advice and suggestions.
[9]{} E. Artin, Theory of braids, [*Math. Ann.*]{} [**48**]{} (1) (1947) 101–126. V. Bargmann, On Unitary Ray Representations of Continuous Groups, [*Math. Ann.*]{} [**59**]{} no. 1 (1954) 1–46. K. Brown, [*The Todd-Coxeter Procedure*]{}, Lecture, Cornell Univ., NY, 2013 <http://www.math.cornell.edu/~kbrown/7350/toddcox.pdf>. H. S. M. Coxeter, W. O. J. Moser, [*G*enerators and relations for discrete groups]{}. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 14, Springer-Verlag, 1980. G.Egan, Applets Gallery (2000) <http://gregegan.customer.netspace.net.au/APPLETS/21/21.html>. J. Humphreys, *Reflection groups and Coxeter groups*, Vol. 29, Cambridge University Press, 1992. B. Palais, Quicktime Movies of Bob Palais demonstrating the Belt Trick and the Plate Trick, <http://www.math.utah.edu/~palais/links.html>. V. Stojanoska, O. Stoytchev, Touching the $\Zz$ in Three-Dimensional Rotations, *Mathematics Magazine* [**81**]{} no. 5 (2008) 345–357 E. Wigner, On Unitary Representations of the Inhomogeneous Lorentz Group, *Math. Ann.* [**40**]{} no. 1 (1939) 149–204.
[^1]: American University in Bulgaria, 2700 Blagoevgrad, Bulgaria; current affiliation: Drexel University, Philadelphia, PA 19104.
[^2]: American University in Bulgaria, 2700 Blagoevgrad, Bulgaria.
|
---
author:
- '**[ Nabil L. Youssef$^{\,1}$ and S. G. Elgendi$^{2,3}$]{}**'
title: '**[Nullity distributions associated with Chern connection]{}**'
---
[$^{1}$Department of Mathematics, Faculty of Science,\
Cairo University, Giza, Egypt]{}
[$^{2}$Department of Mathematics, Faculty of Science,\
Benha University, Benha, Egypt]{}
[$^{3}$Institute of Mathematics, University of Debrecen,\
Debrecen, Hungary]{}
E-mails: nlyoussef@sci.cu.edu.eg, nlyoussef2003@yahoo.fr\
salah.ali@fsci.bu.edu.eg, salahelgendi@yahoo.com
[**Abstract.**]{} The nullity distributions of the two curvature tensors ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ and ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution ${\mathcal{N}}_{R^\ast}$ is proved. Two counterexamples are given: the first shows that ${\mathcal{N}}_{R^\ast}$ does not coincide with the kernel distribution of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$; the second illustrates that ${\mathcal{N}}_{P^\ast}$ is not completely integrable. We give a simple class of a non-Berwaldian Landsberg spaces with singularities.
[**Keywords:** ]{} Klein-Grifone formalism, Chern connection, nullity distribution, kernel distribution, nullity foliation, autoparallel submanifold.\
[**MSC 2010:**]{} 53C60, 53B40, 58B20, 53C12.
Adopting the [pullback approach]{} to Finsler geometry, the nullity distribution has been investigated, for example, in [@akbar.nul3.; @akbar.null.; @ND-Zadeh]. In 2011, Bidabad and Refie-Rad [@bidabad] studied a more general distribution called $k$-nullity distribution. On the other hand, in 1982, Youssef [@Nabil.2; @Nabil.1] studied the nullity distributions of the curvature tensors of Barthel and Berwald connections, adopting the [Klein-Grifone approach ]{}to Finsler geometry. Moreover, Youssef et al. [@ND-cartan] studied the nullity distributions associated to the Cartan connection.
In their paper [@Chern], the present authors investigated the existence and uniqueness of the Chern connection and studied the properties of its curvature tensors following the Klein-Grifone approach. In this paper, we investigate the nullity distributions associated with the Chern connection. We prove the integrability and the autoparallel property of the nullity distribution ${\mathcal{N}}_{R^\ast}$ of the Chern h-curvature ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$. Moreover, we prove the completeness of the nullity foliation associated with ${\mathcal{N}}_{R^\ast}$. We give two interesting counterexamples. The first shows that the nullity distribution ${\mathcal{N}}_{R^\ast}$ does not coincide with the kernel distribution of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ (${\mathcal{N}}_{R^\ast}$ is a proper sub-distribution of $\text{Ker}_{R^\ast}$). The second shows that ${\mathcal{N}}_{P^\ast}$ is not completely integrable. As a by-product, this allows us to give a simple class of non-Berwaldian Landsberg spaces with singularities.
In this section we present a brief account of the basic concepts of Klein-Grifone’s theory of Finsler manifolds. For details, we refer to references [@r21; @r22; @r27; @Nabil.1]. We begin with some notational conventions.
Throughout, $M$ is a smooth manifold of finite dimension $n$. The ${\mathbb R}$-algebra of smooth real-valued functions on $M$ is denoted by $C^\infty(M)$; $\mathfrak{X}(M)$ stands for the $C^\infty(M)$-module of vector fields on $M$. The tangent bundle of $M$ is $\pi_{M}:TM\longrightarrow M$, the subbundle of nonzero tangent vectors to $M$ is $\pi: {{\cal T}}M\longrightarrow M$. The vertical subbundle of $TTM$ is denoted by $V(TM)$. The pull-back of $TM$ over $\pi$ is $P:\pi^{-1}(TM)\longrightarrow {{\cal T}}M$. If $X\in \mathfrak{X}(M)$, $ i_{X}$ and $\mathcal{L}_X$ denote the interior product by $X$ and the Lie derivative with respect to $X$, respectively. The differential of $f\in C^\infty(M) $ is $df$. A vector $\ell$-form on $M$ is a skew-symmetric $C^\infty(M)$-linear map $L:(\mathfrak{X}(M))^\ell\longrightarrow \mathfrak{X}(M)$. Every vector $\ell$-form $L$ defines two graded derivations $i_L$ and $d_L$ of the Grassman algebra of $M$ such that $$i_Lf=0, \,\,\,\, i_Ldf=df\circ L\,\,\,\,\,\, (f\in C^\infty(M)),$$ $$d_L:=[i_L,d]=i_L\circ d-(-1)^{\ell-1}di_L.$$
We have the following short exact sequence of vector bundle morphisms: $$0\longrightarrow
{{\cal T}}M \times_M TM\stackrel{\gamma}\longrightarrow T({{\cal T}}M)\stackrel{\rho}\longrightarrow
{{\cal T}}M \times_M TM\longrightarrow 0.$$ Here $\rho := (\pi_{{{\cal T}}M},\pi_\ast)$, and $\gamma$ is defined by $\gamma (u,v):=j_{u}(v)$, where $j_{u}$ is the canonical isomorphism from $T_{\pi_{M}(v)}M$ onto $ T_{u}(T_{\pi_{M}(v)}M)$. Then, $J:=\gamma\circ\rho$ is a vector $1$-form on $TM$ called the vertical endomorphism. The Liouville vector field on $TM$ is the vector field defined by ${C}:=\gamma\circ\overline{\eta},\,\, \overline{\eta}(u)=(u,u),\, u\in TM.$
A differential form $\omega$ (resp. a vector form $L$) on $TM$ is semi-basic if $ i_{JX}\omega=0$ (resp. $ i_{JX}L=0$ and $ JL=0$), for all $X\in {\mathfrak{X}(TM)}$. A vector $1$-form $G$ on $TM$ is called a [*Grifone connection*]{} if it is smooth on ${{\cal T}}M$, continuous on $TM$ and satisfies $J G=J, \,\, G J=-J $. The vertical and horizontal projectors $v$ and $h$ associated to $G$ are defined by $$v:=\frac{1}{2}
(I-G)\, \,\, \text{and}\,\,\, h:=\frac{1}{2} (I+G).$$
The almost complex structure determined by $G$ is the vector $1$-form $\textbf{F}$ characterized by $\textbf{F}J=h$ and $\textbf{F}h=-J$.
A Grifone connection $G$ induces the direct sum decomposition $$TT M=
V(TM)\oplus H(TM), \,\,\, H(TM):=\emph{{\text{Im}}} ( h).$$ The subbundle $H(TM)$ is called the $G$-horizontal subbundle of $TTM$, the module of its smooth sections will be denoted by $\mathfrak{X}^h({{\cal T}}M)$.
A Grifone connection $G$ is homogeneous if $[C,G]=0$. The torsion and the curvature of $G$ are the vector $2$-forms $t:=\frac{1}{2} [J,G]$ and $\mathfrak{R}:=-\frac{1}{2}[h,h]$, respectively. Note that in the last three equalities the brackets mean Frölicher-Nijenhuis bracket [@r20].
A function $E: TM {\longrightarrow}{\mathbb R}$ is called a [*Finslerian energy*]{} function if it is of class $C^{1}$ on $TM$ and $C^\infty$ on ${{\cal T}}M$; $E(u)>0 $ if $u\in {{\cal T}}M$ and $E(0)=0$; $C\cdot E =2E$, i.e., $E$ is $2^+$-homogeneous; the fundamental $2$-form $\Omega:=dd_{J}E$ has maximal rank. A [*Finsler manifold* ]{} is a manifold together with a Finslerian energy. If $(M,E)$ a Finsler manifold, then
1. there exists a unique spray $S$ for $M$ such that $i_{S}\Omega =-dE$;
2. there exists a unique homogeneous Grifone connection on $TM$ with vanishing torsion, namely $G = [J,S]$, such that $d_hE=0$ ($G$ is conservative’).
We say that $S$ is the [*canonical spray*]{} and $G$ is the [*canonical connection*]{} or [*Barthel connection*]{} of $(M,E)$.
If $(M,E)$ is a Finsler manifold, then the map $\overline{g}$ given by $$\overline{g}(J X,J Y):=\Omega(JX,Y); \,\, \,\, X, Y \in {\mathfrak{X}({{\cal T}}M)}$$ is a metric tensor on $V(TM)$. It can be extended to a metric tensor $g$ on $T(TM)$ by
$$\label{metricg}
g(X,Y):=\overline{g}(JX,JY)+\overline{g}(vX,vY)=\Omega(X,\textbf{F}Y).$$
Now we recall three famous covariant derivative operators on a Finsler manifold, called also connections’. They are the *Berwald connection* ${\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}$, the *Cartan connection* ${D}$ and the *Chern connection* ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}$, given by
$$\label{berwaldconn.}
{\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}_{JX}JY=J[JX,Y],\quad
{\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}_{hX}JY=v[hX,JY],\quad
{\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}\textbf{F}=0;$$
$$\label{cartanconn.}
D_{JX}JY={\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}_{JX}JY+\mathcal{C}(X,Y),\quad
D_{hX}JY={\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}_{hX}JY+\mathcal{C}'(X,Y),\quad
{D}\textbf{F}=0;$$
$$\label{chernconn.}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{JX}JY=J[JX,Y],\quad
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}JY=v[hX,JY]+\mathcal{C}'(X,Y),\quad
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}\textbf{F}=0,$$
($X,Y\in \mathfrak{X}({{\cal T}}M)$). In the formulas (\[cartanconn.\]) and (\[chernconn.\]) $\mathcal{C}$ is the *Cartan tensor*, $\mathcal{C}'$ is the *Landsberg tensor* of $(M,E)$. For their definition, see [@r22], p. 329. The tensors ${\mathcal{C}}$ and $\mathcal{C}'$ are symmetric, semi-basic and for arbitrary semispray $S$ on $TM$, we have $$\label{c(s)}
{\mathcal{C}}(X,S)=\mathcal{C}'(X,S)=0.$$
Let ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ and ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ be the h-curvature and the hv-curvature of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}$, respectively. We list some important identities from [@Chern], which will be needed in the sequel. Below $X$, $Y$, $Z$, $W$ are vector fields, $S$ is a semispray on ${{\cal T}}M$. $$\label{chern.[]}
[hX,hY]=h(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}Y-\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hY}X)-\mathfrak{R}(X,Y);$$ $$\label{eq.1}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)Z=R(X,Y)Z-\mathcal{C}(\textbf{F}\mathfrak{R}(X,Y),Z),$$ where $R$ is the h-curvature of $D$; $$\label{eq.2}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,Y)Z={\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$P$}}(X,Y)Z-(\,\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{JY}\mathcal{C}')(X,Z),$$ where ${\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$P$}}$ is the hv-curvature of ${\raisebox{10pt}{\tiny{$\circ$}}{\kern-7.5pt}\mbox{$D$}}$; $$\label{eq.3}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)S=\mathfrak{R}(X,Y);$$ $$\label{eq.4}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,Y)S={\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(S,Y)X=\mathcal{C}'(X,Y),\,\,\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,S)Z=0;$$ $$\label{eq.5}
\mathfrak{S}_{X,Y,Z}\, \{\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)Z\}=0;$$ $$\label{eq.6}
\mathfrak{S}_{X,Y,Z}\,\{(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}\,\mathfrak{R})(Y,Z)\}=\mathfrak{S}_{X,Y,Z}\{\,
\mathcal{C}'(\textbf{F}\mathfrak{R}(X,Y),Z)\};$$ $$\label{eq.7}
\mathfrak{S}_{X,Y,Z}\{\,(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(Y,Z)\}=\mathfrak{S}_{X,Y,Z}\{\, {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,\textbf{F}\mathfrak{R}(Y,Z))\};$$ $$\label{eq.8}
(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}})(Y,Z)-(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hY}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}})(X,Z)+(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{JZ}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(X,Y)
={\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,\textbf{F}\mathcal{C}'(Y,Z))
-{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(Y,\textbf{F}\mathcal{C}'(X,Z));$$ If $\mathfrak{R}=0$, then $$\label{eq.9}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y,Z,W)=\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Z,W,X,Y),$$ where $ {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y,Z,W):=g(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)Z,JW)$.
In this section, we investigate the nullity distribution of the Chern connection. It should be noted that the nullity distributions of the Barthel, Berwald and Cartan connections have already been studied in [@Nabil.2; @Nabil.1; @ND-cartan], respectively. First, we study the nullity distribution of the h-curvature tensor.
\[nul.chern\] Let ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ be the h-curvature tensor of the Chern connection. The [nullity space]{.nodecor} of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ at a point $z\in TM$ is the subspace of $H_z(TM)$ defined by $${{\mathcal{N}}}_{R^\ast}(z):=\{v\in H_z(TM)| \,\, {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}_z(v,w)=0, \, \,\text{for all}\,\, w\in H_z(TM)\}.$$ The dimension of ${{\mathcal{N}}}_{R^\ast}(z)$, denoted by ${\mu}_{{R^\ast}}(z)$, is the [nullity index]{.nodecor} of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ at $z$. If the nullity index ${\mu}_{R^\ast}$ is constant, then the map ${\mathcal{N}}_{R^\ast}:z\mapsto {{\mathcal{N}}}_{R^\ast}(z) $ defines a distribution ${\mathcal{N}}_{R^\ast}$ of rank ${\mu}_{R^\ast}$, called the [nullity distribution]{.nodecor} of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$. Any smooth section in the nullity distribution ${\mathcal{N}}_{R^\ast}$ is called [a nullity vector field]{.nodecor}. We denote by $\Gamma({{\mathcal{N}}}_{R^\ast})$ the $C^\infty(TM)$-module of the nullity vector fields. We shall assume that $\mu_{R^\ast}\neq 0$ and $\mu_{R^\ast}\neq n$.
Let ${\mathcal{N}}_{R^\ast}(x):=\pi_\ast({\mathcal{N}}_{R^\ast}(z))$ if $\pi(z)=x$. Then ${\mathcal{N}}_{R^\ast}(x)$ isomorphic to ${\mathcal{N}}_{R^\ast}(z)$ via the isomorphism $\pi_\ast\!\upharpoonright_{H_z(TM)}$.
\[ker.chern\] The kernel of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$ at the point $z\in TM$ is defined by $$\emph{\text{Ker}}_{R^\ast}(z):=\{u\in H_z(TM)| \, \, {{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}_z}(v,w) u=0, \, \text{for all}\, \, v,w\in H_z(TM)\}.$$
We have $\emph{\text{Ker}}_{R^\ast}(x)=\pi_\ast(\emph{\text{Ker}}_{R^\ast}(z))$; $x=\pi(z)$.
\[chern.nul\] The nullity distribution ${\mathcal{N}}_{R^\ast}$ has the following properties:
1. ${\mathcal{N}}_{R^\ast}\neq \phi$ and $\emph{\text{Ker}}_{R^\ast}\neq \phi$.
2. ${\mathcal{N}}_{R^\ast}\subseteq \mathcal{N}_\mathfrak{R}$, where $\mathcal{N}_\mathfrak{R}$ is the nullity distribution of the curvature $\mathfrak{R}$ of the Barthel connection.
3. $ {\mathcal{N}}_{R^\ast}\subseteq \emph{\text{Ker}}_{R^\ast}$.
4. If the canonical spray $S$ belongs to $\Gamma({\mathcal{N}}_{R^\ast})$, then $\mathfrak{R}=0$.
5. If $X\in \Gamma({\mathcal{N}}_{R^\ast})$, then $[C,X]\in \Gamma({\mathcal{N}}_{R^\ast})$ and, consequently, $[C,X]\in \Gamma({\mathcal{N}}_{\mathfrak{R}})$.
\(2) Let $X$ be a nullity vector field. Using (\[eq.3\]), we have $$\begin{aligned}
X\in \Gamma({\mathcal{N}}_{R^\ast}) &\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)Z=0\quad \text{for all}\, \, Y,Z \in {\mathfrak{X}(TM)}\\
&\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)S=0\quad \,\text{for all}\, \, Y \in {\mathfrak{X}(TM)}\\
&\Longrightarrow& \mathfrak{R}(X,Y)=0\quad\,\,\,\, \text{for all}\, \, Y \in {\mathfrak{X}(TM)}\\
&\Longrightarrow& X\in \Gamma({\mathcal{N}}_{\mathfrak{R}}).
\end{aligned}$$ (3) Let $Z\in \Gamma(\mathcal{N}_{R^\ast} )$, then, by (\[eq.5\]), we have $\mathfrak{S}_{X,Y,Z}\{\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,Y)Z\}=0.$ Since ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Y,Z)X={\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Z,X)Y=0$, then the result follows.\
(4) This is an immediate consequence of (\[eq.3\]).\
(5) Let $X\in \Gamma({\mathcal{N}}_{R^\ast})$. Since $\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_C\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}=0$ [@Chern] , we get $(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_C\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(X,Y)=0,$ which leads to ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_CX,Y)=0.$ Using (\[chernconn.\]), we have ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}([C,X],Y)=0$. By the homogeneity of $h$, $[C,h]=0$, from which $[C,hX]=h[C,X]$. That is, $[C,hX]$ is horizontal. Hence, $[C,X]\in \Gamma({\mathcal{N}}_{R^\ast})$. Consequently, by (2), $[C,X]\in \Gamma(\mathcal{N}_{\mathfrak{R}})$.
It is important to note that the reverse inclusion in the property (3) of Proposition \[chern.nul\] is not true; that is, ${\text{Ker}}_{R^\ast}\not\subset \mathcal{N}_{R^\ast}$. This is shown by the next example in which the calculations are performed by using [@CFG].
Let $M=\{(x^1,x^2,x^3,x^4)\in \mathbb{R}^4| x^2>0\}$ and\
$U=\{(x^1,...,x^4;y^1,...,y^4)\in \mathbb{R}^4 \times \mathbb{R}^4: \,y^1\neq 0, y^2\neq 0\}\subset TM$. Define $F$ on $U$ by $$F(x,y) := \, ({{{{( x^2)}}^{2}{{(y^1)}}^{4}+{{( y^2)}}^{4}+{{( y^3)}}^{4}+{{( y^4)}}^{4}}})^{1/4}.$$
According to [@appl.], the nullity distribution of the Cartan h-curvature $R$ of $(M,F)$ is $$\label{null.1}
{\mathcal{N}}_R=\{sh_3+th_4\in \mathfrak{X}^h({{\cal T}}M)| \,s,t\in \mathbb{R}\}$$ and the kernel distribution $\ker_R$ of $R$ is $$\label{ker1}
\begin{split}
\ker_R=\left\{s\left(\frac{y^1}{y^2}h_1+h_2+\frac{x^2{(y^1)}^4+{(y^2)}^4+2{(y^3)}^4+2{(y^4)}^4}{y^2{(y^4)}^3}h_4\right)\right. &\\ \left.+t\left(h_3-\frac{{(y^3)}^3}{{(y^4)}^3}h_4\right)\in \mathfrak{X}^h({{\cal T}}M)|s,t\in \mathbb{R}\right\},
\end{split}$$ where $h_i:=\frac{\partial}{\partial x^i}-N^m_i\frac{\partial}{\partial y^m}$ form a basis of $\mathfrak{X}^h({{\cal T}}M)$.
Now, by the NF-package [@CFG], we can perform the following calculations.
[ **Chern h-curvature :**]{}
[ **Nullity distribution of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$:**]{}
Putting ${\it RchernW}^{\it x1 }_{\it x2 x2 }=0$ and ${\it RchernW}^{\it x2 }_{\it x1 x1 }=0$, then we have a system of algebraic equations. The NF-package yields the following solution: $W^1=W^2=0, W^3=s, W^4=t$, where $s,t\in \mathbb{R}$. Then, the nullity distribution is $$\label{nullch.1}
{\mathcal{N}}_{R^\ast}=\{sh_3+th_4\in \mathfrak{X}^h({{\cal T}}M)|\,s,t\in \mathbb{R}\}.$$
[ **Kernel distribution of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$:**]{}
Putting ${\it RchernZ}^{\it x1 }_{\it x1 x2 }=0$, we get $Z^1=\frac{2y1}{y2}r, Z^2=r, Z^3=s, Z^4=t;\,r,s,t\in \mathbb{R}$. Then, the kernel distribution $ \text{Ker}_{R^\ast}$ is $$\label{kerch.1}
\text{Ker}_{R^\ast}={\left\{r\left(\frac{2y1}{y2}h_1+h_2\right)+sh_3+th_4\in \mathfrak{X}^h({{\cal T}}M)|\,r,s,t\in \mathbb{R}\right\}}.$$ Equations (\[nullch.1\]) and (\[kerch.1\]) show that $\text{Ker}_{R^\ast}$ can not be a sub-distribution of ${\mathcal{N}}_{R^\ast}$.
\[coincide\] The nullity distribution ${\mathcal{N}}_{R^\ast}$ of the Chern h-curvature and the nullity distribution ${\mathcal{N}}_R$ of the Cartan h-curvature coincide.
Let $X\in\Gamma({\mathcal{N}}_{R^{\ast}})$. Then, by (\[eq.1\]) and Proposition \[chern.nul\] (2), $X\in\Gamma({\mathcal{N}}_R)$. Hence ${\mathcal{N}}_{R^{\ast}}$ is a subset of $ {\mathcal{N}}_R$. Conversely, let $X\in\Gamma({\mathcal{N}}_R)$. Then, by (\[eq.1\]) and by ${\mathcal{N}}_R\subset{\mathcal{N}}_\mathfrak{R}$ [@ND-cartan], we get $X\in\Gamma({\mathcal{N}}_{R^{\ast}})$, whence, ${\mathcal{N}}_R\subset {\mathcal{N}}_{R^\ast}$.
The above example shows that ${\mathcal{N}}_{R^{\ast}}\subset\text{Ker}_{R^\ast}$ and the reverse inclusion is false by (\[nullch.1\]), (\[kerch.1\]). It also shows that although ${\mathcal{N}}_{R^{\ast}}= {\mathcal{N}}_R$ (see (\[null.1\]) and (\[nullch.1\])), $\text{Ker}_{R^\ast}\neq\text{Ker}_{R}$ by (\[ker1\]), (\[kerch.1\]). In view of the above theorem, the reverse inclusion in [ (2)]{} of Proposition \[chern.nul\] is not true either: $\mathcal{N}_\mathfrak{R}\not\subset \mathcal{N}_R=\mathcal{N}_{R^\ast}$ [@appl.].
The conullity space of the h-curvature tensor at $z$, denoted by ${{\mathcal{N}}_{R^\ast}\!}^\perp(z)$, is the orthogonal complement of ${\mathcal{N}}_{R^\ast}$ in $H_z(TM)$, where the orthogonality is taken with respect to the metric $g$ defined by (\[metricg\]).
For each point $z\in TM$, either $\mu_{R^\ast}(z)=n$ or $\mu_{R^\ast}(z)\leq n-2$. Consequently, $\dim{\emph{\text{Ker}}_{R^\ast}}>n-2$.
If $\mu_{R^\ast}(z)\neq n$, then there is a non-zero horizontal vector $v\notin {\mathcal{N}}_{R^\ast}(z) $. It follows that there is a vector $w\in H_z(TM)$ such that ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}_z(w,v)\neq 0$ and so ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}_z(v,w)\neq 0$. Then $v,w \notin {\mathcal{N}}_{R^\ast}(z)$ and hence $v,w \in {{\mathcal{N}}_{R^\ast}\!}^\perp(z)$. By the antisymmetry of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}$, the vectors $v$ and $w$ are independent. Thus, $\dim {{\mathcal{N}}_{R^\ast}\!}^\perp(z)\geq 2$. Consequently, $\mu_{R^\ast}(z)\leq n-2$.
If $\mathfrak{R}=0$, then *[Im ]{}$(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}) = (J {\mathcal{N}}_{R^\ast} )^\perp$. Consequently, $(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})=n-{\mu}_{R^\ast}$.*
For all $X\in \Gamma({\mathcal{N}}_{R^\ast})$ and $Y,Z,W\in \mathfrak{X}^h({{\cal T}}M)$, we have $$\begin{aligned}
g(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Y,Z)W,JX) &=& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Y,Z,W,X)\\
&=& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(W,X,Y,Z) \,\,\,( \text{by (\ref{eq.9}))}\\
&=& -{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,W,Y,Z)\\
&=& -g(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,W)Y,JZ)\\
&=& 0 \,\,\,( \text{since } X \text{ is a nullity vector field}),\end{aligned}$$ as wanted.
As a direct consequence of Theorem \[coincide\] and the fact that ${\mathcal{N}}_R$ is completely integrable [@ND-cartan], we have the following result.
\[c.i.chern\] Let $\mu_{R^\ast}$ be constant on an open subset $U$ of $TM$. The nullity distribution $z\mapsto \mathcal{N}_{R^\ast}(z)$ is completely integrable on $U$.
According to the Frobenius theorem, there exists a foliation of $M$ by $\mu_{R^\ast}$-dimensional maximal connected submanifolds as leaves, such that the nullity space at a point $x\in M$ is the tangent space to the leaf at $x$. We call the foliation induced by the nullity distribution ${\mathcal{N}}_{R^\ast}$ the nullity foliation and denote it again by ${\mathcal{N}}_{R^\ast}$. So, by Corollary \[c.i.chern\], we have the following result.
\[chern.autoparallel\] The leaves of the nullity foliations $\mathcal{N}_{R^\ast}$ and $\mathcal{N}_\mathfrak{R}$ are auto-parallel submanifolds with respect to the Chern connection.
The fact that ${\mathcal{N}}_{R^\ast}$ is auto-parallel with respect to Chern connection can be proved in a similar manner as the analogous result in [@ND-cartan].
On the other hand, the integrability of the nullity distribution ${\mathcal{N}}_\mathfrak{R}$ of the curvature of Barthel connection has been proved in [@Nabil.2]. We show that if $X,Y\in \Gamma({\mathcal{N}}_\mathfrak{R})$, then $\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_XY\in \Gamma({\mathcal{N}}_\mathfrak{R})$. By (\[eq.6\]), we have $$\mathfrak{S}_{X,Y,Z}\{(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_X\mathfrak{R})(Y,Z)\}=\mathfrak{S}_{X,Y,Z}\{\,
\mathcal{C}'(Z,\textbf{F}\mathfrak{R}(X,Y))\}.$$ Since $X,Y\in \Gamma({\mathcal{N}}_\mathfrak{R})$, $\mathfrak{S}_{X,Y,Z}\{(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_X\mathfrak{R})(Y,Z)\}=0.$ Consequently, $\mathfrak{R}(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_XY,Z)=0 $ for every vector field $Z\in {\mathfrak{X}(TM)}$ and $\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_XY\in \Gamma({\mathcal{N}}_\mathfrak{R})$.
Due to the torsion-freeness of the Levi-Civita connection in Riemannian geometry, the two concepts autoparallel submanifold and totally geodesic submanifold coincide [@kobayashi]. This is not true in Finsler geometry. However, every auto-parallel submanifold is totally geodesic [@E.cartan]. So, we have:
\[leaves\] The leaves of the nullity foliations ${\mathcal{N}}_\mathfrak{R}$ and ${\mathcal{N}}_{R^{\ast}}$ are totally geodesic submanifolds with respect to the Chern connection.
\[coincidenulker\] If $\mathfrak{R}=0$, then the two distributions ${\mathcal{N}}_{R^\ast}$ and $\text{Ker}_{R^\ast}$ coincide.
By Proposition \[chern.nul\] (3), we always have ${\mathcal{N}}_{R^\ast}\subset \text{Ker}_{R^\ast}$. Let $X\in \Gamma(\text{Ker}_{R^\ast})$ and let $Y,Z,W$ be vector fields on $T M$, then by (\[eq.9\]), we have $$\begin{aligned}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Y,Z)X=0 &\Longrightarrow& g({\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Y,Z)X,JW)=0 \\
&\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(Y,Z,X,W)=0\\
&\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,W,Y,Z)=0\\
&\Longrightarrow& g(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,W)Y,JZ)=0\\
&\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(X,W)Y=0\\
&\Longrightarrow& X\in \Gamma({\mathcal{N}}_{R^\ast}),\end{aligned}$$ thus $\text{Ker}_{R^\ast}\subset {\mathcal{N}}_{R^\ast}$.
Let $(M, E)$ be a complete Finsler manifold and $U$ the open subset of M on which $\mu_{R^\ast}$ takes its minimum. If $\mathfrak{R}$ vanishes, then every integral manifold of the nullity foliation $\mathcal{N}_{R^\ast}$ in $U$ is complete.
The proof is inspired by [@akbar.null.], taking into account the fact that the two spaces ${\mathcal{N}}_{R^\ast}(z)$ and ${\mathcal{N}}_{R^\ast}(x)$, $x=\pi(z)$, are isomorphic. Let $N$ be an integral manifold of the nullity foliation $\mathcal{N}_{R^\ast}$ in $U$. To prove that $N$ is complete, it suffices to show that every geodesic $\gamma : [0, c) \rightarrow N $ on $N$ can be extended to a geodesic $ \widetilde{\gamma}: [0,\infty)\rightarrow N $ on $N$. Suppose that such a geodesic extension $ \widetilde{\gamma}$ does not exist. As $N$ is totally geodesic, by Corollary \[leaves\], $\gamma$ is a geodesic on M and thus has a geodesic extension $ \widetilde{\gamma}~:~ [0,\infty)\rightarrow~M $ such that $\gamma=\widetilde{\gamma}\cap N$. It follows that $p:=\widetilde{\gamma}(c)\notin U$. Let $p_0:=\gamma(0)=\widetilde{\gamma}(0)$ and set $r_0:=\mu_{R^\ast}(p_0)$, the dimension of the nullity space $\mathcal{N}_{R^\ast}(p_0)$. Since $\mu_{R^\ast}$ is positive and minimal on $U$, then $\mu_{R^\ast}(p)>r_0>0$. Now, consider a basis $B=\{e_1,..., e_{r_0},e_{r_0+1},...,e_n\}$ for $T_{p_0}M$ such that $\{e_1, ..., e_{r_0}\}$ is a basis for ${\mathcal{N}}_{R^\ast}(p_0)$ and $e_1$ is tangent to $\gamma$ at $p_0=\gamma(0)$. Using the system of differential equations $$\frac{{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}} F_i}{dt}=0,\quad F_i(0)=e_i, \quad i=1,2,...,n,$$ the basis $B$ can be translated into a parallel frame $(F_1, ..., F_{r_0},F_{r_0+1},...,F_n)$ along $\widetilde{\gamma}$. Then $(F_1, ..., F_{r_0})$ is a basis for the nullity space at every point $\widetilde{\gamma}(t)$ in $U\cap V$ for some neighborhood $V$ of $\widetilde{\gamma}(t)$ on $M$. Since $\mu_{R^\ast}(p)>r_0$, there is a vector field $F_a$ along $\widetilde{\gamma}$, for a fixed integer $a$ in the range $r_0+1,...,n$, such that for every $t\in[0,c)$, we have $$\label{contradiction-1}
F_a(\gamma(t))\notin\mathcal{N}_{R^\ast}(\gamma(t)), \,\,\, \,\, F_a(p)\in \mathcal{N}_{R^\ast}(p).$$
Now, let $\widehat{\widetilde{\gamma}}$ be the natural lift of $\widetilde{\gamma}$ to ${{\cal T}}M$ and $\{\widehat{F}_1, ..., \widehat{F}_{r_0},\widehat{F}_{r_0+1},...,\widehat{F}_n\}$ the basis of $H_{\widehat{\widetilde{\gamma}}(t)}TM$ such that $\pi_\ast(\widehat{F}_i)=F_i$. Let $\phi^h_{ijk}$ be the functions defined by $$\label{complete}
{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_i,\widehat{F}_j)\widehat{F}_k=\phi^h_{ijk}\,
\frac{\partial}{\partial y^h}.$$ By (\[eq.7\]), taking into account that $\mathfrak{R}=0$, we have $$(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(Y,Z)+(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hY}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(Z,X)+(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hZ}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(X,Y)=0.$$ Plugging $\widehat{F}_1$, $\widehat{F}_i$ and $\widehat{F}_j$ instead of $X$, $Y$ and $Z$, where $i,j=r_0+1,...,n$, we get $$(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{\widehat{F}_1}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(\widehat{F}_i,\widehat{F}_j)+(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{\widehat{F}_i}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(\widehat{F}_j,\widehat{F}_1)
+(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{\widehat{F}_j}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(\widehat{F}_1,\widehat{F}_i)=0.$$ Since $\widehat{F}_1\in {\mathcal{N}}_{R^\ast}$ and ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$T$}}(hX,hY)=\mathfrak{R}(X,Y)=0$, the last equality takes the form $$\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{\widehat{F}_1}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_i,\widehat{F}_j)+\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_j,[\widehat{F}_1,\widehat{F}_i])
+\,\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_i,[\widehat{F}_j,\widehat{F}_1])=0.$$ Applying the above equation on $\widehat{F}_a$, we get $$\label{drrr}
\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{\widehat{F}_1}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_i,\widehat{F}_j)\widehat{F}_a
+\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_j,[\widehat{F}_1,\widehat{F}_i])\widehat{F}_a+\,\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}}(\widehat{F}_i,[\widehat{F}_j,\widehat{F}_1])\widehat{F}_a=0.$$ Since, $[\widehat{F}_1,\widehat{F}_i]$ is horizontal, it can be written in the form $[\widehat{F}_1,\widehat{F}_i]=\xi^k_{1i}\widehat{F}_k+\xi^\mu_{1i}\widehat{F}_\mu$, where $k=r_0+1,...,n$ and $\mu=1,...,r_0$. Consequently, by (\[complete\]) and (\[drrr\]), noting that $\widehat{F}_\mu$ are null vector fields, we get $$\label{dash}
(\phi^h_{ija})'+\xi^k_{1i}\,\phi^h_{jka}-\xi^k_{1j}\,\phi^h_{ika}=0$$ Since $F_a$ is a nullity vector field at $p$, then for the fixed index $a$, $\phi^h_{lma}(p) = 0$, where $l,m = r_0+1, ..., n$. Hence, the differential equations (\[dash\]) with the initial condition $\phi^h_{lma}(p) = 0 $ imply that the functions $\phi^h_{lma}$ vanish identically. As $\mathfrak{R}=0$, Theorem \[coincidenulker\] and (\[complete\]) give rise to $$\label{final}
F_a(\gamma(t))\in \mathcal{N}_{R^\ast}(\gamma(t)),\,\, \text{for all}\,\, t\in [0,c]$$ Now (\[contradiction-1\]) and (\[final\]) lead to a contradiction. Consequently, $\gamma$ can be extended to a geodesic $\widetilde{\gamma} : [0,\infty) \longrightarrow N$.
In this section we investigate the nullity distribution of the hv-curvature ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ of the Chern connection. We show, by a counterexample, that the nullity distribution ${\mathcal{N}}_{P^\ast}$ is not completely integrable. We find a sufficient condition for ${\mathcal{N}}_{P^\ast}$ to be completely integrable.
Let ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ be the hv-curvature of the Chern connection. The nullity space of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ at a point $z\in TM$ is a subspace of $H_z(TM)$ defined by $$\mathcal{N}_{P^\ast}(z):=\{v\in H_z(TM) | \,\, {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}_z(v,w)=0, \, \,\text{for all}\,\, w\in H_z(TM)\}.$$ The dimension of $\mathcal{N}_{P^\ast}(z)$, denoted by $\mu_{P^\ast}(z)$, is the nullity index of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ at $z$.
\[pchern.nullity\]The nullity distribution of ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ satisfies:
1. $ {\mathcal{N}}_{P^\ast}\neq\phi$.
2. If $X\in \Gamma({\mathcal{N}}_{P^\ast})$, then $[C,X]\in \Gamma({\mathcal{N}}_{P^\ast})$.
3. If $X\in \Gamma({\mathcal{N}}_{P^\ast})$, then $\mathcal{C}'(X,Y)=0,\, \,\text{for all}\,\, Y\in \mathfrak{X}^h({{\cal T}}M).$
4. If $\mu_{P^\ast}=n$, then ${\mathcal{N}}_{{R^\ast}}={\mathcal{N}}_{R^\circ}$,
where ${\mathcal{N}}_{R^\circ}$ is the nullity distribution of the h-curvature of the Berwald connection [@Nabil.1].
A Finsler manifold is said to be Landsbergian if the Landsberg tensor $\mathcal{C}'$ vanishes or, equivalently, $P=0$ [@szilasi]. If the nullity index $\mu_{P^\ast}$ takes its maximum, then by Proposition \[pchern.nullity\] (3), $\mathcal{C}'=0.$ Consequently, a Finsler manifold $(M,E)$ is Landsbergian if the nullity index $\mu_{P^\ast}$ achieves its maximum.
A Finsler manifold $(M,E)$ is Landsbergian if and only if the canonical spray $S$ is a nullity vector field for the the distribution ${\mathcal{N}}_{P^\ast}$.
By (\[eq.4\]), we have $$\begin{aligned}
(M,E)\,\text{is Landsbergian}&\Longleftrightarrow&\mathcal{C}'=0 \\
&\Longleftrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,Y)S=0\,\, \,\text{for all}\,\, X,Y \in {\mathfrak{X}(TM)}\\
&\Longleftrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(S,Y)X=0 \,\,\,\text{for all}\,\, X,Y \in {\mathfrak{X}(TM)}\\
&\Longleftrightarrow& S\in \Gamma({\mathcal{N}}_{P^\ast}),\end{aligned}$$ as was to be shown.
The above theorem shows that the canonical spray $S$ does not belong to the nullity distribution ${\mathcal{N}}_{P^\ast}$ except in the Landsbergian case. This is in contrast to the case of Cartan connection, where the canonical spray always belongs to the nullity distribution of the Cartan hv-curvature $P$.
The nullity distribution ${\mathcal{N}}_{P^\ast}$ is not completely integrable in general, as is illustrated by the following example.
Let $U=\{(x^1,x^2,x^3;y^1,y^2,y^3)\in\mathbb{R}^3\times \mathbb{R}^3:\,y^1, y^2,y^3\neq 0, y^3\neq 4y^2 \}\subset TM$, where $M:= \mathbb{R}^3$. Define $F$ on $U$ by $$F(x,y):=\sqrt [4]{{{ e}^{-{ x^1 x^2}}}{{ (y^1)}}^{2}{{ (y^3)}}^{2}{{e}^{-{\frac {{ y^3}}{{ y^2}}}}}}.$$ By Maple program and NF-package we can perform the following calculations.
[ [[**Barthel connection**]{}]{}]{}
[ [**Chern hv-curvature ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$:**]{}]{}
[ [ **${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$-nullity vectors:**]{}]{}
Putting ${\it PchernW}^{\it h }_{\it ij }=0$, we get a system of algebraic equations. This system has a solution if $y_3=2y_2$ and $x^1>0$: $W^1=s$, $W^2=t$, $W^3=2t$, $
\, s, t\in \mathbb{R}$. Hence, a ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$-nullity vector must have the form $W=sh_1+t(h_2+2h_3)$, where the horizontal basis vector fields $h_1,h_2,h_3$ are given by $h_1=\frac{\partial}{\partial x_1}+\frac{x_2y_1}{2}\frac{\partial}{\partial y_1}$, $h_2=\frac{\partial}{\partial x_2}+\frac{x_1y_2}{2}\frac{\partial}{\partial y_2}$, $h_3=\frac{\partial}{\partial x_3}+\frac{x_1y_2}{2}\frac{\partial}{\partial y_3}$. Now, take $X,Y\in{\mathcal{N}}_{P^\ast}$ such that $X=h_1$, $Y=h_2+2h_3$. Hence, the bracket $[X,Y]=[h_1,h_2+2h_3]= -\frac{y_1}{2}\frac{\partial}{\partial y_1}+\frac{y_2}{2}\frac{\partial}{\partial y_2}+\frac{y_2}{2}\frac{\partial}{\partial y_3}$ is vertical and, consequently, ${\mathcal{N}}_{P^\ast}$ is not completely integrable.
\[p.is.c.i.\] Let $\mu_{P^\ast}$ be constant on an open subset $U$ of $TM$. The nullity distribution ${\mathcal{N}}_{P^\ast}$ is completely integrable on $U$ if and only if $ \mathfrak{R}(X,Y)=0 \,\, \text{and}\,\, (\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{JZ}\,\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(X,Y)=~0$, for all $ X,Y \in \Gamma({\mathcal{N}}_{P^\ast})$.
*Necessity*. Let ${\mathcal{N}}_{P^\ast}$ be completely integrable. Then, if $X,Y\in \Gamma({\mathcal{N}}_{P^\ast})$, the bracket $[hX,hY]$ is horizontal, thus, $\mathfrak{R}(X,Y)=0$. Also, by (\[eq.8\]) and the fact that ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}([hX,hY],Z)=( \,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}})(Y,Z)-(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hY}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}})(X,Z)=0$ (by (\[chern.\[\]\])), we have\
$ (\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{JZ}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(X,Y)=0,\, \text{for all}\,\, X,Y \in \Gamma({\mathcal{N}}_{P^\ast}), \, \text{for all}\,\, Z\in {\mathfrak{X}(TM)}$.
*Sufficiency*. Let $\mathfrak{R}(X,Y)=0$ and $ (\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{JZ}\,\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$R$}})(X,Y)=0$ for all $X, Y\in \Gamma({\mathcal{N}}_{P^\ast})$. As $0=\mathfrak{R}(X,Y)=-v[hX,hY]=-v[X,Y]$, the bracket $[X,Y]$ is horizontal. Making use of (\[chern.\[\]\]) and (\[eq.8\]), we get $$\begin{aligned}
( \,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hX}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}})(Y,Z)-(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{hY}\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}})(X,Z)=0&\Longrightarrow&{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{X}Y-\,{\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$D$}}_{Y}X,Z)=0 \\
&\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}([X,Y]+\mathfrak{R}(X,Y),Z)=0 \\
&\Longrightarrow& {\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}([X,Y],Z)=0\\
&\Longrightarrow&[X,Y]\in \Gamma({\mathcal{N}}_{P^\ast}).
\end{aligned}$$ Hence ${\mathcal{N}}_{P^\ast}$ is completely integrable.
By the property ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(X,Y)Z={\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}(Z,Y)X$ we have the following result.
The nullity distribution ${\mathcal{N}}_{P^\ast}$ and the kernel distribution $\text{Ker}_{P^\ast}$ coincide.
A Finsler manifold in which the Chern hv-curvature tensor ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ vanishes is called a Berwald space [@szilasi]. It is well known that every Berwald space is a Landsberg space, but it is not known whether the converse is true. In [@Shen-Landsberg], Shen introduced a class of non-regular Finsler metrics which is Landsbergian and not Berwaldian. The calculations are not easy, especially, if one wants to study some concrete examples. Here, by using Maple program together with the results of [@Shen-Landsberg] and [@CFG], we give a simple class of proper non-regular non Berwaldian Landsbergian spaces.\
\[ex.3\] Let $M= \mathbb{R}^3$, $U=\{(x^1,x^2,x^3;y^1,y^2,y^3)\in\mathbb{R}^3\times \mathbb{R}^3:\, y^2>0,y^3 >0 \}\subset TM$. Define $F$ on $U$ by $$F(x,y):=f(x^1)\sqrt{{(y^1)}^2+y^2y^3+y^1\sqrt{y^2y^3}}\,e^{\frac{1}{\sqrt{3}}\arctan\Big(\frac{2y^1}{\sqrt{3y^2y^3}}+\frac{1}{\sqrt{3}}\Big)}.$$ The idea is to compute the Landsberg tensor $L_{ijk}$ and the Berwald tensor $G^h_{ijk}$ which are locally given by $$L_{ijk}:=\frac{F}{2}\frac{\partial F}{\partial y^h}G^h_{ijk},\,\,\,\,G^h_{ijk}:=\frac{\partial^3 G^h}{\partial y^i\partial y^j\partial y^k}.$$ Then, we show that the Landsberg tensor vanishes identically while there are some non vanishing components of the Berwald tensor (for simplicity we consider only one nonzero component and check it at a point) .
[ ]{}
By Example \[ex.3\], for any non constant positive smooth function $f$ on $\mathbb{R}$, the Landsberg tensor of $(M,F)$ vanishes (or equivalently, the hv-curvature $P$ of the Cartan connection vanishes) and hence the class is Landsbergian. On the other hand, the hv-curvature ${\raisebox{10pt}{\small$\ast$}{\kern-7.5pt}\mbox{$P$}}$ of the Chern connection does not vanish and hence the class is not Berwaldian. So we can confirm:
There are non-regular Landsberg spaces which are not Berwaldian.
Acknowledgement {#acknowledgement .unnumbered}
===============
The second author would like to express his deep gratitude to Professors József Szilasi, Zoltán Muzsnay and Mr. Dávid Kertész (University of Debrecen) for their valuable discussions and comments.
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|
---
abstract: |
We consider a self-avoiding walk model (SAW) on the faces of the square lattice $\mathbb{Z}^2$. This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles $\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]$ and satisfy the Yang–Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles.
By means of the Yang-Baxter transformation, we show that the 2-point function of the walk in the half-plane does not depend on the rhombic tiling ([*i.e.*]{} on the angles chosen). In particular, this statistic coincides with that of the self-avoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles $\theta$ equal to $\frac{\pi}{3}$.
For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to $1+\sqrt{2}$. We show that the same is true for any choice of angles. In doing so, we also give a new short proof to the fact that the partition function of self-avoiding bridges in a strip of the hexagonal lattice tends to $0$ as the width of the strip tends to infinity. This proof also yields a quantitative bound on the convergence.
author:
- Alexander Glazman
- Ioan Manolescu
bibliography:
- 'SAW.bib'
title: |
Self-avoiding walk on ${\mathbb{Z}}^2$ with Yang–Baxter weights:\
universality of critical fugacity and 2-point function.
---
Self-avoiding walk on ${\mathbb{Z}}^2$ with Yang–Baxter weights {#sec:intro}
===============================================================
In spite of the apparent simplicity of the model, few rigorous results are available for two dimensional self-avoiding walk. The main conjecture is the convergence of plane SAW to a conformally invariant scaling limit. The latter is shown [@LSW] to be equal to $\text{SLE} (8/3)$, provided the scaling limit exists and is conformally invariant. A natural way to attack this problem is via the so-called parafermionic observable (see below for a definition) and its partial discrete holomorphicity. H. Duminil-Copin and S. Smirnov [@DCSmi10] used the parafermionic observable to prove that the connective constant for the hexagonal lattice is equal to $\sqrt{2+\sqrt{2}}$, a result that had beed non-rigorously derived by B. Nienhuis in [@Nie82].
Self-avoiding walk on the square lattice is not believed to be integrable, therefore it is not reasonable to expect any explicit formula for the connective constant in this case, nor the existence of a well-behaved equivalent observable. However, one may study natural variations of the model, such as the weighted version presented here, that render it integrable. By integrability here we mean that the weights satisfy the Yang–Baxter equation. Similar integrable versions exist for all loop $O(n)$ models (see [@N90; @CaIk; @Gl]), we limit ourselves here to $n=0$, that is to self-avoiding walk.
These variations provide a framework to analyse the universality phenomenon, [*i.e.*]{} that the properties of the model at criticality do not depend on the underlying lattice. Though believed to generally occur, the universality of critical exponents on isoradial graphs was established only for the Ising model [@CheSmi12], percolation [@GriMan14] and the random-cluster model [@DumLiMan15]. The current paper is the first step towards universality of the self-avoiding walk.
Here we address the natural question of comparison between the properties of regular self-avoiding walk on the hexagonal lattice and those of weighted self-avoiding walk on a more general rhombic tiling. We show that in the half-plane, the 2-point function between points on the boundary is the identical in the weighted and regular models. A main tool in our proof, as well as in [@GriMan14; @DumLiMan15], is the Yang–Baxter transformation discussed in Section \[sec:YangBaxter\].
Let us now define the model. Consider a series of angles $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$, where $\theta_k\in [\pi/3,2\pi/3]$ for all $k$. Denote by ${\mathbb{H}}(\Theta)$ the right half-plane tiled with columns of rhombi of edge-length $1$ in such a way that all rhombi in the $k$-th column from the left have upper-left angle $\theta_k$. We regard ${\mathbb{H}}(\Theta)$ as a plane graph, and call edges the sides of each rhombus; we will refer to such graphs as rhombic tilings. Embed ${\mathbb{H}}(\Theta)$ so that the origin $0$ is the mid-point of a vertical edge of the boundary. Denote by ${\mathrm{Strip}}_{T}(\Theta)$ the strip consisting of the $T$ leftmost columns of ${\mathbb{H}}(\Theta)$.
A self-avoiding walk on ${\mathbb{H}}(\Theta)$ is a simple curve $\gamma$ starting and ending at midpoints of edges, intersecting edges at right angles and traversing each rhombus in one of the ways depicted in Fig. \[figWeights\]. The weight ${\mathrm{w}}_\Theta(\gamma)$ of a self-avoiding walk $\gamma$ is the product of weights associated to each rhombus; for a rhombus of angle $\theta$ the weight, depending on the configuration of arcs inside it, takes one of the six possible values: 1, $u_1(\theta)$, $u_2(\theta)$, $v(\theta)$, $w_1(\theta)$, $w_2(\theta)$ (see Fig. \[figWeights\] for the correspondence between the local pictures and the weights). These are explicit functions of $\theta$, given below. When it is clear which angles are considered, we will usually omit the subscript $\Theta$ and write ${\mathrm{w}}(\gamma)$.
[.7]{} ![Different ways of passing a rhombus with their weights and an example of a walk of weight $u_1(\theta_1)^2 v(\theta_2)u_1(\theta_2)u_2(\theta_2)w_2(\theta_3)v(\theta_3)u_2(\theta_3)u_1(\theta_4)u_2(\theta_4)$ and length $\frac{3}{\pi}\big[2\theta_1 + 3(\pi - \theta_3) + 7\big]$.[]{data-label="figWeights"}](rhombi_weights_slim.pdf "fig:")
[.3]{} ![Different ways of passing a rhombus with their weights and an example of a walk of weight $u_1(\theta_1)^2 v(\theta_2)u_1(\theta_2)u_2(\theta_2)w_2(\theta_3)v(\theta_3)u_2(\theta_3)u_1(\theta_4)u_2(\theta_4)$ and length $\frac{3}{\pi}\big[2\theta_1 + 3(\pi - \theta_3) + 7\big]$.[]{data-label="figWeights"}](saw_examples2.pdf "fig:")
In 1990, Nienhuis [@N90] computed the set of weights that are coherent with the Yang-Baxter equation for this model (see Section \[sec:YangBaxter\] for details). These are: $$\begin{aligned}
\nonumber
u_1&=\tfrac{\sin(\tfrac{5\pi}{4})\sin(\tfrac{5\pi}{8}+\tfrac{3\theta}{8})}{\sin(\tfrac{5\pi}{4}+\tfrac{3\theta}{8})\sin(\tfrac{5\pi}{8}-\tfrac{3\theta}{8})},&
u_2&=\tfrac{\sin(\tfrac{5\pi}{4})\sin(\tfrac{3\theta}{8})}{\sin(\tfrac{5\pi}{4}+\tfrac{3\theta}{8})\sin(\tfrac{5\pi}{8}-\tfrac{3\theta}{8})},&
v=\tfrac{\sin(\tfrac{5\pi}{8}+\tfrac{3\theta}{8})\sin(-\tfrac{3\theta}{8})}{\sin(\tfrac{5\pi}{4}+\tfrac{3\theta}{8})\sin(\tfrac{5\pi}{8}-\tfrac{3\theta}{8})}\\
w_1&=\tfrac{\sin(\tfrac{5\pi}{8}+\tfrac{3\theta}{8})\sin(\tfrac{5\pi}{4}-\tfrac{3\theta}{8})}{\sin(\tfrac{5\pi}{4}+\tfrac{3\theta}{8})\sin(\tfrac{5\pi}{8}-\tfrac{3\theta}{8})},&
w_2&=\tfrac{\sin(\tfrac{15\pi}{8}+\tfrac{3\theta}{8})\sin(-\tfrac{3\theta}{8})}{\sin(\tfrac{5\pi}{4}+\tfrac{3\theta}{8})\sin(\tfrac{5\pi}{8}-\tfrac{3\theta}{8})}.&
\label{eq_w2c}\end{aligned}$$ Notice that the weights above are all non-negative if and only if $\theta \in [\pi/3,2\pi/3]$. To have a probabilistic interpretation of the model, we limit ourselves to angles in this range. One may more generally define the model on any rhombic tiling, but certain walks may have negative weights (namely $w_1$ and $w_2$ are negative when $\theta > 2 \pi / 3$ and $\theta < \pi/3$, respectively).
Henceforth, we always consider the weights listed above; the associated model will be referred to as the weighted self-avoiding walk. Replacing $\theta$ by $\pi-\theta$, effectively exchanges $u_1$ with $u_2$ and $w_1$ with $w_2$, but does not affect $v$. Hence, there is no ambiguity about which angles parametrise the rhombi.
As explained in [@Gl], if $\theta = \pi/3$, then $w_2 = 0$ and $v = w_1 = u_2 = u_1^2$. Thus, any rhombus may be partitioned into two equilateral triangles, whose intersections with any walk is either void or one arc (see Fig. \[fig:split\]). The weight generated by each rhombus may be computed as the product of two weights associated to the two triangles forming the rhombus, each contributing $1/{\sqrt{2 - \sqrt 2}}$ if traversed by an arc and $1$ otherwise. Thus, if $\Theta$ is the constant sequence equal to $\pi/3$, then each rhombus of $H(\Theta)$ may be partitioned into triangles, and $H(\Theta)$ becomes a triangular lattice (see Fig. \[fig:split\]). The self-avoiding walk model described above becomes that on the hexagonal lattice dual to the triangular one, with weight $1/(\sqrt{2 - \sqrt 2})^{|\gamma|}$ for any SAW $\gamma$ ($|\gamma|$ is the number of edges of $\gamma$). We call this the regular SAW, as it is the most common one.
![A rhombus of angle $\pi/3$ is split into two equilateral triangles. Any triangle contains at most one arc, in which case it contributes $1/{\sqrt{2 - \sqrt 2}}$ to the weight. If all angles are equal to $\pi/3$, all faces of the rhombic tiling (bold black) maybe split into equilateral triangles, and walks may be viewed as regular self-avoiding walks on the hexagonal lattice (gray). []{data-label="fig:split"}](rhombi_to_hex_SAW2.pdf)
In 2009, Cardy and Ikhlef [@CaIk] showed that for these weights, Smirnov’s parafermionic observable (defined later in the text) is partially discretely holomorphic. Employing the original technics developed by Duminil-Copin and Smirnov [@DCSmi10], the first author generalised the calculation of the connective constant to the weighted self-avoiding walk [@Gl]. As a consequence, the weights may be considered critical for the weighted model.
Given two points $a$ and $b$ with integer coordinates on the boundary of the right half-plane, the 2-point function between $a$ and $b$, denoted by $G_\Theta(a,b)$, is the sum of weights of all walks from $a$ to $b$ on ${\mathbb{H}}(\Theta)$ (see Fig. \[fig:theta\_tiling\]): $$\begin{aligned}
G_\Theta(a,b) = \sum_{\gamma \text{ from $a$ to $b$}} {\mathrm{w}}_\Theta(\gamma).\end{aligned}$$ By $G_{\pi/3}(a,b)$ we denote the 2-point function when $\Theta$ is constant, equal to $\pi/3$. As mentioned above, this is the two point function of regular self-avoiding walk on a hexagonal lattice with edge-length $1/\sqrt{3}$.
\[thm-saw-2-point\] Let $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$, where $\theta_k\in [\pi/3,2\pi/3]$ for all $k$. Then $G_\Theta (a,b) = G_{\pi/3}(a,b)$ for any two points $a$ and $b$ on the boundary of the right half-plane.
![ [*Left:*]{} a path contributing to the 2-point funtion $G_{\Theta}(a,b)$. [*Right:*]{} a bridge contributing to $B_{6}(\Theta)$. []{data-label="fig:theta_tiling"}](Theta_tiling.pdf "fig:") ![ [*Left:*]{} a path contributing to the 2-point funtion $G_{\Theta}(a,b)$. [*Right:*]{} a bridge contributing to $B_{6}(\Theta)$. []{data-label="fig:theta_tiling"}](Theta_tiling2.pdf "fig:")
A bridge of width $T$ is a SAW on ${\mathrm{Strip}}_{T}(\Theta)$, starting at $0$ and ending on the right boundary of ${\mathrm{Strip}}_{T}(\Theta)$ (see Fig. \[fig:theta\_tiling\]). The partition function of bridges of width $T$ is $$\begin{aligned}
\label{eq-def-B-theta-t}
B_{T,\Theta} = \sum_{\gamma:\text{ bridge in } {\mathrm{Strip}}_{T}(\Theta)}{{\mathrm{w}}_\Theta(\gamma)}\,,\end{aligned}$$ where the sum is taken over all bridges of width $T$.
For the SAW on the hexagonal lattice it was shown that the total weight of bridges in a strip tends to 0 as the width of the strip tends to infinity [@BBDDG Thm. 10]. We give a new, short proof of this statement which also yields a quantitative bound on the convergence.
\[prop:saw-bridges-hex\] We have $$\begin{aligned}
\label{eq:B_sum}
\sum_{T \geq 1}\frac1T \big(B_T(\pi/3)\big)^3 < \infty.
\end{aligned}$$ As a consequence, the partition function of self-avoiding bridges on the hexagonal lattice vanishes at infinity: $B_T(\pi/3) \xrightarrow[T \to \infty]{} 0$. Moreover $B_T(\pi/3) < 1/(\log T)^{1/3}$ for infinitely many values of $T$.
It is worth mentioning that the proof of the above uses certain symmetries of the hexagonal lattice (most notably the invariance under rotation by $\pi/3$). Hence this proof may not be applied directly to general rhombic tilings $H(\Theta)$. Nevertheless, using Theorem \[thm-saw-2-point\], the part about convergnece of $B_T$ to zero can be extended to weighted self-avoiding walk on any rhombic tiling.
\[thm-saw-bridges\] Let $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$, where $\theta_k\in [\pi/3,2\pi/3]$ for all $k$. Then $B_{T,\Theta} \xrightarrow[T\to \infty]{} 0$.
Our third result refers to self-avoiding walk with fugacity. Weighted self-avoiding walk with surface fugacity may be defined as was done in [@BBDDG] for the regular model. In the half-plane, fugacity rewards (or penalises) walks whenever they approach the boundary by multiplying the weight by some $y\ge 0$. Depending on the value of $y$, a walk chosen with probability proportional to its weight will be either attracted to the boundary or repelled from it. The critical fugacity is the minimal $y$ such that self-avoiding walk with fugacity $y$ “sticks” to the boundary. This description is only illustrative, in fact the total weight of all self-avoiding walks in ${\mathbb{H}}(\Theta)$ is infinite [@Gl Lemma 4.4], and no probability proportional to the weight exists. A precise meaning of critical fugacity will be given below. In order to formally define critical fugacity, we deform the weight of a walk according to its length and its number of visits to the boundary. Let $\Theta = (\theta_k)_{k \geq 1}$ be a family of angles in $[\pi/3,2\pi/3]$ with $\theta_1 = \pi/3$. For a self-avoiding walk $\gamma$ on ${\mathbb{H}}(\Theta)$ define its length $|\gamma|$ as the sum of lengths of each arc, where the lengths of an arc spanning an angle $\theta$ is $\theta \frac{3}{\pi}$ and the length of any straight segment traversing a rhombus is 2. Notice that this definition is such that, when $\Theta$ is constant equal to $\pi/3$, the length of a walk is the number of edges in its representation on the hexagonal lattice. Further write $b(\gamma)$ for the number of times $\gamma$ visits the leftmost column of rhombi as in Fig. \[fig:CR\]. More precisely, recall that each rhombus of the first column may be split into two equilateral triangles, each contributing to ${\mathrm{w}}(\gamma)$ separately. Then $b(\gamma)$ is the number of visits of $\gamma$ to triangles adjacent to the boundary.
Given $x,y\ge 0$, the $x$-deformed weight of a self-avoiding walk $\gamma$ in ${\mathbb{H}}(\Theta)$ with fugacity $y$ is defined as $$\label{eq:weight-modified}
{\mathrm{w}}_\Theta (\gamma;x,y) = {\mathrm{w}}_\Theta (\gamma)\cdot x^{|\gamma|} y^{b(\gamma)}.$$ For $x,y\geq 0$, the partition function of walks in ${\mathbb{H}}(\Theta)$ with fugacity $y$ is defined by: $${\mathrm{SAW}}_\Theta (x,y)= \sum_{\gamma \text{ starts at } 0}{{\mathrm{w}}_\Theta(\gamma; x,y)}\,.$$
The critical fugacity $y_c(\Theta)$ is the positive real number defined by $$\begin{aligned}
\label{eq:yc_def}
y_c(\Theta) = \sup \big\{y \geq 0 \, | \, \forall 0<x<1, {\mathrm{SAW}}_\Theta (x,y) < \infty \big\}\,.\end{aligned}$$
In [@BBDDG] it was proven that the critical fugacity for the regular self-avoiding on the hexagonal lattice is equal to $1+\sqrt{2}$. We prove that the same is true for the self-avoiding walk with integrable weights, given that the rhombi in the first column are of angle $\pi/3$.
\[thm-fugacity\] Let $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$, where $\theta_1= \pi/3$ and $\theta_k\in [\pi/3,2\pi/3]$ for $k>1$. Then $y_c (\Theta) = 1+\sqrt{2}$.
Let us briefly comment on the definition of the critical fugacity. As already mentioned, the partition function of all walks with $x = y =1$ is infinite. This implies directly that ${\mathrm{SAW}}_\Theta (1, y) = \infty$ for all $y >0$. For fixed $y > 0$, write $$x_c(y) = \sup\big\{ x \geq 0 : \ {\mathrm{SAW}}_\Theta (x,y) < \infty \big\}.$$ This definition mimics that of the inverse connective constant for walks with fugacity.
When $y = 1$, that is when no fugacity is added, we have ${\mathrm{SAW}}_\Theta (x,1) < \infty$ for all $x < 1$ (see [@Gl proof of Thm. 1.1]), which is to say $x_c(1) = 1$. The same is true for all $y < y_c(\Theta)$. When $y > y_c(\Theta)$, it follows directly from the definition of the critical fugacity that $x_c(y) <1$.
Thus, a fugacity is supercritical if it affects the value of the “connective constant” of the model. This is exactly the definition of critical fugacity used in [@BBDDG]; we have avoided it here because the connective constant for the weighted model does not appear naturally.
One may also define the critical fugacity $y_c(T, \Theta)$ for a strip ${\mathrm{Strip}}_{T}(\Theta)$ in the same way, simply by replacing ${\mathrm{SAW}}_\Theta(x,y)$ in with the partition function of weighted self-avoiding walks in ${\mathrm{Strip}}_T(\Theta)$: $${\mathrm{SAW}}_{T,\Theta}(x,y) = \sum_{\gamma\,:\, \text{ walk in } {\mathrm{Strip}}_{T}(\Theta)}{{\mathrm{w}}(\gamma;x,y)}.$$
Define also the partition function of weighted bridges by $$B_{T,\Theta}(x,y) = \sum_{\gamma\,:\, \text{ bridge in } {\mathrm{Strip}}_{T}(\Theta)}{{\mathrm{w}}(\gamma;x,y)}.$$ We will consider the above for $x = 1$ as a series in $y$.
\[prop:y\_c\_strip\] Let $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$, where $\theta_1= \pi/3$ and $\theta_k\in [\pi/3,2\pi/3]$ for $k>1$. Then $y_c(T,\Theta)$ is equal to the radius of convergence of $B_{T,\Theta}(1;y)$, and $$y_c(T,\Theta)\xrightarrow[T\to \infty]{} y_c=1+\sqrt{2}.$$
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank Hugo Duminil-Copin for pointing out that the strategy used in Proposition \[prop:saw-bridges-hex\] yields an explicit bound on $B_T$.
The first author was supported by Swiss NSF grant P2GE2\_165093, and partially supported by the European Research Council starting grant 678520 (LocalOrder); he is grateful to the university of Tel-Aviv for hosting him. The second author is member of the NCCR SwissMAP. Part of this work was performed at IMPA (Rio de Janeiro) and at the University of Geneva, we are grateful to both institutions for their hospitality.
Parafermionic observable {#sec:parafermion}
========================
To analyse the behaviour of the self-avoiding walk we will use the *parafermionic observable* introduced by Smirnov in [@Smi_Ising] and its modification to incorporate fugacity introduced in [@BBDDG]. The contour integral of this observable around each rhombus vanishes everywhere except for the part of the boundary where the surface fugacity is inserted. This leads to relations between the partition functions of arcs and bridges that are crucial for our proof.
Observable without fugacity
---------------------------
Fix a sequence $\Theta$ as before. The rows of rhombi of ${\mathbb{H}}(\Theta)$ and ${\mathrm{Strip}}_T(\Theta)$ may be numbered in increasing order by ${\mathbb{Z}}$, with the row $0$ containing the origin on its left boundary. Let ${\mathrm{Rect}}_{T,L}(\Theta)$ be the rectangular-type domain consisting of the rows $-L,\dots, L$ of ${\mathrm{Strip}}_T(\Theta)$ (see Fig. \[fig:CR\]). Denote by $V({\mathrm{Rect}}_{T,L}(\Theta))$ the set of all midpoints of the sides of the rhombi in ${\mathrm{Rect}}_{T,L}(\Theta)$ and by $V(\partial{\mathrm{Rect}}_{T,L}(\Theta))$ the points of $V({\mathrm{Rect}}_{T,L}(\Theta))$ lying on edges of $\partial{\mathrm{Rect}}_{T,L}(\Theta)$ (that is edges of ${\mathrm{Rect}}_{T,L}(\Theta)$ which are only adjacent to one rhombus of ${\mathrm{Rect}}_{T,L}(\Theta)$). Notice that the embedding ensures that $0 \in V(\partial {\mathrm{Rect}}_{T,L}(\Theta))$.
![[*Left:*]{} When $\theta_1 =\pi/3$, the rhombi of the first column may be split into two equilateral triangles. Only visits to the triangles adjacent to the boundary (marked by dots) are counted in $b(\gamma)$. Here $b(\gamma) = 4$. [*Middle:*]{} A rhombus of angle $\theta$ and mid-edges $z_E,z_S,z_N,z_W$. [*Right:*]{} A path ending at $\delta$ in ${\mathrm{Rect}}_{3,2}(\Theta)$. Its winding is $\theta_2$, as for any path ending at this point.[]{data-label="fig:CR"}](fugacity.pdf "fig:") ![[*Left:*]{} When $\theta_1 =\pi/3$, the rhombi of the first column may be split into two equilateral triangles. Only visits to the triangles adjacent to the boundary (marked by dots) are counted in $b(\gamma)$. Here $b(\gamma) = 4$. [*Middle:*]{} A rhombus of angle $\theta$ and mid-edges $z_E,z_S,z_N,z_W$. [*Right:*]{} A path ending at $\delta$ in ${\mathrm{Rect}}_{3,2}(\Theta)$. Its winding is $\theta_2$, as for any path ending at this point.[]{data-label="fig:CR"}](rhombi_change2.pdf "fig:")
The parafermionic observable in the domain ${\mathrm{Rect}}_{T,L}(\Theta)$ (with no fugacity) is the function $F$ defined on $V({\mathrm{Rect}}_{T,L}(\Theta))$ by $$\begin{aligned}
\label{def-parafermion}
F(z)=\sum_{\gamma:0\to z}{{\mathrm{w}}(\gamma)e^{-i\cdot \tfrac{5}{8}\cdot {\mathrm{wind}}(\gamma)}}\qquad \forall z\in V({\mathrm{Rect}}_{T,L}(\Theta)),\end{aligned}$$ where the sum runs over all self-avoiding walks $\gamma$ contained in ${\mathrm{Rect}}_{T,L}(\Theta)$, starting at $0$ and ending at $z$. Above, ${\mathrm{wind}}(\gamma)$ denotes the winding of $\gamma$, i.e. the total angle of rotation of $\gamma$ going from $0$ to $z$ (recall that a walk crosses the sides of rhombi at right angles). For instance, the arc from $z_W$ to $z_N$ in Fig. \[fig:CR\] has winding $\theta$ and the arc from $z_W$ to $z_S$ has winding $\theta - \pi$. Since ${\mathrm{Rect}}_{T,L}(\Theta)$ is a finite region, the sum in the definition of $F$ is finite, hence well-defined.
The value $5/8$ is chosen to render the contour integrals of $F$ null. It is specific to the self-avoiding walk model; similar observables exist for other models, where $5/8$ should be replaced with different values, see [@Smi_Icm06; @CaIk] and [@DCSmi11] for a survey.
The [*partial discrete holomorphicity*]{} stated in the next lemma is a crucial property of the parafermionic observable. Such an observable was first introduced by Smirnov for the FK-Ising model [@Smi_Ising], where it satisfies stronger relations and was used to prove the convergence of interfaces to ${\mathrm{SLE}}$ curves. Later, partial discrete holomorphicity was proved in case of the loop $O(n)$ model [@DCSmi10] on the hexagonal lattice and for the more general loop $O(n)$ model with integrable weights [@CaIk]. Here we state the partial discrete holomorphicity in the form given in [@Gl Lemma 3.1].
\[lem-CR\] The parafermionic observable $F$ satisfies the following relation for each rhombus of ${\mathrm{Rect}}_{T,L}(\Theta)$: $$\begin{aligned}
\label{eq-CR}
F(z_E)-F(z_W)=e^{i\theta}(F(z_S)-F(z_N)),
\end{aligned}$$ where $z_E$, $z_S$, $z_W$ and $z_N$ are the midpoints of the edges of the rhombus, distributed as in Fig. \[fig:CR\].
Equation is reminiscent of the Cauchy–Riemann equations for holomorphic functions; it may also be written as the contour integral of $F$ around any rhombus being null. Summing the real part of over all rhombi in a particular domain yields a relation on the partition function analogous to that of [@DCSmi10]\[Lemma 2\]. Denote the left, right, up and bottom boundaries of ${\mathrm{Rect}}_{T,L}(\Theta)$ by $\alpha$, $\beta$, $\delta$ and $\varepsilon$, respectively. We will use the following notation: $$\begin{aligned}
\label{eq_def_AB}
A_{T,L,\Theta}&=\sum_{\gamma:0\to z\in \alpha}{{\mathrm{w}}(\gamma)}\,,
&B_{T,L,\Theta}&=\sum_{\gamma:0\to z\in \beta}{{\mathrm{w}}(\gamma)}\,,\\
\label{eq_def_DE}
D_{T,L,\Theta}&=\sum_{\gamma:0\to z\in \delta}{\cos(\tfrac{3}{8}{\mathrm{wind}}(\gamma)){\mathrm{w}}(\gamma)}\,,
&E_{T,L,\Theta}&=\sum_{\gamma:0\to z\in \varepsilon}{\cos(\tfrac{3}{8}{\mathrm{wind}}(\gamma)){\mathrm{w}}(\gamma)}\,.\end{aligned}$$ The sums run over all self-avoiding walks in ${\mathrm{Rect}}_{T,L}(\Theta)$ ending at a point in $\alpha$, $\beta$, $\delta$ and $\varepsilon$, respectively. The paths contributing to $A_{T,L,\Theta}$ are called (self-avoiding) arcs.
\[lem\_eq\_rectangle\] For any sequence $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$ of angles between $\tfrac{\pi}{3}$ and $\tfrac{2\pi}{3}$, $$\begin{aligned}
\label{eq_relation_rectangle}
\cos{\tfrac{3\pi}{8}}A_{T,L,\Theta} + B_{T,L,\Theta} + D_{T,L,\Theta} + E_{T,L,\Theta} = 1\,.
\end{aligned}$$
The factor $1$ on the right-hand side of comes from the contribution to $F$ of the empty configuration, which is not accounted for in any of the terms on the left-hand side.
Write $A_{T,\Theta}$ and $ B_{T,\Theta}$ for the partition functions of arcs and bridges, respectively, in ${\mathrm{Strip}}_{T}(\Theta)$.
For any sequence $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$ of angles between $\tfrac{\pi}{3}$ and $\tfrac{2\pi}{3}$, $$\begin{aligned}
\label{eq_relation_strip}
\cos{\tfrac{3\pi}{8}}A_{T,\Theta} + B_{T,\Theta} = 1\,.
\end{aligned}$$
First notice that $A_{T,\Theta} = \lim_{L \to \infty} A_{T,L,\Theta}$ and $B_{T,\Theta} = \lim_{L \to \infty} B_{T,L,\Theta}$. Indeed, any self-avoiding arc of ${\mathrm{Strip}}_{T}(\Theta)$ is contained in a rectangle ${\mathrm{Rect}}_{T,L}(\Theta)$ for $L$ large enough, and hence is accounted for in $A_{T,L,\Theta}$. Since all terms contributing to $A_{T,\Theta}$ are positive, the convergence is proved. The same holds for bridges.
In light of and the above observation, it suffices to prove that $D_{T,L,\Theta}\to 0$ and $E_{T,L,\Theta}\to 0$ as $L \to \infty$. We will prove this for $D_{T,L,\Theta}$, the proof for $E_{T,L,\Theta}$ is identical.
Observe that, any self-avoiding path $\gamma$ contributing to $D_{T,L,\Theta}$ may be completed by at most $T$ steps (that is at most $T$ rhombi with arcs in them) to form a self-avoiding path on ${\mathrm{Strip}}_{T}(\Theta)$, with endpoints $(0,0)$ and $(0,L+1)$. Each rhombus in the completion affects the weight of $\gamma$ by a factor bounded below by some universal constant $c > 0$. Thus using that all angles $\theta_k\in[\pi/3,2\pi/3]$ and hence ${\mathrm{wind}}(\gamma)\geq \pi/3$ we get $$\begin{aligned}
0 \leq D_{T,L,\Theta} \leq c^T \cos\big(\tfrac{\pi}8\big) G_{{\mathrm{Strip}}_T(\Theta)}(0,L+1).
\end{aligned}$$ Finally observe that $$\sum_{L \in {\mathbb{Z}}} G_{{\mathrm{Strip}}_T(\Theta)}(0,L) = A_{T,\Theta} \leq\Big(\cos{\tfrac{3\pi}{8}}\Big)^{-1} < \infty,$$ which implies that $G_{{\mathrm{Strip}}_T(\Theta)}(0,L+1)$ converges to $0$ as $L \to \infty$.
Observable with fugacity
------------------------
The parafermionic observable with fugacity on the boundary was introduced in [@BBDDG] for the hexagonal lattice. It may be adapted easily to our case; we do this below. The observable will be defined inside of rectangles and, for technical reasons, the fugacity will be inserted on the right boundary, rather than on the left. To mark this difference, we add a tilde to all quantities with fugacity on the right.
Let $\Theta = \{\theta_k\}_{k=1}^T$, with $\theta_T= \pi/3$ and $\theta_k\in [\pi/3, 2\pi/3]$ for $1\leq k<T$. Consider ${\mathrm{Rect}}_{T,L}(\Theta)$ and split the rhombi of the last column into equilateral triangles (see Fig. \[fig:split\]). For a SAW $\gamma$ on ${\mathrm{Rect}}_{T,L}(\Theta)$, define its weight as ${\widetilde}{\mathrm{w}}(\gamma;1,y) = {\mathrm{w}}(\gamma)y^{b_r(\gamma)}$, where $b_r(\gamma)$ is the number of visits of $\gamma$ to the triangles adjacent to the right boundary of ${\mathrm{Rect}}_{T,L}(\Theta)$. For $z \in V({\mathrm{Rect}}_{T,L}(\Theta))$, set $$\begin{aligned}
\label{def_parafermion_fug}
{\widetilde}F(z; y)=\sum_{\gamma:0\to z}{{\widetilde}{\mathrm{w}}(\gamma;1,y)e^{-i\cdot\tfrac{5}{8}\cdot {\mathrm{wind}}(\gamma)}}.\end{aligned}$$ It is easy to check (following the same procedure as in [@Gl Lemma 4.1]) that this observable satisfies the same Cauchy-Riemann equation for all rhombi $r$ in columns $1,\dots, T-1$: $$\begin{aligned}
&{\widetilde}F(z_E;y)-{\widetilde}F(z_W;y) - e^{i\theta}({\widetilde}F(z_S;y)-{\widetilde}F(z_N;y)) = 0.\end{aligned}$$ However, for rhombi $r$ in the rightmost column, a “defect” needs to be added to the relation , which thus becomes $$\begin{aligned}
&{\operatorname{Re}}\left[{\widetilde}F(z_E;y)-{\widetilde}F(z_W;y) - e^{i\theta}({\widetilde}F(z_S;y)-{\widetilde}F(z_N;y))\right]
= &&\frac{(y-1)y^*}{y(y^* -1)}\cdot G_{{\mathrm{Rect}}_{T,L}(\Theta)}(0,z_E),\end{aligned}$$ where $$\begin{aligned}
y^* = 1 + \sqrt{2}\,.$$
An analogous of Lemma \[lem\_eq\_rectangle\] may be obtained by summing the real part of the equations above for all rhombi of ${\mathrm{Rect}}_{T,L}(\Theta)$. The result is analogous to [@BBDDG Prop. 4]. We first introduce notation analogous to –; recall that the sides of $R_{T,L}(\Theta)$ are $\alpha,\beta,\delta$ and $\varepsilon$. Set $$\begin{aligned}
{\widetilde}A_{T,L,\Theta}(y)&=\!\!\!\sum_{\gamma:0\to z\in \alpha}\!\!\!{{\widetilde}{\mathrm{w}}(\gamma;1,y)}\,,
&{\widetilde}B_{T,L,\Theta}(y)&=\!\!\!\sum_{\gamma:0\to z\in \beta}\!\!\!{{\widetilde}{\mathrm{w}}(\gamma;1,y)}\,,\\
{\widetilde}D_{T,L,\Theta}(y)&=\!\!\!\sum_{\gamma:0\to z\in \delta}\!\!\!{\cos\big(\tfrac{3}{8}{\mathrm{wind}}(\gamma)\big){\widetilde}{\mathrm{w}}(\gamma;1,y)}\,,
&{\widetilde}E_{T,L,\Theta}(y)&=\!\!\!\sum_{\gamma:0\to z\in \varepsilon}\!\!\!{\cos\big(\tfrac{3}{8}{\mathrm{wind}}(\gamma)\big){\widetilde}{\mathrm{w}}(\gamma;1,y)}\,.\end{aligned}$$
\[lem:rel\_rectangle\_fug\] Let $\Theta= \{\theta_k\}_{1 \leq k \leq T}$, where $\theta_T= \pi/3$ and $\theta_k\in [\pi/3,2\pi/3]$ for $1 \leq k < T$. Then, for any $y >0 $, $$\begin{aligned}
\label{eq:relation_rectangle_fug}
\cos\big(\tfrac{3}{8}\big){\widetilde}A_{T,L,\Theta}(1, y) + \tfrac{y^*-y}{y(y^*-1)}{\widetilde}B_{T,L,\Theta}(1, y) + {\widetilde}D_{T,L,\Theta}(1, y) + {\widetilde}E_{T,L,\Theta}(1, y) = 1\,.
\end{aligned}$$
The proof of this lemma is similar to that of [@BBDDG Prop. 4]; we will not detail it here. The only result of this section that will be used outside of it is the following corollary.
\[cor:small\_y\] Let $\Theta= \{\theta_k\}_{k\in{\mathbb{N}}}$, where $\theta_1 = \frac\pi3$ and $\theta_k\in [\pi/3,2\pi/3]$ for all $k \geq 2$. Assume that $y< 1+\sqrt{2}$. Then $B_{T,\Theta}(y) \leq \tfrac{\sqrt{2}y}{1+\sqrt{2}-y}$.
Fix a sequence $\Theta = (\theta_k)$ as above (with $\theta_1=\pi/3$), a value $T \geq 1$ and $y <1+\sqrt{2}$. Write ${\widetilde}\Theta = (\theta_T,\dots, \theta_1)$ and ${\widetilde}B_{T,{\widetilde}\Theta}(y)$ for the partition function of bridges in ${\mathrm{Strip}}_T({\widetilde}\Theta)$ with fugacity $y$ on the right boundary: $$\begin{aligned}
{\widetilde}B_{T,{\widetilde}\Theta}(y)&=\!\!\!\sum_{\gamma \text{ bridge in ${\mathrm{Strip}}_T({\widetilde}\Theta)$}}\!\!\!{{\widetilde}{\mathrm{w}}_{{\widetilde}\Theta}(\gamma;1,y)(\gamma)}.
\end{aligned}$$ There is an obvious bijection between bridges in ${\mathrm{Strip}}_T({\widetilde}\Theta)$ and those in ${\mathrm{Strip}}_T(\Theta)$: simply reverse the direction of every bridge, mirror it horizontally and shift it vertically so that it starts at row $0$. The weight ${\mathrm{w}}(\gamma)$ of any self-avoiding bridge $\gamma$ is equal to that of its reverse; moreover the winding of any bridge is $0$, whether it is in ${\mathrm{Strip}}_T({\widetilde}\Theta)$ or ${\mathrm{Strip}}_T(\Theta)$. Finally, if bridges in ${\mathrm{Strip}}_T({\widetilde}\Theta)$ are weighted with fugacity $y$ on the right boundary, that corresponds to bridges in ${\mathrm{Strip}}_T(\Theta)$ having fugacity on the left. Thus $$\begin{aligned}
{\widetilde}B_{T,{\widetilde}\Theta}(y) = B_{T,\Theta}(y).
\end{aligned}$$ Next we bound the left-hand side of the above.
Fix some $L > 0$. All walks $\gamma$ in ${\mathrm{Rect}}_{T,L}({\widetilde}\Theta)$ originating at $0$ and with endpoint on $\delta$ and $\varepsilon$ have winding in $[\pi/3,2\pi/3]$ and $[-2\pi/3,-\pi/3]$, respectively. Thus, all terms in are positive when $y<y^*=1+\sqrt{2}$. We find $${\widetilde}B_{T,L,{\widetilde}\Theta}(y) \leq \tfrac{y(y^*-1)}{y^*-y} = \tfrac{\sqrt{2}y}{1+\sqrt{2}-y} \,.$$ Now observe that ${\widetilde}B_{T,{\widetilde}\Theta}(y) = \lim_{L \to \infty} {\widetilde}B_{T,L,{\widetilde}\Theta}(1, y)$. Indeed, any bridge contributing to ${\widetilde}B_{T,{\widetilde}\Theta}(y)$ has a finite vertical span, and is therefore included in ${\widetilde}B_{T,L,{\widetilde}\Theta}(1, y)$ for $L$ large enough. Moreover, all terms in the sum defining ${\widetilde}B_{T,{\widetilde}\Theta}(y)$ are positive. Since the bound for ${\widetilde}B_{T,L,\Theta}(y)$ above is uniform in $L$, it extends to ${\widetilde}B_{T,{\widetilde}\Theta}$ and thus to $B_{T,\Theta}$.
The Yang–Baxter equation {#sec:YangBaxter}
========================
For this section only we will consider a slight generalisation of the model described above. First of all, we will consider rhombi with any angles in $(0,\pi)$. Secondly, we will consider walks on any rhombic tiling; rather than defining this properly, we direct the reader to the examples of figures \[fig\_YB\] and \[fig:saw-YB\]. Finally, we consider also families of walks rather than a single one. For $\gamma_1,\dots, \gamma_n$ a collection of (finite) self-avoiding walks such that all rhombi intersected by $\gamma_1 \cup \dots \cup \gamma_n$ are in one of the settings of Fig. \[figWeights\], define the weight of the family as the product of the weights of each rhombus.
\[prop:YB\] Let ${\mathsf H}$ be a hexagon formed of three rhombi as in Fig. \[fig\_YB\], left diagram. Write $\partial {\mathsf H}$ for the six edges of ${\mathsf H}$ shared by only one rhombus. Let ${\mathsf H}'$ be the rearrangement of the three rhombi that form ${\mathsf H}$, as in Fig. \[fig\_YB\], middle diagram. For any choice of distinct vertices $x_1,y_1,\dots,x_k,y_k$ on the edges of $\partial {\mathsf H}$, $$\sum_{\substack{\gamma_1,\dots, \gamma_k \subset{\mathsf H}\\ \gamma_i \colon x_i\to y_i}} {\mathrm{w}}_{{\mathsf H}}(\gamma_1 \cup \dots \cup \gamma_k)
= \sum_{\substack{\gamma_1,\dots, \gamma_k \subset{\mathsf H}'\\ \gamma_i \colon x_i\to y_i}} {\mathrm{w}}_{{\mathsf H}'}(\gamma_1 \cup \dots \cup \gamma_k)$$ In other words, for any pairs of points on the boundary, the weight of walks connecting these pairs is the same in ${\mathsf H}$ and ${\mathsf H}'$.
The proof consists simply of listing for each choice of $x_1,y_1,\dots,x_k,y_k$ ($k$ is always smaller than $3$) the weights for all possible connections in the two tilings and explicitly computing their sum. The weights were derived in [@N90] to satisfy these equations. Cardy and Ikhlef [@CaIk] found the same weights based on discrete holomorphicity. The connection between the two was explored in [@AB], where the Yang-Baxter equations are explicitly listed.
Equivalent relations may be obtained for any model with loop-weight between $0$ and $2$, with appropriate weight as functions of $n$ (see [@Gl] for the exact formulae). All three papers quoted above deal with general loop-weight; we only treat here the case of null loop-weight.
As a consequence, if a large rhombic tiling contains three rhombi as in Fig. \[fig\_YB\], they may be rearranged without affecting the two point function for pairs of points outside of these three rhombi.
Let $\Omega$ be a rhombic tiling containing a hexagon ${\mathsf H}$ formed of three rhombi as in Proposition \[prop:YB\]. Denote by $\Omega'$ the tiling that coincides with $\Omega$ everywhere except for ${\mathsf H}$, where the three rhombi are rearranged as ${\mathsf H}'$. Then, for any two vertices $a,b$ of $\Omega$ that are not in ${\mathsf H}\setminus\partial{\mathsf H}$, $$\begin{aligned}
\label{eq:HH'}
\sum_{\substack{\gamma\subset\Omega \\ \gamma \colon a\to b}} {\mathrm{w}}_{\Omega}(\gamma)
= \sum_{\substack{\gamma\subset\Omega' \\ \gamma \colon a\to b}} {\mathrm{w}}_{\Omega'}(\gamma).
\end{aligned}$$
We only sketch this. Write the sums in as double sums. First sum over all possible configurations outside $H$ (and $H'$ respectively), then over those inside $H$ (or $H'$) which lead to a single path connecting $a$ to $b$. The inside sum on the right and left hand side is equal due to Proposition \[prop:YB\]; the outside weights are equal in $\Omega$ and $\Omega'$, since the two tilings are identical outside $H$ and $H'$, respectively.
![[*Left:*]{} A hexagon formed of three rhombi of different angles. [*Middle:*]{} The three rhombi may be rearranged to cover the same domain in a different fashion. In the left image, the pairs of points $x_1,y_1$ and $x_2,y_2$ are connected in a single configuration; in the middle image, the same connections are obtained in two distinct configurations. The weight of the left configuration is equal to the sum of the weights of the two middle ones. [*Right:*]{} In a domain, changing three such rhombi does not alter the two point function between points $a$ and $b$.[]{data-label="fig_YB"}](YB_Pentagon2.pdf "fig:") ![[*Left:*]{} A hexagon formed of three rhombi of different angles. [*Middle:*]{} The three rhombi may be rearranged to cover the same domain in a different fashion. In the left image, the pairs of points $x_1,y_1$ and $x_2,y_2$ are connected in a single configuration; in the middle image, the same connections are obtained in two distinct configurations. The weight of the left configuration is equal to the sum of the weights of the two middle ones. [*Right:*]{} In a domain, changing three such rhombi does not alter the two point function between points $a$ and $b$.[]{data-label="fig_YB"}](YB_Pentagon3.pdf "fig:")
Self-avoiding bridges and the 2-point function {#sec:bridges}
==============================================
During the whole section we consider half-space rhombic tilings $H(\Theta)$. Write $H(\pi/3)$ for the tiling with all angles equal to $\pi/3$. Recall that SAW on $H(\pi/3)$ is identical to that on the hexagonal lattice with the weight of a path $\gamma$ given by $(\sqrt{2+\sqrt{2}})^{-|\gamma|}$.
In this section we prove Theorems \[thm-saw-2-point\] and \[thm-saw-bridges\]. Theorem \[thm-saw-2-point\] is shown by means of the Yang-Baxter transformation, which is used to gradually transform the lattice $H(\pi/3)$ into an arbitrary lattice $H(\Theta)$. The relation between the partition functions of arcs and bridges in a strip together with Theorem \[thm-saw-2-point\] may be used to transfer the conclusion of Theorem \[thm-saw-bridges\] from the hexagonal lattice to any lattice $H(\Theta)$. Theorem \[thm-saw-bridges\] for the hexagonal lattice was proven in [@BBDDG]; we provide below a new, shorter proof relying only on the parafermionic observable (see Proposition \[prop:saw-bridges-hex\]), that also provides an explicit (albeit weak) bound on $B_T$.
Proof of Theorem \[thm-saw-bridges\] for the hexagonal lattice.
---------------------------------------------------------------
We will only work here with $H(\pi/3)$. Recall that weighted self-avoiding walk on $H(\pi/3)$ may be viewed as regular self-avoiding walk on a half space hexagonal lattice.
Consider the strip ${\mathrm{Strip}}_{2L+1}(\pi/3)$ with width of $2L+1$ hexagons and inscribe inside it an equilateral triangle ${{\rm Tri}}_{L}$ of side-length $2L+1$ in such a way that the midpoint of its vertical side is 0 (see Fig. \[fig:saw-contour-triangle-strip\]). Let $A^{\Delta}_{2L+1}$ be the partition function of walks starting at $0$, contained in the triangle, and ending on its left side; write $D^{\Delta}_{2L+1}$ for the partition function of walks ending on any of the two other sides of the triangle (see Fig. \[fig:saw-contour-triangle-strip\]).
The partition function $D^{\Delta}_{2L+1}$ is decreasing in $L$ and $$\begin{aligned}
\label{eq:triangle}
B_{2L+1} \leq \cos\left( \tfrac{\pi}{8}\right) D^{\Delta}_{2L+1} .
\end{aligned}$$
By summing the real part of as in the proof of Lemma \[lem\_eq\_rectangle\], we obtain: $$\cos\left( \tfrac{3\pi}{8}\right)A^{\Delta}_{2L+1} + \cos\left( \tfrac{\pi}{8}\right) D^{\Delta}_{2L+1} = 1\,.$$ All walks contributing to $A^{\Delta}_{2L+1}$ also contribute to $A^{\Delta}_{2L+3}$, which implies that $A^{\Delta}_{2L+1}$ is increasing in $L$. By the above equation, $D^{\Delta}_{2L+1}$ is decreasing in $L$.
Moreover, $A^{\Delta}_{2L+1} \leq A_{2L+1} $ since the latter partition function is over a larger set of walks. By eq. : $$\begin{aligned}
B_{2L+1} = 1- \cos\left( \tfrac{3\pi}{8}\right) A_{2L+1} \leq 1- \cos\left( \tfrac{3\pi}{8}\right) A^\Delta_{2L+1} =\cos\left( \tfrac{\pi}{8}\right) D^{\Delta}_{2L+1}.
\end{aligned}$$ This provides the desired conclusion.
![ [*Left:*]{} The strip of width $T=2L+1$ and the equilateral triangle ${{\rm Tri}}_{L}$ of size $2L+1$ inscribed inside it. [*Middle:*]{} The same strip and three examples of walks: one arc contributing to $A_T$ (blue) and two bridges contributing to $B_T$. [*Right:*]{} Three examples of walks in ${{\rm Tri}}_{L}$: one arc contributing to $A^{\Delta}_T$ (blue) and two walks ending on the other sides of the triangle and contributing to $D^{\Delta}_T$. The one ending on the top contributes to ${\mathrm{Ang}}_{L,K}^\Delta$ []{data-label="fig:saw-contour-triangle-strip"}](hex_contour_triang_strip2.pdf "fig:") ![ [*Left:*]{} The strip of width $T=2L+1$ and the equilateral triangle ${{\rm Tri}}_{L}$ of size $2L+1$ inscribed inside it. [*Middle:*]{} The same strip and three examples of walks: one arc contributing to $A_T$ (blue) and two bridges contributing to $B_T$. [*Right:*]{} Three examples of walks in ${{\rm Tri}}_{L}$: one arc contributing to $A^{\Delta}_T$ (blue) and two walks ending on the other sides of the triangle and contributing to $D^{\Delta}_T$. The one ending on the top contributes to ${\mathrm{Ang}}_{L,K}^\Delta$ []{data-label="fig:saw-contour-triangle-strip"}](hex_contour_triang_strip2.pdf "fig:") ![ [*Left:*]{} The strip of width $T=2L+1$ and the equilateral triangle ${{\rm Tri}}_{L}$ of size $2L+1$ inscribed inside it. [*Middle:*]{} The same strip and three examples of walks: one arc contributing to $A_T$ (blue) and two bridges contributing to $B_T$. [*Right:*]{} Three examples of walks in ${{\rm Tri}}_{L}$: one arc contributing to $A^{\Delta}_T$ (blue) and two walks ending on the other sides of the triangle and contributing to $D^{\Delta}_T$. The one ending on the top contributes to ${\mathrm{Ang}}_{L,K}^\Delta$ []{data-label="fig:saw-contour-triangle-strip"}](hex_contour_triang_strip2.pdf "fig:")
We are in the position now to prove Proposition \[prop:saw-bridges-hex\].
By it suffices to show the conclusions of the proposition for $D_{T}^\Delta$ instead of $B_T$.
Recall the notation $G_{\pi/3}(a,b)$ for the 2-point function of walks on $H(\pi/3)$. By , $\lim_{T \to \infty} A_T \leq 1/\cos\left( \tfrac{3\pi}{8}\right)$. The limit above is the partition function of all arcs: $$\lim_{T \to \infty} A_T = \sum_{b \in {\mathbb{Z}}} G(0,k) = 2\sum_{k \geq1} G(0,k).$$ For $L > 0$ and $0 \leq K \leq 2L$, write ${\mathrm{Ang}}^\Delta_{L,K}$ for the partition function of walks in ${{\rm Tri}}_L$, starting at $0$ and ending on the top boundary, $K$ units from the left boundary (see fig \[fig:saw-contour-triangle-strip\]). Then, by vertical symmetry, $$\begin{aligned}
D^\Delta_{2L+1} = 2 \sum_{K=0}^{2L}{\mathrm{Ang}}^\Delta_{L,K}.\end{aligned}$$
Fix $L > 0$. Using concatenations of walks contributing to ${\mathrm{Ang}}^\Delta_{L,K}$ we may construct arcs contributing to $\sum_{b \geq 0} G(-L,b)$ as follows. Divide the right half-plane $H(\pi/3)$ using the lines $arg (z) =\pm \tfrac{\pi}{6}$ into three $\tfrac{\pi}{3}$-angles. For $0 \leq K_3\leq 2K_2\leq 4 K_1\leq 8L$ and walks $\gamma^{(1)},\gamma^{(2)}, \gamma^{(3)}$ contributing to ${\mathrm{Ang}}^\Delta_{L,K_1},{\mathrm{Ang}}^\Delta_{K_1,K_2}$ and ${\mathrm{Ang}}^\Delta_{K_2,K_3}$, respectively, obtain a walk contributing to $G_{-L,K_3}$ by concatenating the translate of $\gamma_1$ by $(0,-L)$, the rotation by $\pi/3$ of the translate of $\gamma^{(2)}$ by $(0,K_1)$, and the rotation by $2\pi/3$ of the translate of $\gamma^{(3)}$ by $(0,K_2)$. By summing over all values of $K_1,K_2,K_3$ we find $$\begin{aligned}
\sum_{K_1=0}^{2L} {\mathrm{Ang}}^\Delta_{L,K_1} \sum_{K_2=0}^{2K_1} {\mathrm{Ang}}^\Delta_{K_1,K_2} \sum_{K_3 = 0}^{2K_2} {\mathrm{Ang}}^\Delta_{K_2,K_3} \leq \sum_{k=L}^{9L}G(0,k).\end{aligned}$$ The sum on the right hand side goes from $L$ to $9L$ since the span of the obtained arc is $K_3 + L$, thus between $L$ and $9L$. Now notice that, by definition, the last sum on the left-hand side is equal to $D^\Delta_{K_2}/2$. This is a decreasing quantity in $K_2$, thus $$\begin{aligned}
\sum_{k=L}^{9L}G(0,k)
&\geq \tfrac12\sum_{K_1=0}^{2L} {\mathrm{Ang}}^\Delta_{L,K_1} \sum_{K_2=0}^{2K_1} {\mathrm{Ang}}^\Delta_{K_1,K_2}\ D^\Delta_{K_2}
\geq \tfrac12\sum_{K_1=0}^{2L} {\mathrm{Ang}}^\Delta_{L,K_1} \sum_{K_2=0}^{2K_1} {\mathrm{Ang}}^\Delta_{K_1,K_2}\ D^\Delta_{2K_1}\\
&= \tfrac14\sum_{K_1=0}^{2L} {\mathrm{Ang}}^\Delta_{L,K_1}\ D^\Delta_{K_1} \ D^\Delta_{2K_1}
\geq \tfrac14\sum_{K_1=0}^{2L} {\mathrm{Ang}}^\Delta_{L,K_1}\ D^\Delta_{2L}\ D^\Delta_{4L}\\
&= \tfrac18 D^\Delta_{L}\ D^\Delta_{2L}\ D^\Delta_{4L}
\geq \tfrac18 \big(D^\Delta_{4L}\big)^3.\end{aligned}$$ Summing the above over $L= 9^k$ we find $$\begin{aligned}
\sum_{k=1}^{\infty}\big(D^\Delta_{4 \cdot 9^k}\big)^3 \leq 8 \sum_{k=1}^{\infty}G(0,k) < \infty.\end{aligned}$$ Now, using the monotonicity in $T$ of $D^\Delta_T$ we may write $$\begin{aligned}
\sum_{k=1}^{\infty}\big(D^\Delta_{4 \cdot 9^k}\big)^3
\geq \sum_{k=1}^{\infty} \frac{1}{32 \cdot 9^k}\sum_{T = 4 \cdot 9^k}^{4 \cdot 9^{k+1} - 1}\big(D^\Delta_{T}\big)^3
\geq \frac{1}{8} \sum_{T = 4}^\infty \frac{1}{T}\big(D^\Delta_{T}\big)^3.\end{aligned}$$ Thus we have proved . This implies in particular that the summand in the last term is smaller than $1/\log T$ for infinitely many values of $T$. More precisely $\big( D ^\Delta_{T} (\log T)^{1/3}\big)_{T}$ contains a subsequence converging to $0$. Finally, since $D^\Delta_{T}$ is decreasing, this implies $D^\Delta_{T} \xrightarrow[T \to \infty]{}0$. The conclusions translate to $B_T$ using .
Proof of Theorem \[thm-saw-2-point\] via the Yang-Baxter equation
-----------------------------------------------------------------
Now we are in the position to prove that the 2-point function is independent of the chosen tiling. First we show that the 2-point function in a strip does not depend on the order of the columns of rhombuses in the tiling. The strategy used here is reminiscent of the use of the Yang-Baxter equation to prove the commutation of transfer matrices, and of the strategy of [@GriMan14].
![ [*Leftmost:*]{} The domain $D_0$ obtained by adding a rhombus $r$ to the rectangle ${\mathrm{Rect}}_{T,L}(\Theta)$. [*Second from the left:*]{} The tiling $D_1$ is the result of the first Yang–Baxter transformation applied in the bold region of $D_0$. [*Third from the left:*]{} After two Yang–Baxter transformations $r$ is pushed down by $2$ units and we obtain $D_2$. [*Rightmost:*]{} After $2L$ repetitions, the rhombus $r$ is pushed all the way to the bottom of ${\mathrm{Rect}}_{T,L}(\Theta)$ and the two columns of rhombi are exchanged. The resulting tiling is $D_{2L}$.[]{data-label="fig:saw-YB"}](YB_transformation.pdf "fig:") ![ [*Leftmost:*]{} The domain $D_0$ obtained by adding a rhombus $r$ to the rectangle ${\mathrm{Rect}}_{T,L}(\Theta)$. [*Second from the left:*]{} The tiling $D_1$ is the result of the first Yang–Baxter transformation applied in the bold region of $D_0$. [*Third from the left:*]{} After two Yang–Baxter transformations $r$ is pushed down by $2$ units and we obtain $D_2$. [*Rightmost:*]{} After $2L$ repetitions, the rhombus $r$ is pushed all the way to the bottom of ${\mathrm{Rect}}_{T,L}(\Theta)$ and the two columns of rhombi are exchanged. The resulting tiling is $D_{2L}$.[]{data-label="fig:saw-YB"}](YB_transformation.pdf "fig:") ![ [*Leftmost:*]{} The domain $D_0$ obtained by adding a rhombus $r$ to the rectangle ${\mathrm{Rect}}_{T,L}(\Theta)$. [*Second from the left:*]{} The tiling $D_1$ is the result of the first Yang–Baxter transformation applied in the bold region of $D_0$. [*Third from the left:*]{} After two Yang–Baxter transformations $r$ is pushed down by $2$ units and we obtain $D_2$. [*Rightmost:*]{} After $2L$ repetitions, the rhombus $r$ is pushed all the way to the bottom of ${\mathrm{Rect}}_{T,L}(\Theta)$ and the two columns of rhombi are exchanged. The resulting tiling is $D_{2L}$.[]{data-label="fig:saw-YB"}](YB_transformation.pdf "fig:") ![ [*Leftmost:*]{} The domain $D_0$ obtained by adding a rhombus $r$ to the rectangle ${\mathrm{Rect}}_{T,L}(\Theta)$. [*Second from the left:*]{} The tiling $D_1$ is the result of the first Yang–Baxter transformation applied in the bold region of $D_0$. [*Third from the left:*]{} After two Yang–Baxter transformations $r$ is pushed down by $2$ units and we obtain $D_2$. [*Rightmost:*]{} After $2L$ repetitions, the rhombus $r$ is pushed all the way to the bottom of ${\mathrm{Rect}}_{T,L}(\Theta)$ and the two columns of rhombi are exchanged. The resulting tiling is $D_{2L}$.[]{data-label="fig:saw-YB"}](YB_transformation.pdf "fig:")
\[prop:saw-YB-strip\] Let ${\mathrm{Strip}}_T(\Theta)$ be a vertical strip tiled with $T$ columns with angles $\theta_1, \dots, \theta_T$. Then for any $a,b$ on the boundary of ${\mathrm{Strip}}_T(\Theta)$ the 2-point function $G(a,b)$ does not depend on the order of angles.
Let ${\mathrm{Strip}}_T(\Theta)$ be a strip as in the statement of the proposition and $a,b$ be two points on its boundary. For $1 \leq i < T$ denote by $\tau_i$ the transposition of $i$ and $i+1$ and by $\Theta \circ \tau_i$ the sequence with $\theta_i$ and $\theta_{i+1}$ transposed: $$\begin{aligned}
\Theta \circ \tau_i = (\theta_1, \dots,\theta_{i-1}, \theta_{i+1},\theta_i,\theta_{i+2},\dots \theta_T).
\end{aligned}$$ In order to prove the proposition, it is sufficient to show that the partition function in $G(a,b)$ in ${\mathrm{Strip}}_T(\Theta)$ is equal to the one in ${\mathrm{Strip}}_T(\Theta\circ \tau_i)$.
This is done by means of the Yang–Baxter transformation, which transforms the rhombic tiling while preserving the partition function (see Section \[sec:YangBaxter\] and references therein for more details).
Fix two points $a$ and $b$ on the boundary of ${\mathrm{Strip}}_T(\Theta)$ and $\varepsilon>0$. For the sake of this proof, if $D$ denotes a part of ${\mathrm{Strip}}_T(\Theta)$ or ${\mathrm{Strip}}_T(\Theta\circ\tau_i)$ (containing $a,b$ on its boundary), write $G_{D}(a,b)$ for the two point function of walks in $D$: $$\begin{aligned}
G_D(a,b) = \sum_{\substack{\gamma \text{ from $x$ to $y$};\\ \gamma \subset D}} {\mathrm{w}}(\gamma).
\end{aligned}$$ First observe that there exists $L >0$ such that $$\begin{aligned}
G_{{\mathrm{Strip}}_T(\Theta)}(a,b) - \varepsilon &\leq G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b) \leq G_{{\mathrm{Strip}}_T(\Theta)}(a,b) \qquad \text{ and }\\
G_{{\mathrm{Strip}}_T(\Theta\circ \tau_i)}(a,b) - \varepsilon &\leq G_{{\mathrm{Rect}}_{T,L}(\Theta\circ \tau_i)}(a,b) \leq G_{{\mathrm{Strip}}_T(\Theta\circ \tau_i)}(a,b).
\end{aligned}$$ (Above we used that the 2-point function is finite, which is the case due to .) Without loss of generality, we may suppose $\theta_i < \theta_{i+1}$ [^1]. Let $D_0$ be the graph obtained by adding a rhombus $r$ to $ {\mathrm{Rect}}_{T,L}(\Theta)$ at the top of the columns $i$ and $i+1$. Precisely, the added rhombus has two sides equal to the top sides of the columns $i$ and $i+1$; the condition $\theta_i < \theta_{i+1}$ ensures that $r$ does not overlap with the rhombi of ${\mathrm{Rect}}_{T,L}(\Theta)$, and $D_0$ is a rhombic tiling (see Fig. \[fig:saw-YB\]). Then we have $$\begin{aligned}
G_{D_0}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b)
= \sum_{\substack{\gamma:x \to y \\ \gamma \text{ uses $r$}}} {\mathrm{w}}(\gamma).
\end{aligned}$$ A path $\gamma$ contributing to the above traverses $r$ only as one arc, hence always has positive weight. In particular, $G_{D_0}(a,b) \geq G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b)$.
On the other hand, to any $\gamma$ as in the sum above, associate the walk $\gamma'$ in ${\mathrm{Rect}}_{T,L+1}(\Theta)$ that connects $a$ to $b$, obtained by keeping the same configuration in ${\mathrm{Rect}}_{T,L}(\Theta)$ as in $D_0$ and replacing the one arc in $r$ by two arcs in the top row of ${\mathrm{Rect}}_{T,L+1}(\Theta)$. Then the ratio of the weight of $\gamma$ and $\gamma'$ is bounded above by some universal constant $c$. Thus $$\begin{aligned}
G_{D_0}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b)
\leq c \big(G_{{\mathrm{Rect}}_{T,L+1}(\Theta)}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b)\big) < c \cdot \varepsilon.
\end{aligned}$$
Apply the Yang-Baxter transformation to the added rhombus and the two rhombi adjacent to it (notice that these indeed form a hexagon). This in effect slides the added rhombus one unit down (see fig. \[fig:saw-YB\]). Call $D_1$ the resulting graph and conclude that $$G_{D_0}(a,b) = G_{D_1}(a,b).$$ The operation may be repeated to slide the added rhombus one more unit downwards. Performing $2L$ such Yang–Baxter transformations leads to $$G_{D_0}(a,b) = G_{D_{2L}}(a,b),$$ where $D_{2L}$ is the rhombic tiling ${\mathrm{Rect}}_{T,L}(\Theta\circ \tau_i)$ with the additional added rhombus at the bottom of columns $i$ and $i+1$.
By the same reasoning as above, $$\begin{aligned}
0 \leq G_{D_{2L}}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta \circ \tau_i)}(a,b)
\leq c \cdot \varepsilon.
\end{aligned}$$ Thus, we conclude that $$\begin{aligned}
c \cdot \varepsilon&> |G_{D_{2L}}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta \circ \tau_i)}(a,b) | \\
&= |G_{D_{0}}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta \circ \tau_i)}(a,b) |\\
&\geq
| G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta \circ \tau_i)}(a,b) | -
|G_{D_{0}}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b) |.
\end{aligned}$$ The last term above is also bounded by $c\cdot\varepsilon$, and we find $$\begin{aligned}
| G_{{\mathrm{Strip}}_{T}(\Theta)}(a,b) - G_{{\mathrm{Strip}}_{T}(\Theta \circ \tau_i)}(a,b) |
& \leq
| G_{{\mathrm{Strip}}_{T}(\Theta)}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b) | \\
&+| G_{{\mathrm{Rect}}_{T,L}(\Theta)}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta \circ \tau_i)}(a,b) |\\
&+| G_{{\mathrm{Strip}}_{T}(\Theta\circ \tau_i)}(a,b) - G_{{\mathrm{Rect}}_{T,L}(\Theta \circ \tau_i)}(a,b) | \leq (2+2c)\varepsilon.
\end{aligned}$$ Since $\varepsilon$ may be chosen arbitrarily small, we find $ G_{{\mathrm{Strip}}_{T}(\Theta)}(a,b) = G_{{\mathrm{Strip}}_{T}(\Theta \circ \tau_i)}(a,b) $, which is the desired conclusion.
Lemma \[prop:saw-YB-strip\] allows us to exchange columns of different angles but it does not permit to change the angles. Next lemma deals with this question and tells us that the 2-point function in a strip decreases when one of the angles is replaced by $\pi/3$.
\[lem:saw-strip-monotonicity\] Let $\Theta = (\theta_1,\dots, \theta_T)$ be a finite sequence of angles with $\theta_k \in [\pi/3,2\pi/3]$ for all $k$. Then for any two points $a,b$ on the left boundary of ${\mathrm{Strip}}_{T}(\Theta)$ we have $$\begin{aligned}
G_{{\mathrm{Strip}}_T(\Theta)} (a,b) \geq G_{{\mathrm{Strip}}_T(\theta_1,\theta_2,\dots,\theta_{T-1},\pi/3)} (a,b).
\end{aligned}$$
Let $\Theta,T, a,b$ be as in the statement. Write $\tilde\Theta $ for the sequence $(\theta_1,\theta_2,\dots,\theta_{T-1},\pi/3)$. We will show that any self-avoiding walk $\gamma$ from $a$ to $b$ in ${\mathrm{Strip}}_T(\Theta)$ has either the same or larger weight than its correspondent walk in ${\mathrm{Strip}}_T(\tilde \Theta)$.
![An arc in ${\mathrm{Strip}}_T(\Theta)$ (left) and the corresponding arc in ${\mathrm{Strip}}_T(\tilde \Theta)$ (right). The difference in weight comes from three types of rhombi depicted in the middle. The first two come in pairs and their combined weight is lowest when $\theta = \pi/3$; the third one has lowest weight when $\theta = \pi/3$. []{data-label="fig:YB_application"}](YB_application.pdf)
Indeed, consider any such walk $\gamma$ in ${\mathrm{Strip}}_T(\Theta)$. The intersection of $\gamma$ with the rightmost column of ${\mathrm{Strip}}_T(\Theta)$ is formed of a family of disjoint arcs, as depicted in Fig. \[fig:YB\_application\]. Write $\chi_1,\dots, \chi_\ell$ for these arcs (take $\ell = 0$ if $\gamma$ does not visit column $T$). The weight of each such arc only depends on $\theta_T$: an arc $\chi_j$ is formed of a rhombus of type $u_1$, a number $k \geq 0$ of rhombi or type $v$ and one rhombus of type $u_2$; its weight is then $$\begin{aligned}
\label{eq:arc_weight}
{\mathrm{w}}_{\theta_T}(\chi_j) =
\frac{{\sin(\frac{5\pi}{4})\sin(\frac{5\pi}{8}+\frac{3\theta_T}{8})}
\big[{\sin(\frac{5\pi}{8}+\frac{3\theta_T}{8})\sin(-\frac{3\theta_T}{8})}\big]^k
{\sin(\frac{5\pi}{4})\sin(\frac{3\theta_T}{8})}}
{\big[\sin(\frac{5\pi}{4}+\frac{3\theta_T}{8})\sin(\frac{5\pi}{8}-\frac{3\theta_T}{8})\big]^{k+2}}
\end{aligned}$$ Moreover, the difference of the weight of $\gamma$ in ${\mathrm{Strip}}_T(\Theta)$ and ${\mathrm{Strip}}_T(\tilde \Theta)$ comes only from the arcs $\chi_1,\dots, \chi_\ell$: $$\begin{aligned}
\frac{{\mathrm{w}}_\Theta(\gamma)}{{\mathrm{w}}_{\tilde \Theta}(\gamma)} = \prod_{j=1}^\ell \frac{{\mathrm{w}}_{\theta_T}(\chi_j)}{{\mathrm{w}}_{\pi/3}(\chi_j)}.
\end{aligned}$$ A direct computation shows that, for any $k \geq 0$, the weight in is minimised when $\theta_T = \pi/3$. Thus, all terms in the right-hand side of the above equality are greater than $1$, and the conclusion is reached.
Let $\Theta = (\theta_1,\dots, \theta_T)$ be a finite sequence of angles with $\theta_k \in [\pi/3,2\pi/3]$ for all $k$. Then for any two points $a,b$ on the left boundary of ${\mathrm{Strip}}_{T}(\Theta)$ we have $$\begin{aligned}
\label{eq:strip_bound1}
G_{{\mathrm{Strip}}_T(\Theta)} (a,b) \geq G_{{\mathrm{Strip}}_T(\pi/3,\theta_1,\theta_2,\dots,\theta_{T-1})} (a,b).
\end{aligned}$$ Additionally, $$\begin{aligned}
\label{eq:strip_bound2}
G_{{\mathrm{Strip}}_T(\Theta)} (a,b) \geq G_{{\mathrm{Strip}}_T(\pi/3)} (a,b),
\end{aligned}$$ where the right hand side is the strip of width $T$ with all angles equal to $\pi/3$.
With the notation above, Lemma \[lem:saw-strip-monotonicity\] states that $$G_{{\mathrm{Strip}}_T(\Theta)} (a,b) \geq G_{{\mathrm{Strip}}_T(\theta_1,\theta_2,\dots,\theta_{T-1},\pi/3)} (a,b).$$ Apply Proposition \[prop:saw-YB-strip\] to deduce that $$G_{{\mathrm{Strip}}_T(\pi/3,\theta_1,\theta_2,\dots,\theta_{T-1})} (a,b) = G_{{\mathrm{Strip}}_T(\theta_1,\theta_2,\dots,\theta_{T-1},\pi/3)} (a,b).$$ This proves the first bound . To obtain it suffices to apply repeatedly .
Now we are ready to prove Theorem \[thm-saw-2-point\].
Recall : $\cos{\tfrac{3\pi}{8}}A_{T,\Theta} = 1 - B_{T,\Theta}$ for any $T$ and sequence $\Theta$. Applying the above to the constant sequence $\pi/3$ and keeping in mind Proposition \[prop:saw-bridges-hex\], we find $$\begin{aligned}
A_{T,\pi/3} \to \Big(\cos{\tfrac{3\pi}{8}}\Big)^{-1}, \qquad \text{ as $T \to \infty$}.
\end{aligned}$$ Now apply to deduce that $$\begin{aligned}
A_{T,\Theta} = \sum_{ L \in {\mathbb{Z}}} G_{{\mathrm{Strip}}_T(\Theta)}(0,L) \geq \sum_{ L \in {\mathbb{Z}}} G_{T,(\pi/3)}(0,L)= A_{T,(\pi/3)}.
\end{aligned}$$ Thus $\lim_{T\to \infty}A_{T,\Theta} \geq \big(\cos{\tfrac{3\pi}{8}}\big)^{-1}$. However, from applied to $\Theta$, we find $A_{T,\Theta} \leq \big(\cos{\tfrac{3\pi}{8}}\big)^{-1}$ for all $T$. Thus $$\begin{aligned}
\sum_{ L \in {\mathbb{Z}}} G_{\Theta}(0,L) = \sum_{ L \in {\mathbb{Z}}} \lim_{T \to \infty } G_{{\mathrm{Strip}}_T(\Theta)}(0,L) =\lim_{T \to \infty }\sum_{ L \in {\mathbb{Z}}} G_{{\mathrm{Strip}}_T(\Theta)}(0,L) &= \Big(\cos{\tfrac{3\pi}{8}}\Big)^{-1} \\
&= \sum_{L\in {\mathbb{Z}}}G_{\pi/3}(0,L).
\end{aligned}$$ Considering that $$G_{\Theta}(0,L)\geq \lim_{T\to \infty}G_{T}(\Theta)(0,L) \geq \lim_{T\to \infty}G_{T,(\pi/3)}(0,L) \geq
G_{\pi/3}(0,L) \qquad \text{ for all $L \in {\mathbb{Z}}$},$$ we conclude that $G_{\Theta}(0,L)= G_{\pi/3}(0,L)$ for all $L$. Finally, using the invariance $G_{\Theta}(a,b) = G_{\Theta}(0,b-a)$, we obtain the desired conclusion.
Proof of Theorem \[thm-saw-bridges\] for general tilings
--------------------------------------------------------
By $$B_{T,\Theta} = 1 - \cos{\tfrac{3\pi}{8}}A_{T,\Theta}.$$ We have shown in the previous proof that $A_{T,\Theta} \to \big(\cos{\tfrac{3\pi}{8}}\big)^{-1}$ as $T\to \infty$, which implies $B_{T,\Theta} \to 0$.
Critical surface fugacity {#sec:fugacity}
=========================
In this section we discuss self-avoiding walks with surface fugacities and prove Theorem \[thm-fugacity\] and Proposition \[prop:y\_c\_strip\]. We split the proof into several steps. First we introduce a slightly different notion of critical fugacity for walks in a strip, denoted $y_c^*(\Theta,T)$; this is then shown to be equal to $y_c(\Theta,T)$ defined in the introduction. Using the Yang–Baxter transformation, we show that the limit of $y_c^*(\Theta,T)$ as $T \to \infty$ does not depend on the sequence $\Theta$; in particular it is equal to that when $\Theta = \pi/3$, which is known to be equal to $1 + \sqrt 2$. Finally, it is shown that the critical fugacity of Theorem \[thm-fugacity\] is indeed equal to $\lim_{T \to \infty }y_c^*(\Theta,T)$.
Critical fugacity in the strip at $x = 1$
-----------------------------------------
When defining the critical fugacity in a strip, one may consider partition functions of walks, arcs or bridges. Below we show that the exact choice has little importance.
For $\Theta = (\theta_k)_{1 \leq k \leq T}$ with $\theta_1 = \pi/3$ and all other angles in $[\pi/3, 2\pi/3]$, recall the notation $${\mathrm{w}}_\Theta (\gamma;x,y) = {\mathrm{w}}_\Theta (\gamma)\cdot x^{|\gamma|}\cdot y^{b(\gamma)}\,, \quad
{\mathrm{SAW}}_{\Theta, T} (x,y) = \sum_{\substack{\gamma \text{ starts at }0 \\ \gamma\subset {\mathrm{Strip}}_T(\Theta) }}{{\mathrm{w}}_\Theta(\gamma;x,y)}\,.$$ where $|\gamma|$ is the length of $\gamma$ and $b(\gamma)$ is the number of visits of $\gamma$ to the left half of the rhombi adjacent to the left boundary of ${\mathrm{Strip}}_T(\Theta)$.
The partition functions of arcs and bridges are defined in a similar way and denoted by $A_{\Theta, T} (x,y)$ and $B_{\Theta, T} (x,y)$. Observe that for any self-avoiding walk $\gamma$ (that is starting and ending at any points of ${\mathrm{Strip}}_T(\Theta)$), its weight ${\mathrm{w}}_{\Theta} (\gamma;x,y)$ may be defined as above.
\[prop:same\_rad\_conv\] Let $\Theta = \{\theta_1, \theta_2,\dots,\theta_T\}$, where $\theta_1 = \tfrac{\pi}{3}$ and $\theta_i \in [\tfrac{\pi}{3},\tfrac{2\pi}{3}]$ for $i>1$. Then the following series (with variable $y$) have the same radius of convergence: $$A_{\Theta, T} (1, y) , \, B_{\Theta, T} ( 1, y) , \, {\mathrm{SAW}}_{\Theta, T} (1, y)\,.$$ Write $y_c^*(T,\Theta)$ for the radius of convergence of the series above.
The set of walks starting at $0$ includes the sets of arcs and bridges. Hence, for any $y >0$, we have: $$\begin{aligned}
&{\mathrm{SAW}}_{\Theta, T} (1, y) \geq A_{\Theta, T} (1, y), \, B_{\Theta, T} (1, y)\, .\end{aligned}$$ Thus, the radius of convergence of ${\mathrm{SAW}}_{\Theta, T} (1, y)$ is smaller than those of $A_{\Theta, T} (1, y)$ and $B_{\Theta, T} (1, y)$.
In order to obtain opposite bounds, we use the decomposition of walks into bridges that was introduced by Hammersley and Welsh [@HW]. We prove the bound only for $B_{\Theta, T} (1, y)$, as for $A_{\Theta, T} (1, y)$ the proof is completely analogous. For $T=1$ the statement is obvious, so below we assume that $T>1$.
Consider a walk $\gamma$ in ${\mathrm{Strip}}_T(\Theta)$ starting at $0$; $\gamma$ will be split into subpaths $\gamma_{-k},\dots, \gamma_\ell$ as described below. The decomposition is illustrated in Fig. \[fig:SAWA\]. Set the lowest (resp. highest) point of $\gamma$ to be the non-empty rhombus with the smallest (resp. largest) second coordinate, and if several such rhombi exists, it is the leftmost (resp. rightmost) among them. Denote these rhombi by $r_{\mathrm{bot}}$ and $r_{\mathrm{top}}$ and let $\gamma_0$ be the subpath of $\gamma$ that links $r_{\mathrm{bot}}$ and $r_{\mathrm{top}}$ ($\gamma_0$ includes $r_{\mathrm{bot}}$ or $r_{\mathrm{top}}$ only if these are endpoints of $\gamma$). Then $\gamma \setminus \gamma_0$ is either empty, or one walk, or a union of two walks, depending on how many of the endpoints of $\gamma$ are contained in $\gamma_0$. If $\gamma = \gamma_0$, the decomposition stops. Otherwise write $\gamma^-$ for the part of $\gamma$ preceding $\gamma_0$ and $\gamma^+$ for the part following $\gamma_0$. We continue by decomposing $\gamma^+$ and $\gamma^-$ in the same fashion: Suppose $\gamma^+$ is not empty and consider its lowest and the highest points. Define $\gamma_{1}$ as the segment between these points. Note that now $\gamma^+ \setminus \gamma_1$ is formed of at most one walk, not two. Continue decomposing $\gamma^+ \setminus \gamma_1$ to obtain $\gamma_2$ etc, until the remaining walk is empty. Apply the same procedure to decompose $\gamma^-$ into $\gamma_{-1}, \gamma_{-2},$ etc.
![[*Top Left:*]{} A walk $\gamma$ in ${\mathrm{Strip}}_T(\Theta)$ with $r_{\mathrm{bot}}$ and $r_{\mathrm{top}}$ marked in gray. [*Top Right:*]{} The decomposition of $\gamma$ in $\gamma^-$ and $\gamma_0$; $\gamma^+$ is void. [*Bottom:*]{} The further decomposition of $\gamma$ into basic pieces. These are completed by the red paths to form bridges.[]{data-label="fig:SAWA"}](SAWA.pdf)
Importantly, in this way $\gamma$ gets split in at most $2T-1$ pieces. Indeed, the left-most points of $\gamma_{0},\gamma_1,\dots, \gamma_\ell$ are each strictly to the right of the preceding one. Thus $\ell < T$. Similarly, the right-most points of $\gamma_{0},\gamma_{-1},\dots, \gamma_{-k}$ are each strictly to the left of the preceding one, and $k <T.$ In general, it is not true that the weight of $\gamma$ is equal to the product of the weights of the pieces obtained above, because the rhombi containing 2 arcs in different pieces contribute $w_1$ (or $w_2$) to the weight of $\gamma$ and $u_1^2$ (or $u_2^2$) to the product of the weights of the pieces. However, since $u_1(\theta)^2 \geq w_1(\theta)$ and $u_2(\theta)^2 \geq w_2(\theta)$ for any $\theta \in [\tfrac{\pi}{3}, \tfrac{2\pi}{3}]$, we obtain the following inequality: $$\label{eq_compare_weights_pieces_walk}
{\mathrm{w}}(\gamma ; 1, y) \leq \prod_{i=-k}^{\ell} {{\mathrm{w}}(\gamma_i ; 1, y)}.$$ Now complement the walks $\gamma_i$ to create bridges by adding straight lines in the rhombi lying to the left (resp. right) of the lower (resp. upper) endpoint of $\gamma_i$ and contained in the same rows as the endpoints (see Fig. \[fig:SAWA\]). Small local modifications may be needed to glue the added paths to $\gamma_i$. Denote the resulting bridges by $\gamma_i^{\mathrm{br}}$. Note that by the choice of $\gamma_i$, the walks $\gamma_i^{\mathrm{br}}$ do not have self-intersections. The walks $\gamma_i$ and $\gamma_i^{\mathrm{br}}$ differ by at most $2T$ rhombi, which are empty for $\gamma_i$ but contain straight lines for $\gamma_i^{\mathrm{br}}$. Thus $${\mathrm{w}}(\gamma_i ; 1, y) \leq \frac{1}{v(\Theta)y} {\mathrm{w}}(\gamma_i^{\mathrm{br}} ; 1, y)\, ,$$ where $v(\Theta)> 0$ is some constant which depends on $T$ and $\Theta$ only. Recall that there are at most $2T-1$ pieces $\gamma_i$. From this, the previous inequality and , we obtain: $$ {\mathrm{w}}(\gamma; 1, y) \leq\frac{1}{[v(\Theta)y]^{2T-1}} \prod_i {{\mathrm{w}}(\gamma_i^{\mathrm{br}}; 1, y)}\, .$$ Sum this inequality over all possible choices of $\gamma$. Using again that there are at most $2T-1$ walks in the decomposition, the right-hand side can be bounded by the partition function of bridges: $$\label{eq:compare_part_func_arc_walk}
{\mathrm{SAW}}_{\Theta, T}(1, y)
\leq \frac{1}{[v(\Theta)y]^{2T-1}}\sum_{\gamma \, : \, 0 \to z, \, \gamma \subset \Omega_T}
\prod_i {{\mathrm{w}}(\gamma_i^{\mathrm{br}};1,y)} \leq \left[\frac{4T}{v(\Theta)y}\right]^{2T-1} (1+B_{\Theta, T}(1, y))^{2T-1},$$ where the additional factor $4T$ in the right hand side is due to the reconstruction cost of $\gamma$ given $(\gamma_i^{\mathrm{br}})_{i\in [-k,\ell]}$.
Hence, the radius of convergence of $B_{\Theta, T}(1, y)$ and ${\mathrm{SAW}}_{\Theta, T}(1, y)$ is the same.
The same strategy may be used to show that $A_{\Theta, T}(1, y)$ and ${\mathrm{SAW}}_{\Theta, T}(1, y)$ have the same radius of convergence. The only difference is that this time the subpaths $\gamma_i$ should be transformed into arcs rather than bridges.
Critical fugacity in the strip: $y_c^*(\Theta,T) = y_c(\Theta,T)$.
-------------------------------------------------------------------
Recall that the critical fugacity in a strip was defined in the introduction as $$y_c(T,\Theta) = \sup \{y\, | \, \forall 0<x<1, \ {\mathrm{SAW}}_{T,\Theta} (x,y) < \infty \}.$$ We show now that the two notions of critical fugacity in a strip, namely $y_c(T,\Theta)$ and $y_c^*(T,\Theta)$, coincide.
\[prop:fug\_2\_def\_strip\] Let $\Theta= \{\theta_k\}_{k=1}^T$, where $\theta_1= \tfrac{\pi}{3}$ and $\theta_k\in [\tfrac{\pi}{3},\tfrac{2\pi}{3}]$ for $k>1$. Then $y_c(T,\Theta) = y_c^*(T,\Theta)$.
We start by a technical lemma which in effect states that a walk in a strip has a positive density of points on the boundary. Such a result is in the spirit of Kesten’s pattern theorem [@kestenone]. For completeness and simplicity, we provide a proof with no reference to Kesten’s result.
Let $\Theta= \{\theta_k\}_{k=1}^T$, where $\theta_1= \tfrac{\pi}{3}$ and $\theta_k\in [\tfrac{\pi}{3},\tfrac{2\pi}{3}]$ for $k>1$. Then there exists a constant $C(T)>0$ which depends only on $T$, such that for any $0<x\leq1$ and $y>1$ $$\begin{aligned}
{\mathrm{SAW}}_{T,\Theta}(x,y) &\leq {\mathrm{SAW}}_{T,\Theta}(xy,1), \label{ineq:fug_xy}\\
{\mathrm{SAW}}_{T,\Theta}(x,x^{-C}y) &\geq {\mathrm{SAW}}_{T,\Theta}(1,y)\,. \label{ineq:fug_x_c_y}\end{aligned}$$
Inequality follows from the fact that the length of a walk is greater than the number of times it visits the boundary.
Inequality is proven by altering arbitrary walks $\gamma$ to form walks $\gamma^{\text{fug}}$ which have a positive density of points on the left boundary. We describe the map $\gamma \mapsto \gamma^{\text{fug}}$ next. Recall the indexing of the rows of ${\mathrm{Strip}}_T(\Theta)$ by $\mathbb Z$. Call *a marked line* of ${\mathrm{Strip}}_T(\Theta)$ the collection of edges separating rows $(k+\tfrac12)T$ and $(k+\tfrac12)T+1$ with $k\in\mathbb{Z}$. Let $\gamma$ be a walk on ${\mathrm{Strip}}_T(\Theta)$ starting at $0$. To define $\gamma^{\text{fug}}$ insert at each marked line two rows of rhombi, containing arcs as described below. Fix a marked line $\ell$, the two rows of rhombi inserted at $\ell$ contain:
for each point in $\gamma\cap \ell$ except the leftmost one, insert two straight vertical arcs of type $v$;
for the leftmost point in $\gamma\cap \ell$, insert a path contained in the two inserted rows that, when viewed from bottom to top, travels left in the lower row, touches the first column turning upwards, then travels back right using the upper row (if the left-most point is in the first column, complete the added rhombi as in the point above);
all rhombi not affected by this procedure are void.
Perform this for all marked lines. Note that when marked lines are not crossed by $\gamma$, the added rows only contain empty rhombi. It is easy to see that the result of this procedure is a self-avoiding walk on ${\mathrm{Strip}}_T(\Theta)$, which we call $\gamma^{\text{fug}}$. See Fig. \[fig:pattern\] for an example.
![[*Left:*]{} A walk $\gamma$ in ${\mathrm{Strip}}_T(\Theta)$ crossing two marked lines (blue). [*Right:*]{} The associated walk $\gamma^{\text{fug}}$; the added rows are marked in gray.[]{data-label="fig:pattern"}](pattern.pdf)
The map $\gamma \mapsto \gamma^{\text{fug}}$ is injective. Indeed it suffices to delete the added rows (whose indices are deterministic) to retrieve $\gamma$ from $\gamma^{\text{fug}}$. Thus $$\begin{aligned}
\label{eq:inj}
{\mathrm{SAW}}_{T,\Theta}(x,x^{-C}y) \geq \sum_{\gamma} {\mathrm{w}}_{\Theta}(\gamma^{\text{fug}};x,x^{-C}y), \qquad \text{for all $C>0$,}\end{aligned}$$ since in the right hand side we only sum the weight of images of walks by the map defined above.
Now observe that, since the length of $\gamma$ inside any rhombus is at most 4, $\gamma$ crosses at least $|\gamma|/(4T^2)$ marked lines. Each marked line generates at least one contribution to the fugacity for $\gamma^{\text{fug}}$, thus $b(\gamma^{\text{fug}}) \geq |\gamma|/(4T^2) $. On the other hand, $\gamma$ visits at most $2|\gamma|/T$ marked lines and for each such line the added rhombi contain a total length of arcs of at most $8T$. Thus $|\gamma^{\text{fug}}| - |\gamma| \leq 16|\gamma|.$ In conclusion $$\begin{aligned}
\frac{b(\gamma^{\text{fug}})}{|\gamma^{\text{fug}}| - |\gamma|} \geq \frac{1}{64 T^2} =:\frac1C.\end{aligned}$$ In particular $$\begin{aligned}
\frac{{\mathrm{w}}_{\Theta}(\gamma^{\text{fug}};x,x^{-C}y)}{{\mathrm{w}}_{\Theta}(\gamma;x,y)}
= x^{|\gamma^{\text{fug}}| - |\gamma| - Cb(\gamma^{\text{fug}}) }y^{b(\gamma^{\text{fug}}) - b(\gamma)}
\geq 1,\end{aligned}$$ since the exponents for $x$ and $y$ are negative and positive, respectively. Inserting this into we find $$\begin{aligned}
{\mathrm{SAW}}_{T,\Theta}(x,x^{-C}y)
&\geq \sum_{\gamma} {\mathrm{w}}_{\Theta}(\gamma^{\text{fug}};x,x^{-C}y)\\
&\geq \sum_{\gamma} {\mathrm{w}}_{\Theta}(\gamma;1,y)
={\mathrm{SAW}}_{T,\Theta}(1,y).
\qedhere\end{aligned}$$
First we show the inequality $y_c(T,\Theta) \ge y_c^*(T,\Theta)$. Take $y>y_c(T,\Theta)$. Then for $x<1$ large enough, ${\mathrm{SAW}}_{T,\Theta}(x,y)$ diverges. By Ineq. , one has that ${\mathrm{SAW}}_{T,\Theta}(xy;1)$ diverges as well. Hence $xy\ge y_c^*(T,\Theta)$. Since $x$ may be arbitrarily close to 1, we proved that $y \ge y_c^*(T,\Theta)$. By choice of $y$ this implies $y_c(T,\Theta) \ge y_c^*(T,\Theta)$.
Let us now show the converse inequality $y_c^*(T,\Theta) \ge y_c(T,\Theta)$. Take $y>y_c^*(T,\Theta)$. Then ${\mathrm{SAW}}_{T,\Theta}(1;y)$ diverges. Use Ineq. to see that ${\mathrm{SAW}}_{T,\Theta}(x,x^{-C(T)}y)$ diverges as well for any $x < 1$. Thus $x^{-C}y \geq y_c(T,\Theta)$ for all $x < 1$, which implies that $y \ge y_c(T,\Theta)$. Since $y>y_c^*(T,\Theta)$ is arbitrary, we proved $y_c^*(T,\Theta) \ge y_c(T,\Theta)$.
Critical fugacities in strips do not depend on $\Theta$
-------------------------------------------------------
Our next goal is to show that $y_c(T,\Theta) \to 1+\sqrt{2}$, i.e. that the critical fugacities on strips of rhombi converge to the critical fugacity on the hexagonal lattice, which corresponds to the case when all rhombi have angle $\pi/3$.
By Proposition \[prop:fug\_2\_def\_strip\], $y_c(T,\Theta)$ is the radius of convergence of ${\mathrm{SAW}}_{T,\Theta} (1,y)$. In the spirit of notation we introduced before, we denote by $y_c(T,\pi/3)$ the radius of convergence of the series ${\mathrm{SAW}}_{T,\pi/3} (1,y)$, i.e. in the case when all rhombi have angle $\tfrac{\pi}{3}$. In the next lemma, it is shown that $y_c(T,\Theta)$ can only increase, when the rightmost column of rhombi is erased, or when all angles of the rhombi are changed to $\tfrac{\pi}{3}$.
\[lem:fug\_monotone\] Let $\Theta= (\theta_k)_{k\geq1}$ be such that $\theta_1= \tfrac{\pi}{3}$ and $\theta_k\in [\tfrac{\pi}{3},\tfrac{2\pi}{3}]$ for $k>1$and $T \geq 2$. Then
$y_c(T,\pi/3) \geq y_c(T,\Theta) $;
$y_c(T-1,\Theta)\geq y_c(T,\Theta)$.
\(i) By Proposition \[prop:same\_rad\_conv\], it is enough to show that for any $y \geq 0$ one has $A_{\Theta, T} (1,y)\ge A_{\pi/3, T} (1,y)$. This inequality was shown in Lemma \[lem:saw-strip-monotonicity\] in the absence of surface fugacities. It is easy to check that the proof adapts straightforwardly when fugacities are added on the left side. Indeed the proof is based on Yang-Baxter transformations that do not affect the left-most column, since this one already has angle $\pi/3$.
\(ii) The inequality $A_{\Theta, T} (y)\ge A_{\Theta, T-1} (y)$ is trivial, since all walks contributing to the right hand side also contribute to the left hand side. The inequality on the radii of convergence follows readily.
Now we are ready to finish the proof of Proposition \[prop:y\_c\_strip\] by showing that $y_c(T,\Theta) \to 1+\sqrt{2}$.
In [@BBDDG] it was shown that the critical surface fugacity on the hexagonal lattice is equal to $1+\sqrt{2}$. In particular, Corollary 8 in [@BBDDG] implies that $y_c^*(\pi/3, T) \to 1+\sqrt{2}$. In Lemma \[lem:fug\_monotone\] it is shown that $y_c^*(\pi/3, T) \geq y_c^*(\Theta, T) $, for any $T$. Hence, $$\lim_{T \to \infty} y_c^*(\Theta, T) \leq 1+\sqrt{2}.$$ The existence of the limit above is ensured by the monotonicity of $y_c^*(\Theta, T)$ in $T$.
The opposite inequality follows directly from Corollary \[cor:small\_y\]. Indeed, suppose that $\lim y_c^*(\Theta, T)<1+\sqrt{2}$. Then for some $T$, one has $y_c^*(\Theta, T)<1+\sqrt{2}$. Consider a value of $y$ between $y_c^*(\Theta, T)$ and $1+\sqrt{2}$ and note that by Corollary \[cor:small\_y\], $B_{T,\Theta} (1,y) = B_{T} (\Theta)(y) \leq \tfrac{\sqrt2 y}{1+\sqrt{2}-y}$. This contradicts the assumption that $y>y_c^*(\Theta, T)$, that is the radius of convergence of $B_{T,\Theta}(1,\cdot)$.
Critical fugacity in half-plane: proof of Theorem \[thm-fugacity\]
------------------------------------------------------------------
In order to prove Theorem \[thm-fugacity\], it remains to show that $y_c = 1+\sqrt{2}$. Recall that $y_c$ is defined as the supremum of all $y$ such that ${\mathrm{SAW}}_\Theta (x,y)$ is finite for all $x<1$.
We will proceed by double inequality. Let $y> 1+\sqrt{2}$. By Proposition \[prop:y\_c\_strip\], there exists $T$ such that $y> y_c(T,\Theta)$. Hence, by the definition of $y_c(T,\Theta)$, there exists $0<x<1$ such that ${\mathrm{SAW}}_{T,\Theta} (x,y) = \infty$. Since ${\mathrm{SAW}}_{T,\Theta} (x,y) \leq {\mathrm{SAW}}_\Theta (x,y)$, the latter diverges as well. This implies that $y\ge y_c$. Recall that $y$ was chosen arbitrarily greater than $1+\sqrt{2}$, thus, $y_c \leq 1+\sqrt{2}$.
The opposite inequality is based on the results obtained through the parafermionic observable with fugacity. Take $1 \leq y<1+\sqrt{2}$. By Corollary \[cor:small\_y\], $B_{T,\Theta}(1,y)<c$, where $c$ is a constant that depends only on $y$. Note that all walks which contribute to $B_{T,\Theta}(1,y)$ have to cross at least $T$ rhombi. Thus, $B_{T,\Theta}(x,y) < x^T\cdot c$, and $\sum_{T \geq 1}B_{T,\Theta}(x,y) < \frac{c}{1-x}< \infty$ for all $x < 1$.
Fix $x < 1$. Let us now prove that ${\mathrm{SAW}}_\Theta(x,y) < \infty$. Write $\Theta'$ for the sequence $(\theta_2,\theta_3,\dots)$. Let $\gamma$ be a walk in ${\mathbb{H}}(\Theta)$. Write $\gamma$ as the concatenation of two walks $\gamma^{(a)}$ and $\gamma^{(w)}$, where $\gamma^{(a)}$ ends at the last visit of $\gamma$ of column $1$. The walk $\gamma^{(w)}$ is contained in columns $2,3,\dots$ and hence does not feel the effect of the fugacity. Thus it may be viewed as a walk in ${\mathbb{H}}(\Theta')$ with weight ${\mathrm{w}}_{\Theta'}(\gamma^{(w)};x,1)$.
Further split $\gamma^{(a)}$ in two walks: $\gamma^{(1)}$ is the walk from the starting point to the first point of $\gamma^{(a)}$ in the right-most column visited by $\gamma^{(a)}$ (write $T$ for the index of this column); $\gamma^{(2)}$ is simply $\gamma^{(a)}\setminus \gamma^{(1)}$. The endpoints of $\gamma^{(1)}$ and $\gamma^{(2)}$ may be modified locally to create two bridges $\gamma^{(b1)}$ and $\gamma^{(b2)}$ in ${\mathrm{Strip}}_T(\Theta)$. Due to the local modifications, there exists a universal constant $\delta>0$ such that $$\begin{aligned}
{\mathrm{w}}_\Theta(\gamma^{(a)};x,y) \le {\mathrm{w}}_\Theta(\gamma^{(1)};x,y){\mathrm{w}}_\Theta(\gamma^{(2)};x,y)
\leq \delta {\mathrm{w}}_\Theta(\gamma^{(b1)};x,y){\mathrm{w}}_\Theta(\gamma^{(b2)};x,y).
\end{aligned}$$ Thus we associated to $\gamma$ a triplet $\gamma^{(b1)}, \gamma^{(b2)}, \gamma^{(w)}$, the first two being bridges in a certain ${\mathrm{Strip}}_T(\Theta)$ and the third being a walk in ${\mathbb{H}}_T(\Theta')$. This operation is clearly injective, and we find $$\begin{aligned}
{\mathrm{SAW}}_\Theta(x,y)
&\leq \sum_{\gamma}{\mathrm{w}}_\Theta(\gamma^{(1)};x,y){\mathrm{w}}_\Theta(\gamma^{(2)};x,y){\mathrm{w}}_\Theta(\gamma^{(w)};x,y)\leq \delta \sum_{T \geq 1} B_{T, \Theta}(x,y)^2 \ {\mathrm{SAW}}_{\Theta'}(x,1)\\
&\leq \delta \Big[\sum_{T \geq 1} B_{T, \Theta}(x,y)\Big]^2 {\mathrm{SAW}}_{\Theta'}(x,1)\leq \delta \Big(\frac{c}{1-x}\Big)^2{\mathrm{SAW}}_{\Theta'}(x,1).
\end{aligned}$$ Finally, since $x<1$, ${\mathrm{SAW}}_{\Theta'}(x,1)< \infty$ which implies ${\mathrm{SAW}}_\Theta(x,y) < \infty$. Since $x < 1$ is arbitrary, this shows that $y < y_c$, and thus that $y_c \geq 1 + \sqrt 2$.
A. Glazman, <span style="font-variant:small-caps;">Tel Aviv University, School of Mathematical Sciences, Tel Aviv, Israel.</span>
`glazman@tauex.tau.ac.il`
I. Manolescu, <span style="font-variant:small-caps;">Universit[é]{} de Fribourg, Fribourg, Switzerland</span>
`ioan.manolescu@unifr.ch`
[^1]: If $\theta_i > \theta_{i+1}$, the rhombus may be added at the bottom and will be slid to the top using Yang–Baxter transformations. If $\theta_i = \theta_{i+1}$ the result is trivial.
|
---
author:
- 'D. Moss'
- 'D. Sokoloff'
- 'A. F. Lanza'
date: 'Received ..... ; accepted .....'
title: 'Polar branches of stellar activity waves: dynamo models and observations '
---
Introduction {#intro}
============
It is widely accepted that the solar activity cycle is more than just a quasiperiodic variation of sunspot number, being rather an activity wave that propagates from mid solar latitudes towards the solar equator. A solar dynamo based on the joint effects of differential rotation and mirror-asymmetric convective motions in the form of the so-called $\alpha$-effect (possibly together with meridional circulation) is considered to be the underlying mechanism for the propagation of activity waves. Indeed, this mechanism gives an equatorwards propagating wave of large-scale magnetic field for a suitable choice of the parameters governing dynamo action. It is natural to expect that such a phenomenon will appear in a variety of stars with convective envelopes, and we might thus be led to expect equatorwards waves of stellar activity.
In fact cyclic activity is known now for many stars of various spectral types, e.g. [@Baliunasetal95], [@Olahetal09]. Clarification of the spatial configuration of the assumed activity wave is a much more delicate undertaking. However contemporary astronomy possesses a range of tools, such as the technique of Doppler Imaging (hereafter DI), with which to investigate the problem. A comprehensive investigation of the problem still remains a desirable milestone for stellar astronomy; however some early results are already available, e.g. [@BerdyuginaHenry07], [@Katsovaetal10]. The point here is that at least some stars exhibit an activity wave that propagates polewards.
For instance, the K-type subgiant component of the RS CVn system has been extensively studied through DI by, e.g., [@Vogtetal99], and shows migration of spots from mid-latitudes towards the rotational poles on a timescale of a few years. Indirect evidence of the same phenomenon is found for several late-type main-sequence stars and young solar analogues, from chromospheric line flux monitoring or photometric optical monitoring, respectively (see Sect. \[observations\]).
In mean-field dynamo models, the direction of migration of the large-scale magnetic field features depends in principle on two key factors – the sign of the $\alpha$ coefficient in the relevant hemisphere and the radial gradient of angular velocity.
The situation is however not so straightforward. The point is that in addition to the equatorwards branch demonstrated by sunspots, the solar activity displays a relatively weak polewards branch, seen in some other tracers, e.g. polar faculae – [@MS89]. It looks implausible a priori that such weak additional branches could be responsible for stellar polewards branches, however the possibility should be recognized.
The aim of this paper is to investigate how the recent and current observational data concerning polewards branches of stellar activity might be connected with ideas from stellar dynamo theory. We appreciate that the observational situation after these pioneering results still remains quite uncertain, and so we study just the most traditional forms of stellar dynamos, i.e. mean-field dynamos based on differential rotation and $\alpha$-effect with simple algebraic quenching.
More recent ideas in solar dynamo theory, such as flux transport dynamos based on meridional circulation, [ e.g. [@BD95], [@CSD95],]{} [@DG06], or dynamical schemes of dynamo saturation, e.g. [@Ketal03], [@SB04], are [ certainly]{} likely to be important. We believe however that a simple initial approach is desirable and therefore we consider a classic dynamo wave model with a simple nonlinearity as the basic model in our research. In this model, the activity wave propagation is primarily associated with the joint action of differential rotation and $\alpha$-effect rather than any effects of meridional circulation. It appears however that some observational features of polewards propagation are difficult to reproduce with this simple model. Therefore, we also investigate briefly some effects of meridional circulation.
[ We also note the possible role of low order dynamo models in elucidating some aspects of stellar magnetic behaviour, e.g. [@WS06], but here we concentrate on models that we feel are more directly interpretable physically. ]{}
Previous theoretical investigations of stellar dynamos have focussed on reproducing the dependence of activity cycle periods on stellar parameters, in particular the rotation period (see, e.g., [@noyesetal84b], [@ossendrijver97], [@jouveetal10] and references therein) or on explaining high-latitude or polar spots that are not observed in the Sun (e.g., [@granzeretal00], [@Holzwarthetal06], [@isiketal07])). We present here for the first time a tentative systematization of the stellar butterfly diagrams that are now emerging from the observations, and try to explain the different behaviours by means of a simple mean-field dynamo model.
[ We stress that our primary aim is not to produce ‘definitive’ mean-field models for any of the observed behaviours. Rather we attempt to illustrate the degree of uncertainty inherent in mean-field parametrizations, and also to show that many behaviours are reproducible by such models. Whilst the fundamental shortcomings of mean-field theory have attracted much interest, rather less attention has been given to, for example, investigating differences in behaviour caused by modest changes to parametrizations. This is, of course, two-edged. It reduces any predictive power of mean-field modelling, but also illustrates the possibility of explaining non-standard behaviours by more unusual regimes.]{}
Butterfly diagrams and polewards activity waves for solar-like stars {#observations}
====================================================================
We introduce some tentative systematization of the information concerning the migration of activity patterns as derived from available observations. Distinct cases are summarized in Table \[observations\_tab\], which lists also our dynamo models with features resembling those observed (see Sect. \[comparison\]), and are briefly described below. They are listed in order of increasing Rossby number ($Ro$), that is the ratio of the rotation period of the star to the convective turnover time at the base of the convection zone; $Ro$ can be roughly related to the dynamo number in the bulk of the convection zone (cf. [@Noyesetal84a]).
Case I:
: Among the RS CVn systems, the K-type component of HR 1099 has a record of DI maps extending over about twenty years with simultaneous coverage in wide-band optical photometry (cf., e.g., [@S09]). [@BerdyuginaHenry07], extending previous work by [@Lanzaetal06], built maps of the distribution of starspots on the active K-type subgiant. Two main active regions were found, one migrating from high latitudes ($\approx 70^{\circ}$) towards mid-latitudes ($\approx
40^{\circ}$), the other from mid-latitudes ($\approx 40^{\circ}$) towards high latitudes ($\approx 70^{\circ}$), these occurring more-or-less simultaneously.
Several other RS CVn binaries show a general behaviour similar to that of HR 1099, although their DI and photometric time series are less extended or have a more limited simultaneous coverage. A characteristic of the active components of RS CVn binary systems is the presence of a polar spot that persists with little modification over timescales of decades, and which may be due to the advection of magnetic flux to the polar region of the star by diffusion or meridional flows, e.g. [@SchrijverTitle01; @Mackayetal04; @Holzwarthetal06]. In contrast, starspots at intermediate and low latitudes seem to follow a cyclic migration that might be associated with an oscillating dynamo, cf. [@S09].
Case II:
: In young, rapidly rotating stars, such as ($P_{\rm rot}=0.51$ days) and ($P_{\rm rot}=1.66$ days), DI has revealed the simultaneous presence of spots at high and low latitudes, as well as a polar spot which has not been observed in all seasons, thus indicating a less persistent feature than in the case of the RS CVn systems (e.g., [@Kovarietal04]). In AB Dor, the spots at low and intermediate latitudes do not appear to migrate significantly ([@Jarvinenetal05; @Jeffersetal07]), in contrast to the case of HR 1099 (case I above).
Case III:
: Solar-like stars with a rotation period of about $5-40$ days have been studied through the long-term monitoring of their chromospheric flux variations, mainly in the framework of the classic Mt. Wilson H&K project and its recent extensions, e.g. [@Baliunasetal95], [@Baliunasetal98], [@HallLockwood04]. [@DonahueBaliunas94] report that 36 stars out of about 100 have several determinations of their rotation period extending over several seasons. Among them, 21 show patterns of rotation that vary with time or with the phase of the activity cycle. Specifically, 12 stars display a pattern that resembles what would be expected from the solar butterfly diagram, although in six of them the rotation period increases as the cycle progresses, in contrast to the solar case. One of the best examples is , see [@DonahueBaliunas92].
The dwarf stars showing anti-solar behaviour seem unlikely to possess an anti-solar pattern of surface differential rotation, i.e. with the poles rotating faster than the equator, because this has never been found from DI observations ([@Barnesetal05]), or from theoretical models of stellar rotation (e.g. [@Rudigeretal98]). Therefore, a plausible explanation is that their active regions migrate polewards rather than equatorwards, which is what we define as case III.
Case IV:
: There is another possibility to produce the phenomenology described in Case III. If stellar activity is not confined to latitudes close to the equator, but is well extended towards the poles, in addition to a polewards dynamo wave there can be another wave propagating from intermediate latitudes towards the equator. Which of the two waves dominates the modulation of the stellar flux depends on the inclination of the rotation axis with respect to the line of sight. If the star is viewed approximately pole-on, the polewards branch will dominate and the observed behaviour is anti-solar, while if the inclination is low, the star shows a solar-like behaviour because the low-latitude branch dominates, as suggested by, e.g., [@MessinaGuinan03].
Cases V and VI:
: Among the stars considered by [@DonahueBaliunas94], there are four stars that seem to reverse the trend of rotation period variation at mid-cycle; six stars that show two narrow, but well separated bands of rotation, suggesting two stationary active latitude belts – we define this as case V; and, finally, stars that have hybrid patterns with one band showing a variation of the rotation period versus the cycle phase, while the other remains fairly constant – we take this behaviour to be representative of case VI. Stars with one or two fixed rotation periods as determined from Ca II H&K monitoring might be characterized by a standing dynamo wave, e.g. [@Baliunasetal06].
Cases III, IV, V, and VI could be different manifestations of the same kind of activity behaviour, which appear distinct, either because of a different inclination of the stellar rotation axis which emphasizes either the polar or the equatorial region of a star, and/or a different intensity of the activity, or a phase shift between the polewards and the equatorwards branches of the butterfly diagram during the activity cycle. The available observations are still too limited to arrive at any sound conclusion on this point. Therefore, we prefer to consider all the suggested behaviours separately because each case makes a specific requirement for the theoretical butterfly diagram.
Case Observational features/example Dynamo interpretation
------ ----------------------------------------------------------- -----------------------------------------
I PW migration from mid-latitude and EW migration
from high latitude (HR 1099) See Sect. \[HR1099\]
II Spot pattern extended in latitude
with high and low latitude spots,
but no definite migration during the cycle (AB Dor) Model 24
III PW migration, i.e. the spot rotation period increases
as the cycle progresses, contrary to the solar case Model 13c
IV EW at low latitudes, PW at high latitudes
(the latter can dominate when the star is viewed pole-on) Model 8
V Two separated narrow activity bands Possible after a further
specialization of stellar hydrodynamics
VI Migration + a standing pattern Model 19
\[observations\_tab\]
Mean-field dynamos
==================
Now we address the problem from the other aspect and discuss how the butterfly diagrams appear in an assortment of stellar and solar-like dynamo models. We discuss here the simplest cases from the viewpoint of dynamo theory, i.e. standard mean-field dynamos, e.g. [@RH], with conventional boundary conditions and numerical implementation, e.g. [@br].
We investigate solutions of the standard mean field dynamo equation: $$\frac{\partial{\vec B}}{\partial t} = \nabla\times({\vec u}\times
{\vec B}+\alpha{\vec B}-\eta\nabla\times{\vec B}), \label{dyneq}$$ where $\eta$ is the turbulent diffusivity and $\alpha$ represents the usual isotropic alpha-effect. The velocity field ${\vec u} =
\Omega \varpi \hat{\phi} + {\vec u}_{\rm m}$, where $\Omega$ is the angular velocity of rotation, $\varpi$ the distance from the rotation axis, $\hat{\phi}$ the unit vector in the azimuthal direction, and ${\vec u}_{\rm m}$ the meridional flow. We restrict our investigation to axisymmetric solutions and solve the dynamo problem in a spherical shell, $r_0\le
r\le 1$, where $r$ is the fractional radius. We adopt $r_0=0.64$ for many of the models, but we also investigate models with a deeper dynamo region with $r_0=0.2$.
When modelling stellar magnetic fields, it is necessary to ensure that the field in the interior joins smoothly on to a force-free field in the external, very low density, region. Splitting the magnetic field into poloidal and toroidal parts, ${\bf B}={\bf
B}_P+{\bf B}_T$, the Lorentz force can be written as $$\begin{aligned}
{\bf L}&=&(\nabla\times{\bf B_P})\times {\bf B}_P+(\nabla\times{\bf B_T})\times {\bf B}_T \nonumber \\
&+&(\nabla\times{\bf B_T})\times {\bf B}_P+(\nabla\times{\bf
B_P})\times {\bf B}_T. \label{lorentz}\end{aligned}$$ For an axisymmetric field, the last term is identically zero, the first two are poloidal vectors and the third is toroidal. In the present case ${\bf B}_T=B_\phi$. The condition ${\bf L}={\bf 0}$ can be satisfied by setting ${\bf
B}_\phi=0$ and $\nabla\times{\bf B}_P={\bf 0}$; this provides the boundary condition on the interior field applied at $r=1$. At the lower boundary we use ’overshoot’ boundary conditions, simulating the decay of the field to zero over a skin depth $\delta$, in the form $\partial g/\partial r=g/\delta$, where $g$ represents the azimuthal component of the vector potential for the poloidal field, or the toroidal field. These boundary conditions have been used before, and have been shown to have no significant effect on the results, except to reduce some field gradients near the base of the convective region.
Models with a solar-like rotation law, based on that derived from helioseismology, and also models with a quasi-cylindrical rotation law appropriate to rapidly rotating lower mass dwarfs, are studied (Fig. \[rotation\]), with a variety of choices for $\alpha(r,\theta)=C_\alpha
f_1(r)f_2(\theta)/[1+({B}/B_{0})^2]$, i.e. a naive $\alpha$-quenching nonlinearity is used, [ where $\theta$ is the colatitude measured from the North pole and $B_{0}$ a reference magnetic field (see below).]{}
Different physical mechanisms can co-operate to produce the $\alpha$-effect and theoretical or observational (e.g. [@Zhang]) knowledge of its radial and colatitudinal distributions, here expressed through the functions $f_{1}(r)$ and $f_{2}(\theta)$, is quite preliminary. Therefore, we adopt only simple parametrizations and explore a number of options to investigate the sensitivity of butterfly diagrams to the underlying assumptions.
The turbulent diffusivity $\eta$ is uniform in the outer part of the convection zone ($r\geq 0.8$), but decreases linearly to one half that value in the domain $r \leq 0.7$. [ Below the CZ proper we expect an overshoot region, where the turbulent intensity, corresponding to the turbulent resistivity, is further reduced. In [@mossbrooke00] (which uses the Malkus-Proctor feedback onto the differential rotation as the sole nonlinearity, as opposed to the algebraic alpha-quenching here), only token recognition of this effect was made for computational reasons. We have followed the same procedure here, recognizing that the gradient in turbulent diffusivity should be larger. Another way of regarding this is to consider the model as having a rather deeper CZ, extending to radius $r_0$. From this viewpoint, the lower boundary condition, that allows a penetration of the field with a skin depth of the order of $\delta$, corresponds to the effect of a strongly reduced diffusivity immediately below the boundary. ]{}
We make the dynamo equation non-dimensional in terms of the stellar radius $R$, the diffusion time $R^{2}/\eta_{0}$, where $\eta_{0}$ is the maximum turbulent diffusivity, and the magnetic field $B_{0}$ defined as in Sect. 3 of [@mossbrooke00]. Thus we introduce the standard dynamo numbers, $C_\alpha=\alpha_0R/\eta_0, C_\omega=\Omega_0R^2/\eta_0$, where $\alpha_0$ is a typical value of $\alpha$ and $\Omega_0$ is the maximum value of the angular velocity.
In the $\alpha\omega$ approximation, we can define the combined dynamo number $D=C_\alpha C_\omega$, and this remains a useful quantity even in $\alpha^2\omega$ models.
To integrate the dynamo equation, we use the code described in [@mossbrooke00], which uses a Runge-Kutta integrator over a standard mesh with 61 points over $r_0\le r\le 1$, and 101 points over $0\le \theta \le \pi$, equally spaced. The results are described in Sect. \[2d\_dynamos\].
A useful simplification of the general axisymmetric mean-field dynamo equation (\[dyneq\]), known as the Parker migratory dynamo, is considered in Sect. \[1D\] and is written in a standard non-dimensional form as: $$\label{eq1} {{\partial B} \over {\partial t}} = Df \sin \theta
{{\partial A} \over {\partial \theta}} + {{\partial^2 B} \over
{\partial \theta^2}} - \mu^2 B, \label{parkerB} \label{1D-eq1}$$ $$\label{eq2} {{\partial A} \over {\partial t}} = \alpha B +
{{\partial^2 A} \over {\partial \theta^2}} - \mu^2 A,
\label{parkerA}$$ where [ $f$ is a factor that allows for a latitudinal variation of the radially averaged angular velocity (see Sect. \[1D\])]{}, $\alpha$ is a nondimensional measure of the $\alpha$ effect, $D=C_\alpha C_\omega$ is the dynamo number, $B$ denotes the toroidal magnetic field and $A$ the toroidal component of the vector potential for the poloidal field. Both latter quantities are averaged in radial direction over the convective shell, [ see [@Baliunasetal06].]{} In other words, here the explicit radial dependence has been removed, and the terms involving $\mu^2$ represent radial diffusion in a spherical shell of thickness approximately $\mu^{-1}$ of the outer radius of the shell – e.g. $\mu\approx 3$ is appropriate for the solar convection zone. [ We solve Eqs. (\[eq1\]) and (\[eq2\]) in the domain $0 \leq \theta \leq \pi$ with $A=0$, $B=0$ at the boundaries. ]{}
Polar branches in the 1D Parker dynamo {#1D}
--------------------------------------
We begin with a simple cartoon which explains the idea of stellar dynamos, i.e. with the 1D Parker (1955) dynamo.
First of all, we estimate the latitudinal variation of $\Omega$, averaged over fractional radii (0.69,1) from a realistic solar rotation law (Fig. 1, left hand panel). Normalized to the polar value of the gradient, the modulation compared to that with no latitudinal variation is modelled by a function $f(\theta)$ (Fig. \[rr\]). Then we ran the Parker dynamo with the usual dynamo number, $D=D_0$ say, replaced by $D=D_0 f(\theta)$, so $f(\theta)=1$ gives the standard case.
We take $\alpha=\cos\theta\sin^m\theta$, with $m=0,2,4$. The larger the value of $m$, the more concentrated around the equator the $\alpha$ effect. This simple parametrization has been used by other authors, e.g. [@rudigeretal03], [@charbonneau10], and we also adopt it here. [ We stress the inherent uncertainty in the spatial dependence of $\alpha$ and refer to, e.g., [@rudigerbrandenburg95] for some justification in the framework of mean field theory with a specific turbulence model. Specifically, they explore an $\alpha$ effect dependence with $m=2$ that is suggested by the extension of their turbulence theory to third order terms in ${\vec \Omega} \cdot {\vec U} \propto \cos \theta$, where $\vec \Omega$ is the angular velocity vector of the star and $\vec U$ the vector of the gradient of the turbulent diffusivity that points in the radial direction. In principle a further extension of the theory to the fifth degree in ${\vec \Omega} \cdot {\vec U}$ would introduce terms proportional to $\sin^{4} \theta $ as we assume in our simple parametrization with $m=4$ (of course we should then also consider expressions for alpha containing a combination of these dependences, but we decided that this was beyond the scope of this paper). Note also that the $\alpha$ effect may have a component arising from magnetostrophic waves excited in the layers where the toroidal field is amplified and stored. This would lead to terms with $m>0$ in the parametrization of the latitudinal dependence of the $\alpha$ effect (see, e.g., [@Schmitt87]).]{}
We emphasize that calculations of $\alpha$ from turbulence models are necessarily severely truncated, and stellar dynamos operate in regimes remote from those in which such calculations are valid. Also, if we consider the potentially more useful, but more problematical, method of obtaining parametrizations of $\alpha$ from analysis of direct numerical simulations, it would be surprising if $m=0$ or $m=2$ or a combination was adequate. However such determinations are currently contentious, in spite of substantial progress, from the early attempts (e.g. Brandenburg & Sokoloff 2002) to the most recent (e.g. Courvoisier & Kim 2009; Brandenburg et al. 2010; Tobias, Dagon & Marston 2011). Realistically, a reliable determination for specific types of stars remains a remote possibility.
With these assumptions on the rotation profile and the alpha effect, we solve the 1D dynamo equations (\[1D-eq1\]) and (\[parkerA\]).
With $D_0<0$ then, as expected, the activity wave propagates equatorwards. The main effect is with the variation of $m$, larger values of $m$ move the migration nearer to the equator.
With $D_0>0$, $m=0$ then, again as expected, there is polewards migration, centred on mid-latitudes, both with $f(\theta)=1$ and with $f(\theta)$ nonuniform. When $f(\theta)$ is nonuniform, the butterfly diagram is concentrated at high latitudes for modestly supercritical $D_0$, while solutions become steady for more supercritical values of $D_0$.
With $m=2$ and $f(\theta)=1$, again behaviour is much as expected, but quite unexpectedly with $f(\theta)$ as in Fig. \[rr\] and $m=2,4,$ the solutions are steady for only marginally supercritical values of $D_0$. These statements are all for solutions with dipole parity, but steady solutions are also found when quadrupole parity is enforced.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Equally spaced isorotation contours for the solar rotation law used (left hand panel) and contours for our quasi-cylindrical law for rapid rotators (right hand panel).[]{data-label="rotation"}](15949f1.ps "fig:"){width="0.3\hsize"} ![Equally spaced isorotation contours for the solar rotation law used (left hand panel) and contours for our quasi-cylindrical law for rapid rotators (right hand panel).[]{data-label="rotation"}](15949f2.ps "fig:"){width="0.55\hsize"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Function $f (=<\frac{d\Omega}{dr}>)$ used in the Parker model (Sect. \[1D\]).[]{data-label="rr"}](15949f3.ps){width="0.92\hsize"}
Except when explicitly stated, all the above experiments were made with a slightly supercritical $|D_0|$ and with a decay term which mimics radial diffusion for the value of the ratio $\mu^{-1}=h/R=1/3$, where $R$ is the stellar radius and $h$ the thickness of the convective zone, e.g. [@Baliunasetal06]. We deduce that, even with this simple dynamo model, behaviour can be remarkably rich and broad conclusions cannot be drawn without a quite careful exploration of the parameter space and the form of $f(\theta)$.
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2D dynamo models {#2d_dynamos}
----------------
### Solar rotation law
Now we turn our attention to 2D models and start by using a solar-like rotation law within fractional radii $0.64\le r \le 1.0$, as shown in the left hand panel of Fig. \[rotation\]. In $r\ge 0.7$ this is an interpolation on MDI data, and is made to match uniform rotation at the lower boundary $r=r_0=0.64$.
A dynamo shell with a lower boundary at $r_0=0.64$ of the stellar radius is a good assumption for G and K type main-sequence stars, but it is not appropriate for a subgiant such as the K1IV active component of HR 1099. Given the current uncertainty on the evolutionary stage of the star, $r_{0}=0.2$ is a reasonable assumption (cf., e.g., [@lanza05]). Because extensive numerical simulations to cover realistic values for this (and many other) dynamo governing parameters are obviously beyond the scope of the paper, we restricted our investigation here to extrapolations that seem reasonable on the basis of available knowledge.
Several forms of the functions $f_1(r), f_2(\theta)$ are used: $f_2(\theta)= \cos\theta \sin^m\theta$, with $m=0,2,4$, and $f_1(r)$ takes the forms shown in Fig. \[alp\_prof\], referred to below as $i_\alpha=0,1,2,3,6,7$, as denoted in the caption. There are thus 18 possible forms of $\alpha(r,\theta)$.
A standard value $C_\omega=\Omega_{\rm eq}R^2/\eta_0 = 6\times 10^4$ was taken. The other dynamo number, $C_\alpha $, was given a slightly supercritical value. The results are summarized in Table 2.
Only odd parity (dipole-like) solutions were studied. Models 1-18 have $C_\alpha<0$, the conventional fix to get near-surface fields migrating in the solar sense. Models 19-30 (not all numbers present) have $C_\alpha>0$. Latitude-time diagrams for the pairs (1,19), (6,24), (8,26), and (12,30), which have the same values of the parameters $i_\alpha$ and $m$ and the same $|C_\alpha|$, are shown in the top four rows of Fig. \[butt1-19\]. Here and below, “near-surface” refers to a fractional radius of ca. $0.93$, and “deep” to radius ca. $r = r_0$.
[ll]{} {width="7cm"} & {width="7cm"} {width="7cm"} & {width="7cm"}{width="7cm"} & {width="7cm"}{width="7cm"} & {width="7cm"}{width="7cm"} & {width="7cm"}{width="7cm"} & {width="7cm"}
Model $i_\alpha$ $m$ Notes
------- ------------ ----- ----------------------------------------------------------------------------------
1 3 4 predominantly EW at low latitudes, weaker PW branch at high latitude
2 0 4 rather strange butterfly, very weak EW at low lats, no migration at high lats
3 2 4 low lat EW, no high lat feature
4 6 4 steady solution
5 7 4 steady solution
6 1 4 predominantly EW at low latitudes, very weak PW branch at high latitude
7 3 2 predominantly EW at low latitudes, weaker PW branch at high latitude
8 0 2 strong PW branch at high lat, EW at low
9 2 2 low lat EW, very weak high lat PW
10 6 2 steady solution
11 7 2 steady solution
12 1 2 EW at low lats, weak PW at high lat
13 3 0 weak EW at low latitudes, much stronger PW branch at high latitude
14 0 0 strong PW at high latitudes, very weak EW branch at low latitude
15 2 0 steady solution
16 6 0 EW at very low lats, no high lat features
17 7 0 steady solution
18 1 0 strong PW at high lat, very weak EW at low lat. Solns steady at large $C_\alpha$
19 3 4 almost no migration - near SW low lats, with very weak PW drift
24 1 4 much as Model 19
26 0 2 EW, extending over most latitudes
30 1 2 mild EW (near SW), over most latitudes
\[solarsummary\]
### Meridional circulation {#mer_circ}
Here we investigate the effects of an arbitrarily imposed meridional circulation, in addition to solar-like differential rotation and $\alpha$-effect, on the butterfly diagram.
We take a circulation determined by a stream function: $$\psi= Rm\frac{1}{2}(r-r_0)^2(r-1)\sin^2\theta\cos\theta,$$ so that $$\begin{aligned}
\label{defcirc}
u_r&=&\frac{1}{r^2\sin\theta}\frac{\partial\psi}{\partial\theta}, \nonumber\\
u_\theta&=& -\frac{1}{r\sin\theta}\frac{\partial \psi}{\partial r},\end{aligned}$$ where $r_0=0.64$ corresponds to the base of the dynamo region. (Taken literally, in the solar case this implies the circulation penetrating into an overshoot region, but limited experimentation with the circulation restricted to $r>r_0=0.7$ suggests little difference). [Here $Rm=U_0R/\eta_0$, where $U_0$ is the maximum value of $u_\theta$ at the surface.]{} This circulation has a single cell in each hemisphere, with polewards flow at the surface if $Rm>0$. The streamlines $\psi={\rm const}$ are shown in Fig. \[merid\].
In our models, the $\alpha$ effects and the maximum of the angular velocity shear are not spatially separated, as in, e.g., flux transport models based on the Babcock-Leighton paradigm (e.g., [@dikpaticharbonneau99]). Therefore, the meridional flow does not play the crucial role it has in those kinds of models, but it can still modify the migration of the activity waves in a given layer when its speed there becomes comparable to the migration speed of the wave in the absence of circulation which is mainly established by the product $|\alpha \partial \Omega / \partial r|$.
![\[merid\] Streamlines of the circulation defined by (\[defcirc\]). With $Rm>0$ the surface flow is polewards. ](15949f19.ps "fig:"){width="8cm"}\
We first look at Model 26 (with a solar-like rotation). In the absence of any meridional circulation, the near-surface migration is equatorwards (Table \[solarsummary\]), but the deep butterfly diagram has polewards and equatorwards branches, the polewards being somewhat stronger, see Fig. \[1099\], top panel. In the presence of our one-cell circulation, polewards at the surface, the near-surface migration remains equatorwards for $Rm\la 40$, but the deep equatorwards branch is strengthened, see the lower panels of Fig. \[1099\]. For larger values of $Rm$, the near-surface migration acquires a weak polewards branch when $Rm\ga
100$, but the deep migration develops two discrete patterns that are close to standing waves. Note also that, here and below, we do not claim to have explored exhaustively the parameter space, but just to have sampled a few, we hope fairly representative, solutions. Other behaviours may well remain to be found.
![\[1099\] Butterfly diagrams for Model 26 (solar rotation, $C_\alpha>0, i_\alpha=0, m=2$) : top - without meridional circulation ($Rm =0$), bottom of the convection zone; middle - $Rm = 20$, sub-surface; bottom - $Rm = 20$, bottom of the convection zone. Solid/broken contours denote positive/negative values of $B_\phi$ respectively. ](15949f20.ps "fig:"){width="7cm"}\
![\[1099\] Butterfly diagrams for Model 26 (solar rotation, $C_\alpha>0, i_\alpha=0, m=2$) : top - without meridional circulation ($Rm =0$), bottom of the convection zone; middle - $Rm = 20$, sub-surface; bottom - $Rm = 20$, bottom of the convection zone. Solid/broken contours denote positive/negative values of $B_\phi$ respectively. ](15949f21.ps "fig:"){width="7cm"}\
![\[1099\] Butterfly diagrams for Model 26 (solar rotation, $C_\alpha>0, i_\alpha=0, m=2$) : top - without meridional circulation ($Rm =0$), bottom of the convection zone; middle - $Rm = 20$, sub-surface; bottom - $Rm = 20$, bottom of the convection zone. Solid/broken contours denote positive/negative values of $B_\phi$ respectively. ](15949f22.ps "fig:"){width="7cm"}
### Quasi-cylindrical rotation law
Here a quasi-cylindrical rotation law, as [@covasetal2005] and shown in the right hand panel of Fig. 1, is used. We take $C_\alpha=-11.7,
C_\omega=1.3\times 10^5$, i.e. significantly supercritical values that might correspond approximately to $\Omega=3\Omega_\odot$. Exceptionally, Models 13 and 19 have $C_\alpha=+11.7>0$. Results are summarized in Table 3 (note that these models are labelled “1c” etc to distinguish them from the models with a solar-type rotation law). Butterfly diagrams for the sub-surface toroidal field for the pairs of Models (1c, 13c), and (7c, 19c) are shown in Fig. \[butt1-19\] in the fifth and sixth rows, respectively. The models in each of these pairs have the same parameters, except that for 13c and 19c, $C_\alpha>0$.
### An intermediate rotation law
Although several stars in our study are quite rapid rotators, the cylindrical rotation law discussed in the previous section is not the only possibility for their rotation. It appears plausible that a regime intermediate between the solar-like and the quasi-cylindrical rotation laws may be more appropriate for such stars. Therefore, we explored the butterfly diagram obtained by adopting such a synthetic rotation curve, i.e., mixing these laws with weights of about 50% (Fig. \[synth\]). Using this $\Omega(r, \theta)$, we obtained butterfly diagrams for $B_\phi$ in both the deep and shallow parts of the convective zone. These are characterized by field migrating from low and high latitudes towards some intermediate latitude, even without including any meridional circulation (Fig. \[1099synth\]). Neither the solar-like nor the quasi-cylindrical rotation law produced such a behaviour.
![Synthetic rotation curve from combining solar-like and quasi-cylindrical rotation laws with 50% weighting.[]{data-label="synth"}](15949f23.ps){width="4cm"}
![\[1099synth\] Butterfly diagrams for the synthetic rotation law (combination of solar-like and quasi-cylindrical): top panel - shallow, lower panel - deep parts of the convective zone.](15949f24.ps "fig:"){width="8cm"}\
![\[1099synth\] Butterfly diagrams for the synthetic rotation law (combination of solar-like and quasi-cylindrical): top panel - shallow, lower panel - deep parts of the convective zone.](15949f25.ps "fig:"){width="8cm"}
### Models with a deep convection zone {#HR1099}
Since $r_0=0.64$ probably gives a CZ that is too shallow for HR 1099, we experimented with models having a deeper dynamo zone ($r_0=0.2$), with both solar-like and quasi-cylindrical rotation laws and also their equally weighted combination. The role of a simple meridional circulation, as parametrized by $Rm$, was also considered (cf. Sect. \[mer\_circ\]).
In general, the deep convective zone appears to favour steady solutions when $Rm
\ne 0$, or standing wave (SW) solutions. The latter may partially be due to the restricted latitudinal extent at the bottom of a deeper convective zone. This conclusion agrees with the previous results of [@mossetal04].
The solar rotation law sometimes gives low latitude equatorwards and high latitude polewards branches simultaneously present near the surface, similar to the models for solar-like stars with $r_0=0.64$ previously discussed. With $C_\alpha<0$, with the quasi-cylindrical rotation law, deep and surface migration was either SW or equatorwards depending on the sign of $Rm$, and becomes steady for large enough $|Rm|$. If $C_\alpha>0$, then solutions with polewards surface migration and SW or vacillatory (V) behaviour deep down were found for $Rm>0$. With $Rm<0$, for the same $C_\alpha$ surface migration patterns were a combination of SW and equatorwards, again with SW in the deep CZ. Again solutions are steady for large enough $|Rm|$. The richest choice of possibilities was obtained for the intermediate rotation law with $C_\alpha$ and $Rm$ varying (Table \[deep\]). It is notable that examples with surface polewards migration were not found. [ These investigations clearly demonstrate that the effects of advection can dominate the basic dynamo wave when $Rm$ is significantly non-zero.]{} We conclude that none of these butterfly diagrams can reproduce the phenomenology observed in HR 1099 (case I of Sect. \[observations\]).
A more promising result (Fig. \[deepCZ\]) can be obtained when using an $\alpha$ profile which changes sign with depth [ (see Fig. \[ialpeq18\]) ]{} which gives clearly opposed waves near top and bottom of the convective zones[^1]. In this case, we only investigated the quasi-cylindrical rotation law for a very limited range of dynamo parameters. This result is consistent with the results of [@ms07] concerning dynamo waves propagating in two separate layers, and allows the behaviour sought with appropriate choices of $C_\alpha$ in the two layers. The separation of the layers, that was assumed by [@ms07] arbitrarily, may be achieved here by exploiting the freedom given by the depth of the convective zone. Of course, this assumes that the deep and surface magnetic fields somehow jointly contribute to the surface activity manifestations (see Sects. \[comparison\] and \[concl\]).
Model $i_\alpha$ $m$ Notes
------- ------------ ----- ---------------------------------------------------------
1c 3 2 strong EW at low latitudes, no high latitude features
2c 0 2 strong EW at low latitudes, no high latitude features
3c 2 2 strong EW at low latitudes, no high latitude features
4c 6 2 strong EW at low latitudes (slightly more extended than
above), no high latitude features
5c 7 2 similar to Model 4c
6c 1 2 similar to Model 4c
7c 3 4 similar to Model 1c
8c 0 4 similar to Model 2c
9c 2 4 again, low lat EW
10c 6 4 similar to Model 9c
11c 7 4 as Model 10c
12c 1 4 as Model 10c
13c 3 2 $C_\alpha>0$. PW at mid-latitudes
19c 3 4 $C_\alpha>0$. PW at low latitudes
\[cylsummary\]
Comparison of the 2D dynamo models with the observations {#comparison}
========================================================
With the solar rotation law, details of migratory patterns are sensitive to the spatial dependence of $\alpha$, and even the general rule that the direction of migration is governed by the sign of $C_{\alpha} C_{\omega}$ appears not always to hold true. The most striking case is that of Models 8 and 26 – see the third row of Fig. \[butt1-19\]. Model 8 (left panel) has $C_{\alpha} < 0$ and shows a strong polewards branch in addition to an equatorwards low-latitude branch. Reversing the sign of $C_{\alpha}$ but maintaining the same rotation law and spatial structure of the $\alpha$ effect, we see only a pronounced equatorwards branch, contrary to simple intuitive expectation. The effect of switching from $C_{\alpha} < 0$ to $C_{\alpha}>0$ is also remarkable and not simply intuitive for Models 6 and 24 (on the second row of Fig. \[butt1-19\]) and for Models 12 and 30 (on the fourth row); these differ in the latitudinal localization of the $\alpha$ effect ($m=4$ and $m=2$, respectively).
With the quasi-cylindrical rotation law, migratory patterns are quite insensitive to the exact form of $\alpha$, the only significant change occurs when the sign of $C_{\alpha} C_{\omega}$ is reversed. The rule linking the sign of $C_{\alpha} C_{\omega}$ to the migration direction appears to hold. Plausibly this is because the spatial structure of $\Omega$ is much simpler than in the solar-like case, and $\alpha$ cannot be ’tweaked’ so as to give extra weight to regions of the envelope with anomalous gradients of $\Omega$, unlike in the solar case.
Our general impression from the above analysis can be summarized as follows. Dynamo models with $r_{0}=0.64$ and a fixed sign of $\alpha$ through the CZ can provide polewards migrating patterns, and observers understand activity in some stars as a manifestation of a polewards propagating pattern (see cases III and IV in Sect. \[observations\]). Models with a solar-like rotation law tend to show both equatorwards and polewards branches, the latter with an intensity that depends on the spatial distribution of the $\alpha$ effect. It is possible to reproduce case IV by, e.g., Model 8, while the case with a stronger equatorwards and a weaker polewards branch, reminissent of the behaviour observed in the Sun, can be compared with Model 6 (see Fig. \[butt1-19\], left hand panel in the second row). The case with polewards migration only, i.e. case III of our classification, is unlikely to be reproduced with a solar-like rotation law, but can be reproduced with a cylindrical law (cf. e.g. Model 13c with $C_{\alpha} > 0$, the right hand panel, fifth row of Fig. \[butt1-19\]). This rotation law may be characteristic of stars rotating significantly faster than the Sun (see, e.g., [@lanza05]).
On the other hand, a pattern of spots extending from the high to the low latitudes without any clear evidence of migration during the activity cycle (case II in Sect. \[observations\]), may be interpreted by, e.g. Model 24 (see Fig. \[butt1-19\], right hand panel in the second row). That model is characterized by a stationary pattern spanning a latitude range from the equator up to $\sim 65^{\circ}-70^{\circ}$. Also Model 19 (Fig. \[butt1-19\], right hand panel, first row) shows a similar stationary pattern, but more localized in latitude. Model 12 (Fig. \[butt1-19\], left hand panel of the fourth row) shows a weak polewards branch that, although probably not capable of producing photospheric spots, could be detectable through the chromospheric Ca II H&K flux modulation. Therefore, it might correspond to our case VI because the pattern localized in latitude gives rise to an almost constant primary periodicity in the chromospheric flux modulation, while the migrating branch produces a secondary periodicity that varies along the activity cycle.
When interpreting the observations, one should consider that the outer contours of the model butterfly diagrams can be affected by the turbulent diffusion. Nevertheless, the direction of migration, which is the relevant information for our study, is not modified, as can immediately be seen by considering the direction of migration given by the interior contours of the diagrams that correspond to increasingly stronger fields. In some of the diagrams, such as those of Model 6 (on the left second row in Fig. \[butt1-19\]) or Model 12 (on the left fourth row) we plot only one contour to represent the polewards branch, but the reality of that branch is confirmed by more detailed investigation. (A single contour merely indicates that the field strength in that branch is relatively small.) Moreover, turbulent diffusion in our model is uniform in the outer layers of the convection zone which implies that it cannot favour a specific direction of migration of the field, i.e. it does not introduce any preference for equatorwards or polewards motion of butterfly contours.
In the case of HR 1099 (case I of Sect. \[observations\]), the point is that the image of polewards patterns extracted from the observations is quite different from that emerging from the theory in all the models with $r_{0}=0.64$ introduced in Sect. \[2d\_dynamos\], irrespective of the adopted rotation law or the inclusion of a meridional circulation. We recall that HR 1099 displays two patterns that migrate in opposite directions, and the migration that begins nearer to the equator is polewards.
If there is simultaneous migration polewards and equatorwards through the same latitudes, then it is difficult to see how any simple mean-field model can reproduce it. We attempt to interpret the two oppositely propagating activity patterns as originating from different spatial volumes (see Sect. \[HR1099\]). Specifically, in the case of HR 1099, we need to consider an $\alpha$ effect that changes sign with radius or the presence of two distinct dynamo layers with opposite signs of $\alpha$. The main difference with respect to the models with $r_{0}=0.64$ is that now we consider the contribution to surface activity from [*both*]{} the deep and the shallow dynamo layers, while in the other cases we assumed that the field pattern at the top of a single dynamo layer directly accounts for the observed butterfly diagram.
We recognize that there are unresolved difficulties with this idea. For example, there is the problem of storing toroidal fields in relatively shallow superadiabatic layers without a too rapid loss through buoyancy instabilities. This issue has been addressed, [*inter alia*]{}, by Brandenburg (2005, 2009). [ We note in passing that in our model the toroidal field strength in the upper region is about 20% of that in the lower. This might assist downward turbulent pumping to stabilize the field by its reduced magnetic buoyancy force (proportional to $B^{2}$), possibly acting preferentially at the base of a stellar supergranular layer analogous to the solar supergranulation (but at a somewhat greater depth because the convection in a subgiant is expected to have larger vertical scales).]{} Turning back to the Sun, other difficulties for a dynamo model operating in the subsurface shear layer may include an incorrect phase difference between poloidal and toroidal fields and a too weak toroidal field due to the strong turbulent diffusion expected in those highly superadiabatic layers, cf. [@Dikpatietal02].
[ Nevertheless, we can attempt to reverse the argument: from both general considerations about mean-field models, and our particular simulations, we are unable to identify a mechanism to produce the simultaneous presence of polewards and equatorwards migration co-located in latitude, other than that discussed above. Thus we conclude that either this or a related mechanism does indeed operate, or that the phenomenon is beyond the scope of mean-field theory.]{}
Note that [@BerdyuginaHenry07] stress the nonaxisymmetric behaviour of HR 1099 and other active stars, whereas we restrict ourselves to axisymmetric models. We fully realize that departures from axisymmetry may play a role in the phenomena displayed by HR 1099, cf. e.g. [@MPS], and seem essential to explain the ’flip-flop’ phenomenon, e.g. [@Moss04]. However nonaxisymmetric dynamo models contain many additional uncertainties and the main features of the observations seem to be reproduced by a simpler model.
near surface deep near surface deep
---------------------------------- ------------------------- ------------------------------- -------------------
weak PW at mid-latitudes, SW V/SW near poles only V/confused
SW at high lats
near SW weak EW SW, concentrated at high lats weak PW, mid-lats
EW almost no field present EW PW
steady steady steady PW
EW EW PW EW
confused butterfly, no migration near SW EW SW
V/confused, near pole only V/confused EW and SW EW and SW
\[deep\]
Discussion and conclusions {#concl}
==========================
Our general conclusion is that it is becoming realistic to construct a framework for classification of [ the very varied]{} stellar dynamo wave behaviours, and to relate this to physical parameters of stellar convective zones. We have presented above a very crude and preliminary outline for such a template.
We confirm that the sign of $D = C_{\alpha} C_{\omega} \sim \alpha
\partial\Omega /\partial r$ is the main quantity which determines the direction of activity wave propagation, [ even though considerable finer detail can be found in the results, e.g. in the form of two branches at low and high latitudes, or of standing wave patterns (cf. Sect. \[comparison\] and Fig. \[butt1-19\]). However,]{} quite unexpectedly, we conclude from the above plots that even if $D>0$ it is quite difficult to excite a pronounced single polewards branch in stars with a solar-like rotation law. It follows that straightforward considerations based on the sign of dynamo number $D$ are inadequate to understand the occurence of polewards branches, and a careful examination of 2D models is important. In our investigation, we consider only the direction of dynamo wave migration and do not attempt to fit the observed spot latitudes. The direction of migration (or standing wave behaviour) appears to be a robust result that does not depend on the details of the butterfly wings (which may be influenced by physical processes that we have not considered).
Observations give a hint that one hemisphere of a star can contain two oppositely propagating activity waves which are pronounced enough to be observable. We find that, at least for some stars, rotation curves and spatial distribution of the other dynamo governing quantities should produce two activity patterns in a hemisphere, of more or less comparable intensity. A phase difference between the equatorwards and the polewards dynamo waves can exist. This idea emerges from looking at the butterfly diagram for Model 1 (top left panel of Fig. \[butt1-19\]), where there is a difference in the field intensity of the two branches as well as a phase shift between the epochs when the low-latitude branch reaches the equator and the high-latitude branch reaches the highest latitude.
The fundamental result of our investigation is that the spot migration observed in main-sequence stars or the presence of standing activity waves can be explained by considering the time evolution of the magnetic field at the upper boundary of a dynamo shell, [ even in a very simple dynamo model. The various]{} behaviours can be reproduced by changing the spatial structure of the $\alpha$-effect and/or the internal rotation law. Also a meridional flow may play a role.
The other important point is that the behaviour of HR 1099 cannot be explained by our simple one-layer models and the inclusion of a meridional circulation does not change this conclusion. On the other hand, a two-layer model, somewhat similar to that introduced by [@ms07], can explain the behaviour of HR 1099 if the two spots migrating in opposite directions are the result of magnetic flux tubes originating in the upper and deeper dynamo layers, respectively. However, this interpretation requires a significant modification of the paradigm used for main-sequence stars. In other words, while for main-sequence stars we use a single-layer dynamo and assume that the observed spots correspond to the field at the upper boundary of the dynamo domain, for the subgiant in HR 1099 (and for subgiant stars in general) we must assume the presence of two dynamo layers separated by an inactive shell, with both layers contributing to the observed spots at the surface.
![Dependence of $\alpha$ on radius at $\theta=\pi/4$ for the case where $\alpha$ changes sign with radius.[]{data-label="ialpeq18"}](15949f26.ps "fig:"){width="7cm"}\
![\[deepCZ\] Butterfly diagrams resembling that of HR 1099 for a deep convective zone with a change of sign for $\alpha$ (Fig. \[ialpeq18\]), quasi-cylindrical rotation, $i_\alpha=0$: sub-surface (upper panel) and deep (bottom panel). ](15949f27.ps "fig:"){width="8cm"}\
![\[deepCZ\] Butterfly diagrams resembling that of HR 1099 for a deep convective zone with a change of sign for $\alpha$ (Fig. \[ialpeq18\]), quasi-cylindrical rotation, $i_\alpha=0$: sub-surface (upper panel) and deep (bottom panel). ](15949f28.ps "fig:"){width="8cm"}
From the viewpoint of dynamo theory, we have shown that by exploiting the considerable range of freedom in choosing ill-known physical quantities, we can produce a wider range of activity wave behaviour than had previously been realized – indeed we can find something approaching the observed range of stellar surface activity waves. Of course, theory cannot at present say which of the models (if any) correspond even loosely to reality. We just make the point that the various observed behaviours are not incompatible with even simple dynamo models. [ The inherent uncertainties of mean field models make it difficult to make a stronger or more useful statement than this.]{} It remains to be clarified, for example, whether there is a correlation of the dynamo wave behaviour with the absolute value of the dynamo number, as suggested by our ordering of the cases in Sect. \[observations\] by increasing values of $Ro$. To take just one point considering a solar-like rotation, when the dynamo number is large (and $Ro$ is small) we generally see two waves with comparable magnetic field strength, starting from mid latitudes and migrating towards the pole and the equator, respectively. On the other hand, when the dynamo number is small (and $Ro$ is large) the equatorwards wave has a stronger field than the polar wave. [ Further observational evidence and characterization of polewards waves would be of special interest and value.]{} Also the simultaneous presence of a standing dynamo wave at a fixed latitude together with a migrating wave is of interest in interpreting the behaviour of some of the Mt. Wilson stars.
Finally, we have of necessity based this exploratory paper on a rather small number of relatively well observed stars (and of a small subset of all possible dynamo models). We thus make the usual plea for more, reliable, data in order to substantiate (or disprove) our attempts at systematization.
The authors are grateful to an anonymous referee for a careful reading and several useful comments on an earlier version of the manuscript. D.S. acknowledges financial support from RFBR project 09-05-00076.
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[^1]: Turbulent convection simulations by [@Kapylaetal09] do indeed show an $\alpha$ effect that changes sign from the bottom to the top of the convective domain (cf. the sign of their tensor coefficient $\alpha_{\rm yy}$, that corresponds to the scalar coefficient $\alpha$ of our simplified mean-field model, in their Figs. 3 and 12). [ This effect also appears when $\alpha$ is determined from some turbulence models – see e.g. [@br] and references therein.]{}
|
---
abstract: |
The unidirectional FM index was introduced by Ferragina and Manzini in 2000 and allows to search a pattern in the index in one direction. The bidirectional FM index (2FM) was introduced by Lam et al. in 2009. It allows to search for a pattern by extending an infix of the pattern arbitrarily to the left or right. The method of Lam et al. can conduct one step in time ${\mathcal{O}}(\sigma)$ while needing space ${\mathcal{O}}(\sigma \cdot n)$ using constant time rank queries on bit vectors. Schnattinger and colleagues improved this time to ${\mathcal{O}}(\log \sigma)$ while using ${\mathcal{O}}(\log \sigma \cdot n)$ bits of space for both, the FM and 2FM index. This is achieved by the use of binary wavelet trees.
In this paper we introduce a new, practical method for conducting an exact search in a uni- and bidirectional FM index in ${\mathcal{O}}(1)$ time per step while using ${\mathcal{O}}(\log \sigma \cdot n) + {o}(\log \sigma \cdot \sigma \cdot n)$ bits of space. This is done by replacing the binary wavelet tree by a new data structure, the *Enhanced Prefixsum Rank dictionary* ([EPR]{}-dictionary).
We implemented this method in the SeqAn C++ library and experimentally validated our theoretical results. In addition we compared our implementation with other freely available implementations of bidirectional indices and show that we are between $\approx 2.6-4.8$ times faster. This will have a large impact for many bioinformatics applications that rely on practical implementations of (2)FM indices e.g. for read mapping. To our knowledge this is the first implementation of a constant time method for a search step in 2FM indices.
author:
- 'Christopher Pockrandt[^1]'
- 'Marcel Ehrhardt[^2]'
- 'Knut Reinert[^3]'
bibliography:
- 'references.bib'
title: 'EPR-dictionaries: A practical and fast data structure for constant time searches in unidirectional and bidirectional FM-indices'
---
FM index, bidirectional, BWT, bit vector, rank queries, read mapping.
Introduction
============
It is seldom that new data structures or algorithms have such a large practical impact as full text indices had for biological sequence analysis. The so-called next-generation sequencing (NGS) allows to produce billions of small DNA strings called *reads*, usually of length 100-250. It is an invaluable technology for a multitude of applications in biomedicine. In many of these applications finding the positions of the DNA strings in a reference genome (i.e., a large string of length $\approx 10^7-10^{10}$) is the first fundamental step preceding downstream analyses. Finding the positions of the reads is commonly referred to as *read mapping*.
Because of sequencing errors and genomic variations not all strings occur exactly in a reference genome. Therefore approximate occurrences must be considered and algorithms for approximate string matching tolerating mismatches, insertions, and deletions must be applied to solve the problem.
This has triggered a plethora of work in the field to implement fast and accurate read mappers. Many of the popular programs like Bowtie2 [@Bowtie2], BWA [@BWA], BWA-Mem [@Li_Aligning_2013], Masai [@Siragusa_Fast_2013], Yara [@Yara2015], and GEM [@Marco-Sola_The] use as their main data structure a version of the FM index [@Ferragina_Opportunistic_2000] that was introduced by Ferragina and Manzini in 2000. The FM index is based on the Burrows-Wheeler transform (BWT) [@Burrows94ablock-sorting] of the given text, i.e., the genomes at hand, and conceptually some lookup tables containing counts of characters in prefixes of the text. In its original form it allows to search exactly for a pattern in *one direction* by matching the characters of the pattern with characters in the BWT [@Burrows94ablock-sorting] (i.e., extending a suffix of the pattern character by character to the left). It was later extended to the 2FM index by Lam et al. [@Lam_High_2009] and Schnattinger et al. [@Schnattinger_Bidirectional_2012]. The 2FM index allows to search in both directions, that means we can extend an infix of a pattern arbitrarily to the left or to the right. In order to reduce its space requirements, the count tables are in practice replaced by efficient bit vector data structures with rank support (see for example [@jacobson1988succinct]). The search method of Lam et al. can conduct one search step in a 2FM index in time ${\mathcal{O}}(\sigma)$ while needing space ${\mathcal{O}}(\sigma \cdot n)$ using constant time rank queries on bit vectors. Schnattinger et al. improved this time to ${\mathcal{O}}(\log \sigma)$ while using ${\mathcal{O}}(\log \sigma \cdot n)$ bits of space for both, the FM and 2FM index. This is achieved by the use of binary wavelet trees introduced by Grossi et al. [@Grossi_High_2003]. In the last years several theoretical results appeared that improved on this. However, none of those has found a way into a practical implementation.
In this paper we introduce a new method for conducting an exact search in a uni- and bidirectional FM index that needs ${\mathcal{O}}(1)$ time per step while using ${\mathcal{O}}(\log \sigma \cdot n) + {o}(\log \sigma \cdot \sigma \cdot n)$ bits of space. This is done by replacing the binary wavelet tree by a new data structure, the *Enhanced Prefixsum Rank dictionary* ([EPR]{}-dictionary). To our knowledge this is the first implementation of a constant time method for 2FM indices. We will show, that the method outperforms other implementations by several factors at the expense of a slight increase in memory usage resulting in a very practical method.
In the following paragraph we will review the concepts of the FM and 2FM index as well as constant time rank queries very shortly (readers unfamiliar with this can find a more detailed description in the appendix).
Introduction to the FM and 2FM Index
------------------------------------
Given a text $T$ of length $n$ over an ordered, finite alphabet $\Sigma = \{c_1, \dots, c_{\sigma}\}$ with $\forall \, 1 \leq i < \sigma: c_{i} <_{lex} c_{i+1}$, let $T[i]$ denote the character at position $i$ in $T$, $\cdot$ the concatenation operator and $T[1..i]$ the prefix of $T$ up to the character at position $i$. $T^{rev}$ represents the reversed text. We assume that $T$ ends with a sentinel character $\$\notin \Sigma$ that does not occur in any other position in $T$ and is lexicographically smaller than any character in $\Sigma$. The FM index needs the Burrows-Wheeler transform (BWT) of $T$. The BWT is the concatenation of characters in the last column of all lexicographically sorted cyclic permutations of the string $T$. We will refer to the BWT as $L$.
In contrast to suffix trees or suffix arrays, where a prefix $P$ of a pattern is extended by characters to the right (referred to as forward search $P \rightarrow Pc$ for $c \in \Sigma$), the FM index can only be searched using backward searches, i.e., extending a suffix $P'$ by characters to the left, $P' \rightarrow cP'$. Performing a single character backward search of $c$ in the FM index will require two pieces of information. First, $C(c)$, the number of characters in $L$ that are lexicographically smaller than $c$, second, $Occ(c, i)$, the number of $c$’s in $L[1..i]$. Given a range $[a, b]$ for $P$; i.e., the range in the sorted list of cyclic permutations that start with $P$, we can compute the range $[a^\prime, b^\prime]$ for $cP$ as follows: $[a^\prime, b^\prime]$ = $[C(c) + Occ(c, a - 1) + 1, C(c) + Occ(c, b)]$.
The 2FM index maintains two FM indices ${\mathcal{I}}$ and ${\mathcal{I}}^{rev}$, one for the original text $T$ and one for the reversed text $T^{rev}$. Searching a pattern left to right on the original text (i.e., conducting a forward search) corresponds to a backward search in ${\mathcal{I}}^{rev}$; searching a pattern right to left in the original text corresponds to a backward search in ${\mathcal{I}}$. The difficulty is to keep both indices *synchronized* whenever a search step is performed. W.l.o.g. we assume that we want to extend the pattern to the right, i.e., perform a forward search $P \rightarrow Pc_j$ for some character $c_j$. First, the backward search $P^{rev} \rightarrow c_jP^{rev}$ is carried out on ${\mathcal{I}}^{rev}$ and its new range $[a^\prime_{rev}, b^\prime_{rev}] = [C(c_j) + Occ(c_j, a_{rev} - 1) + 1, C(c_j) + Occ(c_j, b_{rev})]$ is computed. The new range in ${\mathcal{I}}$ can be calculated using the interval $[a,b]$ for $P$ in ${\mathcal{I}}$ and the range size of the reversed texts index $[a^\prime, b^\prime] = [a+smaller,a+smaller+ b^\prime_{rev} - a^\prime_{rev}]$. To compute $smaller$, e.g. Lam et al. [@Lam_High_2009] propose to perform ${\mathcal{O}}(\sigma)$ backward searches $P^{rev} \rightarrow c_iP^{rev}$ for all $1 \leq i < j$ and sum up the range sizes, i.e., $smaller = \sum_{1 \leq i < j} Occ(c_i, b_{rev}) - \sum_{1 \leq i < j} Occ(c_i, a_{rev} - 1)$ leading to a total running time of ${\mathcal{O}}(\sigma)$.
The implementation of the occurrence table $Occ$ is usually *not* done by storing explicitly the values of the entire table. Instead of storing the entire $Occ : \Sigma \times \{1,\dots,n\} \rightarrow \{1,\dots,n\}$ one uses the more space-efficient *constant time rank dictionary*: for every $c \in \Sigma$ a bit vector $B_c[1..n]$ is constructed such that $B_c[i] = 1$ if and only if $L[i] = c$. Thus the occurrence value equals the number of $1$’s in $B_c[1..i]$, i.e., $Occ(c, i) = rank(B_c, i)$. Jacobson [@jacobson1988succinct] showed that rank queries can be answered in constant time using only ${o}(n)$ additional space per bit vector by employing a sum of two count arrays (i.e., *blocks* and *superblocks*) and a final *in-block* count. Since then many other constant time rank query data structures have been proposed. For an overview we refer the reader to [@Navarro_Fast_2012] containing a comparison of various implementations. For readers unfamiliar with *2-level rank dictionaries*, an explanation is given in the appendix.
Recent improvements on the FM and 2FM index
-------------------------------------------
For large alphabets, it is not practical to maintain for each character a bit vector with rank support. In 2003 Grossi et al. [@Grossi_High_2003] proposed the use of a more space efficient data structure for the FM index, called the *(binary) wavelet tree* (WT) that was later used by Schnattinger [@Schnattinger_Bidirectional_2012] for an implementation of bidirectional FM indices. It is a binary tree of height ${\mathcal{O}}(\log \sigma)$ with a bit vector of length $n$ with rank support at each level. This reduces the space consumption by a factor of ${\mathcal{O}}(\frac{\log \sigma}{\sigma})$ in trade-off for an increased running time of ${\mathcal{O}}(\log \sigma)$. Schnattinger used the fact that not only the rank query for a given character $c$ can be computed in ${\mathcal{O}}(\log \sigma)$ but also the $smaller$ value can be computed in the same asymptotic time which is quite convenient for the 2FM index. Ferragina et al. proposed a new data structure in 2007 [@ferragina2007compressed], the multi-ary wavelet tree, which could be used to speed up the needed rank queries of 2FM indices. In 2013 Belazzougui et al. proposed the first constant-time bidirectional FM index [@belazzougui2013versatile] using minimal perfect hashing, of which to our knowledge no implementation exists (see also [@Belazzougui_Optimal_2015] for an extended version). Our solution is based on bit vectors with rank support, which proved so far to be very fast in practice, in particular due to the [popcount]{} machine operation.
Theoretical results {#sec:main}
===================
In this section we present the main results of this paper. They are based on a simple observation and a new bit vector data structure with rank query support which allows us to improve upon the results of Lam and Schnattinger. Our proposed method runs in constant time per step while using ${\mathcal{O}}(\log \sigma \cdot n) + {o}(\log \sigma \cdot \sigma \cdot n)$ bits of space for small alphabets (i.e., $\sigma < \log(n) / \log\log(n)$) which is in theory inferior in space consumption to the results of mentioned above (see [@belazzougui2013versatile]), but in practice very fast, and presents to our knowledge the first constant time implementation of 2FM indices with this space complexity.
Our first observation is simple. Instead of defining a bit vector for each $c \in \Sigma$ to map characters equal to $c$ in $L$ to $1$’s, we suggest using prefix sum bit vectors $PB_c$, i.e., $PB_c[i] = 1$ if and only if $L[i] \leq_{lex} c$ for $c \in \Sigma$.
\[thm:prefixsum\] A step in a bidirectional search can be performed in time ${\mathcal{O}}(1)$ using ${\mathcal{O}}(\sigma\cdot n)$ bits of space.
We define $\text{\it Prefix-Occ}(c_j, i)=rank(PB_{c_j},i)$; that means it counts the number of occurrences of a character lexicographically smaller or equal than $c_j$ up to position $i$. $\text{\it Prefix-Occ}(c_j, i)$ and thus the $smaller$ value for the 2FM index can now be computed by a single rank query $rank(PB_{c_j}, i)$, the original $Occ(c_j, i)$ value for backward searches needs only two rank queries and a subtraction, namely $Occ(c_j, i)=rank(PB_{c_j}, i) - rank(PB_{c_{j-1}}, i)$ (for the lexicographically smallest character $c_0$ no subtraction is necessary).
Note that the bit vector for the lexicographically largest character can be omitted, since all bits will be set to $1$ and thus $rank(PB_{c_{\sigma}}, i) = i, \;\forall\, 1 \leq i \leq n$.
Our next idea is the main result of this work and will allow us to reduce the space complexity for both the FM and the 2FM index while maintaining the optimal running time of ${\mathcal{O}}(1)$ per search step. Instead of using normal bit vectors we use directly the binary encoding of the BWT (an idea already used by BWT-SW[@Lam_Compressed_2008]). We call our data structure *EPR-dictionary*, short for *Enhanced Prefixsum Rank dictionary*.
The EPR-dictionary {#sec:eprdict}
------------------
The general idea of the EPR-dictionary is as follows. Assuming an ordered alphabet $\Sigma=\{c_1,\ldots,c_{\sigma}\}$, each character $c_i$ is encoded by the binary value $ord_2(c_i)$ of its rank $i$. Conceptually, we use the binary representation of the BWT to derive from it a *spaced* bit vector representation for $PB_c$ for each $s\in\Sigma$. Then we compute the auxiliary data structures (i.e., blocks and superblocks (see also appendix)) for each of those vectors. After we have those auxiliary structures which only need $o(n)$ bits space, we *delete* the bit vectors and only retain the BWT. In practice the blocks and superblocks are computed directly by a linear scan on the BWT. For the last in-block query, we show how to derive the counts $\text{\it Prefix-Occ}(c_j, i)$ from the BWT in constant time using a number of logical and arithmetic operations.
W.l.o.g. we assume that the block length is an even multiple of $\log \sigma$ to avoid case distinctions in the proof. In practice this holds since the in-block query is performed with popcounts on registers the length of which is a power of $2$. All bitmasks used for computing the in-block rank are exactly as long as a block. For a character $c_j$ we define the *rank bitmask* $rb(c_j)$ to be a binary sequence of concatenations of the pattern $0^{\lceil\log\sigma\rceil-1}1\cdot ord_2(c_j)$, i.e., $\lceil\log\sigma\rceil-1$ many bits set to $0$ followed by a $1$ followed by the binary encoding of the character $c_j$. For the DNA alphabet with $\Sigma = \{A, C, G, T\}$ and its binary encodings $\{00, 01, 10, 11\}$ the rank bitmask for $G \in \Sigma$ is for example $rb(G)=01\cdot 10\cdot 01\cdot 10\ldots$.
Counting the characters inside a block is done in two steps. The characters at even and odd positions are counted separately to generate space for an overflow bit. Therefore we need a bitmask $M_E$ masking characters at even positions from the bit vector. $M_E$ has $1$s for each even block of length $\lceil\log\sigma\rceil$, i.e., $M_E=00\cdot 11\cdot 00\cdot 11\cdot 00\cdot 11\ldots$. Finally, we need a bitmask $BM$ which filters out the lowest bit of each odd $\log\sigma$ block, i.e., $BM=01\cdot 00 \cdot 01\cdot 00\cdot 01\cdot 00\ldots$ for $\sigma=4$.
*Step 1.* We first take the characters at odd positions inside the corresponding block of the BWT, subtract it from $rb(c_i)$, which will result in the rightmost bit of even character positions to be set to $1$ if and only if the character to the right is smaller or equal to $c_i$. We then obtain exactly those bits by masking with $BM$.
$$B_{EPR}(c_i)_E = (rb(c_i) - (BWT\&M_E))\&BM$$
*Step 2.* We then take the characters at even positions inside the corresponding block of the BWT by shifting them $\lceil\log\sigma\rceil$ bits to the right and masking with $M_E$. We can now continue as in step 1 by subtracting it from $rb(c_i)$, which will again result in a $1$ bit in the rightmost bit of even character positions to be set to $1$ if and only if the character to the right is smaller or equal to $c_i$. We then apply the bitmask $BM$ to filter only these rightmost bits.
$$B_{EPR}(c_i)_O = (rb(c_i) - ((\gg_{\lceil\log\sigma\rceil} BWT) \& M_E)) \& BM$$
Finally both bit vectors are merged with one of them shifted by $1$ to the left avoiding the rightmost bits to overlap. In practice this is faster than two popcount operations.
$$B_{EPR}(c_i)= B_{EPR}(c_i)_E | (\ll_{1} B_{EPR}(c_i)_O)$$
Since we used the binary encoding of the BWT, note that the underlying rank queries have to be adapted to $\text{\it Prefix-Occ}(c_j, i) = rank(B_{\text{\it EPR}}(c_j), (i-1) \cdot \left\lceil \log \sigma \right\rceil+1)$. It follows directly that $Occ(c_j, i)$ can by computed in constant time by observing that
$$Occ(c_j, i) = \begin{cases}
\text{\it Prefix-Occ}(c_j, i) - \text{\it Prefix-Occ}(c_{j-1}, i) &\mbox{if } j > 0 \\
\text{\it Prefix-Occ}(c_j, i) & \mbox{otherwise}
\end{cases}$$
The EPR-transformed bit vector $B_{\text{\it EPR}}(c_j)$ is now a ”normal” bit vector and thus allows us to compute the prefix sums for a string in constant time. This improves the running time of the 2FM index and makes it optimal in terms of speed.
Let us now take a look at the space consumption. For our exposition we define the block length of $\ell = \left\lfloor \frac{\log n}{2} \right\rfloor$ (if $\ell$ is not a multiple of $\lceil\log\sigma\rceil$ padding strategies can be applied). Given a $B_{\text{\it EPR}}(c_j)$, for the $m$-th superblock we count the number of $1$’s (i.e., the number of occurrences of characters smaller or equal to $c_j$ in the corresponding BWT) from the beginning of $B_{\text{\it EPR}}$ to the end of the superblock in $M'[m][j]=rank(B_{\text{\it EPR}}(c_j), m\cdot \ell^2)$. As there are $\left\lfloor\frac{\lceil\log\sigma\rceil \cdot n}{\ell^2}\right\rfloor$ superblocks and $\sigma$ characters, $M'$ can be stored in $${\mathcal{O}}\left(\sigma\cdot\frac{\log\sigma\cdot n}{\ell^2} \cdot \log n\right)={\mathcal{O}}\left(\sigma\cdot\log\sigma\cdot\frac{n}{\log n}\right)={o}(\sigma \log \sigma \cdot n)$$ bits. For the $m$-th block we count the number of $1$’s from the beginning of the overlapping superblock to the end of the block in $M[m][j]=rank\big(B_{\text{\it EPR}}[1+k\ell..n](c_j), (m-k)\ell\big)$ where $k=\left\lfloor\frac{m-1}{\ell}\right\rfloor\ell$ is the number of blocks left of the overlapping superblock. $M$ has $\left\lfloor\frac{\lceil\log \sigma\rceil \cdot n}{\ell}\right\rfloor$ entries for every character and can be stored in $${\mathcal{O}}\left(\sigma\cdot\frac{\log \sigma\cdot n}{\ell} \cdot \log \ell^2\right)={\mathcal{O}}\left(\sigma\cdot\log \sigma \cdot\frac{ n\cdot\log\log n}{\log n}\right)={o}(\sigma \log \sigma \cdot n)$$ bits.
Let $P$ be a precomputed lookup table such that for each possible infix $V$ of a bit vector $B_{\text{\it EPR}}(c_j)$ of length $\ell$, $i\in\left[1..\left\lfloor\frac{\ell}{\log\sigma}\right\rfloor\right]$ and $c_j \in \Sigma$ holds $P[V][i]=rank(V, (i-1)\cdot\lceil\log\sigma\rceil + 1)$. There are $2^\ell \cdot \left\lfloor\frac{\ell}{\log\sigma}\right\rfloor$ entries of value at most $\left\lfloor\frac{\ell}{\log\sigma}\right\rfloor$ and thus can be stored in $${\mathcal{O}}\left(2^\ell \cdot \frac{\ell}{\log\sigma} \cdot \log\frac{\ell}{\log\sigma}\right)={\mathcal{O}}\left(2^\frac{\log n}{2}\cdot\log (n-\sigma) \cdot \log\log (n-\sigma)\right)=$$ $${\mathcal{O}}\left(\sqrt{n}\cdot \log n \cdot \log\log n \right)={o}(n)$$
bits. Note that we do need this lookup table only *once*, since the position and counting of the bits set to $1$ is the same for all characters.
Equivalent to Theorem 1, we do not need to store blocks and superblocks for $c_{\sigma}$ since $rank(B_{\text{\it EPR}}(c_{\sigma}), i) = i, \;\forall\, 1 \leq i \leq n$.
\[thm:ctr\] One search step in an FM index or 2FM index can be performed in ${\mathcal{O}}(1)$ time using $O(\log \sigma \cdot n) + {o}(\log \sigma\cdot\sigma\cdot n)$ bits of space.
The $BWT$ can be stored in ${\mathcal{O}}(\log \sigma \cdot n)$, all the blocks, superblocks, and lookup table $P$ in ${o}(\log \sigma \cdot \sigma \cdot n)$ bits. A prefix sum rank query requires one superblock and block lookup as well as a constant number of arithmetic and logical operations on the last block which run all in constant time.
Experimental results {#sec:experiments}
====================
In this Section we will conduct computational experiments to validate our theoretical findings and to compare our FM and 2FM indices to another available implementation. All of our tests were conducted on Debian GNU/Linux 7.1 with Intel® Xeon® E5-2667V2 CPUs at fixed frequency of 3.3 GHz to prevent dynamic overclocking effects. All data was stored on tmpfs, a virtual file system in main memory to prevent loading data just on demand during the search and thus effecting the speed of the search by I/O operations.
In the first part of the experiments we will test FM and 2FM indices with our new data structure (EPR) in comparison to the wavelet tree (WT) implementation which was previously the generic standard implementation in SeqAn [@SeqAn08]. Additionally we will run the same benchmarks for other available 2FM implementations, namely the bidirectional wavelet tree by Schnattinger et al. [@Schnattinger_Bidirectional_2012] which we will call 2SCH and the balanced wavelet tree implementation with plain bit vectors and constant-time rank support in the SDSL [@gbmp2014sea] which we will refer to as 2SDSL.
The 2BWT by Lam et al. [@Lam_Compressed_2008] is unfortunately not generic and only works for DNA alphabets. We also were not able to retrieve all hits when switching between forward and backward searches on the same pattern. Unfortunately we couldn’t reach the authors and thus excluded 2BWT from our comparisons.
Another implementation that is worth mentioning is the affix array by Meyer et al. [@Meyer_Structator_2011]. Even though the affix array implementation is generic, the construction algorithm did not terminate for alphabets other than DNA in a reasonable amount of time (several days). Unfortunately the affix array is not stand-alone but part of an application and does not provide a documented interface. Hence we were not able to include the affix array in our tests within a reasonable time frame. Meyer compares the running time of their affix array implementation with 2SCH and states that the affix array is faster by a factor of $1.26$ to $2$. From that we can conclude that our 2FM index implementation using the EPR-dictionary is expected to be faster than the affix array implementation (see below).
Runtime and space consumption
-----------------------------
For the first benchmark we want to make a comparison with alphabets of different sizes to test the predicted independence from $\sigma$ for the EPR implementation. The alphabet sizes are inspired by bioinformatics applications and are of size 4 (DNA), 10 (reduced amino acid alphabet *Murphy10*), 16 (IUPAC alphabet) and 27 (protein alphabet).
We first generated a text of length $10^8$ with a uniform distribution and sampled 1 million queries of length $50$ from this text. The search in the FM and 2FM indices will determine the number of occurrences of the sampled strings. Our sampling will ensure that the text occurs at least once and the stepwise search is never prematurely stopped. This ensures that we have $50$ million single steps in searches. The unidirectional FM indices perform backward searches while for 2FM indices we search the right half of the query first (using forward searches) and then extend the other half of the pattern to the left by backward searches.
In the following we will refer to WT and EPR as unidirectional FM indices and to 2WT and 2EPR as bidirectional FM indices, all part of the SeqAn library.
Table \[tab:runningtime\] gives an overview of the running times of all FM and 2FM index implementations. It shows the absolute runtimes as well as the speedup factor relative to the unidirectional resp. bidirectional wavelet tree implementation. WT, 2WT, 2SCH, 2SDSL are all based on wavelet trees. Our bidirectional wavelet tree implementation 2WT has a similar runtime compared to 2SDSL. It is slightly faster especially for small alphabets.
---------- ----------- ---------- ----------- ---------- ----------- ---------- ----------- ----------
Index time factor Time factor Time factor Time factor
WT 6.59s 1.00 16.97s 1.00 21.61s 1.00 26.87s 1.00
**EPR** **3.63s** **1.82** **5.35s** **3.17** **5.65s** **3.83** **6.20s** **4.34**
2WT 9.32s 1.00 19.15s 1.00 23.44s 1.00 28.83s 1.00
**2EPR** **4.69s** **1.99** **5.78s** **3.31** **5.67s** **4.13** **6.21s** **4.64**
2SDSL 12.21s 0.76 20.58s 0.93 24.43s 0.96 29.76s 0.97
2SCH 14.08s 0.66 22.18s 0.86 26.11s 0.90 31.81s 0.91
---------- ----------- ---------- ----------- ---------- ----------- ---------- ----------- ----------
: Runtimes of various implementations in seconds and their speedup factors with respect to the unidirectional wavelet tree.[]{data-label="tab:runningtime"}
Compared to the wavelet tree implementations the EPR implementation is between 80% (for DNA) and 330% (Protein) faster for unidirectional indices and between 100% (for DNA) and 360% (Protein) faster for bidirectional indices.
Since we anticipate the application of 2EPR to bioinformatics applications, we also compared the runtime of all implementations using the complete human genome sequence. We again searched one million sampled strings of length 50 exactly as described above. The relative results were very similar to the ones in Table \[tab:runningtime\], indeed even slightly better. 2SCH crashed with this data set. 2SDSL was the slowest implementation (15.44s) followed by 2WT (1.6 times as fast) and by 2EPR (2.8 times as fast as 2SDSL) which was again the fastest implementation.
Our experiments also show that we were indeed able to eliminate the $\log \sigma$ factor of wavelet trees in practice, as predicted by Theorem \[thm:ctr\]. While the runtime for the WT implementations grows for larger alphabets with $\log \sigma$ the runtime of EPR and 2EPR increases only slightly for larger alphabets which can be explained by larger indices and therefore more cache misses. This can be seen in the following Figure in which we plot the runtime for EPR for different alphabets and the runtime of WT divided by $\log\sigma$. The resulting times develop very similarly.
coordinates [ ( 4, 4.69) (10, 5.78) (16, 5.67) (27, 6.21) ]{}; coordinates [ ( 4, 4.66) (10, 5.76) (16, 5.86) (27, 6.67) ]{};
When we compare the runtimes of the EPR and 2EPR, they behave as expected, i.e., the unidirectional index is slightly faster, since in each step of the bidirectional index we have to synchronize two indices.
All indices implemented in SeqAn (WT, EPR, 2WT, 2EPR) support up to 3 levels for rank dictionary support: blocks, superblocks and ultrablocks. The tests presented here were performed with a 2-level rank dictionary similar to the one explained in Section \[sec:eprdict\] (or in the Appendix). Table \[tab:space\] illustrates the practical space consumption for all previously discussed indices and of the affix array for DNA (larger alphabets did not finish within several days).
Please note, that the other implementations may use versions of rank dictionaries different to the simple one explained in Section \[sec:eprdict\]. The numbers of FM indices given in Table \[tab:space\] do neither account for storing the text itself nor for storing a compressed suffix array necessary to locate the matches in the text since the libraries use different implementations offering various space-time trade-offs. The running time of the backward and forward searches does not depend on it and the compressed suffix array implementation is independent from the used rank dictionary and thus interchangeable. A typical compressed suffix array implementation as used in the 2SDSL takes $\frac{n}{\eta}\log n$ (when sampling on the text instead of the suffix array). For a sampling rate of $10\%\phantom{..}(\eta = 10)$ the space consumption for our experiments would be $253$ MB and thus still much smaller than the affix array.
Index DNA Murphy10 IUPAC Protein
------- ------ ---------- ------- ---------
EPR 42 156 227 478
2EPR 84 311 454 955
WT 30 51 60 72
2WT 60 102 120 144
2SDSL 68 105 122 145
2SCH 75 108 123 146
AF 2670 - - -
: Space consumption of the rank data structure in Megabyte of various implementations[]{data-label="tab:space"}
The current implementation of the EPR and 2EPR in SeqAn interleaves the bit vector (i.e., the BWT) and precomputed block values but does not interleave superblock values. Reconsidering the design and storing block and superblock values close to the corresponding bit vector region could decrease the number of cache misses for one rank query to one cache miss and thus further improve the running time.
For larger alphabets one might also consider using a 3-level rank dictionary with smaller data types for blocks and superblocks which will reduce the space consumption noticeably at the expense of a slightly higher runtime (i.e., for the protein alphabet we reduce the space consumption from $955$ MB to $581$ MB while increasing the runtime from $6.21$ to $7.67$ seconds). The increased running time is due to another array lookup and thus still constant-time per step.
Effect of the low order terms for space consumption
---------------------------------------------------
In Table \[tab:expchangen\] we show how quickly the ${o}(\log \sigma \cdot \sigma \cdot n)$ data structures for rank queries can be neglected for growing $n$. For the WT and EPR implementations we measured the space needed for both the DNA and the IUPAC alphabet for $n=10^4,10^5,10^6,10^7,10^8,10^9$. We then divided the space consumption of both implementations by the factor in the ${\mathcal{O}}$-term, namely $\log \sigma\cdot n$.
For growing $n$ the ${\mathcal{O}}$-term should dominate the low order ${o}$-term, hence we would expect the resulting number converge to a constant. This is indeed true, as can be seen in Table \[tab:expchangen\]. The EPR implementation converges faster than the WT, which is expected, since our ${o}$-term is larger than the one for the WT implementations. The effect of the ${o}$-terms falls for EPR from $10^5$ to $10^6$ by $35$ resp. $6$ percent, whereas the decline for WT is steeper with $84$ and $123$ percent. From size $10^7$ on, the low order terms are clearly dominated by the ${\mathcal{O}}$-terms.
Method ($\sigma$) $10^4$ $10^5$ $10^6$ $10^7$ $10^8$ $10^9$
------------------- -------- -------- -------- -------- -------- -------- -- --
EPR (4) 2.4000 0.6000 0.4440 0.4292 0.4276 0.4274
EPR (16) 2.0000 1.2400 1.1680 1.1610 1.1602 1.1601
WT (4) 4.4000 0.6400 0.3480 0.3088 0.3056 0.3053
WT (16) 7.0000 0.8000 0.3580 0.3104 0.3057 0.3053
: Influence of the space consumption of the ${o}$-terms with increasing $n$.[]{data-label="tab:expchangen"}
Conclusions {#sec:conclusions}
===========
In this paper we have introduced a new data structure, the EPR-dictionary, that enables constant time prefix sum computations for arbitrary, finite alphabets in ${\mathcal{O}}(\log\sigma\cdot n)+{o}(\log\sigma\cdot\sigma\cdot n)$ bits of space and works directly on the BWT. This allows two important data structures, the FM and 2FM index, to perform single search steps in time ${\mathcal{O}}(1)$. We implemented the dictionary in the C++ library SeqAn and used it for an implementation of an FM and 2FM index. We compared its practical performance with the previous SeqAn implementation using wavelet trees and with other openly available implementations, among them the quasi standard for succinct data structures, the SDSL. We show that the EPR-dictionary implementation supports our theoretical claims, eliminates the $\log\sigma$ factor for searching in bidirectional indices, and performs between $80\%$ and $360\%$ faster than the wavelet tree implementation at the expense of a higher memory consumption. We compared our 2FM implementation against the available, open implementation of Schnattinger et al. (2SCH). Our implementation is between $3$ to $5.1$ times faster than 2SCH and $2.6$ to $4.8$ faster than the 2SDSL. We also showed that the additional space consumption is easily tolerable on normal hardware.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to acknowledge Enrico Siragusa for his previous implementations of the FM index in SeqAn. The first author acknowledges the support of the International Max-Planck Research School for Computational Biology and Scientific Computing (IMPRS-CBSC). We also thank Veli Mäkinen and Simon Gog for very helpful remarks on a previous version of this manuscript during the Dagstuhl seminar 16351 ”Next Generation Sequencing - Algorithms, and Software For Biomedical Applications”.
Appendix {#appendix .unnumbered}
========
In the appendix we give for the reader not familiar with FM and 2FM indices a short introduction.
Introduction to the FM and 2FM Index {#introduction-to-the-fm-and-2fm-index-1 .unnumbered}
------------------------------------
Given a text $T$ of length $n$ over an ordered, finite alphabet $\Sigma = \{c_1, \dots, c_{\sigma}\}$ with $\forall \, 1 \leq i < \sigma: c_{i} <_{lex} c_{i+1}$, let $T[i]$ denote the character at position $i$ in $T$, $\cdot$ the concatenation operator and $T[1..i]$ the prefix of $T$ up to the character at position $i$. $T^{rev}$ represents the reversed text. We assume that $T$ ends with a sentinel character $\$\notin \Sigma$ that does not occur in any other position in $T$ and is lexicographically smaller than any character in $\Sigma$. The FM index needs the Burrows-Wheeler transform (BWT) of $T$. The BWT is the concatenation of characters in the last column of all lexicographically sorted cyclic permutations of the string $T$ (see Figure \[fig:bwt\] for an example). We will refer to the BWT as $L$.
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In contrast to suffix trees or suffix arrays, where a prefix $P$ of a pattern is extended by characters to the right (referred to as forward search $P \rightarrow Pc$ for $c \in \Sigma$), the FM index can only be searched using backward search, i.e., extending a suffix $P'$ by characters to the left, $P' \rightarrow cP'$. Performing a single character backward search of $c$ in the FM index will require two pieces of information. First, $C(c)$, the number of characters in $L$ that are lexicographically smaller than $c$, second, $Occ(c, i)$, the number of $c$’s in $L[1..i]$. Given a range $[a, b]$ for $P$; i.e., the range in the sorted list of cyclic permutations that starts with $P$, we can compute the range $[a^\prime, b^\prime]$ for $cP$ as follows: $[a^\prime, b^\prime]$ = $[C(c) + Occ(c, a - 1) + 1, C(c) + Occ(c, b)]$. We will refer to the BWT together with tables $C$ and $Occ$ as *FM index ${\mathcal{I}}$* (see Figure \[fig:bwt\] for an example of one search step).
The 2FM index maintains two FM indices ${\mathcal{I}}$ and ${\mathcal{I}}^{rev}$, one for the original text $T$ and one for the reversed text $T^{rev}$. Searching a pattern left to right on the original text (i.e., conducting a forward search) corresponds to a backward search in ${\mathcal{I}}^{rev}$; searching a pattern right to left in the original text corresponds to a backward search in ${\mathcal{I}}$. The difficulty is to keep both indices *synchronized* whenever a search step is performed. W.l.o.g. we assume that we want to extend the pattern to the right, i.e., perform a forward search $P \rightarrow Pc_j$ for some character $c_j$. First, the backward search $P^{rev} \rightarrow c_jP^{rev}$ is carried out on ${\mathcal{I}}^{rev}$ and its new range $[a^\prime_{rev}, b^\prime_{rev}] = [C(c_j) + Occ(c_j, a_{rev} - 1) + 1, C(c_j) + Occ(c_j, b_{rev})]$ is computed. The new range in ${\mathcal{I}}$ can be calculated using the interval $[a,b]$ for $P$ in ${\mathcal{I}}$ and the range size of the reversed texts index $[a^\prime, b^\prime] = [a+smaller,a+smaller+ b^\prime_{rev} - a^\prime_{rev}]$. To compute $smaller$, Lam et al. [@Lam_High_2009] propose to perform ${\mathcal{O}}(\sigma)$ backward searches $P^{rev} \rightarrow c_iP^{rev}$ for all $1 \leq i < j$ and sum up the range sizes, i.e., $smaller = \sum_{1 \leq i < j} Occ(c_i, b_{rev}) - \sum_{1 \leq i < j} Occ(c_i, a_{rev} - 1)$ leading to a total running time of ${\mathcal{O}}(\sigma)$ (See Figure \[fig:2fmsearch\] for an illustration).
The implementation of the occurrence table $Occ$ is usually *not* done by storing explicitly the values of the entire table. Instead of storing the entire $Occ : \Sigma \times \{1,\dots,n\} \rightarrow \{1,\dots,n\}$ one uses the more space-efficient *constant time rank dictionary*: for every $c \in \Sigma$ a bit vector $B_c[1..n]$ is constructed such that $B_c[i] = 1$ if and only if $L[i] = c$. Thus the occurrence value equals the number of $1$’s in $B_c[1..i]$, i.e., $Occ(c, i) = rank(B_c, i)$. Jacobson [@jacobson1988succinct] showed that rank queries can be answered in constant time using only ${o}(n)$ additional space per bit vector. Since then many other constant time rank query data structures have been proposed. For an overview we refer the reader to [@Navarro_Fast_2012] containing a comparison of various implementations. Since we will make also use of this technique, we explain the most simple idea, namely the one for *2-level rank dictionaries* in the following paragraph.
Constant time rank queries {#subsec:constrankquery .unnumbered}
--------------------------
In order to store partial prefix sums, the technique uses two levels of lookup table, called *blocks* and *superblocks*. Given a bit vector $B$ of length $n$ we divide it into blocks of length $\ell$ and superblocks of length $\ell^2$ where $$\ell=\left\lfloor\frac{\log n}{2}\right\rfloor.$$ For both, blocks and superblocks we allocate arrays $M$ and $M'$ of sizes $\left\lfloor\frac{n}{\ell}\right\rfloor$ and $\left\lfloor\frac{n}{\ell^2}\right\rfloor$ respectively (see Figure \[fig:2level\] for an illustration).
For the $m$-th superblock we store the number of $1$’s from the beginning of $B$ to the end of the superblock in $M'[m]=rank(B, m \cdot \ell^2)$. As there are $\left\lfloor\frac{n}{\ell^2}\right\rfloor$ superblocks, $M'$ can be stored in ${\mathcal{O}}\left(\frac{n}{\ell^2} \cdot \log n\right)={\mathcal{O}}\left(\frac{n}{\log n}\right)={o}(n)$ bits. For the $m$-th block we store the number of $1$’s from the beginning of the overlapping superblock to the end of the block in $M[m]=rank\big(B[1+k\ell..n],(m-k) \cdot \ell\big)$, where $k=\left\lfloor\frac{m-1}{\ell}\right\rfloor\ell$ is the total number of blocks in all superblocks left of the current superblock. $M$ has $\left\lfloor\frac{n}{\ell}\right\rfloor$ entries and can be stored in ${\mathcal{O}}\left(\frac{n}{\ell} \cdot \log \ell^2\right)={\mathcal{O}}\left(\frac{ n\cdot\log\log n}{\log n}\right)={o}(n)$ bits.
Given a rank query $rank(B, i)$, one can now add the corresponding superblock and block values. But we still have to account for the $1$’s in the block covering position $i$ (in case $i$ is not at the end of a block). Let $P$ be a precomputed lookup table such that for each possible bit vector $V$ of length $\ell$ and $i\in[1..\ell]$ holds $P[V][i]=rank(V,i)$. $V$ has $2^\ell \cdot \ell$ entries of values at most $\ell$ and thus can be stored in $${\mathcal{O}}\left(2^\ell \cdot \ell \cdot \log \ell \right)={\mathcal{O}}\left(2^{\frac{\log n}{2}} \cdot \log n \cdot \log \log n \right)= {\mathcal{O}}\left(\sqrt{n} \cdot \log n \cdot \log\log n \right)={o}(n)$$ bits. We now decompose a rank query into 3 subqueries using the precomputed tables. For a position $i$ we determine the index $p=\left\lfloor\frac{i-1}{\ell}\right\rfloor$ of next block left of $i$ and the index $q=\left\lfloor\frac{p-1}{\ell}\right\rfloor$ of the next superblock left of block $p$. Then it holds: $$rank(B,i)=M'[q]+M[p]+P\big[B[1+p\ell..(p+1)\ell]\big]\big[i-p\ell\big].$$
Since the text $T$ of length $n$ has to be addressed, we assume that a register has at least size $\left\lceil\log n\right\rceil$. Thus $B[1+p\ell..(p+1)\ell]$ fits into a single register and can be determined in ${\mathcal{O}}(1)$ time. Therefore a rank query can be answered in ${\mathcal{O}}(1)$ time. In practice the precomputed lookup table $P$ is replaced by a popcount operation on the CPU register and we have only two lookup operations.
One can now replace the occurrence table by this 2-level dictionary, i.e., by creating a bit vector for every $c \in \Sigma$ and setting it to $1$ if the character occurs in the BWT $L$. This results in ${\mathcal{O}}(\sigma\cdot n)+{o}(\sigma\cdot n)$ bits space requirement.
[^1]: FU Berlin (, <http://reinert-lab.de>).
[^2]: FU Berlin ().
[^3]: FU Berlin ()
|
---
abstract: 'The family of generalized Pseudo-Spectral Time Domain (including the Pseudo-Spectral Analytical Time Domain) Particle-in-Cell algorithms offers substantial versatility for simulating particle beams and plasmas, and well written codes using these algorithms run reasonably fast. When simulating relativistic beams and streaming plasmas in multiple dimensions, they are, however, subject to the numerical Cherenkov instability. Previous studies have shown that instability growth rates can be reduced substantially by modifying slightly the transverse fields as seen by the streaming particles . Here, we offer an approach which completely eliminates the fundamental mode of the numerical Cherenkov instability while minimizing the transverse field corrections. The procedure, numerically computed residual growth rates (from weaker, higher order instability aliases), and comparisons with WARP simulations are presented. In some instances, there are no numerical instabilities whatsoever, at least in the linear regime.'
address:
- 'University of Maryland, College Park, Maryland 20742, USA'
- 'Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA'
- 'Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA'
author:
- 'Brendan B. Godfrey'
- 'Jean-Luc Vay'
bibliography:
- 'C:/Users/Brendan/Documents/LyX/Godfrey.bib'
- 'C:/Users/Brendan/Documents/LyX/Biblio\_JCP\_Godfrey.bib'
title: Improved Numerical Cherenkov Instability Suppression in the Generalized PSTD PIC Algorithm
---
=1
Particle-in-cell ,Pseudo-Spectral Time-Domain,Relativistic beam ,Numerical stability.
Introduction
============
The Pseudo-Spectral Time Domain (PSTD) Particle-in-Cell (PIC) algorithm advances Fourier-transformed electromagnetic fields in time according to the difference equations [@Liu1997], $$\mathbf{E}^{n+1}=\mathbf{E}^{n}-i\mathbf{k}\times\mathbf{B}^{n+\nicefrac{1}{2}}\triangle t-\mathbf{J}^{n+\nicefrac{1}{2}}\triangle t,\label{eq:Eleapfrog}$$ $$\mathbf{B}^{n+\nicefrac{3}{2}}=\mathbf{B}^{n+\nicefrac{1}{2}}+i\mathbf{k}\times\mathbf{E}^{n+1}\triangle t.\label{eq:Bleapfrog}$$ Simulation particles, of course, require fields and produce currents in real, not Fourier, space. Consequently, Fourier transforms of both fields and currents are performed at each time step. The fields and currents in real space typically are located at the nodes of a regular multidimensional mesh, with magnetic fields $\mathbf{B}$ and currents $\mathbf{J}$ offset a half time step from electric fields $\mathbf{E}$.
The PSTD algorithm has been generalized by the authors in two ways [@Vay2014a; @Godfrey2014AAC]. First, the components of $\mathbf{k}$ in Eqs. (\[eq:Eleapfrog\]) and (\[eq:Bleapfrog\]) are replaced by the Fourier transforms of various order finite difference approximations to spatial derivatives on a grid [@Lele1992], which has the effect of narrowing the effective width of the particle interpolation stencils. For instance, $k_{i}$ might be replaced by $$\frac{\sin\left(k_{i}\triangle x_{i}/2\right)}{\triangle x_{i}/2},$$ yielding a pseudo-spectral realization in k-space of the standard Finite-Difference Time Domain (FDTD) algorithm. (In this context the components of $\mathbf{k}$ itself can be viewed as an infinite order approximation.) Second, the field solver is modified to increase the relatively small PSTD Courant time step limit by a factor of *N*, an integer. This modification is equivalent mathematically to sub-cycling the field solver *N* times but is faster computationally. Note that the generalized PSTD algorithm reduces to Haber’s Pseudo-Spectral Analytical Time-Domain (PSATD) algorithm [@HaberICNSP73] in the limit of infinite *N*. Both the details of and the motivation for the generalized PSTD algorithm, dubbed the Pseudo-Spectral Arbitrary Order Time Domain (PSAOTD) algorithm, are described thoroughly in [@Vay2014a].
The numerical Cherenkov instability [@godfrey1974numerical; @godfrey1975canonical] is observed commonly in PIC simulations of relativistic particle beams and streaming plasmas; *e.g.*, [@VayJCP2011; @Spitkovsky:ICNSP2011; @Xu2013], where it can be fast growing and strongly disruptive. At a minimum the instability increases beam emittance [@Cormier-Michel2008]. It arises from nonphysical coupling between the usual electromagnetic modes, possible distorted by numerical effects, and spurious beam modes [@godfrey2013esirkepov; @Godfrey2013PSATD]. The dispersion relation describing the numerical Cherenkov instability has the general form, $$\begin{gathered}
C_{0}+\omega_{p}^{2}\sum_{m_{z}}C_{1}\csc\left[\left(\omega-k_{z}^{\prime}v\right)\frac{\Delta t}{2}\right]+\omega_{p}^{2}\sum_{m_{z}}\left(C_{2x}+\gamma^{-2}C_{2z}\right)\csc^{2}\left[\left(\omega-k_{z}^{\prime}v\right)\frac{\Delta t}{2}\right]\\
+\gamma^{-2}\omega_{p}^{4}\left(\sum_{m_{z}}C_{3z}\csc^{2}\left[\left(\omega-k_{z}^{\prime}v\right)\frac{\Delta t}{2}\right]\right)\left(\sum_{m_{z}}C_{3x}\csc\left[\left(\omega-k_{z}^{\prime}v\right)\frac{\Delta t}{2}\right]\right)=0,\label{eq:drformfull}\end{gathered}$$ where the $C_{i}$ are complicated functions of the instability frequency $\omega$, wave numbers $\mathbf{k}$, time step $\Delta t$, cell size, and alias index $m_{z}$. The particular form of the $C_{i}$ is determined by the details of the numerical algorithm, given in [@Vay2014a] for PSAOTD. The relativistic beam itself is characterized by its normalized energy $\gamma$, axial velocity $v=\left(1-\gamma^{-2}\right)^{\nicefrac{1}{2}}$, and normalized relativistic plasma frequency $\omega_{p}$. Of particular importance are the resonances at $\omega=k_{z}^{\prime}v$, with alias wave numbers $k_{z}^{\prime}=k_{z}+m_{z}\,2\pi/\triangle z$. (The beam propagates along the *z*-axis.) Each alias can trigger an instability, with the most rapid growth typically occurring for $m_{z}=0,-1$. High order interpolation, say cubic, significantly reduces growth rates associated with higher order aliases. Although peak growth rates typically occur at large wave numbers, nontrivial growth rates can occur at quite small wave numbers, especially for $m_{z}=0$ [@godfrey2013esirkepov; @Godfrey2013PSATD]. The nonresonant $m_{z}=0$ instability has two branches [@Godfrey2013PSATD; @Yu2015], the primary branch associated with the $C_{2x}$ term of the dispersion relation, and the secondary branch with the $C_{1}$ term. The former occurs over a wide range of wave numbers, while the latter occurs at $k_{z}\approx\nicefrac{\pi}{2\,\triangle z}$ and small $k_{x}$, if at all. See Fig. \[fig:m0branches\]. Unless otherwise noted, parameters for these and other figures are $\omega_{p}=1$, $\gamma=130$, and $\Delta x=\Delta z=0.3868$. The k-space grid is 65x65, corresponding to a simulation grid of 128x128.
Until recently, the numerical Cherenkov instability was ameliorated by a combination of digital filtering, numerical damping, higher order interpolation (typically cubic), and astute parameter choices [@VayJCP2011; @Yu2014]. Papers within the past few years have offered alternative options, often entailing minor corrections to the transverse electric and magnetic fields as seen by the particles [@Vay2014a; @Godfrey2014AAC; @Godfrey2013PSATD; @Godfrey2014PSATD-TPS]. Specifically, transverse electric and magnetic fields as they are interpolated to the particles are multiplied by $\mathbf{k}$-dependent correction factors $\Psi_{E}$ and $\Psi_{B}$ that vary as $1+\mathcal{O}\left(k^{2}\right)$ for small $\mathbf{k}$. The distinction between such correction factors and the usual digital filters is made clear in [@Godfrey2013PSATD; @GodfreyJCP2014]. The particular analytical correction factors presented in these references are quite effective at ameliorating the first branch of the $m_{z}=0$ instability but ineffective at ameliorating the second, for reasons presented later in this article.
As an alternative, this paper numerically computes correction factors $\Psi_{E}$ and $\Psi_{B}$ just sufficient to completely eliminate both branches of the $m_{z}=0$ numerical Cherenkov instability. The process is straightforward in principle. The $m_{z}=0$ dispersion function is akin to a fifth order polynomial with real coefficients. If it crosses the *$\omega$*-axis in five places (i.e., five real roots), all roots are stable. However, if parameters are varied such that the curve crosses the axis in only three places or one, then the dispersion function has two or four complex roots, respectively, the remaining roots being real. The transition occurs where the curve becomes tangential to the *$\omega$*-axis, *i.e.*, where both the dispersion function and its derivative with respect to $\omega$ vanish. This determines $\Psi_{E}$ and $\Psi_{B}$. In fact, it can be shown that the $m_{z}=0$ portion of Eq. (\[eq:drformfull\]) can be represented as a fifth order polynomial, although this is not required for the argument of this paragraph to be true.
Fig. \[fig:DisFun\], computed for $\nicefrac{v\,\triangle t}{\triangle z}=0.9$ and $k_{z}=k_{x}=5.20$, is illustrative. The left plot displays the dispersion function, Eq. (\[eq:drformfull\]) with $m_{z}=0$ only, for the complete frequency range, $-\nicefrac{\pi}{\triangle t}<\omega<\nicefrac{\pi}{\triangle t}$. Shown are the results for the uncorrected case, $\Psi_{E}=\Psi_{B}=1$ (labeled Base), for the $C_{2x}$ correction factors described in [@Vay2014a] (labeled $C_{2x}$), and for the numerically determined optimal correction factors (labeled Opt). (The $C_{2x}$ correction factors are chosen so that the term $C_{2x}$ vanishes at $\omega=k_{z}v$). The three curves are essentially indistinguishable on this scale, with one crossing at the far left and two or four crossings at the right. The right plot displays a blow-up of the critical frequency range. Each curve has one crossing in this range, and a second crossing is off-scale to the right. However, the Opt curve is seen also to be tangent to the *$\omega$*-axis at a point. Hence, it is stable, and the other two cases are not. The corresponding $m_{z}=0$ instability growth rates are 0.0749, 0.0014, and 0 for Base, $C_{2x}$ , and Opt.
The next section provides sample numerical correction factors and the procedure used to obtain them. The third section then provides corresponding numerical growth rates from Eq. (\[eq:drformfull\]), along with corroborating instability growth rates from WARP two-dimensional simulations [@Warp]. A short concluding section completes the paper.
The analytical and numerical linear growth rate analyses were performed using *Mathematica* [@Mathematica10].
Numerical Procedure for Obtaining $\boldsymbol{\mathbf{\Psi}}_{\mathbf{E}}$ and $\boldsymbol{\Psi}_{\mathbf{B}}$
=================================================================================================================
The factors $\Psi_{E}$ and $\Psi_{B}$ appear linearly in the $m_{z}=0$ dispersion function, obtained from Eq. \[eq:drformfull\] by omitting all but the $m_{z}=0$ terms in the summations, and in its derivative with respect to $\omega$. As a consequence, the dispersion function and its derivative together can be inverted readily to obtain $\Psi_{E}$ and $\Psi_{B}$ as functions of the tangent point, $\omega$, although the actual expressions are quite complicated.
Fig. \[fig:ParamPlot\] (left) is a typical parametric plot of $\Psi_{B}$ vs. $\Psi_{E}$ as $\omega$ is varied. Its parameters are identical to those of Fig. \[fig:DisFun\]. The region below the diagonal curve is stable, above it unstable. (Despite appearances, it bends slightly and does not pass through the origin.) Although any choice of $\Psi_{B}$ and $\Psi_{E}$ along or below this curve stabilizes the $m_{z}=0$ mode, a pair close to (1,1) is to be preferred for minimizing numerical impact on the physics to be simulated. Further constraining the choice by $\Psi_{B}\leq1$, $\Psi_{E}\leq1$ suggests the following procedure:
1. If numerical instability growth is zero at (1,1), set $\Psi_{B}=1$, $\Psi_{E}=1$.
2. Otherwise, search the lines $\Psi_{B}=1$ and $\Psi_{E}=1$ for the point nearest (1,1) at which the dispersion function and its derivative simultaneously vanish. Usually, that point will be on the $\Psi_{E}=1$ line, as illustrated in the left plot of Fig. \[fig:ParamPlot\].
3. Otherwise, search for the point nearest (1,1) at which the dispersion function and its derivative simultaneously vanish in the interior of the $0<\Psi_{B}<1\;\land\;0<\Psi_{E}<1$ region. It is fortunate that this situation is rare, because such points are expensive to find computationally and easy to miss.
4. Otherwise, set $\Psi_{B}=\Psi_{E}=0$. The numerical instability cannot be eliminated, short of excising that portion of $\mathbf{k}$-space.
For completeness, Fig. \[fig:ParamPlot\] (right) provides the parametric plot of $\Psi_{B}$ vs. $\Psi_{E}$ for $k_{z}=4.697,\: k_{x}=0.635$, which lies at the center of the second branch of the $m_{z}=0$ numerical Cherenkov instability in Fig. \[fig:m0branches\]. Stable regions lie above the upper diagonal curve and below the lower diagonal curve. Only a very small reduction from unity in $\Psi_{E}$ is sufficient to eliminate the numerical instability.
Plots of $\Psi_{E}$ and $\Psi_{B}$ for the parameters of Fig. \[fig:DisFun\] are shown in Fig. \[fig:factors\] with the $m_{z}=0$ resonance curve superimposed. $\Psi_{E}$ is equal to unity everywhere but (1) in the small region, barely visible at the bottom center of the plot, where the second branch of the $m_{z}=0$ numerical Cherenkov instability appears in Fig. \[fig:m0branches\], (2) near the resonance curve, where $\Psi_{E}=0$, and (3) at three isolated points where $\Psi_{E}=0$ also. $\Psi_{E}\simeq0.97$ for the second branch of the $m_{z}=0$ numerical instability. In contrast, $\Psi_{B}<1$ everywhere except those $\mathbf{k}$ values where the $m_{z}=0$ growth rate is zero anyway. Note that $1-\Psi_{B}\simeq0$ over a wide range of small $\mathbf{k}$.
The three isolated $\Psi_{E}=\Psi_{B}=0$ points appearing in both plots are associated with a weak instability predicted by Birdsall and Langdon in Problem 5-9a of their well known text [@BirdsallLangdon]. Because this instability has a very narrow bandwidth in k-space, it appears only where the instability band happens to overlap nodes on the k-space numerical grid. Analysis of this Courant-like instability for the generalized PSTD algorithm is provided in [@Vay2014a], and analysis plus growth rates for the PSATD algorithm in [@Godfrey2014PSATD-TPS]. This instability appears to be unimportant in practice except, perhaps, at low $\gamma$.
Applying the correction factors of Fig. \[fig:factors\] for the parameters used to obtain Fig. \[fig:m0branches\] completely eliminates $m_{z}=0$ numerical instability growth but has minimal effect on higher order aliases except for $\mathbf{k}$-space regions where $\Psi_{E}$ or $\Psi_{B}$ are much less than unity. Hence, we choose to truncate fields in $\mathbf{k}$-space according to $$k>\alpha\min\left[\frac{\pi}{\triangle z},\frac{\pi}{v\,\triangle t}\right]\label{eq:clip}$$ in order to minimized peak growth rates of the first branch of the $m_{z}=-1$ instability. Fig. \[fig:ebcor4\] (left) displays the resulting growth rates for $\alpha=0.85$. Truncating $\mathbf{k}$-space according to Eq. (\[eq:clip\]) reduces the peak growth rate from 0.168 to 0.071, and the Fig. \[fig:factors\] correction factors further reduce it to 0.026. Most of the residual growth is associated with the second $m_{z}=-1$ branch. The second branch of the $m_{z}=0$ numerical Cherenkov instability is absent, as expected. In contrast, Eq. (\[eq:clip\]) with $C_{2x}$ correction factors reduces peak growth comparably, but with nontrivial growth rates spread over a larger range in $\mathbf{k}$-space; see Fig. \[fig:ebcor4\] (right). In particular, it causes the second branch of the $m_{z}=0$ numerical instability to shift toward $k_{x}=0$ but does nothing to eliminate it. This is because the topology of the $\Psi_{B}$ vs. $\Psi_{E}$ parametric plot changes rapidly for very small $k_{x}$ near this second instability branch. As a consequence, slightly reducing $\Psi_{E}$ there actually can create an instability where none otherwise exists.
Numerical Instability Growth Rates
==================================
Although $\mathbf{k}$-space truncation with $\alpha=0.85$ yields reasonably effective instability suppression, $\alpha=0.60$ is much better. Additionally, choosing $\alpha=0.60$ better accommodates comparisons with earlier papers [@Godfrey2014PSATD-TPS; @Godfrey2014AAC; @Vay2014a]. Fig. \[fig:peakgrowth\] (left) depicts peak growth rates for $N=1,\,2,\,3,\,4,\,8,\,\infty$, the last being PSATD. Peak growth rates are relatively insensitive to *N*. As always, the rapid variation of peak growth with $\nicefrac{v\,\triangle t}{\triangle z}$ reflects the rapidly changing locations of higher order resonances on the $\mathbf{k}$-space grid. Note that the largest growth rates are associated with $\nicefrac{v\,\triangle t}{\triangle z}$=0.9 , used as the sample case in the preceding Section. No growth whatsoever is predicted for $\nicefrac{v\,\triangle t}{\triangle z}=1.0$, because all aliases coincide there. Peak growth rates for $N=4$ and various order finite difference approximations to $\mathbf{k}$ are shown in Fig. \[fig:peakgrowth\] (right). Results are similar to those in the left plot. Superimposed on the growth rate curves are results from several WARP simulations [@Warp]. Agreement is very good. In addition, a simulation was run to $t=10000$ for $\nicefrac{v\,\triangle t}{\triangle z}=1.0$ with no sign of instability growth. However, at still larger values of time, instability growth at a rate of about 0.003 developed at wave numbers consistent with the second branch of the $m_{z}=0$ numerical Cherenkov instability. Perhaps, a slight change in the beam distribution function over time moved this instability above threshold. If so, reducing $\Psi_{E}$ very slightly there might prevent this very late time growth.
The procedure introduced in this paper remains robust at low $\gamma$. Fig. \[fig:Peak-numerical-instability-3\] presents peak growth rates for $\gamma=3$, $\alpha=0.8$, $N=2,\,3,\,4,\,8,\,\infty$, and infinite order. Results are comparable to the best obtained earlier; see Fig. 7 of [@Godfrey2014PSATD-TPS]. Both here and in [@Godfrey2014PSATD-TPS], $E_{z}$ is offset by one-half cell to suppress the usual quasi-electrostatic numerical instability that occurs at low $\gamma$ [@Birdsall1980; @lewis1972variational; @Langdon1973energy]. The peak in Fig. \[fig:Peak-numerical-instability-3\] at $\nicefrac{v\,\triangle t}{\triangle z}=0.9$ is larger than that in Fig. \[fig:peakgrowth\] only because $\alpha$ is larger.
Conclusion
==========
New field correction factors have been derived that completely eliminate the $m_{z}=0$ numerical Cherenkov instability for the generalized PSTD algorithm, including the PSATD algorithm. When combined with a sharp cutoff digital filter at large $\mathbf{k}$, these correction factors reduce peak growth rates to less (often much less) than 0.01 of the beam’s relativistic plasma frequency. These coefficients are optimal in the sense that they differ from unity only slightly over a broad portion of $\mathbf{k}$-space while eliminating both $m_{z}=0$ numerical instability branches. A disadvantage from the implementation perspective is that the coefficients must be computed numerically for each set of simulation parameters. Software for doing so is available in *Mathematica* CDF format [@WolframCDF] at http://hifweb.lbl.gov/public/BLAST/Godfrey/.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank Irving Haber for suggesting this collaboration and for helpful recommendations. We also are indebted to David Grote for assistance in using the code WARP. This work was supported in part by the Director, Office of Science, Office of High Energy Physics, U.S. Dept. of Energy under Contract No. DE-AC02-05CH11231 and the US-DOE SciDAC ComPASS collaboration, and used resources of the National Energy Research Scientific Computing Center.
This document was prepared as an account of work sponsored in part by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, nor the authors makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California.
References {#references .unnumbered}
==========
![\[fig:m0branches\]Growth rates and resonance curves from PSAOTD dispersion relation for $m_{z}=\left[-2,\,+2\right]$, $v\triangle t/\triangle z=0.9$, and $N=4$. The secondary $m_{z}=0$ branch, although slower growing and confined to a small region in *k*-space, is inconveniently located at small $k_{x}$. ](growContour)
![\[fig:DisFun\]Dispersion function for Base, $C_{2x}$, and Opt correction factors. Left, entire range of frequencies (curves indistinguishable at this scale). Right, narrow range of frequencies near second crossing point. Parameters are as in Fig. \[fig:m0branches\], with $k_{z}=k_{x}=5.20$.](DisFun "fig:")![\[fig:DisFun\]Dispersion function for Base, $C_{2x}$, and Opt correction factors. Left, entire range of frequencies (curves indistinguishable at this scale). Right, narrow range of frequencies near second crossing point. Parameters are as in Fig. \[fig:m0branches\], with $k_{z}=k_{x}=5.20$.](DisFundetail "fig:")
![\[fig:ParamPlot\]$\Psi_{B}-\Psi_{E}$ parametric plots for the parameters of Fig. \[fig:m0branches\]; left, $k_{z}=5.20,\: k_{x}=5.20$, and right, $k_{z}=4.696,\: k_{x}=0.635$. Shaded regions are stable.](ParamPlot1 "fig:")![\[fig:ParamPlot\]$\Psi_{B}-\Psi_{E}$ parametric plots for the parameters of Fig. \[fig:m0branches\]; left, $k_{z}=5.20,\: k_{x}=5.20$, and right, $k_{z}=4.696,\: k_{x}=0.635$. Shaded regions are stable.](ParamPlot2 "fig:")
![\[fig:factors\]$\Psi_{E}$ (left) and $\Psi_{B}$ (right) correction terms for the parameters of Fig. \[fig:m0branches\]. Superimposed at the far upper right of each plot is the $m_{z}=0$ resonance curve.](wexplot "fig:")![\[fig:factors\]$\Psi_{E}$ (left) and $\Psi_{B}$ (right) correction terms for the parameters of Fig. \[fig:m0branches\]. Superimposed at the far upper right of each plot is the $m_{z}=0$ resonance curve.](wbyplot "fig:")
![\[fig:ebcor4\]Growth rates for Fig. (\[fig:m0branches\]) parameters, optimal (left) or $C_{2x}$ (right) correction factors, and $\alpha=0.85$.](growContour4 "fig:")![\[fig:ebcor4\]Growth rates for Fig. (\[fig:m0branches\]) parameters, optimal (left) or $C_{2x}$ (right) correction factors, and $\alpha=0.85$.](growContour1 "fig:")
![\[fig:peakgrowth\]Peak numerical instability growth rates *vs*. $\nicefrac{v\,\triangle t}{\triangle z}$ with for various *N* (left) and for various order finite difference approximations to $\mathbf{k}$ (right); $\alpha=0.6$.](Nall "fig:")![\[fig:peakgrowth\]Peak numerical instability growth rates *vs*. $\nicefrac{v\,\triangle t}{\triangle z}$ with for various *N* (left) and for various order finite difference approximations to $\mathbf{k}$ (right); $\alpha=0.6$.](N4 "fig:")
![\[fig:Peak-numerical-instability-3\]Peak numerical instability growth rates *vs*. $\nicefrac{v\,\triangle t}{\triangle z}$ with $\gamma=3$ and various *N*; $\alpha=0.8$.](g3)
|
---
abstract: |
Time series analysis is used to understand and predict dynamic processes, including evolving demands in business, weather, markets, and biological rhythms. Exponential smoothing is used in all these domains to obtain simple interpretable models of time series and to forecast future values. Despite its popularity, exponential smoothing fails dramatically in the presence of outliers, large amounts of noise, or when the underlying time series changes.
We propose a flexible model for time series analysis, using exponential smoothing cells for overlapping time windows. The approach can detect and remove outliers, denoise data, fill in missing observations, and provide meaningful forecasts in challenging situations. In contrast to classic exponential smoothing, which solves a [*nonconvex*]{} optimization problem over the smoothing parameters and initial state, the proposed approach requires solving a single structured [*convex*]{} optimization problem. Recent developments in efficient convex optimization of large-scale dynamic models make the approach tractable. We illustrate new capabilities using synthetic examples, and then use the approach to analyze and forecast noisy real-world time series. Code for the approach and experiments is publicly available.
bibliography:
- 'references\_sasha.bib'
---
Introduction
============
Exponential smoothing (ES) methods model current and future time series observations as a weighted combinations of past observations, with more weight given to recent data. The word ‘exponential’ reflects the exponential decay of weights for older observations. ES methods have been around since the 1950s, and are still very popular forecasting methods used in business and industry, including supply chain forecasting [@chen2000impact], stock market analysis [@taylor2004volatility; @pindyck1983risk; @brown1961fundamental], weather prediction [@taylor2002neural; @soman2010review], and electricity demand forecasting [@taylor2003short; @moghram1989analysis].
In contrast to many techniques in machine learning, ES provides simple and interpretable models and forecasting capability by assuming a fixed structure for the evolution of the time series. For example, a simple (level only) model is $$\label{eq:simplest}
\hat y_{t+1} = \hat y_t +\alpha(y_t - \hat y_t) = (1-\alpha) \hat y_t + \alpha y_t,$$ where $y_t \in \mathbb{R}$ is an observation at time $t$, and $\hat y_t$ is the estimate of $y_t$ at time $t$ given $(y_1, \dots, y_{t-1})$. The forecast at $t+1$ is adjusted by a fraction $\alpha \in (0,1)$ of the error at time $t$; larger $\alpha$ means greater adjustment. Iterating , we have $$\label{eq:recurr}
\hat y_{t+1} = \alpha\sum_{i = 0}^{t-1} (1-\alpha)^{i} y_{t-i} + (1-\alpha)^t\hat y_1,$$ illustrating the exponential decay.
------------------------------------------------------------
\(a) [**Noisy series fit by HW (green), Robust** ]{}
[**HW (red transparent), and ES cell (red solid).**]{}
------------------------------------------------------------
--------------------------------------------------------------- ------------------------------------------------------------
 
\(b) [**Level** ]{} \(c) [**Trend**]{}
 
\(d) [**Seasonality**]{} \(e) [**Forecast (100 steps)**]{}.
--------------------------------------------------------------- ------------------------------------------------------------
The $\alpha$ in is fit using available data and used across the entire period of interest. More generally, a time series model also includes trend (long-term direction) and periodic seasonality components, with additional smoothing terms ($\beta, \gamma)$ for these components. Classic ES methods use observed data to fit smoothing components, error variance, and initial state, and then use the quantities to provide point forecasts and quantify uncertainty. Every additive ES can be formulated using the compact single source of error (SSOE) representation [@hyndman2002state; @hyndman2008forecasting]: $$\label{eq:ESmodel}
\begin{aligned}
y_{t} &= w^{T} x_{t-1} + \epsilon_{t}\\
x_{t} &= A x_{t-1} + g \epsilon_t
\end{aligned}$$ where $w$ is a linear measurement model, $A$ is a linear transition function and $g$ is the vector of smoothing parameters. In , we have $w = 1$, $x_t = y_t$, and $A = I$, and $g = \alpha$. More generally, $x_t$ tracks the deterministic components of the time series (level, trend, seasonality) while $g$ adjusts for stochastic disturbances.
Flexibility of ES models {#flexibility-of-es-models .unnumbered}
========================
To show how ES models are constructed and transformed into , we compare the simple linear, Holt’s linear, and Holt-Winters models. The simple linear model from tracks the level $l_t\in \mathbb{R}$ using a zero-order polynomial approximation: $$\label{eq:HoltBasicModel}
\begin{array}{l}
y_t = l_{t-1} +\epsilon_t \\
l_t = \alpha y_t + (1-\alpha) l_{t-1}.
\end{array}\quad$$ Holt’s Linear Model uses a first order (tangent) line approximation, and tracks both level $l_t$ and trend $b_t \in \mathbb{R}$: $$\label{eq:HoltLinearModel}
\begin{array}{l}
y_t = l_{t-1} + {\textcolor{blue}{b_{t-1}}} + \epsilon_t \\
l_t = \alpha y_t + (1-\alpha) (l_{t-1} + {\textcolor{blue}{b_{t-1}}})\\
{\textcolor{blue}{b_t = \beta (l_t - l_{t-1}) + (1- \beta) b_{t-1} }}.
\end{array}$$ Finally, the Holt-Winters model adds a seasonality component $s_t \in \mathbb{R}^p$ with known periodicity $p$, giving the augmented state $x_t = (l_t, b_t, s_{t}, s_{t-1}, \dots, s_{t-p-1})$, see .
$$\label{eq:HoltWintersModel}
\begin{array}{l}
y_t = l_{t-1} + {\textcolor{blue}{b_{t-1}}} + {\textcolor{red}{s_{t-p-1}}} + \epsilon_t
\medskip
\\
l_t = \alpha (y_t - {\textcolor{red}{s_{t-p-1}}}) + (1-\alpha) (l_{t-1} + {\textcolor{blue}{b_{t-1}}})\\
{\textcolor{blue}{b_t = \beta (l_t - l_{t-1}) + (1- \beta) b_{t-1} }}\\
{\textcolor{red}{s_t = \gamma (y_t - l_{t-1} - b_{t-1}) +(1-\gamma) s_{t-p-1}}}.
\end{array}$$
To write the Holt-Winters model in form of , take $$\label{eq:SSHoltWinters}
\small
w = \begin{bmatrix} 1\\ 1 \\ 0 \\ \vdots \\ 0 \\ 1\end{bmatrix},
g = \begin{bmatrix} \alpha \\ \beta\\ \gamma \\ 0\\ \vdots \\0 \end{bmatrix},
A = \begin{bmatrix} 1 & 1 & 0 & 0 & \dots & 0 &0\\
0 & 1 & 0 & 0 & \dots & 0 &0 \\
0 & 0 & 0 & 0 & \dots & 0 &1 \\
0 & 0 & 0 & 1 & \dots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots& &\vdots\\
0 & 0 & 0 & 0 & & \ddots & 0\\
0 & 0 & 0 &0 & 0 & \dots & 1
\end{bmatrix}.$$
----------------------------------------------------------------------- -----------------------------------------------------------------------
 
\(a) [**Local estimation by a single ES cell**]{} \(b) [**Overall approach: ES-Cells linked by a dynamic model.**]{}
----------------------------------------------------------------------- -----------------------------------------------------------------------
Limitations of classic ES {#limitations-of-classic-es .unnumbered}
=========================
SSOE models are attractive in their simplicity, since they use a single parameter vector $g$ to model perturbations. However, the same $g$ must applied at every time point; this limits model flexibility and ensures estimates of $g$ are strongly affected by artifacts in the data, including noise and outliers. Consider a toy example with two measurements; $g$ and $x_0$ are found by solving $$\label{eq:simpleObj}
\min_{x_{0},g} (y_{1} - w^Tx_{0})^2 + (y_{2} - w^{T}Ax_{0} + w^{T}g(y_{1} - w^{T}x_{0}))^{2}.$$ If $y_1$ is an outlier, it affects both $x_0$ and $g$ in the fit. Robust statistics are powerless for SSOE models: ignoring $y_1$ necessarily leaves a large $\epsilon_1 = y_1 - w ^Tx_0$, which affects $\epsilon_2$. The propagation is recursive: $$\epsilon_{3} = y_{3} - w^{T}A^{2}x_{0} + w^{T}A g \epsilon_{1} + w^{T} g \epsilon_{2}.$$ Each $\epsilon_t$ appears in all $\epsilon_{t+k}$, $k\geq 1$, so an outlier at any point has to affect $x_0$ and $g$, which control the entire time series. This phenomenon does not occur in other time series formulations, including AR models, where robust methods have been developed, see e.g. [@maronna2006robust]. Earlier work in robust Holt-Winters (RHW) uses M-estimators to filter the observations $y_t$ [@cipra1992robust; @gelper2010robust], and then applies standard HW. In contrast, ES-Cells formulates a single problem to analyze the entire time series, simultaneously denoising, decomposing, and imputing.
The classic approach is nonconvex, and has weak guarantees: stationarity conditions (such as $\nabla f (x) = 0$) do not imply global optimality, and solutions found by iterative methods depend on the initial point. As the level of noise and outliers increases, the ability of black-box optimizers to get reasonable $x_0$ and $(\alpha, \beta,\gamma)$ breaks down. The ES-Cells approach uses a [*strongly convex*]{} formulation; it has a unique global minimum and no other stationary points.
We created a synthetic time series with trend and level shifts, as well as heteroscedastic noise and outliers in the observations. Figure \[fig:noisy\] compares the performance of the HW model[^1] [@holt2004forecasting] and RHW [@gelper2010robust][^2] to the ES-Cells approach[^3]. HW propagates outliers and is adversely affected by heteroscedastic noise, affecting the estimates of level, trend, and especially seasonality components (see Figure \[fig:noisy\] (b-d)). This lack of robustness gives a poor understanding of the overall time series (Figure \[fig:noisy\] (a)) and leads to low forecasting accuracy, as corrupted errors are propagated in future times (see Figure \[fig:noisy\] (e)). For RHW, automatic approaches to find $x_0, \alpha,\beta,\gamma$ failed, and we had to hand-tune parameters; the final result improves on HW but requires a long ‘burn-in’ period, and still produces a somewhat noisy forecast. The ES-Cells approach captures and removes outliers and heteroscedastic noise, and correctly identifies the components.
Time series estimation using ES-Cells {#time-series-estimation-using-es-cells .unnumbered}
=====================================
The ES-Cells approach is constructed from interconnected building blocks. The basic cell consists of local ES estimation over a fixed window, equipped with a convex regularization term (for denoising) and a robust loss function (to guard against outliers), see Fig. \[fig:approach\] (a). The cells are then linked together by the time series dynamics, but allowing discrepancies between $x_t$ and $Ax_{t-1}$, see Fig. \[fig:approach\](b). These differences are treated as samples of $g_t$, analyzed, and used to build forecasting confidence intervals. Fitting the entire ES-Cells model is a convex problem, and can be done at scale using efficient methods for dynamic optimization [@aravkin2016generalized; @JMLR:v14:aravkin13a].
ES cell model
-------------
First we formulate inference for a single ES cell. Given a time point $t$ and an integer $K$, we take a window of size $2K+1$ that includes all the points in the interval $[t-K, t+K]$. Some measurements can be missing; and no time point outside $[0,T]$ has measurements. To model these cases, we introduce indicator variables $$d_t = \begin{cases}
0 & t \not\in[0,T] \; \mbox{or} \; y_t \mbox{ missing } \\
1 & t\in [0,T] \; \mbox{and } y_t \mbox{ observed. }
\end{cases}$$ We also define a unimodal sequence of weights $\alpha$, with $$0 < \alpha_{-K} < \dots < \alpha_{-1} < \alpha_0 > \alpha_1 > \dots \alpha_K > 0.$$ The estimate $\check x_t$ depends only on observations in the times $[\max(t-K,0), \min(t+K, T)]$, and is obtained by propagating the estimate at the start of the window at time $t-K$ to the middle of the window at time $t$, where $\alpha= \max_i\alpha_i = \alpha_0$: $$\label{eq:hatcheck}
\check x_{t} = A^{K}\hat x_{t-K}.$$ The estimate $\hat x_{t-K}$ solves the optimization problem $$\label{eq:singleCellObj}
\min_x \sum_{r = -K}^K d_{t + r}\alpha_r |y_{r+t}-a_{r+K}^T x| + \lambda |b^Tx|,$$ where $a_{r+K} := A^{r+K-1}w$, and $b = [0, 0, 0, 1, -1, 0, \dots 0]^T$ extracts the difference of two seasonality components from the state $x$. The objective function extends the classic ES approach in three respects.
1. The terms $d_t$ keep track of missing observations.
2. The loss used to compare $y_{r+t}$ to $a_{r+K}^T x$ is robust to outliers.
3. The term $|b^Tx|$ adds total variation regularization for the seasonality components.
The objective function is convex, as long as the loss and regularizer are both chosen to be convex. Before presenting the fully linked dynamic model, we rewrite more compactly, avoiding sums. Define $$\begin{aligned}
Y_{t-K} & = \begin{bmatrix} y_{t-K} & \dots & y_{t+K}\end{bmatrix}^T,
\quad {\mathcal{A}}= \begin{bmatrix} a_0 & \dots & a_{2K}\end{bmatrix} \\
D_{t-K} & = \mbox{diag}\left( d_{t-K}\alpha_{-K}, \dots, d_{t}\alpha_{0}, \dots, d_{t+K}\alpha_K\right).
\end{aligned}$$ We can now write as $$\label{eq:singleFull}
\hat x_{t-K} = \arg\min_x \|D_{t-K}(Y_{t-K} - {\mathcal{A}}x)\|_1 + \lambda |b^Tx|.$$
Linking the ES-Cells
--------------------
Each cell estimate only depends on local data. To connect $\check x_t$ and $\check x_{t-1}$, we assume that the estimates satisfy $$\check x_{t} = A \check x_{t-1} + g_t$$ where, in contrast to the error term $g \epsilon_t$, $g_t$ are i.i.d. Gaussian errors. This is equivalent to adding the penalty $$\|g_t\|_2^2 = \|A\check x_{t-1} - \check x_t\|_2^2 = \|A^{K}(A\hat x_{t-(K+1)} - \hat x_{t-K})\|_2^2,$$ see . This links together objectives of form to generate a single objective over the entire sequence\
${\bf x} = \{x_{-K}, \dots, x_{0},\dots, x_{T-K}\}$: $$\label{eq:Fullobj}
\begin{aligned}
\hat {\bf x} &= \arg\min_{{\bf x}} \sum_{t=-K}^{T-K} \|D_{t}(Y_{t} - {\mathcal{A}}x_{t})\|_1 + \lambda_1 |b^Tx_{t}| +\\
& \qquad\qquad \sum_{t = -K}^{T-K-1}\lambda_2\|A^{K}(A x_{t} - x_{t+1})\|_2^2
\end{aligned}$$ The problem in is nonsmooth but [*convex*]{}, and has dynamic structure. It has far more variables than the classic nonconvex ES formulation in . Nonetheless, it can be efficiently solved at scale using recent algorithms for generalized Kalman smoothing [@aravkin2016generalized; @JMLR:v14:aravkin13a]. Given ${\bf \hat x}$, the final time series estimate ${\bf \check x}$ is given by $${\bf \check x} = \left\{ A^K \hat x_{-K}, \dots, A^K \hat x_0, \dots A^K \hat x_{T-K}\right\}.$$
---------------------------------------------------------------------- ---------------------------------------------------------------------
 
\(a) [**Level: Forecast + 99% CI.** ]{} \(b) [**Trend: Forecast + 99% CI.** ]{}
 
\(c) [**Seas.: Forecast + 99% CI.** ]{} \(d) [**TS: Forecast + 99% CI** ]{}.
---------------------------------------------------------------------- ---------------------------------------------------------------------
Time series forecasting using ES-Cells {#time-series-forecasting-using-es-cells .unnumbered}
======================================
ES-Cells capture two main sources of uncertainty that are important for forecasting future values of a time series: uncertainty in the residuals $\epsilon_t$, and in the smoothing parameters $\{ \alpha_t, \beta_t, \gamma_t \}$. ES-Cells track these two sources of uncertainty and can be used to create two separate confidence intervals: one representing the variability of each component of the signal, and the other capturing the structure of the residual.\
Solving the full problem , we obtain the entire sequence ${\bf \hat x}$, as well as corresponding estimates of residuals $\hat \epsilon_t$ and smoothing parameters $\hat g_t$: $$\begin{aligned}
\hat \epsilon_t &= y_{t} - w^T\hat x_{t-1}\\
\hat g_t &= \hat x_t - A \hat x_{t-1}.
\end{aligned}$$ In order to obtain the prediction distribution, we simulate sample paths from the models, using the empirical distribution of $\hat g_t$ and $\hat \epsilon_t$, and conditioned on the final state. This allows us to estimate any desired characteristics of the prediction distribution at a specific forecast horizon, and in particular to estimate confidence intervals that incorporate smoothing parameter and residual uncertainties. We can also incorporate model-based residuals (instead of using the empirical distribution) by generating forecasted $\epsilon_t$ values from any given distribution.
To illustrate the ES forecasting framework, Figure \[fig:forecast\] presents forecasts for the noisy synthetic model introduced in Figure \[fig:noisy\]. In particular, 100 step ahead forecasts and their 99% confidence intervals (using 10000 Monte Carlo runs) are shown for trend and seasonality (panels (a) and (b)). These are obtained by using the empirical $\hat g_t$ distribution. The forecast for the full time series (and a zoomed plot) are shown in panels (c) and (d). The inner 99% CI (strictly inside the shaded region) takes into account only uncertainty in smoothing parameters $g_t$, while the outer CI (the border for the shaded region) takes into account uncertainty in $\epsilon_t$. Since the time series is contaminated by outliers, the outer CI is very wide in this case.
Real world Time Series : Twitter’s user engagement dataset {#real-world-time-series-twitters-user-engagement-dataset .unnumbered}
==========================================================
To test our algorithm we examine an anonymized time series representing user engagement on Twitter. This dataset is publicly available on its official blog[^4] and is fully representative of the challenges tackled in this paper:
- Distinct seasonal patterns due to user behavior across different geographies
- An underlying trend which could be interpreted as organic growth (new people engaging with the social network)
- Anomalies or outliers due to either special events surrounding holidays (christmas, breaking news) or unwanted behavior (bots or spammers)
- Underlying heteroscedastic noise. embodying the variance of the signal.
--------------------------------------------------------- ------------------------------------------------
 
\(a) [**Classic Holt-Winters Analysis and Forecast**]{} \(b) [**ES-Cells Analysis and Forecast.**]{}
--------------------------------------------------------- ------------------------------------------------
![\[fig:mape\] A comparison of the mean absolute percentage error (MAPE), to quantify the improvement of ES-Cells (red) over H-W (purple) for the twitter data in Figure \[fig:twitter\].](plots/PlotPaper/camera_ready/benchmark_mape)
The dataset was originally put online to showcase a robust anomaly detection procedure. With the ES-Cells framework, we can go much further, decomposing the time series into interpretable components, and then forecasting both the components and the entire time series under uncertainty. The original aim (anomaly detection) is easily accomplished by studying the tail of the empirical residual distribution, as discovered by the approach.
The classic Holt-Winters model fits outliers, forecasting sharp growth of engagement, which misses the observed trend (Figure \[fig:twitter\](a)), and finds a very wide 99% confidence interval, In contrast, the ES-Cells approach (Figure \[fig:twitter\](b)) avoids fitting the outliers; the average forecast correctly captures the [*decrease*]{} in the trend, and provides a much tighter asymmetric 99% CI. The improvement can be quantitatively assessed by looking at the Mean Absolute Percentage Error (MAPE) for the forecasted time series over a sliding window of 10 observations, Figure \[fig:mape\]. Traditional Holt-Winters has a higher MAPE than ES-Cell at every time point; moreover, the MAPE of the ES-Cells method is stable over time, while the MAPE of ES increases, illustrating its failure to robustly capture the long term trend of the time series. Trend, level and seasonality are shown in Figure \[fig:twitter\_detail\]. There is a clear decrease in level and trend, which are detected despite the large amounts of noise in the data.
![\[fig:outliers\] Anomaly detection from the ES-Cells fit. The 1.5% most extreme observations are highlighted using red dots. ](plots/PlotPaper/camera_ready/outliers_Y)
Anomaly Detection
-----------------
After fitting the ES procedure, we are left with a residual that we can analyze to understand anomalies in the time series. Figure \[fig:outliers\] shows an example of outlier detection by looking at the 1.5% tails of the residual distribution.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:twitter\_detail\] Twitter dataset: Forecasts and 99% CI for trend (a), level (b) and seasonality (c) obtained by the ES-Cells approach. There is a clear downward direction in level and trend. ](plots/PlotPaper/camera_ready/twittertest_B "fig:")
\(a) [**Trend, Forecast, & 99% CI.** ]{}
![\[fig:twitter\_detail\] Twitter dataset: Forecasts and 99% CI for trend (a), level (b) and seasonality (c) obtained by the ES-Cells approach. There is a clear downward direction in level and trend. ](plots/PlotPaper/camera_ready/twittertest_L "fig:")
\(b) [**Level, Forecast, & 99% CI.** ]{}
![\[fig:twitter\_detail\] Twitter dataset: Forecasts and 99% CI for trend (a), level (b) and seasonality (c) obtained by the ES-Cells approach. There is a clear downward direction in level and trend. ](plots/PlotPaper/camera_ready/twittertest_S1 "fig:")
\(c) [**Seas., Forecast, & 99% CI**]{}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Robust auto completion of missing data
--------------------------------------
The ES-Cell algorithm is also robust to missing observations. Whether the data is missing at random, or in significant contiguous batches, it is automatically filled in by the ES-Cells algorithm. Since the problem is solved globally, nearby outliers do not affect the interpolated values, in contrast to local interpolation methods. Figure \[fig:fill\] shows the result obtained by removing two contiguous chunks of 100 observations each in two distinct parts of the time series. The data is automatically ‘in-filled’ using the ES-Cell procedure.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:scheme\] Classic TS analysis is sequential (a); data are pre-processed, then decomposed into components using models such as Holt-Winters. The approach is limited, because information about underlying structures such as level, trend, and seasonality is not available during the pre-processing. The new approach (b) is global; cleaning, interpolation, and decomposition are done in a unified context. Downstream applications, including anomaly detection and forecasting, significantly improve.](plots/PlotPaper/camera_ready/overview_a "fig:") ![\[fig:scheme\] Classic TS analysis is sequential (a); data are pre-processed, then decomposed into components using models such as Holt-Winters. The approach is limited, because information about underlying structures such as level, trend, and seasonality is not available during the pre-processing. The new approach (b) is global; cleaning, interpolation, and decomposition are done in a unified context. Downstream applications, including anomaly detection and forecasting, significantly improve.](plots/PlotPaper/camera_ready/overview_b "fig:")
\(a) [**Classic approach**]{} \(b) [**Global approach**]{}
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Discussion {#discussion .unnumbered}
==========
ES-Cells is a new model for time series inference, capable of fitting and forecasting data in situations with high noise, frequent outliers, and large contiguous portions of missing data. These features are present in many real-world large-scale datasets. The ES-Cells formulation differs from previous model in its global approach, as shown in Figure \[fig:scheme\]. We simultaneously denoise, impute, and decompose the time series by solving a single convex optimization problem with dynamic structure; then use sampling-based strategies for forecasting and uncertainty quantification. The results are illustrated on simulated and real data, where the proposed method yields a 5-fold improvement in MAPE for the forecasting. The simplicity and versatility of the ES-Cells formulation makes it a superior alternative to Holt-Winters and related time series models. Code for the approach and experiments is publicly available[^5].
![\[fig:fill\] ES-Cells can fill in data missing both at random and in contiguous chunks. In panel (a), two sections of 100 observations each have been deleted. Panel (b) shows the in-filled values (green) plotted over the data from panel (a) (red). The deleted data from panel (a) is shown in blue in panel (b).](plots/PlotPaper/camera_ready/missingchunk_original "fig:")\
(a) [**Deleted data**]{}\
![\[fig:fill\] ES-Cells can fill in data missing both at random and in contiguous chunks. In panel (a), two sections of 100 observations each have been deleted. Panel (b) shows the in-filled values (green) plotted over the data from panel (a) (red). The deleted data from panel (a) is shown in blue in panel (b).](plots/PlotPaper/camera_ready/missingchunk "fig:")\
(b) [**Auto fill-in.**]{}
### Acknowledgements {#acknowledgements .unnumbered}
The work of A. Aravkin was supported by the Washington Research Foundation Data Science Professorship.
[^1]: Implemented in the standard Holt-Winters R module
[^2]: We implemented the approach. In[@gelper2010robust Eqs. 13,14], we take $\sigma_0 = 0.05$, and $\lambda_\sigma = 0.01$. Automated methods for obtaining smoothing parameters and $x_0$ failed in the presence of noise; so for the HW model, we use hand-tuned parameters $\alpha = 0.05$, $\beta = 0.01$, $\gamma = 0.15$, with $x_0$ the first 50 elements of the noisy $Y$.
[^3]: https://github.com/UW-AMO/TimeSeriesES-Cell
[^4]: https://blog.twitter.com/2015/introducing-practical-and-robust-anomaly-detection-in-a-time-series
[^5]: https://github.com/UW-AMO/TimeSeriesES-Cell
|
---
abstract: 'The process of dynamical reduction of the Vlasov-Maxwell equations leads to the introduction of classical [*zitterbewegung*]{} effects in reduced plasma dynamics. These effects manifest themselves in the form of an asymmetric canonical energy-momentum tensor involving the decoupling of the reduced kinetic momentum ${\bf p}$ from the reduced velocity ${\bf u}$ (i.e., ${\bf u}\btimes{\bf p} \neq 0$) as well as reduced polarization and magnetization effects. The reduced intrinsic torque generated by the antisymmetric part of the canonical energy-momentum tensor, which is calculated from the reduced ponderomotive potential, acts as the source for the intrinsic (spin) angular momentum.'
author:
- 'Alain J. Brizard'
title: 'Classical [*zitterbewegung*]{} in reduced plasma dynamics'
---
October 27, 2010
In 1930, Schroedinger [@Schroedinger] provided a simple interpretation for the electron’s spin angular momentum based on Dirac’s theory of the electron [@Dirac]. According to Schroedinger’s [*zitterbewegung*]{} model [@Huang; @Barut_Bracken; @Barut_Zanghi], the intrinsic spin angular momentum $S^{\mu\nu} \equiv \hbar\,\sigma^{\mu\nu}/2$ of the electron can be interpreted as the result of the rapid trembling motion of the electron. The main features of the Lagrangian formulation of [*zitterbewegung*]{} can be summarized by: (I) the decoupling of a particle’s velocity ${\bf v}$ from its momentum ${\bf p}$ (i.e., ${\bf v}\btimes{\bf p} \neq 0)$; (II) the existence of polarization and magnetization effects associated with the decoupling between fast and slow space-time scales; and (III) an asymmetric canonical energy-momentum tensor derived from the Lagrangian density. In the Schroedinger [*zitterbewegung*]{} model, these features are explicitly related to the electron’s spin.
[*Zitterbewegung*]{} effects have recently been investigated in many different physical systems ranging from graphene [@GN_Nature; @Castro_RMP] and photonic materials [@Dreisow_PRL; @Zawadzki] to ultracold atoms [@Vaishnav_PRL] and cold-ion traps [@Lamata_PRL; @Gerritsma_Nature]. The purpose of the present Letter is to show that the reduced Vlasov-Maxwell equations, which are obtained by dynamical reduction, exhibit these same [*zitterbewegung*]{} features derived from the reduced ponderomotive Hamiltonian.
We begin with a brief review of the Schroedinger [*zitterbewegung*]{} model of the electron, which is governed by the Dirac-Maxwell equations [@RQM] $$\left. \begin{array}{rcl}
\gamma^{\mu}\,D_{\mu}\psi & = & mc^{2}\,\psi \\
\ov{D}_{\mu}\ov{\psi}\,\gamma^{\mu} & = & -\,mc^{2}\,\ov{\psi} \\
\partial_{\mu}F^{\mu\nu} & = & (4\pi/c)\,J^{\nu}
\end{array} \right\},
\label{eq:DM_eqs}$$ where $J^{\nu} \equiv (c\varrho, {\bf J}) = ec\,\langle\gamma^{\nu}\rangle$ denotes the electron four-current (expressed in terms of the expectation value $\langle\gamma^{\nu}\rangle \equiv \langle\psi|\gamma^{\nu}|\psi\rangle$ of the Dirac matrix $\gamma^{\nu}$), $D_{\mu} \equiv i\hbar\,\partial_{\mu} - (e/c)\,A_{\mu}$ and $\ov{D}_{\mu} \equiv i\hbar\,\partial_{\mu} + (e/c)\,A_{\mu}$ are the kinetic-momentum operators acting on $\psi$ and its adjoint $\ov{\psi}$, respectively, and $F_{\mu\nu} \equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ denotes the electromagnetic field. We now show that the Lagrangian formulation of the Dirac-Maxwell equations exhibit the three [*zitterbewegung*]{} features (I)-(III).
First, the electron is described in terms of the coordinates ${\bf r}$, its velocity ${\bf v} \equiv d{\bf r}/dt$, and its kinetic momentum ${\bf p}$. While the electron’s velocity has the eigenvalues $\pm\,c$, its expectation value $\langle{\bf v}\rangle$ (expressed in terms of the group velocity of its wavepacket) is less than $c$ in magnitude [@GN_Nature]. The motion of the electron is therefore decomposed into the slow (average) motion of its center of mass and the rapid trembling motion of its [*zitterbewegung*]{} position $\vb{\rho}_{\rm e}$. This rapid motion yields an expectation value for the magnetic dipole-moment of the electron [@Huang; @Barut_mu; @Barut_Zanghi] $$\vb{\mu}_{\rm e} \;\equiv\; \frac{e}{2c}\;\left\langle\vb{\rho}_{\rm e}\btimes\frac{d\vb{\rho}_{\rm e}}{dt}\right\rangle,
\label{eq:mu_electron}$$ whose magnitude $|\vb{\mu}_{\rm e}| \equiv e\hbar/(2mc)$ is the Bohr magneton.
Second, we discuss the electron’s polarization and magnetization and introduce the Gordon decomposition of the electron four-current [@RQM] $$\left. \begin{array}{rcl}
\varrho & \equiv & \varrho_{\rm c} - \nabla\bdot{\bf P} \\
& & \\
{\bf J} & \equiv & {\bf J}_{\rm c} + \partial{\bf P}/\partial t + c\,\nabla\btimes{\bf M}
\end{array} \right\},
\label{eq:rho_J_elec}$$ where $J_{\rm c}^{\nu} \equiv (e/2m)\,[\langle D^{\nu}\rangle - \langle\ov{D}^{\nu}\rangle]$ denotes the ([*spinless*]{}) conduction four-current and the antisymmetric magnetization tensor $$M^{\mu\nu} \;\equiv\; \frac{e\hbar}{2m}\;\langle\sigma^{\mu\nu}\rangle
\label{eq:M_Dirac}$$ is expressed in terms of the spin matrix $\sigma^{\mu\nu} \equiv (i/2)\,(\gamma^{\mu}\gamma^{\nu} - \gamma^{\nu}\gamma^{\mu})$, whose components are the polarization $M^{i0} \equiv c\,P^{i}$ and the magnetization $M^{ij} \equiv \epsilon^{ijk}\,c\,M_{k}$.
Third, the canonical energy-momentum tensor for the Dirac-Maxwell equations $$T^{\mu\nu} \;=\; g^{\mu\nu} \left( \frac{{\sf F}:{\sf F}}{16\pi} \right) \;-\; \frac{1}{4\pi}\;F^{\mu\alpha}\,F_{\alpha}^{\;\;\nu} \;-\;\left\langle
c\gamma^{\mu}\,D^{\nu}\right\rangle
\label{eq:T_Dirac}$$ is derived by Noether method from the Dirac-Maxwell Lagrangian density ${\cal L} = \langle c\gamma^{\mu}D_{\mu} - mc^{2}\rangle + F^{\mu\nu}F_{\nu\mu}/
16\pi$. While the electromagnetic terms in Eq. are symmetric, the Dirac term is not. Indeed, the antisymmetric part $T_{\sf A}^{\mu\nu} \equiv \frac{1}{2} (T^{\mu\nu} - T^{\nu\mu})$ of Eq. is expressed as [@Hilgevoord] $$T^{\mu\nu}_{\sf A} = -\,\frac{c}{2}\,\left\langle \gamma^{\mu}\,D^{\nu} \;-\frac{}{} \gamma^{\nu}\,D^{\mu}\right\rangle \equiv -\,\frac{1}{2}\,
\partial_{\alpha}\left\langle\Sigma^{\alpha[\mu\nu]}\right\rangle,
\label{eq:T_A_Dirac}$$ where the third-rank tensor $\Sigma^{\alpha[\mu\nu]} \equiv -\,\Sigma^{\alpha[\nu\mu]}$: $$\Sigma^{\alpha[\mu\nu]} \;\equiv\; \frac{c}{4}\,\left[ \left( S^{\alpha\mu}\,\gamma^{\nu} - S^{\alpha\nu}\,\gamma^{\mu}\right) + \left(
\gamma^{\mu}\,S^{\nu\alpha} - \gamma^{\nu}\,S^{\mu\alpha}\right)\right]
\label{eq:Sigma_Dirac}$$ is expressed in terms of the intrinsic spin angular momentum $S^{\alpha\beta} \equiv \hbar\,\sigma^{\alpha\beta}/2$. Since $(c\gamma^{i}\,D^{j} -
c\gamma^{j}\,D^{i})$ is the operator-equivalent of the cross-product $\epsilon_{ijk}\,v^{i}\,p_{\rm kin}^{j} \equiv ({\bf v}\btimes{\bf p}_{\rm kin}
)_{k}$, the antisymmetry of the Dirac-Maxwell canonical energy-momentum tensor is, therefore, due to the decoupling of its velocity and kinetic momentum.
Belinfante [@Belinfante_1939; @Belinfante_1940] recognized that the asymmetry of the canonical energy-momentum tensor $T^{\mu\nu}$ could be given a physical interpretation (see also Refs. [@Rosenfeld; @McLennan]), based on the fact that the transformation $$\ov{T}^{\mu\nu} \;\equiv\; T^{\mu\nu} \;+\; \partial_{\alpha}R^{[\alpha\mu]\nu}
\label{eq:ov_T_munu}$$ leaves the energy-momentum conservation law invariant: $$0 \;=\; \partial_{\mu}T^{\mu\nu} \;=\; \partial_{\mu}\ov{T}^{\mu\nu},
\label{eq:em_law}$$ where the third-rank tensor $R^{[\mu\alpha]\nu} \equiv -\,R^{[\alpha\mu]\nu}$ satisfies $\partial^{2}_{\alpha\mu}R^{[\alpha\mu]\nu} \equiv 0$. The condition that the new energy-momentum tensor is symmetric yields the following expression for the antisymmetric part of the canonical energy-momentum tensor $$T_{\sf A}^{\mu\nu} = -\,\frac{1}{2}\;\partial_{\alpha} \left( R^{[\alpha\mu]\nu} \;-\frac{}{} R^{[\alpha\nu]\mu} \right) \equiv -\,\frac{1}{2}\;
\partial_{\alpha}\,\Sigma^{\alpha[\mu\nu]},
\label{eq:T_A_def}$$ and, hence, the antisymmetric part $T_{\sf A}^{\mu\nu}$ of the canonical energy-momentum tensor acts as the source for the antisymmetric second-rank tensor $\Sigma^{\alpha[\mu\nu]}$.
We now show that the third-rank tensor $\Sigma^{\alpha[\mu\nu]}$ is connected to the spin angular-momentum tensor as in the Schroedinger [*zitterbewegung*]{} model \[see Eq. \]. First, we introduce the third-rank orbital-angular-momentum tensor $$L^{\beta[\mu\nu]} \;\equiv\; x^{\mu}\,T^{\beta\nu} \;-\; x^{\nu}\,T^{\beta\mu},
\label{eq:L_ang}$$ and the third-rank spin-angular-momentum tensor $$\begin{aligned}
S^{\beta[\mu\nu]} & \equiv & x^{\mu}\,\partial_{\alpha}R^{[\alpha\beta]\nu} \;-\; x^{\nu}\,\partial_{\alpha}R^{[\alpha\beta]\mu} \nonumber \\
& = & \Sigma^{\beta[\mu\nu]} \;-\; \partial_{\alpha}Q^{[\alpha\beta][\mu\nu]},
\label{eq:S_ang}\end{aligned}$$ where $Q^{[\alpha\beta][\mu\nu]} \equiv R^{[\alpha\beta]\mu}\,x^{\nu} - R^{[\alpha\beta]\nu}\,x^{\mu}$ satisfies $\partial^{2}_{\alpha\beta}
Q^{[\alpha\beta][\mu\nu]} \equiv 0$. Using Eq. , the total angular-momentum tensor $$J^{\beta[\mu\nu]} \;\equiv\; L^{\beta[\mu\nu]} \;+\; S^{\beta[\mu\nu]} \;=\; x^{\mu}\,\ov{T}^{\beta\nu} \;-\; x^{\nu}\,\ov{T}^{\beta\mu}
\label{eq:J_def}$$ satisfies the angular-momentum conservation law $$\partial_{\beta}J^{\beta[\mu\nu]} \;=\; \ov{T}^{\mu\nu} - \ov{T}^{\nu\mu} \;\equiv\; 0,
\label{eq:ang_mom_cons}$$ which follows from the symmetry of the new energy-momentum tensor . Lastly, the equation $\partial_{\beta}L^{\beta[\mu\nu]}$ for the orbital-angular-momentum tensor is $$\partial_{\beta}L^{\beta[\mu\nu]} = T^{\mu\nu} - T^{\nu\mu} \equiv -\,\partial_{\beta}S^{\beta[\mu\nu]} = -\,\partial_{\beta}\Sigma^{\beta[\mu\nu]}.
\label{eq:L_motion}$$ Hence, we see that the asymmetry of the canonical energy-momentum tensor acts as the source of intrinsic (spin) angular momentum.
We note that the antisymmetric energy-momentum tensor $$T_{\sf A}^{\mu\nu} \;=\; -\,\frac{1}{2}\;\varepsilon^{\mu\nu\alpha\beta}\;\partial_{\alpha}\sigma_{\beta} \;\equiv\; \frac{1}{4}\;
\varepsilon^{\mu\nu\alpha\beta}\;\tau_{\alpha\beta}
\label{eq:TA_tau}$$ can be used to define the antisymmetric torque tensor $\tau_{\alpha\beta} \equiv -\,(\partial_{\alpha}\sigma_{\beta} - \partial_{\beta}
\sigma_{\alpha})$, with the spatial components $T_{\sf A}^{ij} \equiv \frac{1}{2}\,\varepsilon^{0ijk}\,\tau_{0k}$ expressed in terms of the intrinsic torque tensor $\tau_{0\mu} \equiv (0, \vb{\tau})$: $$\vb{\tau} \;\equiv\; {\bf v}\btimes{\bf p} \;+\; {\bf E}\btimes{\bf P} \;+\; {\bf B}\btimes{\bf M}.
\label{eq:torque_vec}$$ We therefore see that all three features of the [*zitterbewegung*]{} model are combined in Eq. to act as the source of an intrinsic (spin) angular momentum.
After a brief introduction to the Schroedinger [*zitterbewegung*]{} model and a general discussion of angular-momentum conservation within a Lagrangian perspective, we now discuss the case of the Lagrangian formulation of reduced Vlasov-Maxwell theory. The Lagrangian formulation of the guiding-center and oscillation-center Vlasov-Maxwell equations were developed over 30 years ago [@Dewar; @Cary_K; @PM_85; @PLS; @Boghosian; @Ye_K]. In each case, their respective energy-momentum conservation laws, derived by Noether method, exhibited an asymmetric canonical energy-momentum tensor. Although it was sometimes pointed out that this asymmetry was only apparent [@PLS], based on the knowledge that the conservation of angular momentum required a symmetric physical energy-momentum tensor, Dewar [@Dewar] pointed out that the asymmetry of the canonical energy-momentum tensor could be expressed in terms of an intrinsic spin angular-momentum tensor.
The process of dynamical reduction in single-particle plasma dynamics and plasma kinetic theory [@Brizard_Vlasovia] is associated with the extended near-identity canonical phase-space transformation $\cal{T}_{\epsilon}: \wh{z}^{a} = z^{a} + \epsilon\;\{ S_{1}, z^{a}\} + \epsilon^{2}\,(\{
S_{2}, z^{a}\} + \frac{1}{2}\,\{ S_{1}, \{ S_{1}, z^{a}\}\}) + \cdots$, and its inverse $\cal{T}_{\epsilon}^{-1}: z^{a} = \wh{z}^{a} - \epsilon\;\{
S_{1}, \wh{z}^{a}\} - \epsilon^{2}\,(\{ S_{2}, \wh{z}^{a}\} - \frac{1}{2}\,\{ S_{1}, \{ S_{1}, \wh{z}^{a}\}\}) + \cdots$, generated by the scalar fields $(S_{1}, S_{2},\cdots)$. This near-identity transformation introduces the reduced-displacement vector $$\vb{\rho}_{\epsilon} \;\equiv\; {\sf T}_{\epsilon}^{-1}{\bf x} \;-\; \wh{{\bf x}} \;=\; -\,\epsilon\;G_{1}^{{\bf x}} + \cdots,
\label{eq:rho_epsilon}$$ defined as the difference between the push-forward ${\sf T}_{\epsilon}^{-1}{\bf x}$ of the particle position ${\bf x}$ and the reduced position $\wh{{\bf x}}$. Hence, through the reduced displacement , the dynamical reduction yields the reduced electric-dipole moment [@Brizard_Vlasovia] $$\vb{\pi}_{\epsilon} \;\equiv\; e\;\vb{\rho}_{\epsilon}
\label{eq:pi_def}$$ and the intrinsic magnetic-dipole moment $$\vb{\mu}_{\epsilon} \;\equiv\; \frac{e}{2c}\;\left(\vb{\rho}_{\epsilon}\btimes \frac{d_{\epsilon}\vb{\rho}_{\epsilon}}{dt}\right),
\label{eq:mu_def}$$ which is identical in form to the [*zitterbewegung*]{} expression for the electron’s magnetic-dipole moment.
The extended reduced Vlasov equation for the extended reduced Vlasov distribution $\wh{{\cal F}}$ is expressed as $$0 \;=\; \frac{d_{\epsilon}\wh{{\cal F}}}{dt} \;\equiv\; \frac{d_{\epsilon}\wh{z}^{a}}{dt}\;\pd{\wh{{\cal F}}}{\wh{z}^{a}},
\label{eq:redextVlasov_def}$$ where $$\frac{d_{\epsilon}\wh{x}^{\mu}}{dt} \;=\; \pd{\wh{\cal H}}{\wh{p}_{\mu}} \;\;{\rm and}\;\; \frac{d_{\epsilon}\wh{p}_{\mu}}{dt} \;=\; -\;
\pd{\wh{\cal H}}{\wh{x}^{\mu}}
\label{eq:can_z_dot}$$ denote the canonical Hamilton equations in extended phase space and the reduced Vlasov distribution is defined as $\wh{{\cal F}}(\wh{{\sf z}}) \equiv
c\,\delta[\wh{w} - \wh{H}(\wh{{\bf x}}, \wh{{\bf p}}, t)]\, \wh{F}(\wh{{\bf x}}, \wh{{\bf p}}, t)$, with the reduced extended Hamiltonian satisfying the physical constraint $\wh{{\cal H}} = \wh{H} - \wh{w} \equiv 0$, where the reduced Hamiltonian $$\wh{H} \equiv H - \epsilon\,\frac{dS_{1}}{dt} - \epsilon^{2} \left( \frac{dS_{2}}{dt} - \frac{1}{2} \left\{ S_{1},\; \frac{dS_{1}}{dt}\right\}
\right) + \cdots
\label{eq:ovH_def}$$ is expressed in a form that is the classical equivalent of the Dirac Hamiltonian derived by the Foldy-Wouthuysen (FW) transformation [@RQM].
The dynamical reduction associated with the phase-space transformation $\cal{T}_{\epsilon}$ introduces polarization and magnetization effects into the Maxwell equations, which are transformed into the macroscopic (reduced) Maxwell equations [@Ye_K; @Brizard_Vlasovia] $$\nabla\bdot{\bf D} \;=\; 4\pi\,\wh{\varrho} \;\;{\rm and}\;\;
\nabla\btimes{\bf H} \;-\; \frac{1}{c}\,\pd{{\bf D}}{t} \;=\; \frac{4\pi}{c}\,\wh{{\bf J}},
\label{eq:DH_eq}$$ where the microscopic electric and magnetic fields ${\bf E}$ and ${\bf B}$ are replaced by the macroscopic fields ${\bf D} \equiv {\bf E} + 4\pi\,
{\bf P}_{\epsilon}$ and ${\bf H} \equiv {\bf B} - 4\pi\,{\bf M}_{\epsilon}$, where ${\bf P}_{\epsilon}$ and ${\bf M}_{\epsilon}$ are the reduced polarization and magnetization. We note that the dynamical reduction associated with the phase-space transformation $\cal{T}_{\epsilon}$ has introduced the following expressions for the charge and current densities: $$\left. \begin{array}{rcl}
\varrho & \equiv & \wh{\varrho} \;-\; \nabla\bdot{\bf P}_{\epsilon} \\
{\bf J} & \equiv & \wh{{\bf J}} \;+\; \partial{\bf P}_{\epsilon}/\partial t \;+\; c\,\nabla\btimes{\bf M}_{\epsilon}
\end{array} \right\},
\label{eq:rhoJ_Rpolmag}$$ which are of course similar to the Gordon decomposition observed in the Dirac model.
Lastly, by using the reduced electric-dipole and magnetic-dipole moments -, we construct explicit expressions for the reduced polarization $${\bf P}_{\epsilon} \;\equiv\; \sum\;\int\;\vb{\pi}_{\epsilon}\;\wh{F}\,d^{3}\wh{p},
\label{eq:red_pol}$$ and the reduced magnetization $${\bf M}_{\epsilon} \;\equiv\; \sum\;\int\;\left( \vb{\mu}_{\epsilon} \;+\; \frac{\vb{\pi}_{\epsilon}}{c}\btimes\frac{d_{\epsilon}\wh{\bf x}}{dt}
\right)\;\wh{F}\,d^{3}\wh{p},
\label{eq:red_mag}$$ which combines the intrinsic magnetic-dipole contribution and the moving electric-dipole contribution.
We now show that the reduced Vlasov-Maxwell equations and can be derived from the reduced variational principle $\int d^{4}x\;\delta\wh{\cal{L}} = 0$, where the reduced Lagrangian density is $$\begin{aligned}
\wh{\cal{L}}(x) & \equiv & -\; \sum\;\int d^{4}\wh{p}\;\wh{{\cal F}}(x,\wh{p})\;\wh{{\cal H}}(x,\wh{p}; A, {\sf F}) \nonumber \\
& &+\; \frac{1}{16\pi}\,{\sf F}(x):{\sf F}(x).
\label{eq:redLag_def}\end{aligned}$$ Note that, as a result of the dynamical reduction of the Vlasov equation, the reduced Hamiltonian is expressed in terms of canonical energy-momentum coordinates as $$\begin{aligned}
\wh{H}(\wh{\bf x}, \wh{\bf p},t; A, {\sf F}) & \equiv & \frac{1}{2m}\,|\wh{{\bf p}} - (e/c)\,{\bf A}|^{2} \;+\; e\,\Phi \nonumber \\
& &+\; \Psi_{\epsilon}\left(\wh{{\bf p}} - \frac{e}{c}\,{\bf A}; {\sf F}\right),
\label{eq:ovH_def}\end{aligned}$$ where the reduced [*ponderomotive*]{} potential $\Psi_{\epsilon}$ depends explicitly on the field tensor ${\sf F}_{\mu\nu}$, which once again is the classical equivalent of the Dirac Hamiltonian after the FW transformation. From these field dependences, we define the reduced four-current density $$\wh{J}^{\mu} \;=\; (c\wh{\varrho}, \wh{{\bf J}}) \;\equiv\; \sum\;e\;\int d^{4}\wh{p}\;\wh{{\cal F}}\;\frac{d_{\epsilon}\wh{x}^{\mu}}{dt},
\label{eq:redJ_var}$$ where $d_{\epsilon}\wh{x}^{0}/dt = c$ and $$\frac{d_{\epsilon}\wh{\bf x}}{dt} \;\equiv\; \pd{\wh{{\cal H}}}{\wh{\bf p}} \;=\; \frac{1}{m}\,(\wh{{\bf p}} - \frac{e}{c}\,{\bf A}) +
\pd{\Psi_{\epsilon}}{\wh{{\bf p}}}.
\label{eq:dx_epsilon}$$ The reduced antisymmetric polarization-magnetization tensor [@Boghosian], on the other hand, yields the reduced polarization ${\sf K}_{\epsilon}^{0i} = P_{\epsilon}^{i}$ and the reduced magnetization ${\sf K}_{\epsilon}^{ij} = \epsilon^{ijk}\,M_{\epsilon\;k}$, where $$\left( {\bf P}_{\epsilon},\; {\bf M}_{\epsilon} \right) \;\equiv\; -\;\sum\;\int d^{4}\wh{p}\;\wh{{\cal F}}\;
\left( \pd{\Psi_{\epsilon}}{{\bf E}},\; \pd{\Psi_{\epsilon}}{{\bf B}} \right).
\label{eq:PM_var}$$ Hence, polarization and magnetization are explicitly evaluated in terms of the ponderomotive Hamiltonian $\Psi_{\epsilon}$.
By applying the Nother method, we obtain the reduced canonical energy-momentum tensor $$\begin{aligned}
{\sf T}^{\mu\nu} & \equiv & \frac{1}{4\pi} \left[\; \frac{g^{\mu\nu}}{4}\; {\sf F}:{\sf F} \;-\; \left( {\sf F}^{\mu\sigma} \;+\frac{}{} 4\pi\,
{\sf K}_{\epsilon}^{\mu\sigma}\right)\;{\sf F}_{\sigma}^{\;\;\nu} \;\right] \nonumber \\
& &+\; \sum\;\int d^{4}\wh{p}\; \pd{\wh{{\cal H}}}{\wh{p}_{\mu}}\;\left(\wh{p}^{\nu} - \frac{e}{c}\,A^{\nu} \right)\;\wh{{\cal F}},
\label{eq:Tmunu_def}\end{aligned}$$ which naturally includes the polarization and magnetization and is manifestly asymmetric. The antisymmetric part of the canonical energy-momentum tensor $T_{\sf A}^{ij} \equiv \frac{1}{2}\,\epsilon^{ijk}\,\tau_{k}$ is expressed in terms of the reduced (intrinsic) [*ponderomotive*]{} torque $$\begin{aligned}
\vb{\tau} & \equiv & \sum\,\int\,\left[ \pd{\Psi_{\epsilon}}{\bf E}\btimes{\bf E} \;+\; \pd{\Psi_{\epsilon}}{\bf B}\btimes{\bf B} \right. \nonumber \\
& &\left.\hspace*{0.5in}+\; \pd{\Psi_{\epsilon}}{\wh{\bf p}}\btimes\left(\wh{\bf p} - \frac{e}{c}\,{\bf A} \right) \right]\;\wh{F}\, d^{3}\wh{p}.
\label{eq:torque_pond}\end{aligned}$$ The three [*zitterbewegung*]{} features, which appear explicitly in the expression for the reduced ponderomotive torque, are derived from the reduced ponderomotive potential $\Psi_{\epsilon}$ associated with the process of dynamical reduction.
As an application of the ponderomotive torque , we consider the gyrocenter ponderomotive Hamiltonian [@Brizard_energy] $$\Psi_{\epsilon} \;=\; -\;\frac{mc^{2}}{2\,B_{0}^{2}}\; \left|\,{\bf E}_{\bot} \;+\; \frac{\wh{p}_{\|}\bhat_{0}}{mc}\btimes{\bf B}_{\bot}\,\right|^{2}
\;+\; \mu\;\frac{|{\bf B}_{\bot}|^{2}}{2B_{0}},
\label{eq:Psi_gy}$$ derived in the zero-Larmor-radius (ZLR) limit from the second-order gyrocenter Hamiltonian [@Brizard_Hahm], where a strongly magnetized background plasma is perturbed by the low-frequency electromagnetic fields ${\bf E}_{\bot} \equiv -\nabla_{\bot}\Phi$ and ${\bf B}_{\bot} \equiv \nabla_{\bot}A_{\|}\btimes\bhat_{0}$, which are perpendicular to the background magnetic field ${\bf B}_{0} = B_{0}\,\bhat_{0}$. From the gyrocenter ponderomotive Hamiltonian , we obtain the reduced gyrocenter polarization and magnetization $$\begin{aligned}
\pd{\Psi_{\epsilon}}{{\bf E}_{\bot}} & = & -\;\frac{mc^{2}}{B_{0}^{2}} \left( {\bf E}_{\bot} \;+\; \frac{\wh{p}_{\|}\bhat_{0}}{mc}\btimes{\bf B}_{\bot}
\right) \;\equiv\; -\,\vb{\pi}_{\epsilon}, \label{eq:Psi_gy_E} \\
\pd{\Psi_{\epsilon}}{{\bf B}_{\bot}} & \equiv & -\,\vb{\mu}_{\epsilon} \;-\; \vb{\pi}_{\epsilon}\btimes \frac{\wh{p}_{\|}\bhat_{0}}{mc},
\label{eq:Psi_gy_B}\end{aligned}$$ where $\vb{\mu}_{\epsilon} \equiv -\,\mu\,{\bf B}_{\bot}/B_{0}$ denotes the reduced intrinsic gyrocenter magnetization. These definitions lead to the identity $\partial\Psi_{\epsilon}/\partial{\bf E}_{\bot}\btimes{\bf E}_{\bot} = {\bf E}_{\bot}\btimes\vb{\pi}_{\epsilon} \equiv -\;\partial
\Psi_{\epsilon}/\partial{\bf B}_{\bot}\btimes{\bf B}_{\bot}$, which means that the electromagnetic contribution to the gyrokinetic intrinsic torque vanishes in the ZLR limit. The kinetic part of the gyrokinetic intrinsic torque, on the other hand, involves the ponderomotive velocity $$\begin{aligned}
\pd{\Psi_{\epsilon}}{\wh{\bf p}} & = & -\,\bhat_{0} \left[ \frac{{\bf B}_{\bot}}{B_{0}}\bdot \left( {\bf E}_{\bot}\btimes\frac{c\bhat_{0}}{B_{0}} \;+\; \frac{\wh{p}_{\|}}{m}\;\frac{{\bf B}_{\bot}}{B_{0}} \right) \right] \nonumber \\
& \equiv & -\,\bhat_{0}\;\left|\pd{\Psi_{\epsilon}}{\wh{\bf p}}\right|,
\label{eq:Psi_p}\end{aligned}$$ which vanishes only in the electrostatic limit $({\bf B}_{\bot} \equiv 0)$. Lastly, the intrinsic gyrokinetic torque $$\vb{\tau}_{\rm gy} \;=\; \sum\,\int\,\left( \left|\pd{\Psi_{\epsilon}}{\wh{\bf p}}\right|\;\wh{\bf p}\btimes\bhat_{0} \right)\;\wh{F}\, d^{3}\wh{p}
\label{eq:tau_gy}$$ involves the magnitude of the ponderomotive velocity and the perpendicular component of the gyrocenter kinetic momentum $[\wh{\bf p} - (e/c)\,
A_{\|}\,\bhat_{0}]\btimes\bhat_{0} = \wh{\bf p}\btimes\bhat_{0}$.
In summary, we have shown how classical [*zitterbewegung*]{} effects manifest themselves in reduced Vlasov-Maxwell theory through the reduced ponderomotive Hamiltonian. Here, the ponderomotive torque acts as the source for the intrinsic spin angular momentum. Ongoing work includes the investigation of classical [*zitterbewegung*]{} effects in gyrofluid and gyrokinetic models.
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abstract: 'We derive the inverse spatial Laplacian for static, spherically symmetric backgrounds by solving Poisson’s equation for a point source. This is different from the electrostatic Green function, which is defined on the four dimensional static spacetime, while the equation we consider is defined on the spatial hypersurface of such spacetimes. This Green function is relevant in the Hamiltonian dynamics of theories defined on spherically symmetric backgrounds, and closed form expressions for the solutions we find are absent in the literature. We derive an expression in terms of elementary functions for the Schwarzschild spacetime, and comment on the relation of this solution with the known Green function of the spacetime Laplacian operator. We also find an expression for the Green function on the static pure de Sitter space in terms of hypergeometric functions. We conclude with a discussion of the constraints of the electromagnetic field.'
author:
- Karan Fernandes
- Amitabha Lahiri
title: The inverse spatial Laplacian of spherically symmetric spacetimes
---
Introduction {#intro}
=============
Let us consider a four dimensional, static, spherically symmetric background with metric $$ds^2 = g_{\alpha\beta} dx^{\alpha} dx^{\beta}
= - \lambda^2 dt^2 + h_{\alpha\beta} dx^\alpha dx^\beta\,.
\label{met}$$ By static, we mean that the spacetime admits a hypersurface orthogonal timelike Killing vector $\xi^\mu\,,$ such that $\xi^\mu \xi_\mu = -\lambda^2\,.$ The induced metric on the spacelike hypersurface $\Sigma$ is $h_{\alpha\beta}
= g_{\alpha\beta} + \lambda^{-2} \xi_\alpha\xi_\beta\,,$ and since the hypersurface is assumed to be spherically symmetric, we can write $\lambda = \lambda(r)\,.$
The object of our interest in this paper is the Green function $\widetilde{G}(\vec{x},\vec{y})$ for the induced spatial Laplacian operator, which formally satisfies the equation $$%\mathcal{D}_{\mu}\mathcal{D}^{\mu} \Phi(\vec{x}, \vec{y}) =
%\frac{\delta^3 \left(\vec{x} - \vec{y} \right)}{\sqrt{h}} \equiv
{D}_{\mu} {D}^{\mu} \widetilde{G}(\vec{x}, \vec{y}) =
- 4 \pi \delta \left( \vec{x},\vec{y} \right) \, ,
\label{IL.eq}$$ where $D_\mu$ is the induced covariant derivative compatible with the induced metric, $${D}_{\mu} h_{\alpha \beta} = 0 \,,$$ and the 3-dimensional covariant delta function $\delta\left(\vec{x},
\vec{y} \right)$ is defined by $$\int\limits_\sigma d^3x \sqrt{\det h(\vec{x})} f(\vec{x})\,
\delta\left(\vec{x}, \vec{y} \right) = f(\vec{y})\,,
\label{delta.def}$$ for all well-behaved functions $f(\vec{x})$ if $\sigma\subseteq\Sigma$ includes the point $\vec{y}$, and zero otherwise. This Green function is relevant for the Hamiltonian dynamics of fields on curved backgrounds, as we discuss below.
On the other hand, a different Green function appears in solving for the Coulomb potential in static spherically symmetric spacetimes, and a closed form expression for it is well known. For Maxwell’s equation $$\nabla_{\alpha} F^{\alpha \mu} = -4 \pi J^{\mu} \, ,
\label{max.gen}$$ where $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ is the usual electromagnetic field strength tensor. In what follows, $a_{\mu} = h^{\nu}_{\mu} A_{\nu}$ and $\phi = \xi^{\alpha}A_{\alpha}$ represent the spatial and temporal components of the spacetime field $A_{\mu}$ respectively. By defining the electric field as $$e^{\mu} := \lambda^{-1} \xi_{\alpha}F^{\mu \alpha} \, ,$$ we find that the contraction of Eq. with $\lambda^{-1} \xi_{\mu}$, equivalent to setting $\mu=0$, leads to $$D_{\mu} e^{\mu} = D_{\mu}\left(\lambda^{-1} D^{\mu} \phi - \lambda^{-1} \pounds_{\xi} a^{\mu} \right) = -4 \pi J^{0} \, ,
\label{max.con}$$ where $J^{0} = \lambda^{-1} \xi_{\mu} J^{\mu}$ and $\pounds_{\xi}$ is the Lie derivative with respect to $\xi^{\alpha}$. If we also set $\pounds_\xi \phi = 0 = \pounds_\xi a_\mu\,,$ and take a point charge by setting $J^{0} = \delta\left(\vec{x},\vec{y}\right)$, Eq. reduces to that for the Green function for the electrostatic potential, $$%D^x_{\mu} e^{\mu}(\vec{x}) =
%D^{x}_{\mu}\left(\lambda^{-1}(\vec{x}) D_x^{\mu} \phi(\vec{x}) \right)
D^{x}_{\mu}\left(\lambda^{-1}(\vec{x}) D_x^{\mu} G(\vec{x}, \vec{y}) \right)
= -4 \pi \delta(\vec{x},\vec{y}) \, .
\label{max.gff}$$ The left hand side of Eq. is nothing but the the action of the d’Alembertian on time-independent functions, for which the expansion $$\nabla_{\mu} \nabla^{\mu} \phi = \lambda D_{\mu}\left(\lambda^{-1}D^{\mu}
\phi \right) = D_{\mu} D^{\mu} \phi + \lambda D_{\mu}\left(\lambda^{-1}\right)
D^{\mu} \phi \,
\label{max.4d}$$ reveals that while Eq. and Eq. are the same in flat space, they differ on curved backgrounds where $\lambda$ is not a constant. We will call the Green function corresponding to Eq. the 4d static scalar Green function, and that of Eq. the inverse spatial Laplacian. For the Schwarzschild background, the 4d Green function ${G(\vec{x}, \vec{y})}$ is known in closed form. It can be derived by direct construction of the Hadamard elementary solution [@Cop:1928] and also using the method of multipole expansion [@J.Math.Phys.12.1845; @Hanni:1973fn]. A closed form expression was given in [@Linet:1976sq], which included an additional term missed in [@Cop:1928]. This term accounts for the induced charge behind the horizon of the black hole on the Schwarzschild background, and corresponds to the zero mode contribution in the multipole expansion result. The closed form expression for the static, scalar Green function for the spacetime Laplacian on curved backgrounds has found numerous applications [@Candelas:1980zt; @Isoyama:2012in; @Frolov:2012ip; @Frolov:2012xf; @Casals:2013mpa] predominantly in its use in determining the self force acting on the particle placed on such backgrounds [@Smith:1980tv; @Wiseman:2000rm; @Barack:1999wf; @Beach:2014aba; @Frolov:2014gla; @Casals:2012qq; @Kuchar:2013bla]. Such closed form expressions have additionally been determined for the Reissner-Nördstrom [@Leaute:1976sn], and more recently for Kerr backgrounds [@Ottewill:2012aj].
In contrast, the Green function of Eq. arises in various contexts which involve fields on static foliations of spacetime. These include the gravitational initial value problem [@Gourgoulhon:2007tn; @Dain:2004vi; @Pfeiffer:2004qz], metric fluctuations around solutions of the Einstein field equations [@Bardeen:1980kt; @Antoniadis:2011ib; @Mottola:2016mpl], classical radiation of free-falling charges [@Akhmedov:2010ah], and more recently, renormalization group equations on curved backgrounds [@Rechenberger:2012dt; @Codello:2008vh; @Wetterich:2016ewc], to name a few. This Green function is particularly relevant in the context of Hamiltonian dynamics of fields. The specific context we have in mind is the constrained dynamics of gauge field theories, where this function appears for gravitational [@Deser:1967zzb; @Bonazzola:2003dm; @Fischer:1996qg] and electromagnetic [@physics/9804018; @ashtekar:1996bs; @igarashi:1998hd; @Prescod-Weinstein:2014lua] fields. For example, the Maxwell field has the first class Gauss law constraint $\mathcal{D}_i \pi^i \approx 0\,,$ which implies the existence of redundant or gauge degrees of freedom, which can be eliminated by fixing the gauge and then applying Dirac’s procedure. The resultant Dirac brackets of the fields and their momenta in the radiation gauge on curved backgrounds with horizons involves this Green function [@Fernandes:2016imn]. However, while well motivated in the literature, we found no closed form expressions for them on curved backgrounds. In this work, we consider these functions for spherically symmetric backgrounds, and derive their expressions for the Schwarzschild and pure de Sitter cases. For these backgrounds, the metric of Eq. takes the form $$\begin{aligned}
ds^2 %& = g_{\alpha \beta} dx^{\alpha} dx^{\beta} \notag\\
%% & = - {\lambda(r)}^2 dt^2 + h_{i j} dx^i dx^j \notag\\
& = - {\lambda(r)}^2 dt^2 + \frac{1}{{\lambda(r)}^2} dr^2 + r^2 d\Omega^2 \,.
\label{met}\end{aligned}$$ Both backgrounds possess a horizon, defined by $\lambda=0$.
The organization of our paper is as follows. In Sec. \[GF\], we review the derivation of the static, scalar Green function for the spacetime Laplacian defined on the Schwarzschild background. In Sec. \[Sol\], we derive the solutions of Eq. (\[IL.eq\]) for the Schwarzschild and static pure de Sitter backgrounds. While we were able to determine the closed form expression for the Schwarzschild case in terms of elementary functions, we were unable to find a similar expression for the pure de Sitter background. Finally, in Sec. \[Con\], we discuss the relevance of our result in the constrained quantization of the Maxwell field on spherically symmetric backgrounds.
Derivation of the 4d static, scalar Green function {#GF}
==================================================
The Green function corresponding to Eq. is relevant for Coulomb’s law, as we have seen. Let us briefly review its derivation on the Schwarzschild background following [@J.Math.Phys.12.1845; @Hanni:1973fn], as we will follow a similar procedure for deriving the Green function for Eq..
We take $\lambda^2 = g^{rr} = 1 - \frac{2m}{r}$, and place a [unit]{} charge at $(r', {\theta}', {\phi}')$. With this choice Eq. becomes, in explicit coordinates, $$\begin{aligned}
\sin\theta\partial_{r} \left(r^2 \partial_r G \right) + \frac{1}{\left(1 - \frac{2 m}{r} \right)} \partial_{\theta} \left(\sin \theta \partial_{\theta} G \right) + \frac{1}{\left(1 - \frac{2 m}{r} \right) \text{sin} \theta} \partial^2_{\phi} G
\qquad \notag\\
= - 4 \pi \delta(r - r') \delta(\theta - {\theta}') \delta(\phi - {\phi}') \,,
\label{wgf.me}\end{aligned}$$ where the delta functions are normalized according to $$\begin{aligned}
\int_{2 m}^{\infty} dr \delta(r - r') = 1\,, \quad \int_{0}^{\pi} d\theta \, \delta (\theta - {\theta}')
= 1 \,, \quad \int_{0}^{2 \pi} d \phi \, \delta (\phi - {\phi}') = 1 \,.
\label{wgf.delta}\end{aligned}$$ Away from the point charge, the right hand side of Eq. vanishes, and we can expand $G$ as $$G(\vec{r}, \vec{r}') = \sum_{l=0}^{\infty} R_l(r, r') P_l(\cos \gamma) \, ,
\label{wgf.sep}$$ where $\cos \gamma = \cos \theta \cos {\theta}' + \sin \theta \sin \theta'
\cos \left( \phi - \phi'\right)$. While we could have used the azimuthal symmetry to reduce this to a problem in plane polar coordinates $(r, \theta)$, the calculations are no more complicated for $(r, \theta, \phi)\,,$ so we have chosen to display all coordinates. We note that since $P_l(\cos \gamma)$ is related to the spherical harmonics $Y_{l,m}(\theta, \phi)$ via the Legendre addition theorem (cf. Eqs. (14.30.8), (14.30.9), (14.30.11) of [@NIST]) $$\frac{2l+1}{4 \pi} P_l(\cos \gamma) = \sum_{m=-l}^{l} Y_{l,m}(\theta, \phi)
Y^*_{l,m}(\theta', \phi')\,,
\label{wgf.spher}$$ it further satisfies $$\begin{aligned}
\frac{1}{\sin\theta} \partial_{\theta} \left(\sin\theta \partial_{\theta} P_l(\cos \gamma) \right) + \frac{1}{\sin^2 \theta} \partial_{\phi}^2 P_l(\cos \gamma) = -l(l+1) P_l(\cos\gamma) \,, \label{wgf.lap}\\
\int_{-1}^{1} d \cos \theta \int_{0}^{2 \pi} d \phi P_{l'}
(\cos \gamma) P_{l} (\cos \gamma) = \delta_{ll'}
\frac{4 \pi}{2 l + 1}\,.
\label{wgf.norm}\end{aligned}$$ Substituting Eq. in Eq. away from the source, we find that $R_l(r)$ is a linear combination of two independent solutions, $$R_l(r, r') = A_l(r') g_l(r) + B_l(r') f_l(r) \,,
\label{wgf.rsol}$$ where $g_l(r)$ and $f_l(r)$ are given by [@J.Math.Phys.12.1845; @Israel:1967za] $$\begin{aligned}
g_l(r) & = \begin{cases}
1 \qquad \qquad & \qquad (\text{for} \, l = 0) \, , \\
\frac{2^l l!\, (l-1)!\, m^l}{(2l)!} (r-2 m) \frac{d}{dr} P_l
\left(\frac{r}{m} - 1\right) \, & \qquad (\text{for} \, l \neq 0) \,,
\end{cases}
\label{wgf.wgsol}\\
f_l(r) & = - \frac{ (2 l + 1)!}{2^l (l+1)!\, l!\, m^{l+1}} (r-2 m)
\frac{d}{dr} Q_l \left(\frac{r}{m} - 1\right) \,. \qquad \qquad \qquad \quad
\label{wgf.wsol}\end{aligned}$$ Here $P_l$ and $Q_l$ are the Legendre functions of the first and second kind, respectively. With the exception of $g_0(r) = 1$, the leading term of $g_l(r)$ is proportional to $r^l$ and diverges as $r \to \infty$. Thus this solution is ruled out for large values of $r$. Both $g_l(r)$ and $f_l(r)$ are well behaved at the horizon $r = 2m$. However, $\frac{d}{dr} f_l(r)$ diverges logarithmically as $r \to 2m$, except when $l=0$. On the other hand, the leading behaviour of $f_l(r)$ for large $r$ is proportional to $r^{-l-1}$.
We can thus write Eq. as $$G(\vec{r}, \vec{r}') = \begin{cases}
\displaystyle{\sum_{l=0}^{\infty} A_l(r') f_l(r) P_l(\cos \gamma) \qquad \qquad r > r' } \vspace{0.5em} \\
\displaystyle{\sum_{l=0}^{\infty} B_l(r') g_l(r) P_l(\cos \gamma) \qquad \qquad r < r'}
\end{cases}$$ The continuity of $G$ and discontinuity of $\vec{\nabla}G$ at $\vec{r}=\vec{r}'$ tells us that by defining $r_< = \min(r\,,r')$ and $r_> = \max(r\,,r')$, we can write $G(\vec{r}, \vec{r}')$ as $$G(\vec{r}_<,\vec{r}_>) = \sum_{l=0}^{\infty} g_l(r_<) f_l(r_>) P_l(\cos \gamma) \,.
\label{wgf.res}$$ A bit of algebra now shows that these solutions can be rewritten in the form $$\begin{aligned}
&G(\vec{r},\vec{r}') = \notag \\
&\frac{1}{r r'} \left[\frac{ (r - m) (r' -m) - m^2 \cos \gamma}{\sqrt{(r - m)^2 + (r' - m)^2 - 2 (r - m)(r' - m)\cos \gamma - m^2 \sin^2 \gamma}} + m \right] \,.
\label{wgf.lin}\end{aligned}$$ This expression, found in [@Linet:1976sq], differs from a solution provided many years earlier [@Cop:1928] because of the term $\frac{m}{r r'}\,,$ which accounts for the zero-mode contribution in Eq.. The result in Eq. has been derived recently using the heat kernel method and bi-conformal symmetry in [@Frolov:2014kia].
Inverse Spatial Laplacian of the Schwarzschild background {#Sol}
=========================================================
Now let us get back to the solution of Eq. in the Schwarzschild background. With the source at $(r', {\theta}', {\phi}')$ as before, Eq. takes the form $$\begin{aligned}
\sin \theta \partial_r \left(r^2 \sqrt{1 - \frac{2 m}{r}}
\partial_r \widetilde{G} \right) &+ \frac{1}{\sqrt{1 - \frac{2 m}{r}}}
\partial_{\theta} \left( \sin \theta \partial_{\theta} \widetilde{G} \right)
+ \frac{1}{\sqrt{1 - \frac{2 m}{r}} \sin \theta} \partial_{\phi}^2 \widetilde{G}
\notag \\
&= - 4 \pi \delta(r-r') \delta(\theta -\theta') \delta(\phi -\phi')
\, .
\label{sgf.me}\end{aligned}$$ It will be convenient to make a change of variables from $r$ to $y = \frac{r}{m} -1$. After we find the solution, we can change variables again to express the Green function in terms of the original coordinates.
In terms of $y$, Eq. takes the form $$\begin{aligned}
\sin \theta \left[ \partial_y \left((y+1)^2 \sqrt{\frac{y-1}{y+1}}
\partial_y \widetilde{G} \right) \right.&
\left.
% \right. \notag\\ \qquad \left.
+ \sqrt{\frac{y+1}{y-1}} \left(\frac{1}
{ \sin \theta} \partial_{\theta} \left( \sin \theta \partial_{\theta}
\widetilde{G} \right) + \frac{1}{\sin^2 \theta} \partial_{\phi}^2 \widetilde{G} \right) \right]
\notag\\
&\qquad \qquad = - 4 \pi \frac{\delta(y-y')}{m} \delta(\theta -\theta') \delta(\phi -\phi')
\, ,
\label{sgf.yme}\end{aligned}$$ with the point source located at $(y',\theta',\phi')$ in the new coordinates.
The angular delta functions satisfy the expressions in Eq., while the $y$ delta function now satisfies $$\int_{1}^{\infty} dy \, \delta (y - y') = 1 \, .
\label{sgf.dfn}$$
The first step in our derivation is to consider Eq. far removed from the source. Thus we need to solve the following equation $$0 = \sqrt{\frac{y-1}{y+1}} \partial_y \left((y+1)^2 \sqrt{\frac{y-1}{y+1}} \partial_y\widetilde{G} \right) + \frac{1}{\sin \theta} \partial_{\theta} \left( \sin \theta \partial_{\theta} \widetilde{G} \right) + \frac{1}{ \sin^2 \theta} \partial_{\phi}^2 \widetilde{G} \,.
\label{sgf.hom}$$ Writing $$\widetilde{G} (\vec{y},\vec{y}') = \sum_{l=0}^{\infty} R_l(y,y') P_l(\cos \gamma)\,,
\label{sgf.sep}$$ and substituting Eq. in Eq., we get the differential equation $$\sqrt{\frac{y-1}{y+1}}\frac{d}{dy} \left((y+1)^2 \sqrt{\frac{y-1}{y+1}} \frac{d}{dy} R_l(y,y') \right) -l(l+1) R_l(y,y') =0 \,.
\label{sgf.Req}$$ We have described the solution of Eq. in Appendix \[App\]. The general solution is given in Eq., and it is of the form $$R_l(y,y') = A_l(y') g_l(y) + B_l(y') f_l(y)\,,
\label{sgf.sol}$$ where the functions $g_l(y)$ and $f_l(y)$ involve Legendre polynomials of fractional degree, with the argument $y>1$. Legendre polynomials of fractional degree can be described in terms of hypergeometric functions, for which there exist several representations. A particular representation which we will use is (cf. pp 153-163, Table entry 10 and 28, of [@MO:1966]) $$\begin{aligned}
P_{\nu}^{\mu}(y) & = \frac{\Gamma \left(-\nu - \frac{1}{2} \right)}
{2^{\nu + 1} \sqrt{\pi} \Gamma \left(-\nu - \mu \right)} y^{-\nu + \mu - 1}
(y^2 - 1)^{- \frac{\mu}{2}}\times \notag \\
& \qquad\qquad \times \,_2 F_1 \left(\frac{1 + \nu - \mu}{2},
\frac{2 + \nu - \mu}{2}; \nu + \frac{3}{2}; \frac{1}{y^2} \right) \notag \\
& \qquad\qquad + \frac{2^{\nu} \Gamma \left(\nu + \frac{1}{2} \right)}{ \sqrt{\pi}
\Gamma \left(1 + \nu - \mu \right)} y^{\nu + \mu} (y^2 - 1)^{- \frac{\mu}{2}} \times
\notag \\
& \qquad \qquad
\,_2 F_1 \left(\frac{-\nu - \mu}{2}, \frac{1 -\nu - \mu}{2}; -\nu + \frac{1}{2};
\frac{1}{y^2} \right) \, , \notag \\
e^{-i \pi \mu} Q_{\nu}^{\mu}(y) & = \frac{\sqrt{\pi}
\Gamma\left( 1 + \nu + \mu \right)}{2^{\nu + 1} \Gamma \left(\frac{3}{2}
+ \nu \right)} y^{-\nu - \mu - 1} (y^2 - 1)^{\frac{\mu}{2}} \times \notag \\
& \qquad \qquad \times \,
_2 F_1 \left(\frac{\nu + \mu + 2}{2} , \frac{\nu+ \mu +1}{2}; \nu + \frac{3}{2};
\frac{1}{y^2} \right) \, ,
\label{sgf.ghgf}\end{aligned}$$ The solutions $g_l(y)$ and $f_l(y)$ make use of these solutions for the case of $\mu = \frac{1}{2}$ and $\nu = l$ as shown in Eq. of Appendix A, and can be written as $$\begin{aligned}
g_l(y) & = \frac{1}{\sqrt{y+1}} \left[\frac{1}{2^{l + 1}} y^{-l - \frac{1}{2}} \,_2 F_1 \left(\frac{l + \frac{1}{2}}{2}, \frac{l+ \frac{3}{2}}{2}; l + \frac{3}{2}; \frac{1}{y^2} \right) \right. \notag \\
& \left. \qquad + 2^{l} y^{l+ \frac{1}{2}} \,_2 F_1 \left(\frac{-l - \frac{1}{2}}{2}, \frac{-l + \frac{1}{2}}{2}; -l + \frac{1}{2}; \frac{1}{y^2} \right) \right] \, , \notag \\
f_l(y) & = \sqrt{y - 1} \left[\frac{1}{2^{l}} y^{-l - \frac{3}{2}} \,_2 F_1 \left(\frac{l + \frac{5}{2}}{2} , \frac{l + \frac{3}{2}}{2}; l + \frac{3}{2}; \frac{1}{y^2} \right) \right] \, .
\label{sgf.main}\end{aligned}$$ It turns out that the functions given in Eq. admit expressions in terms of more elementary functions, which we will now describe. These expressions will be relevant in determining the final form of the Green function for the spatial Laplacian. The hypergeometric functions contained in $g_l(y)$ in Eq. are both of the following generic form, with the known representation $$_2 F_1 \left( a , a+ \frac{1}{2}, 2 a + 1, \frac{1}{y^2} \right) =
2^{2a} \left( \frac{ y + \sqrt{y^2 -1}}{y} \right)^{-2 a}\,,$$ where $a$ stands for both $\frac{ l + \frac{1}{2}}{2}$ and $\frac{-l -\frac{1}{2}}{2}$ in the above expression. We can thus write the expression for $g_l(y)$ as $$g_l(y)
= \frac{1}{\sqrt{2}\sqrt{y+1}}
{\left[ \left(y + \sqrt{y^2 - 1} \right)^{-l - \frac{1}{2}} + \left(y + \sqrt{y^2 - 1} \right)^{l + \frac{1}{2}} \right]}\,.
%= \frac{ \left(y + \sqrt{y^2 - 1} \right)^{-l - \frac{1}{2}} + \left(y + \sqrt{y^2 - 1} \right)^{l + \frac{1}{2}}}{ \sqrt{2}\sqrt{y+1}}
\label{sgf.geq}$$ Likewise, the hypergeometric function given in $f_l(y)$ has the following expression in terms of elementary functions, $$_2 F_1 \left( b , b+ \frac{1}{2}, 2 b, \frac{1}{y^2} \right) = \frac{2^{2b - 1} y^{2 b} }{\sqrt{y^2 -1}} \left( y + \sqrt{y^2 -1} \right)^{-2 b + 1}\,,$$ where $b = \frac{l + \frac{3}{2}}{2}$. We can thus write $f_l(y)$ as $$f_l(y) = \sqrt{2} \, \frac{\left( y + \sqrt{y^2 -1} \right)^{-l - \frac{1}{2}}}{\sqrt{y + 1}} \, .
\label{sgf.feq}$$ The calculation below will require the Wronskian of the solutions given in Eq.. Using the above expressions, we readily find that the Wronskian $W(g_l(y),f_l(y),y)$ is given by $$W(g_l(y),f_l(y),y) = - \frac{(2 l + 1)}{(1+y)^{\frac{3}{2}} \sqrt{y - 1}}\,.
\label{sgf.wro}$$ There are two limits to consider of the solutions given in Eq. and Eq., and their derivatives. These are the $y \to 1$ and $y \to \infty$ limits, which correspond to $r \to 2 m$ and $r \to \infty$ respectively. Before describing these, we note that $g_0(y)$ is a special case in that it is a constant, $g_0(y) = 1$ for all values of $y$.
For all the other terms we find the following. As $y \to 1$, both $g_l(y) \to 1$ and $f_l(y) \to 1$ for all values of $l$, i.e. they are both finite. However, all derivatives of $ f_l(y)$ diverge as $y \to 1$, while $\frac{d}{dy} g_l(y) \to l(l+1)$ as $y \to 1$. Thus the near horizon solution must only contain $g_l(y)$, and we must set $B_l(y') = 0$ in Eq. in the region between $(y',\theta',\phi')$ and the event horizon of the black hole.
On the other hand, as $y \to \infty$ , we find that $f_l(y) \to 0$ for all values of $l$, and the derivatives of $f_l(y)$ are also well behaved, but $g_l(y)$ diverges for $l \neq 0$. We must thus set $A_l(y') = 0$ in Eq. to describe the region from $(y',\theta',\phi')$ to $\infty$.
We can therefore write the solution in the following way in the two regions, $$\widetilde{G} \left(\vec{y},\vec{y}'\right) = \begin{cases} \displaystyle{\sum_{l=0}^{\infty}
A_l(y') g_l(y) P_l(\cos \gamma)\,, \qquad \qquad (y < y')} \\
\displaystyle{ \sum_{l=0}^{\infty} B_l(y') f_l(y) P_l(\cos \gamma)\,. \qquad \qquad (y > y')}
\end{cases}$$ Continuity of $\widetilde{G}$ at $y=y'$ implies that $A_l(y') g_l(y') = B_l(y') f_l(y')\,.$ Then we can define a constant $C_l$ such that $$C_l = \frac{A_l(y')}{f_l(y')} = \frac{B_l(y')}{g_l(y')} \, ,$$ using which we can write the solution in the form $$\widetilde{G} \left(\vec{y},\vec{y}'\right)= \begin{cases} \displaystyle{\sum_{l=0}^{\infty}
C_l f_l(y') g_l(y) P_l(\cos \gamma)\,, \qquad \qquad (y < y')} \\
\displaystyle{ \sum_{l=0}^{\infty} C_l g_l(y') f_l(y) P_l(\cos \gamma)\,. \qquad \qquad (y > y')}
\end{cases}
\label{sgf.ssol}$$ We can now determine the constants $C_l$ by appropriately integrating Eq.. To begin with, we insert Eq. into Eq., multiply both sides with $P_{l'}(\cos \gamma)$ and integrate with respect to $\theta$ and $\phi$ to find $$\frac{1}{2 l + 1} \left[\frac{d}{dy} \left((y+1)^2 \sqrt{\frac{y-1}{y+1}}
\frac{d}{dy} R_l(y) \right) - l(l+1) \sqrt{\frac{y+1}{y-1}} R_l(y) \right] =
- \frac{\delta(y-y')}{m}\,.
\label{sgf.int1}$$ Integrating Eq. over an infinitesimal region from $y' - \epsilon$ to $y' + \epsilon$, we get $$\begin{aligned}
- \frac{1}{m} & = \frac{1}{2 l + 1} C_l (y'+1)^2 \sqrt{\frac{y'-1}{y'+1}}\left[g_l(y')
\left.\frac{d f_l(y)}{dy}\right\vert_{y' + \epsilon}
- f_l(y') \left.\frac{d g_l(y)}{dy} \right\vert_{y' - \epsilon} \right] \notag \\
& = \frac{1}{2 l + 1} C_l (y'+1)^{\frac{3}{2}} \sqrt{y'-1} W(g_l(y'),f_l(y'),y') \notag \\
& = - C_l\,,
\label{sgf.con}\end{aligned}$$ where in going from the second to the third equality in Eq., we made use of Eq.. Thus we have determined that $C_l$ is independent of $l$, $$C_l = \frac{1}{m}\,,$$ and we can write the solution of Eq. as $$\widetilde{G}\left(\vec{y}_<\,,\vec{y}_>\right) = \frac{1}{m} \sum_{l=0}^{\infty}
g_l(y_<)\, f_l(y_>) P_l(\cos \gamma)\,,
\label{sgf.csol}$$ where $y_< = \min(y\,,y')$ and $y_> = \max(y\,,y')$. Using Eq. and Eq., we find that the product $g_l(y_<)\, f_l(y_>)$ is given by $$\begin{aligned}
g_l(y_<) f_l(y_>) &= \frac{1}{\sqrt{y_<+1} \sqrt{y_>+1}}
\left[ \left(\frac{y_< + \sqrt{y_<^2 - 1}}{y_> +
\sqrt{y_>^2 - 1}} \right)^{\frac{1}{2} + l} \right. \notag \\
& \qquad + \left.
\left( \left(y_< + \sqrt{y_<^2 - 1} \right) \left(y_>
+ \sqrt{y_>^2 - 1} \right) \right)^{-l - \frac{1}{2}} \right]
\label{sgf.exp}\end{aligned}$$
For the sake of notational convenience, let us define $$\begin{aligned}
A = y_> + \sqrt{y_>^2 - 1} \, \qquad & {\rm and} \qquad B = y_< + \sqrt{y_<^2 - 1}\,. \end{aligned}$$ Using Eq., and the standard expression for the generating function for Legendre polynomials $$\sum_{l=0}^{\infty} t^l P_l(x) = \frac{1}{\sqrt{1 - 2 x t + t^2}}\,,$$ we find that Eq. takes the form $$\begin{aligned}
\widetilde{G}\left(\vec{y}_<\,,\vec{y}_>\right) &= \frac{1}{m} \frac{1}{\sqrt{y_<+1}
\sqrt{y_> + 1}}\, \times \notag \\
& \qquad \times\, \left[\frac{\sqrt{A B}}{\sqrt{A^2 + B^2 - 2 \,
A \, B \,\cos \gamma}} + \frac{\sqrt{A B}}
{\sqrt{A^2\, B^2 + 1 - 2 \, A \, B \,\cos \gamma}} \right] \, .\end{aligned}$$ To write the solution in terms of Schwarzschild coordinates, we simply make the substitution for $y$, and write $$\begin{aligned}
&\widetilde{G}\left(\vec{r}, \vec{r}'\right) = \notag \\
& \frac{1}{\sqrt{r r'}} \left[\frac{\sqrt{\left(\kappa(r) r - m\right) \left(\kappa(r') r' - m\right)}}{\sqrt{\left(\kappa(r) r - m\right)^2 + \left(\kappa(r') r' - m\right) ^2 - 2 \, \left(\kappa(r) r - m\right) \, \left(\kappa(r') r' - m\right) \,\cos \gamma}} \right. \notag\\
& \left. \qquad +\frac{m \sqrt{\left(\kappa(r) r - m\right) \left(\kappa(r') r' - m\right) }}{\sqrt{\left(\kappa(r) r - m\right) ^2\, \left(\kappa(r') r' - m\right) ^2 + m^4 - 2 \, m^2 \left(\kappa(r) r - m\right) \, \left(\kappa(r') r' - m\right) \,\cos \gamma}} \right] \, ,
%&\Phi\left(\vec{r}_<, \vec{r}_>\right) = \notag \\
%& \frac{1}{\sqrt{r_< r_>}} \left[\frac{\sqrt{\left(\kappa(r_<) r_< - m\right) \left(\kappa(r_>) r_> - m\right)}}{\sqrt{\left(\kappa(r_<) r_< - m\right)^2 + \left(\kappa(r_>) r_> - m\right) ^2 - 2 \, \left(\kappa(r_<) r_< - m\right) \, \left(\kappa(r_>) r_> - m\right) \,\cos \gamma}} \right. \notag\\
%& \left. \qquad +\frac{m \sqrt{\left(\kappa(r_<) r_< - m\right) \left(\kappa(r_>) r_> - m\right) }}{\sqrt{\left(\kappa(r_<) r_< - m\right) ^2\, \left(\kappa(r_>) r_> - m\right) ^2 + m^4 - 2 \, m^2 \left(\kappa(r_<) r_< - m\right) \, \left(\kappa(r_>) r_> - m\right) \,\cos \gamma}} \right] \, ,
\label{sgf.sgf}\end{aligned}$$ where we have defined $\kappa(r) = 1 + \lambda(r) = 1 + \sqrt{ 1 - \frac{2 m}{r}}$, and $\kappa(r')$ similarly. As noted earlier, we see that as we take the flat space limit ($m \to 0$), this solution as well as that of Eq. reduce to the Green function of flat space. We also note that just as in the Green function result given in the previous section, this solution is regular at the horizon.
Inverse spatial Laplacian of the de Sitter background
=====================================================
We now turn our attention to writing a closed form expression for the Green function on a de Sitter background. The scalar de Sitter Green function for cosmological de Sitter spacetimes has been derived in [@Bunch:1977sq; @Chernikov:1968zm; @Tagirov:1972vv]. In static coordinates, the thermal Green function for the massless scalar field equation [@Dowker:1977], as well as the Green function for the massive scalar field equation [@Anninos:2011af; @Higuchi:1986ww] are known in the literature. These Green functions correspond to the de Sitter generalization of Eq., whereas we will be concerned with the derivation of the solution of the inverse spatial Laplacian, i.e. of Eq.. The procedure described in this subsection can be used for finding the solution of Eq. as well.
For pure de Sitter space with cosmological constant $\Lambda\,,$ we have $\lambda(r)^2 = 1 - \frac{r^2}{L^2}$, where $L = \sqrt{\frac{3}{\Lambda}}$, working in the quadrant of de Sitter space where the time coordinate increases into the future. We again make a change of coordinates and write $y = \frac{r}{L}$. For this choice, Eq. takes the form $$\begin{aligned}
\sin \theta \left[ \partial_y \left( y^2 \sqrt{1 - y^2} \partial_y \widetilde{G} \right)
+ \frac{1}{\sqrt{1 - y^2}} \right.&\left(\frac{1}{ \sin \theta} \partial_{\theta} \right.
\left( \sin \theta \partial_{\theta} \widetilde{G} \right) +
\left.\left. \frac{1}{\sin^2 \theta}
\partial_{\phi}^2 \widetilde{G} \right) \right] \notag\\
&= - 4 \pi \frac{\delta (y - y')}{L} \delta ( \theta - \theta')\delta (\phi - \phi')\,.
\label{dgf.yme}\end{aligned}$$ The delta functions for the angular variables satisfy Eq., but the $y$ delta function now satisfies $$\int_{0}^{1} dy \delta(y - y') = 1\,.$$ As in the Schwarzschild case, we begin by solving the above equation far away from the source $$\sqrt{1 - y^2} \partial_y \left( y^2 \sqrt{1 - y^2} \partial_y \widetilde{G} \right) + \frac{1}{\sin \theta} \partial_{\theta} \left( \sin \theta \partial_{\theta} \widetilde{G} \right) + \frac{1}{ \sin^2 \theta} \partial_{\phi}^2 \widetilde{G} = 0\,,
\label{dgf.hom}$$ with $$\widetilde{G}(\vec{y},\vec{y}') = \sum_{l=0}^{\infty} R_l(y,y') P_l(\cos \gamma)\,.
\label{dgf.sep}$$ Substituting Eq. in Eq., and using Eq., we get the equation $$\sqrt{1 - y^2} \frac{d}{dy} \left( y^2 \sqrt{1 - y^2}
\frac{d}{dy} R_l(y,y') \right) - l(l+1) R_l(y,y') = 0\,.
\label{dgf.Req}$$ To find the general solution in this case, it will be convenient to express Eq. in terms of $t = \sqrt{1-y^2}$, which results in $$\sqrt{1 - t^2} \frac{d}{dt} \left( (1 - t^2)^{\frac{3}{2}}
\frac{d}{dt} R_l(t,t') \right) - l(l+1) R_l(t,t') = 0\,.
\label{dgf.Req2}$$ Using the ansatz $R_l(t,t') = B_l(t') P_{\mu}^{\nu}(t) A(t)$ as before (see Appendix \[App\]), we find the following general solution $$R_l(t,t') = A'_l(t') (1 - t^2)^{-\frac{1}{4}} P_{\frac{1}{2}}^{l + \frac{1}{2}}(t) + B'_l(t') (1 - t^2)^{-\frac{1}{4}} P_{\frac{1}{2}}^{-l - \frac{1}{2}}(t)\,.
\label{dgf.Req3}$$ The Legendre polynomials described in Eq. can be described in terms of hypergeometric functions. For Legendre polynomials defined in the region between $-1$ and $+1$, we have (cf. p.166 of [@MO:1966]) $$\Gamma(1 - \mu) P_{\nu}^{\mu}(x) = 2^{\mu} ( 1 - x^2 )^{-\frac{\mu}{2}}
\, _2 F_1 \left( \frac{1}{2} + \frac{\nu}{2} - \frac{\mu}{2}, - \frac{\nu}{2} - \frac{\mu}{2} ; 1 - \mu ; 1 - x^2 \right)\,.
\label{dgf.iden}$$ By using the expressions in Eq., and writing the results in terms of the variables $y$ by substituting $1 - t^2 = y^2$, one can find the following general solution $$R_l(y,y') = A_l(y') g_l(y) + B_l(y') f_l(y) \, ,
\label{dgf.Rgen}$$ where $g_l(y)$ and $f_l(y)$ are now given by $$\begin{aligned}
g_l(y) & = y^l \, _2 F_1 \left(\frac{l}{2}, \frac{l}{2} + 1; \frac{3}{2} + l; y^2 \right) \notag \\
f_l(y) & = \frac{1}{y^{l+1}} \, _2 F_1 \left(\frac{-l - 1}{2}, \frac{-l + 1}{2}; \frac{1}{2} - l; y^2 \right)\, .
\label{dgf.Rsol}\end{aligned}$$ Here, $A_l(y')$ and $B_l(y')$ are real coefficients, and the solutions themselves are positive and real in the region between 0 and +1. The Wronskian of the two solutions given in Eq. satisfies the following relation $$W(g_l(y),f_l(y),y) = - \frac{2 l + 1}{y^2 \sqrt{1 - y^2}} \, .
\label{dgf.wro}$$ Unlike in the Schwarzschild case, we were unable to determine a closed form expression of the solutions in terms of elementary functions for arbitrary $l$. The solutions for specific choices of $l$ however can be easily determined. Using the derivative relations satisfied by the hypergeometric functions, we have derived in Appendix \[app2\] the following general form of the $f_l(y)$ solutions $$\begin{aligned}
%g_l(y) &= 1 & (l = 0) \, , \notag\\
%& = \frac{\sqrt{1 - y^2}}{y^{l+1}} \left[ \alpha(y) {\sin}^{-1} (y) + \beta(y) \right] & (l \neq 0) \, ,\notag \\
f_l(y) &= \sum_{n = 0}^{\frac{l-1}{2}} \frac{c_n}{y^{2 n + 2}} & (l \,\text{odd}) \, ,\notag\\
&= \frac{ \sqrt{1 - y^2}}{y} & (l = 0) \, , \notag\\
&= \sqrt{1 - y^2} \sum_{n=1}^{\frac{l}{2}} \frac{c_n}{y^{2 n + 1}} & (l \, \text{even} \, ; l \neq 0) \, ,
\label{dgf.rep} \end{aligned}$$ where $c_{\frac{l-1}{2}} = 1$ for the odd $l$ case, and $c_{\frac{l}{2}} = 1$ for the even $l$ case.
To proceed further, we need to determine the behaviour of the solution in the limit $y\to 0$ and $y\to 1$. As before, $g_0(y) =1$, which follows from $\ _2 F_1 \left(0, 1; \frac{3}{2}; y^2 \right) = 1$, and will not be considered in the following. As $y \to 0$, we can make use of the following derivative relation satisfied by the hypergeometric functions $$\frac{d}{dx} \, _2 F_1 \left( a , b, c, x \right) = \frac{a \, b}{c} \, _2 F_1 \left( a + 1 , b + 1, c + 1, x \right) \, ,
\label{dgf.hypr}$$ as well as $_2 F_1 \left( a , b, c, 0 \right) = 1$, to determine the behaviour of the solutions. We see that $g_l(y)$ and its first derivative vanish, while $f_l(y)$ and its first derivative diverge for all values of $l$, as $y\to 0$. We must thus set $B_l = 0$ in Eq. in the region where $y$ can vanish, in order to have regular solutions.
As $y \to 1$, we need to consider the integral representation of the hypergeometric function to demonstrate that $g_l(y)$ is finite while its first derivative diverges, for $l \neq 0$. This is shown in Appendix \[app2\]. The solutions provided in Eq. tell us the following about the $f_l(y)$ solutions in the limit $y \to 1$. While $f_l(y)$ remains finite for all $l$, and $f_l(1) = 0$ when $l$ is even, the behaviours of the first derivatives differ for even and odd $l$. The first derivative of $f_l(y)$ diverges when $l$ is even, and is finite when $l$ is odd. Regularity of the solutions requires that in the region where $y\to 1$, we not only set $A_l = 0$ for all $l\neq 0$, but also set $B_l = 0$ for even $l$.
We can now determine the general solution $\widetilde{G}(\vec{y},\vec{y}')$ for the point source located at $(y', \theta', \phi')$. Away from the source the solution is given by Eq.. As explained above, in the region $y<y'$ we simply set $B_l(y') = 0$ and sum over all $l$. In the region $y>y'$ we set $A_l(y') = 0$ for all $l \neq 0$ and sum over all odd $l$, but we in addition have the $g_0(y) =1$ term which contributes a constant term. Thus, we can write $$\begin{aligned}
\widetilde{G}(\vec{y},\vec{y}') = \begin{cases} \displaystyle{\sum_{l=0}^{\infty} A_l(y') g_l(y) P_l(\cos \gamma) \quad \qquad \qquad \qquad \qquad (y < y') \, ,}\\
\displaystyle{A'_0 + \sum_{l=0}^{\infty} B_{2l+1}(y') f_{2l+1}(y) P_{2l+1}(\cos \gamma) \, \, \qquad (y>y') \, .} \end{cases}
\label{dgf.sep2}\end{aligned}$$ Finally, we need to match these solutions at $y=y'$. This sets $A_0 = A'_0\,,$ and leads us to define the constant $ C_{2l+1} = \frac{A_{2l+1}(y')}{ f_{2l+1}(y')} =
\frac{B_{2l+1}(y')}{g_{2l+1}(y')}$, and we also find that $A_k$ vanishes for even $k (\neq 0)$. Then we can write $$\widetilde{G}(\vec{y},\vec{y}') = C_0 + R_{2l+1}(y,y') P_{2l+1}(\cos \gamma)\, ,$$ where $C_0 \equiv A_0$ is the constant zero-mode contribution, and $$\begin{aligned}
R_{2l+1}(y,y') = \begin{cases} \displaystyle{ \sum_{l=0}^{\infty} C_{2l+1}
g_{2l+1}(y) f_{2l+1}(y') \, \qquad (y < y') \, ,} \\
\displaystyle{\sum_{l=0}^{\infty} C_{2l+1} f_{2l+1}(y) g_{2l+1}(y') \, \qquad (y>y') \, .}\end{cases}
\label{dgf.ssol}\end{aligned}$$ Multiplying both sides of Eq. with $P_{2l'+1}(\cos \gamma)$ and integrating with respect to $\theta$ and $\phi$, we get $$- \frac{\delta(y-y')}{L} = \frac{1}{4 l + 3} \left[\frac{d}{dy} \left( y^2 \sqrt{1 - y^2}\frac{d}{dy} R_{2l+1}(y,y') \right) - \frac{(2l+1)(2l+3)}{\sqrt{1 - y^2} } R_{2l+1}(y,y') \right] \, ,
\label{dgf.aint}$$ where we have used Eq.. We next integrate over $y$ from $y' - \epsilon$ to $y' + \epsilon$, i.e. over an infinitesimal region about the point source, for which we find $$\begin{aligned}
- \frac{1}{L} & = \frac{1}{4 l + 3} C_{2l+1} {y'}^2 \sqrt{1 - {y'}^2} \,\times \notag \\
& \qquad \times \, \left[g_{2l+1}(y')
\left.\left( \frac{d}{dy} f_{2l+1}(y) \right) \right\vert_{y' + \epsilon}
- f_{2l+1}(y') \left.\left( \frac{d}{dy} g_{2l+1}(y) \right) \right\vert_{y' - \epsilon} \right] \notag \\
& = \frac{1}{4 l + 3} C_{2l+1} {y'}^2 \sqrt{1 - {y'}^2}\, W(g_{2l+1}(y'),f_{2l+1}(y'),y') \notag \\
& = - C_{2l+1} \, ,
\label{dgf.con}\end{aligned}$$ where we have made use of Eq. in going from the second to the third equality in Eq.. Using this, we can write the Green function in the de Sitter case as $$\widetilde{G} \left(\vec{y}_<,\vec{y}_>\right) = \frac{1}{ L} \sum_{l=0}^{\infty} g_{2l+1}(y_<) f_{2l+1}(y_>) P_{2l+1}(\cos \gamma) \, ,
\label{dgf.csol}$$ where $y_< = \text{min}(y,y')$ and $y_> = \text{max}(y,y')$ as before. Unlike in the Schwarzschild case, we have not been able to write this in a simpler form. We can nonetheless substitute for $y$ in Eq. and use this in Eq., by writing $y_< = \frac{r_<}{L}$ and $y_> = \frac{r_>}{L}$, to find the solution in terms of $r\,,$ $$\begin{aligned}
\widetilde{G}\left(\vec{r}_<, \vec{r}_> \right) = \frac{1}{r_>^2} & \sum_{l=0}^{\infty} \left(\frac{r_<}{r_>}\right)^{2l+1} \, _2 F_1 \left(l + \frac{1}{2}, l + \frac{3}{2}, 2l + \frac{5}{2}, \frac{3 r_<^2}{\Lambda} \right)\, \times \notag\\
& \qquad \qquad \times \,_2 F_1 \left(- l-1, -l, -2l - \frac{1}{2}, \frac{3 r_>^2}{\Lambda} \right) P_{2l+1}(\cos \gamma)\,.
\label{dgf.crsol}\end{aligned}$$
Conclusion {#Con}
==========
In this paper, we have discussed a new class of static, scalar Green functions on spherically symmetric spacetimes, those corresponding to the inverse spatial Laplacian defined exclusively on the spatial hypersurface of the spherically symmetric spacetime. Specifically, we have derived the inverse spatial Laplacian in the form of mode solutions for the Schwarzschild and pure de Sitter backgrounds. We have determined the closed form expression for Green function on Schwarzschild spacetime in terms of elementary functions, and on the pure de Sitter space in terms of hypergeometric functions.
As we have mentioned earlier, one of the places where the spatial Laplacian appears is in the constrained quantization of Maxwell fields on static spherical symmetric spacetimes with horizons. Let us now briefly discuss the role of the Green function in that problem; for more details we refer the reader to [@Fernandes:2016imn].
For the Maxwell field, an important distinction between the its treatment on spacetimes with or without horizons is that the Gauss law constraint in the former case picks up additional surface terms from the horizon, $$\Omega(\vec{r}) = - n_\mu \pi^\mu(\vec{r})\delta (r - r_H) +
{D}_\mu \pi^\mu(\vec{r}) \,.
\label{con.con}$$ Here $\pi^\mu$ are the momenta conjugate to the hypersurface projected field $a_\mu\,, r_H$ is the horizon radius, and $n^\mu$ is the outward pointing normal on the horizon.
This is still a first class constraint, and one way of handling it is to fix a gauge and find the corresponding Dirac brackets. An interesting choice of gauge fixing function is one that includes a surface term, $$\label{gf}
\Omega_{gf} = D_\mu a^\mu - n_\mu a^\mu \delta(r - r_H) \,.$$ For this choice, the relevant Dirac bracket becomes $$\left[a_\mu(\vec{r}),\pi^\nu(\vec{y})\right]_{D} = \delta(\vec{r},\vec{y})
\delta_\mu^\nu - {D}_\mu^r {D}_y^\nu
\widetilde{G}\left(\vec{r},\vec{y} \right)\,,
\label{con.isl}$$ where $\delta(\vec{r}, \vec{y})$ is defined in Eq..
A different gauge choice would produce a different set of brackets, for example the gauge choice $\Omega_{gf} = D_\mu (\lambda a^\mu)$ produces Dirac brackets in which $\widetilde G$ is replaced by $G$ of Eq. in the second term on the right hand side, and that term also picks up a factor of $\lambda\,.$
Consider the Dirac bracket in the Schwarzschild background. When one of the arguments is at the horizon, e.g., in the limit $y \to r_H$, we find for the $r - r$ component that $$\begin{aligned}
\left[a_r(\vec{r}),\pi^r(\vec{y})\right]_{D}\Big\vert_{y \to r_H} &= \delta(r,r_H)+ \kappa_H \frac{2r- m(1+ \cos\gamma)}{2 \left(r^2 - mr(1+ \cos\gamma)\right)^{3/2}} \,. \label{db.isl}\end{aligned}$$ For the other gauge choice mentioned above, only the $\delta(r,r_H)$ remains on the right hand side of the above equation in the limit $y \to r_H$. The Dirac brackets comprise one aspect that enters into the quantization of theories. Since the Gauss’ law constraint must be respected by physical states of the theory, the surface term contained in the constraint on these backgrounds will be relevant to states at the horizon. A complete treatment of the quantization of the Maxwell field on static, spherically symmetric backgrounds with horizons lies outside the scope of the present work. In light of the preceding discussion, we can nevertheless expect that the inverse spatial Laplacian will affect the quantization of gauge fields near the horizon.
Since both the de Sitter and Schwarzschild cases admit a mode expansion, where the functions depending on $r$ are ultimately associated Legendre polynomials, it seems plausible to presume that a similar result would hold for the Schwarzschild-de Sitter background. Unfortunately, we have been unable to find a simple transformation for this case since the cubic dependence on $r$ in the lapse function $\lambda$ poses a significant obstacle to the procedure. From the nature of the equation to solve for the Schwarzschild-de Sitter background, it appears that the solution for the corresponding Green function will require a different approach from what was considered here.
Derivation of the general solution of the homogeneous equations {#App}
================================================================
We seek to solve Eq. and Eq., which take the general form $$(1-y^2)\frac{d^2}{dy^2}R_l(y,y') + f(y) \frac{d}{dy} R_l(y,y') + g(y)R_l(y,y') - l(l+1)R_l(y,y') = 0\,.
\label{app.eq}$$ We will solve this equation, for the cases of Eq. and Eq. by making use of the ansatz $R_l(y,y') = B_l(y') P_{\nu}^{\mu}(y)A(y)$. We first recall that the Legendre polynomial is a solution of the following differential equation $$(1 - y^2) \frac{d^2}{dy^2}P_{\nu}^{\mu}(y) - 2y \frac{d}{dy} P_{\nu}^{\mu}(y) + \left[ \nu(\nu +1) - \frac{{\mu}^2}{1 - y^2} \right]P_{\nu}^{\mu}(y) = 0\,.
\label{app.Leq}$$ Expanding Eq., we find $$(1-y^2)\frac{d^2}{dy^2}R_l(y,y') - (2y- 1)\frac{d}{dy}R_l(y,y')+ l(l+1) R(y,y') = 0\,.$$ Substituting the ansatz and making use of Eq., we get $$\begin{aligned}
A(y) &\left[-\left( \nu(\nu +1) - \frac{{\mu}^2}{1 - y^2} \right) P_{\nu}^{\mu}(y)
+ \frac{d}{dy} P_{\nu}^{\mu}(y)\right] \notag \\
& \qquad \qquad + P_{\nu}^{\mu}(y)\left[(1-y^2)\frac{d^2}{dy^2}A(y)
- (2y - 1)\frac{d}{dy}A(y) \right] \notag\\
& \qquad \qquad \qquad + 2(1-y^2)\frac{d}{dy}P_{\nu}^{\mu}(y)\frac{d}{dy}A(y)
+ l(l+1)P_{\nu}^{\mu}(y)A(y) = 0\,.
\label{app.exp}\end{aligned}$$ Collecting terms, we have $$\begin{aligned}
\frac{d}{dy}& P_{\nu}^{\mu}(y) \left[2 (1-y^2) \frac{d}{dy}A(y) + A(y) \right] \notag\\
& \quad + P_{\nu}^{\mu}(y) \left[ (1-y^2)\frac{d^2}{dy^2}A(y) - (2y - 1)\frac{d}{dy}A(y)
\right. \notag \\ & \qquad \qquad \left.
-\left( \nu(\nu +1) - \frac{{\mu}^2}{1 - y^2} - l(l+1) \right) A(y) \right] = 0\,.
\label{app.reo}\end{aligned}$$ We can now explore the simplest possibility which makes Eq. true, namely, that the coefficients of $\frac{d}{dy}P_{\nu}^{\mu}(y)$ and $P_{\nu}^{\mu}(y)$ individually vanish. The coefficient is of $\frac{d}{dy}P_{\nu}^{\mu}(y)$ can be trivially solved to give the following solution for $A(y)$ $$A(y) = \left(\frac{y-1}{y+1}\right)^{\frac{1}{4}} \,.$$ Substituting this solution back in Eq. gives us the following expression $$\left( \frac{1}{4} - {\mu}^2 \right)(y-1)^{-\frac{3}{4}}(y+1)^{-\frac{5}{4}} -\left( \nu (\nu +1) - l(l+1) \right) (y-1)^{\frac{1}{4}}(y+1)^{-\frac{1}{4}} = 0 \, .
\label{app.fin}$$ Eq. holds, provided $\mu = \frac{1}{2}$ and $\nu = l$.
One solution of Eq. is thus $\left(\frac{y-1}{y+1}\right)^{\frac{1}{4}} P_l^{\frac{1}{2}}(y)$. Since our procedure made use of the Legendre polynomials, we would get another solution by simply using $R_l(y,y') = B_l(y')Q_{\nu}^{\mu}(y) A(y)$, with the same solution for $A(y)$. The general solution is thus found to be $$R_l(y,y') = A_l(y') \left(\frac{y-1}{y+1}\right)^{\frac{1}{4}}
P_l^{\frac{1}{2}}\left(y \right) + B_l(y') \left(\frac{y-1}{y+1}\right)^{\frac{1}{4}}
\left( i Q_l^{\frac{1}{2}}\left(y\right) \right)\,.
\label{app.gensol}$$ Eq. is written as it is since $i \, Q_l^{\frac{1}{2}}\left(y\right)$ is a real solution.
This procedure can similarly be used in Eq., which can be written as $$(1-t^2)\frac{d^2}{dt^2} R_l(t,t') - 3 t\frac{d}{dt} R_l(t,t') - \frac{l(l+1)}{(1 - t^2)} R_l(t,t') = 0 \, .$$ Substitution of the ansatz $R_l(t,t') = B_l(t') P_{\nu}^{\mu}(t) A(t)$ now leads to the following equation $$\begin{aligned}
\frac{d}{dt}P_{\nu}^{\mu}(t) & \left[2 \frac{d}{dt}A(t) (1 - t^2) - A(t) t \right] \notag \\
& \qquad + P_{\nu}^{\mu}(t) \left[(1 - t^2) \frac{d^2}{dt^2}A(t) - 3 t \frac{d}{dt}A(t)
\right. \notag \\
& \qquad \qquad - \left. \left( \nu (\nu +1) - \frac{{\mu}^2}{1-t^2} +
\frac{l( l + 1)}{1-t^2} \right) A(t) \right] = 0 \, .
\label{app.ts}\end{aligned}$$ As before, we assume the possibility that the coefficients of the $P_{\nu}^{\mu}(t)$ and $\frac{d}{dt}P_{\nu}^{\mu}(t)$ separately vanish. The coefficient of the latter term vanishing leads to the following simple result for A(t) $$A(t) = (1 - t^2)^{- \frac{1}{4}} \, .$$ Substituting this equation back into Eq. leads to the following result $$- \left[\frac{3}{4} - \nu (\nu +1) \right]t^2 - \left[ \nu (\nu +1) - \frac{1}{2} + l(l+1) - {\mu}^2 \right] = 0 \, ,$$ which is satisfied for the choice of $\nu =\frac{1}{2}$ and $\mu = l + \frac{1}{2}$. Since in this case $\nu \pm \mu$ is an integer but $\mu$ is not, the other independent solution is not $Q^{\mu}_{\nu}$, but rather $P^{-\mu}_{\nu}$. Thus the general solution can be written as $$R_l(t,t') = A_l(t') (1-t^2)^{-\frac{1}{4}} P_{\frac{1}{2}}^{l + \frac{1}{2}}(t) + B_l(t') (1-t^2)^{-\frac{1}{4}} P_{\frac{1}{2}}^{-l - \frac{1}{2}}(t) \, ,$$ which is Eq..
Limits of the de Sitter solutions as $y \to 1$ {#app2}
==============================================
Let us first note that the hypergeometric functions given in Eq. are of the form $\, _2 F_1 \left( a, a+1 ; 2 a + \frac{3}{2} ; y^2 \right)$, where $a = \frac{l}{2}$ and $a = \frac{ - l - 1}{2}$ correspond to the two hypergeometric functions contained in $g_l(y)$ and $f_l(y)$ respectively. There exists a known formula for evaluating the hypergeometric functions at the point $y^2 = 1$. This formula is given by (cf. Eq. (15.4.20) of [@NIST]) $$\, _2 F_1\left(a, b, c, 1 \right) = \frac{\Gamma\left(c\right)\Gamma\left(c - a - b\right)}
{\Gamma\left(c - a\right) \Gamma\left(c - b\right)} \qquad
\Re\left(a + b - c\right) < 0\,; \;c \neq 0 , -1, -2, \dots \,
\label{app2.lim}$$ This formula applies to the hypergeometric functions included in $f_l$ and $g_l$, but not to their derivatives. Let us consider the functions separately to find their derivatives at $y^2 = 1\,.$
$f_l(y)$ solutions and hypergeometric functions
-----------------------------------------------
For the $f_l$ solutions, we need to find the expressions explicitly in order to determine the nature of the derivatives at the point $y = 1$. For the values of $l=0, 1, 2$ and $3$, the corresponding hypergeometric functions are, respectively, $$\begin{aligned}
_2 F_1 \left(-\frac{1}{2}, \frac{1}{2}; \frac{1}{2}; y^2\right) \qquad\qquad & (l=0) \,; \notag\\
%\qquad \qquad
\, _2 F_1 \left(-1, 0; -\frac{1}{2}; y^2\right) \qquad\qquad & (l=1) \,;\notag\\
_2 F_1 \left(- \frac{3}{2}, -\frac{1}{2}; - \frac{3}{2}; y^2\right) \qquad\qquad & (l=2) \,; \notag\\
\qquad \qquad \, _2 F_1 \left(-2, -1; -\frac{5}{2}; y^2\right) \qquad\qquad & (l=3)\,.
\label{app2.4sols}\end{aligned}$$ Here we see that $\, _2 F_1 \left(a, b; c; y^2\right)$ and $\, _2 F_1 \left(a-1, b-1; c-2; y^2\right)$ represent two successive even (odd) solutions when $\left(a, b; c\right) = \,
\left(\frac{-l-1}{2}, \frac{-l+1}{2}; - l + \frac{1}{2}\right)$.
Two hypergeometric functions which are contiguous are related to one another through certain differentiation formulas (cf. Eq.s (15.5.4) and (15.5.9) of [@NIST]). Let us look at the ones relevant to the $f_l$ functions. These are $$\begin{aligned}
%\, _2 F_1 \left(a+n, b+n; c+n; z \right) &= \frac{ (c)_n}{(a)_n (b)_n} \frac{d^n}{dz^n} \,_2 F_1 \left(a, b; c; z \right) \, ,\notag \\
%\, _2 F_1 \left(a, b; c+n; z \right) &= \frac{ (c)_n}{(c - a)_n (c - b)_n} \left(1 - z \right)^{c + n - a - b} \frac{d^n}{dz^n} \left[ \left(1 - z \right)^{a + b - c}\,_2 F_1 \left(a, b; c; z \right) \right] \, . \label{app2.gsol}\\
%~&~ \notag \\
\, _2 F_1 \left(a-n, b-n; c-n; z \right) &= \frac{1}{(c-n)_n}(1-z)^{c+n-a-b} z^{1+n-c}
\frac{d^n}{dz^n}\,\times \notag \\
&\qquad \qquad \times\, \left[ \left(1 - z \right)^{a + b - c} z^{c-1}\,_2 F_1 \left(a, b; c; z \right) \right] \, ,\notag\\
\, _2 F_1 \left(a, b; c-n; z \right) &= \frac{1}{(c-n)_n} z^{1+n-c} \frac{d^n}{dz^n}
\left[ z^{c-1}\,_2 F_1 \left(a, b; c; z \right) \right] \, ,
\label{app2.fsol}\end{aligned}$$ where $(k)_n = \frac{\Gamma \left(k + n\right)}{\Gamma \left( k \right)}$ is Pochhammer’s symbol. Using the two relations in Eq., we can write $$\begin{aligned}
\, _2 F_1 \left(a-n, \right. &\left. b-n; c- 2 n; z \right) \notag\\
& = \frac{z^{1 + 2n - c}}{(c - n)_n (c - 2 n)_n} \frac{d^n}{dz^n}
\left[(1 - z)^{c+n-a-b} \right. \, \times \notag \\
& \qquad \qquad \qquad\left. \times \,
\frac{d^n}{dz^n} \left[z ^{c-1} (1 - z)^{a+b-c} \, _2 F_1 \left(a, b; c; z \right) \right] \right]
\label{app2.f1g}\end{aligned}$$ Using $\, _2 F_1 \left(a, b; c; z \right) = \, _2 F_1 \left(a, a+1; 2a + \frac{3}{2}; z \right)$ in Eq. provides the relevant recursive relation for the $f_l$ hypergeometric functions. To simplify the notation in what follows, let us define $$\, _2 F_1 \left(\frac{-l - 1}{2}, \frac{-l + 1}{2}; - l + \frac{1}{2}; z\right) = F_l(z) \, .
\label{app2.rec}$$ With this definition, the solutions we seek are given by $f_l(y) =
\frac{F_l(y^2)}{y^{l+1}}$. Combining Eq. with Eq., we can write $$\begin{aligned}
F_{l + 2n}(z) = & \frac{z^{2n + \frac{1}{2} + l}}
{\left(l + \left(n -\frac{1}{2}\right)\right)_n \left(l + \left(2 n -\frac{1}{2}\right)\right)_n} \times \notag \\ & \qquad \quad \times
\frac{d^n}{dz^n} \left[(1 - z)^{n+ \frac{1}{2}}
\frac{d^n}{dz^n} \left[z ^{- l - \frac{1}{2}} (1 - z)^{-\frac{1}{2}} F_l(z) \right] \right]\,.
\label{app2.f}\end{aligned}$$ For the recursion relation, we need only consider Eq. with $n=1\,,$ $$F_{l + 2}(z) = \frac{z^{\frac{5}{2} + l}}{\left(l + \frac{1}{2}\right)
\left(l + \frac{3}{2}\right)} \frac{d}{dz} \left[(1 - z)^{\frac{3}{2}}
\frac{d}{dz} \left[z ^{- l - \frac{1}{2}} (1 - z)^{-\frac{1}{2}} F_l(z) \right] \right] \, ,
\label{app2.f1}$$ which upon evaluating the derivatives can be written as $$\begin{aligned}
F_{l + 2}(z) =& \left(1 - \frac{(2l+1)(2l+2) z}{3 + 8l + 4l^2} \right)F_l(z) \notag \\
& \qquad\qquad - \frac{(8l + 4) z - (8l+2)z^2}{3 + 8l + 4l^2} F'_l(z)
- \frac{4(z^3 - z^2)}{3 + 8l + 4l^2}F''_l(z) \, ,
\label{app2.fodd}\end{aligned}$$ where primes denote differentiation with respect to $z$. As we will see below, for even $l$ we can extract a factor of $\sqrt{1-z}$ to write the functions $F_l(z)$ in the form $F_l(z) = \sqrt{1 - z} D_l(z)$. Substituting this in Eq., we find a recursion relation for the functions $D_l(z)\,,$ $$\begin{aligned}
D_{l + 2}(z) =& \left[ \left(1 - \frac{2l (2l+1) z}{3 + 8l + 4l^2} \right)D_l(z) \right.
\notag \\
& \qquad\qquad \left.- \frac{(8l + 4) z - (8l-2)z^2}{3 + 8l + 4l^2} D'_l(z)
- \frac{4(z^3 - z^2)}{3 + 8l + 4l^2}D''_l(z)\right]\,.
\label{app2.fev}\end{aligned}$$
We will now need the lowest order solutions to proceed further. The lowest order solution for even $l$ is $F_0(z)\,,$ which corresponds to $ _2F_1 \left(-\frac{1}{2}, \frac{1}{2}; \frac{1}{2}; z \right)$, as shown in Eq.. The hypergeometric function $\, _2 F_1
\left(\frac{\alpha}{2}, -\frac{\alpha}{2}; \frac{1}{2}; z \right)$ has the following known representation $$\, _2 F_1 \left(\frac{\alpha}{2}, -\frac{\alpha}{2}; \frac{1}{2}; z \right)
= \cos \left( \alpha \sin^{-1} \left(\sqrt{z} \right) \right) \, ,$$ and the case where $\alpha = -1$ is the one we require. The lowest order solution for odd $l$ is $f_1(z)$, which corresponds to $F_1(z) = \ _2 F_1 \left(-1, 0;
-\frac{1}{2}; z \right)$. From the definition of the hypergeometric function $\ _2 F_1 \left(a, b; c; z \right)$ $$\ _2 F_1 \left(a, b; c; z \right) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!} \, ,$$ we know that $\ _2 F_1 \left(0, b; c; z \right) = \ _2 F_1 \left(a, 0; c; z \right) = \ _2 F_1 \left(a, b; c; 0 \right) = 1$. This tells us that the lowest order solutions are simply $$\begin{aligned}
F_0(z) = \ _2 F_1 \left(-\frac{1}{2}, \frac{1}{2}; \frac{1}{2}; z \right) &= \sqrt{1-z} \, , \notag \\
F_1(z) = \ _2 F_1 \left(-1, 0; -\frac{1}{2}; z \right) &= 1\,.\end{aligned}$$ We will use these to derive the expressions for the $f_l(y)$ functions given in Eq.. We begin with the even $l$ solutions. It can be seen that all even $l$ solutions are of the form $F_l(z) = \sqrt{1-z} D_l(z)$. This follows directly from the fact that $F_0(z) = \sqrt{1-z} D_0(z)$, where $D_0(z) = 1\,,$ and Eq.. The operator in Eq. takes a polynomial and produces another polynomial of one order higher. The only exception is $D_0(z)=1\,$ which, when inserted into Eq., produces $D_2(z) = 1\,.$ We can also calculate directly that $F_2(z) = \sqrt{1-z} = \sqrt{1-z}D_2(z)$. It follows from Eq. that $D_{2k}(z)$ is a polynomial of order $(k-1)$ in $z$.
For $F_l(z)$ corresponding to odd $l$, we can use the recursion relation of Eq. directly. For example, substitution of $F_1(z) = 1$ in Eq. leads to $F_3(z) = 1 - \frac{4}{5} z$, substituting this result back in Eq. results in $F_5(z) = 1 - \frac{4}{3} z +
\frac{8}{21} z^2$, etc. Thus $F_{2k+1}(z)$ is a polynomial of order $k$ in $z$. We can now make a change of variable to $z=y^2$ in all the $F_l(z)$ solutions, and consider $f_l(y) = \frac{F_l(y^2)}{y^{l+1}}$ to find the $f_l(y)$ solutions shown in Eq..
Limits of the $g_l(y)$ solutions {#appgl}
--------------------------------
The limit of the $g_l$ hypergeometric functions and their derivatives are most easily determined by making use of the following integral representation for hypergeometric functions (cf. Eq. (15.6.1) of [@NIST]) $$\begin{aligned}
\, _2 F_1\left(a, b, c, x \right) = \frac{\Gamma\left(c \right)}
{\Gamma\left(b \right) \Gamma\left(c - b \right)} &\int_{0}^{1} t^{b - 1}
(1 - t)^{c - b - 1} (1 - x t)^{-a} dt \notag\\ & \qquad \qquad \left( \Re (c) > \Re (b) > 0 \right) \, ,
\label{app2.int}\end{aligned}$$ As we have seen, the $f_l$ hypergeometric functions have $c \leq b$ for all choices of $l$, with the equality holding true for the $l=0$ case. Hence, we could not use Eq. for those functions, and had to make use of the treatment described earlier. For the $g_l$ hypergeometric functions, $a= \frac{l}{2}$, which guarantees that $\, _2 F_1\left(a, a+1, 2a + \frac{3}{2}, y^2 \right)$ always has $c>b>0$. This also holds true for the derivative of this function on account of Eq.. We can thus consider Eq. in terms of the $g_l$ hypergeometric functions we are dealing with, in which case we have $$\, _2 F_1\left(\frac{l}{2},\frac{l}{2}+1, l + \frac{3}{2}, y^2 \right) = \frac{\Gamma\left( l + \frac{3}{2} \right)}{\Gamma\left(\frac{l}{2} + 1\right) \Gamma\left(\frac{l+1}{2} \right)} \int_{0}^{1} \left(\frac{t}{1 - t y^2}\right)^{\frac{l}{2}} (1 - t)^{\frac{l - 1}{2}} dt \, ,
\label{app2.int2}$$ while the derivative of this function takes the form $$\partial_y\left(\, _2 F_1\left(\frac{l}{2}, \frac{l}{2}+1, l + \frac{3}{2}, y^2 \right)\right) = \frac{\Gamma\left(l + \frac{3}{2} \right) l y }{\Gamma\left(\frac{l}{2} + 1\right) \Gamma\left( \frac{l + 1}{2} \right)} \int_{0}^{1} \left(\frac{t}{1 -t y^2}\right)^{\frac{l}{2}+1} (1 - t)^{\frac{l - 1}{2}} dt \, ,
\label{app2.intd}$$
The explicit representation of the $g_l$ hypergeometric functions and their derivatives, in terms of elementary functions, can now be derived using these equations for specific choices of $l$. Since we are interested in the nature of the limit of these functions as $y \to 1$ for any choice of $l$, we can simply take this limit in the above expressions, and then evaluate the integrals. This amounts to the evaluation of standard integrals. We find that Eq. gives us the following finite result $$\, _2 F_1\left(\frac{l}{2},\frac{l}{2}+1, l + \frac{3}{2}, 1\right) = \frac{ 2 \sqrt{\pi}}{(l+1)} \frac{\Gamma \left(l + \frac{3}{2} \right)}{\left(\Gamma \left(\frac{l + 1}{2} \right)\right)^2} \, ,$$ while Eq. diverges for all choices of $l$.
Similarly, the $y \to 0$ limits can also be determined very simply by using substituting $ y=0$ in Eq. and (\[app2.intd\]. One can easily find that $\, _2 F_1\left(\frac{l}{2},\frac{l}{2}+1, l + \frac{3}{2}, 0\right) = 1$, by working out the integral. The derivative of this function at $y=0$ vanishes on account of an overall factor of $y$, and the fact that the integral is finite.
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abstract: |
Consider a time series of measurements of the state of an evolving system, $x(t)$, where $x$ has two or more components. This paper shows how to perform nonlinear blind source separation; i.e., how to determine if these signals are equal to linear or nonlinear mixtures of the state variables of two or more statistically independent subsystems. First, the local distributions of measurement velocities are processed in order to derive vectors at each point in $x \mbox{-space}$. If the data are separable, each of these vectors must be directed along a subspace of $x \mbox{-space}$ that is traversed by varying the state variable of one subsystem, while all other subsystems are kept constant. Because of this property, these vectors can be used to construct a small set of mappings, which must contain the “unmixing” function, if it exists. Therefore, nonlinear blind source separation can be performed by examining the separability of the data after it has been transformed by each of these mappings. The method is analytic, constructive, and model-independent. It is illustrated by blindly recovering the separate utterances of two speakers from nonlinear combinations of their audio waveforms.
source separation, nonlinear signal processing, invariants, sensor, analytic, model-independent
author:
- 'David N. Levin'
title: |
Model-Independent Analytic Nonlinear\
Blind Source Separation
---
Introduction
============
The signals from a process of interest are often contaminated by signals from extraneous processes, which are thought to be statistically independent of the process of interest but are otherwise unknown. This raises the question: can one use the observed signals to determine if two or more independent processes are present, and, if so, can one derive a representation of the evolution of each of them? In other words, if a system is effectively evolving in a closed box, can one process the signals emanating from the box in order to learn the number and nature of the subsystems within it? There is a variety of methods for solving this blind source separation (BSS) problem for the special case in which the signals are linearly related to the underlying independent subsystem states ([@Ans], [@Comon; @Jutten]). However, some observed signals (e.g., from biological or economic systems) may be nonlinear functions of the underlying system states. Computational methods of separating such nonlinear mixtures are limited ([@Jutten], [@Almeida]), even though humans seem to do it in an effortless manner.
Consider an evolving physical system that is being observed by making time-dependent measurements ($x_{k}(t) \mbox{ for } k = 1, \ldots ,N$ where $N \geq 2$), which are coordinates on the system’s state space. In Conclusion, we describe how to choose measurements that comprise such coordinates. The objective of blind source separation is to determine if the measurement time series is separable; i.e., to determine if it can be transformed into another coordinate system, $s$ (called the “source” or “separable” coordinate system), in which the transformed time series describes the evolution of statistically independent subsystems. Specifically, we want to know if there is an invertible, possibly nonlinear, $N \mbox{-component}$ “unmixing” function, $f$, that transforms the measurement time series into a source time series: $$\label{mixture}
s(t) = f[x(t)] ,$$ where the $N$ components of $s(t)$ can be partitioned into statistically independent, possibly multidimensional groups.
This paper utilizes a criterion for “statistical independence” [@Levin-bss-JAP] that differs from the conventional one. Specifically, let $\rho_S(s,\dot{s})$ be the probability density function (PDF) in $(s,\dot{s}) \mbox{-space}$, where $\dot{s}~=~ds/dt$. Namely, let $\rho_S(s,\dot{s}) ds d\dot{s}$ be the fraction of total time that the location and velocity of $s(t)$ are within the volume element $ds d\dot{s}$ at location $(s,\dot{s})$. In this paper, the data are defined to be separable if and only if there is an unmixing function that transforms the measurements so that $\rho_S(s,\dot{s})$ is the product of the density functions of individual components (or groups of components) $$\label{phase space factorization}
\rho_S(s,\dot{s}) = \prod_{a=1,2, \ldots}{\rho_{Sa}(s_{(a)},\dot{s}_{(a)}}) .$$ where $s_{(a)}$ is a subsystem state variable, comprised of one or more of the components of $s$. This criterion for separability is consistent with our intuition that the statistical distribution of the state and velocity of any independent subsystem should not depend on the particular state and velocity of any other independent subsystem.
This criterion for statistical independence should be compared to the conventional criterion, which is formulated in $s \mbox{-space}$ (i.e., state space) instead of $(s,\dot{s}) \mbox{-space}$ (the space of states and state velocities). In particular, let $\rho_S(s)$ be the PDF, defined so that $\rho_S(s) ds$ is the fraction of total time that the trajectory $s(t)$ is located within the volume element $ds$ at location $s$. In some formulations of the BSS problem, the system is said to be separable if and only if there is an unmixing function that transforms the measurements so that $\rho_S(s)$ is the product of the density functions of individual components (or groups of components) $$\label{state space factorization}
\rho_S(s) = \prod_{a=1,2, \ldots}{\rho_{Sa}(s_{(a)})} ,$$ In *every* formulation of BSS, multiple solutions can be created by applying “subsystem-wise” transformations, which transform each subsystem’s components among themselves. These solutions are the same as one another, except for differing choices of the coordinate systems used to describe each subsystem. However, the criterion in (\[state space factorization\]) is so weak that it suffers from a much worse non-uniqueness problem: namely, solutions can almost always be created by mixing the state variables of *different* subsystems of other solutions (see [@Hosseini], [@Jutten], [@Hyvarinen-uniqueness]).
There are at least two reasons why (\[phase space factorization\]) is the preferred way of defining “statistical independence”:
1. If a physical system is comprised of two independent subsystems, we normally expect that there is a unique way of identifying the subsystems. As mentioned above, (\[state space factorization\]) is too weak to meet this expectation. On the other hand, (\[phase space factorization\]) is a much stronger constraint than (\[state space factorization\]). Specifically, (\[state space factorization\]) can be recovered by integrating both sides of (\[phase space factorization\]) with respect to velocity. This shows that the solutions of (\[phase space factorization\]) are a subset of the solutions of (\[state space factorization\]). Therefore, it is certainly possible that (\[phase space factorization\]) reformulates the BSS problem so that it has a unique solution (up to subsystem-wise transformations), although this is not proved in this paper.
2. For all systems that obey the laws of classical physics and are in thermal equilibrium at temperature $T$, the PDF in $(s,\dot{s}) \mbox{-space}$ is proportional to the Maxwell-Boltzmann distribution [@Reif] $$\label{M-B distribution}
e^{-E(s,\dot{s})/(kT)}$$ where $E$ is the system’s energy and $k$ is the Boltzmann constant. If the system consists of two non-interacting subsystems, the system’s energy is the sum of the subsystem energies $$\label{summed energies}
E = E_{1}(s_{(1)},\dot{s}_{(1)}) + E_{2}(s_{(2)},\dot{s}_{(2)})$$ where $s_{(1)}$ and $s_{(2)}$ are subsystem state variables comprised of one or more components of $s$. This demonstrates that, for all classical systems composed of non-interacting subsystems, the system’s PDF in $(s,\dot{s}) \mbox{-space}$ is the product of the subsystem PDFs in $(s,\dot{s}) \mbox{-space}$, as stated in (\[phase space factorization\]).
There are several other ways in which the proposed method of nonlinear BSS differs from methods in the literature:
1. As stated above, in this paper the BSS problem is reformulated in the joint space of states and state velocities. Although there is some earlier work in which BSS is performed with the aid of velocity information ([@Ehsandoust], [@Lagrange]), these papers utilize the *global* distribution of measurement velocities (i.e., the distribution of velocities at all points in state space). In contrast, the method proposed here exploits additional information that is present in the *local* distributions of measurement velocities (i.e., the velocity distributions in each neighborhood of state space).
2. Many investigators have attempted to simplify the BSS problem by assuming prior knowledge of the nature of the mixing function; i.e., they have modelled the mixing function. For example, the mixing function has been assumed to have parametric forms that describe post-nonlinear mixtures [@Taleb], linear-quadratic mixtures [@Merrikh], and other combinations ([@Duarte], [@Yang], [@Tan]). In contrast, the present paper proposes a model-independent method that can be used in the presence of any invertible diffeomorphic mixing function.
3. In many other approaches, nonlinear BSS is reduced to the optimization problem of finding the unmixing function that maximizes the independence of the source signals corresponding to the observed mixtures. This usually requires the use of iterative algorithms with attendant issues of convergence and computational cost (e.g., [@Comon; @Jutten], [@Duarte]). In contrast, the method proposed in this paper is analytic and constructive. Specifically, the observed data are used to construct a small collection of mappings, $\{u(x)\}$, that must contain an unmixing function, if one exists. To perform BSS, it then suffices to determine if any of these functions transforms the measured time series, $x(t)$, into a time series, $u[x(t)]$, having a factorizable PDF. The data are separable if and only if this is the case.
There are two earlier papers ([@Levin-bss-JAP], [@Levin-IEEE-Trans]) that utilize the criterion in (\[phase space factorization\]) in order to perform nonlinear BSS. However, both of these approaches are quite different from the one proposed here. The current paper shows how the measurement time series endows state space with local vectors that contain crucial information about the separability of the data. Specifically, if the data are separable, each of these vectors must be directed along a subspace of $x \mbox{-space}$ that is traversed by varying the state variable of one subsystem, while all other subsystem variables are kept constant. Because of this property, these vectors can be used to determine if the data are separable and, if they are, to determine the transformation to a separable coordinate system. In contrast, the presence of these vectors played no role whatsoever in the methods discussed in [@Levin-IEEE-Trans] and [@Levin-bss-JAP]. Instead,
1. In [@Levin-IEEE-Trans], BSS was performed by deriving a large number of local scalars that must lie in low-dimensional subspaces, if the data were separable. The vectorial structure on state space was not utilized or even recognized.
2. Likewise, local vectors also played no role in [@Levin-bss-JAP]. Instead, the local second-order velocity correlation matrix was taken to define a Riemannian metric on the space of measurements ($x$). Then, nonlinear BSS was performed by using differential geometry to look for a transformation to another coordinate system ($s$), in which this metric was block-diagonal everywhere.
In short, although [@Levin-bss-JAP], [@Levin-IEEE-Trans], and the present paper all utilize the same criterion for statistical independence (i.e., (\[phase space factorization\])), these three approaches differ greatly in how they determine whether this criterion is satisfied by a given signal.
The next section gives a detailed description of the proposed method of nonlinear blind source separation, which is schematically illustrated in Figure \[figure1\]. Section \[experiments\] illustrates the method by using it to blindly recover the utterances of two speakers from nonlinear mixtures of their audio waveforms. The last section discusses the implications of this approach. Note that brief versions of this work were presented in [@Levin; @arXiv], [@Levin; @ITISE], and [@Levin; @LCA-ICA].
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Method
======
For didactic purposes, the next subsection describes a five-step procedure for performing nonlinear BSS of systems with two degrees of freedom. Then, Subsection \[N dimensional systems\] describes how to generalize this procedure so that it can be applied to systems and subsystems having any number of degrees of freedom.
Systems having two degrees of freedom {#two-dimensional systems}
-------------------------------------
*1. The local second- and fourth-order correlations of the measurement velocity ($\dot{x}$) are computed in small neighborhoods of the measurement space. These correlations are used to compute two local vectors ($V_{(i)}(x) \mbox{ for } i = 1,2$)*\
The first step is to construct second-order and fourth-order local correlations of the data’s velocity $$\label{C2 definition}
C_{kl}(x) = \, \langle (\dot{x}_k-\bar{\dot{x}}_k) (\dot{x}_l-\bar{\dot{x}}_l) \rangle_{x}$$ $$\label{C4 definition}
\begin{split}
C_{klmn}(x) = \, \langle (\dot{x}_k-\bar{\dot{x}}_k) & (\dot{x}_l-\bar{\dot{x}}_l) \\
& (\dot{x}_m-\bar{\dot{x}}_m) (\dot{x}_n-\bar{\dot{x}}_n) \rangle_{x}
\end{split}$$ where $\bar{\dot{x}} = \langle \dot{x} \rangle_x$, where the bracket denotes the time average over the trajectory’s segments in a small neighborhood of $x$, and where all subscripts are integers equal to $1$ or $2$. Because $\dot{x}$ is a contravariant vector, $C_{kl}(x)$ and $C_{klmn}(x)$ are local contravariant tensors of rank $2$ and $4$, respectively. The definition of the PDF implies that $C_{kl}(x)$ and $C_{klmn}(x)$ are two of its moments; e.g., $$\label{PDF moment}
C_{kl \ldots}(x) = \frac {\int \rho(x,\dot{x}) (\dot{x}_k-\bar{\dot{x}}_k) (\dot{x}_l-\bar{\dot{x}}_l) \ldots d\dot{x}} {\int \rho(x,\dot{x}) d\dot{x}} ,$$ where $\rho(x,\dot{x})$ is the PDF in the $x$ coordinate system, where “$\ldots$” denotes possible additional subscripts on the left side and corresponding additional factors of $\dot{x}-\bar{\dot{x}}$ on the right side, and where all subscripts are integers equal to $1$ or $2$. Although (\[PDF moment\]) is useful in a formal sense, in practical applications all required correlation functions can be computed directly from local time averages of the data (e.g., (\[C2 definition\])-(\[C4 definition\])), without explicitly computing the data’s PDF. Also, note that velocity “correlations” with a single subscript vanish identically $$\label{C_k=0}
C_k(x)=0 .$$
Next, let $M(x)$ be any local $2 \times 2$ matrix, and use it to define $M \mbox{-transformed}$ velocity correlations, $I_{kl}$ and $I_{klmn}$ $$\label{I2 definition}
I_{kl}(x) = \sum_{1 \leq k', \, l' \leq 2} M_{kk'}(x) M_{ll'}(x) C_{k'l'}(x) ,$$ $$\label{I4 definition}
\begin{split}
I_{klmn}(x) = \sum_{1 \leq k', \, l', \, m', \, n' \leq 2} & M_{kk'}(x) M_{ll'}(x) \\
& M_{mm'}(x) M_{nn'}(x) C_{k'l'm'n'}(x) .
\end{split}$$ Because $C_{kl}(x)$ is generically positive definite, it is possible to find a particular form of $M(x)$ that satisfies $$\label{M definition 1}
I_{kl}(x) = \delta_{kl}$$ $$\label{M definition 2}
\sum_{1 \leq m \leq 2} I_{klmm}(x) = D_{kl}(x) ,$$ where $D(x)$ is a diagonal $2 \times 2$ matrix. Such an $M(x)$ can be constructed from the product of three matrices: 1) a rotation that diagonalizes $C_{kl}(x)$, 2) a diagonal rescaling matrix that transforms this diagonalized correlation into the identity matrix, 3) another rotation that diagonalizes $$\sum_{1 \leq m \leq 2} \tilde{C}_{klmm}(x) ,$$ where $\tilde{C}_{klmn}(x)$ is the fourth-order velocity correlation ($C_{klmn}(x)$) after it has been transformed by the first rotation and the rescaling matrix. As long as $D$ is not degenerate, $M(x)$ is unique, up to arbitrary *local* permutations and/or reflections. In almost all applications of interest, the velocity correlations will be continuous functions of $x$. Therefore, in any neighborhood of state space, there will always be a continuous solution for $M(x)$, and this solution is unique, up to arbitrary *global* permutations and/or reflections.
In any other coordinate system $x'$, the most general solution for $M'$ is given by $$\label{M'}
M'_{kl}(x') = \sum_{1 \leq m, \, n \leq 2} P_{km} M_{mn}(x) \frac{\partial x_n}{ \partial x'_l} ,$$ where $M$ is a matrix that satisfies (\[M definition 1\]) and (\[M definition 2\]) in the $x$ coordinate system and where $P$ is a product of permutation and reflection matrices. This can be proven by substituting this equation into the definition of $I'_{kl}(x')$ and $I'_{klmn}(x')$ and by noting that these quantities satisfy (\[M definition 1\]) and (\[M definition 2\]) in the $x'$ coordinate system because (\[I2 definition\])-(\[I4 definition\]) satisfy them in the $x$ coordinate system. By construction, $M$ is not singular, and, therefore, it has a non-singular inverse.
Notice that (\[M’\]) shows that the rows of $M$ transform as local covariant vectors, up to global permutations and/or reflections. Likewise, the same equation implies that the columns of $M^{-1}$ transform as local contravariant vectors (denoted as $V_{(i)}(x) \mbox{ for } i = 1, 2$), up to global permutations and/or reflections. As shown in the following, these particular vectors contain significant information about the separability of the data. In fact, they can be used to construct a mapping that must be an unmixing function, if one exists.\
*2. The $V_{(i)}(x)$ are used to construct a mapping, $u(x) = (u_{1}(x), u_{2}(x))$.*\
Because we are considering systems with just two degrees of freedom, only one mapping $u(x)$ needs to be constructed. However, to analyze systems with more than two degrees of freedom, a small set of mappings, $\{u(x)\}$, must be constructed, as suggested in Figure \[figure1\] and as described in detail in Subsection \[N dimensional systems\]. Working in the $x$ coordinate system, we begin by picking any point $x_0$. We then find a curve $X(\sigma)$ that passes through $x_0$ and is tangential to the local vector $V_{(1)}(x)$ at each point. Here, $\sigma$ denotes a variable that parameterizes the curve and increases monotonically as the curve is traversed in one direction. Formally, $X(\sigma)$ can be chosen to be a solution of the first-order differential equations $$\label{X 1D}
\frac{dX}{d\sigma} = V_{(1)}(X)$$ that satisfies the boundary condition, $X(0) = x_0$. Then, for each value of $\sigma$, we construct a curve, $Y(\tau)$, which passes through the point $X(\sigma)$ and is tangential to the local vector $V_{(2)}(x)$ at each point. Here, $\tau$ parameterizes this curve, increasing monotonically as it is traversed in one direction. Mathematically, $Y(\tau)$ can be chosen to be a solution of $$\label{Y 2D}
\frac{dY}{d\tau} = V_{(2)}(Y)$$ that satisfies the boundary condition, $Y(0) = X(\sigma)$. Finally, the function $u_{1}(x)$ is defined so that it is constant along each of the $Y$ curves. Specifically, $u_{1}(x) \equiv \sigma$ whenever $x$ is on the $Y$ curve passing through $X(\sigma)$. A function $u_{2}(x)$ can be defined by following an analogous procedure in which the roles of $V_{(1)}(x)$ and $V_{(2)}(x)$ are switched.\
*3. The mapping $u(x)$ is used to transform the measured time series, $x(t)$, into the time series $u[x(t)] = (u_{1}[x(t)],u_{2}[x(t)])$.*\
*4. It is determined if the components of $u[x(t)]$ are statistically independent.*\
This can be done by computing its PDF and determining if it factorizes as $$\label{1D u PDF factorization}
\rho_{U}(u,\dot{u}) = \prod_{a=1,2}{\rho_{Ua}(u_{a},\dot{u}_{a})} .$$ Here, $u$ denotes $u[x(t)]$, and $\dot{u}$ is its time derivative. Alternatively, we can compute a large set of correlations of multiple components of $u[x(t)]$ and then determine if they are products of lower-order correlations, as required by (\[1D u PDF factorization\]).\
*5. The result of step 4 is used to determine if the data are separable and, if they are, to determine an unmixing function. Specifically, if the components of $u[x(t)]$ are found to be statistically independent in step 4, it is obvious that the data are separable and $u(x)$ is an unmixing function. On the other hand, if the components of $u[x(t)]$ are found to be statistically dependent, the data are inseparable in any coordinate system.*\
This last statement is a consequence of the following fact, which is proved in the next two paragraphs: namely, if the data are separable, the constructed mapping, $u(x)$, must be an unmixing function.
Before proving this, we show that the matrix $M$ and the $V_{(i)}(x)$ have simple forms in the separable coordinate system, $s$. In particular, we prove that the following diagonal matrix is the $M$ matrix in the $s$ coordinate system $$\label{diagonal MS}
M_S(s) = \left( \begin{array}{ccc}
C^{-0.5}_{S11}(s_1) & 0 & \\
0 & C^{-0.5}_{S22}(s_2) &
\end{array} \right) ,$$ where $C_{Skl}(s)$ for $k,l = 1, 2$ are the second-order velocity correlations in the $s$ coordinate system. This can be proved by demonstrating that $M_S$ satisfies (\[M definition 1\]) and (\[M definition 2\]) in the $s$ coordinate system. To do this, first note that (\[PDF moment\]), (\[phase space factorization\]), and (\[C\_k=0\]) imply that the second-order velocity correlations are diagonal in the $s$ coordinate system. It follows that (\[M definition 1\]) is satisfied by $M_S$ in the $s$ coordinate system. Furthermore, it is not difficult to show that (\[M definition 2\]) is also satisfied by $M_S$ in the $s$ coordinate system. To see this, substitute (\[diagonal MS\]) into the sum on the left side of (\[M definition 2\]) for $k \neq l$. Because of the diagonality of $M_S$, each term in this summation is proportional to a fourth-order velocity correlation in the $s$ coordinate system that has just one index equal to $1$ (or $2$) and the other three indices all equal to $2$ (or $1$). Each of these terms must vanish because of (\[PDF moment\]), (\[phase space factorization\]), and (\[C\_k=0\]). This completes the proof that $M_S$ satisfies both (\[M definition 1\]) and (\[M definition 2\]) in the $s$ coordinate system, and, therefore, it is the $M$ matrix in the $s$ coordinate system, as asserted above.
Because $M_S$ is diagonal, the local vectors in the $s$ coordinate system, denoted $V_{S(1)}(s)$ and $V_{S(2)}(s)$, are oriented along the unit vectors, $(1,0)$ and $(0,1)$, respectively. Therefore, in the $s$ coordinate system, the curve, $X(\sigma)$, which was used in the definition of $u_{1}(x)$, is a horizontal straight line passing through the point $s[x_{0}]$, Similarly, each $Y$ curve is a vertical straight line passing through $s[X(\sigma)]$ for some value of $\sigma$. This implies that $s_{1}$ is constant along each $Y$ curve, being equal to the value of $s_1$ at its intersection with the $X$ curve. But, recall that $u_{1}(x)$ is also constant along each $Y$ curve, being equal to the value of $\sigma$ at its intersection with the $X$ curve. Therefore, because $\sigma$ is defined to vary monotonically along the $X$ curve and because the values of $s_1$ also vary monotonically along that curve, these paired values must be monotonically related to one another; i.e., $\sigma = h_{1}(s_{1})$ where $h_1$ is a monotonic function. It follows that $u_{1}(x)$ and $s_{1}(x)$ must also be monotonically related at each point; i.e., $u_{1}(x) = h_{1}[s_{1}(x)]$. In a similar manner, it can be shown that $u_{2}(x)$ and $s_{2}(x)$ are also related by some monotonic function. This means that $u_{1}(x)$ and $u_{2}(x)$ are component-wise transformations of $s_{1}(x)$ and $s_{2}(x)$. Because such component-wise transformations do not affect separability, it immediately follows that $u(x)$ is an unmixing function, as asserted above.
Systems having any number of degrees of freedom {#N dimensional systems}
-----------------------------------------------
This subsection describes how the procedure in Subsection \[two-dimensional systems\] can be generalized to perform nonlinear BSS of systems having $N$ degrees of freedom, where $N \geq 2$. The overall strategy is to determine if the system can be separated into two (possibly multidimensional) independent subsystems. If the data cannot be so separated, they are simply inseparable. If such a two-fold separation is possible, the data describing the evolution of each independent subsystem can be examined in order to determine if it can be further separated into two lower-dimensional subsystems. This recursive process can be repeated until each independent subsystem cannot be further divided into lower-dimensional parts. For example, for $N = 3$, we can first determine if the system can be separated into a subsystem with one degree of freedom and a subsystem having two degrees of freedom. If such a separation is possible, the data describing the two-dimensional subsystem can then be examined to determine if it can be further subdivided into two one-dimensional subsystems.
The five-step procedure for performing nonlinear BSS is described below and illustrated in Figure \[figure1\].\
*1. The local second- and fourth-order correlations of the measurement velocity ($\dot{x}$) are computed in small neighborhoods of the measurement space. These correlations are used to compute $N$ local vectors ($V_{(i)}(x) \mbox{ for } i = 1, \ldots ,N$).*\
This is done exactly as in step $1$ in Subsection \[two-dimensional systems\], except for the fact that: 1) each subscript can have any value between $1$ and $N$ (instead of $1$ and $2$); 2) each vector $V_{(i)}(x)$ has $N$ components (instead of two components).\
*2. The $V_{(i)}(x)$ are used to construct a small set of $N \mbox{-component}$ functions, $\{u(x)\}$, each of which is defined to be the union of two functions constructed with fewer components, $u_{(1)}(x)$ and $u_{(2)}(x)$.*\
One such mapping is constructed for each way of partitioning the $V_{(i)}$ into two groups (groups $1$ and $2$), without distinguishing the order of the two groups or the order of vectors within each group. For example, for a three-dimensional system ($N = 3$), three $u(x)$ functions must be constructed, each one corresponding to one of the three distinct ways of partitioning three vectors into two groups: $\{\{V_{1}\}, \{V_{2}, V_{3}\}\}$, $\{\{V_{2}\}, \{V_{1}, V_{3}\}\}$, and $\{\{V_{3}\}, \{V_{1}, V_{2}\}\}$. In contrast, for two-dimensional systems, there is only one way to divide the vectors into two groups, and, therefore, only one function, ${u(x)}$, has to be constructed in order to perform BSS, as described in Subsection \[two-dimensional systems\].
For each grouping, let $N_1$ and $N_2$ denote the number of vectors in groups $1$ and $2$, respectively, and let $G_1$ and $G_2$ denote the collections of values of $i$ for the vectors $V_{(i)}$ in groups $1$ and $2$, respectively. Each mapping, $u(x)$, is comprised of the union of the components of an $N_{1} \mbox{-component}$ function, $u_{(1)}(x)$, and the components of an $N_{2} \mbox{-component}$ function, $u_{(2)}(x)$, which are constructed as described in the next paragraph. For example, for the above-mentioned three-dimensional system, the first mapping to be computed, $u(x)$, has three components, comprised of the single component of $u_{(1)}(x)$ and the two components of $u_{(2)}(x)$.
The construction of $u_{(1)}(x)$ is initiated by picking any point $x_0$ in the $x$ coordinate system. We then find an $N_{1} \mbox{-dimensional}$ curvilinear subspace, consisting of all points that can be reached by starting at $x_0$ and by moving along all linear combinations of the local vectors in group $1$. This subspace can be described by a function $X(\sigma)$, where the components of $\sigma$ ($\sigma_{i} \mbox{ for } i \in G_1$) parameterize the subspace by labelling its points in an invertible fashion. Formally, $X(\sigma)$ can be chosen to be a solution of the differential equations $$\label{X ND}
\frac{\partial{X}}{\partial{\sigma_{i}}} = V_{(i)}(X)$$ for $i \in G_1$ with the boundary condition, $X(0) = x_0$. Then, for each value of $\sigma$, we define an $N_{2} \mbox{-dimensional}$ curvilinear subspace, consisting of all points that can be reached by starting at $X(\sigma)$ and by moving along all linear combinations of the local vectors in group $2$. This subspace can be described by a function $Y(\tau)$, where the components of $\tau$ ($\tau_{j} \mbox{ for } j \in G_2$) parameterize the subspace by labelling its points in an invertible fashion. $Y(\tau)$ can be chosen to be a solution of the differential equations $$\label{Y high dimension}
\frac{\partial{Y}}{\partial{\tau_{j}}} = V_{(j)}(Y)$$ for $j \in G_2$ with the boundary condition, $Y(0) = X(\sigma)$. Finally, the function $u_{(1)}(x)$ is defined so that it is constant on each one of the $Y$ subspaces. Specifically, $u_{(1)}(x) \equiv \sigma$ whenever $x$ is in the $Y$ subspace containing $X(\sigma)$. The function $u_{(2)}(x)$ is defined by following an analogous procedure in which the roles of groups $1$ and $2$ are switched. Finally, the union of the $N_1$ components of $u_{(1)}(x)$ and the $N_2$ components of $u_{(2)}(x)$ is taken to define the mapping, $u(x)$, that corresponds to the chosen grouping of the vectors, $V_{(i)}$, into groups $1$ and $2$.
The foregoing procedure can be illustrated by considering the construction of $u_{(1)}(x)$ from the first grouping of vectors in the three-dimensional case mentioned in the previous paragraph. In that case:
- $X(\sigma)$ describes a curved line that passes through $x_0$, that is parallel to $V_{(1)}(x)$ at each point, and that is parameterized by $\sigma$;
- each function, $Y(\tau)$, describes a curved surface, which intersects that curved line at some value of the parameter $\sigma$ and which is parallel to all linear combinations of $V_{(2)}(x)$ and $V_{(3)}(x)$ at each point;
- along each of these curved surfaces, $u_{(1)}(x)$ is equal to the corresponding value of $\sigma$.
Likewise, for the construction of $u_{(2)}(x)$ in the three-dimensional case:
- $X(\sigma)$ describes a curved surface that passes through $x_0$, that is parallel to all linear combinations of $V_{(2)}(x)$ and $V_{(3)}(x)$ at each point, and that is parameterized by the two components of $\sigma$;
- each function $Y(\tau)$ describes a curved line, which intersects that surface at a value of the parameter $\sigma$ and which is parallel to $V_{(1)}(x)$ at each point;
- along each of these curved lines, $u_{(2)}(x)$ is equal to the corresponding value of $\sigma$.
*3. Each mapping, $u(x)$, is used to transform the time series of measurements, $x(t)$, into a time series of transformed measurements, $u[x(t)]$.*\
For each $u(x)$, the transformed time series, $u[x(t)]$, is the union of the $N_1$ components of $u_{(1)}[x(t)]$ and the $N_2$ components of $u_{(2)}[x(t)]$.\
*4. It is determined if at least one mapping leads to transformed measurements, $u[x(t)]$, having a density function that is the product of the density functions of $u_{(1)}[x(t)]$ and $u_{(2)}[x(t)]$.*\
Specifically, it is determined if at least one transformed time series, $u[x(t)]$, has a PDF that factorizes as $$\label{ND u PDF factorization}
\rho_{U}(u,\dot{u}) = \prod_{a=1,2}{\rho_{Ua}(u_{(a)},\dot{u}_{(a)})} .$$ Here, $u$ denotes $u[x(t)]$, and $\dot{u}$ is its time derivative. Alternatively, we can compute a large set of correlations of multiple components of each transformed time series and then determine if they are products of lower-order correlations of two subsystems, as required by (\[ND u PDF factorization\]).\
*5. The result of step 4 is used to determine if the measurement data are separable and, if they are, to determine an unmixing function. Specifically, if at least one mapping, $u(x)$, produces a factorizable density function, it is obvious that the data are separable and $u(x)$ is an unmixing function. On the other hand, if none of the mappings leads to a factorizable density function, the data are inseparable in any coordinate system.*\
This last statement is a consequence of the following fact, which is proved in the next two paragraphs: namely, if the data are separable, at least one of the mappings, $u(x)$, leads to a density function that is the product of the density functions corresponding to $u_{(1)}$ and $u_{(2)}$.
The only remaining task is to prove the above-mentioned consequence of separability. The first step is to show that the matrix $M$ and the local vectors have simple forms in the separable ($s$) coordinate system. In particular, we prove that the following block-diagonal matrix is the $M$ matrix in the $s$ coordinate system $$\label{block-diagonal MS}
M_S(s) = \left( \begin{array}{ccc}
M_{S1}(s_{(1)}) & 0 & \\
0 & M_{S2}(s_{(2)}) & .
\end{array} \right)$$ Here, each submatrix $M_{Sa}$ is defined to be the $M$ matrix derived from the correlations between components of the corresponding subsystem state variable, $s_{(a)}$. For example, in the case of a separable three-dimensional system, (\[block-diagonal MS\]) asserts that $M_S$ consists of $1 \times 1$ and $2 \times 2$ blocks, which are the $M$ matrices of one-dimensional and two-dimensional subsystems, respectively. In order to prove (\[block-diagonal MS\]), it is necessary to show that $M_S$ satisfies (\[M definition 1\]) and (\[M definition 2\]) in the $s$ coordinate system. To do this, first note that (\[PDF moment\]), (\[phase space factorization\]), and (\[C\_k=0\]) imply that velocity correlations vanish in the $s$ coordinate system if their indices contain a solitary index from any one block. It follows that the second-order velocity correlation in the $s$ coordinate system ($C_{Skl}(s)$) consists of two blocks, each of which contains the second-order velocity correlations of an independent subsystem. This implies that (\[block-diagonal MS\]) satisfies the constraint (\[M definition 1\]), because, by definition, each block of $M_S$ transforms the corresponding block of $C_{Skl}$ into an identity matrix. In order to prove that (\[block-diagonal MS\]) satisfies (\[M definition 2\]), substitute it into the definition of $$\label{ISklmm}
\sum_{1 \leq m \leq N} I_{Sklmm} .$$ Then, note that: 1) when $k$ and $l$ belong to different blocks, each term in this sum vanishes because it factorizes into a product of correlations, one of which has a single index and, therefore, must vanish because of (\[C\_k=0\]); 2) when $k$ and $l$ belong to the same block and are unequal, each term with $m$ in any other block contains a factor equal to $I_{Skl}$, which vanishes for $k ~\neq~ l$, as proved above; 3) when $k$ and $l$ belong to the same block and are unequal, the sum over $m$ in the same block vanishes, because each block of $M_S$ is defined to satisfy (\[M definition 2\]) for the corresponding subsystem. This completes the proof that $M_S$ satisfies (\[M definition 1\]) and (\[M definition 2\]). It follows that $M_S$ is the $M$ matrix in the $s$ coordinate system, as asserted above.
Recall that the local vectors in the $s$ coordinate system are columns of the matrix, $M^{-1}_S$. Because of the block diagonality of $M^{-1}_S$, the local vectors can be sorted into two groups (groups $1$ and $2$) that consist of the columns passing through blocks $1$ and $2$, respectively, and that contain $N_1$ and $N_2$ vectors, respectively. Therefore, at each point $s$, the local vectors in the first group are linear combinations of the unit vectors parallel to the first $N_1$ axes of the $s$ coordinate system, and the local vectors in the second group are linear combinations of the unit vectors parallel to the last $N_2$ axes of the $s$ coordinate system. Hence, in the $s$ coordinate system, the function, $X(\sigma)$, which was used to define $u_{(1)}(x)$, describes the linear subspace that contains the point $s[x_{0}]$ and that is spanned by the first group of unit vectors. Likewise, each $Y(\tau)$, which was used to define $u_{(1)}(x)$, describes a linear subspace that contains $s[X(\sigma)]$ for some value of $\sigma$ and that is spanned by the second group of unit vectors. This implies that the state variable of the first subsystem, $s_{(1)}(x)$, is constant within each $Y$ subspace, being equal to the value of $s_{(1)}(x)$ at the intersection of that $Y$ subspace with the $X$ subspace. But, recall that $u_{(1)}(x)$ is also constant within each $Y$ subspace, being equal to the value of $\sigma$ at its intersection with the $X$ subspace. Therefore, because $\sigma$ is defined to be invertibly related to the points in the $X$ subspace and because the values of $s_{(1)}$ are also invertibly related to the points in the $X$ subspace, these paired values must be invertibly related to one another; i.e., $\sigma = h_{1}(s_{(1)})$ where $h_1$ is an invertible function. It follows that $u_{(1)}(x)$ and $s_{(1)}(x)$ must also be invertibly related at each point; i.e., $u_{(1)}(x) = h_{1}[s_{(1)}(x)]$. In a similar manner, it can be shown that $u_{(2)}(x)$ and $s_{(2)}(x)$ are also related by some invertible function. Because $s_{(1)}$ and $s_{(2)}$ are the state variables of independent subsystems and because $u_{(1)}$ and $u_{(2)}$, respectively, are invertibly related to them, $u_{(1)}$ and $u_{(2)}$ must be subsystem state variables in some other subsystem coordinate systems. This completes the proof of the assertion at the beginning of the previous paragraph: namely, if the data are separable, at least one way of grouping the local vectors (e.g., the grouping corresponding to the above-mentioned blocks) leads to a mapping, $u(x)$, that describes a pair of statistically independent state variables ($u_{(1)}$ and $u_{(2)}$).
Experiments
===========
In this section, the new BSS technique is illustrated by using it to disentangle synthetic nonlinear mixtures of two audio waveforms. The audio waveforms consisted of two thirty-second excerpts from audio books, each one read by a different male speaker. The waveform of each speaker, denoted $s_{k}(t)$ for $k=1 \mbox{ or } 2$, was sampled 16,000 times per second with two bytes of depth. The thick gray lines in Figure \[figure2\] show the two speakers’ waveforms during a short (30 ms) interval. These waveforms were then mixed by the nonlinear functions $$\label{mixing}
\begin{split}
\mu_{1}(s) &= 0.763 s_1 + (958 - 0.0225 s_2)^{1.5} \\
\mu_{2}(s) &= 0.153 s_2 + (3.75 * 10^7-763 s_1 - 229 s_2)^{0.5} ,
\end{split}$$ where $-2^{15} \leq s_1, s_2 \leq 2^{15}$. This is one of a variety of nonlinear transformations that were tried with similar results. The synthetic mixture measurements, $x_{k}(t)$, were taken to be the variance-normalized, principal components of the sampled waveform mixtures, $\mu_{k}[s(t)]$. Figure \[fig\_warped\_grid\] shows how this nonlinear mixing mapped an evenly-spaced Cartesian grid in the $s$ coordinate system onto a warped grid in the $x$ coordinate system. Figure \[fig\_measurements\] shows the distribution of the synthetic measurements created by randomly sampling $x(t)$, and Figure \[figure4\] shows the time course of $x(t)$ during the same short time interval depicted in Figure \[figure2\]. When either waveform mixture ($x_{1}(t)$ or $x_{2}(t)$) was played as an audio file, it sounded like a confusing superposition of two voices, which were quite difficult to understand.
The proposed BSS technique was then applied to these synthetic measurements as follows:
1. The entire set of 500,000 measurements, consisting of $x$ and $\dot{x}$ at each sampled time, was sorted into a $16 \, \times \, 16$ array of bins in $x \mbox{-space}$. Then, the $\dot{x}$ distribution in each bin was used to compute local velocity correlations (see (\[C2 definition\]) and (\[C4 definition\])), and these were used to derive $M$ and $V_{(i)}$ for each bin. Figure \[fig\_V\] shows these local vectors at each point.
2. These vectors were used to construct the mapping, $u(x)$. As described in Method, the first step was to choose some point $x_0$ and then construct the curvilinear line, $X(\sigma)$, that passes through that point and is tangential to the local vector $V_{(1)}(x)$ everywhere. Then, at each point $\sigma$ on this curve, a curvilinear line, $Y(\tau)$, was constructed through it so that it was tangential to the local vector $V_{(2)}(x)$ everywhere. Along each of these $Y$ curves, $u_{1}(x)$ was defined to be a constant equal to the value of $\sigma$ at the curve’s point of intersection with $X(\sigma)$. The mapping, $u_{2}(x)$, was defined by an analogous procedure. In this way, each point $x$ was assigned values of both $u_{1}$ and $u_{2}$, thereby defining the mapping, $u(x)$. One of the groups of thin black lines in Figure \[fig\_warped\_grid\] depicts a family of curves having constant values of $u_1$, which are evenly-spaced and increase as one moves from curve to curve in the family. The other group of thin black lines in Figure \[fig\_warped\_grid\] shows a family of curves having constant values of $u_2$, which are evenly-spaced and increase as one moves from curve to curve in the family.
3. As proved in Method, if the data are separable, $u(x)$ must an unmixing function. Therefore, the separability of the data could be determined by seeing if $u[x(t)]$ has a factorizable density function (or factorizable correlation functions). If the density function does factorize, the data are patently separable, and $u_{1}[x(t)]$ and $u_{2}[x(t)]$ describe the evolution of the independent subsystems. On the other hand, if the density function does not factorize, the data must be inseparable.
In this illustrative example, the separability of the $u$ coordinate system was verified by a more direct method. Specifically, Figure \[fig\_warped\_grid\] shows that the isoclines for increasing values of $u_1$ (or $u_2$) nearly coincide with the isoclines for increasing values of $s_1$ (or $s_2$). This demonstrates that the $u$ and $s$ coordinate systems differ by component-wise transformations of the form: $(u_{1}, u_{2}) = (h_{1}(s_{1}), h_{2}(s_{2}))$ where $h_1$ and $h_2$ are monotonic functions. Because the data are separable in the $s$ coordinate system and because component-wise transformations do not affect separability, the data’s PDF must factorize in the $u$ coordinate system. Therefore, we have accomplished the objectives of BSS: namely, by blindly processing the mixture measurements, $x(t)$, we have determined that the system is separable, and we have computed the transformation, $u(x)$, to a separable coordinate system.
The transformation, $u(x)$, can be applied to the mixture measurements, $x(t)$, to recover the original unmixed waveforms, up to component-wise transformations. The resulting waveforms, $u_{1}[x(t)]$ and $u_{2}[x(t)]$, are depicted by the thin black lines in Figure \[figure2\], which also shows the trajectory of the unmixed waveforms in the $s$ coordinate system. Notice that the two trajectories, $u[x(t)]$ and $s(t)$, are similar except for component-wise transformations along the two axes. A component-wise transformation is especially noticeable as a stretching of $s_{2}(t)$ with respect to $u_{2}[x(t)]$ along the positive $s_2$ axis. When each of the recovered waveforms, $u_{1}[x(t)]$ and $u_{2}[x(t)]$, was played as an audio file, it sounded like a completely intelligible recording of one of the speakers. In each case, the other speaker was not heard, except for a faint “buzzing” sound in the background. Therefore, the component-wise transformations (e.g., the above-mentioned “stretching”), which related the recovered waveforms to the original unmixed waveforms, did not noticeably reduce intelligibility.
Conclusion
==========
This paper describes how to determine the separability of time-dependent measurements of a system, $x(t)$; namely, it shows how to determine if there is a linear or nonlinear function (an unmixing function) that transforms the data into a collection of signals from statistically independent subsystems. First, the measurement time series is shown to endow state space with a local structure, consisting of vectors at each point $x$. If the data are separable, each of these vectors is directed along a subspace traversed by varying the state variable of one subsystem, while all other subsystems are kept constant. Because of this property, these vectors can be used to derive a small number of mappings, $\{u(x)\}$, which must include an unmixing function, if one exists. In other words, the data are separable if and only if one of the $u(x)$ describes a separable coordinate system. Therefore, separability can be determined by testing the separability of the data, after they have been transformed by each of these mappings.
Some comments on this result:
1. The original problem of looking for an unmixing function, $f(x)$, among an *infinite set of functions* was reduced to the simpler problem of constructing a *small number of mappings*, $\{u(x)\}$, and then determining if one of them transforms the data into separable form.
2. The BSS method described in this paper is model-independent in the sense that it can be used to separate data that were mixed by any invertible diffeomorphic mixing function. In contrast, most other approaches to nonlinear BSS are model-dependent because they assume that the mixing function has a specific parametric form ([@Duarte],[@Merrikh],[@Taleb]).
3. Notice that the proposed method is analytic and constructive in the sense that the candidate unmixing functions are constructed directly from the data, by locally manipulating them with linear algebraic techniques. In contrast, many other approaches [@Comon; @Jutten] search for an unmixing function by utilizing more complex techniques, involving neural networks or iterative computations.
4. Theoretically, the proposed method can be applied to measurements described by any diffeomorphic mixing function. However, more data will have to be analyzed in order to handle mixing functions with more pronounced nonlinearities. This is because rapidly varying mixing functions may cause the local vectors ($V_{(i)}$) to vary rapidly in the measurement coordinate system, making it necessary to compute those vectors in numerous small neighborhoods.
5. More data will also be required to apply this method to systems with many degrees of freedom. In Experiments, thirty seconds of data (500,000 samples) were used to recover two audio waveforms from measurements of two nonlinear mixtures. In other experiments, approximately six minutes of data (6,000,000 samples) were used to cleanly recover the waveforms of four sound sources (two speakers and two piano performances) from four signal mixtures. As expected, blind separation for the 4D state space did require more data, but it was not a prohibitive amount.
6. The proposed method does not require unusual computational resources. In any event, the most computationally expensive tasks are the binning of the measurement data and the computation of the local vectors, $V_{(i)}$, in each bin. If necessary, these calculations can be parallelized across multiple CPUs.
7. This paper shows how to perform nonlinear BSS for the case in which the mixture measurements are invertibly related to the state variables of the underlying system. Invertibility can almost be guaranteed by observing the system with a sufficiently large number of independent sensors: specifically, by utilizing at least $2N+1$ independent sensors, where $N$ is the dimension of the system’s state space. In this case, the sensors’ output lies in an $N \mbox{-dimensional}$ subspace embedded within a space of at least $2N+1$ dimensions. Dimensional reduction techniques (e.g., [@Roweis]) can be used to find the subspace coordinates corresponding to the sensor outputs. Because an embedding theorem asserts that this subspace is very unlikely to self-intersect [@Sauer], the coordinates on this subspace are almost certainly invertibly related to the system’s state space, as desired.
8. Separability is an intrinsic or coordinate-system-independent property of data; i.e., if it is true (or false) in one coordinate system, it is true (or false) in all coordinate systems. The local vectors ($V_{(i)}$) also represent a kind of intrinsic structure on state space, and, as mentioned previously, these contain some information about separability, which is available in all coordinate systems. These vectors “mark” state space and are analogous to directional arrows, which mark a physical surface and which can be used as navigational aids, no matter what coordinate system is being used. Many other vectors can be derived from the local velocity distributions of a time series. However, most of them will not have the special property of the $V_{(i)}$: namely, the property of being aligned with the directions traversed by the system when just one subsystem is varied and all others are held constant. For example, the $V_{(i)}$ would not have this critical property if the definition of $M$ (see (\[M definition 1\]) and (\[M definition 2\])) was changed by replacing $\sum_{1 \leq m \leq N} I_{klmm}$ with higher order correlations (e.g., $\sum_{1 \leq m, \, n \leq N} I_{klmmnn}$).
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abstract: 'In a recent publication \[PRL 111, 160405 (2013)\] we proved a version of Heisenberg’s error-disturbance tradeoff. This result was in apparent contradiction to claims by Ozawa of having refuted these ideas of Heisenberg. In a direct reaction \[arXiv:1308.3540\] Ozawa has called our work groundless, and has claimed to have found both a counterexample and an error in our proof. Here we answer to these allegations. We also comment on the submission \[arXiv:1307.3604\] by Rozema [*et al*]{}, in which our approach is unfavourably compared to that of Ozawa.'
author:
- Paul Busch
- Pekka Lahti
- 'Reinhard F. Werner'
title: 'Measurement Uncertainty: Reply to Critics'
---
Introduction
============
Uncertainty relations are a key insight of quantum theory, going back to Heisenberg’s seminal paper [@Heisenberg1927] from 1927. It therefore must come as a surprise to many that over 85 years later there can still be a controversy about them. In fact, there is no controversy about the basic textbook version (“preparation uncertainty”), which forbids quantum states that have too sharp distributions for both position and momentum. Ironically, this aspect is not even discussed in Heisenberg’s paper, who instead considers in his famous microscope example the trade-off between the accuracy of an approximate position measurement and the disturbance of momentum which the measurement causes. This can be generalized to a statement about the impossibility of accurate joint measurements of two complementary observables (“measurement uncertainty”). It is this aspect which is controversial, and while Ozawa [@Ozawa03; @Ozawa04; @Ozawa05] claims to have refuted Heisenberg’s ideas we [@Werner04; @BLW2013a; @BLW2013b; @BLW2013c] have proved a general result confirming his intuition.
This apparent contradiction is not a mathematical one, since the two approaches start from different definitions of “error” and “disturbance”. Ozawa’s approach [@Ozawa03; @Ozawa04; @Ozawa05], which grew out of the work of Arthurs 1965 [@AK65], is based on the expectations of squared differences of certain operators (“noise operators”) in a measurement scheme. These quantities depend on the input state of the measurement. Ozawa claims that a certain inequality written in such terms ( below) is “Heisenberg’s inequality”, and shows that it is wrong in general. The definitions in our approach [@BLW2013c] are characteristic of the measurement scheme, and hence independent of the input state. We prove for these quantities a relation that looks like the standard textbook form of uncertainty relations, even though the $\Delta$-quantities have quite a different meaning. We find that this identifies one aspect of Heisenberg’s paper, which has a general, rigorous quantitative interpretation. In contrast, far from “disproving Heisenberg”, Ozawa has simply failed to identify such a grain of truth.
Ozawa has written an attack on our approach [@Ozawa2013]. This was worded so aggressively, and was so low on scientific quality, that it would seem to be more of an embarrassment to the author than an argument requiring an answer. Nevertheless, as mathematical physicists we have to respond, since he claimed to have found a counterexample to our theorem and also claimed to have spotted an error in our proof. Such a reply is the main content of this note. We also reply to a critique [@Roz13] by one of the experimental groups backing Ozawa.
We have laid out our criticism of the Ozawa approach and, more generally of “noise operator” definitions of error and disturbance in a separate paper [@BLW2013a], which we very briefly summarize and in some ways extend in the next section. For our Theorem with a sketch of proof we refer to [@BLW2013c], for a full proof to [@BLW2013b], and for a discussion of the qubit case, which is the basis of the experiments, to [@BLW2014].
Points of a debate
==================
It is the normal state of affairs that intuitive concepts can be sharpened mathematically in several different ways. Also notoriously non-unique are quantum analogs of classical concepts. All one can do in such a case is to state the options as clearly as possible, and point out contexts in which one may be more appropriate than another. Basically, measurement uncertainty is just another example of this. If you want to discuss operationally defined figures of merit for measurement devices like the “resolution of a microscope”, we recommend our definitions and results. If you hope to get more detailed statements depending on the input states, for example for purposes of cryptography, you may consider entropic ideas [@Furrer], and Ozawa’s might seem a step in the direction of “more detailed” statements. However, there are some requirements which cannot be put aside easily as matters of taste or choice of an appropriate context. Before coming to these, we begin by sketching the context: Ozawa’s and our definitions of the terms in the respective uncertainty relations.
The basic definitions
---------------------
Ozawa considers a model of measurement by which the system is coupled by a unitary operator $U$ to a probe in initial state $\sigma$, which is also a canonical degree of freedom. The position outcome is identified with the position of the probe after the measurement $U^*(\idty\otimes Q)U$. Similarly the momentum of the system before and after the measurement are represented by the operators $P\otimes\idty$ and $U^*(P\otimes\idty)U$, respectively. The Ozawa error $\epsilon_O$ and the Ozawa disturbance $\eta_O$ are then defined by $$\begin{aligned}
\label{errO}
\epsilon_O^2&=& {\mathrm{tr}\left[{\rho}\right]}\otimes\sigma \left(U^*(\idty\otimes Q)U-Q\otimes\idty\right)^2 \\
\eta_O^2&=& {\mathrm{tr}\left[{\rho}\right]}\otimes\sigma \left(U^*(P\otimes\idty)U-P\otimes\idty\right)^2. \label{distO}\end{aligned}$$ Then in general the “uncertainty relation” $$\label{false}
\epsilon_O \eta_O\geq \hbar/2$$ is false.
We consider a general joint measurement device, i.e., any device which outputs both a momentum value and a position value, which in the above setting would be the position of the probe and the momentum of the object particle after the measurement interaction. We say that the position accuracy of such a device is $\delta_Q$, if for arbitrary input state $\rho$ with sharp position at $q$, the distribution of the position outputs has a quadratic deviation at most $\delta_Q^2$ from the “correct” value $q$. Then, for any measuring device $\delta_Q\delta_P\geq\hbar/2$, and equality is attained for a device whose output distribution is the Husimi phase space distribution.
Do Ozawa’s definitions capture Error and Disturbance?
-----------------------------------------------------
If one defines a quantity like “error” of a measurement, then the mathematical concept should share the basic features of the intuitive concept. This is usually tested most clearly on extreme cases: When a device gives an output distribution different from that of an ideal reference, it is clearly in error, and the error quantity defined formally should not assign error zero to this situation. Conversely, a device for which input state and output state are the same can hardly be claimed to have disturbed the system, and so we should expect the definition to assign disturbance zero. Neither criterion is met by Ozawa’s definitions [@BLW2013a]. If anything they express some combination of variances of the device and of the input state. So the criticism in this case is that the words “error” and “disturbance” are misnomers.
A failed quantum generalization
-------------------------------
In the classical case, two random variables $f,g$ for which $(f-g)^2$ has expectation zero coincide with probability one, and hence they have the same distribution. In the quantum case this fails in general, i.e., if for some state $\Psi$ we have $\braket\Psi{(A-B)^2\Psi}=0$ we have $A\Psi=B\Psi$, but when $A$ and $B$ do not commute on the state $\Psi$, the higher moments, and hence the distributions of $A$ and $B$ will not, in general, coincide. This is the reason for some of the failures mentioned in the previous paragraph. Ozawa likes to cite Gauß for justifying the “expected squared difference” expressions and , completely ignoring the non-commutativity. This is a inexcusable oversight, especially in a discussion of quantum uncertainty.
By the rules of the book, measuring $(A-B)^2$ requires setting up a measurement for which the probability for outcomes in an interval is given by the expectation of a spectral projection of $(A-B)$. There is no direct way implementing this given $A$ and $B$, and in any case such a measurement is not compatible with either $A$ or $B$ (unless these observables commute). In other words, the Ozawa error of a microscope refers to a measurement incompatible with the operation of the microscope itself. This is what we mean by saying that his definitions are [*not operational*]{}: they do [*not*]{} reliably indicate the presence, absence, or the extent of differences between the distributions of the target observable and the intended approximating observable. It is still possible to design a measurement for the expectation of $(A-B)^2$, as for any operator, but this does not cure the basic flaw in the definition.
Presumably in response to our non-operationality criticisms in [@Werner04] and [@BuHeLa04], Ozawa [@Ozawa04] offered the so-called 3-state method for determining his error quantity from the statistics of the POVM being measured as an approximate measurement of a given target observable. However, this approach undermines the intended interpretation of Ozawa’s error quantity as being state-specific: the very fact that three distinct states are to be measured in order to obtain its value takes away any justification for associating this quantity with any particular state. This failure of Ozawa’s definition becomes strikingly evident in the qubit case discussed at the end of this note.
State dependence
----------------
The left hand side of is not bounded below by a positive constant. In fact it is easy to come up with examples once one realizes that this failure is typical for state-dependent measures [@Rud2013]. Basically, one can take a standard destructive nearly ideal position measurement followed by the re-preparation of the input state. In this way both error and disturbance can be made arbitrarily small. The disturbance becomes evident on other input states, which is why we consider a supremum over input states at this point. A non-disturbing measurement to us is one which leaves [*every*]{} state unchanged, not one that leaves [*some*]{} state unchanged. So the failure of the error-disturbance product to be bounded below is just due to an unfortunate choice of quantifiers, quite apart from the other flaws of the definitions. Nothing can be learned from this failure about quantum physics.
This has been noted by Appleby [@Appleby1998a; @Appleby1998b] in 1998 for the noise operator approach, who took this as a motivation to move on to a state-independent approach with worst case errors, in this regard much like ours. This was some years before Ozawa started his “refuting Heisenberg” programme. Hence it can be said that in this respect Ozawa’s contribution was a step backwards on Appleby’s insight. To be sure he produced some correction terms which turn into a correct inequality, for which he uses the odd[^1] phrase “universally valid” inequality.
The false inequality plays a key role in Ozawa’s recent writings, namely as “Heisenberg’s” error-disturbance relation. In this way Appleby’s simple observation that it is false becomes Ozawa’s “disproof” of Heisenberg’s paper [@Heisenberg1927], and the experimental verification of this simple fact creates the impression that Heisenberg is in conflict with Nature herself. This created a considerable media-hype in 2012. In order to get a balanced view of this we clearly have to take a look at Ozawa’s justification of blaming Heisenberg for .
Is it Heisenberg’s?
-------------------
So what about historical accuracy? We are perfectly aware that our definitions of error and disturbance are not in Heisenberg’s writings anywhere. Hence our relations are not Heisenberg’s either. They are closer to Heisenberg’s paper than the usual textbook version, since he clearly discusses measurement uncertainty rather than preparation uncertainty. In order to emphasize this, and also to honour the deep intuitions in his paper, we have, on occasion called our relations “Heisenberg relations”. This is covered by the usual naming practices in science.
Ozawa’s case is different, however. His claim of having refuted Heisenberg rests completely on putting words – his own false inequality – into Heisenberg’s mouth. This is absolutely “groundless”. Heisenberg does not even write his relations with an inequality sign but with a mathematically unexplained tilde. He is purposefully vague, in keeping with the program outlined in the introduction of his paper, of giving an intuitive, heuristic understanding of why orbits must be criticised as unobservable in an atom (his discovery, just 2 years old then), but why we can still see tracks in a cloud chamber. This intuition stands as a valid insight and, as we have shown, one can even back it up by a precise quantitative statement. Heisenberg was probably not too interested in rigorous formulations. Although he announces a proof to be given later in the paper (on the basis of “his” commutation relations), he famously never gave one, and probably his grasp of the formalism in 1927 would not have been up to it.
Ozawa’s justification for laying his mistake on Heisenberg is that Heisenberg uses the phrase “mean error” at some point, which in the context hardly means more than the admission that this is not a sharply defined quantity. There is no indication that Heisenberg was thinking in terms of a specific average, or any operator difference for the error. In fact, for modern readers it is clear that he is still quite far from taking abstract operators as basic entities of quantum mechanics, which he considers only through the veil of Dirac’s transformation theory.
Another argument given by Ozawa is his claim that the root-mean-square measurement error (“noise”) and disturbance are “naturally” or “standardly” defined by his $\epsilon_O,\eta_O$ [@Ozawa05]. In his “disproving” manuscript [@Ozawa2013] he implies that his definition is the only possible quantum generalization of Gauß’s classic concept of root-mean-square error. This is reiterated in the supplement of [@Ozawa-etal2014]. Again Ozawa describes the classical case, and Gauß’s remarks on the topic. He shows that formula holds in the case of commuting observables. Then comes a characteristic leap in the argument: The same formula is applied, without even so much as a comment, to the non-commutative case. So even if one is willing to accept Gauß’s definition as part of the Canon, the non-commutative extension is entirely Ozawa’s invention, not justified by him, and, as we have shown, indeed highly problematic. And even if one might hypothesize that Heisenberg, had he cared about making his notion precise, would have tried Gauß’s canonical notion, the claim that he wouldn’t have seen or would have chosen to ignore the typical problems brought in by quantum mechanics, and would thus have translated the “$q_1$” of his paper to Ozawa’s is, at best, an argument by lack of imagination.
Ozawa’s advertising and that of the associated experiments largely rests on the contrast between the false relation consistently referred to as Heisenberg’s on the one hand and the corrected one on the other. This is in stark contrast to the lack of historical argument for this attribution. We have tried [@BLW2013a] to explain some of the richness in the intuitive content of [@Heisenberg1927]. But if later papers [@Vienna2; @Baek13; @Ozawa-etal2014] are any indication we are afraid that Ozawa is not interested, and continues to try to push the point by mechanical repetition. So in contrast to all the argument-free rhetoric we have to insist that the only person ever to advance the false relation for serious consideration is Ozawa. As an aid to understanding what his papers are about we suggest to any potential reader to replace the name Heisenberg by Ozawa, whenever it comes up in relation to . This leads to a rather more adequate representation of the actual scientific content.
Overzealous journalists have taken the false advertising even further by claiming that this “disproof” of Heisenberg’s uncertainty would somehow render quantum cryptography insecure. Indeed, uncertainty relations play a key role in some modern security proofs [@Tomamichel2012; @Berta-etal2010; @Furrer], but the statements used there are proven theorems. How could an error committed elsewhere (by Heisenberg, Ozawa, or whoever) have any bearing on such results?
In any case, despite the severe differences between Ozawa’s and our take on Heisenberg, there is a common core: both parties seem to agree that the logical structure of Heisenberg’s measurement uncertainty principle is that of a mutual trade-off between measurement accuracies (or accuracy vs. disturbance) that is dictated by the incompatibility of the observables under investigation. The focus on disproving Heisenberg surely has helped to bring this fundamental principle once more into the public limelight. However, the scientific question at stake is that of an appropriate, [*operationally significant*]{} and mathematically rigorous, formulation of the principle [*as a consequence of quantum theory*]{}.
Do experiments help?
--------------------
An interesting point of the recent debate is that the failure of Ozawa’s relation has been verified experimentally. This was touted as a confirmation of his theory, a refutation of Heisenberg by Nature herself, and contributed greatly to the hype. But actually an experiment does nothing to “cure” a bad definition. The quantities appearing in Ozawa’s definition may be non-operational in the sense that they do not refer to observables which can be tested in the normal run of microscope measurements. But the linear operators involved have a clear meaning in the formalism and their expectations can obviously be measured in some way. That may be a challenging experiment of independent interest, but the failure of is automatic if the implementation is correct. These experiments certainly do not help to prove Heisenberg wrong. They merely confirm that inequality , which nobody ever claimed to be right, indeed isn’t correct.
We hasten to add that, generally speaking, we are very much in favour of foundational experiments. Only in this case a worthless target was chosen. It is true that sometimes a dubious theoretical calculation (say in condensed matter physics) may be given more credibility by confirming experiments. But in the present case such a correction by experiment simply does not happen. Ozawa’s claims about Heisenberg are exactly as untenable after the experiments as they were before.
How the debate might continue
-----------------------------
We certainly agree with Ozawa that the whole circle of ideas around uncertainty, and especially measurement uncertainty, deserves more rigorous elucidation. There will be many results contributing to this aim, and Ozawa’s corrected relation may play a role there too. We also don’t have a quarrel with state-dependent relations in this context, which may even be just the right thing for improving cryptographical security proofs. Naturally, the false relation is simply to be ignored in this endeavour. We also do not expect that Ozawa’s improved relation will play a very positive role, because it is based on ill-conceived basic notions. There are signs, however, that his group may be contributing after all. In a recent paper [@Buscemi] we see a lot of Ozawa’s old rhetoric and misrepresentation of our work, but also a turn to an information-theoretic framework and state-independent quantities.
With better results appearing on the scene, criticising the noise operator approach may thus lose importance. So apart from Ozawa’s direct attacks on our work (see below) there is only one point which should not go unchallenged: Ozawa’s completely arbitrary attribution of the elementary mistake to Heisenberg.
Reply to Ozawa’s criticism
==========================
The aim of Ozawa’s text [@Ozawa2013] seems not so much to criticize particular aspects of our paper [@BLW2013c] but to brand it as “groundless”, as not covering some cases which “escape it” (despite claims to the contrary), as making “unsupported claims” and giving a “failed proof”. It seems that Ozawa is not prepared to tolerate any countering in the literature to his claim of having refuted Heisenberg. Indicative of this is already the inversion of the title (his “Disproving Heisenberg’s error-disturbance relation” vs. our “Proof of Heisenberg’s error disturbance relation”). It only shows that Ozawa is forgetting elementary logic in his rebuttal rage: Ozawa’s paper is intended as a disproof only of our paper, so for his title to make any sense he would have to identify “Heisenberg’s error-disturbance relation” with our approach, which is surely not what he wants to say.
Ozawa has submitted his paper to a journal whose editors asked us for an open comment to be transmitted to the referees and the author. This open comment is the basis of the current note, so we can assume that Ozawa has been informed of our arguments. Furthermore, in the meantime our paper with the full proof [@BLW2013b] has appeared on the arXiv (in December 2013). Neither text seemed sufficient reason for Ozawa to retract or even modify his arXiv submission [@Ozawa2013].
Ozawa on operationality
-----------------------
Ozawa’s paper does contain a reply to our criticism of his approach as non-operational (see above). This then does not refer to the paper [@BLW2013c], which is the only source he cites: [@BLW2013c] focusses on establishing a positive result rather than criticising other approaches, and so contains nothing of the sort. But possibly Ozawa is referring here to meetings at conferences (RFW in Tokyo around 2002, PL at Växjö in 2013), and perhaps to the papers [@BuHeLa04; @Werner04], where we tried to explain this point to him. His resolution of the problem arising when the momentum before and after the measurement do not commute (p.1, top of first column) is that [*sometimes they do*]{}. Clearly, if that is the strongest answer he can give, his approach would have to relinquish all aspirations to “universality”. Again, this is hardly what he wants to say, but we have to put on record that our criticism remains unanswered.
Linear models
-------------
The particular example he cites here serves also as a “counterexample” to our theorem. Therefore we have to look at it more closely. Indeed, “linear measurement models” have served as a warm-up for the study of measurements since the time of von Neumann, and Ozawa has written several papers about them (cited in his manuscript). These models are relevant in the lab, for example in the quantum optics of field quadratures, where linear transformations are the easiest to perform, or in the mechanical case of a particle by scattering with another particle.
One can also describe this class abstractly as canonical linear transformations of the joint phase space of particle and probe (Ozawa somewhat arbitrarily chooses a three-parameter subclass), which then directly translates into quantum mechanics, because all these transformations can be realized by quadratic Hamiltonians. Let us consider, for definiteness and for getting better assistance from physical intuition, the mechanical case of a particle of mass $m_1$ whose position we determine by scattering of a test particle (“probe”) of mass $m_2$.
Ozawa considers a position measurement, whose outcome is directly the position of the probe. This is a linear function of the particle’s initial position with slope $a=2m_1/(m_1+m_2)$. Now any lab assistant trying to calibrate this measurement would notice this factor, so in order to get a better measurement, one would naturally correct for it. Otherwise, the error on suitably chosen states will become arbitrarily large. In our notation, this translates to an infinite error (“worst case error”) of the position measurement. Including the correction, however, i.e., using an optimized estimator on the same data, one does get a measurement with typically finite error, depending on the initial state of the probe. For momentum one considers the particle’s own momentum (not the probe’s) after the measurement, which will be linear in the input momentum with slope $(m_1-m_2)/(m_1+m_2)$. For the disturbance to be finite we would need this factor to be 1, which contradicts $m_2>0$. Of course, this covers only the trivial part of the disturbance (momentum is different before and after) and not its “uncontrollable” aspect (momentum cannot be retrieved). A suitable correction can again be made and gives measurements with bounded momentum disturbance across all states.
Linear models can thus be used to produce good joint measurements of position and momentum, and as the dilation theory of covariant observables [@screen; @QHA] shows, all phase space covariant measurements can be obtained in this way. These covariant measurements do play a key role in our theory, and indeed the measurement uncertainties are in this case directly related to the preparation uncertainties of the probe state, and with a minimal uncertainty initial probe state one also gets cases of saturation (equality) in our error-disturbance inequality.
Ozawa, however, takes not only these good measurements but also those in which the lab assistant failed to apply the appropriate correction factors. (Of course, as he notes in col2, p.3, the good ones are of Lebesgue measure zero in this set, which to him apparently suffices to brand them as an exotic specialty.) Anyhow, his claim that these models are reasonable and have finite errors, even with an ill chosen calibration factor, is entirely due to the difference between our state-independent and his state-dependent formulation and, in addition, to the ad hoc assumption of finite variances for the initial states. Perhaps we can follow him here by not calling the lab assistant infinitely stupid, but just say that his errors are unbounded. Of course, this in no way invalidates our result.
The indeterminate case
----------------------
In the linear models one can easily produce examples where one uncertainty (in our sense) is zero and the other is infinite. In fact, it suffices to choose an ideal position measurement with a random momentum output, uncorrelated with the input state. Ozawa argues (in his criticism of Appleby, p. 3) that such cases per se invalidate any uncertainty relation, because the product $0\cdot\infty$ “cannot be concluded to be above $\hbar/2$”. Here, as always in mathematics or physics, one has to discuss what indeterminate products mean. The point of the relation is that not both uncertainties can be simultaneously small. From this point of view the only reasonable demand concerning the indeterminate case would be that the theorem should ensure that if one factor is zero the other must be infinite. This is indeed covered by our result since if both uncertainties are finite or one of them zero the inequality holds. Then in whatever way one approaches a situation with one factor zero the other has to diverge, and not just arbitrarily, but in keeping with the finite-finite relation. Thus the indeterminate case requires no extra argument, provided that one does show the relation under the assumption that both factors are finite (including the case of one being zero).
It is perhaps interesting to consult Ozawa’s handling of the indeterminate case in his own relations. Clearly, as the above examples show zero error or disturbance may occur. By his own logic any infinite terms, would invalidate his extended “universally valid” relation. So he just forbids ad hoc any states with infinite variance for position or momentum (at the bottom of p. 2 col. 1). To be sure, he vaguely refers to this as avoidance of “inaccessible resources”, and we could counter his criticism in this spirit by saying that “zero error is an unrealistic idealization” anyhow. But with a proper understanding of the indefinite case no such rhetorical manoeuvers are needed.
An alleged error in our argument
--------------------------------
The last column of Ozawa’s paper is devoted to spotting an error in our argument. Indeed, since he feels that he has presented a counterexample, this is required if he wants to invalidate a proven theorem. Here we have to say that the proof in our paper [@BLW2013c] is clearly labelled as a sketch, in particular the averaging/compactness argument, which is described as somewhat technical. If anyone has doubts about a published proof (or claim thereof) there is a clear procedure dictated by the rules of scientific debate: Either one follows the hints given in the proof, including the hint that a very similar argument for just this part is to be found in an earlier paper [@Werner04]. Or one waits for the announced published full proof [@BLW2013b]. Or, failing the mathematical ability to do the first, and the patience to do the latter one can write to the authors requesting clarification. Ozawa has definitely not pursued one of these routes.
This would still be ok, if even from the sketch one could detect a fatal flaw in the argument. According to Ozawa the “exact point where the argument fails” is in our claim that the set of observables satisfying a certain calibration condition “is convex and compact in a suitable weak topology”. He counters this merely by saying “However, this is not true”. There is no argument whatsoever, beyond the remark that the closure of the set of [*all*]{} observables would contain ones supported on the Stone-Čech-compactification of $\Rl^2$ (actually it could be any other compactification depending on the weak topology which we did not even specify). He could well have learned this from the cited paper [@Werner04], including a proof of how a subset as the one specified may be compact nevertheless. This is analogous to the space of density matrices, whose weak closure contains singular states such as states with sharp position. However, the subset on which the harmonic oscillator has fixed finite energy expectation is indeed weakly compact.
An aspect where Ozawa could have made a point
---------------------------------------------
There is indeed a tension between our claim of an operational measure of errors and the fact that they may turn out infinite quite easily. Of course, this is due to taking the worst case over [*all*]{} quantum inputs over the measuring device. Thus in assessing the resolution of a microscope even objects in a neighbouring galaxy would be tested, which is clearly nonsensical. Ozawa does not make this point in the submitted paper. His claim that our quantities are so often infinite is based on the linear models (see above).
Anyhow, there is a non-operational aspect here in our work, which we intend to address in the near future. What we are currently working to show is that the uncertainty relation as stated is valid even if we limit the calibration to some finite “operating range”. Thus no states with arbitrarily large position or momentum would be involved in the definition of the errors. The relevant condition would be that these operating ranges in position and momentum are large compared to the respective errors. We expect that this would not substantially decrease the uncertainty bounds, with a difference going to zero in the limit of large operating ranges. The relation as stated would thus be interpreted as an idealized version where the operating ranges become infinite. In fact, Appleby [@Appleby1998b] has sketched a similar construction in the noise operator approach.
Even in this more realistic version, however, the distinction made in the first section would persist: It makes no sense to limit the “operating range” to a single state if one aims to define error measures as figures of merit for a measuring device.
Reply to Rozema [*et al*]{} [@Roz13]
====================================
This section comments briefly on the submission [@Roz13] of Rozema [*et al*]{} entitled “A Note on Different Definitions of Momentum Disturbance”. While also apparently conceived as a rebuttal of our paper [@BLW2013c], its tone is rather more conciliatory. We are certainly sympathetic to the comparative approach indicated in the title. However, there are several misunderstandings that need clarification. Our comment was also originally drafted following a request by an editor to comment on the submission [@Roz13] so that we may expect that the authors are informed on our critique. However, the authors have remained quiet, no revision of [@Roz13] has appeared.
Once again: error as figure of merit
------------------------------------
The motivation for building figures of merit for measuring devices (see the introduction) was apparently not appreciated by the authors of [@Roz13]. We consider quantities like the resolution of a microscope, which does not depend on the object we are looking at: It is a promise for accuracy on arbitrary input states. What remains of this in the presentation of [@Roz13] is that we consider the “disturbing power” of an instrument, while the disturbance on a particular state may be smaller. We agree, but that is besides the point for a benchmark quantity. In the same spirit one could propose not to talk about the resolution of a microscope but about its “error generating capacity”, which is presumably an irrelevant quantity because it does not refer just to the one object at hand but is attained at a quite different one.
In this light, the conclusion shortly after equation (2) of [@Roz13], namely that “their relationship does not hold for a given state”, may suggest itself because no state appears in it. Logically, however, since our quantities are maximized over all states, our relation does in fact hold for arbitrary states – it simply does not give the tightest estimate of errors specific to any particular state. Moreover, it does make a statement about every possible joint measurement device, and hence applies to every experiment involving such devices. Of course, if one is interested in measurement error and disturbance measures for a single state, one would turn to state-dependent measures instead of our apparatus benchmark quantities; in fact, our non-maximized quantities are operationally significant as measures of error and disturbance for an individual state – in contrast to Ozawa’s measures.
Maybe because the authors have not fully appreciated the basic nature of our quantities, they misrepresent our result as the conclusion that “any measurement which is capable of achieving a measurement precision of $\Delta X$ on [*some*]{} states must be capable of imparting a momentum disturbance of $\hbar/2\Delta X$ on [*some*]{} (potentially different) state”. (Even though this is presented as a quote, it is not a quote from our paper). The emphasis of the first [*some*]{} is ours: this is plain wrong, it should rather read [*“applicable to all”*]{}. The second [*some*]{} is correct, but the phrase “is capable of” is a very odd rendering of the correct “necessarily imparts”. The parenthetical “potentially different” should be deleted, because it makes no sense when the premise is for [*all*]{} states.
Not all measurements prepare (approximate) eigenstates
------------------------------------------------------
In the last paragraph of description of our work (p. 1, col. 2) Rozema [*et al*]{} convey the impression that our work is mathematically trivial, by saying that we would just need to realize that after a position measurement of accuracy $\Delta X$, the post measurement state would necessarily have a position spread $\Delta X$ (and hence have the claimed momentum uncertainty). But this assumption about the post-measurement state is entirely unwarranted, and we certainly do not make it.
Heisenberg in his 1927 paper has clearly made this identification of position measurement error and the position uncertainty in the final state, in line with the common but limited perspective at the time, that measurements always produce (approximate) eigenstates. It has long since become clear that this identification of the concept of measurement with projective, or von Neumann-Lüders measurements is too narrow. Indeed, for such measurements the preparation uncertainty relation will trivially entail the joint measurement error relation. Interestingly, it turns out that this logical connection does persist in the case of general joint measurements, as we have shown in a variety of examples in [@BLW2013b; @BLW2014].
Comparing Ozawa’s “error” with actual statistical deviations
------------------------------------------------------------
It is quite disconcerting that there has not been any critical conceptual analysis of Ozawa’s measures of disturbance and error among those research groups that use them. Rozema [*et al*]{} share with Ozawa the belief that “a good measure of the disturbance to a state is the root-mean-squared (RMS) difference between $P$ before and after the process, $U$: $$\eta_O(P)=\langle(U^*PU -P)^2\rangle^{\frac 12}.\qquad\qquad (3)\text{''}$$ If Quantum Mechanics were Classical Mechanics this would be fine. But Quantum Mechanics is not Classical Mechanics. In Quantum Mechanics the operationally significant measurement result is not an individual outcome but a distribution of outcomes obtained when the same measurement is performed many times on the same system prepared in the same way (or, if one prefers, when the measurement is performed in a similarly prepared ensemble of (independent) systems). Therefore, to compare the momentum before and after the process, [*one has to compare the two momentum distributions, the momentum distribution before the position measurement and the momentum distribution after the position measurement*]{}.
To this end we use the Wasserstein distance of order 2 [@Villani], which does constitute an operationally significant extension of the classical RMS difference to the noncommutative context. Only in some special cases is Ozawa’s measure $\eta_O(P)$ operationally accessible from the data available (which are the statistics of the approximate measurement at hand and that of an accurate reference measurement performed on an ensemble prepared in the same state), and even in those cases it usually overestimates the disturbance in the momentum distribution (since in the case of the order 2 Wasserstein distance one minimizes the mean of the squared deviation over all joint distributions while for Ozawa’s measure one just picks a particular joint distribution).
This is made evident in the example presented by Rozema [*et al*]{}: the “difference” between the position observable and the space-inverted position observable can be noticed in some states but not in states with a parity symmetric distribution of the position. This means simply that the state change due to a parity operation simply cannot be detected by the position observable in all possible states, and accordingly our measure of the distance between the two (identical) distributions to be compared vanishes appropriately in the symmetric case while Ozawa’s measure gives a misleading nonzero value.
We have expressed our dissatisfaction with Ozawa’s definition only in passing in our paper [@BLW2013c] pointing to a more detailed discussion in preparation, which is now available [@BLW2013a]. Rozema [*et al*]{} cite a 2004 paper [@Werner04] in this regard, which contains the statement that Ozawa’s definition is not operational. It appears that Rozema [*et al*]{} consider this refuted by having measured Ozawa’s quantities. That is, of course, not correct. In principle, any operator, like $ U^*PU - P$, can be measured, be it by realizing an instrument, which determines the outcomes for all spectral projections, or, in the case at hand, in a less detailed way by a weak measurement. The charge is that a meta-experiment is required, and that disturbance and error are not meaningful in terms of the device as given. As noted above, Ozawa has implicitly conceded this point in his “disproving” paper, where he counters this objection only by giving an example of a special case when it does not apply. The criticism, expressed by way of examples also as early as 2004 in [@BuHeLa04], that Ozawa’s error (and similarly then disturbance) can even vanish in states where the input and output distributions are clearly vastly different has never been refuted or directly commented upon up to now. Further striking examples of this kind, including the analogous situation of Ozawa’s disturbance vanishing where the observable in question does get disturbed, have been given in [@BLW2013a].
Returning briefly to the quantity $\eta_O(P)$, the definition recalled above makes it evident that, unless the momentum and the disturbed momentum commute, the quantity $U^* PU-P$ does not commute with either of the two momentum observables, and therefore its measurement is not compatible with the measurement of the undisturbed or disturbed momentum. Hence there is no reason to expect that its values signify the deviation between the undisturbed and disturbed momentum, not any more than measuring kinetic and potential energy separately constitute a measurement of the Hamiltonian of a quantum particle. A similar remark applies to Ozawa’s error.
It is surprising that the experimental groups that succeeded in carefully measuring $\eps_O(A)$ and $\eta_O(B)$ as quantum mechanical expectation values made no attempt to confront these expectation values with the actual error in an approximate measurement of $A$ and the associated actual disturbance of $B$ in the individual state. The interpretation of Ozawa’s quantities is always simply taken for granted – which is understandable to a degree due to their intuitive formal appeal. But it remains puzzling that their deficiencies go unnoticed in the discussion of the experiments although they show up unmistakably (see below).
On trivially vanishing error products (again)
---------------------------------------------
In the same paper [@Werner04] Ozawa’s definition was actually given the benefit of doubt. It is mathematically correct that a state-dependent relation could be stronger by giving more detailed information, from which our “worst case” statement would trivially follow by taking a maximum. The catch here is (back then as now) that the state-dependent analogs are just not true, as Ozawa himself showed, and as was (entirely unsurprisingly) verified by the recent experiments. Indeed the failure of Heisenberg type relations in the Ozawa approach is a trivial consequence of the state dependent definition, and can be realized by models as simple as an exact position measurement, followed by an exact repreparation of the input state. There is nothing physically interesting to be understood from this failure, even if the experiments as such may have been demanding. This “failure” also applies to our operationally significant measures, as witnessed by the model example just referred to [@BLW2013a]. This said, we rush to add that a study of state-dependent error measures, like the Wasserstein-2 distance, is needed, for instance, if one wants to know how much error one has to allow for a measurement in a particular state if one wants to impose a specific bound to the disturbance of some observable in that state.
Ozawa’s inequality destroys its own (intended) interpretation
-------------------------------------------------------------
That the interpretation of $\eta_O(P)$ as a measure of disturbance is ultimately untenable can be seen most strikingly, we believe, in the following example. The quantum theory of measurement asserts the existence of error-free measurement of position with the property that the associated nonselective channel is a constant map, $\rho\mapsto\rho_0$. In this case Ozawa’s inequality \[eq. (5) in [@Roz13]\] reduces to $$\eta_O(P)\ \ge\ \frac\hbar{2\Delta X}.$$ Now, consider the situation where the input state $\rho$ is chosen to be identical to the fixed output state, $\rho=\rho_0$. In that case there is virtually no disturbance for this state, and yet the above inequality shows that $\eta_O(P)$ can be as large as one wishes (by choosing $\rho_0$ with sufficiently small $\Delta X$. By contrast, again our measure of disturbance vanishes in this case. Hence one might say that Ozawa’s inequality invalidates its own (intended) interpretation.
We are pleased to see that, in contrast to Ozawa, Rozema [*et al*]{} do not claim to have read Heisenberg’s mind. Yet they conclude with the statement that although Ozawa’s relation contradicts Heisenberg’s, “we believe that it is closer in spirit to the disturbance [*typically associated*]{} with Heisenberg’s microscope than the definition of Busch et al.” (Our emphasis). We are not so sure about the silent majority, but we trust that the matter can be clarified by scientific debate.
In fact, a curious convergence of the state-independent and state-dependent perspectives on measurement errors emerges in the qubit case. This becomes apparent by a closer look at the Vienna and Toronto experiments, which also serves to illustrate some of the shortcomings of Ozawa’s “error” highlighted above.
What the Vienna and Toronto experiments actually show
-----------------------------------------------------
It must indeed be noted that the experiments of the Vienna [@Erh12] and Toronto [@Roz12] groups are not about the Heisenberg uncertainty relations for canonical conjugate variables, but about certain qubit measurements. The Ozawa relations can be stated in this context, but so can our approach. In fact, it becomes technically much easier. For example, the compactness argument Ozawa tried to take issue with is not necessary at all. To facilitate the comparison we have in the meantime spelled out the finite dimensional version of our approach, particularly for qubit systems [@BLW2014]. We have obtained a joint measurement error trade-off relation of the Heisenberg type, which is of the following generic form: if observables $A,B$ are to be approximated in terms of jointly measurable observables $C,D$, respectively, then the errors $\Delta(A,C)$ and $\Delta(B,D)$ obey an inequality $$\label{qubit-ur}
\Delta(A,C)^2+\Delta(B,D)^2\ge (\text{\rm incompatibility of $A,B$}).$$
Now it turns out that for qubit observables of the form investigated in the experiments, Ozawa’s quantities $\epsilon_O$ and $\eta_O$ do not depend on the input state; this undermines their interpretation as state-specific error or disturbance measures. In both experiments, $\epsilon_O$ and $\eta_O$ are generally bad over-estimates of the actual state-dependent errors. Curiously, the authors of [@Vienna2] do notice the state-independence of $\epsilon_O$, and they do notice that in some states the measured distribution is identical to that of the target observable, even where these observables do not commute; but they do not seem to consider this as an indication of the failure of $\epsilon_O$ as a state-specific error measure.
The quantity $\epsilon_O$ turns out to be directly related to our state-independent error measure and can therefore serve as a rough estimate for it in the qubit case. As a consequence of this fact and , we found further that Ozawa’s quantities obey a similar trade-off relation if applied to the quantities $A,B$ and their jointly measurable approximators $C,D$: $$\label{qubit-oz-ur}
\epsilon_O(A,C)+\epsilon_O(B,D)\ge \tfrac12(\text{\rm incompatibility of $A,B$}).$$ Thus, irrespective of the fact that these quantities fail as faithful state-dependent error measures, they nevertheless give rise to a Heisenberg-type trade-off in the qubit case! This clearly shows that Ozawa’s inequality, rather than constituting a violation of Heisenberg’s error-disturbance relation, simply fails to capture its spirit. The root of this failure can be seen in the unsuitability of the error product to describe such trade-off relations in the case of discrete observables.
We have also shown that the Toronto experiment, which was conceived as a weak measurement, can be interpreted as an approximate joint measurement of smeared versions $C,D$ of qubit observable $A,B$. The values of $\epsilon_O(A,C)$ and $\eta_O(B,D)\equiv\epsilon_O(B,D)$ are there actually equal to our errors, $\Delta(A,C)$ and $\Delta(B,D)$; hence for this measurement scheme, Ozawa’s quantities also satisfy the inequality . Interestingly, for certain parameter settings the Toronto experiment is found to saturate the bound in this inequality. In fact, we would not be surprised if the data for $\epsilon_O(A,C)$ and $\epsilon_O(B,D)$ obtained in the experiments were already sufficient to check this inequality.
[10]{}
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[^1]: For mathematical physicists the attribute “universally valid” normally goes without saying.
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---
abstract: 'We investigate particle accelerators in rotating neutron-star magnetospheres, by simultaneously solving the Poisson equation for the electrostatic potential together with the Boltzmann equations for electrons, positrons and photons on the poloidal plane. Applying the scheme to the three pulsars, Crab, Vela and PSR B1951+32, we demonstrate that the observed phase-averaged spectra are basically reproduced from infrared to very high energies. It is found that the Vela’s spectrum in 10-50 GeV is sensitive to the three-dimensional magnetic field configuration near the light cylinder; thus, a careful argument is required to discriminate the inner-gap and outer-gap emissions using a gamma-ray telescope like GLAST. It is also found that PSR B1951+32 has a large inverse-Compton flux in TeV energies, which is to be detected by ground-based air Cerenkov telescopes as a pulsed emission.'
author:
- Kouichi Hirotani
title: 'High-energy Emission from Pulsar Outer Magnetospheres'
---
Introduction {#sec:intro}
============
Pulsars form the second most numerous class of objects detected in high-energy $\gamma$-rays. To date, six have been detected by the Energetic Gamma Ray Experiment Telescope (EGRET) aboard the Compton Gamma Ray Observatory. The $\gamma$-ray pulsations observed from these objects are particularly important as a direct signature of non-thermal processes in rotating neutron-star magnetospheres, and potentially should help to discriminate among different emission models.
The pulsar magnetosphere can be divided into two zones: The closed zone filled with a dense plasma corotating with the star, and the open zone in which plasma flows along the open field lines to escape through the light cylinder. The last-open field lines form the border of the open magnetic field line bundle. In all the pulsar emission models, particle acceleration takes place in this open zone. In inner-gap (IG) models, which adopts particle acceleration within several neutron-star radii above the polar-cap surface, the energetics and pair cascade spectrum have had success in reproducing the observations (e.g., Daugherty & Harding 1982, 1996). However, the predicted beam size is too small to produce the wide pulse profiles that are observed from high-energy pulsars. Seeking the possibility of a wide hollow cone emission due to flaring field lines, Muslimov and Harding (2004) extended the idea of Arons (1983) and proposed a slot-gap (SG) model, in which emission takes place very close to the last-open field line from the stellar surface to the outer magnetosphere. Since the SG model is an extension of the IG model into the outer magnetosphere, a negative magnetic-field-aligned electric field, ${E_\parallel}$, arises if the magnetic moment vector points in the same hemisphere as the rotation vector. However, the electric current induced by the negative ${E_\parallel}$ does not have a self-consistent closure within the model (Hirotani 2006, hereafter H06).
To contrive an accelerator model that predicts an outward current in the lower latitudes (within the open zone), it is straightforward to extend outer-gap (OG) models (Cheng et al. 1986; Romani & Yadigaroglu 1995, hereafter RY95; Cheng et al. 2000) into the inner magnetosphere. Extending several OG models (Hirotani et al. 2003; Takata et al. 2004), H06 demonstrated that the gap extends from the stellar surface to the outer magnetosphere, that the positive ${E_\parallel}$ extracts ions from the star as a space-charge-limited flow (SCLF), and that most photon emission takes place in the outer magnetosphere because ${E_\parallel}$ is highly screened inside the null surface owing to the discharge of the created pairs. In the present letter, we formulate the scheme (Beskin et al. 1992) in § \[sec:basic\_eqs\], apply it to three rotation-powered pulsars in § \[sec:appl\], and give a brief discussion in § \[sec:summary\].
Gap Electrodynamics {#sec:basic_eqs}
===================
We follow the scheme described in § 2 of H06 to solve the set of Maxwell and Boltzmann equations. The first equation we have to consider is the Poisson equation for the electro-static potential, $\Psi$. As space is limited, we present its Newtonian expression, $ -\nabla^2 \Psi = 4 \pi (\rho-{\rho_{{\rm GJ}}})$. If the real charge density, $\rho$, deviates from the Goldreich-Julian (GJ) charge density, ${\rho_{{\rm GJ}}}$, in some region, an acceleration electric field ${E_\parallel}\equiv -\partial \Psi/\partial s$ arises, where $s$ designates the distance along a magnetic field line.
The second equation we have to consider is the Boltzmann equations for $e^\pm$’s. Assuming a steady state in the frame of reference corotating with the magnetosphere, we obtain $$c\cos\chi \frac{\partial n_\pm}{\partial s}
+\frac{dp} {dt}\frac{\partial n_\pm}{\partial p}
+\frac{d\chi}{dt}\frac{\partial n_\pm}{\partial \chi}
= S_\pm,
\label{eq:BASIC_2}$$ where $c$ denotes the speed of light, $n_+$ (or $n_-$) the dimensionless positronic (or electronic) distribution function normalized by the local GJ number density. The temporal derivatives of momentum and pitch angle, $dp/dt$ and $d\chi/dt$, and the collision term $S_\pm$ are explicitly given in H06. Synchro-curvature radiation-reaction force is included as an external force in $dp/dt$, while the effects of inverse-Compton scatterings (ICS) and (one-photon and two-photon) pair creation processes are in $S_\pm$.
The third equation we have to consider is the Boltzmann equations for photons. Assuming a steady state, and neglecting azimuthal propagations, we obtain $$c\frac{k^r}{\vert\mbox{\boldmath$k$}\vert}
\frac{\partial g}{\partial r}
+c\frac{k^\theta}{\vert\mbox{\boldmath$k$}\vert}
\frac{\partial g}{\partial \theta}
= S_\gamma(r,\theta,c\vert\mbox{\boldmath$k$}\vert,k^r,k^\theta),
\label{eq:BASIC_3}$$ where the wave numbers $k^r$ and $k^\theta$ are given by the ray path, ($r$,$\theta$) are the polar coordinates, and the dimensionless photon distribution function $g$ is normalized by the GJ number density at the stellar surface. ICS, synchro-curvature emission, and the absorption are contained in $S_\gamma$. In H06, the rate of synchrotron emission by secondary pairs created outside the gap, was calculated assuming a constant pitch angle. However, it turns out that only 17 % of the initial particle energy is converted into radiation. In this letter, we corrected this problem by computing the pitch angle evolution of radiating particles, which increases the synchrotron cooling time and hence recovers the time-integrated, radiated energy to 100 % of the initial particle energy. Note that the gap electrodynamics investigated in H06 remains correct, despite the insufficient secondary synchrotron fluxes in H06.
To solve the Poisson equation, we impose $\Psi=0$ at the lower, upper, and inner ($s=0$) boundaries, and ${E_\parallel}=-\partial\Psi/\partial s=0$ at the outer boundary. Ion extraction rate is regulated by the condition ${E_\parallel}=0$ at $s=0$. We parameterize the trans-field thickness of the gap with $h_{\rm m}$. If $h_{\rm m}=1$ the gap exists along all the open field lines, while if $h_{\rm m}\ll 1$ the gap becomes transversely thin.
Application to Individual Pulsars {#sec:appl}
=================================
We apply the theory to three $\gamma$-ray pulsars, Crab, Vela, and B1951+32, focusing on their photon spectra. Even near and outside the light cylinder, photon emission and absorption occur effectively; thus, equations (\[eq:BASIC\_2\]) and (\[eq:BASIC\_3\]) are solved in $0<s<16{\varpi_{\rm LC}}$, where ${E_\parallel}=0$ in $\varpi>0.9{\varpi_{\rm LC}}$; ${\varpi_{\rm LC}}\equiv c/\Omega$ denotes the light cylinder radius, $\Omega$ the stellar rotation frequency, and $\varpi$ the distance from the rotation axis. The field line geometry in $0.9{\varpi_{\rm LC}}<\varpi<2{\varpi_{\rm LC}}$ mimics the aligned dipole in the force-free magnetosphere (Contopoulos et al. 1999; Gruzinov 2005). In $\varpi>2{\varpi_{\rm LC}}$, the field lines are assumed to be straight so that they connect smoothly at $\varpi=2{\varpi_{\rm LC}}$.
Crab pulsar {#sec:Crab}
-----------
We present the results for the Crab pulsar in figure \[fig:spc\_crab\], adopting a magnetic inclination of ${\alpha_{\rm i}}=75^\circ$ and a dipole moment of $\mu=4\times 10^{30}\mbox{\,G\,cm}^3$, which is close to the value ($3.8\times 10^{30}\mbox{\,G\,cm}^3$) deduced from the dipole radiation formula. It follows that the observed pulsed spectrum from IR to VHE can be reproduced, provided that we observe the photons emitted into $75^\circ<\theta<103^\circ$, where $\theta$ denotes the photon propagation direction measured from the rotational axis. Because of the aberration of light, it is reasonable to suppose that photons emitted in a certain range of $\theta$ comes into our line of sight in an obliquely rotating three-dimensional magnetosphere (e.g., RY95). The flux rapidly decreases with decreasing $\theta$ for $75^\circ<\theta<93^\circ$, because ${E_\parallel}$ is highly screened in the inner part of the gap. Nevertheless, an average over $75^\circ < \theta < 103^\circ$, which includes negligible flux between $75^\circ < \theta < 89^\circ$, achieves the current objective, because the spectral normalization can be fitted (within a factor of a few) without changing the spectral shape, by adjusting $h_{\rm m}$.
ICS takes place efficiently near and outside the light cylinder (Aharonian & Bogovalov 2003), because the magnetospheric IR photons, which are emitted along convex field lines, collide with the gap-accelerated positrons near the light cylinder, where the field lines are concave. Most of such upscattered photons, as well as some of the high-energy tail of the curvature component, are absorbed by the $\gamma$-$\gamma$ collisions. As a result, there is a gradual turnover around 10 GeV, which forms a striking contrast with the steep turnover predicted in IG models due to magnetic pair creation. The primary curvature component appears between 100 MeV and 10 GeV, while the secondary synchrotron component appears below a few MeV. Between a few MeV and 100 MeV, the ICS component dominates, because the secondary pairs that have been cooled down below a few hundred MeV efficiently up-scatter magnetospheric UV and X-rays to lose energy. Similar spectral shapes are obtained for different values of ${\alpha_{\rm i}}$, $\mu$, $h_{\rm m}$, provided that the created current is super GJ, by virtue of the negative feedback effects (H06).
Vela pulsar {#sec:Vela}
-----------
Next, we present the results for the Vela pulsar in figure \[fig:spc\_vela\] (left). Taking an angle average over $75^\circ<\theta<103^\circ$ (solid line), we can reproduce the observed pulsed spectrum, except for the RXTE results. The primary curvature component appears between 100 MeV and 10 GeV, while the secondary and tertiary synchrotron components appear below 100 MeV. ICS is negligible for the Vela pulsar because of its weak magnetospheric emission. Similar spectral shapes are obtained for super-GJ solutions, even though we have to adjust $h_{\rm m}$ to obtain an appropriate flux.
In the right panel, we compare the present results with IG (dotted) and OG (dashed) models, where the dash-dotted line denotes the averaged flux for $75^\circ <\theta< 107^\circ$. It follows that the spectrum between 10 and 100 GeV crucially depends on the angles in which the photons that we observe are emitted. Thus, to quantitatively predict the $\gamma$-ray emission from the outer magnetosphere, it is essential to examine the three-dimensional magnetic field structure near and outside the light cylinder.
PSR B1951+32 {#sec:1951}
------------
Thirdly and finally, we present the spectrum of B1951+32 in figure \[fig:spc\_1951\] (left). It follows that the ROSAT and EGRET fluxes are reproduced by taking the flux average in $75^\circ<\theta<103^\circ$ (solid line), except for 17 GeV flux, which was derived from only two photons. Because of its weak magnetic field, less energetic synchrotron photons reduce the absorption of the ICS component. The reduced absorption results in a small synchrotron flux, which further reduces the absorption outside the light cylinder, leading to a large, unabsorbed TeV fluxes. It also follows that the spectrum turns over at lower energy than the IG model prediction (dotted curve in the right panel; Harding 2001).
It should be noted that the predicted ICS flux (above 100 GeV) represents a kind of upper limits, because it is obtained by assuming that all the magnetospheric synchrotron photons illuminate the equatorial region in which gap-accelerated positrons are migrating. Some of such VHE photons materialize as energetic secondary pairs, emitting the synchrotron component between a few keV and 100 MeV. Between 100 MeV and 30 GeV, the primary curvature component dominates, which represents the absolute flux (instead of upper limits). Some of such curvature photons have energies above 10 GeV and are efficiently absorbed to materialize as less energetic pairs, which emit synchrotron radiation below a few keV. Thus, the spectrum below a few keV also represents the absolute flux.
Summary and Discussion {#sec:summary}
======================
To sum up, the self-consistent gap solutions basically reproduce the observed power-law spectra below a few GeV for the three pulsars examined. The cut-off spectra between 10 GeV and 100 GeV strongly reflect the three-dimensional magnetic field configuration near the light cylinder; thus, a discrimination between IG and OG models (e.g., using GLAST) should be carefully carried out. If pulsations are detected above 100 GeV, it undoubtedly indicates that the photons are emitted via ICS near the light cylinder, because VHE emissions cannot be expected in IG models.
Since our analysis is limited within the two-dimensional plane formed by the magnetic field lines that thread the stellar surface on the plane containing both the rotational and magnetic axes, azimuthal structure is still unknown. Thus, we cannot present pulse profiles, phase-resolved spectra, or the polarization angle variations in this letter. Since the gap is most active in the outer part of the magnetosphere (unlike previous OG models, which adopt the vacuum solution of the Poisson equation and hence a uniform emissivity), and since the photons will be emitted along the instantaneous particle motion measured by a static observer (unlike the treatment in the OG model of RY95, who assume a very strong aberration of light near the light cylinder), it is possible that the predicted pulse profiles and so on are quite different from previous OG models. These topics will be discussed in the subsequent paper, which deals with the three-dimensional gap structure near the light cylinder.
The pulsed TeV flux from PSR B1951+32 can be predicted to be above $5 \times 10^{10}$ JyHz, provided that a certain fraction (more than $30\%$) of the magnetospheric soft photons illuminate the equatorial region. However, if the poloidal field lines are more or less straight near the light cylinder, as demonstrated by Spitkovsky (2006, fig. 2) for an oblique rotator, the equatorial region may not be efficiently illuminated. In this case, the VHE flux will decrease from the current prediction. There is room for further investigation how to extend the present analysis into three dimensions, and to combine it with time-dependence three-dimensional force-free electrodynamics. The author is grateful to Drs. N. Otte, R. Taam, J. Takata for helpful suggestions. This work is supported by the Theoretical Institute for Advanced Research in Astrophysics (TIARA) operated under Academia Sinica and the National Science Council Excellence Projects program in Taiwan administered through grant number NSC 95-2752-M-007-006-PAE.
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|
---
abstract: |
A quantum measurement-like event can produce any of a number of macroscopically distinct results, with corresponding macroscopically distinct gravitational fields, from the same initial state. Hence the probabilistically evolving large-scale structure of space-time is not precisely or even always approximately described by the deterministic Einstein equations.
Since the standard treatment of gravitational wave propagation assumes the validity of the Einstein equations, it is questionable whether we should expect all its predictions to be empirically verified. ‘In particular, one might expect the stochasticity of amplified quantum indeterminacy to cause coherent gravitational wave signals to decay faster than standard predictions suggest. This need not imply that the radiated energy flux from gravitational wave sources differs from standard theoretical predictions. An underappreciated bonus of gravitational wave astronomy is that either detecting or failing to detect predicted gravitational wave signals would constrain the form of the semi-classical theory of gravity that we presently lack.
author:
- Adrian
date: 'October 2012 (revised March 2013)'
title: 'Might quantum-induced deviations from the Einstein equations detectably affect gravitational wave propagation?'
---
Introduction
============
One difficulty theoretical physicists currently face is that, as the subject has grown larger and speculative attempts to address fundamental problems have multiplied, our collective knowledge has become increasingly fragmented. Questions which are at the forefront of the attention of one group of people can be pretty much neglected, or not even recognised, by others. Indeed, even individuals may display a version of this: because our attention is selective and trained, we can end up functioning according to a sort of doublethink, in which we note a problem in one context and neglect it in others. I suspect this is actually much more common than we generally realise.
Here, following a broader programme ([@akcausalqt; @aklocalcausality; @salart; @bgqt; @satellite]; see also [@penrose; @diosi; @pearlesquires] for earlier distinct but related ideas in this area) of trying to test underexplored foundational questions relating quantum theory and gravity, I suggest one area — gravitational wave physics — where this phenomenon may be at work. The problem is this. On the one hand, the standard theory of propagating gravitational waves treats them as perturbations of the Einstein gravitational field equations. On the other hand, there is no way of incorporating unpredictable quantum events into a classical theory of gravity described by the Einstein equations.
This is not only because of small corrections arising from an as yet unknown quantum theory of gravity. If this were the only issue, it might be easier to defend the case that the standard perturbative treatment of gravitational waves is likely to be essentially unaltered by quantum corrections, at least in regions where the gravitational field is not too strong. The more immediate problem is that, whenever a quantum measurement type interaction takes place — whether a deliberate measurement by a human observer or a naturally occurring event — it can produce any of a number of macroscopically distinct results from the same initial state. According to the standard understanding of quantum theory, these measurement outcomes are intrinsically probabilistic. Not only can they not be predicted in advance by quantum theory, but there are very strong reasons [@bellbook; @chsh; @cr; @pbr] to believe that no underlying deterministic theory allows us to predict them. These macroscopically distinct outcomes lead to distinct space-times and matter distributions. Each of several possible space-times and matter distributions could thus emerge from the same initial state. If the measurement event is suitably amplified, their differences can be arbitrarily large. Since general relativity is deterministic, it follows that the Einstein equations cannot even approximately describe the large-scale structure of space-time around measurement events.
If we try to describe the space-time physics in the vicinity of the measurement event by some classical stress-energy tensor and metric, it seems that we need to introduce some stochastic source which creates an unpredictable macroscopic perturbation of the metric and tensor, at some point – or, probably more accurately, in some region – in the vicinity of the event. However, not only are the Einstein equations inconsistent with these perturbations, but moreover we know of no generally valid way of incorporating them into some semi-classical theory coupling the metric to a classical quantity derived from the quantum stress-energy tensor, or by some stochastic modification of the Einstein equations.
Looking at current knowledge without a predefined theoretical agenda, then, all we can really say for sure is that general relativity and quantum theory both work well in their respective domains. However, even characterizing precisely those domains of confirmed validity is subtler and harder than it first seems, as the comments above illustrate. It is even harder to justify confidence that we now understand all the principles underlying a unified theory. It’s not evident that mainstream ideas work, and while it’s certainly not evident that comparatively undeveloped alternatives will work either, they do exist.
For example, it is certainly possible to imagine unifications in which both general relativity and quantum theory work as good approximations in their respective domains, and in which gravity is quantized, but nonetheless the structure of space-time is also constrained by additional laws that modify the probability of each space-time and are defined by intrinsically geometric rules that do not follow from any quantum theory [@bgqt]. It’s also possible to imagine theories in which gravity is not quantized at all, and the laws of nature define some probability distribution on classical space-times with quantum matter distributions [@bgqt].
The general hypothesis that gravity and quantum state collapse could be linked [@penrose; @diosi; @pearlesquires], via fundamentally non-unitary dynamical laws is also intriguing, even though it too is hard to make into a precise theory.
Also, even if one of the more currently popular approaches to quantum gravity [*is*]{} correct, it is very unclear how to derive from it a phenomenological higher-level theory that couples microscopic quantum matter with macroscopic events in space-time, or precisely what features such a theory would have.
To reiterate, the point here is not to advocate these comparatively undeveloped alternative ideas, but simply to underline that unifying gravity and quantum theory is an open subject and there are many un(der)explored possibilities. Even in macroscopic regimes with weak gravitational fields, we simply do not have a theory of matter and space-time good enough to fit all observable data. This problem occupies a peculiar status in modern physics: it cannot be denied that the problem is there, but yet most discussions of quantum theory and gravity ignore it. The gap in the literature is so glaring that one almost gets the impression that it is somehow seen as scientifically unsophisticated to look for a theory – even a provisional phenomenological theory – that actually fits the available empirical data.
To sum up: [*the Einstein equations do not actually correctly describe the large-scale structure of space-time and finding equations that do is an open question*]{}. It thus doesn’t seem so obvious that the long-range propagation of gravitational waves is necessarily correctly described by considering them as perturbations of the Einstein equations.
The rest of this paper attempts to flesh out this point mostly by conceptual, rather than mathematical, argument. In mitigation, I would stress again that we know nothing for certain about unifying quantum theory and gravity, and the world possibly already has more than enough mathematically rigorous, but conceptually problematic, and quite likely ultimately irrelevant, calculations based on speculative mathematical ideas about how to solve the problem.
Our ultimate aim, of course, is to test precise mathematical theories against quantitative experimental data, but just at the moment we need new ideas about where to look. It seems to me there are strong reasons to try a different style of scientific reasoning: namely, to look at interesting experiments and observations where we don’t yet know [*for sure*]{} what theory predicts but can – now, or soon – get an empirical answer, and to ask whether there is any semi-plausible phenomenological model or intuition that might contradict the standard expectation (if there is one). Either we verify with certainty interesting features of gravitational physics that were previously either ignored completely or assumed without compelling evidence, or (even better) we learn something new and surprising.
Prior tests of probabilistic semi-classical gravity
===================================================
The Page-Geilker Experiment
---------------------------
The fundamental problem in constructing semi-classical gravity theories was illustrated by a very simple experiment carried out some time ago by Page and Geilker [@pagegeilker]. Page and Geilker’s aim was to refute conclusively the hypothesis that a classical gravitational field couples to quantum matter via the semiclassical Einstein equations $$\label{semiclassical}
G_{\mu \nu} = 8 \pi \langle \psi \, | \, T_{\mu \nu} \, | \, \psi \rangle \, ,$$ where $G_{\mu \nu}$ is the Einstein tensor and $T_{\mu \nu}$ the quantum stress-energy operator. Their hypothesis presupposes an Everettian interpretation of quantum theory, so that the matter field quantum state, $ | \, \psi \rangle$, is supposed to evolve unitarily without collapse.
As Page and Geilker noted, this hypothesis already seemed intrinsically unlikely (perhaps even incredible) before they carried out their experiment, since it is hard to imagine how it could lead to a cosmological theory which accounts for our observation of gravitational fields generally consistent with those predicted by the Einstein equations from the observed positions of astronomical bodies. Page and Geilker’s motivation for their experiment is thus seriously questionable. Perhaps, though, one [*could*]{} imagine some form of theory in which a classical gravitational field couples to the expectation value of quantum matter, while the quantum matter state over time collapses towards definite values for $T_{\mu \nu}$. In any case, it still seems good to have conclusive experimental confirmation even of very solidly based theoretical expectations when, as here, we have no complete theory.
The experiment proceeded by counting the number of decays detected from a radioactive source in a given time interval, and then manually placing large ($\approx 1.5$ kg) lead balls in one of two configurations, with the choice of configuration depending on the decay count. A Cavendish torsion balance, sensitive enough to distinguish between the two configurations, was used to measure the local gravitational field. To good approximation, Eq. (\[semiclassical\]) predicts that in each run the experimenter should (whichever of the two configurations they place the masses in) observe a gravitational field defined by the weighted average of the fields corresponding to the two possible configurations. As most expected, the results were consistent with the hypothesis that the gravitational field is determined by the configuration of the masses chosen in any given run of the experiment, and inconsistent with Eq. (\[semiclassical\]).
We can flesh out the implications of the experiment — or, to capture the historical flow of ideas better, perhaps one should say the implications of the generally held prior intuition that its results would be those which were in fact observed — by looking at three possible solutions to the Einstein equations.
First, consider the classical metric and matter fields, which we denote by $( g^0_{\mu \nu} , \phi^0 )$, in the neighbourhood of some spacelike hypersurface $S$ before the point at which a Page-Geilker experiment is carried out. By “the classical metric and matter fields”, we mean here the fields that would ordinarily be defined by someone trying to model the local space-time neighbourhood using general relativity — for instance, an engineer, trying to predict how small lumps of matter will evolve, and doing so as precisely as is possible without taking quantum theory into account. Let us suppose there is some way of fitting these data to a solution of the Einstein equations, using some well-defined and natural recipe (not necessarily Eq. (\[semiclassical\])) to obtain a classical stress-energy tensor from the quantum matter field, and denote the corresponding spacetime by $\Sigma_0$.
Now consider the classical metric and matter fields, $( g^1_{\mu \nu } , \phi^1 )$ and $( g^2_{\mu \nu } , \phi^2 )$, in the neighbourhood of a spacelike hypersurface $S'$ after the point at which a Page-Geilker experiment is carried out. Let us suppose these data can also be fitted to solutions of the Einstein equations, using the same recipe for a classical stress-energy tensor as before, and denote the corresponding spacetimes by $\Sigma_1$ and $\Sigma_2$.
Since $\Sigma_1$ and $\Sigma_2$ describe macroscopically distinct matter configurations on $S'$, they are not identical, and so cannot both be identical to $\Sigma_0$. In fact, since neither of them is preferred in any way, one would not generally expect [*either*]{} of them to be identical to $\Sigma_0$, assuming that the recipe used to obtain the stress-energy tensors depends in any natural way on the relevant fields. (Obviously, if completely arbitrary recipes are allowed, one could contrive things so that one of them equals $\Sigma_0$. For instance, one could define the recipe for obtaining $\Sigma_0$ to involve first studying the possible experimental outcomes, then constructing $\Sigma_1$ and $\Sigma_2$, and then simply setting $\Sigma_0$ to be equal to one of them.)
To sum up, then, if we can find a way of describing the data before and after the experiment by piecewise continuous solutions of the classical Einstein equations, they will generally be solutions that form part of different spacetimes. In this description (if there is indeed such a description), it is as though some sort of stochastic jump takes place, starting from one solution, and arriving at one of two alternative solutions, both distinct from the original.
Is there such a description? Can the spacetime we actually observe be described by piecewise continuous solutions of the Einstein equations? I don’t think we know for sure: some sort of smoothing could take place in the vicinity of Page-Geilker experiments, for instance. But it seems to be generally tacitly assumed — and it seems to be necessary to assume, in order to explain experimental data — that this is at least approximately the case. For, on the one hand, if it were not the case that large regions of spacetime are well described by a solution to the Einstein equations, we would not be able to account theoretically for any of the confirmed predictions of general relativity. On the other hand, as we have just argued, the sort of macroscopic indeterminacy exhibited by the Page-Geilker experiment implies that spacetime cannot be described globally by a single solution of the Einstein equations.
Probing gravitational non-locality and the Salart et al. experiment
-------------------------------------------------------------------
A more recent proposal [@aklocalcausality] with some related motivations was to look for direct evidence of violations of Bell’s local causality in the gravitational field. Recall that Bell experiments (modulo loopholes) show that any hidden variable theory underlying quantum theory must violate Bell’s condition of local causality. Since non-local hidden variable theories are theoretically uncompelling and difficult to reconcile with relativity, this gives much stronger evidence that the outcomes of quantum experiments are indeed inherently unpredictable. The evidence that the classical gravitational field evolves probabilistically, though, is less direct. While the Page-Geilker experiment [*appears*]{} to confirm that the gravitational field evolves probabilistically, one could still imagine an underlying deterministic semi-classical gravity theory, sensitive to microscopic variables, that predicts each observed evolution, in the same way that de Broglie-Bohm theory and other deterministic hidden variable theories reproduce the predictions of quantum theory. We would like to be able to argue directly that any deterministic phenomenological theory of gravity [*must*]{} be non-local. Since a non-local gravity theory would be very hard to reconcile with either special or general relativity, this would be a compelling reason for abandoning all hope for deterministic semi-classical theories.
Another strong motivation for verifying this point directly is that gravitational collapse models highlight another possible loophole in the interpretation of all Bell experiments to date. This “collapse locality loophole” [@akcausalqt] arises because collapse models suggest that a definite measurement outcome occurs only after macroscopic amplification to a particular scale (which depends on the parameters of the collapse model). If this is correct, to exclude locally causal explanations we need Bell experiments that ensure that collapses, and thus definite measurement outcomes, take place in spacelike separated regions in the two wings. [*No Bell experiment to date ensures this for the full range of collapse model parameters consistent with known experiment*]{} [@akcausalqt].
This motivation is further reinforced by the observation [@akcausalqt] that there [*are*]{} ways in which a consistent theory combining quantum theory and gravity could conceivably produce the outcomes observed in all Bell experiments to date and nonetheless allow only locally causal evolutions of the metric. Of course, models incorporating these ideas have highly non-standard properties, and may be theoretically problematic as well as ad hoc. Still, as with the Page-Geilker experiment, clear experimental data would be much preferable to strong theoretical intuitions and arguments.
A beautiful experiment investigating this possibility was carried out by Salart et al. [@salart], showing that the local causality loophole can indeed be closed at least for gravitational collapse models whose collapse times and scales agree with theoretically motivated estimates proposed by Penrose [@penrose] and Diosi [@diosi]. Further more conclusive experiments have been proposed [@satellite], with the ultimate aim of directly measuring non-locally correlated gravitational fields in such a way that these measurements are themselves completed in space-like separated regions.
Natural measurement-like events
===============================
Of course, quantum measurement-like events with macroscopically distinct consequences take place without any artificial help. On Earth, fissioned particles and cosmic rays leave tracks in mica; frogs can respond to the stimulus of a single photon; a single gamma ray or charged particle can trigger a cancer; the bouncing of photons and cosmic rays off dust particles must from time to time determine the formation or otherwise of a particular macroscopic clumping. Each of these outcomes is effectively a quantum measurement of the position of a particle whose wave function was previously delocalised. On the cosmological scale, quantum fluctuations are believed to have seeded the instabilities that led to galaxy formation.
It would be very interesting indeed to try to quantify the degree to which quantum noise, bubbling up from the microscopic realm, affects the predictability of the macroscopic world in general, and in particular to characterise the degree and type of the resulting deviations from Einstein’s equations. This project is beyond our present scope, though.
For the purposes of the present discussion, we need only take the point that these deviations occur naturally, and presumably have been doing so since very early cosmological times. This is why the outcome of the Page-Geilker experiment was generally (perhaps even universally) anticipated. In other words, while the Page-Geilker experiment is a good illustration of the point that our observed space-time deviated from Einstein’s equations, we do not actually need to appeal to it to make that point.
Quantum gravity: resolution or distraction?
===========================================
According to one school of thought, at this point in the discussion one should throw up one’s hands, regret the fact that we don’t yet have a quantum theory of gravity, and accept that we can’t productively advance the discussion further without one.
It seems to me far from evident that we should heed this counsel of despair. I can see two reasons for optimism.
First, we might not actually need a quantum theory of gravity at all. Second, even if we do, it ought to imply some effective phenomenological theory of classical gravity which incorporates stochastic fluctuations into general relativity. In the first case, we might hope for some [*fundamental*]{} theory which incorporates stochastic fluctuations into general relativity; alternatively, we might hope for a theory based on different principles, which again should imply an effective phenomenological theory of classical gravity of the type just mentioned.
In all these cases, it is reasonable to try to explore how a classical gravity theory with stochastic fluctuations might be probed experimentally. But we don’t have such a theory. Perhaps the best hope, then, is that experiment might guide us to the right theory, if we can at least identify what to look for experimentally.
The following two sub-sections flesh out these arguments in more detail.
Do we need quantum gravity to explain the Page-Geilker experiment?
------------------------------------------------------------------
Embarrassingly, our best theory of gravity, general relativity, has no way of consistently incorporating the results of macroscopically amplified quantum measurement events, whether they occur naturally or are created artificially as in a Page-Geilker experiment. The current conventional wisdom suggests that this embarrassment stems from our failure to devise a consistent quantum theory of gravity.
There is — the standard argument runs — no fully consistent way to couple a classical gravitational field to quantum matter fields: the gravitational field also needs to be quantised. We would expect — the argument proceeds to suggest — that in a full quantum theory of gravity, the gravitational field would evolve so as to be (to very good approximation) correlated with the matter fields in any given branch of the universal wave function. In particular, we would expect a full quantum theory of gravity to predict the observed outcome of the Page-Geilker experiment: this is why Page and Geilker provocatively titled their paper “Indirect Evidence for Quantum Gravity”. More generally, we would expect a full quantum theory of gravity to predict that the gravitational field should be correlated with the positions of astronomical bodies in the way we observe.
Of course, this argument begs several key questions. Aside from the problems of principle in unifying quantum theory and gravity, discussed above, there is the problem of making scientific sense of purely unitary quantum theory. We do not have, despite nearly fifty years’ of effort, any clearly consistent and logically compelling account of how Everett’s original intuition might be fleshed out into a clearly and carefully justified interpretation of a unitarily evolving universal wave function. (State-of-the-art reports and assessments of recent attempts can be found in Ref. [@mwbook].)
Probing an effective theory derivable from quantum gravity
----------------------------------------------------------
What if some version of quantum gravity is correct, though? Suppose, for example, we find some rigorously defined way of carrying out path integrals over gravitational and matter field configurations, and find some evidence that it gives correct answers. In order to understand large-scale gravitational physics, we would [*still*]{} need some (presumably) phenomenological effective theory, derived from our fundamental quantum gravity theory, which characterises the quasiclassical behaviour of matter and gravity that we actually observe. (In Gell-Mann and Hartle’s terminology [@gmh], we would need some way of characterising our own quasiclassical domain within this hypothetical quantum gravitational or quantum cosmological theory.)
In particular, this higher-order theory would need to be consistent with the Page-Geilker experiment and with the observed correlations of gravitational fields and astronomical bodies. It thus seems a reasonable conjecture — suggested by observational evidence, and contradicted by nothing we know about quantum gravity — that we would end up with some sort of stochastically modified version of general relativity, albeit in this case as a derived effective theory rather than a fundamental theory. If so, one might make the further reasonable-looking guess that the propagation of gravitational waves is approximately described by considering them as perturbations of the gravitational field [ *within this higher-order quasiclassical theory*]{}
What happens to gravitational waves in a stochastic modification of general relativity?
=======================================================================================
Without knowing the details, one can only guess. So, without further ado, I shall. A plausible guess, it seems to me, is that stochastic fluctuations break up the coherence of propagating waves. It is difficult to hear someone shouting in a high wind, not only because the noise of the wind drowns out the propagating sound wave, but also because the turbulence causes its amplitude to decay faster than in still air.
If the level of stochastic fluctuations is constant throughout a region in which a wave propagates, the simplest guess would be that the wave amplitude decays by a factor exponential in the region length, in addition to the normal approximately inverse square law decay. Without knowing the theory, one can’t estimate the value of the exponential constant — but if this guess is right, and if gravitational wave astronomy turns out nonetheless to be viable, one might be able to estimate it from observational data, and thereby get quantitative data characterising an important feature of the relation between quantum theory and gravity.
This raises the possibility that the stochastically induced decay of gravitational waves could conceivably prevent gravitational wave astronomy from being viable with presently envisaged gravitational wave detectors. If so, of course, gravitational wave astronomy’s loss would be gravitational theory’s gain.
What about the binary pulsar observations?
==========================================
If one suggests the possibility that the standard account of gravitational wave physics might not be correct, one has to deal with the counter-argument that observations of binary pulsars [@binarypulsar] have already confirmed the standard account to a very impressive degree of precision. This counter-argument has no force against the speculations considered here, though. The suggestion is not that binary pulsars do not emit gravitational waves, and thereby lose energy, as standard theory predicts. The suggestion is, rather, that the gravitational waves lose coherence, and thus decay faster than expected, as they propagate through space, and hence in particular that gravitational wave signals reaching Earth might be weaker than anticipated. Careful observation of a drum vibrating in the distance would reveal that it is losing energy by radiating sound waves; nonetheless, the sound of the drum will not propagate as far in a strong wind. There is no evident inconsistency here.
Comparing Quasiclassical Gravity and Quasiclassical Electrodynamics
===================================================================
To what extent are the problems we raise about our understanding of quasiclassical physics specific to gravity? In particular, are there any reasons to think that classical gravitational waves might behave differently from classical electromagnetic waves?
In considering these questions, it’s helpful first to consider quasiclassical electrodynamics in Minkowski space. Clearly, some of the points made above apply. In particular, we can carry out Bell experiments and ensure that, on each wing, a source of electromagnetic waves behaves differently depending on the measurement setting and outcome on that wing, and that the measurement settings themselves are locally determined by random quantum events. For example, a charged sphere could be move in any of four different ways, depending on the two measurement choices and two possible outcomes, and the measurement choices could be determined, just before the measurements are made, using bits produced by quantum random number generators. Since the outcomes of Bell experiments are nonlocally correlated, we expect this to produce nonlocal correlations in the electromagnetic fields propagating from the regions of the two measurements.
Now, this probably has not been directly tested in experiments to date, and I am not sure we can in principle rigorously exclude models (with very counterintuitive features) that agree with experiments to date but predict that nonlocal correlations of classical electromagnetic fields cannot in fact be observed. Of course, this would be a very surprising outcome indeed. We ignore the possibility here, since our aim is to understand whether one might have possible reasons to look for unexpected behaviour in quasiclassical gravitational physics even if there are no analogous surprises in quasiclassical electrodynamics.
Assuming, then, that nonlocal correlations can be created in macroscopic electromagnetic fields, it follows that quasiclassical electrodynamics in the real world cannot be described by an underlying local deterministic model. Note, though, that the nonlocalities we introduced arise entirely from nonlocal correlations in the motion of sources. Given a description of the motion of each source, we can calculate the subsequent behaviour of the electromagnetic fields it generates. Since electrodynamics is a linear theory, we can obtain a complete solution by superposing the contributions from the various sources. This gives a strategy for building a phenomenological model of quasiclassical electrodynamics in the presence of quantum unpredictability and nonlocality: first apply the predictions of quantum theory to give a model of additional stochastic (and nonlocally correlated) forces acting on the sources, and then solve to obtain the fields. Adding forces that alter the motion of the sources does not affect charge conservation, so in such a model we still have $\partial_{\mu}
J_{\mu} = 0$.
It would be wrong to suggest this gives a rigorous understanding of the relationship between quantum and quasiclassical electrodynamics in Minkowski space. We do not even have a completely rigorous definition of quantum electrodynamics as a non-trivial theory. Nor do we have a precise general prescription for how to obtain quasiclassical equations of motion from quantum theory, either for electrodynamics or for any other physically relevant theory. However, we [*do*]{} at least have an ansatz for dealing with the quasiclassical consequences of quantum experiments with unpredictable and nonlocally correlated outcomes, and this ansatz does not violate the conservation laws necessary for a consistent solution of the electrodynamic equations.
Now compare the situation when we try to model an analogous experiment in which the measurement choices and outcomes of Bell experiments determine the motion of massive objects on each wing, with the measurement choices again locally determined by quantum random number generators. As noted earlier, we cannot model the quasiclassical physics by extrapolating the predictions of general relativity from data on a spacelike hypersurface before the Bell experiment, since general relativity is deterministic and the Bell data are not. So far the analogy with electrodynamics holds, since electrodynamics is also deterministic. However, we run into further problems in this case. To define any consistent solution of the Einstein equations, we need the local conservation of stress-energy, $D_{\nu} T^{\mu \nu}
= 0$. We know of no generally covariant quasiclassical model of the possible outcomes of quantum measurement-like processes that preserves stress-energy and is consistent with general relativity where quantum effects are negligible. (Indeed, even non-relativistic dynamical collapse models [@ghirardi1986unified; @GPR], which might be the best guesses at phenomenological descriptions of the quasiclassical physics emerging from measurement interactions, violate conservation of energy.) Without such a model, it seems our best description of quasiclassical gravitational physics would be by models which generally obey the Einstein equations but have singularities or discontinuities at or in the vicinity of quantum measurement events. And if that [*were*]{} the best possible description, the standard classical derivation of gravitational wave propagation would break down in these regions.
To be sure, there are further uncertainties here. If these discontinuities are physically real, should we expect them to affect the propagation of electromagnetic radiation in the same way as they affect the propagation of gravitational waves? If so, of course, any effect is likelier to be evident in standard (electromagnetic wave observation) astronomy than in gravitational wave astronomy, and the absence of any observed effect to date is a strong constraint. On the other hand, we have a quantum theory of electromagnetism and no quantum theory of gravity. And, if there [*is*]{} a quantum theory of gravity from which quasiclassical solutions obeying Einstein’s equations with discontinuities emerges, we have no clear reason to think that coherent beams of gravitons and photons should scatter similarly from the discontinuities – indeed one might guess that gravitons are more directly affected than photons by a discontinuity in the classical field generated by gravitons.
Some may nonetheless hold the intuition that we should expect exactly the same effects in quasiclassical gravity and quasiclassical electrodynamics. The points made here do not refute this possibility, but they do give significant reasons to query it.
Summary
=======
In this paper, we raised a possibility that does not seem to have been considered: that stochastic corrections to the Einstein equations dissipate gravitational waves. Such stochastic corrections could either arise directly from a fundamental theory or as a phenomenological effect resulting from quantum gravity (or some other presently unknown type of theory). Either way, our guess at their effect on gravitational wave propagation is not provable given the present state of theoretical understanding. But is it obviously wrong, or totally implausible? If, as we suggest, not, it seems a possibility to be kept in mind if and when gravitational wave astronomy produces data, null or otherwise. We hope too that the questions raised here may encourage more attention to be focussed on the problem of finding realistic quasiclassical descriptions of gravitational physics in the presence of quantum measurements, through Bell experiments and otherwise.
Acknowledgements
================
This work was partially supported by a Leverhulme Research Fellowship, a grant from the John Templeton Foundation, and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
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A. Kent. A proposed test of the local causality of spacetime. , pages 369–378, 2009.
D. Salart, A. Baas, JAW Van Houwelingen, N. Gisin, and H. Zbinden. Spacelike separation in a [B]{}ell test assuming gravitationally induced collapses. , 100(22):220404, 2008.
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abstract: 'The synthesis model for the cosmic X–ray background (XRB) –based on the integrated emission of Active Galactic Nuclei (AGNs)– is complemented with new observational results. We adopt the most recent estimates of the AGN X–ray luminosity function and evolution. We adopt the column density distribution of type 2 AGNs observed in the local Universe, instead of describing it as a set of free parameters. We maintain the standard assumptions that type 2 AGNs have the same luminosity function –apart from a constant factor– and the same evolution of type 1s, and that the spectral shapes of both types are independent of redshift. We explore various parametrizations of the data, and in all cases we find that the XRB can be fitted, but the number density ratio of type 2s to type 1s must be higher than the local value, and/or the hard X–ray counts in the 2–10 keV band and the preliminary BeppoSAX counts in the 5–10 keV band are underestimated at a level of a few $\sigma$. These results consistently suggest that type 2 AGNs could undergo a faster evolution than type 1s, or other sources with hard X–ray spectra might contribute to the XRB at intermediate or high redshifts.'
author:
- 'R. Gilli'
- 'G. Risaliti'
- 'M. Salvati'
date: 'Received / Accepted '
title: 'Beyond the standard model for the cosmic X–ray background'
---
Introduction
============
The cosmic X–ray background (XRB) above $\sim 1$ keV is the result of the integrated emission of discrete sources, since the contribution of any intergalactic hot medium must be negligible (Wright et al. 1994). In the soft X–ray band from 0.5 to 2 keV the largest fraction of the XRB has already been resolved into sources (Hasinger et al. 1998), most of which turned out to be broad line active nuclei (Schmidt et al. 1998), i.e. Quasi Stellar Objects (QSOs) and Seyfert galaxies of type 1. The spectra of these sources are however too steep to reproduce also the hard XRB at several tens of keV, where the bulk of the energy resides, and a population of objects with flatter spectra is therefore required.
The most popular synthesis models of the XRB are based on the so–called unification schemes for Active Galactic Nuclei (AGNs), where the orientation of a molecular torus surrounding the nucleus determines the classification of the source. At a zeroth–order approximation level, sources observed along lines of sight free from the torus obscuration should have unabsorbed X–ray spectra and optical broad lines (type 1 AGNs), while sources seen through the torus should have absorbed X–ray spectra and appear as narrow line objects in the optical (type 2 AGNs, e.g. Seyfert 2 galaxies).
In this framework type 2 AGNs provide a natural class of sources with X–ray spectra flattened by absorption. The intrinsic X–ray luminosity function (XLF) of type 2 objects is unknown and has been usually assumed to be the same as the one derived for type 1s (e.g. Boyle et al. 1993), apart from a normalization factor. The cosmological evolution has also been taken identical for type 1s and type 2s. Under these assumptions it has been shown that the broad band 3–100 keV spectrum of the XRB can be reproduced by an appropriate mix of unabsorbed and absorbed AGNs (Matt & Fabian 1994; Madau, Ghisellini & Fabian 1994; Comastri et al. 1995, hereafter Co95). The number ratio $R$ of type 2 to type 1 objects, as well as the distribution of the absorbing column densities $N_{\rm H}$, are key parameters of the models; these have been assumed to be independent of redshift and of intrinsic source luminosity, and have been treated as free parameters in the fitting procedure. Since the overall parameter space of the models is quite large and a good fit to the XRB can be obtained with different set of values, it is important to compare the model predictions with the largest number of observational constraints. Indeed, Co95 showed that the source counts in the 0.5–2 keV and 2–10 keV energy bands, as well as the redshift distributions, could successfully be reproduced by their model.
Very recently an additional set of observational constraints has become available. Deep surveys from ROSAT have extended our knowledge to the low luminosity part of the AGN XLF (Miyaji, Hasinger & Schmidt 1999a, hereafter Mi99a). Contrary to previous results (Boyle et al. 1993; Page et al. 1996; Jones et al. 1997) a pure luminosity evolution (PLE) of AGNs with redshift is no longer consistent with the data, and a luminosity dependent density evolution (LDDE) is required. From the X–ray data of an optically selected sample of Seyfert galaxies Risaliti, Maiolino & Salvati (1999) have determined the $N_{\rm H}$ distribution for local Seyfert 2 galaxies, pointing out that a significant fraction of sources have columns exceeding $N_{\rm H}=10^{25}$ cm$^{-2}$ and are therefore completely thick to Compton scattering. The $R$ ratio between type 2s and type 1s has been determined in the local Universe for low luminosity AGNs, i.e. Seyfert galaxies (Maiolino & Rieke 1995), while the existence of a relevant number of high luminosity absorbed sources, the so–called QSO 2s, which is a basic assumption of previous models, is still uncertain (Akiyama et al. 1998). An observational constraint to the QSO 2 number density can be obtained from the infrared source counts. Indeed, QSO 2s are expected to have strong infrared counterparts, since the dust present in the torus should re–emit in the IR band the nuclear radiation absorbed by the gas. The ultraluminous infrared galaxies (ULIRGs) discovered by IRAS are the only local objects with QSO–like bolometric luminosities (Soifer et al. 1986; Kim & Sanders 1998). Thus, even if all ULIRGs were powered by a hidden AGN, the local QSO 2s could not be more numerous than ULIRGs. Finally, source counts in the 5–10 keV band have been derived for the first time by the BeppoSAX satellite with the HELLAS survey (Giommi et al. 1998; Comastri et al. 1999).
In the present paper we test the standard synthesis model to verify if it remains compatible with the new data. These data leave still some latitude to important parameters of the model, and various choices are possible to fit the XRB equally well. However, in all cases we find moderate but consistent evidence that at least some of the standard assumptions have to be relaxed: extra hard spectrum AGNs are needed at intermediate or high redshifts, in addition to those expected in the usual scenario. The additional sources could be analogous to local Seyfert 2s, if they evolve faster than type 1s, or they could be other astrophysical sources not yet enlisted among the contributors to the XRB. We discuss the observations which could distinguish between the alternatives.
Throughout this paper the deceleration parameter and the Hubble constant are given the values $q_{0}=0.5$ and $H_{0}=50$ km s$^{-1}$ Mpc$^{-1}$.
AGN X–ray properties
====================
The spectra
-----------
After the observations of X–ray satellites like GINGA, ASCA and BeppoSAX, different components have been recognized in the X–ray spectra of AGNs. Starting with Sey 1 galaxies, the basic component is a power law with energy spectral index $\alpha\sim0.9$ (Nandra et al. 1997a) and an exponential cut off at high energies. A mean value for the $e$–folding energy can probably be set at $\sim 300$ keV, although the observed dispersion is very high (Matt 1998). Some of the primary radiation is reprocessed by an accretion disc and/or the torus around the nucleus, producing a flattening of the spectral slope above $\sim 10$ keV, and a strong iron line at 6.4 keV (Nandra & Pounds 1994). Below 1–1.5 keV a radiation excess with respect to the power law emission is detected in a large fraction of Sey 1s (sometimes resulting from a misfit of the “warm absorber” component).
The spectrum of QSO 1s is similar to that of Sey 1s, but there is no evidence for the iron line and the reflection hump to be as common (Lawson & Turner 1997). Assuming that the accretion disc produces most of the line and hump, Nandra et al. (1997b) ascribe these differences to a higher ionization state of the disc in higher accretion rate sources, so that in QSO 1s the spectral features due to photoelectric processes are quenched. Recently, Vignali et al. (1999) have derived a mean spectral slope of $\langle
\alpha \rangle =0.67\pm 0.11$ from a sample of 5 QSO 1s at redshifts above 2. Although the statistics is poor, this result seems to suggest that the spectra of high redshift QSOs are flatter than those of local ones.
In Sey 2 galaxies the power law is cut off by photoelectric absorption at energies increasing with the column density of the intercepted torus. For highly absorbed objects the X–ray luminosity may be dominated by that fraction of the nuclear radiation which is reflected off the torus surface towards the observer. When $N_{\rm H}>10^{25}$ cm$^{-2}$ the obscuring medium is completely thick to Compton scattering and the spectrum is a pure reflection continuum as described by Lightman & White (1988), with a 2–10 keV luminosity about two orders of magnitude lower than that of Sey 1s (Maiolino et al. 1998). On the contrary, when $N_{\rm H}<10^{24}$ cm$^{-2}$ the medium is Compton–thin and the spectrum is dominated by the component transmitted through the torus. In the range $10^{24}<N_{\rm H}<10^{25}$ cm$^{-2}$ both a transmitted and a reflected component contribute to the observed luminosity, the Circinus galaxy (Matt et al. 1999) being a typical example. Also Sey 2 galaxies often have soft emission in excess of the absorbed power law (Turner et al. 1997). These soft excesses are however two orders of magnitude weaker than those of Sey 1s of the same intrinsic luminosity and their nature is still unclear (probably scattered or starburst radiation).
The XLF and cosmological evolution
----------------------------------
The most recent results about the AGN XLF and cosmological evolution have been obtained by Mi99a by combining data from several ROSAT surveys. Down to a limiting flux of $10^{-15}$ erg s$^{-1}$ cm$^{-2}$, reached by the deep survey in the Lockman Hole, they collected a sample of about 670 sources, which is the largest X–ray selected sample of AGNs presently available. The local XLF is described with a smoothed double power law of the following form:
$$\phi(L_{\rm x})=\frac{{\rm d}\,\Phi(L_{\rm x})}{{\rm
d\,log}L_{\rm x}}={A}\,\left[(L_{\rm x}/{L_*})
^{{\gamma_1}}
+(L_{\rm x}/{L_*})^{{\gamma_2}}
\right]^{-1},$$
where $L_{\rm x}$ is the observed 0.5–2 keV X–ray luminosity, ranging from $10^{41.7}$ to $10^{47}$ erg s$^{-1}$. The best fit values for the cosmology adopted here are: $A=(1.57 \pm 0.11)\times 10^{-6}$ Mpc$^{-3}$, $L_*=0.57^{+0.33}_{-0.19} \times 10^{44}$ erg s$^{-1}$, $\gamma_1=0.68\pm
0.18$ and $\gamma_2=2.26\pm 0.95$.
The XLF has been found to evolve from redshift 0 up to $z_{cut}=1.51\pm
0.15$, with an evolution rate which drops at low luminosities according to the factor: $$\begin{aligned}
e(z,L_{\rm x})= \left\{
\begin{array}{ll}
(1+z)^{\max(0,{p1}-{\alpha}({\rm log}\;
{L_{\rm a}} - {\rm log}\;L_{\rm x}))}
& L_{\rm x}<L_{\rm a} \\
(1+z)^{{p1}}
& L_{\rm x}\ge L_{\rm a}\;;\\
\end{array}
\right.
\nonumber\end{aligned}$$
here $p1=5.4\pm 0.4$, $\alpha = 2.3\pm 0.8$ and ${\rm log} L_{\rm a}=44.2$ (fixed). The X–ray AGNs have been observed at redshifts up to $z=4.6$ and there is no evidence for a decline in their space density beyond $z\sim 3$, unlike what is found in optical (Schmidt, Schneider & Gunn 1995) and radio surveys (Shaver et al. 1997).
We note that the XLF parametrization of Mi99a is a preliminary result and is not a unique representation of the ROSAT data. The extrapolation of the high redshift XLF into the low luminosity range, where few data are available, is not well constrained. Indeed, the number of low luminosity, high redshift AGNs could be higher than the Mi99a representation (Hasinger et al. 1999). Another cause of uncertainty is the possible presence of type 2 AGNs in the Mi99a sample: unlike previous works, where only (optical) type 1 AGNs were included, Mi99a do not discriminate between type 1s and type 2s; some of the latter could then appear in the ROSAT bandpass because of their soft excesses and, for sources at high redshifts, because of the $K$–correction. Objects with type 2 optical spectra are relatively rare in the ROSAT sample (Hasinger et al. 1999). As for the X–ray spectral type, within any given model the raw counts can be corrected for the contribution of the absorbed sources, and these can be subtracted from the XLF: in the following we consider also this approach, and investigate the robustness of our conclusions with respect to the correction. In general, the correspondence between optical and X–ray spectral classification is broadly verified in the local Universe, albeit with some blurring (see Section 2.3); at high redshifts the question is still unsettled.
Previous works about the XLF of AGNs used a PLE model both in the soft X rays, by combining observations from ROSAT and Einstein (Boyle et al. 1993, 1994; Jones et al. 1997), and in the hard X rays from ASCA data (Boyle et al. 1998a). In the Mi99a data the fit with a PLE model is rejected with a high significance. A pure density evolution model is marginally rejected, and LDDE models are preferred, even if several variants are still being discussed.
The number and column densities of local type 2 AGNs
----------------------------------------------------
In the local Universe 5–10% of the galaxies show Seyfert activity (Maiolino & Rieke 1995; Ho, Filippenko & Sargent 1997). From a sample of $\sim 90$ nearby Seyfert galaxies limited in the B magnitude of the host galaxy, Maiolino & Rieke (1995) derived an estimate for the local ratio $R$ of type 2 to type 1 Seyferts. From our point of view Seyfert types 1.8, 1.9 and 2, which have flat X–ray spectra due to absorption by cold gas, can be grouped as type 2 objects, while types 1, 1.2 and 1.5, which have steep X–ray spectra without significant cold absorption ($N_{\rm H}<10^{21}$ cm$^{-2}$), can be grouped as type 1s. Here it is noted that the relation between Seyfert type and X–ray absorption is not univocal. By observing with ROSAT the complete sample of Piccinotti et al. (1982), Schartel et al. (1997) showed that at least a fraction of type 1 AGNs suffer from X–ray absorption by more than $N_{\rm H}=10^{21}$ cm$^{-2}$. However this fraction (grouping Seyfert 1, 1.2 and 1.5) is only 20%, and on average their $N_{\rm H}$ does not exceed $10^{22}$ cm$^{-2}$. The inclusion of some moderate–absorption type 1s should not change significantly our results.
By considering Seyfert types 1.8, 1.9 and 2 as type 2s, and Seyfert types 1, 1.2 and 1.5 as type 1s, Maiolino & Rieke found $R$=4.0$\pm$0.9, in agreement with the results of Osterbrock & Martel (1993) and, more recently, Ho et al. (1997).
From the Maiolino & Rieke sample Risaliti et al. (1999) have derived a distribution of X–ray column densities for Sey 2s. The selection of the sample by means of optical narrow emission lines, rather than in the X–rays, should avoid biases against X–ray absorbed sources. It turned out that most of the sources are affected by strong absorption, $\sim 75\%$ of the objects having $N_{\rm H}>10^{23}$ cm$^{-2}$. Furthermore, a significant fraction of sources ($>25\%$) are absorbed by $N_{\rm H}>10^{25}$ cm$^{-2}$. Their results are shown in Fig. 1 and compared with the $N_{\rm H}$ distribution assumed by Co95.
In the limited luminosity range of Seyfert galaxies Risaliti et al. (1999) did not find evidence of a correlation between absorption and luminosity. In the wider luminosity domain which includes the QSOs the evidence is contradictory. Recent results from the IRAS 1–Jy survey (Kim & Sanders 1998) show that the space density of ULIRGs at $z\stackrel{<}{_{\sim}}
0.1$ is similar to that of optically selected QSOs of comparable bolometric luminosity. By considering that the number of obscured QSOs cannot exceed that of ULIRGs, and assuming that every ULIRG is powered only by nuclear activity, we can set a conservative upper limit of 2 to the $R$ ratio at high luminosities. The actual value of $R$ might be significantly lower. Indeed, there is evidence that the fraction of AGN–powered ULIRGs decreases from 50% for IR luminosities $L_{\rm
IR}\stackrel{>}{_{\sim}} 1.7\times 10^{46}$ erg s$^{-1}$ to 15% below this value (Lutz et al. 1998), the remaining ones being dominated by starburst activity.
The model
=========
The XRB spectrum
----------------
Our models are completely analogous to those of the canonical lineage, the only differences arising from updated input parameters. A key set of such parameters is the one referring to the XLF and its cosmological evolution, and for model A1 we adopt the results of Mi99a. Strictly speaking, the XLF of Mi99a refers to the observed 0.5–2 keV luminosities and could be considered as the 0.5–2 keV XLF in the rest frame only by assuming simple power law spectra with energy index $\alpha=1$ (i.e. zero $K$–correction). Indeed, in the rest frame energy range seen by ROSAT at different redshifts, the spectra of type 1 AGNs assumed in our models do not differ significantly from a power law with $\alpha=1$. Since we refer the Mi99a XLF to unabsorbed AGNs, without correcting for the contribution of absorbed AGNs to the ROSAT counts, model A1 might be biased in favor of a soft XRB. We discuss the strength of this bias in connection with models A2 and B in the following.
The absorption distribution in type 2 AGNs is no longer derived from best fitting, instead it is taken equal to the local one, as measured by Risaliti et al. (1999). The objects for which only a lower limit is available, $N_{\rm H}>10^{24}$ cm$^{-2}$, have been assigned to the bin $10^{24}<N_{\rm H}<10^{25}$ cm$^{-2}$.
Because of the evidence that the $R$ ratio decreases with the intrinsic luminosity of the AGNs, at least locally, we introduce a change with respect to the canonical scenario: the XLF is divided in two luminosity regions as follows:
$$\phi(L_{\rm x})=\phi(L_{\rm x})e^{-\frac{L_{\rm x}}{L_s}}+\phi(L_{\rm
x})(1-e^{-\frac{L_{\rm x}}{L_s}})\;,$$
with the 0.5–2 keV $e$–folding luminosity set equal to $L_s$=10$^{44.3}$ erg s$^{-1}$, following Miyaji, Hasinger & Schmidt (1999b; hereafter Mi99b). The first and second term represent the XLF of Sey 1s and QSO 1s, respectively, and, apart from the exponential factors, are equal to the Mi99a functions as given in Section 2.2. The XLF of Sey 2s and QSO 2s are $R_{\rm S}$ and $R_{\rm Q}$ times the XLF of the corresponding type 1 objects. In this parametrization we can explore various hypotheses, including for instance the effects of eliminating altogether the QSO 2s.
Following Co95, and the experimental evidence referred to in Section 2.1, we assume that the basic spectrum for type 1 AGNs is a power law with energy index $\alpha=0.9$ and exponential cut off with $e$–folding energy $E_c=320$ keV. Below 1.5 keV the soft excess is modeled with a power law of index 1.3. A reflection component from the accretion disc has been included for Sey 1s with relative normalization $f_d=1.29$ (Co95). Beside the disc, we have also included for Sey 1s a torus reflection component which is normalized in accordance with the prescriptions of Ghisellini, Haardt & Matt (1994). In type 1 AGNs the relative contribution of the torus at 30 keV is 29% and 55% for $N_{\rm H}=10^{24}$ and $10^{25}$ cm$^{-2}$, respectively. If we assume that the column density of the torus is approximately the same for all obscured lines of sight, from the measured $N_{\rm H}$ distribution we find that the torus contributes on the average 28% at 30 keV. The same disc and torus reflection components of Sey 1s have been included also in the QSO spectra: this is against the evidence at low redshifts, but mimics the harder power law seen at high redshifts (Vignali et al. 1999), where most of the XRB is produced. If anything, this assumption tends to reduce the need for additional hard spectrum sources, thus strengthening our results.
Sey 2 spectra have been computed for different amounts of intrinsic absorption (log$N_{\rm H}$=21.5, 22.5, 23.5, 24.5, 25.5) to cover all the observed column densities. In the Compton thin regime the adopted spectrum is that of Sey 1s with a photoelectric cut off and a lower amount of disc reflection ($f_d=0.88$, Co95). In this regime the component reflected by the torus does not contribute significantly to the observed radiation (5% at 30 keV for log$N_{\rm H}$=23.5, inclusive of orientation effects). For the sources with log$N_{\rm H}$=25.5 we have adopted a pure reflection continuum. The normalization of the spectrum is determined so as to reproduce the contribution of thick tori to the flux of Sey 1s (55% at 30 keV) after correcting for orientation effects (Ghisellini, Haardt & Matt, 1994). This approach predicts that the 2–10 keV continuum luminosity of completely Compton thick sources is about 2% of the typical luminosity of Sey 1s, in agreement with the results of Maiolino et al. (1998). A composite reflected/transmitted spectrum has been considered for Circinus–like sources with log$N_{\rm H}$=24.5, where the reflected and transmitted components have been normalized in analogy with the previous cases. [^1]
We have modeled the soft excess of Sey 2s with a power law of index 1.3 and a normalization at 1 keV which is 3% of the primary de–absorbed power law. In analogy with type 1s, the spectra of QSO 2s –if at all present– are assumed to be identical to those of Sey 2s. Finally, we have added to the input spectra an iron emission line at 6.4 keV. Following Gilli et al. (1999) we have considered lines with different equivalent widths according to the spectral absorption, and have not included the iron line in the spectra of QSOs.
In our model A1 we assume $R_{\rm Q}=0$, i.e. we do not include QSO 2s. The Mi99a XLF are integrated in the range $10^{41}<L_{\rm x}<10^{49}$ erg s$^{-1}$ up to $z_{max}=4.6$. The contribution of clusters of galaxies has been included by considering thermal bremsstrahlung spectra with a distribution of temperatures. We have adopted the 2–10 keV luminosity vs temperature relation of David et al. (1993), and the 2–10 keV X–ray luminosity function of Ebeling et al. (1997). The cluster XLF is assumed not to evolve, and is integrated in the range $10^{42}<L_{\rm x}<10^{47}$ erg s$^{-1}$ up to $z_{max}=2$.
The overall XRB spectrum resulting from the model is shown in Fig. 2 as a solid line, which is the sum of the contibutions of the other labeled curves. In order to fit the observed XRB spectrum we need a ratio $R_{\rm S}=4.2$, in good agreement with the local value. Above $\sim 1$ keV, where the XRB is completely extragalactic, the model provides a good fit to the data from ASCA (Gendreau et al. 1995) and the compilation of Gruber (1992) based on HEAO–1 A2 measurements. The contribution of the AGN iron line to the model XRB is found to be less than 7% at $\sim 6.4/(1+z_{cut})$ in agreement with the results of Gilli et al. (1999) obtained in a different framework (PLE). Clusters of galaxies are found to contribute to the model XRB by $\sim 12\%$ at 1 keV, in agreement with the results of Oukbir, Bartlett & Blanchard (1997).
The X–ray source counts
-----------------------
We now compare the predictions of model A1 with the observed source counts in different X–ray bands. The results in the soft 0.5–2 keV band are shown in Fig. 3. The expected AGN counts, which are dominated by unabsorbed sources, agree with the data of Mi99a; the expected cluster counts agree with the data of Jones et al. (1998), Rosati et al. (1995), and De Grandi et al. (1999). Since the XLF and its evolution are derived from the ROSAT counts, this is no more than a self–consistency check; the slight overprediction at low fluxes ($\sim$30% at $\sim 2\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$) is due to the $K$–displaced type 2 objects, as anticipated previously.
In the hard 2–10 keV band the predictions of the model have to be compared with the results of HEAO–1 A2 (Piccinotti et al. 1982), ASCA (Cagnoni, Della Ceca & Maccacaro 1998; Ogasaka et al. 1998; Ueda et al. 1998) and BeppoSAX (Giommi et al. 1998). At the flux limit of $S\simeq 3\times 10^{-11}$ erg s$^{-1}$ cm$^{-2}$ Piccinotti et al. (1982) found that AGNs and clusters of galaxies have the same surface density of $1.1\times 10^{-3}$ deg$^{-2}$. However, the AGN density found by these authors is likely to be overestimated by $\sim
20\%$ due to the local supercluster (Co95). As shown in Fig. 4, after the Piccinotti et al. point is corrected by 20%, the model is in agreement with the data within $1\sigma$. On the contrary the disagreement cannot be solved at fainter fluxes. At $S\sim 2\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ the AGN surface density expected in our model is about a factor of 2 lower than the measurement of Cagnoni et al. (1998). This corresponds to a $\sim 2\sigma$ discrepancy. When comparing the model with the data of the ASCA Large Sky Survey (Ueda et al. 1998) the discrepancy is even larger.
The situation is worse still in the 5–10 keV band. The only available counts in this band are from the HELLAS survey performed by BeppoSAX (Giommi et al. 1998; Comastri et al. 1999). At the flux of $\sim 2\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$ the observed surface density is $2.7\pm0.7$ deg$^{-2}$, which is a factor of 4 ($\sim 3\sigma$) above the predictions (Fig. 5).
Correction for absorbed sources
-------------------------------
One of the main assumptions of model A1 is that the XLF and evolution derived by Mi99a refer to unabsorbed AGNs. However, as discussed in Sect. 2.2, this is likely not the case. In order to evaluate the effects of our assumption, we have computed a different variant, A2, which adopts the XLF and LDDE of Mi99b. These authors allow self–consistently within their model for the $K$–correction, and for the absorbed sources which, especially at faint fluxes and high redshifts, appear in the ROSAT counts; thus the parameters they provide refer to unabsorbed sources only. Of course, our model is different from theirs, and the self–consistency is lost; however, this is likely to be a higher order effect, and for a first order estimate we can include their parameters in our computation. The results are shown in Fig. 6. Note that around 1 keV the unabsorbed sources produce 30% of the XRB, exactly as in Mi99b. Note also that in order to fit the XRB spectrum a ratio $R_{\rm S}=13$ is now required, which is much higher than the local value and implies additional hard spectrum sources. Since the contribution of type 2s has increased with respect to A1, in order to make up for the reduced contribution of type 1s, the mean spectrum of the population producing the XRB is harder, and the discrepancies between the model predictions and the hard counts are somewhat reduced (though not completely eliminated).
Discussion
==========
The main difference between models A1 and A2 is the fractional contribution of type 1 AGNs to the XRB; this contribution is dominated by objects close to the XLF break at redshifts close to $z_{cut}$, and is not well constrained by the data. In the former model 60% of the 1–keV XRB is due to type 1s, so the local value of the type ratio $R_{\rm S}$ is sufficient to account for the entire XRB; the average spectrum, though, is too soft, and the softness shows up in a marginal discrepancy with the XRB spectrum at $>40$ keV (Fig. 2), and unacceptable discrepancies with the hard counts (Figs. 4 and 5). In the latter model the type 1s account for only 30% of the 1–keV XRB, and making up the entire XRB requires an $R_{\rm S}$ much larger than the local value; now the average spectrum is harder, the shape discrepancy disappears and the count discrepancies are reduced (Fig. 6). By extrapolating from these two models we can make qualitative predictions on still different parametrizations of the Mi99a sample: for instance, models adopting density evolution with a dependence on luminosity weaker than Mi99a would predict a type 1 soft X–ray contribution higher than 60%, and would miss the hard counts by factors larger than Figs. 4 and 5. Density evolution with a dependence on luminosity stronger than Mi99b (if at all acceptable) would require $R_{\rm S}>13$, which is already three times the local value.
The main result of our analysis is precisely this one: no matter which variant is adopted for the XLF and the evolution, the models which incorporate the most recent observations within the standard prescriptions always produce some discrepancy. The discrepancy may appear as an underprediction of the observed hard counts, or a type 2 to type 1 ratio higher than the observed local value, but in all cases it points to additional hard spectrum sources at intermediate or high redshifts.
For the sake of completeness, we present in the Appendix a PLE model with $R_{\rm Q}=R_{\rm S}$ (model B): this region of the parameter space is not favored by the most recent data, but was adopted in practically all previous works on the XRB. Note that model B has the same type 1 soft X–ray contribution as model A2 (30%), but the type 2 contribution here is due to higher luminosity sources, which show up in higher flux bins: indeed, the XRB spectrum is well fitted, the counts in the ASCA band are matched, and the discrepancy with the HELLAS counts is reduced to $2\sigma$. The cost to be paid is a number density $R_{\rm Q}=7.7$, which –again– is definitely higher than the local upper limit $R_{\rm Q}<2$. Irrespective of the plausibility of PLE and QSO 2s, we stress that even model B results in a discrepancy, and the discrepancy is concordant with the results of the other, less controversial variants.
A population of absorbed or hard spectrum AGNs evolving more rapidly than the type 1s could accomodate all the problems discussed above. In this context one should be reminded that the hard counts already resolve $\sim$30% of the XRB at fluxes $\sim~5\times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$, so they must converge rapidly just below these values. The optical identifications of the counts in the 2–10 keV and 5–10 keV bands are still largely incomplete. Up to now 34 X–ray sources detected in the ASCA LSS survey (Ueda et al. 1998) have been identified (Akiyama et al. 1998), and 28 objects turned out to be AGNs. They are 22 broad line AGNs (type 1–1.5) with $0<z<1.7$, and 6 type 2 AGNs with $0<z<0.7$. The number of identified sources of the BeppoSAX HELLAS survey is lower, but the distribution of the AGNs seems similar to the ASCA one: 7 broad line QSOs with $0.2<z<1.3$ and 5 Seyferts 1.8–1.9 with $0.04<z<0.34$ (Fiore et al. 1999).
If one accepts these low redshift type 2 identifications, one has to find a physical reason for a convergence so recent in comparison with all other AGNs (BL–Lacs excepted) and star forming galaxies. Alternatively, one could rely on the poor statistics to maintain that the hard counts are mostly due to optically empty fields, containing very absorbed, very powerful sources at redshifts $>1$.
There are prospective candidates in both scenarios. In the low–$z$ hypothesis one could assume that “normal” Seyfert 2 galaxies evolve more rapidly than type 1s, so that $R_{\rm S}$ increases with redshift up to the required value. Not only the number ratio, but also the $N_{
\rm H}$ distribution could change with cosmic time (Franceschini et al. 1993). Local Sey 2s are associated with a star formation activity higher than Sey 1 and normal galaxies ( Maiolino et al. 1995; Rodríguez-Espinoza et al. 1986), so this assumption would have interesting implications on the star formation history. One could also invoke Advection Dominated Accretion Flows (ADAFs, Di Matteo et al. 1998) whose luminosity is proportional to $\dot M^2$, where $\dot M$ is the mass accretion rate, and which become normal QSOs at large $\dot M$: thus, they should evolve more rapidly than normal AGNs at intermediate redshifts, and should undergo a change of class at high redshifts. In the high–$z$ hypothesis one should resort to ULIRGs, which indeed are absorbed and powerful, and appear to evolve as fast as required \[$(1+z)^{7.6\pm3.2}$, Kim & Sanders 1998\]. As mentioned before there is evidence that the IR emission of ULIRGs is powered both by starburst and AGN processes; Kim, Veilleux & Sanders (1998) and Lutz et al. (1998) find that the fraction of AGN–powered infrared luminous galaxies increases with the bolometric luminosity, and reaches 30–50% in the ULIRG range. While normal starbursts are inefficient emitters in the hard X–rays, obscured AGNs in ULIRGs could easily explain the hard counts.
Finally, it should be noted that the optical identifications of the hard X–ray counts, scanty as they are, suggest that at the given X–ray flux the type 1s are more numerous than the type 2s. Concordant evidence is provided by recent BeppoSAX observations of the Marano field and the Lockman hole (Hasinger et al. 1999): most of the BeppoSAX sources have ROSAT counterparts, which in most cases are optically identified with type 1 AGNs. This type composition is in agreement with the predictions of, for instance, model A1 (Figs. 4 and 5). But if the hard counts in excess of the model are attributed entirely to obscured AGNs, then the predicted type ratio is reversed, with the type 2s more numerous than the type 1s. The numbers involved are too small to draw any conclusion, however they seem to suggest that some of the hard counts are due to flat X–ray spectrum sources with type 1 optical spectra; a few similar sources might have been found already in the ASCA LSS (Akiyama et al. 1998).
Clearly, a decisive progress in this area will require more numerous and more secure identifications of hard X–ray counts; given the various hypotheses, counterparts should be looked for not only at optical wavelengths, but also in the infrared and submillimeter domains, where AGN–dominated ULIRGs should be conspicuous.
Summary and conclusions
=======================
In this paper we have shown that the standard prescriptions for synthesizing the XRB from the integrated emission of AGNs are not consistent with a number of recent observational constraints, and some of them must be relaxed.
We have worked out models (A1 and A2) which take into account detailed input spectra of AGNs, the $N_{\rm H}$ distribution observed in local Seyfert 2s, and the XLF and evolution newly determined from the largest ROSAT sample. The latter data do not define a unique parametrization, and the two models explore different variants. As prescribed by the standard model, the XLF and evolution of type 2 AGNs are taken from type 1s, and the spectra of both types are taken independent of redshift; the only fitting parameter is the number ratio $R$ of type 2s to type 1s. We find that model A1 reproduces the XRB and the soft counts with a ratio $R$ compatible with the local value, but underestimates the hard counts. Model A2 is less discrepant as far as the counts are concerned, but requires a ratio $R$ definitely larger than observed locally. We have also computed a model adopting a canonical pure luminosity evolution (model B). In agreement with the results of Co95, model B can reproduce the XRB, the soft X–ray counts and the ASCA hard counts in the 2–10 keV band. It is also consistent within 2$\sigma$ (or discrepant at 2$\sigma$) with the preliminary BeppoSAX counts in the 5–10 keV band. Nevertheless, it requires a number of type 2 QSOs much higher than the local upper limit, and perhaps already ruled out by the deep X–ray surveys.
The discrepancies found in all models are to some extent model dependent, but all of them point in the same direction, and suggest that hard spectrum sources at intermediate or high redshifts are needed in addition to the predictions of the standard scenario. The X–ray spectrum of these additional sources could be flattened by absorption, or could be intrinsically hard. In the former hypothesis reasonable candidate counterparts could be rapidly evolving, “normal” Seyfert 2s. One should also note that a fraction of ULIRGs seem to be powered by AGNs, and their cosmological evolution seems faster than that of unabsorbed QSOs. The alternate hypothesis could instead require the presence of ADAFs. Optical identifications of the hard X–ray sources are still largely incomplete and do not allow yet to decide between the various possibilities.
We are grateful to A. Comastri and G. Zamorani for a careful reading of the manuscript, and to T. Miyaji G. Hasinger and M. Schmidt for permission to use their LDDE model in advance of publication. Our presentation was greatly improved by the comments of the referee, Prof. G. Hasinger. This work was partly supported by the Italian Space Agency (ASI) under grant ARS–98–116/22 and by the Italian Ministry for University and Research (MURST) under grant Cofin98–02–32.
Comparison with a PLE model
===========================
We have computed a canonical synthesis model of the XRB by adopting the XLF and PLE of Jones et al. (1997); since only AGNs with broad optical lines are included, there is no need to correct for the contribution of type 2 AGNs. This sample is smaller than Mi99a, and presumably low luminosity sources at high redshifts are underrepresented. Our model B assumes the XLF and PLE indicated above, includes QSO 2s as numerous as the Sey 2s, and adopts the absorption distribution of Risaliti et al. (1999) at all redshifts and all luminosities. In the cosmology adopted here, model B makes only $\sim 30\%$ of the soft XRB with type 1 AGNs, and one needs $R_{\rm S}=R_{\rm Q}=7.7$ to fit the overall background (Fig. A1). Due to the large contribution of type 2 AGNs, the model XRB spectrum is very hard. Furthermore, due to the high “effective” luminosity implied by QSO 2s, the ASCA counts are reproduced. The discrepancy with the data in the 5–10 keV band, on the contrary, is not eliminated, although it is reduced to a $2\sigma$ level. Because of the preliminary nature of the HELLAS data one might debate about its significance. At any rate, one should stress that this marginal result can be obtained only by assuming a strong (a factor $>$4) differential evolution of QSO 2s with respect to QSO 1s, so that at $z_{cut}$ the former would outnumber the latter by a factor $\sim$8.
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[^1]: For log $N_{\rm H}$=24.5 the effects of Compton scattering begin to be important. We have checked the error introduced by our approximation with respect to the MonteCarlo simulations of Matt, Pompilio & La Franca (1999). The counts remain unaffected, while the model XRB at 30 keV should be reduced by 10–15%, and even more type 2s should be included in order to maintain the agreement with the data.
|
---
abstract: |
[Recent work on F-theory GUTs has shown that the predicted masses, and magnitudes of the mixing matrix elements in the quark and lepton sectors are in close accord with experiment. In this note we estimate the CP violating phase of the mixing matrices by considering the Jarlskog invariant. We find by carefully treating certain cancellations in the computation of the Jarlskog invariant that $\left\vert J_{\text{quark}} \right\vert \sim \alpha^{3}_{GUT} \sim 6 \times10^{-5}$, and that the CP violating phase of the quark sector is large, in accord with experiment. Moreover, we predict (up to order one factors) that $\left\vert
J_{\text{lepton}} \right\vert \sim \alpha_{GUT} \sim 4 \times10^{-2}$ and that the CP violating phase of the lepton sector is also large.]{}
author:
- 'Jonathan J. Heckman'
- Cumrun Vafa
title: 'CP Violation and F-theory GUTs'
---
Introduction
============
CP violating effects provide an important probe of physics of the Standard Model and its minimal extensions. In recent work, F-theory GUTs [@BHVI; @BHVII; @DWI; @DWII] have been proposed as a framework for making contact between string theory and phenomenology. More recently it has been shown that this framework naturally realizes flavor hierarchies in both the quark and lepton sectors which are in accord with experiment [@HVCKM; @BHSV] (see also [@FontIbanez; @RANDSD]). Up to order one complex numbers multiplying each matrix entry, the up and down Yukawas are:$$\lambda_{u}\sim\left(
\begin{array}
[c]{ccc}\varepsilon_{u}^{8} & \varepsilon_{u}^{6} & \varepsilon_{u}^{4}\\
\varepsilon_{u}^{6} & \varepsilon_{u}^{4} & \varepsilon_{u}^{2}\\
\varepsilon_{u}^{4} & \varepsilon_{u}^{2} & 1
\end{array}
\right) \text{, }\lambda_{d}\sim\left(
\begin{array}
[c]{ccc}\varepsilon_{d}^{5} & \varepsilon_{d}^{4} & \varepsilon_{d}^{3}\\
\varepsilon_{d}^{4} & \varepsilon_{d}^{3} & \varepsilon_{d}^{2}\\
\varepsilon_{d}^{3} & \varepsilon_{d}^{2} & 1
\end{array}
\right) \text{,}\label{quarks}$$ while the charged lepton and neutrino Yukawas are:$$\lambda_{l}\sim\left(
\begin{array}
[c]{ccc}\varepsilon_{l}^{8} & \varepsilon_{l}^{6} & \varepsilon_{l}^{4}\\
\varepsilon_{l}^{6} & \varepsilon_{l}^{4} & \varepsilon_{l}^{2}\\
\varepsilon_{l}^{4} & \varepsilon_{l}^{2} & 1
\end{array}
\right) \text{, }\lambda_{\nu}\sim\left(
\begin{array}
[c]{ccc}\varepsilon_{\nu}^{2} & \varepsilon_{\nu}^{3/2} & \varepsilon_{\nu}\\
\varepsilon_{\nu}^{3/2} & \varepsilon_{\nu} & \varepsilon_{\nu}^{1/2}\\
\varepsilon_{\nu} & \varepsilon_{\nu}^{1/2} & 1
\end{array}
\right) \text{,}\label{leptons}$$ to leading order in the small expansion parameters $\varepsilon$. As a first approximation, $\varepsilon_{u,d,l,\nu}\sim\alpha_{GUT}^{1/2}\sim0.2$, although the specific value of each $\varepsilon$ depends on the details of the geometry. Remarkably, these crude order of magnitude estimates yield masses and mixing angles which match with experiment. For example, the magnitudes of the mixing matrix elements are given up to order one coefficients as [@HVCKM; @BHSV]:$$\begin{aligned}
\left\vert V_{CKM}^{F-th}\right\vert & \sim\left(
\begin{array}
[c]{ccc}1 & \alpha_{GUT}^{1/2} & \alpha_{GUT}^{3/2}\\
\alpha_{GUT}^{1/2} & 1 & \alpha_{GUT}\\
\alpha_{GUT}^{3/2} & \alpha_{GUT} & 1
\end{array}
\right) \sim\left(
\begin{array}
[c]{ccc}1 & 0.2 & 0.008\\
0.2 & 1 & 0.04\\
0.008 & 0.04 & 1
\end{array}
\right) \label{CKMPMNS}\\
\left\vert V_{PMNS}^{F-th}\right\vert & \sim\left(
\begin{array}
[c]{ccc}U_{e1} & \alpha_{GUT}^{1/4} & \alpha_{GUT}^{1/2}\\
\alpha_{GUT}^{1/4} & U_{\mu2} & \alpha_{GUT}^{1/4}\\
\alpha_{GUT}^{1/2} & \alpha_{GUT}^{1/4} & U_{\tau3}\end{array}
\right) \sim\left(
\begin{array}
[c]{ccc}0.87 & 0.45 & 0.2\\
0.45 & 0.77 & 0.45\\
0.2 & 0.45 & 0.87
\end{array}
\right) \text{,}\label{NEUTNEUT}$$ where the $U$’s appearing in $V_{PMNS}$ are constrained by the requirement that the norm of each row and column vector is one. Moreover, in the case of neutrinos this also leads to the prediction that the as yet undetected $(1,3)$ element of the mixing matrix should be close to the current experimental bound. This is to be compared with the observed values:$$\left\vert V_{CKM}^{\text{obs}}\right\vert \sim\left(
\begin{array}
[c]{ccc}0.97 & 0.23 & 0.004\\
0.23 & 0.97 & 0.04\\
0.008 & 0.04 & 0.99
\end{array}
\right) \text{, }\left\vert V_{PMNS}^{\text{obs}}\right\vert \sim\left(
\begin{array}
[c]{ccc}0.77-0.86 & 0.50-0.63 & 0-0.22\\
0.22-0.56 & 0.44-0.73 & 0.57-0.80\\
0.21-0.55 & 0.4-0.71 & 0.59-0.82
\end{array}
\right) \text{,}$$ where $V_{CKM}$ is taken from [@PDG], and the $3\sigma$ values of $V_{PMNS}$ are from [@GG].
In [@HVCKM; @BHSV], it was assumed that since the Yukawas are only known up to multiplication by order one complex numbers, it is natural to expect CP violating effects to be present. In fact based on this it was assumed that one could not reliably estimate the CP violating phases of the quark and lepton mixing matrices $V_{CKM}$ and $V_{PMNS}$. However the CP violating phase is very special, and one might have thought that an asymmetric hierarchy of the leptonic Yukawas would lead to a more predictive structure. We will investigate this below. In terms of the standard mixing angle parameterization, CP violation stems from the phases in the mixing matrix: $$V_{\text{mix}}=\left(
\begin{array}
[c]{ccc}c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} & c_{12}c_{13}-s_{12}s_{23}s_{13}e^{i\delta} & s_{23}c_{13}\\
s_{12}s_{13}-c_{12}c_{23}s_{13}e^{i\delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} & c_{23}c_{13}\end{array}
\right) \cdot D_{\alpha}\text{,}\label{VMIX}$$ where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$ for mixing angles $\theta_{ij}$, and $D_{\alpha}$ is the identity for quark and Dirac neutrino mixing, and for Majorana neutrinos, $D_{\alpha}=$ diag$(e^{i\alpha_{1}/2},e^{i\alpha_{2}/2},1)$. A parameterization independent measure of CP violation is given by the Jarlskog invariant $J$ [@JAR], which for Hermitian Yukawas $\lambda$ and $\lambda^{\prime}$ is given as:$$\det\left[ \lambda,\lambda^{\prime}\right] =2iJ\underset{j=1}{\overset{3}{{\displaystyle\prod}
}}\left( \lambda_{j}-\lambda_{j-1}\right) (\lambda_{j}^{\prime}-\lambda_{j-1}^{\prime})\text{,}\label{detcom}$$ where $\lambda_{j}$ denotes the eigenvalues of $\lambda$ such that $\lambda_{1}<\lambda_{2}<\lambda_{3}\equiv\lambda_{0}$ with similar conventions for the $\lambda^{\prime}$’s, and the pair of matrices $(\lambda,\lambda^{\prime})$ refers to the Yukawa pairs $(\lambda_{u},\lambda_{d})$ or $(\lambda_{l},\lambda_{\nu})$. The masses are related to the eigenvalues as:$$(m_{1},m_{2},m_{3})=v\cdot(\lambda_{1},\lambda_{2},\lambda_{3})\text{,}$$ with $v$ a suitable Higgs vev. In terms of the mixing angles and $\delta$, $J$ is given by:$$J=s_{12}s_{23}s_{13}c_{12}c_{23}c_{13}^{2}\sin\delta\text{.}\label{OURJAR}$$
CP Violation Estimates
======================
We now compute the values of the Jarlskog invariants in F-theory GUTs. Although the Yukawas in equations (\[quarks\]) and (\[leptons\]) are not in general Hermitian, as noted in [@FJ], the polar decomposition theorem ensures that any Yukawa can be written as $\lambda=\lambda_{H}\cdot U$, where $\lambda_{H}$ is Hermitian and $U$ is unitary. Since it can be shown that the $U$’s do not play a role in the Jarlskog invariant, it is enough to consider the case of Hermitian Yukawas. Further note that the hierarchy of the Hermitian matrix $\lambda_{H}$ is the same as that of the original Yukawa.
We now proceed to estimate the value of the Jarlskog invariant in both the quark and lepton sectors using equation (\[detcom\]). Expanding in powers of the $\varepsilon$’s, the leading order behavior of the commutators $\left[
\lambda_{u},\lambda_{d}\right] $ and $\left[ \lambda_{l},\lambda_{\nu
}\right] $ is:$$\begin{aligned}
\left[ \lambda_{u},\lambda_{d}\right] & \sim\left(
\begin{array}
[c]{ccc}\varepsilon_{u}^{4}\varepsilon_{d}^{3} & \varepsilon_{u}^{4}\varepsilon
_{d}^{2}+\varepsilon_{u}^{2}\varepsilon_{d}^{3} & \varepsilon_{u}^{4}+\varepsilon_{d}^{3}\\
\varepsilon_{u}^{4}\varepsilon_{d}^{2}+\varepsilon_{u}^{2}\varepsilon_{d}^{3}
& \varepsilon_{u}^{2}\varepsilon_{d}^{2} & \varepsilon_{u}^{2}+\varepsilon
_{d}^{2}\\
\varepsilon_{u}^{4}+\varepsilon_{d}^{3} & \varepsilon_{u}^{2}+\varepsilon
_{d}^{2} & \varepsilon_{u}^{2}\varepsilon_{d}^{2}\end{array}
\right) \text{,}\label{COMMTAT}\\
\left[ \lambda_{l},\lambda_{\nu}\right] & \sim\left(
\begin{array}
[c]{ccc}\varepsilon_{l}^{4}\varepsilon_{\nu} & \varepsilon_{l}^{4}\varepsilon_{\nu
}^{1/2}+\varepsilon_{l}^{2}\varepsilon_{\nu} & \varepsilon_{l}^{4}+\varepsilon_{\nu}\\
\varepsilon_{l}^{4}\varepsilon_{\nu}^{1/2}+\varepsilon_{l}^{2}\varepsilon
_{\nu} & \varepsilon_{l}^{2}\varepsilon_{\nu}^{1/2} & \varepsilon_{l}^{2}+\varepsilon_{\nu}^{1/2}\\
\varepsilon_{l}^{4}+\varepsilon_{\nu} & \varepsilon_{l}^{2}+\varepsilon_{\nu
}^{1/2} & \varepsilon_{l}^{2}\varepsilon_{\nu}^{1/2}\end{array}
\right) \text{.}$$ Here, we note that the $(3,3)$ component of each commutator is suppressed by powers of $\varepsilon$ because the order one component cancels out. The most naive estimate would be to estimate this determinant by taking the product of diagonal entries. Note, however, that the off-diagonal elements are sometimes larger than their diagonal neighbors. One might therefore be tempted to consider the largest monomial in $\varepsilon$ contributing to the determinant, assuming no further cancellations between these monomials. This would lead to the estimate $\det \left[ \lambda_{u},\lambda_{d}\right] \sim \varepsilon^{10}$ and $\det \left[ \lambda_{l},\lambda_{\nu}\right] \sim \varepsilon^{9/2}$. But even this is not without subtleties because non-trivial cancellations between terms at the same order in an expansion in $\varepsilon$ can, and indeed *will* occur. As can be checked using for example Mathematica, the leading order behavior of the determinant in the two cases of interest are:$$\begin{aligned}
\det\left[ \lambda_{u},\lambda_{d}\right] & \sim\varepsilon_{u}^{4}\varepsilon_{d}^{9} + \varepsilon_{u}^{6}\varepsilon_{d}^{7}\\
\det\left[ \lambda_{l},\lambda_{\nu}\right] & \sim\varepsilon_{l}^{4}\varepsilon_{\nu}^{3}\text{.}$$ Estimating the product of eigenvalue differences appearing on the righthand side of equation (\[detcom\]) as $\left( \lambda_{2}\lambda_{2}^{\prime
}\right) \left( \lambda_{3}\lambda_{3}^{\prime}\right) ^{2}$, and approximating all $\varepsilon$’s by the same parameter, it now follows that the magnitude of the Jarlskog invariant in the quark and lepton sectors is roughly given as:$$\begin{aligned}
\left\vert J_{\text{quark}}\right\vert & \sim\frac{\varepsilon^{13}}{\varepsilon^{7}}\sim\varepsilon^{6}\label{JQUARK}\\
\left\vert J_{\text{lepton}}\right\vert & \sim\frac{\varepsilon^{7}}{\varepsilon^{5}}\sim\varepsilon^{2}\text{.}\label{JLEP}$$
Based on this estimate, we can also extract the value of $\left\vert
\sin\delta\right\vert $. Using the form of the mixing matrix in equation (\[VMIX\]) and extracting estimates for the magnitudes for each $\cos
\theta_{ij}$ and $\sin\theta_{ij}$, the Jarlskog invariants for the quark and neutrino sector can then be written as:$$\begin{aligned}
\left\vert J_{\text{quark}}\right\vert & \sim\varepsilon^{6}\left\vert
\sin\delta_{\text{quark}}\right\vert \\
\left\vert J_{\text{lepton}}\right\vert & \sim\varepsilon^{2}\left\vert
\sin\delta_{\text{lepton}}\right\vert \text{.}$$ Returning to equations (\[JQUARK\]) and (\[JLEP\]), it follows that in both cases we have:$$\left\vert \sin\delta_{\text{quark}}\right\vert \sim\left\vert \sin
\delta_{\text{lepton}}\right\vert \sim1,$$ which correspond to order one numbers (which must be less than one).
We now determine the numerical value of the Jarlskog invariants for the quarks and leptons. Since CP violation is a feature of the mixing matrix, we shall use the rough estimate $\varepsilon\sim\alpha_{GUT}^{1/2}\sim 0.2$ appearing in equations (\[CKMPMNS\]) and (\[NEUTNEUT\]). Equations (\[JQUARK\]) and (\[JLEP\]) then yield:$$\begin{aligned}
\left\vert J_{\text{quark}}^{F-th}\right\vert & \sim \alpha^{3}_{GUT} \sim 6\times10^{-5}\label{JTAB}\\
\left\vert J_{\text{lepton}}^{F-th}\right\vert & \sim \alpha_{GUT} \sim 4\times10^{-2}\text{.}\label{JTABNU}$$ While the value of $J_{\text{lepton}}$ is still not known, the observed value of $J_{\text{quark}}$ is [@PDG]:$$J_{\text{quark}}^{\text{obs}}\sim3.08\times10^{-5}\text{,}$$ which is remarkably close to $\left\vert J_{\text{quark}}^{F-th}\right\vert $!
As mentioned previously, we expect large CP violation in both the quark and neutrino sectors, so that$$\left\vert \sin\delta^{F-th}\right\vert \sim1\text{.}$$ This is to be compared with the observed value:$$\sin\delta_{\text{quark}}^{\text{obs}}\sim0.93.$$ Finally, the fact that $\sin \delta_{\text{lepton}}$ is not suppressed in this scenario suggests it may be possible to experimentally measure it soon.
*Acknowledgements* We thank G. Feldman for helpful discussions. We also thank Y. Nir for alerting us to an erroneous conclusion in a previous version of this paper. The work of the authors is supported in part by NSF grant PHY-0244821.
=1.6pt
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|
---
author:
- Mihai Gradinaru and Emeline Luirard
title: 'Diffusive behaviour for some time-inhomogeneous stochastic kinetic models'
---
[[**Abstract:**]{} In this paper, we consider a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian random force. More precisely, we study the position $X_t$ of a particle having the velocity as solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form $t^{-\beta}F(v)$, $F$ satisfying some homogeneity condition and $\beta>0$. The diffusive behaviour of the position in large time is proven and the precise rate of convergence is pointed out by using stochastic analysis tools.]{}\
[[**Key words:**]{} kinetic stochastic equation; time-inhomogeneous diffusions; explosion times; scaling transformations; asymptotic distributions.]{}\
[[**MSC2010 Subject Classification:**]{} Primary 60J60; Secondary 60H10; 60J65; 60F05.]{}
Introduction
============
It is classical that the kinetic Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle submitted to some drag and random forces. Moreover the Feynman-Kac formula allows to make the link between the kinetic Fokker-Planck equation and stochastic differential equation driven by a Brownian motion called Langevin equation. In the simple linear case the solution of the Langevin equation is a well-known Gaussian process, the Ornstein-Uhlenbeck process.
Some models in several domains as fluids dynamics, statistical mechanics, biology, are based on the Fokker-Planck and Langevin equations in their classical form or on generalisations, for instance non-linear or driven by other random noises than Brownian motion. The behaviour in large time of the solution to the corresponding stochastic differential equation is one of the usual questions when studying these models. Although the tools of partial differential equations allowed to ask of this kind of questions, since these models are probabilistic, tools based on stochastic processes could be used. For instance, in [@CCM] the persistent turning walker model was introduced, inspired from the modelling of fish motion. An associated diffusion solves a kinetic Fokker-Planck equation based on an Ornstein-Uhlenbeck Gaussian process and the authors studied the large time diffusive behaviour of this model by using appropriate tools from stochastic analysis.
In the last decade the asymptotic study of solutions of non-linear Langevin’s type was the subject of an important number of papers, see for instance [@CNP], [@EG], [@FT1]. For instance in [@FT1] the following system is studied $$V_t=v_0+B_t-\frac{\rho}{2}\int_{0}^{t}F(V_s){\mathop{}\!\mathrm{d}}s
\quad\mbox{ and }\quad X_t=x_0+\int_{0}^{t}V_s {\mathop{}\!\mathrm{d}}s.$$ In other words one considers a particle moving such that its velocity is a diffusion with an invariant measure behaving like $(1+|v|^2)^{-\rho/2}$, as $|v|\to\infty$. The authors prove that for large time, after a suitable rescaling, the position process behaves as a Brownian motion or other stable processes, following the values of $\rho$. It should be noted that in these cited papers the standard tools associated to time-homogeneous equations are used: invariant measure, scale function, speed measure and so on. Several of these tools will not be available when the drag force is depending explicitly on time.
Let us first describe our problem: consider a one-dimensional time-inhomogeneous stochastic kinetic model driven by a Brownian motion. We denote by $(X_t)_{t\geq0}$ the one-dimensional process describing the position of a particle at time $t$ having the velocity $V_t$. The velocity process $(V_t)_{t\geq 0}$ is supposed to follow a Brownian dynamics in a potential $U(t,v)$, varying in time : $$\label{position}
{\mathop{}\!\mathrm{d}}V_t={\mathop{}\!\mathrm{d}}B_t-\dfrac{1}{2}\partial_vU(t,V_t){\mathop{}\!\mathrm{d}}t\quad
\mbox{ and }\quad
X_t=X_0+\int_{0}^{t}V_s{\mathop{}\!\mathrm{d}}s.$$ In the present paper the potential is of the form $t^{-\beta}\int_{0}^{v}F(u){\mathop{}\!\mathrm{d}}u$, with $\beta>0$ and $F$ satisfying some homogeneity condition. A natural question is to understand the behaviour of the position process in large time. More precisely we look for the limit in distribution of $v({\varepsilon})X_{t/{\varepsilon}}$, as ${\varepsilon}\to 0$, where $v({\varepsilon})$ is some rate of convergence. We will see that the position process has a diffusive behaviour and we give the precise rate and variance of the limit, by studying a Brownian martingale. The strategy to tackle this problem is based on the following idea: use a scaling transformation in order to get a stochastic differential equation having a negligible term, as $t$ is large, in order to compare with a simpler stochastic differential equation (see also [@GO] for a similar reasoning).
Our paper is organised as follows: in the next section we introduce notations and we state our main result. Existence and non-explosion of solutions are studied in Section 3 and the proof of our main result is given in Section 4. We collect in the Appendix our technical results.
Notations and main result
=========================
Assume first that the system velocity - position is given, for $t\geq t_0>0$, by $$\label{equation time}\tag{SKE}
{\mathop{}\!\mathrm{d}}V_t= {\mathop{}\!\mathrm{d}}B_t-t^{-{\beta}}F(V_t){\mathop{}\!\mathrm{d}}t,\ V_{t_0}=v_0>0,
\quad\mbox{ and }\quad
{\mathop{}\!\mathrm{d}}X_t=V_t {\mathop{}\!\mathrm{d}}t, \ X_{t_0}=x_0\in \mathbb{R}.$$ where $\beta>0$ and $(B_t)_{t\geq0}$ is a standard Brownian motion. $F$ is supposed to satisfy either which is a homogeneity condition or when it is dominated by a positive function satisfying the homogeneity condition. That is, $F$ satisfies either $$\tag{$H_1\alpha$}\label{hyp1}
\mbox{\sl for some }\alpha \in \mathbb{R},\ \forall x \in \mathbb{R},\ \lambda>0, \ F(\lambda x)=\lambda^{\alpha}F(x),$$ or $$\tag{$H_2\alpha$}\label{hyp2}
\begin{gathered} \forall x\in \mathbb{R}, \ \forall\lambda>0,\ \operatorname{sgn}(F(\lambda x))=\operatorname{sgn}(F(x)) \\\mbox{and }F\mbox{ is dominated by a positive function }G\mbox{ satisfying \eqref{hyp1}.}
\end{gathered}$$ Obviously is a generalization of . Nevertheless, we keep both assumptions since some proofs are simpler written under and are similar under . As an example of function satisfying one can keep in mind $F:v\mapsto \operatorname{sgn}(v){\left\lvertv\right\rvert}^{\alpha}$ (see also [@GO]), and as an example of function satisfying (with $\alpha=0$) $F:v\mapsto \nicefrac{v}{(1+v^2)}$ (see also [@FT1]).
\[pi\] If a function $\pi$ satisfies , then $$\label{pi1-1} \pi(x)=\begin{cases}
\pi(1)x^{\alpha} & \mbox{ if } x> 0,\\
0 & \mbox{ if } x=0,\\
\pi(-1){\left\lvertx\right\rvert}^{\alpha} & \mbox{ if } x< 0.
\end{cases}$$
Let us state our main result which holds under both assumptions.
\[main\_thm\_inh\] Consider $\alpha\geq0$ and $\beta >\frac{\alpha+1}{2}$. Let $(V_t,X_t)_{t\geq t_0}$ be a solution to . If $\alpha< 1$ or $\big(F(-1),F(1)\big) \in \mathbb{R}^-\times \mathbb{R}^+$. Then, as ${\varepsilon}\to 0$, $$\label{change_expo}
({\varepsilon}^{\nicefrac{3}{2}}X_{t/{\varepsilon}})_{t\ge t_0}\stackrel{\text{}}{\Longrightarrow}(\mathcal{B}_{t^3/3})_{t\geq t_0},$$ in the space of continuous functions ${\mathcal{C}}([t_0,\infty))$ endowed by the uniform topology, were $(\mathcal{B}_t)_{t\geq0}$ is a standard Brownian motion.
\[naive\_approach\]
1. When $\alpha=1$ and $\big(F(-1),F(1)\big) \notin \mathbb{R}^-\times \mathbb{R}^+$ the result is still true by assuming some technical conditions on $t_0$, $F(1)$ and $F(-1)$ (see Remark \[alpha=1\]).
2. By trying to adapt naively the proof of [Theorem 1a) p. 2 in [@FT1]]{}, we are led to find a solution to the Poisson equation $\,\frac{1}{2}\partial^2_{xx}g(s,x)+\partial_sg(s,x)- F(x)s^{-\beta}\partial_xg(s,x)=-x$. This PDE does not admit an evident solution and seems to be ill-posed. Thus, due to the time-dependence of the stochastic differential equation satisfied by the velocity process, one has to proceed quite differently.
Existence and non-explosion of solution
=======================================
In the following, suppose $\alpha\geq 0$ and $\beta>\frac{\alpha+1}{2}$ and set $\Omega=\overline{\mathcal{C}}([t_0,\infty))$, the set of continuous functions $\omega:[t_0,\infty)\to\mathbb{R}\cup\{\infty\}$ which equals $\infty$ after their explosion time (possibly infinite). Following the idea used in [@GO], we first perform a change of time in in order to produce at least one time-homogeneous coefficient in the transformed equation. For every ${\mathcal{C}}^2$-diffeomorphism $\phi: [0,t_1)\to [t_0,\infty)$, let introduce the scaling transformation $\Phi_{\phi}$ given, for $\omega \in \Omega $, by $$\Phi_{\phi}(\omega)(s):=\dfrac{\omega(\phi(s))}{\sqrt{\phi'(s)}}\text{, with }s\in [0,t_1).$$
The result containing the change of time transformation is given in [@GO], Proposition 2.1, p. 187. For the sake of completeness we state and sketch the proof in our context.
\[changeoftime\] If $V$ is a solution to equation , then $V^{(\phi)}$ is a solution to $$\label{change of time equation}
{\mathop{}\!\mathrm{d}}V^{(\phi)}_s={\mathop{}\!\mathrm{d}}W_s -\dfrac{\sqrt{\phi'(s)}}{\phi(s)^{\beta}}F(\sqrt{\phi'(s)}V_s^{(\phi)}){\mathop{}\!\mathrm{d}}s -\dfrac{\phi''(s)}{\phi'(s)}\dfrac{V_s^{(\phi)}}{2}{\mathop{}\!\mathrm{d}}s, \ V_{0}^{(\phi)}=\dfrac{V_{\phi(0)}}{\sqrt{\phi'(0)}},$$ where $V^{(\phi)}=\Phi_{\phi}(V)$ and $W_t:=\medint\int_{0}^{t}\dfrac{{\mathop{}\!\mathrm{d}}B_{\phi(s)}}{\sqrt{\phi'(s)}}$.
If $V^{(\phi)}$ is a solution to , then $V$ is a solution to equation , where $V=\Phi_{\phi}^{-1}(V^{(\phi)})$ and $B_t-B_{t_0}:=\medint\int_{t_0}^{t}\sqrt{(\phi'\circ \phi^{-1})(s)}{\mathop{}\!\mathrm{d}}W_{\phi^{-1}(s)}$.
Furthermore uniqueness in law, pathwise uniqueness or strong existence hold for equation if and only if they hold for equation .
Let $V$ be a solution to equation . Thanks to Lévy’s characterization theorem of the Brownian motion, $W$ is a standard Brownian motion. Then, by a change of variable $t=\phi(s)$, one gets $$V_{\phi(t)}-V_{\phi(0)}=\int_{0}^{t}\sqrt{\phi'(s)}{\mathop{}\!\mathrm{d}}W_s-\int_{0}^{t}\dfrac{F(V_{\phi(s)})}{\phi(s)^{\beta}}\phi'(s){\mathop{}\!\mathrm{d}}s.$$ The integration by parts formula yields $${\mathop{}\!\mathrm{d}}\left(\dfrac{V_{\phi(s)}}{\sqrt{\phi'(s)}}\right)={\mathop{}\!\mathrm{d}}W_s-\dfrac{\sqrt{\phi'(s)}}{\phi(s)^{\beta}}F(V_{\phi(s)}){\mathop{}\!\mathrm{d}}s -\dfrac{\phi''(s)}{2\phi'(s)}\dfrac{V_{\phi(s)}}{\sqrt{\phi'(s)}}{\mathop{}\!\mathrm{d}}s.$$ From which follows . The proof of the second part is similar.
In the following, we will use two particular changes of time, depending on which term of should become time-homogeneous:
- [*exponential change of time*]{}: denoting $\phi_e: t \mapsto t_0 e^t$, the exponential scaling transformation is given by $\Phi_e(\omega):s\in \mathbb{R}^{+}\mapsto \dfrac{\omega_{t_0 e^s}}{\sqrt{t_0}e^{\nicefrac{s}{2}}}$, for $\omega\in \Omega $. Set $V^{(e)}:=\Phi_e(V)$. Thanks to \[changeoftime\], the process $(V^{(e)})_{t\geq0}$ satisfies the equation $$\label{equation Ve}
{\mathop{}\!\mathrm{d}}V_s^{(e)}={\mathop{}\!\mathrm{d}}W_s-\dfrac{V_s^{(e)}}{2}{\mathop{}\!\mathrm{d}}s- t_0^{\nicefrac{1}{2}-\beta} e^{(\nicefrac{1}{2}-\beta)s}F\big(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}\big){\mathop{}\!\mathrm{d}}s,$$ where $(W_t)_{t\geq0}$ is a standard Brownian motion.
- [*power change of time*]{}: setting $\gamma:= \frac{2\beta}{\alpha+1}>1$, consider $\phi_{\gamma}\in \mathcal{C}^2([0,t_1))$ the solution to the Cauchy problem $$\phi_{\gamma}'=\phi_{\gamma}^{\gamma}, \ \phi_{\gamma}(0)=t_0.$$ Clearly $\phi_{\gamma}(t)=\big(t_0^{1-\gamma}+(1-\gamma)t\big)^{\nicefrac{1}{(1-\gamma)}}$: the maximal time $t_1$ satisfies $(\gamma-1)t_1=t_{0}^{1-\gamma}$ and the power scaling transformation is given by $\Phi_\gamma(\omega):s\in \mathbb{R}^{+}\mapsto \dfrac{\omega(\phi_{\gamma}(s))}{\phi_{\gamma}(s)^{\nicefrac{\gamma}{2}}}$. The process $V^{(\gamma)}:=V^{(\phi_{\gamma})}$ satisfies the equation $$\label{power hyp2}
{\mathop{}\!\mathrm{d}}V_s^{(\gamma)} ={\mathop{}\!\mathrm{d}}W_s-\rho \phi_{\gamma}^{-\nicefrac{\alpha \beta}{(\alpha+1)}}(s) F\Big(\sqrt{\phi'_{\gamma}(s)}V_s^{(\gamma)}\Big){\mathop{}\!\mathrm{d}}s-\gamma \phi_{\gamma}^{\gamma-1}(s)\dfrac{V_s^{(\gamma)}}{2}{\mathop{}\!\mathrm{d}}s,$$ where $(W_t)_{t\geq0}$ is a standard Brownian motion.
In the following we will study the existence and the behaviour of the solution to , first under the homogeneity assumption and then under the domination assumption .
Study under
------------
In the following we assume . Then the process $V^{(\gamma)}$ satisfies the equation $$\label{power}
{\mathop{}\!\mathrm{d}}V_s^{(\gamma)} ={\mathop{}\!\mathrm{d}}W_s- F(V_s^{(\gamma)}){\mathop{}\!\mathrm{d}}s-\gamma \phi_{\gamma}^{\gamma-1}(s)\dfrac{V_s^{(\gamma)}}{2}{\mathop{}\!\mathrm{d}}s,\ s \in [0,t_1),$$ which can be written, by using the expression of $\phi_{\gamma}$, as $$\label{power2}
{\mathop{}\!\mathrm{d}}V_s^{(\gamma)} ={\mathop{}\!\mathrm{d}}W_s- F(V_s^{(\gamma)}){\mathop{}\!\mathrm{d}}s-\delta\dfrac{V_s^{(\gamma)}}{t_1-s}{\mathop{}\!\mathrm{d}}s, \ s \in [0,t_1),$$ where $\delta=\dfrac{\gamma}{2(\gamma-1)}$. [Proposition 3.2, p. 188, in [@GO]]{} can be stated in the present situation:
\[existence hyp1\] For $\alpha>-1$, there exists a pathwise unique strong solution to , defined up to the explosion time.
We sketch the proof in our context. Note that, since $\alpha>-1$, $x\mapsto {\left\lvertx\right\rvert}^{\alpha}$ is locally integrable. Leaving out the third term on the right-hand side of , one gets $$\label{homogene}
{\mathop{}\!\mathrm{d}}H_s ={\mathop{}\!\mathrm{d}}W_s- F(H_s){\mathop{}\!\mathrm{d}}s, \ s \in [0,t_1).$$ By using [Proposition 2.2, p. 28, in [@cherny]]{}, there exists a unique weak solution $H$ to the time-homogeneous equation defined up to the explosion time. Moreover, the Girsanov transformation induces a linear bijection between weak solutions defined up to the explosion time to equations and . It follows that there exists a unique weak solution $V^{(\gamma)}$ to equation . Therefore, by using \[changeoftime\], there exists a unique weak solution $V$ to equation . Besides, by using [Corollary 3.4 and Proposition 3.2, pp. 389-390, in [@RY]]{}, pathwise uniqueness holds for the equation . The conclusion follows from [Theorem 1.7, p. 368, in [@RY]]{}.
When $\alpha=1$, drift and diffusion are Lipschitz and satisfy locally linear growth. The existence and non-explosion of $V$ follow from [Theorem 2.9, p. 289, in [@KS]]{}.
\[explosion time\]
- When $\alpha\leq 1$ or $(F(-1),F(1)) \in \mathbb{R}^-\times \mathbb{R}^+$, the explosion time of $V$ is a.s. infinite.
- Else, i.e. if $\alpha>1$ and $(F(-1),F(1))\in (\mathbb{R}^*_+\times \mathbb{R}_+)\cup (\mathbb{R}\times \mathbb{R}_-)$, $\mathbb{P}(\tau_{\infty}=\infty)\in (0,1)$, where $\tau_{\infty}$ denotes the explosion time of $V$.
We split the proof in several steps.
<span style="font-variant:small-caps;">Step 1.</span> Assume first that $\alpha\leq 1$ or $(F(-1),F(1)) \in \mathbb{R}^-\times \mathbb{R}^+$. We will use [Theorem 10.2.1, p. 254, in [@stroock]]{}. Call $\mathcal{L}_t$ the time-inhomogeneous infinitesimal generator of $V$, given by $$\mathcal{L}_t:= \dfrac{1}{2}\dfrac{\partial^2}{\partial x^2} - \dfrac{F(x)}{t^{\beta}}\dfrac{\partial}{\partial x}.$$ Let $\varphi$ be a twice continuous differentiable positive function such that $\varphi(x)=1+x^2$ for all ${\left\lvertx\right\rvert}\geq 1$, $\varphi(x)=1$ for all ${\left\lvertx\right\rvert}\leq \frac{1}{2}$ and $\varphi\geq1$. Note that $\varphi$ does not depend on time. Hence $\big(\partial_t+\mathcal{L}_t\big)\varphi=\mathcal{L}_t\varphi$.\
Let $T\geq t_0$ and call $c_T$ the supremum of $\mathcal{L}_t\varphi$ on $[t_0,T]\times [-1,1]$. Then, for all ${\left\lvertx\right\rvert}\leq 1$ and $t\in [t_0, T]$, $$\mathcal{L}_t\varphi(x)\leq c_T \leq c_T\varphi(x).$$ Moreover, for all ${\left\lvertx\right\rvert}> 1$ and $t\in [t_0, T]$, for $C$ a positive constant, $$\mathcal{L}_t\varphi(x)= -2 x\dfrac{F(x)}{t^{\beta}}+1 \leq \begin{cases}
1 \leq \varphi(x), \qquad\qquad\quad \mbox{if } (F(-1),F(1)) \in \mathbb{R}^-\times \mathbb{R}^+, \\2
\max({\left\lvertF(1)\right\rvert},{\left\lvertF(-1)\right\rvert})x^2+1 \leq C\varphi(x), \quad \mbox{if }\,\alpha\leq1.
\end{cases}$$ So, by using [Theorem 10.2.1, p. 254, in [@stroock]]{}, we deduce that $\tau_{\infty}$ is infinite a.s.
<span style="font-variant:small-caps;">Step 2.</span> Assume now the contrary, that is $\alpha>1$ and $(F(-1),F(1))\in \big(\mathbb{R}^*_+\times \mathbb{R}_+\big)\cup \big(\mathbb{R}\times \mathbb{R}_-\big)$. We follow the ideas of the proof of [Proposition 3.7, pp. 191-192, in [@GO]]{}. We first show that $\mathbb{P}(\tau_{\infty}=\infty)>0$. Let $V^{(\gamma)}$ be the pathwise unique strong solution to equation . Also denote by $b$, the $\delta$-Brownian bridge, the pathwise unique strong solution to equation $$\label{brown bridge}
{\mathop{}\!\mathrm{d}}b_s={\mathop{}\!\mathrm{d}}W_s -\delta \dfrac{b_s}{t_1-s}{\mathop{}\!\mathrm{d}}s,\ b_0=x_0, \ s\in [0,t_1).$$ Note that the equation is obtained from by omitting the second term on the right-hand side. Denote by $\tau_{\infty}^{(\gamma)}$ the explosion time of $V^{(\gamma)}$, clearly, $\tau_{\infty}^{(\gamma)}\in [0,t_1]\cup \{\infty\}$ a.s. and $\{\tau_{\infty}^{(\gamma)}\geq t_1\}=\{\tau_{\infty}=\infty\}$. Note that $b$ becomes continuous on $[0,t_1]$, with $b_{t_1}=0$ a.s.
Fix $n\geq 1$, for all $s\in [0,t_1]$, define $$T_n:=\inf\left\{s\in [0,t_1), {\left\lvertV_s^{(\gamma)}\right\rvert}\geq n \right \}, \ \sigma_n:=\inf\{ s\in[0,t_1], {\left\lvertb_s\right\rvert} \geq n \},$$ and $$\mathcal{E}(s):= \exp\left(\int_{0}^{s}- F(b_u){\mathop{}\!\mathrm{d}}W_u -\dfrac{1}{2}\int_{0}^s F(b_u)^2 {\mathop{}\!\mathrm{d}}u \right).$$ Then, one has, since $\alpha\geq 0$, $$\begin{aligned}
\mathbb{E}\left[\exp\left(\dfrac{1}{2}\int_{0}^{s\wedge \sigma_n} F(b_u)^2 {\mathop{}\!\mathrm{d}}u \right)\right]&\leq \mathbb{E}\left[\exp\left(\dfrac{1}{2}\int_{0}^{s\wedge \sigma_n} n^{2\alpha}\max(F(1)^2,F(-1)^2) {\mathop{}\!\mathrm{d}}u \right)\right] \\ & \leq \exp\left(\dfrac{t_1}{2} n^{2\alpha}\max(F(1)^2,F(-1)^2) \right),
\end{aligned}$$ so Novikov’s condition applies to $(\mathcal{E}_{s \wedge \sigma_n})_{s\geq0}$. By using the Girsanov transformation between $b$ and $V^{(\gamma)}$, we can write for every integer $n\geq1$, $s\in[0,t_1]$ and $A\in \mathcal{F}_s$, $$\mathbb{E}\left[ \mathbb{1}_A\left(V^{(\gamma)}_{\bullet \wedge T_n} \right)\mathbb{1}_{T_n>s} \right]=\mathbb{E}\big[ \mathbb{1}_A\left(b_{\bullet \wedge \sigma_n} \right)\mathcal{E}(s\wedge \sigma_n)\mathbb{1}_{\sigma_n>s} \big].$$
Letting $n\to \infty$, we obtain by the dominated convergence theorem and Fatou’s lemma, $$\mathbb{E}\left[ \mathbb{1}_A\left(V^{(\gamma)} \right)\mathbb{1}_{\tau_{\infty}^{(\gamma)}>s} \right]\geq\mathbb{E}\left[ \mathbb{1}_A\left(b \right)\mathcal{E}(s) \right].$$ Hence, $\mathbb{P}(\tau_{\infty}=\infty)=\mathbb{P}(\tau_{\infty}^{(\gamma)}\geq t_1)\geq \mathbb{E}[\mathcal{E}(t_1)]>0$. <span style="font-variant:small-caps;">Step 3.</span> We will show that $\mathbb{P}(\tau_{\infty}=\infty)<1$ when $F(1)>0$ and $F(-1)>0$. Our strategy is to apply [Theorem 10.2.1, p. 254, in [@stroock]]{}. Let $T>t_0$ and choose $a\in(1,\alpha)$. Also, one can choose $k\geq 1$ such that $a(a-1)^{-1}
<k(T-t_0)$. Introduce the continuous differentiable negative function $f_1:x\mapsto\dfrac{-1/2}{1+{\left\lvertx\right\rvert}^a}$, and, for $\mu>0$, the bounded twice continuous differentiable function $$G_{1,\mu}(x)=\exp\left(\mu \int_{-\infty}^{x}f_1(y){\mathop{}\!\mathrm{d}}y \right), \ x\in \mathbb{R}.$$ For all $t\in [t_0,T]$ and $x\in \mathbb{R}$, $$\begin{aligned}
\big(\partial_t+\mathcal{L}_t\big)G_{1,\mu}(x)=\mathcal{L}_tG_{1,\mu}(x)&=
\mu G_{1,\mu}(x)\left[ F(x)t^{-\beta}{\left\lvertf_1(x)\right\rvert}+\dfrac{1}{2}f'_1(x)+\dfrac{\mu}{2}f_1^2(x)\right] \\ & \geq \mu G_{1,\mu}(x)\left[ F(x)T^{-\beta}{\left\lvertf_1(x)\right\rvert}+\dfrac{1}{2}f'_1(x)+\dfrac{\mu}{2}f_1^2(x)\right].
\end{aligned}$$ Since ${\left\lvertf_1(x)\right\rvert}\underset{{\left\lvertx\right\rvert}\to \infty}{\sim}\frac{1}{2}{\left\lvertx\right\rvert}^{-a}$, $\lim\limits_{{\left\lvertx\right\rvert}\to \infty}F(x){\left\lvertf_1(x)\right\rvert}= +\infty$, and using that $\lim\limits_{{\left\lvertx\right\rvert}\to \infty}f'_1(x)=0$, there exists $r\geq 1$ such that, for all $\mu>0$, $$\big(\partial_t+\mathcal{L}_t\big)G_{1,\mu}(x)
\geq\mu G_{1,\mu}(x)\left[ F(x)T^{-\beta}{\left\lvertf_1(x)\right\rvert}+\dfrac{1}{2}f'_1(x)\right] \geq k\mu G_{1,\mu}(x) \text{ on } [t_0,T]\times [-r,r]^c.$$ Moreover, since $f_1^2$ is bounded away from zero, while ${\left\lvertf'_1\right\rvert}$ is bounded on $[-r,r]$, there exists $\mu_0$, such that, since $F$ is non-negative, $$\big(\partial_t+\mathcal{L}_t\big)G_{1,\mu_0}(x)
\geq\mu_0 G_{1,\mu_0}(x)\left[\dfrac{1}{2}f'_1(x)+\dfrac{\mu_0}{2}f_1^2(x)\right] \geq k\mu_0 G_{1,\mu_0}(x) \text{ on } [t_0,T]\times [-r,r].$$ Hence, for all $t\in[t_0,T]$ and $x\in \mathbb{R}$, $
\big(\partial_t+\mathcal{L}_t\big)G_{1,\mu_0}(x)
\geq k\mu_0G_{1,\mu_0}(x)$. Besides, since ${\left\lvertf_1(x)\right\rvert}\leq 1 \wedge {\left\lvertx\right\rvert}^{-a}$, $$\int_{-\infty}^{x_0}\big(-f_1(x)\big){\mathop{}\!\mathrm{d}}x\leq \int_{\mathbb{R}}\big(1 \wedge {\left\lvertx\right\rvert}^{-a}\big){\mathop{}\!\mathrm{d}}x=a(a-1)^{-1}<k(T-t_0).$$ Thus, $G_{1,\mu_0}(x_0) > e^{-k\mu_0(T-t_0)}\geq e^{-k\mu_0(T-t_0)}\sup_{x\in\mathbb{R}} G_{1,\mu_0}(x)$. Therefore, [Theorem 10.2.1, p. 254, in [@stroock]]{} applies and $V$ explodes in finite time with positive probability.
When $F(-1)<0$ and $F(1)<0$, one can work in the same way, using instead $G_{1,\mu}$ the function $x\mapsto \exp\left( \mu \int_{x}^{+\infty} f_1(y) {\mathop{}\!\mathrm{d}}y \right)$, in order to get that $\mathbb{P}(\tau_{\infty}=\infty)<1$.
<span style="font-variant:small-caps;">Step 4.</span> It remains to show that $\mathbb{P}(\tau_{\infty}=\infty)<1$ when $F(1)<0 $ and $F(-1)>0$. As in the previous step, we choose $a\in(1,\alpha)$ and for any $T>t_0$, one can choose again $k\geq 1$ such that $a(a-1)^{-1}
<k(T-t_0)$. Moreover, one can see that there exists a continuous differentiable odd function $f_2$, defined on $\mathbb{R}$, vanishing only at $x=0$, such that ${\left\lvertf_2(x)\right\rvert}\leq 1 \wedge {\left\lvertx\right\rvert}^{-a}$, and $$f_2(x):=kx, \ x\in\left[-\frac{1}{2k}, \frac{1}{2k}\right], \ \lim\limits_{{\left\lvertx\right\rvert}\to \infty}{\left\lvertx\right\rvert}^{\alpha}{\left\lvertf_2(x)\right\rvert}=\infty\ \text{and}\ \lim\limits_{{\left\lvertx\right\rvert}\to \infty}f'_2(x)=0.$$ For $\mu>0$, we introduce the bounded twice continuous differentiable function $$G_{2,\mu}(x):=\exp\left( \mu \int_{0}^x f_2(y) {\mathop{}\!\mathrm{d}}y \right), \ x\in \mathbb{R}.$$ Note that for all $x\in \mathbb{R}$ and $t\in [t_0,T]$, $$\begin{aligned}
\big(\partial_t+\mathcal{L}_t\big)G_{2,\mu}(x)=\mathcal{L}_tG_{2,\mu}(x)&=
\mu G_{2,\mu}(x)\left[ \dfrac{{\left\lvertF(x)f_2(x)\right\rvert}}{t^{\beta}}+\dfrac{1}{2}f'_2(x)+\dfrac{\mu}{2}f_2^2(x)\right] \\& \geq \mu G_{2,\mu}(x)\left[\rho \dfrac{{\left\lvertx\right\rvert}^{\alpha}{\left\lvertf_2(x)\right\rvert}}{t^{\beta}}+\dfrac{1}{2}f'_2(x)+\dfrac{\mu}{2}f_2^2(x) \right],
\end{aligned}$$ where $\rho= \min\big\{{\left\lvertF(1)\right\rvert}, {\left\lvertF(-1)\right\rvert}\big\}>0$. One can conclude, using the same argument as in the proof of [Proposition 3.7, p. 13, in [@GO]]{}.
Study under
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We assume now . Since, the equation doesn’t have any time-homogeneous term, the previous method cannot be used to conclude to the existence up to explosion. Instead, one will use the exponential change of time process to get
If $\alpha\geq0$, there exists a pathwise unique strong solution to , defined up to the explosion time.
The proof is identical to that of \[existence hyp1\], by considering $V^{(e)}$ instead of $V^{(\gamma)}$.
- When $\alpha\leq 1$ or $(F(-1),F(1)) \in \mathbb{R}^-\times \mathbb{R}^+$, the explosion time of $V$ is a.s. infinite.
- Else, i.e. $\alpha>1$ and $(F(-1),F(1))\in \big(\mathbb{R}^*_+\times \mathbb{R}_+\big)\cup \big(\mathbb{R}\times \mathbb{R}_-\big)$, $\mathbb{P}(\tau_{\infty}=\infty)>0$, where $\tau_{\infty}$ denotes the explosion time of $V$.
The proof is identical to that of \[explosion time\] by considering $G$ instead of ${\left\lvertF\right\rvert}$.
Asymptotic behaviour of the solution
====================================
Let observe the SDE satisfied by $V^{(e)}$: leaving out the last term, it yields the equation of the Ornstein-Ulhenbeck process: $${\mathop{}\!\mathrm{d}}U_s={\mathop{}\!\mathrm{d}}W_s-\dfrac{U_s}{2}{\mathop{}\!\mathrm{d}}s.$$
<span style="font-variant:small-caps;">Step 1.</span> Firstly, for all $t\geq0$,
$$\begin{aligned}
\label{relation2}
X_t^{(e)}&:=\notag \int_{0}^{t}V_s^{(e)}{\mathop{}\!\mathrm{d}}s=\int_{t_0}^{t_0 e^t}\dfrac{V_u}{u^{\nicefrac{3}{2}}}{\mathop{}\!\mathrm{d}}u\stackrel{IBP}{=}\dfrac{X_{t_0 e^t}}{t_0^{\nicefrac{3}{2}}e^{\nicefrac{3t}{2}}}-\dfrac{X_{t_0}}{t_0^{\nicefrac{3}{2}}}+\dfrac{3}{2}\int_{t_0}^{t_0 e^t}\dfrac{X_s}{s^{\nicefrac{5}{2}}}ds\\
&=X_{t_0 e^t}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}-X_{t_0}t_0^{\nicefrac{-3}{2}}+\dfrac{3}{2}\int_0^tX_{t_0 e^u}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3u}{2}}{\mathop{}\!\mathrm{d}}u.
\end{aligned}$$
The behaviour of the third term on the right-hand side is unknown. However, by setting $G:t\mapsto \medint\int_0^tX_{t_0 e^u}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3u}{2}}{\mathop{}\!\mathrm{d}}u$, may be written as $$G'(t)+\dfrac{3}{2}G(t)=X_t^{(e)}+X_{t_0}t_0^{\nicefrac{-3}{2}}, \ G(0)=0.$$ This ODE can be solved :$$\begin{aligned}
G:t\mapsto& e^{\nicefrac{-3t}{2}}\int_{0}^{t}\left(X_s^{(e)}+X_{t_0}t_0^{\nicefrac{-3}{2}} \right)e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s= e^{\nicefrac{-3t}{2}}\int_{0}^{t}X_s^{(e)}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s+\dfrac{2}{3}X_{t_0}t_0^{\nicefrac{-3}{2}}(1-e^{\nicefrac{-3t}{2}}).
\end{aligned}$$ Hence, using the two equalities of $G'$, one obtains that, for all $t\geq0$, $$X_{t_0 e^t}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}=X_t^{(e)}-\dfrac{3}{2}e^{\nicefrac{-3t}{2}}\int_{0}^{t}X_s^{(e)}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s+X_{t_0}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}.$$
<span style="font-variant:small-caps;">Step 2.</span> We now study the second term on the right-hand side. Since for $u\geq0$, $X_{u}^{(e)}=\medint\int_{0}^{u}V_{s}^{(e)}{\mathop{}\!\mathrm{d}}s$, by Itô’s formula one gets, $$\dfrac{3}{2}e^{\nicefrac{-3t}{2}}\int_{0}^{t}X_{u}^{(e)}e^{\nicefrac{3u}{2}}{\mathop{}\!\mathrm{d}}u = X_{t}^{(e)}- e^{\nicefrac{-3t}{2}}\int_{0}^{t}V_{s}^{(e)}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s.$$ Replacing in the preceding equality, $$\label{prem method}
X_{t_0 e^t}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}=X_{t_0}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}+e^{\nicefrac{-3t}{2}}\int_{0}^{t}V_{s}^{(e)}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s.$$ Moreover, applying again Itô’s formula, $$\begin{aligned}
V_{t}^{(e)}e^{\nicefrac{3t}{2}} &=V_0^{(e)}+\dfrac{3}{2}\int_{0}^{t}V_s^{(e)}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s+\int_{0}^{t}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}V_s^{(e)} \\&=V_0^{(e)}+\int_{0}^{t}V_s^{(e)}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}s + \int_{0}^{t}e^{\nicefrac{3s}{2}}{\mathop{}\!\mathrm{d}}W_s-\int_{0}^{t}e^{\nicefrac{3s}{2}} t_0^{\nicefrac{1}{2}-\beta} e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}){\mathop{}\!\mathrm{d}}s.
\end{aligned}$$ Hence, $$\begin{aligned}
X_{t_0 e^t}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}= e^{\nicefrac{-3t}{2}}(X_{t_0}t_0^{\nicefrac{-3}{2}}-V_0^{(e)})+V_t^{(e)}&-\int_{0}^{t}e^{\nicefrac{-3(t-s)}{2}}{\mathop{}\!\mathrm{d}}W_s \\&+ t_0^{\nicefrac{1}{2}-\beta} \int_{0}^{t}e^{\nicefrac{-3(t-s)}{2}}e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}){\mathop{}\!\mathrm{d}}s.
\end{aligned}$$ One can express $V^{(e)}$ with the help of Itô’s formula again: $$\label{formuleVe}
V_t^{(e)}= V_0^{(e)}e^{\nicefrac{-t}{2}}+ \int_{0}^{t}e^{\nicefrac{-(t-s)}{2}}{\mathop{}\!\mathrm{d}}W_s- \int_{0}^{t} e^{\nicefrac{-(t-s)}{2}}t_0^{\nicefrac{1}{2}-\beta} e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}){\mathop{}\!\mathrm{d}}s.$$ Hence, it becomes $$\begin{gathered}
X_{t_0 e^t}t_0^{\nicefrac{-3}{2}}e^{\nicefrac{-3t}{2}}= e^{\nicefrac{-3t}{2}}(X_{t_0}t_0^{\nicefrac{-3}{2}}-V_0^{(e)})+V_0^{(e)}e^{\nicefrac{-t}{2}}+\int_{0}^{t}\left[e^{-\frac{(t-s)}{2}}-e^{-\frac{3(t-s)}{2}}\right]{\mathop{}\!\mathrm{d}}W_s \\- t_0^{\nicefrac{1}{2}-\beta} \int_{0}^{t}\left[e^{-\frac{(t-s)}{2}}-e^{-\frac{3(t-s)}{2}}\right]e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}){\mathop{}\!\mathrm{d}}s.
\end{gathered}$$ It follows that, for all $u\geq t_0$, and ${\varepsilon}>0$, $$\begin{gathered}
{\varepsilon}^{\nicefrac{3}{2}}X_{u/{\varepsilon}}={\varepsilon}^{\nicefrac{3}{2}}(X_{t_0}-V_0^{(e)}t_0^{\nicefrac{3}{2}})+V_0^{(e)}\sqrt{t_0{\varepsilon}}u+u^{\nicefrac{3}{2}}\int_{0}^{\ln(\nicefrac{u}{t_0{\varepsilon}})}\left[e^{-\frac{(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}-e^{-\frac{3(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}\right]{\mathop{}\!\mathrm{d}}W_s \\ - u^{\nicefrac{3}{2}} t_0^{\nicefrac{1}{2}-\beta} \int_{0}^{\ln(\nicefrac{u}{t_0{\varepsilon}})}\left[e^{-\frac{(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}-e^{-\frac{3(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}\right]e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}){\mathop{}\!\mathrm{d}}s.
\end{gathered}$$ Set, for $t\geq 0$, the continuous local martingale, vanishing at $0$, $$M_t:=\medint\int_{0}^{t}\left(e^{\nicefrac{3t}{2}}e^{\nicefrac{-(t-s)}{2}}-e^{\nicefrac{3s}{2}}\right){\mathop{}\!\mathrm{d}}W_s.$$ The quadratic variation of $(M_t)_{t\geq0}$ is $\langle M,M\rangle_t= \nicefrac{(e^t-1)^3}{3}$. Hence $\langle M, M\rangle_{\infty}=\infty$, so by the Dambis-Dubins-Schwarz theorem ([Theorem 1.6, p. 181, in [@RY]]{}), there exists a standard Brownian motion $(\mathcal{B}_t)_{t\geq0}$ such that $M_t=\mathcal{B}_{\nicefrac{(e^t-1)^3}{3}}$. Then one can write $$\begin{gathered}
\label{equation finale}
{\varepsilon}^{\nicefrac{3}{2}}X_{u/{\varepsilon}}={\varepsilon}^{\nicefrac{3}{2}}(X_{t_0}-V_0^{(e)}t_0^{\nicefrac{3}{2}})+V_0^{(e)}\sqrt{t_0{\varepsilon}}u+(t_0 {\varepsilon})^{\nicefrac{3}{2}}M_{\ln(\nicefrac{u}{t_0{\varepsilon}})} \\ - u^{\nicefrac{3}{2}} t_0^{\nicefrac{1}{2}-\beta} \int_{0}^{\ln(\nicefrac{u}{t_0{\varepsilon}})}\left[e^{-\frac{(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}-e^{-\frac{3(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}\right]e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s{(e)}){\mathop{}\!\mathrm{d}}s.
\end{gathered}$$
<span style="font-variant:small-caps;">Step 3.</span> Letting ${\varepsilon}\to 0$ in , the first two terms on the right-hand side converge to 0 a.s. and uniformly on compact sets, while the third term converges to 0 a.s. uniformly on compact sets, by the dominated convergence theorem. We fix $T\geq t_0$.
For all $s\geq 0$, $$\begin{gathered}
\sup_{u\in[t_0,T] }{\left\lvert\mathbb{1}_{[0,\ln(\nicefrac{u}{t_0{\varepsilon}})]}u^{\nicefrac{3}{2}} t_0^{\nicefrac{1}{2}-\beta} \left[e^{-\frac{(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}-e^{-\frac{3(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}\right]e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)}) \right\rvert}\\
\leq \mathbb{1}_{\mathbb{R}^+}(s) t_0^{\nicefrac{1}{2}-\beta} \left[T\sqrt{{\varepsilon}t_0}e^{\nicefrac{s}{2}}+(t_0{\varepsilon})^{\nicefrac{3}{2}}e^{\nicefrac{3s}{2}}\right]e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)})\underset{{\varepsilon}\to 0}{\longrightarrow}0 \text{ a.s.}
\end{gathered}$$ For all ${\varepsilon}>0$ and $s\geq 0$, $$\begin{gathered}
\sup_{u\in[t_0,T]}{\left\lvert\mathbb{1}_{[0,\ln(\nicefrac{u}{t_0{\varepsilon}})]}u^{\nicefrac{3}{2}} t_0^{\nicefrac{1}{2}-\beta} \left[e^{-\frac{(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}-e^{-\frac{3(\ln(\nicefrac{u}{t_0{\varepsilon}})-s)}{2}}\right]e^{(\nicefrac{1}{2}-\beta)s}F(\sqrt{t_0}e^{\nicefrac{s}{2}}V_s^{(e)})\right\rvert}
\\\leq\begin{cases}
2 T^{\nicefrac{3}{2}} t_0^{\nicefrac{(\alpha+1)}{2}-\beta}e^{(\frac{\alpha+1}{2}-\beta)s}(2\ln(s))^{\nicefrac{\alpha}{2}}{\left\lvertF\left(\dfrac{V_s^{(e)}}{\sqrt{2\ln(s)}}\right)\right\rvert}\mathbb{1}_{\mathbb{R}^+}(s) & \mbox{under \eqref{hyp1},}\\
2 T^{\nicefrac{3}{2}} t_0^{\nicefrac{(\alpha+1)}{2}-\beta}e^{(\frac{(\alpha+1)}{2}-\beta)s}(2\ln(s))^{\nicefrac{\alpha}{2}}G\left(\dfrac{V_s^{(e)}}{\sqrt{2\ln(s)}}\right)\mathbb{1}_{\mathbb{R}^+}(s) & \mbox{under \eqref{hyp2}.}
\end{cases}
\end{gathered}$$ Recall that $\alpha\geq0$, then $F$ is a continuous function on $\mathbb{R}$. Thus, thanks to \[bounded\], $\Big|F\Big(\frac{V_s^{(e)}}{\sqrt{2\ln(s)}}\Big)\Big|$ is bounded for $s$ large enough. Using the continuity of $G$, the same argument applies under .
Finally, we obtain $${\varepsilon}^{\nicefrac{3}{2}}X_{u/{\varepsilon}}=Y_u^{{\varepsilon},t_0}+(t_0{\varepsilon})^{\nicefrac{3}{2}}\mathcal{B}_{\nicefrac{(\frac{u}{t_0{\varepsilon}}-1)^3}{3}}\stackrel{d}{= }Y_u^{{\varepsilon},t_0}+\mathcal{B}_{\nicefrac{(u-t_0{\varepsilon})^3}{3}}.$$ where $\sup_{u\in[t_0,T] }Y_u^{{\varepsilon},t_0}\underset{{\varepsilon}\to 0}{\longrightarrow}0$ almost surely, for all $T\geq t_0$. Thus, as ${\varepsilon}\to 0$, $${\mathop{}\!\mathrm{d}}\left({\varepsilon}^{\nicefrac{3}{2}}X_{u/{\varepsilon}},(t_0{\varepsilon})^{\nicefrac{3}{2}}\mathcal{B}_{\nicefrac{(\frac{u}{t_0{\varepsilon}}-1)^3}{3}}\right)\stackrel{\mathbb{P}}{\longrightarrow}0,$$ where $$\displaystyle {\mathop{}\!\mathrm{d}}:f,g\in C([t_0,+\infty))\mapsto \sum_{n=1}^{+\infty}\dfrac{1}{2^n}\min\Big(1,\sup_{[t_0,n]}{\left\lvertf(t)-g(t)\right\rvert}\Big)\text{ is a metric on }C([t_0,+\infty)).$$Indeed, fix $a >0$ and choose $N>0$ such that $\sum_{n=N+1}^{+\infty}\frac{1}{2^n}\leq \frac{a}{2}$, then, $${\mathop{}\!\mathrm{d}}\left({\varepsilon}^{\nicefrac{3}{2}}X_{u/{\varepsilon}},(t_0{\varepsilon})^{\nicefrac{3}{2}}\mathcal{B}_{\nicefrac{(\frac{u}{t_0{\varepsilon}}-1)^3}{3}}\right)\leq \frac{a}{2}+\sum_{n=1}^{N}\dfrac{1}{2^n}\sup_{[t_0,n]}{\left\lvertY_u^{\epsilon,t_0}\right\rvert}.$$ It follows that $$\mathbb{P}\left( {\mathop{}\!\mathrm{d}}\left({\varepsilon}^{\nicefrac{3}{2}}X_{u/{\varepsilon}},(t_0{\varepsilon})^{\nicefrac{3}{2}}\mathcal{B}_{\nicefrac{(\frac{u}{t_0{\varepsilon}}-1)^3}{3}}\right) >a \right) \leq \sum_{n=1}^{N}\mathbb{P}\left(\sup_{[t_0,n]}{\left\lvertY_u^{\epsilon,t_0}\right\rvert}>a' \right)\underset{\epsilon \to 0}{\longrightarrow}0,$$ where $a'=a(\sum_{n\geq N}^{+\infty}\nicefrac{1}{2^n})^{-1}$. The proof will be complete by applying [Theorem 3.1, p. 27, in [@Bill]]{}.
Appendix {#appendix .unnumbered}
========
We collect in this section some technical results. First, we discuss results of existence, explosion and behaviour of some time-homogeneous processes $V^+$ and $V^-$. For $\alpha>-1$, let $\pi$ be a non-negative function satisfying $$\forall x\in \mathbb{R}, \lambda>0, \ \pi(\lambda x)=\lambda^{\alpha}\pi(x).$$ Recall that $\pi$ satisfies . Under we take $\pi={\left\lvertF\right\rvert}$ and under , we take $\pi=G$.
Define the pathwise unique strong solution (up to the explosion time) to the time-homogeneous equation $$\label{Vpm}
{\mathop{}\!\mathrm{d}}V_s^{\pm}={\mathop{}\!\mathrm{d}}W_s - \dfrac{V_s^{\pm}}{2}{\mathop{}\!\mathrm{d}}s \pm t_0^{\frac{\alpha+1}{2}-\beta}\pi(V_s^{\pm})\mathbb{1}_{\{\pm F(V_s^{\pm})< 0\}} {\mathop{}\!\mathrm{d}}s.$$
\[explosion Vpm\] Set $\tau^{\pm}_{\infty}$, $\tau_{\infty}$, respectively the explosion time of $V^{\pm}$ and $V$.
1. If $\alpha\leq 1$ or $F(1)\geq 0$, then $\tau_{\infty}^+=\infty$ a.s.
2. If $\alpha\leq 1$ or $F(-1)\leq 0$, then $\tau_{\infty}^-=\infty$ a.s.
3. If $\alpha>1$ and $F(1)<0$, then $\mathbb{P}(\tau_{\infty}^+=\infty)=0$.
4. If $\alpha>1$ and $F(-1)>0$, then $\mathbb{P}(\tau_{\infty}^-=\infty)=0$.
<span style="font-variant:small-caps;">Step 1.</span> Firstly let us prove that $V^-\leq V^{(e)}\leq V^+$ almost surely. Indeed, if we denote $${\tt b}(t,x)=-\dfrac{x}{2}- t_0^{\nicefrac{1}{2}-\beta}e^{(\nicefrac{1}{2}-\beta)t}F(\sqrt{t_0}e^{\nicefrac{t}{2}}x)
\;\mbox{ and }\;{\tt b}^+(x)=-\dfrac{x}{2}+ t_0^{\nicefrac{(\alpha+1)}{2}-\beta}\pi(x)\mathbb{1}_{\{F(x)\leq0\}},$$ we can write, for all $t\geq 0$ and all $x\in\mathbb R$, $${\tt b}(t,x)\leq{\tt b}^{+}(x)
\Longleftrightarrow
\, - e^{(1/2-\beta)t}F(\sqrt{t_0}e^{t/2}x) \leq t_0^{\frac{\alpha}{2}}\pi(x)\mathbb{1}_{\{F(x)\leq0\}}.$$ This inequality holds true by the choice of $\pi$ and the assumption $\nicefrac{(\alpha+1)}{2}-\beta <0$. By using a comparison theorem (see [Theorem 1.1, Chap. VI p.437, in [@Ikeda]]{}) one gets, $ V^{(e)}\leq V^+$, almost surely. The other inequality can be obtained in the same way.
<span style="font-variant:small-caps;">Step 2.</span> Call $\tau_{\infty}^{(e)}$ the explosion time of $V^{(e)}$, then $\{\tau_{\infty}^{(e)}=\infty\}=\{\tau_{\infty}=\infty\}$. So, $\{ \tau_{\infty}^-=\infty \} \cap \{\tau_{\infty}^+=\infty\} \subset \{ \tau_{\infty}=\infty \} $.\
We give the detailed proof for (i) and (iii), the other parts could be obtained by changing “$+$” and “$-$” in the reasoning. First, we prove (i). The scale function of $V^+$ is given, for $x\in \mathbb{R}$, by $${\tt p}^+(x):=\int_{0}^x \exp\left(\frac{y^2}{2}-2 t_0^{\frac{(\alpha+1)}{2}-\beta}\medint\int_{0}^y\pi(z)\mathbb{1}_{\{F(z)\leq0\}}{\mathop{}\!\mathrm{d}}z \right){\mathop{}\!\mathrm{d}}y.$$ Note that, if $x<0$, $\displaystyle -{\tt p}^+(x)\geq \medint\int_{x}^{0}e^{\nicefrac{y^2}{2}}{\mathop{}\!\mathrm{d}}y$. Thus ${\tt p}^+(-\infty)=-\infty$. Suppose that $F(1)\geq0$, then for $x\geq 0$, $\displaystyle {\tt p}^+(x) =\medint\int_{0}^{x}e^{\nicefrac{y^2}{2}}{\mathop{}\!\mathrm{d}}y$, so ${\tt p}^+(\infty)=\infty$. By [Proposition 5.22, p. 345, in [@KS]]{}, the conclusion follows. Assume now that $\alpha< 1$ and $F(1)<0$. Then, for $x\geq0$, $\displaystyle {\tt p}^+(x)=\medint\int_{0}^x \exp\Big(\frac{y^2}{2}-2 t_0^{\frac{\alpha+1}{2}-\beta}\pi(1)\frac{y^{\alpha+1}}{\alpha+1}\Big){\mathop{}\!\mathrm{d}}y$, so ${\tt p}^+( \infty)= \infty$. Using the same result in [@KS], the conclusion follows. If $\alpha=1$, the drift has linear growth and the conclusion is clear.
<span style="font-variant:small-caps;">Step 3.</span> We proceed with the proof of (iii). Assume $\alpha>1$ and $F(1)<0$. As previously, ${\tt p}^+(-\infty)=-\infty$. Besides, ${\tt p}^+(\infty)<\infty$. Denote ${\tt m}^+:y\mapsto 2/({\tt p}^+)'(y)$ the speed measure of $V^+$. Fix $y>0$, then, setting ${\tt c}=2t_0^{\frac{\alpha+1}{2}-\beta}\pi(1)>0$, one can apply integration by parts to get: $$\begin{aligned}
\left(\tt p^+(\infty)- \tt p^+(y) \right){\tt m}^+(y) =& 2\exp\left(-\dfrac{y^2}{2}+{\tt c}\frac{y^{\alpha+1}}{\alpha+1}\right)\int_{y}^{+\infty} \exp\left(\dfrac{z^2}{2}-{\tt c}\frac{z^{\alpha+1}}{\alpha+1}\right) {\mathop{}\!\mathrm{d}}z \\ =&\dfrac{2}{{\tt c}y^{\alpha}-y}+ 2\exp\left(-\dfrac{y^2}{2}+{\tt c}\frac{y^{\alpha+1}}{\alpha+1}\right)\int_{y}^{\infty}e^{\frac{z^2}{2}-{\tt c}\frac{z^{\alpha+1}}{\alpha+1}}\dfrac{1-{\tt c}\alpha z^{\alpha-1}}{(z-{\tt c}z^{\alpha})^2}{\mathop{}\!\mathrm{d}}z.
\end{aligned}$$ One can deduce, by integrating small $o$, that $$\left(\tt p^+(\infty)- \tt p^+(y) \right){\tt m}^+(y)\underset{y\to \infty}{\sim}\dfrac{2}{{\tt c}y^{\alpha}-y}$$ which is an integrable function at $\infty$. The conclusion follows from [Theorem 5.29, p. 348, in [@KS]]{}.
We turn now to the study of the growth rate of the velocity process $V^{(e)}$.
\[Motoo Vpm\] When $\alpha< 1$ or $(F(-1),F(1)) \in \mathbb{R}^-\times \mathbb{R}^+$, $$\label{magnitude_Vpm}
\limsup_{t\to \infty}\dfrac{V^+_t}{\sqrt{2\ln(t)}}\leq 1 \text{ a.s.,}\quad\text{ and }\quad
\limsup_{t\to \infty}\dfrac{-V^-_t}{\sqrt{2\ln(t)}}\leq 1 \text{ a.s.}$$ Moreover, $$\label{magnitude_Ve}
\limsup_{t\to \infty}\dfrac{{\left\lvertV_t^{(e)}\right\rvert}}{\sqrt{2\ln(t)}}\leq1 \text{ a.s.}$$
\[bounded\] As a consequence of , when $\alpha< 1$ or $(F(-1),F(1)) \in \mathbb{R}^-\times \mathbb{R}^+$, for $t$ large enough, ${\left\lvertV_t^{(e)}\right\rvert}\leq C \sqrt{2\ln(t)}$ for some positive constant $C$.
\[Details\] Assuming , one gets writting $$\limsup_{t\to \infty}\dfrac{V^{(e)}_t}{\sqrt{2\ln(t)}}\leq \limsup_{t\to \infty}\dfrac{V^+_t}{\sqrt{2\ln(t)}}\leq 1 \text{ a.s.}\
$$ \_[t]{}\_[t]{}1 Before, proving let us state Motoo’s theorem which proof is given in [@motoo].
\[motoo\] Let $X$ be a regular continuous strong Markov process in $(a,\infty)$, $a\in [-\infty,\infty)$. Assume also that $X$ is time-homogeneous, with scale function ${\tt s}$. For every real positive increasing function $h$, $$\mathbb{P}\left( \limsup_{t\to \infty} \dfrac{X_t}{h(t)}\geq 1\right)=0 \text{ or }1 \text{ according to whether } \int^{\infty}\dfrac{{\mathop{}\!\mathrm{d}}t}{{\tt s}(h(t))}<\infty \text{ or } =\infty .$$
Motoo’s theorem yields for all ${\varepsilon}>0$, $$\mathbb{P}\Big(\limsup_{t\to \infty}\dfrac{V^+_t}{\sqrt{2\ln(t)}}\geq 1+{\varepsilon}\Big)=0 \text{ and }\mathbb{P}\Big(\limsup_{t\to \infty}\dfrac{-V^-_t}{\sqrt{2\ln(t)}}\geq 1+{\varepsilon}\Big)=0.$$ Indeed, define $\tilde{V}^-:=-V^-$, then $${\mathop{}\!\mathrm{d}}\tilde{V_s}^{-}=-{\mathop{}\!\mathrm{d}}W_s - \dfrac{\tilde{V_s}^{-}}{2}{\mathop{}\!\mathrm{d}}s + t_0^{\nicefrac{(\alpha+1)}{2}-\beta}\pi(-\tilde{V_s}^{-})\mathbb{1}_{\{F(-\tilde{V_s}^{-})> 0\}} {\mathop{}\!\mathrm{d}}s.$$ Fix $y_0>0$. The scale function of $V^+$ and $\tilde{V}^-$ is given, for $y\geq y_0$, by $${\tt s}^{\pm}(y)= \kappa\medint\int_{y_0}^{y}\exp\Big(\frac{z^2}{2}-2C^{\pm}\frac{z^{\alpha+1}}{\alpha+1}\Big){\mathop{}\!\mathrm{d}}z.$$ Here and elsewhere $\kappa$ denotes positive constants which can change of value from line to line, and $$C^+:=t_0^{\frac{\alpha+1}{2}-\beta} \pi(1)\mathbb{1}_{\{F(1)<0\}}\;\;\mbox{ for } V^+\quad\mbox{ and }\quad
C^-:=t_0^{\frac{\alpha+1}{2}-\beta} \pi(-1)\mathbb{1}_{\{F(-1)>0\}}\;\;\mbox{ for } \tilde{V}^-.$$ Let $\epsilon>0$. Define the positive increasing function $h:t\mapsto (1+\epsilon)\sqrt{2\ln(t)}$. We will show that $\nicefrac{1}{{\tt s}(h)}$ is integrable at infinity. Firstly, remark that $$\int_{y_0}^{+\infty} \dfrac{1}{{\tt s}(h(t))}{\mathop{}\!\mathrm{d}}t = \int_{h(y_0)}^{+\infty} \dfrac{1}{{\tt s}(y)}\dfrac{{\mathop{}\!\mathrm{d}}y }{h'(h^{-1}(y))}=\int_{h(y_0)}^{+\infty} \dfrac{1}{{\tt s}(y)}\dfrac{y \exp\big(\nicefrac{y^2}{\big(2(1+\epsilon)^2\big)}\big) }{(1+{\varepsilon})^2}{\mathop{}\!\mathrm{d}}y .$$ It remains to find an equivalent of ${\tt s}$ at infinity. In the following “$\asymp$” means equality up to a multiplicative positive constant. Fix $y>y_0$, integrating by parts, one gets, $$\begin{aligned}
{\tt s}(y)&\asymp \int_{y_0}^y \exp\Big(\frac{z^2}{2}-2C^{\pm}\,\frac{z^{\alpha+1}}{\alpha+1}\Big)\big(z-2C^{\pm}\,z^{\alpha}\big)\cdot\dfrac{1}{z-2C^{\pm}\,z^{\alpha}}{\mathop{}\!\mathrm{d}}z \\ &\asymp \dfrac{\exp\big(\frac{y^2}{2}-2C^{\pm}\,\frac{y^{\alpha+1}}{\alpha+1}\big)}{y-2C^{\pm}\,y^{\alpha}}-\kappa+\int_{y_0}^{y}\dfrac{1-2\alpha C^{\pm}\,z^{\alpha-1}}{(z-2C^{\pm}\,z^{\alpha})^2}\exp\Big(\frac{z^2}{2}-2C^{\pm}\,\frac{z^{\alpha+1}}{\alpha+1}\Big){\mathop{}\!\mathrm{d}}z.
\end{aligned}$$ Since $\alpha>-1$, $\displaystyle\lim_{y\to\infty}\dfrac{1-2\alpha C^{\pm}\,y^{\alpha-1}}{(y-2C^{\pm}\,y^{\alpha})^2}=0$, except when $\alpha=1$ and $C^{\pm}=\frac{1}{2}$. Moreover the function $y\mapsto\exp\Big(\frac{y^2}{2}-2C^{\pm}\,\frac{y^{\alpha+1}}{\alpha+1}\Big)$ is not integrable at infinity, when $C^{\pm}=0$, or $\alpha<1$, or $\alpha=1$ and $C^{\pm}<\frac{1}{2}$. In these cases one gets, by integration, $${\tt s}(y)\underset{y\to \infty}{\sim}\kappa \dfrac{\exp\big(\frac{y^2}{2}-2C^{\pm}\,\frac{y^{\alpha+1}}{\alpha+1}\big)}{y-2C^{\pm}\,y^{\alpha}}.$$ Hence, $$\dfrac{1}{{\tt s}(y)}\dfrac{y \exp\big(\frac{y^2}{2(1+{\varepsilon})^2}\big)}{(1+{\varepsilon})^2}\underset{y\to \infty}{\sim} \kappa\big(y^2-2C^{\pm}\,y^{\alpha+1}\big)\exp\Big(-\frac{y^2}{2}\Big(1-\frac{1}{(1+{\varepsilon}^2)}\Big)+2C^{\pm}\,\frac{y^{\alpha+1}}{\alpha+1}\Big).$$ which is integrable if $C^{\pm}=0$ or $\alpha<1$, or $\alpha=1$ and $C^{\pm}<\frac{1}{2}$. One can conclude using Motoo’s theorem.
\[alpha=1\] The preceding proof shows that the result of \[Motoo Vpm\] is still true for $\alpha=1$ and some condition depending on $\pi(1)$ and $\pi(-1)$ which ensure that $C^{\pm}<\frac{1}{2}$. Consequently, our main result is also true under the same condition.
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---
address: 'Samuel Pocchiola — Département de mathématiques d’Orsay, bâtiment 425, Faculté des sciences d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex, france'
author:
- Samuel Pocchiola
title: ' Lie algebras of infinitesimal automorphisms for the model manifolds of general classes ${\sf II}$, ${\sf III_2}$ and ${\sf IV_2}$'
---
abstract {#abstract .unnumbered}
========
We determine the Lie algebra of infinitesimal CR-automorphisms of the models of general classes ${\sf II}$, ${\sf III_2}$ and ${\sf IV_2}$ through Cartan’s equivalence method.
Introduction
============
The classification of CR-manifolds up to dimension $5$ has highlighted the existence of $6$ non-trivial classes of CR-manifolds, which have been referred to as general classes ${\sf I}$, ${\sf II}$, ${\sf III}_1$, ${\sf III}_2$, ${\sf IV}_1$ and ${\sf IV}_2$ [@MPS]. Each of these classes entails a distinguished manifold, the model, whose Lie algebra of infinitesimal CR-automorphisms is of maximal dimension. It plays a special role, as CR-manifolds belonging to the same class can be viewed as its deformations, generally by the way of Cartan connection. The aim of this paper is to determine the Lie algebra of infinitesimal CR-automorphisms of the models for general classes ${\sf II}$, ${\sf III}_2$ and ${\sf IV}_2$. This is already known [@BES-2007; @Kaup-Zaitsev] for general classes ${\sf II}$ (Engel manifolds) and ${\sf IV}_2$ (2-nondegenerate, $5$-dimensional CR-manifolds of constant Levi rank $1$), but is unknown, to our knowledge, in the case of general class ${\sf III}_2$. In our view, the main interest of this paper is to provide a unified treatment for the $3$ classes through the use of Cartan’s equivalence method, in the spirit of [@Olver-1995]. Cartan’s equivalence method has indeed been employed recently to solve the equivalence problem for general classes ${\sf II}$, ${\sf III}_2$ and ${\sf IV_2}$ [@pocchiola; @pocchiola2; @pocchiola3]. For each of these classes, the solution to the equivalence problem for the model has been of a great help for the treatment of the general case, as a similar structure of normalizations of the group parameters occurs in both cases.
For general class ${\sf II}$, the model is provided by Beloshapka’s cubic in ${\mathbb{C}}^3$, which is the CR-manifold defined by the equations: $${\sf B}: \qquad \qquad
\begin{aligned}
w_1 & = {\overline}{w_1} + 2 \, i \, z {\overline}{z}, \\
w_2 & = {\overline}{w_2} + 2 \, i \, z {\overline}{z} \left( z + {\overline}{z} \right).
\end{aligned}$$ For general class ${\sf III}_2$, the model is the $5$-dimensional submanifold ${\sf N} \subset {\mathbb{C}}^4$ defined by: $${\sf N}: \qquad \qquad
\begin{aligned}
w_1 & = {\overline}{w_1} + 2 \, i \, z {\overline}{z}, \\
w_2 & = {\overline}{w_2} + 2 \, i \, z {\overline}{z} \left( z + {\overline}{z} \right), \\
w_3 & = {\overline}{w_3} + 2 i \, z {{\overline}{z}}( z^2 + \frac{3}{2} \, z {{\overline}{z}}+ {{\overline}{z}}^2 ).
\end{aligned}$$ For general class ${\sf IV}_2$, the model is provided by the tube over the future light cone, ${\sf LC} \subset {\mathbb{C}}^3$, defined by: $${\sf LC}: \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\left( {\sf Re} \,z_1 \right)^2 -\left( {\sf Re} \,z_2 \right)^2
- \left( {\sf Re} \,z_3 \right)^2
=
0, \qquad \qquad {\sf Re} \,z_1 > 0
.$$
A Cartan connection has been constructed for CR-manifolds belonging to general class ${\sf II}$ [@BES-2007; @pocchiola2] and ${\sf III}_2$ [@pocchiola3]. The equivalence problem for manifolds belonging to general class ${\sf IV}_2$ has been solved either by the determination of an absolute parallelism [@Isaev-Zaitsev; @pocchiola], or the construction of a Cartan connection [@Medori-Spiro]. We use Cartan’s equivalence method for which we refer to [@Olver-1995] as a standard reference.
Class ${\sf II}$
================
This section is devoted to the determination of the Lie algebra of CR-automorphisms of Beloshapka’s cubic in ${\mathbb{C}}^3$, which is the CR-manifold defined by the equations: $${\sf B}: \qquad \qquad
\begin{aligned}
w_1 & = {\overline}{w_1} + 2 \, i \, z {\overline}{z}, \\
w_2 & = {\overline}{w_2} + 2 \, i \, z {\overline}{z} \left( z + {\overline}{z} \right).
\end{aligned}$$ It is the model manifold for generic $4$-dimensional CR-manifolds of CR dimension $1$ and real codimension $2$, i.e. CR-manifolds belonging to class ${\sf II}$, in the sense that any such manifold might be viewed as a deformation of Beloshapka’s cubic by the way of a Cartan connection [@BES-2007; @pocchiola2]. The main result of this section is:
\[thm:Bc\] Beloshapka’s cubic, $${\sf B}: \qquad \qquad
\begin{aligned}
w_1 & = {\overline}{w_1} + 2 \, i \, z {\overline}{z}, \\
w_2 & = {\overline}{w_2} + 2 \, i \, z {\overline}{z} \left( z + {\overline}{z} \right),
\end{aligned}$$ has a ${\bf 5}$-dimensional Lie algebra of CR-automorphisms. A basis for the Maurer-Cartan forms of ${\sf aut_{CR}}({\sf B})$ is provided by the $5$ differential $1$-forms $\sigma$, $\rho$, $\zeta$, ${\overline}{\zeta}$, $\alpha$, which satisfy the structure equations: $$\begin{aligned}
d \sigma &= 3 \left. \alpha \wedge \sigma \right. + \left. \rho \wedge \zeta \right.
+ \left. \rho \wedge {\overline}{\zeta} \right., \\
d \rho & = 2 \left. \alpha \wedge \rho \right. + i \, \left. \zeta \wedge {\overline}{\zeta} \right., \\
d \zeta &= \left. \alpha \wedge \zeta \right., \\
d {\overline}{\zeta} &= \left. \alpha \wedge {\overline}{\zeta} \right., \\
d \alpha & = 0.
\end{aligned}$$
Initial G-structure.
--------------------
The vectors field ${\mathcal{L}_1}$ defined by: $${\mathcal{L}_1}:=
\frac{\partial}{\partial z}
+
i \, {{\overline}{z}}\,
\frac{\partial}{\partial u_1}
+ i \,
\left( 2 z {{\overline}{z}}+ {{\overline}{z}}^2 \right) \, \frac{\partial}{\partial u_2},$$ together with its conjugate:
$${\overline{\mathcal{L}_1}}:= \frac{\partial}{\partial {{\overline}{z}}}
- i \, z \,
\frac{\partial}{\partial u_1}
- i \,
\left( 2 z {{\overline}{z}}+ z^2 \right) \, \frac{\partial}{\partial u_2},$$
constitute a basis of $T^{1,0}_p {\sf B}$ at each point $p$ of ${\sf B}$. Moreover the vector fields ${\mathcal{T}}$ and ${\mathcal{S}}$ defined by: $${\mathcal{T}}: = i \, \big[{\mathcal{L}_1}, {\overline{\mathcal{L}_1}}]
,$$ and $${\mathcal{S}}: = \big[{\mathcal{L}_1}, {\mathcal{T}}\big],$$ complete a frame on ${\sf B}$: $$\big\{\mathcal{S},\,
\mathcal{T},\,\mathcal{L}, \,
\overline{\mathcal{L}}
\big\}
.$$ The expressions of ${\mathcal{T}}$ and ${\mathcal{S}}$ are:
$$\begin{aligned}
{\mathcal{T}}& = 2 \, \frac{\partial}{\partial u_1} + \left( 4z + 4 {{\overline}{z}}\right) \frac{\partial}{\partial u_2}, \\
{\mathcal{S}}& = 4 \, \frac{\partial}{\partial u_2}
.\end{aligned}$$
The dual coframe $ \big(\sigma_0,\,
\rho_0,
\,
\zeta_0,
\,
\overline{\zeta_0}
\big)$ is thus given by:
$$\begin{aligned}
\sigma_0 &= \frac{i}{4} \, {{\overline}{z}}^2 \, dz -
\frac{i}{4} \, z^2 \, d {{\overline}{z}}- \left( \frac{1}{2} \, z + \frac{1}{2} \, {{\overline}{z}}\right) du_1 + \frac{1}{4} \, du_2
,\\
\rho_0 &= - \frac{i}{2} \, {{\overline}{z}}\, dz + \frac{i}{2} \, z \, d{{\overline}{z}}+ \frac{1}{2} \, du_1, \\
\zeta_0 &= dz,\\
{\overline}{\zeta_0} &= d{{\overline}{z}}.
\end{aligned}$$
We deduce the structure equations enjoyed by $
\big(\sigma_0,\,
\rho_0,\,
\zeta_0, \,
\overline{\zeta_0},
\big)
$: $$\label{eq:B}
\begin{aligned}
d \sigma_0 & = \rho_0 \wedge \zeta_0 + \rho_0 \wedge {\overline}{\zeta_0}, \\
d \rho_{0} & = i \, \zeta_0 \wedge {\overline}{\zeta_0},\\
d \zeta_0 & = 0, \\
d {\overline}{ \zeta_{0}} & = 0.
\end{aligned}$$
As the torsion coefficients of these structure equations are constants, we have the following result:
Beloshapka’s cubic is locally isomorphic to a Lie group whose Maurer-Cartan forms satisfy the structure equations $(\ref{eq:B})$.
The matrix Lie group which encodes suitably the equivalence problem for Beloshapka’s cubic (see [@pocchiola2]) is the $10$-dimensional Lie group $G_1$ whose elements $g$ are of the form: $$g := \begin{pmatrix}
{{{\sf a}}^2} {{\overline}{{\sf a}}}& 0 & 0 & 0 \\
{{\sf c}}& {{\sf a}}{{\overline}{{\sf a}}}& 0 & 0 \\
{{\sf d}}& {{\sf b}}& {{\sf a}}& 0 \\
{{\sf e}}& {{\overline}{{\sf b}}}& 0 & {{\overline}{{\sf a}}}\end{pmatrix}.$$
With the notations: $$\begin{aligned}
{1}
\omega_{0} :=
\begin{pmatrix}
\sigma_{0} \\ \rho_0 \\ \zeta_{0} \\ {\overline}{\zeta}_0
\end{pmatrix}, & \qquad
\omega:=
\begin{pmatrix}
\sigma \\ \rho \\ \zeta \\ {\overline}{\zeta}
\end{pmatrix},\end{aligned}$$ we introduce the $G_1$-structure $P^1$ on ${\sf B}$ constituted by the coframes $\omega$ which satisfy the relation:
$$\omega:= g \cdot \omega_0.$$
The proof of theorem (\[thm:Bc\]) relies on successive reductions of $P^1$ through Cartan’s equivalence method.
Normalization of [[[a]{}]{}]{}
------------------------------
The structure equations for the lifted coframe $\omega$ are related to those of the base coframe $\omega_0$ by the relation: $$\label{eq:str}
d \omega = dg \cdot g^{-1} \wedge \omega + g \cdot d \omega_{0}.$$ The term $ dg \cdot g^{-1} \wedge \omega$ depends only on the structure equations of $G_1$ and is expressed through its Maurer-Cartan forms. The term $ g \cdot d \omega_{0}$ contains the so-called torsion coefficients of the $G_1$-structure. We can compute it easily in terms of the forms $\sigma$, $\rho$, $\zeta$, ${\overline}{\zeta}$, by a simple multiplication by $g$ in the formulae $(\ref{eq:B})$ and a linear change of variables. The Maurer-Cartan forms for the group $G_1$ are given by the linearly independent entries of the matrix $dg \cdot g^{-1}$, which are: $$\begin{aligned}
\alpha^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\alpha^2 &:= -{\frac {{{\sf b}}{ d{{\sf a}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}+{\frac {{ d{{\sf b}}}}{{{\sf a}}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^3 &:= -{\frac {{{\sf c}}{ d{{\sf a}}}}{{ {{\overline}{{\sf a}}}}\,{{{\sf a}}}^{3}}}-{\frac {{{\sf c}}{ d{{\overline}{{\sf a}}}}}{{{ {{\overline}{{\sf a}}}}}^{2}{{{\sf a}}}^{2}}} + { \frac {{ d{{\sf c}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^4 &: =-{\frac { \left( {{\sf d}}{{\sf a}}{ {{\overline}{{\sf a}}}}-{{\sf b}}{{\sf c}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{4}{
{ {{\overline}{{\sf a}}}}}^{2}}}-{\frac {{{\sf c}}{ d{{\sf b}}}}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}}}+{\frac {{
d{{\sf d}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^5 & :=-{\frac { \left( {{\sf e}}{{\sf a}}{ {{\overline}{{\sf a}}}}-{ {{\overline}{{\sf b}}}}\,{{\sf c}}\right) { d{{\overline}{{\sf a}}}}
}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{3}}}-{\frac {{{\sf c}}{ d{{\overline}{{\sf b}}}}}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}}}+
{\frac {{ d{{\sf e}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}},
\end{aligned}$$ together with their conjugates.
The first structure equation is given by:
$$d \sigma = 2 \left.\alpha^1 \wedge \sigma \right. + \left.{\overline}{\alpha^1} \wedge \sigma \right.
+ \left( \frac{{{\sf e}}}{{{\sf a}}\aa^2} + \frac{{{\sf d}}}{{{\sf a}}^2 {{\overline}{{\sf a}}}} \right) \left.\sigma \wedge \rho \right.
- \frac{{{\sf c}}}{{{\sf a}}^2 {{\overline}{{\sf a}}}} \, \left.\sigma \wedge \zeta \right.
- \frac{{{\sf c}}}{{{\sf a}}{{\overline}{{\sf a}}}^2} \, \left. \sigma \wedge {\overline}{\zeta} \right. + \left.\rho \wedge \zeta \right. +
\frac{{{\sf a}}}{{{\overline}{{\sf a}}}} \, \left. \rho \wedge {\overline}{\zeta} \right..$$
from which we immediately deduce that $\frac{{{\sf a}}}{{{\overline}{{\sf a}}}}$ is an essential torsion coefficient which might be normalised to $1$ by setting: $${{\sf a}}= {{\overline}{{\sf a}}}.$$
Normalizations of ${{\sf b}}$ and ${{\sf c}}$
---------------------------------------------
We have thus reduced the $G_{1}$ equivalence problem on ${\sf B}$ to a $G_2$ equivalence problem, where $G_2$ is the $9$ dimensional real matrix Lie group whose elements are of the form $$g := \begin{pmatrix}
{{{\sf a}}}^3 & 0 & 0 & 0 \\
{{\sf c}}& {{\sf a}}^2 & 0 & 0 \\
{{\sf d}}& {{\sf b}}& {{\sf a}}& 0 \\
{{\sf e}}& {{\overline}{{\sf b}}}& 0 & {{\sf a}}\end{pmatrix},
\qquad \qquad {{\sf a}}\in {\mathbb{R}}.$$ The Maurer-Cartan forms of $G_2$ are given by: $$\begin{aligned}
\beta^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\beta^2 &:= -{\frac {{{\sf b}}d{{\sf a}}}{{{{\sf a}}}^{3}}}+{\frac {{ d{{\sf b}}}}{{{\sf a}}^2}}, \\
\beta^3 &:=- 2 \, {\frac {{{\sf c}}d{{\sf a}}}{{{{\sf a}}}^{4}}} + { \frac {{ d{{\sf c}}}}{{{{\sf a}}}^{3}}}, \\
\beta^4 & :=-{\frac { \left( {{\sf d}}{{\sf a}}^2 -{{\sf b}}{{\sf c}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{6}}}-{\frac {{{\sf c}}{ d{{\sf b}}}}{{{{\sf a}}}^{5}}}+{\frac {{
d{{\sf d}}}}{{{{\sf a}}}^{3}}}, \\
\beta^5 & :=-\frac { \left( {{\sf e}}{{\sf a}}^2-{ {{\overline}{{\sf b}}}}\,{{\sf c}}\right) d{{\sf a}}}{{{\sf a}}^6}-\frac {{{\sf c}}{ d{{\overline}{{\sf b}}}}}{{{\sf a}}^{5}}+
\frac { d{{\sf e}}}{{{\sf a}}^{3}}, \\
\end{aligned}$$ together with ${\overline}{\beta^2}$, ${\overline}{\beta^3}$, ${\overline}{\beta^4}$, ${\overline}{\beta^5}$. Using formula $(\ref{eq:str})$, we get the structure equations for the lifted coframe $(\sigma, \rho, \zeta, {\overline}{\zeta})$ from those of the base coframe $(\sigma_0, \rho_0, \zeta_0, {\overline}{\zeta_0})$ by a matrix multiplication and a linear change of coordinates, as in the first step:
$$\begin{gathered}
d \sigma = 3 \, \beta^1 \wedge \sigma \\
+
U^{\sigma}_{\sigma \rho} \left. \sigma \wedge \rho \right.
+
U^{\sigma}_{\sigma \zeta} \left. \sigma \wedge \zeta \right.
+
U^{\sigma}_{\sigma {\overline}{\zeta}} \left. \sigma \wedge {\overline}{\zeta} \right.
+
\rho \wedge \zeta
+
\rho \wedge {\overline}{\zeta},\end{gathered}$$
$$\begin{gathered}
d \rho
=
2 \beta^{1} \wedge \rho + \beta^3 \wedge \sigma \\
+
U^{\rho}_{\sigma \rho} \, \sigma \wedge \rho
+
U^{\rho}_{\sigma \zeta} \, \sigma \wedge \zeta
+
U^{\rho}_{\sigma {\overline}{\zeta}} \, \sigma \wedge {\overline}{\zeta} \\
+
U^{\rho}_{\rho \zeta} \, \rho \wedge \zeta
+
U^{\rho}_{\rho {\overline}{\zeta}} \, \rho \wedge {\overline}{\zeta}
+
i \, \zeta \wedge {\overline}{\zeta}
,\end{gathered}$$
$$\begin{gathered}
d \zeta
=
{\beta}^{1} \wedge \zeta + {\beta}^{2} \wedge \rho + {\beta}^{4} \wedge \sigma \\
+
U^{\zeta}_{\sigma \rho} \, \sigma \wedge \rho
+
U^{\zeta}_{\sigma \zeta} \, \sigma \wedge \zeta
+
U^{\zeta}_{\sigma {\overline}{\zeta}} \, \sigma \wedge {\overline}{\zeta}
+
U^{\zeta}_{\rho \zeta} \, \rho \wedge \zeta \\
+
U^{\zeta}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
U^{\zeta}_{\zeta \overline{\zeta}} \, \zeta \wedge \overline{\zeta}
.\end{gathered}$$
We now proceed with the absorption phase. We introduce the modified Maurer-Cartan forms: $$\widetilde{\beta}^i= \beta^i - y_{\sigma} \, \sigma - y_{\rho}^i \, \rho - y_{\zeta}^i \, \zeta \,
- y_{\overline{\zeta}}^i \, \overline{\zeta},$$ such that the structure equations rewrite:
$$\begin{gathered}
d \sigma = 3 \left. \widetilde{\beta}^1 \wedge \sigma \right. \\
+
\left( U^{\sigma}_{\sigma \rho} - 3 \, y^1_{\rho} \right) \, \left. \sigma \wedge \rho \right.
+
\left( U^{\sigma}_{\sigma \zeta} -3 \,y^1_{\zeta} \right) \, \left. \sigma \wedge \zeta \right. \\
+
\left( U^{\sigma}_{\sigma {\overline}{\zeta}} -3 \,y^1_{{\overline}{\zeta}} \right) \, \left. \sigma \wedge {\overline}{\zeta} \right.
\rho \wedge \zeta
+
\rho \wedge {\overline}{\zeta},\end{gathered}$$
$$\begin{gathered}
d \rho
=
2 \widetilde{\beta}^{1} \wedge \rho + \widetilde{\beta}^3 \wedge \sigma \\
+
\left( U^{\rho}_{\sigma \rho} + 2 \, y^1_{\sigma} - y^3_{\rho} \right) \, \left. \sigma \wedge \rho \right.
+
\left( U^{\rho}_{\sigma \zeta} - y^3_{\zeta} \right) \, \left. \sigma \wedge \zeta \right. \\
+
\left( U^{\rho}_{\sigma {\overline}{\zeta}} - y^3_{{\overline}{\zeta}} \right) \, \left. \sigma \wedge {\overline}{\zeta} \right.
+
\left( U^{\rho}_{\rho \zeta} - 2 \, y^1_{\zeta} \right) \, \left. \rho \wedge \zeta \right. \\
+
\left( U^{\rho}_{\rho {\overline}{\zeta}} - 2 \, y^1_{{\overline}{\zeta}} \right) \, \left. \rho \wedge {\overline}{\zeta} \right.
+
i \, \left. \zeta \wedge {\overline}{\zeta} \right.
,\end{gathered}$$
$$\begin{gathered}
d \zeta
=
\widetilde{\beta^{1}} \wedge \zeta + \widetilde{\beta^{2}} \wedge \rho + \widetilde{\beta^{4}} \wedge \sigma \\
+
\left( U^{\zeta}_{\sigma \rho} + y^2_{\sigma}- y^4_{\rho} \right) \, \left. \sigma \wedge \rho \right.
+
\left( U^{\zeta}_{\sigma \zeta} + y^1_{\sigma} - y^4_{\zeta} \right) \, \left. \sigma \wedge \zeta \right. \\
+
\left( U^{\zeta}_{\sigma {\overline}{\zeta}} - y^4_{{\overline}{\zeta}} \right) \, \left. \sigma \wedge {\overline}{\zeta} \right.
+
\left( U^{\zeta}_{\rho \zeta} + y^1_{\rho} - y^2_{\zeta} \right) \, \left. \rho \wedge \zeta \right. \\
+
\left( U^{\zeta}_{\rho \overline{\zeta}} - y^2_{{\overline}{\zeta}} \right) \, \left. \rho \wedge \overline{\zeta} \right.
+
\left( U^{\zeta}_{\zeta \overline{\zeta}} - y^1_{{\overline}{\zeta}} \right) \, \left. \zeta \wedge \overline{\zeta} \right.
.\end{gathered}$$
We get the following absorbtion equations:
$$\begin{aligned}
{3}
3 \, y^1_{\rho} & = U^{\sigma}_{\sigma \rho}, & \qquad \qquad
3 \,y^1_{\zeta} & = U^{\sigma}_{\sigma \zeta}, & \qquad \qquad
3 \,y^1_{{\overline}{\zeta}} & = U^{\sigma}_{\sigma {\overline}{\zeta}}, \\
-2 \, y^1_{\sigma} + y^3_{\rho} &= U^{\rho}_{\sigma \rho}, &\qquad \qquad
y^3_{\zeta} &= U^{\rho}_{\sigma \zeta} , &\qquad \qquad
y^3_{{\overline}{\zeta}} &= U^{\rho}_{\sigma {\overline}{\zeta}}, \\
2 \, y^1_{\zeta} &= U^{\rho}_{\rho \zeta}, & \qquad \qquad
2 \, y^1_{{\overline}{\zeta}} &= U^{\rho}_{\rho {\overline}{\zeta}}, & \qquad \qquad
-y^2_{\sigma} + y^4_{\rho} & = U^{\zeta}_{\sigma \rho}, \\
-y^1_{\sigma} + y^4_{\zeta} & = U^{\zeta}_{\sigma \zeta}, & \qquad \qquad
y^4_{{\overline}{\zeta}} & = U^{\zeta}_{\sigma {\overline}{\zeta}}, & \qquad \qquad
-y^1_{\rho} + y^2_{\zeta} & = U^{\zeta}_{\rho \zeta}, \\
y^2_{{\overline}{\zeta}} & = U^{\zeta}_{\rho \overline{\zeta}}, & \qquad \qquad
y^1_{{\overline}{\zeta}} & = U^{\zeta}_{\zeta \overline{\zeta}}.\end{aligned}$$
Eliminating $y^1_{{\overline}{\zeta}}$ among the previous equations leads to: $$U^{\zeta}_{\zeta {\overline}{\zeta}} = \frac{1}{2} \, U^{\rho}_{\rho {\overline}{\zeta}} = \frac{1}{3} \, U^{\sigma}_{\sigma {\overline}{\zeta}}
,$$ that is: $${\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}} = \frac{1}{2} \, \left( {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}-{\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}} \right)
=- \frac{1}{3} \, {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}
,$$ from which we easily deduce that $${{\sf b}}= {{\sf c}}= 0.$$
Normalizations of ${{\sf d}}$ and ${{\sf e}}$
---------------------------------------------
We have thus reduced the group $G_2$ to a new group $G_3$, whose elements are of the form $$g := \begin{pmatrix}
{{{\sf a}}}^3 & 0 & 0 & 0 \\
0 & {{\sf a}}^2 & 0 & 0 \\
{{\sf d}}& 0 & {{\sf a}}& 0 \\
{{\sf e}}& 0 & 0 & {{\sf a}}\end{pmatrix}.$$
The Maurer Cartan forms of $G_3$ are: $$\begin{aligned}
\gamma^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\gamma^2 & :=-{\frac {{{\sf d}}{ d{{\sf a}}}}{{{{\sf a}}}^{4}}} + {\frac {{
d{{\sf d}}}}{{{{\sf a}}}^{3}}}, \\
\gamma^3 & :=-\frac { {{\sf e}}d {{\sf a}}}{{{\sf a}}^4} +
\frac { d{{\sf e}}}{{{\sf a}}^{3}}.
\end{aligned}$$
The third loop of Cartan’s method is straightforward. We get the following structure equations: $$\begin{aligned}
d \sigma &= 3 \left.
\gamma^1 \wedge \sigma \right.
+ \frac{{{\sf d}}+ {{\sf e}}}{{{\sf a}}^4} \left. \sigma \wedge \rho \right. + \left. \rho \wedge \zeta \right.
+ \left. \rho \wedge {\overline}{\zeta} \right., \\
d \rho & = 2 \left. \gamma^1 \wedge \rho \right. + i \, \frac{{{\sf e}}}{{{\sf a}}^3} \, \left. \sigma \wedge \zeta \right.
- i \, \frac{{{\sf d}}}{{{\sf a}}^3} \left. \sigma \wedge {\overline}{\zeta} \right. + i \, \left. \zeta \wedge {\overline}{\zeta} \right., \\
d \zeta &= \left. \gamma^1 \wedge \zeta \right.
+ \left. \gamma^2 \wedge \sigma \right. + \frac{{{\sf d}}\left( {{\sf d}}+ {{\sf e}}\right)}{{{\sf a}}^6}
\left. \sigma \wedge \rho \right.
+
\frac{{{\sf d}}}{{{\sf a}}^3} \left. \rho \wedge \zeta \right. + \frac{{{\sf d}}}{{{\sf a}}^3} \left. \rho \wedge {\overline}{\zeta} \right., \\
d {\overline}{\zeta} &= \left. \gamma^1 \wedge {\overline}{\zeta} \right. + \left. \gamma^3 \wedge \sigma \right.
+ \frac{{{\sf e}}\left( {{\sf d}}+ {{\sf e}}\right)}{{{\sf a}}^6} \left. \sigma \wedge \rho \right.
+
\frac{{{\sf e}}}{{{\sf a}}^3} \left. \rho \wedge \zeta \right. + \frac{{{\sf e}}}{{{\sf a}}^3} \left. \rho \wedge {\overline}{\zeta} \right.
,\end{aligned}$$ from which we deduce that we can perform the normalizations: $${{\sf e}}= {{\sf d}}= 0.$$
With the $1$-dimensional group $G_4$ whose elements $g$ are of the form: $$g := \begin{pmatrix}
{{{\sf a}}}^3 & 0 & 0 & 0 \\
0 & {{\sf a}}^2 & 0 & 0 \\
0 & 0 & {{\sf a}}& 0 \\
0 & 0 & 0 & {{\sf a}}\end{pmatrix},$$ and whose Maurer-Cartan form is given by $$\alpha:= \frac{d {{\sf a}}}{{{\sf a}}},$$ we get the following structure equations: $$\begin{aligned}
d \sigma &= 3 \left. \alpha \wedge \sigma \right. + \left. \rho \wedge \zeta \right.
+ \left. \rho \wedge {\overline}{\zeta} \right., \\
d \rho & = 2 \left. \alpha \wedge \rho \right. + i \, \left. \zeta \wedge {\overline}{\zeta} \right., \\
d \zeta &= \left. \alpha \wedge \zeta \right., \\
d {\overline}{\zeta} &= \left. \alpha \wedge {\overline}{\zeta} \right.
.\end{aligned}$$ No more normalizations are allowed at this stage. We thus just perform a prolongation by adjoining the form $\alpha$ to the structure equations, whose exterior derivative is given by: $$d \alpha = 0.$$ This completes the proof of Theorem \[thm:Bc\].
Class ${\sf III_{2}}$
=====================
This section is devoted to the determination of the Lie algebra of CR-automorphisms of the model manifold of class ${\sf III_{2}}$ which is defined by the equations: $${\sf N}: \qquad \qquad
\begin{aligned}
w_1 & = {\overline}{w_1} + 2 \, i \, z {\overline}{z}, \\
w_2 & = {\overline}{w_2} + 2 \, i \, z {\overline}{z} \left( z + {\overline}{z} \right), \\
w_3 & = {\overline}{w_3} + 2 i \, z {{\overline}{z}}( z^2 + \frac{3}{2} \, z {{\overline}{z}}+ {{\overline}{z}}^2 ).
\end{aligned}$$ It is the model manifold for CR-manifolds belonging to class ${\sf III_2}$, in the sense that any such manifold might be viewed as a deformation of ${\sf N}$ by the way of a Cartan connection ( [@pocchiola3]). The main result of this section is the following:
\[thm:N\] The model of the class ${\sf III_2}$[:]{} $${\sf N}: \qquad \qquad
\begin{aligned}
w_1 & = {\overline}{w_1} + 2 \, i \, z {\overline}{z}, \\
w_2 & = {\overline}{w_2} + 2 \, i \, z {\overline}{z} \left( z + {\overline}{z} \right), \\
w_3 & = {\overline}{w_3} + 2 i \, z {{\overline}{z}}( z^2 + \frac{3}{2} \, z {{\overline}{z}}+ {{\overline}{z}}^2 ),
\end{aligned}$$ has a ${\bf 6}$-dimensional Lie algebra of CR-automorphisms. A basis for the Maurer-Cartan forms of ${\sf aut_{CR}}({\sf N})$ is provided by the $6$ differential $1$-forms $\tau$, $\sigma$, $\rho$, $\zeta$, ${\overline}{\zeta}$, $\alpha$, which satisfy the structure equations: $$\begin{aligned}
d \tau &= 4 \left. \alpha \wedge \tau \right.
+ \left.\sigma \wedge \zeta \right.
+ \left. \sigma \wedge {\overline}{\zeta} \right., \\
d \sigma &= 3 \left. \alpha \wedge \sigma \right. + \left. \rho \wedge \zeta \right.
+ \left. \rho \wedge {\overline}{\zeta} \right., \\
d \rho & = 2 \left. \alpha \wedge \rho \right. + i \, \left. \zeta \wedge {\overline}{\zeta} \right., \\
d \zeta &= \left. \alpha \wedge \zeta \right., \\
d {\overline}{\zeta} &= \left. \alpha \wedge {\overline}{\zeta} \right.,\\
d \alpha &= 0.
\end{aligned}$$
Initial $G$-structure
---------------------
The vector fields : $$\mathcal{L}
:=
\frac{\partial}{\partial z}
+
i\overline{z}\,\frac{\partial}{\partial u_1}
+
i\big(2z\overline{z}+\overline{z}^2\big)\,
\frac{\partial}{\partial u_2}
+
i\big(3z^2\overline{z}
+
3z\overline{z}^2
+
\overline{z}^3
\big)\,
\frac{\partial}{\partial u_3},$$ with its conjugate: $$\overline{\mathcal{L}}
:=
\frac{\partial}{\partial\overline{z}}
-
iz\,\frac{\partial}{\partial u_1}
-
i\big(2z\overline{z}+z^2\big)\,
\frac{\partial}{\partial u_2}
-
i\big(
3z\overline{z}^2
+
3z^2\overline{z}
+
z^3
\big)\,
\frac{\partial}{\partial u_3},$$ constitute a basis of $T^{1,0}_p {\sf N}$ and of $T^{0,1}_p {\sf N}$ at each point $p$ of ${\sf N}$. Moreover the vector fields ${\mathcal{T}}$, ${\mathcal{S}}$ and $\mathcal{R}$ defined by: $${\mathcal{T}}: = i \, \big[\mathcal{L}, {\overline{\mathcal{L}_1}}]
,$$
$${\mathcal{S}}: = \big[{\mathcal{L}_1}, {\mathcal{T}}\big],$$
and
$$\mathcal{R} := \big[{\mathcal{L}_1}, {\mathcal{S}}\big],$$
complete a frame on ${\sf N}$: $$\big\{\mathcal{R}, \mathcal{S},\,
\mathcal{T},\,\mathcal{L}, \,
\overline{\mathcal{L}}
\big\}
.$$ The expressions of ${\mathcal{T}}$, ${\mathcal{S}}$ and $\mathcal{R}$ are:
$$\begin{aligned}
\mathcal{T}
:=
&
\,\,2\,\frac{\partial}{\partial u_1}
+
\big(4z+4\overline{z}\big)
\frac{\partial}{\partial u_2}
+
\big(6z^2+12z\overline{z}+6\overline{z}^2\big)\,
\frac{\partial}{\partial u_3}, \\
{\mathcal{S}}:=
&
\,\,4\,\frac{\partial}{\partial u_2}
+
\big(12z+12\overline{z}\big)\,
\frac{\partial}{\partial u_3},
\\
\mathcal{R}
:=
&
\,\,12\,\frac{\partial}{\partial u_3}.
\end{aligned}$$
The dual coframe $ \big\{ \tau_0,\,\sigma_0,\,
\rho_0,\,\overline{\zeta_0},\,
\zeta_0
\big\}$ is thus given by:
$$\begin{aligned}
\tau_0 &= - \frac{i}{12} \, {{\overline}{z}}^3 \, dz
+ \frac{i}{12} \, z^3 \, d {{\overline}{z}}+ \left( \frac{1}{4}
\, z^2 + \frac{1}{2} \, z {{\overline}{z}}+ \frac{1}{4} \, {{\overline}{z}}^2 \right) du_1
- \left( \frac{1}{4} \, z + \frac{1}{4} \, {{\overline}{z}}\right) du_2 + \frac{1}{12} \, du_3, \\
\sigma_0 &= \frac{i}{4} \, {{\overline}{z}}^2 \, dz -
\frac{i}{4} \, z^2 \, d {{\overline}{z}}- \left( \frac{1}{2}
\, z + \frac{1}{2} \, {{\overline}{z}}\right) du_1 + \frac{1}{4} \, du_2, \\
\rho_0 &= - \frac{i}{2} \, {{\overline}{z}}\, dz + \frac{i}{2} \, z \, d{{\overline}{z}}+ \frac{1}{2} \, du_1 ,\\
\zeta_0 &= dz, \\
{\overline}{\zeta_0} &= d{{\overline}{z}}.
\end{aligned}$$
We deduce the structure equations enjoyed by the base coframe $
\big\{ \tau_0,\,\sigma_0,\,
\rho_0,\,\overline{\zeta_0},\,
\zeta_0
\big\}
$: $$\label{eq:N}
\begin{aligned}
d \tau_{0} & = \sigma_0 \wedge \zeta_0 + \sigma_0 \wedge {\overline}{\zeta_0}, \\
d \sigma_0 & = \rho_0 \wedge \zeta_0 + \rho_0 \wedge {\overline}{\zeta_0}, \\
d \rho_{0} & = i \, \zeta_0 \wedge {\overline}{\zeta_0},\\
d \zeta_0 & = 0, \\
d {\overline}{ \zeta_{0}} & = 0.
\end{aligned}$$
As the torsion coefficients of these structure equations are constants, we have the following result:
The model of the class ${\sf III_2}$ is locally isomorphic to a Lie group whose Maurer-Cartan forms satisfy the structure equations $(\ref{eq:N})$.
The matrix Lie group which encodes suitably the equivalence problem for the model of class ${\sf III_2}$ (see [@pocchiola3]) is the $18$-dimensional Lie group $G_1$ whose elements $g$ are of the form: $$g := \begin{pmatrix}
{{{\sf a}}^3} {{\overline}{{\sf a}}}& 0 & 0 & 0 & 0 \\
{{\sf f}}& {{\sf a}}^2 {{\overline}{{\sf a}}}& 0 & 0 & 0 \\
{{\sf g}}& {{\sf c}}& {{\sf a}}{{\overline}{{\sf a}}}& 0 & 0 \\
{{\sf h}}& {{\sf d}}& {{\sf b}}& {{\sf a}}& 0 \\
{{\sf k}}& {{\sf e}}& {{\overline}{{\sf b}}}& 0 & {{\overline}{{\sf a}}}\end{pmatrix}.$$
With the notations:
$$\begin{aligned}
{1}
\omega_{0} :=
\begin{pmatrix}
\tau_0 \\ \sigma_0 \\ \rho_0 \\ \zeta_{0} \\ {\overline}{\zeta}_0
\end{pmatrix}, & \qquad
\omega:=
\begin{pmatrix}
\tau \\ \sigma \\ \rho \\ \zeta \\ {\overline}{\zeta}
\end{pmatrix},\end{aligned}$$
we introduce the $G_1$-structure $P^1$ on ${\sf N}$ constituted by the coframes $\omega$ which satisfy the relation:
$$\omega:= g \cdot \omega_0.$$
As in the case of Beloshapka’s cubic, the proof of theorem (\[thm:N\]) relies on successive reductions of $P^1$ through Cartan’s equivalence method.
Normalization of [[[a]{}]{}]{}
------------------------------
The Maurer-Cartan forms of $G_1$ are given by: $$\begin{aligned}
\alpha^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\alpha^2 &:= -{\frac {{{\sf b}}{ d{{\sf a}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}+{\frac {{ d{{\sf b}}}}{{{\sf a}}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^3 &:=-{\frac {{{\sf c}}{ d{{\sf a}}}}{{ {{\overline}{{\sf a}}}}\,{{{\sf a}}}^{3}}}-{\frac {{{\sf c}}{ d{{\overline}{{\sf a}}}}}{{{ {{\overline}{{\sf a}}}}}^{2}{{{\sf a}}}^{2}}} + { \frac {{ d{{\sf c}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^4 & :=-{\frac { \left( {{\sf d}}{{\sf a}}{ {{\overline}{{\sf a}}}}-{{\sf b}}{{\sf c}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{4}{
{ {{\overline}{{\sf a}}}}}^{2}}}-{\frac {{{\sf c}}{ d{{\sf b}}}}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}}}+{\frac {{
d{{\sf d}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^5 & :=-{\frac { \left( {{\sf e}}{{\sf a}}{ {{\overline}{{\sf a}}}}-{ {{\overline}{{\sf b}}}}\,{{\sf c}}\right) { d{{\overline}{{\sf a}}}}
}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{3}}}-{\frac {{{\sf c}}{ d{{\overline}{{\sf b}}}}}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}}}+
{\frac {{ d{{\sf e}}}}{{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}}}, \\
\alpha^6 & :=-2\,{\frac {{{\sf f}}{ d{{\sf a}}}}{{ {{\overline}{{\sf a}}}}\,{{{\sf a}}}^{4}}}-{\frac {{{\sf f}}{
d{{\overline}{{\sf a}}}}}{{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}}}+{\frac {{ d{{\sf f}}}}{{ {{\overline}{{\sf a}}}}\,{{{\sf a}}}^{3}}}, \\
\alpha^7 & :=-{\frac { \left( {{\sf g}}{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}-{{\sf c}}{{\sf f}}\right) { d{{\sf a}}}}{{{
{{\overline}{{\sf a}}}}}^{2}{{{\sf a}}}^{6}}}-{\frac { \left( {{\sf g}}{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}-{{\sf c}}{{\sf f}}\right) {
d{{\overline}{{\sf a}}}}}{{{ {{\overline}{{\sf a}}}}}^{3}{{{\sf a}}}^{5}}}-{\frac {{{\sf f}}{ d{{\sf c}}}}{{{{\sf a}}}^{5}{{ {{\overline}{{\sf a}}}}}
^{2}}}+{\frac {{ d{{\sf g}}}}{{ {{\overline}{{\sf a}}}}\,{{{\sf a}}}^{3}}}, \\
\alpha^8 & :=-{\frac { \left( {{\sf h}}{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}- {{\sf d}}{{\sf f}}{{\sf a}}{ {{\overline}{{\sf a}}}}-{{\sf b}}{{\sf g}}{{{\sf a}}}
^{2}{ {{\overline}{{\sf a}}}}+{{\sf b}}{{\sf c}}{{\sf f}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{7}{{ {{\overline}{{\sf a}}}}}^{3}}}-{\frac {
\left( {{\sf g}}{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}-{{\sf c}}{{\sf f}}\right) { d{{\sf b}}}}{{{{\sf a}}}^{6}{{ {{\overline}{{\sf a}}}}}^{3}}}-
{\frac {{{\sf f}}{ d{{\sf d}}}}{{{{\sf a}}}^{5}{{ {{\overline}{{\sf a}}}}}^{2}}}+{\frac {{ d{{\sf h}}}}{{ {{\overline}{{\sf a}}}}
\,{{{\sf a}}}^{3}}},\\
\alpha^9 & :=-{\frac { \left( {{\sf k}}{{{\sf a}}}^{3}{{ {{\overline}{{\sf a}}}}}^{2}-{{\sf e}}{{\sf f}}{{\sf a}}{ {{\overline}{{\sf a}}}}-{
{{\overline}{{\sf b}}}}\,{{\sf g}}{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}+{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf f}}\right) { d{{\overline}{{\sf a}}}}}{{{{\sf a}}}^{6}{{ {{\overline}{{\sf a}}}}
}^{4}}}-{\frac { \left( {{\sf g}}{{{\sf a}}}^{2}{ {{\overline}{{\sf a}}}}-{{\sf c}}{{\sf f}}\right) { d{{\overline}{{\sf b}}}}}{{{{\sf a}}}^{6}
{{ {{\overline}{{\sf a}}}}}^{3}}}-{\frac {{{\sf f}}{ d{{\sf e}}}}{{{{\sf a}}}^{5}{{ {{\overline}{{\sf a}}}}}^{2}}}+{\frac {{
d{{\sf k}}}}{{ {{\overline}{{\sf a}}}}\,{{{\sf a}}}^{3}}},
\end{aligned}$$ together with their conjugates.
The first structure equation is given by: $$\begin{gathered}
d \tau = 3 \left.\alpha^1 \wedge \tau \right. + \left.{\overline}{\alpha^1} \wedge \tau \right. \\
+ T^{\tau}_{\tau \sigma} \left. \tau \wedge \sigma \right.
+ T^{\tau}_{\tau \rho} \left. \tau \wedge \rho \right.
+ T^{\tau}_{\tau \zeta} \left. \tau \wedge \zeta \right. \\
+ T^{\tau}_{\tau {\overline}{\zeta}} \left. \tau \wedge {\overline}{\zeta} \right.
+ T^{\tau}_{\sigma \rho} \left.\sigma \wedge \rho \right.
+ \left.\sigma \wedge \zeta \right.
- \frac{{{\sf a}}}{{{\overline}{{\sf a}}}} \, \left. \sigma \wedge {\overline}{\zeta} \right.
,\end{gathered}$$ from which we immediately deduce that $\frac{{{\sf a}}}{{{\overline}{{\sf a}}}}$ is an essential torsion coefficient which shall be normalized to $1$ by setting: $${{\sf a}}= {{\overline}{{\sf a}}}.$$
We thus have reduced the $G_{1}$ equivalence problem to a $G_2$ equivalence problem, where $G_2$ is the $10$ dimensional real matrix Lie group whose elements are of the form $$g = \begin{pmatrix}
{{{\sf a}}^4} & 0 & 0 & 0 & 0 \\
{{\sf f}}& {{\sf a}}^3 & 0 & 0 & 0 \\
{{\sf g}}& {{\sf c}}& {{\sf a}}^2 & 0 & 0 \\
{{\sf h}}& {{\sf d}}& {{\sf b}}& {{\sf a}}& 0 \\
{{\sf k}}& {{\sf e}}& {{\overline}{{\sf b}}}& 0 & {{\sf a}}\end{pmatrix}
,$$
Normalizations of ${{\sf f}}$, ${{\sf b}}$ and ${{\sf c}}$
----------------------------------------------------------
The Maurer-Cartan forms of $G_2$ are given by: $$\begin{aligned}
\beta^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\beta^2 &:= -{\frac {{{\sf b}}d{{\sf a}}}{{{{\sf a}}}^{3}}}+{\frac {{ d{{\sf b}}}}{{{\sf a}}^2}}, \\
\beta^3 &:=- 2 \, {\frac {{{\sf c}}d{{\sf a}}}{{{{\sf a}}}^{4}}} + { \frac {{ d{{\sf c}}}}{{{{\sf a}}}^{3}}}, \\
\beta^4 &: =-{\frac { \left( {{\sf d}}{{\sf a}}^2 -{{\sf b}}{{\sf c}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{6}}}-{\frac {{{\sf c}}{ d{{\sf b}}}}{{{{\sf a}}}^{5}}}+{\frac {{
d{{\sf d}}}}{{{{\sf a}}}^{3}}}, \\
\beta^5 & :=-\frac { \left( {{\sf e}}{{\sf a}}^2-{ {{\overline}{{\sf b}}}}\,{{\sf c}}\right) d{{\sf a}}}{{{\sf a}}^6}-\frac {{{\sf c}}{ d{{\overline}{{\sf b}}}}}{{{\sf a}}^{5}}+
\frac { d{{\sf e}}}{{{\sf a}}^{3}}, \\
\beta^6 & := -3 \,{\frac {{{\sf f}}{ d{{\sf a}}}}{{{{\sf a}}}^{5}}} + {\frac {{ d{{\sf f}}}}{{{{\sf a}}}^{4}}}, \\
\beta^7 & :=- 2 \, {\frac { \left( {{\sf g}}{{{\sf a}}}^{3}-{{\sf c}}{{\sf f}}\right)
{ d{{\sf a}}}}{{{\sf a}}^{8}}} -{\frac {{{\sf f}}{ d{{\sf c}}}}{{{{\sf a}}}^{7}}}+{\frac {{ d{{\sf g}}}}{{{{\sf a}}}^{4}}}, \\
\beta^8 & :=-{\frac { \left( {{\sf h}}{{{\sf a}}}^{5}- {{\sf d}}{{\sf f}}{{\sf a}}^2-{{\sf b}}{{\sf g}}{{{\sf a}}}
^{3}+{{\sf b}}{{\sf c}}{{\sf f}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{10}}}-{\frac {
\left( {{\sf g}}{{{\sf a}}}^{3}-{{\sf c}}{{\sf f}}\right) { d{{\sf b}}}}{{{{\sf a}}}^{9}}}-
{\frac {{{\sf f}}{ d{{\sf d}}}}{{{\sf a}}^{7}}}+{\frac {{ d{{\sf h}}}}{
{{{\sf a}}}^{4}}},\\
\beta^9 & :=-{\frac { \left( {{\sf k}}{{{\sf a}}}^{5}-{{\sf e}}{{\sf f}}{{\sf a}}^2-{
{{\overline}{{\sf b}}}}\,{{\sf g}}{{{\sf a}}}^{3}+{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf f}}\right) { d{{\sf a}}}}{{{{\sf a}}}^{10}}}-{\frac { \left( {{\sf g}}{{{\sf a}}}^{3}-{{\sf c}}{{\sf f}}\right) { d{{\overline}{{\sf b}}}}}{{{{\sf a}}}^{9}}}
-
{\frac {{{\sf f}}{ d{{\sf e}}}}{{{{\sf a}}}^{5}{{ {{\overline}{{\sf a}}}}}^{2}}}+{\frac {{
d{{\sf k}}}}{{{{\sf a}}}^{4}}},
\end{aligned}$$ together with ${\overline}{\beta^i}, \,\,\, i=2 \dots 9$.
Using formula (\[eq:str\]), we get the structure equations for the lifted coframe $(\tau, \sigma, \rho, \zeta, {\overline}{\zeta})$ from those of the base coframe $(\tau_0, \sigma_0, \rho_0, \hat{\zeta}_0, {\overline}{\hat{\zeta}_0})$ by a matrix multiplication and a linear change of coordinates, as in the first step:
$$\begin{gathered}
d \tau
=
4 \, \beta^1 \wedge \tau \\
+
U^{\tau}_{\tau \sigma} \, \tau \wedge \sigma
+
U^{\tau}_{\tau \rho} \, \tau \wedge \rho
+
U^{\tau}_{\tau \zeta} \, \tau \wedge \zeta
+
U^{\tau}_{\tau {\overline}{\zeta}} \, \tau \wedge {\overline}{\zeta} \\
+
U^{\tau}_{\sigma \rho} \, \sigma \wedge \rho
+
\sigma \wedge \zeta
+
\sigma \wedge {\overline}{\zeta},\end{gathered}$$
$$\begin{gathered}
d \sigma = 3 \, \beta^1 \wedge \sigma + \beta^6 \wedge \tau \\
+
U^{\sigma}_{\tau \sigma} \left. \tau \wedge \sigma \right.
+
U^{\sigma}_{\tau \rho} \left. \tau \wedge \rho \right.
+
U^{\sigma}_{\tau \zeta} \left. \tau \wedge \zeta \right. \\
+
U^{\sigma}_{\tau {\overline}{\zeta}} \left. \tau \wedge {\overline}{\zeta} \right.
+
U^{\sigma}_{\sigma \rho} \left. \sigma \wedge \rho \right.
+
U^{\sigma}_{\sigma \zeta} \left. \sigma \wedge \zeta \right. \\
+
U^{\sigma}_{\sigma {\overline}{\zeta}} \left. \sigma \wedge {\overline}{\zeta} \right.
+
\rho \wedge \zeta
+
\rho \wedge {\overline}{\zeta},\end{gathered}$$
$$\begin{gathered}
d \rho
=
2 \beta^{1} \wedge \rho + \beta^3 \wedge \sigma + \beta^7 \wedge \tau \\
+
U^{\rho}_{\tau \sigma} \, \tau \wedge \sigma
+
U^{\rho}_{\tau \rho} \, \tau \wedge \rho
+
U^{\rho}_{\tau \zeta} \, \tau \wedge \zeta
+
U^{\rho}_{\tau \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
U^{\rho}_{\sigma \rho} \, \sigma \wedge \rho \\
+
U^{\rho}_{\sigma \zeta} \, \sigma \wedge \zeta
+
U^{\rho}_{\sigma {\overline}{\zeta}} \, \sigma \wedge {\overline}{\zeta}
+
U^{\rho}_{\rho \zeta} \, \rho \wedge \zeta
+
U^{\rho}_{\rho {\overline}{\zeta}} \, \rho \wedge {\overline}{\zeta}
+
i \, \zeta \wedge {\overline}{\zeta}
,\end{gathered}$$
$$\begin{gathered}
d \zeta
=
{\beta}^{1} \wedge \zeta + {\beta}^{2} \wedge \rho + {\beta}^{4} \wedge \sigma + \beta^8 \wedge \tau \\
+
U^{\zeta}_{\tau \sigma} \, \tau \wedge \sigma
+
U^{\zeta}_{\tau \rho} \, \tau \wedge \rho
+
U^{\zeta}_{\tau \zeta} \, \tau \wedge \zeta
+
U^{\zeta}_{\tau {\overline}{\zeta}} \, \tau \wedge {\overline}{\zeta} \\
+
U^{\zeta}_{\sigma \rho} \, \sigma \wedge \rho
+
U^{\zeta}_{\sigma \zeta} \, \sigma \wedge \zeta
+
U^{\zeta}_{\sigma {\overline}{\zeta}} \, \sigma \wedge {\overline}{\zeta}
+
U^{\zeta}_{\rho \zeta} \, \rho \wedge \zeta \\
+
U^{\zeta}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
U^{\zeta}_{\zeta \overline{\zeta}} \, \zeta \wedge \overline{\zeta}
.\end{gathered}$$
We now proceed with the absorption phase. We introduce the modified Maurer-Cartan forms: $$\widetilde{\beta}^i= \beta^i -y_{\tau}^i \, \tau - y_{\sigma} \, \sigma - y_{\rho}^i \, \rho - y_{\zeta}^i \, \zeta \,
- y_{\overline{\zeta}}^i \, \overline{\zeta}.$$ The structure equations rewrite: $$\begin{gathered}
d \tau
=
4 \, \widetilde{\beta}^1 \wedge \tau \\
+
\left( U^{\tau}_{\tau \sigma} - 4 \, y^1_{\sigma} \right) \, \left. \tau \wedge \sigma \right.
+
\left( U^{\tau}_{\tau \rho} - 4 \, y^1_{\rho} \right) \, \left. \tau \wedge \rho \right. \\
+
\left(U^{\tau}_{\tau \zeta} - 4 \, y^1_{\zeta} \right) \, \left. \tau \wedge \zeta \right.
+
\left(U^{\tau}_{\tau {\overline}{\zeta}} - 4 \, y^1_{{\overline}{\zeta}} \right) \, \left. \tau \wedge {\overline}{\zeta} \right. \\
+
U^{\tau}_{\sigma \rho} \, \left. \sigma \wedge \rho \right.
+
\left.\sigma \wedge \zeta \right.
+
\left. \sigma \wedge {\overline}{\zeta} \right.,\end{gathered}$$
$$\begin{gathered}
d \sigma = 3 \, \left. \widetilde{\beta}^1 \wedge \sigma \right. + \left. \widetilde{\beta}^6 \wedge \tau \right. \\
+
\left( U^{\sigma}_{\tau \sigma} + 3 \, y^1_{\tau} - y^6_{\sigma} \right) \, \left. \tau \wedge \sigma \right.
+
\left( U^{\sigma}_{\tau \rho} - y^6_{\rho} \right) \, \left. \tau \wedge \rho \right. \\
+
\left( U^{\sigma}_{\tau \zeta} - y^6_{\zeta} \right) \, \left. \tau \wedge \zeta \right.
+
\left( U^{\sigma}_{\tau {\overline}{\zeta}} - y^6_{{\overline}{\zeta}} \right) \, \left. \tau \wedge {\overline}{\zeta} \right. \\
+
\left( U^{\sigma}_{\sigma \rho} - 3 \, y^1_{\rho} \right) \, \left. \sigma \wedge \rho \right.
+
\left( U^{\sigma}_{\sigma \zeta} -3 \,y^1_{\zeta} \right) \, \left. \sigma \wedge \zeta \right. \\
+
\left( U^{\sigma}_{\sigma {\overline}{\zeta}} -3 \,y^1_{{\overline}{\zeta}} \right) \, \left. \sigma \wedge {\overline}{\zeta} \right.
+
\rho \wedge \zeta
+
\rho \wedge {\overline}{\zeta},\end{gathered}$$
$$\begin{gathered}
d \rho
=
2 \widetilde{\beta}^{1} \wedge \rho + \widetilde{\beta}^3 \wedge \sigma + \widetilde{\beta}^7 \wedge \tau \\
+
\left( U^{\rho}_{\tau \sigma} + y^3_{\tau} - y^7_{\sigma} \right) \, \left. \tau \wedge \sigma \right.
+
\left( U^{\rho}_{\tau \rho} + 2 \, y^1_{\tau} - y^7_{\rho} \right) \, \left. \tau \wedge \rho \right. \\
+
\left( U^{\rho}_{\tau \zeta} - y^7_{\zeta} \right) \, \left. \tau \wedge \zeta \right.
+
\left( U^{\rho}_{\tau \overline{\zeta}} - y^7_{{\overline}{\zeta}} \right) \, \left. \rho \wedge \overline{\zeta} \right. \\
+
\left( U^{\rho}_{\sigma \rho} + 2 \, y^1_{\sigma} - y^3_{\rho} \right) \, \left. \sigma \wedge \rho \right.
+
\left( U^{\rho}_{\sigma \zeta} - y^3_{\zeta} \right) \, \left. \sigma \wedge \zeta \right. \\
+
\left( U^{\rho}_{\sigma {\overline}{\zeta}} - y^3_{{\overline}{\zeta}} \right) \, \left. \sigma \wedge {\overline}{\zeta} \right.
+
\left( U^{\rho}_{\rho \zeta} - 2 \, y^1_{\zeta} \right) \, \left. \rho \wedge \zeta \right. \\
+
\left( U^{\rho}_{\rho {\overline}{\zeta}} - 2 \, y^1_{{\overline}{\zeta}} \right) \, \left. \rho \wedge {\overline}{\zeta} \right.
+
i \, \left. \zeta \wedge {\overline}{\zeta} \right.
,\end{gathered}$$
$$\begin{gathered}
d \zeta
=
\widetilde{\beta^{1}} \wedge \zeta +
\widetilde{\beta^{2}} \wedge \rho + \widetilde{\beta^{4}} \wedge \sigma + \widetilde{\beta^8} \wedge \tau \\
+
\left( U^{\zeta}_{\tau \sigma} + y^4_{\tau}- y^8_{\sigma} \right) \, \left. \tau \wedge \sigma \right.
+
\left( U^{\zeta}_{\tau \rho} + y^2_{\tau} - y^8_{\rho} \right) \, \left. \tau \wedge \rho \right. \\
+
\left( U^{\zeta}_{\tau \zeta} + y^1_{\tau} - y^8_{\zeta} \right) \, \left. \tau \wedge \zeta \right.
+
\left( U^{\zeta}_{\tau {\overline}{\zeta}} - y^8_{{\overline}{\zeta}} \right) \, \left. \tau \wedge {\overline}{\zeta} \right. \\
+
\left( U^{\zeta}_{\sigma \rho} + y^2_{\sigma}- y^4_{\rho} \right) \, \left. \sigma \wedge \rho \right.
+
\left( U^{\zeta}_{\sigma \zeta} + y^1_{\sigma} - y^4_{\zeta} \right) \, \left. \sigma \wedge \zeta \right. \\
+
\left( U^{\zeta}_{\sigma {\overline}{\zeta}} - y^4_{{\overline}{\zeta}} \right) \, \left. \sigma \wedge {\overline}{\zeta} \right.
+
\left( U^{\zeta}_{\rho \zeta} + y^1_{\rho} - y^2_{\zeta} \right) \, \left. \rho \wedge \zeta \right. \\
+
\left( U^{\zeta}_{\rho \overline{\zeta}} - y^2_{{\overline}{\zeta}} \right) \, \left. \rho \wedge \overline{\zeta} \right.
+
\left( U^{\zeta}_{\zeta \overline{\zeta}} - y^1_{{\overline}{\zeta}} \right) \, \left. \zeta \wedge \overline{\zeta} \right.
.\end{gathered}$$
We get the following absorbtion equations:
$$\begin{aligned}
{3}
4 \, y^1_{\sigma} &=U^{\tau}_{\tau \sigma},
& \qquad \qquad 4 \, y^1_{\rho} &= U^{\tau}_{\tau \rho} ,
& \qquad \qquad 4 \, y^1_{\zeta} &= U^{\tau}_{\tau \zeta} , \\
4 \, y^1_{{\overline}{\zeta}} &= U^{\tau}_{\tau {\overline}{\zeta}},
& \qquad \qquad - 3 \, y^1_{\tau} + y^6_{\sigma} & =U^{\sigma}_{\tau \sigma}, &
\qquad \qquad y^6_{\rho} & = U^{\sigma}_{\tau \rho} , \\
y^6_{\zeta} &= U^{\sigma}_{\tau \zeta}, & \qquad \qquad
y^6_{{\overline}{\zeta}} & = U^{\sigma}_{\tau {\overline}{\zeta}}, & \qquad \qquad
3 \, y^1_{\rho} & = U^{\sigma}_{\sigma \rho}, \\
3 \,y^1_{\zeta} & = U^{\sigma}_{\sigma \zeta}, & \qquad \qquad
3 \,y^1_{{\overline}{\zeta}} & = U^{\sigma}_{\sigma {\overline}{\zeta}}, & \qquad \qquad
-y^3_{\tau} + y^7_{\sigma} & = U^{\rho}_{\tau \sigma}, \\
-2 \, y^1_{\tau} + y^7_{\rho} &= U^{\rho}_{\tau \rho}, & \qquad \qquad
y^7_{\zeta} & = U^{\rho}_{\tau \zeta}, &\qquad \qquad
y^7_{{\overline}{\zeta}} &= U^{\rho}_{\tau \overline{\zeta}}, \\
-2 \, y^1_{\sigma} + y^3_{\rho} &= U^{\rho}_{\sigma \rho}, &\qquad \qquad
y^3_{\zeta} &= U^{\rho}_{\sigma \zeta} , &\qquad \qquad
y^3_{{\overline}{\zeta}} &= U^{\rho}_{\sigma {\overline}{\zeta}}, \\
2 \, y^1_{\zeta} &= U^{\rho}_{\rho \zeta}, & \qquad \qquad
2 \, y^1_{{\overline}{\zeta}} &= U^{\rho}_{\rho {\overline}{\zeta}}, & \qquad \qquad
-y^4_{\tau} + y^8_{\sigma} & = U^{\zeta}_{\tau \sigma},\\
-y^2_{\tau} + y^8_{\rho} & = U^{\zeta}_{\tau \rho},& \qquad \qquad
- y^1_{\tau} + y^8_{\zeta} & = U^{\zeta}_{\tau \zeta} , & \qquad \qquad
y^8_{{\overline}{\zeta}} & =U^{\zeta}_{\tau {\overline}{\zeta}} , \\
-y^2_{\sigma} + y^4_{\rho} & = U^{\zeta}_{\sigma \rho}, & \qquad \qquad
-y^1_{\sigma} + y^4_{\zeta} & = U^{\zeta}_{\sigma \zeta}, & \qquad \qquad
y^4_{{\overline}{\zeta}} & = U^{\zeta}_{\sigma {\overline}{\zeta}}, \\
-y^1_{\rho} + y^2_{\zeta} & = U^{\zeta}_{\rho \zeta}, & \qquad \qquad
y^2_{{\overline}{\zeta}} & = U^{\zeta}_{\rho \overline{\zeta}}, & \qquad \qquad
y^1_{{\overline}{\zeta}} & = U^{\zeta}_{\zeta \overline{\zeta}}.\end{aligned}$$
Eliminating $y^1_{{\overline}{\zeta}}$ among the previous equations leads to: $$U^{\zeta}_{\zeta {\overline}{\zeta}} = \frac{1}{2} \, U^{\rho}_{\rho {\overline}{\zeta}} =
\frac{1}{3} \, U^{\sigma}_{\sigma {\overline}{\zeta}} = \frac{1}{4} \, U^{\tau}_{\tau {\overline}{\zeta}}
,$$ that is: $${\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}} = \frac{1}{2} \, \left( {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}-{\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}} \right)
=- \frac{1}{3} \,\left( {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}+{\frac {{{\sf f}}}{{{{\sf a}}}^{4}}} \right) =-\frac{1}{4} \,{\frac {{{\sf f}}}{{{{\sf a}}}^{4}}},$$ from which we easily deduce that $${{\sf b}}= {{\sf c}}= {{\sf f}}= 0.$$
We have thus reduced the group $G_2$ to a new group $G_3$, whose elements are of the form $$g := \begin{pmatrix}
{{{\sf a}}^4} & 0 & 0 & 0 & 0 \\
0 & {{\sf a}}^3 & 0 & 0 & 0 \\
{{\sf g}}& 0 & {{\sf a}}^2 & 0 & 0 \\
{{\sf h}}& {{\sf d}}& 0 & {{\sf a}}& 0 \\
{{\sf k}}& {{\sf e}}& 0 & 0 & {{\sf a}}\end{pmatrix}.$$
Normalization of ${{\sf g}}$, ${{\sf d}}$ and ${{\sf e}}$
---------------------------------------------------------
The Maurer Cartan forms of $G_3$ are: $$\begin{aligned}
\gamma^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\gamma^2 & :=-{\frac {{{\sf d}}{ d{{\sf a}}}}{{{{\sf a}}}^{4}}} + {\frac {{
d{{\sf d}}}}{{{{\sf a}}}^{3}}}, \\
\gamma^3 &: =-\frac { {{\sf e}}d {{\sf a}}}{{{\sf a}}^4} +
\frac { d{{\sf e}}}{{{\sf a}}^{3}}, \\
\gamma^4 &: =- 2 \, \frac { {{\sf g}}d {{\sf a}}}{{{\sf a}}^5} +
\frac { d{{\sf g}}}{{{\sf a}}^{4}}, \\
\gamma^5 &: =- \frac { {{\sf h}}d {{\sf a}}}{{{\sf a}}^5} +
\frac { d{{\sf h}}}{{{\sf a}}^{4}}, \\
\gamma^6 &: =- \frac { {{\sf k}}d {{\sf a}}}{{{\sf a}}^5} +
\frac { d{{\sf k}}}{{{\sf a}}^{4}}.
\end{aligned}$$
We get the following structure equations: $$d \tau = 4 \left. \gamma^1 \wedge \tau \right.
+ V^{\tau}_{\tau \sigma} \, \left. \tau \wedge \sigma \right. + \sigma \wedge \zeta + \sigma \wedge {\overline}{\zeta},$$
$$d \sigma = 3 \left. \gamma^1 \wedge \sigma \right.
+
V^{\sigma}_{\tau \rho} \left. \tau \wedge \rho \right.
+
V^{\sigma}_{\tau \zeta} \left. \tau \wedge \zeta \right.
+
V^{\sigma}_{\tau {\overline}{\zeta}} \left. \tau \wedge {\overline}{\zeta} \right.
+
V^{\sigma}_{\sigma \rho} \left. \sigma \wedge \rho \right.
+
\left. \rho \wedge \zeta \right.
+
\left. \rho \wedge {\overline}{\zeta} \right.
,$$
$$d \rho
=
2 \left. \gamma^{1} \wedge \rho \right. + \left. \gamma^4 \wedge \tau \right.
+
V^{\rho}_{\tau \sigma} \left. \tau \wedge \sigma \right.
+
V^{\rho}_{\tau \zeta} \left. \tau \wedge \zeta \right.
+
V^{\rho}_{\tau \overline{\zeta}} \left. \rho \wedge \overline{\zeta} \right.
+
V^{\rho}_{\sigma \zeta} \left. \sigma \wedge \zeta \right.
+
V^{\rho}_{\sigma {\overline}{\zeta}} \left. \sigma \wedge {\overline}{\zeta} \right.
+
i \, \left. \zeta \wedge {\overline}{\zeta} \right.
,$$
$$\begin{gathered}
d \zeta
=
\left. \gamma^{1} \wedge \zeta \right.+ \left. {\gamma}^{2} \wedge \sigma \right.
+ \left. \gamma^5 \wedge \tau \right. \\
+
V^{\zeta}_{\tau \sigma} \, \tau \wedge \sigma
+
V^{\zeta}_{\tau \rho} \, \tau \wedge \rho
+
V^{\zeta}_{\tau \zeta} \, \tau \wedge \zeta
+
V^{\zeta}_{\tau {\overline}{\zeta}} \, \tau \wedge {\overline}{\zeta} \\
+
V^{\zeta}_{\sigma \rho} \, \sigma \wedge \rho
+
V^{\zeta}_{\sigma \zeta} \, \sigma \wedge \zeta
+
V^{\zeta}_{\sigma {\overline}{\zeta}} \, \sigma \wedge {\overline}{\zeta}
+
V^{\zeta}_{\rho \zeta} \, \rho \wedge \zeta \\
+
V^{\zeta}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
,\end{gathered}$$
and $$\begin{gathered}
d {\overline}{\zeta}
=
\left. \gamma^{1} \wedge {\overline}{\zeta} \right.+ \left. \gamma^{3} \wedge \sigma \right.
+ \left. \gamma^6 \wedge \tau \right. \\
+
V^{{\overline}{\zeta}}_{\tau \sigma} \, \tau \wedge \sigma
+
V^{{\overline}{\zeta}}_{\tau \rho} \, \tau \wedge \rho
+
V^{{\overline}{\zeta}}_{\tau \zeta} \, \tau \wedge \zeta
+
V^{{\overline}{\zeta}}_{\tau {\overline}{\zeta}} \, \tau \wedge {\overline}{\zeta} \\
+
V^{{\overline}{\zeta}}_{\sigma \rho} \, \sigma \wedge \rho
+
V^{{\overline}{\zeta}}_{\sigma \zeta} \, \sigma \wedge \zeta
+
V^{{\overline}{\zeta}}_{\sigma {\overline}{\zeta}} \, \sigma \wedge {\overline}{\zeta}
+
V^{{\overline}{\zeta}}_{\rho \zeta} \, \rho \wedge \zeta \\
+
V^{{\overline}{\zeta}}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
.\end{gathered}$$
From these equations, we immediately see that $V^{\sigma}_{\tau \zeta}$, $V^{\zeta}_{\rho {\overline}{\zeta}}$ and $V^{{\overline}{\zeta}}_{\rho \zeta}$ are essential torsion coefficients. As we have: $$\begin{aligned}
{3}
V^{\sigma}_{\tau \zeta} &= - \frac{{{\sf g}}}{{{\sf a}}^4}, \qquad \qquad & V^{\zeta}_{\rho {\overline}{\zeta}} & = \frac{{{\sf d}}}{{{\sf a}}^3} , \qquad \qquad &
V^{{\overline}{\zeta}}_{\rho \zeta} &=\frac{{{\sf e}}}{{{\sf a}}^3},\end{aligned}$$ we obtain the new normalizations: $${{\sf d}}= {{\sf e}}= {{\sf g}}= 0.$$
The reduced group $G_4$ is of the form: $$g := \begin{pmatrix}
{{{\sf a}}^4} & 0 & 0 & 0 & 0 \\
0 & {{\sf a}}^3 & 0 & 0 & 0 \\
0 & 0 & {{\sf a}}^2 & 0 & 0 \\
{{\sf h}}& 0 & 0 & {{\sf a}}& 0 \\
{{\sf k}}& 0 & 0 & 0 & {{\sf a}}\end{pmatrix}.$$ Its Maurer-Cartan forms are given by: $$\begin{aligned}
\delta^1 &: = {\frac {{ d{{\sf a}}}}{{{\sf a}}}}, \\
\delta^2 &: =- \frac { {{\sf h}}d {{\sf a}}}{{{\sf a}}^5} +
\frac { d{{\sf h}}}{{{\sf a}}^{4}}, \\
\delta^3 & :=- \frac { {{\sf k}}d {{\sf a}}}{{{\sf a}}^5} +
\frac { d{{\sf k}}}{{{\sf a}}^{4}}.
\end{aligned}$$ The structure equations are easily computed as:
$$\begin{aligned}
d \tau &= 4 \left. \delta^1 \wedge \tau \right. + \frac{{{\sf h}}+ {{\sf k}}}{{{\sf a}}^4} \left. \tau \wedge \sigma \right.
+ \left.\sigma \wedge \zeta \right.
+ \left. \sigma \wedge {\overline}{\zeta} \right., \\
d \sigma &= 3 \left. \delta^1 \wedge
\sigma \right. + \frac{{{\sf h}}+ {{\sf k}}}{{{\sf a}}^4} \left. \tau \wedge \rho \right. + \left. \rho \wedge \zeta \right.
+ \left. \rho \wedge {\overline}{\zeta} \right., \\
d \rho & = 2 \left. \delta^1 \wedge \rho \right. + i \, \frac{{{\sf k}}}{{{\sf a}}^4} \, \left. \tau \wedge \zeta \right.
- i \, \frac{{{\sf h}}}{{{\sf a}}^4} \left. \tau \wedge {\overline}{\zeta} \right. + i \, \left. \zeta \wedge {\overline}{\zeta} \right., \\
d \zeta &= \left. \delta^1 \wedge \zeta
\right. + \left. \delta^2 \wedge \tau \right. + \frac{{{\sf h}}\left( {{\sf h}}+ {{\sf k}}\right)}{{{\sf a}}^8} \left. \tau \wedge \sigma \right.
+
\frac{{{\sf h}}}{{{\sf a}}^4} \left. \sigma \wedge \zeta \right. + \frac{{{\sf h}}}{{{\sf a}}^4} \left. \sigma \wedge {\overline}{\zeta} \right., \\
d {\overline}{\zeta} &= \left. \delta^1 \wedge {\overline}{\zeta} \right. + \left. \delta^3 \wedge \tau \right.
+ \frac{{{\sf k}}\left( {{\sf h}}+ {{\sf k}}\right)}{{{\sf a}}^8} \left. \tau \wedge \sigma \right.
+
\frac{{{\sf k}}}{{{\sf a}}^4} \left. \sigma \wedge \zeta \right. + \frac{{{\sf k}}}{{{\sf a}}^4} \left. \sigma \wedge {\overline}{\zeta} \right.
.\end{aligned}$$
We deduce from these equations that we can perform the normalization: $${{\sf h}}= {{\sf k}}= 0.$$ With the $1$-dimensional group $G_5$ of the form: $$g := \begin{pmatrix}
{{{\sf a}}^4} & 0 & 0 & 0 & 0 \\
0 & {{\sf a}}^3 & 0 & 0 & 0 \\
0 & 0 & {{\sf a}}^2 & 0 & 0 \\
0 & 0 & 0 & {{\sf a}}& 0 \\
0 & 0 & 0 & 0 & {{\sf a}}\end{pmatrix},$$ whose Maurer-Cartan form is given by $$\alpha:= \frac{d {{\sf a}}}{{{\sf a}}},$$ we get the following structure equations: $$\begin{aligned}
d \tau &= 4 \left. \alpha \wedge \tau \right.
+ \left.\sigma \wedge \zeta \right.
+ \left. \sigma \wedge {\overline}{\zeta} \right., \\
d \sigma &= 3 \left. \alpha \wedge \sigma \right. + \left. \rho \wedge \zeta \right.
+ \left. \rho \wedge {\overline}{\zeta} \right., \\
d \rho & = 2 \left. \alpha \wedge \rho \right. + i \, \left. \zeta \wedge {\overline}{\zeta} \right., \\
d \zeta &= \left. \alpha \wedge \zeta \right., \\
d {\overline}{\zeta} &= \left. \alpha \wedge {\overline}{\zeta} \right..
\end{aligned}$$ No more normalizations are allowed at this stage. We thus just perform a prolongation by adjoining the form $\alpha$ to the structure equations, whose exterior derivative is given by: $$d \alpha = 0.$$ This completes the proof of Theorem \[thm:N\] .
Class ${\sf IV_{2}}$
====================
Class ${\sf IV_{2}}$ is constituted by the $5$-dimensional real hypersurfaces $M^5 \subset {\mathbb{C}}^3$ which are of CR-dimension $2$, whose Levi form is of constant rank $1$ and which are $2$-nondegenerate, i.e. their Freeman forms are non-zero. The most symmetric manifold of this class is the tube over the future light cone, which is defined by the equation:
$${\sf LC}: \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\left( {\sf Re} \,z_1 \right)^2 -\left( {\sf Re} \,z_2 \right)^2
- \left( {\sf Re} \,z_3 \right)^2
=
0, \qquad \qquad {\sf Re} \,z_1 > 0
.$$
This section is devoted to the determination of the Lie algebra ${\sf aut_{CR}}({\sf LC})$ of infinitesimal CR-automorphisms of ${\sf LC}$. This has been done before by Kaup and Zaitsev [@Kaup-Zaitsev]. We prove the following result:
\[thm:LC\] The tube over the future light cone: $${\sf LC}: \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\left( {\sf Re} \,z_1 \right)^2 -\left( {\sf Re} \,z_2 \right)^2
- \left( {\sf Re} \,z_3 \right)^2
=
0, \qquad \qquad {\sf Re} \,z_1 > 0
.$$ has a ${\bf 10}$-dimensional Lie algebra of CR-automorphisms. A basis for the Maurer-Cartan forms of ${\sf aut_{CR}}({\sf LC})$ is provided by the $10$ differential $1$-forms $\rho$, $\kappa$, $\zeta$, ${\overline}{\kappa}$, ${\overline}{\zeta}$, $\pi^1$, $\pi^2$, ${\overline}{\pi^1}$, ${\overline}{\pi^2}$, $\Lambda$, which satisfy the Maurer-Cartan equations:
$$\label{eq:structure}
\begin{aligned}
d \rho & = \pi^1 \wedge \rho + {\overline}{\pi^1} \wedge \rho + i \, \kappa \wedge {\overline}{\kappa}, \\
d \kappa & = \pi^1 \wedge \kappa + \pi^2 \wedge \rho + \zeta \wedge {\overline}{\kappa}, \\
d \zeta & = i \, \pi^2 \wedge \kappa + \pi^1 \wedge \zeta - {\overline}{\pi^1} \wedge \zeta, \\
d {\overline}{\kappa} & = {\overline}{\pi^1} \wedge {\overline}{\kappa} + {\overline}{\pi^2} \wedge \rho - \kappa \wedge {\overline}{\zeta}, \\
d {\overline}{\zeta} & = - i \, {\overline}{\pi^2} \wedge {\overline}{\kappa} + {\overline}{\pi^1} \wedge {\overline}{\zeta} - {\pi^1} \wedge {\overline}{\zeta}, \\
d \pi^1 & = \Lambda \wedge \rho + i \, \kappa \wedge {\overline}{\pi^2} + \zeta \wedge {\overline}{\zeta}, \\
d \pi^2 & = \Lambda \wedge \kappa + \zeta \wedge {\overline}{\pi^2} + \pi^2 \wedge {\overline}{\pi^1}, \\
d {\overline}{\pi^1} & = \Lambda \wedge \rho - i \, {\overline}{\kappa} \wedge \pi^2 - \zeta \wedge {\overline}{\zeta}, \\
d {\overline}{\pi^2} & = \Lambda \wedge {\overline}{\kappa} + {\overline}{\zeta} \wedge \pi^2 - \pi^1 \wedge {\overline}{\pi^2}, \\
d \Lambda & = - \pi^1 \wedge \Lambda + i\, \pi^2 \wedge {\overline}{\pi^2} - \pi^1 \wedge {\overline}{\Lambda}.
\end{aligned}$$
Geometric set-up
----------------
In order to motivate our subsequent notations, it is convenient to introduce some general results on CR-manifolds belonging to class ${\sf IV_{2}}$, for which we refer to [@pocchiola] for a proof.
Let $M \subset {\mathbb{C}}^3$ be a smooth hypersurface locally represented as a graph over the $5$-dimensional real hyperplane ${\mathbb{C}}_{ z_1} \times {\mathbb{C}}_{ z_2} \times {\mathbb{R}}_v$: $$u
=
F\big(z_1,z_2,\overline{z_1},\overline{z_2},v\big),$$ where $F$ is a local smooth function depending on $5$ arguments. We assume that $M$ is a CR-submanifold of CR dimension $2$ which is $2$-nondegenerate and whose Levi form is of constant rank $1$. The two vector fields ${\mathcal{L}_1}$ and $\mathcal{L}_2$ defined by:
$$\begin{aligned}
\mathcal{L}_j &
=
\frac{\partial}{\partial z_j}
+
A^j\,
\frac{\partial}{\partial v},
&& A^j := - i \, \frac{F_{z_j}}{1 + i \, F_v},
&&& j= 1,2,\end{aligned}$$
constitute a basis of $T^{1,0}_pM$ at each point $p$ of $M$ and thus provide an identification of $T^{1,0}_pM$ with ${\mathbb{C}}^2$ at each point. Moreover, the real $1$-form $\sigma$ defined by: $$\sigma := dv - A^1 \, dz_1 - A^2 \, dz_2 - {\overline}{A^1} \, d{\overline}{z_1}
- {\overline}{A^2} \, d {\overline}{z_2},$$ satisfies $$\{ \sigma =0 \} = T^{1,0}M \oplus T^{0,1}M,$$ and thus provides an identication of the projection $${\mathbb{C}}\otimes T_pM \longrightarrow \left. {\mathbb{C}}\otimes T_pM \right.\big/
\left( T^{1,0}_pM \oplus T^{0,1}_pM \right)$$ with the map $\sigma_p$: ${\mathbb{C}}\otimes T_pM \longrightarrow {\mathbb{C}}$. With these two identifications, the Levi form $LF$ can be viewed at each point $p$ as a skew hermitian form on ${\mathbb{C}}^2$ represented by the matrix: $$LF=
\begin{pmatrix}
\sigma_p \left( i \, \big[ {\mathcal{L}_1}, {\overline{\mathcal{L}_1}}\big] \right)
& \sigma_p \left( i \,\big[ \mathcal{L}_2, {{\overline{\mathcal{L}_1}}} \big] \right)\\
\sigma_p \left( i \,\big[ {\mathcal{L}_1}, {\overline}{\mathcal{L}_2} \big] \right)
& \sigma_p \left( i \,\big[ \mathcal{L}_2, {\overline}{\mathcal{L}_2} \big] \right) \\
\end{pmatrix}
.$$ The fact that $LF$ is supposed to be of constant rank $1$ ensures the existence of a certain function $k$ such that the vector field $${\mathcal{K}}:= k \,
{\mathcal{L}_1}+ \mathcal{L}_2$$ lies in the kernel of $LF$. Here are the expressions of ${\mathcal{K}}$ and $k$ in terms of the graphing function $F$: $${\mathcal{K}}= k \, \partial_{z_1} + \partial_{z_2} - \frac{i}{1 + i \, F_v}
\left( k \, F_{z_1} + F_{z_2} \right) \partial_v,$$ [$$k = - \frac{F_{z_2, {\overline}{z_1}} + F_{z_2, {\overline}{z_1}}
\, F_{v}^2 - i \, F_{{\overline}{z_1}} \, F_{z_2, v} - F_{{\overline}{z_1}}
\, F_{v} \, F_{v, z_2} + i \, F_{z_2} \, F_{{\overline}{z_1}} \,
F_{v,v} - F_{z_2} \, F_v \, F_{v, {\overline}{z_1}}}{
F_{z_1, {\overline}{z_1}} + F_{z_1, {\overline}{z_1}} \, F_v^2 - i \, F_{{\overline}{z_1}}
\, F_{z_1, v} - F_{{\overline}{z_1}} \, F_v \, F_{ z_1, v} + i \, F_{z_1}
\, F_{{\overline}{z_1}, v}
+ F_{z_1} \, F_{{\overline}{z_1}} \, F_{v, v} -F_{z_1} \, F_v \, F_{v,{\overline}{z_1}}}
.$$]{} From the above construction, the four vector fields ${\mathcal{L}_1}$, ${\mathcal{K}}$, ${\overline{\mathcal{L}_1}}$, ${{\overline}{\mathcal{K}}}$ constitute a basis of $T^{1,0}_pM \oplus T^{0,1}_pM$ at each point $p$ of $M$. It turns out that the vector field ${\mathcal{T}}$ defined by: $${\mathcal{T}}:= i \, \big[ {\mathcal{L}_1},{\overline{\mathcal{L}_1}}\big]$$ is linearly independant from ${\mathcal{L}_1}$, ${\mathcal{K}}$, ${\overline{\mathcal{L}_1}}$, ${{\overline}{\mathcal{K}}}$.
It is well known (see [@Fels-Kaup-2007; @Merker-2003]) that the tube over the future light cone is locally biholomorphic to the graphed hypersurface: $$u = \frac{z_1 {\overline}{z_1} + \frac{1}{2} z_1^2 {\overline}{z_2} + \frac{1}{2} {\overline}{z_1^2} z_2}{1 - z_2 {\overline}{z_2}}.$$ The five vector fields ${\mathcal{L}_1}$, ${\mathcal{K}}$, ${\overline{\mathcal{L}_1}}$, ${{\overline}{\mathcal{K}}}$ and ${\mathcal{T}}$, wich constitute a local frame on ${\sf LC}$, have thus the following expressions:
$${\mathcal{L}_1}:=
\frac{\partial}{\partial z_1}
-
i \frac{{\overline}{z_1} + z_1 {\overline}{z_2}}{1 - z_2 {\overline}{z_2}}
\frac{\partial}{\partial v}
,$$
$${\mathcal{K}}:= - \frac{{\overline}{z_1} + z_1 {\overline}{z_2}}{1 - z_2
{\overline}{z_2}} \, \frac{\partial}{\partial z_1} + \frac{\partial}{\partial z_2} +
\frac{i} {2} \frac{{\overline}{z_1}^2 +2 z_1 {\overline}{z_1} {\overline}{z_2} + z_1^2 {\overline}{z_2}^2}{\left(1 - z_2 {\overline}{z_2}\right)^2} \,
\frac{\partial}{\partial v},$$
and
$${\mathcal{T}}:= -\frac{2}{1 - z_2 {\overline}{z_2}} \, \frac{\partial}{\partial v}.$$
Moreover the function $k$ is given by
$$k:= - \frac{{\overline}{z_1} + z_1 {\overline}{z_2}}{1 - z_2 {\overline}{z_2}}
.$$
Let $(\rho_0, \kappa_0, \zeta_0, {\overline}{\kappa_0}, {\overline}{\zeta}_0)$ be the dual coframe of $({\mathcal{T}}, {\mathcal{L}_1}, {\mathcal{K}}, {\overline{\mathcal{L}_1}}, {{\overline}{\mathcal{K}}})$. We have:
\_0 =- ( [z\_1]{} + z\_1 [z\_2]{} ) dz\_1 - dz\_2 + ( z\_1 + [z\_1]{} z\_2 ) d [z\_1]{} + dz\_2 + (-1 + z\_2 [ z\_2]{} ) dv
,
\_0 = dz\_1 + dz\_2
,
\_0 = dz\_2
,
[\_0]{} = d [z\_1]{} + d [z\_2]{}
,
[\_0]{} = d [z\_2]{}
.
A direct computation gives the structure equations enjoyed by the coframe $(\rho_0, \kappa_0, \zeta_0, {\overline}{\kappa_0}, {\overline}{\zeta}_0)$:
$$\label{eq:StEq}
\begin{aligned}
d \rho_{0} & = \frac{{\overline}{z_2}}{1-z_2 {\overline}{z_2}} \left. \rho_0 \wedge \zeta_0 \right.
+ \frac{z_2}{1-z_2 {\overline}{z_2}} \left. \rho_0 \wedge {\overline}{\zeta_0} \right. + i \left. \kappa_0 \wedge {\overline}{\kappa_0} \right., \\
d \kappa_{0} & = \frac{{\overline}{z_2}}{1-z_2 {\overline}{z_2}} \, \left. \kappa_0 \wedge \zeta_0 \right.
- \frac{1}{1- z_2 {\overline}{z_2}} \, \left. \zeta_0 \wedge {\overline}{\kappa}_0 \right.
, \\
d \zeta_0 & = 0, \\
d {\overline}{\kappa}_0 & =
\frac{1}{1- z_2 {\overline}{z_2}} \, \left. \kappa_0 \wedge
{\overline}{\zeta_0} \right. + \frac{z_2}{1-z_2 {\overline}{z_2}} \, \left. {\overline}{\kappa_0} \wedge {\overline}{\zeta_0} \right.
,\\
d {\overline}{ \zeta_{0}} & = 0.
\end{aligned}$$
The matrix Lie group which encodes the equivalence problem for ${\sf LC}$ is the $10$ dimensional Lie group $G_1$ whose elements are of the form: $$g := \begin{pmatrix}
{\sf c}\overline{\sf c} & 0 & 0 & 0 & 0 \\
{\sf b} & {\sf c} & 0 & 0 & 0 \\
{\sf d} & {\sf e} & {\sf f} & 0 & 0 \\
\overline{\sf b} & 0 & 0 & \overline{\sf c} & 0 \\
\overline{\sf d} & 0 & 0 & \overline{\sf e} & \overline{\sf f}
\end{pmatrix},$$ where ${{\sf c}}$ and ${{\sf f}}$ are non-zero complex numbers whereas ${{\sf b}}$, ${{\sf d}}$ and ${{\sf e}}$ are arbitrary complex numbers (see [@pocchiola; @MPS]). We introduce the $5$ new one-forms $\rho$, $\kappa$, $\zeta$, $\overline{ \kappa}$, $\overline{ \zeta}$ by the relation:
$$\begin{pmatrix}
\rho
\\
\kappa
\\
\zeta
\\
\overline{\kappa}
\\
\overline{\zeta}
\end{pmatrix}
:=
g \cdot
\begin{pmatrix}
\rho_0
\\
\kappa_0
\\
\zeta_0
\\
\overline{\kappa_0}
\\
\overline{\zeta_0}
\end{pmatrix}
,$$
which we abbreviate as: $$\omega := g \cdot \omega_0.$$
The coframes $\omega$ define a $G_1$ structure $P^1$ on ${\sf LC}$. The rest of this section is devoted to reduce $P^1$ to an absolute parallelism on ${\sf LC}$ through Cartan equivalence method.
Normalization of ${{\sf f}}$
----------------------------
The Maurer Cartan forms of $G_1$ are the following: $$\begin{aligned}
\alpha^1 &: = \frac{ d {{\sf c}}}{{{\sf c}}}, \\
\alpha^2 &:= \frac{ d {{\sf b}}}{{{\sf c}}{{\overline}{{\sf c}}}} - \frac{{{\sf b}}\, d {{\sf c}}}{{{\sf c}}^2}{{{\overline}{{\sf c}}}}, \\
\alpha^3 &:= \frac{ d {{\sf d}}}{{{\sf c}}{{\overline}{{\sf c}}}} - \frac{{{\sf b}}\,
d {{\sf e}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}} + \frac{\left( -{{\sf d}}{{\sf c}}+ {{\sf e}}{{\sf b}}\right) d {\sf f}}{{{\sf c}}^2 {{\overline}{{\sf c}}}{\sf f}}, \\
\alpha^4 & := \frac{d {{\sf e}}}{{{\sf c}}} - \frac{{{\sf e}}\, d {\sf f}}{{{\sf c}}\sf f}, \\
\alpha^5 &:= \frac{d {\sf f}}{\sf f}.
\end{aligned}$$
The structure equations read as:
d = \^1 +\
+ T\^\_ + T\^\_ + T\^\_ + T\^\_ + i ,
d = \^[1]{} + \^[2]{}\
+ T\^\_ + T\^\_ + T\^\_ + T\^\_ + T\^\_ + T\^\_ + T\^\_ ,
d = \^[3]{} + \^[4]{} + \^[5]{}\
+ T\^\_ + T\^\_ + T\^\_ + T\^\_ . . + T\^\_ + T\^\_ + T\^\_ ,
where the expressions of the torsion coefficients $T^{{{\scriptscriptstyle{\bullet}}}}_{{{\scriptscriptstyle{\bullet}}}{{\scriptscriptstyle{\bullet}}}}$ are given in the appendix.
We now proceed with the absorption step of Cartan’s method. We introduce the modified Maurer-Cartan forms $\widetilde{\alpha}^i$, which are a related to the $1$-forms $\alpha^i$ by the relations: $$\widetilde{\alpha}^i := \alpha^i - x_{\rho}^i \, \rho \, -
x_{\kappa}^i \, \kappa - x_{\zeta}^i \, \zeta \, - \, x_{\overline{\kappa}}^i \, \overline{\kappa} \, - \,
x_{\overline{\zeta}}^i \, \overline{\zeta},$$ where $x^1$, $x^2$, $x^3$, $x^4$ and $x^5$ are arbitrary complex-valued functions. The previously written structure equations take the new form: $$\begin{gathered}
d \rho = \widetilde{\alpha}^1 \wedge \rho + \overline{\widetilde{\alpha}^1} \wedge \rho \\
+ \left( T_{\rho \kappa}^{\rho}- x_\kappa^1 -
x_{\overline{\kappa}}^1 \right) \rho \wedge \kappa \, + \, \left(T_{\rho \zeta}^{\rho} - x_{\kappa}^1 -
\overline{ x_{\overline{\zeta}}^1} \right)
\rho \wedge \zeta \\ \, + \, \left(T_{\rho \overline{\kappa}}^{\rho} - x_{\overline{\kappa}}^1-
\overline{ x_{\kappa}^1} \right) \rho \wedge \overline{\kappa}
\, + \, \left( T_{\rho \overline{\zeta}}^{\rho} - x_{\zeta}^1 - x_{\overline{\zeta}}^1 \right)
\rho \wedge \overline{\zeta} \\ + i \, \kappa \wedge \overline{\kappa},\end{gathered}$$ $$\begin{gathered}
d \kappa = \widetilde{\alpha}^{1} \wedge \kappa + \widetilde{\alpha}^{2} \wedge \rho \\
+ \, \left(T_{\rho \kappa}^{\kappa} - x_\kappa^2 + x_\rho^1
\right) \rho \wedge \kappa \, + \, \left(T_{\rho \zeta}^{\kappa} - x_{\kappa}^2 \right)
\, \rho \wedge \zeta \\
+ \, \left( T_{\rho \overline{\kappa}}^{\kappa} - x_{\overline{\kappa}}^2 \right) \rho \wedge \overline{\kappa} +
\left( T^{\kappa}_{\rho \overline{\zeta}} - x_{\overline{\zeta}}^2 \right) \rho \wedge \overline{\zeta} \\
+ \, \left( T_{\kappa \zeta}^{\kappa} + x_{\zeta}^1
\right)\, \kappa \wedge \zeta \,
+ \, \left( T_{\kappa \overline{\kappa}}^{\kappa} - x_{\overline{\kappa}}^1 \right) \kappa \wedge \overline{\kappa} \\
+ \, T_{\zeta \overline{\kappa}}^{\kappa} \, \zeta \wedge \overline{\kappa} \, + \, \left( T_{\kappa \overline{\zeta}}^1
- x_{\kappa \overline{\zeta} }^1 \right) \, \kappa \wedge \zeta,\end{gathered}$$ $$\begin{gathered}
d \zeta = \widetilde{\alpha}^3 \wedge \rho + \widetilde{\alpha}^4 \wedge \kappa + \widetilde{\alpha}^5 \wedge \zeta \\
+ \left( T_{\rho \kappa}^{\zeta}
- x_{\kappa}^3 + x_{\rho}^4 \right) \rho \wedge \kappa + \left( T_{\rho \zeta}^{\zeta} - x_{\zeta}^3 + x_{\rho}^5
\right) \rho \wedge \zeta \\
+ \left(T_{\rho \kappa}^{\zeta} - x_{\overline{\kappa}}^3 \right) \left. \rho \wedge \overline{\kappa} \right. +
\left( T_{\rho \overline{\zeta}}^{\zeta} -x_{\overline{\zeta}}^3 \right) \rho \wedge \overline{\zeta} \\
+ \, \left( T_{\kappa \overline{\kappa}}^{\zeta} -
x_{\overline{\kappa}}^4 \right) \kappa \wedge \overline{\kappa} +
\left( T_{\zeta
\overline{\kappa}}^{\zeta} - x_{\overline{\kappa}}^5 \right) \zeta \wedge \overline{\kappa} \\
+ \left( x_{\kappa}^5 - x_{\zeta}^4 \right) \kappa \wedge \zeta - x_
{\overline{\kappa}}^4 \,
\kappa \wedge \overline{\kappa} \\
\, + \, \left(x_{\overline{\kappa}}^5 - x_{\overline{\zeta}}^4 \right) \overline{\kappa} \wedge \zeta - x_{\overline{\zeta}}^5
\, \zeta \wedge \overline{\zeta}.\end{gathered}$$
We then choose $x^1$, $x^2$, $x^3$, $x^4$ and $x^5$ in a way that eliminates as many torsion coefficients as possible. We easily see that the only coefficient which can not be absorbed is the one in front of $\zeta \wedge {\overline}{\kappa}$ in $d \kappa$, because it does not depend on the $x^i$’s. We choose the normalization $$T_{\zeta \overline{\kappa}}^{\kappa} = 1,$$ which yields to : $$\sf{f} = - \frac{c}{\overline{c}} \, \frac{1}{1- z_2 {\overline}{z_2}}.$$ We notice that the absorbed structure equations take the form: $$\begin{aligned}
d \rho & = \widetilde{\alpha}^1 \wedge
\rho + \overline{\widetilde{\alpha}^1} \wedge \rho + i \, \kappa \wedge \overline{\kappa}, \\
d \kappa & = \widetilde{\alpha}^{1} \wedge \kappa + \widetilde{\alpha}^{2} \wedge \rho + \zeta \wedge \overline{\kappa}, \\
d \zeta & = \widetilde{\alpha}^3 \wedge \rho + \widetilde{\alpha}^4 \wedge \kappa + \widetilde{\alpha}^5 \wedge \zeta. \end{aligned}$$
The normalization of ${\sf f}$ gives the new relation : $$\begin{pmatrix}
\rho \\
\kappa \\
\zeta \\
\overline{\kappa}\\
\overline{\zeta}
\end{pmatrix}
=
\begin{pmatrix}
{\sf c} \overline{\sf c} & 0 &0 &0 &0 \\
{\sf b} & {\sf c} & 0 & 0 & 0 \\
{\sf d} & {\sf e} & \frac{c}{\overline{c}} \, \frac{1}{-1+ z_2 {\overline}{z_2}} & 0 & 0 \\
{\sf \overline{b}} & 0 & 0 & {\sf \overline{c}} & 0 \\
0 & 0 & {\sf \overline{d}} & {\sf \overline{e}} & \frac{{{\overline}{{\sf c}}}}{{{\sf c}}} \, \frac{1}{-1+ z_2 {\overline}{z_2}}
\end{pmatrix}
\cdot
\begin{pmatrix}
\rho_0 \\
\kappa_0 \\
\zeta_0 \\
\overline{\kappa}_0 \\
\overline{\zeta}_0
\end{pmatrix}
.$$ We thus introduce the new one-form $$\hat{\zeta}_{0} = - \frac{1}{1- z_2 {\overline}{z_2}} \cdot \zeta_0
,$$ such that the previous equation rewrites : $$\begin{pmatrix}
\rho \\
\kappa \\
\zeta \\
\overline{\kappa}\\
\overline{\zeta}
\end{pmatrix}
=
\begin{pmatrix}
{\sf c} \overline{\sf c} & 0 &0 &0 &0 \\
{\sf b} & {\sf c} & 0 & 0 & 0 \\
{\sf d} & {\sf e} & \frac{\sf c}{\overline{\sf c}} & 0 & 0 \\
{\sf \overline{b}} & 0 & 0 & {\sf \overline{c}} & 0 \\
0 & 0 & {\sf \overline{d}} & {\sf \overline{e}} & \frac{\sf c}{\sf \overline{c}}
\end{pmatrix}
\cdot
\begin{pmatrix}
\rho_0 \\
\kappa_0 \\
\hat{\zeta_0} \\
\overline{\kappa}_0 \\
\overline{\hat{\zeta}}_0
\end{pmatrix}.$$
We have reduced the $G_{1}$ equivalence problem to a $G_2$ equivalence problem, where $G_2$ is the $8$ dimensional real matrix Lie group whose elements are of the form $$g = \begin{pmatrix}
{\sf c} \overline{\sf c} & 0 &0 &0 &0 \\
{\sf b} & {\sf c} & 0 & 0 & 0 \\
{\sf d} & {\sf e} & \frac{\sf c}{\overline{\sf c}} & 0 & 0 \\
{\sf \overline{b}} & 0 & 0 & {\sf \overline{c}} & 0 \\
0 & 0 & {\sf \overline{d}} & {\sf \overline{e}} & \frac{\sf c}{\sf \overline{c}}
\end{pmatrix}.$$ We determine the new structure equations enjoyed by the base coframe $(\rho_0, \kappa_0, \hat{\zeta_0}, \kappa_0, \overline{\hat{\zeta_0}})$. We get :
d \_[0]{} =- [z\_2]{} \_[0]{} \_0 - z\_2 \_0 + i \_0 ,
d\_[0]{}= -[z\_2]{} \_0 + \_[0]{} [\_[0]{}]{} ,
d \_0 = z\_2 \_0 [\_0]{}
.
Normalization of ${{\sf b}}$
----------------------------
The Maurer forms of the $G_{2}$ are given by the independant entries of the matrix $d g \cdot g^{-1}$. We have: $$dg \cdot g^{-1}
=
\begin{pmatrix}
\beta^1 + {\overline}{\beta^1} & 0 &0 &0 &0 \\
\beta^2 & \beta^1 & 0 & 0& 0 \\
\beta^3 & \beta^4 & \beta^1 - {\overline}{\beta^1} & 0 &0 \\
{\overline}{\beta^2} & 0 & 0 & {\overline}{\beta^1} & 0 \\
{\overline}{\beta^3} & 0 & 0 & {\overline}{\beta^4} & - \beta^1 + {\overline}{\beta^1}
\end{pmatrix}
,$$ where the forms $\beta^1$, $\beta^2$, $\beta^3$ and $\beta^4$ are defined by $$\begin{gathered}
\beta^1 : = \frac{d {\sf c}}{\sf c}, \\
\beta^2 := \frac{d {{\sf b}}}{{{\sf c}}{{\overline}{{\sf c}}}}- \frac{{{\sf b}}d {{\sf c}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}}, \\
\beta^3 := \frac{\left( - {{\sf d}}{{\sf c}}+ {{\sf e}}{{\sf b}}\right) d {{\sf c}}}{{{\sf c}}^3 {{\overline}{{\sf c}}}}
- \frac{\left(- {{\sf d}}{{\sf c}}+ {{\sf e}}{{\sf b}}\right) d {{\overline}{{\sf c}}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}^2} + \frac{d {{\sf d}}}{{{\sf c}}{{\overline}{{\sf c}}}} -
\frac{{{\sf b}}d {{\sf e}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}}, \\
\beta^4 := - \frac{{{\sf e}}d {{\sf c}}}{{{\sf c}}^2} + \frac{{{\sf e}}d {{\overline}{{\sf c}}}}{{{\overline}{{\sf c}}}{{\sf c}}} + \frac{d {{\sf e}}}{{{\sf c}}}.\end{gathered}$$ Using formula (\[eq:str\]), we get the structure equations for the lifted coframe $(\rho, \kappa, \zeta, {\overline}{\kappa}, {\overline}{\zeta})$ from those of the base coframe $(\rho_0, \kappa_0, \hat{\zeta}_0, {\overline}{\kappa_0}, {\overline}{\hat{\zeta}_0})$:
$$\begin{gathered}
d \rho
=
\beta^{1} \wedge \rho + \overline{\beta^{1}} \wedge \rho \\
+
U^{\rho}_{\rho \kappa} \, \rho \wedge \kappa
+
U^{\rho}_{\rho \zeta} \, \rho \wedge \zeta
+
U^{\rho}_{\rho \overline{\kappa}} \, \rho \wedge \overline{\kappa} \\
+
U^{\rho}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
i \, \kappa \wedge \overline{\kappa}
,\end{gathered}$$
$$\begin{gathered}
d \kappa
=
\beta^{1} \wedge \kappa + \beta^{2} \wedge \rho \\
+
U^{\kappa}_{\rho \kappa} \, \rho \wedge \kappa
+
U^{\kappa}_{\rho \zeta} \, \rho \wedge \zeta
+
U^{\kappa}_{\rho \overline{\kappa}} \, \rho \wedge \overline{\kappa}
+
U^{\kappa}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta} \\
+
U^{\kappa}_{\kappa \zeta} \, \kappa \wedge \zeta
+
U^{\kappa}_{\kappa \overline{\kappa}} \, \kappa \wedge \overline{\kappa}
+
\zeta \wedge \overline{\kappa}
,\end{gathered}$$
$$\begin{gathered}
d \zeta
=
\beta^{3} \wedge \rho + \beta^{4} \wedge \kappa + \beta^{1} \wedge \zeta - \overline{\beta^{1}} \wedge \zeta \\
+
U^{\zeta}_{\rho \kappa} \, \rho \wedge \kappa
+
U^{\zeta}_{\rho \zeta} \, \rho \wedge \zeta
+
U^{\zeta}_{\rho \overline{\kappa}} \, \rho \wedge \overline{\kappa} \\
+
U^{\zeta}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
U^{\zeta}_{\kappa \zeta} \, \kappa \wedge \zeta
+
U^{\zeta}_{\kappa \overline{\kappa}} \, \kappa \wedge \overline{\kappa} \\
+
U^{\zeta}_{\kappa \overline{\zeta}} \, \kappa \wedge \overline{\zeta}
+
U^{\zeta}_{\zeta \overline{\kappa}} \, \zeta \wedge \overline{\kappa}
+
U^{\zeta}_{\zeta \overline{\zeta}} \, \zeta \wedge \overline{\zeta}
.\end{gathered}$$
We introduce the modified Maurer-Cartan forms $\widetilde{\beta}^i$ which differ from the $\beta^i$ by a linear combination of the $1$-forms $\rho$, $\kappa$, $\zeta$, $\overline{\kappa}$, $\overline{\zeta}$, i.e. that is : $$\widetilde{\beta}^i= \beta^i -
y_{\rho}^i \, \rho \, - y_{\kappa}^i \, \kappa - y_{\zeta}^i \, \zeta \, - \, y_{\overline{\kappa}}^i \, \overline{\kappa} \, - \,
y_{\overline{\zeta}}^i \, \overline{\zeta}.$$ The structure equations rewrite:
d = \^1 + [\^1]{}\
+ ( U\_\^ - y\_\^1 - [y]{}\^1\_ ) + ( U\_\^ - y\_\^1 - [y]{}\^1\_ ) + ( U\_\^ - y\_\^1 - [y]{}\^1\_ ) + ( U\_\^ - y\_\^1 - [y]{}\^1\_ ) + i ,
d = \^1 + \^2\
+ ( U\_\^ + y\^1\_ - y\^2\_ ) + ( U\_\^ - y\^2\_ ) + ( U\_\^ - y\^2\_ )\
+ ( U\_\^ - y\^2\_ ) + ( U\_\^ - y\^1\_ ) + ( U\_\^ - y\^1\_ ) - y\^1\_ + ,
d = \^[3]{} + \^[4]{} + \^[1]{} -\
+ ( U\_\^ - y\^3\_ + y\^4\_ ) + ( U\_\^ - y\^3\_ + y\^1\_ - [y]{}\^1\_ ) + ( U\_\^ - y\^3\_ ) .. + ( U\_\^ - y\^4\_ + y\^1\_-[y]{}\^1\_ ) + ( U\_[[ ]{}]{}\^ - y\^4\_ ) + ( U\_\^ - y\^4\_ ) + ( U\_[[ ]{}]{}\^ - y\^1\_ + [y]{}\^1\_ ) + ( U\_\^ - y\^1\_ + [y]{}\^1\_ ) .
We get the following absorbtion equations: $$\begin{aligned}
{3}
y_{\kappa}^1 + {\overline}{y}^1_{{\overline}{\kappa}} &= U_{\rho \kappa}^{\rho},
& \qquad \qquad y_{\zeta}^1 + {\overline}{y}^1_{{\overline}{\zeta}} &= U_{\rho \zeta}^{\rho},
& \qquad \qquad y_{{\overline}{\kappa}}^1 + {\overline}{y}^1_{\kappa} &= U_{\rho {\overline}{\kappa}}^{\rho}, \\
y_{{\overline}{\zeta}}^1 + {\overline}{y}^1_{\zeta} & = U_{\rho \zeta}^{\rho},
& \qquad \qquad - y^1_{\rho} + y^2_{\kappa}& = U_{\rho \kappa}^{\kappa}, &
\qquad \qquad y^2_{\zeta} & = U_{\rho \zeta}^{\kappa}, \\
y^2_{{\overline}{\kappa}} &=U_{\rho {\overline}{\kappa}}^{\kappa}, & \qquad \qquad
y^2_{{\overline}{\zeta}}& = U_{\rho {\overline}{\zeta}}^{\kappa}, & \qquad \qquad
y^1_{\zeta} &= U_{\kappa \zeta}^{\kappa}, \\
y^1_{{\overline}{\kappa}} & = U_{\kappa {\overline}{\kappa}}^{\kappa}, & \qquad \qquad
y^1_{{\overline}{\zeta}} &=0, & \qquad \qquad
y^3_{\kappa} - y^4_{\rho} & = U_{\rho \kappa}^{\zeta}, \\
y^3_{\zeta} - y^1_{\rho} + {\overline}{y}^1_{\rho} &= U_{\rho \zeta}^{\zeta}, & \qquad \qquad
y^3_{{\overline}{\kappa}} & = U_{\rho {\overline}{\kappa}}^{\zeta}, &\qquad \qquad
y^4_{\zeta} - y^1_{\kappa} + {\overline}{y}^1_{\kappa} &= U_{\kappa \zeta}^{\zeta}, \\
y^4_{{\overline}{\kappa}} &= U_{\kappa {\overline}{ \kappa}}^{\zeta}, &\qquad \qquad
y^4_{{\overline}{\zeta}} &= U_{\kappa {\overline}{\zeta}}^{\zeta}, &\qquad \qquad
y^1_{{\overline}{\kappa}} - {\overline}{y}^1_{\kappa} &= U_{\zeta {\overline}{ \kappa}}^{\zeta}, \\
y^1_{{\overline}{\zeta}} - {\overline}{y}^1_{\zeta} &= U_{\zeta {\overline}{\zeta}}^{\zeta}. \end{aligned}$$ Eliminating the $y^{\bullet}_{\bullet}$ among theses equations leads to the following relations between the torsion coefficients : $$\begin{aligned}
U_{\rho {\overline}{\kappa}}^{\rho} & = {\overline}{U_{\rho \kappa}^{\rho}}, \\
U_{\rho {\overline}{\zeta}}^{\rho} & = {\overline}{U_{\rho \zeta}^{\rho}}, \\
U_{\rho \zeta}^{\rho}& = U_{\kappa \zeta}^{\kappa},\\
U_{\zeta {\overline}{ \zeta}}^{\zeta}& = - U_{\rho{\overline}{ \zeta}}^{\rho}, \\
2 \, U_{\kappa {\overline}{\kappa}}^{\kappa} & = U_{\zeta {\overline}{\kappa}}^{\zeta} + U_{\rho {\overline}{\kappa}}^{\rho}. \end{aligned}$$ We verify easily that the first four equations do not depend on the group coefficients and are already satisfied. However, the last one does depend on the group coefficients. It gives us the normalization of ${{\sf b}}$ as it rewrites : $$\label{eq:nb}
{{\sf b}}= - i \, {{\overline}{{\sf c}}}{{\sf e}}.$$ The absorbed structure equations rewrite: $$\begin{aligned}
d \rho &= \widetilde{\beta}^1 \wedge \rho + {\overline}{\widetilde{\beta}^1} \wedge \rho + i\, \kappa \wedge {\overline}{\kappa}, \\
d \kappa &= \widetilde{\beta}^1 \wedge \kappa + \widetilde{\beta}^2 \wedge \rho
+
\zeta \wedge {\overline}{\kappa}
,\\
d \zeta &=
\widetilde{\beta}^{3} \wedge \rho + \widetilde{\beta}^{4} \wedge \kappa
+ \widetilde{\beta}^{1} \wedge \zeta - \overline{\widetilde{\beta}^{1}} \wedge \zeta +
\left( U_{\zeta {\overline}{\kappa}}^{\zeta} + U_{\rho {\overline}{\kappa}}^{\rho} -2 \, U_{\kappa {\overline}{\kappa}}^{\kappa} \right)
\, \zeta \wedge {\overline}{\kappa}.\end{aligned}$$
Normalization of ${{\sf d}}$
----------------------------
We have thus reduced the group $G_2$ to a new group $G_3$, whose elements are of the form $${\sf g} = \begin{pmatrix}
{{\sf c}}{{\overline}{{\sf c}}}& 0 & 0 & 0 & 0 \\
- i\, {{\sf e}}{{\overline}{{\sf c}}}& {{\sf c}}& 0 & 0 &0 \\
{{\sf d}}& {{\sf e}}& \frac{{{\sf c}}}{{{\overline}{{\sf c}}}} & 0 & 0 \\
i\, {{\overline}{{\sf e}}}{{\sf c}}& 0 & 0 & {{\overline}{{\sf c}}}& 0 \\
{{\overline}{{\sf d}}}& 0 & 0 & {{\overline}{{\sf e}}}& \frac{{{\overline}{{\sf c}}}}{{{\sf c}}}
\end{pmatrix}
.$$ It is a six-dimensional real Lie group. We compute its Maurer Cartan forms with the usual formula $$d {\sf g} \cdot {\sf g}^{-1}
=
\begin{pmatrix}
\gamma^1 + {\overline}{\gamma}^1 & 0 & 0 & 0 &0 \\
\gamma^2 & \gamma^1 & 0 & 0 & 0 \\
\gamma^3 & i\, \gamma^2 & \gamma^1 - {\overline}{\gamma}^1 & 0 & 0 \\
{\overline}{\gamma}^2 & 0 & 0 & {\overline}{\gamma}^1 & 0 \\
- \gamma^3 & 0 & 0 &- i\, {\overline}{\gamma}^2 & - \gamma^1 + {\overline}{\gamma}^1 \\
\end{pmatrix},$$ where $$\gamma^1 := \frac{d {{\sf c}}}{{{\sf c}}},$$ $$\gamma^2 := i \, {{\sf e}}\frac{d {{\sf c}}}{{{\sf c}}^2} - i\, \frac{{{\sf e}}\, d {{\overline}{{\sf c}}}}{{{\sf c}}{{\overline}{{\sf c}}}} - i\, \frac{d {{\sf e}}}{{{\sf c}}},$$ and $$\gamma^3 := \left(\frac{{{\sf d}}{{\sf c}}+ i\, {{\sf e}}^2 {{\overline}{{\sf c}}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}}\right) \left(\frac{d {{\overline}{{\sf c}}}}{{{\overline}{{\sf c}}}} - \frac{ d {{\sf c}}}{{{\sf c}}} \right) + \frac{d {{\sf d}}}{{{\sf c}}{{\overline}{{\sf c}}}} +
i \, \frac{{{\sf e}}d {{\sf e}}}{{{\sf c}}^2}
.$$
As the normalization of ${{\sf b}}$ does not depend on the base variables, the third loop of Cartan’s method is straightforward. We get the following structure equations:
$$\begin{gathered}
d \rho
=
\gamma^{1} \wedge \rho + \overline{\gamma^{1}} \wedge \rho \\
+
V^{\rho}_{\rho \kappa} \, \rho \wedge \kappa
+
V^{\rho}_{\rho \zeta} \, \rho \wedge \zeta
+
V^{\rho}_{\rho \overline{\kappa}} \, \rho \wedge \overline{\kappa} \\
+
V^{\rho}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
i \, \kappa \wedge \overline{\kappa}
,\end{gathered}$$
$$\begin{gathered}
d \kappa
=
\gamma^{1} \wedge \kappa + \gamma^{2} \wedge \rho \\
+
V^{\kappa}_{\rho \kappa} \, \rho \wedge \kappa
+
V^{\kappa}_{\rho \zeta} \, \rho \wedge \zeta
+
V^{\kappa}_{\rho \overline{\kappa}} \, \rho \wedge \overline{\kappa} \\
+
V^{\kappa}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta}
+
V^{\kappa}_{\kappa \zeta} \, \kappa \wedge \zeta
+
V^{\kappa}_{\kappa \overline{\kappa}} \, \kappa \wedge \overline{\kappa}
+
\zeta \wedge \overline{\kappa}
,\end{gathered}$$
$$\begin{gathered}
d \zeta
=
\gamma^{3} \wedge \rho + i \, \gamma^{2} \wedge \kappa + \gamma^{1} \wedge
\zeta - \overline{\gamma^{1}} \wedge \zeta \\
+
V^{\zeta}_{\rho \kappa} \, \rho \wedge \kappa
+
V^{\zeta}_{\rho \zeta} \, \rho \wedge \zeta
+
V^{\zeta}_{\rho \overline{\kappa}} \, \rho \wedge \overline{\kappa}
+
V^{\zeta}_{\rho \overline{\zeta}} \, \rho \wedge \overline{\zeta} \\
+
V^{\zeta}_{\kappa \zeta} \, \kappa \wedge \zeta
+
V^{\zeta}_{\kappa \overline{\kappa}} \, \kappa \wedge \overline{\kappa}
+
V^{\zeta}_{\kappa \overline{\zeta}} \, \kappa \wedge \overline{\zeta}
+
V^{\zeta}_{\zeta \overline{\kappa}} \, \zeta \wedge \overline{\kappa} \\
+
V^{\zeta}_{\zeta \overline{\zeta}} \, \zeta \wedge \overline{\zeta}
.\end{gathered}$$
We now start the absorption step. We introduce: $$\widetilde{\gamma}^i := \gamma^i - z^i_{\rho} \, \rho - z^i_{\kappa}
\, \kappa - z^i_{\zeta} \, \zeta - z^i_{{\overline}{\kappa}} \, {\overline}{\kappa} - z^i_{{\overline}{\zeta}} \, {\overline}{\zeta}
.$$
The structure equations are modified accordingly:
d = \^[1]{} +\
+ ( V\_\^ - z\^1\_- [z\^1\_]{} ) . . + ( V\_\^ - z\^1\_ - [z\^1\_]{} ) . . + ( V\_ - z\^1\_ -[z\^1\_]{} ) . . + ( V\^\_ - [z\^1\_]{} - z\^1\_ ) . . ,
d = \^[1]{} + \^[2]{}\
+ ( V\_\^ - z\^2\_ + z\^\_ ) . . + ( V\_\^ - z\^2\_ ) . . + ( V\^\_ - z\^2\_ ) . . + ( V\_- z\^2\_ ) . . + ( V\_\^ - z\^1\_ ) . . + ( V\^\_ - z\^1\_ ) . . + - z\^1\_ . . ,
and
d = \^[3]{} + i \^[2]{} + \^[1]{} -\
+ ( V\^\_ - z\^3\_ + i z\_\^2 ) . . + ( V\^\_ + z\^1\_ - z\^3\_- [\^1\_]{} ) . . + ( V\^\_ - z\^3\_ ) . . + ( V\^\_ - z\^3\_ ) . . + ( V\^\_ - i z\^2\_ ) . . + ( V\^\_ - i z\^2\_ ) . . + ( V\^\_ - z\^1\_ + [z\^1\_]{} ) . . .
We thus want to solve the system of linear equations :
$$\begin{aligned}
{3}
z^1_{\kappa} + {\overline}{z^1_{{\overline}{\kappa}}} &= V^{\rho}_{\rho \kappa},
& \qquad \qquad z^1_{{\overline}{\kappa}} + {\overline}{z^1_{\kappa}} &= V^{\rho}_{\rho {\overline}{\kappa}},
& \qquad \qquad z^1_{\zeta} + {\overline}{z^1_{{\overline}{\zeta}}} &= V^{\rho}_{\rho \zeta}, \\
{\overline}{z^1_{\zeta}} + z^1_{{\overline}{\zeta}} &= V^{\rho}_{\rho {\overline}{\zeta}},
& \qquad \qquad z^2_{\kappa} - z^1_{\rho} & = V^{\kappa}_{\rho \zeta},
& \qquad \qquad z^2_{{\overline}{\kappa}} &= V^{\kappa}_{\rho {\overline}{\kappa}}, \\
z^2_{\zeta}& = V^{\kappa}_{\rho \zeta},
& \qquad \qquad z^2_{{\overline}{\zeta}} &= V^{\kappa}_{\rho {\overline}{\zeta}},
& \qquad \qquad z^1_{\zeta}& = V^{\kappa}_{\kappa \zeta}, \\
z^1_{{\overline}{\zeta}} &= 0,
& \qquad \qquad z^1_{{\overline}{\kappa}} &= V^{\kappa}_{\kappa {\overline}{\kappa}},
& \qquad \qquad z^3_{\kappa} - i \, z^2_{\rho} &= V^{\zeta}_{\rho \kappa}, \\
- z^1_{\rho} + {\overline}{z^1_{\rho}} + z^3_{\zeta} &= V^{\zeta}_{\rho \zeta},
& \qquad \qquad z^1_{\kappa} - {\overline}{z^1_{{\overline}{\kappa}}} - i\, z^2_{\zeta} &= - V^{\zeta}_{\kappa \zeta},
& \qquad \qquad i \, z^2_{{\overline}{\kappa}} &= V^{\zeta}_{\kappa {\overline}{\kappa}}, \\
z^3_{{\overline}{\kappa}}& = V^{\zeta}_{\rho {\overline}{\kappa}},
& \qquad \qquad z^3_{{\overline}{\zeta}} &= V^{\zeta}_{\rho {\overline}{\zeta}},
& \qquad \qquad i \, z^2_{{\overline}{\zeta}} &= V^{\zeta}_{\kappa {\overline}{\zeta}}, \\
z^1_{\kappa} - {\overline}{z^1_{\kappa}} &= V^{\zeta}_{\zeta {\overline}{\kappa}},
& \qquad \qquad z^1_{{\overline}{\zeta}} - {\overline}{z^1_{\zeta}} &= V^{\zeta}_{\zeta {\overline}{\zeta}}.\end{aligned}$$
This is easily done as: $$\left\{
\begin{aligned}
z^1_{\kappa} & = {\overline}{V^{\rho}_{\rho {\overline}{\kappa}}}, \\
z^1_{{\overline}{\kappa}} & = V^{\kappa}_{\kappa {\overline}{\kappa}}, \\
z^1_{\zeta} & = V^{\rho}_{\rho \zeta}, \\
z^1_{{\overline}{\zeta}} & = 0, \\
z^2_{{\overline}{\kappa}} & = V^{\kappa}_{\rho {\overline}{\kappa}}, \\
z^2_{{\overline}{\zeta}} & = V^{\kappa}_{\rho {\overline}{\zeta}}, \\
z^2_{\zeta} & = V^{\kappa}_{\rho \zeta}, \\
z^3_{{\overline}{\kappa}} & = V^{\zeta}_{\rho {\overline}{\kappa}}, \\
z^3_{{\overline}{\zeta}} & = V^{\zeta}_{\rho {\overline}{\zeta}}, \\
z^3_{\zeta} & = V^{\zeta}_{\rho \zeta} + z^1_{\rho} - z^1_{\rho}, \\
z^3_{\kappa} & = V^{\zeta}_{\rho \kappa} + i\, z^2_{\rho}, \\
z^2_{\kappa} & = V^{\kappa}_{\rho \zeta} + z^1_{\rho},
\end{aligned}
\right.$$ where $z^1_{\rho}$ and $z^2_{\rho}$ may be choosen freely. Eliminating the $z^{\bullet}_{\bullet}$ we get the following additional conditions on the $V^{\bullet}_{\bullet \bullet}$ : $$\label{eq:31}
\left\{
\begin{aligned}
V^{\rho}_{\rho {\overline}{\kappa}} &= {\overline}{V^{\rho}_{\rho \kappa}}, \\
V^{\rho}_{\rho {\overline}{\zeta}} &= {\overline}{V^{\rho}_{\rho \zeta}}, \\
V^{\rho}_{\rho \zeta} &= V^{\kappa}_{\kappa \zeta}, \\
i \, V^{\kappa}_{\rho {\overline}{\zeta}} &= V^{\zeta}_{\kappa {\overline}{\zeta}}, \\
V^{\rho}_{\rho \zeta} & = - {\overline}{V^{\zeta}_{\zeta {\overline}{\zeta}}}, \\
2 \, V^{\kappa}_{\kappa {\overline}{\kappa}} & = V^{\rho}_{\rho {\overline}{\kappa}} + V^{\zeta}_{\zeta {\overline}{\kappa}},
\end{aligned}
\right.$$ and $$\left\{
\begin{aligned}
i \, V^{\kappa}_{\rho {\overline}{\kappa}} & = V^{\zeta}_{\kappa {\overline}{\kappa}}, \\
V^{{\overline}{\zeta}}_{\kappa {\overline}{\zeta}} + V^{\zeta}_{\kappa \zeta} & = i \, V^{\kappa}_{\rho \zeta}.
\end{aligned}
\right.$$
We easily verify that the equations (\[eq:31\]) are indeed satisfied. However the remaining two equations are not and they provide two essential torsion coefficients, namely $i \, V^{\kappa}_{\rho {\overline}{\kappa}} - V^{\zeta}_{\kappa {\overline}{\kappa}}$ and $V^{{\overline}{\zeta}}_{\kappa {\overline}{\zeta}} + V^{\zeta}_{\kappa \zeta} - i \, V^{\kappa}_{\rho \zeta}$, which will give us at least one new normalization of the group coefficients. Indeed we have $$i \, V^{\kappa}_{\rho {\overline}{\kappa}} - V^{\zeta}_{\kappa {\overline}{\kappa}}= - 2i \, \frac{{{\sf d}}}{{{\sf c}}{{\overline}{{\sf c}}}} + \frac{{{\sf e}}^2}{{{\sf c}}^2}
.$$ Setting this expression to $0$, we get the normalization of the parameter ${{\sf d}}$: $${{\sf d}}= - i\, \frac{1}{2} \, \frac{{{\sf e}}^2 {{\overline}{{\sf c}}}}{ {{\sf c}}}.$$
Prologation of the $G_4$ structure
----------------------------------
We have reduced the previous $G_3$-structure to a $G_4$-structure, where $G_4$ is the four dimensional matrix Lie group whose elements are of the form : $$\begin{pmatrix}
{{\sf c}}{{\overline}{{\sf c}}}& 0 & 0 & 0 & 0 \\
-i \, {{\sf e}}{{\overline}{{\sf c}}}& {{\sf c}}& 0 & 0 & 0 \\
- \frac{i}{2} \, \frac{{{\sf e}}^2 {{\overline}{{\sf c}}}}{{{\sf c}}} & {{\sf e}}& \frac{{{\sf c}}}{{{\overline}{{\sf c}}}} & 0 & 0 \\
i \, {{\overline}{{\sf e}}}{{\sf c}}& 0 & 0 & {{\overline}{{\sf c}}}& 0 \\
\frac{i}{2} \, \frac{{{\overline}{{\sf e}}}^2 {{\sf c}}}{{{\overline}{{\sf c}}}} &0 & 0 & {{\overline}{{\sf e}}}& \frac{{{\overline}{{\sf c}}}}{{{\sf c}}} \\
\end{pmatrix}
.$$
The basis for the Maurer-Cartan forms of $G_4$ is provided by the four forms $$\delta^1 := \frac{d {{\sf c}}}{{{\sf c}}} \quad, \quad \delta^2 := i \, {{\sf e}}\frac{d {{\sf c}}}{{{\sf c}}^2}
- i\, \frac{{{\sf e}}\, d {{\overline}{{\sf c}}}}{{{\sf c}}{{\overline}{{\sf c}}}} - i\, \frac{d {{\sf e}}}{{{\sf c}}} \quad ,
\quad {\overline}{\delta^1} \quad , \quad {\overline}{\delta^2}.$$
Now we just substitute the parameter ${{\sf d}}$ by its normalization in the structure equations at the third step. We have to take into account the fact that $d {{\sf d}}$ is modified accordingly. Indeed we have: $$d {{\sf d}}= - i {{\sf e}}\frac{{{\overline}{{\sf c}}}}{{{\sf c}}} - \frac{i}{2} \,
\frac{{{\sf e}}^2 {{\overline}{{\sf c}}}}{{{\sf c}}} \left( \frac{ d {{\overline}{{\sf c}}}}{{{\overline}{{\sf c}}}} - \frac{d {{\sf c}}}{{{\sf c}}} \right)
.$$ The forms $\gamma^1$ and $\gamma^2$ are not modified as they do not involve terms in $d {{\sf d}}$, but this is not the case for $\gamma^3$ which is transformed into: $$\begin{aligned}
\gamma^3 & = \frac{d {{\sf d}}}{{{\sf c}}{{\overline}{{\sf c}}}} +
i \, \frac{{{\sf e}}}{{{\sf c}}^2} - \frac{{{\sf d}}\, d {{\sf c}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}^2}
- i \, {{\sf e}}^2 \frac{d {{\sf c}}}{{{\sf c}}^3} + \frac{{{\sf d}}\, d {{\overline}{{\sf c}}}}{{{\sf c}}{{\overline}{{\sf c}}}^2} +
i \,\frac{{{\sf e}}^2 \, d{{\overline}{{\sf c}}}}{{{\overline}{{\sf c}}}{{\sf c}}^2} \\
&= 0.\end{aligned}$$ The expressions of $d \rho$ and $d \kappa$ are thus unchanged from the expressions given by the structure equations at the third step, except the fact that we shall replace ${{\sf d}}$ by $ - \frac{i}{2} \frac{{{\sf e}}^2 {{\overline}{{\sf c}}}}{{{\sf c}}} + i \, \frac{{{\sf c}}}{{{\overline}{{\sf c}}}} \, H$ in the expression of each torsion coefficient $V_{\bullet \bullet}^{\bullet}$, which we rename $W_{\bullet \bullet}^{\bullet}$, and the fact that the forms $\gamma^1$ and $\gamma^1$ shall be replaced by the forms $\delta^1$ and $\delta^2$, that is:
d = \^1 +\
+ W\^\_ + W\^\_ + W\^\_ + W\^\_ + i ,
and
d = \^[1]{} + \^[2]{}\
+ W\^\_ + W\^\_ + W\^\_ . . + W\^\_ . . + W\^\_ + W\^\_ + .
The expression of $d \zeta$ is obtained in the same way, setting $\gamma_3$ to zero, and renaming $W^{\bullet}_{\bullet \bullet}$ the coefficients $V_{\bullet \bullet}^{\bullet}$ in which one performs the substitution $d= - i \frac{1}{2} \, \frac{{{\sf e}}^2 {{\overline}{{\sf c}}}}{{{\sf c}}}$:
d = i \_[2]{} + \_[1]{} -\
+ W\^\_ + W\^\_ + W\^\_ + W\^\_ + W\^\_ . . + W\^\_ + W\^\_ + W\^\_ + W\^\_ .
Let us now proceed with the absorption phase. We make the two substitutions: $$\begin{aligned}
& \delta^1 : = \widetilde{\delta}^1 +
w^1_{\rho} \, \rho + w^1_{\kappa} \, \kappa + w^1_{\zeta} \, \zeta +
w^1_{{\overline}{\kappa}} \, {\overline}{\kappa} + w^1_{{\overline}{\zeta}} \, {\overline}{\zeta}, \\
& \delta^2 : = \widetilde{\delta}^2 +
w^2_{\rho} \, \rho + w^2_{\kappa} \, \kappa + w^2_{\zeta} \, \zeta + w^2_{{\overline}{\kappa}} \, {\overline}{\kappa} + w^2_{{\overline}{\zeta}} \, {\overline}{\zeta},
\end{aligned}$$ in the previous equations. We get:
d = \^1 +\
+ ( W\_\^ - w\^1\_- [w\^1\_]{} ) . . + ( W\_\^ - w\^1\_ - [w\^1\_]{} ) . . + ( W\_ - w\^1\_ -[w\^1\_]{} ) . . + ( W\^\_ - [w\^1\_]{} - w\^1\_ ) . .,
d = \^[1]{} + \^[2]{}\
+ ( W\_\^ - w\^2\_ + w\^[1]{}\_ ) . . + ( W\_\^ - w\^2\_ ) . . + ( W\^\_ - w\^2\_ ) . . + ( W\_- w\^2\_ ) . . + ( W\_\^ - w\^1\_ ) . . + ( W\^\_ - w\^1\_ ) . . + - w\^1\_ . .,
and
d = i + -\
+( W\^\_ + i w\_\^2 ) . . + ( W\^\_ + w\^1\_ - [w\^1\_]{} ) . . + W\^\_ . . + W\^\_ . . + ( W\^\_ - i w\^2\_ ) . . + ( W\^\_ - i w\^2\_ ) . . + ( W\^\_ - w\^1\_ + [w\^1\_]{} ) . . .
From the last equation, we immediately see that $ W^{\zeta}_{\rho {\overline}{\kappa}}$ and $W^{\zeta}_{\rho {\overline}{\zeta}} $ are two new essential torsion coefficients. We find the remaining ones by solving the set of equations:
$$\begin{aligned}
{3}
w^1_{\kappa} + {\overline}{w^1_{{\overline}{\kappa}}} &= W^{\rho}_{\rho \kappa},
& \qquad \qquad w^1_{{\overline}{\kappa}} + {\overline}{w^1_{\kappa}} &= W^{\rho}_{\rho {\overline}{\kappa}},
& \qquad \qquad w^1_{\zeta} + {\overline}{w^1_{{\overline}{\zeta}}} &= W^{\rho}_{\rho \zeta}, \\
{\overline}{w^1_{\zeta}} + w^1_{{\overline}{\zeta}} &= W^{\rho}_{\rho {\overline}{\zeta}},
& \qquad \qquad w^2_{\kappa} - w^1_{\rho} & = W^{\kappa}_{\rho \kappa},
& \qquad \qquad w^2_{{\overline}{\kappa}} &= W^{\kappa}_{\rho {\overline}{\kappa}}, \\
w^2_{\zeta}& = W^{\kappa}_{\rho \zeta},
& \qquad \qquad w^2_{{\overline}{\zeta}} &= W^{\kappa}_{\rho {\overline}{\zeta}},
& \qquad \qquad w^1_{\zeta}& = W^{\kappa}_{\kappa \zeta}, \\
w^1_{{\overline}{\zeta}} &= 0,
& \qquad \qquad w^1_{{\overline}{\kappa}} &= W^{\kappa}_{\kappa {\overline}{\kappa}},
& \qquad \qquad - i \, w^2_{\rho} &= W^{\zeta}_{\rho \kappa}, \\
- w^1_{\rho} + {\overline}{w^1_{\rho}} &= W^{\zeta}_{\rho \zeta},
& \qquad \qquad w^1_{\kappa} - {\overline}{w^1_{{\overline}{\kappa}}} - i\, w^2_{\zeta} &= - W^{\zeta}_{\kappa \zeta},
& \qquad \qquad i \, w^2_{{\overline}{\kappa}} &= W^{\zeta}_{\kappa {\overline}{\kappa}}, \\
w^1_{\kappa} - {\overline}{w^1_{\kappa}} &= W^{\zeta}_{\zeta {\overline}{\kappa}},
& \qquad \qquad i \, w^2_{{\overline}{\zeta}} &= W^{\zeta}_{\kappa {\overline}{\zeta}},
& \qquad \qquad w^1_{{\overline}{\zeta}} - {\overline}{w^1_{\zeta}} &= W^{\zeta}_{\zeta {\overline}{\zeta}},\end{aligned}$$
which lead easily as before to: $$\label{absorption}
\left\{
\begin{aligned}
w^1_{\kappa} & = {\overline}{W^{\rho}_{\rho {\overline}{\kappa}}}, \\
w^1_{{\overline}{\kappa}} & = W^{\kappa}_{\kappa {\overline}{\kappa}}, \\
w^1_{\zeta} & = W^{\rho}_{\rho \zeta}, \\
w^1_{{\overline}{\zeta}} & = 0, \\
w^2_{{\overline}{\kappa}} & = W^{\kappa}_{\rho {\overline}{\kappa}}, \\
w^2_{{\overline}{\zeta}} & = W^{\kappa}_{\rho {\overline}{\zeta}}, \\
w^2_{\zeta} & = W^{\kappa}_{\rho \zeta}, \\
w^2_{\kappa} & = W^{\kappa}_{\rho \kappa} + w^1_{\rho}, \\
w^2_{\rho}& = W^{\zeta}_{\rho \kappa}, \\
-w^1_{\rho} + {\overline}{w^1_{\rho}} & = W_{\rho, \zeta}^{\zeta}.
\end{aligned}
\right.$$ Eliminating the $w^{\bullet}_{\bullet}$ from (\[absorption\]), we get one additionnal condition on the $W^{\bullet}_{\bullet \bullet}$ which has not yet been checked, namely that $W_{\rho, \zeta}^{\zeta}$ shall be purely imaginary. The computation of $W_{\rho, \zeta}^{\zeta}$, $ W^{\zeta}_{\rho {\overline}{\kappa}}$ and $W^{\zeta}_{\rho {\overline}{\zeta}} $ gives: $$W_{\rho, \zeta}^{\zeta} = i \, \frac{{{\sf e}}{{\overline}{{\sf e}}}}{{{\sf c}}{{\overline}{{\sf c}}}}
- \frac{i}{2} \, \frac{{{\sf e}}^2 {{\overline}{{\sf c}}}}{{{\sf c}}^3} \, {\overline}{z_2} - \frac{i}{2} \, \frac{{{\overline}{{\sf e}}}^2 {{\sf c}}}{{{\overline}{{\sf c}}}^3} \, z_2
,$$ $$W^{\zeta}_{\rho {\overline}{\kappa}}=0,$$ and $$W^{\zeta}_{\rho {\overline}{\zeta}}=0
,$$ which indicates that no further normalizations are allowed at this stage and that we must perform a prolongation of the problem. Let us introduce the modified Maurer Cartan forms of the group $G_4$, namely : $$\left\{
\begin{aligned}
& \hat{\delta}^1 : = \delta^1 - w^1_{\rho}
\, \rho - w^1_{\kappa} \, \kappa - w^1_{\zeta} \, \zeta - w^1_{{\overline}{\kappa}} \, {\overline}{\kappa} - w^1_{{\overline}{\zeta}} \, {\overline}{\zeta}, \\
& \hat{\delta}^2 : = \delta^2 - w^2_{\rho}
\, \rho - w^2_{\kappa} \, \kappa - w^2_{\zeta} \, \zeta - w^2_{{\overline}{\kappa}} \, {\overline}{\kappa} - w^2_{{\overline}{\zeta}} \, {\overline}{\zeta}
,\end{aligned}
\right.$$ where $w^i_{\rho}$, $w^i_{\kappa}$, $w^i_{\zeta}$, $w^i_{{\overline}{\kappa}}$, $w^i_{{\overline}{\zeta}}$, $i = 1 , \, 2 $, are the solutions of the system of equations (\[absorption\]) corresponding to $w^1_{\rho} + {\overline}{w^1_{\rho}} = 0,$ that is : $$\label{eq:deltatilde}
\left\{
\begin{aligned}
& \hat{\delta}^1 : = \delta^1 + \frac{1}{2} \, V^{\zeta}_{\rho \zeta} \, \rho
- {\overline}{V^{\rho}_{\rho \kappa}} \, \kappa - V^{\rho}_{\rho \zeta} \, \zeta - V^{\kappa}_{\kappa {\overline}{\kappa}} \, {\overline}{\kappa}, \\
& \hat{\delta}^2 : = \delta^2 -
V^{\zeta}_{\rho \kappa} \, \rho - \left( V^{\kappa}_{\rho \kappa} - \frac{1}{2} V^{\zeta}_{\rho \zeta} \right) \kappa
- V^{\kappa}_{\rho \zeta} \, \zeta - V^{\kappa}_{\rho {\overline}{\kappa}} \, {\overline}{\kappa} - V^{\kappa}_{\rho {\overline}{\zeta}} \, {\overline}{\zeta}
.\end{aligned}
\right.$$ We also introduce the modified Maurer Cartan forms which correspond to solutions of the system (\[absorption\]) when ${\sf Re}(w^1_{\rho})$ is not necessarily set to zero, namely : $$\label{eq:pi}
\left\{
\begin{aligned}
& \pi^1 : =\hat{ \delta}^1 - \Re(w^1_{\rho}) \, \rho, \\
& \pi^2 : = \hat{\delta}^2 - \Re(w^1_{\rho}) \, \kappa.
\end{aligned}
\right.$$ Let $P^9$ be the nine dimensional $G_4$-structure constituted by the set of all coframes of the form $(\rho, \kappa, \zeta, {\overline}{\kappa}, {\overline}{\zeta})$ on $M^5$. The initial coframe $(\rho_0, \kappa_0, \zeta_0,
{\overline}{\kappa}_0, {\overline}{\zeta}_0)$ gives a natural trivialisation $P^9 \stackrel{p} \rightarrow M^5 \times G_{4}$ which allows us to consider any differential form on $M^5$ or $G^4$ as a differential form on $P^9$. If $\omega$ is a differential form on $M^5$ for example, we just consider $p^*( pr_1^*(\omega))$, where $pr_1$ is the projection on the first component $M^5 \times G_4 \stackrel{pr_1} \rightarrow M^5$. We still denote this form by $\omega$ in the sequel. Following [@Olver-1995], we introduce the two coframes $(\rho, \kappa, \zeta, {\overline}{\kappa}, {\overline}{\zeta}, \delta^1, \delta^2, {\overline}{\delta^1}, {\overline}{\delta^2})$ and $(\rho, \kappa, \zeta, {\overline}{\kappa},
{\overline}{\zeta}, \pi^1, \pi^2, {\overline}{\pi^1}, {\overline}{\pi^2})$ on $P^9$. Setting $ {\sf t}:= - \Re(w^1_{\rho})$, they relate to each other by the relation: $$\begin{pmatrix}
\rho \\
\kappa \\
\zeta \\
{\overline}{\kappa} \\
{\overline}{\zeta} \\
\pi^1 \\
\pi^2 \\
{\overline}{\pi^1} \\
{\overline}{\pi^2} \\
\end{pmatrix}
= g_{{\sf t}}
\cdot
\begin{pmatrix}
\rho \\
\kappa \\
\zeta \\
{\overline}{\kappa} \\
{\overline}{\zeta} \\
\delta^1 \\
\delta^2 \\
{\overline}{\delta^1} \\
{\overline}{\delta^2} \\
\end{pmatrix}
,$$ where $g_{\sf t}$ is defined by $$g_{\sf t}:=\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
t & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & t & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
t & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & t & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}
.$$ The set $\left \{ g_{\sf t}, {\sf t} \in \mathbb{R} \right\}$ defines a one- dimensional Lie group, whose Maurer Cartan form is given by $d {\sf t}$, which we rename $\Lambda$ in the sequel. We now start the reduction step in the equivalence problem on $P^9$. From the definition of $\pi^1$ and $\pi^2$ as the solutions of the absorption equations $(\ref{absorption})$, the five first structure equations read as $$\label{steq}
\begin{aligned}
d \rho & = \pi^1 \wedge \rho + {\overline}{\pi^1} \wedge \rho + i \, \kappa \wedge {\overline}{\kappa}, \\
d \kappa & = \pi^1 \wedge \kappa + \pi^2 \wedge \rho + \zeta \wedge {\overline}{\kappa}, \\
d \zeta & = i \, \pi^2 \wedge \kappa + \pi^1 \wedge \zeta - {\overline}{\pi^1} \wedge \zeta, \\
d {\overline}{\kappa} & = {\overline}{\pi^1} \wedge {\overline}{\kappa} + {\overline}{\pi^2} \wedge \rho - \kappa \wedge {\overline}{\zeta}, \\
d {\overline}{\zeta} & = - i \, {\overline}{\pi^2} \wedge {\overline}{\kappa} + {\overline}{\pi^1} \wedge {\overline}{\zeta} - {\pi^1} \wedge {\overline}{\zeta}. \\
\end{aligned}$$ We could obtain the expressions of $d \pi^1$ and $d \pi^2$ by taking the exterior derivative of the previous five equations. But for now, as we have explicit expressions of $\pi^1$ and $\pi^2$ given by formulae $(\ref{eq:deltatilde})$ and $(\ref{eq:pi})$, we can perform an actual computation:
$$\begin{gathered}
d \pi^1 = d {\sf t} \wedge \rho \\
+ X^1_{\rho \kappa} \,
\rho \wedge \kappa + X^1_{\rho \zeta} \,
\rho \wedge \zeta + X^1_{\rho {\overline}{\kappa}} \, \rho \wedge {\overline}{\kappa} + X^1_{\rho {\overline}{ \zeta}} \,
\rho \wedge {\overline}{\zeta} \\
+ X^1_{\rho \pi^1} \, \rho \wedge \pi^1 + X^1_{\rho \pi^2} \, \rho \wedge \pi^2
+ X^1_{\rho {\overline}{\pi^1}} \, \rho \wedge {\overline}{\pi^1} \\ + X^1_{\rho {\overline}{\pi^2}} \, \rho \wedge {\overline}{\pi^2}
+ i \, \kappa \wedge {\overline}{\pi^2} + \zeta \wedge {\overline}{\zeta}
,\end{gathered}$$
and $$\begin{gathered}
d \pi^2 = d{\sf t} \wedge \kappa \\
+ X^2_{\rho \kappa} \,
\rho \wedge \kappa + X^2_{\kappa \zeta} \,
\kappa \wedge \zeta + X^2_{\kappa {\overline}{\kappa}} \, \kappa \wedge {\overline}{\kappa} + X^2_{\kappa {\overline}{ \zeta}} \,
\kappa \wedge {\overline}{\zeta} \\
+ X^2_{\kappa \pi^1} \, \kappa \wedge \pi^1 + X^2_{\kappa \pi^2} \, \kappa \wedge \pi^2
+ X^2_{\kappa {\overline}{\pi^1}} \, \kappa \wedge {\overline}{\pi^1} \\ + X^2_{\kappa {\overline}{\pi^2}} \, \kappa \wedge {\overline}{\pi^2}
+ \zeta \wedge {\overline}{\pi^2} + \pi^2 \wedge {\overline}{\pi^1}
.\end{gathered}$$
From these equations, we see that the absorption is straightforward and that there remain no nonconstant essential torsion term. Indeed if we define the absorbed form $\Lambda$ by: $$\Lambda = d {\sf t} - X^2_{\rho \kappa} \, \rho -
X^1_{\rho \kappa} \,\kappa - \sum_{\nu = \zeta, \pi^1, \cdots, {\overline}{\pi^2}} X^1_{\rho \nu} \, \nu,$$ the previous two equations become: $$d \pi^1 = \Lambda \wedge \rho + i \, \kappa \wedge {\overline}{\pi^2} + \zeta \wedge {\overline}{\zeta},$$ and $$d \pi^2 = \Lambda \wedge \kappa
+ \zeta \wedge {\overline}{\pi^2} + \pi^2 \wedge {\overline}{\pi^1}.$$ A straightforward computation gives the expression of $d \Lambda$: $$d \Lambda = - \pi^1 \wedge \Lambda + i\, \pi^2 \wedge {\overline}{\pi^2} - \pi^1 \wedge {\overline}{\Lambda}.$$ Let us summarize the results that we have obtained so far: The ten $1$-forms $\rho$, $\kappa$, $\zeta$, ${\overline}{\kappa}$, ${\overline}{\zeta}$, $\pi^1$, $\pi^2$, ${\overline}{\pi^1}$, ${\overline}{\pi^2}$, $\Lambda$ satisfies the structure equations given by $(\ref{eq:structure})$. This completes the proof ofh Theorem \[thm:LC\].
Torsion coefficients for the $G$-structures on ${\sf B }$
=========================================================
Coefficients $U^{\bullet}_{\bullet \bullet}$
--------------------------------------------
$$U^{\sigma}_{\sigma \rho} = {\frac {{{\sf e}}}{{{{\sf a}}}^{3}}}+{\frac {{{\sf d}}}{{{{\sf a}}}^{3}}}
,$$
$$U^{\sigma}_{\sigma \zeta} = -{\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}
,$$
$$U^{\sigma}_{\sigma {\overline}{\zeta}} = -{\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}
,$$
$$U^{\rho}_{\sigma \rho} = {\frac {{{\sf c}}{{\sf e}}}{{{{\sf a}}}^{6}}}+{\frac {{{\sf c}}{{\sf d}}}{{{{\sf a}}}^{6}}} - {\frac {i{{\sf b}}{{\sf e}}}{{{{\sf a}}}^{5}}}+{\frac {i{{\sf d}}{ {{\overline}{{\sf b}}}}}
{{{{\sf a}}}^{5}}}
,$$
$$U^{\rho}_{\sigma \zeta} = {\frac {i{{\sf e}}}{{{{\sf a}}}^{3}}}-{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}}{{{{\sf a}}}^{5}}}-{\frac {{{{\sf c}}}^{2}}
{{{{\sf a}}}^{6}}}
,$$
$$U^{\rho}_{\sigma {\overline}{\zeta}} = {\frac {i{{\sf b}}{{\sf c}}}{{{{\sf a}}}^{5}}}-{\frac {{{{\sf c}}}^{2}}{{{{\sf a}}}^{6}}}-{\frac {i{{\sf d}}}{{{{\sf a}}}^{3}}
}
,$$
$$U^{\rho}_{\rho \zeta} = {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}+{\frac {i{ {{\overline}{{\sf b}}}}}{{{{\sf a}}}^{2}}}
,$$
$$U^{\rho}_{\rho {\overline}{\zeta}} = {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}-{\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}}
,$$
$$U^{\zeta}_{\sigma \rho} = {\frac {{{{\sf d}}}^{2}}{{{{\sf a}}}^{6}}}+{\frac {i{
{{\overline}{{\sf b}}}}\,{{\sf d}}{{\sf b}}}{{{{\sf a}}}^{7}}}-{\frac {i{{\sf e}}{{{\sf b}}}^{2}}{{{{\sf a}}}^{7}}}+{\frac {{{\sf d}}{{\sf e}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\zeta}_{\sigma \zeta} = {\frac {i{{\sf b}}{{\sf e}}}{{{{\sf a}}}^{5}}}-{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf b}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf c}}{{\sf d}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\zeta}_{\sigma {\overline}{\zeta}} = -{\frac {{{\sf c}}{{\sf d}}}{{{{\sf a}}}^{6}}}-{\frac {i{{\sf d}}{{\sf b}}}{{{{\sf a}}}^{5}}}+{
\frac {i{{{\sf b}}}^{2}{{\sf c}}}{{{{\sf a}}}^{7}}}
,$$
$$U^{\zeta}_{\rho \zeta} = {\frac {{{\sf d}}}{{{{\sf a}}}^{3}}}+{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf b}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\zeta}_{\rho {\overline}{\zeta}}= {\frac {{{\sf d}}}{{{{\sf a}}}^{3}}}-{\frac {i{{{\sf b}}}^{2}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\zeta}_{\zeta \overline{\zeta}} = {\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}}
.$$
Torsion coefficients for the $G$-structures on ${\sf N}$
========================================================
Coefficients $U^{\bullet}_{\bullet \bullet}$
---------------------------------------------
$$U^{\tau}_{\tau \sigma} = \frac{{{\sf h}}}{{{\sf a}}^4} - \frac{{{\overline}{{\sf b}}}{{\sf g}}}{{{\sf a}}^6} - \frac{{{\sf b}}{{\sf g}}}{{{\sf a}}^6} + \frac{{{\sf k}}}{{{\sf a}}^6}
,$$
$$U^{\tau}_{\tau \rho} = {\frac {{{\sf b}}{{\sf f}}}{{{{\sf a}}}^{6}}}+{\frac {{ {{\overline}{{\sf b}}}}\,{{\sf f}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\tau}_{\tau \zeta} = -{\frac {{{\sf f}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\tau}_{\tau {\overline}{\zeta}} = -{\frac {{{\sf f}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\tau}_{\sigma \rho} = -{\frac {{{\sf b}}}{{{{\sf a}}}^{2}}}-{\frac {{ {{\overline}{{\sf b}}}}}{{{{\sf a}}}^{2}}}
,$$
$$U^{\sigma}_{\tau \sigma} = {\frac {{{\sf g}}{{\sf e}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf h}}{{\sf c}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf k}}{{\sf c}}}{{{{\sf a}}}^{7}}}+{
\frac {{{\sf g}}{{\sf d}}}{{{{\sf a}}}^{7}}}+{\frac {{{\sf f}}{{\sf k}}}{{{{\sf a}}}^{8}}}+{\frac {{{\sf f}}{{\sf h}}}{{{{\sf a}}}^{8}}}-{
\frac {{{\sf f}}{ {{\overline}{{\sf b}}}}\,{{\sf g}}}{{{{\sf a}}}^{10}}}-{\frac {{{\sf f}}{{\sf b}}{{\sf g}}}{{{{\sf a}}}^{10}}}
,$$
$$U^{\sigma}_{\tau \rho} =
{\frac {{ {{\overline}{{\sf b}}}}\,{{{\sf f}}}^{2}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf b}}{{{\sf f}}}^{2}}{{{{\sf a}}}^{10}}}-{
\frac {{{\sf f}}{{\sf e}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf f}}{{\sf d}}}{{{{\sf a}}}^{7}}}+{\frac {{{\sf k}}}{{{{\sf a}}}^{4}}}+{
\frac {{{\sf h}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\sigma}_{\tau \zeta} = -{\frac {{{\sf g}}}{{{{\sf a}}}^{4}}}+{\frac {{{\sf c}}{{\sf f}}}{{{{\sf a}}}^{7}}}-{\frac {{{{\sf f}}}^{2}}{{{{\sf a}}}^{8}}}
,$$
$$U^{\sigma}_{\tau {\overline}{\zeta}} = -{\frac {{{\sf g}}}{{{{\sf a}}}^{4}}}+{\frac {{{\sf c}}{{\sf f}}}{{{{\sf a}}}^{7}}}-{\frac {{{{\sf f}}}^{2}}{{{{\sf a}}}^{8}}}
,$$
$$U^{\sigma}_{\sigma \rho} = {\frac {{{\sf e}}}{{{{\sf a}}}^{3}}}+{\frac {{{\sf d}}}{{{{\sf a}}}^{3}}}-{\frac {{{\sf b}}{{\sf f}}}{{{{\sf a}}}^{6}}}-{
\frac {{ {{\overline}{{\sf b}}}}\,{{\sf f}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\sigma}_{\sigma \zeta} = -{\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}+{\frac {{{\sf f}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\sigma}_{\sigma {\overline}{\zeta}} = -{\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}+{\frac {{{\sf f}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\rho}_{\tau \sigma} = {\frac {-i{{\sf e}}{{\sf b}}{{\sf g}}}{{{{\sf a}}}^{9}}}-{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf h}}}{{{{\sf a}}}^{9}}}+{\frac {i{{\sf d}}{
{{\overline}{{\sf b}}}}\,{{\sf g}}}{{{{\sf a}}}^{9}}}+{\frac {i{{\sf b}}{{\sf c}}{{\sf k}}}{{{{\sf a}}}^{9}}}+{\frac {{{\sf e}}{{\sf g}}{{\sf c}}}{{{{\sf a}}}^{10}}}+
{\frac {{{\sf d}}{{\sf g}}{{\sf c}}}{{{{\sf a}}}^{10}}}-{\frac {i {{\sf d}}{{\sf k}}}{{{{\sf a}}}^{7}}}+{\frac {i{{\sf e}}{{\sf h}}}{{{{\sf a}}}^{7}}}-
{\frac {{{{\sf c}}}^{2}{{\sf h}}}{{{{\sf a}}}^{10}}}-{\frac {{{{\sf c}}}^{2}{{\sf k}}}{{{{\sf a}}}^{10}}}-{\frac {{{{\sf g}}}^
{2}{{\sf b}}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf g}}{{\sf k}}}{{{{\sf a}}}^{8}}}+{\frac {{{\sf g}}{{\sf h}}}{{{{\sf a}}}^{8}}}-{\frac {{
{{\sf g}}}^{2}{ {{\overline}{{\sf b}}}}}{{{{\sf a}}}^{10}}}
,$$
$$U^{\rho}_{\tau \rho} = -{\frac {{{\sf c}}{{\sf d}}{{\sf f}}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf f}}{{\sf b}}{{\sf g}}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf f}}{ {{\overline}{{\sf b}}}}\,{{\sf g}}}
{{{{\sf a}}}^{10}}}-{\frac {{{\sf c}}{{\sf e}}{{\sf f}}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf h}}{{\sf c}}}{{{{\sf a}}}^{7}}}+{\frac {{{\sf k}}{{\sf c}}}{
{{{\sf a}}}^{7}}}-{\frac {i{{\sf b}}{{\sf k}}}{{{{\sf a}}}^{6}}}+{\frac {i{{\sf b}}{{\sf e}}{{\sf f}}}{{{{\sf a}}}^{9}}}-{\frac {i{{\sf d}}{
{{\overline}{{\sf b}}}}\,{{\sf f}}}{{{{\sf a}}}^{9}}}+{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf h}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\rho}_{\tau \zeta} = {\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf f}}}{{{{\sf a}}}^{9}}}-{\frac {i{{\sf e}}{{\sf f}}}{{{{\sf a}}}^{7}}}-{\frac {i{
{{\overline}{{\sf b}}}}\,{{\sf g}}}{{{{\sf a}}}^{6}}}+{\frac {{{{\sf c}}}^{2}{{\sf f}}}{{{{\sf a}}}^{10}}}-{\frac {{{\sf g}}{{\sf c}}}{{{{\sf a}}}^{7}}}+{
\frac {i{{\sf k}}}{{{{\sf a}}}^{4}}}-{\frac {{{\sf g}}{{\sf f}}}{{{{\sf a}}}^{8}}}
,$$
$$U^{\rho}_{\tau \overline{\zeta}} = {\frac {-i{{\sf b}}{{\sf c}}{{\sf f}}}{{{{\sf a}}}^{9}}}+{\frac {i{{\sf d}}{{\sf f}}}{{{{\sf a}}}^{7}}}+{\frac {i{{\sf b}}{{\sf g}}}{{{{\sf a}}}^{6}}}
+{\frac {{{{\sf c}}}^{2}{{\sf f}}}{{{{\sf a}}}^{10}}}-{\frac {{{\sf g}}{{\sf c}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf g}}{{\sf f}}}{{{{\sf a}}}^{8
}}}-{\frac {i{{\sf h}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\rho}_{\sigma \rho} = {\frac {{{\sf c}}{{\sf e}}}{{{{\sf a}}}^{6}}}+{\frac {{{\sf c}}{{\sf d}}}{{{{\sf a}}}^{6}}}-{\frac {{{\sf g}}{{\sf b}}}{{{{\sf a}}}^{6}}}-{
\frac {{{\sf g}}{ {{\overline}{{\sf b}}}}}{{{{\sf a}}}^{6}}}-{\frac {i{{\sf b}}{{\sf e}}}{{{{\sf a}}}^{5}}}+{\frac {i{{\sf d}}{ {{\overline}{{\sf b}}}}}
{{{{\sf a}}}^{5}}}
,$$
$$U^{\rho}_{\sigma \zeta} = {\frac {i{{\sf e}}}{{{{\sf a}}}^{3}}}-{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}}{{{{\sf a}}}^{5}}}-{\frac {{{{\sf c}}}^{2}}
{{{{\sf a}}}^{6}}}+{\frac {{{\sf g}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\rho}_{\sigma {\overline}{\zeta}} = {\frac {i{{\sf b}}{{\sf c}}}{{{{\sf a}}}^{5}}}-{\frac {{{{\sf c}}}^{2}}{{{{\sf a}}}^{6}}}-{\frac {i{{\sf d}}}{{{{\sf a}}}^{3}}
}+{\frac {{{\sf g}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\rho}_{\rho \zeta} = {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}+{\frac {i{ {{\overline}{{\sf b}}}}}{{{{\sf a}}}^{2}}}
,$$
$$U^{\rho}_{\rho {\overline}{\zeta}} = {\frac {{{\sf c}}}{{{{\sf a}}}^{3}}}-{\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}}
,$$
U\^\_ = -++++-+--[ ]{}+--+ ,
$$U^{\zeta}_{\tau \rho} = {\frac {{{\sf k}}{{\sf d}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf d}}{{\sf e}}{{\sf f}}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf h}}{ {{\overline}{{\sf b}}}}\,{{\sf f}}}{{{{\sf a}}}^{10}}}+{\frac {{{\sf h}}{{\sf b}}{{\sf f}}}{{{{\sf a}}}^{10}}}-{\frac {i{{\sf k}}{{{\sf b}}}^{2}}{{{{\sf a}}}^{8}}}-{\frac {
i{ {{\overline}{{\sf b}}}}\,{{\sf d}}{{\sf f}}{{\sf b}}}{{{{\sf a}}}^{11}}}+{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf h}}{{\sf b}}}{{{{\sf a}}}^{8}}}+{\frac {i{{\sf e}}{{\sf f}}{{{\sf b}}}^{2}}{{{{\sf a}}}^{11}}}+{\frac {{{\sf d}}{{\sf h}}}{{{{\sf a}}}^{7}}}-{\frac {{{{\sf d}}}^{2}{{\sf f}}}{{{{\sf a}}}^{10}}
}
,$$
$$U^{\zeta}_{\tau \zeta} = -{\frac {{{\sf g}}{{\sf d}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf f}}{{\sf h}}}{{{{\sf a}}}^{8}}}-{\frac {i{{\sf b}}{{\sf e}}{{\sf f}}}{{{{\sf a}}}^{9}}}-{
\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf g}}{{\sf b}}}{{{{\sf a}}}^{8}}}+{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf f}}{{\sf b}}}{{{{\sf a}}}^{11}}}+{
\frac {{{\sf c}}{{\sf d}}{{\sf f}}}{{{{\sf a}}}^{10}}}+{\frac {i{{\sf b}}{{\sf k}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\zeta}_{\tau {\overline}{\zeta}}= -{\frac {{{\sf g}}{{\sf d}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf f}}{{\sf h}}}{{{{\sf a}}}^{8}}}+{\frac {i{{\sf d}}{{\sf f}}{{\sf b}}}{{{{\sf a}}}^{9}}}+{
\frac {{{\sf c}}{{\sf d}}{{\sf f}}}{{{{\sf a}}}^{10}}}-{\frac {i{{\sf h}}{{\sf b}}}{{{{\sf a}}}^{6}}}+{\frac {i{{{\sf b}}}^{2}{{\sf g}}}{{{{\sf a}}}^{
8}}}-{\frac {i{{{\sf b}}}^{2}{{\sf c}}{{\sf f}}}{{{{\sf a}}}^{11}}}
,$$
$$U^{\zeta}_{\sigma \rho} = {\frac {{{{\sf d}}}^{2}}{{{{\sf a}}}^{6}}}-{\frac {{{\sf h}}{ {{\overline}{{\sf b}}}}}{{{{\sf a}}}^{6}}}+{\frac {i{
{{\overline}{{\sf b}}}}\,{{\sf d}}{{\sf b}}}{{{{\sf a}}}^{7}}}-{\frac {i{{\sf e}}{{{\sf b}}}^{2}}{{{{\sf a}}}^{7}}}+{\frac {{{\sf d}}{{\sf e}}}{{{{\sf a}}}^{6}}}-
{\frac {{{\sf h}}{{\sf b}}}{{{{\sf a}}}^{6}}}
,$$
$$U^{\zeta}_{\sigma \zeta} = {\frac {i{{\sf b}}{{\sf e}}}{{{{\sf a}}}^{5}}}-{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf c}}{{\sf b}}}{{{{\sf a}}}^{7}}}-{\frac {{{\sf c}}{{\sf d}}}{{{{\sf a}}}^{6}}}+{\frac {{{\sf h}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\zeta}_{\sigma {\overline}{\zeta}} = -{\frac {{{\sf c}}{{\sf d}}}{{{{\sf a}}}^{6}}}-{\frac {i{{\sf d}}{{\sf b}}}{{{{\sf a}}}^{5}}}+{\frac {{{\sf h}}}{{{{\sf a}}}^{4}}}+{
\frac {i{{{\sf b}}}^{2}{{\sf c}}}{{{{\sf a}}}^{7}}}
,$$
$$U^{\zeta}_{\rho \zeta} = {\frac {{{\sf d}}}{{{{\sf a}}}^{3}}}+{\frac {i{ {{\overline}{{\sf b}}}}\,{{\sf b}}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\zeta}_{\rho {\overline}{\zeta}}= {\frac {{{\sf d}}}{{{{\sf a}}}^{3}}}-{\frac {i{{{\sf b}}}^{2}}{{{{\sf a}}}^{4}}}
,$$
$$U^{\zeta}_{\zeta \overline{\zeta}} = {\frac {i{{\sf b}}}{{{{\sf a}}}^{2}}}.$$
Torsion coefficients for the $G$-structures on ${\sf LC}$
=========================================================
Coefficients $T^{\bullet}_{\bullet \bullet}$
--------------------------------------------
$$T^{\rho}_{\rho \kappa} = i\, {\frac { \overline{\sf b}}{{\sf c}\overline{ \sf c}}} - {\frac {\sf e }{{\sf c}{ \sf f}}}\,
\frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}},$$
$$T^{\rho}_{\rho \zeta} = \frac{1}{{{\sf f}}} \, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2} },$$
$$T^{\rho}_{\rho \overline{\kappa}}
=
-i \, {\frac {\sf b}{{\sf c} \overline{ \sf c}}}
+
{\frac {\overline{\sf e}}{\overline{ \sf c}\overline{\sf f}}} \, - \frac{z_2}{1-z_2 {\overline}{z_2}},$$
$$T^{\rho}_{\rho \overline{\zeta}} = \frac {1}{{{\overline}{{\sf f}}}} \, \frac{z_2}{1-z_2 {\overline}{z_2}}
,$$
$$T^{\kappa}_{\rho \kappa}
=
\frac {{\sf e}
\overline{\sf b} }{{\sf c} \overline{ \sf c}^2{\sf f}} \, \frac{1}{1-z_2 {\overline}{z_2}}
+
\frac {{\sf d} }{{\sf c}\overline{ \sf c}{\sf f}}\, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}} +
i\, {\frac
{{\sf b}\overline{\sf b}}{{\sf c}^{2} \overline{ \sf c} ^{2}}}
-
{\frac {{\sf e} {\sf b}}{{\sf c}^{2}\overline{ \sf c}{\sf f}}}\, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}}$$
$$T^{\kappa}_{\rho \zeta}
=
-{\frac {\overline{\sf b}}{
\overline{ \sf c} ^{2}{\sf f} }}\,\frac{1}{1-z_2 {\overline}{z_2}}
,$$
$$T^{\kappa}_{\rho \overline{\kappa}}
=
\frac {\sf d }{\overline{ \sf c}
^2 {{\sf f}}}\,\frac{1}{1-z_2 {\overline}{z_2}}
-
\frac {{\sf e} {\sf b}}{{\sf c}
\overline{ {\sf c}} ^2 {{\sf f}}}\,\frac{1}{1-z_2 {\overline}{z_2}}
-
i\, {\frac {{\sf b}^{2}}{{\sf c}^{2}
\overline{ \sf c} ^{2}}}
+
{\frac {{\sf b}\overline{\sf e} }{{\sf c} \overline{ \sf c} ^{2}\overline{\sf f}}\, - \frac{z_2}{1-z_2 {\overline}{z_2}}
}
,$$
$$T^{\kappa}_{\rho \overline{\zeta}}
=
\frac { \sf b}{{\sf c}\overline{ \sf c}
\overline{\sf f}} \, \frac{z_2}{1-z_2 {\overline}{z_2}},$$
$$T^{\kappa}_{\kappa \zeta}
=
\frac{1}{{{\sf f}}} \, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}}
,$$
$$T^{\kappa}_{\kappa \overline{\kappa}}
=
{\frac {\sf e}{\overline{ \sf c} {{\sf f}}}}\,\frac{1}{1-z_2 {\overline}{z_2}}
+
i\, {\frac {\sf b}{
{\sf c}\overline{ \sf c}}}
,$$
$$T^{\kappa}_{\zeta \overline{\kappa}}
=
-\frac {{\sf c}}{\overline{ \sf c} {{\sf f}}}\,\frac{1}{1-z_2 {\overline}{z_2}}
,$$
$$T^{\zeta}_{\rho \kappa}
=
\frac {{\sf
e}^{2}\overline{\sf b}}{{\sf c}^{2}
\overline{ \sf c} ^2 {{\sf f}}} \, \frac{1}{1-z_2 {\overline}{z_2}}
+
i\, {\frac {{\sf d}\overline{\sf b}}{{\sf c}^{2}
\overline{ \sf c} ^2}}
,$$
$$T^{\zeta}_{\rho \zeta}
=
-
{\frac { {\sf e}\overline{\sf b}}{{\sf c} \overline{ \sf c} ^{2} {{\sf f}}}}\, \frac{1}{1-z_2 {\overline}{z_2}}
-
{\frac { {\sf e} {\sf b}}{{\sf c}^{2}
\overline{ \sf c}{\sf f}}} \, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}}
+
{\frac {{ \sf d} }{{\sf c}\overline{ \sf c}{\sf f}}}\, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}}
,$$
$$T^{\zeta}_{\rho \overline{\kappa}}
=
\frac { {\sf e}{ \sf d}}{{{\sf c}}{{\overline}{{\sf c}}}^2 {{\sf f}}}\, \frac{1}{1-z_2 {\overline}{z_2}}
-
{\frac {{\sf e}^{2}{\sf b}}{{\sf c}^{2
}\overline{ \sf c}^{2} {{\sf f}}}} \, \frac{1}{1-z_2 {\overline}{z_2}}
-
i\,{\frac {{\sf b}{\sf d}}{{\sf c}^{2}
\overline{ \sf c} ^{2}}}
-
{\frac {{ \sf d}\overline{\sf e}}{{\sf c} \overline{ \sf c}^{2}\overline{\sf f}} \, \frac{z_2}{1-z_2 {\overline}{z_2}}
}
,$$
$$T^{\zeta}_{\rho \overline{\zeta}}
=
{\frac {{ \sf d}}{{\sf c}\overline{ \sf c}
\overline{\sf f}}}\, \frac{z_2}{1 - z_2 {\overline}{z_2}}
,$$
$$T^{\zeta}_{\kappa \zeta}
=
+{\frac {{ \sf e}}{{\sf c}\overline{ \sf c}
\overline{\sf f}}}\, \frac{{\overline}{z_2}}{1 - z_2 {\overline}{z_2}}
,$$
$$T^{\zeta}_{\kappa \overline{\kappa}}
=
{\frac {{\sf e}^{2}}{{\sf c}\overline{ \sf c}{\sf f}}}\,\frac{1}{1-z_2 {\overline}{z_2}}
+
{
i\,\frac {\sf d}{{\sf c}\overline{ \sf c}}}
,$$
$$T^{\zeta}_{\zeta \overline{\kappa}}
=
-\frac { {{\sf e}}}{ {{\overline}{{\sf c}}}{{\sf f}}} \, \frac{1}{1-z_2 {\overline}{z_2}}
.$$
Coefficients $U^{\bullet}_{\bullet \bullet}$
--------------------------------------------
$$U^{\rho}_{\rho \kappa} = i\, {\frac {\overline{\sf b}}{{\sf c}\overline{ \sf c}}}
+
{\frac { {\sf e}\overline{ \sf c}}{{\sf c}^{2}}}\,{\overline}{z_2}
,$$
$$U^{\rho}_{\rho \zeta} = -
{\frac {\overline{ \sf c}}{{\sf c}}}\, {\overline}{z_2}
,$$
$$U^{\rho}_{\rho \overline{\kappa}}
=
-i {\frac {{\sf b}}{{\sf c}\overline{ \sf c}}}
+
{\frac {\overline{\sf e}{\sf c}}{ \overline{ \sf c}^2}} \, z_2
,$$
$$U^{\rho}_{\rho \overline{\zeta}} =
-
\frac{{\sf c}}{\overline{ \sf c}} \, z_2
,$$
$$U^{\kappa}_{\rho \kappa}
=
-
{\frac { {\sf e}
\overline{\sf b}}{{\sf c}^{2}\overline{ \sf c}}}
-
{\frac {{ \sf d}}{{\sf c}^
{2}}} \, {\overline}{z_2}
+
i\, {\frac {{\sf b}\overline{\sf b}}{{\sf c}^{2}
\overline{ \sf c}^2}}
+
{\frac {{\sf b} {\sf e}}{{\sf c}
^{3}}}\,{\overline}{z_2}
,$$
$$U^{\kappa}_{\rho \zeta}
= \frac{{{\overline}{{\sf b}}}}{{{\sf c}}{{\overline}{{\sf c}}}}
,$$
$$U^{\kappa}_{\rho \overline{\kappa}}
=
-{\frac {\sf d}{{\sf c}\overline{ \sf c}}}
+
{\frac { {\sf e}{\sf b}}{{\sf c}^{2}\overline{ \sf c}}}
-
i\, {\frac {
{\sf b}^{2}}{{\sf c}^{2} \overline{ \sf c}^2}}
+
{\frac {{\sf b}
\overline{\sf e}}{
\overline{ \sf c} ^{3}}}\, z_2
,$$
$$U^{\kappa}_{\rho \overline{\zeta}}
=
-{\frac {\sf b}{
\overline{ \sf c}^{2}}}\, z_2
,$$
$$U^{\kappa}_{\kappa \zeta}
=
-{\frac {\overline{ \sf c}}{{\sf c}}}\,{\overline}{z_2}
,$$
$$U^{\kappa}_{\kappa \overline{\kappa}}
=
-{\frac {\sf e}{\sf c}}+i\, {\frac {{\sf b}}{{\sf c}\overline{ \sf c}}}
,$$
$$U^{\zeta}_{\rho \kappa}
=
-
{\frac { {\sf e}\overline{\sf d}
}{{\sf c} \overline{ \sf c}
^{2}}}\, z_2
+
{\frac { {{\overline}{{\sf b}}}{\sf e}\overline{\sf e}} {
\overline{ \sf c} ^{3}{\sf c}}} \, z_2
-
{\frac {
{\sf e}^{2}\overline{\sf b}}{\overline{ \sf c}{\sf c}^{3}}}
+
i\, {\frac {{ \sf d}\overline{\sf b}}{{\sf c}
^{2} \overline{ \sf c}^2}}
,$$
$$U^{\zeta}_{\rho \zeta}
=
{\frac {\overline{\sf d}}{
\overline{ \sf c}^2}} \, z_2
-
{
\frac {\overline{\sf e} \overline{\sf b}}{ \overline{ \sf c}^{3}}}\, z_2
+
\frac{{{\sf e}}{{\overline}{{\sf b}}}}{{{\sf c}}^2 {{\overline}{{\sf c}}}}
+
{\frac {{\sf b}{\sf e}}{{\sf c}
^{3}}}\, {\overline}{z_2}
-
{\frac {{ \sf d}}{{\sf c}
^{2}}}\, {\overline}{z_2}
,$$
$$U^{\zeta}_{\rho \overline{\kappa}}
=
2\,{\frac {\overline{\sf e}{ \sf d}}{
\overline{ \sf c}^{3}}} \, z_2
-
{\frac { {\sf e}\overline{\sf e} {\sf b}}{
\overline{ \sf c}^{3}{\sf c}}} \, z_2
-
{\frac { {\sf e}{\sf d}}{{\sf c}^{2}\overline{ \sf c}}}
+
{\frac {{\sf e}^{2}{\sf b}}{\overline{ \sf c}{\sf c}
^{3}}}
-
i\, {\frac {{\sf d}{\sf b}}{{\sf c}^{2} \overline{ \sf c}^2}}
,$$
$$U^{\zeta}_{\rho \overline{\zeta}}
=
-2\,{\frac { {\sf d} }{
\overline{ \sf c}^{2}}} \, z_2
+
{\frac {{\sf e} {\sf b}
}{{\sf c } \overline{ \sf c}^{2}}} \, z_2
,$$
$$U^{\zeta}_{\kappa \zeta}
=
-
{\frac { {\sf e}\overline{ \sf c}}{{\sf c}^{2}}} \, {\overline}{z_2}
,$$
$$U^{\zeta}_{\kappa \overline{\kappa}}
=
{\frac { {\sf e}\overline{\sf e} }{ \overline{ \sf c}^2
}} \, z_2
-
{\frac {{\sf e}^{2}}{{\sf c}^{2}
}}
+
i\, {\frac {\sf d}{{\sf c}\overline{ \sf c}}}
,$$
$$U^{\zeta}_{\kappa \overline{\zeta}}
=
- {\frac { {\sf e}}{\overline{ \sf c}}}\, z_2
,$$
$$U^{\zeta}_{\zeta \overline{\kappa}}
=
- {\frac {\overline{\sf e}{\sf c}}{
\overline{ \sf c}^2}} \, z_2
+
{\frac {\sf e}{\sf c}}
,$$
$$U^{\zeta}_{\zeta \overline{\zeta}}
=
{\frac {{\sf c}}{\overline{ \sf c}}}\,z_2
.$$
Coefficients $V^{\bullet}_{\bullet \bullet}$
---------------------------------------------
$$V^{\rho}_{\rho \kappa}
=
-
\frac {\overline {\sf e}} {\overline{\sf c}}
+
{\frac {{\sf e} \overline{\sf c} } {{{\sf c}}^{2}
}} \, {\overline}{z_2}
,$$
$$V^{\rho}_{\rho \zeta}
=
-
{\frac {{\overline{\sf c}} }{{\sf c} } \, {\overline}{z_2}}
,$$
$$V^{\rho}_{\rho \overline{\kappa}}
=
-{\frac {{\sf e}} {{\sf c}}}
+
{\frac {{\overline{\sf e}} {\sf c} }{{{\overline{\sf c}
}}^{2}}} \, z_2
,$$
$$V^{\rho}_{\rho \overline{\zeta}}
=
-\frac{\sf c}{\overline{\sf c}}
\, z_2
,$$
$$V^{\kappa}_{\rho \kappa}
=
-
{\frac {{ \sf d}}{{\sf c}^
{2}}} \, {\overline}{z_2}
- i \,
{\frac {{{\sf e}}^2 {{\overline}{{\sf c}}}}{{\sf c}
^{3}}}\,{\overline}{z_2}$$
$$V^{\kappa}_{\rho \zeta}
=
i \, \frac{\overline{\sf e}}{\overline{\sf c}}
,$$
$$V^{\kappa}_{\rho \overline{\kappa}}
=
-
{\frac {{\sf d}}{{\sf
c} {{{\overline}{{\sf c}}}}}}
-
i \, \, \frac{{\sf e} \overline{\sf e} }{\overline{\sf c}^{2}} \, z_2
,$$
$$V^{\kappa}_{\rho \overline{\zeta}}
=
i \, \frac {{\sf e} }{\overline{\sf c}} \, z_2
,$$
$$V^{\kappa}_{\kappa \zeta}
=
- \frac{\overline{\sf c}}{\sf c} \, {\overline}{z_2}
,$$
$$V^{\kappa}_{\kappa \overline{\kappa}}
=0
,$$
$$V^{\zeta}_{\rho \kappa}
=
- \frac{{{\overline}{{\sf d}}}{{\sf e}}}{{{\sf c}}{{\overline}{{\sf c}}}^2} \, z_2
-
{\frac {{\sf d}{\overline{\sf e}}}{{\sf c}
{{\overline{\sf c}}}^{2}}}
+
i \, \frac {{\sf e} \overline{\sf e}^{2}} {\overline{\sf c}^{3}} \, z_2
-
i \, \frac{{\sf e}^{2} \overline{\sf e}} {{\sf c}^{2}{\overline{\sf c}}}
,$$
$$V^{\zeta}_{\rho \zeta}
=
-\frac{{{\overline}{{\sf d}}}}{{{\overline}{{\sf c}}}^2} \, z_2
+
i \, {\frac {{\sf e}\,{\overline{\sf e}}}{{\sf c}
\,{\overline{\sf c}}}}
-
i \, \frac{{{\overline}{{\sf c}}}{{\sf e}}^2} {{{\sf c}}^3} \, {\overline}{z_2}
-
i \, \frac{{{\sf c}}{{\overline}{{\sf e}}}^2}{{{\overline}{{\sf c}}}^3} \, z_2
-
\frac{\sf d}{{\sf c}^{2}} \, {\overline}{z_2}
,$$
$$V^{\zeta}_{\rho \overline{\kappa}}
=
2 \, \frac{{\sf d} \overline{\sf e}} {{\overline{\sf c}}^{3}} \, z_2
+
i \, \frac{\overline{\sf e} {\sf e}^{2}} {
\overline{\sf c}^{2}{\sf c}} \, z_2
-
2 \, {\frac {{\sf e} {\sf d}}{{\overline{\sf c}} {{\sf c
}}^{2}}}
-
i \, {\frac {{{\sf e}}^{3}}{{{\sf c}}^{3}}}
,$$
$$V^{\zeta}_{\rho \overline{\zeta}}
=
-2 \, {\frac {{\sf d}}{{{
\overline{\sf c}}}^{2}}} \, z_2
-
i \, {\frac {{{\sf e}}^{2}
}{{\sf c}\,{\overline{\sf c}}}} \, z_2
,$$
$$V^{\zeta}_{\kappa \zeta}
=
-
{\frac {{\sf e}\,{\overline{\sf c}}}{{{\sf c}}^{2}}} \, {\overline}{z_2}
,$$
$$V^{\zeta}_{\kappa \overline{\kappa}}
=
{\frac {{\sf e}{\overline{\sf e}}}
{{{\overline{\sf c}}}^{2}}} \, z_2
-
{\frac {{{\sf e}}^{2}}{{{
\sf c}}^{2}}}
+
i \, {\frac {{\sf d}}{{\sf c}\,{\overline{\sf c}}}}
,$$
$$V^{\zeta}_{\kappa \overline{\zeta}}
=
- \frac{\sf e}{{\overline{\sf c}}} \, z_2
,$$
$$V^{\zeta}_{\zeta \overline{\kappa}}
=
-\frac{ \overline{\sf e} {\sf c}} { \overline{\sf c}^{2} } \, z_2
+
{\frac {{\sf e}}{{\sf c}}}
,$$
$$V^{\zeta}_{\zeta \overline{\zeta}}
=
{\frac{\sf c}{\overline{\sf c}}} \, z_2
.$$
Coefficients $X^{\bullet}_{\bullet \bullet}$
--------------------------------------------
$$X^1_{\rho \kappa}= - \frac{1}{2} \, {\sf t} \, \frac{{{\overline}{{\sf c}}}{{\sf e}}}{{{\sf c}}} \, {\overline}{z_2}
- \frac{3}{8} \, i \, \frac{{{\sf e}}^2 {{\overline}{{\sf e}}}}{{{\sf c}}^3} \, {\overline}{z_2} + \frac{1}{2} \, {\sf t} \frac{{{\overline}{{\sf e}}}}{{{\overline}{{\sf c}}}}
+ \frac{1}{8} \, i \, \frac{{{\sf e}}{{\overline}{{\sf e}}}^2}{{{\sf c}}{{\overline}{{\sf c}}}^2}
\, z_2 {\overline}{z_2} + \frac{1}{8} \, i \, \frac{{{\sf e}}^3 {{\overline}{{\sf c}}}^2}{{{\sf c}}^5} \, {\overline}{z_2}^2 +
\frac{1}{4} \, i \, \frac{{{\overline}{{\sf e}}}^2 {{\sf e}}}{{{\sf c}}{{\overline}{{\sf c}}}^2}
,$$
$$X^1_{\rho \zeta} = - \frac{1}{4} \, i \, \frac{{{\overline}{{\sf e}}}^2}{{{\overline}{{\sf c}}}^2} \, + \frac{1}{2} \, i \, \frac{{{\sf e}}{{\overline}{{\sf e}}}}{{{\sf c}}^2} \, {\overline}{z_2} -
\frac{1}{4} \, i \, \frac{{{\overline}{{\sf c}}}^2 {{\sf e}}^2}{{{\sf c}}^4} \, {\overline}{z_2}^2
,$$
$$X^1_{\rho {\overline}{\kappa}} = {\overline}{X^1_{\rho \kappa}}
,$$
$$X^1_{\rho {\overline}{\zeta}} = {\overline}{X^1_{\rho \zeta}}
,$$
$$X^1_{\rho \pi^1} = - {\sf t}
,$$
$$X^1_{\rho \pi^2} = \frac{1}{2} \, \frac{{{\overline}{{\sf e}}}}{{{\overline}{{\sf c}}}} + \frac{1}{2} \, \frac{{{\sf e}}{{\overline}{{\sf c}}}}{{{\sf c}}^2} \, {\overline}{z_2}
,$$
$$X^1_{\rho {\overline}{\pi^1}} = - {\sf t}
,$$
$$X^1_{\rho {\overline}{\pi^2}} = {\overline}{X^1_{\rho \pi^2}}
,$$
$$X^2_{\rho \kappa} = - \frac{1}{4} \, \frac{{{\sf e}}{{\overline}{{\sf e}}}^3}{{{\overline}{{\sf c}}}^4} \, z_2 - \frac{1}{4} \frac{{{\sf e}}^3 {{\overline}{{\sf e}}}}{{{\overline}{{\sf c}}}^4} \, {\overline}{z_2}
+ \frac{1}{8} \, \, \frac{{{\sf e}}^2 {{\overline}{{\sf e}}}^2}{{{\sf c}}^2 {{\overline}{{\sf c}}}^2}
\, {\overline}{z_2} + \frac{1}{4} \, \frac{{{\sf e}}^2 {{\overline}{{\sf e}}}^2}{{{\sf c}}^2 {{\overline}{{\sf c}}}^2} + \frac{1}{16} \, \frac{{{\overline}{{\sf e}}}^4 {{\sf c}}^2}{{{\overline}{{\sf c}}}^6} \, z_2^2
+ \frac{1}{16} \, \frac{{{\sf e}}^4 {{\overline}{{\sf c}}}^2}{{{\sf c}}^6} \, {\overline}{z_2}^2 - {\sf t}^2,$$
$$X^2_{\kappa \nu } = X^1_{\rho \nu} \, \, \, \, \, \, \, \, \, \, \, \text{ for} \, \, \, \nu= \zeta, \pi^1, \cdots, {\overline}{\pi^2}.$$
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|
=1
Introduction
============
The Jack polynomials form a parametrized basis of symmetric polynomials. A special case of these consists of the Schur polynomials, important in the character theory of the symmetric groups. By means of a commutative algebra of differential-difference operators the theory was extended to nonsymmetric Jack polynomials, again a parametrized basis but now for all polynomials in $N$ variables. These polynomials are orthogonal for several different inner products, and in each case they are simultaneous eigenfunctions of a commutative set of self-adjoint operators. These inner products are invariant under permutations of the coordinates, that is, the symmetric group. One of these inner products is that of $L^{2}\big(\mathbb{T}^{N},K_{\kappa}(x) \mathrm{d}m(x)\big)$, where $$\begin{gathered}
\mathbb{T}^{N} :=\big\{ x\in\mathbb{C}^{N}\colon \vert x_{j} \vert =1,\, 1\leq j\leq N\big\} ,\\
\mathrm{d}m(x) =(2\pi) ^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N},
\qquad x_{j}=\exp( \mathrm{i}\theta _{j}) , \qquad -\pi<\theta_{j}\leq\pi, \qquad 1\leq j\leq N,\\
K_{\kappa}(x) =\prod_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert ^{2\kappa}, \qquad \kappa>-\frac{1}{N};\end{gathered}$$ defining the $N$-torus, the Haar measure on the torus, and the weight function respectively. Beerends and Opdam [@Beerends/Opdam1993] discovered this orthogonality property of symmetric Jack polynomials. Opdam [@Opdam1995] established orthogonality structures on the torus for trigonometric polynomials associated with Weyl groups; the nonsymmetric Jack polynomials form a special case.
Griffeth [@Griffeth2010] constructed vector-valued Jack polynomials for the family $G(n,p,N) $ of complex reflection groups. These are the groups of permutation matrices (exactly one nonzero entry in each row and each column) whose nonzero entries are $n^{\rm th}$ roots of unity and the product of these entries is a $(n/p)^{\rm th}$ root of unity. The symmetric groups and the hyperoctahedral groups are the special cases $G(1,1,N) $ and $G(2,1,N)$ respectively. The term “vector-valued” means that the polynomials take values in irreducible modules of the underlying group, and the action of the group is on the range as well as the domain of the polynomials. The author [@Dunkl2010] together with Luque [@Dunkl/Luque2011] investigated the symmetric group case more intensively. The basic setup is an irreducible representation of the symmetric group, specified by a partition $\tau$ of $N$, and a parameter $\kappa$ restricted to an interval determined by the partition, namely $-1/h_{\tau}<\kappa<1/h_{\tau}$ where $h_{\tau}$ is the maximum hook-length of the partition $\tau$. More recently [@Dunkl2016] we showed that there does exist a positive matrix measure on the torus for which the nonsymmetric vector-valued Jack polynomials (henceforth NSJP’s) form an orthogonal set. The proof depends on a matrix-version of Bochner’s theorem about the relation between positive measures on a compact abelian group and positive-definite functions on the dual group, which is a discrete abelian group. In the present situation the torus is the compact (multiplicative) group and the dual is $\mathbb{Z}^{N}$. By using known properties of the NSJP’s we produced a positive-definite matrix function on $\mathbb{Z}^{N}$ and this implied the existence of the desired orthogonality measure. Additionally we showed that the part of the measure supported by $\mathbb{T}_{\rm reg}^{N}:=\mathbb{T}^{N}\backslash\bigcup_{i<j} \{ x\colon x_{i}=x_{j} \} $ is absolutely continuous with respect to the Haar measure $\mathrm{d}m$ and satisfies a first-order differential system. In this paper we complete the description of the measure by proving there is no singular part. The idea is to use the functional equations satisfied by the inner product to establish a correspondence to the differential system. The main reason for the argument being so complicated is that the “obvious” integration-by-parts argument which works smoothly for the scalar case with $\kappa>1$ has great difficulty with the singularities of the measure of the form $\vert x_{i}-x_{j}\vert ^{-2\vert \kappa\vert }$. We use a Cauchy principal-value argument based on a weak continuity condition across the faces $\{x\colon x_{i}=x_{j}\} $ (as an over-simplified one-dimensional example consider the integral $\int_{-1}^{1}\frac{\mathrm{d}}{\mathrm{d}x}f(x) \mathrm{d}x$ with $f(x) =\vert 2x+x^{2}\vert ^{-1/4}$: the integral is divergent but the principal value $\lim\limits_{\varepsilon\rightarrow0_{+}}\big\{ \int_{-1}^{-\varepsilon}+\int_{\varepsilon}^{1}\big\} f^{\prime}(x) \mathrm{d}x=f(1) -f(-1) +\lim\limits_{\varepsilon \rightarrow0_{+}}\{ f(-\varepsilon) -f(( \varepsilon))\} $ and $f(-\varepsilon) -f(\varepsilon) =O\big( \varepsilon^{3/4}\big) $ hence the limit exists).
The differential system is a two-sided version of a Knizhnik–Zamolodchikov equation (see [@Felder/Veselov2007]) modified to have solutions homogeneous of degree zero, that is, constant on circles $\{( ux_{1},\ldots$, $ux_{N}) \colon \vert u\vert =1\} $. The purpose of the latter condition is to allow solutions analytic on connected components of $\mathbb{T}_{\rm reg}^{N}$. Denote the degree of $\tau$ by $n_{\tau}$. The solutions of the differential system are locally analytic $n_{\tau}\times n_{\tau}$ matrix functions with initial condition given by a constant matrix. That is, the solution space is of dimension $n_{\tau}^{2}$ but only one solution can provide the desired weight function. Part of the analysis deals with conditions specifying this solution – they turn out to be commutation relations involving certain group elements. In the subsequent discussion it is shown that the weight function property holds for a very small interval of $\kappa$ values if these relations are satisfied. This is combined with the existence theorem of the positive-definite matrix measure to finally demonstrate that the measure has no singular part for any $\kappa$ in $-1/h_{\tau}<\kappa<1/h_{\tau}$.
In a subsequent development [@Dunkl2017] it is shown that the square root of the matrix weight function multiplied by vector-valued symmetric Jack polynomials provides novel wavefunctions of the Calogero–Sutherland quantum mechanical model of identical particles on a circle with $1/r^{2}$ interactions.
Here is an outline of the contents of the individual sections:
- Section \[mods\]: a short description of the representation of the symmetric group associated to a partition; the definition of Dunkl operators for vector-valued polynomials and the definition of nonsymmetric Jack polynomials (NSJP’s) as simultaneous eigenvectors of a commutative set of operators; and the Hermitian form given by an integral over the torus, for which the NSJP’s form an orthogonal basis.
- Section \[difsys\]: the definition of the linear system of differential equations which will be demonstrated to have a unique matrix solution $L(x) $ such that $L(x) ^{\ast}L(x) \mathrm{d}m(x) $ is the weight function for the Hermitian form; the proof that the system is Frobenius integrable and the analyticity and monodromy properties of the solutions on the torus.
- Section \[byparts\]: the use of the differential equation to relate the Hermitian form to $L(x) ^{\ast}L(x) $ by means of integration by parts; the result of this is to isolate the role of the singularities in the process of proving the orthogonality of the NSJP’s with respect to $L^{\ast}L\mathrm{d}m$.
- Section \[locps\]: deriving power series expansions of $L(x) $ near the singular set $\bigcup_{i<j}\big\{ x\in\mathbb{T}^{N}\colon$ $x_{i}=x_{j}\big\}$, in particular near the set $\{x\colon x_{N-1}=x_{N}\} $; description of commutation properties of the coefficients with respect to the reflection $(N-1,N) $; the behavior of $L$ across the mirror $\{ x\colon x_{N-1}=x_{N}\} $.
- Section \[bnds\]: the derivation of global bounds on $L(x)$ and local bounds on the coefficients of the power series, needed to analyze convergence properties of the integration by parts.
- Section \[suffco\]: the proof of a sufficient condition for the validity of the Hermitian form; the condition is partly that $\kappa$ lies in a small interval around $0$ and that the boundary value of $L(x)$ satisfies a commutativity condition; the proof involves very detailed analysis of bounds on $L$, since the local bounds have to be integrated over the entire torus.
- Section \[orthmu\]: further analysis of the orthogonality measure constructed in [@Dunkl2016], in particular the proof of the formal differential system satisfied by the Fourier–Stieltjes (Laurent) series of the measure; this is used to show that the measure has no singular part on the open faces, such as $$\begin{gathered}
\big\{ \big( e^{\mathrm{i}\theta_{1}},e^{\mathrm{i}\theta_{2}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\theta_{N-1}}\big)\colon \theta_{1}<\theta_{2}<\cdots<\theta_{N-2}<\theta_{N-1}<\theta_{1}+2\pi\big\} ;\end{gathered}$$ in turn this property is shown to imply the validity of the sufficient condition set up in Section \[suffco\].
- Section \[anlcmat\]: analyticity properties of the solutions of matrix equations with analytic coefficients; the results are used to extend the validity of the Hermitian form to the desired interval $-1/h_{\tau}<\kappa<1/h_{\tau}$ from the smaller interval found in Section \[suffco\].
Modules of the symmetric group {#mods}
==============================
The *symmetric group* $\mathcal{S}_{N}$, the set of permutations of $\{1,2,\ldots,N\} $, acts on $\mathbb{C}^{N}$ by permutation of coordinates. For $\alpha\in\mathbb{Z}^{N}$ the norm is $\vert \alpha\vert :=\sum\limits_{i=1}^{N}\vert \alpha_{i}\vert $ and the monomial is $x^{\alpha}:= \prod\limits_{i=1}^{N} x_{i}^{\alpha_{i}}$. Denote $\mathbb{N}_{0}:=\{0,1,2,\ldots\}$. The space of polynomials $\mathcal{P}:= \operatorname{span}_{\mathbb{C}}\big\{ x^{\alpha}\colon \alpha \in\mathbb{N}_{0}^{N}\big\} $. Elements of $\operatorname{span}_{\mathbb{C}}\big\{ x^{\alpha}\colon \alpha\in\mathbb{Z}^{N}\big\} $ are called *Laurent* polynomials. The action of $\mathcal{S}_{N}$ is extended to polynomials by $wp(x) =p(xw) $ where $( xw) _{i}=x_{w(i) }$ (consider $x$ as a row vector and $w$ as a permutation matrix, $[w] _{ij}=\delta_{i,w(j)}$, then $xw=x[w] $). This is a representation of $\mathcal{S}_{N}$, that is, $w_{1}(w_{2}p) (x) =(w_{2}p) (xw_{1}) =p(xw_{1}w_{2}) =(w_{1}w_{2}) p(x) $ for all $w_{1},w_{2}\in\mathcal{S}_{N}$.
Furthermore $\mathcal{S}_{N}$ is generated by reflections in the mirrors $\{x\colon x_{i}=x_{j}\} $ for $1\leq i<j\leq N$. These are *transpositions*, denoted by $(i,j)$, so that $x(i,j) $ denotes the result of interchanging $x_{i}$ and $x_{j}$. Define the $\mathcal{S}_{N}$-action on $\alpha\in\mathbb{Z}^{N}$ so that $(xw) ^{\alpha}=x^{w\alpha}$ $$\begin{gathered}
(xw) ^{\alpha}=\prod_{i=1}^{N}x_{w(i) }^{\alpha_{i}}=\prod_{j=1}^{N}x_{j}^{\alpha_{w^{-1}(j) }},\end{gathered}$$ that is $(w\alpha) _{i}=\alpha_{w^{-1}(i) }$ (take $\alpha$ as a column vector, then $w\alpha=[w] \alpha$).
The *simple reflections* $s_{i}:=(i,i+1)$, $1\leq i\leq N-1$, generate $\mathcal{S}_{N}$. They are the key devices for applying inductive methods, and satisfy the *braid* relations: $$\begin{gathered}
s_{i}s_{j} =s_{j}s_{i},\qquad \vert i-j\vert \geq2;\\
s_{i}s_{i+1}s_{i} =s_{i+1}s_{i}s_{i+1}.\end{gathered}$$
We consider the situation where the group $\mathcal{S}_{N}$ acts on the range as well as on the domain of the polynomials. We use vector spaces, called $\mathcal{S}_{N}$-modules, on which $\mathcal{S}_{N}$ has an irreducible unitary (orthogonal) representation: $\tau\colon \mathcal{S}_{N}\rightarrow
O_{m}(\mathbb{R}) $ $\big(\tau(w) ^{-1}=\tau\big( w^{-1}\big) =\tau(w) ^{T}\big)$. See James and Kerber [@James/Kerber1981] for representation theory, including a modern discussion of Young’s methods.
Denote the set of *partitions* $$\begin{gathered}
\mathbb{N}_{0}^{N,+}:=\big\{ \lambda\in\mathbb{N}_{0}^{N}\colon \lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{N}\big\} .\end{gathered}$$ We identify $\tau$ with a partition of $N$ given the same label, that is $\tau\in\mathbb{N}_{0}^{N,+}$ and $\vert\tau\vert =N$. The length of $\tau$ is $\ell(\tau) :=\max \{ i\colon \tau_{i}>0 \} $. There is a Ferrers diagram of shape $\tau$ (also given the same label), with boxes at points $(i,j) $ with $1\leq i\leq\ell(\tau) $ and $1\leq j\leq\tau_{i}$. A *tableau* of shape $\tau$ is a filling of the boxes with numbers, and a *reverse standard Young tableau* (RSYT) is a filling with the numbers $\{1,2,\ldots,N\} $ so that the entries decrease in each row and each column. We exclude the one-dimensional representations corresponding to one-row $(N) $ or one-column $(1,1,\ldots,1) $ partitions (the trivial and determinant representations, respectively). We need the important quantity $h_{\tau}:=\tau_{1}+\ell(\tau) -1$, the maximum hook-length of the diagram (the *hook-length* of the node $(i,j) \in \tau$ is defined to be $\tau_{i}-j+\# \{ k\colon i<k\leq\ell(\tau) \&j\leq\tau_{k} \} +1$). Denote the set of RSYT’s of shape $\tau$ by $\mathcal{Y}(\tau)$ and let $$\begin{gathered}
V_{\tau}=\operatorname{span} \{ T\colon T\in\mathcal{Y}(\tau)\}\end{gathered}$$ (the field is $\mathbb{C}(\kappa) $) with orthogonal basis $\mathcal{Y}(\tau) $. For $1\leq i\leq N$ and $T\in \mathcal{Y}(\tau) $ the entry $i$ is at coordinates $( rw(i,T) ,cm(i,T)) $ and the *content* is $c(i,T) :=cm(i,T) -rw(i,T) $. Each $T\in\mathcal{Y}(\tau) $ is uniquely determined by its *content vector* $[ c(i,T)] _{i=1}^{N}$. Let $S_{1}(\tau) =\sum\limits_{i=1}^{N}c(i,T) $ (this sum depends only on $\tau$) and $\gamma:=S_{1}(\tau)/N$. The $\mathcal{S}_{N}$-invariant inner product on $V_{\tau}$ is defined by $$\begin{gathered}
\langle T,T^{\prime}\rangle _{0}:=\delta_{T,T^{\prime}}\times \prod_{\substack{1\leq i<j\leq N,\\ c(i,T) \leq c(j,T) -2}}\left( 1-\frac{1}{( c(i,T) -c(j,T)) ^{2}}\right) , \qquad T,T^{\prime}\in\mathcal{Y} (\tau) .\end{gathered}$$ It is unique up to multiplication by a constant.
The *Jucys–Murphy* elements $\sum\limits_{j=i+1}^{N}(i,j)$ satisfy $\sum\limits_{j=i+1}^{N}\tau((i,j)) T=c(i,T) T$ and thus the central element $\sum\limits_{1\leq i<j\leq N}(i,j) $ satisfies $\sum\limits_{1\leq i<j\leq N} \tau((i,j)) T=S_{1}(\tau) T$ for each $T\in\mathcal{Y}(\tau) $. The basis is ordered such that the vectors $T$ with $c(N-1,T) =-1$ appear first (that is, $cm(N-1,T) =1$, $rw(N-1,T) =2$). This results in the matrix representation of $\tau((N-1,N)) $ being $$\begin{gathered}
\left[
\begin{matrix}
-I_{m_{\tau}} & O\\
O & I_{n_{\tau}-m_{\tau}}
\end{matrix}
\right] ,\end{gathered}$$ where $n_{\tau}:=\dim V_{\tau}=\#\mathcal{Y}(\tau) $ and $m_{\tau}$ is given by $\operatorname{tr}( \tau( (N-1,N))) =n_{\tau}-2m_{\tau}$. From the sum $\sum\limits_{i<j}\tau ((i,j)) =S_{1}(\tau) I$ it follows that $\binom{N}{2}\operatorname{tr}( \tau((N-1,N))) =S_{1}(\tau) n_{\tau}$ and $m_{\tau}= n_{\tau}\big( \frac{1}{2}-\frac{S_{1}(\tau) }{N(N-1) }\big)$. (The transpositions are conjugate to each other implying the traces are equal.)
Jack polynomials
----------------
The main concerns of this paper are measures and matrix functions on the torus associated to $\mathcal{P}_{\tau}:=\mathcal{P}\otimes V_{\tau}$, the space of $V_{\tau}$-valued polynomials, which is equipped with the $\mathcal{S}_{N}$ action: $$\begin{gathered}
w\left( x^{\alpha}\otimes T\right) =(xw) ^{\alpha}\otimes\tau(w) T, \qquad \alpha\in\mathbb{N}_{0}^{N}, \qquad T\in \mathcal{Y}(\tau) ,\\
wp(x) =\tau(w) p(xw), \qquad p\in\mathcal{P}_{\tau},\end{gathered}$$ extended by linearity. There is a parameter $\kappa$ which may be generic/transcendental or complex.
The *Dunkl* and *Cherednik–Dunkl* operators are ($1\leq i\leq N$, $p\in\mathcal{P}_{\tau}$) $$\begin{gathered}
\mathcal{D}_{i}p(x) :=\frac{\partial}{\partial x_{i}}p(x) +\kappa\sum_{j\neq i}\tau((i,j)) \frac{p(x) -p(x(i,j))}{x_{i}-x_{j}},\\
\mathcal{U}_{i}p(x) :=\mathcal{D}_{i}( x_{i}p(x)) -\kappa\sum_{j=1}^{i-1}\tau( (i,j)) p(x(i,j)) .\end{gathered}$$
The commutation relations analogous to the scalar case hold: $$\begin{gathered}
\mathcal{D}_{i}\mathcal{D}_{j} =\mathcal{D}_{j}\mathcal{D}_{i}, \qquad \mathcal{U}_{i}\mathcal{U}_{j}=\mathcal{U}_{j}\mathcal{U}_{i}, \qquad 1\leq i,j\leq
N;\\
w\mathcal{D}_{i} =\mathcal{D}_{w(i) }w, \qquad \forall\, w\in \mathcal{S}_{N}; \qquad s_{j}\mathcal{U}_{i}=\mathcal{U}_{i}s_{j}, \qquad j\neq i-1,i;\\
s_{i}\mathcal{U}_{i}s_{i} =\mathcal{U}_{i+1}+\kappa s_{i}, \qquad \mathcal{U}_{i}s_{i}=s_{i}\mathcal{U}_{i+1}+\kappa, \qquad \mathcal{U}_{i+1}s_{i}=s_{i}\mathcal{U}_{i}-\kappa.\end{gathered}$$ The simultaneous eigenfunctions of $\{\mathcal{U}_{i}\} $ are called (vector-valued) nonsymmetric Jack polynomials (NSJP). For generic $\kappa$ these eigenfunctions form a basis of $\mathcal{P}_{\tau}$ (this property fails for certain rational numbers outside the interval $-1/h_{\tau}<\kappa<1/h_{\tau}$). There is a partial order on $\mathbb{N}_{0}^{N}\times\mathcal{Y}(\tau) $ for which the NSJP’s have a triangular expression with leading term indexed by $(\alpha,T) \in\mathbb{N}_{0}^{N}\times\mathcal{Y}(\tau) $. The polynomial with this label is denoted by $\zeta_{\alpha,T}$, homogeneous of degree $\sum\limits_{i=1}^{N}\alpha_{i}$ and satisfies $$\begin{gathered}
\mathcal{U}_{i}\zeta_{\alpha,T} =\left( \alpha_{i}+1+\kappa c\left(r_{\alpha}(i) ,T\right) \right) \zeta_{\alpha,T}, \qquad 1\leq i\leq N,\\
r_{\alpha}(i) :=\#\{ j\colon \alpha_{j}>\alpha_{i}\}+\# \{ j\colon 1\leq j\leq i,\alpha_{j}=\alpha_{i} \} ;\end{gathered}$$ the rank function $r_{\alpha}\in\mathcal{S}_{N}$ and $r_{\alpha}=I$ if and only if $\alpha$ is a partition. The vector $$\begin{gathered}
[ \alpha_{i}+1+\kappa c( r_{\alpha}(i) ,T)] _{i=1}^{N}\end{gathered}$$ is called the spectral vector for $(\alpha,T) $. The NSJP structure can be extended to Laurent polynomials. Let $e_{N}:=\prod\limits_{i=1}^{N}x_{i}$ and $\boldsymbol{1}:=(1,1,\ldots,1) \in\mathbb{N}_{0}^{N}$, then $r_{\alpha+m\boldsymbol{1}}=r_{\alpha}$ for any $\alpha\in\mathbb{N}_{0}^{N}$ and $m\in\mathbb{Z}$. The commutation $\mathcal{U}_{i}( e_{N}^{m}p) =e_{N}^{m}( m+\mathcal{U}_{i}) p$ for $1\leq i\leq
N$ and $p\in\mathcal{P}_{\tau}$ imply that $e_{N}^{m}\zeta_{\alpha,T}$ and $\zeta_{\alpha+m\boldsymbol{1},T}$ have the same spectral vector for any $m\in\mathbb{N}_{0}$. They also have the same leading term (see [@Dunkl2016 Section 2.2]) and hence $e_{N}^{m}\zeta_{\alpha,T}=\zeta_{\alpha
+m\boldsymbol{1},T}$ for $\alpha\in\mathbb{N}_{0}^{N}$. This fact allows the definition of $\zeta_{\alpha,T}$ for any $\alpha\in\mathbb{Z}^{N}$: let $m=-\min_{i}\alpha_{i}$ then $\alpha+m\boldsymbol{1}\in\mathbb{N}_{0}^{N}$ and set $\zeta_{\alpha,T}:=e_{N}^{-m}\zeta_{\alpha+m\boldsymbol{1},T}$.
For a complex vector space $V$ a Hermitian form is a mapping $\langle \cdot,\cdot\rangle \colon V\otimes V\rightarrow\mathbb{C}$ such that $\langle u,cv\rangle =c\langle u,v\rangle $, $\langle u,v_{1}+v_{2}\rangle =\langle u,v_{1}\rangle+\langle u,v_{2}\rangle $ and $\langle u,v\rangle
=\overline{\langle v,u\rangle }$ for $u,v_{1},v_{2}\in V$, $c\in\mathbb{C}$. The form is positive semidefinite if $\langle u,u\rangle \geq0$ for all $u\in V$. The concern of this paper is with a particular Hermitian form on $\mathcal{P}_{\tau}$ which has the properties (for all $f,g\in\mathcal{P}_{\tau},T,T^{\prime}\in\mathcal{Y}(\tau)$, $w\in\mathcal{S}_{N}$, $1\leq i\leq N$): $$\begin{gathered}
\langle 1\otimes T,1\otimes T^{\prime} \rangle =\langle T,T^{\prime} \rangle _{0},\label{admforms}\\
\langle wf,wg \rangle =\langle f,g\rangle,\nonumber\\
\langle x_{i}\mathcal{D}_{i}f,g\rangle =\langle f,x_{i}\mathcal{D}_{i}g\rangle ,\nonumber\\
\langle x_{i}f,x_{i}g\rangle =\langle f,g\rangle.\nonumber\end{gathered}$$ The commutation $\mathcal{U}_{i}=\mathcal{D}_{i}x_{i}-\kappa\sum\limits_{j<i}(i,j) =x_{i}\mathcal{D}_{i}+1+\kappa\sum\limits_{j>i}(i,j) $ together with $\langle (i,j) f,g\rangle =\langle f,(i,j)g\rangle $ show that $\langle \mathcal{U}_{i}f,g\rangle= \langle f,\mathcal{U}_{i}g\rangle $ for all $i$. Thus uniqueness of the spectral vectors (for all but a certain set of rational $\kappa$ values) implies that $\langle \zeta_{\alpha,T},\zeta_{\beta,T^{\prime}
}\rangle =0$ whenever $(\alpha,T) \neq(\beta,T^{\prime}) $. In particular polynomials homogeneous of different degrees are mutually orthogonal, by the basis property of $\{ \zeta_{\alpha,T}\} $. For this particular Hermitian form, multiplication by any $x_{i}$ is an isometry for all $1\leq i\leq N$. This involves an integral over the torus. The equations (\[admforms\]) determine the form uniquely (up to a multiplicative constant if the first condition is removed).
Denote $\mathbb{C}_{\times}:=\mathbb{C}\backslash\{0\} $ and $\mathbb{C}_{\rm reg}^{N}:=\mathbb{C}_{\times}^{N}\backslash\bigcup\limits_{i<j}\{x\colon x_{i}=x_{j}\}$. The torus is a compact multiplicative abelian group. The notations for the torus and its Haar measure in terms of polar coordinates are$$\begin{gathered}
\mathbb{T}^{N} :=\big\{ x\in\mathbb{C}^{N}\colon \vert x_{j} \vert =1,\, 1\leq j\leq N\big\} ,\\
\mathrm{d}m(x) =(2\pi) ^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}, \qquad x_{j}=\exp ( \mathrm{i}\theta _{j}) , \qquad -\pi<\theta_{j}\leq\pi, \qquad 1\leq j\leq N.\end{gathered}$$
Let $\mathbb{T}_{\rm reg}^{N}:=\mathbb{T}^{N}\cap\mathbb{C}_{\rm reg}^{N}$, then $\mathbb{T}_{\rm reg}^{N}$ has $(N-1) !$ connected components and each component is homotopic to a circle (if $x$ is in some component then so is $ux= ( ux_{1},\ldots,ux_{N} ) $ for each $u\in\mathbb{T}$).
Let $\omega:=\exp\frac{2\pi\mathrm{i}}{N}$ and $x_{0}:=\big( 1,\omega,\ldots,\omega^{N-1}\big)$. Denote the connected component of $\mathbb{T}_{\rm reg}^{N}$ containing $x_{0}$ by $\mathcal{C}_{0}$.
Thus $\mathcal{C}_{0}$ is the set consisting of $\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) $ with $\theta_{1}<\theta_{2}< \cdots<\theta_{N}<\theta_{1}+2\pi$.
In [@Dunkl2016] we showed that if $-1/h_{\tau}<\kappa<1/h_{\tau}$ then there exists a positive matrix-valued measure $\mathrm{d}\mu$ on $\mathbb{T}^{N}$ such that for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big) $, $w\in\mathcal{S}_{N}$, $1\leq i\leq N$, $$\begin{gathered}
\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x)g(x) =\int_{\mathbb{T}^{N}}f(xw) ^{\ast}\tau(w) ^{-1}\mathrm{d}\mu(x) \tau(w) g(xw) ,\\
\int_{\mathbb{T}^{N}}( x_{i}\mathcal{D}_{i}f(x))^{\ast}\mathrm{d}\mu(x) g(x) =\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x)x_{i}\mathcal{D}_{i}g(x) ,\\
\int_{\mathbb{T}^{N}}(1\otimes T) ^{\ast}\mathrm{d}\mu (x) (1\otimes T) =\langle T,T\rangle_{0}, \qquad T\in\mathcal{Y}(\tau) .\end{gathered}$$ We introduced the notation $$\begin{gathered}
f(x) ^{\ast}\mathrm{d}\mu(x) g(x):=\sum\limits_{T,T^{\prime}\in\mathcal{Y}(\tau) }\overline {f(x) _{T}}g(x) _{T^{\prime}}\mathrm{d}\mu_{T,T^{\prime}}(x),\end{gathered}$$ where $f,g\in\mathcal{P}_{\tau}$ have the components $(f_{T}),(g_{T}) $ with respect to the orthonormal basis $$\begin{gathered}
\big\{ \langle T,T\rangle _{0}^{-1/2}T\colon T\in\mathcal{Y}(\tau)\big\}.\end{gathered}$$ Thus the Hermitian form $\langle f,g\rangle =\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x) g(x) $ satisfies (\[admforms\]). Furthermore we showed that $$\begin{gathered}
\mathrm{d}\mu(x) =\mathrm{d}\mu_{s}(x) +L(x) ^{\ast}H(\mathcal{C}) L(x)\mathrm{d}m(x) ,\end{gathered}$$ where the singular part $\mu_{s}$ is the restriction of $\mu$ to $\mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}$, $H(\mathcal{C}) $ is constant and positive-definite on each connected component $\mathcal{C}$ of $\mathbb{T}_{\rm reg}^{N}$ and $L(x) $ is a matrix function solving a system of differential equations. That system is the subject of this paper. In a way the main problem is to show that $\mu$ has no singular part.
The differential system {#difsys}
=======================
Consider the system (with $\partial_{i}:=\frac{\partial}{\partial x_{i}}$, $1\leq i\leq N$) for $n_{\tau}\times n_{\tau}$ matrix functions $L(x) $ $$\begin{gathered}
\partial_{i}L(x) =\kappa L(x) \left\{\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\tau((i,j))-\frac{\gamma}{x_{i}}I\right\} ,\qquad 1\leq i\leq N,\label{Lsys}\\
\gamma :=\frac{S_{1}(\tau) }{N}=\frac{1}{2N}\sum_{i=1}^{\ell(\tau) }\tau_{i}(\tau_{i}-2i+1) .\nonumber\end{gathered}$$ The effect of the term $\frac{\gamma}{x_{i}}I$ is to make $L(x)$ homogeneous of degree zero, that is, $\sum\limits_{i=1}^{N}x_{i}\partial_{i}L(x) =0$. The differential system is defined on $\mathbb{C}_{\rm reg}^{N}$, Frobenius integrable and analytic, thus any local solution can be continued analytically to any point in $\mathbb{C}_{\rm reg}^{N}$. Different paths may produce different values; if the analytic continuation is done along a closed path then the resultant solution is a constant matrix multiple of the original solution, called the monodromy matrix, however if the closed path is contained in a simply connected subset of $\mathbb{C}_{\rm reg}^{N}$ then there is no change.
Integrability means that $\partial_{i}( \kappa L(x) A_{j}(x)) =\partial_{j}( \kappa L(x)A_{i}(x)) $ for $i\neq j$, writing the system as $\partial_{i}L(x) =\kappa L(x) A_{i} (x) $, $1\leq i\leq N$ (where $A_{i}(x) $ is defined by equation (\[Lsys\])). The condition becomes$$\begin{gathered}
\kappa^{2}L(x) A_{i}(x) A_{j}(x) +\kappa L(x) \partial_{i}A_{j}(x) =\kappa^{2}L(x) A_{j}(x) A_{i}(x) +\kappa L(x) \partial_{j}A_{i}(x) .\end{gathered}$$ Since $\partial_{i}A_{j}(x) =\frac{\tau((i,j)) }{( x_{i}-x_{j}) ^{2}}=\partial_{j}A_{i}(x) $ it suffices to show that $A_{i}(x)^{\prime}:=\sum\limits_{k\neq i}\frac{\tau(( i,k)) }{x_{i}-x_{k}}$ and $A_{j}(x) ^{\prime}:=\sum\limits_{\ell\neq j}\frac{\tau((j,\ell)) }{x_{j}-x_{\ell}}$ commute with each other. The product $A_{i}(x)^{\prime}A_{j}(x) ^{\prime}$ is a sum of $-\frac{1}{(x_{i}-x_{j}) ^{2}}I$, terms of the form $\frac{\tau((i,k) ( j,\ell)) }{( x_{i}-x_{k})( x_{j}-x_{\ell}) }+\frac{\tau(( i,\ell)( j,k)) }{( x_{i}-x_{\ell})(x_{j}-x_{k}) }$ for $\{ i,k\} \cap\{ j,\ell\}
=\varnothing$, and terms involving the $3$-cycles $(i,j,k) $ and $(j,i,k) $ occurring as $$\begin{gathered}
\frac{\tau ( (i,j) (j,k)) }{(x_{i}-x_{j}) (x_{j}-x_{k}) }+\frac{\tau((i,k) (j,i)) }{(x_{i}-x_{k})
(x_{j}-x_{i}) }+\frac{\tau ( (i,k) (j,k)) }{(x_{i}-x_{k}) (x_{j}-x_{k})}\\
\qquad{} =\frac{\tau((i,j,k)) }{( x_{i}-x_{k}) (x_{j}-x_{k}) }+\frac{\tau((j,i,k)) }{(x_{i}-x_{k})( x_{j}-x_{k}) },\end{gathered}$$ (because $(i,j) (j,k) =(i,k) (j,i) =(i,j,k) $ and $(i,k)(j,k) =(j,i,k) $) and the latter two terms are symmetric in $i$, $j$.
We consider only fundamental solutions, that is, $\det L(x)\neq0$. Recall Jacobi’s identity: $$\begin{gathered}
\frac{\partial}{\partial t}\det F(t) =\operatorname{tr}\left(\operatorname{adj}(F(t)) \frac{\partial}{\partial t}F(t) \right),\end{gathered}$$ where $F(t) $ is a differentiable matrix function and $\operatorname{adj}(F(t)) F(t) =\det F(t) I$, that is, $\operatorname{adj} ( F(t)) =\{ \det F(t) \} F(t) ^{-1}$ when $F(t) $ is invertible; thus $$\begin{gathered}
\partial_{i}\det L(x) =\operatorname{tr}\big( \{ \det L(x)\} L(x) ^{-1}\partial_{i}L(x)
\big) =\kappa\det L(x) \operatorname{tr}A_{i}(x) .\end{gathered}$$ This can be solved: from $\sum\limits_{i<j}\tau((i,j)) =S_{1}(\tau) I$ it follows that $\operatorname{tr}( \tau((i,j))) =\binom{N}{2}^{-1}S_{1}(\tau) n_{\tau}=\frac{2}{N-1}\gamma n_{\tau}$ (and $n_{\tau}=\#\mathcal{Y}(\tau) $). We obtain the system $$\begin{gathered}
\partial_{i}\det L(x) =\kappa\gamma n_{\tau}\left\{ \frac {2}{N-1}\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}-\frac{1}{x_{i}}\right\} \det
L(x) , \qquad 1\leq i\leq N.\end{gathered}$$ By direct verification $$\begin{gathered}
\det L(x) =c\prod\limits_{1\leq i<j\leq N}\left( -\frac{(x_{i}-x_{j}) ^{2}}{x_{i}x_{j}}\right) ^{\kappa\lambda/2},\qquad \lambda:=\frac{\gamma n_{\tau}}{2(N-1) }=\operatorname{tr}(\tau(( 1,2))) ,\end{gathered}$$ is a local solution for any $c\in\mathbb{C}_{\times}$, if $x_{k}=e^{\mathrm{i}\theta_{k}}$, $1\leq k\leq N$ then $-\frac{( x_{i}-x_{j})^{2}}{x_{i}x_{j}}=4\sin^{2}\frac{\theta_{i}-\theta_{j}}{2}$ (with the principal branch of the power function, positive on positive reals). This implies $\det L(x) \neq0$ for $x\in\mathbb{C}_{\rm reg}^{N}$ (and of course $\det L(x) $ is homogeneous of degree zero).
\[L(xw)\]If $L(x) $ is a solution of in some connected open subset $U$ of $\mathbb{C}_{\rm reg}^{N}$ then $L(xw)\tau(w) ^{-1}$ is a solution in $Uw^{-1}$.
First let $w=(j,k) $ for some fixed $j$, $k$. If $i\neq j,k$ then replace $x$ by $x(j,k) $ in $\partial_{i}L$ to obtain$$\begin{gathered}
\partial_{i}L(x(j,k)) \tau(( j,k)) =\kappa L(x(j,k)) \left\{ \sum_{\ell\neq
i,j,k}\frac{\tau((i,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,j)) }{x_{i}-x_{k}}+\frac
{\tau((i,k)) }{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} \tau(j,k) \\
\hphantom{\partial_{i}L(x(j,k)) \tau(( j,k))}{} =\kappa L(x(j,k)) \tau(j,k)
\left\{ \sum_{\ell\neq i,j,k}\frac{\tau ( (i,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,k))
}{x_{i}-x_{k}}+\frac{\tau((i,j)) }{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} ,\end{gathered}$$ because $(i,j) (j,k) =(j,k) (i,k) $. Next let $w=(i,j) $, then $\partial_{i}[L(x(i,j))] =(\partial_{j}L)(x(i,j)) $ and $$\begin{gathered}
\partial_{i}[ L(x(i,j)) \tau((i,j))] =\kappa L( x(i,j)) \left\{ \sum_{\ell\neq i,j}\frac{\tau ( (j,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,j))
}{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} \tau( (i,j)) \\
\hphantom{\partial_{i}[ L(x(i,j)) \tau((i,j))]}{} =\kappa L(x(i,j)) \tau( (i,j)) \left\{ \sum_{\ell\neq i,j}\frac{\tau((i,\ell)) }{x_{i}-x_{\ell}}+\frac{\tau((i,j)) }{x_{i}-x_{j}}-\frac{\gamma}{x_{i}}I\right\} ,\end{gathered}$$ by use of $(j,\ell) (i,j) =(i,j)(i,\ell) $. Arguing by induction suppose $L(xw) \tau(w) ^{-1}$ is a solution then so is $L( x(j,k) w) \tau(w) ^{-1}\tau((j,k)) =L( x(j,k) w) \tau((j,k) w) ^{-1}$, for any$(j,k) $, that is, the statement holds for $w^{\prime}=(j,k) w$.
Let $w_{0}:= ( 1,2,3,\ldots, N ) = ( 12 )(23)\cdots(N-1,N) $ denote the $N$-cycle and let $\langle w_{0}\rangle $ denote the cyclic group generated by $w_{0}$. There are two components of $\mathbb{T}_{\rm reg}^{N}$ which are set-wise invariant under $\langle w_{0}\rangle $ namely $\mathcal{C}_{0}$ and the reverse $\{ \theta_{N}<\theta_{N-1}<\cdots<\theta_{1}<\theta _{N}+2\pi\} $. Indeed $\langle w_{0}\rangle $ is the stabilizer of $\mathcal{C}_{0}$ as a subgroup of $\mathcal{S}_{N}$.
Henceforth we use $L(x) $ to denote the solution of (\[Lsys\]) in $\mathcal{C}_{0}$ which satisfies $L(x_{0}) =I$.
Suppose $x\in\mathcal{C}_{0}$ and $m\in\mathbb{Z}$ then $L( xw_{0}^{m}) =\tau(w_{0}) ^{-m}L(x) \tau(w_{0}) ^{m}$.
Consider the solution $L(xw_{0}) \tau(w_{0}) ^{-1}$ which agrees with $\Xi L(x) $ for all $x\in \mathcal{C}_{0}$ for some fixed matrix $\Xi$. In particular for $x=x_{0}$ where $x_{0}w_{0}=\big( \omega,\ldots,\omega^{N-1},1\big) =\omega x_{0}$ (recall $(xw) _{i}=x_{w(i) }$) we obtain $\Xi L(x_{0}) =L ( x_{0}w_{0} ) \tau(w_{0})
^{-1}=L(\omega x_{0}) \tau(w_{0}) ^{-1}=L(x_{0}) \tau(w_{0}) ^{-1}$; because $L(x) $ is homogeneous of degree zero. Thus $\Xi=\tau(w_{0}) ^{-1}$. Repeated use of the relation shows $L( xw_{0}^{m}) =\tau(w_{0}) ^{-m}L(x) \tau(w_{0}) ^{m}$.
Because of its frequent use denote $\upsilon:=\tau(w_{0}) $ (the letter $\upsilon$ occurs in the Greek word *cycle*).
For $w\in\mathcal{S}_{N}$ set $\nu(w) :=\upsilon^{1-w(1) }$. For any $x\in\mathbb{T}_{\rm reg}^{N}$ there is a unique $w_{x}$ such that $w_{x}(1) =1$ and $xw_{x}^{-1}\in\mathcal{C}_{0}$. Set $M(w,x) :=\nu(w_{x}w) $.
As a consequence $\nu ( w_{0}^{m}w ) =\upsilon^{-m}\nu(w) $ for any $w\in\mathcal{S}_{N}$ and $m\in\mathbb{Z}$; since $w_{0}^{m}w(1) -1=(w(1) +m-1) \operatorname{mod}N$. Also $M(I,x) =I$. There is a 1-1 correspondence $w\mapsto\mathcal{C}_{0}w$ between $ \{ w\in \mathcal{S}_{N}\colon w(1) =1 \} $ and the connected components of $\mathbb{T}_{\rm reg}^{N}$.
For any $w_{1},w_{2}\in\mathcal{S}_{N}$ and $x\in\mathbb{T}_{\rm reg}^{N}$$$\begin{gathered}
M( w_{1}w_{2},x) =M( w_{2},xw_{1}) M (w_{1},x) .\end{gathered}$$
By definition $M(w_{1}w_{2},x) =\nu(w_{x}w_{1}w_{2}) $ and $M(w_{1},x) =\nu(w_{x}w_{1})
=\upsilon^{-m}$ where $w_{x}w_{1}(1) =m+1$. Let $w_{3}=w_{xw_{1}}$, that is, $w_{3}(1) =1$ and $xw_{1}w_{3}^{-1}\in\mathcal{C}_{0}$. From $\big( xw_{x}^{-1}\big) \big( w_{x}w_{1}w_{3}^{-1}\big)$ $\in\mathcal{C}_{0}$ it follows that $w_{x}w_{1}w_{3}^{-1}\in\langle w_{0}\rangle $, in particular $w_{x}w_{1}w_{3}^{-1}=w_{0}^{m}$ because $w_{x}w_{1}w_{3}^{-1}(1) =w_{x}w_{1}(1) =m+1=w_{0}^{m}(1) $. Thus $M(w_{2},xw_{1}) =\nu( w_{3}w_{2}) =\nu\big( w_{0}^{-m}w_{x}w_{1}w_{2}\big) =\upsilon^{m}\nu(w_{x}w_{1}w_{2})$, and $\upsilon^{m}=M(w_{1},x) ^{-1}$. This completes the proof.
Suppose $w\in\mathcal{S}_{N}$ and $x\in\mathbb{T}_{\rm reg}^{N}$ then $M\big(w^{-1},xw\big) =M(w,x) ^{-1}$.
Indeed $M\left( w^{-1},xw\right) M(w,x) =M\big(ww^{-1},x\big) =I$.
We can now extend $L(x) $ to all of $\mathbb{T}_{\rm reg}^{N}$ from its values on $\mathcal{C}_{0}$.
\[DefL(x)T\]For $x\in\mathbb{T}_{\rm reg}^{N}$ let$$\begin{gathered}
L(x) :=L\big( xw_{x}^{-1}\big) \tau(w_{x}) .\end{gathered}$$
\[L(xw)M\]For any $x\in\mathbb{T}_{\rm reg}^{N}$ and $w\in\mathcal{S}_{N}$ $$\begin{gathered}
L(xw) =M(w,x) L(x) \tau (w) .\end{gathered}$$
Let $w_{1}=w_{xw}$, that is, $w_{1}(1) =1$ and $xww_{1}^{-1}\in\mathcal{C}_{0}$, then by definition $L(xw) =L\big(xww_{1}^{-1}\big) \tau(w_{1}) $ and $L\big( xw_{x}^{-1}\big) =L(x) \tau(w_{x}) ^{-1}$. Let $m=w_{x}w(1) -1$. Since $w_{x}ww_{1}^{-1}$ fixes $\mathcal{C}_{0}$ and $w_{x}ww_{1}^{-1}(1) =w_{x}w(1) =m+1$ it follows that $w_{x}ww_{1}^{-1}=w_{0}^{m}$. Thus $w_{1}=w_{0}^{-m}w_{x}w$, $$\begin{gathered}
L\big( xww_{1}^{-1}\big) \tau(w_{1}) =L\big(xw_{x}^{-1}w_{0}^{m}\big) \tau\big( w_{0}^{-m}w_{x}w\big)
=\upsilon^{-m}L\big( xw_{x}^{-1}\big) \upsilon^{m}\tau\big( w_{0}^{-m}w_{x}w\big) \\
\hphantom{L\big( xww_{1}^{-1}\big) \tau(w_{1})}{} =\upsilon^{-m}L\big( xw_{x}^{-1}\big) \tau(w_{x}w) =\upsilon^{-m}L(x) \tau(w)\end{gathered}$$ and $M(w,x) =\nu(w_{x}w) =\upsilon^{-m}$.
The adjoint operation on Laurent polynomials and $\boldsymbol{L(x)}$
--------------------------------------------------------------------
The purpose is to define an operation which agrees with taking complex conjugates of functions and Hermitian adjoints of matrix functions when restricted to $\mathbb{T}^{N}$, and which preserves analyticity. The parameter $\kappa$ is treated as real in this context even where it may be complex (to preserve analyticity in $\kappa$). For $x\in\mathbb{C}_{\times}^{N}$ define $\phi x:=\big( x_{1}^{-1},x_{2}^{-1},\ldots,x_{N}^{-1}\big) $, then $\phi(xw) =(\phi x) w$ for all $w\in \mathcal{S}_{N}$.
\[defadj\]
1. If $f(x) =\sum\limits_{\alpha\in\mathbb{Z}^{N}}c_{\alpha}x^{\alpha}$ is a Laurent polynomial then $f^{\ast}(x) :=\sum\limits_{\alpha\in\mathbb{Z}^{N}}\overline{c_{\alpha}}x^{-\alpha}$.
2. If $f(x) =\sum\limits_{\alpha\in \mathbb{Z}^{N}}A_{\alpha}x^{\alpha}$ is a Laurent polynomial with matrix coefficients then $f^{\ast}(x) :=\sum\limits_{\alpha \in\mathbb{Z}^{N}}A_{\alpha}^{\ast}x^{-\alpha}$.
3. if $F(x) $ is a matrix-valued function analytic in an open subset $U$ of $\mathbb{C}_{\times}^{N}$ then $F^{\ast}(x) :=\overline{( F(\overline{\phi x})) }^{T}\!$ and $F^{\ast}$ is analytic on $\phi U$, that is, if $F(x)\! =\![ F_{ij}(x) ]_{i,j=1}^{N}$ then $F^{\ast}(x)\! =\! [ \overline{F_{ji}( \overline{\phi x}) }] _{i,j=1}^{N}\!$ (for example if $F_{12}(x) =c_{1}\kappa x_{1}x_{3}^{-1}+c_{2}x_{2}^{2}x_{3}^{-1}x_{4}^{-1}$ then $F_{21}^{\ast}(x) =\overline{c_{1}}\kappa x_{1}^{-1}x_{3}+\overline{c_{2}}x_{2}^{-2}x_{3}x_{4}$).
Loosely speaking $F^{\ast}(x) $ is obtained by replacing $x$ by $\phi x$, conjugating the complex constants and transposing. The fundamental chamber $\mathcal{C}_{0}$ is mapped by $\phi$ onto $\big\{ \big( e^{\mathrm{i}\theta_{j}}\big) _{j=1}^{N}\colon \theta_{1}>\theta_{2}>\cdots$ $>\theta_{N}>\theta_{1}-2\pi\big\} $, again set-wise invariant under $w_{0}$. Using $\frac{\mathrm{d}}{\mathrm{d}t}\big\{ f\big( \frac{1}{t}\big) \big\}
=-\frac{1}{t^{2}}\big( \frac{d}{dt}f\big) \big( \frac{1}{t}\big)$ we obtain the system $$\begin{gathered}
\partial_{i}L(\phi x) =\kappa L(\phi x) \left\{\sum_{j\neq i}\frac{x_{j}}{x_{i}}\frac{\tau ( (i,j)) }{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\} , \qquad 1\leq i\leq N.\end{gathered}$$ Transposing this system leads to (note $\tau(w) ^{T}=\tau (w) ^{\ast}=\tau\big( w^{-1}\big) $) $$\begin{gathered}
\partial_{i}L(\phi x) ^{T}=\kappa\left\{ \sum_{j\neq i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j)) }
{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\} L(\phi x)^{T}, \qquad 1\leq i\leq N.\end{gathered}$$ Now use part (3) of Definition \[defadj\] and set up the system whose solution of $$\begin{gathered}
\partial_{i}L^{\ast}(x) =\kappa\left\{ \sum_{j\neq i}\frac{x_{j}}{x_{i}}\frac{\tau((i,j)) }{x_{i}-x_{j}}+\frac{\gamma}{x_{i}}\right\} L^{\ast}(x) , \qquad 1\leq
i\leq N.\label{L*sys}\end{gathered}$$ satisfying $L^{\ast}(x_{0}) =I$ is denoted by $L^{\ast}(x)$. The constants in the system are all real so replacing complex constants by their complex conjugates preserves solutions of the system. The effect is that $L(x) ^{\ast}$ agrees with the Hermitian adjoint of $L(x) $ for $x\in\mathcal{C}_{0}$ (for real $\kappa$). The goal here is to establish conditions on a constant Hermitian matrix $H$ so that $K(x) :=L^{\ast}(x) HL(x) $ has desirable properties, such as $K(xw) =\tau(w)^{-1}K(x) \tau(w) $ and $K(x) \geq0$ (i.e., positive definite).
Similarly to the above $\tau((i,j)) L^{\ast}(x(i,j)) $ is also a solution of (\[L\*sys\]), implying that $\tau(w) L^{\ast}(xw) $ is a solution for any $w\in\mathcal{S}_{N}$, the inductive step is $$\begin{gathered}
\tau((i,j)) \tau(w) L( x(i,j) w) ^{\ast}=\tau((i,j)w) L(x(i,j)w) ^{\ast}.\end{gathered}$$ Also $L^{\ast}(x_{0}w_{0}) =L^{\ast}\big( \omega^{-1}x_{0}\big) =L^{\ast} (x_{0}) =I$ (thus there is a matrix $\widetilde{\Xi}$ such that $\tau(w_{0}) L^{\ast}(xw_{0}) =L^{\ast} ( \phi x) \widetilde{\Xi}$ for all $x\in\mathcal{C}_{0}$, and $\widetilde{\Xi}=\tau(w_{0}) =\upsilon$. In analogy to $L$ for $x\in\mathbb{T}_{\rm reg}^{N}$ and the same $w_{x}$ as above let $L ( \phi x_{0}) ^{T}=I$, $L(\phi x) ^{T}:=\tau ( w_{x}) ^{-1}L\big( \phi xw_{x}^{-1} \big) ^{T}$ (since $\phi xw_{x}^{-1}\in\phi\mathcal{C}_{0})$. Then $L(\phi xw) ^{T}=\tau(w) ^{-1}L(\phi x) ^{T}M(w,x)^{-1}$.
For any nonsingular constant matrix $C$ the function $CL(x) $ also satisfies (\[Lsys\]) and the function $K(x) :=L^{\ast}(x) C^{\ast}CL(x) $ satisfies the system $$\begin{gathered}
x_{i}\partial_{i}K(x) =\kappa\sum_{j\neq i}\left\{ \frac{x_{j}}{x_{i}-x_{j}}\tau((i,j)) K(x) +K(x) \tau((i,j)) \frac{x_{i}
}{x_{i}-x_{j}}\right\} , \qquad 1\leq i\leq N.\label{Kdieq}\end{gathered}$$ This formulation can be slightly generalized by replacing $C^{\ast}C$ by a Hermitian matrix $H$ (not necessarily positive-definite) without changing the equation.
For the purpose of realizing the form (\[admforms\]) we want $K$ to satisfy $K(xw) =\tau(w) ^{-1}K(x)\tau(w) $, that is, $$\begin{gathered}
K(xw) =\tau(w) ^{-1}L^{\ast}(x)M(w,x) ^{-1}HM(w,x) L(x) \tau(w) \\
\hphantom{K(xw)}{} =\tau(w) ^{-1}L^{\ast}(x) \upsilon^{m}H\upsilon^{-m}L(x) \tau(w)\end{gathered}$$ (from Proposition \[L(xw)M\]), where $m=w_{x}w(1) -1$. The condition is equivalent to $$\begin{gathered}
\upsilon H=H\upsilon,\end{gathered}$$ which is now added to the hypotheses, summarized here:
\[hypoLH\] $L(x) $ is the solution of such that $L(x_{0}) =I$ and $L(x) =L\big( xw_{x}^{-1}\big) \tau(w_{x}) $ for $x\in\mathbb{T}_{\rm reg}^{N}$ where $w_{x}(1) =1$ and $xw_{x}^{-1}\in\mathcal{C}_{0}$; $L^{\ast}(x) $ is the solution of satisfying $L^{\ast}(x_{0}) =I$, $K(x) =L^{\ast }(x) HL(x) $ is a solution of and $H$ satisfies $H^{\ast}=H$, $\upsilon H=H\upsilon$.
Integration by parts {#byparts}
====================
In this section we establish the relation between the differential system and the abstract relation $\langle x_{i}\mathcal{D}_{i}f,g\rangle
=\langle f,x_{i}\mathcal{D}_{i}g \rangle $ holding for $1\leq i\leq N$ and $f,g\in C^{1}\big( \mathbb{T}^{N};V_{\tau}\big) $. We demonstrate how close $L$ is to providing the desired inner product, by performing an integration-by-parts over an $\mathcal{S}_{N}$-invariant closed set $\subset\mathbb{T}_{\rm reg}^{N}$. Here $L(x) $ and $H$ satisfy the hypotheses listed in Condition \[hypoLH\] above. We use the identity $x_{i}\partial_{i}f^{\ast}(x) =-( x_{i}\partial_{i}f) ^{\ast}(x) $. For $\delta>0$ let $$\begin{gathered}
\Omega_{\delta}:=\Big\{ x\in\mathbb{T}^{N}\colon \min\limits_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert \geq\delta\Big\} .\end{gathered}$$ This set is invariant under $\mathcal{S}_{N}$ and $K(x) $ is bounded and smooth on it. Thus the following integrals exist.
\[xdfKg-fKxdg\]Suppose $H$ satisfies Condition [\[hypoLH\]]{} then for $f,g\in C^{1}\big( \mathbb{T}^{N};V_{\tau}\big) $ and $1\leq i\leq N$ $$\begin{gathered}
\int_{\Omega_{\delta}}\big\{ {-}( x_{i}\mathcal{D}_{i}f(x)) ^{\ast}K(x) g(x) +f(x) ^{\ast}K(x) x_{i}\mathcal{D}_{i}g(x)\big\} \mathrm{d}m(x) \\
\qquad {} =\int_{\Omega_{\delta}}x_{i}\partial_{i}\{ f(x) ^{\ast}K(x) g(x)\} \mathrm{d}m(x).\end{gathered}$$
By definition $$\begin{gathered}
x_{i}\mathcal{D}_{i}g(x) =x_{i}\partial_{i}g(x) +\kappa\sum_{j\neq i}\frac{x_{i}}{x_{i}-x_{j}}\tau((i,j))( g(x) -g( x(i,j))), \\
( x_{i}\mathcal{D}_{i}f(x) ) ^{\ast}
=-x_{i}\partial_{i}f(x) ^{\ast}+\kappa\sum_{j\neq i}\frac{x_{j}}{x_{j}-x_{i}}( f(x) ^{\ast}-f( x(i,j)) ^{\ast}) \tau((i,j)) .\end{gathered}$$ Thus $$\begin{gathered}
-( x_{i}\mathcal{D}_{i}f(x) ) ^{\ast}K(x) g(x) +f(x) ^{\ast}K(x)x_{i}\mathcal{D}_{i}g(x) \nonumber\\
\qquad{} =x_{i}\partial_{i}f(x) ^{\ast}+x_{i}\partial_{i}g(x) \nonumber\\
\qquad\quad{} +\kappa f(x) ^{\ast}\sum_{j\neq i}\left\{ \frac{x_{j}}{x_{i}-x_{j}}\tau((i,j)) K(x)+K(x) \tau((i,j)) \frac{x_{i}}{x_{i}-x_{j}}\right\} g(x) \nonumber\\
\qquad\quad{} -\kappa\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\big\{ x_{j}f( x(i,j)) ^{\ast}\tau((i,j)) K(x) g(x) +x_{i}f(x) ^{\ast}K(x) \tau((i,j)) g( x(i,j))\big\} \nonumber\\
\qquad{} =x_{i}\partial_{i}\{ f(x) ^{\ast}K(x)g(x) \} \label{dfKg}\\
\qquad\quad{} -\kappa\sum_{j\neq i}\frac{1}{x_{i}-x_{j}}\big\{ x_{j}f ( x (i,j )) ^{\ast}\tau((i,j)) K(x) g(x) +x_{i}f(x) ^{\ast}K(x) \tau((i,j)) g( x(i,j))\big\}.\nonumber\end{gathered}$$ For each pair $\{i,j\} $ the term inside $\{\cdot\} $ is invariant under $x\mapsto x(i,j)$, because $K(x(i,j)) =\tau( (i,j)) K(x) \tau((i,j)) $, and $x_{i}-x_{j}$ changes sign under this transformation. Thus $$\begin{gathered}
\int_{\Omega_{\delta}}\frac{x_{j}f(x(i,j)) ^{\ast}\tau((i,j)) K(x) g(x) +x_{i}f(x) ^{\ast}K(x) \tau((i,j)) g(x(i,j)) }{x_{i}-x_{j}}\mathrm{d}m(x) =0\end{gathered}$$ for each $j\neq i$ because $\Omega_{\delta}$ and $\mathrm{d}m$ are invariant under $(i,j)$.
Observe the value of $\kappa$ is not involved in the proof. Since $x_{j}\partial_{j}=-\mathrm{i}\frac{\partial}{\partial\theta_{j}}$ when $x_{j}=e^{\mathrm{i}\theta_{j}}$ and $\mathrm{d}m(x) = (2\pi) ^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}$ one step of integration can be directly evaluated. Consider the case $i=N$ and for a fixed $(N-1) $-tuple $( \theta_{1},\ldots,\theta_{N-1}) $ with $\theta_{1}<\theta_{2}<\cdots <\theta_{N-1}<\theta_{1}+2\pi$ such that $\big\vert e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{i}}\big\vert \geq\delta$ the integral over $\theta_{N}$ is over a union of closed intervals. These are the complement of $\bigcup\limits_{1\leq j\leq N-1} \{ \theta\colon \theta_{j}-\delta^{\prime}<\theta<\theta_{j}
+\delta^{\prime}\} $ in the circle, where $\sin\frac{\delta^{\prime}}{2}=\frac{\delta}{2}$. This results in an alternating sum of values of $f^{\ast}Kg$ at the end-points of the closed intervals. Analyzing the resulting integral (over $( \theta_{1},\ldots,\theta_{N-1}) $ with respect to $\mathrm{d}\theta_{1} \cdots \mathrm{d}\theta_{N-1}$) is one of the key steps in showing that a given $K$ provides the desired inner product. In other parts of this paper we find that $H$ must satisfy another commuting relation.
Local power series near the singular set {#locps}
========================================
In this section assume $\kappa\notin\mathbb{Z+}\frac{1}{2}$. We consider the system (\[Lsys\]) in a neighborhood of the face $ \{ x\colon x_{N-1}=x_{N} \} $ of $\mathcal{C}_{0}$. We use a coordinate system which treats the singularity in a simple way. For a more concise notation define $$\begin{gathered}
x(u,z) =( x_{1},x_{2},\ldots,x_{N-2},u-z,u+z)\in\mathbb{C}_{\times}^{N}$$
We consider the system in terms of the variable $x(u,z) $ subject to the conditions that the points $x_{1},x_{2},\ldots,x_{N-2},u$ are pairwise distinct and $\vert z\vert <\min\limits_{1\leq j\leq N-2}\vert x_{j}-u \vert $, also $\vert z\vert <\vert u\vert$, $\operatorname{Im}\frac{z}{u}>0$ (these conditions imply $\arg(u-z) <\arg(u+z) $). This allows power series expansions in $z$.
For $z_{1},z_{2}\in\mathbb{C}_{\times}$ let $$\begin{gathered}
\rho(z_{1},z_{2}) :=\left[
\begin{matrix}
z_{1}I_{m_{\tau}} & O\\
O & z_{2}I_{n_{\tau}-m_{\tau}} \end{matrix}
\right] .\end{gathered}$$ Let $\sigma:=\tau((N-1,N)) =\rho( -1,1) $. We analyze the local solution $L( x(u-z,u+z)) $ with an initial condition specified later. We obtain the differential system (using $\partial_{z}:=\frac{\partial}{\partial z}$, $\partial_{u}:=\frac{\partial}{\partial u}$) $$\begin{gathered}
\partial_{z}L(x) =\partial_{N}L-\partial_{N-1}L\\
\hphantom{\partial_{z}L(x)}{} =\kappa L\left\{ \sum_{j=1}^{N-2}\left( \frac{\tau((j,N)) }{u-x_{j}+z}-\frac{\tau( (j,N-1)) }{u-x_{j}-z}\right) +\frac{\tau(N-1,N) }{z}-\frac{\gamma}{u+z}I+\frac{\gamma}{u-z}I\right\} ,
\\
\partial_{u}L(x) =\partial_{N}L+\partial_{N-1}L\\
\hphantom{\partial_{u}L(x)}{} =\kappa L\left\{ \sum_{j=1}^{N-2}\left( \frac{\tau((j,N)) }{u-x_{j}+z}+\frac{\tau( (j,N-1)) }{u-x_{j}-z}\right) -\frac{\gamma}{u+z}I-\frac{\gamma}{u-z}I\right\} ,
\\
\partial_{j}L(x) =\kappa L(x) \left\{\sum_{i=1,i\neq j}^{N-2}\frac{\tau((i,j)) }{x_{j}-x_{i}}-\frac{\gamma}{x_{j}}I+\frac{\tau( (j,N-1)) }{x_{j}-u+z}+\frac{\tau( ( j,N)) }{x_{j}-u-z}\right\} , \\
\hphantom{\partial_{j}L(x) =}{} 1\leq j\leq N-2.\end{gathered}$$
Using the expansion $\frac{1}{t-z}=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{t^{n+1}}$ for $\vert z\vert <\vert t\vert $ we let $$\begin{gathered}
\beta_{n}(x(u,0)) :=\sum_{j=1}^{N-2}\frac {\tau((j,N)) }{(u-x_{j}) ^{n+1}}\end{gathered}$$ for $n=0,1,2,\ldots$and express the equations as (since $\sigma\tau (( j,N)) \sigma=\tau((j,N-1))$) $$\begin{gathered}
\partial_{z}L(x) =\kappa L(x) \left\{
\sum_{n=0}^{\infty}\big\{ (-1) ^{n}\beta_{n}( x(u,0)) -\sigma\beta_{n}(x(u,0)) \sigma\big\} z^{n}+\frac{\sigma}{z}-\frac{\gamma}{u+z}I+\frac{\gamma}{u-z}I\right\}, \\
\partial_{u}L(x) =\kappa L(x) \left\{\sum_{n=0}^{\infty}\big\{ (-1) ^{n}\beta_{n}( x(u,0)) +\sigma\beta_{n}(x(u,0))\sigma\big\} z^{n}-\frac{\gamma}{u+z}I-\frac{\gamma}{u-z}I\right\} ,\\
\partial_{j}L(x) =\kappa L(x) \left\{\sum_{i=1,i\neq j}^{N-2}\frac{\tau((i,j)) }{x_{j}-x_{i}}-\frac{\gamma}{x_{j}}I-\sum_{n=0}^{\infty}\frac{\tau(
(j,N-1)) +(-1) ^{n}\tau((j,N)) }{(u-x_{j}) ^{n+1}}z^{n}\right\} ,\\
\hphantom{\partial_{j}L(x) =}{} 1\leq j\leq N-2.\end{gathered}$$ Set $$\begin{gathered}
B_{n}(x(u,0)) =(-1) ^{n}\beta _{n}(x(u,0)) -\sigma\beta_{n}( x(u,0)) \sigma, \qquad n=0,1,2,\ldots.\end{gathered}$$ Note $\sigma B_{n}x(u,0) \sigma=(-1) ^{n+1}B_{n}(x(u,0)) $. Suggested by the relation $$\begin{gathered}
\frac{\partial}{\partial z}\rho \big( z^{-\kappa},z^{\kappa}\big)=\frac{\kappa}{z}\rho\big({}-z^{-\kappa},z^{\kappa}\big)=\frac{\kappa }{z}\rho\big(z^{-\kappa},z^{\kappa}\big) \sigma\end{gathered}$$ we look for a solution of the form $$\begin{gathered}
L(x) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big)
\sum_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n},\label{Lzseries}\end{gathered}$$ where each $a_{n}(x(u,0)) $ is matrix-valued and analytic in $x(u,0) $, and the initial condition is $\alpha _{0}\big( x^{(0) }\big) =I$, where $x^{(0) }$ is a base point, chosen as $\big( 1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big) $ (that is, $u=\omega^{-3/2}$, $z=0$), where $\omega:=e^{2\pi\mathrm{i}/N}$. Implicitly restrict $(x_{1},\ldots,x_{N-1},u) $ to a simply connected open subset of $\mathbb{C}_{\rm reg}^{N-1}$ containing $\big( 1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2}\big)$. Substitute (\[Lzseries\]) in the $\partial_{z}$ equation (suppressing the $x(u,0) $ argument in the $\alpha_{n}$’s) $$\begin{gathered}
\partial_{z}L =\kappa\gamma\left( \frac{1}{u+z}-\frac{1}{u-z}\right)\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)
^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}z^{n}\\
\hphantom{\partial_{z}L =}{} +\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\frac{\kappa}{z}\rho\big( z^{-\kappa},z^{\kappa}\big) \sigma\sum_{n=0}^{\infty}\alpha_{n}z^{n}\\
\hphantom{\partial_{z}L =}{} +\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}\\
\hphantom{\partial_{z}L}{}
=\kappa\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \\
\hphantom{\partial_{z}L =}{}
\times\sum_{n=0}^{\infty}\alpha_{n}z^{n}\left\{ \sum_{m=0}^{\infty}B_{m}(u) z^{m}+\frac{\sigma}{z}-\gamma\left( \frac{1}{u+z}-\frac{1}{u-z}\right) \right\} ,\end{gathered}$$ which simplifies to $$\begin{gathered}
\frac{\kappa}{z}\sum_{n=0}^{\infty}( \sigma\alpha_{n}-\alpha_{n}\sigma ) z^{n}+\sum_{n=1}^{\infty}n\alpha_{n}z^{n-1}=\kappa\sum
_{n=0}^{\infty}\alpha_{n}z^{n}\sum_{m=0}^{\infty}B_{m} ( x (u,0 ) ) z^{m}.\label{eqnDz}\end{gathered}$$ The equations for $\partial_{u}$ and $\partial_{j}$ simplify to $$\begin{gathered}
\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\left\{ \sum_{n=0}^{\infty}\partial_{u}\alpha_{n}z^{n}
-\kappa\gamma\left( \frac{1}{u+v}+\frac{1}{u-v}\right) \sum_{n=0}^{\infty }\alpha_{n}z^{n}\right\} \\
\qquad{} =\kappa\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right)^{-\gamma\kappa}\sum_{n=0}^{\infty}\alpha_{n}z^{n}\\
\qquad\quad{} \times\left\{ \sum_{m=0}^{\infty}\big\{ (-1) ^{m}\beta
_{m}(x(u,0)) +\sigma\beta_{m}( x(u,0)) \sigma\big\} z^{m}-\frac{\gamma}{u+v}I-\frac{\gamma}{u-v}I\right\} ,\end{gathered}$$ leading to (with $1\leq j\leq N-2$) $$\begin{gathered}
\sum_{n=0}^{\infty}\partial_{u}\alpha_{n}(x(u,0))z^{n} =\kappa\sum_{n=0}^{\infty}\alpha_{n}( x(u,0)) z^{n}\sum_{m=0}^{\infty}\big\{ (-1) ^{m}\beta_{m}(x(u,0)) +\sigma\beta_{m} ( x (u,0)) \sigma\big\} z^{m},\\
\sum_{n=0}^{\infty}\partial_{j}\alpha_{n}(x(u,0))z^{n} =\kappa\sum_{n=0}^{\infty}\alpha_{n}( x(u,0)) z^{n}\\
\hphantom{\sum_{n=0}^{\infty}\partial_{j}\alpha_{n}(x(u,0))z^{n} =}{}
\times\left\{ \sum_{i=1,i\notin j}^{N-2}\frac{\tau( (i,j)) }{x_{j}-x_{i}}-\sum_{m=0}^{\infty}\frac{\tau((j,N-1)) +(-1) ^{m}\tau((j,N)) }{(u-x_{j}) ^{m+1}}z^{m}\right\}.\end{gathered}$$ We only need the equations for $\alpha_{0}( x(u,0)) $ (that is, the coefficient of $z^{0}$) to initialize the $\partial_{z}$ equation (this is valid because the system is Frobenius integrable): $$\begin{gathered}
\partial_{u}\alpha_{0}(x(u,0)) =\kappa \alpha_{0}(x(u,0)) \big\{ \beta_{0}(x(u,0)) +\sigma\beta_{0}( x(u,0)) \sigma\big\} ,\label{dua0x}\\
\partial_{j}\alpha_{0}(x(u,0)) =\kappa \alpha_{0}(x(u,0)) \left\{ \sum_{i=1,i\notin j}^{N-2}\frac{\tau((i,j)) }{x_{j}-x_{i}} -\frac{\tau((j,N-1)) +\tau((
j,N)) }{(u-x_{j}) }\right\} ,\nonumber\\
\hphantom{\partial_{j}\alpha_{0}(x(u,0)) =}{} 2\leq j\leq N-2.\nonumber\end{gathered}$$
$\sigma\alpha_{0}(x(u,0)) \sigma=\alpha_{0}(x(u,0)) $ and $\alpha_{0}( x(u,0)) $ is invertible.
By hypothesis $\alpha_{0}\big( x^{(0) }\big) =I$. The right hand sides of the system are invariant under the transformation $Q\mapsto \sigma Q\sigma$ thus $\alpha_{0}(x(u,0)) $ and $\sigma\alpha_{0}(x(u,0)) \sigma$ satisfy the same system. They agree at the base-point $x^{(0) }$, hence everywhere in the domain. By Jacobi’s identity the determinant satisfies (where $\lambda:=\operatorname{tr}(\sigma) =n_{\tau}-2m_{\tau}$) $$\begin{gathered}
\partial_{u}\det\alpha_{0}(x(u,0)) =\kappa\det\alpha_{0}(x(u,0)) \operatorname{tr} \{ \beta_{0}(x(u,0)) +\sigma\beta
_{0}(x(u,0)) \sigma \} \\
\hphantom{\partial_{u}\det\alpha_{0}(x(u,0))}{} =2\kappa\det\alpha_{0}(x(u,0)) \mathrm{\lambda }\sum_{j=1}^{N-2}\frac{1}{(u-x_{j}) },\\
\partial_{j}\det\alpha_{0}(x(u,0)) =\kappa\lambda\det\alpha_{0}(x(u,0)) \left\{\sum_{i=1,i\notin j}^{N-2}\frac{1}{x_{j}-x_{i}}-\frac{2}{u-x_{j}}\right\},\qquad 1\leq j\leq N-2,\\
\det\alpha_{0}(x(u,0)) =\prod_{1\leq i<j\leq N-2}\left( \frac{x_{i}-x_{j}}{x_{i}^{(0) }-x_{j}^{(0) }}\right) ^{\lambda\kappa}\prod_{i=1}^{N-2}\left( \frac{x_{i}-u}{x_{i}^{(0) }-x_{N-1}^{(0) }}\right)^{2\lambda\kappa},\end{gathered}$$ the multiplicative constant follows from $\alpha_{0}\big( x^{(0) }\big) =I$. Thus $\alpha_{0}(x(u,0))$ is nonsingular in its domain.
We turn to the inductive definition of $\{ \alpha_{n}( x(u,0) ) \colon n\geq1\} $.
In terms of the block decomposition $( m_{\tau}+( n_{\tau}-m_{\tau})) \times( m_{\tau}+( n_{\tau}-m_{\tau})) $ (henceforth called the $\sigma$-*block decomposition*) of a matrix $$\begin{gathered}
\alpha=\left[
\begin{matrix}
\alpha_{11} & \alpha_{12}\\
\alpha_{21} & \alpha_{22}\end{matrix}
\right]\end{gathered}$$ $\sigma\alpha\sigma=\alpha$ if and only if $\alpha_{12}=O=\alpha_{21}$ and $\sigma\alpha\sigma=-\alpha$ if and only if $\alpha_{11}=O=\alpha_{22}$. Write the $\sigma$-block decomposition of $\alpha_{n}(u) $ as $$\begin{gathered}
\alpha_{n}=\left[
\begin{matrix}
\alpha_{n,11} & \alpha_{n,12}\\
\alpha_{n,21} & \alpha_{n,22}\end{matrix}
\right]\end{gathered}$$ then the coefficient of $z^{n-1}$ on the left side of equation (\[eqnDz\]) is $$\begin{gathered}
\kappa ( \sigma\alpha_{n}-\alpha_{n}\sigma ) +n\alpha_{n}=\left[
\begin{matrix}
n\alpha_{n,11} & (n-2\kappa) \alpha_{n,12}\\
( n+2\kappa) \alpha_{n,21} & n\alpha_{n,22}
\end{matrix}
\right] ,\end{gathered}$$ and on the right side it is $$\begin{gathered}
\kappa S_{n}(x(u,0)) :=\kappa\sum_{i=0}^{n-1}\alpha_{n-1-i}B_{i}(x(u,0)) ,\end{gathered}$$ for $n\geq1$. Arguing inductively suppose $\sigma\alpha_{m}\sigma= (-1 ) ^{m}\alpha_{m}$ for $0\leq m\leq n$, then $\sigma S_{n}\sigma
=\sum\limits_{i=0}^{n-1} ( \sigma\alpha_{n-1-i}\sigma ) ( \sigma B_{i}\sigma ) =\sum\limits_{i=0}^{n-1}(-1) ^{n-1-i+i-1}\alpha_{n-1-i}B_{i}$ and thus $\sigma S_{n}(u) \sigma= (-1) ^{n}S_{n}(u) $. In terms of the $\sigma$-block decomposition $\left[
\begin{matrix}
S_{n,11} & S_{n,12}\\
S_{n,21} & S_{n,22}\end{matrix}
\right] $ of $S_{n}(x(u,0)) $ this condition implies $S_{n,12}=O=S_{n,21}$ when $n$ is even, and $S_{n,11}=O=S_{n,22}$ when $n$ is odd. This implies (for $n=1,2,3,\ldots$) $$\begin{gathered}
\alpha_{2n}(x(u,0)) =\frac{\kappa}{2n}S_{2n}(x(u,0)) ,\label{recurS}\\
\alpha_{2n-1} ( x ( u,0 )) =\rho\left( \frac{\kappa}{2n-1-2\kappa},\frac{\kappa}{2n-1+2\kappa}\right) S_{2n-1}(x(u,0)) ,\nonumber\end{gathered}$$ and thus $\sigma\alpha_{n}(x(u,0)) \sigma= ( -1) ^{n}\alpha_{n}(x(u,0)) $. In particular $$\begin{gathered}
\alpha_{1}(x(u,0)) =\rho\left( \frac{\kappa }{1-2\kappa},\frac{\kappa}{1+2\kappa}\right) \alpha_{0}( x(u,0)) B_{0}(x(u,0)) ,\end{gathered}$$ and all the coefficients are determined; by hypothesis $\kappa\notin \mathbb{Z+}\frac{1}{2}$ and the denominators are of the form $2m+1\pm2\kappa$.
Henceforth denote the series (\[Lzseries\]), solving (\[Lsys\]) with the normalization $\alpha_{0}\big( x^{(0) }\big) =I$ by $L_{1}(x) $. It is defined for all $x(u,z) \in\mathcal{C}_{0}$ subject to $\vert z\vert <\min\limits_{1\leq j\leq N-2}\vert x_{j}-u \vert $, also $\vert z\vert <\vert u\vert$, $\operatorname{Im}\frac{z}{u}>0$. The radius of convergence depends on $x(u,0) $. Return to using $L ( x) $ to denote the solution from Definition \[DefL(x)T\] (on all of $\mathbb{T}_{\rm reg}^{N}$ and $L(x_{0}) =I$). In terms of $x (u,z ) $ the point $x_{0}$ corresponds to $u=\frac{1}{2} \big(\omega^{-1}+\omega^{-2} \big) $, $z=\frac{1}{2}\big( \omega^{-1}-\omega^{-2}\big) $, $x(u,z) =\big( 1,\omega,\ldots,\omega^{N-3},u-z,u+z\big) $, then $\min\limits_{1\leq j\leq N-2} \vert u-x_{j}\vert =\sin\frac{\pi}{N}\big( 5+4\cos\frac{2\pi}{N}\big) $ and $\vert z\vert =\sin\frac{\pi}{N}$ (also $\frac{z}{u}=\mathrm{i}\tan\frac{\pi}{N}$) and $x_{0}$ is in the domain of convergence of the series $L_{1}(x) $. Thus the relation $L_{1} (x ) =L_{1}(x_{0}) L(x) $ holds in the domain of $L_{1}$ in $\mathcal{C}_{0}$. This implies the important fact that $L_{1}(x_{0}) $ is an analytic function of $\kappa$, to be exploited in Section \[anlcmat\].
Behavior on boundary
--------------------
The term $\rho ( z^{-\kappa},z^{\kappa} ) $ implies that $L_{1}(x) $ is not continuous at $z=0$, that is, on the boundary $ \{ x\colon x_{N-1}=x_{N}\}$. However there may be a weak type of continuity, specifically $$\begin{gathered}
\lim\limits_{x_{N-1}-x_{N}\rightarrow0} (K(x) -K( x(N-1,N))) =0.\end{gathered}$$ With the aim of expressing the desired $K(x) $ in the form $L(x) ^{\ast}C^{\ast}CL(x) $ (and $C$ is unknown at this stage) we consider $CL(x) $ in series form, that is $CL_{1}(x_{0}) ^{-1}L_{1}(x) $ (recall $\det L(x) \neq0$ in $\mathcal{C}_{0}$). We analyze the effect of $C$ on the weak continuity condition. Denote $C^{\prime}:=CL_{1}( x_{0}) ^{-1}$.
From Proposition \[L(xw)M\] $L(x(N-1,N)) =\nu((N-1,N)) L(x) \tau( (N-1,N)) =L(x) \sigma$, because $w(1) =1$ for $w=(N-1,N) $, \[for the special case $N=3$, $\tau=(2,1) $, $\mathbb{T}_{\rm reg}^{3}$ has two components and we define $L(x) =L( x(2,3)) \sigma$ for the component $\neq\mathcal{C}_{0}$\]. By use of $x(u,z)(N-1,N) =x(u,-z) $ it follows that $$\begin{gathered}
CL( x(u,z) (N-1,N)) =CL(x(u,z) ) \sigma =C^{\prime}\left( x_{N}x_{N-1}\right) ^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(u)z^{n}\sigma\\
\hphantom{CL( x(u,z) (N-1,N))}{} =C^{\prime}\sigma ( x_{N}x_{N-1} ) ^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(u)
(-1) ^{n}z^{n},\end{gathered}$$ because $\sigma\alpha_{n}(u) \sigma=(-1)^{n}\alpha_{n}(u) $ and $\sigma=\rho(-1,1) $.
Recall $L^{\ast}(x) $ is defined as $L(\phi x)^{T}$ with complex constants replaced by their conjugates. Then $\phi x(u,z) =\big( x_{1}^{-1},x_{2}^{-1},\ldots,x_{N-2}^{-1},\frac{1}{u-z},\frac{1}{u+z}\big)$. To compute $L_{1} ( \phi x (u,z )) $ replace $u$ by $u^{\prime}=\frac{u}{(u+z)
(u-z) }$ and replace $z$ by $z^{\prime}=-\frac{z}{(u+z) (u-z) }$. When restricted to the torus $u^{\prime}=\frac{1}{2}\big( \frac{1}{x_{N-1}}+\frac{1}{x_{N}}\big) =\overline{u}$ and $z^{\prime}=\frac{1}{2}\big( \frac{1}{x_{N-1}}-\frac{1}{x_{N}}\big) =\overline{z}$. The terms $\beta_{n}(u) :=\sum\limits_{j=1}^{N-2}\frac{\tau((j,N)) }{( u-x_{j}) ^{n+1}}$ in the intermediate formulae for $L_{1}$ are replaced by their complex conjugates when $x(u,z) \in\mathbb{T}^{N}$. Similarly $\widetilde{\beta}_{k}:=\sum\limits_{m=0}^{\infty}\sum\limits_{j=1}^{N-2}\frac{\tau((j,N)) }{(u_{0}-x_{j}) ^{k+1}}$ transforms to $\overline{( \widetilde{\beta}_{k}) }$ because the constant $u_{0}$ is conjugated. Thus for $x(u,z) \in\mathbb{T}_{\rm reg}^{N}$ $$\begin{gathered}
L_{1}(x(u,z)) ^{\ast}=\sum_{m=0}^{\infty}\alpha_{m}(u) ^{\ast}\overline{z}^{m}\rho \big( \overline {z}^{-\kappa},\overline{z}^{\kappa}\big) C^{^{\prime}\ast} (\overline{x_{N}x_{N-1}}) ^{-\gamma\kappa};\end{gathered}$$ $\alpha_{m}(u) ^{\ast}$ denotes the adjoint of the matrix $\alpha_{m}(u) $. Then $$\begin{gathered}
L_{1}( x(u,z) (N-1,N)) ^{\ast}=\sum_{m=0}^{\infty}(-1) ^{m}\alpha_{m}(u) ^{\ast
}\overline{z}^{m}\rho( \overline{z}^{-\kappa},\overline{z}^{\kappa}) \sigma C^{^{\prime}\ast}( \overline{x_{N}x_{N-1}})
^{-\gamma\kappa}.\end{gathered}$$ Furthermore (recall $K(x(N-1,N)) =\sigma K(x) \sigma$ by definition) $$\begin{gathered}
K(x(u,z)) =\sum_{m,n=0}^{\infty}\overline
{z}^{m}z^{n}\alpha_{m}(u) ^{\ast}\rho\big( \overline
{z}^{-\kappa},\overline{z}^{\kappa}\big) C^{\prime\ast}C^{\prime}\rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{n}(u) ,\\
K(x(u,-z)) =\sum_{m,n=0}^{\infty}\overline
{z}^{m}z^{n}(-1) ^{m+n}\alpha_{m}(u) ^{\ast}\rho\big( \overline{z}^{-\kappa},\overline{z}^{\kappa}\big) \sigma
C^{\prime\ast}C^{\prime}\sigma\rho\big( z^{-\kappa},z^{\kappa}\big)\alpha_{n}(u) .\end{gathered}$$ The term of lowest order in $z$ in $K(x(u,z)) -K( x(u,-z)) $ is $$\begin{gathered}
\alpha_{0}(u) ^{\ast}\rho\big( \overline{z}^{-\kappa
},\overline{z}^{\kappa}\big) \big\{ C^{\prime\ast}C^{\prime}-\sigma
C^{\prime\ast}C^{\prime}\sigma\big\} \rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{0}(u) .\end{gathered}$$ In terms of the $\sigma$-block decomposition, with$$\begin{gathered}
C^{\prime\ast}C^{\prime}=
\begin{bmatrix}
c_{11} & c_{12}\\
c_{12}^{\ast} & c_{22}
\end{bmatrix}
,\qquad \alpha_{0}(u) =
\begin{bmatrix}
a_{11}(u) & O\\
O & a_{22}(u)
\end{bmatrix}\end{gathered}$$ the expression equals $$\begin{gathered}
2
\begin{bmatrix}
O & \left( \dfrac{z}{\overline{z}}\right) ^{\kappa}a_{11}(u)
^{\ast}c_{12}a_{22}(u) \\
\left( \dfrac{\overline{z}}{z}\right) ^{\kappa}a_{22}(u)
^{\ast}c_{12}^{\ast}a_{11}(u) & O
\end{bmatrix},\end{gathered}$$ which tends to zero as $z\rightarrow0$ if and only if $c_{12}=0$, that is, $\sigma C^{\ast}C\sigma=C^{\ast}C$.
Suppose $C^{\prime\ast}C^{\prime}$ commutes with $\sigma$ then $$\begin{gathered}
K(x(u,z)) -K( x(u,z) (N-1,N)) =O\big( \vert z\vert ^{1-2\vert\kappa\vert }\big).\end{gathered}$$
The hypothesis implies $C^{\prime\ast}C^{\prime}$ commutes with $\rho(z^{-\kappa},z^{\kappa}) $, thus $$\begin{gathered}
K(x(u,z)) -K ( x(u,z)(N-1,N)) \\
\qquad{} =\sum_{m,n=0}^{\infty}\overline{z}^{m}z^{n}\big( 1-(-1)^{m+n}\big) \alpha_{m}(u) ^{\ast}\rho\big(\vert z \vert ^{-2\kappa},\vert z\vert ^{2\kappa}\big)C^{\prime\ast}C^{\prime}\alpha_{n}(u) \\
\qquad{} =2z\alpha_{0}(u) ^{\ast}\rho\big( \vert z\vert^{-2\kappa},\vert z\vert ^{2\kappa}\big) C^{\prime\ast}
C^{\prime}\alpha_{1}(u) +2\overline{z}\alpha_{1}(u) ^{\ast}\rho\big( \vert z\vert ^{-2\kappa}, \vert z \vert ^{2\kappa}\big) C^{^{\prime}\ast}C^{\prime}\alpha_{0} (u) \\
\qquad\quad{} +\sum_{m+n\geq2}^{\infty}\overline{z}^{m}z^{n}\big( 1-(-1)^{m+n}\big) \alpha_{m}(u) ^{\ast}\rho\big(\vert
z \vert ^{-2\kappa},\vert z\vert ^{2\kappa}\big)C^{\prime\ast}C^{\prime}\alpha_{n}(u) .\end{gathered}$$ The dominant terms come from $m=0,n=1$ and $m=1,n=0$; both of order $O\big(\vert z\vert ^{1-2\vert \kappa\vert }\big) $.
We will see later for purpose of integration by parts, that the change in $K$ between the points $\big( x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta}, e^{\mathrm{i} ( \theta-\varepsilon ) }\big) $ and $\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}( \theta+\varepsilon) }\big) $ is a key part of the analysis; this uses the relation $K\big( \big( x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta },e^{\mathrm{i}( \theta-\varepsilon) }\big) \big) =\sigma
K\big( \big( x_{1},\ldots,x_{N-2},e^{\mathrm{i} ( \theta -\varepsilon ) },e^{\mathrm{i}\theta}\big) \big) \sigma$.
Bounds {#bnds}
======
In this section we derive bounds on $L(x) $ of global and local type. Throughout we adopt the normalization $L(x_{0}) =I$. The operator norm on $n_{\tau}\times n_{\tau}$ complex matrices is defined by $\Vert M\Vert =\sup\{ \vert Mv\vert \colon \vert v \vert =1 \} $.
\[Lbnd\]There is a constant $c$ depending on $\kappa$ such that $\Vert L(x) \Vert \leq c\prod\limits_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert ^{-\vert \kappa\vert }$ for each $x\in\mathbb{T}_{\rm reg}^{N}$.
The proof is a series of steps starting with a general result which applies to matrix functions satisfying a linear differential equation in one variable.
\[bdsM\]Suppose $M(0) =I$, $\frac{d}{dt}M(t)
=M(t) F(t) $ and $ \Vert F(t) \Vert \leq f(t) $ for $0\leq t\leq1$ then $ \Vert M(t) -I\Vert \leq\exp\int_{0}^{t}f(s)\mathrm{d}s-1$ and $\Vert M(1)\Vert \leq\exp\int_{0}^{1}f(s) \mathrm{d}s$.
Let $\ell(t) := \Vert M(t) -I \Vert $ then the equation $M (t) -I=\int_{0}^{t}M(s) F(s) \mathrm{d}s$ and the inequalities $\Vert M(t) \Vert \leq\Vert M(t) -I \Vert + \Vert I \Vert $ (and $ \Vert I\Vert =1$) imply that $\ell(t) \leq\int_{0}^{t}(\ell(s) +1) f(s) \mathrm{d}s$. Define differentiable functions $b(t) $ and $h(t) $ by $$\begin{gathered}
h(t) :=\exp\int_{0}^{t}f(s) \mathrm{d}s,\\
b(t) h(t) =\int_{0}^{t}( \ell (s) +1) f(s) \mathrm{d}s+1.\end{gathered}$$ Apply $\frac{d}{dt}$to the latter equation:$$\begin{gathered}
b^{\prime}(t) h(t) +b(t) f (t) h(t) =( \ell(t) +1)f(t) ,\\
b^{\prime}(t) h(t) =f(t)\left\{ \ell(t) +1-\int_{0}^{t} ( \ell(s)+1) f(s) \mathrm{d}s-1\right\} \\
\hphantom{b^{\prime}(t) h(t)}{} =f(t) \left\{ \ell(t) -\int_{0}^{t}(\ell(s) +1) f(s) \mathrm{d}s\right\}\leq0.\end{gathered}$$ Hence $b^{\prime}(t) \leq0$ and $b(t) \leq b(0) =1$ which implies $$\begin{gathered}
\ell(t) \leq\int_{0}^{t} ( \ell(s) +1 ) f(s) \mathrm{d}s=b(t) h(t) -1\leq h(t) -1.\end{gathered}$$ Finally $ \Vert M(1) \Vert \leq$ $ \Vert M ( 1 ) -I \Vert +1\leq\exp\int_{0}^{1}f(s) \mathrm{d}s$.
Next we set up a differentiable path $p(t) = ( p_{1} ( t ) ,\ldots,p_{N}(t)) $ in $\mathbb{C}_{\rm reg}^{N}$ starting at $x_{0}$ and obtain the equation $$\begin{gathered}
\frac{\mathrm{d}}{\mathrm{d}t}L(p(t)) =\kappa L( p(t)) \sum_{i=1}^{N}\left\{ \sum_{j\neq i}\frac{p_{i}^{\prime}(t) }{p_{i}(t) -p_{j}(t) }
\tau((i,j)) -\gamma\frac{p_{i}^{\prime}(t) }{p_{i}(t) }I\right\} \\
\hphantom{\frac{\mathrm{d}}{\mathrm{d}t}L(p(t))}{} =\kappa L(p(t)) \left\{ \sum_{1\leq i<j\leq
N}\frac{p_{j}^{\prime}(t) -p_{i}^{\prime}(t)}{p_{j}(t) -p_{i}(t) }\tau ( (i,j)) -\gamma\sum_{i=1}^{N}\frac{p_{i}^{\prime} (t) }{p_{i}(t) }I\right\} .\end{gathered}$$ Suppose $x=\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) \in\mathcal{C}_{0}$ and $\theta_{1}<\theta_{2}<\cdots<\theta
_{N}<\theta_{1}+2\pi$. Define the path $p(t) =\big( e^{\mathrm{i}g_{1}(t) },\ldots,e^{\mathrm{i}g_{N}(t) }\big) $ where $g_{j}(t) =(1-t)
\frac{2(j-1) \pi}{N}+t\theta_{j}$ for $1\leq j\leq N$. Then $p(t) \in\mathcal{C}_{0}$ for $0\leq t\leq1$ because $g_{i+1}(t) -g_{i}(t) =(1-t)
\frac{2\pi}{N}+t ( \theta_{i+1}-\theta_{i} ) >0$ for $1\leq i<N$ and $2\pi+g_{1}(t) -g_{N}(t) =2\pi+t\theta_{1}-(1-t) \frac{2(N-1) \pi}{N}-t\theta
_{N}=(1-t) \frac{2\pi}{N}+t ( 2\pi+\theta_{1}-\theta _{N}) >0$. The factor of $\tau((i,j)) $ in the equation is $$\begin{gathered}
\mathrm{i}\frac{g_{j}^{\prime}(t) e^{\mathrm{i}g_{j} (t) }-g_{i}^{\prime}(t) e^{\mathrm{i}g_{i}(t) }}{e^{\mathrm{i}g_{j}(t) }-e^{\mathrm{i}g_{i}(t) }}=
\frac{1}{2}\left\{( g_{j}^{\prime}(t) -g_{i}^{\prime}(t)) \frac{\cos\big( \frac{1}{2}(g_{j}(t) -g_{i}(t) ) \big) }{\sin\big(
\frac{1}{2}( g_{j}(t) -g_{i}(t) )\big) }+\mathrm{i} ( g_{j}^{\prime}(t) +g_{i}^{\prime
}(t) ) \right\} .\end{gathered}$$ We will apply Lemma \[bdsM\] to $\widetilde{L}(x) =\prod\limits_{j=1}^{N}x_{j}^{\gamma\kappa}L(x) $; this only changes the phase and removes the $\sum\limits_{i=1}^{N}\frac{p_{i}^{\prime} (t) }{p_{i}(t) }$ term. In the notation of Lemma \[bdsM\] $$\begin{gathered}
f(t) =\vert \kappa\vert \sum\limits_{i<j}\left\vert \mathrm{i}\frac{g_{j}^{\prime}(t) e^{\mathrm{i}g_{j}(t) }-g_{i}^{\prime}(t)
e^{\mathrm{i}g_{i}(t) }}{e^{\mathrm{i}g_{j}(t)}-e^{\mathrm{i}g_{i}(t) }}\right\vert\end{gathered}$$ (since $\Vert\tau((i,j))\Vert =1$). To set up the integral $\int_{0}^{1}f(t) \mathrm{d}t$ let $$\begin{gathered}
\phi_{ij} (t) =\frac{1}{2}( g_{j}(t) -g_{i}(t)) =\frac{1}{2}\left\{ (1-t) \frac{2(j-i)
\pi}{N}+t(\theta_{j}-\theta_{i}) \right\}\end{gathered}$$ so that $\phi
_{ij}^{^{\prime}}(t) =\frac{1}{2}\big( \theta_{j}-\theta_{i}+\frac{2(j-i) \pi}{N}\big) $ and $0<\phi_{ij}(t) <\pi$ for $i<j$ and $0\leq t\leq1$. The terms $\vert \mathrm{i} ( g_{j}^{\prime}(t) +g_{i}^{\prime}(t) )\vert \leq 4\pi$ provide a simple bound (no singularities off $\mathbb{T}_{\rm reg}^{N}$). The dominant terms come from $\int_{0}^{1}\frac{\vert \phi_{ij}^{\prime}\cos\phi_{ij}(t)\vert }{\sin\phi_{ij}(t) }\mathrm{d}t$. There are two cases. Let $\phi_{0},\phi_{1}$ satisfy $0<\phi_{0},\phi_{1}<\pi$ and let $\phi(t) =(1-t) \phi_{0}+t\phi_{1}$. The antiderivative $\int\frac{\phi^{\prime}\cos\phi(t) }{\sin \phi(t) }\mathrm{d}t=\log\sin\phi(t) $. The first case applies when either $0<\phi_{0},\phi_{1}\leq\frac{\pi}{2}$ or $\frac{\pi
}{2}\leq\phi_{0},\phi_{1}<\pi$ (assign $\phi_{0}=\frac{\pi}{2}=\phi_{1}$ to the first interval); then $\phi^{\prime}\cos\phi(t) \geq0$ if $0<\phi_{0}\leq\phi_{1}\leq\frac{\pi}{2}$ or $\frac{\pi}{2}\leq\phi_{1}\leq\phi_{0}<\pi$ and $\phi^{\prime}\cos\phi(t) <0$ otherwise. These imply $$\begin{gathered}
\int_{0}^{1}\frac{ \vert \phi^{\prime}\cos\phi(t)\vert }{\sin\phi(t) }\mathrm{d}t=\left\vert \log\frac{\sin\phi_{1}}{\sin\phi_{0}}\right\vert .\end{gathered}$$ The second case applies when either $0<\phi_{0}<\frac{\pi}{2}<\phi_{1}<\pi$ (thus $\phi^{\prime}>0$) or $0<\phi_{1}<\frac{\pi}{2}<\phi_{0}<\pi$. Let $\phi(t_{0}) =\frac{\pi}{2}$ (that is, $t_{0}=\frac{\pi /2-\phi_{0}}{\phi_{1}-\phi_{0}})$. In the first situation $$\begin{gathered}
\int_{0}^{1}\frac{\vert \phi^{\prime}\cos\phi(t)\vert }{\sin\phi(t) }\mathrm{d}t =\int_{0}^{t_{0}}
\frac{\phi^{\prime}\cos\phi(t) }{\sin\phi(t)}\mathrm{d}t-\int_{t_{0}}^{1}\frac{\phi^{\prime}\cos\phi(t)
}{\sin\phi(t) }\mathrm{d}t =-\log\sin\phi_{0}-\log\sin\phi_{1},\end{gathered}$$ since $\log\sin\frac{\pi}{2}=0$; and the same value holds for the second situation. We obtain $$\begin{gathered}
\int_{0}^{1}f(t) \mathrm{d}t\leq\vert \kappa\vert \sum_{1\leq i<j\leq N}\left\{ -\log\sin\frac{\theta_{j}-\theta_{i}}{2}-\log\sin\frac{(j-i) \pi}{N}+4\pi\right\} .\end{gathered}$$ Taking exponentials and using the lemma (recall $\big\vert e^{\mathrm{i}\phi_{1}}-e^{\mathrm{i}\phi_{2}}\big\vert =2\sin\big\vert \frac{\phi
_{1}-\phi_{2}}{2}\big\vert $) we obtain$$\begin{gathered}
\Vert L(x)\Vert \leq c\prod\limits_{1\leq i<j\leq N}\vert x_{i}-x_{j}\vert ^{-\vert \kappa\vert }.\end{gathered}$$ This bound applies to all of $\mathbb{T}_{\rm reg}^{N}$ when $L ( x_{0}) $ commutes with $\upsilon$ and $L$ is extended to $\mathbb{T}_{\rm reg}^{N}$ as in Definition \[DefL(x)T\]. This completes the proof of Theorem \[Lbnd\].
Next we find bounds on the series expansion from (\[Lzseries\]) $$\begin{gathered}
L_{1}(x) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big)
\sum_{n=0}^{\infty}\alpha_{n}(x(u,0)) z^{n},\end{gathered}$$ where $\vert z\vert <\delta_{0}:=\min\limits_{1\leq j\leq N-2}\left\vert u-x_{j}\right\vert $ and $\operatorname{Im}\frac{z}{u}>0$. Recall the recurrence (\[recurS\]) $$\begin{gathered}
S_{n} :=\sum_{i=0}^{n-1}\alpha_{n-1-i}\big\{ (-1) ^{i}\beta_{i}-\sigma\beta_{i}\sigma\big\} , \qquad \beta_{i}:=\sum_{j=0}^{N-2}\frac{\tau((j,N)) }{(u-x_{j}) ^{i+1}},\\
\alpha_{2n}(x(u,0)) =\frac{\kappa}{2n}S_{2n},\\
\alpha_{2n+1}(x(u,0)) =\rho\left(\frac{\kappa}{2n+1-2\kappa},\frac{\kappa}{2n+1+2\kappa}\right) S_{2n+1}.\end{gathered}$$
Suppose $\vert \kappa\vert \leq\kappa_{0}<\frac{1}{2}$ and $\lambda:=( N-2) \kappa_{0}$ then for $n\geq0$ $$\begin{gathered}
\Vert \alpha_{2n}(x(u,0)) \Vert \leq \Vert \alpha_{0}(x(u,0)) \Vert \frac{(\lambda) _{n}\big( \lambda+\frac{1}{2}-\kappa
_{0}\big) _{n}}{n!\big( \frac{1}{2}-\kappa_{0}\big) _{n}}\delta _{0}^{-2n},\label{bndan}\\
\Vert \alpha_{2n+1}(x(u,0)) \Vert \leq \Vert \alpha_{0}(x(u,0)) \Vert \frac{(\lambda) _{n+1}\big( \lambda+\frac{1}{2}-\kappa
_{0}\big) _{n}}{n!\big( \frac{1}{2}-\kappa_{0}\big) _{n+1}}\delta _{0}^{-2n-1}.\nonumber\end{gathered}$$
Suppose $n\geq1$ then $ \Vert S_{n} \Vert \leq\sum\limits_{i=0}^{n-1-i} \Vert \alpha_{n-1-i} \Vert ( 2N-4) \delta
_{0}^{-i-1}$ (since $ \Vert \tau((j,N-1))\Vert =1$). Furthermore, since $\big\vert \frac{\kappa}{n\pm2\kappa }\big\vert \leq\frac{\kappa_{0}}{n-2\kappa_{0}}$ for $n\geq2$, we find $$\begin{gathered}
\Vert \alpha_{2n+1}\Vert \leq\frac{2\lambda}{2n+1-2\kappa_{0}}\sum\limits_{i=0}^{2n} \Vert \alpha_{2n-i}\Vert \delta_{0}^{-i-1},\\
\Vert \alpha_{2n}\Vert \leq\frac{\lambda}{n}\sum\limits_{i=0}^{2n-1} \Vert \alpha_{2n-1-i}\Vert \delta_{0}^{-i-1}.\end{gathered}$$ To set up an inductive argument let $t_{n}$ denote the hypothetical bound on $\Vert \alpha_{n}(x(u,0))\Vert $ and set $v_{n}=\sum\limits_{i=0}^{n-1}t_{n-1-i}\delta_{0}^{-i-1}$; then $v_{n}=\delta_{0}^{-1}( t_{n-1}+v_{n-1}) $ for $n\geq2$. Setting $t_{2n}=\frac{\lambda}{n}v_{2n}$ and $t_{2n+1}=\frac{2\lambda}{2n+1-2\kappa_{0}}v_{2n+1}$ the recurrence relations become $$\begin{gathered}
t_{2n} =\frac{\lambda}{n}\left( t_{2n-1}+\frac{2n-1-2\kappa_{0}}{2\lambda }t_{2n-1}\right) \delta_{0}^{-1}=\frac{2\lambda+2n-1-2\kappa_{0}}{2n}t_{2n-1}\delta_{0}^{-1},\\
t_{2n+1} =\frac{2\lambda}{2n+1-2\kappa_{0}}\left( t_{2n}+\frac{n}{\lambda }t_{2n}\right) \delta_{0}^{-1}=\frac{2\lambda+2n}{2n+1-2\kappa_{0}}t_{2n}\delta_{0}^{-1}.\end{gathered}$$ Starting with $ \Vert \alpha_{1} \Vert \leq\frac{2\lambda}{1-2\kappa_{0}} \Vert \alpha_{0} \Vert \delta_{0}^{-1}=t_{1}$ the stated bounds are proved inductively.
By use of Stirling’s formula for $\frac{\Gamma ( n+a ) }{\Gamma(n+b) }\sim n^{a-b}$ we see that $t_{n}$ behaves like (a multiple of) $n^{2\lambda-1}$ for large $n$. Also there is a constant $c^{\prime}$ depending on $N$ and $\kappa_{0}$ such that $$\begin{gathered}
\sum_{n=2}^{\infty}\Vert \alpha_{n}(x(u,0)) \Vert \vert z\vert ^{n}\leq c^{\prime}\Vert \alpha_{0}(x(u,0)) \Vert \left( \frac{\vert z\vert }{\delta_{0}}\right) ^{2}\left( 1-\frac{\vert z\vert }{\delta_{0}}\right) ^{-2\lambda-2}.\label{bndan2z}\end{gathered}$$
We also need to analyze the effect of small changes in $u$. Fix a point $x( \widetilde{u},0) $ and consider series expansions of $\alpha_{n}( x( \widetilde{u},0)) $ in powers of $ ( u-\widetilde{u} ) $. Let $\delta_{1}:=\min\limits_{1\leq j\leq N-2}\vert \widetilde{u}-x_{j}\vert $. Recall equation (\[dua0x\]) $$\begin{gathered}
\partial_{u}\alpha_{0}(x(u,0)) =\kappa\alpha _{0}(x(u,0)) \{ \beta_{0}( x ( u,0) ) +\sigma\beta_{0}(x(u,0))\sigma\} ,\end{gathered}$$ and solve this in the form $$\begin{gathered}
\alpha_{0}(x(u,0)) =\sum_{n=0}^{\infty}\alpha_{0,n} ( x ( \widetilde{u},0 ) ) (u-\widetilde{u}) ^{n}.\end{gathered}$$ This leads to the recurrence (suppressing the arguments $x(\widetilde{u},0) $) $$\begin{gathered}
\sum_{n=1}^{\infty}n\alpha_{0,n} ( u-\widetilde{u} ) ^{n-1} \\
\qquad{} =\kappa\sum_{n=0}^{\infty}\alpha_{0,n} ( u-\widetilde{u} ) ^{n}
\sum_{m=0}^{\infty}(-1) ^{m} ( u-\widetilde {u} ) ^{m}\sum_{j=0}^{N-2}\frac{\tau ( (j,N-1) ) +\tau((j,N)) }{( \widetilde{u}-x_{j}) ^{m+1}},\\
( n+1) \alpha_{0,n+1} =\kappa\sum_{m=0}^{n}\alpha_{0,n-m}\widetilde{\beta}_{m}( x( \widetilde{u},0) ),\\
\widetilde{\beta}_{m}( x( \widetilde{u},0) ):=(-1) ^{m}\sum_{j=0}^{N-2}\frac{\tau((j,N-1)) +\tau((j,N)) }{(\widetilde{u}-x_{j}) ^{m+1}}.\end{gathered}$$ Thus $\Vert \widetilde{\beta}_{m}\Vert \leq\frac{2(N-2) }{\delta_{1}^{m+1}}$ and by a similar method as above we find $$\begin{gathered}
\Vert \alpha_{0,n} ( x ( \widetilde{u},0 ))\Vert \leq\frac{( 2\lambda) _{n}}{n!}\delta_{1}^{-n}\Vert \alpha_{0}( x( \widetilde{u},0))\Vert, \label{a0nbd}\end{gathered}$$ where $\lambda=( N-2) \kappa_{0}$. From $$\begin{gathered}
\alpha_{1}(x(u,0)) =\rho\left( \frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right) \alpha_{0}( x(u,0) ) \sum_{j=0}^{N-2}\frac{\tau( (j,N-1)
) -\tau((j,N)) }{(u-x_{j})}\\
\hphantom{\alpha_{1}(x(u,0))}{} =\rho\left( \frac{\kappa}{1-2\kappa},\frac{\kappa}{1+2\kappa}\right)\sum_{n=0}^{\infty}\alpha_{0,n} ( u-\widetilde{u} )^{n}\\
\hphantom{\alpha_{1}(x(u,0))=}{} \times\sum_{m=0}^{\infty}(-1) ^{m} ( u-\widetilde {u}) ^{m}\sum_{j=0}^{N-2}\frac{\tau( (j,N-1)) -\tau((j,N)) }{(\widetilde
{u}-x_{j}) ^{m+1}},\end{gathered}$$ we can derive a recurrence for the coefficients in $\alpha_{1}( x ( u,0)) =\sum\limits_{n=0}^{\infty}\alpha_{1,n}( x(\widetilde{u},0))( u-\widetilde{u}) ^{n}$. Also $$\begin{gathered}
\Vert \alpha_{1,n}( x( \widetilde{u},0) )\Vert \leq\frac{2\lambda}{1-2\kappa_{0}}\Vert \alpha_{0}(x( \widetilde{u},0) )\Vert \sum_{j=0}^{n}\frac{(2\lambda) _{n-j}}{(n-j) !}\delta_{1}^{j-n}\delta_{1}^{-j-1}\\
\hphantom{\Vert \alpha_{1,n}( x( \widetilde{u},0) )\Vert}{}
=\frac{2\lambda(2\lambda+1) _{n}}{( 1-2\kappa_{0}) n!}\delta_{1}^{-n-1}\Vert \alpha_{0}( x(\widetilde{u},0))\Vert ;\end{gathered}$$ note $2\lambda(2\lambda+1) _{n}=(2\lambda) _{n+1}$.
Essentially we are setting up bounds on behavior of $L( x (u,z)) $ for points near $x( \widetilde{u},0) $ in terms of $ \Vert \alpha_{0} (x(\widetilde{u},0))\Vert $ which is handled by the global bound.
In the series $$\begin{gathered}
\sum_{n=0}^{\infty}\alpha_{n}(x(u,0)) z^{n}=\sum_{m,n=0}^{\infty}\alpha_{n,m}( x( \widetilde{u},0)) z^{n}( u-\widetilde{u}) ^{m}\end{gathered}$$ the first order terms are$$\begin{gathered}
\alpha_{00}( x( \widetilde{u},0) ) +\alpha
_{0,1}( x( \widetilde{u},0)) ( u-\widetilde{u}) +\alpha_{1,0}( x( \widetilde{u},0)) z,\end{gathered}$$ and the bounds (\[bndan\]) for the omitted terms $$\begin{gathered}
\sum_{n=2}^{\infty}\Vert \alpha_{n}(x(u,0))\Vert \vert z\vert ^{n}\leq c^{\prime}\Vert \alpha_{0}(x(u,0))\Vert \left( \frac{\vert z\vert }{\delta_{0}}\right) ^{2}\left( 1-\frac{\vert z\vert}{\delta_{0}}\right) ^{-2\lambda-2},\nonumber\\
\sum\limits_{n=2}^{\infty}\Vert \alpha_{0,n}( x(\widetilde{u},0) ) \Vert \vert u-\widetilde{u}\vert ^{n} \leq(2\lambda) _{2}\left( \frac
{\vert u-\widetilde{u}\vert }{\delta_{1}}\right) ^{2}\left(
1-\frac{\vert u-\widetilde{u}\vert }{\delta_{1}}\right)
^{-2\lambda-2}\Vert \alpha_{0}( x( \widetilde{u},0))\Vert ,\label{dblseries}\\
\vert z\vert \sum_{n=1}^{\infty}\Vert \alpha_{1,n}(
x( \widetilde{u},0)) \Vert \vert
u-\widetilde{u}\vert ^{n} \leq\frac{(2\lambda) _{2}
}{1-2\kappa_{0}}\left( \frac{\vert z( u-\widetilde{u})
\vert }{\delta_{1}^{2}}\right) \left( 1-\frac{\vert
u-\widetilde{u}\vert }{\delta_{1}}\right) ^{-2\lambda-2}\Vert
\alpha_{0}( x( \widetilde{u},0))\Vert.\nonumber\end{gathered}$$ Note there is a difference between $\delta_{0}$ and $\delta_{1}$: $\delta _{0},\delta_{1}$ are the distances from the nearest $x_{j}$ ($1\leq j\leq
N-2)$ to $u,\widetilde{u}$ respectively; thus the double series converges in $\vert z\vert +\vert u-\widetilde{u}\vert <\delta_{1}$ because this implies $\vert z\vert <\delta_{1}- \vert u-\widetilde{u} \vert \leq\delta_{0}$, by the triangle inequality: $\delta_{1}\leq\vert u-\widetilde{u}\vert +\delta_{0}$ .
Sufficient condition for the inner product property {#suffco}
===================================================
In this section we will use the series $$\begin{gathered}
L_{1}(y,u-z,u+z) =\left( \prod\limits_{j=1}^{N}x_{j}\right) ^{-\gamma\kappa}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty
}\alpha_{n}(x(u,0)) z^{n},\end{gathered}$$ normalized by $\alpha_{0}\big( x^{(0) }\big) =I$ where $x^{(0) }=\big( 1,\omega,\omega^{2},\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big) $, $\omega=e^{2\pi\mathrm{i}/N}$. The hypothesis is that there exists a Hermitian matrix $H$ such that $\upsilon H=H\upsilon$ (recall $\upsilon:=\tau(w_{0})$) and the matrix $H_{1}$ defined by $$\begin{gathered}
L_{1}(x) ^{\ast}H_{1}L_{1}(x) =L(x) ^{\ast}HL(x) \label{LHL}\end{gathered}$$ commutes with $\sigma=\tau(N-1,N) $ (recall $L (x_{0}) =I$). Setting $x=x_{0}$ we find that $H=L_{1}(x_{0}) ^{\ast}H_{1}L_{1}(x_{0}) $. The analogous condition has to hold for each face of $\mathcal{C}_{0}$ and any such face can be obtained from $\{ x_{N-1}=x_{N}\} $ by applying $x\mapsto xw_{0}^{m}$ with suitable $m$. For notational simplicity we will work out only the $\{ x_{N-2}=x_{N-1}\} $ case. From the general relation $w(i,j) w^{-1}=(w(i),w(j))$ we obtain $w_{0}^{-1}(N-1,N) w_{0}=(N-2,N-1)$. A matrix $M$ commutes with $\tau(N-2,N-1)$ if and only if $\upsilon M\upsilon^{-1}$ commutes with $\sigma$. Let $$\begin{gathered}
x^{\prime} =\big( x_{1}^{\prime},\ldots,x_{N-3}^{\prime},u-z,u+z,x_{N}^{\prime}\big) ,\\
x^{\prime}w_{0}^{-1} =\big( x_{N}^{\prime},x_{1}^{\prime},\ldots ,x_{N-3}^{\prime},u-z,u+z\big) =x,\\
L_{2}( x^{\prime}) :=\upsilon^{-1}L_{1}\big( x^{\prime}w_{0}^{-1}\big) \upsilon =\left( \prod\limits_{j=1}^{N}x_{j}\right) ^{-\gamma\kappa}\upsilon
^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(y,u) \upsilon z^{n}.\end{gathered}$$ This is a solution of (\[Lsys\]) by Proposition \[L(xw)\]. This has the analogous behavior to $L_{1}$; writing $$\begin{gathered}
\upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \alpha_{n} (y,u) \upsilon=\big\{ \upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa
}\big) \upsilon\big\} \big\{ \upsilon^{-1}\alpha_{n} (y,u) \upsilon\big\}\end{gathered}$$ implies the relations $$\begin{gathered}
\tau(N-2,N-1) \big\{ \upsilon^{-1}\rho\big( z^{-\kappa },z^{\kappa}\big) \upsilon\big\} =\big\{ \upsilon^{-1}\rho\big(
z^{-\kappa},z^{\kappa}\big) \upsilon\big\} \tau(N-2,N-1),\\
\tau(N-2,N-1) \big\{ \upsilon^{-1}\alpha_{n} (y,u) \upsilon\big\} \tau(N-2,N-1) = (-1) ^{n}\big\{ \upsilon^{-1}\alpha_{n}(y,u)
\upsilon\big\} ,\qquad n\geq0.\end{gathered}$$ We claim that the Hermitian matrix $H_{2}$ defined by $$\begin{gathered}
L_{2}(x) ^{\ast}H_{2}L_{2}(x) =L(x)^{\ast}HL(x) \label{L2H2L2}\end{gathered}$$ commutes with $\tau(N-2,N-1) $. There is a subtle change: the base point $x^{(0)}=\big( 1,\omega,\ldots,\omega^{N-2}$, $\omega^{-3/2},\omega^{-3/2}\big) $ is replaced by $\big( \omega ,\ldots,\omega^{N-2},\omega^{-3/2},\omega^{-3/2},1\big) $ and now $\omega x_{0}=\big( \omega,\ldots,\omega^{N-1},1\big) $ is in the domain of convergence of $L_{2}$. Set $x=\omega x_{0}$ in (\[L2H2L2\]) to obtain $$\begin{gathered}
L_{2}(\omega x_{0}) =\upsilon^{-1}L_{1}\big( \omega x_{0}w_{0}^{-1}\big) \upsilon=\upsilon^{-1}L_{1}(x_{0})\upsilon,\\
H_{2} =( L_{2}(\omega x_{0}) ^{\ast})^{-1}HL_{2}(\omega x_{0}) ^{-1}=\upsilon^{-1}(L_{1}(x_{0}) ^{\ast}) ^{-1}\upsilon H\upsilon^{-1}L_{1}(x_{0}) ^{-1}\upsilon\\
\hphantom{H_{2}}{} =\upsilon^{-1} ( L_{1}(x_{0}) ^{\ast} )^{-1}HL_{1}(x_{0}) ^{-1}\upsilon=\upsilon^{-1}H_{1}\upsilon,\end{gathered}$$ because $H$ commutes with $\upsilon$ (and $L(\omega x_{0}) =L(x_{0}) =I$ by the homogeneity). Thus $H_{2}$ commutes with $\tau(N-2,N-1) $.
From Theorem \[Lbnd\] we have the bound $$\begin{gathered}
\Vert L(x) ^{\ast}HL(x)\Vert \leq c\prod\limits_{i<j}\vert x_{i}-x_{j} \vert ^{-2\vert \kappa\vert }.
\end{gathered}$$ Denote $K(x) =L(x) ^{\ast}HL(x) $. We will show that there is an interval $-\kappa_{1}<\kappa<\kappa_{1}$ where $\kappa_{1}$ depends on $N$ such that $$\begin{gathered}
\int_{\mathbb{T}^{N}}\big\{ ( x_{N}\mathcal{D}_{N}f ) ^{\ast}(x) K(x) g(x) -f^{\ast} (x t) K(x) x_{N}\mathcal{D}_{N}g(x) \big\}\mathrm{d}m(x) =0,\end{gathered}$$ for each $f,g\in C^{1}\big( \mathbb{T}^{N};V_{\tau}\big) $. Consider the Haar measure of $\big\{ x\colon \min\limits_{i<j}\vert x_{i}-x_{j}\vert
<\varepsilon\big\} $; let $\sin\frac{\varepsilon^{\prime}}{2} =\frac{\varepsilon}{2}$ and $i<j$ then $m\big\{ x\colon \vert x_{i}-x_{j}\vert \leq\varepsilon\big\} =\frac{1}{\pi}\varepsilon^{\prime}$, thus $m\big\{ x\colon \min\limits_{i<j}\vert x_{i}-x_{j}\vert <\varepsilon\big\} \leq \binom{N}{2}\frac{\varepsilon^{\prime}}{\pi}$. The integral is broken up into three pieces. The aim is to let $\delta \rightarrow0$; where $\delta$ satisfies an upper bound $\delta<\min\big(\big( 2\sin\frac{\pi}{N}\big) ^{2},\frac{1}{9}\big)$; the first term comes from the maximum spacing of $N$ points on $\mathbb{T}$ and the second is equivalent to $3\delta<\delta^{1/2}$. Also $\delta^{\prime}:=2\arcsin\frac{\delta}{2}$.
1. $\min\limits_{i<j}\vert x_{i}-x_{j}\vert <\delta$, done with the integrability of $\prod\limits_{i<j}\vert x_{i}-x_{j}\vert
^{-2\vert \kappa\vert }$ for $\vert \kappa\vert <\frac{1}{N}$ (from the Selberg integral $\int_{\mathbb{T}^{N}}\prod \limits_{i<j}\vert x_{i}-x_{j}\vert ^{-2 \vert \kappa \vert }\mathrm{d}m(x) =\frac{\Gamma 1-N \vert \kappa\vert) }{\Gamma( 1-\vert \kappa\vert) ^{N}}$), and the measure of the set is $O( \delta) $. The limit as $\delta\rightarrow0$ is zero by the dominated convergence theorem.
2. $\delta\leq\min\limits_{1\leq i<j\leq N-1}\vert x_{i}-x_{j}\vert <\delta^{1/2}$ and $\delta\leq\min\limits_{1\leq i\leq N-1}\vert x_{i}-x_{N}\vert $; this case uses the same bound on $K$ and the $\mathbb{T}^{N-1}$-Haar measure of $\big\{ x\in \mathbb{T}^{N-1} \colon \min\limits_{1\leq i<j\leq N-1} \vert x_{i} -x_{j} \vert <\delta^{1/2}\big\} ;$
3. $\min\limits_{1\leq i<j\leq N-1}\vert x_{i}-x_{j}\vert \geq\delta^{1/2}$ and $\min\limits_{1\leq i\leq N-1}\vert x_{i}-x_{N}\vert \geq\delta$. This is done with a detailed analysis using the double series from (\[dblseries\]).
The total of parts (2) and (3), that is, the integral over $\Omega_{\delta}$, equals $$\begin{gathered}
\int_{\Omega_{\delta}}x_{N}\partial_{N}\{ f(x)^{\ast}K(x) g(x)\} \mathrm{d}m (x).\end{gathered}$$ We use the coordinates $x_{j}=e^{\mathrm{i}\theta_{j}}$, $1\leq j\leq N$; thus $x_{N}\partial_{N}=-\mathrm{i}\frac{\partial}{\partial
\theta_{N}}$. For fixed $( \theta_{1},\ldots,\theta_{N-1}) $ the condition $x\in\Omega_{\delta}$ implies that the set of $\theta_{N}$-values is a union of disjoint closed intervals (it is possible there is only one, in the extreme case $\theta_{j}=j\delta^{\prime}$ for $1\leq j\leq N-1$ the interval is $N\delta^{\prime}\leq\theta_{N}\leq2\pi$). In case (2) the $\theta_{N}$-integration results in a sum of terms $( f^{\ast}Kg) \big(e^{\mathrm{i}\theta_{1}}, \ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\phi }\big) $ with coefficients $\pm1$ where $\min\limits_{1\leq i\leq N-1} \vert \phi-\theta_{i} \vert =\delta$. Each such sum is bounded by $2(N-1) c\Vert f\Vert _{\infty}\Vert g \Vert _{\infty}\delta^{-N(N-1) \vert \kappa \vert }$, because $\prod\limits_{i<j}\vert x_{i}-x_{j}\vert \geq\delta^{N(N-1) /2}$ on $\Omega_{\delta}$. Thus the integral for part (2) is bounded by $$\begin{gathered}
2(N-1)\Vert f\Vert _{\infty}\Vert g\Vert _{\infty}\delta^{-N(N-1) \vert \kappa\vert }\binom{N-1}{2}\left( 2\arcsin\frac{\delta^{1/2}}{2}\right) \leq c^{\prime
}\delta^{1/2-N(N-1) \vert \kappa\vert },\end{gathered}$$ for some finite constant $c^{\prime}$ (depending on $f$, $g$). This term tends to zero as $\delta\rightarrow0$ if $\vert \kappa\vert <\frac {1}{2N(N-1) }$.
In part (3) the intervals $[ \theta_{i}-\delta^{\prime},\theta_{i}+\delta^{\prime}] $ are pairwise disjoint because $\vert\theta_{i}-\theta_{j}\vert \geq3\delta^{\prime}$ for $i\neq j$ (recall $\sqrt{\delta}>3\delta$). To simplify the notation assume $\theta_{1}<\theta_{2}<\cdots$ (the other cases follow from the group invariance of the setup). Then the $\theta_{N}$-integration yields $$\begin{gathered}
(2\pi) ^{1-N}\sum_{j=1}^{N-1}\int_{R_{\delta}}\left\{
\begin{matrix}
( f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots
,e^{\mathrm{i}\theta_{j}},\ldots,e^{\mathrm{i} ( \theta_{j}-\delta^{\prime} ) }\big) \\
- ( f^{\ast}Kg ) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{j}},\ldots,e^{\mathrm{i} ( \theta_{j}+\delta^{\prime} ) }\big)
\end{matrix}
\right\} \mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N-1},\end{gathered}$$ where $R_{\delta}:=\big\{ ( \theta_{1},\ldots,\theta_{N-1} ) :\theta_{1}<\theta_{2}<\cdots<\theta_{N-1}<\theta_{1}+2\pi,\min \big\vert
e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i}\theta_{k}}\big\vert \geq\sqrt{\delta }\big\}$. It suffices to deal with the term with $j=N-1$; this allows the use of the double series. It is fairly easy to show that $ ( f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}( \theta_{N-1}-\delta^{\prime}) }\big) - (f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}( \theta_{N-1}+\delta^{\prime})}\big) $ tends to zero with $\delta$ but this is not enough to control the integral. The idea is to show that $$\begin{gathered}
\big\vert ( f^{\ast}Kg ) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}-\delta^{\prime} ) }\big) - (f^{\ast}Kg) \big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime} ) }\big) \big\vert \\
\qquad{} \leq c''\delta^{1/2-2\vert \kappa\vert }\big\vert (f^{\ast}Kg) \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime} )}\big) \big\vert\end{gathered}$$ for some constant $c''$. This can then be bounded using the $ \Vert K\Vert $ bound for sufficiently small $\vert \kappa\vert $. Fix $y=\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-2}}\big) $ and let $x ( y,u-v,u+v ) $ denote $\big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-2}},u-z,u+z\big)$. We will use the form $K=L_{1}^{\ast}H_{1}L_{1}$ from (\[LHL\]) with two pairs of values along with $\widetilde{u}=e^{\mathrm{i}\theta_{N-1}}$, and set $\zeta=e^{\mathrm{i\delta}^{\prime}}$
1. $\eta^{(1) }=x ( y,u_{1}-z_{1},u_{1}+z_{1} ) =x\big( y,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta
_{N-1}+\delta^{\prime} ) }\big) $, then $u_{1}=\frac{1}{2}\widetilde{u} ( 1+\zeta ) $, $z_{1}=\frac{1}{2}\widetilde{u} (\zeta-1 ) $, $u_{1}-\widetilde{u}=z_{1}$, $ \vert z_{1} \vert =\delta$,
2. $\eta^{(2) }=x ( y,u_{2}-z_{2},u_{2}+z_{2} ) =x\big( y,e^{\mathrm{i} ( \theta_{N-1}-\delta^{\prime} ) },e^{\mathrm{i}\theta_{N-1}}\big) $, then $u_{2}=\frac{1}{2}\widetilde {u}\big( 1+\zeta^{-1}\big) $, $z_{2}=\frac{1}{2}\widetilde{u}\big(1-\zeta^{-1}\big) =\zeta^{-1}z_{1}$, $u_{2}-\widetilde{u}=-z_{2}$, $ \vert z_{2} \vert =\delta$.
Let $\eta^{(3) }\!=\!x\big( y,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}-\delta^{\prime} ) }\big)\! =\!\eta^{(2) }(N-1,N) $, then by construction $K\big( \eta^{(3) }\big) \!=\!\sigma K\big( \eta^{(2) }\big) \sigma$. We start by disposing of the $f$ and $g$ factors: by uniform continuous differentiability there is a constant $c^{\prime \prime\prime}$ such that $\big\Vert f\big( \eta^{(1)}\big) -f\big( \eta^{(3) }\big) \big\Vert \leq c^{\prime\prime\prime}\delta^{\prime}$ and $\big\Vert g\big( \eta^{(1) }\big) -g\big( \eta^{(3) }\big) \big\Vert \leq c^{\prime\prime\prime}\delta^{\prime}$ (same constant for all of $\mathbb{T}^{N}$). So the error made by assuming $f$ and $g$ are constant is bounded by $c^{\prime\prime\prime}\delta^{\prime}\big( \big\Vert K\big(\eta^{(1) }\big) \big\Vert +\big\Vert K\big(\eta^{(2) }\big) \big\Vert \big) $. The problem is reduced to bounding $K\big( \eta^{(1) }\big) -\sigma K\big( \eta^{(2) }\big) \sigma$. To add more detail about the effect of the $\ast$-operation on $u$ and $z$ we compute $$\begin{gathered}
z^{\ast}=\frac{1}{2}\left( \frac{1}{u+z}-\frac{1}{u-z}\right) =-\frac{z}{u^{2}-z^{2}}, \qquad u^{\ast}=\frac{1}{2}\left( \frac{1}{u+z}+\frac{1}{u-z}\right)=\frac{u}{u^{2}-z^{2}}\end{gathered}$$ and if $u-z=e^{i\theta_{N-1}}$, $u+z=e^{i\theta_{N}}$ then $$\begin{gathered}
z =\frac{1}{2}e^{\mathrm{i} ( \theta_{N-1}+\theta_{N} )
/2}\big( e^{\mathrm{i} ( \theta_{N}-\theta_{N-1} ) /2}-e^{\mathrm{i} ( \theta_{N-1}-\theta_{N} ) /2}\big)
=\mathrm{i}e^{\mathrm{i} ( \theta_{N-1}+\theta_{N} ) /2}\sin \frac{\theta_{N}-\theta_{N-1}}{2},\\
z^{\ast} =\mathrm{i}e^{-\mathrm{i} ( \theta_{N-1}+\theta_{N} )/2}\sin\frac{\theta_{N-1}-\theta_{N}}{2}=\overline{z},\\
u =\frac{1}{2}e^{\mathrm{i} ( \theta_{N-1}+\theta_{N} ) /2}\big( e^{\mathrm{i} ( \theta_{N}-\theta_{N-1} ) /2}+e^{\mathrm{i} ( \theta_{N-1}-\theta_{N} ) /2}\big) =e^{\mathrm{i}( \theta_{N-1}+\theta_{N}) /2}\cos\frac{\theta_{N}-\theta_{N-1}}{2},\\
u^{\ast} =e^{-\mathrm{i}( \theta_{N-1}+\theta_{N}) /2}\cos\frac{\theta_{N-1}-\theta_{N}}{2}=\overline{u};\end{gathered}$$ the $\ast$-operation agrees with complex conjugate on the torus and $\rho( z^{-\kappa},z^{\kappa}) ^{\ast}=\rho( \overline{z}^{-\kappa},\overline{z}^{\kappa}) $. The reason for this is to emphasize that $L(x) ^{\ast}$ is an analytic function agreeing with the (Hermitian) adjoint of $L(x)$. Thus $$\begin{gathered}
K\big( \eta^{(1) }\big) =\sum_{n,m=0}^{\infty}\alpha_{n}( x(u_{1},0)) ^{\ast}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big) ^{\ast}H_{1}\rho\big( z_{1}^{-\kappa
},z_{1}^{\kappa}\big) \alpha_{m} ( x(u_{1},0) ) ( z_{1}^{\ast} ) ^{m}z_{1}^{n}\nonumber\\
\hphantom{K\big( \eta^{(1) }\big)}{} =\sum_{n,m=0}^{\infty}\alpha_{n} ( x(u_{1},0) )
^{\ast}H_{1}\rho \big( \vert z_{1} \vert ^{-2\kappa}, \vert z_{1} \vert ^{2\kappa}\big) \alpha_{m} ( x(u_{1},0) ) \overline{z_{1}}^{m}z_{1}^{n},\label{Keta1}\end{gathered}$$ because $H_{1}$ commutes with $\sigma$ and hence with $\rho ( z_{1}^{-\kappa},z_{1}^{\kappa}) $, and $$\begin{gathered}
K\big( \eta^{(3) }\big) =\sigma K\big( \eta^{(2) }\big) \sigma\nonumber\\
\hphantom{K\big( \eta^{(3) }\big)}{} =\sum_{n,m=0}^{\infty}(-1)
^{m+n}\alpha_{n}(x(u_{2},0)) ^{\ast}H_{1}\rho\big(\vert z_{2}\vert ^{-2\kappa},\vert z_{2}\vert ^{2\kappa}\big) \alpha_{m}(x( u_{2},0)) \overline{z_{2}}^{m}z_{2}^{n},\label{Keta3}\end{gathered}$$ because $\sigma\alpha_{n}(x(u_{2},0)) \sigma=(-1) ^{n}\alpha_{n}(x(u_{2},0)) $ for $n\geq0$. Now we use the expansion in powers of $(u-\widetilde{u}) ^{n}$ to evaluate $K\big( \eta^{(1)}\big) -K\big( \eta^{(3) }\big) $. From the inequality (\[bndan2z\]) $$\begin{gathered}
\sum_{n=2}^{\infty} \Vert \alpha_{n}(x(u,0))
\Vert \vert z\vert ^{n} \leq c^{\prime} \Vert
\alpha_{0}(x(u,0)) \Vert \left(
\frac{\vert z\vert }{\delta_{0}}\right) ^{2}\left( 1-\frac
{\vert z\vert }{\delta_{0}}\right) ^{-2\lambda-2}\nonumber\\
\hphantom{\sum_{n=2}^{\infty} \Vert \alpha_{n}(x(u,0)) \Vert \vert z\vert ^{n}}{} =c^{\prime}\delta \Vert \alpha_{0}(x(u,0))
\Vert \big( 1-\delta^{1/2}\big) ^{-2\lambda-2}\label{bnd1}\end{gathered}$$ with $\delta_{0}=\min\limits_{1\leq j\leq N-2} \vert u-x_{j} \vert =\delta^{1/2}$ we can restrict the problem to $0\leq n,m\leq1$. The omitted terms in $K\big( \eta^{(1) }\big) -K\big( \eta^{(3) }\big) $ are bounded by $c^{\prime\prime}\delta^{1-2 \vert \kappa \vert } \Vert B_{1} \Vert \Vert \alpha_{0} (x( u_{2},x) ) \Vert ^{2}$, for some constant $c^{\prime\prime}$. Then $$\begin{gathered}
L_{1}\big(\eta^{(1)}\big) =\left( \prod \limits_{j=1}^{N}x_{j}^{(1) }\right) ^{-\gamma\kappa}\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big) \left\{
\begin{matrix}
\alpha_{00}(x(\widetilde{u},0)) +\alpha
_{0,1}(x(\widetilde{u},0)) ( u_{1}-\widetilde{u}) \\
+\alpha_{1,0}(x(\widetilde{u},0)) z_{1}+O(\delta)
\end{matrix}
\right\} ,\\
\sigma L_{1}\big(\eta^{(2)}\big) \sigma =\left(\prod\limits_{j=1}^{N}x_{j}^{(2) }\right) ^{-\gamma\kappa}\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) \left\{
\begin{matrix}
\alpha_{00}(x(\widetilde{u},0)) +\alpha_{0,1}(x(\widetilde{u},0)) ( u_{2}-\widetilde{u}) \\
-\alpha_{1,0}(x(\widetilde{u},0)) z_{2}+O(\delta)
\end{matrix}
\right\} ,\end{gathered}$$ because $\sigma\alpha_{1}(x(u,0)) \sigma= (-1) ^{n}\alpha_{1}(x(u,0)) $. The terms $O(\delta) $ correspond to the bound in (\[bnd1\]). Drop the argument $x(\widetilde{u},0) $ for brevity. Combining these with (\[Keta1\]) and (\[Keta3\]) we obtain $$\begin{gathered}
K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) = \{ \alpha_{0,1} ( u_{1}-\widetilde{u} )
+\alpha_{1,0}z_{1} \} ^{\ast}H_{1}\rho\big( \vert z_{1}\vert ^{-2\kappa},\vert z_{1}\vert ^{2\kappa}\big)\alpha_{00}\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{} +\alpha_{00}^{\ast}H_{1}\rho\big(\vert z_{1}\vert ^{-2\kappa
}, \vert z_{1} \vert ^{2\kappa}\big) \{ \alpha_{0,1} (u_{1}-\widetilde{u}) +\alpha_{1,0}z_{1}\} \\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
+ \{ \alpha_{0,1} ( u_{1}-\widetilde{u} ) +\alpha_{1,0}z_{1} \} ^{\ast}H_{1}\rho\big(\vert z_{1}\vert ^{-2\kappa},\vert z_{1}\vert ^{2\kappa}\big)\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
\times \{ \alpha_{0,1}(u_{1}-\widetilde{u}) +\alpha_{1,0}z_{1}\} \\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
- \{ \alpha_{0,1} ( u_{2}-\widetilde{u} ) -\alpha_{1,0}z_{2} \} ^{\ast}H_{1}\rho\big(\vert z_{2}\vert ^{-2\kappa},\vert z_{2}\vert ^{2\kappa}\big) \alpha_{00}\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
-\alpha_{00}^{\ast}H_{1}\rho\big( \vert z_{2}\vert ^{-2\kappa},\vert z_{2}\vert ^{2\kappa}\big) \{ \alpha_{0,1}(
u_{1}-\widetilde{u}) -\alpha_{1,0}z_{1}\} \\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
- \{ \alpha_{0,1} ( u_{2}-\widetilde{u} ) +\alpha_{1,0}z_{2}\} ^{\ast}H_{1}\rho\big(\vert z_{2}\vert ^{-2\kappa
},\vert z_{2}\vert ^{2\kappa}\big)\\
\hphantom{K\big(\eta^{(1)}\big) -K\big( \eta^{(3)}\big) =}{}
\times \{ \alpha_{0,1}(u_{2}-\widetilde{u}) +\alpha_{1,0}z_{2}\} +O(\delta) .\end{gathered}$$ The key fact is that the $\alpha_{00}^{\ast}H_{1}\rho\big( \vert z_{1} \vert ^{-2\kappa}, \vert z_{1} \vert ^{2\kappa}\big) \alpha_{00}$ terms cancel out ($ \vert z_{1} \vert = \vert z_{2} \vert $).
From $\Vert \alpha_{1,0}(x(\widetilde{u},0)) \Vert \leq\frac{(2\lambda) _{2}}{( 1-2\kappa_{0}) }\delta_{1}^{-1}\Vert \alpha_{0}( x(\widetilde{u},0)) \Vert $ and $\Vert \alpha_{0,1}(x(\widetilde{u},0)) \Vert\leq2\lambda\delta_{1}^{-1} \Vert \alpha_{0} ( x ( \widetilde {u},0))\Vert $ (from (\[a0nbd\])) $\delta_{1}=\delta_{0}-\vert u\vert =\delta^{1/2}-\delta=\delta^{1/2} (1-\delta^{1/2}) $. Thus the sum of the first order terms in $K\big(
\eta^{(1) }\big) -K\big( \eta^{(3)}\big)$ is bounded by $c^{\prime\prime\prime} \Vert \alpha_{0} ( x (\widetilde{u},0 ) ) \Vert ^{2}\delta^{1/2-2\vert
\kappa \vert }\big(1-\delta^{1/2}\big) ^{-1} \Vert H_{1}\Vert $, where the constant $c^{\prime\prime\prime}$ is independent of $x(\widetilde{u},0) $ (but is dependent on $\kappa_{0}$ and $N$). Note $ \vert u_{1}-\widetilde{u} \vert = \vert u_{2}-\widetilde{u}\vert =\vert z_{1}\vert =\vert z_{2}\vert =\delta$. The second last step is to relate $\Vert \alpha_{0}(x(\widetilde{u},0))\Vert $ to $\Vert L_{1}\big(\eta^{(1)}\big) \Vert $; indeed $$\begin{gathered}
L_{1}\big(\eta^{(1)}\big) =\left( \prod\limits_{j=1}^{N}x_{j}^{(1) }\right) ^{-\gamma\kappa}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big) \left\{ \alpha_{0}( x (\widetilde{u},0)) +\sum_{n=1}^{\infty}\alpha_{n}(x(\widetilde{u},0)) z_{1}^{n}\right\} .\end{gathered}$$ Similarly to (\[bnd1\]) $$\begin{gathered}
\sum_{n=1}^{\infty}\Vert \alpha_{n}( x( \widetilde{u},0) ) \Vert \vert z\vert ^{n} \leq
c^{\prime}\Vert \alpha_{0}( x(\widetilde{u},0) ) \Vert \left( \frac{\vert z\vert }{\delta_{0}}\right) \left( 1-\frac{\vert z\vert }{\delta_{0}}\right)
^{-2\lambda-1}\\
\hphantom{\sum_{n=1}^{\infty}\Vert \alpha_{n}( x( \widetilde{u},0) ) \Vert \vert z\vert ^{n}}{}
=c^{\prime} \Vert \alpha_{0} ( x(\widetilde{u},0)) \Vert \delta^{1/2}\big(1-\delta^{1/2}\big) ^{-2\lambda-1}\leq c^{\prime\prime}\Vert \alpha_{0}( x( \widetilde{u},0) ) \Vert \delta^{1/2},\end{gathered}$$ (if $\delta<\frac{1}{9}$ then $1-\delta^{1/2}>\frac{2}{3}$); thus $$\begin{gathered}
\Vert \alpha_{0}(x(\widetilde{u},0)) \Vert \big( 1-c^{\prime\prime}\delta^{1/2}\big) \leq\delta^{-\vert \kappa\vert } \big\Vert L_{1}\big( \eta^{(1)}\big) \big\Vert .\end{gathered}$$ By Theorem \[Lbnd\] $$\begin{gathered}
\big\Vert L\big(\eta^{(1)}\big) \big\Vert \leq
c\prod\limits_{1\leq i<j\leq N-1}\vert x_{i}-x_{j}\vert
^{-\vert \kappa\vert }\prod\limits_{j=1}^{N-2}\big\vert
e^{\mathrm{i}\theta_{j}}-e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime
} ) }\big\vert ^{-\vert \kappa\vert }\big\vert
e^{\mathrm{i}\theta_{N-1}}-e^{\mathrm{i} ( \theta_{N-1}+\delta^{\prime}) }\big\vert ^{-\vert \kappa\vert }\\
\hphantom{\big\Vert L\big(\eta^{(1)}\big) \big\Vert}{}
\leq c\delta^{-\vert \kappa\vert \{ (N+1)(N-2) /2+1\} },\end{gathered}$$ because the first two groups of terms satisfy the bound $\vert x_{i}-x_{j} \vert \geq\delta^{1/2}$. Combining everything we obtain the bound $$\begin{gathered}
\big\Vert K\big(\eta^{(1)}\big) -K\big( \eta^{(3) }\big) \big\Vert \leq c^{\prime\prime\prime} \Vert
\alpha_{0}(x(\widetilde{u},0)) \Vert^{2}\delta^{1/2-2\vert \kappa\vert }\big(1-\delta^{1/2}\big)
^{-1} \Vert B_{1} \Vert \\
\hphantom{\big\Vert K\big(\eta^{(1)}\big) -K\big( \eta^{(3) }\big) \big\Vert }{}
\leq c^{\prime\prime} \Vert H_{1} \Vert \delta^{1/2-2 \vert \kappa \vert -\vert \kappa\vert \{ (N+1)(N-2) +2\} }.\end{gathered}$$ The constant is independent of $\eta^{(1) }$ and the exponent on $\delta$ is $\frac{1}{2}-\vert \kappa\vert \big( N^{2}-N+2\big)$. Thus the integral of part (3) goes to zero as $\delta \rightarrow0$ if $\vert \kappa\vert <\big( 2\big(N^{2}-N+2\big) \big) ^{-1}$. This is a crude bound, considering that we know everything works for $-1/h_{\tau}<\kappa<1/h_{\tau}$, but as we will see, an open interval of $\kappa$ values suffices.
\[suffctH\]If there exists a Hermitian matrix $H$ such that $$\begin{gathered}
\upsilon H=H\upsilon \qquad \text{and} \qquad ( L_{1}(x_{0}) ^{\ast})^{-1}HL_{1}(x_{0}) ^{-1}
\end{gathered}$$ commutes with $\sigma$, and $-\big(2\big( N^{2}-N+2\big) \big) ^{-1}<\kappa<\big( 2\big( N^{2}-N+2\big) \big) ^{-1}$ then $$\begin{gathered}
\int_{\mathbb{T}^{N}}\{ ( x_{i}\mathcal{D}_{i}f(x)
) ^{\ast}L(x) ^{\ast}HL(x) g(x) -f(x) ^{\ast}L(x) ^{\ast}HL(x) x_{i}\mathcal{D}_{i}g(x) \} \mathrm{d}m(x) =0\end{gathered}$$ for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big)$ and $1\leq i\leq N$.
It is important that we can derive uniqueness of $H$ from the relation, because the conditions $\langle wf,wg\rangle = \langle f,g\rangle $, $\langle x_{i}f,x_{i}g\rangle =\langle f,g \rangle $, and $ \langle x_{i}\mathcal{D}_{i}f,g \rangle = \langle f,x_{i}\mathcal{D}_{i}g \rangle $ for $w\in\mathcal{S}_{N}$ and $1\leq i\leq N$ determine the Hermitian form uniquely up to multiplication by a constant. Thus the measure $K(x) \mathrm{d}m(x)$ is similarly determined, by the density of Laurent polynomials.
The orthogonality measure on the torus {#orthmu}
======================================
At this point there are two logical threads in the development. On the one hand there is a sufficient condition implying the desired orthogonality measure is of the form $L^{\ast}HL\mathrm{d}m$, specifically if $H$ commutes with $\upsilon$, $( L_{1}(x_{0}) ^{\ast})^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$, and $\vert \kappa\vert <( 2( N^{2}-N+2))^{-1}$. However we have not yet proven that $H$ exists. On the other hand in [@Dunkl2016] we showed that there does exist an orthogonality measure of the form $\mathrm{d}\mu=\mathrm{d}\mu_{S}+L^{\ast}HL\mathrm{d}m$ where $\operatorname{spt}\mu_{S}\subset\mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}$, $H$ commutes with $\upsilon$, and $-1/h_{\tau}<\kappa<1/h_{\tau}$ (the support of a Baire measure $\nu$, denoted by $\operatorname{spt}\nu$, is the smallest compact set whose complement has $\nu$-measure zero). In the next sections we will show that $( L_{1}(x_{0}) ^{\ast}) ^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$ and that $H$ is an analytic function of $\kappa$ in a complex neighborhood of this interval. Combined with the above sufficient condition this is enough to show that there is no singular part, that is, $\mu_{S}=0$. The proof involves the formal differential equation satisfied by the Fourier–Stieltjes series of $\mu$, which is used to show $\mu_{S}=0$ on $\big\{ x\in\mathbb{T}^{N}\colon \#\{x_{j}\} _{j=1}^{N}=N-1\big\} $ (that is, $x$ has at least $N-1$ distinct components). In turn this implies $( L_{1}^{\ast}(x_{0})) ^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$. The proofs unfortunately are not short. In the sequel $H$ refers to the Hermitian matrix in the formula for $\mathrm{d}\mu$ and $K$ denotes $L^{\ast}HL$. Also $H$ is positive-definite since the measure $\mu$ is positive (else there exists a vector $v$ with $Hv=0$ and then the $C^{(1) }\big( \mathbb{T}_{\rm reg}^{N};V_{\tau}\big) $ function given by $f(x) :=L(x) ^{-1}vg(x) $ where $g$ is a smooth scalar nonnegative function with support in a sufficiently small neighborhood of $x_{0}$, has norm $ \langle f,f \rangle =0$, a contradiction). Thus $H$ has a positive-definite square root $C$ which commutes with $\upsilon$. Now extend $CL(x) $ from $\mathcal{C}_{0}$ to all of $\mathbb{T}_{\rm reg}^{N}$ by Definition \[DefL(x)T\] and so $K(x) =L^{\ast}(x) C^{\ast}CL(x) $ for all $x\in\mathbb{T}_{\rm reg}^{N}$ (this follows from $K(xw) =\tau(w) ^{-1}K(x) \tau(w) $).
Furthermore $\int_{\mathbb{T}^{N}} \Vert K(x)\Vert \mathrm{d}m(x) <\infty$ because $K\mathrm{d}m$ is the absolutely continuous part of the finite Baire measure $\mu$.
We will show that $(L_{1}^{\ast}(x_{0})) ^{-1}C^{\ast}CL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$. The proof begins by establishing a recurrence relation for the Fourier coefficients of $K(x) $, which comes from equation (\[Kdieq\]). For $F(x) $ integrable on $\mathbb{T}^{N}$, possibly matrix-valued, and $\alpha\in\mathbb{Z}^{N}$ let $\widehat{F}_{\alpha}=\int_{\mathbb{T}^{N}}F(x) x^{-\alpha}\mathrm{d}m(x)$. Clearly $\int_{\mathbb{T}^{N}}x^{\beta}F (x) x^{-\alpha}\mathrm{d}m(x) =\widehat{F}_{\alpha-\beta}$; and if $\partial_{i}F(x) $ is also integrable then (integration-by-parts) $$\begin{gathered}
\int_{\mathbb{T}^{N}}x_{i}\partial_{i}F(x) x^{-\alpha}\mathrm{d}m(x) =\alpha_{i}\int_{\mathbb{T}^{N}}F (x) x^{-\alpha}\mathrm{d}m(x) .\label{diffFC}\end{gathered}$$ For a subset $J\subset\{1,2,\ldots,N\} $ let $\varepsilon_{J}\in\mathbb{N}_{0}^{N}$ be defined by $( \varepsilon_{J}) _{i}=1$ if $i\in J$ and $=0$ otherwise; also $\varepsilon_{i}:=\varepsilon_{\{i\} }$. For $1\leq i\leq N$ let $$\begin{gathered}
E_{i} :=\{1,2,\ldots,N\} \backslash\{i\} ,\qquad E_{ij} :=E_{i}\backslash\{j\} ,\\
p_{i}(x) :=\prod\limits_{j\neq i}(x_{i}-x_{j}) =\sum\limits_{\ell=0}^{N-1}(-1) ^{\ell}x_{i}^{N-1-\ell}\sum\limits_{J\subset E_{i},\#J=\ell}x^{\varepsilon_{J}}.\end{gathered}$$ Equation (\[Kdieq\]) can be rewritten as $$\begin{gathered}
p_{i}(x) x_{i}\partial_{i}K(x) =\kappa\sum_{j\neq i}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) \{ x_{j}\tau((i,j)) K(x) +K(x) \tau((i,j)) x_{i}\};\label{Kdiffeq1}\end{gathered}$$ this is a polynomial relation which shows that $p_{i}(x) x_{i}\partial_{i}K(x) $ is integrable and which has implications for the Fourier coefficients of $K$.
For $1\leq i\leq N$ and $\alpha\in\mathbb{Z}^{N}$ the Fourier coefficients $\widehat{K}$ satisfy $$\begin{gathered}
\sum_{\ell=0}^{N-1}(-1) ^{\ell}(\alpha_{i}+\ell) \sum\limits_{J\subset E_{i},\, \#J=\ell}\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{J}}\nonumber\\
\qquad{} =\kappa\sum_{j\neq i}\sum_{\ell=0}^{N-2}(-1) ^{\ell}\sum_{J\subset E_{ij},\, \#J=\ell}\big\{ \tau ( (i,j)) \widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1) \varepsilon
_{i}-\varepsilon_{J}}\tau((i,j))\big\}.\label{FCrec}\end{gathered}$$
Multiply both sides of (\[Kdiffeq1\]) by $x_{i}^{1-N}$; this makes the terms homogeneous of degree zero. Suppose $j\neq i$ then $$\begin{gathered}
x_{i}^{1-N}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) =\prod\limits_{\ell\neq i,j}\left( 1-\frac{x_{\ell}}{x_{i}}\right)
=\sum_{\ell=0}^{N-2}(-1) ^{\ell}x_{i}^{-\ell}\sum_{J\subset E_{ij},\, \#J=\ell}x^{\varepsilon_{J}}.\end{gathered}$$ Multiply the right side by $x^{-\alpha}\mathrm{d}m(x) $ and integrate over $\mathbb{T}^{N}$ to obtain$$\begin{gathered}
\kappa\sum_{j\neq i}\sum_{\ell=0}^{N-2}(-1) ^{\ell}\sum_{J\subset E_{ij},\, \#J=\ell}\big\{ \tau( (i,j)) \widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{j}-\varepsilon_{J}}+\widehat{K}_{\alpha+(l+1) \varepsilon_{i}-\varepsilon_{J}}\tau((i,j)) \big\} .\end{gathered}$$ The sum is zero unless $\alpha\in\boldsymbol{Z}_{N}$ where $\boldsymbol{Z}_{N}:=\Big\{ \alpha\in\mathbb{Z}^{N}\colon \sum\limits_{j=1}^{N}\alpha_{j}=0\Big\} $, by the homogeneity. For the left side start with (\[diffFC\]) applied to $x_{i}^{1-N}p_{i}(x) x_{i}\partial_{i}K(x) $ $$\begin{gathered}
(\alpha_{i}+N-1) \int_{\mathbb{T}^{N}}p_{i}(x)K(x) x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x) \\
\qquad{} =\int_{\mathbb{T}^{N}}\big\{ ( x_{i}\partial_{i}p_{i}(
x) ) K(x) +p_{i}(x) (x_{i}\partial_{i}K(x) ) \big\} x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x) ,\\
\int_{\mathbb{T}^{N}}p_{i}(x)( x_{i}\partial _{i}K(x)) x_{i}^{1-N}x^{-\alpha}\mathrm{d}m (x) \\
\qquad{} =\int_{\mathbb{T}^{N}}( (\alpha_{i}+N-1) p_{i} ( x) -x_{i}\partial_{i}p_{i}(x) ) K(x)
x_{i}^{1-N}x^{-\alpha}\mathrm{d}m(x) \\
\qquad{} =\int_{\mathbb{T}^{N}}\sum_{\ell=0}^{N-1}(-1) ^{\ell} (\alpha_{i}+\ell ) x_{i}^{-\ell}\sum\limits_{J\subset E_{i},\#J=\ell
}x^{\varepsilon_{J}}K(x) x^{-\alpha}\mathrm{d}m(x)\\
\qquad{} =\sum_{\ell=0}^{N-1}(-1) ^{\ell} ( \alpha_{i}+\ell ) \sum\limits_{J\subset E_{i},\#J=\ell}\widehat{K}_{\alpha+\ell\varepsilon_{i}-\varepsilon_{J}}.\end{gathered}$$
Combining the two sides finishes the proof. If $\alpha\notin\boldsymbol{Z}_{N}$ then both sides are trivially zero.
This system of recurrences has the easy (and quite undesirable) solution $\widehat{K}_{\alpha}=I$ for all $\alpha\in\boldsymbol{Z}_{N}$ and $0$ otherwise. The right side becomes $2\kappa\sum\limits_{j\neq i}\tau ((i,j)) \sum\limits_{\ell=0}^{N-2}(-1) ^{\ell}\binom {N-2}{\ell}=0$ (for $N\geq3$, an underlying assumption), and the left side is $\sum\limits_{\ell=0}^{N-1}(-1) ^{\ell}(\alpha_{i}+\ell) \binom{N-1}{\ell}I=0$. This $\widehat{K}$ corresponds to the measure $\frac {1}{2\pi}\mathrm{d}\theta$ on the circle $\big\{ e^{\mathrm{i}\theta} (1,\ldots,1 ) \colon -\pi<\theta\leq\pi\big\}$. Next we show that $\widehat{\mu}_{\alpha}:=\int_{\mathbb{T}^{N}}x^{-\alpha}\mathrm{d}\mu (x ) $ satisfies the same recurrences. Proposition 5.2 of [@Dunkl2016] asserts that if $\alpha,\beta\in\mathbb{N}_{0}^{N}$ and $\sum\limits_{j=1}^{N}(\alpha_{j}-\beta_{j}) =0$ then $$\begin{gathered}
( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta}
=\kappa\sum_{\alpha_{j}>\alpha_{i}}\sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\tau((i,j)) \widehat{\mu}_{\alpha+\ell(
\varepsilon_{i}-\varepsilon_{j}) -\beta}\nonumber\\
\hphantom{( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta} }{}
-\kappa\sum_{\alpha_{i}>\alpha_{j}}\sum_{\ell=0}^{\alpha_{i}-\alpha_{j}
-1}\tau((i,j)) \widehat{\mu}_{\alpha+\ell(
\varepsilon_{j}-\varepsilon_{i}) -\beta} -\kappa\sum_{\beta_{j}>\beta_{i}}\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell( \varepsilon_{i}-\varepsilon_{j})
-\beta}\tau((i,j)) \nonumber\\
\hphantom{( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta} }{}
+\kappa\sum_{\beta_{i}>\beta_{j}}\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i})
-\beta}\tau((i,j)) .\label{A(a-b)}\end{gathered}$$ The relation $\tau(w) ^{\ast}\widehat{\mu}_{w\alpha}\tau (w) =\widehat{\mu}_{\alpha}$ is shown in [@Dunkl2016 Theorem 4.4]. Introduce Laurent series $\sum\limits_{\alpha\in\boldsymbol{Z}_{N}}B_{\alpha}^{(i,j) }x^{\alpha}$ ($i\neq j$) satisfying $$\begin{gathered}
B_{\alpha}^{(i,j) }-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) } =\widehat{\mu}_{\alpha},\qquad
B_{\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) } =0,\end{gathered}$$ note $$\begin{gathered}
\alpha-\alpha_{j}(\varepsilon_{j}-\varepsilon_{i}) = \big(\ldots,\overset{i}{\alpha_{i}+\alpha_{j}},\ldots,\overset{j}{0},\ldots
,\overset{\ell}{\alpha_{\ell}},\ldots \big) , \qquad \ell\neq i,j.\end{gathered}$$ The purpose of the definition is to produce a formal Laurent series satisfying $$\begin{gathered}
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\alpha}B_{\alpha}^{(i,j) }x^{\alpha}=\sum_{\alpha}\widehat{\mu}_{\alpha}x^{\alpha}.\end{gathered}$$ The ambiguity in the solution is removed by the second condition (note that $\sum_{\alpha}\big( B_{\alpha}^{(i,j) }-cI\big) x^{\alpha}$ also solves the first equation for any constant $c$).
\[Bij-Bji\]Suppose $i\neq j$ and $\alpha\in\boldsymbol{Z}_{N}$ then $B_{\alpha}^{(i,j)}\tau((i,j))=\tau((i,j))B_{(i,j)\alpha}^{(j,i)}$.
Start with $\widehat{\mu}_{\alpha}\tau((i,j)) =\tau((i,j)) \widehat{\mu}_{(i,j) \alpha}$ and the defining relations $$\begin{gathered}
B_{\alpha}^{(i,j) }\tau((i,j))-B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\tau((i,j)) =\widehat{\mu}_{\alpha}\tau((i,j)) ,\\
\tau((i,j)) B_{(i,j) \alpha}^{(j,i) }-\tau((i,j)) B_{(i,j) \alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i)}=\tau((i,j)) \widehat{\mu}_{(i,j)\alpha};\end{gathered}$$ subtract the second equation from the first: $$\begin{gathered}
B_{\alpha}^{(i,j) }\tau((i,j))-\tau((i,j)) B_{(i,j) \alpha}^{(j,i) }=B_{\alpha+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\tau((i,j)) -\tau((i,j)) B_{(i,j) \alpha+\varepsilon_{j}-\varepsilon_{i}}^{(j,i) }.\end{gathered}$$ By two-sided induction $$\begin{gathered}
B_{\alpha}^{(i,j) }\tau((i,j))-\tau((i,j)) B_{(i,j) \alpha}^{(j,i) }=B_{\alpha+s( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }\tau((i,j))-\tau((i,j)) B_{(i,j)
\alpha+s(\varepsilon_{j}-\varepsilon_{i}) }^{(j,i)}\end{gathered}$$ for all $s\in\mathbb{Z}$, in particular for $s=\alpha_{j}$ where the right hand side vanishes by definition.
For $\gamma\in\boldsymbol{Z}_{N}$ and $1\leq i\leq N$ $$\begin{gathered}
\gamma_{i}\widehat{\mu}_{\gamma}=\kappa\sum_{j\neq i}\big\{{-}\tau ((i,j)) B_{\gamma}^{(j,i) }+B_{\gamma}^{(i,j) }\tau((i,j)) \big\}.\label{Btomu}\end{gathered}$$
The proof involves a number of cases (for each $(i,j) $ whether $\gamma_{i}\geq0$ or $\gamma_{i}<0$, $\gamma_{j}\geq0$ or $\gamma_{j}<0$). Consider equation (\[A(a-b)\]), in the terms on the first line (with $\tau((i,j))$ acting on the left) use the substitution $\widehat{\mu}_{\delta}=B_{\delta}^{(j,i)}-B_{\delta-\varepsilon_{i}+\varepsilon_{j}}^{(j,i) }$, and for the terms on the second line (with $\tau((i,j))$ acting on the right) use the substitution $\widehat{\mu}_{\delta}=B_{\delta }^{(i,j) }-B_{\delta+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}$. Set $\alpha_{\ell}=\max(\gamma_{\ell},0) $ and $\beta_{\ell}=\max ( 0,-\gamma_{\ell} ) $ for $1\leq\ell\leq N$, thus $\gamma=\alpha-\beta$. The left hand side is $( \alpha_{i}-\beta_{i}) \widehat{\mu}_{\alpha-\beta}=\gamma_{i}\widehat{\mu}_{\gamma}$. We consider two possibilities separately: (i) $\alpha_{i}\geq0$, $\beta_{i}=0$; (ii) $\alpha_{i}=0$, $\beta_{i}>0$; and describe the typical $\tau((i,j)) $ terms. The sums over $\ell$ telescope. In the following any term of the form $\tau((i,j)) \widehat{\mu}_{\cdot}$ or $\widehat{\mu}_{\cdot}\tau((i,j)) $ not mentioned explicitly is zero. Proposition \[Bij-Bji\] is used in each case. For case (i) and $\alpha_{j}>\alpha_{i}$ $$\begin{gathered}
\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}=\tau((i,j)) \sum_{\ell=1}^{\alpha
_{j}-\alpha_{i}}\big( B_{\gamma+\ell( \varepsilon_{i}-\varepsilon_{j}) }^{(j,i) }-B_{\gamma+\ell ( \varepsilon_{i}-\varepsilon_{j}) -\varepsilon_{i}+\varepsilon_{j}}^{(j,i) }\big) \\
\hphantom{\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}}{}
=\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha
_{i}}\big( B_{\gamma+\ell ( \varepsilon_{i}-\varepsilon_{j} )
}^{(j,i) }-B_{\gamma+(\ell-1) (\varepsilon_{i}-\varepsilon_{j}) }^{(j,i) }\big)\\
\hphantom{\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}}{}
=\tau((i,j)) \big( B_{\gamma+ ( \alpha_{j}-\alpha_{i})( \varepsilon_{i}-\varepsilon_{j})}^{(j,i) }-B_{\gamma}^{(j,i) }\big) \\
\hphantom{\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}-\alpha_{i}}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{i}-\varepsilon_{j} )-\beta}}{}
=\tau((i,j)) \big( B_{(i,j)\gamma}^{(j,i) }-B_{\gamma}^{(j,i) }\big)=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j))\end{gathered}$$ For case (i) and $\alpha_{i}>\alpha_{j}\geq0=\beta_{j}$ $$\begin{gathered}
-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{j}-\varepsilon_{i}) -\beta} =-\tau((i,j)) \sum
_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\big( B_{\gamma+\ell (\varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }-B_{\gamma+( \ell+1)( \varepsilon_{i}-\varepsilon
_{j}) }^{(j,i) }\big) \\
\hphantom{-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{j}-\varepsilon_{i}) -\beta}}{}
=-\tau((i,j)) \big( B_{\gamma}^{(j,i) }-B_{(i,j) \gamma}^{(j,i) }\big)\\
\hphantom{-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-\alpha_{j}-1}\widehat{\mu}_{\alpha+\ell ( \varepsilon_{j}-\varepsilon_{i}) -\beta}}{}
=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) ,\end{gathered}$$ note $\gamma+(\alpha_{i}-\alpha_{j}) ( \varepsilon_{j}-\varepsilon_{i}) =(i,j) \gamma$. For case (i) and $\alpha_{i}>\alpha_{j}=0>-\beta_{j}$ $$\begin{gathered}
-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta} =-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-1}\big( B_{\gamma+\ell ( \varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }-B_{\gamma+(\ell+1) (\varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }\big) \\
\hphantom{-\tau((i,j)) \sum_{\ell=0}^{\alpha_{i}-1}\widehat{\mu}_{\alpha+\ell(\varepsilon_{j}-\varepsilon_{i})-\beta}}{}
=-\tau((i,j)) \big( B_{\gamma}^{(j,i)}-B_{\gamma+\gamma_{i}( \varepsilon_{j}-\varepsilon_{i}) }^{(j,i) }\big) ,
\\
-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau((i,j))=
-\sum_{\ell=1}^{\beta_{j}}\big( B_{\gamma-\ell ( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }-B_{\gamma-\ell(\varepsilon_{i}-\varepsilon_{j})+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\big) \tau((i,j)) \\
\hphantom{-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau((i,j))}{}
=-\sum_{\ell=1}^{\beta_{j}}\big( B_{\gamma-\ell ( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }-B_{\gamma-(\ell-1) ( \varepsilon_{i}-\varepsilon_{j}) }^{(
i,j) }\big) \tau((i,j))\\
\hphantom{-\sum_{\ell=1}^{\beta_{j}}\widehat{\mu}_{\alpha-\ell(\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau((i,j))}{}
=\big(B_{\gamma}^{(i,j) }-B_{\gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }\big) \tau((i,j))\end{gathered}$$ let $\delta=\gamma+\gamma_{i}(\varepsilon_{j}-\varepsilon_{i}) $ then $\delta_{k}=\gamma_{k}$ for $k\neq i,j$, $\delta_{i}=0$, and $\delta_{j}=\gamma_{i}+\gamma_{j}$; also $(i,j) \delta=\gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j})$. Thus the sum of the terms for this case is $$\begin{gathered}
-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) +\tau((i,j)) B_{\delta}^{(j,i)}+B_{(i,j) \delta}^{(i,j) }\tau( (i,j))\\
\qquad{} =-\tau((i,j)) B_{\gamma}^{(j,i) }+B_{\gamma}^{(i,j) }\tau((i,j)) .\end{gathered}$$ For case (ii) and $\beta_{j}=-\gamma_{j}>\beta_{i}=-\gamma_{i}>0$ $$\begin{gathered}
-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j)) =-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\big( B_{\gamma-\ell (\varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }-B_{\gamma-(\ell-1) ( \varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }\big) \tau((i,j)) \\
\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j))}{}
=\big( {-}B_{\gamma- ( \gamma_{i}-\gamma_{j} ) (\varepsilon_{i}-\varepsilon_{j}) }^{(i,j) }+B_{\gamma}^{(i,j) }\big) \tau((i,j))\\
\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j))}{}
=\big( {-}B_{(i,j) \gamma}^{(i,j) }+B_{\gamma}^{(i,j) }\big) \tau((i,j))\\
\hphantom{-\sum_{\ell=1}^{\beta_{j}-\beta_{i}}\widehat{\mu}_{\alpha-\ell (\varepsilon_{i}-\varepsilon_{j}) -\beta}\tau( (i,j))}{}
=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) .\end{gathered}$$ For case (ii) and $\beta_{i}>\beta_{j}=-\gamma_{j}\geq0$ (and $\alpha_{j}=0$) $$\begin{gathered}
\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j)) =\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\big( B_{\gamma-\ell(\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }-B_{\gamma-(\ell+1)( \varepsilon_{j}-\varepsilon
_{i}) }^{(i,j) }\big) \tau ( (i,j)) \\
\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j))}{}
=\big( B_{\gamma}^{(i,j) }-B_{\gamma- ( \gamma_{j}-\gamma_{i}) (\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }\big) \tau((i,j))\\
\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j))}{}
=\big(B_{\gamma}^{(i,j) }-B_{(i,j) \gamma}^{(i,j) }\big) \tau((i,j)) \\
\hphantom{\sum_{\ell=0}^{\beta_{i}-\beta_{j}-1}\widehat{\mu}_{\alpha-\ell (\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j))}{}
=-\tau((i,j)) B_{\gamma}^{(j,i)}+B_{\gamma}^{(i,j) }\tau((i,j)) .\end{gathered}$$ For case (ii) and $-\beta_{i}=\gamma_{i}<0<\gamma_{j}=\alpha_{j}$ (and $\beta_{j}=0$) $$\begin{gathered}
\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}}\widehat{\mu}_{\alpha+\ell( \varepsilon_{i}-\varepsilon_{j})
-\beta}+\sum_{\ell=0}^{\beta_{i}-1}\widehat{\mu}_{\alpha-\ell(\varepsilon_{j}-\varepsilon_{i}) -\beta}\tau( (i,j)) \\
\qquad{} =\tau((i,j)) \sum_{\ell=1}^{\alpha_{j}}\big(B_{\gamma+\ell ( \varepsilon_{i}-\varepsilon_{j} ) }^{(j,i) }-B_{\gamma+(\ell-1) ( \varepsilon
_{i}-\varepsilon_{j}) }^{(j,i) }\big) \\
\qquad\quad{} +\sum_{\ell=0}^{\beta_{i}-1}\big( B_{\gamma-\ell ( \varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }-B_{\gamma-(\ell+1) (\varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }\big) \tau((i,j)) \\
\qquad{} =\tau((i,j)) \big( B_{\gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j}) }^{(j,i)}-B_{\gamma}^{(j,i) }\big) +\big( B_{\gamma}^{(i,j)}-B_{\gamma+\gamma_{i}( \varepsilon_{j}-\varepsilon_{i}) }^{(i,j) }\big) \tau ( (i,j)) \\
\qquad{} =-\tau((i,j)) B_{\gamma}^{(j,i) }+B_{\gamma}^{(i,j) }\tau((i,j)) ,\end{gathered}$$ because $(i,j) ( \gamma+\gamma_{j}( \varepsilon_{i}-\varepsilon_{j}) ) =\gamma+\gamma_{i}( \varepsilon_{j}-\varepsilon_{i}) $. In the trivial case $\gamma_{i}=\gamma_{j}$ so that $(i,j) \gamma=\gamma$ where are no nonzero $\tau ((i,j)) $ terms the equation $-\tau( (i,j)) B_{\gamma}^{(j,i) }-B_{\gamma}^{(
i,j) }\tau((i,j)) =0$ applies. Thus in each case and for each $j\neq i$ the right hand side contains the expression $-\kappa\big( \tau((i,j)) B_{\gamma}^{(
j,i) }-B_{\gamma}^{(i,j) }\tau( (i,j)) \big)$.
In the following there is no implied claim about convergence, because any term $x^{\alpha}$ appears only a finite number of times in the equation.
For $1\leq i\leq N$ the formal Laurent series $F(x) :=\sum\limits_{\alpha\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\alpha}x^{\alpha}$ satisfies the equation $$\begin{gathered}
p_{i}(x) x_{i}\partial_{i}F(x) =\kappa\sum_{j\neq i}\prod\limits_{\ell\neq i,j}(x_{i}-x_{\ell}) \{ x_{j}\tau((i,j)) F(x) +F(x) \tau((i,j)) x_{i}\}
.\label{diffKLs}\end{gathered}$$
Start with multiplying equation (\[Btomu\]) by $x_{i}^{1-N}p_{i}(x) x^{\gamma}$ and sum over $\gamma\in\boldsymbol{Z}_{N}$ to obtain $$\begin{gathered}
\prod\limits_{j=1,\, j\neq i}^{N}\left( 1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}
=\kappa\sum_{j\neq i}\prod\limits_{k\neq i,j}\left( 1-\frac{x_{k}}{x_{i}}\right) \left( 1-\frac{x_{j}}{x_{i}}\right) \\
\hphantom{\prod\limits_{j=1,j\neq i}^{N}\left( 1-\frac{x_{j}}{x_{i}}\right)\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}}{}
\times\left\{ -\tau((i,j)) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i) }x^{\gamma}+\sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j) }x^{\gamma}\tau((i,j)) \right\} .\end{gathered}$$ By construction $$\begin{gathered}
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j) }x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\big( B_{\gamma}^{(i,j) }-B_{\gamma+\varepsilon_{i}-\varepsilon_{j}}^{(i,j) }\big) x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}\end{gathered}$$ and $$\begin{gathered}
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i) }x^{\gamma}=-\frac{x_{j}}{x_{i}}\left(1-\frac{x_{i}}{x_{j}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(j,i) }x^{\gamma}=-\frac{x_{j}}{x_{i}}\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}.\end{gathered}$$ Thus the equation becomes $$\begin{gathered}
\prod\limits_{j=1,\, j\neq i}^{N}\left( 1-\frac{x_{j}}{x_{i}}\right)
\sum_{\gamma\in\boldsymbol{Z}_{N}}\gamma_{i}\widehat{\mu}_{\gamma}x^{\gamma}\\
\qquad{} =\kappa\sum_{j\neq i}\prod\limits_{k\neq i,j}\left( 1-\frac{x_{k}}{x_{i}}\right) \left\{ \frac{x_{j}}{x_{i}}\tau((i,j))
\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}+\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}\tau((i,j))\right\} .\end{gathered}$$ This completes the proof.
The coefficients $\{ \widehat{\mu}_{\alpha}\} $ satisfy the same recurrences as $\big\{ \widehat{K}_{\alpha}\big\}$ in .
Maximal singular support
------------------------
Above we showed that $\mu$ and $K$ satisfy the same Laurent series differential systems (\[Kdiffeq1\]) and (\[diffKLs\]), thus the singular part $\mu_{S}$ also satisfies this relation. The singular part $\mu_{S}$ is the restriction of $\mu$ to $\bigcup\limits_{i<j}\big\{ x\in\mathbb{T}^{N} \colon x_{i}=x_{j}\big\} $, a closed set. For each pair $ \{k,\ell\} $ let $E_{k\ell}=\big\{ x\in\mathbb{T}^{N}\colon x_{k}\neq x_{\ell}\big\} $, an open subset of $\mathbb{T}^{N}$. For $i\neq j$ let $$\begin{gathered}
T_{i,j}=\big\{ x\in\mathbb{T}^{N}\colon x_{i}=x_{j}\big\} \cap\bigcap \limits_{\{ k,\ell \} \cap\{i,j\} =\varnothing} \{ E_{k\ell}\cap E_{ik}\cap E_{jk}\};\end{gathered}$$ this is an intersection of a closed set and an open set, hence $T_{i,j}$ is a Baire set and the restriction $\mu_{i,j}$ of $\mu$ to $T_{i,j}$ is a Baire measure. Informally $T_{i,j}=\big\{ x\in\mathbb{T}^{N}\colon x_{i}=x_{j},\# \{ x_{k} \} =N-1\big\}$. We will prove that $\mu_{i,j}=0$ for all $i\neq j$. That is, $\mu_{S}$ is supported by $\big\{ x\in \mathbb{T}^{N}\colon \# \{ x_{k} \} \leq N-2\big\} $ (the number of distinct coordinate values is $\leq N-2$). In [@Dunkl2016 Corollary 4.15] there is an approximate identity $$\begin{gathered}
\sigma_{n}^{N-1}(x) :=\sum_{k=0}^{n}\frac{(-n) _{k}}{( 1-n-N) _{k}}\sum_{\alpha\in\boldsymbol{Z}_{N},\, \vert \alpha \vert =2k}x^{\alpha},\end{gathered}$$ which satisfies $\sigma_{n}^{N-1}(x) \geq0$ and $\sigma _{n}^{N-1}\ast\nu\rightarrow\nu$ as $n\rightarrow\infty$, in the weak-$\ast$ sense for any finite Baire measure $\nu$ on $\mathbb{T}^{N}/\mathbb{D}$ (referring to functions and measures on $\mathbb{T}^{N}$ homogeneous of degree zero as Laurent series). The set $T_{i,j}$ is pointwise invariant under $(i,j)$ thus $\mathrm{d}\mu_{i,j}(x)=\mathrm{d}\mu_{i,j}(x(i,j))=\tau ((i,j)) \mathrm{d}\mu_{i,j}(x)\tau((i,j))$.
The density of Laurent polynomials in $C^{(1) }\big(\mathbb{T}^{N}\big) $ can be shown by using an approximate identity, for example: $u_{n}(x) =\Big\{ \frac{1}{n+1}\sum\limits_{j=-n}^{n}( n- \vert j \vert +1) x_{1}^{j}\Big\} \sigma _{n}^{N-1}(x) $; for any $\alpha\in\mathbb{Z}^{N}$ the coefficient of $x^{\alpha}$ in $u_{n}(x) $ tends to $1$ as $n\rightarrow\infty$ (express $\alpha=(\alpha_{1}-m) \varepsilon_{1}+( -m,\alpha_{2},\ldots,\alpha_{N}) $ where $m=\sum\limits_{j=2}^{N}\alpha_{j}$). Then $f\ast u_{n}\rightarrow f$ in the $C^{(1) }\big( \mathbb{T}^{N}\big) $ norm.
Let $K_{n}^{s}=\sigma_{n}^{N-1}\ast\mu_{S}$ (convolution), a Laurent polynomial, fix $\ell$ in $1\leq\ell\leq N$, and consider the functionals $F_{\ell,n}$, $G_{\ell,n}$ on scalar functions $p\in C^{(1)}\big( \mathbb{T}^{N}\big) $ $$\begin{gathered}
F_{\ell,n}(p) :=\int_{\mathbb{T}^{N}}p(x)
\prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) x_{\ell}\partial_{\ell}K_{n}^{s}(x) \mathrm{d}m(x), \\
G_{\ell,n}(p) :=\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \left\{ \frac{x_{i}}{x_{\ell}}\tau\left( \ell,i\right) K_{n}^{s}(x) +K_{n}^{s}(x) \tau( \ell,i)\right\} \mathrm{d}m(x) .\end{gathered}$$ By construction the functionals annihilate $x^{\alpha}$ for $\alpha\notin\mathbf{Z}_{N}$. For a fixed $\alpha\in\mathbf{Z}_{N}$ the value $F_{\ell,n}( x^{-\alpha}) -G_{\ell,n}( x^{-\alpha})$ is $$\begin{gathered}
\alpha_{\ell}A_{\alpha}b_{n}(\alpha) +\sum_{i=1}^{N-1} (-1) ^{i}\sum_{J\subset E_{\ell},\, \#J=i}( \alpha_{\ell}+i)A_{\alpha+i\varepsilon_{\ell}-\varepsilon_{J}}b_{n}( \alpha+i\varepsilon_{\ell}-\varepsilon_{J}) \\
-\kappa\sum_{j=1,\, j\neq\ell}^{N}\sum_{i=0}^{N-2}(-1) ^{\ell}\sum_{J\subset E_{\ell,j},\, \#J=i}\left\{
\begin{matrix}
\tau(\ell,j) A_{\alpha+(i+1) \varepsilon_{\ell
}-\varepsilon_{j}-\varepsilon_{J}}b_{n}( \alpha+(i+1)
\varepsilon_{\ell}-\varepsilon_{j}-\varepsilon_{J}) \\
{} +A_{\alpha+i\varepsilon_{\ell}-\varepsilon_{J}}\tau(\ell,j)
b_{n}( \alpha+i\varepsilon_{\ell}-\varepsilon_{J})
\end{matrix}
\right\},\end{gathered}$$ where $b_{n}(\gamma) :=\frac{(-n) _{\vert \gamma\vert /2}}{(1-N-n) _{\vert \gamma\vert/2}}$ (from the Laurent series of $\sigma_{n}^{N-1})$, and $A_{\gamma}:=\int_{\mathbb{T}^{N}}x^{-\gamma}\mathrm{d}\mu_{S}$ . Thus for fixed $\alpha$ the coefficients $b_{n}(\cdot) \rightarrow1$ as $n\rightarrow
\infty$ and the expression tends to the differential system \[Kdiffeq1\] and$$\begin{gathered}
\lim_{n\rightarrow\infty}\big( F_{\ell,n} ( x^{-\alpha} )-G_{\ell,n}( x^{-\alpha}) \big) =0.\end{gathered}$$ This result extends to any Laurent polynomial by linearity. From the approximate identity property $$\begin{gathered}
\lim_{n\rightarrow\infty}G_{\ell,n}(p) =\kappa\sum_{i\neq\ell
}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left(1-\frac{x_{j}}{x_{\ell}}\right) \left\{ \frac{x_{i}}{x_{\ell}}\tau (\ell,i) \mathrm{d}\mu_{S}(x) +\mathrm{d}\mu_{S}(x) \tau( \ell,i) \right\} ,\end{gathered}$$ and $$\begin{gathered}
\Vert G_{\ell,n}(p)\Vert \leq M\sup_{x,\, i}\left\vert p(x) \prod\limits_{j\neq\ell,i}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\vert ,\end{gathered}$$ where $M$ depends on $\mu_{S}$. Also $$\begin{gathered}
F_{\ell,n}(p) =-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{ p(x) \prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\} K_{n}(x) \mathrm{d}m(x) ,\end{gathered}$$ and $$\begin{gathered}
\lim_{n\rightarrow\infty}F_{\ell,n}(p) =-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{ p(x) \prod\limits_{j\neq\ell
}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\} \mathrm{d}\mu_{S}(x)\end{gathered}$$ for Laurent polynomials $p$. By density of Laurent polynomials in $C^{(1) }\big( \mathbb{T}^{N}\!/\mathbb{D}\big) $ ($\mathbb{D}\! =\!\{( u,u,\ldots,u) \colon\!$ $\vert u\vert =1 \} $ thus functions homogeneous of degree zero on $\mathbb{T}^{N}$ can be considered as functions on the quotient group $\mathbb{T}^{N}/\mathbb{D}$) we obtain $$\begin{gathered}
-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{ p(x)\prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\}\mathrm{d}\mu_{S}(x)\nonumber\\
\qquad{} =\kappa\sum_{i\neq\ell}\int_{\mathbb{T}^{N}}p(x) \prod\limits_{j\neq\ell,i}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \left\{
\frac{x_{i}}{x_{\ell}}\tau\left( \ell,i\right) \mathrm{d}\mu_{S}(x) +\mathrm{d}\mu_{S}(x) \tau( \ell,i)\right\} ,\label{Tijformula}\end{gathered}$$ for all $p\in C^{(1) }\big( \mathbb{T}^{N}/\mathbb{D}\big)$.
For $1\leq i<j\leq N$ the restriction $\mu_{S}|T_{i,j}=0$.
It suffices to take $i=1,j=2$. Let $E$ be an open neighborhood of a point in $T_{1,2}$ such that if $x\in\overline{E}$ (the closure) and $x_{i}=x_{j}$ for some pair $i<j$ then $i=1$ and $j=2$. Let $f(x) \in C^{(1) }\big( \mathbb{T}^{N}/\mathbb{D}\big) $ have support $\subset E$. Thus $f(x) =0=\partial_{1}f(x) $ at each point $x$ such that $x_{i}=x_{j}$ for some pair $\{i,j\} \neq \{1,2\} $ ($f=0$ on a neighborhood of $\bigcup\limits_{i<j}\{
x\colon x_{i}=x_{j} \} \backslash T_{1,2}$). Then in formula (\[Tijformula\]) (with $\ell=1$) applied to $f$ the measure $\mu_{S}$ can be replaced with $\mu_{1,2}$. Evaluate the derivative $$\begin{gathered}
x_{1}\partial_{1}\left\{ f(x) \prod\limits_{j\neq1}\left(
1-\frac{x_{j}}{x_{1}}\right) \right\} =\left( 1-\frac{x_{2}}{x_{1}}\right) f(x) x_{1}\partial_{1}\left\{ \prod\limits_{j>2}
\left( 1-\frac{x_{j}}{x_{1}}\right) \right\} \\
\qquad{} +f(x) \frac{x_{2}}{x_{1}}\prod\limits_{j>2}\left( 1-\frac
{x_{j}}{x_{1}}\right) +\left( x_{1}\partial_{1}f(x) \right) \left( 1-\frac{x_{2}}{x_{1}}\right) \prod\limits_{j>2}\left(
1-\frac{x_{j}}{x_{1}}\right) .\end{gathered}$$ Each term vanishes on $\bigcup\limits_{i<j}\{x\colon x_{i}=x_{j}\} \backslash T_{1,2}$, and restricted to $T_{1,2}$ the value is $f(x) \prod\limits_{j>2}\big( 1-\frac{x_{j}}{x_{1}}\big) $. Thus $$\begin{gathered}
-\int_{\mathbb{T}^{N}}x_{1}\partial_{1}\left\{ f(x)
\prod\limits_{j\neq1}\left( 1-\frac{x_{j}}{x_{1}}\right) \right\} \mathrm{d}\mu_{S}(x) =-\int_{\mathbb{T}^{N}}x_{1}
\partial_{1}\left\{ f(x) \prod\limits_{j\neq1}\left(1-\frac{x_{j}}{x_{1}}\right) \right\} \mathrm{d}\mu_{1,2}(x)
\\
\hphantom{-\int_{\mathbb{T}^{N}}x_{1}\partial_{1}\left\{ f(x)
\prod\limits_{j\neq1}\left( 1-\frac{x_{j}}{x_{1}}\right) \right\} \mathrm{d}\mu_{S}(x)}{}
=-\int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left(1-\frac{x_{j}}{x_{1}}\right) \mathrm{d}\mu_{1,2}(x).\end{gathered}$$ The right hand side of the formula reduces to $$\begin{aligned}
\kappa\int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left(1-\frac{x_{j}}{x_{1}}\right) \left\{ \frac{x_{2}}{x_{1}}\tau (1,2) \mathrm{d}\mu_{1,2}(x) +\mathrm{d}\mu_{1,2} (x) \tau(1,2) \right\} \\
\qquad {} =2\kappa\tau(1,2) \int_{\mathbb{T}^{N}}f(x) \prod\limits_{j>2}\left( 1-\frac{x_{j}}{x_{1}}\right) \mathrm{d}\mu_{1,2}(x) ,\end{aligned}$$ since $\mathrm{d}\mu_{1,2}(x) \tau(1,2) =\tau(1,2) \mathrm{d}\mu_{1,2}(x) $. Thus the integral is a matrix $F(f) $ such that $$\begin{gathered}
( I+2\kappa \tau(1,2)) F(f) =0,\end{gathered}$$ which implies $F(f) =0$ provided $\kappa\neq\pm\frac{1}{2}$. Replacing $f(x) $ by $f(x) \prod\limits_{j>2}\big(1-\frac{x_{j}}{x_{1}}\big) ^{-1}$ shows that $\mu_{1,2}=0$, since $E$ was arbitrarily chosen.
Boundary values for the measure
-------------------------------
In this subsection we will show that $K$ satisfies the weak continuity condition $$\begin{gathered}
\lim\limits_{x_{N-1}-x_{N}\rightarrow0} ( K(x)-K(x(N-1,N))) =0\end{gathered}$$ at the faces of $\mathcal{C}_{0}$ and then deduce that $H_{1}$ commutes with $\sigma$ (as described in Theorem \[suffctH\]). The idea is to use the inner product property of $\mu$ on functions supported in a small enough neighborhood of $x^{(0)}=\big(1,\omega,\ldots,\omega^{N-3},\omega^{-3/2},\omega^{-3/2}\big)$ where $\mu_{S}$ vanishes, so that only $K$ is involved, then argue that a failure of the continuity condition leads to a contradiction.
Let $0<\delta\leq\frac{2\pi}{3N}$ and define the boxes $$\begin{gathered}
\Omega_{\delta} =\big\{ x\in\mathbb{T}^{N}\colon \big\vert x_{j} -x_{j}^{(0) }\big\vert \leq2\sin\tfrac{\delta}{2},\, 1\leq j\leq
N\big\} ,\\
\Omega_{\delta}^{\prime} =\big\{ x\in\mathbb{T}^{N-1}\colon \big\vert x_{j}-x_{j}^{(0) }\big\vert \leq2\sin\tfrac{\delta}{2},\, 1\leq
j\leq N-1\big\}\end{gathered}$$ (so if $x_{j}=e^{\mathrm{i\theta}_{j}}$ then $\big\vert \theta_{j}-\frac{2\pi(j-1) }{N}\big\vert \leq\delta$, for $1\leq j\leq N-2$ and $\big\vert \theta_{j}-\frac{( 2N-3) \pi}{2}\big\vert $ for $N-1\leq j\leq N$). Then $x\in\Omega_{\delta}$ implies $ \vert x_{i}-x_{j} \vert \geq2\sin\frac{\delta}{2}$ for $1\leq i<j\leq N$ except for $i=N-1$, $j=N$ (that is, $ \vert \theta_{i}-\theta_{j} \vert \geq\delta $). Further $\Omega_{\delta}$ is invariant under $(N-1,N)$, while $\Omega_{\delta}\cap\Omega_{\delta}(i,N) =\varnothing$ for $1\leq i\leq N-2$. For brevity set $\phi_{0}=\frac{(2N-3) \pi
}{2}$, $e^{\mathrm{i\phi}_{0}}=\omega^{-3/2}$. We consider the identity $$\begin{gathered}
\int_{\mathbb{T}^{N}} ( x_{N}\mathcal{D}_{N}f(x)) ^{\ast}\mathrm{d}\mu(x) g(x) -\int_{\mathbb{T}^{N}}f(x) ^{\ast}\mathrm{d}\mu(x) x_{N}\mathcal{D}_{N}g(x) =0\end{gathered}$$ for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big) $ whose support is contained in $\Omega_{\delta}$. Then $\operatorname{spt}((x_{N}\mathcal{D}_{N}f(x) ) ^{\ast}g(x)) \subset\Omega_{\delta}$ and $\operatorname{spt}( f(x) ^{\ast}x_{N}\mathcal{D}_{N}g(x)) \subset \Omega_{\delta}$.
The support hypothesis and the construction of $\Omega_{\delta}$ imply that $\Omega_{\delta}\cap\big( \mathbb{T}^{N}\backslash\mathbb{T}_{\rm reg}^{N}\big) \subset T_{N-1,N}$ and thus $\mathrm{d}\mu$ can be replaced by $K(x) \mathrm{d}m(x) $ in the formula. Recall the general identity (\[dfKg\]) $$\begin{gathered}
-( x_{N}\mathcal{D}_{N}f(x)) ^{\ast}K(x) g(x) +f(x) ^{\ast}K(x)x_{N}\mathcal{D}_{N}g(x) \\
\qquad{} =x_{N}\partial_{N} \{ f(x) ^{\ast}K(x)g(x) \} -\kappa\sum_{1\leq j\leq N-1}\frac{1}{x_{N}-x_{j}}\big\{ x_{j}f(x(j,N)) ^{\ast}\tau((j,N))
K(x) g(x) \\
\qquad\quad{} +x_{N}f(x) ^{\ast}K(x) \tau((j,N)) g( x( j,N)) \big\} .\end{gathered}$$
Specialize to $\operatorname{spt}(f) \subset\Omega_{\delta}$ and $\operatorname{spt}(g) \subset\Omega_{\delta}$ and $x\in\Omega_{\delta}$ then only the $j=N-1$ term in the sum remains, and this term changes sign under $x\mapsto x(N-1,N) $.
For $\varepsilon>0$ let $\Omega_{\delta,\varepsilon}=\big\{ x\in \Omega_{\delta}\colon \vert x_{N-1}-x_{N} \vert \geq2\sin\frac {\varepsilon}{2}\big\} $, then $$\begin{gathered}
\int_{\Omega_{\delta,\varepsilon}}\big\{ x_{N}\partial_{N} ( f (x ) ^{\ast}K(x) g(x) ) \\
\qquad{} + (x_{N}\mathcal{D}_{N}f(x) ) ^{\ast}K(x)g(x) -f(x) ^{\ast}K(x) x_{N}\mathcal{D}_{N}g(x) \big\} \mathrm{d}m(x)=0,\end{gathered}$$ because $\Omega_{\delta,\varepsilon}$ is $(N-1,N) $-invariant (similar argument to Proposition \[xdfKg-fKxdg\]). By integrability $$\begin{gathered}
\lim_{\varepsilon\rightarrow0_{+}}\int_{\Omega_{\delta,\varepsilon}}\big\{ ( x_{N}\mathcal{D}_{N}f(x) ) ^{\ast}K(x) g(x) -f(x) ^{\ast}K(x)
x_{N}\mathcal{D}_{N}g(x) \mathrm{d}m(x) \big\} =0,\end{gathered}$$ hence $$\begin{gathered}
\lim_{\varepsilon\rightarrow0_{+}}\int_{\Omega_{\delta,\varepsilon}}x_{N}\partial_{N} ( f(x) ^{\ast}K(x) g (x )) \mathrm{d}m(x) =0.\end{gathered}$$ Now we use iterated integration. For fixed $\theta_{N-1}$ the ranges for $\theta_{N}$ are obtained by inserting suitable gaps into the interval $ [ \phi_{0}-\delta,\phi_{0}+\delta ] $ (as usual, $x=\big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) $):
1. $\phi_{0}-\delta\leq\theta_{N-1}\leq\phi_{0}-\delta+\varepsilon \colon [\theta_{N-1}+\varepsilon,\phi_{0}+\delta] $,
2. $\phi_{0}-\delta+\varepsilon\leq\theta_{N-1}\leq\phi_{0}+\delta-\varepsilon\colon [ \phi_{0}-\delta,\theta_{N-1}-\varepsilon ]\cup [ \theta_{N-1}+\varepsilon,\phi_{0}+\delta ] $,
3. $\phi_{0}+\delta-\varepsilon\leq\theta_{N-1}\leq\phi_{0}+\delta\colon [\phi_{0}-\delta,\theta_{N-1}-\varepsilon]$.
From $x_{N}\partial_{N}=-\mathrm{i}\frac{\partial}{\partial\theta_{N}}$ it follows that $$\begin{gathered}
\frac{1}{2\pi}\int_{a}^{b}x_{N}\partial_{N}(f^{\ast}Kg)
\big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) \big) \mathrm{d}\theta_{N}\\
\qquad{} =\frac{1}{2\pi\mathrm{i}}\big\{ (f^{\ast}Kg) \big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}b}\big)\big) -(f^{\ast}Kg) \big( \big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}a}\big) \big) \big\} .\end{gathered}$$ Since $f$ and $g$ are at our disposal we can take their supports contained in $\Omega_{\delta/2}$ then for $0<\varepsilon\leq\frac{\delta}{4}$ the $\mathrm{d}\theta_{N}$-integrals for (1) and (3) vanish and the integrals in (2) have the value $$\begin{gathered}
\frac{1}{2\pi\mathrm{i}}\big\{ (f^{\ast}Kg) \big( \big(e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i} ( \theta_{N-1}-\varepsilon) }\big) \big) - ( f^{\ast}Kg ) \big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}( \theta_{N-1}+\varepsilon ) }\big)
\big) \big\} . $$
We use the power series (from (\[Lzseries\])) $$\begin{gathered}
L_{1}(x(u,z)) =\left( \big( u^{2}-z^{2}\big) \prod_{j=1}^{N-2}x_{j}\right) ^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n} ( x (u,0 ) ) z^{n},\end{gathered}$$ with the notation $x(u,z) = ( x_{1},\ldots,x_{N-2},u-z,u+z ) $ for $x\in\Omega_{\delta}$. Recall $\alpha_{n} (x(u,0)) $ is analytic for a region including $\Omega_{\delta}$ and $\sigma\alpha_{n}(x(u,0)) \sigma=(-1) ^{n}\alpha_{n}(x(u,0))$. Also $\alpha_{0}(x(u,0)) $ is invertible. As in Section \[locps\] define $C_{1}:=CL_{1}(x_{0}) ^{-1}$ so that $L_{1}(x) ^{\ast}C_{1}^{\ast}C_{1}L_{1}(x) =L(x) ^{\ast}HL(x) $ on their common domain, and set $H_{1}=C_{1}^{\ast}C_{1}$. It suffices to use the approximation $\sum\limits_{n=0}^{\infty}\alpha_{n}(x(u,0))z^{n}=\alpha_{0}(x(u,0)) +O (\vert z \vert)$, uniformly in $\Omega_{2\pi/3N}$.
Let $$\begin{gathered}
\eta^{(1) }(x,\theta,\varepsilon) :=\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta+\varepsilon)}\big),\\
\eta^{(2) }(x,\theta,\varepsilon) :=\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}(\theta-\varepsilon)},e^{\mathrm{i}\theta}\big),\\
\eta^{(3) }(x,\theta,\varepsilon) :=\big(x_{1},\ldots,x_{N-2},e^{\mathrm{i}\theta},e^{\mathrm{i}(\theta-\varepsilon)}\big)\end{gathered}$$ with $\eta^{(1) },\eta^{(2) }\in\Omega_{\delta}\cap\mathcal{C}_{0}$ and $\eta^{(3) }=\eta^{(2)}(N-1,N) $. Set $\zeta=e^{\mathrm{i}\varepsilon}$. Then $$\begin{gathered}
\eta^{(1) } =x( u_{1}-z_{1},u_{1}+z_{1}), \qquad u_{1}=\frac{1}{2}e^{\mathrm{i}\theta} ( 1+\zeta ) , \qquad z_{1}=\frac
{1}{2}e^{\mathrm{i}\theta}(\zeta-1) ,\\
\eta^{(2) } =x ( u_{2}-z_{2},u_{2}+z_{2} ), \qquad u_{2}=\frac{1}{2}e^{\mathrm{i}\theta}\big( 1+\zeta^{-1}\big), \qquad z_{2}=\frac{1}{2}e^{\mathrm{i}\theta}\big( 1-\zeta^{-1}\big) =\zeta^{-1}z_{1}.\end{gathered}$$
The invariance properties of $K$ imply $K\big( \eta^{(3) }\big) =\sigma K\big(\eta^{(2)}\big) \sigma$. Then $$\begin{gathered}
K\big(\eta^{(1)}\big) =\alpha_{0} ( x (u_{1},0 ) ) ^{\ast}\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa }\big) ^{\ast}H_{1}\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big) \alpha_{0}( x(u_{1},0)) +O\big( \vert z_{1} \vert ^{1-2\vert \kappa\vert }\big) ,\\
\sigma K\big(\eta^{(2)}\big) \sigma =\alpha_{0} (x(u_{2},0) ) ^{\ast}\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) ^{\ast}\sigma H_{1}\sigma\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) \alpha_{0}(x(u_{2},0)) +O\big( \vert z_{2} \vert ^{1-2\vert \kappa\vert}\big),\end{gathered}$$ because $\sigma\alpha_{0}(x(u,0)) \sigma =\alpha_{0}(x(u,0)) $ and $\sigma=\rho(-1,1) $ commutes with $\rho( z_{2}^{-\kappa},z_{2}^{\kappa}) $. To express $K\big(\eta^{(1)}\big) -\sigma K\big(\eta^{(2)}\big) \sigma$ let $$\begin{gathered}
A_{1}=\rho\big( z_{1}^{-\kappa},z_{1}^{\kappa}\big)^{\ast}H_{1}\rho\big(z_{1}^{-\kappa},z_{1}^{\kappa}\big)=O\big(\vert z_{1}\vert^{-2\vert \kappa\vert}\big),\\
A_{2} =\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) ^{\ast}\sigma H_{1}\sigma\rho\big( z_{2}^{-\kappa},z_{2}^{\kappa}\big) =O\big(\vert z_{2}\vert^{-2\vert \kappa\vert }\big),\end{gathered}$$ then $$\begin{gathered}
K\big(\eta^{(1)}\big) -\sigma K\big( \eta^{(2) }\big) \sigma\\
\qquad{} =\alpha_{0} ( x(u_{1},0) ) ^{\ast}A_{1}\alpha_{0}( x(u_{1},0)) -\alpha_{0}(x(u_{2},0)) ^{\ast}A_{2}\alpha_{0}(x(u_{2},0)) +O\big(\vert z_{1}\vert ^{1-2\vert \kappa\vert}\big) .\end{gathered}$$ Also $u_{2}-u_{1}=\frac{1}{2}x_{N-1}\xi_{1}\big( \zeta^{-1}-\zeta\big) =O(\vert z_{1}\vert) $ (since $\vert z_{1} \vert = \vert 1-\zeta \vert $) thus $\alpha_{0} ( x(u_{1},0) ) -\alpha_{0}( x(u_{2},0)) =O( \vert z_{1}\vert) $ and $$\begin{gathered}
K\big(\eta^{(1)}\big) -\sigma K\big( \eta^{(2)}\big) \sigma=\alpha_{0} ( x(u_{1},0) ) ^{\ast}( A_{1}-A_{2}) \alpha_{0}( x(u_{1},0)) +O\big(\vert z_{1}\vert ^{1-2\vert \kappa \vert }\big) .\end{gathered}$$ Using the $\sigma$-decomposition write $$\begin{gathered}
H_{1}:=\begin{bmatrix}
H_{11} & H_{12}\\
H_{12}{}^{\ast} & H_{11}\end{bmatrix}
,\end{gathered}$$ then $$\begin{gathered}
A_{1}-A_{2} =
\begin{bmatrix}
H_{11}\vert z_{1}\vert ^{-2\kappa} & H_{12}\left( \dfrac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}\\
H_{12}{}^{\ast}\left( \dfrac{z_{1}}{\overline{z_{1}}}\right) ^{-\kappa} &
H_{11}\vert z_{1}\vert ^{2\kappa}\end{bmatrix}
-
\begin{bmatrix}
H_{11}\vert z_{2}\vert ^{-2\kappa} & -H_{12}\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\\
-H_{12}{}^{\ast}\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{-\kappa} &
H_{11}\vert z_{2}\vert ^{2\kappa}\end{bmatrix}
\\
\hphantom{A_{1}-A_{2}}{} =
\begin{bmatrix}
O & H_{12}\left\{ \left( \dfrac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa
}+\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\right\} \\
H_{12}{}^{\ast}\left\{ \left( \dfrac{z_{1}}{\overline{z_{1}}}\right)
^{-\kappa}+\left( \dfrac{z_{2}}{\overline{z_{2}}}\right) ^{-\kappa}\right\}
& O
\end{bmatrix},\end{gathered}$$ because $ \vert z_{2} \vert = \vert z_{1} \vert $. Next $$\begin{gathered}
\frac{z_{1}}{\overline{z_{1}}} =e^{2\mathrm{i}\theta}\frac{e^{\mathrm{i}\varepsilon}-1}{e^{-\mathrm{i}\varepsilon}-1}=-e^{\mathrm{i} (
2\theta+\varepsilon ) },\qquad \frac{z_{2}}{\overline{z_{2}}}=e^{2\mathrm{i}\theta}\frac{1-e^{-\mathrm{i}\varepsilon}}{1-e^{\mathrm{i}\varepsilon}
}=-e^{\mathrm{i} ( 2\theta-\varepsilon ) },\\
\left( \frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa} =\big( {-}e^{2\mathrm{i}\theta
}\big) ^{\kappa}\big\{ e^{\mathrm{i}\varepsilon\kappa}+e^{-\mathrm{i}\varepsilon\kappa}\big\} =2\big({-}e^{2\mathrm{i}\theta}\big) ^{\kappa
}\cos\varepsilon\kappa,\end{gathered}$$ where some branch of the power function is used (the interval where it is applied is a small arc of the unit circle), and $\phi_{0}-\delta_{1}<\theta<\phi_{0}-\delta_{1}$.
We show that $H_{12}=O$, equivalently $H_{1}$ commutes with $\sigma$. By way of contradiction suppose some entry $h_{ij}\neq0$ ($1\leq i\leq m_{\tau}<j\leq
n_{\tau}$). There exist $r>0$, $\delta_{1}\geq\delta_{2}>0$ and $c\in \mathbb{C}$ with $\vert c\vert =1$ such that $$\begin{gathered}
\operatorname{Re}\big( 2c\big( {-}e^{2\mathrm{i}\theta}\big) ^{\kappa}h_{ij}\big) >r\end{gathered}$$ for $\phi_{0}-\delta_{2}\leq\theta\leq\phi_{0}+\delta_{2}$. Let $p (x) \in C^{(1) }\big( \mathbb{T}^{N}\big) $ such that $\operatorname{spt}(p) \subset\Omega_{\delta_{2}/2}$, $0\leq p (x) \leq1$ and $p(x) =1$ for $x\in\Omega_{\delta_{2}/4}$. Let $f(x) =p(x) \alpha_{0}( x(u,0))^{-1}\varepsilon_{i}$ and $g(x) =cp ( x) \alpha_{0}(x(u,0)) ^{-1}\varepsilon _{j}$ (for $x\in\Omega_{\delta_{2}/2}$). Also impose the bound $0<\varepsilon <\frac{\delta_{2}}{4}$. Then $$\begin{gathered}
f\big(\eta^{(1)}\big) ^{\ast}K\big( \eta^{(1) }\big) g\big(\eta^{(1)}\big) =p\big(\eta^{(1) }\big) ^{2}c\left( \frac{z_{1}}{\overline{z_{1}}
}\right) ^{\kappa}h_{ij}+O\big( \vert z_{1}\vert ^{1-2 \vert \kappa \vert }\big), \\
f\big( \eta^{(3) }\big) ^{\ast}\sigma K\big(\eta^{(2) }\big) \sigma g\big( \eta^{(3)}\big) =-p\big( \eta^{(3) }\big) ^{2}c\left(
\frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}h_{ij}+O\big( \vert z_{2} \vert ^{1-2\vert \kappa\vert }\big) ,\end{gathered}$$
Suppose $x=\big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N}}\big) \in\Omega_{\delta_{2}/2}$ then $p(x) =1$ for $\phi_{N-1}-\frac{\delta_{2}}{4}\leq\theta_{N-1},\theta_{N}\leq\phi _{N-1}-\frac{\delta_{2}}{4}$, thus $p\big( \eta^{(1) } (x,\theta,\varepsilon) \big) =1$ for $\phi_{0}-\frac{\delta_{2}}{4}\leq\theta\leq\phi_{0}-\frac{\delta_{2}}{4}-\varepsilon$ and $p\big( \eta^{(3) }(x,\theta,\varepsilon) \big) =1$ for $\phi_{0}-\frac{\delta_{2}}{4}+\varepsilon\leq\theta\leq\phi_{0}-\frac {\delta_{2}}{4}$. By the continuous differentiability it follows that for $\phi_{0}-\frac{\delta_{2}}{4}\leq\theta\leq\phi_{0}-\frac{\delta_{2}}{4}$ both $p\big(\eta^{(1)}\big) =1+O( \varepsilon) $ and $p\big( \eta^{(3) }\big) =1+O(\varepsilon)$. Thus $$\begin{gathered}
p\big(\eta^{(1)}\big) f\big( \eta^{(1)}\big) ^{\ast}K\big(\eta^{(1)}\big) g\big(\eta^{(1)}\big) p\big(\eta^{(1)}\big)
-p\big( \eta^{(3) }\big) f\big(\eta^{(3)}\big) ^{\ast}\sigma K\big(\eta^{(2)}\big) \sigma g\big( \eta^{(3) }\big) p\big(\eta^{(3)}\big) \\
\qquad {} =p\big(\eta^{(1)}\big) ^{2}c\left\{ \left(
\frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\right\} h_{ij}+O\big(\vert z_{1}\vert ^{1-2\vert \kappa\vert }\big) +O(\varepsilon) .\end{gathered}$$ By construction $$\begin{gathered}
\operatorname{Re}\left( c\left\{ \left( \frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa
}\right\} h_{ij}\right) >r\cos\varepsilon\kappa,\end{gathered}$$ multiply the inequality by $p\big( \big( e^{\mathrm{i}\theta_{1}},\ldots,e^{\mathrm{i}\theta_{N-1}},e^{\mathrm{i}\theta_{N-1}}\big) \big)^{2}$ and integrate over the $(N-1) $-box $\Omega_{\delta_{2}}^{\prime}$ with respect to $\mathrm{d}m_{N-1}=\big( \frac{1}{2\pi}\big) ^{N-1}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N-1}$. This integral dominates the integral over $\Omega_{\delta_{2}/4}^{\prime}$, thus $$\begin{gathered}
\operatorname{Re}\int_{\Omega_{\delta_{2}}^{\prime}}p\big( \eta^{(1) }\big) ^{2}c\left\{ \left( \frac{z_{1}}{\overline{z_{1}}}\right) ^{\kappa}+\left( \frac{z_{2}}{\overline{z_{2}}}\right) ^{\kappa}\right\} h_{ij}\mathrm{d}m_{N-1}\geq r\cos\varepsilon\kappa\left(\frac{\delta_{2}}{2\pi}\right) ^{N-1}.\end{gathered}$$ This contradicts the limit of the integral being zero as $\varepsilon\rightarrow0$. The ignored parts of the integral are $O\big( \varepsilon
^{1-2\vert \kappa\vert }\big) $ and $ \vert \kappa \vert <\frac{1}{2}$. We have proven the following:
\[H1comm\]For $-1/h_{\tau}<\kappa<1/h_{\tau}$ the matrix $H_{1}= (L_{1}^{\ast}(x_{0})) ^{-1}HL_{1}(x_{0})^{-1}$ commutes with $\sigma$.
Analytic matrix arguments {#anlcmat}
=========================
In this section we set up some tools from linear algebra dealing with matrices whose entries are analytic functions of one variable. The aim is to establish the existence of an analytic solution for the matrices described in Theorem \[H1comm\]. The key fact is that the solution $L_{1}(x;\kappa)$ of (\[Lsys\]) is analytic in $\kappa$ for $\vert \kappa\vert <\frac{1}{2}$, in fact for $\kappa\in\mathbb{C}\backslash ( \mathbb{Z}+\frac{1}{2})$; the series expansion in (\[Lzseries\]) does not apply to $\kappa\in\mathbb{Z}+\frac{1}{2}$ and a logarithmic term has to be included for this case. Set $b_{N}:= ( 2 ( N^{2}-N+2 )) ^{-1}$, the bound from Section \[suffco\]. One would like use analytic continuation to extend the inner product property of $L^{\ast}HL$ from the interval $-b_{N}<\kappa<b_{N}$ to $-1/h_{\tau}<\kappa<1/h_{\tau}$ but the Bochner theorem argument for the existence of $\mu$ does not allow $\kappa$ to be a complex variable. The following arguments work around this obstacle.
\[Matrixeq\]Suppose $M(\kappa) $ is an $m\times n$ complex matrix with $m\geq n-1$ such that the entries are analytic in $\kappa\in
D_{r}:= \{ \kappa\in\mathbb{C}\colon \vert \kappa\vert <r \}$, some $r>0$ and $\operatorname{rank}(M(\kappa)) =n-1$ for a real interval $-r_{1}<\kappa<r_{1}$ then $\operatorname{rank}( M (\kappa)) =n-1$ for all $\kappa\in D_{r}$ except possibly at isolated points $\lambda$ where $\operatorname{rank} ( M(\lambda)) <n-1$, and there is a nonzero vector function $v (\kappa) $, analytic on $D_{r}$ such that $M(\kappa)v(\kappa) =0$ and $v(\kappa) $ is unique up to multiplication by a scalar function.
Let $M^{\prime}(\kappa) $ be any $n\times n$ submatrix of $M(\kappa) $ (when $m\geq n$), that is, $M^{\prime}$ is composed of $n$ rows of $M(\kappa) $, then $\det M^{\prime} (\kappa) $ is analytic for $\kappa\in D_{r}$ and by hypothesis $\det M^{\prime}(\kappa) =0$ for $-r_{1}<\kappa<r_{1}$. This implies $\det M^{\prime}(\kappa) \equiv0$ for all $\kappa$, by analyticity. Thus $\operatorname{rank}(M(\kappa)) \leq n-1$ for all $\kappa\in D_{r}$. For each subset $J= \{ j_{1},\ldots,j_{n-1} \} $ with $1\leq j_{1}<\cdots<j_{n-1}\leq m$ let $M_{J,k}(\kappa) $ be the $(n-1) \times(n-1) $ submatrix of $M(\kappa) $ consisting of rows $\#$ $j_{1},\ldots,j_{n-1}$ and deleting column $\#k$, and $X_{J} (\kappa) :=[ \det M_{J,1}(\kappa) ,\ldots,\det M_{J,n}(\kappa)] $, an $n$-vector of analytic functions. There exists at least one set $J$ for which $X_{J}(0) \neq [ 0,\ldots,0 ] $ otherwise $\operatorname{rank} ( M (0)) <n-1$. By continuity there exists $\delta>0$ such that at least one $\det M_{J,k}(\kappa) \neq0$ for $ \vert \kappa \vert <\delta$ and $v(\kappa) = \big[(-1)^{k-1}\det M_{J,k}(\kappa) \big] _{k=1}^{n}$ is a nonzero vector in the null-space of $M(\kappa)$ (by Cramer’s rule and the rank hypothesis). The analytic equation $M(\kappa) v(\kappa) =0$ holds in a neighborhood of $\kappa=0$ and thus for all of $D_{r}$. If for isolated points $\kappa_{1},\ldots,\kappa_{\ell}$ in $\vert \kappa\vert \leq r_{2}<r$ then $v(\kappa) $ can be multiplied by $\prod \limits_{j=1}^{\ell}\big( 1-\frac{\kappa}{\kappa_{j}}\big) ^{-a_{j}}$ for suitable positive integers $a_{1},\ldots,a_{\ell}$ to produce a solution never zero in $\vert \kappa\vert \leq r_{2}<r$. (It may be possible that there are infinitely many zeros in the open set $D_{r}$.)
We include the parameter in the notations for $L$ and $L_{1}$. The $\ast$ operation replaces $x_{j}$ by $x_{j}^{-1}$, the constants by their conjugates, and transposing, but $\kappa$ is unchanged to preserve the analytic dependence, see Definition \[defadj\]. For $x\in\mathbb{T}_{\rm reg}^{N}$ and real $\kappa$ the Hermitian adjoint of $L_{1} (x_{0};\kappa) $) agrees with $L_{1}(x_{0};\kappa) ^{\ast}$. The matrix $M(\kappa) $ is implicitly defined by the linear system with the unknown $B_{1}$ $$\begin{gathered}
B_{1} =\sigma B_{1}\sigma,\nonumber\\
\upsilon L_{1}^{\ast}(x_{0};\kappa) B_{1}L_{1} (x_{0};\kappa ) =L_{1}^{\ast}(x_{0};\kappa) B_{1}L_{1}(x_{0};\kappa) \upsilon.\label{Mkeqn}\end{gathered}$$ (Recall $\upsilon=\tau(w_{0}) $.) The entries of $M (\kappa) $ are analytic in $\vert \kappa\vert <\frac{1}{2}$. The equation $B_{1}=\sigma B_{1}\sigma$ implies that $B_{1}$ has $n:=m_{\tau}^{2}+ ( n_{\tau}-m_{\tau} ) ^{2}$ possible nonzero entries, by the $\sigma$-block decomposition. The number of equations $m= n_{\tau}^{2}-\dim \{ A\colon A\upsilon=\upsilon A \} $. Because $w_{0}$ and $(N-1,N) $ generate $\mathcal{S}_{N}$ and $\tau$ is irreducible $A\sigma=\sigma A$ and $A\upsilon=\upsilon A$ imply $A=cI$ for $c\in \mathbb{C}$ by Schur’s lemma. This implies $n\geq m-1$ (else there are two linearly independent solutions). By a result of Stembridge [@Stembridge1989 Section 3] $n$ can be computed from the following: (recall $\omega:= \exp\frac{2\pi\mathrm{i}}{N}$) for $0\leq j\leq N-1$ set $e_{j}$ equal to the multiplicity of $\omega^{j}$ in the list of the $n_{\tau}$ eigenvalues of $\upsilon$ and set $F_{\tau}(q) :=q^{e_{0}}+q^{e_{1}}+\dots+q^{e_{N-1}}$ then $$\begin{gathered}
F_{\tau}(q) =\left\{ q^{n(\tau) }\prod_{i=1}^{N}\big( 1-q^{i}\big) \prod_{(i,j) \in\tau}\big(1-q^{h(i,j) }\big) ^{-1}\right\} \operatorname{mod}\big(1-q^{N}\big),\end{gathered}$$ where $n(\tau) :=\sum\limits_{i=1}^{\ell(\tau) }(i-1) \tau_{i}$ and $h(i,j) $ is the hook length at $(i,j) $ in the diagram of $\tau$ (note $F_{\tau} (1) =n_{\tau}$). Thus $\dim \{ A\colon A\upsilon=\upsilon A \} =\sum\limits_{j=0}^{N-1}e_{j}^{2}$. For example let $\tau=(4,2) $ then $n_{\tau}=9$, $m_{\tau}=3$ and $n=45$ while $F_{\tau}(q) =2+q+2q^{2}+q^{3}+2q^{4}+q^{5}$ and $\dim \{ A\colon A\upsilon=\upsilon A\} =15$, $m=66$.
For $-1/h_{\tau}<\kappa<1/h_{\tau}$ there exists a unique Hermitian matrix $H$ such that $\mathrm{d}\mu=L^{\ast}HL\mathrm{d}m$. Also $( L_{1}(x_{0}) ^{\ast}) ^{-1}HL_{1}(x_{0}) ^{-1}$ commutes with $\sigma$.
For any Hermitian $n_{\tau}\times n_{\tau}$ matrix $B$ define the Hermitian form $$\begin{gathered}
\langle f,g\rangle _{B}:=\int_{\mathbb{T}^{N}}f(x)^{\ast}L(x) ^{\ast}BL(x) g(x)\mathrm{d}m(x)\end{gathered}$$ for $f,g\in C^{(1) }\big( \mathbb{T}^{N};V_{\tau}\big) $. If the form satisfies $\langle wf,wg\rangle _{B}=\langle f,g \rangle _{B}$ for all $w\in\mathcal{S}_{N}$ and $ \langle x_{i}\mathcal{D}_{i}f,g\rangle _{B}$ $= \langle f,x_{i}\mathcal{D}_{i}g \rangle _{B}$ for $1\leq i\leq N$ then $B$ is determined up to multiplication by a constant. This follows from the density of the span of the nonsymmetric Jack (Laurent) polynomials in $C^{(1) }\big(\mathbb{T}^{N};V_{\tau}\big)$. By Theorem \[H1comm\] there exists a nontrivial solution of the system (\[Mkeqn\]) for $-1/h_{\tau}<\kappa<1/h_{\tau}$. Thus $\operatorname{rank}M(\kappa) \leq n-1$ in this interval. Now suppose that $B_{1}$ is a nontrivial solution of (\[Mkeqn\]) for some $\kappa$ such that $-b_{N}<\kappa<b_{N}$. Then both $B^{(1) }:=B_{1}+B_{1}^{\ast}$ and $B^{(2) }:=\mathrm{i}(B_{1}-B_{1}^{\ast}) $ are also solutions (by the invariance of (\[Mkeqn\]) under the adjoint operation). Let $H^{(i) }:=L_{1}^{\ast}(x_{0};\kappa) B^{(i)}L_{1}(x_{0};\kappa) $ for $i=1,2$ then by Theorem \[suffctH\] the forms $\langle \cdot,\cdot\rangle _{H^{(1) }}$ and $\langle \cdot,\cdot\rangle _{H^{(2) }}$ satisfy the above uniqueness condition. Hence either $B_{1}$ is Hermitian or $B^{(1) }=rB^{(2) }$ for some $r\neq0$ which implies $B_{1}=\frac{1}{2}(r-\mathrm{i}) B^{(2) }$, that is, $B_{1}$ is a scalar multiple of a Hermitian matrix. Thus there is a unique (up to scalar multiplication) solution of (\[Mkeqn\]) which implies $\operatorname{rank}M(\kappa) \geq n-1$ in $-b_{N}<\kappa<b_{N}$.
Hence the hypotheses of Theorem \[Matrixeq\] are satisfied, and there exists a nontrivial solution $B_{1}(\kappa) $ which is analytic in $\vert \kappa\vert <\frac{1}{2}$. Since the Hermitian form is positive definite for $-1/h_{\tau}<\kappa<1/h_{\tau}$ we can use the fact that $B_{1}(\kappa) $ is a multiple of a positive-definite matrix when $\kappa$ is real (in fact, of the matrix $H_{1}$ arising from $\mu$ as in Section \[orthmu\]) and its trace is nonzero (at least on a complex neighborhood of $\{ \kappa\colon -1/h_{\tau}<\kappa<1/h_{\tau}\} $ by continuity). Set $B_{1}^{\prime}(\kappa) :=\Big( n_{t}/\sum\limits_{i=1}^{n_{\tau}}B_{1}(\kappa) _{ii}\Big) B_{1}(\kappa) $, analytic and $\operatorname{tr}( B_{1}^{\prime}(\kappa)) =1$ thus the normalization produces a unique analytic (and Hermitian for real $\kappa$) matrix in the null-space of $M(\kappa) $. Let $H(\kappa) =L_{1}(x_{0};\kappa) ^{\ast}B_{1}(\kappa) L_{1}(x_{0};\kappa) $ then for fixed $f,g\in C^{(1) }\big(\mathbb{T}^{N};V_{\tau}\big) $ and $1\leq i\leq N$ $$\begin{gathered}
\int_{\mathbb{T}^{N}}\left\{
\begin{matrix}
( x_{i}\mathcal{D}_{i}f(x) ) ^{\ast}L^{\ast}(x;\kappa) H(\kappa) L(x;\kappa) g(x) \\
-f(x) ^{\ast}L^{\ast}(x;\kappa) H(\kappa) L(x;\kappa) x_{i}\mathcal{D}_{i}g(x)
\end{matrix}\right\} \mathrm{d}m(x)\end{gathered}$$ is an analytic function of $\kappa$ which vanishes for $-b_{N}<\kappa<b_{N}$ hence for all $\kappa$ in $-1/h_{\tau}<\kappa<1/h_{\tau}$; this condition is required for integrability. This completes the proof.
By very complicated means we have shown that the torus Hermitian form for the vector-valued Jack polynomials is given by the measure $L^{\ast}HL\mathrm{d}m$. The orthogonality measure we constructed in [@Dunkl2016] is absolutely continuous with respect to the Haar measure. We conjecture that $L^{\ast} ( x;\kappa) H(\kappa) L(x;\kappa)$ is integrable for $-1/\tau_{1}<\kappa<1/\ell(\tau) $ but $H(\kappa)$ is not positive outside $\vert \kappa \vert <1/h_{\tau}$ (the length of $\tau$ is $\ell(\tau) :=\max\{ i\colon \tau_{i}\geq1\}$). In as yet unpublished work we have found explicit formulas for $L^{\ast}HL$ for the two-dimensional representations $(2,1)$ and $(2,2) $ of $\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ respectively, using hypergeometric functions. It would be interesting to find the normalization constant, that is, determine the scalar multiple of $H(\kappa) $ which results in $\langle 1\otimes T,1\otimes T \rangle _{H(\kappa)}= \langle T,T \rangle _{0}$ (see (\[admforms\])) the “initial condition” for the form. In [@Dunkl2016 Theorem 4.17(3)] there is an infinite series for $H(\kappa)$ but it involves all the Fourier coefficients of $\mu$.
Acknowledgement {#acknowledgement .unnumbered}
---------------
Some of these results were presented at the conference “Dunkl operators, special functions and harmonic analysis” held at Universität Paderborn, Germany, August 8–12, 2016.
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---
abstract: 'In this paper we study an $s$-wave topological superconductor (SC) with a square vortex-lattice. We proposed a topological Majorana lattice model to describe this topological state which was supported by the numerical calculations. We found that the Majorana lattice model is really a “topological SC” on the parent topological SC. Such hierarchy structure becomes a new holographic feature of the topological state.'
author:
- Jiang Zhou
- 'Ya-Jie Wu'
- 'Rong-Wu Li'
- Jing He
- 'Su-Peng Kou [^1]'
date: 'November 16,2012'
title: 'Hierarchical Topological Superconductor – a Majorana Vortex Lattice Model'
---
Introduction
============
Recently, the search for exotic states supporting Majorana fermions (modes) has attracted increasing interests due to their potential applications in fault-tolerant quantum computations[@YT; @SDS; @LFu; @NR; @MS; @JDS; @RML; @DAI; @IPR; @WBI; @JAL; @CNA; @SDAS; @STE; @AKI]. A creative proposal is the proximity effect between $s$-wave superconductor (SC) and topological insulator[@LFu]. This system exhibits non-trivial topological properties, including the nontrivial Chern number in the momentum space, the chiral Majorana edge states, in particular, the Majorana mode around the $\pi$-flux vortex. Another possible example of such quantum exotic states is the chiral $p+ip$ topological superconductors[@NR]. The quantized magnetic vortex ($\pi$-flux) in the chiral $p+ip$ topological SC hosts the Majorana zero modes and obeys non-Abelian statistics. When there are two $\pi$-fluxes nearby, the intervortex quantum tunneling effect occurs and the Majorana modes on two $\pi$-fluxes couple. The tunneling amplitude is determined by the overlap of the wave function of the localized Majorana bound states[@MCH2; @MCH]. Thus, such tunneling must be taken into account when the average distance between localized $\pi$-fluxes becomes the order of the Majorana bound state decay length. For the topological quantum computation based on the non-Abelian anyons, the tunneling effect would split the zero-energy bound states and lift the ground state degeneracy. Beside, the sign of energy splitting is important for understanding the many-body collective states[@CNA; @CG; @AF].
However, till now people have not identify the experimental realizations of $p$-wave SC in condensed matter physics[@DAI; @ASTE]. Instead, people pointed out that the chiral $p$-wave SC may be realized in ultra-cold fermionic atom systems[@VGU; @TMI; @CZH; @NRC; @YNS]. Theoretically, $p$-wave SC may be realized via a p-wave Feshbach resonance in experiment. Due to huge particle-loss[@STE], this idea also has not been realized. For this reason, people proposed a more advantageous scenario based on $s$-wave SC of cold fermionic atoms with laser-field-generated effective spin-orbit (SO) interactions and a large Zeeman field[@MS]. In this scenario, there is a non-Abelian topological phase that is different from $p$-wave SC, in which the SO interaction plays the role of the $p$-wave SC order parameter.
In this paper, we start from this $s$-wave topological SC model and show that there exists the Majorana mode hosted by the $\pi$-flux (quantized magnetic vortex). Then we focus on the super-lattice of $\pi$-fluxes. Because each $\pi$-flux traps a Majorana mode, we can have a lattice model of the Majorana modes that is called the Majorana lattice model. We took into account of zero mode tunneling that couples the vortex sites. We found that this model shows nontrivial topological properties, including a nonvanishing Chern number, chiral Majorana edge state[@VILL]. In this sense, the Majorana lattice model is really a “topological SC” on the parent topological SC. Such hierarchy relationship between the Majorana lattice model and the $s$-wave topological SC model with vortex-lattice is a new holographic feature of the topological states.
The paper is organized as follow: We first introduce the topological $s$-wave SC model on a square lattice in Sec. II, then study the topological properties of this model. In Sec. III, the Majorana mode around a $\pi$-flux is obtained and the intervortex quantum tunneling effect is also studied. We also study the topological SC with a square vortex-lattice numerically. In Sec. IV we write down a Majorana lattice model to describe the coupling effect between Majorana modes trapped in vortices, and the topological properties of this Majorana lattice model are also analyzed. Finally we draw the conclusion in Sec. V.
The s-wave pairing topological superconductor with Rashba spin-orbital coupling
===============================================================================
As a starting point, the $s$-wave pairing SC with Rashba SO coupling is defined on a square lattice[@MS], which is described by $$\mathcal{H}=\mathcal{H}_{k}+\mathcal{H}_{so}+\mathcal{H}_{sc}$$ where the kinetic term $\mathcal{H}_{k}$, the Rashba spin-orbital (SO) coupling term $\mathcal{H}_{so}$, and the superconducting pairing term $\mathcal{H}_{sc}$ are given as $$\begin{split}
\mathcal{H}_{k}= & -t_{s}\sum_{j\sigma}\sum_{\mu=\vec{x},\vec{y}}(c_{j+\mu \sigma}^{\dag}c_{j\sigma}+c_{j-\mu \sigma}^{\dag}c_{j\sigma})\\
& -u\sum_{j\sigma}c_{j\sigma}^{\dag}c_{j\sigma}-h\sum_{j\sigma}c_{j\sigma
}^{\dag}\sigma^{z}c_{j\sigma},\\
\mathcal{H}_{so}= & -\lambda \sum_{j}[(c_{j-\vec{x}\downarrow}^{\dag
}c_{j\uparrow}-c_{j+\vec{x}\downarrow}^{\dag}c_{j\uparrow})\\
& +i(c_{j-\vec{y}\downarrow}^{\dag}c_{j\uparrow}-c_{j+\vec{y}\downarrow
}^{\dag}c_{j\uparrow})]+H.c,\\
\mathcal{H}_{sc}= & -\Delta \sum_{j}(c_{j\uparrow}^{\dag}c_{j\downarrow
}^{\dag}+H.c)
\end{split}$$ where $c_{j\sigma}$ ($c_{j\sigma}^{\dag}$) annihilates (creates) a fermion at site $j=(j_{x},j_{y})$ with spin $\sigma=(\uparrow,\downarrow)$, $\mu=\vec{x}$ or $\vec{y}$, which is a basic vector for the square lattice. $\lambda$ serves as the SO coupling constant and $\Delta$ as the SC pairing order parameter. In addition, in order to observe nontrivial phases supporting Majorana modes excitation, the chemical potential $u$ and the Zeeman term $h$ also be included in this model.
The Hamiltonian $\mathcal{H}$ can be transformed into momentum space from $c_{j}=1/\sqrt{L}\sum_{k}c_{k}e^{-ikR_{j}}$. Writing $c_{j}$ in the particle-hole basis $\psi_{k}^{\dag}=(c_{k\uparrow},c_{k\downarrow
},c_{-k\uparrow}^{\dag},c_{-k\downarrow}^{\dag})$, we obtain its Bogoliubov-de Gennes(BDG) form $$\mathcal{H}=\sum_{BZ}\psi_{k}^{\dag}\mathcal{H}(k)\psi_{k}d^{2}k,$$ where the Bloch Hamiltonian $\mathcal{H}(k)$ is a $4\times4$ matrix $$\mathcal{H}(k)=\left(
\begin{array}
[c]{cccc}\epsilon(k)-h & g(k) & 0 & -\Delta \\
g^{\ast}(k) & \epsilon(k)+h & \Delta & 0\\
0 & \Delta & -\epsilon(k)+h & g^{\ast}(k)\\
-\Delta & 0 & g(k) & -\epsilon(k)-h
\end{array}
\right)$$ with $$\begin{split}
\epsilon(k) & =-2t_{s}(\cos k_{x}+\cos k_{y})-u\\
g(k) & =-2\lambda(\sin k_{y}+i\sin k_{x}).
\end{split}$$ Then the energy spectrum $\epsilon_{\pm}$ is obtained by diagonalizing $\mathcal{H}(k)$ as $$\epsilon_{\pm}=\pm \lbrack m^{2}(k)+|g(k)|^{2}+h^{2}\pm2\sqrt{|g(k)|^{2}\epsilon^{2}(k)+h^{2}m^{2}(k)}]^{\frac{1}{2}}$$ where $m(k)=\sqrt{\epsilon^{2}(k)+\Delta^{2}}$.
In Fig.\[phase\], we plot the global phase diagram. The blue lines are the boundaries between the topological SC and the SC with trivial topological properties. In the topological SC, the ground state has a nontrivial topological invariant (Chern-number) $\mathcal{Q}=1$[@MS]. Such topological invariant is robust, for the Rashba SO coupling can be mapped into a $p+ip$ SC gap through a suitable unitary transformation (see details in Ref.[@MS]). We also study the density of state (DOS) in the topological superconductor numerically and show the results in Fig.\[dosa\]. From this result one can see that there exists a finite energy gap of the topological SC state. In other regions of the phase diagram, the ground states are the SC state with trivial topological properties.
The operators of the Majorana edge states of this topological SC as shown in Fig.3 satisfy $\gamma_{0}^{\dag}=\gamma_{0}$ , which indicates the edge states are really Majorana fermions. Indeed, we can see that the particle-hole operator $\mathcal{C}=\sigma^{x}\otimes \mathbf{1}$ acts on $\mathcal{H}(k)$ as $$\mathcal{C}\mathcal{H}(k)\mathcal{C}^{-1}=-\mathcal{H}^{\ast}(-k)$$ implies that the Bogoliubov quasi-particle operator follows $\gamma_{-k}^{\dag}=\gamma_{k}$ . From Fig.3 (a), one can find that the Majorana edge modes cross zero energy at $k_{y}=0$ for the topological SC with $u=-4t_{s}$, $\lambda=0.5t_{s}$, $h=0.8t_{s}$. The odd number of crossings leads to the topological protected Majorana zero modes. For the trivial SC with $u=-4t_{s}$, $\lambda=0.5t_{s}$, $h=0.4t_{s},$ there is no such crossing (Fig.3 (b)).
Majorana zero modes around vortices of the topological SC
=========================================================
Majorana zero modes around a pair of vortices
---------------------------------------------
We now start with the discussion on the Majorana zero modes which associate with a pair of $\pi$-flux-vortices. The particle density distribution of the electrons is shown in Fig.\[pden\]. The main result shows that there appear two approximately zero modes in the presence of two well-separated $\pi
$-flux-vortices. When two $\pi$-fluxes are well separated, the quantum tunneling effect can be ignored and we have two quantum states with zero energy. On the other hand, for the small spatial distance $D$, the vortices interaction becomes stronger and the energy splitting can not be neglected. As shown in Fig.\[spl\], the energy splitting $\delta E$ as the function of $D$, oscillates and decreases exponentially. Likewise, a pair of $\pi
$-flux-vortices in the topological $p$-wave SC or quasi-hole in the Moore-Read state has the qualitative similar behavior[@MCH; @MCH2; @MBA].
Topological properties of the topological SC with a square vortex-lattice
-------------------------------------------------------------------------
Next, we study the topological SC with a square vortex-lattice numerically. The illustration of the vortex-lattice ($\pi$-flux-lattice) was shown in Fig.\[vlat\].
The tunneling effect between vortices would lead to the coupling between two Majorana fermions around the vortices. Thus there exists a mid-gap energy band for the Majorana fermions. We study the density of state (DOS) in the topological SC with a square vortex-lattice numerically and the results are shown in Fig.\[dos1\]. From Fig.\[dos1\], one can see that besides the energy bands of the paired electrons there exists a mid-gap energy band in parent topological SC. In particular, the mid-gap energy band has finite energy gap. It means that this mid-gap system as shown in Fig.\[dos1\] may be a topological state associated with a non-trivial topological number intuitively. To check the topological properties of the mid-gap system, we calculate its edge states. We consider a system on a cylinder with $12$ super-unitcells along $x$-direction while periodic boundary along $y$-direction. Thus, $k_{y}$ is still a good quantum number and permits the Fourier transformation $c_{k_{y}}(j_{x})=\frac{1}{\sqrt{L_{y}}}\sum_{j_{y}}c(j_{x},j_{y})e^{ik_{y}j_{y}}.$ The spectral flow of this system on a cylinder is plotted in Fig.\[sfl\]. From Fig.\[sfl\], one can see that two gapless chiral edge states are localized at the boundaries. On the other hand, we also plot the spectral flow in Fig.\[sf2\]. When the parent SC is a non-topological SC, the edge states disappear. As a result, we conclude that the mid-gap system shown in Fig.\[dos1\] is indeed a topological state. From Fig.9, we also observe the disappear of the mid-gap band which is induced by the vortex-lattice. So the Majorana mode around the $\pi$-flux is protected by topological invariant of the parent $s$-wave SC**.**
The topological Majorana lattice model
======================================
In the last section we found a topological mid-gap system of the s-wave topological SC with a square vortex-lattice numerically. In this section we will learn the nature of this topological mid-gap system analytically.
We propose an effective description of the $s$-wave topological SC with a square vortex-lattice, of which each vortex traps a Majorana zero mode and two Majorana zero modes couple with each other by a short range interaction. The interaction strength is just the energy splitting $\delta E$ from the intervortex quantum tunneling. We call this effective description as the Majorana lattice model, of which the tight-binding Hamiltonian can be written as $$\mathcal{H}_{m.f}=i\sum_{(j,k)}s_{jk}t_{jk}\gamma_{k}\gamma_{j}$$ where $t_{jk}$ is the hopping amplitude from $j$ to $k$, and satisfies $t_{jk}=t_{jk}^{\ast}$. $\gamma_{j}$ is a Majorana operator ($\gamma_{j}^{\dag}=\gamma_{j}$) obeying anti-commutate relation $\{ \gamma_{j},\gamma
_{k}\}=2\delta_{jk}$. $s_{ij}=-s_{ji}$ is a gauge factor. The pair $(j,k)$ denotes the summation that runs over all the nearest neighbor (NN) pairs (with hopping amplitude $t$) and all the next-nearest neighbor (NNN) pairs (with hopping amplitude $t^{\prime}$). From the polygon rule proposed in Ref.[@EGR], each triangular plaquette possesses $\pi/2$ quantum flux effectively. This Hamiltonian allows a $Z_{2}$ gauge choice $s_{jk}=\pm1$. For our Majorana lattice model, one possible gauge is given by Fig.\[gag\]. Thus, the total number of Majorana modes $N$ must be even and then we can divide the Majorana lattice into two sublattices.
Then the Majorana modes can be combined pairwise to create $N/2$ complex fermionic states by pairing the Majorana operators as $$\gamma_{2j-1}^{a}=a_{j}+a_{j}^{\dag},\quad \gamma_{2j}^{b}=(a_{j}-a_{j}^{\dag
})/i$$ where $a_{j}$ ($a_{j}^{\dag}$) annihilates (creates) a fermion at link $j$. In terms of operator $a_{j}$, the Majorana Hamiltonian takes the form of a “*topological SC*” state as follows: $$\begin{aligned}
\mathcal{H}_{m.f} & =\sum_{\mathbf{x}}\big[t(a_{\mathbf{x}}^{\dag
}a_{\mathbf{x}+\mathbf{j}}-a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{j}}^{\dag
})+t(a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{j}}-a_{\mathbf{x}}^{\dag
}a_{\mathbf{x}+\mathbf{j}}^{\dag})\nonumber \\
& +2t(a_{\mathbf{x}}^{\dag}a_{\mathbf{x}+\mathbf{i}}-a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{i}}^{\dag})-2it(a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{i}}+a_{\mathbf{x}}^{\dag}a_{\mathbf{x}+\mathbf{i}}^{\dag})\nonumber \\
& +t^{\prime}(a_{\mathbf{x}}^{\dag}a_{\mathbf{x}+\mathbf{i}-\mathbf{j}}-a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{i}-\mathbf{j}}^{\dag}-a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{i}-\mathbf{j}}+a_{\mathbf{x}}^{\dag}a_{\mathbf{x}+\mathbf{i}-\mathbf{j}}^{\dag})\nonumber \\
& +t^{\prime}(a_{\mathbf{x}}^{\dag}a_{\mathbf{x}+\mathbf{i}+\mathbf{j}}-a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{i}+\mathbf{j}}^{\dag}+a_{\mathbf{x}}a_{\mathbf{x}+\mathbf{i}+\mathbf{j}}-a_{\mathbf{x}}^{\dag}a_{\mathbf{x}+\mathbf{i}+\mathbf{j}}^{\dag})\nonumber \\
& -t(a_{\mathbf{x}}^{\dag}a_{\mathbf{x}}-a_{\mathbf{x}}a_{\mathbf{x}}^{\dag
})\big]\end{aligned}$$ where $\mathbf{i}$ and $\mathbf{j}$ are two orthogonal unit vectors (Fig.\[gag\]).
Performing a Fourier transformation $a_{k}=\frac{1}{\sqrt{L_{\mathbf{x}}}}\sum_{\mathbf{x}}a_{\mathbf{x}}e^{-ikR_{\mathbf{x}}},$ the Hamiltonian becomes $$\mathcal{H}_{m.f}=\sum_{k,a}\psi_{k}^{\dag}d^{a}(k)\tau_{a}\psi_{k}$$ in the basis $\psi_{k}^{\dag}=(a_{k}^{\dag},a_{-k})$, where $a=x,y,z$, $\tau_{a}$ is the Pauli matrix, and $$\begin{aligned}
\begin{split}
d^{z}(k) & =-2t\sin^{2}k_{y}+4t^{\prime}\cos k_{x}\cos^{2}k_{y}\\
d^{y}(k) & =-\sin2k_{y}(t+2t^{\prime}\cos k_{x})\\
d^{x}(k) & =-2t\sin k_{x}\end{split}\end{aligned}$$ See the detailed calculations in Appendix. Then we get the energy spectrum $E(k)$ of the Majorana lattice model as $$E(k)=\pm \sqrt{\sum_{a=x,y,z}\left \vert d_{a}(k)\right \vert ^{2}}.$$ From this result we can derive that there always exists an **** energy gap of the Majorana lattice model as long as $t^{\prime}\neq0$. Then we calculate the DOS of the Majorana lattice model and show the result in Fig.\[dos2\]. One may compare the DOS of the Majorana lattice model in Fig.\[dos2\] and the DOS of the mid-gap system in Fig.\[dos1\] and find the similarity between them.
To characterize the topological properties of the Majorana lattice model, we define the nontrivial topological invariant - the Chern number $$\mathcal{Q}=\frac{1}{4\pi}\int \int_{BZ}d^{2}k\frac{1}{|\mathbf{d}(k)|^{3}}\mathbf{d}(k)\cdot \frac{\partial \mathbf{d}(k)}{\partial k_{x}}\times
\frac{\partial \mathbf{d}(k)}{\partial k_{y}}$$ which measures that the unit vector $\mathbf{d}(k)/|\mathbf{d}(k)|$ maps the Brillouin zone boundary onto sphere $S^{2}$ via the Chern number $\mathcal{Q}$. In the presence of NNN hopping $t^{\prime}$, we have $\mathcal{Q}=\pm1$. Furthermore, we calculate the edge states of the Majorana lattice model and present the result in Fig.\[edge2\]. One may compare the spectrum of the edge states of the Majorana lattice model in Fig.\[edge2\] with that of the mid-gap system in Fig.\[sfl\] and also find the similarity between them. Now we can conclude that the (topological) Majorana lattice model captures the key low energy physics of the topological SC with a square vortex-lattice.
In addition, we finish this section by a brief discussion of the $s$-wave topological SC with triangle vortex-lattice. If in the region of $|t^{\prime
}/t|<<1$, we observe that the model is characterized by the winding number $\mathcal{Q}=\pm1$, and this may stem from the redundant NN hopping (which behave as the NNN hopping of the square Majorana lattice model) of the triangular lattice. Moreover, the exotic phase $\mathcal{Q}=\pm3$ can be reached by varying of the ratio $t^{\prime}/t$. In the region of $|t^{\prime
}/t|>>1$, the triangular Majorana lattice model has a $\mathcal{Q}=\pm3$ phase. It’s worth to point out that, for the triangular Majorana lattice model in $p+ip$ SC, or interacting vortex-lattice in Kitaev’s honeycomb model, people may find similar phase diagram[@VILL; @p].
Conclusion
==========
In the end, we draw a conclusion. In this paper we studied the properties of an $s$-wave topological SC with a square vortex-lattice. Because each vortex traps a Majorana zero mode, when we took into account of zero mode tunneling that couples the vortex sites, the Majorana zero modes of the vortex-lattice form a Majorana lattice model. We found that this Majorana lattice model shows nontrivial topological properties, including a nonvanishing Chern number, chiral Majorana edge state. In this sense, the Majorana lattice model is really a “topological SC” on the parent topological SC. And the “topological SC” induced by the vortex lattice is topologically protected by the topological invariant of the parent $s$-wave SC. Such correspondence between the topological properties of the Majorana lattice model and the topological properties of the $s$-wave topological SC model with vortex-lattice is another holographic feature of the topological state. In addition, we also had used the numerical approach to study the $s$-wave topological SC with a square vortex-lattice and got similar results.
Furthermore, the Majorana lattice model for chiral $p+ip$ topological superconductor with vortex-lattice and that for the coupled system due to the proximity effect between $s$-wave SC and three dimensional topological insulator with vortex-lattice have the same topological properties to our case. Thus, this approach paves a new way to observe the Majorana modes for the topological SC.
This work is supported by National Basic Research Program of China (973 Program) under the grant No. 2011CB921803, 2012CB921704 and NFSC Grant No.11174035.
Fourier transformation of the Hamiltonian for the Majorana lattice model
========================================================================
The Hamiltonian of the Majorana lattice model can be obtained by Fourier transformation $a_{\mathbf{x}}=1/\sqrt{L_{\mathbf{x}}}\sum_{\mathbf{k}}a_{\mathbf{k}}e^{-i\mathbf{k}R_{\mathbf{x}}},$$$\begin{aligned}
\mathcal{H}_{m.f} & =1/L_{\mathbf{x}}\sum_{\mathbf{k},\mathbf{k^{\prime}}}\sum_{\mathbf{x}}\nonumber \\
& \{t(a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}e^{i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{j}}}-a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}^{\dag}e^{-i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{j}}})\nonumber \\
& +t(a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}e^{-i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{j}}}-a_{\mathbf{k}}^{\dag
}a_{\mathbf{k^{\prime}}}^{\dag}e^{i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime
}}R_{\mathbf{x}+\mathbf{j}}})\nonumber \\
& +2t^{\prime}(a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}e^{i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}}}-a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}^{\dag}e^{-i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}}})\nonumber \\
& -2it(a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}e^{-i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}}}+a_{\mathbf{k}}^{\dag
}a_{\mathbf{k^{\prime}}}^{\dag}e^{i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime
}}R_{\mathbf{x}+\mathbf{i}}})\nonumber \\
& +t^{\prime}(a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}e^{i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}-\mathbf{j}}}-a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}^{\dag}e^{-i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}-\mathbf{j}}})\nonumber \\
& -t^{\prime}(a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}e^{-i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}-\mathbf{j}}}-a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}^{\dag}e^{i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}-\mathbf{j}}})\nonumber \\
& +t^{\prime}(a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}e^{i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}+\mathbf{j}}}-a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}^{\dag}e^{-i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}+\mathbf{j}}})\nonumber \\
& +t^{\prime}(a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}e^{-i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}+\mathbf{j}}}-a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}^{\dag}e^{i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}+\mathbf{i}+\mathbf{j}}})\nonumber \\
& -t(a_{\mathbf{k}}^{\dag}a_{\mathbf{k^{\prime}}}e^{i\mathbf{k}R_{\mathbf{x}}-i\mathbf{k^{\prime}}R_{\mathbf{x}}}-a_{\mathbf{k}}a_{\mathbf{k^{\prime}}}^{\dag}e^{-i\mathbf{k}R_{\mathbf{x}}+i\mathbf{k^{\prime}}R_{\mathbf{x}}})\}\end{aligned}$$ where $R_{\mathbf{x}+\mathbf{\delta}}=R_{\mathbf{x}}+\mathbf{\delta}$, ($\mathbf{\delta}=\mathbf{i}, \mathbf{j}, \mathbf{i}+\mathbf{j}$). Using the identity $\delta_{\mathbf{k},\mathbf{k}^{\prime}}=1/L_{\mathbf{x}}\sum_{\mathbf{x}}e^{i(\mathbf{k}-\mathbf{k}^{\prime})R_{\mathbf{x}}}$, we have $$\begin{aligned}
\mathcal{H}_{m.f} & =\sum_{\mathbf{k}}\{t(a_{k}^{\dag}a_{k}e^{-ik_{y}}-a_{k}a_{k}^{\dag}e^{ik_{y}})\nonumber \\
& +t(a_{k}a_{-k}e^{ik_{y}}-a_{k}^{\dag}a_{-k}^{\dag}e^{-ik_{y}})\nonumber \\
& +2t^{\prime}(a_{k}^{\dag}a_{k}e^{-ik_{x}}-a_{k}a_{k}^{\dag}e^{ik_{x}})\nonumber \\
& -2it(a_{k}a_{-k}e^{ik_{x}}+a_{k}^{\dag}a_{-k}^{\dag}e^{-ik_{x}})\nonumber \\
& +t^{\prime}(a_{k}^{\dag}a_{k}e^{-i(k_{x}-k_{y})}-a_{k}a_{k}^{\dag
}e^{i(k_{x}-k_{y})})\nonumber \\
& -t^{\prime}(a_{k}a_{-k}e^{i(k_{x}-k_{y})}-a_{k}^{\dag}a_{-k}^{\dag
}e^{-i(k_{x}-k_{y})})\nonumber \\
& +t^{\prime}(a_{k}^{\dag}a_{k}e^{-i(k_{x}+k_{y})}-a_{k}a_{k}^{\dag
}e^{i(k_{x}+k_{y})})\nonumber \\
& +t^{\prime}(a_{k}a_{-k}e^{i(k_{x}+k_{y})}-a_{k}^{\dag}a_{-k}^{\dag
}e^{-i(k_{x}+k_{y})})\nonumber \\
& -t(a_{k}^{\dag}a_{k}-a_{k}a_{k}^{\dag})\}\end{aligned}$$ which is $$\begin{aligned}
\mathcal{H}_{m.f} & =(t\cos k_{y}+2t^{\prime}\cos k_{x}+2t^{\prime}\cos
k_{x}\cos k_{y})a_{k}^{\dag}a_{k}\nonumber \\
& -(t\cos k_{y}+2t^{\prime}\cos k_{x}+2t^{\prime}\cos k_{x}\cos k_{y})a_{k}a_{k}^{\dag}\nonumber \\
& -(2t\sin k_{x}+it\sin k_{y}+2it^{\prime}\sin k_{y}\cos k_{x})a_{-k}a_{k}\nonumber \\
& -(2t\sin k_{x}-it\sin k_{y}-2it^{\prime}\sin k_{y}\cos k_{x})a_{k}^{\dag
}a_{-k}^{\dag}$$ or $$\mathcal{H}_{m.f}=\sum_{\mathbf{k}}\sum_{a=x,y,z}\psi_{\mathbf{k}}^{\dag}d^{a}(\mathbf{k})\sigma_{a}\psi_{\mathbf{k}}.$$ where $\psi_{\mathbf{k}}^{\dag}=(a_{\mathbf{k}}^{\dag},a_{-\mathbf{k}}).$
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[^1]: Corresponding author Email: spkou@bnu.edu.cn
|
---
abstract: 'We explore the possibility of probing the nucleon’s transversity distribution $\delta q(x)$ through the final state interaction between two mesons ($\pi^+\pi^-$, $\pi K$, or $K\overline K$) produced in transversely polarized nucleon-nucleon collisions. We present a single spin asymmetry and estimate its magnitude under some assumptions for the transversity distribution function and the unknown interference fragmentation function.'
address: |
[ ]{}Center for Theoretical Physics\
Laboratory for Nuclear Science\
and Department of Physics\
Massachusetts Institute of Technology\
Cambridge, Massachusetts 02139\
[ ]{}
author:
- Jian Tang
date: 'MIT-CTP-2769 hep-ph/xxxxxxx Submitted to [*Physical Review D*]{} July 1998'
title: 'Probing the Nucleon’s Transversity Via Two-Meson Production in Polarized Nucleon-Nucleon Collisions[^1]'
---
\#1
The quark transversity distribution in the nucleon $\delta q(x)$ measures the probability difference to find a quark polarized along versus opposite to the polarization of a nucleon polarized transversely to its direction of motion [@ralston79; @jaffe91; @artru93; @cortes92]. Along with unpolarized and longitudinally polarized quark distributions, it completely characterizes the state of quarks in the nucleon at leading twist in high-energy processes. While the other two have been studied extensively in the past through various high-energy experiments, very little is known about the transversity distribution $\delta q(x)$ since it decouples from hard QCD processes at the leading twist due to its chiral-odd property. For example, it is suppressed like ${\cal O}(m_q/Q)$ in totally-inclusive deep inelastic scattering (DIS).
In our recent works[@jjt1], we have studied the semi-inclusive production of two mesons ([*e.g.*]{} $\pi^+\pi^-$, $\pi K$, or $K\overline K$) in the current fragmentation region in DIS on a transversly polarized nucleon. We have shown that the interference effect between the $s-$ and $p-$wave of the two-meson system around the $\rho$ (for pions), $K^*$ (for $\pi K$), or $\phi$ (for kaons) provides a single spin asymmetry which may be sensitive to the quark transversity distribution in the nucleon. Such interference allows the quark’s polarization information to be carried through the quantity $\vec k_+
\times \vec k_- \cdot \vec S_\perp$, where $\vec k_+$, $\vec k_-$, and $\vec S_\perp$ are the three-momenta of $\pi^+$ ($K$), $\pi^-$ ($\overline K$), and the nucleon’s transverse spin, respectively. This effect appears at the leading twist level, and the production rates for pions and kaons are large in DIS. However, it would vanish by T-invariance in the absence of final state interactions, or by C-invariance if the two-meson state were an eigenstate of C-parity. Hence there is no effect in the regions of the two-meson mass dominated by a single resonance. However, both suppressions are evaded in the $\rho$ ($\pi^+\pi^-$), $K^*$ ($\pi K$), and $\phi$ ($K\overline K$) mass regions where both $s-$ and $p-$wave production channels are active.
In this paper, we extend our study to discuss the possibility of probing the quark transversity distribution in the nucleon via two-meson semi-inclusive production in transversely polarized nucleon-nucleon collisions. Various processes in transversely polarized nucleon-nucleon collisions have been suggested to measure the nucleon’s transversity distribution since it was first introduced about two decades ago[@ralston79], among which are transversely polarized Drell-Yan [@ralston79] and two-jet production[@ji92; @jaffe-saito]. However, the Drell-Yan cross section is small and requires an antiquark transversity distribution, which is likely to be quite small. The asymmetry obtained in two-jet production is rather small due to the lack of a gluon contribution[@ji92; @jaffe-saito]. On the other hand, in the process described here, the gluon-quark scattering dominates and only one beam need be transversely polarized. Unless the novel interference fragmentation function is anomalously small, this will provide a feasible way to probe the nucleon’s transversity distribution. The results of our analysis are summarized by Eq. (\[asymmetry\]) where we present the asymmetry for $\pi^+\pi^- (\pi K, K\overline{K})$ production in transversely polarized nucleon-nucleon collisions.
We consider the semi-inclusive nucleon-nucleon collision process with two-pion final states being detected: $N\vec N_\bot\rightarrow
\pi^+\pi^- X$. (The analysis to follow applies as well to $\pi K$ or $K\overline{K}$ production.) One of the nucleon beams is transversely polarized with polarization vector $S_\mu$, and momentum $P_\mu^A$. The other is unpolarized, with momentum denoted by $P_\mu^B$. The experimentally observable invariant variables are defined as $s\equiv (P_A + P_B)^2,\hspace*{0.5cm}t\equiv (P_h - P_A)^2,
\hspace*{0.5cm} u\equiv (P_h-P_B)^2$, and the invariants for the underlying partonic processes are $\hat{s}\equiv (p_a + p_b)^2,\hspace*{0.5cm} \hat{t}
\equiv (p_c - p_a)^2,
\hspace*{0.5cm} \hat{u}\equiv (p_c-p_b)^2$, where $P_h$ is the total momentum of the two-pion system, $p_a$, $p_b$, $p_c$, $p_d$ are the momenta for the underlying partonic scattering processes (see Fig. \[fig1\]). The longitudinal momentum fractions $x_a$, $x_b$ and $z$ are given by $p_a=x_aP_A$, $p_b=x_bP_B$ and $P_h=zp_c $. The $\sigma [(\pi\pi)^{I=0}_{l=0}]$ and $\rho [(\pi\pi)^{I=1}_{l=1}]$ resonances are produced with momentum $P_h$. We recognize that the $\pi\pi$ $s$-wave is not resonant in the vicinity of the $\rho$ and our analysis does not depend on a resonance approximation. For simplicity we refer to the non-resonant $s$-wave as the “$\sigma$”.
The invariant squared mass of the two-pion system is $m^2 = (k_++k_-)^2$, with $k_+$ and $k_-$ the momentum of $\pi^+$ and $\pi^-$, respectively. The decay polar angle in the rest frame of the two-meson system is denoted by $\Theta$, and the azimuthal angle $\phi$ is defined as the angle of the normal of two-pion plane with respect to the polarization vector $\vec S_\perp$ of the nucleon, $\cos\phi = {{\vec k_+}\times{\vec k_-}\cdot\vec S_\perp / |\vec k_+\times
\vec k_-||\vec S_\perp|}$. This is the analog of the “Collins angle” defined by the $\pi^+\pi^-$ system [@collins94].
Since we are only interested in a result at the leading twist, we follow the helicity density matrix formalism developed in Refs. [@jaffe95; @jaffe96], in which all spin dependence is summarized in a [*double*]{} helicity density matrix. We factor the process at hand into basic ingredients (See Fig. \[fig1\]): the $N\rightarrow q$ (or $N\rightarrow g$) distribution function, the hard partonic $q_aq_b\rightarrow q_cq_d$ cross section, the $q \rightarrow (\sigma, \rho)$ fragmentation, and the decay $(\sigma, \rho)\rightarrow \pi^+\pi^-$, all as density matrices in helicity basis: $$\begin{aligned}
\left[{{d^7\sigma(N\vec N_\perp\rightarrow \pi^+\pi^- X)}
\over{dx_a\, dx_b\, d\hat{t}\, dz\, dm^2\, d\cos\Theta\,
d\phi}}\right]_{H'H}&&
\nonumber
\\*[14.4pt]
&&\hspace*{-2.8cm}=\left[{\cal F}(x_a)\otimes {{d^3\sigma(q_a
q_b\rightarrow q_cq_d)}
\over{dx_a\, dx_b\, d\hat{t}}}\otimes
{{d^2\hat{\cal M}}\over{dz\,
dm^2}}\otimes
{{d^2{\cal D}}\over{d\cos\Theta\, d\phi}}\otimes {\cal F}(x_b)\right]_{H'H}
%\nonumber\end{aligned}$$ where $H(H')$ are indices labeling the helicity states of the polarized nucleon. In order to incorporate the final state interaction, we have separated the $q\rightarrow \pi^+\pi^-$ fragmentation process into two steps. First, the quark fragments into the resonance ($\sigma$, $\rho$) , then the resonance decays into two pions, as shown in the middle of the Fig. \[fig1\].
The $s-p$ interference fragmentation functions describe the emission of a $\rho(\sigma)$ from a parton, followed by absorption of $\sigma(\rho)$ forming a parton. Imposing various symmetry (helicity, parity and time-reversal) restrictions, the interference fragmentation can be cast into a double density matrix notation[@jjt1] $$\begin{aligned}
{d^2 \hat{\cal M}\over dz\, dm^2}
= &&
\Delta_0(m^2)\left\{I\otimes \bar\eta_0\,
\hat{q}_{_I}(z)
+\left(\sigma_+\otimes \bar\eta_-
+ \sigma_-\otimes \bar\eta_+\right)\delta\hat{q}_{_I}(z)\right\}
\Delta^*_1(m^2)
\nonumber
\\
& &
+\Delta_1(m^2)\left\{I\otimes \eta_0\,
\hat{q}_{_I}(z)
+\left(\sigma_-\otimes \eta_+
+ \sigma_+\otimes\eta_-\right)\delta\hat{q}_{_I}(z)
\right\}\Delta^*_0(m^2)\ ,
\label{fragmentation}\end{aligned}$$ where $\sigma_\pm\equiv (\sigma_1\pm i\sigma_2)/2$ with $\{\sigma_i\}$ the usual Pauli matrices. The $\eta$’s are $4\times 4$ matrices in $(\sigma, \rho)$ helicity space with nonzero elements only in the first column, and the $\bar\eta$’s are the transpose matrices ($\bar\eta_0 = \eta_0^T, \bar\eta_+=\eta_-^T,
\bar\eta_-=\eta_+^T$), with the first rows $(0,0,1,0)$, $(0,0,0,1)$, and $(0,1,0,0)$ for $\bar\eta_0$, $\bar\eta_+$, and $\bar\eta_-$, respectively. The explicit definition of the fragmentation functions will be given in Ref. [@jjt3].
The final state interactions between the two pions are included explicitly in $$\Delta_0(m^2)=-i \sin\delta_0
e^{i\delta_0}\ ,\hspace*{1cm}
\Delta_1(m^2)=-i \sin\delta_1
e^{i\delta_1}\ ,
\label{propagators}$$ where $\delta_0$ and $\delta_1$ are the strong interaction $\pi\pi$ phase shifts which can be determined by the $\pi\pi\: {\cal T}$-matrix [@scattheory]. Here we have suppressed the $m^2$ dependence of the phase shifts for simplicity.
The decay process, $(\sigma,\rho)\rightarrow \pi^+\pi^-$, can be easily calculated and encoded into the helicity matrix formalism. The result for the interference part is [@jjt1] $${d^2 {\cal D}\over d\cos\Theta\, d\phi}
={\sqrt{6}\over 8\pi^2 m}
\, \sin\Theta \left[ie^{-i\phi}\left(\eta_--\bar\eta_-\right)
+ie^{i\phi}\left(\eta_+-\bar\eta_+\right)
-\sqrt{2}\cot\Theta \left(\bar\eta_0 +\eta_0\right)\right]\ .$$ Here we have adopted the customary conventions for the $\rho$ polarization vectors, $\vec{\epsilon}_{\pm} = {\mp}(\hat{e}_1\pm i\hat{e}_2)/\sqrt{2}$ and $\vec{\epsilon}_0=\hat{e}_3$ in its rest frame with $\hat e_j$’s the unit vectors.
In the double density matrix notation, the quark distribution function ${\cal F}_q(x)$ in the nucleon can be expressed as [@jaffe96] $${\cal F}_q(x) = {1\over 2} q(x)~I\otimes I + {1\over 2} \Delta
q(x)~\sigma_3 \otimes
\sigma_3+{1\over 2} \delta q(x)~
\left(\sigma_+\otimes\sigma_-+\sigma_-\otimes\sigma_+\right)\ ,
\label{calf}$$ where the first matrix in the direct product is in the nucleon helicity space and the second in the quark helicity space. Here $q(x)$, $\Delta q(x)$, and $\delta q(x)$ are the spin average, helicity difference, and transversity distribution functions, respectively, and their dependences on $Q^2$ have been suppressed.
The gluon distribution function ${\cal F}_g(x)$ in the nucleon can be written as $${\cal F}_g(x) = {1\over 3} G(x)~I\otimes I_g + {1\over 3} \Delta
G(x)~\sigma_3 \otimes
S_g^3\ ,
\label{calg}$$ where $I_g$ and $S_g^3$ are $3\times 3$ matrices in gluon helicity space with nonzero elements only on the diagonal: diag($I_g$)=$\{1,1,1\}$ and diag($S_g^3$)=$\{1,0,-1\}$ . Here $G(x)$ and $\Delta G(x)$ are the spin average and helicity difference gluon distributions in the nucleon, respectively, and, just like in Eq. (\[calf\]), their dependences on $Q^2$ have been suppressed. Note that there is no gluon transversity distribution $\delta G(x)$ in the nucleon at the leading twist due to helicity conservation. This is one of the reasons why transverse asymmetries in two-jet production are typically small, as pointed out by Ji[@ji92], Jaffe and Saito[@jaffe-saito].
Several hard partonic processes contribute here, as shown in the middle of Fig. \[fig1\]. The cross sections can be written as follows (here we list only the relevant parts, i.e. spin-average and transversity-dependent ones), $${d^3\sigma(q_aq_b\rightarrow q_cq_d)\over dx_a\, dx_b d\hat{t}}=
{{\pi\alpha_s^2}\over{2\hat{s}^2}}\,
I_b\otimes\left[{\hat{\bar{\sigma}}}_{ab}^{cd}\, I_a\otimes I_c
+ 4 \delta\hat{\sigma}_{ab}^{cd}\,
\left(\sigma_a^+\otimes\sigma_c^-+\sigma_c^-\otimes\sigma_a^+\right)\right]\ ,
\label{sigmahel}$$ where subscripts $a$, $b$, $c$ means that the helicity matrices above are in $a$, $b$, $c$ parton helicity spaces, respectively (See Fig. \[fig1\]). $\hat{\bar{\sigma}}_{ab}^{cd}$ and $\delta\hat{\sigma}_{ab}^{cd}$ are the spin-average and transversity-dependent cross sections for the underlying partonic processes $q_aq_b\rightarrow q_cq_d$, respectively, which are shown in the Table I.
Combining all the above ingredients together, and integrating over $\Theta$ to eliminate the $\hat q_I$ dependence, we obtain a single spin asymmetry as follows,
$$\begin{aligned}
{\cal A}_{\bot\top}&\equiv& {d\sigma_\bot -d\sigma_\top
\over d\sigma_\bot +d\sigma_\top}
%\nonumber\\
=-\frac {\sqrt{6}\pi}{4}\sin\delta_0\sin\delta_1\sin(\delta_0-\delta_1)
\cos\phi\nonumber \\
&&\otimes {{[\delta q(x_a)\otimes G(x_b)\otimes\delta\hat{q}_I(z)]
\delta\hat{\sigma}_{qg}^{qg}
+[\delta\bar{q}(x_a)\otimes G(x_b)\otimes\delta\hat{\bar{q}}_I(z)]
\delta\hat{\sigma}_{\bar{q}g}^{\bar{q}g}
+...
%+[\delta q\otimes\delta\hat{q}_I\otimes q]\delta\sigma_{qq}^{qq}
}\over{\{[G(x_a)\otimes G(x_b)]\hat{\bar\sigma}_{gg}^{q\bar q}+
[q(x_a)\otimes G(x_b)]
\hat{\bar\sigma}_{qg}^{qg}
%[q\otimes q]\sigma_{qq}^{qq}
+...\}\otimes[\sin^2\delta_0\hat{q}_0(z)+
\sin^2\delta_1\hat{q}_1(z)]}}
%\nonumber
\label{asymmetry}\end{aligned}$$
where $\hat{q}_0(z)$ and $\hat{q}_1(z)$ are spin-average fragmentation functions for the $\sigma$ and $\rho$ resonances, respectively, and the summation over flavor is suppressed for simplicity. The terms denoted by $...$ include quark quark and quark antiquark scattering contributions. This asymmetry can be measured either by flipping the target transverse spin or by binning events according to the sign of the crucial azimuthal angle $\phi$ (See Fig. \[fig4\]). The “figure of merit” for this asymmetry, $\sin\delta_0\sin\delta_1\sin(\delta_0-\delta_1)$, is shown in Fig. \[fig2\].
The flavor content of the asymmetry ${\cal A}_{\bot\top}$ can be revealed by using isospin symmetry and charge conjugation restrictions. For $\pi^+\pi^-$ production, isospin symmetry gives $\delta\hat{u}_{I} = - \delta\hat{d}_{I}$ and $\delta\hat{s}_{I} =
0$. Charge conjugation implies $\delta \hat{q}^a_{I}=-\delta
\hat{\bar q}^a_{I}$. Thus there is only one independent interference fragmentation function for $\pi^+\pi^-$ production, and it may be factored out of the asymmetry, e.g. $\sum_a \delta q_a\delta\hat{q}_{I}^a=
[ (\delta u - \delta\bar u)- (\delta d - \delta\bar
d)]\delta\hat{u}_{I}$. Similar application of isospin symmetry and charge conjugation to the $\rho$ and $\sigma$ fragmentation functions that appear in the denominator of Eq. (\[asymmetry\]) leads to a reduction in the number of independent functions: $\hat{u}_i =\hat{d}_i=\hat{\bar u}_i=\hat{\bar
d}_i$ and $\hat{s}_i=\hat{\bar s}_i$ for $i=\{0,1\}$. For other systems the situation is more complicated due to the relaxation of the Bose symmetry restriction. For example, for the $K\overline K$ system, $\delta \hat{q}^a_I=-\delta
\hat{\bar q}^a_{I}$ still holds, but $\delta\hat{u}_{I}$, $\delta\hat{d}_{I}$, and $\delta\hat{s}_I$, are in general independent. We also note that application of the Schwartz inequality puts an upper bound on the interference fragmentation function, $\delta\hat{q}_I^2\leq
4\hat{q}_0\hat{q}_1/3$ for each flavor.
The size of the asymmetry ${\cal A}_{\bot\top}$ critically depends upon the ratio of the $s-p$ interference fragmentation function and $\rho$ and $\sigma$ fragmentation functions, which is unknown at present. In order to estimate the magnitude of ${\cal A}_{\bot\top}$, we saturate the Schwartz inequality and replace the interference fragmentation with its upper bound, i.e. $\delta\hat{q}_{I}^2=
4\hat{q}_0\hat{q}_1/3$ for each flavor. Meanwhile, we assume the $\sigma$ and $\rho$ fragmentation functions are equal to each other. Thus, the fragmentation function dependences cancel out in ${\cal A}_{\bot\top}$. We also assume that the transversity $\delta q(x, Q^2)$ saturates the Soffer inequality[@soffer]: $2 |\delta q(x, Q^2)|= q(x, Q^2)+\Delta q(x, Q^2)$. We use the polarized structure functions obtained by Gehrmann and Stirling through next-to-leading order analysis of experimental data[@gs]. We also go to the region $m=0.83{\rm GeV}$, around which the phase factor $|\sin\delta_0\sin\delta_1\sin(\delta_0-\delta_1)|$ is large (See Fig. \[fig2\]), and let $\cos\phi = 1$. The asymmetry as function of $p_T^{\rm jet}$ at $\sqrt{s}=500 {\rm GeV}$ and $\sqrt{s}=200 {\rm GeV}$ for pseudo-rapidity $\eta=0.0$ and $\eta=0.35$ is shown in Fig. \[fig3\], where $p_T^{\rm jet}$ is the transverse momentum of the jet. The size of asymmetry is about $12-15\%$ at $p_T^{\rm jet}=120 {\rm GeV}$ for $\sqrt{s}=500 {\rm GeV}$ and about $17-20\%$ at $p_T^{\rm jet}=90 {\rm GeV}$ for $\sqrt{s}=200 {\rm GeV}$, which would be measurable at RHIC.
A few comments can be made about our numerical results. Firstly, under the above approximations, the asymmetry is independent of $z$. Of course, the experiment may not be able to determine $p_T^{\rm jet}$ — the transverse momentum of the jet, so direct comparison between our asymmetry and experimental data will require event simulation. Secondly, because we don’t know the sign of the unknown quantities yet, we can not determine the sign of the asymmtery, the asymmetry shown in Fig. \[fig3\] should only be taken as its magnitude. Finally, in order to estimate the asymmetry, we have made very optimistic assumptions about the novel interference fragmentation functions and transversity distribution functions, so our estimates here should be regarded as “the high side”.
To summarize, we have studied the possibility of probing the quark transversity distribution in the nucleon via two-meson semi-inclusive production of (only one beam) transversely polarized nucleon-nucleon collisions. We obtained a single spin asymmetry that is sensitive to the quark transversity and estimated its magnitude.
I would like to thank Bob Jaffe for encouraging and enlightening conversations relating to this subject. I would also like to thank Xuemin Jin for helpful discussions. I am also grateful to N. Saito and M. Baker for informations related to RHIC.
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=3.0in
=3.5in
=3.0in
=3.0in
-------------------------------------------- --------------------------------------------------------------- ----------------------------------------------------
Partonic process Spin Average Transversity Dependent
$ab\rightarrow cd$ Cross Section$-\hat{\bar\sigma}_{ab}^{cd}$ Cross Section$-\delta\hat{\sigma}_{ab}^{cd}$
$qg\rightarrow qg$ ${{\hat{s}^2+\hat{u}^2}\over {\hat{t}^2}}-{\frac 49} ${{\hat{s}\hat{u}}\over {\hat{t}^2}}-{\frac 49}$
{{\hat{s}^2+\hat{u}^2}\over {\hat{s}\hat{u}}}$
\[0.5cm\] $\bar{q}g\rightarrow \bar{q}g$ ${{\hat{s}^2+\hat{u}^2}\over {\hat{t}^2}}-{\frac 49} ${{\hat{s}\hat{u}}\over {\hat{t}^2}}-{\frac 49}$
{{\hat{s}^2+\hat{u}^2}\over {\hat{s}\hat{u}}}$
\[0.5cm\] $qq\rightarrow qq$ ${\frac 49}\left({{\hat{s}^2+\hat{u}^2} ${\frac 4{27}}{{\hat{s}}\over{\hat{t}}}-{\frac 49}
\over {\hat{t}^2}}+ {{\hat{s}\hat{u}}\over{\hat{t}^2}}$
{{\hat{s}^2+\hat{t}^2}\over {\hat{u}^2}}\right)-{\frac 8{27}}
{{\hat{s}^2}\over{\hat{u} \hat{t}} }$
\[0.5cm\] $qq'\rightarrow qq'$ ${\frac 49}{{\hat{s}^2+\hat{u}^2}\over{\hat{t}^2}}$ $-{\frac 49}{{\hat{s}\hat{u}}\over{\hat{t}^2}}$
\[0.5cm\] $q\bar{q}\rightarrow q\bar{q}$ ${\frac 49}\left({{\hat{s}^2+\hat{u}^2}\over {\hat{t}^2}}+ ${\frac 8{27}}{{\hat{u}}\over{\hat{t}}}-{\frac 49}
{{\hat{u}^2+\hat{t}^2}\over {\hat{s}^2}}\right)-{\frac 8{27}} {{\hat{s}\hat{u}}\over{\hat{t}^2}}$
{{\hat{u}^2}\over {\hat{s}\hat{t}}}$
\[0.5cm\] $q\bar{q}'\rightarrow q\bar{q}'$ ${\frac 49}{{\hat{s}^2+\hat{u}^2}\over{\hat{t}^2}}$ $-{\frac 49}{{\hat{s}\hat{u}}\over{\hat{t}^2}}$
\[0.5cm\] $gg\rightarrow q\bar{q}$ ${\frac 16}{{\hat{t}^2+\hat{u}^2} $---$
\over{\hat{t}\hat{u}}}-{\frac 38}{{\hat{t}^2+\hat{u}^2}
\over{\hat{s}^2}}$
\[0.5cm\]
-------------------------------------------- --------------------------------------------------------------- ----------------------------------------------------
: Partonic cross sections for $q_aq_b\rightarrow q_cq_d$ (only spin-average and transversity-dependent parts are shown here).
\[table\]
[^1]: This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement \#DF-FC02-94ER40818.
|
---
abstract: 'We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term *multiplexed* means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,\ldots,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,\ldots,d\}\times\{1,\ldots,d\}$. This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term *random environment* means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates. This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (*i.e.*the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere.'
address: |
Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, United Kingdom\
Mikhail.Menshikov@durham.ac.uk\
Institut de Recherche Mathématique, Université de Rennes I and CNRS UMR 6625, 35042 Rennes Cedex, France\
Dimitri.Petritis@univ-rennes1.fr\
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, CEP 05508-090, São Paulo SP, Brasil\
popov@ime.usp.br
author:
- Mikhail Menshikov
- Dimitri Petritis
- 'Serguei Popov[^1]'
bibliography:
- 'rwre.bib'
- 'matrix.bib'
- 'petritis.bib'
- 'cstar.bib'
title: Bindweeds or random walks in random environments on multiplexed trees and their asympotics
---
Introduction {#sec_intro}
============
On the generality of the random walk in a random environment on a tree {#ssec_genericity}
----------------------------------------------------------------------
Markov chains on denumerable graphs enter in the modelling of a rich variety of phenomena; among such graphs, trees play a basic and generic rôle in the sense that they can encode simultaneously
- the topological structure of a vast class of general (*i.e.* not necessarily tree-like) denumerable graphs,
- the combinatorial structure of paths on these general graphs, and
- the probability structure generated on the trajectory space of the Markov chain evolving on the these general graphs.
In the sequel we give a short explanation of the reason trees play such a generic rôle among denumerable graphs. We know, after Kolmogorov, that in order to describe a randov variable $X$ with space of outcomes a discrete measurable set $({\mathbb{X}},{\mathcal{X}})$, of law ${\mathbb{P}}_X$, one has to use an abstract probability space $(\Omega,{\mathcal{F}},{\mathbb{P}})$ on which the randov variable is defined. However, the choice of this space is not unique. Among the infinite possible choices, there exists a “minimal” one, given by $\Omega={\mathbb{X}}$, realising the random variable as the identity map $X(\omega)=\omega$ and ${\mathbb{P}}={\mathbb{P}}_X$. Similarly, when $(X_n)_{n\in{\mathbb{N}}}$ is a sequence of *independent* and identically distributed random variables with space of outcomes the discrete space $({\mathbb{X}}, {\mathcal{X}})$, the minimal realisation of the abstract probability space carrying the whole sequence is the *space of trajectories or full shift* $\Omega=
{\mathbb{X}}^{\mathbb{N}}$ and the sequence is realised by the canonical projection $X_n(\omega)=\omega_n$, for $n\in{\mathbb{N}}$. However for a sequence of *dependent* random variables the space of trajectories may be not minimal. The reader can easily convince herself by considering the example of ${\mathbb{X}}=\{a,b,\ldots, z\}$ and $(X_n)_{n\in{\mathbb{N}}}$ being the sequence of letters appearing in a natural language. Then the occurance $(X_n, X_{n+1}, X_{n+2})=rzt$ never appears in a language like English or French.
For the special case of Markovian dependence, the natural space for the realisation of the sequence $(X_n)_{n\in{\mathbb{N}}})$ is the so called unilateral or bilateral *subshift spaces* obtained from thh full shift by deleting all sequences containing forbidden subwords. For Markovian sequences, defined through a stochastic matrix $P$, subshift spaces can also be obtained in terms of the adjacency matrix of the sequence $A:{\mathbb{X}}\times{\mathbb{X}}\rightarrow \{0, 1\}$, given by the formula $A(x,y)=1$ whenever $P(x,y)>0$ and zero otherwise.
Suppose now we are given an arbitrary directed graph having a finite or denumerable set of vertices and being such that only a finite number of edges is emitted out of every vertex, not having any sources or sinks, and having (complex) weights attached on its edges. Since the vertex set is at most countable, it is in bijection with an at most countable alphabet. We say that a sequence of letters from the alphabet is a path of the graph, if the corresponding sequence of vertices is such that any ordered pair of subsequent vertices is an element of the edge set. The set of paths of arbitrary length is called *path space of the graph* (see [@KumPasRaeRen] for precise definitions.) Notice that the condition that two subsequent vertices must be an allowed edge prevents some sequences of vertices from being a path. The adjacency matrix defines a subshift space for the trajectories of a symbolic unilateral or bilateral Markov chain that is identifed with the path space of the graph. It is easy to show that the path space of any graph has a natural tree structure [@CamPet-wien], giving thus a first hint that trees play a prominent rôle among graphs. Their importance does not stop here however. We can in fact define a so-called *evaluation map* from the path space of a graph into some set ${\mathbb{A}}$. Depending on the precise algebraic structure of the set ${\mathbb{A}}$ and of the evaluation map, a vast class of objects can be defined. Instead of giving precise definitions, let us give the following
[(An elementary case)]{} Start with the finite complete graph on 4 vertices, denote them by the letters E, N, W, and S for definiteness, and consider the unilateral path space of the graph, *i.e.* the set of words of arbitrary length on the alphabet of these 4 letters. Choose for the space ${\mathbb{A}}$ the Abelian group ${\mathbb{Z}}^2$ and for the evaluation map the function taking the value $n_E {{\bf e}}_1+n_N {{\bf e}}_2+ n_W (-{{\bf e}}_1)+ n_S (-{{\bf e}}_2)\in {\mathbb{Z}}^2$, where $n_E,n_N, n_W, n_S$ denote the occurrences of the letters E, N, W, and S in a given word, and ${{\bf e}}_1,{{\bf e}}_2$ are the unit vectors of ${\mathbb{Z}}^2$. The evaluation map can be thought as the physical obervable “position of the random walker” when the history of the individual directions of movement is kept in memory; individual directions define a path in the path space and the evaluation map computes the actual position of the walker which has done $n_E$ eastward movements, $n_W$ westward movements, etc. The graph that is generated in this way is the graph that coincides with the Cayley graph of the Abelian group ${\mathbb{Z}}^2$. The evaluation map being many-to-one, its multiplicity encodes the combinatorics of the path space while the product of the graph weights gives the relative weight of individual paths in the path space of this graph. When the weights are probability vectors, this weight coincides with the probability of the trajectory of the simple random walk on ${\mathbb{Z}}^2$. Notice however that the weights need not to be probability vectors, allowing us to consider both random and quantum grammars.
The above example serves as a basic paradigm; by appropriately changing the evaluation map, it can be generalised in numerous ways to produce configuration space for DNA strands (when the path space coincides with the path space of the complete graph over 4 letters), non-Abelian groups like the free group on 2 generators, Cuntz-Krieger $C^*$-algebras (*i.e* non-commutative operator algebras attaching non-zero partial isometries on every edge; see [@Pet-GDG] for additional details) naturally arising in some problems of quantum information. Infinite trees provide a rich variety of mathematical problems, particularly connected to their non-amenability; beyond their mathematical interest they arise as more or less realistic models in several applied fields like random search algorithms in large data structures, Internet traffic, random grammars and probabilistic Turing machines, DNA coding, interacting random strings and automated languages, etc. The previously exposed ideas, already lurking in [@Mal-ISS; @Mal-SEGG], have been exemplified in [@FlaSed; @CamPet-wien], and will be further exploited in [@Pet-GDG].
Random environment is a means to introduce context-dependence in the language. Again, due to the generality of trees as underlying graphs, random walks in random environment on trees provide interesting context-dependent results for random walks on a huge class of more general graphs.
Random walks in random environments on various types of graphs are known to display a behaviour dramatically differing from the one for ordinary random walks on the same graph. See [@Sol; @Sin; @KesKozSpi; @Kes; @LyoPem; @ComMenPop; @Mal-RG; @MenPet-rwre; @ComPop; @CamPet-rwrol] etc.for a very partial list of known results and models.
Rough statement of the main results for bindweeds
-------------------------------------------------
As a simple example, consider a rooted tree with constant branching $b$, and let us construct a random walk in random environment on it by sampling the transition probabilities (or transition rates, for continuous time) in each vertex from a given distribution, independently. When studying the question of (positive) recurrence of such a walk, one naturally arrives (see [@LyoPem; @MenPet-rwre]) at the following model. On each edge $a$ of the tree, we place an independent copy of a positive random variable ${\hat \xi}_a$. Then, for each vertex ${{\bf v}}$ denote ${\hat \xi}[{{\bf v}}]={\hat \xi}_{a_1}\ldots{\hat \xi}_{a_n}$, where $a_1,\ldots,a_n$ is the (unique) path connecting the root to ${{\bf v}}$. Models of this type are called [*multiplicative cascades*]{} (see e.g. [@KahPey; @LiuRou]), and, to study the positive recurrence of the corresponding random walk in random environment, one has to answer the question whether the sum of ${\hat \xi}[{{\bf v}}]$ is finite. It turns out (see e.g. [@LyoPem]) that the classification parameter for this problem is $$\label{class_param}
\hat\lambda = \inf_{s\in[0,1]}{\mathbb{E}}{\hat \xi}_a^s,$$ which is then compared to $1/b$ (in fact, in [@LyoPem] the case of general tree was considered) and the following result is established:
- if $\hat\lambda b<1$ then the random walk for almost all environments is positive recurrent
- if $\hat\lambda b>1$ then the random walk for almost all environments is transient.
The critical case $\hat\lambda b=1$ is more complicated.
In [@MenPetPop], we introduced a model of random walk with internal degrees of freedom whose multiplicative cascade counterpart is expressed by a model placing random matrices on the edges in place of scalar random variables. The main classification parameter, $\lambda$, will be defined in the formula (\[eq-k(s)\]) below, and the main results are Theorems \[th-mmc-ergodicity\] and \[th-mmc-transience\] (a lot of preliminary work is required, however, before formulating these results). This model is also equivalent to random walk in random environment on a multiplexed tree, a process we term *bindweed* in the sequel. However, for trees with average branching $b$, the rough statement of our result is as above with $\hat\lambda$ replaced by $\lambda$.
It was remarked in [@MenPet-rwre] that asymptotic properties like recurrence/transience of random walk on trees with constant branching $b$ are intimately connected to the existence of non-trivial solutions for the so-called multiplicative chaos equation of order $b$, first introduced as a simple turbulence model in [@Man]. The simplest variant of the multiplicative chaos equation is the following: let $(\xi_i)_{i=1,\ldots, b}$, with $b\in{\mathbb{N}}$, be a finite family of non-negative random variables having known joint distribution and $(Y'_i)_{i=1,\ldots, b}$ and $Y$ be a family of $b+1$ independent non-negative random variables distributed according the same unknown law and verifying $$Y{\stackrel{\text{\tiny law}}{=}}\sum_{i=1}^b Y'_i \xi_i.$$ The multiplicative chaos problem consists in determining under which conditions on the joint distribution of the $\xi$’s the above equation has a non-trivial solution. This scalar problem is thoroughly studied in the literature, see e.g. [@ColKou; @DurLig; @Liu97]. As we remark later in this paper, the matrix multiplicative chaos equation may be an interesting problem to study as well.
Notation {#ssec-notations}
========
In this section we give the formal definitions concerning trees, in particular, we define the notions of the growth rate and the branching number.
We denote ${\mathbb{R}}_+=[0,\infty[$, ${{\mathbb{Z}}_+ }=\{0,1,2,\ldots\}$, ${{\mathbb{N}}}=\{1,2,3,\ldots\}$, and for every $n\in {{\mathbb{N}}}$, ${\mathbb{N}}_n=\{1,2,\ldots,n\}$, while ${\mathbb{N}}_0=\emptyset$. Let ${\mathcal{A}}\equiv{\mathcal{A}}^1$ be a finite or infinite denumerable set, called the *alphabet*. Define ${\mathcal{A}}^0=\{\emptyset\}$ and for every $n\in {{\mathbb{N}}}$ denote $${\mathcal{A}}^n=\{{{\mbox{\boldmath $ \alpha $}}}=\alpha_1\cdots\alpha_n:
\alpha_i\in {\mathcal{A}}\ \textrm{for}\ i\in{\mathbb{N}}_n\}$$ the set of words of length $n$ (*i.e.* having $n$ letters), $${\mathcal{A}}^*=\cup_{n\in{{\mathbb{Z}}_+ }}{\mathcal{A}}^n$$ the set of words of arbitrary (finite) length, and $$\partial{\mathcal{A}}^*\equiv{\mathcal{A}}^\infty=
\{{{\mbox{\boldmath $ \alpha $}}}=\alpha_1\alpha_2\cdots: \alpha_i\in {\mathcal{A}}\ \textrm{for}\
i\in{{\mathbb{N}}}\}$$ the set of infinite words. Finally, denote ${{\overline{{\mathcal{A}}^*}}}={\mathcal{A}}^*\cup\partial{\mathcal{A}}^*$ and ${{\stackrel{\circ}{{\mathcal{A}}^*}}}={\mathcal{A}}^*\setminus {\mathcal{A}}^0$.
For every ${{\mbox{\boldmath $ \alpha $}}}\in{\mathcal{A}}^*$, there exists $n\in{{\mathbb{Z}}_+ }$ such that ${{\mbox{\boldmath $ \alpha $}}}\in {\mathcal{A}}^n$; in this situation $|{{\mbox{\boldmath $ \alpha $}}}|:=n$ denotes the *length* of the word ${{\mbox{\boldmath $ \alpha $}}}$ with the convention $|\emptyset|=0$. Consistently, for every ${{\mbox{\boldmath $ \alpha $}}}\in\partial{\mathcal{A}}^*$, we have $|{{\mbox{\boldmath $ \alpha $}}}|=\infty$. For ${{\mbox{\boldmath $ \alpha $}}}\in{{\overline{{\mathcal{A}}^*}}}$ with $|{{\mbox{\boldmath $ \alpha $}}}|\geq n$ we denote by ${{\mbox{\boldmath $ \alpha $}}}{{\upharpoonright}_{n}}=
\alpha_1\cdots\alpha_n\in{\mathcal{A}}^n$ the *restriction* of ${{\mbox{\boldmath $ \alpha $}}}$ to its $n$ first letters with the convention ${{\mbox{\boldmath $ \alpha $}}}{{\upharpoonright}_{0}}=\emptyset$. For every ${{\mbox{\boldmath $ \alpha $}}}\in{{\stackrel{\circ}{{\mathcal{A}}^*}}}$, the *ancestor* $\hat{{{\mbox{\boldmath $ \alpha $}}}}$ of ${{\mbox{\boldmath $ \alpha $}}}$ is defined by $\hat{{{\mbox{\boldmath $ \alpha $}}}}={{\mbox{\boldmath $ \alpha $}}}{{\upharpoonright}_{|{{\mbox{\boldmath $ \alpha $}}}|-1}}$. For ${{\mbox{\boldmath $ \alpha $}}}\in{\mathcal{A}}^*$ and ${{\mbox{\boldmath $ \beta $}}}\in{{\overline{{\mathcal{A}}^*}}}$, the *concatenation* of ${{\mbox{\boldmath $ \alpha $}}}$ followed by ${{\mbox{\boldmath $ \beta $}}}$ is the word ${{\mbox{\boldmath $ \alpha $}}}{{\mbox{\boldmath $ \beta $}}}=
\alpha_1\cdots\alpha_{|{{\mbox{\boldmath $ \alpha $}}}|}\beta_1\beta_2\cdots$ and for ${{\mbox{\boldmath $ \alpha $}}},{{\mbox{\boldmath $ \beta $}}}\in {\mathcal{A}}^*$, their *common radix* ${{\mbox{\boldmath $ \alpha $}}}\wedge{{\mbox{\boldmath $ \beta $}}}$ is the longest word ${{\mbox{\boldmath $ \gamma $}}}\in{\mathcal{A}}^*$ such that ${{\mbox{\boldmath $ \alpha $}}}={{\mbox{\boldmath $ \gamma $}}}{{\mbox{\boldmath $ \alpha' $}}}$ and ${{\mbox{\boldmath $ \beta $}}}={{\mbox{\boldmath $ \gamma $}}}{{\mbox{\boldmath $ \beta' $}}}$ for some words ${{\mbox{\boldmath $ \alpha' $}}},{{\mbox{\boldmath $ \beta' $}}}\in {\mathcal{A}}^*$. We write ${{\mbox{\boldmath $ \alpha $}}}\leq{{\mbox{\boldmath $ \beta $}}}$ if ${{\mbox{\boldmath $ \alpha $}}}={{\mbox{\boldmath $ \alpha $}}}\wedge{{\mbox{\boldmath $ \beta $}}}$.
[**Remark: **]{}
Notice that, consistently with the above notation, the symbol ${\mathbb{N}}^*$ denotes the set of finite words on the alphabet ${\mathbb{N}}$, contrary to some tradition (especially the French one) where this symbol is used to denote what we call here ${{\mathbb{N}}}$.
A mapping $B:{\mathbb{N}}^*\rightarrow{{\mathbb{Z}}_+ }$ is called a *branching function*.
To each branching function corresponds a uniquely determined rooted tree ${\mathbb{T}}=({\mathbb{V}},{\mathbb{A}})$ with vertex set ${\mathbb{V}}\equiv{\mathbb{V}}^*(B)\subseteq{\mathbb{N}}^*$ and edge set ${\mathbb{A}}={{\stackrel{\circ}{{\mathbb{V}}}}}$ defined as follows: ${\mathbb{V}}^*(B)=\cup_{n\in{\mathbb{N}}}{\mathbb{V}}^n(B)$ where ${\mathbb{V}}^0(B)=\{\emptyset\}=
\{\textrm{root}\}\equiv\{{{\bf 0}}\}$ and for $n\in{{\mathbb{N}}}$, $${\mathbb{V}}^n(B)=\{{{\bf v}}=v_1\cdots v_n: v_l\in {\mathbb{N}}_{B({{\bf v}}{{\upharpoonright}_{l-1}})},
\ \textrm{for}\ l=1,\ldots, n\}.$$ The branching function is said to be *without extinction* if the corresponding tree has non-trivial boundary $\partial{\mathbb{V}}$. The edge set is the subset of unordered pairs of vertices $[{{\bf u}},{{\bf v}}]=[{{\bf v}},{{\bf u}}]$ such that either ${{\bf v}}=\hat{{{\bf u}}}$ or ${{\bf u}}=\hat{{{\bf v}}}$. Since every vertex has a unique ancestor, every edge is indexed by its outmost vertex, *i.e.* for every ${{\bf v}}\in{{\stackrel{\circ}{{\mathbb{V}}}}}$, the corresponding edge is $a({{\bf v}})=[\hat{{{\bf v}}},{{\bf v}}]$, showing thus that ${\mathbb{A}}\simeq {{\stackrel{\circ}{{\mathbb{V}}}}}$.
If ${{\bf u}},{{\bf v}}\in{\mathbb{V}}$ and ${{\bf u}}\leq{{\bf v}}$ we define the *path* $[{{\bf u}},{{\bf v}}]$ as the collection of the $|{{\bf v}}|-|{{\bf u}}|$ edges $[{{\bf u}}, {{\bf v}}{{\upharpoonright}_{|{{\bf u}}|+1}}],\ldots, [\hat{{{\bf v}}},{{\bf v}}]$, and if ${{\bf u}}=\emptyset$ then we simply denote by $[{{\bf v}}]$ the path $[\emptyset,{{\bf v}}]$ for every ${{\bf v}}\in{{\stackrel{\circ}{{\mathbb{V}}}}}$. In the sequel we shall consider only *branching functions without extinction*.
\[def-grr\] Let $\kappa_n={\mathop{\rm card}}{\mathbb{V}}^n(B)$ denote the cardinality of the $n^{\textrm{th}}$ generation of the tree defined by the branching function without extinction $B$. We call *lower growth rate* of the tree $${\underline{\mathsf{gr}}({\mathbb{V}})}=\liminf_n \kappa_n^{1/n},$$ *upper growth rate* of the tree $${{{\overline{\mathsf{gr}}}}({\mathbb{V}})}=\limsup_n \kappa_n^{1/n},$$ and, if ${\underline{\mathsf{gr}}({\mathbb{V}})}={{{\overline{\mathsf{gr}}}}({\mathbb{V}})}$, we call the common value *growth rate* $${\mathsf{gr}({\mathbb{V}})}=\lim_n \kappa_n^{1/n}.$$
For ${{\bf u}},{{\bf v}}\in\partial{\mathbb{V}}$, define $\delta({{\bf u}},{{\bf v}})=\exp(-|{{\bf u}}\wedge{{\bf v}}|)$. It can be shown that $\delta$ is a distance on $\partial{\mathbb{V}}$. Moreover if $\|B\|_\infty=\sup_{{{\bf v}}\in{\mathbb{V}}} B({{\bf v}})<\infty$ then the space $(\partial{\mathbb{V}},\delta)$ is compact and we can define its Hausdorff dimension $\dim_H\partial{\mathbb{V}}$ as usual (see [@Falconer] for instance).
\[def-brr\] For a tree ${\mathbb{V}}$ generated by a branching function $B$ with $\|B\|_\infty<\infty$, we define its *branching rate* $${\mathsf{br}({\mathbb{V}})}=\exp(\dim_H\partial{\mathbb{V}}).$$
It is shown in [@GraMauWil; @LyoPem] that ${\mathsf{br}({\mathbb{V}})}=\sup\{\lambda: \inf\sum_{v\in C}\lambda^{-|{{\bf v}}|}>0\}$ where the infimum is evaluated over all cutsets $C$ of ${\mathbb{V}}$. We have in general that ${\mathsf{br}({\mathbb{V}})}\leq {\underline{\mathsf{gr}}({\mathbb{V}})}$.
Matrix multiplicative cascades and the corresponding results {#ssec-mmc}
============================================================
Let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be some abstract probability space which carries all the random variables that will be needed in the model. Let $({\mathbb{V}},{\mathbb{A}})$ be the rooted tree associated with a given branching function $B$. Let $G$ be the topological group ${\mathsf{GL}(d,{\mathbb{R}})}$, ${\mathcal{G}}$ its Borel $\sigma$-algebra and $\mu$ a probability on $(G,{\mathcal{G}})$. Denote by $\sigma_\mu={\mathop{\sf supp}}\mu\subset G$ the support of the measure $\mu$ and by $\Sigma_\mu$ the semi-group generated by $\sigma_\mu$. On $(\Omega,{\mathcal{F}},{\mathbb{P}})$, define an edge-indexed family of independent $G$-valued random variables $(\xi_a)_{a\in{\mathbb{A}}}$ identically distributed according to $\mu$, *i.e.*$${\mathbb{P}}(\xi_a\in {\mathop{\rm d}}g)=\mu({\mathop{\rm d}}g), \mbox{ for all } a\in{\mathbb{A}}.$$ For ${{\bf u}}, {{\bf v}}\in{\mathbb{V}}$ with ${{\bf u}}\leq{{\bf v}}$ define $$\xi[{{\bf u}},{{\bf v}}]\equiv\prod\limits^{\leftarrow}_{a\in[{{\bf u}},{{\bf v}}]}\xi_a,$$ where $\prod\limits^{\leftarrow}$ denotes the product in reverse order, *i.e.* if $[{{\bf u}},{{\bf v}}]=a_1\cdots a_k$ then $\xi[{{\bf u}},{{\bf v}}]=\xi_{a_k}\cdots \xi_{a_1}$ with the convention $\xi[{{\bf v}},{{\bf v}}]=e$ where $e$ is the neutral element of $G$. We introduce the following $G$-valued random processes: the *matrix-multiplicative cascade process* $$\psi_n=\sum_{{{\bf v}}\in{\mathbb{V}}^n} \xi[{{\bf v}}], \quad n \in {\mathbb{N}}$$ and the *integrated matrix-multiplicative cascade process* $$\zeta_n=\sum_{k=1}^n \psi_k, \quad n \in{{\mathbb{N}}}.$$
For a fixed ${{\bf v}}\in\partial {\mathbb{V}}$ and all $n\in{\mathbb{N}}$ $$X_n\equiv X_n({{\bf v}})= \xi[{{\bf v}}{{\upharpoonright}_{n}}].$$ It is immediate to see (cf. [@Revuz]) that $(X_n)_{n\in{{\mathbb{N}}}}$ is a $G$-valued multiplicative Markov chain with stochastic kernel $$P(g,{\mathop{\rm d}}g')\equiv{\mathbb{P}}(X_{n+1}\in {\mathop{\rm d}}g'|X_n=g)=\mu\star\delta_g({\mathop{\rm d}}g'),
\quad n\in {{\mathbb{N}}},$$ where for two measures $\mu,\mu'$ on $(G,{\mathcal{G}})$ their convolution $\mu\star\mu'$ is defined by its dual action on $L^1(G)$ via $${\langle\,}\mu\star\mu', f{\,\rangle}\equiv \int_G f(g) \mu\star\mu'({\mathop{\rm d}}g)=
\int_G\int_G f(gg')\mu({\mathop{\rm d}}g)\mu'({\mathop{\rm d}}g'),$$ for all $f\in L^1(G)$. We equip ${\mathop{\sf End}}({\mathbb{R}}^d)\equiv {\mathsf{GL}(d,{\mathbb{R}})}=G$ with the operator norm, denoted $\|\cdot\|$, stemming from the $l_1$ norm of the vector space ${\mathbb{R}}^d$.
In order to be able to apply the results of [@GuiLeP] to our special case, we require the following conditions on $\mu$:
**Condition 1 (Integrability):**For all $s\in{\mathbb{R}}_+$, $$\int_G \|g\|^s \mu({\mathop{\rm d}}g)<\infty.$$
**Condition 2 (Strong irreducibility):** We assume that the set $\Sigma_\mu$ is strongly irreducible, *i.e.* there is no finite $\Sigma_\mu$-invariant family of proper subspaces.
**Condition 3 (Strict positivity):**We assume that $\sigma_\mu\subseteq {{\overline{G}}}_+$ and that $\mu({{\overline{G}}}_+\setminus G_+)=0$.
For $s\geq 0$, define $$\label{def-k(s)}
k(s)=\lim_n\left(\int_G \|g\|^s\mu^{\star n}({\mathop{\rm d}}g)\right)^{1/n}.$$ (By virtue of theorem 1 of [@GuiLeP], under conditions 1–3 this limit exists in ${\mathbb{R}}_+$ and defines a log-convex function. As a matter of fact, in [@GuiLeP] a weaker condition than 3, called proximality, is needed to prove this result.) We define in the sequel the quantity $\lambda$, that turns out to be the main classification parameter for the matrix multiplicative cascades model, by $$\label{eq-k(s)}
\lambda= \inf_{s\in[0,1]} k(s)$$ (compare (\[eq-k(s)\]) with (\[class\_param\]).)
We are now in the position to state our main results.
\[th-mmc-ergodicity\] Let $({\mathbb{V}},{\mathbb{A}})$ be some tree defined in terms of a given branching function $B$ and ${{{\overline{\mathsf{gr}}}}({\mathbb{V}})}$ and $\lambda$ defined as in definition \[def-grr\] and equation (\[eq-k(s)\]) respectively. Under the conditions 1, 2, and 3, $$\lambda {{{\overline{\mathsf{gr}}}}({\mathbb{V}})}<1 \Rightarrow
\zeta_{\infty, ij}<\infty
\textrm{ almost surely, for all}\ i,j=1,\ldots,d.$$
\[th-mmc-transience\] Let ${\mathsf{br}({\mathbb{V}})}$ and $\lambda$ be the quantities introduced in definition \[def-brr\] and equation \[eq-k(s)\] respectively and let $\chi\in {\mathbb{R}}^d$ be the vector having all its components equal to 1: $\chi_i=1$, for all $i=1,\ldots, d$. Let $({\mathbb{V}},{\mathbb{A}})$ be some tree defined in terms of a given branching function $B$ without extinction. Under the conditions 1, 2, and 3’, $$\lambda {\mathsf{br}({\mathbb{V}})}>1\Rightarrow Z_\infty := (\chi, \zeta_\infty\chi)=
\infty \ \textrm{almost surely.}$$
[**Remark: **]{}
Similarly to [@LyoPem], there is a gap between Theorems \[th-mmc-ergodicity\] and \[th-mmc-transience\], since in general the branching number need not be equal to the growth rate. However, this is not very important, because in most of the practical examples these quantities do coincide.
[**Remark: **]{}
As mentioned above, the classification parameter for this problem is $\lambda=\inf_{s\in[0,1]} k(s)$. This parameter is not explicitly computable in general since it involves the infinite product of matrices. However for some particular cases this quantity can be computed explicitly as stated in the following proposition.
\[prop-lambda=rho\] Suppose that the measure $\mu$ is such that $g_{ij}<1/d$ almost surely for all $i,j=1,\ldots,d$. Then $\lambda$ is the largest eigenvalue of the matrix ${\mathbb{E}}g$.
[**Remark: **]{}
It is interesting to consider the *chaos equation* for the case of matrix-valued random variables and constant branching $b$: $$\label{chaos_eq}
Y {\stackrel{\text{\tiny law}}{=}}\sum_{j=1}^b Y'_j\xi_j,$$ where $Y,Y'_j,\xi_j$ are $G$-valued random variables, and $\xi_j$ (which are not necessarily independent) are distributed according to $\mu$; $Y'_j$, $j=1,\ldots,b$, are i.i.d. and have the same (unknown) law as $Y$. Analogously to [@MenPet-rwre] we can get that (at least in the case when $\xi_1$ satisfies conditions 1,2,3) $\lambda d = 1$ is a necessary condition for the existence of solution of (\[chaos\_eq\]). It is an open problem whether this condition is sufficient.
[**Remark: **]{}
The condition of independence of the random variables $\xi_a$ can be relaxed; what is important is
1. if $\xi_a$ and $\xi_b$ are not adjacent to the same vertex then they must be independent, and
2. the $\xi$’s that belong to any path emanating from the root must be independent.
The bindweed model {#ssec-rw}
==================
In this section we introduce a model describing an evolution of a random string in random environment on a tree (which is somewhat similar to the model studied in [@ComMenPop]) which we call the bindweed model. Then, we show that its classification from the point of view of positive recurrence can be obtained by using theorems \[th-mmc-ergodicity\] and \[th-mmc-transience\].
Let ${\mathcal{S}}=\{1,\ldots,d\}$ be a finite alphabet and denote, in accordance with the notations introduced in Section \[ssec-notations\], ${\mathcal{S}}^{n+1}=\{{{\mbox{\boldmath $ \sigma $}}}=\sigma_0 \cdots \sigma_n: \sigma_i\in{\mathcal{S}}\}$ the set of words of length $n+1$ composed from the symbols of the alphabet ${\mathcal{S}}$, ${\mathcal{S}}^0$ the set containing only the empty word and ${\mathcal{S}}^*$ the set of words of arbitrary length. Suppose that a branching function $B$ is given on ${\mathbb{N}}^*$ and denote ${\mathbb{V}}^n\equiv{\mathbb{V}}^n(B)$ the corresponding generations of the tree determined by $B$. Therefore, the rooted tree ${\mathbb{T}}=({\mathbb{V}},{\mathbb{A}})$ is uniquely defined.
Now we are going to construct a continuous-time Markov chain with state space $\mathfrak{S}$, defined by $$\mathfrak{S} = \{\hat{\emptyset}\}\cup \bigcup_{n=1}
({\mathbb{V}}^n\times{\mathcal{S}}^{n+1}),$$ where $\hat{\emptyset}$ is a special state to be defined later. In fact, what happens is the following: we place a word ${{\mbox{\boldmath $ \sigma $}}}=\sigma_0\ldots \sigma_n$ on the tree ${\mathbb{T}}$ in such a way that the $0^{\textrm{th}}$ symbol of the word is placed on the root ${{\bf 0}}$, for any $i=1,\ldots,n$ the $i^{\textrm{th}}$ symbol of the word is placed somewhere in ${\mathbb{V}}^i$, and, if the $i^{\textrm{th}}$ symbol $\sigma_i$ is placed on vertex ${{\bf u}}$, and $\sigma_{i+1}$ on ${{\bf v}}$, then ${{\bf u}}<{{\bf v}}$ and $[{{\bf u}},{{\bf v}}]\in{\mathbb{A}}$ (see figure \[fig\_bindweed\]). The state ${\hat \emptyset}$ means that nothing is placed on the tree.
![[]{data-label="fig_bindweed"}](mmc-fig1.eps){width="50.00000%"}
Now, let us define the dynamics of the bindweed model. Suppose that for any $a\in{\mathbb{A}}$ two collections of positive numbers $(\nu_{yz}(a), y,z \in {\mathcal{S}})$, $(\mu_y(a), y \in {\mathcal{S}})$ are given. If the bindweed model is in the state $({{\bf u}},{{\mbox{\boldmath $ \sigma $}}})$, where ${{\mbox{\boldmath $ \sigma $}}}=\sigma_0\ldots \sigma_{n-1}y$, ${{\bf u}}\in{\mathbb{V}}^n$, then
- for $n\geq 0$ it jumps to the state $({{\bf v}}, \sigma_0\ldots \sigma_{n-1}y z)$ with rate $\nu_{y z}(a({{\bf v}}))$, for all ${{\bf v}}\in{\mathbb{V}}: {{\bf u}}=\hat{{{\bf v}}}$;
- for $n\geq 1$ it jumps to the state $(\hat{{{\bf u}}}, \sigma_0\ldots
\sigma_{n-1})$ with rate $\mu_y(a({{\bf u}}))$.
For any $\sigma_0\in{\mathcal{S}}$ the transitions ${\hat \emptyset}\to ({{\bf 0}},\sigma_0)$ and $({{\bf 0}},\sigma_0)\to {\hat \emptyset}$ occur with rate $1$. Thus, we have defined a continuous-time Markov chain with state space $\mathfrak{S}$.
Let us describe now how to choose the transition rates. Let $\rho$ be any probability measure on ${\mathbb{R}}^{d^2+d}_+$. Suppose that for any $a\in{\mathbb{A}}$ the vector $\Xi(a)=(\nu_{yz}(a),y,z\in{\mathcal{S}},
\mu_y,y\in{\mathcal{S}})$ is random, having distribution $\rho$, and $(\Xi(a), a\in{\mathbb{A}})$ are independent and identically distributed. Fix a realisation of that collection of random vectors and consider the bindweed model with the transition rates ruled by that realisation. So, the model that we constructed is a continuous-time Markov chain in a quenched random environment.
Now, we are interested in obtaining a classification of this Markov chain with respect to positive recurrence. For $({{\bf v}},{{\mbox{\boldmath $ \sigma $}}})\in\mathfrak{S}$ denote by $\pi({{\bf v}},{{\mbox{\boldmath $ \sigma $}}})$ the stationary measure. For any $a\in{\mathbb{A}}$ let $\xi_a$ be a $d\times d$ matrix whose matrix elements are defined in the following way: $\xi_{a, xy}=\nu_{xy}(a)/\mu_y(a)$, $x,y\in{\mathcal{S}}$. It is not difficult to see that we have a reversible Markov chain, so it is clear that $\pi({\hat \emptyset}) = \pi({{\bf 0}},x)$, for all $x\in{\mathcal{S}}$, and, for any ${{\bf v}}\in{\mathbb{V}}^n$, $n\geq 1$, and $x,y,\sigma_0,\ldots,\sigma_{n-2}\in{\mathcal{S}}$, we can formally write $$\begin{aligned}
\pi({{\bf v}},\sigma_0\ldots \sigma_{n-2}x y)
& = & \frac{\nu_{xy}(a({{\bf v}}))}{\mu_y(a({{\bf v}}))}
\pi(\hat{{{\bf v}}},\sigma_0\ldots \sigma_{n-2}x)\nonumber \\
&=& \xi_{a({{\bf v}}),xy} \pi(\hat{{{\bf v}}},\sigma_0\ldots \sigma_{n-2}x)
\label{eq-revers}.\end{aligned}$$
Then it is shown in [@MenPetPop] that $$\sum_{\substack{{{\bf v}}\in{\mathbb{V}}^n\\{{\mbox{\boldmath $ \sigma $}}}\in {\mathcal{S}}^n}}
\pi({{\bf v}},{{\mbox{\boldmath $ \sigma $}}}) = \pi({\hat \emptyset})
\left(\chi, \sum_{{{\bf v}}\in{\mathbb{V}}^n} \xi[{{\bf v}}] \chi\right)$$ where $\chi$ is the vector of order $d$ with all its coordinates equal to $1$. Thus $\sum_{({{\bf v}},{{\mbox{\boldmath $ \sigma $}}})
\in\mathfrak{S}}\pi({{\bf v}},{{\mbox{\boldmath $ \sigma $}}})$ is finite if and only if $Z_{\infty}$ is finite. Thus, theorems \[th-mmc-ergodicity\] and \[th-mmc-transience\] allow us to obtain the classification of the bindweed model in random environment from the point of view of positive recurrence, in the following way:
Suppose that the distribution of the random matrix $\xi_a$ is such that the Conditions 1, 2, and 3 are satisfied. Let $\lambda$ be the quantity defined as in Section \[ssec-mmc\]. Then
- if $\lambda{{{\overline{\mathsf{gr}}}}({\mathbb{V}})}< 1$, then the bindweed model is positive recurrent;
- if $\lambda{\mathsf{br}({\mathbb{V}})}> 1$, then the bindweed model is not positive recurrent.
Open problems and further developments
======================================
We demonstrated a close relationship between matrix multiplicative cascades and random walks in random environment on multiplexed trees. In particular it is proven in [@MenPetPop] that both systems are classified by the same parameter. However, the critical region remains out of reach for the moment. Firstly it is not known whether, for sufficiently recular trees so that ${{{\overline{\mathsf{gr}}}}({\mathbb{V}})}={\mathsf{br}({\mathbb{V}})}=b$, the walk is null recurrent for $\lambda b=1$ or some additional condition is needed on $\mu$ as is the case for scalar multiplicative chaos [@Liu97] and for random walk [@LyoPem]. Returning to the general tree where ${{{\overline{\mathsf{gr}}}}({\mathbb{V}})}\not={\mathsf{br}({\mathbb{V}})}$, we obtain a gap in the space of classifying parameters. It is however conjectured in [@Mal-ISS] that in general, the set of critical values is of zero Lebesgue measure for string problems. We expect the same phenomenon to occur here. Nevertheless, whether the critical value of the parametre is ${\mathsf{br}({\mathbb{V}})}$ or ${{{\overline{\mathsf{gr}}}}({\mathbb{V}})}$ or some intermediate value is unknown for the moment.
An important step towards understanding these problem should be made if conditions for the existence of non-trivial fixed points of the functional equation (\[chaos\_eq\]) were obtained. This remains for the moment an open problem although under investigation.
[^1]: Partially supported by CNPq (302981/02-0)
|
---
abstract: |
Motivated by modern applications of light detection and ranging (LIDAR), we study the model of an optical receiver based on an avalanche photo-diode (APD), followed by electronic circuitry for detection of reflected optical signals and estimation of their delay. This model is known to be quite complicated as it consists of at least three different types of noise: thermal noise, shot noise, and multiplicative noise (excess noise) that stems from the random gain associated with the photo-multiplication of the APD. Consequently, the derivation of the optimal likelihood ratio test (LRT) associated with signal detection is a non–trivial task, which has no apparent exact closed–form solution. We consider instead a class of relatively simple detectors, that are based on correlating the noisy received signal with a given deterministic waveform, and our purpose is to characterize the optimal waveform in the sense of the best trade–off between the false–alarm (FA) error exponent and the missed–detection (MD) error exponent. In the same spirit, we also study the problem of estimating the delay on the basis of maximizing the correlation between the received signal and a time–shifted waveform, as a function of this time shift. We characterize the optimal correlator waveform that minimizes the mean square error (MSE) in the regime of high signal–to–noise ratio (SNR). The optimal correlator waveforms for detection and for estimation turn out to be different, but their limiting behavior is the same: when the thermal Gaussian noise is dominant, the optimal correlator waveform becomes proportional to the clean signal, but when the thermal noise is negligible compared to the other noises, then it becomes logarithmic function of the clean signal, as expected.\
[**Index Terms:**]{} optical detection, likelihood–ratio test, range finding, delay estimation, shot noise, error exponent.
author:
- Neri Merhav
title: Optimal Correlators for Detection and Estimation in Optical Receivers
---
The Andrew & Erna Viterbi Faculty of Electrical Engineering\
Technion - Israel Institute of Technology\
Technion City, Haifa 32000, ISRAEL\
E–mail: [merhav@ee.technion.ac.il]{}\
Introduction and Background
===========================
The concept of light detection and ranging (LIDAR), which can be thought of as the optical analogue of radar, is by no means new, and during the many years it has been in use, it has found an extremely large variety of applications in a wide spectrum of areas and disciplines, including: agriculture, archaeology, biology, astronomy, geology and soil sciences, forestry, meteorology, and military applications, just to name a few. Most notably, there are several modern technologies that involve LIDAR, such as autonomous vehicles, space flight devices, robots of many kinds, systems with GRID based processing, and in the future, and face recognition in biometric systems (e.g., at airports).
This background sets the stage and motivates renewed interest in optical signal detection and estimation. The customary model of a direct–detection optical receiver (or detector) consists of a photo-diode (PIN diode or avalanche photo-diode), that converts the intensity of the received optical (laser) signal, modeled as the rate function of a variable–rate Poisson process, into a train of current impulses generated by the photo–electrons at random time instants, pertaining to the Poisson arrivals. This current is then fed into some electronic circuitry, whose first stage is normally a trans-impedance amplifier (TIA), that amplifies the current signal and converts it into a relatively strong voltage signal, but this amplification comes at the cost of some distortion as well as thermal noise associated with the electronic circuitry.
The challenge in the development of a solid theory of detection and estimation, for such a signal–plus–noise system, is that it is has a rather complicated model due to the various types of noise involved. The combination of shot noise, due to the photo-diode, and the thermal electronic noise is already not trivial. In the case of an avalanche photo–diode (APD), which is the more relevant case, there is an additional, third type of noise, namely, the [*excess noise*]{}, induced by the APD, which is actually a multiplicative noise process pertaining to fluctuations in the random gain associated with the avalanche mechanism, as each primary electron–hole pair (generated by a photon absorbed in the photo-diode), may generate secondary electron–holes, which in turn can generate additional electron–holes, and so on. For more details, and additional aspects of the problem area, the interested reader is referred to some earlier work, e.g., [@BL90], [@Einarsson96], [@EH81], [@FGS75], [@GJN09], [@GP87], [@MS76], [@Olsson89], [@Personick71a], [@Personick71b], [@Personick08], [@Salz85], [@SSH18], [@TM94], and [@YLC19], which is by no means an exhaustive list of relevant articles (and a book).
To provide just a rough, preliminary view on the problem and to fix ideas, we now give an informal presentation of the model and explain the difficulties more concretely. The received signal is modeled as $$\label{sigmod}
y(t)=\sum_{k=1}^K g_kh(t-t_k)+n(t),~~~~~~0\le t < T,$$ where $K$ is the number of photo–electrons generated during the time interval $[0,T)$, $h(\cdot)$ is the current pulse contributed by a single electron (which is nearly equal to the charge of the electron multiplied by the Dirac delta function), $\{t_k\}$ are the random Poisson arrival times induced by the optical signal, $\lambda(t)$, sensed by the APD, $\{g_k\}$ are the random gains induced by the APD photo-multiplication, and $\{n(t)\}$ is thermal noise, modeled here, and in earlier works, to be white Gaussian noise with spectral density $N_0/2$.[^1] The signal detection problem, in its basic form, is about binary hypothesis testing. The null hypothesis is that $y(t)=n(t)$, whereas the alternative is as in (\[sigmod\]). Had $K$, $\{g_k\}$ and $\{t_k\}$ been known to the receiver, the likelihood ratio (LR) would have been readily given by (see, e.g., [@FGS75]): $$\begin{aligned}
\label{lr}
L&=&\frac{\exp\left\{-\frac{1}{N_0}\int_0^T\left[y(t)-\sum_{k=1}^Kg_kh(t-t_k)\right]^2\mbox{d}t\right\}}
{\exp\left\{-\frac{1}{N_0}\int_0^Ty^2(t)\mbox{d}t\right\}}\nonumber\\
&=&\exp\left\{\frac{2}{N_0}\sum_{k=1}^K\int_0^Ty(t)h(t-t_k)\mbox{d}t-
\frac{1}{N_0}\sum_{k=1}^K\sum_{l=1}^Kg_kg_l
R(t_k-t_l)\right\},\end{aligned}$$ where $R(\tau)=\int_0^Th(t)h(t-\tau)\mbox{d}t$. Since these random parameters are unknown, the actual LR must be obtained by taking the expectation of $L$ with respect to (w.r.t.) their randomness. Deriving this expectation appears to be notoriously difficult, mainly due to the second term at the exponent, i.e., the double sum over $k$ and $l$.
It is this difficulty that triggered many researchers in the field to harness their wisdom in the quest for satisfactory solutions, and accordingly, there is rich literature on the subject, dating back many years into the past. As far as general guidelines go, a possible approach to alleviate this difficulty is the [*estimator–correlator approach*]{} [@Kailath69], which asserts that the expected LR of detection of a random signal in Gaussian additive white noise is given by the same expression as if the desired signal, $\sum_kg_kh(t-t_k)$, was known (i.e., the same as if $K$, $\{g_k\}$ and $\{t_k\}$ were known), except that it is replaced by its causal, minimum mean–square error (MMSE) estimator given $\{y(t)\}$. The caveat, however, is clear: deriving this MMSE causal estimator is an extremely difficult problem on its own.
To the best of the author’s knowledge, the first article that is directly relevant to this kind of study, for the above described specific signal model, is the article by Foschini, Gilbert and Salz [@FGS75]. Their approach was to view the factor associated with the double sum in the second line of (\[lr\]), namely, the term, $\exp\left\{-\frac{1}{N_0}\sum_{k,l}g_kg_lR(t_k-t_l)\right\}$, as the characteristic function of the random variable $\sum_k g_kx(t_k)$ (for given $K$, $\{g_k\}$ and $\{t_k\}$), where $\{x(t)\}$ is an auxiliary zero–mean, stationary Gaussian process with auto-correlation function $R(\tau)$. At the next step, the expectation over the randomness of $\{x(t)\}$ was commuted with the expectations over $K$, $\{t_k\}$ and $\{g_k\}$, which are easier to carry out for a given realization of $\{x(t)\}$. The result is a more compact expression of the LR, but even after this simplification, it is not explicit enough to be implementable in practice, or to analyze its performance in full generality. At this point, the approach taken in [@FGS75] was to carry out a series of approximations, yielding explicit asymptotic forms of the optimal detector and its performance at least in the limits of very low and very high signal–to–noise ratio (SNR). The resulting approximate LRT for high SNR, however, was still rather complicated to implement. Also, the behavior for moderate SNR was left open.
A year later, Mazo and Salz [@MS76] studied the performance of integrate–and–dump filters and also obtained exact formulas for the random gain of the APD on the basis of the earlier study by Personick [@Personick71a], [@Personick71b]. See also [@TM94]. Kadota [@Kadota88] has also derived an approximate LR test for a model like (\[sigmod\]). His approximation approach was different from that of [@FGS75]. It was based on neglecting the effect of overlaps between localized noise elements, which basically amounts to ignoring the cross terms of the double summation in the exponent of (\[lr\]) on the ground that $R(\cdot)$ is a very narrow function (see also [@Hero91] who used the same approximation for the purpose of estimation). More recently, Helstorm and Ho [@HH92] and Ho [@Ho95] have applied saddle–point integration and thereby studied the behavior of certain pulse shapes at the optical receiver in terms of the performance of the decoder. Other studies are guided by the approach of approximating the distribution of the shot noise of the photo-diode by the Gaussian distribution, owing to considerations in the spirit of the central limit theorem (CLT), see e.g., [@Einarsson96 Subsections 5.6.3, 5.8.4], [@EH81]. In this context, the well–known [*optical matched filter*]{} (see, e.g., [@GP87], [@Hero91]) is the main building block of the optimal detector that simply maximizes the SNR at the sampling time, $t=T$. This Gaussian approximation approach, however, raises some concerns since the CLT is not valid for assessing the tails of the distribution and in particular, error exponents, which are the relevant players when probabilities of large deviations events, like (the asymptotically rare) FA and MD error events, are studied.
In this paper, we take a different approach. Motivated by considerations of the desired simplicity of optical detectors for LIDAR systems (especially when they need to be implemented on mobile devices), we consider the class of optical signal detectors that are based on correlating the noisy received signal with a given deterministic waveform, and we characterize the waveform with the best trade–off between the false–alarm (FA) probability and the missed–detection (MD) probability. More precisely, our derivation addresses the trade-off between the asymptotic error exponents of the FA and MD probabilities using Chernoff bounds, without resorting to Gaussian approximations. We also provide numerical results that compare the performance of the best correlator to that of the optical matched filter (or, more precisely, the matched correlator), which is coherent with the above–mentioned Gaussian approximation approach. It is demonstrated that the proposed optimal correlator outperforms the optical matched correlator, in terms of the trade–off between the FA and the MD error exponents. It should be pointed out that in addition to the random fluctuations of the APD photo-multiplier, our model also incorporates the effect of dark current that exists even under the null hypothesis.
In the same spirit and with a similar motivation, we also study, for the same type of signal model, the problem of estimating the delay of a received signal on the basis of maximizing the correlation between the received signal and a time–shifted waveform, as a function of this shift. We characterize the optimal correlator waveform that minimizes the mean square error (MSE) in the regime of high SNR, as an extension of the analysis provided by Bar-David [@BarDavid69], who analyzed the high–SNR MSE of the maximum likelihood (ML) estimator for the pure Poissonian regime (i.e., without thermal noise). Once again, the emphasis is on simplicity and therefore, the performance of this estimator cannot be compared to the much more complicated, approximate MAP estimator due to Hero [@Hero91], which is based on approximating the likelihood function, using the same approach as Kadota [@Kadota88].
The optimal correlator waveforms for detection and for estimation turn out to be different, but their limiting behavior is the same in both detection and estimation problems: when the thermal Gaussian noise is dominant, the optimal correlator waveform becomes proportional to the clean signal (like the classical matched filter for additive white Gaussian noise), but when the thermal noise is negligible compared to the other noises, then it becomes logarithmic function of the clean signal, as expected in view of [@BarDavid69].
The outline of the remaining part of the paper is as follows. In Section \[thesignalmodel\], we present the model under discussion in full detail. In Section \[detection\], we address the signal detection problem, first and foremost, for the case of zero dark current. The case of positive dark current, which follows the same general ideas (but more complicated), is also outlined, but relatively briefly. Finally, in Section \[estimation\], we address the problem of time delay estimation.
The Signal Model {#thesignalmodel}
================
In this section, we provide a formal presentation of the signal model, that was briefly described in the Introduction. As mentioned before, we consider the model, $$\label{model}
y(t)=\sum_{k=1}^K g_k h(t-t_k)+n(t),~~~~~~~~~~~~0\le t < T,$$ whose various ingredients are described as follows. The variable $K$ is a Poissonian random variable, distributed according to $$\mbox{Pr}\{K=\kappa\}=e^{-\Lambda}\frac{\Lambda^\kappa}{\kappa!},
~~~~~~\kappa=0,1,2\ldots,~~~~\Lambda=\int_0^T\lambda(t)\mbox{d}t,$$ where $\lambda(t)$ is the a rate function that depends upon the intensity of the received optical signal. In particular, $$\lambda(t)=\frac{\eta{{\cal P}}(t)}{\hbar\omega}+\lambda_{\mbox{\tiny d}},$$ where $\eta$ is the quantum efficiency of the APD, ${{\cal P}}(t)$ is the instantaneous power of the optical signal, $\hbar$ is Planck’s constant, $\omega$ is the angular frequency of the light wave, and $\lambda_{\mbox{\tiny d}}$ is the dark current. The variables $\{g_k\}$ are independently identically distributed (i.i.d.) positive integer random variables that designate the avalanche gains. According to Personick [@Personick71a], [@Personick71b], the distribution of these random variables depends on the physics of the APD, and its characteristic function obeys a certain implicit equation, which is solvable in closed form when only the electrons (and not holes) cause ionizing collisions. In this case, the distribution of each $g_k$ is geometric: $$\label{geom}
\mbox{Pr}\{g_k=g\}=(e^\zeta-1)\cdot e^{-\zeta g},~~~~~~g=1,2,\ldots,~~\zeta > 0.$$ For the sake of concreteness, we will henceforth adopt the assumption of this geometric distribution. It also includes the case of a deterministic gain ($g_k =1$ with probability one), which corresponds to the case of the PIN diode, by taking the limit $\zeta\to\infty$. The function $h(t)$ is the current pulse contributed by the passage of a single photo–electron and hence its integral must be equal to the electric charge of the electron, $q_{\mbox{\tiny e}}$. Naturally, this is a very narrow pulse, which for most practical purposes, can be approximated by $h(t)\approx
q_{\mbox{\tiny e}}\delta(t)$, where $\delta(t)$ is the Dirac delta function. However, $h(t)$ can also be understood to include the convolution with some front–end filter, which is part of the electronic circuitry (e.g., the TIA). The times $\{t_k\}$ are the random Poissonian photon arrival times, taking on values in $[0,T)$ and being induced by the optical waveform, $\lambda(t)$. Finally, $\{n(t)\}$ is Gaussian white noise with spectral density $N_0/2$, which is assumed to be independent of $K$, $\{g_k\}_{k=1}^K$ and $\{t_k\}_{k=1}^K$. Also, given $K$, $\{g_k\}$ are statistically independent of $\{t_k\}$. Recall that for the underlying Poissonian process defined, conditioned on the event $K=\kappa$, the unordered random arrival times, $t_1,\ldots,t_\kappa$, are i.i.d. and their common density function is given by $f(t)=\lambda(t)/\Lambda$, for $0\le t\le T$, and $f(t)=0$ elsewhere.
Observe that if we present $g_k$ as $\bar{g}+\Delta g_k$, where $\bar{g}={\mbox{\boldmath $E$}}\{g_k\}$ and then $\Delta g_k$ designates the fluctuation, then the random input signal can be represented as $$\begin{aligned}
\sum_k(\bar{g}+\Delta g_k)h(t-t_k)&=&\bar{g}\sum_kh(t-t_k)+\sum_k\Delta
g_kh(t-t_k)\nonumber\\
&=&\bar{g}[q_{\mbox{\tiny e}}\lambda(t)+n_{\mbox{\tiny s}}(t)]+\sum_k\Delta
g_kh(t-t_k)\nonumber\\
&=&\bar{g}q_{\mbox{\tiny e}}\lambda(t)+\bar{g}n_{\mbox{\tiny
s}}(t)+\sum_k\Delta g_kh(t-t_k).\end{aligned}$$ The first term, $\bar{g}q_{\mbox{\tiny e}}\lambda(t)$, is the desired clean signal (plus dark current), the second term, defined as $$\bar{g}n_{\mbox{\tiny s}}(t)=\bar{g}\left[\sum_kh(t-t_k)-q_{\mbox{\tiny e}}\lambda(t)\right],$$ is shot noise (amplified by $\bar{g}$), and the last term is multiplicative noise. Thus, together with the Gaussian noise, $n(t)$, of (\[model\]), there are three types of noise in this model, as already mentioned in the Introduction.
Signal Detection {#detection}
================
In this section, we study the signal detection problem. We begin with the case $\lambda_{\mbox{\tiny d}}=0$ (no dark current), which is considerably simpler, and then outline the extension to the more general case, $\lambda_{\mbox{\tiny d}} > 0$. Before, we move into the technical details, a comment is in order, and it applies even to the case of no dark current. Consider then the signal detection problem of deciding between the two hypotheses: $$\begin{aligned}
& &{{\cal H}}_0:~y(t)=n(t)\\
& &{{\cal H}}_1:~y(t)=\sum_{k=1}^K g_k h(t-t_k)+n(t),\end{aligned}$$ using a detector, that is based on a correlator, that is, calculating the quantity $$\int_0^T w(t)y(t)\mbox{d}t$$ and comparing it to a threshold, $\theta T$. Here, $\{w(t),~0\le t\le T\}$ is a deterministic waveform to be optimized, and $\theta > 0$ is a threshold parameter that controls the trade-off between the FA and MD probabilities. Clearly, if the noise was purely Gaussian white noise, under both hypotheses, the optimal choice of $w(t)$ would have been matched to the desired signal, i.e., $w(t)=\lambda(t)$. Here, however, as explained in Section \[thesignalmodel\], there are two additional types of non–Gaussian noise under ${{\cal H}}_1$. Since the classical matched correlator, $w(t)=\lambda(t)$, is no longer necessarily optimal under non–Gaussian noise, and since we would still be interested in a detector that is relatively easy to implement, the natural question is whether there is a waveform $w(t)$ better than $w(t)=\lambda(t)$. If so, then what is the optimal waveform $w_*(t)$ for this detection problem? In this context, we should also mention again the notion of the optical matched filter (see e.g., [@GP87], [@Hero91]), the optical analogue of the classical matched filter, which maximizes the SNR at the sampling time, $t=T$, taking into account that the intensity of the shot noise is proportional to the desired signal, $\lambda(t)$ (unlike the case of pure thermal noise). But the relevance of the SNR as the only parameter that counts for detection performance is valid only under the Gaussian regime, so the optical matched filter is applied either under the Gaussian approximation, or when only second moments are important. The Gaussian approximation, under this model, is largely justified by CLT considerations. Here, however, we wish to avoid CLT considerations, as we are interested in the FA and MD error exponents, which are, in fact, given by large deviations rate functions. As is well known, the CLT is not valid for the large-deviations regime and for tails of distributions. Indeed, the optimal $w_*(t)$ that we derive below will be different from the optical matched filter.
The Case of No Dark Current
---------------------------
Consider the hypothesis testing problem defined at the beginning of the introductory part of this section, which is for the case of no dark current. Let $E{\stackrel{\Delta} {=}}\int_0^Tw^2(t)\mbox{d}t$ and $P{\stackrel{\Delta} {=}}E/T$. Then, the FA probability is given by $$\begin{aligned}
P_{\mbox{\tiny FA}}&=&\mbox{Pr}\left\{\int_0^T w(t)n(t)\mbox{d}t > \theta
T\right\}\nonumber\\
&=&Q\left(\frac{\theta T}{\sqrt{N_0E/2}}\right)\nonumber\\
&{\stackrel{\cdot} {=}}&\exp\left\{-\frac{\theta^2T^2}{N_0E}\right\}\nonumber\\
&=&\exp\left\{-\frac{\theta^2T}{N_0P}\right\}\nonumber\\
&{\stackrel{\Delta} {=}}&e^{-E_{\mbox{\tiny FA}}(\theta)T}\end{aligned}$$ where the notation ${\stackrel{\cdot} {=}}$ designates asymptotic equivalence in the exponential scale, i.e., for two positive functions of $T$, $a(T)$ and $b(T)$, the assertion $a(T){\stackrel{\cdot} {=}}b(T)$, stands for the assertion that $\lim_{T\to\infty}\frac{1}{T}\log\frac{a(T)}{b(T)}=0$. Since the FA error exponent, $E_{\mbox{\tiny FA}}(\theta)=\theta^2/N_0P$, depends on the waveform $w(\cdot)$ only via its power, $P$, it is obvious that the maximization of the MD error exponent for given FA error exponent is equivalent to its maximization subject to a power constraint imposed on $w(\cdot)$. Denoting $Z{\stackrel{\Delta} {=}}\int_0^T w(t)n(t)\mbox{d}t$, we next assess the MD error exponent using the Chernoff bound, assuming that $h(t)=q_{\mbox{\tiny e}}\delta(t)$. $$\begin{aligned}
P_{\mbox{\tiny MD}}&=&\mbox{Pr}\left\{\int_0^T w(t)y(t)\mbox{d}t \le \theta
T\right\}\nonumber\\
&=&\mbox{Pr}\left\{q_{\mbox{\tiny e}}\sum_{k=1}^Kg_kw(t_k)+Z \le \theta
T\right\}\nonumber\\
&\le&\inf_{s\ge 0} e^{s\theta
T}{\mbox{\boldmath $E$}}\left\{\exp\left[-sq_{\mbox{\tiny e}}\sum_{k=1}^Kg_kw(t_k)-sZ\right]\right\}\nonumber\\
&=&\inf_{s\ge 0} \exp\left\{s\theta
T+\frac{s^2N_0PT}{4}\right\}{\mbox{\boldmath $E$}}\left\{\exp
\left[-sq_{\mbox{\tiny e}}\sum_{k=1}^Kg_kw(t_k)\right]\right\}\nonumber\\
&=&\inf_{s\ge 0} \exp\left\{s\theta
T+\frac{s^2N_0PT}{4}\right\}{\mbox{\boldmath $E$}}\left\{\prod_{k=1}^Ke^{-sq_{\mbox{\tiny e}}g_kw(t_k)}\right\}.\end{aligned}$$ To calculate the last expectation, we proceed in two steps. First, we average each factor over $g_k$, assuming that it is geometrically distributed as in (\[geom\]). This gives $$\begin{aligned}
\sum_{g=1}^\infty P(g)e^{-gsq_{\mbox{\tiny e}}w(t_k)}&=&(e^\zeta-1)
\sum_{g=1}^\infty e^{-g[sq_{\mbox{\tiny e}}w(t_k)+\zeta]}\nonumber\\
&=&\frac{e^{\zeta}-1}{e^{sq_{\mbox{\tiny e}}w(t_k)+\zeta}-1}.\end{aligned}$$ As a second step, we average over $K$ and $\{t_k\}_{k=1}^K$, and get $$\begin{aligned}
{\mbox{\boldmath $E$}}\left\{\prod_{k=1}^Ke^{-sq_{\mbox{\tiny e}}g_kw(t_k)}\right\}&=&{\mbox{\boldmath $E$}}\left\{\prod_{k=1}^K\frac{e^{\zeta}-1}
{e^{sq_{\mbox{\tiny e}}w(t_k)+\zeta}-1}\right\}\nonumber\\
&=&\exp\left\{\int_0^T\lambda(t)\left[\frac{e^{\zeta}-1}
{e^{sq_{\mbox{\tiny e}}w(t)+\zeta}-1}-1\right]\mbox{d}t\right\}\nonumber\\
&=&\exp\left\{-e^{\zeta}\int_0^T\lambda(t)\cdot\frac{e^{sq_{\mbox{\tiny e}}w(t)}-1}
{e^{sq_{\mbox{\tiny e}}w(t)+\zeta}-1}\mbox{d}t\right\},\end{aligned}$$ where we have used the fact [@BarDavid69 eqs. (8)–(13)] that for an arbitrary positive function $f$, $${\mbox{\boldmath $E$}}\left\{\prod_{k=1}^K f(t_k)\right\}=\exp\left\{\int_0^T\lambda(t)[f(t)-1]\mbox{d}t\right\}.$$ Thus, $$P_{\mbox{\tiny MD}}\le
\exp\left\{-T\sup_{s\ge
0}\left[\frac{e^{\zeta}}{T}\int_0^T\lambda(t)\cdot\frac{e^{sq_{\mbox{\tiny e}}w(t)}-1}
{e^{sq_{\mbox{\tiny e}}w(t)+\zeta}-1}\mbox{d}t-s\theta-s^2\frac{N_0P}{4}\right]\right\}.$$ Consider first the special case where $g_k\equiv 1$ with probability one, which is obtained in the limit $\zeta\to\infty$. In this case, the above simplifies to $$P_{\mbox{\tiny MD}}\le
\exp\left\{-T\sup_{s\ge 0}\left[\frac{1}{T}\int_0^T\lambda(t)[1-e^{-sq_{\mbox{\tiny e}}w(t)}]
\mbox{d}t-s\theta-s^2\frac{N_0P}{4}\right]\right\}.$$
[*Remark.*]{} Note that had the channel been purely Gaussian, that is, without the shot noise, and the desired signal was $q_{\mbox{\tiny e}}\lambda(t)$, we would have obtained $$P_{\mbox{\tiny MD}}\le
\exp\left\{-T\sup_{s\ge 0}\left[\frac{1}{T}\int_0^Tq_{\mbox{\tiny e}}\lambda(t)\cdot sw(t)
\mbox{d}t-s\theta-s^2\frac{N_0P}{4}\right]\right\}.$$ This means that the difference $$\frac{1}{T}\int_0^T\lambda(t)\cdot[sq_{\mbox{\tiny e}}w(t)-(1-e^{-sq_{\mbox{\tiny e}}w(t)})]\mbox{d}t$$ designates the loss due to the additional shot noise.[^2]
We therefore need to maximize the exponent over both $s$ and $\{w(t),~0\le t\le T\}$. For a given $s$, the optimal $\{w(t)\}$ minimizes $\int_0^T\lambda(t)e^{-sq_{\mbox{\tiny e}}
w(t)}\mbox{d}t$ subject to the power constraint $\int_0^Tw^2(t)\mbox{d}t\le PT$, which is equivalent to minimizing $$\int_0^T\lambda(t)e^{-sq_{\mbox{\tiny e}}w(t)}\mbox{d}t+
\frac{s^2q_{\mbox{\tiny e}}^2}{2c}\left[\int_0^Tw^2(t)\mbox{d}t-PT\right],$$ where $s^2q_{\mbox{\tiny e}}^2/2c > 0$ is a Lagrange multiplier.[^3] Finding the optimal function $w(\cdot)$ is a standard problem in calculus of variations, whose solution is characterized as follows. Let the function $p[\cdot]$ denote the inverse of the monotonically increasing function $b[x]{\stackrel{\Delta} {=}}xe^x$, $x\ge 0$, i.e., $p[y]$ is the solution $x$ to the equation $xe^x=y$, $y\ge 0$. The optimal $w(t)$ is given by $$w_*(t)=\frac{1}{sq_{\mbox{\tiny e}}}p[c\cdot\lambda(t)],$$ where $c > 0$ is chosen such that $\int_0^Tp^2[c\cdot\lambda(t)]\mbox{d}t=s^2q_{\mbox{\tiny e}}^2PT$. The MD exponent is then given by $$E_{\mbox{\tiny MD}}(\theta)=\sup_{s\ge
0}\left[\frac{1}{T}\int_0^T\lambda(t)(1-\exp\{-p[c\cdot\lambda(t)]\})
\mbox{d}t-s\theta-s^2\frac{N_0P}{4}\right],$$ where it should be kept in mind that $c$ depends on $s$.
Note that if the optimal $s$ is very small (which is the case when $N_0$ and/or $\theta$ are large, then the solution to the equation $sq_{\mbox{\tiny e}}w(t)e^{sq_{\mbox{\tiny e}}w(t)}=c\cdot\lambda(t)$ is found near the origin, where $p[x]\approx x$, which means that $w_*(t)$ is nearly proportional to $\lambda(t)$, namely, the classical matched correlator. If, on the other hand, the optimal $s$ is very large (which is the case when $N_0$ and $\theta$ are both small), then the solution is found away from the origin, where $p[x]\approx \ln x$. In this case, $w_*(t)\approx\frac{1}{sq_{\mbox{\tiny e}}}\ln[c\cdot\lambda(t)]$, in agreement of optimal photo–counting detector (see, e.g., [@BarDavid69]), which is obtained in the absence of Gaussian noise.\
[*Example 1.*]{} Consider the frequently–encountered case where $\lambda(t)$ is a two–level signal, where half of the time $\lambda(t)=\lambda_1$ and in the other half, $\lambda(t)=\lambda_2$. Then, $w_*(t)$ must also be a two–level signal. Owing to the power constraint, we may denote these two levels by $w$ and $\sqrt{2P-w^2}$, respectively, and it remains to maximize the exponent over $s$ and $w$ alone. Specifically, we have $$E_{\mbox{\tiny MD}}(\theta)=\sup_{s\ge
0}\max_{0\le w\le\sqrt{2P}}\left[\frac{\lambda_1}{2}(1-e^{-sq_{\mbox{\tiny e}}w})+
\frac{\lambda_2}{2}(1-e^{-sq_{\mbox{\tiny e}}\sqrt{2P-w^2}})
-s\theta-s^2\frac{N_0P}{4}\right].$$ Fig. \[graph1\] compares the exponent of the optimal waveform, $w_*$, to that of the optical matched filter [@GP87 eq. (8)], $$w_{\mbox{\tiny omf}}(t)=\frac{\lambda(t)}{\lambda(t)+N_0/(2q_{\mbox{\tiny e}}^2\overline{g^2})},$$ for the following values of the parameters of the problem: $P=10$, $N_0/q_{\mbox{\tiny e}}^2=0.0001$, $\lambda_1=1$ and $\lambda_2=10$. The numerical value of the spectral density of the thermal noise was deliberately chosen extremely small, in order to demonstrate a situation where the noise is far from being Gaussian, and thereby examine sharply the validity of the Gaussian approximation. In this case, $w_{\mbox{\tiny omf}}(t)$ is nearly equal to unity for all $t$, which means a pure, unweighted integrator [@GP87 p. 1291, Remark 2]. As can be seen, the optimal correlator $w_*$ improves upon the optical matched correlator fairly significantly, especially for large values of the threshold parameter, $\theta$. This concludes Example 1. $\Box$
![MD error exponents of the optical matched correlator (dashed blue curve) and the optimal correlator $w_*$ (solid red curve) as functions of $\theta$, for a deterministic gain ($\zeta\to\infty$), $P=10$, $N_0/q_{\mbox{\tiny e}}^2=0.0001$, $\lambda_1=1$ and $\lambda_2=10$.[]{data-label="graph1"}](md-exponent-dgain-opticalmf.eps){width="8.5cm" height="8.5cm"}
Returning to the case of a general, finite $\zeta$, and carrying out a similar optimization, we find that the optimal $w_*(t)$ is now given by $$w_*(t)=\frac{1}{sq_{\mbox{\tiny e}}}\cdot p_\zeta[c\cdot\lambda(t)],$$ where $p_\zeta$ is the inverse of the function $$b_\zeta[x]=\frac{x(e^{x+\zeta}-1)^2}{e^{x+2\zeta}-e^{x+\zeta}},$$ and where, once again, $c$ is chosen such that $$\int_0^Tp_\zeta^2[c\cdot\lambda(t)]\mbox{d}t=s^2q_{\mbox{\tiny e}}^2PT.$$ Here too, if $N_0$ and/or $\theta$ are large, then $s$ must be small, and then due to the power constraint, $c$ must be small too, which means that the functions $p_\zeta$ and $b_\zeta$ operate near the origin, where they are roughly linear, as $$b_\zeta[x]\approx x\cdot\frac{(e^\zeta-1)^2}{e^{2\zeta}-e^\zeta}.$$ At the other extreme, on the other hand, $p_\zeta$ and $b_\zeta$ operate away from the origin, where $$b_\zeta[x]\approx
\frac{xe^{2x+2\zeta}}{e^{x+2\zeta}-e^{x+\zeta}}=\frac{xe^{x+\zeta}}{e^\zeta-1},$$ which is again, nearly exponential, and so, $p_\zeta$ is approximately logarithmic, as before. The MD exponent is therefore given by $$E_{\mbox{\tiny MD}}(\theta)=
\sup_{s\ge
0}\left[\frac{e^{\zeta}}{T}\int_0^T\lambda(t)\cdot\frac{\exp\{p_\zeta[c\cdot\lambda(t)]\}-1}
{\exp\{p_\zeta[c\cdot\lambda(t)]+\zeta\}-1}\mbox{d}t-s\theta-s^2\frac{N_0P}{4}\right].$$
[*Example 2.*]{} Consider again the setting of Example 1 above, except that here $\zeta$ is finite. Specifically, Fig. 2 displays a comparison analogous to that of Fig. 1 for the case of a random gain with parameter $\zeta=0.1$, where for the red solid curve, $s$ and $w$ are chosen to maximize $$E_{\mbox{\tiny MD}}(\theta)=\sup_{s\ge
0}\max_{0\le
w\le\sqrt{2P}}\left[\frac{\lambda_1e^\zeta}{2}\cdot\frac{e^{sq_{\mbox{\tiny e}}w}-1}
{e^{sq_{\mbox{\tiny e}}w+\zeta}-1}+
\frac{\lambda_2e^\zeta}{2}\cdot\frac{e^{sq_{\mbox{\tiny e}}\sqrt{2P-w^2}}-1}
{e^{sq_{\mbox{\tiny e}}\sqrt{2P-w^2}+\zeta}-1}
-s\theta-s^2\frac{N_0P}{4}\right],$$ and for the blue, dashed curve, $w$ is chosen to be the optical matched correlator as before. As can be seen, here too, the optimal $w$ improves upon the optical matched correlator and the gap is rather considerable, especially as $\theta$ grows.
![MD error exponents of the optical matched correlator (blue dashed curve) and the optimal correlator (red solid curve) as functions of $\theta$, for a random gain ($\zeta=0.1$), $P=10$, $N_0/q_{\mbox{\tiny e}}^2=0.0001$, $\lambda_1=1$ and $\lambda_2=10$.[]{data-label="graph2"}](md-exponent-rgain-opticalmf.eps){width="8.5cm" height="8.5cm"}
The Case of Positive Dark Current
---------------------------------
The case of $\lambda_{\mbox{\tiny d}} > 0$ is analyzed on the basis of similar ideas, and we therefore cover it relatively briefly, highlighting mostly the points where there is a substantial difference relative to the zero dark–current case.
Extending the analysis to the case of positive dark current, a similar derivation yields the following FA exponent for a given correlator waveform, $w(t)$: $$E_{\mbox{\tiny FA}}(\theta,w)=\sup_{0\le s<
\zeta/w_{\max}}\left[s\theta-\frac{s^2N_0P}{4}-\frac{\lambda_{\mbox{\tiny d}}
e^\zeta}{T}\int_0^T\frac{e^{sq_{\mbox{\tiny e}}w(t)}-1}{e^\zeta-e^{sq_{\mbox{\tiny e}}w(t)}}\mbox{d}t\right],$$ where $w_{\max}=\sup_{0\le t\le T}w(t)$. Here, the trade–off between the FA and the MD error exponents is somewhat more involved than in the zero dark–current case, since the FA error exponent depends on $w(\cdot)$ in a more complicated manner than just via its power $P$. In particular, we would now like to find the pair $(\theta,w)$ that maximizes $E_{\mbox{\tiny MD}}(\theta,w)$ over all pairs such that $E_{\mbox{\tiny FA}}(\theta,w)\ge S$, for a given $S >
0$ that designates the target FA exponent.[^4] Equivalently, we would like to solve the problem $$\sup_{\theta,w}\left\{E_{\mbox{\tiny
MD}}(\theta,w)+\mu E_{\mbox{\tiny FA}}(\theta,w)\right\},$$ where $\mu$ is a Lagrange multiplier chosen to meet the constraint, $E_{\mbox{\tiny FA}}(\theta,w)\ge S$. More specifically, using Chernoff bounds for both error exponents, this amounts to solving the problem, $$\begin{aligned}
& &\sup_{s\ge 0}\sup_{\sigma\ge 0}\sup_{\theta\ge 0}\sup_{w:~w_{\max} < \zeta/s}\left[
\frac{e^{\zeta}}{T}\int_0^T\lambda(t)\cdot\frac{e^{\sigma q_{\mbox{\tiny e}} w(t)}-1}
{e^{\sigma q_{\mbox{\tiny e}}
w(t)+\zeta}-1}\mbox{d}t-\sigma\theta-\sigma^2\frac{N_0P}{4}+\right.\nonumber\\
& &\left.\mu s\theta-\frac{\mu s^2N_0P}{4}-\frac{\mu\lambda_0
e^\zeta}{T}\int_0^T\frac{e^{sq_{\mbox{\tiny e}}w(t)}-1}{e^\zeta-
e^{sq_{\mbox{\tiny e}}w(t)}}\mbox{d}t\right]\nonumber\\
&=&\sup_{s\ge 0}\sup_{\sigma\ge 0}\sup_{\theta\ge 0}\sup_{w:~w_{\max} <
\zeta/s}\left[
\frac{e^{\zeta}}{T}\int_0^T\left(\lambda(t)\cdot\frac{e^{\sigma q_{\mbox{\tiny e}}w(t)}-1}
{e^{\sigma q_{\mbox{\tiny e}}
w(t)+\zeta}-1}-\mu\lambda_0
\frac{e^{sq_{\mbox{\tiny e}}w(t)}-1}{e^\zeta-e^{sq_{\mbox{\tiny e}}w(t)}}\right)\mbox{d}t+\right.\nonumber\\
& &\left.(\mu s-\sigma)\theta-(\sigma^2+\mu s^2)\frac{N_0P}{4}\right].\end{aligned}$$ We will not continue any further to the full, detailed solution of this problem, beyond the following comment which applies to the case of a deterministic gain. In the limit of $\zeta\to\infty$, the above trade-off yields $$w_*(t)=\ln\left[\frac{\sigma}{\lambda_{\mbox{\tiny d}}\mu s}\cdot\lambda(t)\right],$$ in other words, in the presence of dark-current, the relation is always logarithmic.
Time Delay Estimation {#estimation}
=====================
In this section, we consider the problem of time delay estimation. The underlying model is the same as before, except that the optical signal is time–shifted, i.e., $\lambda(t-\theta)$, where $\theta$ is the delay. It is assumed that the support of $\lambda(t-\theta)$ in included in the interval $[0,T]$, for every $\theta$ in the range of uncertainty. As mentioned in the Introduction, this is a relevant problem in LIDAR systems, where distances to certain objects have to be estimated, similarly as in classical radar systems. Here too, our basic building block is a correlator. Consider an estimator of the form, $$\hat{\theta}=\arg\max_\theta Q(\theta),$$ where $$Q(\theta)=\int_0^Ty(t)w(t-\theta)\mbox{d}t,$$ and where $w(\cdot)$ is a twice–differentiable waveform to be optimized, with the property that the temporal cross–correlation function, $$R_{w\lambda}(\tau)=\int_0^T\lambda(t)w(t-\tau)\mbox{d}t$$ achieves its maximum at $\tau=0$. This implies $$\dot{R}_{w\lambda}(0)=\int_0^T\lambda(t)\dot{w}(t)\mbox{d}t=0,$$ where dotted functions designate derivatives. Similarly as the assumption concerning $\lambda(\cdot)$, we also assume that the support of the waveform $w(t-\theta)$ is fully included in $[0,T]$ for the entire range of search of the estimated delay.
Our analysis begins similarly as in [@BarDavid69], which assumes high SNR and small estimation errors. Accordingly, consider the Taylor series expansion of first derivative, $\dot{Q}(\theta)$, around the true parameter value, $\theta_0$: $$0=\dot{Q}(\hat{\theta})\approx\dot{Q}(\theta_0)+(\hat{\theta}-\theta_0)\ddot{Q}(\theta_0),$$ where $\ddot{Q}(\theta)$ is the second derivative of $Q(\theta)$. This yields $$\begin{aligned}
\label{approxerror}
\hat{\theta}-\theta_0&\approx&-\frac{\dot{Q}(\theta_0)}{\ddot{Q}(\theta_0)}\nonumber\\
&=&-\frac{\bar{g}q_{\mbox{\tiny
e}}\int_0^T\lambda(t-\theta_0)\dot{w}(t-\theta_0)\mbox{d}t+\int_0^Tn_T(t)\dot{w}(t-\theta_0)\mbox{d}t}
{\bar{g}q_{\mbox{\tiny e}}\int_0^T\lambda(t-\theta_0)\ddot{w}(t-\theta_0)\mbox{d}t
+\int_0^Tn_T(t)\ddot{w}(t-\theta_0)\mbox{d}t}\nonumber\\
&=&-\frac{\bar{g}q_{\mbox{\tiny
e}}\int_0^T\lambda(t)\dot{w}(t)\mbox{d}t+\int_0^Tn_T(t)\dot{w}(t-\theta_0)\mbox{d}t}
{\bar{g}q_{\mbox{\tiny e}}\int_0^T\lambda(t)\ddot{w}(t)\mbox{d}t
+\int_0^Tn_T(t)\ddot{w}(t-\theta_0)\mbox{d}t}\nonumber\\
&=&-\frac{
\int_0^Tn_T(t)\dot{w}(t-\theta_0)\mbox{d}t}
{\bar{g}q_{\mbox{\tiny e}}\cdot\int_0^T\lambda(t)\ddot{w}(t)\mbox{d}t
+\int_0^Tn_T(t)\ddot{w}(t-\theta_0)\mbox{d}t}\nonumber\\
&{\stackrel{\Delta} {=}}&\frac{U_n}{A+U_d},\end{aligned}$$ where $n_T(t)$ is the total noise, composed of the shot noise, the multiplicative noise and the thermal noise, i.e., $$n_{\mbox{\tiny T}}(t)=n_{\mbox{\tiny s}}(t)+n_{\mbox{\tiny m}}(t)+n(t),$$ where $$n_{\mbox{\tiny s}}(t)=\bar{g}\cdot\left[\sum_kh(t-t_k)-q_{\mbox{\tiny
e}}\lambda(t-\theta)\right],$$ and $$n_{\mbox{\tiny m}}(t)=\sum_k\Delta g_kh(t-t_k).$$ Thus, $${\mbox{\boldmath $E$}}\{(\hat{\theta}-\theta)^2\}\approx\frac{{\mbox{\boldmath $E$}}\{U_n^2\}}{A^2},$$ where we have made a further approximation by neglecting the contribution of the random variable $U_d$ relative to the deterministic constant $A$ (at the denominator of the last line of (\[approxerror\])) since $A$ is proportional to $T$, whereas the standard deviation of $U_d$ is proportional to $\sqrt{T}$ (see also [@BarDavid69 eqs. (38)–(40)] for a more detailed justification of a similar approximation). It is easy to see that all three noise components are uncorrelated with each other. The auto-correlation function of the thermal noise is $R_n(t,s)=\frac{N_0}{2}\delta(t-s)$. The auto-correlation function of the (amplified) shot noise is $$\begin{aligned}
R_{\mbox{\tiny
s}}(t,s)&=&\bar{g}^2\cdot{\mbox{\boldmath $E$}}\left\{\left[\sum_kh(t-t_k)-q_{\mbox{\tiny
e}}\lambda(t)\right]\cdot\left[\sum_kh(s-t_k)-q_{\mbox{\tiny
e}}\lambda(s)\right]\right\}\nonumber\\
&=&\bar{g}^2\cdot{\mbox{\boldmath $E$}}\left\{\sum_{k,l}h(t-t_k)h(s-t_l)\right\}-\bar{g}^2q_{\mbox{\tiny
e}}^2\lambda(t)\lambda(s)\nonumber\\
&=&\bar{g}^2\cdot{\mbox{\boldmath $E$}}\left\{\sum_{k}h(t-t_k)h(s-t_k)\right\}+
\bar{g}^2\cdot{\mbox{\boldmath $E$}}\left\{\sum_{k\ne l}h(t-t_k)h(s-t_l)\right\}
-\bar{g}^2q_{\mbox{\tiny
e}}^2\lambda(t)\lambda(s)\nonumber\\
&=&\bar{g}^2\cdot{\mbox{\boldmath $E$}}\{K\}\frac{1}{\Lambda}
\int_0^T\lambda(\tau-\theta)h(t-\tau)h(s-\tau)\mbox{d}\tau
+\nonumber\\
& &\bar{g}^2\cdot{\mbox{\boldmath $E$}}\{K(K-1)\}\frac{1}{\Lambda^2}
\left[\int_0^T\lambda(\tau-\theta)h(t-\tau)\mbox{d}\tau\right]
\left[\int_0^T\lambda(\tau-\theta)h(s-\tau)\mbox{d}\tau\right]-\bar{g}^2q_{\mbox{\tiny
e}}^2\lambda(t)\lambda(s)\nonumber\\
&=&\bar{g}^2\cdot\int_0^T\lambda(\tau-\theta)h(t-\tau)h(s-\tau)\mbox{d}\tau.\end{aligned}$$ Similarly, $$\begin{aligned}
R_{\mbox{\tiny
m}}(t,s)&=&{\mbox{\boldmath $E$}}\{n_{\mbox{\tiny m}}(t)n_{\mbox{\tiny m}}(s)\}\nonumber\\
&=&{\mbox{\boldmath $E$}}\left\{\sum_{k,l}\Delta g_k\Delta g_lh(t-t_k)h(s-t_l)\right\}\nonumber\\
&=&\mbox{Var}\{g\}\cdot{\mbox{\boldmath $E$}}\left\{\sum_{k}h(t-t_k)h(s-t_k)\right\}+
{\mbox{\boldmath $E$}}\left\{\sum_{k\ne l}\Delta g_k\Delta g_lh(t-t_k)h(s-t_l)\right\}\nonumber\\
&=&\mbox{Var}\{g\}\cdot{\mbox{\boldmath $E$}}\left\{\sum_{k}h(t-t_k)h(s-t_k)\right\}\nonumber\\
&=&\mbox{Var}\{g\}\cdot\int_0^T\lambda(\tau-\theta)h(t-\tau)h(s-\tau)\mbox{d}\tau.\end{aligned}$$ It follows that $$R_{\mbox{\tiny T}}(t,s){\stackrel{\Delta} {=}}R_{\mbox{\tiny
s}}(t,s)+R_{\mbox{\tiny
m}}(t,s)+R_n(t,s)=\overline{g^2}\cdot\int_0^T\lambda(\tau-
\theta)h(t-\tau)h(s-\tau)\mbox{d}\tau+\frac{N_0}{2}\delta(t-s).$$ Thus, $$\begin{aligned}
{\mbox{\boldmath $E$}}\{U_n^2\}&=&\int_0^T\int_0^T\mbox{d}t\mbox{d}sR_{\mbox{\tiny
T}}(t,s)\dot{w}(t-\theta)\dot{w}(s-\theta)\nonumber\\
&=&\frac{N_0}{2}\int_0^T\dot{w}^2(t-\theta)\mbox{d}t+\overline{g^2}
\int_0^T\int_0^T\mbox{d}t\mbox{d}s\dot{w}(t-\theta)\dot{w}(s-\theta)\int_0^T
\lambda(\tau-\theta)h(t-\tau)h(s-\tau)\mbox{d}\tau\nonumber\\
&=&\frac{N_0}{2}\int_0^T\dot{w}^2(t-\theta)\mbox{d}t+\overline{g^2}
\int_0^T\mbox{d}\tau\lambda(\tau-\theta)\left[\int_0^T\mbox{d}t\dot{w}(t-\theta)h(t-\tau)\right]
\left[\int_0^T\mbox{d}s\dot{w}(s-\theta)h(s-\tau)\right]\nonumber\\
&\approx&\frac{N_0}{2}\int_0^T\dot{w}^2(t-\theta)\mbox{d}t+\overline{g^2}q_{\mbox{\tiny
e}}^2\int_0^T\lambda(t)\dot{w}^2(t)\mbox{d}t\nonumber\\
&=&\int_0^T\left[\frac{N_0}{2}+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)\right]\dot{w}^2(t)\mbox{d}t,\end{aligned}$$ which yields $${\mbox{\boldmath $E$}}\{(\hat{\theta}-\theta)^2\}\approx
\frac{\int_0^T\left[\frac{N_0}{2}+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)\right]\dot{w}^2(t)\mbox{d}t}{\overline{g}^2q_{\mbox{\tiny
e}}^2\left[\int_0^T\lambda(t)\ddot{w}(t)\mbox{d}t\right]^2}.$$ It is desired to find the optimal waveform $\{w(t)\}$. First, observe that the MSE is invariant to scaling of $\{w(t)\}$, so the problem is equivalent to maximizing the absolute value of $$\int_0^T\lambda(t)\ddot{w}(t)\mbox{d}t$$ subject to a given value of $$\label{cons}
\int_0^T\left[\frac{N_0}{2}+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)\right]\dot{w}^2(t)\mbox{d}t.$$ Let $\omega(t)=\dot{w}(t)$ and $$v(t)=\omega(t)\cdot\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}.$$ Then, the problem is equivalent to that of maximizing the absolute value of $$\int_0^T\lambda(t)\cdot\left[\frac{v(t)}{\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}}\right]'\mbox{d}t,$$ or equivalently, $$\lambda(t)\cdot\frac{v(t)}{\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}}\Bigg|_0^T-
\int_0^T\dot{\lambda}(t)\cdot\frac{v(t)}{\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}}\mbox{d}t,$$ subject to a given value of (\[cons\]). Now, assuming that $\lambda(0)=\lambda(T)=0$, the contribution of the first term in the last expression vanishes, and so, we wish to maximize the absolute value of $$\int_0^T\dot{\lambda}(t)\cdot\frac{v(t)}{\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}}\mbox{d}t$$ for a given energy of $\{v(t)\}$. The maximum is achieved for $$v(t)=\frac{\dot{\lambda}(t)}{\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}},$$ which yields $$\omega(t)=\frac{v(t)}{\sqrt{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)}}=\frac{\dot{\lambda}(t)}{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(t)},$$ and so, finally, $$w_*(t)=\int_0^t\frac{\dot{\lambda}(s)\mbox{d}s}{N_0/2+\overline{g^2}q_{\mbox{\tiny
e}}^2\lambda(s)}=\frac{1}{\overline{g^2}q_{\mbox{\tiny e}}^2}
\ln\left[\frac{N_0}{2}+\overline{g^2}q_{\mbox{\tiny e}}^2\lambda(t)\right]+c,$$ where $c$ is an integration constant (not to be confused with the earlier defined constant, $c$). Equivalently, after a simple manipulation of the integration constant, $w_*(t)$ can be presented as $$w_*(t)\propto\ln\left[1+\frac{2\overline{g^2}q_{\mbox{\tiny
e}}^2}{N_0}\cdot\lambda(t)\right].$$ As can be seen, for very large $N_0$, $w(t)$ is approximately proportional to $\lambda(t)$, as expected. For very small $N_0$, $w(t)$ is proportional to $\ln\lambda(t)$, which is also expected. Note that if $\lambda(t)$ includes a dark–current component, $\lambda_{\mbox{\tiny d}}$, then it simply adds to the spectral density of the thermal noise, $N_0/2$. In other words, there is no distinction here between the dark–current shot noise and the Gaussian thermal noise. This should not be surprising, because when calculating the MSE, only second moments are relevant, and not finer details of the distributions of the various kinds of noise.
Let us compare the MSE of the optimal correlator $w_*$ to that of the ordinary matched correlator, $w(t)=\lambda(t)$. Denote $\lambda_0=2\overline{g^2}q_{\mbox{\tiny e}}^2/N_0$. Then, the MSE of the latter is given by $$\mbox{MSE}
=\frac{1}{\lambda_0}\cdot\frac{\int_0^T[1+\lambda(t)/\lambda_0]\dot{\lambda}^2(t)\mbox{d}t}
{\left[\int_0^T\dot{\lambda}^2(t)\mbox{d}t\right]^2},$$ whereas the former is given by $$\mbox{MSE}_{\mbox{\tiny opt}}
=\frac{1}{\lambda_0}\cdot\frac{1}{\int_0^T\frac{\dot{\lambda}^2(t)\mbox{d}t}{1+\lambda(t)/\lambda_0}}.$$ It is easy to see that $\mbox{MSE}_{\mbox{\tiny opt}}$ is indeed smaller than $\mbox{MSE}$, due to the Schwartz–Cauchy inequality, as $$\begin{aligned}
\left[\int_0^T\dot{\lambda}^2(t)\mbox{d}t\right]^2&=&
\left[\int_0^T\frac{\dot{\lambda}(t)}{\sqrt{1+\lambda(t)/\lambda_0}}\cdot
\dot{\lambda}(t)\sqrt{1+\frac{\lambda(t)}{\lambda_0}}\mbox{d}t\right]^2\nonumber\\
&\le&\left[\int_0^T\frac{\dot{\lambda}^2(t)}{1+\lambda(t)/\lambda_0}\mbox{d}t\right]\cdot
\left[\int_0^T\dot{\lambda}^2(t)\left(1+\frac{\lambda(t)}{\lambda_0}\right)\mbox{d}t\right].\end{aligned}$$ The difference becomes smaller when $\dot{\lambda}(t)\sqrt{1+\lambda(t)/\lambda_0}$ becomes closer to be proportional to $\dot{\lambda}(t)/\sqrt{1+\lambda(t)/\lambda_0}$, namely, when $\lambda(t)\ll \lambda_0$, as expected.
This high–SNR analysis can be extended to handle non–white noise and even non–stationary thermal noise of auto-correlation function $R_n(s,t)={\mbox{\boldmath $E$}}\{n(s)n(t)\}$. The only difference is that now the numerator of (\[approxerror\]) will be replaced by $$\int_0^T\dot{w}(s)\dot{w}(t)[R_n(s,t)+\overline{g^2}q_{\mbox{\tiny
e}}^2\sqrt{\lambda(s)\lambda(t)}\delta(t-s)]\mbox{d}s\mbox{d}t
{\stackrel{\Delta} {=}}\int_0^T\dot{w}(s)\dot{w}(t)R_0(s,t)\mbox{d}s\mbox{d}t.$$ The resulting optimum correlator would then be given by $$w_*(t)=\int_0^t\mbox{d}\tau\int_0^T\mbox{d}s\dot{\lambda}(s)R_0^{-1}(s,\tau),$$ where $R_0^{-1}(s,\tau)$ is the formal inverse of the kernel $R_0(s,t)$ (provided that it exists), namely, the kernel that satisfies $$\int_0^TR_0(s,t_1)R_0^{-1}(s,t_2)\mbox{d}s=\delta(t_1-t_2).$$
The other side of the coin of high–SNR estimation performance is the probability of anomaly (see, e.g., [@WJ65 Chap. 8]). The common practice in assessing this probability is to divide the interval $[0,T]$ into small bins whose sizes are about the pulse width and to apply a union bound on the (essentially identical) probabilities of the pairwise events that the maximum $Q(\theta)$ in each bin exceeds the one that contains the correct delay. Since the number of bins grows only linearly in $T$, the exponent of this probability is the same as that of each individual bin. The analysis of each pairwise event can be carried out very similarly to the earlier analysis of the FA probability (just with a few minor twists), whose dependence on the mismatched correlator $w$ is different that of the above derived approximated MSE. Thus, the signal design should seek a compromise that trades off the weak–noise MSE with the probability of anomaly. In the high–SNR regime, a logarithmic correlator is good both for the MSE and for keeping the anomaly probability small.
Acknowledgment {#acknowledgment .unnumbered}
==============
Interesting discussions with Moshe Nazarathy, at the early stages of this work, are acknowledged with thanks.
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[^1]: The flat spectrum assumption is adopted here mainly for the sake of simplicity of the exposition. The extension to colored noise is not difficult.
[^2]: The integrand is, of course, non–negative due to the inequality $x\ge 1-e^{-x}$.
[^3]: This form of the Lagrange multiplier is adopted as it allows a convenient representation of the solution. It is legitimate since there is complete freedom to control its value by the choice of the constant $c > 0$.
[^4]: Previously, $S$ was given by $\theta^2/N_0P$, so for a given $P$, $S$ was proportional to $\theta^2$.
|
---
abstract: 'We show that the quantum automorphism group of the Clebsch graph is $SO_5^{-1}$. This answers a question by Banica, Bichon and Collins from 2007. More general for odd $n$, the quantum automorphism group of the folded $n$-cube graph is $SO_n^{-1}$. Furthermore, we show that if the automorphism group of a graph contains a pair of disjoint automorphisms this graph has quantum symmetry.'
address: 'Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany'
author:
- Simon Schmidt
bibliography:
- 'Clebschgraph.bib'
title: Quantum automorphisms of folded cube graphs
---
[^1]
Introduction {#introduction .unnumbered}
============
The concept of quantum automorphism groups of finite graphs was introduced by Banica and Bichon in [@QBan; @QBic]. It generalizes the classical automorphism groups of graphs within the framework of compact matrix quantum groups. We say that a graph has no quantum symmetry if the quantum automorphism group coincides with its usual automorphism group. A natural question is: When does a graph have no quantum symmetry? This has been studied in [@Che] for some graphs on $p$ vertices, $p$ prime, and more recently the author showed in [@QAutPetersen] that the Petersen graph does not have quantum symmetry. Also Lupini, Mančinska and Roberson proved that almost all graphs have trivial quantum automorphim group in [@Nonlocal], which implies that almost all graphs do not have quantum symmetry.
In this article we develop a tool for detecting quantum symmetries namely we show that a graph has quantum symmetry if its automorphism group contains a pair of disjoint automorphisms (Theorem \[noncomm\]). As an example, we apply it to the Clebsch graph and obtain that it does have quantum symmetry (Corollary \[C\]).
We even go further and prove that the quantum automorphism group of the Clebsch graph is $SO_5^{-1}$, the $q$-deformation at $q=-1$ of $SO_5$, answering a question from [@survey]. For this we use the fact that the Clebsch graph is the folded 5-cube graph. This can be pushed further to more general folded $n$-cube graphs: In [@hyperoctahedral], two generalizations of the hyperoctahedral group $H_n$ are given, one of them being $O_n^{-1}$ as quantum symmetry group of the hypercube graph. To prove that $O_n^{-1}$ is the quantum symmetry group of the hypercube graph, Banica, Bichon and Collins used the fact that the hypercube graph is a Cayley graph. It is also well known that the folded cube graph is a Cayley graph. We use similar techniques as in [@hyperoctahedral] to show that for odd $n$, the quantum symmetry group of the folded $n$-cube graph is $SO_n^{-1}$ which is the quotient of $O_n^{-1}$ by some quantum determinant condition (Theorem \[main\]). This constitutes our main result.
Preliminaries
=============
Compact matrix quantum groups
-----------------------------
We start with the definition of compact matrix quantum groups which were defined by Woronowicz [@CMQG1; @CMQG2] in 1987. See [@Nesh; @Tim] for recent books on compact quantum groups.
A *compact matrix quantum group* $G$ is a pair $(C(G),u)$, where $C(G)$ is a unital (not necessarily commutative) $C^*$-algebra which is generated by $u_{ij}$, $1 \leq i,j \leq n$, the entries of a matrix $u \in M_n(C(G))$. Moreover, the \*-homomorphism $\Delta: C(G) \to C(G) \otimes C(G)$, $u_{ij} \mapsto \sum_{k=1}^n u_{ik} \otimes u_{kj}$ must exist, and $u$ and its transpose $u^{t}$ must be invertible.
An important example of a compact matrix quantum group is the quantum symmetric group $S_n^+$ due to Wang [@WanSn]. It is the compact matrix quantum group, where $$\begin{aligned}
C(S_n^+) := C^*(u_{ij}, \, 1 \leq i,j \leq n \, | \, u_{ij} = u_{ij}^* = u_{ij}^2, \, \sum_{l} u_{il} = \sum_{l} u_{li} =1).\end{aligned}$$
An action of a compact matrix quantum group on a $C^*$-algebra is defined as follows ([@Pod; @WanSn]).
Let $G=(C(G), u)$ be a compact matrix quantum group and let $B$ be a $C^*$-algebra. A *(left) action* of $G$ on $B$ is a unital \*-homomorphism $\alpha: B \to B\otimes C(G)$ such that
- $(\mathrm{id} \otimes \Delta ) \circ \alpha = (\alpha \otimes \mathrm{id}) \circ \alpha$
- $\alpha(B)(1 \otimes C(G))$ is linearly dense in $B \otimes C(G)$.
In [@WanSn], Wang showed that $S_n^+$ is the universal compact matrix quantum group acting on $X_n = \{1,\dots,n\}$. This action is of the form $\alpha: C(X_n) \to C(X_n) \otimes C(S_n^+)$, $$\begin{aligned}
\alpha(e_i) = \sum_{j} e_j \otimes u_{ji}. \end{aligned}$$
Quantum automorphism groups of finite graphs
--------------------------------------------
In 2005, Banica [@QBan] gave the following definition of a quantum automorphism group of a finite graph.
Let $\Gamma =(V, E)$ be a finite graph with $n$ vertices and adjacency matrix $\varepsilon \in M_n(\{0,1\})$. The *quantum automorphism group* ${G_{aut}^+}(\Gamma)$ is the compact matrix quantum group $(C({G_{aut}^+}(\Gamma)),u)$, where $C({G_{aut}^+}(\Gamma))$ is the universal $C^*$-algebra with generators $u_{ij}, 1 \leq i,j \leq n$ and relations $$\begin{aligned}
&u_{ij} = u_{ij}^* = u_{ij}^2 &&1 \leq i,j \leq n,\label{QA1}\\
&\sum_{l=1}^n u_{il} = 1 = \sum_{l=1}^n u_{li}, &&1 \leq i \leq n,\label{QA2}\\
&u \varepsilon = \varepsilon u \label{QA3},\end{aligned}$$ where is nothing but $\sum_ku_{ik}{\varepsilon}_{kj}=\sum_k{\varepsilon}_{ik}u_{kj}$.
There is another definition of a quantum automorphism group of a finite graph by Bichon in [@QBic], which is a quantum subgroup of ${G_{aut}^+}(\Gamma)$. But this article concerns ${G_{aut}^+}(\Gamma)$. See [@SWe] for more on quantum automorphism groups of graphs.
The next definition is due to Banica and Bichon [@BanBic]. We denote by ${G_{aut}}(\Gamma)$ the usual automorphism group of a graph $\Gamma$.
Let $\Gamma = (V,E)$ be a finite graph. We say that $\Gamma$ has *no quantum symmetry* if $C({G_{aut}^+}(\Gamma))$ is commutative, or equivalently $$\begin{aligned}
C({G_{aut}^+}(\Gamma)) = C({G_{aut}}(\Gamma)).\end{aligned}$$ If $C({G_{aut}^+}(\Gamma))$ is non-commutative, we say that $\Gamma$ has *quantum symmetry*.
Note that ${G_{aut}}(\Gamma) {\subseteq}{G_{aut}^+}(\Gamma)$, so in general a graph $\Gamma$ has more quantum symmetries than symmetries.
Compact matrix quantum groups acting on graphs
----------------------------------------------
An action of a compact matrix quantum group on a graph is an action on the functions on the vertices, but with additional structure. This concept was introduced by Banica and Bichon [@QBan; @QBic].
Let $\Gamma = (V,E)$ be a finite graph and $G$ be a compact matrix quantum group. An *action of $G$ on $\Gamma$* is an action of $G$ on $C(V)$ such that the magic unitary matrix $(v_{ij})_{1 \leq i,j \leq |V|}$ associated to the formular $$\begin{aligned}
\alpha(e_i) = \sum_{j=1}^{|V|} e_j \otimes v_{ji}\end{aligned}$$ commutes with the adjacency matrix, i.e $v\varepsilon = \varepsilon v$.
If $G$ acts on a graph $\Gamma$, then we have a surjective \*-homomorphism ${\varphi}: C({G_{aut}^+}(\Gamma)) \to C(G)$, $u \mapsto v$.
The following theorem shows that commutation with the magic unitary $u$ yields invariant subspaces.
\[preserve\] Let $\alpha: C(X_n) \to C(X_n) \otimes C(G), \alpha(e_i) = \sum_{j} e_j \otimes v_{ji}$ be an action, where $G$ is a compact matrix quantum group and let $K$ be a linear subspace of $C(X_n)$. The matrix $(v_{ij})$ commutes with the projection onto $K$ if and only if $\alpha(K) {\subseteq}K \otimes C(G)$.
Looking at the spectral decomposition of the adjacency matrix, we see that this action preserves the eigenspaces of the adjacency matrix.
\[Eigen\] Let $\Gamma=(V,E)$ be an undirected finite graph with adjacency matrix $\varepsilon$. The action $\alpha: C(V) \to C(V) \otimes C({G_{aut}^+}(\Gamma))$, $\alpha(e_i) = \sum_{j} e_j \otimes u_{ji}$, preserves the eigenspaces of $\varepsilon$, i.e. $\alpha(E_\lambda) {\subseteq}E_\lambda \otimes C({G_{aut}^+}(\Gamma))$ for all eigenspaces $E_\lambda$.
It follows from the spectral decomposition that every projection $P_{E_{\lambda}}$ onto $E_{\lambda}$ is a polynomial in $\varepsilon$. Thus it commutes with the fundamental corepresentation $u$ and Theorem \[preserve\] yields the assertion.
Fourier transform {#Fourier}
-----------------
One can obtain a $C^*$-algebra from the group ${\mathbb Z}_2^n$ by either considering the continuous functions $C({\mathbb Z}_2^n)$ over the group or the group $C^*$-algebra $C^*({\mathbb Z}_2^n)$. Since ${\mathbb Z}_2^n$ is abelian, we know that $C({\mathbb Z}_2^n) \cong C^*({\mathbb Z}_2^n)$ by Pontryagin duality. This isomorphism is given by the Fourier transform and its inverse. We write $$\begin{aligned}
{\mathbb Z}_2^n&= \{t_1^{i_1} \dots t_n^{i_n} | i_1, \dots ,i_n \in \{0,1\}\},\\
C({\mathbb Z}_2^n) &= \mathrm{span}( e_{t_1^{i_1} \dots t_n^{i_n}} \, | \, t_1^{i_1} \dots t_n^{i_n} \in {\mathbb Z}_2^n),\\
C^*({\mathbb Z}_2^n) &= C^*(t_1, \dots , t_n \, | \,t_i = t_i^*, t_i^2 = 1, t_i t_j = t_j t_i),\\
\intertext{where} e_{t_1^{i_1} \dots t_n^{i_n}} : {\mathbb Z}_2^n &\to {\mathbb C}, \qquad e_{t_1^{i_1} \dots t_n^{i_n}}({t_1^{j_1} \dots t_n^{j_n}}) = \delta_{i_1 j_1} \dots \delta_{i_n j_n}.\end{aligned}$$
The proof of the following proposition can be found in [@quizzy] for example.
The \*-homomorphisms $$\begin{aligned}
{\varphi}: C({\mathbb Z}_2^n) &\to C^*({\mathbb Z}_2^n), \qquad e_{t_1^{i_1}\dots t_n^{i_n}} \to \frac{1}{2^n} \sum_{j_1, \dots, j_n=0}^1 (-1)^{i_1 j_1 + \dots + i_n j_n} t_1^{j_1} \dots t_n^{j_n}
\intertext{and}
\psi: C^*({\mathbb Z}_2^n) &\to C({\mathbb Z}_2^n), \qquad t_1^{i_1} \dots t_n^{i_n}\to \sum_{j_1, \dots, j_n=0}^1 (-1)^{i_1 j_1 + \dots + i_n j_n}e_{t_1^{j_1}\dots t_n^{j_n}}, \end{aligned}$$ where $i_1, \dots, i_n \in \{0,1\}$, are inverse to each other. The map ${\varphi}$ is called Fourier transform, the map $\psi$ is called inverse Fourier transform.
A criterion for a graph to have quantum symmetry
================================================
In this section, we show that a graph has quantum symmetry if the automorphism group of the graph contains a certain pair of permutations. For this we need the following definition.
Let $V = \{1,\dots, r\}$. We say that two permutations $\sigma:V \to V$ and $\tau: V \to V$ are *disjoint*, if $\sigma(i) \neq i$ implies $\tau(i) =i$ and vice versa, for all $i \in V$.
\[noncomm\] Let $\Gamma=(V,E)$ be a finite graph without multiple edges. If there exist two non-trivial, disjoint automorphisms $\sigma, \tau \in {G_{aut}}(\Gamma)$, $\mathrm{ord}(\sigma) = n, \mathrm{ord}(\tau)=m$, then we get a surjective \*-homomorphism ${\varphi}: C({G_{aut}^+}(\Gamma)) \to C^*({\mathbb Z}_n * {\mathbb Z}_m)$. In particular, $\Gamma$ has quantum symmetry.
Let $\sigma, \tau \in {G_{aut}}(\Gamma)$ be non-trivial disjoint automorphisms with $\mathrm{ord}(\sigma) = n, \mathrm{ord}(\tau)=m$. Define $$\begin{aligned}
A&:=C^*(p_1, \dots, p_n, q_1, \dots, q_m| p_k=p_k^* = p_k^2, q_l=q_l^*=q_l^2, \sum_{k=1}^n p_k =1= \sum_{l=1}^m q_l)\\
&\cong C^*({\mathbb Z}_n * {\mathbb Z}_m).\end{aligned}$$ We want to use the universal property to get a surjective \*-homomorphism ${\varphi}: C({G_{aut}^+}(\Gamma)) \to A$. This yields the non-commutativity of ${G_{aut}^+}(\Gamma)$, since $p_k,q_l$ do not have to commute. In order to do so, define $$\begin{aligned}
u' := \sum_{l=1}^m \tau^l \otimes q_l + \sum_{k=1}^n \sigma^k \otimes p_k - \mathrm{id}_{\mathrm{M_r}(\mathbb{C}) \otimes A} \in \mathrm{M_r}(\mathbb{C}) \otimes A \cong \mathrm{M_r}(A),\end{aligned}$$ where $\tau^l, \sigma^k$ denote the permutation matrices corresponding to $\tau^l, \sigma^k \in {G_{aut}}(\Gamma)$. This yields $$\begin{aligned}
u'_{ij} = \sum_{l=1}^m \delta_{j\tau^l(i)} \otimes q_l + \sum_{k=1}^n \delta_{j\sigma^k(i)} \otimes p_k - \delta_{ij} \in {\mathbb C}\otimes A \cong A.\end{aligned}$$ Now, we show that $u'$ does fulfill the relations of $u \in \mathrm{M_r}(\mathbb{C}) \otimes A$, the fundamental representation of ${G_{aut}^+}(\Gamma)$. Since we have $\tau^l, \sigma^k \in {G_{aut}}(\Gamma)$, it holds $\tau^l \varepsilon = \varepsilon \tau^l$ and $\sigma^k \varepsilon = \varepsilon \sigma^k$ for all $1 \leq l \leq m$, $1 \leq k \leq n$, where $\varepsilon$ denotes the adjacency matrix of $\Gamma$. Therefore, we have$$\begin{aligned}
u' (\varepsilon \otimes 1) &= \left(\sum_{l=1}^m \tau^l \otimes q_l + \sum_{k=1}^n \sigma^k \otimes p_k - \mathrm{id}_{\mathrm{M_r}(\mathbb{C}) \otimes A}\right) (\varepsilon \otimes 1)\\
&=\sum_{l=1}^m \tau^l \varepsilon \otimes q_l + \sum_{k=1}^n \sigma^k \varepsilon \otimes p_k - (\varepsilon \otimes 1)\\
&=\sum_{l=1}^m \varepsilon \tau^l \otimes q_l + \sum_{k=1}^n \varepsilon \sigma^k \otimes p_k - (\varepsilon \otimes 1)\\
&= (\varepsilon \otimes 1) \left(\sum_{l=1}^m \tau^l \otimes q_l + \sum_{k=1}^n \sigma^k \otimes p_k - \mathrm{id}_{\mathrm{M_r}(\mathbb{C}) \otimes A}\right)\\
&=(\varepsilon \otimes 1) u'.\end{aligned}$$ Furthermore, it holds $$\begin{aligned}
\sum_{i=1}^r u'_{ji} &=\sum_{i=1}^r \left(\sum_{l=1}^m \delta_{i\tau^l(j)} \otimes q_l + \sum_{k=1}^n \delta_{i\sigma^k(j)} \otimes p_k\right) - 1\otimes 1\\
&= 1 \otimes \left(\sum_{l=1}^m q_l \right)+ 1 \otimes \left(\sum_{k=1}^n p_k\right) - 1 \otimes 1\\
&= 1 \otimes 1.\end{aligned}$$ A similar computation shows $\sum_{i=1}^r u'_{ij} = 1 \otimes 1$. Since $\tau$ and $\sigma$ are disjoint, we have $$\begin{aligned}
u'_{ij} = \sum_{l=1}^m \delta_{j\tau^l(i)} \otimes q_l + \sum_{k=1}^n \delta_{j\sigma^k(i)} \otimes p_k - \delta_{ij} =
\begin{cases}
\sum_{k \in N_{ij}} p_k, \text{ if } \sigma(i) \neq i\\
\sum_{l \in M_{ij}} q_l, \text{ if } \tau(i) \neq i\\
\delta_{ij}, \text{ otherwise,}
\end{cases}\end{aligned}$$ where $N_{ij} = \{ k \in \{1 \dots n\}; \, \sigma^k(i)=j\}, M_{ij} = \{l \in \{1 \dots m\}; \, \tau^l(i) =j\}$. Thus, all entries of $u'$ are projections. By the universal property, we get a \*-homomorphism ${\varphi}: C({G_{aut}^+}(\Gamma)) \to A, u \mapsto u'.$
It remains to show that ${\varphi}$ is surjective. Since we know that $\mathrm{ord}(\sigma) = n, \mathrm{ord}(\tau)=m$, there exist $s,t \in V$ where $\tau^{l_1}(s) \neq \tau^{l_2}(s)$ for all $l_1 \neq l_2$, $l_1, l_2 \in \{1, \dots ,m\}$ and $\sigma^{k_1}(t) \neq \sigma^{k_2}(t)$ for all $k_1 \neq k_2$, $k_1, k_2 \in \{1 ,\dots, n\}$. Therefore, we have ${\varphi}(u_{s\tau^l(s)}) =u'_{s\tau^l(s)} = q_l$ and ${\varphi}(u_{t\sigma^k(t)}) =u'_{t\sigma^k(t)} = p_k$ for all $l \in \{1 ,\dots, m\}, k \in \{1 ,\dots ,n\}$ and since $A$ is generated by $p_k$ and $q_l$, ${\varphi}$ is surjective.
Let $K_4$ be the full graph on 4 points. We know that ${G_{aut}}(K_4) = S_4$ and ${G_{aut}^+}(K_4)=S_4^+$. We have disjoint automorphisms in $S_4$, where for example $\sigma=(12), \tau=(34) \in S_4$ give us the well known surjective \*-homomorphism $$\begin{aligned}
\varphi:C(S_4^+) &\to C^*(p,q \, | \, p=p^*=p^2, q = q^* =q^2), \\
u &\mapsto \begin{pmatrix} p&1-p&0&0\\1-p&p&0&0\\0&0&q&1-q\\0&0&1-q&q\end{pmatrix}, \end{aligned}$$ yielding the non-commutativity of $S_4^+$.
Let $\Gamma=(V,E)$ be a finite graph without multiple edges, where there exist two non-trivial, disjoint automorphisms $\sigma, \tau \in {G_{aut}}(\Gamma)$. To show that $\Gamma$ has quantum symmetry it is enough to see that we have the surjective \*-homomorphism $$\begin{aligned}
{\varphi}': C({G_{aut}^+}(\Gamma)) &\to C^*(p,q \, | \,p=p^*=p^2, q = q^* =q^2), \\
u &\mapsto \sigma \otimes p + \tau \otimes q + \mathrm{id}_{M_r({\mathbb C})} \otimes (1-q-p).\end{aligned}$$
At the moment, we do not have an example of a graph $\Gamma$, where ${G_{aut}}(\Gamma)$ does not contain two disjoint automorphisms but the graph has quantum symmetry.
The Clebsch graph has quantum symmetry
======================================
As an application of Theorem \[noncomm\], we show that the Clebsch graph does have quantum symmetry. In Section \[sect\], we will study the quantum automorphism group of this graph.
(18:1cm) – (162:1cm) – (306:1cm) – (90:1cm) – (234:1cm) – cycle; (18:3cm) – (90:3cm) – (162:3cm) – (234:3cm) – (306:3cm) – cycle; (18:1cm) – (54:2cm) –(90:1cm) – (126:2cm) – (162:1cm) – (198:2cm) – (234:1cm) – (270:2cm) – (306:1cm) – (342:2cm) – cycle; (18:3cm) – (126:2cm) – (234:3cm)– (342:2cm) – (90:3cm) – (198:2cm) – (306:3cm) – (54:2cm) – (162:3cm) – (270:2cm) – cycle; in [18,90,162,234,306]{} [(:1cm) – (:3cm); (:3cm) circle (2pt); (:1cm) circle (2pt);]{} in [54, 126, 198, 270, 342]{} [(:2cm) circle (2pt); (:0cm) circle (2pt); (:0cm) – (:2cm);]{}
The Clebsch graph has disjoint automorphisms.
We label the graph as follows
in [18,90,162,234,306]{} [(:3cm) circle (2pt); (:1cm) circle (2pt);]{} in [54, 126, 198, 270, 342]{} [(:2cm) circle (2pt); (:0cm) circle (2pt);]{} (1) at (90:3cm); (2) at (18:3cm); (3) at (198:2cm); (4) at (306:3cm); (5) at (162:3cm); (6) at (162:1cm); (7) at (270:2cm); (8) at (306:1cm); (9) at (90:1cm); (10) at (234:1cm); (11) at (126:2cm); (12) at (234:3cm); (13) at (54:2cm); (14) at (18:1cm); (15) at (90:0cm); (16) at (342:2cm);
Then we get two non-trivial disjoint automorphisms of this graph $$\begin{aligned}
\sigma &= (2\,3)(6\,7)(10\,11)(14\,15),\\
\tau &= (1\,4)(5\,8)(9\,12)(13\,16).\end{aligned}$$
\[C\] The Clebsch graph does have quantum symmetries, i.e. $C(G_{aut}^+(\Gamma_{Clebsch}))$ is non-commutative.
By Theorem \[noncomm\], we get that $C(G_{aut}^+(\Gamma_{Clebsch}))$ is non-commutative. Looking at the proof of Theorem \[noncomm\], we get the surjective \*-homomorphism $\varphi:C(G_{aut}^+(\Gamma_{Clebsch})) \to C^*(p,q \, | \, p = p^* = p^2, q = q^* = q^2)$, $$\begin{aligned}
u &\mapsto u'=\begin{pmatrix} u''&0&0&0\\ 0&u''&0&0\\0&0&u''&0\\0&0&0&u''\end{pmatrix},
\intertext{where}
u'' &=\begin{pmatrix} q&0&0&1-q\\0&p&1-p&0\\0&1-p&p&0\\1-q&0&0&q\end{pmatrix}.\end{aligned}$$
- The Clebsch graph is the folded $5$-cube graph, which will be introduced in Section \[sect\]. There we will study the quantum automorphism group for $(2m+1)$-folded cube graphs going far beyond Corollary \[C\].
- Using Theorem \[noncomm\], it is also easy to see that the folded cube graphs have quantum symmetry, but this will also follow from our main result (Theorem \[main\]).
The quantum group $SO_n^{-1}$
=============================
Now, we will have a closer look at the quantum group $SO_n^{-1}$, but first we define $O_n^{-1}$, which appeared in [@hyperoctahedral] as the quantum automorphism group of the hypercube graph. For both it is immediate to check that the comultiplication $\Delta$ is a \*-homomorphism.
We define $O_n^{-1}$ to be the compact matrix quantum group $(C(O_n^{-1}), u)$, where $C(O_n^{-1})$ is the universal $C^*$-algebra with generators $u_{ij}$, $1 \leq i,j \leq n$ and relations $$\begin{aligned}
&u_{ij} = u_{ij}^*, &&1\leq i,j \leq n, \label{7.1}\\
&\sum_{k=1}^n u_{ik}u_{jk} = \sum_{k=1}^n u_{ki}u_{kj} = \delta_{ij}, && 1\leq i,j \leq n,\label{7.2}\\
&u_{ij}u_{ik} = - u_{ik}u_{ij}, u_{ji}u_{ki} = -u_{ki}u_{ji}, &&k\neq j,\label{7.3}\\
&u_{ij}u_{kl}=u_{kl}u_{ij}, &&i\neq k, j \neq l.\label{7.4}\end{aligned}$$
For $n=3$, $SO_n^{-1}$ appeared in [@4points], where Banica and Bichon showed $SO_3^{-1} = S_4^+$. Our main result in this paper is that for $n$ odd, $SO_n^{-1}$ is the quantum automorphism group of the folded $n$-cube graph.
We define $SO_n^{-1}$ to be the compact matrix quantum group $(C(SO_n^{-1}), u)$, where $C(SO_n^{-1})$ is the universal $C^*$-algebra with generators $u_{ij}$, $1 \leq i,j \leq n$, Relations – and $$\begin{aligned}
\sum_{\sigma \in S_n} u_{\sigma(1)1}\dots u_{\sigma(n)n} =1.\label{7.5}\end{aligned}$$
\[sumzero\] Let $(u_{ij})_{1 \leq i,j \leq n}$ be the generators of $C(SO_n^{-1})$. Then $$\begin{aligned}
\sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n-1)n-1}u_{\sigma(n)k} =0\end{aligned}$$ for $k \neq n$.
Let $1 \leq k \leq n-1$. Using Relations and we get $$\begin{aligned}
u_{\sigma(1)1} \dots u_{\sigma(k)k} \dots u_{\sigma(n-1)n-1}u_{\sigma(n)k} &= - u_{\sigma(1)1} \dots u_{\sigma(n)k} \dots u_{\sigma(n-1)n-1}u_{\sigma(k)k}\\
&=-u_{\tau(1)1} \dots u_{\tau(k)k} \dots u_{\tau(n-1)n-1}u_{\tau(n)k}\end{aligned}$$ for $\tau = \sigma \circ(k \, n) \in S_n$. Therefore the summands corresponding to $\sigma$ and $\tau$ sum up to zero. We have $\sigma \neq \tau$ for all those pairs since $k \neq n$ and for $\sigma_1 \neq \sigma_2$ we get $\tau_1= \sigma_1\circ(k \, n) \neq \sigma_2 \circ(k \, n)= \tau_2$. Thus summing over those pairs yields the assertion.
The next lemma gives an equivalent formulation of Relation . One direction is a special case of [@Dualpairs Lemma 4.6].
\[SO\] Let $A$ be a $C^*$-algebra and let $u_{ij} \in A$ be elements that fulfill Relations $\eqref{7.1}-\eqref{7.4}$. Let $j \in \{1,\dots,n\}$ and define $$\begin{aligned}
I_j = \{ (i_1,\dots ,i_{n-1}) \in \{1,\dots ,n\}^{n-1} \, | \, i_a \neq i_b \text{ for } a \neq b, i_s \neq j \text{ for all } s\}.\end{aligned}$$ The following are equivalent
- We have $$\begin{aligned}
1&=\sum_{\sigma \in S_n} u_{\sigma(1)1}\dots u_{\sigma(n)n}.\intertext{
\item[(ii)] It holds} u_{jn} &= \sum_{(i_1,\dots,i_{n-1}) \in I_j} u_{i_1 1}\dots u_{i_{n-1}n-1}, \quad 1\leq j\leq n.
\end{aligned}$$
We first show that (ii) implies (i). It holds $$\begin{aligned}
1 = \sum_{j=1}^n u_{jn}^2 = \sum_{j=1}^n \sum_{(i_1,\dots,i_{n-1}) \in I_j} u_{i_1 1}\dots u_{i_{n-1}n-1}u_{jn}, \end{aligned}$$ where we used Relation and (ii). Furthermore, we have $$\begin{aligned}
\sum_{j=1}^n \sum_{(i_1,\dots,i_{n-1}) \in I_j} u_{i_1 1}\dots u_{i_{n-1}n-1} u_{jn}
&= \sum_{\substack{i_1, \dots, i_n;\\ i_a \neq i_b \text{ for } a\neq b}} u_{i_1 1} \dots u_{i_n n}\\
&= \sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n)n}\end{aligned}$$ and thus (ii) implies (i).
Now we show that (i) implies (ii). We have $$\begin{aligned}
u_{jn} = \sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n)n}u_{jn} = \sum_{k=1}^n \sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n-1)n-1}u_{\sigma(n)k}u_{jk}, \end{aligned}$$ since $\sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n-1)n-1}u_{\sigma(n)k}u_{jk} =0$ for $k \neq n$ by Lemma \[sumzero\]. We get $$\begin{aligned}
\sum_{k=1}^n \sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n-1)n-1} u_{\sigma(n)k}u_{jk} &= \sum_{\sigma \in S_n} u_{\sigma(1)1} \dots u_{\sigma(n-1)n-1}\sum_{k=1}^n u_{\sigma(n)k}u_{jk}\\
&= \sum_{\sigma \in S_n} u_{\sigma(1)1}\dots u_{\sigma(n-1)n-1}\delta_{\sigma(n)j}\\
&= \sum_{(i_1,\dots,i_{n-1}) \in I_j} u_{i_1 1}\dots u_{i_{n-1}n-1}, \end{aligned}$$ where we used Relation and we obtain $u_{jn} = \sum_{(i_1,\dots,i_{n-1}) \in I_j} u_{i_1 1}\dots u_{i_{n-1}n-1}$.
We now discuss representations of $SO_{2m+1}^{-1}$. For definitions and background for this proposition, we refer to [@hopfgalois; @subgroups; @schauenburg].
The category of corepresentations of $SO_{2m+1}^{-1}$ is tensor equivalent to the category of representations of $SO_{2m+1}$.
We first show that $C(SO_{2m+1}^{-1})$ is a cocycle twist of $C(SO_{2m+1})$ by proceeding like in [@subgroups Section 4]. Take the unique bicharacter $\sigma: {\mathbb Z}_2^{2m} \times {\mathbb Z}_2^{2m} \to \{\pm 1\}$ with $$\begin{aligned}
&\sigma(t_i, t_j) = - 1 = - \sigma(t_j, t_i) ,&& \text{for } 1 \leq i < j \leq 2m,\\
&\sigma(t_i, t_i) = (-1)^m, && \text{for } 1 \leq i \leq 2m+1,\\
&\sigma(t_i, t_{2m+1}) = (-1)^{m-i} = - \sigma(t_{2m+1}, t_i), && \text{for } 1 \leq i \leq 2m,\end{aligned}$$ where we use the identification ${\mathbb Z}_2^{2m} = \langle t_1, \dots, t_{2m+1} \, | \, t_i^2=1, t_i t_j = t_j t_i, t_{2m+1} = t_1\dots t_{2m}\rangle$. Let $H$ be the subgroup of diagonal matrices in $SO_{2m+1}$ having $\pm 1$ entries. We get a surjective \*-homomorphism $$\begin{aligned}
\pi : C(SO_{2m+1}) &\to C^*({\mathbb Z}_2^{2m})\\
u_{ij} &\mapsto \delta_{ij} t_i\end{aligned}$$ by restricting the functions on $SO_{2m+1}$ to $H$ and using Fourier transform. Thus we can form the twisted algebra $C(SO_{2m+1})^{\sigma}$, where we have the multiplication $$\begin{aligned}
[u_{ij}] [u_{kl}] = \sigma(t_i, t_k) \sigma^{-1}(t_j, t_l) [u_{ij}u_{kl}] =\sigma(t_i, t_k) \sigma(t_j, t_l) [u_{ij}u_{kl}].\end{aligned}$$ We see that the generators $[u_{ij}]$ of $C(SO_{2m+1})^{\sigma}$ fulfill the same relations as the generators of $C(SO_{2m+1}^{-1})$ and thus we get an surjective \*-homomorphism ${\varphi}: C(SO_{2m+1}^{-1}) \to C(SO_{2m+1})^{\sigma}, \,u_{ij} \mapsto [u_{ij}].$ This is an isomorphism for example by using Theorem 3.5 of [@Kassel]. Now, Corollary 1.4 and Proposition 2.1 of [@hopfgalois] yield the assertion.
Quantum automorphism groups of folded cube graphs {#sect}
=================================================
In what follows, we will introduce folded cube graphs $FQ_n$ and show that for odd $n$, the quantum automorphism group of $FQ_n$ is $SO_n^{-1}$.
The folded $n$-cube graph $FQ_n$
--------------------------------
The folded $n$-cube graph $FQ_n$ is the graph with vertex set $V= \{ (x_1, \dots, x_{n-1}) \, | \, x_i \in \{0,1\}\}$, where two vertices $(x_1,\dots, x_{n-1})$ and $ (y_1, \dots,y_{n-1})$ are connected if they differ at exactly one position or if $(y_1, \dots,y_{n-1}) = (1-x_1, \dots, 1-x_{n-1})$.
To justify the name, one can obtain the folded $n$-cube graph by identifing every opposite pair of vertices from the $n$-hypercube graph.
The folded cube graph as Cayley graph
-------------------------------------
It is known that the folded cube graphs are Cayley graphs, we recall this fact in the next lemma.
The folded $n$-cube graph $FQ_n$ is the Cayley graph of the group ${\mathbb Z}_2^{n-1}= \langle t_1, \dots t_n \rangle$, where the generators $t_i$ fulfill the relations $t_i^2=1, t_i t_j = t_j t_i, t_n = t_1\dots t_{n-1}$.
Consider the Cayley graph of ${\mathbb Z}_2^{n-1}= \langle t_1, \dots t_n \rangle$. The vertices are elements of ${\mathbb Z}_2^{n-1}$, which are products of the form $g = t_1^{i_1}\dots t_{n-1}^{i_{n-1}}$. The exponents are in one to one correspondence to $(x_1, \dots, x_{n-1}), \, x_i \in \{0,1\}$, thus the vertices of the Cayley graph are the vertices of the folded $n$-cube graph. The edges of the Cayley graph are drawn between vertices $g,h$, where $g = h t_i$ for some $i$. For $k \in \{1,\dots, n-1\}$, the operation $h \to h t_k$ changes the $k$-th exponent to $1-i_k$, so we get edges between vertices that differ at exactly one expontent. The operation $h \to h t_n$ takes $t_1^{j_1}\dots t_{n-1}^{j_{n-1}}$ to $t_1^{1-j_1}\dots t_{n-1}^{1-j_{n-1}}$, thus we get the remaining edges of $FQ_n$.
Eigenvalues and Eigenvectors of $FQ_n$
--------------------------------------
We will now discuss the eigenvalues and eigenvectors of the adjacency matrix of $FQ_n$.
\[EV\] The eigenvectors and corresponding eigenvalues of $FQ_n$ are given by $$\begin{aligned}
w_{i_1\dots i_{n-1}}&= \sum_{j_1,\dots,j_{n-1}=0}^1(-1)^{i_1j_1 + \dots +i_{n-1}j_{n-1}}e_{t_1^{j_1}\dots t_{n-1}^{j_{n-1}}}\\
\lambda_{i_1\dots i_{n-1}} &= (-1)^{i_1}+ \ldots +(-1)^{i_{n-1}}+(-1)^{i_1+\ldots+i_{n-1}},\end{aligned}$$ when the vector space spanned by the vertices of $FQ_n$ is identified with $C({\mathbb Z}_2^{n-1})$.
Let $\varepsilon$ be the adjacency matrix of $FQ_n$. Then we know for a vertex $p$ and a function $f$ on the vertices that $$\begin{aligned}
\varepsilon f(p) = \sum_{q ; (q,p) \in E} f(q).\end{aligned}$$ This yields $$\begin{aligned}
\varepsilon e_{t_1^{j_1}\dots t_{n-1}^{j_{n-1}}} = \sum_{k=1}^n e_{t_kt_1^{j_1}\dots t_{n-1}^{j_{n-1}}} = e_{t_1^{j_1 +1}\dots t_{n-1}^{j_{n-1}}} + \dots+ e_{t_1^{j_1}\dots t_{n-1}^{j_{n-1}+1}} + e_{t_1^{j_1+1}\dots t_{n-1}^{j_{n-1}+1}}.\end{aligned}$$ For the vectors in the statement we get$$\begin{aligned}
\varepsilon w_{i_1 \dots i_{n-1}} &= \sum_{j_1,\dots,j_{n-1}} (-1)^{i_1j_1 + \dots +i_{n-1}j_{n-1}}\varepsilon e_{t_1^{j_1}\dots t_{n-1}^{j_{n-1}}}\\
&=\sum_{s=1}^{n-1} \sum_{j_1,\dots,j_{n-1}} (-1)^{i_1j_1 + \dots +i_{n-1}j_{n-1}}e_{t_1^{j_1}\dots t_s^{j_s+1}\dots t_{n-1}^{j_{n-1}}}\\
&\qquad + \sum_{j_1,\dots,j_{n-1}} (-1)^{i_1j_1 + \dots +i_{n-1}j_{n-1}}e_{t_1^{j_1+1}\dots t_{n-1}^{j_{n-1}+1}}.
\intertext{Using the index shift $j_s' = j_s +1\bmod 2$, for $s \in \{1, \dots n-1\}$, we get}
\varepsilon w_{i_1 \dots i_{n-1}}&=\sum_{s=1}^{n-1} \sum_{j_1,\dots j_s', \dots j_{n-1}} (-1)^{i_1j_1 + \dots + i_s(j_s'+1)+ \dots +i_{n-1}j_{n-1}}e_{t_1^{j_1}\dots t_s^{j_s'}\dots t_{n-1}^{j_{n-1}}}\\
&\qquad + \sum_{j_1',\dots,j_{n-1}'} (-1)^{i_1(j_1'+1) + \dots +i_{n-1}(j_{n-1}'+1)}e_{t_1^{j_1'}\dots t_{n-1}^{j_{n-1}'}}\\
&=\sum_{s=1}^{n-1} \sum_{j_1,\dots, j_s', \dots, j_{n-1}}(-1)^{i_s}(-1)^{i_1j_1 + \dots +i_{n-1}j_{n-1}}e_{t_1^{j_1}\dots t_s^{j_s'}\dots t_{n-1}^{j_{n-1}}} \\
&\qquad + \sum_{j_1',\dots,j_{n-1}'} (-1)^{i_1+ \dots + i_{n-1}}(-1)^{i_1j_1' + \dots +i_{n-1}j_{n-1}'}e_{t_1^{j_1'}\dots t_{n-1}^{j_{n-1}'}}\\
&= ((-1)^{i_1}+ \ldots +(-1)^{i_{n-1}}+(-1)^{i_1+\ldots+i_{n-1}})w_{i_1\dots i_{n-1}}\\
&= \lambda_{i_1\dots i_{n-1}}w_{i_1\dots i_{n-1}}.\end{aligned}$$ Since those are $2^{n-1}$ vectors that are linearly independent, the assertion follows.
The following lemma shows what the eigenvectors look like if we identify the vector space spanned by the vertices of $FQ_n$ with $C^*({\mathbb Z}_2^{n-1})$.
\[ev\] In $C^*({\mathbb Z}_2^{n-1})= C^*(t_1, \dots, t_n \, | \, t_i^2=1, t_i t_j = t_j t_i, t_n = t_1\dots t_{n-1})$ the eigenvectors of $FQ_n$ are $$\begin{aligned}
\hat{w}_{i_1\dots i_{n-1}} = t_1^{i_1} \dots t_{n-1}^{i_{n-1}}\end{aligned}$$ corresponding to the eigenvalues $\lambda_{i_1\dots i_{n-1}}$ from Lemma \[EV\].
We obtain $\hat{w}_{i_1\dots i_{n-1}} $ by using the Fourier transform (see Section \[Fourier\]) on $w_{i_1\dots i_{n-1}}$ from Lemma \[EV\].
Note that certain eigenvalues in Lemma \[EV\] coincide. We get a better description of the eigenvalues and eigenspaces of $FQ_n$ in the next lemma.
\[ES\] The eigenvalues of $FQ_n$ are given by $\lambda_k = n-2k$ for $k\in 2{\mathbb Z}\cap \{0,\dots,n\}$. The eigenvectors $t_1^{i_1} \dots t_{n-1}^{i_{n-1}}$ corresponding to $\lambda_k$ have word lengths $k$ or $k-1$ and form a basis of $E_{\lambda_k}$. Here $E_{\lambda_k}$ denotes the eigenspace to the eigenvalue $\lambda_k$.
Let $k\in 2{\mathbb Z}\cap \{0,\dots,n\}$. By Lemma \[EV\] and Lemma \[ev\], we get that an eigenvector $t_1^{i_1} \dots t_{n-1}^{i_{n-1}}$ of word length $k$ (here $k\neq n$, if $n$ is even) with respect to $t_1, \dots, t_{n-1}$ corresponds to the eigenvalue $$\begin{aligned}
(-1)^{i_1}+ \ldots +(-1)^{i_{n-1}}+(-1)^{i_1+\ldots+i_{n-1}} = -k + (n-1-k) + 1 = n-2k.\end{aligned}$$ Now consider an eigenvector $t_1^{i_1} \dots t_{n-1}^{i_{n-1}}$ of word length $k-1$. Then we get the eigenvalue $$\begin{aligned}
(-1)^{i_1}+ \ldots +(-1)^{i_{n-1}}+(-1)^{i_1+\ldots+i_{n-1}} = -(k-1) + (n-k) -1 = n-2k.\end{aligned}$$ We go through all the eigenvectors of Lemma \[ev\] in this way and we obtain exactly the eigenvalues $\lambda_k = n-2k$. Since the eigenvectors of word lengths $k$ or $k-1$ are exactly those corresponding to $\lambda_k$, they form a basis of $E_{\lambda_k}$.
The quantum automorphism group of $FQ_{2m+1}$
---------------------------------------------
For the rest of this section, we restrict to the folded $n$-cube graphs, where $n =2m +1$ is odd. We show that in this case, the quantum automorphism group is $SO_n^{-1}$. We need the following lemma.
\[P\] Let $\tau_1, \dots ,\tau_n$ be generators of $C^*({\mathbb Z}_2^{n-1})$ with $\tau_i^2=1, \tau_i \tau_j = \tau_j \tau_i, \tau_n = \tau_1\dots \tau_{n-1}$ and let A be a $C^*$-algebra with elements $u_{ij} \in A$ fulfilling Relations – . Let $(i_1, \dots, i_l) \in \{ 1,\dots, n\}^l$ with $i_a \neq i_b$ for $a \neq b$, where $1 \leq l \leq n$. Then $$\begin{aligned}
\sum_{j_1, \dots, j_l=1}^n \tau_{j_1}\dots\tau_{j_l} \otimes u_{j_1i_1} \dots u_{j_l i_l} = \sum_{\substack{j_1, \dots, j_{l};\\j_a\neq j_b \text{ for } a\neq b}} \tau_{j_1}\dots\tau_{j_{l}}\otimes u_{j_1i_1}\dots u_{j_{l}i_l}.\end{aligned}$$
Let $j_{s} = j_{s+1} =k$ and let the remaining $j_l$ be arbitrary. Summing over $k$, we get $$\begin{aligned}
\sum_{k=1}^n \tau_{j_1} \dots \tau_{j_{s-1}} &\tau_k^2 \tau_{j_{s+2}}\dots \tau_{j_l} \otimes u_{j_1i_1}\dots u_{ki_s}u_{ki_{s+1}}\dots u_{j_li_l} \\
&= \tau_{j_1} \dots \tau_{j_{s-1}}\tau_{j_{s+2}}\dots \tau_{j_l}\otimes u_{j_1i_1}\dots \left(\sum_{k=1}^n u_{ki_s}u_{ki_{s+1}}\right)\dots u_{j_li_l}\\
&=0\end{aligned}$$ by Relation since $i_s \neq i_{s+1}$. Doing this for all $s \in \{1, \dots, l-1\}$ we get $$\begin{aligned}
\sum_{j_1, \dots, j_l=1}^n \tau_{j_1}\dots\tau_{j_l} \otimes u_{j_1i_1} \dots u_{j_l i_l} =\sum_{j_1\neq\dots \neq j_l} \tau_{j_1}\dots\tau_{j_l} \otimes u_{j_1i_1} \dots u_{j_l i_l}.\end{aligned}$$ Now, let $j_s = j_{s+2} = k$ and let $j_1 \neq \dots \neq j_l$. Since $k=j_s \neq j_{s+1}$ and $i_a \neq i_b$ for $a \neq b$, we have $u_{k i_s} u_{j_{s+1}i_{s+1}} = u_{j_{s+1}i_{s+1}}u_{k i_s}$. We also know that $\tau_k \tau_{j_{s+1}} = \tau_{j_{s+1}} \tau_k$ and thus $$\begin{aligned}
\sum_{k=1}^n &\tau_{j_1} \dots \tau_{j_{s-1}} \tau_k \tau_{j_{s+1}} \tau_k \tau_{j_{s+3}}\dots \tau_{j_l} \otimes u_{j_1i_1}\dots u_{ki_s}u_{j_{s+1}i_{s+1}}u_{ki_{s+2}}\dots u_{j_li_l}\\
&= \tau_{j_1} \dots \tau_{j_{s-1}} \tau_{j_{s+1}} \tau_{j_{s+3}}\dots \tau_{j_l} \otimes u_{j_1i_1}\dots u_{j_{s+1}i_{s+1}}\left(\sum_{k=1}^nu_{ki_s}u_{ki_{s+2}}\right)\dots u_{j_li_l}\\
&=0\end{aligned}$$ by Relation since $i_s \neq i_{s+2}$. This yields $$\begin{aligned}
\sum_{j_1, \dots, j_l=1}^n \tau_{j_1}\dots\tau_{j_l} \otimes u_{j_1i_1} \dots u_{j_l i_l}=\sum_{\substack{j_1, \dots, j_{l};\\j_a\neq j_b \text{ for } 0< |a-b| \leq 2}} \tau_{j_1}\dots\tau_{j_{l}}\otimes u_{j_1i_1}\dots u_{j_{l}i_l}\end{aligned}$$ The assertion follows after iterating this argument $l$ times.
We first show that $SO_n^{-1}$ acts on the folded $n$-cube graph.
\[act\] For $n$ odd, the quantum group $SO_n^{-1}$ acts on $FQ_n$.
We need to show that there exists an action $$\begin{aligned}
\alpha: C(V_{FQ_n}) \to C(V_{FQ_n}) \otimes C(SO_n^{-1}), \qquad \alpha(e_i) = \sum_{j=1}^{|V_{FQ_n}|} e_j \otimes v_{ji}\end{aligned}$$ such that $(v_{ij})$ commutes with the adjacency matrix of $FQ_n$. By Fourier transform, this is the same as getting an action $$\begin{aligned}
\alpha: C^*({\mathbb Z}_2^{n-1}) \to C^*({\mathbb Z}_2^{n-1}) \otimes C(SO_n^{-1}),\end{aligned}$$ where we identify the functions on the vertex set of $FQ_n$ with $C^*({\mathbb Z}_2^{n-1})$. We claim that $$\begin{aligned}
\alpha(\tau_i) = \sum_{j=1}^n \tau_j \otimes u_{ji}\end{aligned}$$ gives the answer, where $$\begin{aligned}
\tau_i = t_1\dots \check{t_i}\dots t_{n-1} \text{ for } 1 \leq i \leq n-1, \qquad \tau_n = t_n\end{aligned}$$ for $t_i$ as in Lemma \[ev\] and $(u_{ij})$ is the fundamental corepresentation of $SO_n^{-1}$. Here $\check{t_i}$ means that $t_i$ is not part of the product. These $\tau_i$ generate $C^*({\mathbb Z}_2^{n-1})$, with relations $\tau_i = \tau_i^*, \tau_i^2=1, \tau_i\tau_j = \tau_j\tau_i$ and $\tau_n = \tau_1\dots\tau_{n-1}$. Define $$\begin{aligned}
\tau_i' = \sum_{j=1}^n \tau_j \otimes u_{ji}.\end{aligned}$$
To show that $\alpha$ defines a \*-homomorphism, we have to show that the relations of the generators $\tau_i$ also hold for $\tau_i'$. It is obvious that $(\tau_i')^* = \tau_i'$. Using Relations – it is straightforward to check that $(\tau'_i)^2=1$ and $\tau_i'\tau_j'=\tau_j'\tau_i'$. Now, we show $\tau'_n = \tau'_1\dots\tau'_{n-1}$. By Lemma \[P\], it holds $$\begin{aligned}
\tau'_1\dots\tau'_{n-1} &= \sum_{\substack{i_1, \dots, i_{n-1};\\i_a\neq i_b \text{ for } a\neq b}} \tau_{i_1}\dots\tau_{i_{n-1}}\otimes u_{i_11}\dots u_{i_{n-1}n-1}\\
&= \sum_{j=1}^n \sum_{(i_1,\dots,i_{n-1}) \in I_j} \tau_{i_1}\dots\tau_{i_{n-1}}\otimes u_{i_11}\dots u_{i_{n-1}n-1},\end{aligned}$$ where $I_j =\{ (i_1,\dots, i_{n-1}) \in \{1,\dots, n\}^{n-1} \, | \, i_a \neq i_b \text{ for } a \neq b, i_s \neq j \text{ for all } s\}$ like in Lemma \[SO\]. For all $(i_1,\dots,i_{n-1}) \in I_j$, we know that $\tau_{i_1}\dots\tau_{i_{n-1}} = \tau_1\dots \check{\tau_j}\dots \tau_n$. Using $\tau_n = \tau_1\dots\tau_{n-1}$ and $\tau_i^2=1$, we get $\tau_1\dots \check{\tau_j}\dots \tau_n= \tau_j$ and thus $$\begin{aligned}
\sum_{j=1}^n \sum_{(i_1,\dots,i_{n-1}) \in I_j} &\tau_{i_1}\dots\tau_{i_{n-1}}\otimes u_{i_11}\dots u_{i_{n-1}n-1}\\
&= \sum_{j=1}^n \left(\tau_j \otimes \sum_{(i_1,\dots,i_{n-1}) \in I_j}u_{i_11}\dots u_{i_{n-1}n-1}\right).\end{aligned}$$ The equivalent formulation of Relation in Lemma \[SO\] yields $$\begin{aligned}
\tau'_1\dots\tau'_{n-1} = \sum_{j=1}^n \left( \tau_j \otimes \sum_{(i_1,\dots,i_{n-1}) \in I_j}u_{i_11}\dots u_{i_{n-1}n-1} \right)
= \sum_{j=1}^n \tau_j \otimes u_{jn} = \tau'_n.\end{aligned}$$ Summarising, the map $\alpha$ exists and is a \*-homomorphism. It is straightforward to check that $\alpha$ is unital and since $u$ is a corepresentation, $\alpha$ is coassociative.
Now, we show that $\alpha(C^*({\mathbb Z}_2^{n-1}))(1 \otimes C(SO_n^{-1}))$ is linearly dense in $C^*({\mathbb Z}_2^{n-1}) \otimes C(SO_n^{-1})$. It holds $$\begin{aligned}
\sum_{i=1}^n \alpha(\tau_i)(1 \otimes u_{ki}) = \sum_{j=1}^n \left( \tau_j \otimes \sum_{i=1}^n u_{ji}u_{ki} \right)= \sum_{j=1}^n \tau_j \otimes \delta_{jk} = \tau_k \otimes 1,\end{aligned}$$ thus $(\tau_k \otimes 1) \in \alpha(C^*({\mathbb Z}_2^{n-1}))(1 \otimes C(SO_n^{-1}))$ for $1 \leq k \leq n$. Since $\alpha$ is unital, we also get $1 \otimes C(SO_n^{-1}) {\subseteq}\alpha(C^*({\mathbb Z}_2^{n-1}))(1 \otimes C(SO_n^{-1}))$. By a standard argument, see for example [@SWe Section 4.2], we get that $\alpha(C^*({\mathbb Z}_2^{n-1}))(1 \otimes C(SO_n^{-1}))$ is linearly dense in $C^*({\mathbb Z}_2^{n-1}) \otimes C(SO_n^{-1})$.
It remains to show that the magic unitary matrix associated to $\alpha$ commutes with the adjacency matrix of $FQ_n$. We want to show that $\alpha$ preserves the eigenspaces of the adjacency matrix, i.e. $\alpha(E_\lambda) {\subseteq}E_\lambda \otimes C(SO_n^{-1})$ for all eigenspaces $E_\lambda$, then Theorem \[preserve\] yields the assertion. Since it holds $t_j = \tau_j \tau_n$, by Lemma \[ev\] we have eigenvectors $$\begin{aligned}
\hat{w}_{i_1 \dots i_{n-1}} = t_1^{i_1} \dots t_{n-1}^{i_{n-1}} = \begin{cases} \tau_1^{i_1} \dots \tau_{n-1}^{i_{n-1}} &\text{ for } \sum_{k=1}^{n-1} i_k \text{ even} \\ \tau_1^{1-i_1} \dots \tau_{n-1}^{1-i_{n-1}}&\text{ for } \sum_{k=1}^{n-1} i_k \text{ odd}\end{cases}\end{aligned}$$ corresponding to the eigenvalues $\lambda_{i_1 \dots i_{n-1}}$ as in Lemma \[EV\]. Using Lemma \[ES\], we see that the eigenspaces $E_{\lambda_k}$ are spanned by eigenvectors $\tau_1^{i_1} \dots \tau_{n-1}^{i_{n-1}}$ of word lengths $k$ or $n-k$, where we consider the word length with respect to $\tau_1, \dots, \tau_{n-1}$.
Let $1 \leq l \leq n-1$. By Lemma \[P\], we have for $i_1,\dots, i_l$, $i_a \neq i_b$ for $a \neq b$: $$\begin{aligned}
\alpha(\tau_{i_1} \dots \tau_{i_l}) = \sum_{\substack{j_1, \dots, j_{l};\\j_a\neq j_b \text{ for } a\neq b}} \tau_{j_1}\dots \tau_{j_l} \otimes u_{j_1 i_1} \dots u_{j_l i_l}.\end{aligned}$$ For $\tau_{j_1} \dots \tau_{j_l}$, where $j_s \neq n$ for all $s$, we immediately get that this is in the same eigenspace as $\tau_{i_1} \dots \tau_{i_l}$ since $\tau_{j_1} \dots \tau_{j_l}$ has the same word length as $\tau_{i_1} \dots \tau_{i_l}$. Take now $\tau_{j_1} \dots \tau_{j_l}$, where we have $j_s =n$ for some $s$. We get $$\begin{aligned}
\tau_{j_1}\dots \tau_{j_l} &= \tau_{j_1}\dots \check{\tau_{j_s}}\dots \tau_{j_l}\tau_n\\
&= \tau_{j_1}\dots \check{\tau_{j_s}}\dots \tau_{j_l}\tau_1 \dots \tau_{n-1}, \end{aligned}$$ which has word length $n-1-(l-1) = n-l$, thus it is in the same eigenspace as $\tau_{i_1} \dots \tau_{i_l}$. This yields $$\begin{aligned}
\alpha(E_\lambda) {\subseteq}E_\lambda \otimes C(SO_n^{-1}),\end{aligned}$$ for all eigenspaces $E_\lambda$ and thus $SO_n^{-1}$ acts on $FQ_n$ by Theorem \[preserve\].
Now, we can prove our main theorem.
\[main\] For $n$ odd, the quantum automorphism group of the folded $n$-cube graph $FQ_n$ is $SO_n^{-1}$.
By Lemma \[act\] we get a surjective map $C(G_{aut}^+(FQ_n)) \to C(SO_n^{-1})$. We have to show that this is an isomorphism between $C(SO_n^{-1})$ and $C(G_{aut}^+(FQ_n))$. Consider the universal action on $FQ_n$ $$\begin{aligned}
\beta: C^*({\mathbb Z}_2^{n-1}) \to C^*({\mathbb Z}_2^{n-1}) \otimes C(G_{aut}^+(FQ_n)).\end{aligned}$$ Consider $\tau_1, \dots ,\tau_n $ like in Lemma \[act\]. They have word length $n-2$ or $n-1$ with respect to $t_1, \dots, t_{n-1}$ and they form a basis of $E_{-n+2}$ by Lemma \[ES\]. Therefore, we get elements $x_{ij}$ such that $$\begin{aligned}
\beta(\tau_i)= \sum_{j=1}^n \tau_j \otimes x_{ji}\end{aligned}$$ by Corollary \[Eigen\]. Similar to [@hyperoctahedral] one shows that $x_{ij}$ fulfill Relations –. It remains to show that Relation holds. Applying $\beta$ to $\tau_n = \tau_1\dots \tau_{n-1}$ and using Lemma \[P\] yields $$\begin{aligned}
\sum_{j} \tau_j \otimes x_{jn} = \beta(\tau_n)&= \sum_{\substack{i_1, \dots ,i_{n-1};\\i_a\neq i_b \text{ for } a\neq b}}\tau_{i_1}\dots \tau_{i_{n-1}}\otimes x_{i_11}\dots x_{i_{n-1}n-1}\\
&=\sum_{j=1}^n \sum_{(i_1,\dots,i_{n-1}) \in I_j} \tau_{i_1}\dots\tau_{i_{n-1}}\otimes x_{i_11}\dots x_{i_{n-1}n-1}.\end{aligned}$$ As in the proof of Lemma \[act\], we have $\tau_{i_1}\dots\tau_{i_{n-1}}=\tau_j$ for $(i_1, \dots, i_{n-1}) \in I_j$ and we get $$\begin{aligned}
\sum_{j=1}^n \tau_j \otimes x_{jn} = \sum_{j=1}^n \left(\tau_j \otimes \sum_{(i_1,\dots,i_{n-1}) \in I_j}x_{i_11}\dots x_{i_{n-1}n-1}\right).\end{aligned}$$ We deduce $$\begin{aligned}
x_{jn} = \sum_{(i_1,\dots,i_{n-1}) \in I_j}x_{i_11}\dots x_{i_{n-1}n-1},\end{aligned}$$ which is equivalent to Relation by Lemma \[SO\]. Thus, we also get a surjective map $C(SO_n^{-1}) \to C(G_{aut}^+(FQ_n))$ which is inverse to the map $C(G_{aut}^+(FQ_n)) \to C(SO_n^{-1})$.
- It was asked in [@survey] by Banica, Bichon and Collins to investigate the quantum automorphism group of the Clebsch graph. Since the $5$-folded cube graph is the Clebsch graph we get $G_{aut}^+(\Gamma_{Clebsch}) = SO_5^{-1}$.
- The 3-folded cube graph is the full graph on four points, thus our theorem yields $S_4^+ = SO_3^{-1}$, as already shown in [@4points].
We do not have a similar theorem for folded cube graphs $FQ_n$ with $n$ even, since the eigenspace $E_{-n+2}$ behaves different in the odd case.
[^1]: The author is supported by the DFG project *Quantenautomorphismen von Graphen*. The author is grateful to his supervisor Moritz Weber for proofreading the article and for numerous discussions on the topic. He also wants to thank Julien Bichon and David Roberson for helpful comments and suggestions.
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abstract: 'In globular clusters, dynamical evolution produces luminous X-ray emitting binaries at a rate about 200 times greater than in the field. If globular clusters also produce SNe Ia at a high rate, it would account for much of the SN Ia events in early type galaxies and provide insight into their formation. Here we use archival HST images of nearby galaxies that have hosted SNe Ia to examine the rate at which globular clusters produce these events. The location of the SN Ia is registered on an HST image obtained before the event or after the supernova faded. Of the 36 nearby galaxies examined, 21 had sufficiently good data to search for globular cluster hosts. None of the 21 supernovae have a definite globular cluster counterpart, although there are some ambiguous cases. This places an upper limit to the enhancement rate of SN Ia production in globular clusters of about 42 at the 95% confidence level, which is an order of magnitude lower than the enhancement rate for luminous X-ray binaries. Even if all of the ambiguous cases are considered as having a globular cluster counterpart, the upper bound for the enhancement rate is 82 at the 95% confidence level, excluding an enhancement rate of 200. Barring unforeseen selection effects, we conclude that globular clusters are not responsible for producing a significant fraction of the SN Ia events in early-type galaxies.'
author:
- 'Pearce C. Washabaugh and Joel N. Bregman'
title: The Production Rate of SN Ia Events in Globular Clusters
---
Introduction
============
Not only are SN Ia events of interest as endpoints of stellar evolution, they have taken on a special importance because of their central role as standard candles. It was with these objects that the acceleration of the universe was first detected and they will likely play a central role in accurately measuring the acceleration parameters over cosmological time (e.g., @riess07 [@suzu12]). This greater accuracy requires not only a large number of SNe Ia, but a good understanding of the supernova events. An important part of the picture is the set of conditions that cause white dwarfs to undergo a SN Ia event.
Within galaxies, close binary systems are rare and they are generally not created through dynamical interaction but are formed with a sufficiently small separation. However, a different situation occurs in globular clusters because the dynamical interaction time is typically 10$^{{\rm 8}}$-10$^{{\rm 9}}$ yr. These interactions cause binaries to harden to the point that the constituent stars become close enough for mass transfer (e.g., @hut92 [@pool03]). When this occurs, the star that expands beyond its Roche lobe experiences Roche lobe overflow and subsequent mass transfer to the more compact object. When the compact object is a neutron star, we obtain a low mass X-ray binary (LMXB), which can be quite luminous in X-rays, making them easy to identify. Luminous X-ray binaries are seen in early-type galaxies, and many occur in globular clusters [@ang01; @saraz03; @irwin05; @kundu07; @siva07].
The rate enhancement of LMXBs in globular clusters may be a critical quantity when estimating the number of Ia SNe that might be anticipated in these systems [@ivan06; @pfahl09]. In the Milky Way, about 1/10 of luminous X-ray binaries are in globular clusters (typical X-ray luminosities exceeding $10^{36}$ erg sec$^{-1}$; e.g., @verb06), and if one sums the globular clusters by mass [@mand91] and compares it to the stellar mass for the Milky Way [@binney1998], one finds that LMXBs occur about 200 times more commonly than in an equivalent mass of field stars (the mass fraction in globular clusters is 0.05% based on these resources). A similar enhancement can be determined for early-type galaxies, which are the galaxies considered in our sample. There is a difference compared to the Milky Way, because at the distances of even nearby early-type galaxies, the binaries must be significantly more luminous in order to be detectable. These LMXBs have X-ray luminosities usually exceeding 10$^{37.5} erg sec^{-1}$ [@irwin05]. In these galaxies, about half of the X-ray binaries are in globular clusters [@saraz03; @siva07]. Globular clusters are typically more common in early-type galaxies, relative to the Milky Way, which can be expressed by the specific frequency of globular clusters, $S_N$ [@harris91]. The specific frequency is 0.5 for the Milky way, but is 2.6 for E/S0 in small groups and 5.4 for E/S0 galaxies in Virgo and Fornax [@harris91]; most of the galaxies in the Sivakoff sample were in the Virgo cluster with the specific frequency in the range 3.5-14. Using the data given in the Sivakoff sample, for a median $S_N$ = 5, an order of magnitude greater than the Milky Way, the stellar mass fraction in globular cluster stars is about 0.5%. For about half of the LMXBs to be in globular clusters, this implies an enhancement rate of about 200 for sources more luminous than $2 \times 10^{38}$ erg sec$^{-1}$ in the 0.3-10 keV band. We conclude that the enhancement rate of producing a LMXB in a globular cluster is about 200, for X-ray luminosities in excess of $10^{36}$ erg sec$^{-1}$
If the fraction of SNe Ia that are hosted by globular clusters was the same as the fraction of LMXBs hosted by globular clusters, we would expect to find about 20-50% of SN Ia events in globular clusters, which is the proposition that we test here. The differences between LMXBs and SN Ia progenitors are significant in that SN Ia events are associated with binaries involving a white dwarf (or two white dwarfs), rather than a neutron star (or black hole) plus a non-degenerate star, so the enhancement rate may be very different. However, the frequency of white dwarf mass-transfer binaries also appears to be enhanced in globular clusters, based on X-ray observations [@heink03; @pool06]. They are fainter than neutron star binaries (due to the difference in the depth of the potential well), but deep observations of nearby Galactic globular clusters indicates a large number of hard white dwarf binaries (Cataclysmic Variables). Since globular clusters appear to be factories for the production of mass-transfer binaries, it is only natural to wonder if they might produce a significant fraction of the SNe Ia seen in early-type galaxies. In this paper we investigate the frequency with which such nearby SNe Ia are associated with globular clusters. We accomplish this by examining the location of these supernovae on *Hubble Space Telescope* (HST) images that are sufficiently deep to detect globular cluster in external galaxies. In the next section we discuss the criteria by which we chose our targets. The following two sections give a summary of the results and notes on individual objects. The final section is a discussion of the results and implications for related efforts. Recently, but after our initial presentation of results [@wash09], @voss12 published a similar effort, which we discuss in more detail in the final section.
Target Selection
================
In the selection of targets, it is crucial to work with systems that have well-defined positions for the SN. Early-type galaxies, the host sample, often have thousands of globular clusters (GCs) so the chance that the SN position will overlap with a random GC (or background source) unassociated with the SN becomes significant if the positional uncertainty is $r > 1\arcsec$ (positional uncertainties were taken from the NASA Extragalactic Database). If there are GCs in the possible region that the SN went off and the error circle is large because of a high positional uncertainty, then the event can at best be considered ambiguous because of the chance of random superposition. Also, the region must have at least two guide stars whose positions are well known so that the HST image can be accurately aligned.
Another criteria is that the system has to be close enough that one can detect most of the stars that lie in GCs. This does not mean that all globular clusters be detected, as the less luminous clusters add little to the total number of globular cluster stars. For a luminosity function that is log normal, as found for the Milky Way and M31, the center of the Gaussian is at $M_V$ = $-$7.5 and with a dispersion of $\sigma$ = 1.25 (e.g., @binney1998). For these parameters, 73% of all globular cluster stars are in clusters brighter than $M_V$ =$ -$8 and 93% in clusters brighter than $M_V$ =$-$7. So, if they are brighter than about V = 25 mag, HST can obtain good photometric measurements in modest observing times. This requirement implies a distance modulus of 32.5 or less, or a distance limit of 32 Mpc, which we used as a selection criteria. Ground based images do not have the ability to reach the resolution and magnitude required for galaxies of this distance, making them inadequate for this project. We note that with sufficient S/N, the extended nature of most GCs will be detected at HST resolution [@jord05], subtending $\sim 0.1-0.2\arcsec$ in radius.
A third consideration is that the search region of the host galaxy must be suitable for the detection of GCs. A galaxy with a very high surface brightness will compromise the ability to detect GCs and to determine that they are extended. In practical terms, one needs to avoid regions where the galaxy surface brightness exceeds 17 magnitudes per square arcsecond, otherwise the S/N of a typical globular cluster ($M_V = -7.5$) in a typical exposure falls below the detection threshold (discussed further in section 4). Also, the density of GCs increases close to the core approximately like a de Vaucouleurs law, so a search region near the core has a much higher chance of superposition and such matches are given as ambiguous detections. For nearby early-type galaxies, these restrictions mean that there is significant incompleteness for Ia SNe within about 10-20$\arcsec$ of the galaxy center. Unfortunately, this requirement is often at odds with archival HST programs, which generally target the central part of galaxies, although there are a few directed programs to detect particular supernovae. There were some HST observations to follow the light curve of particular SNe, and when there were also pre-SNe images (or 3+ yr post images), we could identify the position very accurately and these objects have the smallest error circles, typically 0.3$\arcsec$ or less (see below).
Finally, enough time needs to have passed since the SN Ia event so that it is fainter than the globular cluster we are trying to study. The peak absolute magnitude of a SN Ia is about $-$18 (in either B, V, or I) and it has to become fainter than $-$6, fading by at least 12 magnitudes. Based on known light curves (e.g., @ries99 [@salv01; @fran02]) and extrapolation from the known dimming due to radioactive half lives (or using the SN 1987a data), a dimming of 12 magnitudes occurs in a little less than 3 years in the optical bands.
To sum up, the important factors governing the selection of targets are:
- Positional uncertainty ( $>$1 arc sec may be ambiguous)
- At least two guide stars in image
- Galaxy is within 32 Mpc
- SN Ia occurred within suitable location in galaxy: away from high surface brightness regions
- The images must have been taken before or at least three years after the SN event occurred.
- The host is an early-type galaxy
With these criteria, we searched all known historical Ia SNe through 2007 as listed in the IAU Central Bureau for Astronomical Telegrams,\
http://www.cbat.eps.harvard.edu/lists/Supernovae.html. This resulted in the 36 Ia SNe events listed in Tables 1 and 2.
Observational Results
=====================
Of the 36 galaxies that satisfied our target criterion, only 21 had met our observational criteria, including: the SN Ia fell within the HST footprint and not in a CCD gap (this is the primary criteria for excluding 15 events); the observation was taken either before the supernova event at least three years after the event; astrometric objects are present on the image; and the supernova position had adequate positional accuracy (Tables 1, 2). Aside from one STIS image, where SN 1996X in NGC 5061 was observed in the 50CCD mode (MIRVIS spectral element), the rest of the images were either from the ACS or the WFPC2 (Tables 3, 5, 6). For the WFPC2 and STIS materials, we used the data products from the *Hubble Legacy Archive*, while for the ACS data, we used the standard drizzled image that is one of the data products, available through MAST. See Table 3 for exposure times and filters.
To determine whether a SN Ia event corresponded to a globular cluster, we needed to align the position of the SN with the HST image by translating and rotating the image so that at least two guide stars matched with positions given in the 2MASS survey. There can be an inaccuracy in the absolute position of an HST image that can lead to improper identifications, so we pursued a few avenues to obtain the best registration of the SN position on the HST image. In a few cases, there was an HST image in which the SN was visible and another one, usually taken before the SN occurred. In these cases, an alignment to 0.1-0.3$\arcsec$ was achieved, which are our most accurate results (e.g., SN 2003hv in NGC 1201). When two HST images were not available, we had to rely on the accuracy of the optical discovery images, which are often quoted to an accuracy of less than 1$\arcsec$. In the discovery images, the positions for some of the SNe are measured relative to the center of the galaxy or to background point sources with known positions. We checked the accuracy of the galaxy center being used, and when HST images included the galaxy center, we could use this information to position the error circle location. More often, we performed astrometry corrections to the HST image for the alignment with the optical SN position. For the astrometry, we used the locations of stars in the field that occur in the 2MASS and the USNO-B1 catalogue, and the galaxy center, when available. The size of the resulting error circle is the incoherent sum of the errors from the SNe discovery position plus the astrometry uncertainty. See Table 4 for error circle radii.
A source is a non-detection if no point source is found within or on the error circle. A clear detection would be the result of a single object, detected at above 5$\sigma$, lying within the error circle, with a low probability that this occurs by chance from contaminating sources. An ambiguous result refers to when either (1) the source within the error circle is in the 3-5$\sigma\ $ range, (2) there are multiple objects within the error circle, (3) or when the density of sources is sufficiently high that a object can easily occur within the error circle by chance. With these criteria, in the sample of 21 objects, there were 16 clear non-detections and 5 that were ambiguous. A more detailed object-by-object discussion is given below. The limiting magnitudes (Table 4) are often dictated by the brightness of the stellar background, which was measured directly from the image local to the SN position. We calculated the limiting magnitudes by using these backgrounds, along with the exposure time, with the exposure time calculators provided at the HST site. The median limiting magnitude was 24.4 mag, with a median distance modulus of 31.6.
Object-By-Object Discussion
---------------------------
We describe each event in our sample, listed chronologically.
SN 1919A, NGC 4486: The SN positional uncertainty is 5.0$\arcsec$ since the SN occurred in 1919. This uncertainty is too large for our purposes.
SN 1960R, NGC 4382: None of the WFPC2 and ACS images contain the SN position.
SN 1961H, NGC 4564: The supernova occurs only 6$\arcsec$ from the galaxy center, so the galaxy surface brightness is high as is the number of globular clusters; chance coincidences are likely. There is a dim object in the error circle (radius=1.7$\arcsec$) in the ACS images. Due to the large number of sources, we consider this association to be ambiguous. The limiting magnitude is 23.0 in the F850LP band. See fig. 3
SN 1976K, NGC 3226: A very precise ground-based position was provided by [@klemola1986], with an uncertainty of only $\sim$0.1$\arcsec$. Using the ACS images, no object was found within the SN error circle (radius=0.4$\arcsec$) at our limiting magnitude of $25.1$ in the F814W band.
SN 1980i, NGC 4374: None of the WFPC2 and ACS images contain the SN position.
SN 1982W, NGC 5485: None of the WFPC2 and ACS images contain the SN position.
SN 1983G, NGC 4753: None of the WFPC2 and ACS images contain the SN position.
SN 1983O, NGC 4220: The region where the SN occurred in lies in the galactic disk and has a relatively large number of globular clusters. The SN has a positional uncertainty ($\approx$1$\arcsec$), thus our error circle (1.8$\arcsec$) encompassed several objects. This constitutes a high level of contamination, making it impossible to associate the SN position with a globular cluster. See fig. 2
SN 1986G, NGC 5128: The SN is off of the only ACS image, but lies on three WFPC2 images. There are at least four objects in the SN error circle $(radius = 1.7\arcsec)$ and the region is in an area of heavy dust extinction. The chance of super-position with unrelated sources is extremely high. The uncertainty of the SN position is high enough that it is impossible to identify the SN with an underlying optical counterpart. See fig. 4
SN 1991bg, NGC 4374: Three point sources in the WFPC2 and ACS images were used for astrometry and led to a good absolute positions for the HST images, but the uncertainty in the discovery images dominates the size of the error circle. There were no objects in the error circle (radius = 1.7$\arcsec$) in any images at our limiting magnitude of $24.4$.
SN 1991F, NGC 3485: Using the SN position given by [@gomez95], we determined that no WFPC2 or ACS images show a source at the location of the supernova to our limiting magnitude of $24.4$ in the F606W band.
SN 1992A, NGC 1380: The ACS images was taken $12 - 14$ years after the SN and the astrometric sources aligned well with their nominal positions, so a positional shift was is unnecessary. This SN has a higher than average positional uncertainty ($1.5\arcsec$), leading to an error circle a radius of $2.5\arcsec$. Despite this relatively large error circle, no optical counterpart was found within the circle to our limiting magnitude of $24.0$ in the F555W band.
SN 1994D, NGC 4526: Since the SN position is known to a high degree of accuracy (0.1$\arcsec$) we were able to constrain the error circle to only 0.4$\arcsec$. No objects were found in the error circle at the limiting magnitude of $23.2$ in the F850LP band.
SN 1995D, NGC 2962: No WFPC2 or ACS images contain the SN position.
SN 1996X, NGC 5061: The SN is visible in every WFPC2 image and appears very faint by 2000, the most recent WFPC2 image of the region. There is a STIS image of the region taken in 2002. We were able to align this image with the WFPC2 images so that the error circle has radius $0.2\arcsec$. The STIS image showed that there was no point source at the SN location to our limiting magnitude of 26.0 (the 50CCD setting was used).
SN 1996bk, NGC 5308: This is a flattened early-type galaxy in which the supernova lies in the midplane, about 20$\arcsec$ from the center and the surface brightness is moderately high. There are several objects within the error circle on the ACS image, and there would be multiple objects within the $2.0\arcsec$ error circle if it were place almost anywhere along the disk of the galaxy. Thus, there is severe contamination, making it impossible to determine if an optical object is associated with the supernova. See fig. 5
SN 1997X, NGC 4691: No HST images of this object exist.
SN 1999gh, NGC 2986: There are two WFPC2 images of this object. One shows a potential cluster at the location of the SN, but the other does not confirm this result, implying that this source is a cosmic ray. Our error circle has a radius of $0.9\arcsec$. The limiting magnitude is $25.3$ and the filter was F702W.
SN 2000cx, NGC 524: All WFPC2 images contain the SN when it was visible. The only ACS image, taken with the HRC, does not include the location of the SN. Thus there is no HST image of the region taken after the supernova has faded.
SN 2000ds, NGC 2768: No object appears within our error circle (radius = 2.0$\arcsec$) to a limiting magnitude of $24.2$ with the F658N.
SN 2001A, NGC 4261: The WFPC2 and ACS images were offset from each other by a few tenths of an arcsecond. Using the galactic center for astrometry and relative centering of the images, we found no optical counterpart inside the error circles (radius=1.2$\arcsec$) to our limiting magnitude of $26.2$ with the F606W.
SN 2001fu, M-03-23-011: There are no HST images of this object.
SN 2002hl, NGC 3665: No WFPC2 or ACS images of this object exist. There is a NICMOS image, however the SN does not lie within the image.
SN 2003gs, NGC 936: Using the ACS images, we were able to align the position of the SN with WFPC2 images taken 7 years before the SN, and another ACS image taken 2 years after so that the error circle had radius $0.2\arcsec$. None of these show any object within the SN error circle to the limiting magnitude of $24.6$ in the F555W.
SN 2003hv, NGC 1201: There are many ACS images of the supernova, but none of the region after the SN faded. Only one WFPC2 image was taken seven years before the SN. To find the SN region, we measured the offset of the SN from nearby point sources in the ACS image. We transferred this offset to the WFPC2 image. This yielded a very small error circle radius of $0.1\arcsec$. At that location, we found a dim object with a S/N $\approx$ 3-4 and an apparent magnitude of 22.6 in the F814W (Vegamag system); the exposure was only 160 seconds. Assuming $V-I=1.1$ and a distance modulus of 31.53, the object has an absolute magnitude of $M_I= -8.9$ or $M_V=-7.8$, which would be consistent with that of a typical globular cluster. See fig. 1
SN 2003hx, NGC 2076: No HST images have been taken of this object.
SN 2003if, NGC 1302: No HST images have been taken of this object.
SN 2004W, NGC 4649: The ACS image fails to show an object within the error circle (radius = 1.0$\arcsec$) at our limiting magnitude of 25.3 in the F850LP. The SN position did not fall on any WFPC2 images.
SN 2005cf, M-01-39-03: All HST images of the galaxy contain the SN location and show the SN before it faded. There are no images of the SN location after the SN faded.
SN 2005cn, NGC 5061: No HST images overlap with the SN position.
SN 2005cz, NGC 4589: In the ACS images, there were no objects within our error circle (radius = 0.9$\arcsec$) to a limiting magnitude of 25.1 in the F814W filter.
SN 2006dd, NGC 1316: There were no objects visible in our error circle (radius = 1.2$\arcsec$) at our limiting magnitude of 24.7 with the F814W. We note that the region is in an area of significant dust extinction, which might obscure a low optical luminosity cluster on the far side of the galaxy.
SN 2006dy, NGC 5587: There is a small patch of slightly brighter pixels within the error circle (radius =1.3$\arcsec$), but this is not a significant detection. The limiting magnitude is 25.3 in the F814W.
SN 2006E, NGC 5338: No HST images of this object exist.
SN 2007gi, NGC 4036: WFPC2 images include this SN. No optical counterpart was found within the error circle (radius = 0.9$\arcsec$) to our limiting magnitude of 21.8 using the F658N filter.
SN 2007on, NGC 1404: This target was observed with ACS images and from this data set we determined an error circle of radius = 1.0$\arcsec$ and a limiting magnitude of 25.0 with the F850LP filter. We concur with the thorough treatment of @voss08 that there is no optical counterpart at the location of the supernova. Initially, @voss08 claimed that the supernova had an X-ray precursor, but with improved data, they determined that the faint X-ray source is not aligned with the SN Ia event [@roel08].
Discussion and Conclusions
==========================
Our goal was to determine whether SN Ia events are produced through dynamical interactions in globular clusters, since globular clusters often host most of the close binary systems composed of a neutron star and a normal star. To address this, we examined 36 galaxies that have had SN Ia events and are sufficiently close that most globular clusters can be detected with HST images. A number of galaxies proved to be unsuitable in that the SN Ia event fell outside of the image, or the images include the bright supernova, which prevented us from detecting an underlying globular cluster (there was no pre-event image or an image after the supernova had faded). 21 of the galaxies fulfilled all of our selection criteria and were fully considered. Four of the objects had positional uncertainties too high to be of use, leading to multiple objects within the error circle. Of the systems with adequate data and astrometry, we found 16 SN events without a definitive optical counterpart. In only one event, SN 2003hv in NGC 1201, is there a possible optical counterpart found with low positional uncertainty. The optical counterpart is not a high S/N detection, so the surface brightness profile cannot be determined nor can a color be calculated. If the optical counterpart is at the distance of NGC 1201, it would have a magnitude of about $M_V=-7.7$, which is a typical globular cluster. A deeper observation would show whether this is a reliable source and if it has the colors and extent expected from a globular cluster.
We can consider how these non-detections and a possible detection relate to the number of detections that might be theoretically predicted. If globular clusters host SN Ia events at the same rate as the rest of the galaxy, we would not have expected any detections unless we used a sample size approximately 10 times larger. We take the conservative assumption that all of the ambiguous or contamination cases are not true associations of a GC with a SN Ia. Based on this assumption, one detection in our sample of $21$ systems will imply a rate enhancement of about 10 for SN Ia events in GCs over field populations. This calculation was made by dividing the total luminosity of the observed GCs by the total luminosities of their respective host galaxies. Total system GC luminosities were found by multiplying the number of GCs in each galaxy by the mean luminosity given by the Milky Way Globular Cluster Luminosity Function (see Table 7 for data on the GC systems and the data sources). When the number of globular clusters in a galaxy was not available in the literature, we estimated the number by counting the number of GCs in the systems using WFPC2 images.
Not all globular clusters could be detected with the available images due to the depth of the image, which is determined by a combination of the exposure time, filter used, and stellar background, which rises toward the center of the galaxy. Keep in mind that this method of estimation is inherently biased, as we only have access to data on the GC light that falls on the HST field of view. For each SN, we determined the local background in the image and calculated the limiting 4$\sigma$ detection threshold. Using this threshold, we calculated the accessible fraction of the stars given by the globular cluster luminosity function, using the luminosity function for the Milky Way and M31, corrected to the relevant wavelength band [@binney1998; @ashman1995]. The typical object has observations that included more than 95% of the globular cluster stars with only a few exceptions in which the observation time is short (less than 200 seconds; the median observation is 900 seconds) or the SN occurs close to the inner part of the galaxy where the central surface brightness is large. Due to a few objects where less than 60% of the globular cluster stars are detectable, 85% of globular cluster stars would have been detected. We include this incompleteness correction in formulating the statistics below.
This approach yielded similar numbers for galaxies with known globular cluster data @pfahl09. If none of the ambiguous cases have an optical counterpart, then there are 0 globular clusters at the locations of SNe Ia for 21 events. This result implies a correspondence rate (Poisson mean) of $\mu$ = 0.0 - 0.18 between SNe Ia and globular clusters, at the 95% confidence level (using the Wilson score interval, @wilson27). If one ambiguous object has a globular cluster counterpart (SN 2003hv is the most likely), then the implied correspondence rate is $\mu$ = 0.048, with a 95% confidence bound of $\mu$=0.01-0.23. A value for $\mu$ of 0.18 (the upper bound to the confidence range for zero coincidences) is a rate enhancement of $33$. A value for $\mu$ of 0.23 (the upper bound to the confidence range for one coincidence) is a rate enhancement of $42$.
For the ambiguous cases that we consider less likely, two detections would result in a correspondence rate of $\mu$=.095 with a 95% confidence bound of $\mu$=.03-.29 which will result in a maximum rate enhancement of $53$. Three detections would result in a correspondence rate of $\mu$=.14 with a 95% confidence bound of $\mu$=.05-.34 which will result in a maximum rate enhancement of $62$. Four detections would result in a correspondence rate of $\mu$=.14 with a 95% confidence bound of $\mu$=.07-.40 which will result in a maximum rate enhancement of $73$. Five detections would result in a correspondence rate of $\mu$=.24 with a 95% confidence bound of $\mu$=.10-.45 which will result in a maximum rate enhancement of $82$.
In a recent work, @voss12 also examined the relationship between SNe Ia and globular clusters. They included events out to a distance of 100 Mpc and performed both a literature search as well as using archival HST data. They did not restrict their study to early-type galaxies and most of the galaxies in their sample are spirals. There were 35 SNe Ia examined, with 18 systems close enough to identify most of the globular clusters. For the remaining 17, only the brightest globular clusters would have been visible. They do not find a globular cluster counterpart to any of the SNe Ia. Unlike our study, no ambiguous cases are reported. They present results as a function of galaxy type, so for the early-type galaxies with normal SN Ia events, the subsample corresponding to our sample, they have an effective 7.25 objects, and they place an upper limit to the enhancement rate of 46-90 (90% confidence) and 82-163 (99% confidence). These results are consistent with ours, although we have a larger sample of SN Ia events in early-type galaxies.
Our limits on the rate enhancement is much less than that for luminous X-ray binaries ($> 10^{35} erg s^{-1}$), where the enhancement for globular clusters in early type galaxies is about 200 (see §1). If the rate enhancement of 200 also applied to SN Ia events, approximately one-third of these supernovae would be associated with globular clusters. We can compare our results to those from theoretical modeling. In their theoretical stellar evolution modeling of white dwarfs in binary systems, @ivan06 find that globular clusters can produce as many SN Ia events as the rest of the galaxy. For the galaxy and globular cluster parameters chosen, this corresponds to an enhancement rate of about 200. @shara02 used N-body simulations and found that the double-degenerate channel for SN Ia formation would be enhanced in globular clusters by a factor of 10 or more. @pfahl09 discuss this issue, and based on a variety of calculations, they suggest an enhancement of 10 for the production of SN Ia events in globular clusters. Our results are in conflict with a high enhancement rate, such as suggested by @ivan06, but are consistent within a rate enhancement of an order of magnitude.
The rate at which SNe are associated with globular clusters may depend on radius within the galaxy, which can lead to a bias in our results. The globular cluster distribution does not necessarily follow the light of the galaxy in the sense that there are relatively fewer globular clusters in the inner region, as is the case in the giant elliptical galaxy NGC 4486 (M 87), where the core radius for the distribution is 56$\arcsec$, compared to 6$\arcsec$ for the underlying stellar component [@kundu99]. If the relative deficit of globular clusters in the inner part of early-type galaxies were due to dynamical destruction (e.g., @hut92), a potential SN Ia progenitor, originally created in a globular cluster, would appear as part of the underlying stellar distribution. Because the *HST* observations exist primarily for the inner 1$^{\prime}$ of the galaxies in our sample, this could lead to an underestimation of the true association rate between globular clusters and SNe.
While this might seem like a plausible scenario, observational tests of globular cluster destruction do not provide support. Destruction mechanisms should act at different rates for low and high mass globular clusters, which should modify the globular cluster luminosity function [@agui88; @murray92]. This would be manifest as a radial change in the luminosity function, but none is seen [@kundu99]. The other element needed to cause a bias is that globular clusters are destroyed after SN Ia progenitors are formed. However, the typical time since destruction is several Gyr while the binary formation mechanism in globular clusters occurs continuously [@ivan06] and on the much shorter relaxation time of 10$^8$-10$^9$ yr. We conclude that there is not an obvious central galaxy bias for finding GC-SN associations, but a more definitive statement could be made by accumulating statistics on SN Ia events occurring more than 1$^{\prime}$ from the galaxy center, for which there is currently little *HST* data.
If globular clusters had proven to be SN Ia factories, it would have solved a problem with the production rate of these systems. In his review, @maoz10 summarizes the arguments that for SN Ia in old populations, the double-degenerate channel is the likely path. However, the present-day rate for the double-degenerate path in the field stars of early-type galaxies falls an order of magnitude short of the observed rate of SN Ia events. Now that we have shown that globular clusters are not responsible for most SN Ia events (barring unforeseen bias effects), the difficulty of forming enough of these events persists.
If the enhancement rate for SNe Ia in globular clusters is about 10, it should be possible to improve the size of the sample by at least a factor of two and confirm this prediction. Within our sample, a dozen targets have no image data, and with current instruments on HST, the needed imaging can be accomplished with only modest time investments. Other targets will become available in the near future as the SN Ia events fade. Perhaps the most urgent observation is to resolve the tentative association of a supernova with an optical counterpart in NGC 1201. As this has only a single 160 second exposure, substantially improved images can be obtained easily and two-color images would determine the color and morphology of the object. This is a program worth pursuing, because identifying even a single SN Ia with a globular cluster gives us the age and metallicity of the progenitor [@pfahl09], information that is otherwise unobtainable. Such information not only gives insight into the formation mechanism of SNe Ia for old populations, it might help reduce systematics when using SNe Ia for cosmology.
Acknowledgements
================
The authors would like to thank Mario Hamuy, Maximilian Stritzinger, Carlos Contreras, Ted Dobosz, Stefano Benetti, Mark Philips, and Chris Smith for providing us with unpublished images and information on the supernovae used in this work. Also, we would like to thank Jimmy Irwin, Lou Strolger, and Roger Chevalier for their comments and insight. This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. Also, this research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This work has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory. Finally, we would like to acknowledge financial support from NASA for these activities through a ADP grant. We gratefully acknowledge financial support for this project from NASA through an ADAP grant.
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[rrrrrrr]{} 1961 H& NGC4564& 12h36m27.265s &11d26m24.62s&0&5N& Ea\
1976 K& NGC3226&10h23m28.826s &19d54m15.14s&28E&21N& E2: pec\
1983 O& NGC4220& 12h16m13.204s&47d52m45.16s&19E& 15S& SA0$^{+}$(r)\
1986 G& NGC5128& 13h25m36.458s&-43d01m53.65s&120E&60S& S0 pec\
1991 bg& NGC4374& 12h25m03.698s&12d52m15.64s&2E&57S& E1\
1991 F& NGC3458& 10h56m15.099s &57d06m54.34s&13.4E&12.1S& SAB0\
1992 A& NGC1380&03h36m27.414s &-34d57m31.39s&3W&62N& SA0\
1994 D& NGC4526&12h34m02.395s &07d42m05.70s&9W&7N & SAB0$^{0}$(s)\
1996 X& NGC5061&13h18m01.130s &-26d50m45.30s&52W&31S& E0\
1996 bk& NGC5308& 13h46m57.980s&60d58m12.90s&17.9W&10.5S& S0\
1999 gh& NGC2986& 09h44m19.750s &-21d16m25.00s&52W&15.8N& E2\
2000 ds& NGC2768& 09h11m36.240s &60d01m42.20s&9.1W&32.1S& E6\
2001 A& NGC4261&12h19m23.010s&05d49m40.50s&3.1W&10.7N& E2-3\
2003 gs& NGC 936&02h27m38.360s &-01d09m35.40s&13.4E&14.6S& SB0$^{+}$(rs)\
2003 hv& NGC1201& 03h04m09.320s &-26d05m7.5s&17.2E&56.7S& SA0$^{0}$(r)\
2004 W& NGC4649&12h43m36.520s &11d31m50.80s&51.6W&78.7S& E2\
2005 cz& NGC4589& 12h37m27.850s&74d11m24.50s&12E&6S& E2\
2006 dd& NGC1316&03h22m41.620s &-37d12m13.00s&0.3W&16N& SAB0$^{0}$(s) pec\
2006 dy& NGC5587& 14h22m11.450s &13d55m14.20s&10E&9N& S0/a\
2007 gi& NGC4036 &12h01m23.420s &61d53m33.80s &23E&11S& S0$^{-}$\
2007 on& NGC1404&03h38m50.900s & -35d34m30.00s&12E&68N& E1\
[rrrrrrr]{} 1919 A& NGC4486 (M87)& 12h30m48.64s &12d25m04.2s&15W & 100N & cD-1 pec\
1960 R& NGC4382&12h25m24.834s &18d09m19.35s&8E&132S& SA0$^{+}$(s) pec\
1980 I& NGC4374& 12h25m38.117s &12d53m23.50s&454E&20N& E1\
1982 W& NGC5485& 14h07m08.958s&55d01m07.38s&19W&63N& SA0 pec\
1983 G& NGC4753& 12h52m20.787s &00d39m36.72s&17W&14S& I0\
1995 D& NGC2962& 09h40m54.753s &05d08m26.16s&11E& 90.5S & (R)SAB0$^{+}$(rs)\
1997 X& NGC4691& 12h48m14.280s &-03d19m58.50s&7.2E&0.3N& (R)SB0/a(s) pec\
2000 cx& NGC 524& 01h24m46.150s &09d30m30.90s&23.0W&109S& SA0$^{+}$(rs)\
2001 fu& M-03-23-11& 08h52m16.580s&-17d44m29.80s&25.1W&10.6N& (R’)SB0$^{0}$(s)\
2002 hl& NGC3665& 11h24m40.120s &38d46m03.00s&42.0W&16.6N& SA0$^{0}$(s)\
2003 hx& NGC2076& 05h46m46.970s&-16d47m00.60s&5.2W&2.6S& S0$^{+}$\
2003 if& NGC1302&03h19m52.610s &-26d03m50.50s&19.3E&11.9S& (R)SB0/a(r)\
2005 cf& M-01-39-03&15h21m32.210s &-07d24m47.50s &15.7W&123N& S0 pec\
2005 cn& NGC5061&13h18m00.460s &-26d48h33.10s&62W&103N& E0\
2006 E& NGC5338 & 13h53m28.650s&05d12m22.80s &31.1E&6.3S& SB0\
[rrrrrrr]{} 1961 H& NGC4564&ACS& WFC &jfs22011&1120&F850LP\
1976 K& NGC3226&ACS& WFC &j6jt02y2q&350&F814W\
1983 O& NGC4220& ACS & WFC &j8mx74010&700&F658N\
1986 G& NGC5128& WFPC2 &1 &u4100206b&753&F555W\
1991 bg& NGC4374& WFPC2 & 1&u34k0103t&4320&F814W/F658N/F547M\
1991 F& NGC3458& WFPC2 & 1&u67n3202b&160&F606W\
1992 A& NGC1380& WFPC2 &1 &u3gh0702b &2600&F555W\
1994 D& NGC4526&ACS & WFC &jfs08011&1120&F850LP\
1996 X& NGC5061&STIS& MIRVIS &o6fz12010&900&MIRVIS\
1996 bk& NGC5308& ACS& WFC&j9ew04030&3480&F814W\
1999 gh& NGC2986& WFPC2&1 &u3cm1402r&700&F702W\
2000 ds& NGC2768& ACS & WFC&j6jt08011&1700&F658N\
2001 A& NGC4261& ACS &HRC&j8da04031 &782&F606W\
2003 gs& NGC 936& WFPC2&1&u2jf0303t&140&F555W\
2003 hv& NGC1201& WFPC2 &1& u2tv 3501b&160&F814W\
2004 W& NGC4649& ACS & WFC& j8fs03011&1120&F850LP\
2005 cz& NGC4589& ACS & WFC&j9ar03030&1600&F814W\
2006 dd& NGC1316& ACS & WFC&j6n201030&2200&F814W\
2006 dy& NGC5587& WFPC2 &1&u41v0704m&890&F814W/F450W\
2007 gi& NGC4036 & WFPC2 &1&u3780303b&700&F658N\
2007 on& NGC1404&ACS & WFC&j90x3010&1130&F850LP\
[rrrr]{} 1961 H& NGC4564&1.7& 23.0\
1976 K& NGC3226&.1& 25.1\
1983 O& NGC4220& 1.8& 22.3\
1986 G& NGC5128& 1.7 & 23.2\
1991 bg& NGC4374& 1.7 & 24.4\
1991 F& NGC3458& 1.0 & 24.4\
1992 A& NGC1380& 2.5 &24.0\
1994 D& NGC4526&.4 & 23.2\
1996 X& NGC5061&.2& 26.0\
1996 bk& NGC5308& 2.0& 25.3\
1999 gh& NGC2986& .9& 25.3\
2000 ds& NGC2768& 2.0 & 24.2\
2001 A& NGC4261& 1.2 &26.2\
2003 gs& NGC 936& .2&24.6\
2003 hv& NGC1201& .1 &22.6\
2004 W& NGC4649& 1.0& 25.3\
2005 cz& NGC4589& .9 & 25.1\
2006 dd& NGC1316& 1.2 & 24.7\
2006 dy& NGC5587& 1.3 &25.3\
2007 gi& NGC4036 & .9 &21.8\
2007 on& NGC1404& 1.0 & 25.0\
[rrrr]{} 1961 H& NGC4564& J8FS22ETQ &7/7/03\
1976 K& NGC3226 &J6JT02021,J6JT02011& 3/8/2003, 3/8/2003\
1980 I& NGC4374 &J8FS06O0Q, J8FS06011, J8FS06021 &1/21/03, 1/21/03,1/21/03\
1982 W& NGC5485 &J8MXC5010, J8MXC5FDQ &11/8/03, 11/8/03\
1986 G& NGC5128& None &N/A\
1991 F& NGC3458& None &N/A\
1991 bg& NGC4374 &J8FS06O0Q, J8FS06011, J8FS06021 &1/21/03, 1/21/03,1/21/03\
1992 A& NGC1380 &J90X04010, J90X04020, J90X04DJQ &9/6/04, 9/7/04, 9/6/04\
1994 D& NGC4526 &J8FS08LNQ& 7/12/03\
1996 X& NGC5061 &O6FZ12010 (STIS image, no ACS available)& 5/5/02\
1999 gh& NGC2986 &None& N/A\
2000 ds& NGC2768 &J6JT08011, J6JT08021, J8DT02011, & 1/14/03, 1/14/03, 5/31/02,\
2001 A& NGC4261 &J9OB01010, J9OB01020& 12/25/06, 12/25/06\
2003 gs& NGC 936 &J8Z457011& 9/29/04\
2003 hv& NGC1201& J8Z465021& 7/12/04\
2004 W& NGC4649 &J8FS03011, J8FS03021& 6/17/03, 6/17/03\
2005 cz& NGC4589 &J9AR03011, J9AR03021, J9AR03031& 11/11/06, 11/1106, 11/11/06\
2006 dd& NGC1316 &J6N202010, J6N201010, J6N201030& 3/4/03, 3/7/03, 3/7/03\
2006 dy& NGC5587 &None& N/A\
2007 gi& NGC4036 &None& N/A\
2007 on& NGC1404& J90X03010, J90X03020, J90X03AXQ & 9/10/04, 9/10/04, 9/10/4\
[rrrr]{} 1961 H& NGC4564 &U3HY0301B& 6/18/99\
1976 K& NGC3226 &None &N/A\
1980 I& NGC4374& U27L4P01B, U34K0103B, U34K0106B &4/4/1994, 3/4/96, 4/3/96\
1982 W& NGC5485 &None &N/A\
1986 G& NGC5128 &U4100201B, U4100206B, U4100209B& 10/1/98,10/1/98,10/1/98\
1991 F& NGC3458 &U67N3202B& 6/24/01\
1991 bg& NGC4374 &U27L4P01B, U34K0103B, U34K0106B &4/4/1994, 3/4/96, 4/3/96\
1992 A& NGC1380& U3GH0702B &9/4/96\
1994 D& NGC4526 &U3HY0801B& 6/18/99\
1996 X& NGC5061& U3GH0801B, U3GH0803B, U3GH0805B& 9/1/97, 9/1/97, 9/1/97\
1999 gh& NGC2986 &U3CM1402B &1/18/99\
2000 ds& NGC2768 &U3M71605B, U3M71608B, U2TV1801B& 20/5/98, 20/5/98, 4/12/95\
2001 A& NGC4261 &U2I50205B &13/12/94\
2003 gs& NGC 936 &None &N/A\
2003 hv& NGC1201& U2TV3501B& 27/12/96\
2004 W& NGC4649 &None& N/A\
2005 cz& NGC4589 &U2BM0B03B, U3CM7302B, U67G7701B &14/05/94, 1/10/99, 6/23/01\
2006 dd& NGC1316 &None &N/A\
2006 dy& NGC5587 &U41V0702B &13/7/97\
2007 gi& NGC4036 &U3780303B &12/5/97\
2007 on& NGC1404 &U34M0408B, U34M0402B, U29R5J01B& 4/3/96, 4/3/96, 12/06/95\
[rrrr]{} 1961 H& NGC4564 & 2700& @kiss97\
1976 K& NGC3226 & 480 & @kiss97\
1986 G& NGC5128 & 1500& @har98\
1991 bg& NGC4374& 3040& @kiss97\
1992 A& NGC1380& 2300& @kiss971\
1996 X& NGC5061& 84& @geb99\
2000 ds& NGC2768 & 118& @kun01b\
2001 A& NGC4261 & 321& @gio05\
2003 hv& NGC1201& 77& @kun01b\
2004 W& NGC4649 & 5100& @kiss97\
2005 cz& NGC4589 & 179 & @kun01a\
2006 dd& NGC1316 & 1496& @goo04\
2007 on& NGC1404 & 138& @geb99\
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abstract: 'The multiyear problem of a two-body system consisting of a Reissner-Nordström black hole and a charged massive particle at rest is here solved by an exact perturbative solution of the full Einstein-Maxwell system of equations. The expressions of the metric and of the electromagnetic field, including the effects of the electromagnetically induced gravitational perturbation and of the gravitationally induced electromagnetic perturbation, are presented in closed analytic formulas.'
address:
- 'Istituto per le Applicazioni del Calcolo “M. Picone”, CNR I-00161 Rome, Italy'
- |
International Center for Relativistic Astrophysics - I.C.R.A.\
University of Rome “La Sapienza”, I-00185 Rome, Italy
- 'Physics Department, University of Rome “La Sapienza”, I-00185 Rome, Italy'
author:
- 'D. Bini'
- 'A. Geralico'
- 'R. Ruffini'
title: 'On the equilibrium of a charged massive particle in the field of a Reissner-Nordström black hole'
---
,
,
Einstein-Maxwell systems ,black hole physics 04.20.Cv
The study of a massive charged particle in equilibrium in a Reissner-Nordström black hole background presents a variety of conceptual issues still widely open after more than twenty years of research, ranging from the classical aspects of general relativity to the quantum aspects of black hole tunneling processes (see e.g. Parikh and Wilczek [@PW]).
The problem of the interaction of a charged particle, neglecting its mass contribution, with a Reissner-Nordström black hole was addressed by Leaute and Linet [@leaute]. They extended previous results obtained in the case of a Schwarzschild spacetime by Hanni [@hannijth], Cohen and Wald [@CoW], Hanni and Ruffini [@HR] and Linet [@linet] himself. Their study was done in the test field approximation neglecting the backreaction both of the mass and of the charge of the particle on the background electromagnetic and gravitational fields.
We here approach the complete problem of a massive charged particle of mass $m$ and charge $q$ at rest in the field of a Reissner-Nordström black hole with mass $\mathcal{M}$ and charge $Q$. The full Einstein-Maxwell system of equations are solved taking into account the backreaction on the background fields due to the presence of the charged massive particle. The source terms of the Einstein equations contain the energy-momentum tensor associated with the particle’s mass, the electromagnetic energy-momentum tensor associated with the background field as well as additional interaction terms, first order in $m$ and $q$. Such terms are proportional to the product of the square of the charge $Q$ of the background geometry and the mass $m$ of the particle ($\sim Q^2m$) and to the product of the charges of both the particle and the black hole ($\sim qQ$). These terms originate from the electromagnetically induced gravitational perturbation” [@jrz2]. On the other hand, the source terms of the Maxwell equations contain the electromagnetic current associated with the particle’s charge as well as interaction terms proportional to the product of the black hole’s charge $Q$ and the mass $m$ of the particle ($\sim Qm$), originating the gravitationally induced electromagnetic perturbation” [@jrz1].
We summarize here the main results based on the first order perturbation approach formulated by Zerilli [@Zerilli] using the tensor harmonic expansion of both the gravitational and electromagnetic fields. Details will be found in [@bgrPRD].
The Reissner-Nordström black hole metric is given by $$\begin{aligned}
\label{RNmetric}
ds^2&=&- f(r)dt^2 + f(r)^{-1}dr^2+r^2(d\theta^2 +\sin ^2\theta d\phi^2)\ ,
\nonumber\\
f(r)&=&1 - \frac {2\mathcal{M}}{r}+\frac{Q^2}{r^2}\ ,\end{aligned}$$ with associated electromagnetic field $$\label{RNemfield}
F=-\frac{Q}{r^2}dt\wedge dr\ ,$$ and the horizon radii are $r_\pm={\mathcal M}\pm\sqrt{{\mathcal M}^2-Q^2}={\mathcal M}\pm\Gamma$. We consider the case $ |Q|\leq {\mathcal M} $ and the region $r>r_+$ outside the outer horizon. The extreme” charged hole corresponds to $|Q|={\mathcal M}$.
The particle is assumed to be at rest at the point $r=b $ on the polar axis $\theta=0$. The only nonvanishing components of the stress-energy tensor and of the current density are given by $$\begin{aligned}
\label{sorgenti}
T_{{00}}^{\rm{part}}&=&{\frac {m}{2\pi {b}^{2}}}f(b)^{3/2}\delta \left( r-b \right) \delta \left( \cos \theta -1 \right)\ ,\nonumber\\
J^{{0}}_{\rm{part}}&=&{\frac {q}{2\pi {b}^{2}}}\delta \left( r-b \right) \delta \left( \cos \theta -1 \right)\ ,\end{aligned}$$ and the combined Einstein-Maxwell equations are thus $$\begin{aligned}
\label{EinMaxeqs}
\tilde G_{\mu \nu }&=&8\pi \left(T_{\mu \nu }^{\rm{part}} + \tilde T_{\mu \nu }^{\rm em}\right)\ ,\nonumber\\
\tilde F^{\mu \nu }{}_{;\,\nu }&=& 4\pi J^{\mu }_{\rm{part}}\ , \quad {}^* \tilde F^{\alpha\beta}{}_{;\beta}=0\ .\end{aligned}$$ The quantities denoted by a tilde refer to the total electromagnetic and gravitational fields, to first order of the perturbation $$\begin{aligned}
\label{pertrelations}
\tilde g_{\mu \nu }&=&g_{\mu \nu } + h_{\mu \nu }\ ,\nonumber\\
\tilde F_{\mu \nu }&=&F_{\mu \nu }+ f_{\mu \nu }\ ,\nonumber\\
\tilde T_{\mu \nu }^{\rm em}&=&\frac1{4\pi}\left[\tilde g^{\rho \sigma }\tilde F_{\rho \mu }\tilde F_{\sigma \nu } - \frac14\tilde g_{\mu \nu }\tilde F_{\rho \sigma }\tilde F^{\rho \sigma }\right]\ ,\nonumber\\
\tilde G_{\mu \nu }&=&\tilde R_{\mu \nu }-\frac12\tilde g_{\mu \nu }\tilde R\ .\end{aligned}$$ Note that the covariant derivative operation makes use of the perturbed metric $\tilde g_{\mu \nu }$. The corresponding quantities without a tilde refer to the background Reissner-Nordström metric (\[RNmetric\]) and electromagnetic field (\[RNemfield\]). Following Zerilli’s [@Zerilli] procedure we expand the fields $h_{\mu \nu }$ and $f_{\mu \nu }$ as well as the source terms (\[sorgenti\]) in tensor and scalar harmonics respectively (see Tables I, II, III and V in Ref. [@Zerilli]). The perturbation equations are then obtained from the system (\[EinMaxeqs\]), keeping terms to first order in the mass $m$ of the particle and its charge $q$. The axial symmetry of the problem about the $z$ axis ($\theta=0$) allows to put the azimuthal parameter equal to zero in the expansion, leading to a great simplification. Furthermore, it is sufficient to consider only electric-parity perturbations, since there are no magnetic sources [@jrz2; @jrz1; @Zerilli].
The Regge-Wheeler gauge [@ReggeW] leads to the following set of gravitational and electromagnetic perturbation functions: $K$, $H_0$, $H_2$, $\tilde f_{{01}} $ and $\tilde f_{{02}}$. The Einstein-Maxwell field equations (\[EinMaxeqs\]) give rise to the following system of radial equations for values $l\geq2$ of the multipoles $$\begin{aligned}
\label{eq1RN}
0&=&{e^{2\nu}} \left[ 2K{}''-\frac2rW{}'+\left(\nu{}'+\frac6r\right) { K{}'}- 4\left(\frac1{r^2}+\frac{\nu{}'}r\right)W \right]-\frac{2\lambda e^{\nu}}{r^2}(W+K)\nonumber\\
&& -2{\frac {{Q}^{2}{e^{\nu}}W }{{r}^{4}}}-4{\frac{Q{e^{\nu}}\tilde f_{{01}}}{{r}^{2}}}+A_{{00}}\ , \nonumber\\
\label{eq2RN}
0&=&\frac 2rW{}'-\left(\nu{}'+\frac2r\right)K{}' -\frac{2\lambda e^{-\nu}}{r^2}(W-K)- 2{\frac{{Q}^{2}{e^{-\nu}}W }{{r}^{4}}}+4{\frac {Q{e^{-\nu}}\tilde f_{{01}} }{{r}^{2}}}\ ,\nonumber\\
\label{eq3RN}
0&=&K{}''+\left(\nu{}'+\frac2r\right)K{}'- W{}'' -2\left(\nu{}'+\frac1r\right)W{}'\nonumber\\
&&+\left(\nu{}''+{\nu{}'}^2+\frac{2\nu{}'}r\right)(K-W) -2{\frac{{Q}^{2}{e^{-\nu}}K}{{r}^{4}}}
+\frac{4Q{e^{-\nu}}}{{r}^{2}} \tilde f_{{01}}\ ,\nonumber\\
\label{eq4RN}
0&=&-W{}' + K{}' -\nu{}' W +4{\frac {Q{e^{-\nu}}\tilde f_{{02}} }{{r}^{2}}}\ ,\nonumber\\
\label{eq5RN}
0&=&\tilde f_{{01}}{}'+\frac2{r}\tilde f_{{01}} -{\frac {l\left( l+1 \right) {e^{-\nu}}\tilde f_{{02}} }{{r}^{2}}}-{\frac {Q}{{r}^{2}}}K{}' +4\pi v\ ,\nonumber\\
\label{eq6RN}
0&=&\tilde f_{{01}} - \tilde f_{{02}}{}'\ ,\end{aligned}$$ where $\lambda=\frac12 \left( l-1 \right) \left( l+2 \right)$, $H_0= H_2\equiv W$ and $e^{\nu}=f(r)$ is Zerilli’s notation; a prime denotes differentiation with respect to $r$. The quantities $$\label{sorgexp}
A_{{00}}=8\sqrt{\pi} \frac{m\sqrt {2l+1}}{b^2}f(b)^{3/2}\delta \left( r-b \right)\ , \qquad
v=\frac1{2\sqrt{\pi}} \frac{q\sqrt {2l+1}}{b^2}\delta \left( r-b \right)\$$ come from the expansion of the source terms (\[sorgenti\]).
We have a system of $6$ coupled ordinary differential equations for $4$ unknown functions: $K$, $W$, $\tilde f_{{01}} $ and $\tilde f_{{02}}$. The compatibility of the system requires that the following stability condition holds $$\label{bonnoreqcond}
m=qQ\frac{b f(b)^{1/2}}{{\mathcal M}b-Q^2}\ ,$$ involving the black hole and particle parameters as well as their separation distance $b$. This condition coincides with the equilibrium condition for a test particle of mass $m$ and charge $q$ in the field of a Reissner-Nordström black hole given by Bonnor [@bonnor]. There he simply considered the classical expression for the equation of motion of the particle $$m U^\alpha \nabla_\alpha U^\beta =q F^\beta{}_\mu U^\mu\ ,$$ with 4-velocity $U =f(r)^{-1/2}\partial_t$, neglecting all the feedback terms, and obtained Eq. (\[bonnoreqcond\]) as the equilibrium condition. The coincidence of these results is quite surprising, since our gravitational and electromagnetic fields including all the feedback terms are quite different from those used by Bonnor.
If the black hole is extreme”, then from Eq. (\[bonnoreqcond\]) follows that also the particle must have $q/m=1$, and equilibrium exists independent of the separation. In the general non-extreme case $Q/\mathcal{M}<1$ there is instead only one position of the particle which corresponds to equilibrium, for any given value of the charge-to-mass ratios of the bodies. In this case the particle charge-to-mass ratio must satisfy the condition $q/m>1$.
We now give the general expression for both the perturbed gravitational and electromagnetic fields in closed analytic form by summing over all multipoles of the Zerilli expansion [@bgrPRD]. The perturbed metric is given by $$\label{lineelemnonextr}
d{\tilde s}^2=-[1-{\bar {\mathcal H}}]f(r)dt^2 + [1+{\bar {\mathcal H}}][f(r)^{-1}dr^2+r^2(d\theta^2 +\sin ^2\theta d\phi^2)]\ ,$$ where $$\label{barH}
{\bar {\mathcal H}}=2\frac{m}{br}f(b)^{-1/2}\frac{(r-{\mathcal M})(b-{\mathcal M}) -\Gamma^2\cos\theta}{{\bar {\mathcal D}}}\ ,$$ with $$\label{barD}
{\bar {\mathcal D}} = [(r-{\mathcal M})^2+(b-{\mathcal M})^2 - 2(r -{\mathcal M})(b-{\mathcal M})\cos\theta- \Gamma^2\sin^2\theta]^{1/2}\ .$$ Note that in the extreme case $Q/\mathcal{M}=q/m=1$ this solution reduces to the linearized form of the well known exact solution by Majumdar and Papapetrou [@maj; @pap] for two extreme Reissner-Nordström black holes. The asymptotic mass measured at large distances by the Schwarzschild-like behaviour of the metric of the whole system consisting of black hole and particle is given by $$M_{\rm eff}={\mathcal M}+m+E_{\rm int}\ ,$$ where the interaction energy turns out to be $$E_{\rm int}=-m\left[1-\left(1-\frac{{\mathcal M}}{b}\right)f(b)^{-1/2}\right]\ .$$ It can be shown that this perturbed metric is spatially conformally flat; moreover, the solution remains valid as long as the condition $|{\bar {\mathcal H}}|\ll1$ is satisfied.
The nonvanishing components of the perturbed electric field are given by $$\begin{aligned}
\label{Zeremtensorpertnonextr}
E_r&=&\frac{q}{r^3}\frac{{\mathcal M}r-Q^2}{{\mathcal M}b-Q^2}\frac1{{\bar {\mathcal D}}}\bigg\{-\bigg[{\mathcal M}(b-{\mathcal M})+\Gamma^2\cos\theta\nonumber\\
&&+[(r-{\mathcal M})(b-{\mathcal M})-\Gamma^2\cos\theta]\frac{Q^2}{{\mathcal M}r-Q^2}\bigg]\nonumber\\
&&+\frac{r[(r-{\mathcal M})(b-{\mathcal M})-\Gamma^2\cos\theta]}{{\bar {\mathcal D}}^2}[(r-{\mathcal M}) -(b-{\mathcal M})\cos\theta]\bigg\}\ ,
\nonumber\\
E_{\theta}&=&q\frac{{\mathcal M}r-Q^2}{{\mathcal M}b-Q^2}
\frac{b^2f(b)f(r)}{{\bar {\mathcal D}}^3}\sin\theta\ . \end{aligned}$$ The total electromagnetic field to first order of the perturbation is then $$\label{RNemfieldpertnonextr}
\tilde F=-\left[\frac{Q}{r^2}+E_r\right]dt\wedge dr - E_{\theta}dt\wedge d\theta\ .$$ The total perturbed electrostatic potential is given by $$\label{relperttest}
V_{\rm tot}=V_{\rm test}+\left[1+\frac12\left(1-\frac{r}b\right){\bar {\mathcal H}}+\frac{qQ}{{\mathcal M}b-Q^2}\left(1-\frac{{\mathcal M}}b\right)\right]V^{\rm{BH}}\ ,$$ where $V^{\rm{BH}}=Q/r$ is the black hole electrostatic potential, while $V_{\rm test}$ denotes the electrostatic potential of the particle obtained within the test-field approximation by Leaute and Linet [@leaute] $$\label{solRNpot}
V_{\rm test} = \frac q{b r} \frac{(r-{\mathcal M})(b-{\mathcal M})
-\Gamma^2\cos\theta}{{\bar {\mathcal D}}} + \frac{q{\mathcal M}}{b r}\ .$$ The second and third terms in the bracketed expression of (\[relperttest\]) represent the gravitationally induced” and electromagnetically induced” electrostatic potential respectively and the equilibrium condition (\[bonnoreqcond\]) has been conveniently used.
The Zerilli’s procedure of expansion of both the gravitational and electromagnetic fields in tensor harmonics is largely used in the literature to study linear perturbations of spherically symmetric spacetimes due to some external source. We have given here the analytic solution for a problem which has raised much interest and discussions for many years. We have obtained closed form expressions for both perturbed metric and electromagnetic field due to a charged massive particle at rest in the field of a Reissner-Nordström black hole, taking advantage of the static character of the perturbation as well as of the axial symmetry of the configuration. The infinite sum of multipoles converges to an analytic form.
In addition to its theoretical significance, this result can become an important tool in testing the validity of numerical investigations addressing the dynamics of many body solutions in Einstein-Maxwell systems.
[00]{}
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|
---
author:
- 'Martin Morin[^1]'
- Pontus Giselsson
title: Sampling and Update Frequencies in Proximal Variance Reduced Stochastic Gradient Methods
---
\[section\]
[^1]: Dept. Automatic Control, Lund University, Lund, Sweden (, ).
|
---
abstract: 'We present results based on X-ray, optical, and radio observations of the massive galaxy cluster CIZA J0107.7+5408. We find that this system is a post core passage, dissociative, binary merger, with the optical galaxy density peaks of each subcluster leading their associated X-ray emission peaks. This separation occurs because the diffuse gas experiences ram pressure forces while the effectively collisionless galaxies (and presumably their associated dark matter halos) do not. This system contains double peaked diffuse radio emission, possibly a double radio relic with the relics lying along the merger axis and also leading the X-ray cores. We find evidence for a temperature peak associated with the SW relic, likely created by the same merger shock that is powering the relic radio emission in this region. Thus, this system is a relatively rare clean example of a dissociative binary merger, which can in principle be used to place constraints on the self-interaction cross-section of dark matter. Low frequency radio observations reveal ultra-steep spectrum diffuse radio emission that is not correlated with the X-ray, optical, or high frequency radio emission. We suggest that these sources are radio phoenixes, which are preexisting non-thermal particle populations that have been re-energized through adiabatic compression by the same merger shocks that power the radio relics. Finally, we place upper limits on inverse Compton emission from the SW radio relic.'
author:
- 'S. W. Randall, T. E. Clarke, R. J. van Weeren, H. T. Intema, W. A. Dawson, T. Mroczkowski, E. L. Blanton, E. Bulbul, S. Giacintucci'
bibliography:
- 'references.bib'
title: 'Multi-wavelength Observations of the Dissociative Merger in the Galaxy Cluster CIZA J0107.7+5408'
---
Introduction {#sec:intro}
============
According to the standard model of hierarchical structure formation, galaxy clusters largely grow through cluster mergers. Major mergers are the most energetic events in the Universe since the Big Bang, and can release $\ga 10^{64}$ erg in gravitational binding energy. Such mergers significantly disturb the intracluster medium (ICM) of galaxy clusters, and are useful for the study of ICM plasma physics, cluster scaling relations, non-thermal particle populations, the growth of large scale structure, and the physics of dark matter.
Dissociative mergers occur when the ICM of a merging subcluster experiences sufficient ram pressure forces to displace it from the gravitational potential minimum of its dark matter halo. Such offsets are observed as separations between the X-ray surface brightness peaks and/or the optical galaxy density and gravitational lensing mass peaks, where the effectively collisionless galaxies are expected to trace their dark matter halos. Such systems, e.g., the well-known Bullet Cluster [@2002ApJ...567L..27M], have been used to demonstrate the existence of dark matter, and to place limits on the self-interaction cross-section of dark matter, since self-interacting dark matter would experience ram-pressure-like forces during a merger, potentially leading to an offset between the optical galaxies and the dark matter peak [e.g., @2004ApJ...606..819M; @2006ApJ...648L.109C; @2006ApJ...652..937B; @2007NuPhS.173...28C; @2008ApJ...679.1173R; @2012ApJ...747L..42D; @2015Sci...347.1462H].
Merging or otherwise dynamically disturbed systems often host diffuse radio sources. These sources can be broadly divided into at least three categories: radio relics, radio halos, and radio phoenixes . Relics are typically found in the outskirts of clusters, and often have large sizes ($\ga 1$ Mpc) and polarized emission. In contrast, halos are centered on cluster cores, and are typically not strongly polarized. The location and morphology of radio phoenixes varies widely, although they are typically smaller than relics and halos. Relics, halos, and phoenixes are thought to be powered by cluster mergers, through direct shock acceleration, merger induced turbulence, and adiabatic compression by shocks, respectively . Observationally, ultra-steep spectrum (USS) diffuse radio sources with radio spectral indices $\alpha \la -1.5$[^1] [such as those described by @2001AJ....122.1172S] represent a particular subset of sources. It is currently unknown whether these sources are typically weaker “classical” radio relics associated with more minor mergers, radio phoenixes, “AGN relics” , or some mix of these three. New low-frequency radio survey instruments such as LOFAR are expected to find large numbers of such diffuse USS sources [@2011JApA...32..557R].
Here we present results from [*Chandra,*]{} X-ray; VLA, WSRT, and GMRT radio; and INT optical observations of the galaxy cluster CIZA J0107.7+5408 (hereafter CIZA 0107), at a redshift of $z=0.1066$ [@2002ApJ...580..774E], which contains both high and low frequency diffuse radio emission. The primary aims of this paper are to characterize the dynamical state of CIZA 0107; search for merger signatures (such as disturbed morphology and shock fronts); compare X-ray, optical, and radio observations to investigate the nature of the diffuse radio emission; and to evaluate the potential for this system to probe ICM and dark matter physics. When corrected to the reference frame defined by the CMB[^2], the redshift corresponds to a luminosity distance of $D_{\rm L} = 470$ Mpc and a scale of 1.86 kpc arcsec$^{-1}$ for a cosmology with $\Omega_0 = 0.3$, $\Omega_{\Lambda} = 0.7$, and $H_0 =
73$ [$\rm km\ s^{-1}$]{} Mpc$^{-1}$. All uncertainty ranges are 68% confidence intervals ([i.e.]{}, 1$\sigma$), unless otherwise stated.
Data Analysis {#sec:data}
=============
Chandra X-ray Observations {#sec:chandra}
--------------------------
CIZA 0107 was observed by [*Chandra*]{} for 23 ks on 2013 June 26 (ObsID 15152). The aimpoint was on the front-side illuminated ACIS-I CCD. All data were reprocessed from the level 1 event files using [CIAO]{} and [CALDB 4.6.7]{}. CTI and time-dependent gain corrections were applied. [lc\_clean]{} was used to check for periods of background flares[^3]. The mean event rate was calculated from a source free region using time bins within 3$\sigma$ of the overall mean, and bins outside a factor of 1.2 of this mean were discarded. There were no periods of strong background flares, such that the cleaned exposure time was 23 ks. To model the background we used the [ CALDB[^4]]{} blank sky background files appropriate for this observation, normalized to match the 10-12 keV count rate in our observations to account for variations in the particle background. Point sources were detected using the [CIAO]{} tool [wavdetect]{}, and source regions were checked by eye and subsequently excluded from the analysis. The background subtracted, exposure corrected [*Chandra*]{} image of the full ACIS-I field of view (FOV) is shown in Figure \[fig:chandra\].
Unless otherwise specified, all spectra were fitted in the 0.7-8.0 keV band using [xspec]{}, with an absorbed [apec]{} model. Spectra were grouped with a minimum of 40 counts per energy bin. @1998SSRv...85..161G abundance ratios and [AtomDB 2.0.2]{} [@2012ApJ...756..128F] were used throughout.
![ [*Chandra*]{} 0.3–8.0 keV X-ray image of the full ACIS-I FOV, background subtracted and exposure corrected, and smoothed with a 3 radius Gaussian. \[fig:chandra\] ](ciza0107_chandra.pdf){width="1.1\columnwidth"}
WSRT Radio Observations {#sec:wrst}
-----------------------
We use the 1.4 GHz WSRT radio observations presented in . The 1.4 GHz image of the cluster has a noise levels of 29 $\mu$Jy beam$^{-1}$ and a resolution of 21$\times$17. In addition, we make use of a lower-resolution image with a resolution of 60 and with emission from compact sources subtracted to better bring out the diffuse emission. For radio imaging, we used a Briggs weighting scheme with a robust parameter of 0.5 for the full image and natural weighting with a UV-taper for the source subtracted image. For more details about these observations and the data reduction the reader is referred to .
GMRT Radio Observations {#sec:gmrt}
-----------------------
On 2009 November 23, CIZA 0107 was observed with the GMRT [@1991ASPC...19..376S] simultaneously at 240 and 610 MHz within the scope of a larger project to study newly discovered radio relics (project code 17\_049; PI van Weeren). The total time on-source was about 200 minutes. For reducing the data, we used the SPAM package and a standard data reduction recipe as described in . We used the primary calibrator 3C48 for deriving all instrumental calibrations, adopting the @2012MNRAS.423L..30S flux scale. The effective bandwidth used for imaging was 6.5 and 32.8 MHz at 240 and 610 MHz, respectively. The final images were made at resolutions of 13.3$\times$10.4 and 5.7$\times$4.1, respectively. The achieved background RMS noises as measured in the center of the beam were 0.57 and 0.054 mJy beam$^{-1}$, respectively. We complemented these targeted observations with a 150 MHz image made from archival TGSS[^5] survey data (project code 18\_031, pointing R04D65), reprocessed by us in exactly the same way as described above. Using 16 MHz of bandwidth, and using 3C147 as our primary calibrator, we obtained an image with resolution of 29.5$\times$21.9, and a local background RMS noise near the target of 13 mJy beam$^{-1}$. All GMRT images were created using Briggs weighting with a robust parameter of -1 to achieve a more stable PSF.
VLA Radio Observations
----------------------
We use the 73.8 MHz Very Large Array Low-frequency Sky Survey redux [VLSSr, @2014MNRAS.440..327L] to measure the low frequency emission in CIZA 0107. We used the standard VLSSr image, which has a spatial resolution of 75 and a measured noise level of $\sigma_{\rm rms}=78$ mJy beam$^{-1}$ in the field around CIZA 0107. We refer the reader to @2014MNRAS.440..327L for details of the image calibration and processing. All flux measurements include a “CLEAN bias” of 0.66$\sigma_{\rm
rms}$ beam$^{-1}$ for all regions with fluxes above 3$\sigma_{\rm
rms}$, following the estimates of @2014MNRAS.440..327L.
Optical Observations {#sec:optical}
--------------------
Optical images of CIZA 0107 were taken with the Wide Field Camera (WFC) on the 2.5 m Isaac Newton Telescope (INT) in the $V$, $R$, and $I$ bands. For more details on the data reduction see . We compute galaxy iso-density contours by counting the number of galaxies per unit area on the sky. For this we first created a catalog of objects with [sextractor]{} . We excluded all unresolved objects from the catalog to avoid the numerous foreground stars in the field. To select cluster members we only counted objects with $V-R$ and $R-I$ colors within 0.5 magnitude from the average cD colors ($V-R = 1.0$, $R-I=0.75$). In addition, we required these objects to be fainter than the cDs but brighter than magnitude 22 to reduce contamination from more distant galaxies. This selection gave 220 cluster member galaxies in the region spanned by the galaxy density contours shown in Figure \[fig:optical\].
The Structure of the ICM
========================
Global ICM Properties {#sec:global}
---------------------
We measured the global properties of the ICM by fitting the X-ray spectrum extracted from within a 645 kpc radius with an absorbed [apec]{} model. As a CIZA cluster [Clusters in the Zone of Avoidance, @2002ApJ...580..774E; @2007ApJ...662..224K], CIZA 0107 is at a relatively low Galactic latitude of $b \approx -8.6583$[$^\circ$]{}. Due to the relatively high absorption close to the Galactic plane, we allowed $N_{\rm H}$ to vary as a free parameter. This provided an acceptable fit, with $N_{\rm H} = 3.5^{+0.1}_{-0.1}\times 10^{21}$ cm$^{-2}$, $kT =
7.8^{+0.4}_{-0.3}$ keV, and $Z = 0.31^{+0.05}_{-0.05} \, Z_\odot$, and with a $\chi^2$ per degree of freedom of $\chi^2_\nu = 390/378 =
1.03$. This fitted value for $N_{\rm H}$ is significantly larger than the weighted average value from the Leiden/Argentine/Bonn (LAB) survey of $N_{\rm H} = 2.3\times 10^{21}$ cm$^{-2}$. We find that fixing $N_{\rm H}$ at this value gives a much worse fit, with $kT =
11.9^{+0.6}_{-0.6}$ keV and $\chi^2_\nu = 493/379 = 1.30$. We note that there is some systematic uncertainty at low energies associated with modeling the contamination build-up on the ACIS detector, which will in principle affect our absorption and temperature measurements, however this effect is relatively small, $\la10$%[^6]. We conclude that $N_{\rm H}$ can vary significantly (by at least 50% or so) on angular scales smaller than what is resolved in the LAB survey, (which has a resolution of about 0.6[$^\circ$]{}, larger than the [*Chandra*]{} ACIS-I FOV of 0.27[$^\circ$]{} and much larger than [ *Chandra’s*]{} resolution of $\sim$0.5), consistent with results from some more recent, higher angular resolution survey maps [e.g., @2011ApJS..194...20P].
We estimated $r_{\rm 500}$, the radius within which the mean density is equal to 500 times the critical density at the cluster redshift, and $M_{\rm 500}$, the total mass within $r_{\rm 500}$, using the $M_{\rm 500}-T_X$ scaling relation of [@2009ApJ...692.1033V]. The radius was iteratively determined by estimating $r_{\rm 500}$, measuring the temperature by fitting a spectrum extracted with $r <
0.15 \, r_{\rm 500}$ excluded, determining $M_{\rm 500}$ (and $r_{\rm
500}$) from the $M_{\rm 500}-T_X$ relation, and repeating until the values converged. We find $T_{\rm 500} = 7.8^{+0.5}_{-0.5}$ keV, $M_{\rm 500} = 7.8^{+0.8}_{-0.7} \times 10^{14}$ M$_\odot$, and $r_{\rm 500} = 1.35^{+0.05}_{-0.04}$ Mpc. We note that these uncertainty ranges only include statistical uncertainties. Systematic uncertainties will likely increase the total uncertainty. Furthermore, cluster mergers can temporarily boost the global temperature, leading to biased temperature measurements and inferred masses [@2001ApJ...561..621R; @2002ApJ...577..579R]. Our derived value for $M_{\rm 500}$ is consistent with the value derived from [*Planck*]{} Sunyaev–-Zel’dovich observations of $M_{\rm 500, SZ} = 5.8^{+0.3}_{-0.3} \times 10^{14}$ M$_\odot$ within 3$\sigma$ .
The X-ray Image {#sec:image}
---------------
The smoothed [*Chandra*]{} image of CIZA 0107 is shown in Figure \[fig:chandra\_smo\]. Point sources have been excised, and the resulting gaps filled in by sampling from Poisson distributions matched to local annular background regions. The ICM shows a clear elongation from NE to SW, and no very bright peaks to suggest the presence of a cool core, indicating a dynamically unrelaxed merging system. The central bright X-ray emission is non-circular, and also elongated from NE to SW. A close up view of this “central bar” is shown in Figure \[fig:core\]. The morphology is suggestive of two distinct peaks, with a possible faint extension to the SW.
To better show subtle structure in the central bar, we created the unsharp-masked image shown in Figure \[fig:core\]. This image was derived by separately smoothing the X-ray image with 10and 30radius Gaussians and taking the difference of the results [see @2006MNRAS.366..417F]. The two bright peaks are clearly visible, as is a third peak to the SW. We extracted the surface brightness profile across the box region in Figure \[fig:core\], from NE to SW. The result is shown in Figure \[fig:bar\_prof\]. The bright central peak is visible at $\sim 50$ kpc, and the fainter NE peak at $\sim -125$ kpc. Given the errors, the NE peak may also be consistent with a flattened tail. There is a hint of the faint SW peak between 200-300 kpc, although this feature is not statistically significant.
Thermal Structure {#sec:tmaps}
-----------------
To map the thermal structure of the ICM, we generated smoothed spectral maps using the method described in [@2008ApJ...688..208R]. Spectra were extracted from circular regions, centered on each temperature map pixel and containing $\sim$2000 source counts in the 0.7-8.0 keV band, and fitted with an absorbed [apec]{} model. Galactic absorption was fixed at the global value derived in Section \[sec:global\]. Pixel values were derived from the fitted model parameters. For comparison, we also constructed a temperature map using the contour binning method developed by [@2006MNRAS.371..829S], where extraction regions are defined based on surface brightness contours, using a signal-to-noise (S/N) ratio of 44 per region.
The [*Chandra*]{} temperature, pseudo-entropy, and pseudo-pressure maps are shown in Figure \[fig:tmaps\]. Both the smoothed and the contour binned temperature maps show a high temperature region to the SW, beyond the X-ray surface brightness peaks, and in the region of the SW optical galaxy density peak and the bright SW radio relic (Section \[sec:radio\]). This region appears as a pressure peak in the pseudo-pressure map. The pseudo-entropy map shows a central low entropy region that is extended from NE to SW, consistent with the presence of two partially disrupted, merging cluster cores. Thus, the spectral maps support a roughly head-on merger scenario with a NE-SW merger axis, with two subcluster cores and the presence of high-temperature, shock-heated gas to the SW.
[[ =.5 ]{}]{}
To determine the significance of the SW temperature peak, we measured the projected temperature profile in four directions (see the angular sectors shown in Figure \[fig:profiles\]). Due to the limited number of counts, we chose only four bins per sector. We note that the amount of absorption can affect temperature measurements, with underestimated $N_{\rm H}$ biasing the temperature high. Since $N_{\rm H}$ is relatively large in this region of the sky, and may change across the field, the absorption was allowed to vary in each fit. The abundance was also allowed to vary. The resulting temperature profiles are shown in Figure \[fig:ktprof\]. The highest temperature by far is to the SW, in the region of the temperature peak seen in the temperature map, with $kT = 18.5 \pm 6.0$ keV. Compared with other sectors in the same radial range, this feature is significant at about 1.8$\sigma$ (see Section \[sec:dynamical\]). Since $N_{\rm H}$ is allowed to vary, its uncertainty is folded into the temperature uncertainty. If we fix $N_{\rm H}$ at our best fitting global ICM value of $3.5\times 10^{21}$ cm$^{-2}$, we find a statistically equivalent fit with $kT = 14.6^{+2.4}_{-3.1}$ keV for the SW temperature peak. In this case, the temperature increase is significant at 2.3$\sigma$ as compared with other sectors in the same radial range also with $N_{\rm
H}$ fixed at the global value. If we instead fix $N_{\rm H}$ at the LAB survey value of $2.3\times 10^{21}$ cm$^{-2}$ we find a somewhat worse statistical fit (with a null hypothesis probability of 2.8% versus 5.1% for the free $N_{\rm H}$ case) with $kT = 30^{+9}_{-6}$ keV. For every annular bin, the best-fitting value of $N_{\rm H}$ is within 2$\sigma$ of the best-fitting global ICM value. Although the SW temperature peak is only marginally statistically significant, based on its location on the merger axis and correlation with the SW radio relic, as expected for a merger bow shock (see Section \[sec:dynamical\]), we conclude that it is likely a real feature.
We note that, given the high temperature, lack of strong emission lines, and limited number of counts, we cannot confirm the thermal nature of this emission (as compared with, e.g., inverse Compton emission from the interaction of the CMB with the radio emitting particles in this region, or emission from unresolved point sources associated with the nearby BCG). This issue is discussed further in Section \[sec:ic\].
Diffuse Radio Emission {#sec:radio}
======================
The radio emission is shown in Figure \[fig:radio\], ranging from 74 MHz to 1.4 GHz. In , the diffuse 1.4 GHz radio emission was classified as a radio halo with a north-south extension, based on the continuous structure of the source and the rough correspondence with the low resolution [ *ROSAT*]{} X-ray image. The new [*Chandra*]{} images reveal a NE-SW extension of the ICM, with a doubly peaked core (Section \[sec:image\]). In light of these new data, we suggest that the diffuse radio emission represents a two-sided radio relic, possibly with radio halo emission between the relics, as has been observed in other systems , although other interpretations are possible. For a detailed discussion, see Section \[sec:relic\]. For the purposes of this paper, we refer to the diffuse 1.4 GHz emission as a radio relic throughout.
The 1.4 GHz radio flux profile extracted from the box region shown in Figure \[fig:radio\] is compared with the X-ray surface brightness profile in the same region in Figure \[fig:bar\_prof\]. The radio profile shows two clear peaks (in contrast with the single central peak expected for radio halos), which are at larger radii than the two bright X-ray peaks, in the regions of the BCGs and optical galaxy density peaks. The SW radio peak is roughly coincident with the SW temperature peak identified in Section \[sec:tmaps\]. The observations are therefore consistent with what is expected for radio relics in a late stage merging system, where the relics trace ICM merger shocks, which lead the X-ray cores. The dynamical state of CIZA 0107 is discussed further in Section \[sec:dynamical\].
The 74 MHz image reveals an elongated structure, NW of the southern subcluster center, and a fainter secondary peak to the SE (Figure \[fig:radio\], lower right panel and blue contours). The NW feature is seen as an extended, possibly double peaked source at 150 MHz and 240 MHz, but is not visible at 610 MHz or 1.4 GHz (see the contours in Figure \[fig:radio\]). It is not coincident with the brighter regions of the relic at 1.4 GHz, suggesting that it may be an unrelated feature. Similarly, the SE 74 MHz peak is seen at 150 MHz, and possibly 240 MHz, but not at higher frequencies. Neither feature is clearly associated with a bright X-ray or optical source (see Figure \[fig:optical\_radio\]).
In Table \[tab:radio\], we give radio fluxes at 1.4 GHz, 240 MHz, 150 MHz, and 74 MHz, along with some spectral indices, in three different regions: a larger region containing all of the diffuse emission detected at all frequencies, a smaller region corresponding to the NW USS source at 74 MHz, and a slightly smaller region corresponding to the SW USS source at 74 MHz. Fluxes were not calculated at 610 MHz since we expect some flux loss on the scale of the diffuse emission due to the lack of short baselines. To account for calibration uncertainties, we include a systematic error of 5% at 1.4 GHz, 10% at 240 MHz and 150 MHz, and 15% at 74 MHz. We follow the calibration uncertainties adopted by @2014MNRAS.440..327L for the VLSSr data. For the WSRT and the GMRT, our adopted calibration uncertainties are based on our experience with working with data from these observatories, as we have done elsewhere [e.g., @2014ApJ...781L..32V]. For the total emission in the region of the relic, we find spectral indices from 74 MHz to 150 MHz and 74 MHz to 240 MHz of $\alpha^{150}_{74} = -1.2$ and $\alpha^{240}_{74} = -1.4$. For the NW, elongated structure detected at 74 MHz (Figure \[fig:radio\]), we find significantly steeper emission, with $\alpha^{150}_{74} = -2.3$ and $\alpha^{240}_{74} = -2.1$. Similarly, for the SE peak at 74 MHz we find $\alpha^{150}_{74} = -2.2$ and $\alpha^{240}_{74} = -2.1$ (see Figure \[fig:radio\_spec\]). These features are discussed further in Section \[sec:uss\].
[lcccccc]{} Total&$72\pm4.5$&$329\pm41$&$759\pm163$&$1779\tablenotemark{a}\pm637$&$-1.4\pm0.3$&$-1.2\pm0.6$\
NW USS Source&$29\pm1.9$&$174\pm21$&$420\pm80$&$2126\pm412$&$-2.1\pm0.2$&$-2.3\pm0.4$\
SE USS Source&$6.8\pm0.9$&$66\pm10$&$159\pm50$&$778\pm223$&$-2.1\pm0.3$&$-2.2\pm0.6$
Discussion {#sec:discuss}
==========
Cluster Dynamical State {#sec:dynamical}
-----------------------
X-ray and radio observations of CIZA 0107 show that it is a non-dynamically relaxed merger system, with the merger axis likely along the NE-SW direction. The overall X-ray emission is extended along this axis, with two X-ray surface brightness peaks along the same line, consistent with two merging subclusters. There is a high temperature region to the SW, along the merger axis, consistent with the presence of a merger bow shock leading the SW core. There is a hint of an excess in the X-ray surface brightness at this location (Figures \[fig:core\] & \[fig:bar\_prof\]), consistent with the presence of a shock, although this feature is only marginally significant. The radio emission is also extended from NE to SW, at multiple frequencies, with two bright peaks at 1.4 GHz roughly 200 kpc from the overall cluster center (the center of the annular bin regions shown in Figure \[fig:profiles\]). The SW radio peak is coincident with the SW shock region. This is consistent with a double radio relic, as seen in other merging systems, where particles in the ICM are (re)accelerated at the merger shock associated with each subcluster and emit synchrotron radiation.
The optical image is shown in Figure \[fig:optical\], with the [ *Chandra*]{} X-ray surface brightness contours overlaid in red, and the optical flux-weighted cluster galaxy density contours overlaid in blue. This image shows two BCGs, indicated with green circles, consistent with two merging subclusters. The SW BCG was identified by [@1995MNRAS.274...75C], with a reported redshift of $z=0.109$ (although this redshift is described as “provisional” due to poor S/N). Both of the optical galaxy density peaks and both BCGs lie on the merger axis, at larger radii than the X-ray peaks, and separated from one another by roughly 500 kpc in projection. We conclude that CIZA 0107 is likely a dissociative post-merger system, similar to the famous Bullet cluster [@2002ApJ...567L..27M; @2007PhR...443....1M].
In this scenario, the diffuse ICM in the subcluster cores experiences ram pressure drag forces during the cluster merger, while the effectively collisionless galaxies and dark matter (DM) halos do not, leading to a separation between the ICM and galaxy/DM peaks. Therefore, the ICM is expected to trail the galaxies roughly along the merger axis [although the gas may lead the galaxies during the later stages of a merger due to the ram pressure slingshot effect, see @2007PhR...443....1M]. The ICM and galaxy density peak offsets indicate that this is a post core passage merging system. Thus we expect the NE X-ray and galaxy peaks to be moving NE, and their SW counterparts to the SW (in projection). As in the Bullet cluster, the SW galaxy density peak is at roughly the location of the SW leading bow shock. The lack of complicated structure in the ICM along with the subcluster cores, the merger shock, the radio relics, the optical galaxy density peaks, and the BCGs all lying roughly along the same line (i.e., the merger axis) suggest that this is a low impact parameter merger (at least in projection), also similar to the Bullet cluster.
Figure \[fig:ktprof\] shows a temperature increase between 200-400 kpc to the SW as compared with other sectors, which all have $kT \approx 7.5$ keV. Averaging the NW, NE, and SE sectors together in this radial range, we find a temperature of $kT = 7.5 \pm
0.6$ keV, as compared with $kT = 18.5 \pm 6.0$ keV in the SW. Applying the standard Rankine-Hugoniot shock jump conditions for an ideal gas with a constant ratio of specific heats of $\gamma = 5/3$, this corresponds to a Mach number of $M = 2.3 \pm 0.4$. In principle, this is a lower limit on the true Mach number, since projection effects will tend to bias the measured temperature increase low. For a 7.5 keV gas, this Mach number corresponds to a relative ICM velocity of 3250 [$\rm km\ s^{-1}$]{}. As shown by [@2007MNRAS.380..911S], the Mach number of an ICM merger shock can overestimate the relative velocity of the subcluster DM halos, as is likely the case for the Bullet cluster.
When viewed along a tangent line to the Mach cone, ICM shocks are expected to appear as sharp surface brightness edges in high angular resolution X-ray observations. However, no such edge is clearly visible in the region of the SW shock, neither in the X-ray image nor the unsharp-masked image. To further test for an edge in this region, we extracted the integrated emissivity profiles in the SW and NE sectors. These profiles were generated assuming the radial temperature profile in each sector followed the projected temperature profile (Figure \[fig:ktprof\]) with the abundance fixed at $Z = 0.3
\, Z_\odot$. The results are shown fitted with a 3D $\beta$-model density profile in Figure \[fig:emprofs\]. Both sectors are reasonably well-described by the model, with no clear evidence for an edge feature, although the SW profile shows weak evidence for a dip in the profile at $\sim370$ kpc, at the outer edge of the SW radio relic and high temperature region.
We modeled this dip using a discontinuous double power-law density profile of the form $$\label{eq:nr}
n_e(r) = \left\{
\begin{array}{ll}
n_0 ( \frac{r}{r_{\rm br}} )^{-k_1} & (r \le r_{\rm br})\\
n_1 ( \frac{r}{r_{\rm br}} )^{-k_2} & (r > r_{\rm br}) \\
\end{array}
\right. ,$$ where $r_{\rm br}$ is the break radius. This model is shown fitted to the data in Figure \[fig:sw\_edge\]. If we fix the break radius at $r_{\rm br}=370$ kpc, we find evidence for a weak edge, corresponding to a density jump factor of $1.2^{+0.1}_{-0.1}$. However, if the location of the break radius is allowed to vary the profile is also well-fit by a model with no discontinuous jump and a change in slope at a smaller radius. The lack of an apparent sharp surface brightness edge may indicate that there is a line of sight component to this merger, such that the edge is diminished by projection effects, while the hot, shock-heated gas is still visible in projection, as has been reported for other systems, (e.g., [A2443, @2013ApJ...772...84C]; [A2744, @2011ApJ...728...27O]). The angle between the merger axis and the plane of the sky needn’t be very large to obfuscate shock edges [see Figure 18 in @2011ApJ...728...27O]. Additionally, it may be difficult to pick out edges by eye in this relatively shallow (23 ks) observation, especially in fainter regions where the image must be smoothed to show ICM structure. The radius of curvature of merger shock fronts is generally not expected to be centered on the cluster itself. Since we are unable to identify the location of the edge, we cannot match the radius of curvature, center, and bin boundary location of our profile extraction regions to the edge. This will effectively blur the edge in the extracted profile, making it more difficult to detect. We conclude that deeper observations are required to confirm or rule out the presence of a surface brightness edge in this region.
The BCG of each subcluster was observed with the Wide Field Grism Spectrograph 2 on the University of Hawaii 2.2-meter Mauna Kea telescope as part of the CIZA survey [@2002ApJ...580..774E; @2007ApJ...662..224K]. They find redshifts of 0.103 for both of the BCGs (D. Kocevski private communication). While the exact uncertainty of these redshifts are not reported, they add to the supporting X-ray, radio, and optical imaging evidence that CIZA 0107 is composed of two nearby clusters that are undergoing a major merger.
Dark Matter Self-Interaction Cross Section {#sec:sidm}
------------------------------------------
[@2004ApJ...606..819M] outline four methods for placing constraints on the self-interaction cross-section of DM ($\sigma_{\rm
DM}$) using dissociative mergers. Placing such constraints on $\sigma_{\rm DM}$ requires lensing observations to map the total mass distribution, as well as optical spectroscopy to identify subcluster member galaxies, optical centroids, and subcluster line-of-sight velocities, neither of which are currently available for CIZA 0107. Nevertheless, we can use simple estimates to show that this system can, in principle, provide competitive constraints on $\sigma_{\rm
DM}$. To this end, let us assume that the dark matter halos are centered on the optical galaxy density peaks shown in Figure \[fig:optical\]. The offsets between the DM peaks and gas peaks indicates that the scattering depth of the DM particles cannot be much larger than 1, otherwise the DM subhaloes would experience drag forces similar to the gas and there would be no offset. Following [@2004ApJ...606..819M], we can write the subcluster DM scattering depth as $$\label{eq:tau}
\tau_s = \frac{\sigma_{\rm DM}}{m_{\rm DM}} \Sigma_s,$$ where $\Sigma_s$ is the DM surface mass density. To estimate $\Sigma_s$, we assume that the gas is isothermal and follows a single $\beta-$model density profile (which is roughly consistent with observations, see Figures \[fig:ktprof\] & \[fig:emprofs\]), and that it is in hydrostatic equilibrium. The latter assumption is not strictly true for this merging system, but suffices for the simple estimate we make here. In this case, the total density profile can be written as $$\rho(r) = \frac{3 \beta kT}{4 \pi G \mu m_p} \left[ \frac{x^2(3+x^2)}{(1+x^2)^2} \right],$$ where $x \equiv r/r_c$, $r_c$ and $\beta$ are the core radius and index that describe the gas density profile, $kT$ is the gas temperature, and $\mu m_p$ is the average mass per gas particle. Integrating over a circular aperture with projected radius $x_0$ and along the line of sight, and dividing by the area of the aperture, we find that the projected surface mass density is $$\begin{aligned}
\Sigma_s & = \frac{4}{x_0^2} \int_{0}^{x_0}x \int_{x}^{\infty}\frac{r
\rho(r)}{\sqrt{r^2 - x^2}} dr \, dx \\ & = \frac{3 \beta kT}{2 G \mu m_p
\sqrt{x_0^2 + r_c^2}}.
\end{aligned}$$ From the IEM profile fits in Section \[sec:dynamical\], we find $\beta
\approx 1.8$ and $r_c \approx 630$ kpc. For $kT = 8$ keV, this gives $\Sigma_s \approx 0.26$ g cm$^{-2}$ within a fiducial projected radius of $x_0 = 100$ kpc. From Equation \[eq:tau\] we then find that $\sigma_{\rm DM}/m_{\rm DM} \la 4$ cm$^2$ g$^{-1}$. This can be compared with a similar estimate based on observations of the Bullet cluster, but with $\Sigma_s$ measured from lensing observations rather than estimated based on scaling relations, which gives $\sigma_{\rm DM}/m_{\rm DM} \la 5$ cm$^2$ g$^{-1}$ [@2004ApJ...606..819M].
We stress that, given the lack of gravitational lensing observations to locate the DM peaks, we do not consider our limit on $\sigma_{\rm
DM}/m_{\rm DM}$ to be meaningful (here, we have assumed that the DM peaks are coincident with the optical centroids, which would imply $\sigma_{\rm
DM}=0$). Rather, the intent is to demonstrate that this system can potentially provide constraints on $\sigma_{\rm
DM}/m_{\rm DM}$ that are comparable to those derived for the Bullet cluster using similar methods, which currently provides the tightest constraints of any individual system known. Optical lensing and galaxy spectroscopy observations of CIZA 0107 will allow more accurate constraints to be placed on the DM self-interaction cross-section using a variety of methods, as has been done previously for the Bullet cluster [@2004ApJ...606..819M; @2008ApJ...679.1173R].
The Nature of the Diffuse Radio Emission {#sec:diffuse}
----------------------------------------
### High Frequency Diffuse Radio Emission {#sec:relic}
The radio image at 1.4 GHz shows two distinct peaks in the diffuse emission along the merger axis (Section \[sec:radio\], Figure \[fig:radio\]). In principle, each peak may represent a radio relic associated with a merger shock, or a radio halo associated with a subcluster core. Double (as opposed to single) radio relics are not uncommon , while double radio halos are extremely rare but not undocumented . There exists an empirical correlation between the radio halo power at 1.4 GHz, $P_{\rm 1.4 GHz}$, and the 0.1–2.4 keV X-ray luminosity within $r_{\rm 500}$, $L_{X,500}$. For the entire system, including both radio peaks, we find $P_{\rm 1.4 GHz} = 1.9 \times
10^{24}$ W Hz$^{-1}$ and $L_{X,500} = 3.9 \times
10^{44}$ erg s$^{-1}$. This places it somewhat above the $P_{\rm 1.4
GHz}$ – $L_{X,500}$ relation of [@2013ApJ...777..141C], although it is within the range of the scatter about this relation. Thus, we can not rule out a double radio halo interpretation of this based on the radio and X-ray powers. We note that we can likely exclude an AGN origin for this diffuse emission on Mpc scales since the synchrotron loss time is too short for electrons to have time to diffuse over these large regions (hence, in-situ particle acceleration is needed).
The SW radio peak is coincident with the high temperature, presumably shock-heated gas to the SW (Figures \[fig:tmaps\] & \[fig:ktprof\]). It is offset from the SW X-ray peak by $\sim200$ kpc (Figure \[fig:bar\_prof\]), roughly coincident with the SW BCG (Figure \[fig:radio\]), inconsistent with what is expected for a radio halo, which should remain centered on the ICM of its host cluster. There is a hint of a small increase in X-ray surface brightness at the location of the SW radio peak (also seen as the faint peak to the SW in Figure \[fig:core\]), but this feature is too faint to be one of the main subcluster cores, and is instead likely due to ICM compression at the shock front. Given the coincidence with the high temperature region, a possible local increase in the X-ray surface brightness, its placement along the merger axis, and its separation from the X-ray surface brightness peak, we suggest that the SW radio peak may be a radio relic that has been energized by a local merger shock.
For the NE radio peak, the interpretation is less clear. The offset from the NE X-ray peak is smaller and less significant (Figure \[fig:bar\_prof\]), and there is no indication of an X-ray shock in this region. Thus, we conclude that this system is most likely either a double radio relic, possibly with a faint radio halo between the relics, or a SW radio relic and NE radio halo. Due to the relative rarity of these configurations, the double radio relic interpretation is probably more likely. The somewhat flocculent appearance of the relic, in contrast with the sharp, linear morphology of some other relics , could in principle be due to a small inclination angle between the merger axis and the plane of the sky, as suggested by the lack of a sharp shock front edge in the X-ray (Section \[sec:dynamical\]). For simplicity, we refer to the high frequency radio structure as a double radio relic, or simply “the relic”, throughout.
### Ultra-Steep Spectrum Radio Sources {#sec:uss}
The 74 MHz radio image (Figure \[fig:radio\]) shows two distinct peaks near the SW subcluster (an elongated structure to the NW, and a separate peak to the SE) that are not aligned with the merger axis, nor do they correlate with the brighter 1.4 GHz emission. The peaks become less prominent with increasing frequency, and are no longer visible at $\nu \ga 610$ MHz. This is a reflection of the very steep spectral indices of these sources, with $\alpha_{\rm NE}
\approx \alpha_{\rm SW} \approx -2$ (see Section \[sec:radio\]). Neither source is clearly associated with a distinct radio, optical, or X-ray source (Figure \[fig:optical\_radio\]).
The radio spectra in the regions of the two USS sources are shown in Figure \[fig:radio\_spec\], with lines of $\alpha = -2.1$ normalized at 74 MHz overlaid. The spectra are consistent with power-laws at low frequency, but lie significantly above the extrapolated models at 1.4 GHz. This is consistent with the interpretation of the USS sources as unrelated to the radio relic, which is most clearly visible at 1.4 GHz, as suggested by the very different morphologies at 74 MHz and 1.4 GHz. Assuming a typical spectral index of $\alpha = -1$ for the radio relic, we find that the observations are consistent with a non-detection at 74 MHz in the VLSSr.
Such diffuse, ultra-steep spectrum radio sources have been identified in other clusters , but they are relatively rare, and the nature of these sources is not fully understood. Although we are unable to fit detailed models to the radio spectra of these sources since we only detect them at three frequencies, their steep spectra are consistent with old non-thermal electron populations that have been re-energized by a shock (i.e., radio phoenixes). According to this model, a pre-existing electron population (from, e.g., an old radio radio lobe) is re-energized by the passage of an ICM shock. Due to the high sound speed in the lobe, the plasma is not shocked, but rather compressed adiabatically . This model can explain the lack of correlation between the low and high frequency radio emission, and the fact that the USS sources are displaced from the merger axis. The morphology of these sources depends on the spatial distribution of the pre-existing electron population, as opposed to classical radio relics which directly trace ICM shocks. The lack of obvious host galaxies for these sources is also consistent with this interpretation, since older radio lobes would have had time to detach from their hosts and fade before being re-energized by a merger shock.
### Discussion {#sec:radio_discuss}
We conclude that the most likely interpretation of the observations is that the diffuse radio emission is from at least two distinct types of components: classical radio relics, where electrons are accelerated by a shock through, for example, diffusive shock acceleration [@1983RPPh...46..973D; @2001RPPh...64..429M], and radio phoenixes, where old radio structures are re-energized by passing shocks through adiabatic compression. Both processes are driven by the same merger event. Thus, CIZA 0107 provides a relatively rare case of radio relics and radio phoenixes observed in the same system. USS sources have been identified as radio phoenixes in other merging systems [e.g., @2013ApJ...772...84C; @2015MNRAS.448.2197D], and therefore may provide a means of identifying dynamically disturbed systems from low frequency radio observations.
It is currently unclear whether relics and phoenixes represent different phases of a general “life-cycle” of non-thermal particles in the ICM, or whether they arise from distinct particle populations. I.e., radio lobes and phoenixes may eventually break apart, releasing their particles to the ICM and providing a diffuse non-thermal particle component that may be re-accelerated by ICM shocks to create radio relics. Alternatively, radio relic particles may have a different origin, e.g., they may be accelerated directly from the thermal pool. However, it seems unlikely that diffusive shock acceleration is efficient enough to accelerate particles from the thermal pool to the required energies . Furthermore, not all strong merger shocks are associated with radio relics [@2011MNRAS.417L...1R], suggesting that the presence of a preexisting non-thermal particle population is required. High resolution, low frequency radio observations will help confirm the USS sources as radio phoenixes, which often have complicated, filamentary morphologies [e.g., see @2001AJ....122.1172S].
Inverse Compton Emission {#sec:ic}
------------------------
In principle, the high temperature peak to the SW seen in Figures \[fig:tmaps\] & \[fig:ktprof\] could arise from the contribution of a non-thermal component that is not included in our spectral model. In particular, we expect some level of inverse Compton (IC) emission due to the interaction of the synchrotron radio emitting electrons in the radio relic with the CMB. Despite this expectation, diffuse IC emission has yet to be conclusively detected in galaxy clusters [for recent results and reviews see @2012RAA....12..973O; @2014ApJ...792...48W; @2015ApJ...800..139G]. To test for the presence of IC emission, we added a power-law component to the model for the high temperature bin roughly 330 kpc to the SW shown in Figure \[fig:ktprof\]. We tried both allowing the temperature of the thermal component to vary and fixing it at the typical value at this radius of 8 keV. In neither case did including a power-law component significantly improve the fit.
To place a conservative limit on IC emission from this region, we fit the spectrum with an absorbed power-law, with no thermal component. This model provided a statistically equivalent fit to the single temperature thermal model, with an F-test probability of 72% (we consider an F-test probability of $\la 5$% to indicate a significant improvement). This degeneracy is due to our fairly shallow exposure, and to the lack of strong emission lines from such high temperature thermal plasma. The best-fit photon index was $\Gamma = -1.5^{+0.1}_{-0.1}$, which is close to the spectral slope of the radio emission $\alpha^{240}_{74} =
-1.3$ (Section \[sec:radio\]). Assuming that the synchrotron and IC emitting particles are the same population, and that these particles follow a power-law distribution in energy, the synchrotron and IC spectral slopes are expected to be equal [@1977OISNP..89.....P]. This model gives a total 2–10 keV flux of $2.4^{+0.1}_{-0.1} \times
10^{-12}$ erg cm$^{-2}$ s$^{-1}$.
@2001ApJ...557..560P gives a convenient expression relating the monochromatic IC X-ray and synchrotron radio fluxes to the implied magnetic field strength, $$\label{eq:flux_ratio}
\begin{aligned}
R \equiv \frac{f_{\rm IC}(kT)}{f_{\rm sync}(\nu)} & = 1.86 \times 10^{-8}
\left( \frac{\rm photons}{\rm cm^{2}\, s\, keV\, Jy} \right) \\ &
\times \left (\frac{kT}{20\, {\rm
keV}} \right)^{-\Gamma}\left( \frac{\nu}{\rm{GHz}} \right)^{\Gamma - 1} \\ &
\times \left( \frac{T_{\rm CMB}}{2.8 {\rm K}} \right)^{\Gamma + 2} \left(\frac{B}{\mu
{\rm G}} \right)^{-\Gamma}c(p),
\end{aligned}$$ where $\Gamma = (p + 1)/2$, $p$ is the power-law slope of the electron energy distribution $N(E) \propto E^{-p}$, $f_{\rm IC}(kT)$ is the IC flux density at energy $kT$, $f_{\rm sync}(\nu)$ is the synchrotron flux density at frequency $\nu$, $T_{\rm CMB}$ is the CMB temperature at the cluster redshift, and $c(p)$ is a normalization factor that is a complicated function of $p$ [with values $10 < c(p) < 1000$ for typical values of $p$, see @1979rpa..book.....R]. For $2 \la p \la 5$, $c(p)$ can be approximated as $c(p) \approx e^{1.42 p - 0.51}$. Using this approximation with Equation \[eq:flux\_ratio\] and solving for the magnetic field, one finds $$\label{eq:bfield}
\begin{aligned}
&B = \left( \frac{20 {\rm keV}}{kT} \right) \left( \frac{\nu}{{\rm
GHz}} \right)^{(p-1)/(p+1)} e^{\frac{2.84(p-r)}{p+1}} \mu \rm{G}, \\ &
r = 0.7 \ln \left[ \frac{R_{\rm obs}(kT, \nu)}{1.11 \times 10^{-8}}
\right] .
\end{aligned}$$ Note that there is a typo in the exponential term in the equivalent expression in @2001ApJ...557..560P (their Equation 7). Our power-law fits in this region give an IC flux density of $F_{\rm IC}(1 \, {\rm keV}) = 4.8 \times 10^{-4}$ photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$, while the radio observations give $F_{\rm sync}(1.4 \, {\rm GHz}) = 1.35 \times 10^{-2}$ Jy (for the latter, bright point sources have been removed to give the flux of the diffuse emission only). For $\Gamma=1.55$, this gives a magnetic field strength of $B =
0.01$ $\mu{\rm G}$.
Since the X-ray surface brightness does not correlate with the radio emission in detail, and since diffuse IC flux has yet to be conclusively detected in deep observations of other galaxy clusters, we conclude that the thermal model is much more likely, despite the fact that the thermal and non-thermal models provide statistically equivalent fits. Since the non-thermal model assumes that all of the emission in this region is IC emission, whereas it is in fact very likely dominated by thermal ICM emission, the above flux is a very conservative upper limit on the true IC flux, and thus $B >
0.01$ $\mu{\rm G}$ is a very conservative lower limit on the magnetic field strength. For comparison, the equipartition magnetic field strength implied by the total diffuse 1.4 GHz flux is roughly 1.6 $\mu{\rm G}$, assuming a ratio of energy in protons to energy in electrons of $k=1$ and a minimum Lorentz factor of $\gamma_{\rm min}
= 100.$
Summary
=======
We present results from X-ray, optical and radio observations of the massive galaxy cluster CIZA J0107.7+5408. Observations at all three wavelengths show a double-peaked morphology, with all peaks lying along roughly the same axis. The optical and 1.4 GHz radio peaks are at larger cluster radii than the X-ray peaks. The X-ray temperature map reveals a high temperature peak to the SW, roughly coincident with the SW radio peak at 1.4 GHz. We conclude that this system is a post core passage dissociative merger. The X-ray peaks lag the optical galaxy density peaks due to ram pressure forces on the ICM. Merger shocks lead the merging subclusters, giving rise to the possible doubly peaked radio relic and the shock heated gas in the region of the SW relic. The SW temperature rise implies a shock Mach number of at least $M = 2.3 \pm 0.4$. Rough estimates suggest that follow up optical lensing and spectroscopic observations may allow interesting limits to be placed on the self-interaction cross-section of dark matter, as has been done for other dissociative merging systems.
Low frequency radio observations reveal diffuse, ultra-steep spectrum radio emission, with $\alpha \approx -2$. This emission shows two peaks near the SW subcluster, although the peaks do not correlate with the merger axis, the X-ray emission, or the high frequency radio emission ([i.e.]{}, the radio relics). We suggest that these features are radio phoenixes, formed when old but relatively cohesive radio structures (likely created by radio galaxies) are re-energized due to adiabatic compression by passing merger shocks. Thus, CIZA 0107 is a relatively rare case containing clear examples of both classical radio relics and USS radio phoenixes.
Finally, we use the X-ray observations to place very conservative upper limits on the IC flux, and lower limits on the ICM magnetic field strength, in the region of the SW relic.
Acknowledgments {#acknowledgments .unnumbered}
===============
Support for this work was partially provided by the Chandra X-ray Center through NASA contract NAS8-03060, the Smithsonian Institution, and by the [*Chandra*]{} X-ray Observatory grant GO3-14134X. Basic research in radio astronomy at the Naval Research Laboratory is supported by 6.1 Base funding. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We thank the staff of the GMRT who have made these GMRT observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We thank Dale Kocevski for providing spectroscopic redshifts for the BCGs, and Paul Nulsen for useful discussions.
[^1]: Where the radio flux $F_\nu \propto \nu^\alpha$.
[^2]: The NASA/IPAC Extragalactic Database (NED) is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[^3]: <http://asc.harvard.edu/contrib/maxim/acisbg/>
[^4]: <http://cxc.harvard.edu/caldb/>
[^5]: <http://tgss.ncra.tifr.res.in/>
[^6]: http://cxc.harvard.edu/ciao/why/acisqecontam.html
|
---
abstract: 'The connectivity of the potential energy landscape in supercooled atomic liquids is investigated through the calculation of the instantaneous normal modes spectrum and a detailed analysis of the unstable directions in configuration space. We confirm the hypothesis that the mode-coupling critical temperature is the $T$ at which the dynamics crosses over from free to activated exploration of configuration space. We also report the observed changes in the local connectivity of configuration space sampled during aging, following a temperature jump from a liquid to a glassy state.'
address: 'Dipartimento di Fisica, Universita’ di Roma “La Sapienza” and Istituto Nazionale per la Fisica della Materia, Piazzale Aldo Moro 2, I-00185, Roma, Italy'
author:
- 'Claudio Donati, Francesco Sciortino and Piero Tartaglia'
date: 'Revised version LM7663: '
title: 'Role of the unstable directions in the equilibrium and aging dynamics of supercooled liquids.'
---
epsf
Understanding the microscopic mechanism for the incredible slowing down of the dynamics in supercooled glass forming liquids is one of the hot topics in condensed matter physics. In recent years, the combined effort of high level experimental techniques[@exp], computational analysis[@num; @sri] and sophisticated theoretical approaches[@goetze1; @teo1; @teo2] has provided an enormous amount of novel information. In particular, it appears more and more clearly that in addition to the melting temperature and the calorimetric glass transition temperature, $T_g$, another temperature plays a relevant role. This temperature, located between 1.2 and 2 $T_g$ depending on the fragility of the liquid, signals a definitive change in the microscopic processes leading to structural relaxation. Mode Coupling Theory (MCT)[@goetze1] was first in identifying the role of this cross-over temperature, $T_c$. According to MCT, for $T \ge T_c$ the molecular dynamics is controlled by the statistics of the orbits in phase space [@orbits], while below $T_c$, the dynamics becomes controlled by phonon assisted processes[@goetze1]. Studies based on disordered mean field $p$-spin models have also stressed the role of such a cross-over temperature[@thirumalai]. Computer simulation studies of realistic models of liquids have addressed the issue of the structure of configuration space in supercooled states [@sri]. Two different techniques have provided relevant information on phase space structure: the instantaneous normal mode approach (INM)[@keyes1], which focuses on the properties of the finite temperature Hessian, allowing the calculation of the curvature of the potential energy surface (PES) along $3N$ independent directions and (ii) the inherent structure (IS) approach[@stillinger], which focuses on the local minima of the potential energy. Different from mean field models, computer simulation analysis provides a description based on the system’s potential energy, the free energy entering only via the equilibrium set of analyzed configurations.
In principle, the INM approach should be well suited for detecting a change in the structure of the PES visited above and below $T_c$. A plot of the $T$-dependence of the fraction of directions in configuration space with negative (unstable) curvature, $f_u$, should reveal the presence of $T_c$. Unfortunately, as it was soon found out[@keyes2], anharmonic effects play a non negligible role and several of the negative curvature directions are observed even in crystalline states, where diffusivity is negligible. A similar situation is seen in p-spin models, where the number of negative eigenvalues of the Hessian is not zero at $T_c$ [@biroli]. Bembenek and Laird[@laird] suggested inspecting the energy profile along the unstable directions to partition the negative eigenmodes in shoulder modes ($sh$) (i.e. anharmonic effects) and double-well ($dw)$ modes (i.e. directions connecting different basins). While for a particular molecular system the fraction of double well modes $f_{dw}$ has been shown to go to zero close to $T_c$[@water], for atomic system, $f_{dw}$ is significantly different from zero even in crystalline and glassy states[@laird]. The existence of $dw$ directions in thermodynamic conditions where the diffusivity is zero, i.e. in situations where the system is constrained in a well defined basin, strongly suggests that the two minima joined by the unstable double-well directions may lead to the same inherent structure IS. Such possibility has been demonstrated very clearly in Ref.[@gezelter].
The aim of this Letter is to present a detailed evaluation of: (i) the number of escape directions, $N_{escape}$, leading to a basin [*different*]{} from the starting one; (ii) the number of distinct basins, $N_{distinct}$, which are connected on average to each configuration via a $dw$ direction. This analysis, based on a computationally demanding procedure, shows that indeed the PES regions visited in thermal equilibrium above and below $T_c$ are clearly different. We also study the evolution of the sampled PES as a function of time following a quench from above to below $T_c$. We show that, in analogy with the equilibrium case, two qualitatively different dynamical regimes exist during aging, related to different properties of the sampled PES.
The system we study is composed of a binary (80:20) mixture of $N=1000$ Lennard Jones atoms[@parameter]. The dynamics of this system is well described by MCT, with a critical temperature $T_c$ equal to $0.435$[@kob].
We begin by considering the equilibrium case. At each $T$, for each of the 50 analyzed configurations, we calculate the INM spectrum and — by rebuilding the potential energy profile along straight paths following directions with negative curvature— we classify the unstable modes into $dw$ and $sh$. It is important to notice that the classification is done by studying the shape of the PES along one eigenvector, i.e. beyond the point in configuration space where the eigenvector was calculated. In principle, to identify a $dw$ or a $sh$ mode, one should follow a curvilinear path. Indeed the use of straight paths guarantees only the identification of the $dw$ modes whose one-dimensional saddle energy is close to the potential energy of the system. As discussed in more length in [@discussion] those $dw$ modes are the ones relevant for describing motion in configuration space.
Figure \[fig:fdw\] shows the $T$ dependence of the unstable and $dw$ modes for the studied system. In agreement with the analogous calculation of Ref.[@laird], even at the lowest temperature where equilibration is feasible within our computer facilities, $f_{dw}$ is significantly different from zero. To estimate roughly the number of double wells which do not contribute to diffusion, we have calculated the IS associated with the T=0.446 equilibrium configurations, and we have heated them back to various temperatures below $T_c$. The corresponding $f_u$ and $f_{dw}$ are also shown in Fig.\[fig:fdw\]. Both these quantities depend linearly on the temperature. If we extrapolate $f_{dw}$ for the equilibrium and for these non-diffusive cases, we find that the two curves cross close to the MCT critical temperature $T_c$. Thus $T_c$ seems to be the temperature at which the number of directions leading to different basins goes to zero, leaving only local activated processes as residual channels for structural relaxation[@precisazione].
To estimate in a less ambiguous way the number of different basins which can be accessed from each configuration we apply the following procedure: (i) calculate the $dw$ directions via INM calculation; (ii) follow each straight $dw$ direction climbing over the potential energy barrier and down on the other side until a new minimum is found; (iii) perform a steepest descent path starting from the new minimum found along the $dw$ direction, as first suggested by Gezelter et al.[@gezelter]; (iv) save the resulting $IS$ configuration; (v) repeat (ii-iv) for each $dw$ mode. This procedure produces a list of $IS$s which can be reached from the initial configuration crossing a one-dimensional $dw$. We note that our procedure only guarantees that the two starting configurations for the quenches are on different sides of the double well, since — as discussed in [@discussion] — there is some arbitrariness in the location of the two minima.
We calculate the relative distance $d_{ij}$
$$d_{ij}=\sqrt{\frac{1}{N}\sum_{l=1}^N (x_l^i-x^j_l)^2+(y_l^i-y^j_l)^2+
(z_l^i-z^j_l)^2}
\label{matrix}$$
between all IS pairs in the list to determine the number of [*distinct*]{} basins, $N_{distinct}$, connected to the starting configuration. Here $x_m^k$, $y_m^k$, and $z_m^k$ are the coordinates of the $m$-th particle in the $k$-th IS of the list ($IS_k$). We also calculate the distance $d_{0i}$ — where $0$ indicates the IS associated to the starting configuration — to enumerate the number of escape directions, $N_{escape}$, leading to a basin different from the original one ($IS_0$).
In Fig. \[fig:doj\] (upper panel) we show a plot of the distributions of the distances $d_{0i}$ between the starting IS and the minima identified with this procedure for different temperatures. At high temperatures, the distribution has a single peak centered approximately at $d=0.3$. For lower temperatures, the peak moves to the left and decreases in height. At the same time, a second but distinct peak, centered around $d\approx 10^{-4}$, appears. Since our sample is composed of $1000$ particles, an average distance of the order of $10^{-4}$ means that, also in the case that only one particle has a different position between the two different configurations, this particle has moved less that $0.003$ interparticle distance. Thus, we consider the IS $i$ and $j$ as coincident if $d_{ij}<
10^{-2.5}$.
The upper panel of Fig.\[fig:NdvsT\] reports the $T$-dependence of the number of $dw$ directions $N_{dw}$, the number of escape directions and the number of distinct basins which can be reached by following a $dw$ direction. We find that at high temperature nearly all $dw$ directions leads to a different IS, thus fully contributing to the diffusion process. On lowering the temperature, a large fraction of the directions leads to the same minimum, and, close to $T_c$, almost all $dw$ directions lead to intra-basin motion. A basin change becomes a rare event. Data in Fig.\[fig:NdvsT\] support the view that $T_c$ is associated to a change in the PES sampled by the system, and, consequently, in the type of dynamics that the system experiences. Above $T_c$ the system can change basins in configuration space by moving freely along an accessible $dw$ direction, while below $T_c$ the thermally driven exploration of configuration space favors the exploration of the interior of the basins. Diffusion in configuration space, i.e. basin changes, requires an local activated process.
We next turn to the study of out-of-equilibrium dynamics. Configurations are equilibrated at high temperature ($T=5.0$) and at time $t_w=0$ are brought to low temperature $T_f=0.2$ by instantaneously changing the control temperature of the thermostat. Within 100 molecular dynamics steps, the kinetic energy of the system reaches the value corresponding to $T_f$. Configurations are recorded for different waiting times $t_w$ after the quench. We then repeat the same analysis done for the equilibrium configurations, as described before.
Figure \[fig:doj\] (lower panel) shows the distribution of the distances $d_{0i}$ between the minima $IS_0$ and $IS_i$. For short $t_w$ the distribution has a single peak centered approximately at $d_{0i}=0.3$, as found in the equilibrium configurations at high temperature. As the system ages, the peak shifts to smaller distances and a second distinct broad peak centered at approximately $d_{0i}=5\times 10^{-4}$ appears. The observed behavior is similar to the equilibrium one, on substituting $T$ with $t_w$.
The lower panel of Fig. \[fig:NdvsT\] shows $N_{dw}$, $N_{escape}$ and $N_{distinct}$ as functions of $t_w$. At small $t_w$, following a double well direction always brings to a basin that is distinct from the original one, while at long times all $dw$ directions leads to intra-basin motion. Again, the behavior is very reminiscent of the equilibrium one, on substituting $T$ with $t_w$. From these results, we suggest that two qualitatively different dynamical processes control the aging processes, respectively for short and long $t_w$[@kobag]. For short waiting times the system is always close to saddle points that allow it to move from one basin to a distinct one. As the system ages, it is confined to wells that are deeper and deeper, and the number of basins that it can visit by following a $dw$ direction decays to zero. Thus, for long $t_w$ the dynamics of aging proceeds through local activated processes, that allow the system to pass over potential energy barriers. As shown in Fig. \[fig:NdvsT\], beyond $t_w= 10^5$ the number of distinct basins is almost zero, which, in analogy with the equilibrium results, supports the view that a cross-over from MCT-like dynamics to hopping dynamics may take place at a cross-over time during the aging process.
In summary, this Letter confirms the hypothesis that $T_c$ can be considered as the $T$ at which dynamics crosses over from the free-exploration to the locally activated exploration of configuration space. In this respect, the analogy between the slowing down of dynamics of liquids and the slowing down of the dynamics in mean field $p$-spin model is strengthened. This Letter also shows that changes in dynamical processes associated with changes in the explored local connectivity of configuration space are observed even during the aging process, in the time window accessed by molecular dynamics experiments. MCT-based models of aging[@latz] could be able to describe the early part of the aging process — i.e. the saddle-dominated dynamics — but may not be adequate for describing the locally activated dynamical region.
We acknowledge financial support from the INFM PAIS 98, PRA 99 and [*Iniziativa Calcolo Parallelo*]{} and from MURST PRIN 98. We thank W. Kob, G. Parisi and T. Keyes for discussions.
See, for example, R. Torre [*et al.*]{}, Phys. Rev E [**57**]{}, 1912 (1998); F. Sette [*et al.*]{}, Science, [**280**]{}, 1550, (1998); L. Schneider [*et al.*]{}, J. Non-Cryst. Solids [**235-237**]{}, 173 (1998).
K. Binder [*et al.*]{}, in [*Complex Behavior of Glassy Systems*]{} M. Rubi and C. Perez-Vicente Eds. (Springer Berlin 1997); W. Kob, J. Phys.: Cond. Matter [**11**]{}, R85 (1999).
S. Sastry [*et al.*]{}, Nature, [**393**]{}, 554 (1998); A. Heuer, Phys. Rev. Lett. [**78**]{}, 4051 (1997); F. Sciortino [*et al.*]{} Phys. Rev. Lett. [**83**]{} 3214 (1999); T. Schröder [*et al.*]{}, J. Chem. Phys. [112]{},9834 (2000); A. Scala [*et al.*]{}, [*Nature*]{} in press (2000).
W. Götze and L. Sjögren, Rep. Prog. Phys. [**55**]{}, 241 (1992); W. Götze, J .Phys.: Condens. Matter [**11**]{}, A1 (1999).
M. Mezard and G. Parisi, J. Phys.: Cond.Matter, [**11**]{}, A157 (1999); M. Cardenas [*et al.*]{}, J. Chem. Phys. [**110**]{}, 1726 (1999). R. J. Speedy, J. Chem. Phys. [**110**]{}, 4559 (1999). M.R. Mayr [*et al.*]{}, J. Non-Crys. Solids [**235**]{}-[**237**]{}, 71, (1998)
T. R. Kirkpatrick and D. Thirumalai, Phys. Rev. Lett. [**58**]{}, 2091 (1987); A. Crisanti and H.-J. Sommers, Z. für Physik B [**87**]{}, 341 (1992).
T. Keyes, J. Phys. Chem. [**101**]{}, 2921 (1997).
F. H. Stillinger and T. A. Weber, Phys. Rev. A [**28**]{}, 2048 (1983); J. Chem. Phys. [**80**]{}, 4434 (1984). P. G. Debenedetti [*et al.*]{}, J. Phys. Chem. B[**103**]{}, 10258 (1999). T. Keyes, J. Chem. Phys. [**101**]{}, 5081 (1994). M. C. C. Ribeiro, P. A. Madden, J. Chem. Phys. [**108**]{}, 3256 (1998).
G. Biroli, J. Phys. Math. Gen. [**32**]{} 8365 (1999).
S. Bembenek and B. Laird, Phys. Rev. Lett. [**74**]{}, 936 (1995); J. Chem. Phys. [**104**]{}, 5199 (1996).
F. Sciortino and P. Tartaglia, Phys. Rev. Lett. [**78**]{}, 2385 (1997); La Nave [*et al,*]{} Phys. Rev. Lett. [**84**]{}, 4605 (2000).
J. D. Gezelter [*et al*]{},. J. Chem. Phys. [**107**]{}, 4618 (1997).
The interaction parameters are given by: $\epsilon_{AA}=1.0$, $\epsilon_{AB}=1.5$, $\epsilon_{BB}=0.5$, $\sigma_{AA}=1.0$, $\sigma_{AB}=0.8$, $\sigma_{BB}=0.88$. Lengths are defined in units of $\sigma_{AA}$, temperature $T$ in units of $\epsilon_{AA}/k_B$, and time in units of $\sqrt{ m \sigma_{\rm AA}^2
/48\epsilon_{\rm AA}}$. Both atoms have unitary mass $m$. The simulations have been run for up to $5 \cdot 10^6$ MD steps in the NVT ensemble (Nose-Hoover thermostat). One MD step is 0.02.
W. Kob and H.C. Andersen, Phys. Rev. Lett. [**73**]{}, 1376 (1994); Phys. Rev. E [**52**]{}, 4134 (1995).
Motion along a straight direction in configuration space will always be associated with a fast rise of the energy profile, since every direction will always describe pairs of atoms moving close enough to probe the repulsive part of the pair potentials [@keyes2]. In systems with steep repulsive potentials, such an unphysical rise of the potential energy profile along the straight eigenmode direction sets in very early. In other words, the eigenvector direction changes very rapidly as the system moves in configuration space and very soon the energy profile differs from the profile evaluated along the straight eigenvector approximation. The major effects of such an artificial rise in energy are[@keyes2] (i) the transformation of $dw$ direction in $sh$ directions and (ii) the arbitrary location of the two minima along the $dw$ direction and the extremely low value of the one-dimensional barriers along the $dw$. In particular, effect (i) sets in when, in the studied direction, the system is located far from the saddle and effect (ii) sets in when the system is located close to the saddle point \[T. Keyes and W.-X. Li, J. Chem. Phys. [**111**]{}, 5503 (1999)\]. Notwithstanding these potential pitfalls in the classification procedure, the classification helps in understanding the dynamical changes taking place in the liquid as a function of $T$ and $\rho$. Indeed the only saddles which are relevant for the dynamical behavior of the system are the ones which are explored by the system, i.e. the ones located in a potential energy range $V \pm \Delta V$ — where $V$ is the average potential energy of the system and $\Delta V$ is a measure of the potential energy fluctuations. The accessed saddles are included in the type (ii) classification in the above list. As a result, the classification performed along straight directions retains its validity, even if no physical meaning can be attributed to the calculated one-dimensional energy profile. Of course, it would be more appropriate to re-calculate the eigenvectors at each step and follow the curved one-dimensional energy profile. Such a procedure could still suffer from the frequent mode-crossing events which are known to characterize the liquid dynamics \[M. Buchener and T. Dorfmüller, J. Mols. Liquids, [**65/66**]{}, 157 (1995).\].
Such estimate of the free-exploration to activated dynamics cross-over temperature could be biased by the arbitrary choice of the $IS$ configuration (the $T=0.446$ configurations).
W. Kob [*et al.*]{}, Europhys. Letts. [**49**]{}, 590 (2000).
A. Latz, J. Phys.: Condens. Matter, in the press.
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abstract: 'The spectral line shapes for hydrogen–like heavy ion emitters embedded in strongly correlated two–component electron–ion plasmas are investigated with numerical simulations. For that purpose the microfield fluctuations are calculated by molecular dynamics simulations where short range quantum effects are taken into account by using a regularized Coulomb potential for the electron–ion interaction. The microfield fluctuations are used as input in a numerical solution of the time–dependent Schrödinger equation for the radiating electron. In distinction to the standard impact and quasistatic approximations the method presented here allows to account for the correlations between plasma ions and electrons. The shapes of the Ly$_{\alpha}$–line in Al are investigated in the intermediate regime. The calculations are in good agreement with experiments on the Ly$_{\alpha}$–line in laser generated plasmas.'
author:
- 'H. B. Nersisyan'
- 'C. Toepffer'
- 'G. Zwicknagel'
title: 'Microfield Fluctuations and Spectral Line Shapes in Strongly Coupled Two–Component Plasmas'
---
Introduction {#sec:1}
============
Measurements of emission and absorption spectra of atoms and ions are one of the most important tools in plasma diagnostics [@gri97; @sal98; @dat]. They allow for investigations of the properties of various laboratory and space plasmas. In particular, spectral line shapes may be analyzed to yield a wealth of information on the plasma parameters provided, however, that the data are compared to accurate computations of the spectral line broadening.
Traditionally, in the theory of spectral line broadening in plasmas ion effects in most cases were calculated within the quasistatic approximation, while the electron perturbation was believed to satisfy the impact approximation. The description of this method (Standard Theory (ST)) is given, e.g., in Refs. [@gri97; @sal98; @dat]. The separation of perturbations into ion and electron parts, in general, cannot be made without a loss of accuracy, although it is argued [@ale96] that in many cases it is justified. A more serious problem, however, is that each of the ion and electron parts often needs to be considered beyond the limits of the quasistatic and impact approximations. In particular, the ion motion in plasma leads to the so–called ion dynamics effects. It was first shown theoretically [@duf69; @fri71] and soon found in experiments [@kel73; @gru77; @gru78], that the ion dynamics can be responsible for significant corrections to the spectral line widths. In order to advance the calculations beyond the ST, several numerical methods have been developed. Among the first is the method developed in Ref. [@sta83], where a computer code was used to simulate the ion motion along straight paths, while the electron contribution was calculated using the impact approximation. The method was further improved in Ref. [@sta86] by using molecular–dynamics (MD) simulations for the ions, thus accounting for interactions between the radiators and the ion perturbers. In Refs. [@gig87; @heg88; @car89; @gig96] the motion of both ions and electrons was numerically simulated. The particle motion was simulated using straight path trajectories, which is applicable when the correlations between the perturbers and radiator are neglected. Later, the area of applicability was extended by using hyperbolic [@ale97] and, recently, exact paths for the perturbers (see, e.g., Refs. [@mar03; @mar04; @sta06] and references therein). In Refs. [@mar03; @mar04] the model was based on one–component plasmas (OCPs) treating the full spectrum as a superposition of the electronic and ionic contributions and thus neglects the influence of the attractive interactions between electrons and ions. This is well justified for weakly coupled plasma where the ionic and electronic fields can be handled separately for spectra modeling. But there is an increasing number of experiments of interest which are far beyond such parameter regimes (see, e.g., Refs. [@sae99; @eid00; @and02; @eid03]). In such cases a simple superposition of the electronic and ionic fields becomes insufficient due to strong nonlinear effects and the total field in a two–component plasmas (TCP) should serve as the starting point for spectra modeling. The related microfield distribution (MFD) and the line shapes including the full attractive electron–ion interaction has thus been attracting more and more attention and has already been studied, e.g., in Refs. [@sad09; @ner05; @ner06; @ner08; @cal07; @fer07; @sta07; @tal02].
The present paper is the continuation of Ref. [@mar04] but we treat here the ions and electrons on an equal footing by concentrating on the TCP. For that purpose we perform MD simulations which span the entire range between the impact and the quasistatic approximations. We solve the time–dependent Schrödinger equation for the radiator in the fluctuating microfield generated by the plasma particles. The MD simulations in conjunction with the Schrödinger equation to study the line shapes in a TCP have been previously considered in Ref. [@sta06]. Our model is thus similar to that considered in [@sta06] but, in addition, it allows to treat the interaction of the emitted photons with radiating electron.
MD simulations {#sec:2}
==============
A two–component classical plasma of electrons and ions (with the charge $Ze$) is in equilibrium with the temperature $T$ completely described by the coupling parameters $\Gamma _{\alpha \beta }$ with $\alpha $,$\beta =e$,$i$. Introducing the mean electron–electron ($a_{e}$), ion–ion ($a_{i}$) and electron–ion ($a$) distances through the relations, $a_{\alpha}^{-3}=4\pi n_{\alpha}/3$, $a^{-3}=4\pi n/3$ (where $n=n_{e}+n_{i}$ is the total plasma density with $n_{e}=Zn_{i}$) these parameters are defined as [@ner05; @ner06; @ner08] $$\Gamma _{\alpha\alpha}=\frac{q^{2}_{\alpha} e_{S}^{2}}{k_{B}T a_{\alpha}} \ ,
\quad \Gamma _{ei}=\frac{Ze_{S}^{2}}{k_{B}T a} \ .
\label{eq:11}$$Here $q_{e}=-1$, $q_{i}=Z$, $e_{S}^{2}=e^{2}/4\pi \varepsilon _{0}$ and $\varepsilon _{0}$ is the permittivity of the vacuum.
It is well known [@kel63; @deu81] that, to avoid the collapse of the classical system of electrons and ions, the Coulomb electron–ion interaction potential must be replaced by the pseudopotential with a regularized short–range behavior. In this paper we consider the electron–ion pair interaction potential $-e_{S}^{2} q_{\beta}u_{ei}(r)$, where $\beta =i,R$, $q_{R}e$ is the charge of the radiator (throughout this paper the index $R$ refers to the radiators) and $$u_{ei}( r) =\frac{1}{r}( 1-e^{-r/\delta}) ,
\label{eq:13}$$which is regularized at small distances. The cutoff parameter $\delta$ may be qualitatively thought of as a classical emulation of the electron thermal de Broglie length. For large distances $r>\delta$ the potential becomes Coulomb, while for $r<\delta$ the Coulomb singularity is removed and $u_{ei}(0)=1/\delta$. By this the short range effects based on the uncertainty principle are included [@tal02; @kel63; @deu81].
For a classical description of a plasma the electron degeneracy parameter $\Theta _{e}$, i.e., the ratio of the thermal energy and the Fermi energy must fulfill $\Theta _{e}=k_{B}T/E_{F}>1$. Since an ion is much heavier than an electron this condition is usually fulfilled for ions. Therefore one can expect that the regularization given by Eq. is less important for ions than for electrons. Furthermore, scattering of any two particles is classical for impact parameters that are large compared to the de Broglie wavelengths $\lambdabar _{\alpha\beta}=( 2\pi\hbar^{2}/\mu_{\alpha\beta }k_{B}T)%
^{1/2}$, where $\mu_{\alpha\beta }$ is the reduced mass of the particles $\alpha $ and $\beta $. Typical impact parameters are given by the Landau lengths, $\lambda _{L\alpha \beta }=e_{S}^{2}\vert q_{\alpha }q_{\beta }\vert /k_{B}T$. Its ratio to the de Broglie wavelengths is given by $$\sigma _{\alpha \beta }=\frac{\Gamma _{ei} \left\vert q_{\alpha }q_{\beta }\right\vert }{Z}%
\frac{a}{\lambdabar _{\alpha \beta }} \ .
\label{eq:14}$$Note that $\sigma _{ee}<\sigma _{ei}\ll \sigma _{ii}$. A classical description of the scattering events in the TCP is valid if $\sigma _{ee}>1$.
A collective length scale is given by the Debye screening radius, for a TCP $\lambda_{D}=a/(3\Gamma _{ei})^{1/2}$. The plasma frequencies for electrons and ions $\omega _{p\alpha }=(4\pi q_{\alpha }^{2}%
n_{\alpha }e_{S}^{2}/m_{\alpha })^{1/2}$ with $\alpha =e,i$ set the collective time scales $\omega _{pe}^{-1}$ and $\omega _{pi}^{-1}$ for electronic and ionic subspecies, respectively. Due to their large mass ratio the electrons and the ions move on very different timescales. Moreover, for a nonrelativistic treatment the thermal energy of the particles must be smaller than their rest energy. Since this is important only for electrons we require that $k_{B}T\ll m_{e}c^{2}$. Also the validity of the dipole approximation for plasma–radiator interaction used in Sec. \[sec:3\] requires that the characteristic length scale of the plasma microfield must be larger than the effective atomic length scale $a_{Z}=a_{B}/Z$, where $a_{B}$ is the Bohr radius. Since this length is $\simeq a$ the dipole approximation is valid when $a\gtrsim a_{Z}$ which is usually fulfilled for heavy radiators. In this paper we consider hydrogen–like ions as radiators in a completely ionized TCP, i.e. we assume that $q_{R}=Z_{R}-1$, where $Z_{R}e$ is the charge of the nucleus of the radiating ion.
Let us now briefly discuss the limitations of the MD model arising from the classical description of the electrons, i.e., from the neglect of the quantum effects in the short–range electron–ion interaction. As shown in Ref. [@fis01] two constraints on the parameter $\delta$ determining the regularized potential must be considered. In the parameter regime when a significant fraction of the simulated electrons is found in the quasibound states the simulated $Z$ and $Z_R$ are effectively reduced and the MD simulations are not adequate. Thus, the parameter $\delta$ must be chosen large enough to suppress the formation of the classical bound states of electrons. On the other hand the condition $\delta\lesssim a$ must be fulfilled so not to affect the free electron density at $r\sim a$. The probability to found an electron within a volume $r\lesssim\delta $ from an ion is estimated by $W\simeq (4\pi\delta^{3} /3)n_{e}g_{ei}(0)$, where the electron–ion radial distribution function $g_{ei}(r)$ can be approximated by the nonlinear Debye–Hückel expression $g_{ei}(r)\simeq \exp{[\beta_{e} Ze^{2}_{S}u_{ei}(r)]}$ (see, e.g., Refs. [@ner05; @tal02]) with $\beta_{e} =1/k_ {B}T$. Note that in the case when $Z\gg 1$, the capture of an electron on the quasibound orbit reduce the effective $Z$ and $\Gamma _{ei}$ for the next one, so a significant fraction of electrons can stay free even for $\Gamma _{ei}a/\delta \gg 1$. The minimal value of $W(\delta )$ occurs at $\delta =\Gamma_{ei}a/3$. It is thus clear that for $\Gamma_{ei} \gtrsim 0.5$ the capture of the electrons onto classical orbits becomes important and the significant contribution from the classically bound electrons cannot be avoided in the MD models with point–like classical electrons.
{width="165mm"}
The electric microfield distribution (MFD) $P(E)$ plays a central role for the line shape. Models for this distribution exist in the limits of an ideal plasma [@hol19], a weakly coupled plasma [@hoo68] and for very strongly coupled plasmas [@may47]. For intermediate cases an effective independent–particle model known as Adjustable Parameter Exponential (APEX) approximation has been developed in Refs. [@igl83; @duf85; @boe87] for an ionic OCP. It rests essentially on the pair distribution function and has been tested by comparison with MD and Monte–Carlo simulations. Recently in Refs. [@ner05; @ner06; @ner08] we have suggested the theoretical models named PMFEX (Potential of Mean Force Exponential approximation) and PMFEX+ which turn out to be a very reliable approaches for calculating the MFD of a TCP with attractive interaction. In order to cover the entire range from small to large plasma parameters we use here classical MD simulations which have been described in detail in Ref. [@zwi99] (see also Refs. [@ner05; @ner06; @ner08]). As an example the normalized MFDs from PMFEX and MD are compared in Fig. \[fig:1\] where the electric microfields are scaled in units of the Holtsmark field $E_{H}$ for a TCP (see [ner05]{} for details) $$E_{H}=\frac{C\mathcal{Z}e}{4\pi \varepsilon _{0}a^{2}},\qquad \mathcal{Z}=%
\left[ \frac{Z\left( 1+Z^{1/2}\right) }{Z+1}\right] ^{2/3}
\label{eq:15}$$with an effective charge $\mathcal{Z}$ and $C=(8\pi /25)^{1/3}$. These distributions were obtained from ensembles of fields taken at a charged reference point which is chosen to be one of the plasma ions.
The MFDs for Al$^{13+}$ TCP with temperature $500$ eV and with coupling parameters $\Gamma _{ee}=0.037$, $\Gamma _{ii}=2.65$ and $\Gamma _{ee}=0.046$, $\Gamma _{ii}=3.34$ are shown in left and right panels of Fig. \[fig:1\], respectively, for different values of $\delta$. The density of plasma electrons is measured in units of $n_0
=5\times 10^{23}$ cm$^{-3}$. The dot–dashed and dotted curves are, for comparison, the Holtsmark MFDs for a TCP with regularized Coulomb potential. Here these MFDs depend on $Z$ and $\delta$ as discussed in [@ner05]. To demonstrate the importance of the attractive interactions we also plotted the MFDs $P_{0}(E)$ resulting from the corresponding electronic and ionic OCPs with $\Gamma _{ee}$ and $\Gamma _{ii} $, respectively (open circles). To that end the distribution $Q_{0}(\mathbf{E})=P_{0}(E)/(4\pi E^2)$ of the total field $\mathbf{E}=\mathbf{E}_{1}+\mathbf{E}_{2}$ is calculated as $$Q_{0}(\mathbf{E}) =\int d\mathbf{E}_{1}d\mathbf{E}_{2} \delta \left( \mathbf{%
E}-\mathbf{E}_{1}-\mathbf{E}_{2}\right) Q_{e} (\mathbf{E}_{1}) Q_{i} (%
\mathbf{E}_{2})
\label{eq:16}$$from the MFD of the ionic OCP at a charged point $Q_{i}(\mathbf{E}_{2})$ and of the electronic OCP at a neutral reference point $Q_{e}(\mathbf{E}_{1})$. The distribution $Q_{0}(\mathbf{E})$ thus represents the MFD in a TCP assuming that the ion–electron attractive interaction is absent. Here $Q_{e}(\mathbf{E}_{1})$ and $Q_{i}(\mathbf{E}_{2})$ are taken from MD simulations of an OCP. As the thermal motion of the particles is suppressed with increasing coupling the distributions $P(E)$ and the mean electric fields are shifted towards smaller values as shown in Fig. \[fig:1\]. This figure also shows the importance of the attractive interactions in plasmas. The behavior of the MFD with respect to the variation of the parameter $\delta$ is particularly noteworthy. For fixed coupling parameters the maximum of the MFD shifts only slightly to lower field strengths $E$ with increasing $\delta$ (Fig. \[fig:1\]), while the maximum itself increases with $\delta$. This is related to the largest possible single–particle field $|E_e(0)| =e /(8\pi\varepsilon_{0}\delta^2 )$, which an electron can produce at the ion. Thus the nearest neighbor electronic MFD vanishes for electric fields larger than $|E_e(0)|$, and smaller $\delta$ will result in larger contributions to $P(E)$ at higher fields $E$ with a corresponding reduction of $P(E)$ at small fields. Therefore, the formation of the tails in the MFD and enhancement of the electric microfield at small $\delta$ may have important influence on the spectral line shapes of the radiating particles. Further examples for charged and neutral radiators, together with a detailed discussion of the limits of the PMFEX treatment at increasing coupling, are given in Refs. [@ner05; @ner06; @ner08].
Wave equation for a radiator {#sec:3}
============================
In this section we describe the solution of the wave equation for a hydrogen–like ion coupled to the time–dependent electric microfield. The microfield fluctuations in the plasma are calculated by the MD simulations as discussed in Sec. \[sec:2\].
We consider a hydrogen–like ion in a time–dependent electric microfield. The Hamiltonian is the sum of $\hat{H}_{0}$ describing the unperturbed ion and a dipole term $\hat{H}_{\mathrm{int}}=e\mathbf{r}\cdot \mathbf{E}(t)$ for the interaction between the bound electron (distance $\mathbf{r}$ from nucleus) and the microfield $\mathbf{E}(t)$, $$\hat{H}=\hat{H}_{0}+\hat{H}_{\mathrm{int}} \ .
\label{eq:1}$$The electron moves in the potential of a nucleus with charge $Z_{R}e$. In the present application it turns out that it suffices to start from the non–relativistic Schrödinger equation $$\hat{H}_{0}\left\vert \alpha \right\rangle =\left( \frac{\hat{\mathbf{p}}^{2}%
}{2m_{e}}-\frac{Z_{R}e_{S}^{2}}{r}\right) \left\vert \alpha \right\rangle =\hbar
\omega _{\alpha }\left\vert \alpha \right\rangle
\label{eq:2}$$for the time–independent electronic state $\vert \alpha \rangle $ with energy $E_{\alpha }=\hbar \omega _{\alpha }$. Here $\hat{\mathbf{p}}$ is the momentum operator and $\alpha $ is a multiindex including radial, angular momentum and spin quantum numbers. The present calculations are done in the configuration space corresponding to the solutions of Eq. . In order to discretize the continuum a boundary condition $\langle \mathbf{r}|\alpha \rangle =0$ is imposed at a radius $r=R_{0}$, which is chosen sufficiently large in order to avoid an influence on the final results. The radial wave functions with this boundary condition are still confluent hypergeometric functions, but the radial quantum numbers of bound states are not integers any more [@mar03]. In order to obtain a finite basis the (former) continuum states are cut off at sufficiently large quantum numbers. Alternatively the continuum could be handled by forming wave packets with a width that must be adjusted appropriately [@mul94]. In Refs. [@mar03; @mar04] the time–dependent equation $\hat{H}\Psi (t)=%
i\hbar \dot{\Psi}(t)$ with the Hamiltonian has also been solved on a grid for the electron wave function [@cal00; @rei00]. This is more advantageous for the description of the continuum and it is easier to implement the interactions between the radiator and the plasma particles beyond the dipole term in Eq. . On the other hand spatially extended states require very large simulation boxes. Quite generally in the present context the solution on the grid is more expensive numerically than working in configuration space. We adopted the latter for the subsequent calculations.
At high $Z_{R}$ relativistic corrections must be considered and also the spin should be treated as a dynamical variable. It turns out that in the cases considered here it suffices to include the first–order fine–structure shift [@fri90] $$\Delta E_{nlj}=-\frac{Z_{R}^{2}\alpha _{S}^{2}\left\vert E_{n}\right\vert }{n%
}\left( \frac{1}{\gamma _{lj}}-\frac{3}{4n}\right) \ ,
\label{eq:3}$$where $\gamma _{lj}=j+1/2$ and $\gamma _{lj}=1$ at $l\geqslant 1$ and $l=0$, respectively. Here $n$ is the principal quantum number of the hydrogen–like ion, $E_{n}=-Z_{R}^{2}E_{B}/n^{2}$ is the corresponding energy ($E_{B}=e_{S}^{2}/2a_{B}$ is the Bohr energy), $j=l\pm 1/2$ is the total angular momentum quantum number and $\alpha _{S}\simeq 1/137$ is the Sommerfeld constant.
The dipole interaction $e\mathbf{r}\cdot \mathbf{E}(t)$ between the radiator and the plasma is time–dependent and possibly strong. Going beyond the second order treatment of Ref. [@jun00] we use the interaction picture with the unperturbed basis states given by Eq. . The time–dependent Schrödinger equation with the total Hamiltonian (\[eq:1\]) can be solved using Dirac’s method. The perturbed electron wave function is represented as a sum of wave functions of the unperturbed Hamiltonian with time–dependent coefficients $c_{\alpha }(t)$ $$\Psi (t)=\sum_{\alpha }c_{\alpha }(t)e^{-i\omega _{\alpha }t}\left\vert
\alpha \right\rangle \ .
\label{eq:4}$$A substitution of Eq. into the time–dependent Schrödinger equation and orthogonality of the spatial wave functions, i.e., $\langle \alpha |\beta \rangle %
=\delta _{\alpha \beta }$, gives the set of coupled ordinary and linear differential equations $$\dot{c}_{\alpha }(t)=-\frac{ie}{\hbar }\mathbf{E}\left( t\right) \cdot
\sum_{\beta }e^{i\omega _{\alpha \beta }t}c_{\beta }(t)\left\langle \alpha
\left\vert \mathbf{r}\right\vert \beta \right\rangle
\label{eq:5}$$which is solved iteratively. Here $\hbar \omega _{\alpha \beta }$ is the transition energy between atomic states $\alpha $ and $\beta $, i.e., $\omega _{\alpha \beta } %
=\omega _{\alpha }-\omega _{\beta }$. Within the dipole approximation the transition rate per unit time and energy interval $\mathcal{I}(\omega )$ for the emission of photons is proportional to the power spectrum of the dipole operator [@jac98]. It is defined as the square of the absolute value of the Fourier transform of the expectation value of the dipole operator. Hence we introduce $$\mathcal{I}\left( \omega \right) =\frac{2e_{S}^{2}\omega ^{3}}{3\pi c^{3}\hbar
^{2}\tau }\left\vert \int_{0}^{\tau }\mathbf{d}(t)e^{i\omega t}dt\right\vert ^{2}
\label{eq:6}$$with $\tau \to \infty $ and the expectation value of the dipole moment $$\mathbf{d}(t)=\sum_{\beta ,\alpha }e^{i\omega _{\beta \alpha }t}c_{\beta
}^{\ast }(t)c_{\alpha }(t)\left\langle \beta |\mathbf{r}|\alpha
\right\rangle .
\label{eq:7}$$
Let us now consider a transition $\alpha \to g$ downwards to a state $\vert g\rangle $ which is nearly filled, i.e. $c_{\beta }(t)=\delta _{\beta g}$. Then the dipole moment in Eq. is calculated with respect to the state $\vert g\rangle $ and $$\mathcal{I}\left( \omega \right) =\frac{2e_{S}^{2}\omega ^{3}}{3\pi c^{3}\hbar
^{2}\tau }\left\vert \sum_{\alpha }\left\langle g\left\vert \mathbf{r}%
\right\vert \alpha \right\rangle \int_{0}^{\tau }c_{\alpha }(t)e^{i\left(
\omega -\omega _{\alpha g}\right) t}dt\right\vert ^{2}.
\label{eq:8}$$ Our numerical model includes also the interaction $\hat{H}_{\mathrm{e\gamma }}$ of the radiating electron with the emitted photons which, however, has been neglected in Eq. . In this case as shown in Refs. [@mar03; @mar04] the total radiated power given in Eq. is underestimated for the excited radiators where $\vert c_{g}\vert ^{2}<1$. This can be compensated by dividing through the time–averaged occupation probability of the lower state $$\mathcal{I}\left( \omega \right) \to \frac{\mathcal{I}\left( \omega \right) }{%
\langle \left\vert c_{g}(t)\right\vert ^{2} \rangle _{t}} \ .
\label{eq:9}$$The subsequent calculations will be done in this dipole power spectrum approximation (DPSA). We have tested the validity of this approximation in the wide range of plasma and radiator parameters by comparing explicitly $\mathcal{I}(\omega )$ with the spectrum obtained from the total Hamiltonian $\hat{H}^{\prime }=\hat{H}_{0}+\hat{H}_{\mathrm{int}}%
+\hat{H}_{\mathrm{e\gamma }}$. We have found that in the parameter regime considered in Sec. \[sec:4\] the DPSA is justified as the emission of radiation through the interaction $\hat{H}_{\mathrm{e\gamma }}$ changes the occupation probabilities of the radiator’s states on a much slower scale than the fluctuating electric microfields. However, our preliminary results show that the electron–photon interaction $\hat{H}_{\mathrm{e\gamma }}$ may have an important contribution to the wings of the line especially in the case of light emitters. We intend to take up further studies on this issue in a separate paper.
At this stage we have neglected the feedback of the radiator’s excitation to the plasma. In this respect the plasma particles move as if they had an infinite mass. As they have a finite velocity it appears as if the radiating electron is embedded in a plasma of infinite temperature. Accordingly the time evolution of the total system will lead to an equal population of all electronic states. As the time–dependent feedback could be implemented only at a very great expense in the MD simulations we enforce a canonical equilibrium state of the plasma and the radiating electron by modifying the interaction in Eq. according to $$e\mathbf{r}\cdot \mathbf{E}(t)\rightarrow e^{-\beta_{e} \hat{H}_{0}/2}e\mathbf{r}%
\cdot \mathbf{E}(t)e^{\beta_{e} \hat{H}_{0}/2} .
\label{eq:10}$$The time–dependent equation describing the coupling of the microfield to the radiator is then solved for an ensemble of typically thirty independent microfields which yields the mean emission as well as the statistical error.
Results {#sec:4}
=======
Using the theoretical background introduced so far we present in this Section calculations of the shape of the Ly$_{\alpha }$–line of Al$^{12+}$ radiating ion embedded in a Al$^{13+}$–TCP in a wide range of plasma parameters. As we have mentioned in Sec. \[sec:3\] for heavy ions the relativistic corrections, i.e. the fine structure of the levels must be accounted for. Using the fine structure shift in Eq. the unperturbed Ly$_{\alpha }$ transition energy becomes $$\hbar \omega _{\mathrm{Ly}_{\alpha }}=\frac{3}{4}Z_{R}^{2}E_{B}\left(
1+C_{j}Z_{R}^{2}\alpha _{S}^{2}\right)
\label{eq:Ly}$$with $C_{1/2}=\frac{11}{48}$, $C_{3/2}=\frac{5}{16}$ and for Al$^{12+}$ ions $\hbar\omega_{\mathrm{Ly}_{\alpha }}\simeq 1728.1$ eV and $\hbar \omega _{\mathrm{Ly}%
_{\alpha }}\simeq 1729.4$ eV with $j=1/2$ and $j=3/2$, respectively.
We start from the line as it is broadened by the Al$^{13+}$ ions and electrons in the plasma. Then we fold with the weighted fine structure shift Eq. which is $\hbar \Delta \omega \simeq 1.29$ eV according to $$\mathcal{I}_{\mathrm{LS}}(\omega ) =\frac{1}{3}\mathcal{I}\left( \omega +\frac{2}{3}\Delta \omega
\right) +\frac{2}{3}\mathcal{I}\left( \omega -\frac{1}{3}\Delta \omega \right)
\label{eq:17}$$and account for the Doppler effect [@gri97; @sal98] which broadens a line with the unperturbed frequency $\omega _{0}$ according to a Gaussian distribution $$\mathcal{D}\left( \Delta \omega \right) =\frac{1}{\sqrt{2\pi }\sigma }\exp %
\left[ -\frac{1}{2}\left( \frac{\Delta \omega }{\sigma }\right) ^{2}\right] ,
\label{eq:18}$$where $\Delta \omega =\omega -\omega _{0}$ and $$\sigma ^{2}=\frac{\omega _{W}^{2}}{8\ln 2}=\omega _{0}^{2}\frac{k_{B}T}{%
M_{R}c^{2}}.
\label{eq:19}$$Here $\hbar \omega _{W}$ is the full width at half maximum (FWHM) of the line and $M_{R}$ is the radiating ion mass. Some values of FWHM $\hbar\omega _{W}$ for aluminum are shown in Table \[tab:1\]. Finally, we fold to account for the experimental resolution.
$k_{B}T$ (eV) 10 10$^{2}$ $5\times 10^{2}$ 10$^{3}$ 10$^{4}$ $10^{5}$
--------------- ------ ---------- ------------------ ---------- ---------- ----------
FWHM (eV) 0.08 0.26 0.57 0.81 2.57 8.12
: Doppler broadening (FWHM) of the Ly$_{\alpha }$–line in an Al plasma.[]{data-label="tab:1"}
We discuss now the simulated Ly$_{\alpha }$–line profiles at solid state densities $n_{0}\leqslant n_{e}\leqslant 4n_{0}$ and at $k_{B}T=500$ eV. Some results for the Ly$_{\alpha }$–line shape without Doppler broadening and LS coupling are shown in Figs. \[fig:2\]-\[fig:5\]. In all cases the ratio $a_{Z}/a\ll 1$ is small and the use of the dipole approximation in Eq. is fulfilled. The line broadening and shift towards lower photon energies (redshift) is clearly visible in Fig. \[fig:2\], which shows the line profile at fixed temperature 500 eV and different densities. Here the regularization parameter $\delta =\lambdabar_{ei}$ is determined as the thermal wavelength. With increasing density the influence of the plasma effects on the line shape becomes more pronounced. At very large densities from $n_{e}=2.6n_{0}$ (dashed line) and up to the value $n_{e}=4n_{0}$ (dash–dotted line) there is hardly any broadening and the line is only redshifted towards lower photon energies by the plasma effects. In this high density regime the fluctuating electric fields become sufficiently strong to cause asymmetric shapes due to nonlinear coupling.
To gain more insight we now fix the plasma temperature (500 eV) and the density ($n_{e}=n_{0}$) and show in Fig. \[fig:3\] the line profile for different regularization parameters $\delta $ (the lines without symbols), $\delta =0.08a$ (solid line), $\delta =0.1a$ (dashed line) and $\delta =0.4a$ (dotted line). Here the thermal wavelengths of the electrons are chosen as the relevant lengths $\delta $ at which a smoothing of the ion–electron interaction due to quantum diffraction becomes effective. For comparison we also calculate the line shapes for non–isothermic plasma with different electronic and ionic temperatures $T_{e}=500$ eV and $T_{i}=50$ eV, respectively (the lines with symbols). As shown in Fig. \[fig:3\] the width of the lines decreases with increasing parameter $\delta $, i.e. by “softening” of the ion–electron interaction. Besides, keeping the electron temperature unchanged and decreasing the ionic temperature leads to an additional broadening of the lines and this is visible for more Coulomb–like interactions with small $\delta $.
![Simulated Ly$_{\alpha }$–spectra of a Al$^{12+}$ radiating ion embedded in a Al$^{13+}$-TCP of a temperature of 500 eV and solid state densities $n_{e}=n_{0}$ (solid line), $n_{e}=1.6n_{0}$ (dotted line), $%
n_{e}=2.6n_{0}$ (dashed line) and $n_{e}=4n_{0}$ (dash-dotted line). The spectra are normalized to the area under the curves. (Online color:www.cpp-journal.org).[]{data-label="fig:2"}](fig2.eps){width="80mm"}
![Simulated Ly$_{\alpha }$–spectra of a Al$^{12+}$ radiating ion embedded in a Al$^{13+}$-TCP of a density of $n_{e}=n_{0}$. The lines without and with symbols correspond to the equilibrium ($T_{e}=T_{i}=500$ eV) and non-equilibrium ($T_{e}=500$ eV, $T_{i}=50$ eV) TCPs, respectively. The regularization parameter is $\delta =0.08a$ (solid line), $\delta =0.1a$ (dashed lines) and $\delta =0.4a$ (dotted lines). The spectra are normalized to the area under the curves. (Online color:www.cpp-journal.org).[]{data-label="fig:3"}](fig3.eps){width="80mm"}
{width="165mm"}
In Fig. \[fig:4\] we put together the results obtained for two values of plasma densities ($n_{0}$ and $2n_{0}$) and the parameter $\delta $. In the top and bottom panels we take $\delta =1.5a_{Z}$ and $\delta =7.6a_{Z}$, respectively, where $a_{Z}$ is the effective Bohr radius of Al$^{12+}$. Note that for two plasma densities $n_{0}$ and $2n_{0}$ the chosen values of $\delta$ in units of $a_Z$ are equivalent to $0.08a$, $0.1a$ and $0.4a$, $0.5a $, respectively, in units of the Wigner–Seitz radius $a$ of a TCP. We also demonstrate the influence of the electron–ion attractive interaction on the spectral line shapes plotting the spectra $\mathcal{I}_{0}(\omega)$ resulting from a superposition of the electronic and ionic OCPs (dashed lines). $\mathcal{I}_{0}(\omega)$ is calculated by folding the spectra $\mathcal{I}_{e}(\omega)$ and $\mathcal{I}_{i}(\omega)$ which are obtained from simulations of the radiative transitions of a radiator embedded in an electronic OCP and of the ionic OCP, respectively. The microfields $\mathbf{E}(t)$ in an electronic and ionic OCPs are simulated at a neutral and charged reference points, respectively. The spectrum $\mathcal{I}_{0}(\omega)$ thus represents the line shape in a TCP assuming that the ion–electron attractive interaction is switched off. As shown in Fig. \[fig:4\] the width of the line now turns out to be highly sensitive to both the choice of $\delta $ and the density $n_{e}$, with a much stronger dependence on $n_{e}$ for smaller $\delta $. The influence of the high–electric field tails in the MFDs at small $\delta$ on the spectral lines is now clearly shown in Fig. \[fig:4\]. Smaller $\delta$ results in higher electric fields which broaden the spectral lines and reduce the peak intensity. More precisely we observed that the line width behave approximately as $\hbar\Delta\omega \sim n_{e}/\delta$.
In the following we will compare our calculations with the results of experiments performed in Garching [@and02] where an Al plasma is created by the irradiation of the target with laser pulses of 150 fs duration at an intensity of a few $10^{17}$ W/cm$^2$. The systematic investigations carried out in Refs. [@mar03; @mar04] show that the standard (quasistatic and impact) approximations become doubtful if the plasma density reaches that of the solid state. In the last years experiments have approached this regime, see, e.g., [@sae99; @eid00; @and02]. We note that the theoretical model discussed so far assume a homogeneous equilibrium plasma. Obviously this is not the state in which the laser leaves the target after the irradiating pulse. In particular self–absorption due to plasma inhomogeneities leads to an additional line broadening which is difficult to analyze. Fortunately there has been considerable experimental progress to reduce the self–absorption [@and02].
Earlier experiments on the Ly$_{\alpha }$–line in Al$^{12+}$ at solid state density [@sae99; @eid00] were subject to self–absorption in the cooler and less dense surface regions of the target. This can be prevented by using thin (to reduce absorption) target layers with sharp boundaries (to enhance homogeneity). For that purpose a 25 nm Al target layer was embedded in solid carbon at depths ranging from $d=25$ nm to $d=400$ nm [@and02]. With increasing depth the expansion of the Al layer is suppressed and the homogeneity of the Al plasma is improved. In Fig. \[fig:5\] we compare our simulations with the experimental results (filled circles) for $d=400$ nm and $k_{B}T=500$ eV from which the underground has been subtracted. The fine structure and the Doppler broadening are taken into account as described above. Then the simulated Ly$_{\alpha }$–lines are folded with the experimental resolution (0.9 eV, FWHM) and compared with the experimental line assuming densities $5\times 10^{23}$ cm$^{-3}$ (dashed line) and $10^{24}$ cm$^{-3}$ (thin solid line). At these two densities the plasma parameters for the Al$^{13+}$–TCP are $\Gamma _{ii}=2.65$, $\Gamma _{ee}=0.04 $ and $\Gamma _{ii}=3.34$, $\Gamma _{ee}=0.05$, respectively. All curves in Fig. \[fig:5\] are normalized to the peak intensity. Finally, the position of the simulated line must be redshifted by 2 eV. This is the dense plasma line shift (DPLS) Ref. [@gri97] due to the screening of the electron–nucleus interaction by hot background electrons. Assuming a Debye–screened interaction instead of the $r^{-1}$–Coulomb potential first–order perturbation theory yields a shift of the required magnitude. A comparison of the two simulated curves in Fig. \[fig:5\] allows to conclude that the remaining uncertainty in the determination of the density of the target is of order $10^{23}$ cm$^{-3}$. Our results show that the quantum mechanics of close electron–ion collisions is important over and above the plasma redshift. If the quantum diffraction parameter $\delta $ is fixed at physically reasonable values near the effective Bohr radius $a_{Z}$, our calculations favor a somewhat larger density than $n_{0}$ as proposed in Ref. [@and02]. Clearly a more quantum mechanical treatment of the electron component in the plasmas is desirable.
![Comparison of the experimental line [@and02] (filled circles) with our simulation results (dashed and solid lines), i.e. with the solid curves from the top panels of Fig. \[fig:4\] after taking into account the Doppler broadening, the LS-coupling, the experimental resolution and a redshift (see text). The experimental line is subtracted by the underground. Here the curves are normalized to the peak intensity. (Online color:www.cpp-journal.org).[]{data-label="fig:5"}](fig5.eps){width="80mm"}
Conclusions {#sec:5}
===========
In this paper we have presented a model for spectral lines that works without some assumptions which underly the conventional impact and quasi–static approximations. In particular we (i) consider two–component plasma (TCP) with attractive interactions between electrons and ions, (ii) account for the strong Coulomb correlations between plasma particles, (iii) account for radiator states including the continuum, which are not directly involved in the transition, (iv) allow for a non–perturbative treatment.
We have compared our model with recent experiments on Al targets and found good agreement for the Ly$_{\alpha}$–transitions. The more exact treatments beyond the standard approximations will become highly desirable in connection with experiments at higher densities and temperatures at the planned (X)FEL facilities.
A critical discussion of our results suggests further improvements. (i) The dipole approximation in Eq. for the interaction of the microfield with the radiator suffices for the present experiments [@and02]. In even denser plasmas one must account for close collisions between the radiator and the plasma particles with a quadrupole term in the expansion of the interaction and finally with an exact treatment [@jun00]. (ii) Relativistic and spin effects beyond the simple fine structure given by Eq. can be taken into account by treating the radiator with the Dirac equation [@mul94]. (iii) The major He–like satellite is well separated from the Ly$_{\alpha }$–line in the experiment [@and02]. However, there will be closer satellites due to spectator electrons in higher configurations, which may affect the “red” shoulder of the line. For spectators in the continuum this effect merges into the DPLS. The satellites impose a challenge as they offer an additional tool to determine the temperature of the plasma, see, e.g. Ref. [@sal98]. For that purpose one has to solve the multi–electron wave equation, for example in the relativistic case the Dirac equation [@dya89]. (iv) At high densities of the laser–produced plasmas the self–absorption is likely to occur. In general, the consideration of radiative transfer in dense plasmas is involved. To estimate the effect of the self–absorption in the calculation of the line profile one–dimensional monolayer model can be used which has already been successfully applied in Ref. [@lor08] (see also references therein).
This work was supported by the Deutsche Forschungsgemeinschaft (DFG-TO 91/5-3), the Bundesministerium für Bildung und Forschung (BMBF, 06ER128) and by the Gesellschaft für Schwerionenforschung (GSI, ER/TOE).
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abstract: 'We derive a boundary monotonicity formula for a class of biharmonic maps with Dirichlet boundary conditions. A monotonicity formula is crucial in the theory of partial regularity in super-critical dimensions. As a consequence of such a boundary monotonicity formula, one is able to show partial regularity for variationally biharmonic maps and full boundary regularity for minimizing biharmonic maps.'
author:
- 'Serdar Altuntas[^1]'
date:
title: '**Derivation of a boundary monotonicity inequality for variationally biharmonic maps**'
---
Introduction
============
Over the last decades it has turned out that a monotonicity formula is necessary in super-critical dimensions to show partial regularity. Before the study of weakly biharmonic maps has begun, one has considered weakly harmonic maps. Let $\mathcal{M}$ be a smooth Riemannian manifold of dimension $m\in {\mathbb{N}}$ with or without boundary and $\mathcal{N}\subset {\mathbb{R}}^n$ be a compact Riemannian manifold without boundary. We call a map $u\in W^{1,2}(\mathcal{M},\mathcal{N})$ weakly harmonic iff it is a critical point of the so-called Dirichlet-energy $$\begin{aligned}
E_1(u)=\int_{\mathcal{M}}\vert Du\vert^2{\text{d}}\mu_{\mathcal{M}},{\addtocounter{equation}{1}\tag{\theequation}}\label{1}\end{aligned}$$ for variations of the form $u_t={\pi_{\mathcal{N}}}(u+tV)$ for $V\in C^{\infty}_0(\mathcal{M},{\mathbb{R}}^n)$. Here, ${\pi_{\mathcal{N}}}$ denotes the nearest point projection. Critical points of $E_1$ satisfy a nonlinear system of second order equations $$\begin{aligned}
\Delta u=\text{tr}(A\circ u)(Du\otimes Du){\addtocounter{equation}{1}\tag{\theequation}}\label{1b}\end{aligned}$$ in the sense of distribution with a critically nonlinear right-hand side where $\text{tr}(A)$ denotes the trace of the second fundamental form of $\mathcal{N}$. There are several regularity results of weakly harmonic maps. In 1948 C.B. Morrey [@22] showed that every minimizing map $u\in W^{1,2}(\mathcal{M},\mathcal{N})$ is $C^{\infty}$ for a manifold of dimension $\dim{\mathcal{M}}=m\leq2$. For $m=2$, F. Héléin [@23] proved that any weakly harmonic map $u\in W^{1,2}(\mathcal{M},\mathcal{N})$ is smooth inside $\mathcal{M}$. The right-hand side is a priori just in $L^1(\mathcal{M},\mathcal{N})$. Therefore, the information from is not enough to get some regularity results in dimensions $m>2$. A counter-example of T. Riviére [@24] illustrates this fact. In 1995 he constructed an everywhere discontinuous weakly harmonic map. Therefore, one has to consider stationary harmonic maps which are weakly harmonic and in addition critical points of $E_1$ for inner variations. A useful property of stationary harmonic maps is that they fulfil an energy monotonicity formula which is crucial to show partial regularity in super-critical dimensions. The first result of partial regularity for stationary harmonic maps in arbitrary compact manifolds was shown by Bethuel [@25] which is a generalisation of Evans work in [@6] where he considered maps from a subset of the Euclidean space into the unit sphere $\mathcal{N}=\mathbb{S}^{n-1}$. Another class of harmonic maps are energy minimizing harmonic maps. We call $u\in W^{1,2}(\mathcal{M},\mathcal{N})$ a minimizing harmonic map if $E_1(u)\leq E_1(v)$ for all $v\in W^{1,2}(\mathcal{M},\mathcal{N})$ such that $u-v\in W_0^{1,2}(\mathcal{M},\mathcal{N})$. R. Schoen und K. Uhlenbeck [@26; @27] established interior partial regularity and boundary regularity for minimizing harmonic maps. An analogy to weakly harmonic maps are (extrinsically[^2]) weakly biharmonic maps which are critical points of the so-called bienergy or Hessian energy $$\begin{aligned}
E_2(u)=\int_{\mathcal{M}}\vert \Delta u\vert^2{\text{d}}\mu_{\mathcal{M}}.{\addtocounter{equation}{1}\tag{\theequation}}\label{1c}\end{aligned}$$ They were firstly studied by S.-Y. A. Chang, L. Wang and P. C. Yang in [@4] in domains of dimension greater than or equal four into spheres. Again, a monotonicity formula for stationary biharmonic maps in super-critical dimensions was crucial to show interior partial regularity. However, they derived this monotonicity formula only for sufficiently regular maps. G. Angelsberg [@1] gave a rigorous proof of this monotonicity formula for stationary biharmonic maps $u\in W^{2,2}(B_r,\mathcal{N})$. A monotonicity formula for intrinsically stationary biharmonic maps was derived by R. Moser [@30]. In the case of minimizing maps, M.-C. Hong and C. Wang [@28] showed that any minimizing biharmonic map for $\mathcal{N}=\mathbb{S}^{n-1}$ is smooth off a singular set $\Sigma$ whose Hausdorff dimension is at most $m-5$, where $m\in {\mathbb{N}}_{\geq 5}$. C. Scheven [@3] showed that for an arbitrary target manifold $\mathcal{N}$ the singular set of a minimizing biharmonic map has Hausdorff dimension at most $m-5$. A boundary regularity theory for stationary biharmonic maps was initiated by H. Gong, T. Lamm and C. Wang in [@7]. They derived a boundary monotonicity inequality for biharmonic maps of class $ W^{4,2}(\Omega,\mathcal{N})$, where $\Omega=B_R^+(a)$ is a half ball with Euclidian metric. Both assumptions are not natural. The first assumption ’$W^{4,2}$’ trivializes the regularity problem. The second assumption ’$\Omega=B_R^+(a)$’ excludes curved parts of the boundary. Therefore, a flattening of the boundary will change the bienergy functional $E_2$ by lower order terms. Furthermore, K. E. Mazowiecka [@29] proved recently in her dissertation that minimizing biharmonic maps are smooth in a full neighborhood of the boundary under the assumption that there exists a boundary monotonicity formula. However, the proof of the boundary monotonicity inequality is missing and this turns out to be technically very demanding. We derive in a boundary monotonicity inequality for a class of biharmonic maps in the function space $W^{2,2}(B_R^+,\mathcal{N})$ and close this gap in Mazowiecka’s dissertation. In this sense, we provide the last missing ingredient for the proof of the full boundary regularity of minimizing biharmonic maps. We also include the case of a curved boundary. We proceed as in [@17 Theorem 2], i.e. we consider variations of the form $u_t={\pi_{\mathcal{N}}}(u\circ \varphi_t-g\circ \varphi_t+g)$ and use the methods in [@1]. Since we allow slightly more general variations than in the case of stationary biharmonic maps, we call our maps *variationally biharmonic maps* similarly to [@18]. For the derivation of the boundary monotonicity inequality we need at first a differential equation which we derive in .\
Now, we introduce our setting and give some definitions: Let $\Omega\subset {\mathbb{R}}^m$ be a bounded domain with smooth boundary of dimension $m\geq5$ equipped with a smooth Riemannian metric $\gamma$ and $\mathcal{N}$ be a smooth, compact Riemannian manifold without boundary which is isometrically embedded in Euclidean space ${\mathbb{R}}^n$. For $$u\in W^{2,2}(\Omega,\mathcal{N}):=\left\lbrace u\in W^{2,2}(\Omega,{\mathbb{R}}^n): u(x)\in \mathcal{N} \text{ for a.e. } x\in \Omega\right\rbrace$$ satisfying Dirichlet boundary conditions $$\left(u,Du\right)\vert_{\partial \Omega}=\left(g,Dg\right)\vert_{\partial \Omega}$$ in the sense of trace for given boundary data $g\in C^3(\Omega,\mathcal{N})$ the so-called extrinsic bienergy functional is defined as $$\begin{aligned}
E(u)=\int_{\Omega}\vert \Delta_{\gamma}u\vert^2{\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{2}\end{aligned}$$ Here, $\Delta_{\gamma}:=\gamma^{ij}\left(\partial_i\partial_j-\Gamma_{ij}^k\partial_k\right)$ denotes the *Laplace-Beltrami-operator* and $\mu_{\gamma}:=\mathcal{L}^m\llcorner\sqrt{\gamma}$ stands for the Riemannian measure on $\Omega$, where $\sqrt{\gamma}:=\sqrt{\det{(\gamma_{ij})}}$ and $\Gamma_{ij}^k:=\frac{1}{2}\gamma^{kl}(\partial_i\gamma_{jl}-\partial_l\gamma_{ij}+\partial_j\gamma_{il})$ are the Christoffel-symbols of the second kind.
The *Riemannian gradient* $\operatorname{grad}_{\gamma} f(x)$ of $f\in C^1(\Omega,{\mathbb{R}})$ is defined by $\gamma\left(\operatorname{grad}_{\gamma}f(x),X\right)=X(f)$ for all $x\in \Omega$ and every vector field $X=X^i\partial_i\in C^1(\Omega,{\mathbb{R}}^n)$. In coordinates we have $\operatorname{grad}_{\gamma}f(x)=\gamma^{ij}(x)\partial_if(x)\partial_j$. The *Riemannian divergence* $\div_{\gamma}$ of a vector field $X\in C^1(\Omega,{\mathbb{R}}^n)$ is defined as the trace of the map $Y\mapsto \nabla_{Y}X$, where $\nabla$ denotes the covariant derivative. In coordinates, $\div_{\gamma}{X}=\dfrac{1}{\sqrt{\gamma}}\partial_k\left( \sqrt{\gamma}X^k\right)$.
For $\delta>0$, let $V_{\delta}$ be a neighborhood of $\mathcal{N}$, which is given by $V_{\delta}:=\left\lbrace p\in {\mathbb{R}}^n: \text{dist}(p,\mathcal{N})<\delta\right\rbrace $. Since $\mathcal{N}$ is smooth and compact, there are sufficiently small $\delta>0$, so that for all $p\in V_{\delta}$ a unique point ${\pi_{\mathcal{N}}}(p)\in\mathcal{N}$ with $\vert p-{\pi_{\mathcal{N}}}(p)\vert =\text{dist}(p,\mathcal{N})$ exists. The map ${\pi_{\mathcal{N}}}:V_{\delta}\rightarrow \mathcal{N}$ is called *nearest point projection*. The total derivative of ${\pi_{\mathcal{N}}}$ in $p\in \mathcal{N}$ is the *orthogonal projection* onto the tangential space in $p$, i.e. $D{\pi_{\mathcal{N}}}: {\mathbb{R}}^n\rightarrow T_p\mathcal{N}$. For more details see for example Moser [@15 chapter 3] or [@12 chapter 2.12.3].
A map $u\in W^{2,2}(\Omega,\mathcal{N})$ is said to be *weakly biharmonic* if and only if it satisfies $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0} E(u_t)=0{\addtocounter{equation}{1}\tag{\theequation}}\label{2b}\end{aligned}$$ for all variations of the form $u_t={\pi_{\mathcal{N}}}(u+t\psi)$ with $\psi\in C^{\infty}_0(\Omega,{\mathbb{R}}^n)$. A weakly biharmonic map $u\in W^{2,2}(\Omega,\mathcal{N})$ is called *stationary biharmonic* if it satisfies additionally for variations of the form $u_t(x)=u(x+t\xi(x))$ with $\xi\in C^{\infty}_0(\Omega,{\mathbb{R}}^m)$. We say that $u\in W^{2,2}(\Omega,\mathcal{N})$ is a *minimizing biharmonic* map if and only if $E(u)\leq E(v)$ for all $v\in W^{2,2}(\Omega,\mathcal{N})$ with $u-v\in W_0^{2,2}(\Omega,\mathcal{N})$. Clearly, minimizing biharmonic maps are stationary biharmonic. We give another class of biharmonic maps in the following
\[def\] We name a map $u\in W^{2,2}(\Omega,\mathcal{N})$ **variationally biharmonic with respect to the Dirichlet boundary conditions** $\left(u,Du\right)\vert_{\partial \Omega}=\left(g,Dg\right)\vert_{\partial \Omega}$ if it is weakly biharmonic and satisfies for variations of the form $u_t(x)={\pi_{\mathcal{N}}}(u\circ \varphi_t-g\circ \varphi_t+g)$. Here, $\varphi_t$ is a $C^{\infty}$-family of diffeomorphisms from $\Omega$ into $\Omega$ that satisfy $\varphi(\partial\Omega)\subset \partial\Omega$, $\varphi_0=id_{\Omega}$, $\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\varphi_t(x)=\xi(x)\in C^{\infty}(\Omega,{\mathbb{R}}^m)$.
Throughout, we use the following notations $$\begin{aligned}
\int_{\partial B_r^+\setminus \partial B_{\rho}^+}f:=\int_{\partial B_r^+}f-\int_{\partial B_{\rho}^+}f,\quad & \int_{S_{r}^+\cup S_{\rho}^+}f:=\int_{S_{r}^+}f+\int_{S_{\rho}^+}f.\end{aligned}$$
Now, we state our main result:
\[thm\] For $m\in {\mathbb{N}}_{\geq5}$, let $u\in W^{2,2}(B^+,\mathcal{N})$ be a variationally biharmonic map from the half-ball $B^+:=B_R^+(a):=B_R(a)\cap {\mathbb{R}}^{m-1}\times[0,\infty)$ with center $a\in {\mathbb{R}}^{m-1}\times \left\lbrace 0\right\rbrace $ and radius $R>0$ to a Riemannian manifold $\mathcal{N}\subset {\mathbb{R}}^n$. Let $B^+$ be equipped with a general smooth Riemannian metric $\gamma$, where the metric satisfies $\gamma_{ij}(a)=\delta_{ij}$ for $1\leq i,j\leq n$ and for an ellipticity constant $G\geq1$ and a constant $H\geq 0$ the conditions $$\begin{aligned}
G^{-1}\vert \theta\vert^2\leq \sum_{i,j=1}^m\gamma_{ij}(x)\theta^i\theta^j\leq G\vert \theta\vert^2, \qquad \vert \gamma_{ij}(x)\vert\leq G, \qquad \vert \partial_k\gamma_{ij}(x)\vert,\vert \partial_l\partial_k\gamma_{ij}(x)\vert\leq H {\addtocounter{equation}{1}\tag{\theequation}}\label{1}\end{aligned}$$ for all $x\in B^+$, $\theta=(\theta^1,\ldots,\theta^m)\in {\mathbb{R}}^m$. Furthermore, we denote the curved and flat part of $\partial B^+$ by $S_R^+:=\partial B_R\cap \left\lbrace x\in {\mathbb{R}}^m:x^m>0\right\rbrace$ and $T_R:=\partial B_R^+\cap \left\lbrace x\in {\mathbb{R}}:x^m=0\right\rbrace$. Suppose that the Dirichlet boundary conditions $\left(u,Du\right)\vert_{T_R}=\left(g,Dg\right)\vert_{T_R}$ hold for given boundary data $g\in C^3(B^+,\mathcal{N})$. Then, there are constants $\chi=\chi(\mathcal{N},G,H,\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B^+_1)})\geq0$, $\mathsf{C}_1,\ldots,\mathsf{C}_7\geq0$, so that $$\begin{aligned}
&{\text{e}}^{\chi r}r^{4-m}\int_{B_r^+(a)}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}-{\text{e}}^{\chi\rho}\rho^{4-m}\int_{B_{\rho}^+(a)}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}+\mathsf{C}_1 r\\
&\quad+\mathsf{C}_2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{5-m}\int_{B^+_{\tau}(a)}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+ \mathsf{C}_3\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}(a)} \vert Du\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad +\mathsf{C}_4\int_{ B^+_{r}(a)\setminus B^+_{\rho}(a)}{\text{e}}^{\chi\vert x-a\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x-a\vert^{m-5}}{\text{d}}\mu_{\gamma}+\mathsf{C}_5\int_{ B^+_{r}(a)\setminus B^+_{\rho}(a)}{\text{e}}^{\chi\vert x-a\vert}\dfrac{\vert Du\vert^2 }{\vert x-a\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+\mathsf{C}_6\int_{S_{r}^+(a)\cup S_{\rho}^+(a)}{\text{e}}^{\chi\vert x-a\vert}\dfrac{\vert D^2u\vert^2 }{\vert x-a\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\mathsf{C}_7\int_{S_{r}^+(a)\cup S_{\rho}^+(a)}{\text{e}}^{\chi\vert x-a\vert}\dfrac{\vert Du\vert^2}{\vert x-a\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}\\
&\geq 4\int_{B^+_r(a)\setminus B^+_{\rho}(a)}{\text{e}}^{\chi\vert x-a\vert}\left( \dfrac{ (u_i+u_{ij}(x-a)^j)^2}{\vert x-a\vert^{m-2}}+\dfrac{(m-2)\vert Du\cdot (x-a)\vert^2}{\vert x-a\vert^{m}}\right){\text{d}}\mu_{\gamma}\label{57}\tag{M}\\
&\quad+2\int_{S^+_r(a)\setminus S^+_{\rho}(a)}{\text{e}}^{\chi\vert x-a\vert}\left(-\dfrac{ u_{i} u_{ij}(x-a)^j}{\vert x-a\vert^{m-3}}+ 2\dfrac{ \vert Du\cdot (x-a)\vert^2}{\vert x-a\vert^{m-1}}-2\dfrac{ \vert Du\vert^2}{\vert x-a\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\end{aligned}$$ holds for a.e. $0<\rho<r<R$, where $\mathsf{C}_1,\ldots,\mathsf{C}_7$ depend on $m,\mathcal{N}, G, H$ and $\Vert Dg\Vert_{C^2}$ and $\mathsf{C}_1,\ldots,\mathsf{C}_5$ additionally on $\Vert u\Vert_{L^{\infty}(B^+_1)}$. Moreover, $\chi$ and $\mathsf{C}_1$ to $\mathsf{C}_7$ vanish for $Dg\rightarrow0$ in $C^2$ and for constant metric $\gamma$.
There are two known consequences of such a boundary monotonicity inequality similar to . The first one was shown by H. Gong, T. Lamm and C. Wang [@7]. They obtained the following result: if $u$ is a stationary biharmonic map that satisfies a certain boundary monotonicity inequality, then there exists a closed subset $\Sigma\subset\bar{\Omega}$, with ${\mathcal{H}}^{m-4}(\Sigma)=0$, such that $u\in C^{\infty}(\bar{\Omega}\setminus \Sigma,\mathcal{N})$. The second one was established by K. Mazowiecka [@29]. She proved that every minimizing biharmonic map which satisfies a certain boundary inequality is smooth on a full neighborhood of the boundary $\partial\Omega$. In both proofs, $\Omega$ is a subset of ${\mathbb{R}}^m$, $m\geq5$, with Euclidean metric and $\left(u,\dfrac{\partial u}{\partial \nu}\right)\bigg\vert_{\partial\Omega}=\left(\phi,\dfrac{\partial \phi}{\partial \nu}\right)\bigg\vert_{\partial\Omega}$ for a given $\phi\in C^{\infty}(\Omega_{\delta},\mathcal{N})$ where $\Omega_{\delta}=\left\lbrace x\in \bar{\Omega}: \text{dist}(x,\partial\Omega)<\delta\right\rbrace $ for some $\delta>0$. Here, $\nu$ denotes the outer normal vector.
Differential equation for variational biharmonic maps {#sec2}
=====================================================
The starting point for our derivation of the boundary monotonicity inequality is the differential equation in the following
\[lemma1\] Let $u\in W^{2,2}(\Omega,\mathcal{N})$ be a variational biharmonic with respect to the Dirichlet boundary conditions $\left(u,Du\right)\vert_{\partial \Omega}=\left(g,Dg\right)\vert_{\partial \Omega}$, then the following differential equation holds for all $\xi\in C^{\infty}(\Omega,{\mathbb{R}}^m)$ with $\xi\in T_x(\partial \Omega)$ for every $x\in \partial\Omega$: $$\begin{aligned}
&\int_{\Omega}\left(4\Delta_{\gamma}u\cdot D^2u\operatorname{grad}_{\gamma}\xi+2\Delta_{\gamma}u\cdot Du\Delta_{\gamma}\xi-\vert \Delta_{\gamma}u\vert^2\div_{\gamma}\xi\right){\text{d}}\mu_{\gamma}\label{7}\tag{D} \\
&=\int_{\Omega}\left(2\Delta_{\gamma}u\cdot \Delta_{\gamma}\left[D{\pi_{\mathcal{N}}}(u)\left(Dg\xi\right)\right]+2\Delta_{\gamma}u\cdot \partial_l\gamma^{ij}\xi^l\partial_i\partial_ju-2\Delta_{\gamma}u\cdot \partial_l\left(\gamma^{ij}\Gamma_{ij}^k\right)\xi^l\partial_ku\right){\text{d}}\mu_{\gamma}\end{aligned}$$
Here, ’$\partial_i$’ denotes partial derivation with respect to $x^i$.
Let $\varphi_t$ be as in with $\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\varphi_t=\xi\in C^{\infty}(\Omega,{\mathbb{R}}^m)$. We consider the variation ${\mathcal{U}}_t(x):=u\circ \varphi_t(x)-g\circ \varphi_t(x)+g(x)$ in ${\mathbb{R}}^n$. For $x\in \partial\Omega$ it holds ${\mathcal{U}}_t(x)=g(x)$ and $\partial_l{\mathcal{U}}_t(x)=\partial_lg(x)$ for all $1\leq l\leq m$. So, ${\mathcal{U}}_t$ satisfies the boundary conditions, and it holds ${\mathcal{U}}_0=u$. Since $u\circ \varphi_t(x)\in \mathcal{N}$, the image of ${\mathcal{U}}_t(x)$ is for sufficiently small $\vert t\vert$ in a neighborhood of $\mathcal{N}$, i.e. in the domain of $\pi_{\mathcal{N}}$. Thus, we consider the variation $x\mapsto {\pi_{\mathcal{N}}}({\mathcal{U}}_t(x))=:u_t(x)$ and therefore the following functional $$\begin{aligned}
\int_{\Omega}\vert \Delta_{\gamma} u_t(x)\vert^2{\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{8}\end{aligned}$$ With the transformation $x\mapsto \varphi^{-1}_t(x)$ we get $$\begin{aligned}
\int_{\Omega}\vert \Delta_{\gamma} u_t(x)\vert^2{\text{d}}\mu_{\gamma}&=\int_{\Omega}\vert \Delta_{\gamma}u_t\circ\varphi_t^{-1}(x)\vert^2 \det{D\varphi_t^{-1}(x)}\sqrt{\gamma\circ\varphi_t^{-1}(x)}{\text{d}}\mathcal{L}^m(x)\\
&:=\int_{\Omega}f(t,x){\text{d}}\mathcal{L}^m.{\addtocounter{equation}{1}\tag{\theequation}}\label{9}\end{aligned}$$ To derive the equation we differentiate the functional with respect to $t$ and evaluate the result at $t=0$. Since we consider variational biharmonic maps, it holds $$\begin{aligned}
0&=\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\int_{\Omega}f(t,x){\text{d}}\mathcal{L}^m.{\addtocounter{equation}{1}\tag{\theequation}}\label{9b}\end{aligned}$$ For the sake of clarity we omit the argument ’$x$’. Now, it holds $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}f(t,x)&=2 \Delta_{\gamma}u\cdot \left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\Delta_{\gamma}u_t\circ\varphi_t^{-1}\right) \sqrt{\gamma}+\vert \Delta_{\gamma}u\vert^2\cdot \left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0} \det{D\varphi_t^{-1}} \right)\sqrt{\gamma}\\
&\quad+\vert \Delta_{\gamma}u\vert^2\cdot \left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\sqrt{\gamma\circ\varphi_t^{-1}}\right){\addtocounter{equation}{1}\tag{\theequation}}\label{10}.\end{aligned}$$ In six steps we compute the following three terms, $
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\Delta_{\gamma}u_t\circ\varphi_t^{-1}$, $\displaystyle \dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0} \det{D\varphi_t^{-1}} $ and $\displaystyle \dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\sqrt{\gamma\circ\varphi_t^{-1}}$:\
*Step 1*. We have $$\begin{aligned}
\Delta_{\gamma}u_t=\gamma^{ij}\left(\partial_i\partial_ju_t-\Gamma^k_{ij}\partial_ku_t \right){\addtocounter{equation}{1}\tag{\theequation}}\label{12}\end{aligned}$$ where $$\begin{aligned}
\partial_ku_t=\partial_k\left[{\pi_{\mathcal{N}}}({\mathcal{U}}_t)\right]=D{\pi_{\mathcal{N}}}({\mathcal{U}}_t)(\partial_k{\mathcal{U}}_t){\addtocounter{equation}{1}\tag{\theequation}}\label{13}\end{aligned}$$ with $$\begin{aligned}
\partial_k{\mathcal{U}}_t=\left( \partial_l(u-g)\circ \varphi_t \right)\partial_k\varphi_t^l+\partial_kg{\addtocounter{equation}{1}\tag{\theequation}}\label{14}\end{aligned}$$ and $$\begin{aligned}
\partial_i\partial_ju_t&=\partial_i\partial_j\left[{\pi_{\mathcal{N}}}({\mathcal{U}}_t)\right] = D^2{\pi_{\mathcal{N}}}({\mathcal{U}}_t)(\partial_i{\mathcal{U}}_t,\partial_j{\mathcal{U}}_t)+D{\pi_{\mathcal{N}}}({\mathcal{U}}_t)(\partial_i\partial_j{\mathcal{U}}_t){\addtocounter{equation}{1}\tag{\theequation}}\label{15}
\end{aligned}$$ with $$\begin{aligned}
\partial_i\partial_j{\mathcal{U}}_t&=\left( \partial_k\partial_l(u-g)\circ \varphi_t \right)\partial_i\varphi_t^l\partial_j\varphi_t^k+\left( \partial_k(u-g)\circ \varphi_t \right)\partial_i\partial_j\varphi_t^k+\partial_i\partial_jg.{\addtocounter{equation}{1}\tag{\theequation}}\label{16}
\end{aligned}$$ Now, we get from $$\begin{aligned}
\Delta_{\gamma}u_t\circ\varphi_t^{-1}&=\gamma^{ij}\circ \varphi_t^{-1}\left(\partial_i\partial_ju_t\circ \varphi_t^{-1}-\Gamma^k_{ij}\circ \varphi_t^{-1}\partial_ku_t\circ \varphi_t^{-1} \right).{\addtocounter{equation}{1}\tag{\theequation}}\label{17}\end{aligned}$$ Due to and we have $$\begin{aligned}
\partial_ku_t\circ \varphi_t^{-1}=D{\pi_{\mathcal{N}}}({\mathcal{U}}_t\circ \varphi_t^{-1})(\partial_k{\mathcal{U}}_t\circ \varphi_t^{-1}){\addtocounter{equation}{1}\tag{\theequation}}\label{18}\end{aligned}$$ and $$\begin{aligned}
\partial_i\partial_ju_t\circ \varphi_t^{-1}&= D^2{\pi_{\mathcal{N}}}({\mathcal{U}}_t\circ \varphi_t^{-1})(\partial_i{\mathcal{U}}_t\circ \varphi_t^{-1},\partial_j{\mathcal{U}}_t\circ \varphi_t^{-1})\\
&\quad+D{\pi_{\mathcal{N}}}({\mathcal{U}}_t\circ \varphi_t^{-1})(\partial_i\partial_j{\mathcal{U}}_t\circ \varphi_t^{-1}).{\addtocounter{equation}{1}\tag{\theequation}}\label{19}
\end{aligned}$$
*Step 2*. From we obtain by using the product rule, $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\Delta_{\gamma}u_t\circ\varphi_t^{-1}&=\left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\gamma^{ij}\circ \varphi_t^{-1}\right)\cdot\left(\partial_i\partial_ju-\Gamma_{ij}^k\partial_ku\right)+\gamma^{ij}\cdot\left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{i}\partial_{j}u_t\circ \varphi_t^{-1}\right)\\
&\quad-\gamma^{ij}\cdot\left[ \left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\Gamma^k_{ij}\circ \varphi_t^{-1}\right) \cdot \partial_ku+\Gamma^k_{ij}\cdot \left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{k}u_t\circ \varphi_t^{-1} \right)\right].{\addtocounter{equation}{1}\tag{\theequation}}\label{20}\end{aligned}$$
*Step 3*. Next, we compute $\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{j }u_t\circ \varphi_t^{-1} $ and $ \dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{i}\partial_{j}u_t\circ \varphi_t^{-1}$. Due to we have $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{k}u_t\circ \varphi_t^{-1} &=D^2{\pi_{\mathcal{N}}}(u)\left(\partial_ku,\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}{\mathcal{U}}_t\circ\varphi_t^{-1}\right)+D{\pi_{\mathcal{N}}}(u)\left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_k{\mathcal{U}}_t\circ\varphi_t^{-1}\right){\addtocounter{equation}{1}\tag{\theequation}}\label{21}.\end{aligned}$$ Since ${\mathcal{U}}_t\circ\varphi_t^{-1}=(u-g)+g\circ \varphi_t^{-1}$, it holds $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}{\mathcal{U}}_t(\varphi_t^{-1})&=Dg\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\varphi_t^{-1}=-Dg\xi,{\addtocounter{equation}{1}\tag{\theequation}}\label{22}\end{aligned}$$ where we used in the last step that $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\varphi_t^{-1}=-\xi,{\addtocounter{equation}{1}\tag{\theequation}}\label{24}\end{aligned}$$ which is a consequence of the chain rule and the fact $\dfrac{{\text{d}}}{{\text{d}}t}\varphi_t=\xi$. Equation yields $$\begin{aligned}
\partial_k{\mathcal{U}}_t\circ\varphi_t^{-1}=\left( \partial_l(u-g) \right)\partial_k\varphi_t^l\circ\varphi_t^{-1}+\partial_kg\circ\varphi_t^{-1}{\addtocounter{equation}{1}\tag{\theequation}}\label{26}.\end{aligned}$$ Consequently, we get with the equation $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_k{\mathcal{U}}_t\circ\varphi_t^{-1}&=\left( \partial_l(u-g) \right)\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\left(\partial_k\varphi_t^l\circ\varphi_t^{-1}\right)+\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\left(\partial_kg\circ\varphi_t^{-1}\right)\\
&=\partial_l(u-g) \dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\left(\partial_k\varphi_t^l\circ\varphi_t^{-1}\right)-\partial_l\partial_kg\xi^l\\
&=\partial_l(u-g)\partial_k\xi^l-\partial_l\partial_kg\xi^l=D(u-g)\partial_k\xi-\partial_kDg\xi{\addtocounter{equation}{1}\tag{\theequation}}\label{29}.\end{aligned}$$ We put the equations and into , and obtain $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{k }u_t\circ \varphi_t^{-1} &=-D^2{\pi_{\mathcal{N}}}(u)\left(\partial_ku,Dg\xi\right)+D{\pi_{\mathcal{N}}}(u)\left(D(u-g)\partial_k\xi-\partial_kDg\xi\right)\\
&=-D^2{\pi_{\mathcal{N}}}(u)\left(\partial_ku,Dg\xi\right)+Du\partial_k\xi-D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_k\xi+\partial_kDg\xi\right).
{\addtocounter{equation}{1}\tag{\theequation}}\label{30}\end{aligned}$$ The second equality in yields because of $D{\pi_{\mathcal{N}}}(u)(Du\partial_k\xi)=Du\partial_k\xi$, since $Du\cdot v\in T_u\mathcal{N}$ for a.e. $x\in \Omega$ and all $v\in {\mathbb{R}}^m$. Analogue to the above computations, we get from equation with and : $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{i}\partial_{j}u_t\circ \varphi_t^{-1}&= -D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right)\\
&\quad+D^2{\pi_{\mathcal{N}}}(u)\left(D(u-g)\partial_i\xi-\partial_iDg\xi,\partial_ju\right)\\
&\quad+D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,D(u-g)\partial_j\xi-\partial_jDg\xi\right)\\
&\quad- D^2{\pi_{\mathcal{N}}}(u)\left(\partial_i\partial_ju,Dg\xi\right)+D{\pi_{\mathcal{N}}}(u)\left(\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_i\partial_j{\mathcal{U}}_t\circ \varphi_t^{-1} \right){\addtocounter{equation}{1}\tag{\theequation}}\label{31}\end{aligned}$$ Due to equation we have $$\begin{aligned}
\partial_i\partial_j{\mathcal{U}}_t\circ\varphi^{-1}_t&= \partial_k\partial_l(u-g)\left(\partial_i\varphi_t^l\circ\varphi^{-1}_t\right)\left(\partial_j\varphi_t^k\circ\varphi^{-1}_t\right)\\
&\quad+\left( \partial_k(u-g)\circ \varphi_t\circ\varphi^{-1}_t \right)\partial_i\partial_j\varphi_t^k\circ\varphi^{-1}_t+\partial_i\partial_jg\circ\varphi^{-1}_t.{\addtocounter{equation}{1}\tag{\theequation}}\label{32}
\end{aligned}$$ Moreover, it holds $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0} \partial_i\partial_j\varphi^k_t\circ \varphi^{-1}_t&=\partial_i\partial_j\xi^k.{\addtocounter{equation}{1}\tag{\theequation}}\label{33}\end{aligned}$$ So, we obtain with the equations and that $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_i\partial_j{\mathcal{U}}_t\circ \varphi_t^{-1}&=\partial_{k}\partial_{l}(u-g)\partial_i\xi^{l}\delta_{jk}+\partial_{k}\partial_{l}(u-g)\partial_j\xi^{k}\delta_{il}+\partial_k(u-g)\partial_i\partial_j\xi^k-\partial_{i}\partial_j\partial_kg\xi^k\\
&=\partial_jD(u-g)\partial_i\xi+\partial_iD(u-g)\partial_j\xi+D(u-g)\partial_i\partial_j\xi-\partial_i\partial_jDg\xi{\addtocounter{equation}{1}\tag{\theequation}}\label{34},\end{aligned}$$ whereby equation becomes $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\partial_{i}\partial_{j}u_t\circ \varphi_t^{-1}&= -D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right)\\
&\quad+D^2{\pi_{\mathcal{N}}}(u)\left(D(u-g)\partial_i\xi-\partial_iDg\xi,\partial_ju\right)\\
&\quad+D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,D(u-g)\partial_j\xi-\partial_jDg\xi\right)\\
&\quad- D^2{\pi_{\mathcal{N}}}(u)\left(\partial_i\partial_ju,Dg\xi\right)\\
&\quad+D{\pi_{\mathcal{N}}}(u)\left(\partial_iD(u-g)\partial_j\xi+\partial_jD(u-g)\partial_i\xi\right)\\
&\quad-D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\partial_j\xi+\partial_i\partial_jDg\xi\right)+Du\partial_i\partial_j\xi{\addtocounter{equation}{1}\tag{\theequation}}\label{36}\end{aligned}$$ where we used $D\pi(u)(Du\partial_i\partial_j\xi)=Du\partial_i\partial_j\xi$ in the last line. Since $Du\cdot v\in T_u\mathcal{N}$ for a.e. $x\in \Omega$ and all $v\in {\mathbb{R}}^m$, it holds $Du\partial_i\xi=D{\pi_{\mathcal{N}}}(u)(Du\partial_i\xi)$. Differentiating this with respect to $x^j$, we get $$\begin{aligned}
\partial_jDu\partial_i\xi+Du\partial_i\partial_j\xi&=D^2{\pi_{\mathcal{N}}}(u)(Du\partial_i\xi,\partial_ju)+D{\pi_{\mathcal{N}}}(u)(\partial_jDu\partial_i\xi)\\
&\quad+D{\pi_{\mathcal{N}}}(u)(Du\partial_i\partial_j\xi).{\addtocounter{equation}{1}\tag{\theequation}}\label{38}\end{aligned}$$ Equation becomes due to the identity $D{\pi_{\mathcal{N}}}(u)(Du\partial_i\partial_j\xi)=Du\partial_i\partial_j\xi$, $$\begin{aligned}
\partial_jDu\partial_i\xi&=D^2{\pi_{\mathcal{N}}}(u)(Du\partial_i\xi,\partial_ju)+D{\pi_{\mathcal{N}}}(u)(\partial_jDu\partial_i\xi){\addtocounter{equation}{1}\tag{\theequation}}\label{39}.\end{aligned}$$ Hence, the equation reduces to $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\left(\partial_{i}\partial_{j}u_t\circ \varphi_t^{-1} \right)&= -D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right)\\
&\quad-D^2{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\xi+\partial_iDg\xi,\partial_ju\right)-D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,Dg\partial_j\xi+\partial_jDg\xi\right)\\
&\quad- D^2{\pi_{\mathcal{N}}}(u)\left(\partial_i\partial_ju,Dg\xi\right)\\
&\quad-D{\pi_{\mathcal{N}}}(u)\left(\partial_iDg\partial_j\xi+\partial_jDg\partial_i\xi\right)-D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\partial_j\xi+\partial_i\partial_jDg\xi\right)\\
&\quad+\partial_jDu\partial_i\xi+\partial_iDu\partial_j\xi+Du\partial_i\partial_j\xi.{\addtocounter{equation}{1}\tag{\theequation}}\label{40}\end{aligned}$$
*Step 4*. Furthermore, we have because of , $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\gamma^{ij}\circ \varphi_t^{-1}=-\partial_l\gamma^{ij}\xi^l\text{ and }\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\Gamma^k_{ij}\circ \varphi_t^{-1} =-\partial_l\Gamma_{ij}^k\xi^l{\addtocounter{equation}{1}\tag{\theequation}}\label{41}.\end{aligned}$$ Putting , , and into yields the following equation, $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\Delta_{\gamma}u_t\circ\varphi_t^{-1}&=-\partial_l\gamma^{ij}\xi^l\cdot\left(\partial_i\partial_ju-\Gamma_{ij}^k\partial_ku\right)+\gamma^{ij}\cdot\partial_l\Gamma_{ij}^k\xi^l\partial_ku\\
&\quad+\gamma^{ij}\cdot\left(\partial_jDu\partial_i\xi+\partial_iDu\partial_j\xi+Du\partial_i\partial_j\xi\right)\\
&\quad+\gamma^{ij}\cdot \Gamma^k_{ij}\cdot D^2{\pi_{\mathcal{N}}}(u)\left(\partial_ku,Dg\xi\right)-Du\gamma^{ij}\cdot \Gamma^k_{ij}\partial_k\xi\\
&\quad+ \gamma^{ij}\cdot \Gamma^k_{ij}\cdot D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_k\xi+\partial_kDg\xi\right)\\
&\quad-\gamma^{ij}\cdot D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right)\\
&\quad-\gamma^{ij}\cdot D^2{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\xi+\partial_iDg\xi,\partial_ju\right)\\
&\quad-\gamma^{ij}\cdot D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,Dg\partial_j\xi+\partial_jDg(x)\xi\right)\\
&\quad-\gamma^{ij}\cdot D^2{\pi_{\mathcal{N}}}(u)\left(\partial_i\partial_ju,Dg\xi\right)\\
&\quad-\gamma^{ij}\cdot D{\pi_{\mathcal{N}}}(u)\left(\partial_iDg\partial_j\xi+\partial_jDg\partial_i\xi\right)\\
&\quad-\gamma^{ij}\cdot D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\partial_j\xi+\partial_i\partial_jDg\xi\right).{\addtocounter{equation}{1}\tag{\theequation}}\label{43}\end{aligned}$$
*Step 5*. Now, we continue by determining $\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\left( \det{D\varphi_t^{-1}} \right)$. For an $(m\times m)$-matrix $(z_{\alpha\beta})_{\alpha,\beta=1}^m$ let be $\text{ad}_{jk}\left((z_{\alpha\beta})_{\alpha,\beta=1}^m\right):=(-1)^{j+k}\det \left((z_{\alpha\beta})_{\alpha,\beta=1}^m\right)_{\alpha\neq j,\beta\neq k}$ the $(m-1)\times (m-1)$-minors for all $1\leq j,k\leq m$. With the Laplacian expansion theorem we deduce $\dfrac{\partial}{\partial z_{jk}}\det((z_{\alpha\beta})_{\alpha,\beta=1}^m)=\text{ad}_{jk}((z_{\alpha\beta})_{\alpha,\beta=1}^m)$. Due to $\text{ad}_{jk}(\text{id})=\delta_{jk}$ we obtain using the chain rule and equation , $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\text{det }(D\varphi_t^{-1})&=-\div{ \xi}.{\addtocounter{equation}{1}\tag{\theequation}}\label{46}\end{aligned}$$ *Step 6*. Finally, we get by using chain rule and equation once again, $$\begin{aligned}
\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}\sqrt{\gamma\circ\varphi_t^{-1}}=-\partial_k\left(\sqrt{\gamma}\right)\xi^k.{\addtocounter{equation}{1}\tag{\theequation}}\label{47}\end{aligned}$$ Now, we put , and into , and summarize suitably. Then, we obtain $$\begin{aligned}
\dfrac{1}{\sqrt{\gamma}}\dfrac{{\text{d}}}{{\text{d}}t}\bigg\vert_{t=0}f(t,x)&=2 \Delta_{\gamma}u\cdot \partial_l\left(\gamma^{ij}\Gamma_{ij}^k\right)\xi^l\partial_ku-2 \Delta_{\gamma}u \cdot\partial_l\gamma^{ij}\xi^l\partial_i\partial_ju\\
&\quad-\vert \Delta_{\gamma}u\vert^2 \div{\xi}-\vert \Delta_{\gamma}u\vert^2 \dfrac{1}{\sqrt{\gamma}}\partial_k\left(\sqrt{\gamma}\right)\xi^k-2 \Delta_{\gamma}u\cdot\gamma^{ij} \Gamma^k_{ij} Du\partial_k\xi\\
&\quad+2 \Delta_{\gamma}u\cdot\gamma^{ij}\left(\partial_jDu\partial_i\xi+\partial_iDu\partial_j\xi+Du\partial_i\partial_j\xi\right)\\
&\quad+2 \Delta_{\gamma}u\cdot\gamma^{ij}\Gamma^k_{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_ku,Dg\xi\right)\\
&\quad+ 2 \Delta_{\gamma}u\cdot\gamma^{ij}\Gamma^k_{ij}D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_k\xi+\partial_kDg\xi\right)\\
&\quad-2 \Delta_{\gamma}u\cdot\gamma^{ij}D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right)\\
&\quad-2 \Delta_{\gamma}u\cdot\gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\xi+\partial_iDg\xi,\partial_ju\right)\\
&\quad-2 \Delta_{\gamma}u\cdot\gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,Dg\partial_j\xi+\partial_jDg\xi\right)\\
&\quad-2 \Delta_{\gamma}u\cdot \gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_i\partial_ju,Dg\xi\right)\\
&\quad-2 \Delta_{\gamma}u\cdot\gamma^{ij} D{\pi_{\mathcal{N}}}(u)\left(\partial_iDg\partial_j\xi+\partial_jDg\partial_i\xi\right)\\
&\quad-2 \Delta_{\gamma}u\cdot \gamma^{ij} D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\partial_j\xi+\partial_i\partial_jDg\xi\right).{\addtocounter{equation}{1}\tag{\theequation}}\label{48}\end{aligned}$$ Notice that $\div_{\gamma}{\xi}=\div{\xi}+\dfrac{1}{\sqrt{\gamma}}\partial_k\left(\sqrt{\gamma}\right)\xi^k=\div{\xi}+\Gamma^l_{kl}\xi^k$. Furthermore, we can rewrite $2 \Delta_{\gamma}u\cdot\gamma^{ij}\left(\partial_jDu\partial_i\xi+\partial_iDu\partial_j\xi\right)$ into $4\Delta_{\gamma}u\cdot\gamma^{ij}\partial_iDu\partial_j\xi$ and with the Riemannian Gradient $\operatorname{grad}_{\gamma}f=\gamma^{ij}\partial_if\partial_j$ into $4\Delta_{\gamma}u\cdot D^2u\operatorname{grad}_{\gamma}{\xi}$. Moreover, we have $ \gamma^{ij}Du\partial_i\partial_j\xi-\gamma^{ij}\Gamma^k_{ij}Du\partial_k\xi=Du\Delta_{\gamma}\xi$. With this abbreviations and putting into we obtain the *differential equation for variationally biharmonic maps* after reformulations, $$\begin{aligned}
&\int_{\Omega}\left( 4\Delta_{\gamma}u\cdot D^2u\operatorname{grad}_{\gamma}{\xi}+2\Delta_{\gamma}u\cdot Du\Delta_{\gamma}\xi-\vert \Delta_{\gamma}u\vert^2 \div_{\gamma}{\xi}\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&=\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad+\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\xi+\partial_iDg\xi,\partial_ju\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m(x)\\
&\quad+\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,Dg\partial_j\xi+\partial_jDg\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad+\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_i\partial_ju,Dg\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad+\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D{\pi_{\mathcal{N}}}(u)\left(\partial_iDg\partial_j\xi+\partial_jDg\partial_i\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad+\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_i\partial_j\xi+\partial_i\partial_jDg\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad-\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} \Gamma^k_{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_ku,Dg\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad-\int_{\Omega} 2 \Delta_{\gamma}u\cdot\gamma^{ij} \Gamma^k_{ij} D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_k\xi+\partial_kDg\xi\right)\sqrt{\gamma}{\text{d}}\mathcal{L}^m\\
&\quad-\int_{\Omega} 2 \Delta_{\gamma}u\cdot\partial_l\left(\gamma^{ij}\Gamma_{ij}^k\right)\xi^l\partial_ku\sqrt{\gamma}{\text{d}}\mathcal{L}^m+\int_{\Omega} 2 \Delta_{\gamma}u\cdot \partial_l\gamma^{ij}\xi^l\partial_i\partial_ju\sqrt{\gamma}{\text{d}}\mathcal{L}^m{\addtocounter{equation}{1}\tag{\theequation}}\label{49}.\end{aligned}$$ It is straightforward to see that equation is equivalent to . This concludes the proof.
Notice that equation takes the form of the equation in Lemma 1 in [@1] for Euclidean metric and constant boundary values, since the right-hand side is identical to zero in this case.
Derivation of a boundary monotonicity inequality {#sec3}
================================================
Before we start with the derivation, we want to mention the following
In our estimates we take care to produce ’good-natured’ error terms (integrals). We say that an error term is ’good-natured’ if the dimension of integration region minus number of derivatives on $u$ is greater than $\vert x\vert$-powers in the denominator. If an error term is good-natured then it vanishes for small radii.
We derive the boundary monotonicity formula in 8 steps. All constants appearing in the proof may depend on $m,\mathcal{N},G,H$. Further dependecies will be indicated in parentheses, e.g. $C_1(\Vert Dg\Vert_{L^{\infty}})$.\
*Step 1.* We set $\Omega=B^+$ in from the proof of . Now, we form the right-hand side of so that no second order derivatives of $\xi$ appear on the right-hand side of . Moreover, we split $\displaystyle\int_{B^+}2\Delta_{\gamma}u\cdot Du \Delta_{\gamma}\xi{\text{d}}\mu_{\gamma}$ into $$\int_{B^+}2\Delta_{\gamma}u\cdot Du \gamma^{ij}\partial_i\partial_j\xi{\text{d}}\mu_{\gamma}-\int_{B^+}2\Delta_{\gamma}u\cdot Du\gamma^{ij}\Gamma^k_{ij}\partial_k\xi{\text{d}}\mu_{\gamma}$$ and $\displaystyle\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \div_{\gamma}{\xi}{\text{d}}\mu_{\gamma}$ into $\displaystyle\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \div{\xi}{\text{d}}\mu_{\gamma}+\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \Gamma^l_{kl}\xi^k{\text{d}}\mu_{\gamma}$. Then, we bring the second summand respectively on the right-hand side of . So, we get the following equation, $$\begin{aligned}
&\int_{B^+}\left( 4\Delta_{\gamma}u\cdot D^2u \operatorname{grad}_{\gamma}{\xi}+2\Delta_{\gamma}u\cdot Du \gamma^{ij}\partial_i\partial_j\xi-\vert \Delta_{\gamma}u\vert^2 \div{\xi}\right){\text{d}}\mu_{\gamma}\\
&\quad-\int_{B^+}\left( 2 \Delta_{\gamma}u\cdot D{\pi_{\mathcal{N}}}(u)\left(Dg\gamma^{ij}\partial_i\partial_j\xi\right)\right){\text{d}}\mu_{\gamma}\\
&=\int_{B^+} 2 \Delta_{\gamma}u\cdot\gamma^{ij} D^3{\pi_{\mathcal{N}}}(u)\left(\partial_iu,\partial_ju,Dg\xi\right){\text{d}}\mu_{\gamma}+\int_{B^+} 2 \Delta_{\gamma}u\cdot D^2{\pi_{\mathcal{N}}}(u)\left(\Delta_{\gamma}u,Dg\xi\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{B^+} 4 \Delta_{\gamma}u\cdot\gamma^{ij} D^2{\pi_{\mathcal{N}}}(u)\left(\partial_iu,Dg\partial_j\xi+\partial_jDg\xi\right){\text{d}}\mu_{\gamma}+\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \Gamma^l_{kl}\xi^k{\text{d}}\mu_{\gamma}\\
&\quad+\int_{B^+} 4 \Delta_{\gamma}u\cdot\gamma^{ij} D{\pi_{\mathcal{N}}}(u)\left(\partial_iDg\partial_j\xi\right){\text{d}}\mu_{\gamma}+\int_{B^+}2\Delta_{\gamma}u\cdot Du\gamma^{ij}\Gamma^k_{ij}\partial_k\xi{\text{d}}\mu_{\gamma}\\
&\quad+\int_{B^+} 2 \Delta_{\gamma}u\cdot D{\pi_{\mathcal{N}}}(u)\left(\Delta_{\gamma}(Dg)\xi\right){\text{d}}\mu_{\gamma}-\int_{B^+} 2 \Delta_{\gamma}u\cdot\gamma^{ij} \Gamma^k_{ij} D{\pi_{\mathcal{N}}}(u)\left(Dg\partial_k\xi\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{B^+} 2 \Delta_{\gamma}u\cdot \partial_l\gamma^{ij}(x)\xi^l\partial_i\partial_ju{\text{d}}\mu_{\gamma}-\int_{B^+} 2 \Delta_{\gamma}u\cdot\partial_l\left(\gamma^{ij}\Gamma_{ij}^k\right)\xi^l\partial_ku{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{58}.\end{aligned}$$ *Step 2*. Next, we estimate the left-hand side of by the right-hand side of and abbreviate the ’left-hand side of’ by $LHS$. We obtain, $$\begin{aligned}
\vert LHS\eqref{58}\vert&\leq 2C_1\int_{B^+}\vert \Delta_{\gamma}u\vert\vert Du\vert^2 \vert\xi\vert {\text{d}}\mu_{\gamma}+ C_2\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \vert\xi\vert {\text{d}}\mu_{\gamma}+ 4C_3\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert \vert D\xi\vert {\text{d}}\mu_{\gamma} \\
&\quad+4C_4\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert \vert \xi\vert {\text{d}}\mu_{\gamma}+C_5\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \vert\xi\vert {\text{d}}\mu_{\gamma}+4C_6\int_{B^+}\vert \Delta_{\gamma}u\vert \vert D\xi\vert {\text{d}}\mu_{\gamma}\\
&\quad+2C_7\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert \vert D\xi\vert {\text{d}}\mu_{\gamma}
+2C_8\int_{B^+}\vert \Delta_{\gamma}u\vert \vert \xi\vert {\text{d}}\mu_{\gamma}+2C_9\int_{B^+}\vert \Delta_{\gamma}u\vert \vert D\xi\vert {\text{d}}\mu_{\gamma}\\
&\quad+2C_{10}\int_{B^+}\vert \Delta_{\gamma}u\vert\vert D^2u\vert \vert \xi\vert {\text{d}}\mu_{\gamma}+2C_{11}\int_{B^+}\vert \Delta_{\gamma}u\vert\vert Du\vert \vert \xi\vert {\text{d}}\mu_{\gamma} {\addtocounter{equation}{1}\tag{\theequation}}\label{59},
\end{aligned}$$ where $C_1=C_1(\Vert Dg\Vert_{\infty})$, $C_2=C_2(\Vert Dg\Vert_{\infty})$, $C_3=C_3(\Vert Dg\Vert_{\infty})$, $C_4=C_4(\Vert Dg\Vert_{C^1})$, $C_6=C_6(\Vert Dg\Vert_{C^1})$, $C_8=C_8(\Vert Dg\Vert_{C^2})$ and $C_9=C_9(\Vert Dg\Vert_{\infty})$.
For all $\nu\in {\mathbb{N}}$ we choose a function $\psi_{\nu}\in C^{\infty}({\mathbb{R}}_+,[0,1])$ with $\psi_{\nu}\equiv1$ on $[0,1-\frac{1}{\nu}]$, $\psi_{\nu}\equiv0$ on $[1,\infty)$, $\psi_{\nu}'\leq0$ and $\displaystyle\int_{{\mathbb{R}}_+}\vert \psi_{\nu}'\vert=1$. Thereby, we define for $0<\tau<1$ and $a\in {\mathbb{R}}^{m-1}\times \left\lbrace 0\right\rbrace $ the cut-off function $\xi(x):={\psi_{\nu,\tau}}(x)\cdot (x-a)=\psi_{\nu}\left(\dfrac{\vert x-a\vert}{\tau}\right)\cdot (x-a)$. We assume without loss of generality that $a=0$. Thus, we have $\vert \xi\vert\leq\vert x\vert {\psi_{\nu,\tau}}$ and $\vert D\xi\vert \leq \dfrac{\vert x\vert}{\tau}\vert {\psi_{\nu,\tau}}'(x)\vert+{\psi_{\nu,\tau}}(x)$ where ${\psi_{\nu,\tau}}'(x):=\psi_{\nu}'\left(\dfrac{\vert x\vert}{\tau}\right)$. Therefore, we get for the right-hand side of , $$\begin{aligned}
&RHS\eqref{59}\\
&\leq 2C_1(\Vert Dg\Vert_{\infty})\int_{B^+}\vert \Delta_{\gamma}u\vert\vert Du\vert^2 \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+C_{12}(\Vert Dg\Vert_{\infty})\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+ 4C_{13}(\Vert Dg\Vert_{\infty})\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+ \dfrac{4}{\tau} C_{13}(\Vert Dg\Vert_{\infty})\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert \vert x\vert \vert{\psi_{\nu,\tau}}'\vert {\text{d}}\mu_{\gamma}\\
&\quad+4C_{14}(\Vert Dg\Vert_{C^1})\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+2C_{15}(\Vert Dg\Vert_{C^1})\int_{B^+}\vert \Delta_{\gamma}u\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{2}{\tau} C_{15}(\Vert Dg\Vert_{C^1})\int_{B^+}\vert \Delta_{\gamma}u\vert \vert x\vert \vert {\psi_{\nu,\tau}}'\vert {\text{d}}\mu_{\gamma}+2C_8(\Vert Dg\Vert_{C^2})\int_{B^+}\vert \Delta_{\gamma}u\vert \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+2C_{10}\int_{B^+}\vert \Delta_{\gamma}u\vert\vert D^2u\vert \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&=\int_{B^+}\left(2C_1\vert \Delta_{\gamma}u\vert\vert Du\vert^2 +4C_{14}\vert \Delta_{\gamma}u\vert \vert Du\vert+2C_8\vert \Delta_{\gamma}u\vert +2C_{10}\vert \Delta_{\gamma}u\vert\vert D^2u\vert+C_{12}\vert \Delta_{\gamma}u\vert^2\right)\vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+\int_{B^+}\left(4C_{13}\vert \Delta_{\gamma}u\vert \vert Du\vert +2C_{15}\vert \Delta_{\gamma}u\vert\right) {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{1}{\tau} \int_{B^+}\left(4C_{13}\vert \Delta_{\gamma}u\vert \vert Du\vert +2C_{15} \vert \Delta_{\gamma}u\vert\right) \vert x\vert \vert{\psi_{\nu,\tau}}'\vert {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{60}\\
&=:I+II+III,\end{aligned}$$ where $C_{12}=C_{12}(\Vert Dg\Vert_{\infty}):=C_2(\Vert Dg\Vert_{\infty})+C_5 $, $C_{13}=C_{13}(\Vert Dg\Vert_{\infty}):=C_3(\Vert Dg\Vert_{\infty})+\dfrac{1}{2} C_7$, $C_{14}=C_{14}(\Vert Dg\Vert_{C^1}):=C_4(\Vert Dg\Vert_{C^1})+\frac{1}{2}C_{11}$ and $C_{15}=C_{15}(\Vert Dg\Vert_{C^1}):=2C_6(\Vert Dg\Vert_{C^1})+C_9(\Vert Dg\Vert_{\infty})$. Now, we estimate $I$, $II$ and $III$ as follows. With the hepl of Young’s inequality, we estimate $$\begin{aligned}
I&\leq C_{16}\int_{B^+}\vert \Delta_{\gamma}u\vert^2\vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+C_{17}\int_{B^+}\vert Du\vert^4 \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma} + C_{10}\int_{B^+}\vert D^2u\vert^2 \vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+C_{18}\int_{B^+}\vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma} {\addtocounter{equation}{1}\tag{\theequation}}\label{61}\end{aligned}$$ where $C_{16}=C_{16}(\Vert Dg\Vert_{C^2}):=C_1+C_8+C_{10}+C_{12}+2C_{14}$, $C_{17}=C_{17}(\Vert Dg\Vert_{C^1}):=C_1+C_{14}$ and $C_{18}=C_{18}(\Vert Dg\Vert_{C^2}):=C_8+C_{14}$. Further, we obtain $$\begin{aligned}
&II\\
&\leq C_{19}\int_{B^+}\vert \Delta_{\gamma}u\vert^2\vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+C_{13}\int_{B^+}\vert Du\vert^4\vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+\int_{B^+}\left(\dfrac{C_{13}}{\vert x\vert}+\dfrac{C_{15}}{\vert x\vert^3}\right) {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{62}\end{aligned}$$ with $C_{19}=C_{19}(\Vert Dg\Vert_{C^1}):=2C_{13}+C_{15}$. Moreover, we get due to $\vert \Delta_{\gamma}u\vert \leq G\vert D^2u\vert+C_5\vert Du\vert$ and applying Young’s inequality, $$\begin{aligned}
&III\\
&\leq \dfrac{1}{\tau} \int_{B^+}\left(C_{20}\vert D^2u\vert^2\vert x\vert+2C_{13}G\dfrac{\vert Du\vert^2}{\vert x\vert}+C_{21}\vert Du\vert^2+\dfrac{C_{15}G}{\vert x\vert}+C_{15}C_5 \right) \vert x\vert \vert{\psi_{\nu,\tau}}'\vert {\text{d}}\mu_{\gamma},{\addtocounter{equation}{1}\tag{\theequation}}\label{63}\end{aligned}$$ where $C_{20}=C_{20}(\Vert Dg\Vert_{C^1}):=(2C_{13}+C_{15})G$ and $C_{21}=C_{21}(\Vert Dg\Vert_{C^1}):=(4C_{13}+C_{15})C_5$. Together with , and we obtain, since $\vert x\vert \leq 1$ on the domain of integration $$\begin{aligned}
&\vert LHS\eqref{58}\vert\leq I+II+III\\
&\leq \int_{B^+}\left(C_{22}\vert \Delta_{\gamma}u\vert^2+C_{23}\vert Du\vert^4+C_{10}\vert D^2u\vert^2\right)\vert x\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+\int_{B^+}\left(\dfrac{C_{13}+C_{15}+C_{18}}{\vert x\vert^3}\right) {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{1}{\tau} \int_{B^+}\left(C_{20}\vert D^2u\vert^2\vert x\vert+(2C_{13}G+C_{21})\dfrac{\vert Du\vert^2}{\vert x\vert}+\dfrac{C_{15}(G+C_5)}{\vert x\vert} \right) \vert x\vert \vert{\psi_{\nu,\tau}}'\vert {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{64}\\
&=:IV+V+VI\end{aligned}$$ with $C_{22}=C_{22}(\Vert Dg\Vert_{C^2}):=C_{16}+C_{19}$ and $C_{23}=C_{23}(\Vert Dg\Vert_{C^1}):=C_{13}+C_{17}$. Now, we multiply the inequality by ${\text{e}}^{\chi\tau}\tau^{3-m}$ and integrate over $[\rho,r]$, $$\begin{aligned}
\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\vert LHS(58)\vert{\text{d}}\tau\leq \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}(IV+V+VI){\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{65},\end{aligned}$$ where $0<\rho<r<R$. ${\psi_{\nu,\tau}}(x)$ converge to the characteristic function of $B^+_{\tau}$ as $\nu\rightarrow\infty$. Thus, applying the dominated convergence theorem and estimating $\vert x\vert<\tau$, we obtain $$\begin{aligned}
\lim_{\nu\rightarrow\infty}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}(IV+V){\text{d}}\tau&\leq \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\left(C_{22}\vert \Delta_{\gamma}u\vert^2+C_{23}\vert Du\vert^4+C_{10}\vert D^2u\vert^2\right) {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+\tilde{C}_{24}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}}\dfrac{1}{\vert x\vert^3} {\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{66},\end{aligned}$$ for a.e. $0<\rho<r< R$ where $\tilde{C}_{24}:=\tilde{C}_{24}(\Vert Dg\Vert_{C^2})$. The square roots of the eigenvalues $\lambda_l$ of $(\gamma_{ij})$ lie in $(G^{-1/2},G^{1/2})$. Hence, $\sqrt{\gamma}=\sqrt{\det{\gamma_{ij}(x)}}=\displaystyle\prod_{l=1}^m\lambda_l^{1/2}\leq G^{m/2}$. Thus, it follows for the last integral in , $$\begin{aligned}
\tilde{C}_{24}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}}\dfrac{1}{\vert x\vert^3} {\text{d}}\mu_{\gamma}{\text{d}}\tau&\leq C_{24}\cdot (r-\rho) {\addtocounter{equation}{1}\tag{\theequation}}\label{68}\end{aligned}$$ with $C_{24}=C_{24}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}})$. Furthermore, it holds by Gagliardo-Nirenberg’s interpolation inequality (cf. [@21], page 125 and 126) for $j=1$, $p=4$, $k=r=2$, $\alpha=\dfrac{1}{2}$ and $q,s=\infty$ $$\begin{aligned}
\Vert Dv\Vert_{L^4}\leq \tilde{C}_1 \Vert D^2v\Vert_{L^2}^{1/2}\Vert v\Vert_{L^{\infty}}^{1/2}+\tilde{C}_2\Vert v\Vert_{L^{\infty}}{\addtocounter{equation}{1}\tag{\theequation}}\label{70}.\end{aligned}$$ Rescaling from $B_1^+$ to $B_{\tau}^+$, we obtain the following version of this estimate: $$\begin{aligned}
\tau^{4-m}\int_{{B_{\tau}^+}}\vert Du\vert^4{\text{d}}\mu_{\gamma}&\leq C_{25}(B^+_1)\Vert u\Vert^2_{L^{\infty}(B^+_1)}\tau^{4-m}\int_{{B_{\tau}^+}}\vert D^2u\vert^2{\text{d}}\mu_{\gamma}+C_{26}(B^+_1)\Vert u\Vert^4_{L^{\infty}(B^+_1)}{\addtocounter{equation}{1}\tag{\theequation}}\label{72}\end{aligned}$$ For the right-hand side of we get with and the following estimate, $$\begin{aligned}
RHS\eqref{66}&\leq C_{22} \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert^2{\text{d}}\mu_{\gamma}{\text{d}}\tau+C_{27}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+C_{28}\cdot (r-\rho){\addtocounter{equation}{1}\tag{\theequation}}\label{73}\end{aligned}$$ where $C_{27}=C_{27}(\Vert Dg\Vert_{C^1},B^+_1,\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{10}+C_{23}\cdot C_{25}\Vert u\Vert^2_{L^{\infty}(B_1^+)}$ and\
$C_{28}=C_{28}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)} ):=C_{24}+C_{23}\cdot C_{26}\Vert u\Vert^4_{L^{\infty}(B_1^+)}{\text{e}}^{\chi R}$. Thanks to Lemma 2 in the appendix of [@1] we obtain for a.e. $0<\rho<r<R$: $$\begin{aligned}
&\lim_{\nu\rightarrow\infty}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VI{\text{d}}\tau\\
&=\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(C_{20}\dfrac{\vert D^2u\vert^2}{\vert x\vert^{m-5}} +(2C_{13}G+C_{21})\dfrac{\vert Du\vert^2}{\vert x\vert^{m-3}} +\dfrac{C_{15}(G+C_5)}{\vert x\vert^{m-3}}\right) {\text{d}}\mu_{\gamma}\\
&\leq \int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(C_{20}\dfrac{\vert D^2u\vert^2}{\vert x\vert^{m-5}} +(2C_{13}G+C_{21})\dfrac{\vert Du\vert^2}{\vert x\vert^{m-3}}\right) {\text{d}}\mu_{\gamma}+C_{29}\cdot (r-\rho){\addtocounter{equation}{1}\tag{\theequation}}\label{78}\end{aligned}$$ with $C_{29}=C_{29}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}}):=G^{m/2}{\mathcal{H}}^{m-1}(S_1^+){\text{e}}^{\chi R}C_{15}(G+C_{5})R^2$. Together with and we have, $$\begin{aligned}
&\lim_{\nu\rightarrow\infty} \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\vert LHS\eqref{58}\vert {\text{d}}\tau\leq \lim_{\nu\rightarrow\infty} \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}(IV+V+VI) {\text{d}}\tau\\
&\leq C_{22} \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert^2{\text{d}}\mu_{\gamma}{\text{d}}\tau+C_{27}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+C_{30}\cdot (r-\rho)\\
&\quad+\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(C_{20}\dfrac{\vert D^2u\vert^2}{\vert x\vert^{m-5}} +C_{31}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-3}}\right) {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{79}\end{aligned}$$ where $C_{30}=C_{30}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}}):=C_{28}+C_{29}$ and $C_{31}=C_{31}(\Vert Dg\Vert_{C^1}):=2C_{13}G+C_{21}$.\
*Step 3.* Notice that it holds $$\begin{aligned}
\lim_{\nu\rightarrow\infty}\bigg\vert \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m} LHS\eqref{58} {\text{d}}\tau\bigg\vert \leq\lim_{\nu\rightarrow\infty} \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\vert LHS\eqref{58}\vert {\text{d}}\tau\leq RHS\eqref{79}{\addtocounter{equation}{1}\tag{\theequation}}\label{80}\end{aligned}$$ because of the monotonicity for integrals. We will find an estimate for $\displaystyle\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m} LHS\eqref{58} {\text{d}}\tau.$ For that purpose we rewrite $LHS\eqref{58}$ as follows, $$\begin{aligned}
LHS\eqref{58}&=\int_{B^+}\left(4\Delta_{\gamma}u\cdot\gamma^{ij}\partial_iDu\partial_j\xi+2\Delta_{\gamma}u\cdot Du\Delta\xi-\vert \Delta_{\gamma}u\vert^2\div{\xi}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{B^+}\left(\Delta_{\gamma}u\cdot Du(\gamma^{ij}-\delta_{ij})\partial_i\partial_j\xi -\Delta_{\gamma}u\cdot D{\pi_{\mathcal{N}}}(u)\left( Dg\cdot\gamma^{ij}\partial_i\partial_j\xi \right)\right){\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{81}\\
&=:VII+VIII\end{aligned}$$ For the sake of clarity we use $f_i$ for partial derivatives $\partial_if$ and write $f_{ij}$ instead of $\partial_i\partial_jf$. Putting $\xi^k_j(x)=\dfrac{1}{\tau}{\psi_{\nu,\tau}}'\dfrac{x^jx^k}{\vert x\vert}+{\psi_{\nu,\tau}}\delta_{jk}$, $\div{\xi(x)}=\dfrac{\vert x\vert}{\tau}{\psi_{\nu,\tau}}'+m{\psi_{\nu,\tau}}$ and $\Delta \xi^k(x)=\dfrac{1}{\tau^2}{\psi_{\nu,\tau}}''x^k+\dfrac{m+1}{\tau}{\psi_{\nu,\tau}}'\dfrac{x^k}{\vert x\vert}$ where ${\psi_{\nu,\tau}}:=\psi_{\nu}\left(\dfrac{\vert x\vert}{\tau}\right)$, ${\psi_{\nu,\tau}}':=\psi_{\nu}'\left(\dfrac{\vert x\vert}{\tau}\right)$ and ${\psi_{\nu,\tau}}'':=\psi_{\nu}''\left(\dfrac{\vert x\vert}{\tau}\right)$ into $VII$ we obtain $$\begin{aligned}
VII&=4\int_{B^+}\Delta_{\gamma}u\cdot\gamma^{ik}u_{ik} {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}-m\int_{B^+}\vert \Delta_{\gamma}u\vert^2 {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{4}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot\gamma^{ij}u_{ik}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}+\dfrac{2(m+1)}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{2}{\tau^2}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k {\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}-\dfrac{1}{\tau}\int_{B^+}\vert \Delta_{\gamma}u\vert^2\vert x\vert {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{82}\end{aligned}$$ Using the Laplace-Beltrami-operator $\Delta_{\gamma}u=\gamma^{ik}u_{ik}-\gamma^{ik}\Gamma_{ik}^lu_{l}$ we can rewrite the first integral in as follows, $$\begin{aligned}
4\int_{B^+}\Delta_{\gamma}u\cdot\gamma^{ik}u_{ik} {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}=4\int_{B^+}\vert\Delta_{\gamma}u\vert^2{\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma} +4\int_{B^+}\Delta_{\gamma}u\cdot\gamma^{ik}\Gamma_{ik}^lu_{l} {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{83}\end{aligned}$$ Moreover, we rewrite the third integral in with $\gamma^{ij}=\delta^{ij}+(\gamma^{ij}-\delta^{ij})$ as follows, $$\begin{aligned}
\dfrac{4}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot\gamma^{ij}u_{ik}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}&=\dfrac{4}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot u_{jk}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{4}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot(\gamma^{ij}-\delta^{ij})u_{ik}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{84}\end{aligned}$$ So, we have with and for , $$\begin{aligned}
VII&=(4-m)\int_{B^+}\vert \Delta_{\gamma}u\vert^2 {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}-\dfrac{1}{\tau}\int_{B^+}\vert \Delta_{\gamma}u\vert^2\vert x\vert {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}+\dfrac{4}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot u_{jk}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{2(m+1)}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}+\dfrac{2}{\tau^2}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k {\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}\\
&\quad+4\int_{B^+}\Delta_{\gamma}u\cdot\gamma^{ik}\Gamma_{ik}^lu_{l} {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}+\dfrac{4}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot(\gamma^{ij}-\delta_{ij})u_{ik}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{85}\end{aligned}$$ Multiplying $VII$ by ${\text{e}}^{\chi\tau}\tau^{3-m}$ and integrating over $[\rho,r]$ yields $$\begin{aligned}
&\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VII{\text{d}}\tau\\
&=\int_{\rho}^r{\text{e}}^{\chi\tau}(4-m)\tau^{3-m}\int_{B^+}\vert \Delta_{\gamma}u\vert^2 {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\text{d}}\tau-\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\vert \Delta_{\gamma}u\vert^2 \vert x\vert{\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{4\Delta_{\gamma}u\cdot u_{jk}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2(m+1)\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{1-m}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k{\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}{\text{d}}\tau+4\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\Delta_{\gamma}u\cdot\gamma^{ik}\Gamma_{ik}^lu_{l} {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+4\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot(\gamma^{ij}-\delta_{ij})u_{ik}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{86}.\end{aligned}$$ We set $I_{\nu}(\tau):=\displaystyle\tau^{4-m}\int_{{\mathbb{R}}^m_+}\vert \Delta_{\gamma} u\vert^2{\psi_{\nu,\tau}}{\text{d}}x$. It holds $$\begin{aligned}
I_{\nu}'(\tau)&=(4-m)\tau^{3-m}\int_{B^+}\vert \Delta_{\gamma} u\vert^2{\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}-\tau^{2-m}\int_{B^+}\vert \Delta_{\gamma} u\vert^2\vert x\vert{\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{87}.\end{aligned}$$ According to Lemma 2 in the appendix of [@1] and the dominated convergence theorem, $\displaystyle\int_{\rho}^r{\text{e}}^{\chi \tau} I_{\nu}'(\tau){\text{d}}\tau$ tends for $\nu\rightarrow\infty$ to $$\begin{aligned}
\int_{\rho}^r{\text{e}}^{\chi \tau}\left((4-m)\tau^{3-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma} u\vert^2{\text{d}}x+\tau^{4-m}\int_{\partial B^+_{\tau}}\vert \Delta_{\gamma} u\vert^2{\text{d}}x\right){\text{d}}\tau=\int_{\rho}^r{\text{e}}^{\chi \tau}I'(\tau){\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{88}\end{aligned}$$ for all $\rho,r$. Since $\frac{{\text{d}}}{{\text{d}}\tau}({\psi_{\nu,\tau}}')=-\frac{1}{\tau^2}{\psi_{\nu,\tau}}''\vert x\vert$ we get by using Fubini’s theorem and applying integration by parts $$\begin{aligned}
&2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{1-m}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k{\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&=-2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k\dfrac{{\text{d}}}{{\text{d}}\tau}({\psi_{\nu,\tau}}')\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&=-2\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k\left(\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\dfrac{{\text{d}}}{{\text{d}}\tau}({\psi_{\nu,\tau}}'){\text{d}}\tau\right)\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}\\
&=-2\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k\left({\text{e}}^{\chi r}r^{3-m}\psi'_{\nu,r}-{\text{e}}^{\chi \rho}\rho^{3-m}\psi'_{\nu,\rho}\right)\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}\\
&\quad+2(3-m)\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k{\psi_{\nu,\tau}}'\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2\chi\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k {\psi_{\nu,\tau}}'\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{89}\end{aligned}$$ for a.e. $0<\rho<r<R$. Furthermore, it holds for the last two integrals in , $$\begin{aligned}
&4\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\Delta_{\gamma}u\cdot\gamma^{ik}\Gamma_{ik}^lu_{l} {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+4\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot(\gamma^{ij}-\delta_{ij})u_{ik}x^jx^k}{\vert x\vert} {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\leq 4C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+4H\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\vert \Delta_{\gamma}u\vert \vert D^2u\vert \vert x\vert^2 \vert {\psi_{\nu,\tau}}'\vert{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{90}.\end{aligned}$$ Here, we have used the inequality $$\begin{aligned}
\vert \gamma^{ij}(x)-\delta^{ij}\vert \leq \vert D\gamma^{ij}\vert \vert x\vert\leq H\vert x\vert{\addtocounter{equation}{1}\tag{\theequation}}\label{91}\end{aligned}$$ which follows from the assumption that $\gamma_{ij}(0)=\delta_{ij}$. We obtain with , and the following estimate for the right-hand side of , $$\begin{aligned}
RHS\eqref{86}&\leq \int_{\rho}^r{\text{e}}^{\chi\tau}I_{\nu}'(\tau){\text{d}}\tau+\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\left(\dfrac{4\Delta_{\gamma}u\cdot u_{jk}x^jx^k}{\vert x\vert}+\dfrac{8\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert}\right) {\psi_{\nu,\tau}}'{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad-2\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k\left({\text{e}}^{\chi r}r^{3-m}\psi'_{\nu,r}-{\text{e}}^{\chi \rho}\rho^{3-m}\psi'_{\nu,\rho}\right)\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\Delta_{\gamma}u\cdot u_k x^k {\psi_{\nu,\tau}}'\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+4C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\vert \Delta_{\gamma}u\vert \vert Du\vert {\psi_{\nu,\tau}}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+4H\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\vert \Delta_{\gamma}u\vert \vert D^2u\vert \vert x\vert^2 \vert {\psi_{\nu,\tau}}'\vert{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{92}.\end{aligned}$$ Thanks to Lemma 2 in the appendix of [@1], the dominated convergence theorem and Lebesgue’s differentiation theorem we obtain together with , $$\begin{aligned}
&\lim_{\nu\rightarrow\infty}RHS\eqref{92}\\
&=\int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau-2\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}\\
&\quad+\int_{S^+_r\setminus S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{2\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert^{m-3}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\left(\dfrac{4\Delta_{\gamma}u\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert^{m-2}}\right) {\text{d}}\mu_{\gamma}\\
&\quad+4C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert \vert Du\vert {\text{d}}\mu_{\gamma}{\text{d}}\tau+4H\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert \Delta_{\gamma}u\vert \vert D^2u\vert}{ \vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{93}.\end{aligned}$$ for a.e. $0<\rho<r<R$. With $2\vert \Delta_{\gamma}u\vert \vert Du\vert\leq \tau \vert \Delta_{\gamma}u\vert^2+\dfrac{1}{\tau} \vert Du\vert^2$ we estimate the second to last integral in as follows, $$\begin{aligned}
&4C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert \vert Du\vert {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\leq 2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+_{\tau}} \vert Du\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\leq2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+ C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}} \vert Du\vert^4 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+C_7{\text{e}}^{\chi R}G^{m/2}\mathcal{L}^m(B_1^+)\cdot (r-\rho) {\addtocounter{equation}{1}\tag{\theequation}}\label{95}\end{aligned}$$ where we applied Young’s inequality in the last step. Applying the interpolation inequality yields $$\begin{aligned}
RHS\eqref{95}&\leq C_7C_{25}\Vert u\Vert^2_{L^{\infty}(B_1^+)}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+C_{32}\cdot (r-\rho){\addtocounter{equation}{1}\tag{\theequation}}\label{96}\end{aligned}$$ where $C_{32}=C_{32}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$. Hence, $$\begin{aligned}
RHS\eqref{93}&\leq 2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+C_7C_{25}\Vert u\Vert^2_{L^{\infty}(B_1^+)}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+C_{32}\cdot (r-\rho){\addtocounter{equation}{1}\tag{\theequation}}\label{97}.\end{aligned}$$ For the last integral in we obtain due to $\vert \Delta_{\gamma} u\vert\leq G\vert D^2u\vert+C_7\vert Du\vert$, $$\begin{aligned}
&4H\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert \Delta_{\gamma}u\vert \vert D^2u\vert}{ \vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}\\
&\leq 8C_5\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2u\vert^2 }{ \vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}+24C_5^2\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert \vert D^2u\vert}{ \vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}\\
&\leq (8C_5+12C_5^2)\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2u\vert^2 }{ \vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}+12C_5^2R^2\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert^2}{ \vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{99}\end{aligned}$$ We get the second inequality in due to $\vert x\vert<R$. Additonally, we rewrite the third and fourth integral in with $\Delta_{\gamma}u=\Delta u+(\gamma^{ij}-\delta^{ij})u_{ij}-\gamma^{ij}\Gamma_{ij}^lu_l=:\Delta u+\Delta' u$ into $$\begin{aligned}
&\int_{S^+_r\setminus S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{2\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert^{m-3}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\left(\dfrac{4\Delta_{\gamma}u\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert^{m-2}}\right) {\text{d}}\mu_{\gamma}\\
&=\int_{S^+_r\setminus S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{2\Delta u\cdot u_k x^k}{\vert x\vert^{m-3}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\left(\dfrac{4\Delta u\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8\Delta u\cdot u_k x^k}{\vert x\vert^{m-2}}\right) {\text{d}}\mu_{\gamma}\\
&\quad+\int_{S^+_r\setminus S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{2\Delta'u\cdot u_k x^k}{\vert x\vert^{m-3}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\left(\dfrac{4\Delta'u\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8\Delta'u\cdot u_k x^k}{\vert x\vert^{m-2}}\right) {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{100}.\end{aligned}$$
*Step 4.* Next, we prove the following
\[lem2.6\] For arbitrary maps $u\in W^{2,2}(B^+)$ and $g\in C^3(B^+)$ with $(u,Du)\vert_{T_R}=(g,Dg)\vert_{T_R}$ in the sense of trace it holds $$\begin{aligned}
&\int_{S^+_{r}\setminus S^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2u_{ii}\cdot u_kx^k}{\vert x\vert^{m-3}}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{4 u_{ii}\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8 u_{ii}\cdot u_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\leq-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}+C_{36}(m,G,\Vert Dg\Vert_{C^1},R)\cdot (r-\rho)\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}+\dfrac{u_{ii} u_{k}x^k}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad+ 4C_5\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+12C_5\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{101},\end{aligned}$$ for a.e. $0<\rho<r<R$.
To prove we have to apply integration by parts. Hereby, derivatives of third order appear temporarily in intermediate steps. But $u$ is a $W^{2,2}$-map. Thus, we approximate $u$ by $C^{3}(B^+_{R-\varepsilon}) \ni u^{\varepsilon}:=g+\left[\zeta(x^m)\cdot (u-g)\right]\ast\eta_{\varepsilon}$, where $\eta_{\varepsilon}(x):=\varepsilon^{-m}\eta(\dfrac{x}{\varepsilon})$ , $\eta\in C^{\infty}({\mathbb{R}}^m, [0,\infty))$ with ${\text{supp}}({\eta})\subset \overline{B}_1(0)$, $\displaystyle\int_{{\mathbb{R}}^n}\eta {\text{d}}x=1$ is a mollifier and $\zeta$ is a cut-off function with $\zeta=0$ on $[0,2\varepsilon]$, $\zeta=1$ on $[3\varepsilon,\infty)$ and $\vert D\zeta\vert\leq \dfrac{2}{\varepsilon}$, $\vert D^2\zeta\vert\leq \dfrac{c}{\varepsilon^2}$. $u^{\varepsilon}$ satisfies the boundary conditions $u^{\varepsilon}=g$, $Du^{\varepsilon}=Dg$ and $D^2u^{\varepsilon}=D^2g$ on $T_{R-\varepsilon}\times [0,2\varepsilon]$. From standard properties of mollifications and Poincaré’s inequality we infer $(u^{\varepsilon}-g)\rightarrow (u-g)$ in $W^{2,2}$.
We proceed as in [@1] page 291 and approximate $u$ by $u^{\varepsilon}$ as already mentioned. We start with reformulation of the following boundary integral: $$\begin{aligned}
&\int_{S^+_{r}\setminus S^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2 u^{\varepsilon}_{ii} u^{\varepsilon}_kx^k}{\vert x\vert^{m-3}}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&=\int_{\partial B^+_{r}\setminus \partial B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2 u^{\varepsilon}_{ii} u^{\varepsilon}_kx^k}{\vert x\vert^{m-3}}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{T_{r}\setminus T_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2 g_{ii} g_kx^k}{\vert x\vert^{m-3}}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\leq \int_{S^+_{r}\setminus S^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2 u^{\varepsilon}_{ii} u^{\varepsilon}_kx^kx^j}{\vert x\vert^{m-2}}\nu^{j}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+2\Vert D^2g\Vert_{\infty}\Vert Dg\Vert_{\infty}G^{m/2}\int_{T_{r}\setminus T_{\rho}}\dfrac{{\text{e}}^{\chi \vert x\vert}}{\vert x\vert^{m-4}}{\text{d}}{\mathcal{H}}^{m-1}\\
&= \int_{B^+_{r}\setminus B^+_{\rho}}\left({\text{e}}^{\chi \vert x\vert}\dfrac{2 u^{\varepsilon}_{ii} u^{\varepsilon}_kx^kx^j}{\vert x\vert^{m-2}}\sqrt{\gamma}\right)_j{\text{d}}\mathcal{L}^{m}+2\Vert D^2g\Vert_{\infty}\Vert Dg\Vert_{\infty}G^{m/2}{\mathcal{H}}^{m-1}(T_1)\left({\text{e}}^{\chi r}r^3 -{\text{e}}^{\chi \rho}\rho^{3}\right)\\
&\leq \int_{B^+_{r}\setminus B^+_{\rho}}\left({\text{e}}^{\chi \vert x\vert}\dfrac{2 u^{\varepsilon}_{ii} u^{\varepsilon}_kx^kx^j}{\vert x\vert^{m-2}}\sqrt{\gamma}\right)_j{\text{d}}\mathcal{L}^{m}+C_{33}\cdot \left(r-\rho\right){\addtocounter{equation}{1}\tag{\theequation}}\label{102}\end{aligned}$$ where $C_{33}=C_{33}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$. Applying product and chain rule we obtain $$\begin{aligned}
&\int_{ B^+_{r}\setminus B^+_{\rho}}\left({\text{e}}^{\chi \vert x\vert}\dfrac{2u^{\varepsilon}_{ii} u^{\varepsilon}_kx^kx^j}{\vert x\vert^{m-2}}\sqrt{\gamma}\right)_j{\text{d}}\mathcal{L}^m\\
&=\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{2u^{\varepsilon}_{iij}x^j u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}+\dfrac{2u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{6u^{\varepsilon}_{ii} u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{2\chi u^{\varepsilon}_{ii} u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-3}}+\dfrac{2u^{\varepsilon}_{ii} u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}=:IX_a+IX_b {\addtocounter{equation}{1}\tag{\theequation}}\label{103}.\end{aligned}$$ Hence, $$\begin{aligned}
&IX_a-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{4 u^{\varepsilon}_{ii}\cdot u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8 u^{\varepsilon}_{ii}\cdot u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}-\dfrac{u^{\varepsilon}_{iij}x^j u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{ii} u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}=:X{\addtocounter{equation}{1}\tag{\theequation}}\label{104}.\end{aligned}$$ Now, we compute using integration by parts $$\begin{aligned}
&\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=\int_{\partial B^+_r\setminus \partial B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\nu^i\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ijk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{ 2u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ (2-m)u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad-\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma},{\addtocounter{equation}{1}\tag{\theequation}}\label{105}\end{aligned}$$ a.e. $\rho$ and $r$, where $\nu^i$ denotes the $i$-th component of the unit normal vector. We have $\nu_S=\dfrac{x}{\vert x\vert}$ on $S^+_R$ and $\nu_T=-e_{m}=-(0,\ldots,0,1)$ on $T_R$. We split the boundary integral in into flat and curved part, and keep the boundary conditions in mind. It holds $$\begin{aligned}
&\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{T_r\setminus T_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ g_{m} g_{jk}x^jx^k}{\vert x\vert^{m-2}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ijk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{ 2u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ (2-m)u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad-\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{106}\end{aligned}$$ Next, we apply integration by parts on ’$\displaystyle -\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ijk}x^jx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}$’ in : $$\begin{aligned}
&-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ijk}x^jx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=-\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+\int_{T_r\setminus T_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ g_{i} g_{ik}x^kx^m}{\vert x\vert^{m-2}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ij}x^j u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+3\dfrac{ u^{\varepsilon}_{i} u_{ik}x^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{107}\end{aligned}$$ Using , equation becomes $$\begin{aligned}
&\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}-\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\int_{T_r\setminus T_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ g_{i} g_{ik}x^kx^m}{\vert x\vert^{m-2}}-\dfrac{ g_{m} g_{jk}x^jx^k}{\vert x\vert^{m-2}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ij}x^j u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ (m-2)u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad-\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{108}\end{aligned}$$ The integral over the flat part $T_r\setminus T_{\rho}$ of the boundary can be bounded from below by $ -2C_{33}\cdot (r-\rho)$. Hence, we obtain $$\begin{aligned}
&\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\geq \int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}-\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-2C_{33}\cdot (r-\rho)\\
&\quad+\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ij}x^j u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ (m-2)u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad-\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i}x^i u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{110}\end{aligned}$$ We continue as follows using again integration by parts, $$\begin{aligned}
&\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( -\dfrac{u^{\varepsilon}_{iij}x^j u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( -\dfrac{u^{\varepsilon}_{ij}x^ix^j u^{\varepsilon}_kx^k}{\vert x\vert^{m-1}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+\int_{T_r\setminus T_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{g_{mj}x^j g_kx^k}{\vert x\vert^{m-2}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{u^{\varepsilon}_{ij}x^j u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{ii} u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{i} u^{\varepsilon}_{ij}x^j}{\vert x\vert^{m-2}}-\dfrac{(m-2)u^{\varepsilon}_{ij}x^ix^j u^{\varepsilon}_kx^k}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{ij}x^ix^j u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_{ij}x^ju^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\geq \int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( -\dfrac{u^{\varepsilon}_{ij}x^ix^j u^{\varepsilon}_kx^k}{\vert x\vert^{m-1}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-C_{33}\cdot (r-\rho)\\
&\quad+\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{u^{\varepsilon}_{ij}x^j u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{ii} u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{i} u^{\varepsilon}_{ij}x^j}{\vert x\vert^{m-2}}-\dfrac{(m-2)u^{\varepsilon}_{ij}x^ix^j u^{\varepsilon}_kx^k}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{ij}x^ix^j u^{\varepsilon}_{k}x^k}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_{ij}x^ju^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{113}\end{aligned}$$ Thus, we have $$\begin{aligned}
-\dfrac{1}{2}X&=\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ii} u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}-\dfrac{u^{\varepsilon}_{iij}x^j u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{ii}u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\geq \int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( -\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-3C_{33}\cdot (r-\rho)\\
&\quad+2\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ u^{\varepsilon}_{ij}x^j u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{u^{\varepsilon}_{ii}u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u^{\varepsilon}_{ij}x^ju^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}-\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{114}\end{aligned}$$ Moreover, it holds $$\begin{aligned}
&\int_{ B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{u^{\varepsilon}_{ii}u^{\varepsilon}_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&=\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u^{\varepsilon}_{i}x^i)^2}{\vert x\vert^{m-1}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{T_r\setminus T_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ g_mg_{k}x^k}{\vert x\vert^{m-2}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{u^{\varepsilon}_{i}u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}+\dfrac{(u^{\varepsilon}_{i})^2}{\vert x\vert^{m-2}}-\dfrac{(m-2)(u^{\varepsilon}_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad-\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi (u^{\varepsilon}_{i} x^i)^2}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{115}.\end{aligned}$$ The integral over the flat part $T_r\setminus T_{\rho}$ of the boundary can be bounded from below by $ -2C_{34}\cdot (r-\rho)$ where $C_{34}=C_{34}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$. Thereby and with follow $$\begin{aligned}
-\dfrac{1}{2}X\geq RHS\eqref{114}&\geq\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( 2\dfrac{ (u^{\varepsilon}_{i}x^i)^2}{\vert x\vert^{m-1}}-\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-C_{35}\cdot (r-\rho)\\
&\quad+2\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u^{\varepsilon}_{ij}x^j)^2}{\vert x\vert^{m-2}}-\dfrac{(u^{\varepsilon}_{i})^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u^{\varepsilon}_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad+\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u^{\varepsilon}_{ij}x^ju^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}-\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\chi (u^{\varepsilon}_{i} x^i)^2}{\vert x\vert^{m-1}}+\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma},{\addtocounter{equation}{1}\tag{\theequation}}\label{117}\end{aligned}$$ where $C_{35}=C_{35}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=3C_{33}+C_{34}$. In addition, we get by Gauss’s integration theorem (cf. [@1], page 292) $$\begin{aligned}
0&=-\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( 2\dfrac{ (u^{\varepsilon}_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+2\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( 2\dfrac{ u^{\varepsilon}_iu^{\varepsilon}_{ij}x^j}{\vert x\vert^{m-2}}+2\dfrac{(u^{\varepsilon}_{i})^2}{\vert x\vert^{m-2}}+\dfrac{\chi (u^{\varepsilon}_{i})^2}{\vert x\vert^{m-3}}+\dfrac{(u^{\varepsilon}_i)^2}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{118},\end{aligned}$$ which we add to . Hence, after suitbale reformulations we obtain the following inequality $$\begin{aligned}
X&\leq-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u^{\varepsilon}_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u^{\varepsilon}_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+2C_{35}\cdot (r-\rho)\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u^{\varepsilon}_i+u^{\varepsilon}_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u^{\varepsilon}_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u^{\varepsilon}_{i} u^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u^{\varepsilon}_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u^{\varepsilon}_{i})^2}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u^{\varepsilon}_{ij}x^ju^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}-\dfrac{u^{\varepsilon}_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u^{\varepsilon}_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}+2\dfrac{(u^{\varepsilon}_i)^2}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}+2\dfrac{u^{\varepsilon}_iu^{\varepsilon}_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}.{\addtocounter{equation}{1}\tag{\theequation}}\label{119}\end{aligned}$$ Thus, from standard properties of mollification we get $$\begin{aligned}
&\lim_{\varepsilon\searrow0}\left(IX_b+X\right)+C_{33}\cdot (r-\rho)\\
&=\int_{S^+_{r}\setminus S^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2u_{ii}\cdot u_kx^k}{\vert x\vert^{m-3}}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{4 u_{ii}\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8 u_{ii}\cdot u_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\leq-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{36}\cdot (r-\rho)\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}+\dfrac{u_{ii} u_{k}x^k}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u_{ij}x^ju_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}-\dfrac{u_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u_iu_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}+2\dfrac{(u_i)^2}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(2\dfrac{u_iu_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}+\dfrac{u_{ii} u_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma} {\addtocounter{equation}{1}\tag{\theequation}}\label{120}\end{aligned}$$ for a.e. $\rho$ and $r$ where $C_{36}=C_{36}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=2C_{35}+C_{33}$. We estimate the last three integrals in as follows, $$\begin{aligned}
&\quad-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u_{ij}x^ju_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}-\dfrac{u_{jk}x^jx^k}{\vert x\vert^{m-2}}\cdot\dfrac{u_{i}(\sqrt{\gamma})_i}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad-2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{u_iu_{ik}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}+2\dfrac{(u_i)^2}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(2\dfrac{u_iu_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{(\sqrt{\gamma})_i}{\sqrt{\gamma}}+\dfrac{u_{ii} u_{k}x^k}{\vert x\vert^{m-2}}\cdot\dfrac{x^j(\sqrt{\gamma})_j}{\sqrt{\gamma}}\right){\text{d}}\mu_{\gamma} \\
&\leq 8C_5\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{\vert D^2 u\vert \vert Du\vert }{\vert x\vert^{m-4}}+\dfrac{ \vert Du\vert^2 }{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\leq 4C_5\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+12C_5\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{121}.\end{aligned}$$ Altogether, we have $$\begin{aligned}
&\int_{S^+_{r}\setminus S^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{2u_{ii}\cdot u_kx^k}{\vert x\vert^{m-3}}\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{4 u_{ii}\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8 u_{ii}\cdot u_kx^k}{\vert x\vert^{m-2}}\right){\text{d}}\mu_{\gamma}\\
&\leq-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{36}\cdot (r-\rho)\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}+\dfrac{u_{ii} u_{k}x^k}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad +4C_5\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+12C_5\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{122}.\end{aligned}$$ This concludes the proof of .
*Step 5*. We continue with an estimate of the last integral in . Since $\vert \Delta'u\vert \leq H\vert x\vert \vert D^2u\vert+C_7\vert Du \vert$, it holds $$\begin{aligned}
&-\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\left(\dfrac{4\Delta'u\cdot u_{jk}x^jx^k}{\vert x\vert^{m-2}}+\dfrac{8\Delta'u\cdot u_k x^k}{\vert x\vert^{m-2}}\right) {\text{d}}\mu_{\gamma}\\
&\leq \int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{4(H\vert x\vert \vert D^2u\vert +C_7\vert Du\vert ) \vert D^2u\vert }{\vert x\vert^{m-4}}+\dfrac{8(H\vert x\vert \vert D^2u\vert +C_7\vert Du\vert ) \vert Du\vert }{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\leq C_{37}\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{ \vert D^2u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+C_{38}\int_{B_{r}^+\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{124}\end{aligned}$$ where $C_{37}:=8H+2C_7$ and $C_{38}:=4H+6C_7$. Furthermore, we estimate the second to last integral in . Due to $$\begin{aligned}
\vert \Delta' u\vert=\vert(\gamma^{ij}-\delta^{ij})u_{ij}-\gamma^{ij}\Gamma_{ij}^lu_l\vert\leq H\vert x\vert \vert D^2u\vert+C_7\vert Du\vert{\addtocounter{equation}{1}\tag{\theequation}}\label{125}\end{aligned}$$ we get the following estimate, $$\begin{aligned}
&\int_{S^+_r\setminus S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{2\Delta'u\cdot u_k x^k}{\vert x\vert^{m-3}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\leq \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{2(H\vert x\vert \vert D^2u\vert +C_7\vert Du\vert ) \vert Du\vert }{\vert x\vert^{m-4}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\leq H\int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+(2C_7+H)\int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}{\addtocounter{equation}{1}\tag{\theequation}}\label{126},\end{aligned}$$ where we used $2\vert D^2u\vert \vert Du\vert\leq \vert D^2u\vert^2\vert x\vert+ \vert Du\vert^2/\vert x\vert$ in the last step. Further, we rewrite the second integral in with $\Delta_{\gamma}=\Delta+\Delta'$ as follows, $$\begin{aligned}
-2\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta_{\gamma}u\cdot u_k x^k}{\vert x\vert^{m-3}}&=-2\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{u_{ii} u_k x^k}{\vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}\underbrace{-2\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta'u\cdot u_k x^k}{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}}_{=:XI}{\addtocounter{equation}{1}\tag{\theequation}}\label{127}.\end{aligned}$$ It follows due to : $$\begin{aligned}
XI &\leq 2H\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{\vert D^2u\vert \vert Du\vert }{\vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}+2C_7\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-4}} {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{128}\end{aligned}$$ Using $2\vert D^2u\vert \vert Du\vert\leq \vert D^2u\vert^2+ \vert Du\vert^2$ yields $$\begin{aligned}
RHS\eqref{128}&\leq H\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}+H\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{ \vert Du\vert^2 }{\vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}\\
&\quad+2C_7\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-4}} {\text{d}}\mu_{\gamma}\\
&\leq H\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}+(2C_7R+HR^2)\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert }\dfrac{ \vert Du\vert^2 }{\vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{130}.\end{aligned}$$ Because of , , , , , and the following estimate holds, $$\begin{aligned}
&\lim_{\nu\rightarrow\infty}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VII{\text{d}}\tau\leq RHS\eqref{93}\\
&\leq \int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau\bcancel{-2\chi\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{u_{ii} u_k x^k}{\vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}}+C_{39}\cdot (r-\rho)\\
&\quad-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}+\bcancel{\dfrac{u_{ii} u_{k}x^k}{\vert x\vert^{m-3}}}\right){\text{d}}\mu_{\gamma}\\
&\quad +C_{40}\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+C_{41}\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+H\int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+(2C_7+H)\int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+C_{42}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{131}\end{aligned}$$ where $C_{39}=C_{39}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{32}+C_{36}$, $C_{40}=C_{40}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=8C_5+12C_5^2+H\chi+C_{37}$, $C_{41}=C_{41}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=(2C_7R+HR^2)\chi+12C_5^2R^2+C_{38}$ and $C_{42}=C_{42}(\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_7C_{25}\Vert u\Vert^2_{L^{\infty}(B_1^+)}$.
*Step 6*. Observe that it holds $$\begin{aligned}
\xi_{ij}^k=\dfrac{x^ix^jx^k}{\tau^2\vert x\vert^2}{\psi_{\nu,\tau}}''+\dfrac{1}{\tau}\left(\delta_{jk}x^i+\delta_{ik}x^j+\delta_{ij}x^k-\dfrac{x^ix^jx^k}{\vert x\vert^2}\right){\psi_{\nu,\tau}}'\cdot\dfrac{1}{\vert x\vert}{\addtocounter{equation}{1}\tag{\theequation}}\label{132}\end{aligned}$$ where ${\psi_{\nu,\tau}}'':=\psi_{\nu}''\left(\dfrac{\vert x\vert}{\tau}\right)$. Putting into $VIII$ yields $$\begin{aligned}
VIII&=2\int_{B^+}\Delta_{\gamma}u\cdot\underbrace{\left( (\gamma^{ij}-\delta_{ij})u_k- \gamma^{ij} D{\pi_{\mathcal{N}}}(u)\left( g_k \right)\right)}_{=:w_k^{ij}} \xi^k_{ij}{\text{d}}\mu_{\gamma}=2\int_{B^+}\Delta_{\gamma}u\cdot w_k^{ij} \xi^k_{ij}{\text{d}}\mu_{\gamma}\\
&=\dfrac{2}{\tau^2}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}\\
&\quad+\dfrac{2}{\tau}\int_{B^+}\Delta_{\gamma}u\cdot \left(w^{ik}_kx^i+w^{kj}_kx^j+w^{jj}_kx^k\right) {\psi_{\nu,\tau}}'\cdot \dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}\\
&\quad-\dfrac{2}{\tau}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w_k^{ij}x^ix^jx^k}{\vert x\vert^2}{\psi_{\nu,\tau}}'\cdot \dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}
{\addtocounter{equation}{1}\tag{\theequation}}\label{133}.\end{aligned}$$ We multiply with ${\text{e}}^{\chi\tau}\tau^{3-m}$ and integrate over $[\rho,r]$, i.e., $$\begin{aligned}
\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VIII{\text{d}}\tau&=2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{1-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\Delta_{\gamma}u\cdot \left(w^{ik}_kx^i+w^{kj}_kx^j+w^{jj}_kx^k\right) {\psi_{\nu,\tau}}'\cdot \dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad-2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w_k^{ij}x^ix^jx^k}{\vert x\vert^2}{\psi_{\nu,\tau}}'\cdot \dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{134}.\end{aligned}$$ We reform the first integral in with Fubini and integration by parts as follows, $$\begin{aligned}
&2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{1-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}''{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&=-2\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} \left({\text{e}}^{\chi r}r^{3-m}\psi'_{\nu,r}-{\text{e}}^{\chi \rho}\rho^{3-m}\psi'_{\nu,\rho}\right)\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}\\
&\quad+2(3-m)\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}'\cdot\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2\chi \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}'\cdot\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{135}.\end{aligned}$$ Putting into yields $$\begin{aligned}
&\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VIII{\text{d}}\tau\\
&=-2\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} \left({\text{e}}^{\chi r}r^{3-m}\psi'_{\nu,r}-{\text{e}}^{\chi \rho}\rho^{3-m}\psi'_{\nu,\rho}\right)\cdot\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}\\
&\quad+2(2-m)\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}'\cdot\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2\chi \int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^2} {\psi_{\nu,\tau}}'\cdot\dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad+2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{2-m}\int_{B^+}\Delta_{\gamma}u\cdot \left(w^{ik}_kx^i+w^{kj}_kx^j+w^{jj}_kx^k\right) {\psi_{\nu,\tau}}'\cdot \dfrac{1}{\vert x\vert}{\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{136}.\end{aligned}$$ We obtain by Lebesgue’s differentiation theorem and Lemma 2 in the appendix of [@1] as $\nu\rightarrow\infty$ $$\begin{aligned}
&\lim_{\nu\rightarrow\infty}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VIII{\text{d}}\tau\\
&=-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^{m-1}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+2(2-m)\int_{ B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^{m}} {\text{d}}\mu_{\gamma}\\
&\quad+2\chi \int_{ B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta_{\gamma}u\cdot w^{ij}_kx^ix^jx^k}{\vert x\vert^{m-1}} {\text{d}}\mu_{\gamma}+2\int_{ B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\dfrac{\Delta_{\gamma}u\cdot \left(w^{ik}_kx^i+w^{kj}_kx^j+w^{jj}_kx^k\right)}{\vert x\vert^{m-2}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{137}\end{aligned}$$ for a.e. $0<\rho<r<R$. Thus, the following estimate holds $$\begin{aligned}
RHS\eqref{137}&\leq 2\int_{S^+_r\cup S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert \Delta_{\gamma}u\vert \vert w^{ij}_k\vert}{\vert x\vert^{m-4}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+2(m+1)\int_{ B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert \Delta_{\gamma}u\vert \vert w^{ij}_k\vert }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert \Delta_{\gamma}u\vert \vert w^{ij}_k\vert }{\vert x\vert^{m-4}}{\text{d}}\mu_{\gamma} {\addtocounter{equation}{1}\tag{\theequation}}\label{140}.\end{aligned}$$ Since $\vert w^{ij}_k\vert\leq H\vert Du\vert \vert x\vert +C_{46}$ where $C_{46}=C_{46}(\Vert Dg\Vert_{\infty}):=G\cdot \sup_{B^+}\vert D{\pi_{\mathcal{N}}}\circ u\vert \Vert Dg\Vert_{\infty}$ and with $\vert \Delta_{\gamma}u\vert\leq G\vert D^2u\vert +C_7\vert Du\vert$ we obtain from $$\begin{aligned}
&\lim_{\nu\rightarrow\infty}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}VII{\text{d}}\tau\leq RHS\eqref{140}\\
&\leq C_{47}\int_{S^+_r\cup S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{48}\int_{S^+_r\cup S_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-4}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+C_{49}\int_{ B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+C_{50}\int_{ B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+C_{45}\cdot (r+\rho)+C_{51}\cdot (r-\rho){\addtocounter{equation}{1}\tag{\theequation}}\label{143}\end{aligned}$$ where $C_{45}=C_{45}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$, $C_{47}=C_{47}(\Vert Dg\Vert_{\infty})$, $C_{48}=C_{48}(\Vert Dg\Vert_{\infty})$, $C_{49}=C_{49}( \Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$, $C_{50}=C_{50}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$ and $C_{51}=C_{51}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)})$. Altogether, it holds because of and (cf. ) $$\begin{aligned}
&\lim_{\nu\rightarrow\infty}\int_{r}^{\rho}{\text{e}}^{\chi \tau}\tau^{3-m}LHS\eqref{58}{\text{d}}\tau\\
&\leq \int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau+2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau+C_{53}\cdot (r-\rho)+C_{45}\cdot (r+\rho)\\
&\quad+C_{42}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad +C_{54}\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+C_{55}\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+C_{56}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{57}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}{\addtocounter{equation}{1}\tag{\theequation}}\label{144}.\end{aligned}$$ where $C_{53}=C_{53}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{39}+C_{51}$, $C_{54}=C_{54}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{40}+C_{49}$, $C_{55}=C_{55}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{41}+C_{50}$, $C_{56}=C_{56}(\Vert Dg\Vert_{\infty}):=H+C_{47}$ and $C_{57}=C_{57}(\Vert Dg\Vert_{\infty}):=C_{48}+2C_7+H$. Now, from we infer the chain of inequalities $$\begin{aligned}
RHS\eqref{144}\geq \lim_{\nu\rightarrow\infty}\int_{r}^{\rho}{\text{e}}^{\chi \tau}\tau^{3-m}LHS\eqref{58}{\text{d}}\tau\geq -RHS\eqref{81}{\addtocounter{equation}{1}\tag{\theequation}}\label{145},\end{aligned}$$ i.e. $$\begin{aligned}
&\int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau+2C_7\int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau+C_{42}\int_{\rho}^r{\text{e}}^{\chi\tau}J(\tau){\text{d}}\tau+2C_{45}\cdot r\\
&\quad-2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad +C_{54}\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+C_{55}\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+C_{56}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{57}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}\\
&\geq -C_{22} \int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau-C_{27}\int_{\rho}^r{\text{e}}^{\chi\tau}J(\tau){\text{d}}\tau-C_{30}\cdot (r-\rho)\\
&\quad-C_{20}\int_{B^+_r\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2}{\vert x\vert^{m-5}} {\text{d}}\mu_{\gamma}-C_{31}\int_{B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-3}} {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{146}\end{aligned}$$ where we set $J(\tau):=\displaystyle\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}$ and used $C_{53}\cdot (r-\rho)+C_{45}\cdot (r+\rho)\leq 2C_{45}\cdot r$, since $\rho < r$. This inequality can be rewritten to $$\begin{aligned}
&\int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau+(2C_7+C_{22})\int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau+(C_{30}+C_{32}+2C_{45})\cdot r-(C_{30}+C_{32})\cdot \rho\\
&\quad+(C_{42}+C_{27})\int_{\rho}^r{\text{e}}^{\chi\tau}J(\tau){\text{d}}\tau\\
&\quad +(C_{20}+C_{54})\cdot \int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+(C_{31}+C_{55})\cdot \int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+C_{56}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{57}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}\\
&\geq 4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad-2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{147}.\end{aligned}$$
*Step 7*. Next, we recast $J(\tau)=\displaystyle\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}$ with the aid of integration by parts. For that purpose we approximate $u$ by $u^{\varepsilon}$ as in . It holds $$\begin{aligned}
&\tau^{4-m}\int_{B^+_{\tau}}\vert D^2u^{\varepsilon}\vert^2 {\text{d}}\mu_{\gamma}\\
&=\tau^{4-m}\int_{\partial B^+_{\tau}} u^{\varepsilon}_i u^{\varepsilon}_{ij}\nu^j \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\tau^{4-m}\int_{B^+_{\tau}}u^{\varepsilon}_iu^{\varepsilon}_{ijj} {\text{d}}\mu_{\gamma}-\tau^{4-m}\int_{B^+_{\tau}} u^{\varepsilon}_i u^{\varepsilon}_{ij}\left(\sqrt{\gamma}\right)_j {\text{d}}\mathcal{L}^m\\
&=\tau^{4-m}\int_{\partial B^+_{\tau}} \left(u^{\varepsilon}_i u^{\varepsilon}_{ij}\nu^j-u^{\varepsilon}_i\nu^iu^{\varepsilon}_{jj}\right) \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\tau^{4-m}\int_{B^+_{\tau}}\left(u^{\varepsilon}_i\dfrac{\left(\sqrt{\gamma}\right)_i}{\sqrt{\gamma}} u^{\varepsilon}_{jj}- u^{\varepsilon}_i u^{\varepsilon}_{ij}\dfrac{\left(\sqrt{\gamma}\right)_j}{\sqrt{\gamma}}\right) {\text{d}}\mu_{\gamma}+\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta u^{\varepsilon}\vert^2 {\text{d}}\mu_{\gamma}\\
&=\int_{S^+_{\tau}} \left(\dfrac{u^{\varepsilon}_i u^{\varepsilon}_{ij}x^j}{\vert x\vert^{m-3}}-\dfrac{u^{\varepsilon}_ix^iu^{\varepsilon}_{jj}}{\vert x\vert^{m-3}}\right) \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}-\tau^{4-m}\int_{T_{\tau}} \left(g_i g_{im}-g_mg_{jj}\right) \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+\tau^{4-m}\int_{B^+_{\tau}}\left(u^{\varepsilon}_i\dfrac{\left(\sqrt{\gamma}\right)_i}{\sqrt{\gamma}} u^{\varepsilon}_{jj}- u^{\varepsilon}_i u^{\varepsilon}_{ij}\dfrac{\left(\sqrt{\gamma}\right)_j}{\sqrt{\gamma}}\right) {\text{d}}\mu_{\gamma}+\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta u^{\varepsilon}\vert^2 {\text{d}}\mu_{\gamma}\\
&\leq \int_{S^+_{\tau}} \left(\dfrac{u^{\varepsilon}_i u^{\varepsilon}_{ij}x^j}{\vert x\vert^{m-3}}-\dfrac{u^{\varepsilon}_ix^iu^{\varepsilon}_{jj}}{\vert x\vert^{m-3}}\right) \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1} +C_{58}R^3\\
&\quad+\tau^{4-m}\int_{B^+_{\tau}}\left(u^{\varepsilon}_{jj}u^{\varepsilon}_i\Gamma^l_{il} - u^{\varepsilon}_i u^{\varepsilon}_{ij}\Gamma^l_{jl}\right) {\text{d}}\mu_{\gamma}+\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta u^{\varepsilon}\vert^2 {\text{d}}\mu_{\gamma} {\addtocounter{equation}{1}\tag{\theequation}}\label{148}\end{aligned}$$ where $C_{58}=C_{58}(\Vert Dg\Vert_{C^1}):=2\Vert Dg\Vert_{\infty}\Vert D^2g\Vert_{\infty}G^{m/2}{\mathcal{H}}^{m-1}(T_1)$. As $\varepsilon\searrow0$, we obtain $$\begin{aligned}
\tau^{4-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}&\leq \int_{S^+_{\tau}} \left(\dfrac{u_i u_{ij}x^j}{\vert x\vert^{m-3}}-\dfrac{u_ix^iu_{jj}}{\vert x\vert^{m-3}}\right) \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1} +C_{59}\\
&\quad+\tau^{4-m}\int_{B^+_{\tau}}\left(u_{jj}u_i\Gamma^l_{il} - u_i u_{ij}\Gamma^l_{jl}\right) {\text{d}}\mu_{\gamma}+\tau^{4-m}\int_{B^+_{\tau}}\vert \Delta u\vert^2 {\text{d}}\mu_{\gamma} {\addtocounter{equation}{1}\tag{\theequation}}\label{149}\end{aligned}$$ for a.e. $\tau\in (0,R)$ where $C_{59}(\Vert Dg\Vert_{C^1},R):=C_{58}R^3$. Notice that $\Delta u=\Delta_{\gamma}u-(\gamma^{ij}-\delta^{ij})u_{ij}+\gamma^{ij}\Gamma^k_{ij}u_k$ holds. Thus, $\vert \Delta u\vert^2\leq 3\vert\Delta_{\gamma}u\vert^2 + 3H^2\vert x\vert^2 \vert D^2u\vert^2 +3C_7^2\vert Du\vert^2$. Moreover, we estimate for the second to last integral in with $2\vert D^2u\vert \vert Du\vert\leq \tau \vert D^2u\vert^2 +\dfrac{1}{\tau}\vert Du\vert^2$ as follows, $$\begin{aligned}
&\tau^{4-m}\int_{B^+_{\tau}}\left(u_{jj}u_i\Gamma^l_{il} - u_i u_{ij}\Gamma^l_{jl}\right) {\text{d}}\mu_{\gamma}\\
&\leq C_5\tau^{5-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}+C_5\tau^{3-m}\int_{B^+_{\tau}} \vert Du\vert^2 {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{152}.\end{aligned}$$ Thus, it holds $$\begin{aligned}
&\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{4-m}\int_{B^+_{\tau}} \vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\leq \int_{B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert} \left(\dfrac{u_i u_{ij}x^j}{\vert x\vert^{m-3}}-\dfrac{u_ix^iu_{jj}}{\vert x\vert^{m-3}}\right) {\text{d}}\mu_{\gamma}+ 3\int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau +C_{59}{\text{e}}^{\chi R}\cdot (r-\rho)\\
&\quad+C_{60}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{5-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+ C_{61}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}} \vert Du\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau{\addtocounter{equation}{1}\tag{\theequation}}\label{153}\end{aligned}$$ where $C_{60}:=C_5+3H^2$ and $C_{61}:=C_5+3C_7^2$.\
*Step 8*. We put $\chi=\chi(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}}):=2C_7+C_{22}+3C_{27}+3C_{42}$. Then, we obtain from with the aid of the inequality $$\begin{aligned}
&\int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau+\chi\int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau+\tilde{C}_{62}\cdot r-\tilde{C}_{63}\cdot \rho\\
&\quad+(C_{27}+C_{42})\left(C_{60}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{5-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+ C_{61}\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}} \vert Du\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\right)\\
&\quad +\tilde{C}_{60}\cdot\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+\tilde{C}_{61}\cdot \int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+C_{56}\cdot\int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+C_{57}\cdot \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}\\
&\geq 4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}\\
&\quad+2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}-2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}+2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\quad-(C_{27}+C_{42})\int_{B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert} \left(\dfrac{u_i u_{ij}x^j}{\vert x\vert^{m-3}}-\dfrac{u_ix^iu_{jj}}{\vert x\vert^{m-3}}\right) {\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{154}\end{aligned}$$ where $\tilde{C}_{60}=\tilde{C}_{60}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{20}+C_{54}$, $\tilde{C}_{61}=\tilde{C}_{61}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{31}+C_{55}$, $\tilde{C}_{62}=\tilde{C}_{62}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{30}+C_{32}+2C_{45}+(C_{27}+C_{42})C_{59}{\text{e}}^{\chi R}$, $\tilde{C}_{63}=\tilde{C}_{63}(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}(B_1^+)}):=C_{30}+C_{32}+(C_{27}+C_{42})C_{59}{\text{e}}^{\chi R}$ with $\tilde{C}_{62}\geq \tilde{C}_{63}$. Beyond that, it holds $$\begin{aligned}
-2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}+2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\geq -2\dfrac{ \vert Du\vert^2 \vert x \vert^2}{\vert x\vert^{m-1}}+2\dfrac{ \vert Du\vert^2}{\vert x\vert^{m-3}}=0{\addtocounter{equation}{1}\tag{\theequation}}\label{155}.\end{aligned}$$ Hence, applying Young’s inequality we obtain $$\begin{aligned}
&2\chi\int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\left(\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}-2\dfrac{ (u_{i} x^i)^2}{\vert x\vert^{m-1}}+2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right){\text{d}}\mu_{\gamma}\\
&\geq -\chi \int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{ \vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}-\chi \int_{ B^+_{r}\setminus B^+_{\rho}}{\text{e}}^{\chi \vert x\vert}\dfrac{ \vert D u\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{156}.\end{aligned}$$ Analogously, we obtain for the last integral on the left-hand side of the estimate $$\begin{aligned}
&-(C_{27}+C_{42})\int_{B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi\vert x\vert} \left(\dfrac{u_i u_{ij}x^j}{\vert x\vert^{m-3}}-\dfrac{u_ix^iu_{jj}}{\vert x\vert^{m-3}}\right) {\text{d}}\mu_{\gamma}\\
&\geq -(C_{27}+C_{42}) \int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{ \vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}-(C_{27}+C_{42}) \int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{ \vert D u\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}{\addtocounter{equation}{1}\tag{\theequation}}\label{157}.\end{aligned}$$ Moreover, observe that it holds $$\begin{aligned}
\int_{\rho}^r{\text{e}}^{\chi\tau}I'(\tau){\text{d}}\tau+\chi\int_{\rho}^r{\text{e}}^{\chi\tau}I(\tau){\text{d}}\tau&=\int_{\rho}^r\dfrac{{\text{d}}}{{\text{d}}\tau}\left({\text{e}}^{\chi\tau}I(\tau)\right){\text{d}}\tau={\text{e}}^{\chi r}I(r)-{\text{e}}^{\chi\rho}I(\rho){\addtocounter{equation}{1}\tag{\theequation}}\label{159}.\end{aligned}$$ Because of , , and $\rho < r$ we obtain from the inequality $$\begin{aligned}
&{\text{e}}^{\chi r}r^{4-m}\int_{B_r^+}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}-{\text{e}}^{\chi\rho}\rho^{4-m}\int_{B_{\rho}^+}\vert \Delta_{\gamma}u\vert^2 {\text{d}}\mu_{\gamma}+\mathsf{C}_1\cdot r\\
&\quad+\mathsf{C}_2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{5-m}\int_{B^+_{\tau}}\vert D^2u\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau+ \mathsf{C}_2\int_{\rho}^r{\text{e}}^{\chi\tau}\tau^{3-m}\int_{B^+_{\tau}} \vert Du\vert^2 {\text{d}}\mu_{\gamma}{\text{d}}\tau\\
&\quad +\mathsf{C}_3\int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2 u\vert^2 }{\vert x\vert^{m-5}}{\text{d}}\mu_{\gamma}+\mathsf{C}_4\cdot \int_{ B^+_{r}\setminus B_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2 }{\vert x\vert^{m-3}}{\text{d}}\mu_{\gamma}\\
&\quad+\mathsf{C}_6\int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert D^2u\vert^2 }{\vert x\vert^{m-6}} \sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}+\mathsf{C}_7 \int_{S_{r}^+\cup S_{\rho}^+}{\text{e}}^{\chi \vert x\vert}\dfrac{\vert Du\vert^2}{\vert x\vert^{m-4}}\sqrt{\gamma} {\text{d}}{\mathcal{H}}^{m-1}\\
&\geq 4\int_{B^+_r\setminus B^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left( \dfrac{ (u_i+u_{ij}x^j)^2}{\vert x\vert^{m-2}}+\dfrac{(m-2)(u_{i}x^i)^2}{\vert x\vert^{m}}\right){\text{d}}\mu_{\gamma}\\
&\quad+2\int_{S^+_r\setminus S^+_{\rho}}{\text{e}}^{\chi\vert x\vert}\left(-\dfrac{ u_{i} u_{ik}x^k}{\vert x\vert^{m-3}}+ 2\dfrac{ (u_{i}x^i)^2}{\vert x\vert^{m-1}}-2\dfrac{ (u_{i})^2}{\vert x\vert^{m-3}}\right)\sqrt{\gamma}{\text{d}}{\mathcal{H}}^{m-1}{\addtocounter{equation}{1}\tag{\theequation}}\label{161}\end{aligned}$$ where we have set $\mathsf{C}_1=\mathsf{C}_1(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}}):=\tilde{C}_{62}$, $\mathsf{C}_2=\mathsf{C}_2(\Vert Dg\Vert_{C^1},\Vert u\Vert_{L^{\infty}}):=(C_{27}+C_{42})C_{60}$, $\mathsf{C}_3=\mathsf{C}_3(\Vert Dg\Vert_{C^1},\Vert u\Vert_{L^{\infty}}):=(C_{27}+C_{42})C_{61}$, $\mathsf{C}_4=\mathsf{C}_4(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}}):=\tilde{C}_{60}+\chi+C_{27}+C_{42}$, $\mathsf{C}_5=\mathsf{C}_5(\Vert Dg\Vert_{C^2},\Vert u\Vert_{L^{\infty}}):=\tilde{C}_{61}+\chi+C_{27}+C_{42}$, $\mathsf{C}_6=\mathsf{C}_6(\Vert Dg\Vert_{\infty}):=C_{56}$ and $\mathsf{C}_7=\mathsf{C}_7(\Vert Dg\Vert_{\infty}):=C_{57}$. So, we have the boundary monotonicity inequality for $a=0$. This concludes the proof of .
**Acknowledgements.** I would like to thank Prof. Dr. Christoph Scheven for his many helpful advices.
[l]{} Angelsberg, G.: A monotonicity formula for stationary biharmonic maps, Math. Z. **252** (2006), pp. 287-293. Bethuel, F.: On the singular set of stationary harmonic maps, Manuscripta Math. 78, 417-443 (1993). Chang, S.-Y.A., Wang, L., Yang, P.: A regularity theory for biharmonic maps. Comm. Pure Appl. Math. **52**, 1113-1137 (1999). Evans, L.C.: Partial regularity for stationary harmonic maps into spheres. Arch. Rational Mech. Anal. 116 (1991) 101-113. Gong, H., Lamm, T., Wang, C.: Boundary regularity of stationary biharmonic maps. Calc. Var. (2012) 45:165-191. Große-Brauckmann, K.: Interior and boundary monotonicity formulas for stationary harmonic maps, manuscripta math. 77, 89-95 (1992). Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris Sér. I Math., 312(8):591–596, 1991. Hong, M.-C., Wang, C.: Regularity and relaxed problems of minimizing biharmonic maps into spheres. Calc. Var. Partial Differential Equations, 23(4):425–450, 2005. Mazowiecka, K.E.: Singularities of harmonic and biharmonic maps into compact manifolds, 2017. PhD dissertation. Morrey, C.B. Jr.: The problem of Plateau on a Riemannian manifold. Ann. Math. 49, 807-951, 1948. Moser, R.: A Variational Problem Pertaining to Biharmonic Maps, Communications in Partial Differential Equations, 33:9, 1654-1689. Moser, R.: Partial Regularity for Harmonic Maps and Related Problems, World Scientific Publishing, Hackensack, NJ, 2005. Nirenberg, L: On elliptic partial differential equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 3 : Volume 13 (1959) no. 2 , p. 115-162. Scheven, C.: Dimension reduction for the singular set of biharmonic maps. Adv. Calc. Var **1** (2008), 53-91. Scheven, C.: Variationally harmonic maps with general boundary conditions: Boundary regularity. Calc. Var (2006) 25(4), 409-429. Schoen, R., Uhlenbeck, K.: A regularity theory of harmonic maps. J. Diff. Geom. 17 (1982) 307-335. Schoen, R., Uhlenbeck, K.: Boundary Regularity and The Dirichlet Problem for Harmonic Maps. J. Diff. Geom. 18 (1983) 353-268. Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps, ETH Lecture Notes, Birkhäuser, Zürich, 1996. Rivière, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175, 197-226 (1995).
[^1]: Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany
[^2]: One distinguishes between extrinsically and intrisically biharmonic maps. We say that a map is intrinsically biharmonic iff it is a critical point of $\mathcal{E}(u)=\int_{\mathcal{M}}\vert \nabla Du\vert^2{\text{d}}\mu_{\mathcal{M}}$. The energy $\mathcal{E}$ does not depend on the embedding $\mathcal{N}\hookrightarrow {\mathbb{R}}^n$ while $E_2$ does. Therefore, the distinction extrinsically and intrinsically.
|
---
abstract: 'Shielding charged particle beams from transverse magnetic fields is a common challenge for particle accelerators and experiments. We demonstrate that a magnetic field cloak is a viable solution. It allows for the use of dipole magnets in the forward regions of experiments at an Electron Ion Collider (EIC) and other facilities without interfering with the incoming beams. The dipoles can improve the momentum measurements of charged final state particles at angles close to the beam line and therefore increase the physics reach of [these]{} experiments. In contrast to other magnetic shielding options (such as active coils), a cloak requires no external powering. We discuss the design parameters, fabrication, and limitations of a magnetic field cloak and demonstrate that cylinders made from 45 layers of YBCO high-temperature superconductor, combined with a ferromagnetic shell made from epoxy and stainless steel powder, shield more than 99% of a transverse magnetic field of up to 0.45 T (95 % shielding at 0.5 T) at liquid nitrogen temperature. The ferromagnetic shell reduces field distortions caused by the superconductor alone by 90% at 0.45 T.'
address:
- 'Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA'
- 'Superconducting Magnet Division, Brookhaven National Laboratory, Upton, NY 11973, USA'
- 'Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA'
- 'Department of Physics, University of Washington, Seattle, WA 98195, USA'
- 'Department of Physics, University of Virginia, Charlottesville, VA 22904, USA'
- 'Collider Accelerator Department, Brookhaven National Laboratory, Upton, NY 11973, USA'
- 'Physics Department, Northeastern University, Boston, MA 02115, USA'
author:
- 'K. G. Capobianco-Hogan'
- 'R. Cervantes'
- 'A. Deshpande'
- 'N. Feege'
- 'T. Krahulik'
- 'J. LaBounty'
- 'R. Sekelsky'
- 'A. Adhyatman'
- 'G. Arrowsmith-Kron'
- 'B. Coe'
- 'K. Dehmelt'
- 'T. K. Hemmick'
- 'S. Jeffas'
- 'T. LaByer'
- 'S. Mahmud'
- 'A. Oliveira'
- 'A. Quadri'
- 'K. Sharma'
- 'A. Tishelman-Charny'
bibliography:
- 'magcloak\_references.bib'
title: A Magnetic Field Cloak For Charged Particle Beams
---
Electron Ion Collider ,Interaction region ,Magnetic field shielding ,Magnetic field cloaking
Introduction
============
Magnetic fields are routinely used to steer charged particle beams and to analyze the momenta of charged particles produced in fixed-target and collider experiments. The field component transverse to the trajectory of a charged particle deflects it and, in the case of a polarized beam crossing a field gradient, depolarizes it. Beams at particle collider facilities need adequate shielding from fields that would cause disturbances. Established designs of magnetic field shields use cylinders made from low-temperature superconductors [@Martin:1972xd]. Magnetic flux lines incident on a superconducting cylinder induce screening currents, and the magnetic fields generated by these currents counteract the external field. As a result, the inside of the cylinder remains field-free, while the field on the outside is distorted. This distortion can be corrected by adding a ferromagnetic shell around the superconductor. Unlike the superconductor, a ferromagnetic shell [pulls in magnetic flux lines. The combination]{} of superconductor and ferromagnet forms a magnetic field cloak (see Fig. \[fig:conceptual\_cloak\]). The ferromagnet of a superconductor-ferromagnet bilayer effectively contains all field distortions caused by the superconductor if its magnetic permeability $\mu_r$ is tuned to $$\begin{aligned}
\mu_r = \frac{R_2^2 + R_1^2}{R_2^2 -R_1^2},
\label{eqn:permeability}\end{aligned}$$ where $R_1$ and $R_2$ are the inner and outer radius of the ferromagnet ($R_1$ is also the outer radius of the superconductor) [@gomory2012]. Thus, a cloak can provide a field-free tunnel without disturbing the external field.
Magnetic field cloaks are topics of active research [@Giunchi:2016ncd; @3dcloak2015]. We want to demonstrate that our design, which uses high-temperature superconductor (HTS) cylinders, is a viable solution to cloak charged particle beams at future particle accelerator facilities such as the Electron Ion Collider (EIC). Such a facility would require a cloak that shields a magnetic field of at least 0.5 T over a length of 1 m. Section \[sec:scshielding\] briefly summarizes the basics of shielding magnetic fields with superconductors, Sec. \[sec:prototypes\] explains the fabrication of our superconductor shields and cloak prototypes, Sec. \[sec:setups\] describes our test setups, Sec. \[sec:results\] presents the results of magnetic field shielding and cloaking measurements with our prototypes, and Sec. \[sec:conclusion\] gives our conclusions.
Shielding magnetic fields with high temperature superconductors {#sec:scshielding}
===============================================================
The response of a type-II superconductor to magnetic fields is characterized by two critical fields: $B_{c1}$ and $B_{c2}$. If a cylinder made from such a material is exposed to a transverse magnetic field below $B_{c1}$, the flux lines are bent around the cylinder. Above a certain threshold field, the field penetrating the superconductor has an approximately logarithmic time dependence [@RevModPhys.68.911]. Between $B_{c1}$ and $B_{c2}$, flux vortices form and allow the field to partially seep through the superconductor. In this field range, stacking multiple layers of superconductor improves the overall magnetic field shielding. Above $B_{c2}$, the superconductivity is destroyed and the field passes through the shield. The critical fields of a superconductor become larger the lower the superconductor’s temperature falls below its critical temperature. High-temperature superconductors (HTS) are especially convenient because they can be cooled below their critical temperature with liquid nitrogen (rather than liquid helium), resulting in relatively modest cryogenic costs and fast prototyping. Although various experiments have characterized HTS shields [@Denis2007; @Fagnard2009; @Kvitkovic2011; @Karthikeyan1994; @Fagnard2010; @Kvitkovic2009], we are not aware of any substantial shielding past 10 mT with such materials, especially for transverse fields. Ref. [@Kvitkovic2009] comes closest, but the authors only shield approximately 60% of the field with an applied field of amplitude 50 mT and frequency 10 Hz. If HTS can be demonstrated to shield fields above 0.5 T, they can replace low temperature superconductor shields (such as NbTi sheets [@nippon2002]).
Prototype construction {#sec:prototypes}
======================
1 m long / 2-layer and a 4.5 inch long / 4-layer HTS shield {#sec:shieldbnl}
-----------------------------------------------------------
We use 46 mm wide superconductor wire insert manufactured by American Superconductor Inc ([www.amsc.com](www.amsc.com)) to fabricate superconducting cylinders. This width is an intermediate stage on their production line and, due to low demand, not currently supported as a product [@Rupich2010]. The insert is made from a YBCO ceramic with a critical temperature of about 90 K. The ceramic is deposited on an oxide-buffered Ni-W alloy substrate and coated with silver. The width and flexibility of this superconductor allow us to combine two strips to a cylinder of up to approximately 1 inch diameter. The orientation of the strips along the cylinder axis facilitates the induction of supercurrents that generate a magnetic dipole field. Therefore, it is most effective for shielding transverse magnetic fields [@nouri2013]. The maximum field that a cylinder can shield increases with the number of layers as long as the field does not exceed the second critical field of the superconductor [@cervantes2015].
Figure \[fig:sc\_bnl\] shows our HTS shield prototype consisting of four 1 m long superconductor strips attached to a 60 inch long stainless steel tube with 1 inch outer diameter. The superconductor strips form a double layer on the top and the bottom of the pipe and overlap at the connecting sides. We use Kapton$\textsuperscript{\textregistered}$ tape and zip ties to hold the superconductor strips in place. In addition, we test a 4.5 inch long HTS shield prototype with four layers of superconductor strips attached to an aluminum tube.
4.5 inch long / 45-layer HTS shield {#sec:shield45}
-----------------------------------
Using a fabrication process based on [@Martin:1972xd], we build a HTS shield with 45 layers of 46 mm wide American Superconductor HTS wire insert. The process uses a die-and-mandrel setup heated in an oven to laminate multiple layers of superconductor wire and solder. Removing excess superconductor and solder on the sides with a milling machine creates half-cylinders. We combine two of these half cylinders to form a full shielding tube. The left panel of Fig. \[fig:sc\_45layers\] shows the two halves of our 4.5 inch long, 1 inch outer diameter HTS shield with 45 layers. [The thickness of this shield varies between 0.22 inch and 0.26 inch along its circumference.]{} Based on an extrapolation method described in [@cervantes2015] and measurements for single layer shielding, we expect this 45-layer prototype to shield more than 99% of a transverse magnetic field up to 0.5 T.
4.5 inch long / 4-layer and 4.5 inch long / 45-layer HTS cloak {#sec:cloak45}
--------------------------------------------------------------
To fabricate a ferromagnetic shell, we mix 430 stainless steel powder (magnetic permeability $\mu \approx 1000$ [@oxley2009]) with commercial epoxy and pour the mixture into a tubular mold. [ We keep the mold upright to help air bubbles accumulate at the top and invert it every minute for 30 minutes to prevent the steel powder from setting while the epoxy is hardening. When placing the hardened cylinder in a 30 mT homogenous magnetic field perpendicular to its axis, we observe maximum field shielding at the center of the cylinder and a symmetric shielding profile around it (Fig. \[fig:sbu\_bvz\] illustrates the same kind of shielding measurement performed on superconducting cylinders). This confirms the uniform distribution of steel powder in the ferromagnetic shell.]{} Controlling the amount of stainless steel powder allows us to control the permeability of the mixture [@rozanov2008] (see Sec. \[sec:fmtuning\] for more details). [ We use a $4.5$ inch long ferromagnetic shell with inner radius $R_1 = 0.533(2)$ inch and outer radius $R_2 = 0.805(1)$ inch to assemble cloak prototypes with the 4-layer and 45-layer HTS shields. Given these dimensions, Eqn. \[eqn:permeability\] predicts that a magnetic permeability of $\mu_r \; = \; 2.56(2)$ yields perfect cloaking.]{} The right panel of Fig. \[fig:sc\_45layers\] shows our fully assembled magnetic field cloak prototype using the 45-layer HTS shield.
Test setups {#sec:setups}
===========
The Van de Graaff Facility at Brookhaven National Laboratory
------------------------------------------------------------
The Tandem Van de Graaff Facility at Brookhaven National Laboratory provides users with a variety of ion beams. [We use the facility to test how well our 1 m HTS shield (see Sec. \[sec:shieldbnl\]) shields ion beams from a magnetic dipole field.]{} Figure \[fig:bnlsetup\] shows a cross section of the test setup. The stainless steel core of our shield connects to the beam line, and a five-sided aluminum box insulated with 1 inch thick extruded polystyrene foam plates holds liquid nitrogen to cool the superconductor. A steering dipole magnet (a square arrangement of four coils with iron yokes, an inner opening of 4 inch by 4 inch, and field variations of less than 0.5 mT over a 1 inch diameter area in the center) is placed around the liquid nitrogen bath so that it creates a vertical magnetic field w.r.t. the beam line. We use Lakeshore’s cryogenic transverse hall sensor (HGCT3020) and Model 425 Gaussmeter to measure the vertical component of the magnetic field in the center of the magnet as a function of the magnet current, as well as the field at different positions along the beam line at a fixed magnet current. In addition, we measure the deflection of $^{7}_{3}\mbox{Li}^{3+}$ and $^{16}_{8}\mbox{O}^{3+}$ ion beams as a function of the magnet current by recording the beam position with a zinc sulfide screen, which sits in a target chamber downstream of our HTS shield, and a digital camera.
![The beam test setup in the BNL Van de Graaff beam line (beam entering from the left) showing the stainless steel tube with superconductor shield inside the aluminum box filled with liquid nitrogen, the steering dipole magnet in the center, and the target chamber with zinc sulfide screen on the right.[]{data-label="fig:bnlsetup"}](figures_publish/Figure4_illustr_bnl_setup_cut){width="75.00000%"}
Helmholtz coil setup
--------------------
We use the homogenous magnetic field generated by a pair of Helmholtz coils (GMW Model 5451, manufacturer quoted field uniformity $\Delta B/B$ less than $\pm$200ppm over a 30 mm sphere) to measure the permeability of ferromagnetic cylinders and to test magnetic field cloaking up to fields of 50 mT. A custom rig holds the prototype under test (superconductor cylinder, ferromagnetic cylinder, or full cloak) and a Hall sensor. The rig allows us to move the Hall sensor independently in three Cartesian directions to map the magnetic field inside of and around the prototype. We cool the prototypes by immersing them in a bath of liquid nitrogen confined in a box made from extruded polystyrene foam. Figure \[fig:helmholtz\_setup\] shows the setup.
The 4 Tesla Magnet Facility at Argonne National Laboratory
----------------------------------------------------------
The 4 Tesla Magnet Facility at Argonne National Laboratory gives users access to an MRI magnet to test detector components in a very homogenous field from 0 to 4 T. [We use it to measure the permeability of ferromagnetic cylinders and to test magnetic field cloaking up to fields of 0.5 T.]{} Figure \[fig:mri\_setup\] shows our test setup using this magnet. An open aluminum box (10.3 inch high, 10 inch wide, 10 inch deep) surrounded by 1 inch thick extruded polystyrene foam plates sits on a rail system inside the magnet and holds liquid nitrogen for cooling. Our 45-layer HTS shield, a ferromagnet cylinder, and a full cloak can be held inside this box by a mount fixed to its base. We use a translational stage (three Cartesian degrees of freedom) on the rails to position a Hall sensor connected to a long rod to measure the magnetic field inside our prototypes and to map the magnetic field around them.
Results {#sec:results}
=======
Magnetic field shielding
------------------------
We use all three setups described in Sec.\[sec:setups\] to test different aspects of shielding magnetic fields with our HTS shields. In the Van de Graaff setup, the 1 m long, 2-layer HTS shield extends significantly past the magnet. Therefore, we expect only minimal field leaking in through the ends of the shield. Figure \[fig:bnl\_bvz\] shows the vertical component [$B_T$]{} of the magnetic field in this setup measured with a Hall sensor at different positions along the beam axis. [The steering magnet is set to a nominal field of 30 mT. An offset of 0.1 mT is added to all measurements]{} to make negative values visible on the logarithmic scale. The figure compares measurements at room temperature and with liquid nitrogen cooling. This shows that the 1 m long, 2-layer HTS tube shields most of the dipole field when it is in its superconducting state. We attribute the field distortions inside the shield to mechanical imperfections (which allow field to leak through the shield), as well as artifacts caused by background fields trapped inside the superconductor during cool-down. In addition, the ferromagnetic substrate of the superconductor causes field distortions both at room temperature and at cryogenic temperatures.
![Vertical component [$B_T$]{} of the magnetic field measured in the Van de Graaff setup at different positions along the axis of the 1 m long, 2-layer HTS shield at room temperature (‘no SC shielding‘) and with liquid nitrogen cooling (‘with SC shielding‘) at a nominal steering dipole field of 30 mT. The vertical lines indicate the extension of the HTS shield. The ordinate uses logarithmic scale. In addition, an offset of 0.1 mT (indicated by the horizontal grey line) is added to each measurement.[]{data-label="fig:bnl_bvz"}](figures_publish/Figure7_shield_BvsZ_vdg2layer){width="75.00000%"}
Figure \[fig:sbu\_bvz\] shows the same type of measurement for the 4.5 inch, 4-layer HTS and 4.5 inch, 45-layer HTS shields placed inside the Helmholtz coils setup. The distortions inside these shields are smaller. However, a significant fraction of the field leaks into the shields because the Helmholtz coils extend beyond the ends of these shields. This effect is stronger for the 4-layer HTS shield because its inner diameter is larger than the inner diameter of the 45-layer shield.
![Magnetic field component [$B_T$]{} measured at various positions along the axis of the 4.5 inch long, 4-layer HTS and 4.5 inch long, 45-layer HTS shields in the Helmholtz coils setup at a nominal field of 40 mT. The vertical lines indicate the extension of the HTS shields. The ordinate uses logarithmic scale.[]{data-label="fig:sbu_bvz"}](figures_publish/Figure8_shield_BvZ_Helmholtz){width="75.00000%"}
Figure \[fig:B\_v\_time\] shows the field measured with a Hall sensor in the center of the 2-layer, 1 m long superconductor shield in the Van de Graaff setup as a function of time for nominal steering dipole fields of $B_{a}$ = 11 mT and $B_{a}$ = 45 mT. At the lower dipole setting, the field inside this shield is stable, while at the higher dipole setting it increases approximately logarithmically with time. As mentioned in Sec. \[sec:scshielding\], such a time dependence above a certain threshold field is an expected behavior for HTS shields.
Figure \[fig:bnl\_bvi\] shows the magnetic field shielding performance of the 1 m long, 2-layer HTS shield in the Van de Graaff setup. The top panel presents the deflection of a 8.14 MeV $^{7}_{3}\mbox{Li}^{3+}$ beam as a function of the steering dipole field $B_{a}$, both at room temperature (superconductor does not shield field) and at liquid nitrogen temperature (superconductor shields field). The superconducting shield reduces the beam deflection by about 94% (see [@kyle_conference] for a comparison of our $^{7}_{3}\mbox{Li}^{3+}$ and $^{16}_{8}\mbox{O}^{3+}$ results). The bottom panel shows the field measured with a Hall sensor in the center of the shield as a function of steering dipole fields. The open markers indicate field measurements that show the previously mentioned increase over time, i.e. measurements for which the mean reading [in the last quarter is more than two standard deviations higher than the mean]{} reading in the first quarter of a ten minute measurement.
![Top panel: Displacement $\Delta x_{Li7}$ of the $^{7}_{3}\mbox{Li}^{3+}$ beam passing through the 1 m long, 2-layer superconductor shield in the Van de Graaff setup (at room temperature and in its superconducting state) as a function of the nominal steering dipole field [$B_{a}$]{}. Bottom panel: Mean field [$B_T$]{} measured in the center of the shield as a function of the steering dipole field. Open symbols indicate time dependence of the measured field.[]{data-label="fig:bnl_bvi"}](figures_publish/Figure10_shielding_vdg2layer){width="75.00000%"}
Figure \[fig:sc\_shield\_mri\] shows the shielding performance of our 45-layer HTS shield prototype in the MRI magnet in linear scale (top) and logarithmic scale (bottom). The open markers indicate field measurements showing an increase over time. The prototype shields more than 99% of an external field up to 0.45 T, and 95% up to 0.50 T external field.
Ferromagnetic shells for magnetic field cloaking {#sec:fmtuning}
------------------------------------------------
Figure \[fig:fm\_mu\_b\] shows the magnetic permeability of two 4.5 inch long ferromagnetic epoxy / steel powder cylinders with different fractional masses $f_M$ of steel powder in the mixture at room temperature in the field of the MRI magnet as a function of the applied field $B_a$. The fractional mass $f_M$ is the mass of the steel powder used in the mixture divided by the mass of steel power plus epoxy. We determine the magnetic permeability from $$\begin{aligned}
B_{T}(r < R_1) = \frac{4 \mu_r R_2^2}{(\mu_r +1)^2 R_2^2 - (\mu_r-1)^2 R_1^2} B_{a},
\label{eqn:findmu}\end{aligned}$$ where [$B_{T}$]{} and $B_{a}$ are the magnetic field measured in the center of the cylinder and the nominal MRI field, and $R_1$ and $R_2$ are the inner and outer radius of the ferromagnetic cylinder [@zangwill2012]. The magnetic permeability increases with higher $f_M$ and decreases with increasing applied field without saturating completely within the tested field range.
Figure \[fig:fm\_mu\_fmass\] summarizes the relative magnetic permeability measured at liquid nitrogen temperature with the Helmholtz coil setup at an applied field of 40 mT for 4.5 inch long ferromagnetic cylinders of various $f_{M}$. We empirically find that the function $$\begin{aligned}
\mu_r (f_M) \; = \; \frac{p0}{\tan(p1 \, \cdot \, f_M \, + \, p2)} \, + \, p3
\label{eqn:calcmu}\end{aligned}$$ describes the relation between $\mu_r$ and $f_M$ in these cylinders with the parameters listed in table \[tab:fit\_mur\_fm\] and a $\chi^2/\mbox{DOF}$ of 2.4. This plot and function allow to select $f_M$ to fabricate a ferromagnetic cylinder of desired $\mu_r$. The magnetic permeability at liquid nitrogen temperature is smaller than at room temperature, which we attribute to an increase in the anisotropy constant [@oxley2009]. This needs to be accounted for when selecting $f_{M}$.
--------------------------- ---------------- --------------- ---------------
p0 p1 p2 p3
\[0.5ex\] 0.70 $\pm$ 0.05 -2.2 $\pm$ 0.4 1.9 $\pm$ 0.4 1.1 $\pm$ 0.2
\[1ex\]
--------------------------- ---------------- --------------- ---------------
: Fit parameters for Eqn. \[eqn:calcmu\] and the line shown in Fig. \[fig:fm\_mu\_fmass\].[]{data-label="tab:fit_mur_fm"}
Magnetic field cloaking
-----------------------
Based on the diameters of our 4.5 inch long cloak prototypes, Eqn. \[eqn:permeability\] predicts perfect cloaking with a ferromagnetic shell [ that has a relative magnetic permeability of 2.56(2)]{}. Figures \[fig:cloak\_bvx\_1\] and \[fig:cloak\_bvz\_1\] show maps of the Helmholtz coils magnetic field at a nominal applied field of 40 mT for the coils itself, the coils with a 4.5 inch long, 4-layer HTS shield, and the coils with a full cloak (also using the 4.5 inch long, 4-layer HTS shield). [The ferromagnetic shell used for this cloak has a $\mu_r$ of 2.43(4) (which is 5% below the theoretically ideal value of 2.56(2))]{} at liquid nitrogen temperature and at 40 mT applied field. These figures demonstrate that the cloak significantly reduces the field disturbances that the superconducting cylinder alone would cause. Because the field of the Helmholtz coils extend beyond the ends of the cloak, the figures show fringe effects near the ends of the cloak.
Shielding and cloaking higher fields requires switching to the 4.5 inch long, 45-layer HTS cylinder. Due to imperfections in the fabrication, this cylinder is not perfectly cylindrical and leaves a gap of about 0.1 inch when inserted into the ferromagnetic shell. Using this superconductor with the ferromagnet with [$\mu_r$ = 2.43(4)]{} yields increased field disturbances around the cloak (see Fig. \[fig:cloak\_diff\_sc\]). We find that we can compensate for some of these effects by selecting a lower permeability ferromagnet, or using this ferromagnet at higher fields, which effectively reduces its $\mu_r$. In addition, the 45-layer HTS shield has a gap where the two half-shells connect. Figure \[fig:cloak\_angle\] shows the field distortions near the cloak caused by different alignments of the gap with respect to the applied field.
To demonstrate magnetic field cloaking in the MRI magnet, we map the magnetic flux [$B_{T}$]{} in the direction of the MRI axis near the 4.5 inch long, 45-layer HTS shield and a full cloak (same HTS shield and a ferromagnetic shell with [$\mu_r$ = 2.43(4)]{}). Figure \[fig:cloak\_mri\_1d\_front\] shows that at 0.45 T nominal MRI field, the HTS shield causes a field distortion with an amplitude of 16% of the field at 1 cm distance, while the distortion near the cloak is only 2% of the applied field. The shape of this distortion hints at a misalignment of the superconductor with respect to the MRI field (see Fig. \[fig:cloak\_angle\]). At 0.5 T nominal MRI field, the distortion amplitude is 13% of the field near the the HTS shield alone and 2% near the cloak. Figure \[fig:cloak\_mri\_3d\] illustrates the field distortions caused by the HTS shield at up to 10 cm from the superconductor and that the cloak significantly mitigates these distortions.
Conclusion {#sec:conclusion}
==========
Our tests demonstrate that a magnetic field cloak is a viable option to shield charged particle beams from transverse magnetic fields of up to at least 0.5 T. This allows for new designs of collider experiments and enables, for example, the use of dipole magnets in the forward regions of an EIC detector to improve the momentum measurement of charged final state particles at angles close to the beam line. The number of HTS layers used affects the maximum shielded field. At the same time, cooling the cloak with liquid helium instead of liquid nitrogen would significantly increase the maximum shielded field for a given number of layers, or would reduce the number of layers needed to shield a given field. The cloaking performance can be maximized by carefully tuning the permeability of the ferromagnet and precisely fabricating the HTS cylinder. Moreover, ensuring proper alignment of the HTS shield w.r.t the magnetic field or using a gap-free HTS cylinder (e.g. by evaporating the superconductor directly onto a core) would help to minimize field distortions around the cloak. Fringe field effects at the ends of the cloak can be mitigated by extending the cloak beyond the field. The design parameters, fabrication procedures, and limitations of a magnetic field cloak established in this paper pave the way to realize such a device. This, in turn, opens up new possibilities to design future collider facilities and experiments.
Acknowledgements
================
This work was supported by the DOE Office of Nuclear Physics’ Electron-Ion Collider Detector R&D initiative administered by Brookhaven National Laboratory (Contract No. DE-AC02-98CH10886). Special thanks go to the program’s advisory committee for their helpful guidance and facilitation. We also thank B. Parker, R. Gupta, V. Ptitsyn, Y. Goto, K. Boyle, I. Nakagawa, and J. Seele for valuable discussions, as well as American Superconductors for making their 46 mm wide superconductor wire insert available to us. In addition, we gratefully acknowledge the support of the staff of the Tandem Van de Graaff Facility at Brookhaven National Laboratory, the 4 Tesla Magnet Facility at Argonne National Laboratory, and the Stony Brook Physics Department machine shop. We thank S. Karthas, R. Lefferts, A. Lipski, and I. Yoon for all their valued help. A significant portion of this research project was carried out by undergraduate students, and we thank D. Aviles, G. Bello Portmann, D. Bhatti, I. Bromberg, R. Bruce, J. Chang, B. Chase, E. Jiang, P. Karpov, Y. Ko, Y. Kulinich, R. Losacco, E. Michael, J. Nam, H. Powers, V. Shetna, S. Thompson, H. Van Nieuwenhuizen, and N. Ward for their contributions. Finally, we thank A. Chhugani, and the Simons Summer Research Program for enabling her to participate in this project as a high school student.
References {#references .unnumbered}
==========
|
---
abstract: 'We formulate a Stackelberg game between an attacker and a defender of a power system. The attacker attempts to alter the load setpoints of the power system covertly and intelligently, so that the voltage stability margin of the grid is reduced, driving the entire system towards a voltage collapse. The defender, or the system operator, aims to compensate for this reduction by retuning the reactive power injection to the grid by switching on control devices, such as a bank of shunt capacitors. A modified Backward Induction method is proposed to find a cost-based Stackelberg equilibrium (CBSE) of the game, which saves the players’ costs while providing the optimal allocation of both players’ investment resources under budget and covertness constraints. We analyze the proposed game extensively for the IEEE 9-bus power system model and present an example of its performance for the IEEE 39-bus power system model. It is demonstrated that the defender is able to maintain system stability unless its security budget is much lower than the attacker’s budget.'
author:
- 'Lu An, Aranya Chakrabortty, and Alexandra Duel-Hallen[^1]'
bibliography:
- 'ref\_all.bib'
- 'ref\_SG.bib'
- 'ref\_power.bib'
title: |
**A Stackelberg Security Investment Game for Voltage Stability\
of Power Systems**
---
Stackelberg game, voltage stability, load attacks, security investment, power systems
Introduction
============
Over the past decade, significant research has been done on cyber-security of power systems [@anu] with applications in state estimation [@peng], volt/VAr control [@dan3], automatic generation control [@ashok], etc. In this paper we address an equally important and yet less-visited cyber-attack scenario in power systems - namely, covert attacks on loads causing degradation of voltage stability. Unlike other papers, such as [@rad], that report detection and control problems for load attacks, our goal is to formulate an investment strategy that power system operators can adopt to secure the grid when an attacker tries to drive it to voltage collapse by manipulating a chosen set of load setpoints. This manipulation can be done in a covert way for each individual load, so that the user does not feel any difference in consumption, but when hundreds of such loads are tweaked simultaneously, the cumulative effect can still result in severe degradation of voltage stability [@simpson2016voltage].
We use a Stackelberg game (SG) [@Osborne1994] to formulate this security investment, considering the defender as the leader and the attacker as the follower [@Basar2019]. Game theory has been a common tool for analyzing security problems in cyber-physical systems [@7011006; @Basar2019]. Cooperative and non-cooperative games have also been proposed for non-attack scenarios such as load balancing and voltage stability in [@Avraam2018VoltageCS; @8107250]. To the best of our knowledge, no research has been done to explore how game theory pertains to security investment for load attacks. Moreover, most game-theoretic security investment research employs dynamic games [@Basar2019], including stochastic games and games that utilize learning, where the players repeatedly update their investment strategies in response to the opponents’ actions. However, these games are not practical when long-term, fixed security investment is desired.
The SG for our problem is set up as follows. The attacker plans to hack covertly into a set of loads and modify their setpoints to increase the system-wide voltage instability index [@simpson2016voltage]. The defender switches on control devices to compensate the reactive power balance in the grid proactively, so that the instability index remains close to its safe value if an attack occurs in the future. Both players are subject to budget constraints. Additionally, the attacker aims to remain covert. We modify the standard backward induction [@Osborne1994] for SG to choose a Stackelberg equilibrium (SE) [@amir1999stackelberg] that reduces the players’ costs while retaining their payoffs. The resulting cost-based Stackelberg equilibrium (CBSE) provides guidelines to the system operator for fixed, long-term grid protection against voltage stability attacks. We validate our results using the IEEE 9-bus and 39-bus power system models and demonstrate that voltage stability can be maintained unless the defender’s security resources are much more limited than the attacker’s budget.
Power System Model {#sec:sysmod}
==================
We first recall the definition of voltage instability index from [@simpson2016voltage], which will be used as the central metric for evaluation of our game. Consider a power system with $M \geq 1$ generators, and $K \geq 1$ loads, where the load buses are indexed as the first $K$ buses, followed by $M$ generator buses. Let the steady-state voltage magnitudes at the load buses be stacked as ${{\mathbf{V}}_L} = [V_1,\cdots,V_K] \in {\mathbb R^K}$, and at the generator buses as ${{\mathbf{V}}_G} = [V_{K+1},\cdots,V_{K+M}] \in {\mathbb R^M}$. Let the admittance matrix of the network be denoted as ${\mathbf{Y}}={\mathbf{G}}+j{\mathbf{B}}$, where ${\mathbf{B}}$ is referred to as the susceptance matrix. We partition the susceptance matrix ${\mathbf{B}} \in \mathbb R^{(K +M) \times (K +M)}$ into four block matrices as: $$\label{eq:B}
{\bf{B}} = \left( {\begin{array}{*{20}{c}}
{{{\bf{B}}_{LL}}}&{{{\bf{B}}_{LG}}}\\
{{{\bf{B}}_{GL}}}&{{{\bf{B}}_{GG}}}
\end{array}} \right),$$ where ${{\bf{B}}_{LL}}$ contains the interconnections among loads, and ${{\bf{B}}_{LG}}={{\bf{B}}_{GL}}^{T}$ represents the interconnections between loads and generators. Following the derivations in [@simpson2016voltage], one can then define the open-circuit load voltage vector as: $$\label{eq:VLo}
{\bf{V}}_L^* = - {\bf{B}}_{LL}^{ - 1}{{\bf{B}}_{LG}}{{\bf{V}}_G},$$ and, subsequently, the symmetric stiffness matrix as: $$\label{eq:Q_cirt}
{{\bf{Q}}_{cirt}} \triangleq \frac{1}{4}{\rm{diag}}({\bf{V}}_L^*) \cdot {{\bf{B}}_{LL}} \cdot {\rm{diag}}({\bf{V}}_L^*),$$ where $\rm{diag}(\cdot)$ denotes the diagonal matrix.
Using (\[eq:VLo\]) and (\[eq:Q\_cirt\]), the *voltage instability index* of the system can be defined as: $$\label{eq:delta}
\Delta = ||{\mathbf{Q}}_{cirt}^{ - 1}{{\mathbf{Q}}_L}|{|_\infty },$$ where ${{\mathbf{Q}}_L} = [Q_1,\cdots,Q_K] \in \mathbb R^K$ is a $K$-dimensional real vector that represents the *reactive power setpoints* at the load buses. Here, $||\cdot||_\infty$ refers to the $\ell_{\infty}$-norm, which picks the absolute value of the element with the largest magnitude in a vector. The $k^{\text{th}}$ entry of the matrix-vector product ${\mathbf{Q}}_{cirt}^{ - 1}{{\mathbf{Q}}_L}$ captures the stability stress on load $k$, with $||\cdot||_\infty$ identifying the maximally stressed node. According to Theorem 1 in [@simpson2016voltage], the power flow equation will have a unique, stable solution if $\Delta<1$. Equivalently, $\Delta\geq1$ indicates that at least one load bus in the system is overly stressed and can be responsible for a voltage collapse. We refer to $1-\Delta$ as the *voltage stability margin* [@van2007voltage]. The larger the value of $\Delta$, the narrower the stability margin is and the closer the power system is to a voltage collapse. Denote the *nominal voltage stability index* ${\Delta ^0}$ as the value of $\Delta$ computed from (\[eq:delta\]) using the *nominal reactive power setpoints* ${{\mathbf{Q}}_L^0}$ (over a certain period of time assuming that the setpoints are constant over this period).
According to Proposition 3 in supplementary note 6 of [@simpson2016voltage], $\mathbf{Q}_{cirt}^{-1}$ has negative elements and ${\mathbf{Q}_L}$ has positive elements. Thus, if some elements of ${\mathbf{Q}_L}$ increase, the $\ell_{\infty}$-norm in (4) also increases. Therefore, the voltage instability index $\Delta$ in (4) increases as the reactive power demands of the loads grow. An attacker can increase the reactive power demands at appropriately chosen load buses by adding an incremental vector ${{\mathbf{q}}_a} = [q_a^1,\cdots,q_a^K] \in \mathbb R^K$ to ${{\mathbf{Q}}_L^0}$ and thus easily narrow down the voltage stability margin. Since only the reactive power setpoints are tampered with, and not the active power setpoints, the user may not feel any difference in her consumption pattern, which makes this type of attack unobservable to a large extent. The attacker can further make this attack covert by designing the entries of ${{\mathbf{q}}_a}$ small enough that they maintain the load bus voltages to be within their usual allowable range of 0.9 per unit (pu) to 1.1 pu while still pushing $\Delta$ towards 1. To prepare for possible future attacks, the operator, or the defender, can switch on voltage control devices, such as shunt capacitors and power electronic converters, to compensate for the potential increase in consumption in advance. These control devices may or may not be located at the load bus. If they are not, their equivalent contribution of reactive power at the $K$ load buses can be obtained by simple network reduction. Let this equivalent $K$-dimensional reactive power compensation vector be denoted as ${{\mathbf{q}}_d}=[q_d^1,\cdots,q_d^K] \in \mathbb R^K$. When an attack happens, the overall reactive power balance becomes ${{\bf{Q}}_L^{'}}={{{\bf{Q}}_L^0} + {\bf{q}}_a - {{\bf{q}}_d}}$. The goal of the defender is to compensate for the attacker’s actions and to avoid the voltage collapse by maintaining the post-attack $\Delta$ as close as possible to the nominal $\Delta^0$. We assume that the players have full knowledge of the system model and each other’s parameters. Thus, this investigation characterizes ideal game performance. We plan to extend it to uncertain scenarios in future work.
The Cost-based Stackelberg Game {#sec:SG}
===============================
In the proposed [*Stackelberg game (SG)*]{}, the *actions* of the attacker, $\mathbf{a} \in \mathbb R^K$, and the defender, $\mathbf{d} \in \mathbb R^K$, correspond to a finite number of discrete *investment levels* into the $K$ loads and $K$ control devices, respectively. A higher value of each element $a_k$ (or $d_k$) indicates a greater chance of successful attack (or protection) of the $k^{\text{th}}$ load. Given an investment pair (${\mathbf{a}}$, ${\mathbf{d}}$), the *utilities*, or *payoffs*, of the attacker and the defender are termed ${U^a}({\mathbf{a}},{\mathbf{d}})$ and ${U^d}({\mathbf{a}},{\mathbf{d}})$, respectively, expressed in terms of the instability index $\Delta$. The attacker aims to maximize $\Delta$ (thus degrading the system performance) while the defender aims to reduce it. In this zero-sum game [@Osborne1994], ${U^d}({\mathbf{a}},{\mathbf{d}}) = - {U^a}({\mathbf{a}},{\mathbf{d}})$. The defender is the *leader*, who establishes its investment profile first. Given a defenders’ strategy ${\mathbf{d}}$, the attacker *follows* by choosing its action ${{\mathbf{a}}}=g({\mathbf{d}})=\mathop {\arg \max }\limits_{{\mathbf{a}}} {U^a}({{\mathbf{a}}},{{\mathbf{d}}})$, a best response to ${\mathbf{d}}$. Thus, the defender chooses a strategy ${{\mathbf{d}}^*}$ that maximizes its utility given the attacker’s best responses $g({\mathbf{d}})$ to all its actions. A resulting Stackelberg equilibrium (SE) [@Osborne1994] (${{\mathbf{a}}^*}$, ${{\mathbf{d}}^*}$), where ${{\mathbf{a}}^*}=g({{\mathbf{d}}^*})$, optimizes the utility of each player in an SG. Finally, we modify the standard *Backward Induction* (BI) method [@Osborne1994] for computing an SE and develop the *cost-based Stackelberg game* (CBSG) that saves the players’ costs without compromising their payoffs.
Players’ Actions and Cost Constraints
-------------------------------------
The attacker’s actions are denoted as ${\mathbf{a}}=[a_1,\cdots, a_k, \cdots, a_K] \in \mathbb R^K$, where $a_k
\in \{0,1/(L_a-1), \allowbreak 2/(L_a-1),\cdots,1\}$ is a discrete level of investment into load $k$, and $L_a$ denotes the number of attacker’s investment levels. We assume each load is equipped with protective software. The value of $a_k$ denotes the probability of successfully hacking into load $k$, which is determined by attacker’s investment level, or the amount of resources allocated to hacking this load. Thus, for any attack action ${\mathbf{a}}$, there are $2^K$ possible outcomes. Define the $i^{\text{th}}$ outcome of attack at all loads by a binary $K$-tuple $\mathbf{O}^i = [o_1^i, \cdots, o_k^i, \cdots, o_K^i], \forall i=1,\cdots,2^K$, where $o_k^i=1$ if attack at node $k$ is successful and $o_k^i=0$ if it fails. Given an attacker’s action vector ${\mathbf{a}}$, the probability of outcome $\mathbf{O}^i$ is given by: $$\label{eq:Pr_Oi}
P_{{\mathbf{a}}}(\mathbf{O}^i) = \prod\limits_{k:{\forall o_k^i} = 1} {{a_k}} \prod\limits_{k:{\forall o_k^i} = 0} {\left( {1 - {a_k}} \right)}.$$
In addition, we assume that if the attacker successfully hacks into load $k$, the nominal reactive power demand $Q_k$ of this load will be increased by $q_a^k$, where $Q_k$ is the $k^{\text{th}}$ element of ${{\mathbf{Q}}_L}$ in (\[eq:delta\]). The combined incremental demand for outcome $\mathbf{O}^i$ is represented by a $1 \times K$ vector given by: $$\label{eq:q_a_i}
{\mathbf{q}}_a^i = \mathbf{O}^i \odot {{\mathbf{q}}_a},$$ where ${{\mathbf{q}}_a}=[q_a^1,\cdots,q_a^k,\cdots,q_a^K]$ and $\odot$ indicates element-wise multiplication.
Next, we define the defender’s actions as $\mathbf{d}=[d_1,\cdots,d_k,\cdots,d_K] \in \mathbb R^K$, where $d_k\in\{0,1/(L_d-1),2/(L_d-1),\cdots,1\}$ denotes the defender’s investment level on load $k$, or equivalently, the control device of that load, and $L_d$ is the number of defender’s investment levels. Let us assume the maximum reactive power that the defender is able to compensate on load $k$ is $q_d^{k,\max}$ when the level $d_k=1$, where $q_d^{k,\max}$ is selected so that the voltage at that load bus does not exceed 1.1 pu. For the level $d_k$, the defender’s compensation is $q_d^k= d_k q_d^{k,\max}$. The reactive power demand compensation for all loads is specified by the $1 \times K$ vector: $$\label{eq:q_d}
{{\mathbf{q}}_d}=[q_d^1,\cdots,q_d^k,\cdots,q_d^K].$$
Finally, we assume both players’ investments are subject to the following constraints. The *attacker’s constraints* include:
1. **Cost constraint**: Assume attack on load $k$ at full effort (i.e., when $a_k=1$) has cost $\gamma_a \in \mathbb R$. Scaling this cost by the level of effort $a_k$ and summing over all loads, we obtain the following constraint on the total cost of the attacker: $$\label{eq:a_ac}
\gamma_a{||{{\mathbf{a}}}||_1} \le 1,$$ where $||\cdot||_1$ denotes the $\ell_1$-norm, which is given by the sum of the magnitudes of all elements of the vector.
2. **Covertness constraint**: Considering that the voltage at every load bus is mandated to be within an operating range of 0.9 to 1.1 pu, the attacker must be covert in the sense that it cannot change the demand at any target bus $k$ beyond a limit $q_a^{k,\max}$ as that may violate this voltage range, leading to the attack being caught by the operator. This covertness constraint is, therefore, modeled as: $$\label{eq:a_cc}
q_a^k \leq q_a^{k,\max}, \forall k.$$ Note that $q_a^{k,\max}$ will be different for different $k$ due to physical variabilities of the loads.
The *defender* has the following *constraint*:
1. **Cost of protection**: Assuming full protection (i.e. $d_k=1$) for load $k$ costs $\gamma_d \in \mathbb R$, the defender’s budget constraint is given by: $$\label{eq:d_pc}
\gamma_d{||{{\mathbf{d}}}||_1} \le 1.$$
In (\[eq:a\_ac\]) and (\[eq:d\_pc\]), we assumed without loss of generality that the total cost of each player is bounded by 1. Thus, the scalars $\gamma_a$ and $\gamma_d$ represent scaled costs per load of the attacker and defender, respectively.
Players’ Utility Functions
--------------------------
Prior to the attack, the instability index $\Delta = \Delta^0$. The attacker aims to increase $\Delta$, but not exceed $\Delta=1$ since the latter results in system voltage collapse and any additional investment wastes the attacker’s resources. Moreover, to save its cost, the defender invests only to compensate for the attacker’s action, i.e. it aims to reduce $\Delta$ while maintaining $\Delta\geq \Delta^0$. Thus, the utilities of the players are defined in terms of the *deviation* $\Delta - {\Delta ^0}$.
Given the attacker’s and defender’s actions ${\mathbf{a}}$ and ${\mathbf{d}}$, respectively, the reactive power demand vector for the $i^{\text{th}}$ outcome $\mathbf{O}^i$ is computed as ${{\bf{Q}}_L^{i'}}={{{\bf{Q}}_L^0} + {\bf{q}}_a^i - {{\bf{q}}_d}}$. The *attacker’s utility* for the $i^{\text{th}}$ outcome $\mathbf{O}^i$ is given by: $$\label{eq:Ua_i}
U_i^a({\bf{d}}) = Clip\left( {{{\left\| {{\bf{Q}}_{cirt}^{ - 1}{\bf{Q}}_L^{i'}} \right\|}_\infty }}; (\Delta^0, 1) \right) - {\Delta ^0},$$ where $$\label{eq:clip}
Clip(x;({\Delta ^0},1)) = \left\{ {\begin{array}{*{20}{c}}
{{\Delta ^0}}&{x \le {\Delta ^0}}\\
x&{{\Delta ^0} < x < 1}\\
1&{x \ge 1}
\end{array}} \right..$$
Given the strategy pair $({{\mathbf{a}}}, {{\mathbf{d}}})$ under the attacker’s constraints (\[eq:a\_ac\]) and (\[eq:a\_cc\]), the attacker’s utility is represented as the expectation of (\[eq:Ua\_i\]) over all outcomes: $$\begin{aligned}
\label{eq:Ua}
{U^a}({\bf{a}},{\bf{d}}) &= {\rm E}\left( {U_i^a({\bf{d}})} \right) = \sum\limits_i^{{2^K}} {{P_{{\mathbf{a}}}(\mathbf{O}^i)U_i^a({\bf{d}})} } , \hfill \\
\mbox{s.t.}\;\;\;{\kern 1pt} & \gamma_a{||{{\mathbf{a}}}||_1} \le 1, \;\;
q_a^k \leq q_a^{k,\max}, \forall k. \nonumber\end{aligned}$$ In the proposed zero-sum game, the *defender’s utility* under the constraint (\[eq:d\_pc\]) is given by: $$\begin{aligned}
\label{eq:Ud}
{U^d}({\bf{a}},{\bf{d}}) & = -{U^a({\bf{a}},{\bf{d}})}, \hfill \\
\mbox{s.t.}\;\;\;{\kern 1pt} & {\gamma_d}||{{\mathbf{d}}}||_1 \le 1.\nonumber\end{aligned}$$
Finally, we make the following realistic assumptions: (i) the attacker is able to cause voltage collapse when it has unlimited resources and the defender is inactive, and (ii) the defender is able to compensate fully for the attacks when both players have unlimited budgets.
Cost-based Stackelberg Equilibrium (CBSE)
-----------------------------------------
An SE is usually found using the *Backward Induction* (BI) algorithm [@Osborne1994]. Since multiple SEs are possible in an SG, we modify the BI method to select an SE that saves both players’ costs. The *Cost-based Backward Induction* (CBBI) algorithm is described below:\
**Step 1:** **(a)** For each defender’s action ${\mathbf{d}}$ that satisfies (\[eq:d\_pc\]), the attacker determines the set of its best responses $\mathcal{G}({\mathbf{d}})$, where ${g}({{\mathbf{d}}}) \in \mathcal{G}({\mathbf{d}})$ if $$\begin{aligned}
\label{eq:a_r1}
{g}({{\mathbf{d}}}) &= \mathop {\arg \max }\limits_{{\mathbf{a}}} {U^a}({{\mathbf{a}}},{{\mathbf{d}}}), \hfill \\
\mbox{s.t.}\;\;\;{\kern 1pt} & \gamma_a{||{{\mathbf{a}}}||_1} \le 1, \; q_a^k \leq q_a^{k,\max}, \forall k, \nonumber \end{aligned}$$
**(b)** For any ${\mathbf{d}}$, if there are multiple attacker’s best responses in $\mathcal{G}({\mathbf{d}})$, the attacker chooses a response with the *smallest cost*: $$\label{eq:a_r2}
g_o({\bf{d}}) = \mathop {\arg \min }\limits_{{g}({\bf{d}}) \in \mathcal{G}({\mathbf{d}})} ||{g}({\bf{d}})||_1.$$ **Step 2: (a)** The defender determines the set of investment strategies $\mathcal{D}$ that maximize its payoff where ${{\mathbf{d}}^*} \in \mathcal{D}$ if: $$\begin{aligned}
\label{eq:d_r1}
{{\mathbf{d}}^*} &= \mathop {\arg \max }\limits_{\mathbf{d}} {U^d}({g_o}({\mathbf{d}}),{\mathbf{d}}), \hfill\\
\mbox{s.t.}\;\;\;{\kern 1pt} & {\gamma_d}||{{\mathbf{d}}}||_1 \le 1. \nonumber\end{aligned}$$
**(b)** If multiple solutions exist in $\mathcal{D}$, a strategy with the *smallest cost* is chosen: $$\label{eq:d_r2}
{{\mathbf{d}}_o^*} = \mathop {\arg \min }\limits_{{{\mathbf{d}}^*}\in \mathcal{D}} ||{{\mathbf{d}}^*}||_1.$$ Denote $$\label{eq:a_r3}
\mathbf{a}_o^* = g_o({\mathbf{d}}_o^*).$$
The strategy pair $({\mathbf{a}_o^*},{\mathbf{d}_o^*})$ in (\[eq:d\_r2\]) and (\[eq:a\_r3\]) is a *cost-based Stackelberg equilibrium (CBSE)*, and the corresponding game is termed the *cost-based Stackelberg game (CBSG)*. The following Theorem summarizes several properties of SGs and of the proposed CBSG.
\
**(a)** An SE exists in a finite two-player SG.\
**(b)** All SEs of a zero-sum SG have the same payoffs.\
**(c)** In a CBSG, given $L_a$ and $L_d$, the utility of each player is non-increasing with its cost per load when the opponent’s cost per load is fixed.\
**(d)** Given $L_a$ and $L_d$, there exist $\epsilon>0$ and $\theta>0$ such that when $\gamma_a<\epsilon$ while $\gamma_d>\theta$, the attacker’s utility at CBSE $U^a({\mathbf{a}_o^*},{\mathbf{d}_o^*}) = 1-\Delta^0$ (i.e. voltage collapse occurs). Moreover, there exists an $\alpha>0$ such that when $\gamma_d<\alpha$, the attacker’s utility at CBSE $U^a({\mathbf{a}_o^*},{\mathbf{d}_o^*}) = 0$ (i.e. $\Delta=\Delta^0$).\
**(e)** When $L_d$ (or $L_a$) is increased to a number of investment levels $L_d^{'}$ (or $L_a^{'}$) that satisfies $L_d^{'}-1=n(L_d-1)$ (or $L_a^{'}-1=n(L_a-1)$), where $n$ is a positive integer, the defender’s (or attacker’s) utility does not decrease if the costs and the opponent’s number of investment levels $L_a$ (or $L_d$) are fixed.
Please refer to [@AnCDCsupp], [@An2020thesis Appx.B].
From Theorem 1, CBBI selects an SE with reduced costs of both players while providing the payoff of any other SE.
Instead of defining a zero-sum game with hard cost constraints, a general-sum SG, where the costs and covertness are incorporated into in the utility functions [@8815018], can be investigated. Note that Theorem 1(b) does not hold for this non-zero sum game. We expect the performance trends of this game to resemble those of the proposed zero-sum SG.
Numerical Results {#sec:cs}
=================
Game Analysis for the IEEE 9-bus System
---------------------------------------
The IEEE 9-bus system has 6 load buses, which are potential targets for the players in the proposed SG. The nominal voltage instability index for this system is computed as $\Delta^0 = 0.1935$. In the simulation, $q_a^{k,\max}$ is determined by the covertness constraint (\[eq:a\_cc\]), and we set $q_d^{k,\max}=2$ pu, $\forall k$. It was verified that these compensations do not violate the $[0.9, 1.1]$ pu voltage range for any bus.
First, we examine the *dependency of the proposed CBSG on the players’ costs*. Fig. \[fig:Ua9\_La3Ld3\] shows the attacker’s utility $U^a({\mathbf{a}_o^*},{\mathbf{d}_o^*})$ (\[eq:Ua\]) at CBSE while Fig. \[fig:ga0\_9\] and \[fig:gd1\_9\] illustrate the players’ strategies for varying scaled costs of attack $\gamma_a$ and protection $\gamma_d$ assuming three investment levels for each player. We observe the performance trends described in Theorem 1 (a)$\sim$(d). In Fig. \[fig:Ua9\_La3Ld3\], the largest attacker’s utility is $1-\Delta^0=0.8065$ (voltage collapse), which occurs when $\gamma_a\leq0.15$ and $\gamma_d\geq0.75$. In this case, the defender’s cost per load greatly exceeds that of the attacker’s, so the attacker is able to increase its reactive power demand to achieve $\Delta=1$ while the defender cannot compensate due to its limited resources. On the other hand, when the defender’s cost is small ($\gamma_d\leq 0.15$), implying the defender has sufficient resources to compensate for any level of attack, the resulting $U^d({\mathbf{a}_o^*},{\mathbf{d}_o^*})=-U^a({\mathbf{a}_o^*},{\mathbf{d}_o^*})=0$ or $\Delta = \Delta^0$. Finally, we found that as $\gamma_d \rightarrow \infty$ (not shown), the defender becomes inactive. In this case, voltage collapse happens if $\gamma_a\leq 0.3$ while $\Delta=\Delta^0$ is achieved only if the attacker is also inactive ($\gamma_a>2$). By comparing these results with Fig. \[fig:Ua9\_La3Ld3\], we conclude that strategic protection is necessary for maintaining a reliable instability index $\Delta$.
![Attacker’s utility at CBSE vs. $\gamma_a$ and $\gamma_d$ for $L_a=L_d=3$[]{data-label="fig:Ua9_La3Ld3"}](figures/Ua9_La3Ld3.pdf){width="35.00000%"}
[|>m[9mm]{}||>m[14mm]{}|>m[12mm]{}||>m[14mm]{}|>m[12mm]{}|]{} & &\
& $\Delta-\Delta^0$ & Ranking & $\Delta^0-\Delta$ & Ranking\
4 & 0.2947 & 4 & 0.0892 & 4\
5 & 0.2825 & 6 & 0.2379 & 1\
6 & 0.3040 & 1 & 0.2101 & 2\
7 & 0.2871 & 5 & 0.0584 & 5\
8 & 0.2987 & 3 & 0.1364 & 3\
9 & 0.3025 & 2 & 0.0257 & 6\
\[tab:importance\_9a\]
Next, to illustrate the players’ strategy choices, we list the *“importance" ranking of loads* for both players in Table \[tab:importance\_9a\]. First, we show the increment of the instability index $\Delta-\Delta^0$ (assuming the initial value $\Delta^0$) when the reactive power demands of individual loads are increased by the maximum allowed covertness limit $q_a^{k,\max}$. The greater the increment for an individual load, the more “important" that load is to the attacker. In addition, we illustrate the “importance" order of the loads from the defender’s perspective by examining the decrement $\Delta^0-\Delta$ when the initial value is $\Delta^0$ and the defender compensates a fixed $q_d^{k,\max}=1$ pu, $\forall k$, on a single load. Similarly, the greater the decrement, the more “important" that load is to the defender. While Tables \[tab:importance\_9a\] shows the “importance" ranking of the loads before the attack, i.e. when the initial $\Delta=\Delta^0$, we found that the “importance" ranking does not depend on the initial value of $\Delta$.
In general, *multiple SEs* are possible for any choice of game settings. In this example, multiple SEs occur in two regions in the range of costs shown in Fig. \[fig:Ua9\_La3Ld3\]. First, for $\gamma_d \leq 0.15$, the defender is able to invest into all loads, resulting in $\Delta=\Delta^0$, but best responses of the attacker vary, creating multiple SEs. The CBSE occurs when the attacker chooses not to act to save its cost as shown in the first row of Fig. \[fig:ga0\_9\]. Second, in the region $\gamma_d\geq 0.75$, $\gamma_a \leq 0.15$, the attacker is very strong while the defender is severely resources-limited. Thus, voltage collapse cannot be avoided. While multiple SEs exist, e.g. the attacker invests fully into all loads and the defender invests into some “important" loads, the CBSE corresponds to the cases illustrated in the bottom row of Fig. \[fig:ga0\_9\] and top row of Fig. \[fig:gd1\_9\]. By choosing the CBSE, the defender saves cost by not acting since it cannot avoid voltage collapse while the attacker invests into its three “important" loads, sufficient to achieve voltage collapse.
![Player’s strategies at CBSE when $\gamma_a\leq 0.15$, $L_a=L_d=3$[]{data-label="fig:ga0_9"}](figures/ga0_9.pdf){width="47.00000%"}
![Player’s strategies at CBSE when $\gamma_d\geq0.75$, $L_a=L_d=3$[]{data-label="fig:gd1_9"}](figures/gd1_9.pdf){width="47.00000%"}
Next, we examine *the effect of cost constraints on the players’ investment strategies*. In Fig. \[fig:ga0\_9\] the attacker has plentiful resources ($\gamma_a\leq 0.15$). As the cost of defense per load $\gamma_d$ increases, the defender protects fewer loads and/or reduces the level of protection, thus reducing the utility $U^d({\mathbf{a}_o^*},{\mathbf{d}_o^*})$ at CBSE, or increasing $\Delta$. When $\gamma_d=0.2$, the defender targets its “important" loads (Table \[tab:importance\_9a\]), i.e. 4, 5, 6, and 8 are fully protected ($d_k=1$) while less “important" loads 7 and 9 are protected at half-strength ($d_k=0.5$). When $\gamma_d=0.4$, the defender’s budget tightens further, and only the most “important" four loads are protected although only the third ranked load is protected fully, revealing limitations of the load-ranking method in Table \[tab:importance\_9a\]. The latter ranking is based on attacking or defending a single load and thus is imprecise for multiple-load attack or protection scenarios due to nonlinearity of (\[eq:delta\]). In Fig. \[fig:gd1\_9\], we illustrate the players’ strategies at CBSE when the defender is resource-limited. For $\gamma_a\leq 0.15$, the defender is able to reduce $\Delta$ by protecting the “important" loads 5, 6 and/or 8 at the level $d_k=0.5$. These strategies correspond to the defender’s best effort under limited resources. The attacker also chooses to attack its “important" loads, but tries to avoid investing into the loads protected by the defender. These choices are caused by the proposed game hierarchy and the nonlinear nature of the payoff function.
Next, we examine *the dependency of the players’ payoffs on the levels of investment* $L_a$, $L_d$. In Fig. \[fig:La2\], we illustrate the attacker’s utility at CBSE as the defender’s number of investment levels $L_d$ varies while fixing $L_a=2$. Similar simulations were performed for other scenarios, where the number of levels of one player is fixed while the other player’s number of investment levels varies, and the results confirm the conclusion in Theorem 1(e). Since the game complexity scales as $L_a^K \times L_d^K$ and becomes very large even for the 9-bus system ($K=6$) as $L_a$ or $L_d$ grows, more thorough analysis of this dependency will be addressed in future work.
![Attacker’s utility at CBSE vs. $\gamma_a$ when $L_a=2$ and $L_d$ varies, $\gamma_d=0.5$[]{data-label="fig:La2"}](figures/La2.pdf){width="30.00000%"}
Finally, we compared the game described in this Section with the Individual Optimization (IO) method, where the players do not take into account the opponent’s actions or the game hierarchy [@AnCDCsupp; @An2020thesis]. Significant losses in cost and up to 18% loss in payoff were observed for some cost pairs for each player when using the IO method, thus underscoring the importance of strategic investment.
CBSG for the IEEE 39-bus System
-------------------------------
One reason for performing a detailed analysis of our game on a relatively small power system model, such as the IEEE 9-bus system, was to demonstrate that the attacker’s and defender’s investment choices are mostly limited to the top few “important" loads (from Table \[tab:importance\_9a\] and Fig. \[fig:ga0\_9\]-\[fig:gd1\_9\]). Moreover, for the IEEE 9 bus system, we compared the game above, where all loads were used (Fig. 1-4), with the SG where each player targets only its top four “important" loads in Table I. We found that when $\gamma_a>0.5$ and $\gamma_d>0.5$, the SEs and, thus, the playoffs of the two games are exactly the same. Outside this cost range, the difference between the payoffs of the two games is at most 0.18. This result can be explained by observing that in the region $\gamma_a>0.5$ and $\gamma_d>0.5$, the attacker (or defender) has sufficient resources for attacking (or protecting) only 4 loads fully or partially with 3-level investment, and, thus, concentrates on the top four ‘important" loads even if other loads are included in its action set. Taking a hint from this observation, we can apply the proposed game to any larger-scale power system model over a subset of loads that includes the most “important" loads. We expect the resulting performance to approximate closely that of the full-scale game (over all system loads) except for the cost region where one of the players is not resource-constrained. This approach reduces computational complexity significantly in practical resource-limited scenarios. We next validate our game using important loads of the 39-bus model. This model has 29 loads, and the nominal voltage stability index is computed as $\Delta^0=0.5560$. We first determine the most “important" loads of the IEEE 39-bus system using the approach described in Sec.IV.A (Table \[tab:importance\_9a\]). We found that the five most “important" loads for the attacker are $11>6>5>10>13$ while the five most “important" loads to the defender are $7>8>5>6>11$, where $A>B$ indicates that load $A$ is ranked higher than load $B$.
![Attacker’s utility at CBSE vs. $\gamma_a$ and $\gamma_d$ for the IEEE 39-bus system, $L_a=L_d=2$[]{data-label="fig:Ua39_La2Ld2"}](figures/Ua39_La2Ld2_new.pdf){width="35.00000%"}
Based on the above analysis, we construct a CBSG over the selected subset of loads in the IEEE 39-bus system: $\{5,6,7,8,10,11,13\}$, which includes both players’ five most important loads. Moreover, we assume both players employ $L_a=L_d=2$ in this pilot study since a higher number of levels increases complexity significantly. We note that in the region $\gamma_a>0.2$ and $\gamma_d>0.2$, only 5 loads can be attacked or protected with 2-level investment. Thus, in the latter region, each player concentrates most of the time on the top five “important" loads, and the simpler SG where only the top five “important" loads of each player are employed in the action sets is expected to closely approximate the complex game where all 29 loads are targeted. Fig. \[fig:Ua39\_La2Ld2\] shows the attacker’s utility at CBSE for varying scaled costs $\gamma_a$ and $\gamma_d$. We observe the same performance trends as in Fig. \[fig:Ua9\_La3Ld3\] and Theorem 1. Note that only the region $\gamma_a>0.2$ and $\gamma_d>0.2$ provides accurate estimation of the full-scale game. Although the payoffs are inaccurate in the shaded region of Fig. \[fig:Ua39\_La2Ld2\] ($\gamma_a<0.2$ or $\gamma_d<0.2$), we are certain that voltage collapse ($\Delta=1$) occurs in this region according to Theorem 1(d), but the exact location of voltage collapse would require a full-scale game. Note that small values $\gamma_a<0.2$ or $\gamma_d<0.2$ indicate very large resources of one player, which is unlikely in practice. Finally, we observe that in the IEEE 39-bus system, the attacker is successful in raising the instability index $\Delta$ over a larger range of the cost region than in Fig. \[fig:Ua9\_La3Ld3\] since the nominal instability index of the 39-bus model is $0.5560$, which is much higher than that for the IEEE 9-bus system ($0.1935$). In other words, the 39-bus model is more “stressed" than the 9-bus model. Nevertheless,voltage collapse is expected to occur only when the attacker has very small cost $\gamma_a$ and the defender’s cost $\gamma_d$ is large. We conclude that for both examples, voltage collapse can be successfully prevented unless the defender’s security resources are disproportionately limited relative to the attacker’s budget.
Conclusion
==========
We proposed a cost-based Security Investment Stackelberg game for voltage stability of a power system. In the proposed game, investment resources are allocated strategically to optimize the players’ performance objectives and to save costs. It is demonstrated that voltage stability is maintained unless the defender’s security budget is much lower than the attacker’s budget. Future work will focus on extending the proposed game to power system models with uncertainties, scenarios where a player has limited knowledge of the opponent’s resources, as well as to numerical approaches that scale well with the size of the system model.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Pratishtha Shukla for helpful discussion on the game formulation and analysis.
[^1]: The authors are with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695. (e-mail: [lan4@ncsu.edu]{}, [achakra2@ncsu.edu]{}, [sasha@ncsu.edu]{})
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abstract: 'The LHCb detector with its unique pseudorapidity coverage allows to perform soft-QCD measurements in the kinematic forward region where QCD models have large uncertainties. Selected analyses related to soft-Diffraction will be summarised in these proceedings. Energy flow and charged particle multiplicity have been measured separately in different event classes. They give input for modelling the underlying event in *pp* collisions. Prompt hadron ratios are important for hadronisation models, while the $\overline p/p$ ratio is a good observable to test models of baryon number transport.'
author:
- |
[*Marco Meissner$^1$, on behalf of the LHCb collaboration*]{}\
$^1$Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, INF 226, 69120 Heidelberg, Germany\
title: 'Results of Soft-Diffraction at LHCb'
---
Introduction
============
The LHCb experiment at the Large Hadron Collider is a dedicated experiment to study CP-violating processes and rare decays of hadrons containing beauty and charm quarks. The detector is a single-arm forward spectrometer [@LHCb] designed to efficiently detect the decay products of B-hadrons in a pseudorapidity range of approximately $2<\eta<5$. This also allows LHCb to make soft-QCD measurements in a kinematic region which is hardly accessible by the general purpose detectors. The analyses presented in these proceedings are selected soft-QCD measurements in the context of soft-diffraction. The data used for these analyses are Proton-Proton (*pp*) collisions at centre-of-mass energies of $\sqrt{s}=0.9$ and $7\;$TeV recorded with minimum bias triggers in the low luminosity running phase in 2010. Important for the presented analyses are the tracking system, which is composed of a high precision Silicon Vertex Locator (VELO) surrounding the interaction point and the main tracking stations located downstream of a dipole magnet. Particle identification is performed by two Ring Imaging Cherenkov (RICH) detectors which allow separation of charged particles in a momentum range of $2-100\;$GeV/c.
Forward Energy Flow
===================
For a particular pseudorapidity interval $\Delta\eta$ the total energy flow is defined as $$\frac{1}{N_{int}} \frac{dE_{tot}}{d\eta}=\frac{1}{\Delta\eta}\left( \frac{1}{N_{int}}\sum_{i=1}^{N_{part,\eta}} E_{i,\eta} \right),$$ where $N_{int}$ is the number of inelastic *pp* interactions. Measuring the energy flow (EF) at large pseudorapidities directly probes multi-parton interactions (MPI) and parton radiation which contribute to the underlying event in proton-proton collisions. The measurement has been performed in 4 different event classes, an (1) inclusive minimum bias sample which requires to have at least one reconstructed track with a momentum $p$ greater than $2\;$GeV/c in the forward acceptance ($1.9<\eta<4.9$). Further there is a (2) hard scattering sub-sample which implies at least one high $p_{T}$ track per event ($p_{T}>3\;$GeV/c). By exploiting the additional backwards coverage of the VELO it was possible to obtain a (3) diffractive enriched and a (4) non-diffractive enriched sample of events. These were selected by looking for backward tracks in the pseudorapidity range of $-3.5<\eta<-1.5$. This selection exploits the fact, that a large rapidity gap is an experimental signature to identify diffractive processes.
[l]{}[0.61]{} {width="30.00000%"} {width="30.00000%"}\
{width="30.00000%"} {width="30.00000%"}
The measured total EF, which is the sum of charged and neutral EF, is depicted in Fig.\[EF\_1\], superimposed with different PYTHIA generator predictions. The EF in the four event samples increases from the diffractive sample to the inclusive minimum bias and non-diffractive sample up to the hard scattering sample. The errors are dominated by systematic uncertainties, like model dependence for correcting detector effects, uncertainties for the track finding and residual pile-up. These uncertainties decrease towards larger $\eta$ which is the most interesting region for studying MPI phenomena. In all event classes, the PYTHIA 6 tunes underestimate the EF especially at larger pseudorapidities but overestimate at lower $\eta$. The default PYTHIA 8 prediction (8.135) is in better agreement except for the hard scattering sample. The energy flow in diffractive enriched events is well described by PYTHIA 8. The measurement was also compared to predictions of cosmic ray generators (details see [@LHCb_EF]) which were not tuned to LHC data. The EPOS and SYBILL generators show a good agreement with data in the minimum bias and non-diffractive sample while QGSJET predictions are best for hard scattering. The EF in diffractive events seem to be underestimated by all cosmic ray generators.
Charged Particle Multiplicity
=============================
The multiplicity of primary produced charged particles has been measured [@LHCb_CPM] for *pp* collisions at $\sqrt{s}=7\;$TeV. Primary particles are defined as either directly produced in the *pp* collision or from short lived decays ($\tau<10\;ps$). For this measurement only information from the VELO has been used. As there is a negligible influence from the magnetic field in this sub-detector the measurement has no explicit momentum cut-off for low energetic particles.
[r]{}[0.61]{} {width="30.00000%"} {width="30.00000%"}
Further it allows to measure particle production in a small backward $\eta$ range ($-2.5<\eta<-2.0$) in addition to the regular forward coverage ($2<\eta<4.5$). The measured multiplicity distribution (Fig. \[CPM\_1\]) for events with at least one track in the forward acceptance shows that all generators, namely PYTHIA 6, PYTHIA 8 and PHOJET underestimate the amount of promptly produced charged particles. However, testing PYTHIA 6 tunes for which diffractive processes were switched off at generator level seem to give an accidentally better agreement with the measured data. Studying a sub-sample of hard QCD events by requiring at least one high $p_{T}$-track with $p_{T}>1\;$GeV/c results in an increase of charged particles. At least for this hard event sample, some PYTHIA 6 tunes provide a reasonable description of the data.
Prompt Hadron Ratios
====================
The LHCb collaboration measured prompt hadron production ratios [@LHCb_PHR] as a function of pseudorapidity in three different $p_{T}$-bins for *pp* collisions at centre-of-mass energies of $\sqrt{s}=0.9$ and $7\;$TeV. The measured anti-particle/particle ratios $K^{-}/K^{+}$, $\pi^{-}/\pi^{+}$ and $\overline p/p$ as well as the different-particle ratios $(p+\overline p)/(\pi^{+}+\pi^{-})$, $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ and $(p+\overline p)/(K^{+}+K^{-})$ are probes for hadronisation models implemented in Monte Carlo event generators. Further, some of these ratios can be used to test models of baryon to meson and strangeness suppression. A crucial ingredient in measuring these ratios is a good particle identification which is provided by the two RICH detectors. The PID efficiencies were directly determined from data using decays of resonances like $\Lambda \rightarrow p\pi^{-}$, $\phi \rightarrow K^{+}K^{-}$ and $K_{S}^{0} \rightarrow \pi{+}\pi{-}$. The dominant systematic uncertainty remains the PID efficiency because of the limited size of the calibration sample. Comparing the measured hadron ratios to different PYTHIA 6 tunes shows that no tune is able to describe the entire set of measurements. Only each type of hadron ratio can be described by at least one single tune. Of special interest is the $\overline p/p$ ratio, which is sensitive to baryon number transport. At $\sqrt{s}=0.9\;$TeV the $\overline p/p$ ratio has a significant $\eta$ dependence, which is qualitatively described by all PYTHIA 6 tunes. But only the Perugia NOCR tune, which favours an extreme model of baryon transport, is able to give a quantitatively good prediction while other generator tunes underestimate baryon transport.
[l]{}[0.5]{} {width="50.00000%"}
However, at $\sqrt{s}=7\;$TeV the Perugia NOCR model tends to now overestimate baryon transport. The same ratio can also be studied as function of rapidity loss $\Delta y=y_{beam}-y_{particle}$, defined as the difference of the rapidity of the beam and the considered particles. This representation allows to compare measurements of experiments at different centre-of-mass energies, as it is shown in Fig. \[PHR\_1\]. The LHCb measurement covers a wider range in rapidity loss and improves previous measurements with a better precision. Combining the LHCb data points and the complementary ALICE measurement [@Alice] allows to perform a fit within in the Regge model [@Regge]. In this model, baryon production at high energies is driven by Pomeron exchange and baryon transport by string junction exchange. In this picture, the gained fit parameters are related to contributions from these two mechanisms. The fit result of a low string junction contribution with low intercept point allows to draw conclusions about the associated standard Reggeon or the Odderon.\
[99]{} LHCb collaboration, A. A. Alves Jr. [*et al.*]{}, JINST [**3**]{} S08005 (2008). LHCb collaboration, R. Aaij [*et al.*]{}, Eur. Phys. J. [**C73**]{} 2421 (2013). LHCb collaboration, R. Aaij [*et al.*]{}, Eur. Phys. J. [**C72**]{} 1947 (2012). LHCb collaboration, R. Aaij [*et al.*]{}, Eur. Phys. J. [**C72**]{} 2168 (2012). The ALICE collaboration, Phys. Rev. Lett. [**105**]{} 072002 (2010). D. Karzeev, Phys. Lett. B [**378**]{} 238 (1996).
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abstract: 'We show how the early data runs of the LHC can provide valuable checks of the different components of the formalism used to predict the cross sections of central exclusive processes. The ‘soft’ rapidity gap survival factor can be studied in electroweak processes, such as $W$+gaps events, where the bare amplitude is well known. The generalized gluon distribution, in the appropriate kinematic region, can be probed by exclusive $\Upsilon$ production. The perturbative QCD effects, especially the Sudakov-like factor, can be probed by exclusive two- and three-jet production. We discuss the possible role of enhanced absorptive corrections which would violate the soft-hard factorization implied in the usual formalism, and suggest ways that the LHC may explore their presence.'
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plus 2mm minus 2mm 23.0cm 17.0cm -1.0in -42pt
IPPP/08/07\
DCPT/08/14\
4 March 2008\
[**Early LHC measurements to check predictions for central exclusive production**]{}
V.A. Khoze$^{a,b}$, A.D. Martin$^a$ and M.G. Ryskin$^{a,b}$\
$^a$ Institute for Particle Physics Phenomenology, University of Durham, Durham, DH1 3LE\
$^b$ Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg, 188300, Russia
Introduction
============
Central exclusive production is now recognized as an important search scenario for new physics at the LHC, see for instance, [@KMRProsp] - [@cr] and references therein. The experimental studies of such processes are at the heart of the FP420 project [@LOI; @cox] which proposes to complement the CMS and ATLAS experiments at the LHC by installing additional forward proton detectors 420m away from the interaction region. In particular, these detectors will allow the measurement of the exclusive production of new heavy particles, such as Higgs bosons. As demonstrated in [@KKMRext] - [@fghpp] such measurements will be able to provide valuable information on the Higgs sector of MSSM and other popular BSM scenarios.
Indeed, central exclusive processes are very interesting both from the viewpoint of theory, since they contain a mixture of soft and hard QCD effects, and of experiment, as they provide a clean environment to measure the quantum numbers and masses of new objects which may be seen at the LHC. Moreover, the $J_z=0$ selection rule ($J_z$ is the projection of the total angular momentum along the proton beam direction), arising in central exclusive diffractive processes [@KMRmm; @DKMOR], provides a unique possibility to study directly the coupling of the Higgs-like bosons to the bottom quarks, because the LO QCD background is strongly suppressed by the $J_z=0$ rule. As is well known, the determination of the $Hb\bar {b}$ Yukawa coupling appears to be very difficult for other search channels at the LHC.
The theoretical formalism [@KMR; @KMRProsp; @KMRsoft] needed to describe a central exclusive diffractive process of a system $A$ contains quite distinct parts, shown symbolically in Fig. \[fig:parts\]. In brief, we first have to calculate the $gg \to A$ hard subprocess, $H$, convoluted with the gluon distributions $f_g$. Next we must account for the higher loop corrections which reflect the absence of additional QCD radiation in the hard subprocess – that is, for the Sudakov suppression factor $T$. Finally we must enter soft physics to calculate the survival probability $S^2$ of the rapidity gaps either side of $A$ – that is the probability that the exclusive nature of the process will not be destroyed by the secondaries produced by the rescattering of the incoming particles.
![A symbolic diagram for the central exclusive production of a system $A$.[]{data-label="fig:parts"}](parts.eps){height="6cm"}
Objective
=========
The uncertainties associated with the prediction of the rate of a central exclusive process are potentially not small. Each of the above stages has its own uncertainties. Therefore, it is important to perform checks of the approach using processes with appreciable cross sections that will be experimentally accessible in the first data runs of the LHC with integrated luminosities in the range 100 ${\rm pb}^{-1}$ to 1 ${\rm fb}^{-1}$. Here, our aim is to identify processes where the different ingredients of the formalism used to calculate central exclusive production can be tested experimentally, more or less independently.
The outgoing protons in the forward regions of the main LHC detectors will be measured by the proton tagging detectors (Roman Pots). At 220 m on either side of the CMS detector there exists the roman pots of the TOTEM experiment [@TOTEM; @CMS-TOTEM]. ATLAS will have the ALFA detector [@ALFA] at 240m as well as the proposed RP220 detector [@RP220] at 220m. As we already mentioned, FP420 project [@LOI; @cox] proposes to install forward proton taggers for both ATLAS and CMS. Note that RP220 and FP420 aim to operate at high luminosities. However it is quite likely that for the first 1-2 years of LHC running the forward proton detectors will not be operational. We therefore first consider measurements which do not rely on tagging the forward outgoing protons.
Even without proton tagging, diffractive measurements at the LHC can be performed through the detection of rapidity gaps. This is a well known technique used extensively at HERA and the Tevatron. A summary of the forward detectors instrumented around ATLAS and CMS is given, for instance, in [@forw]. The central detectors (CD) of CMS and ATLAS have an acceptance in pseudorapidity $\eta$ of roughly $|\eta |<2.5$ for tracking information and $|\eta |<5$ for calorimeter information. We will discuss a situation where a heavy system $A$ is detected in the region $|\eta |<2.5$, and where the calorimeters in $2<\eta <5$ interval, are used to select events with rapidity gaps.
In the present paper the word [*gap*]{} means a rapidity interval $\Delta\eta$ devoid of hadronic activity – for the charged particles we assume a ‘track veto’, while the absence of neutrals should be checked by the calorimeters. The amplitudes of all the processes discussed below are infrared stable and are not affected by the possibility of soft gluon emission.
Due to angular ordering, which originates from coherence, in those processes where the gap is provided by $W$-boson exchange (as in Fig. \[fig:WZ\](a,c) below), the only possible soft gluons occur at the edge of the gap and arise from the corresponding quark jet. These gluons should be accounted for in the jet searching algorithm. In cases where the gaps are associated with colour-singlet, two-gluon exchange (as in central exclusive dijet production), the presence of Sudakov-like $T$-factors in the unintegrated gluon distributions (see eq.(\[eq:rat6\]) below) guarantee that there is no emission of any additional soft gluons.
The selection of rapidity gap events by a ‘veto’ trigger can be used up to rather large luminosities, when the mean number, $N$, of interactions per bunch crossing is sizeable. However at larger luminosities the efficiency of the trigger is reduced by a factor $e^{-N}$ – that is, by the probability to have no additional ‘pile-up’ inelastic interaction in the bunch crossing. This probability can be measured independently in the same experiment.
Of course, the proposal to use calorimeters in the $2<\eta <5$ interval to select the events with a rapidity gap does not mean that we will only consider gaps with $\Delta \eta<3$. First, part of the gap can be at smaller $\eta$ and, secondly, extensive additions are foreseen to enlarge the coverage in the forward regions[^1]. Possible “holes” in particle observation in small rapidity intervals between different calorimeters do not affect our predictions very much. The probability to produce extra soft hadrons in the hole region only is proportional to the effective Pomeron-Pomeron cross section which is rather small, according to the triple-Regge analysis of UA8 data [@UA8]. We evaluate the correction to be less than 3$\%$ for a hole of size $\Delta \eta_{\rm hole} \sim 1.5$.
The main uncertainties of the predictions for exclusive processes are associated with the calculation of
- the probability $S^2$ that additional soft secondaries will not populate the gaps separating the centrally produced system $A$ from the outgoing protons (or the products of their dissociations)[^2];
- the probability to find the appropriate gluons that are given by generalized unintegrated distributions $f_g(x,x',Q_t^2)$;
- the higher order QCD corrections to the hard subprocess amplitude, where the most important is the so-called Sudakov suppression caused (in the Feynman gauge) by the double-log loop, denoted $T$ in Fig. \[fig:parts\], see [@KMR];
- the so-called semi-enhanced absorptive corrections (see, for example, [@KKMR; @bbkm; @KMRln]) and other effects which may violate the soft-hard factorization (which is implied by Fig. \[fig:parts\]).
We discuss below, in turn, possible checks of the various ingredients of the calculation of the exclusive cross sections. To be precise, we address the uncertainties (i)-(iv) in Sections 3-6 respectively.
Gap survival probability $S^2$
==============================
As a rule, the gap survival probability, that is the probability that the secondaries produced in additional soft interactions do not populate the rapidity gaps, is calculated within a multichannel eikonal approach. This method of calculation implicitly assumes a factorization between the soft and hard parts of the process. The probability of elastic $pp$ rescattering, shown symbolically by $S^2$ in Fig. \[fig:parts\], can be calculated in a model independent way once the elastic $pp$ amplitude is known, that is, the elastic cross section $d\sigma_{\rm el}/dt$ is measured at the LHC. However there may be some excited states (corresponding to $N^*$ resonances and low mass diffractive dissociation) between the blob $S$ and the perturbative QCD amplitude on the right-hand-side of Fig. \[fig:parts\]. The presence of such additional states enlarges the absorptive correction. Usually this effect is calculated using a Good-Walker formalism [@GW] with a two- or three-channel eikonal. In order to experimentally check the role of this effect, we need to consider a process with a bare cross section that can be reliably calculated theoretically. Good candidates are the production of $W$ or $Z$ bosons with rapidity gaps on either side.
$W$ production with rapidity gaps
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![Diagrams for (a) $W$ production with 2 rapidity gaps, (b) inclusive $W$ production, and (c) $Z$ production with 2 rapidity gaps.[]{data-label="fig:WZ"}](WZ.eps){height="6cm"}
In the case of ‘$W$+gaps’ production the main contribution comes from the diagram shown in Fig. \[fig:WZ\](a) [@KMRphoton]. One gap, $\Delta \eta_1$, is associated with photon exchange, while the other, $\Delta \eta_2$ , is associated with $W$ exchange. The cross section of this process can be calculated straightforwardly. It is proportional to the quark distribution in the proton at a large scale and not too small $x$, where the uncertainties of the parton densities are small. To select these events at the expected luminosities of the early data runs, we can use the rapidity gap veto trigger combined with a high $p_t$ decay lepton or jet trigger. The momentum transferred through the photon is typically small and the rapidity gap $\Delta \eta_1$ runs from the rapidity of the $W$ boson, $y_W$, up to the maximum rapidity measured. The gap, $\Delta \eta_2$, corresponding to $W$ exchange, is limited by the rapidity of the quark jet, which balances the transverse momentum of the $W$ boson (which has a broad distribution with $q_{tW} \sim M_W/2$ since it is driven by the $t$-channel $W$ propagator).
At first sight, the probability for soft rescattering in such a process is rather small and we would expect $S^2 \sim 1$. The reason is that the transverse momentum, $k_t$, distribution of the exchanged photon is given by the logarithmic integral , for which the dominant contribution comes from the low $k_t^2$ region. In other words, the main contribution comes from the region of large impact parameters, $b_t$, where the opacity of the proton is small. However the minimum value of $|t|$, |t\_[min]{}| \[eq:tmin\] is not negligibly small. Note that the momentum fraction $x_p=1-\xi$ associated with the upper proton can be measured with sufficient accuracy[^3], [*even without the tagging of the forward protons*]{}, by summing the momentum fractions \_i = e\^[y\_i]{}/ \[eq:xi\] of the outgoing $W$ and the hadrons observed in the calorimeters. As long as the gap $\Delta \eta_2$ is large, the dominant contribution to the sum $\xi=\sum \xi_i$ comes from the decay products of the $W$ boson[^4]. For example, for $\eta_W=2.3(-2.3)$, we expect an $\xi$ distribution centred about $\xi \sim 0.1(0.001)$.
. We use the superscript + to indicate that the cross sections correspond to the configuration where the photon is emitted by the proton going in the + direction. The event rate observed in the central detector will be twice as large as that shown.[]{data-label="fig:W1"}](W1.eps){height="9cm"}
![The rapidity gap survival factor $S^2$ as a function of $\xi$ calculated using the global soft model of [@KMRsoft]. []{data-label="fig:W2"}](W2.eps){height="9cm"}
![The rapidity gap survival factor $S^2$ as a function of $\xi$ calculated using the global soft model of [@KMRsoft], assuming that the valence (sea) quarks are associated with the weak (strong) absorptive components. The small spread of the predictions arising from the different partonic content of the diffractive eigenstates mean that $W$+gaps events offer a meaningful test of the $S^2$ factor. Note that $S^2$ for the $W^+$ signal is larger since it has a bigger valence quark contribution.[]{data-label="fig:W3"}](W3.eps){height="9cm"}
The predictions are shown in Figs. \[fig:W1\]–\[fig:W3\]. First, in Fig. \[fig:W1\], we neglect the absorptive correction, and show the cross section for $W$+gaps events for different choices of the gap size $\Delta \eta_2$. We see cross sections of up to 1 pb, depending on the gap size and on $\xi$. The rescattering decreases the cross section by the factor $S^2$ shown in Figs. \[fig:W2\] and \[fig:W3\]. Already, at this stage, we face the possibility of a violation of soft-hard factorization. We use the Good-Walker formalism [@GW] to describe multichannel eikonal rescattering. That is, we use the diffractive eigenstates $\phi_k$ which diagonalize the ‘nucleon’-Pomeron couplings $\beta_{ij}$, which describe the transition from nucleon excited state $i$ to state $j$. In Ref. [@KMRsoft] it was assumed that each eigenstate has the same size and the same partonic composition. The continuous curve in Fig. \[fig:W2\] was calculated under this assumption. On the other hand, in the extreme cases where we assume that the partons which participate in the process of Fig. \[fig:WZ\](a) are concentrated in the diffractive eigenstate with the lowest (largest) absorptive cross section, we obtain the results shown by the dashed (dotted) curves. A rather more realistic scenario (see, for example, [@KKMR]) is to allocate the valence quark to the component with the smallest absorption and the sea quark to the component with largest cross section. The corresponding predictions are shown in Fig. \[fig:W3\]. Since the valence quark contribution is more important for $W^+$ production and for the configuration with the largest gap size $\Delta\eta_2$, the expected gap survival factor $S^2$ is found to be larger. Now, the spread of predictions caused by the different partonic content, shown in Fig. \[fig:W3\], is rather small. With a more realistic assumption (in which each diffractive state contains some part of the valence and some part of the sea) the spread will be even smaller.
In the first LHC data runs it may be difficult to measure the absolute value of the cross section with sufficient accuracy. Most probably the ratio ($W$+gaps/$W$ inclusive) will be measured first. In this case, the inclusive $W$ production process (in the same kinematic region, Fig. \[fig:WZ\](b)) plays the role of the luminosity monitor. Note that the cross section for inclusive $W$ production is much larger than that with rapidity gaps. The reason is an inclusive $W$ is produced directly by $q\bar{q}$ fusion, which is prohibited for gap events since the colour flow produced by the $t$-channel quarks populates the $\Delta \eta_{1,2}$ rapidity gaps.
Of course, the survival factor $S^2$ measured in $W$+gaps events is quite different from that for exclusive Higgs production, which comes from smaller values of $b_t$. Nevertheless this measurement is a useful check of the model for soft rescattering.
$Z$ production with rapidity gaps
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It would appear that a good way to study the low $b_t$ region directly is to observe $Z$ boson production via $WW$ fusion, see Fig. \[fig:WZ\](c). Here, both of the rapidity gaps originate from heavy boson exchange and the corresponding $b_t$ region is similar to that for central exclusive Higgs production[^5]. The expected $Z$+gaps cross section is of the order of 0.2 pb, and $S^2$=0.3 for $\Delta \eta_{1,2} > 3$ and for quark jets with $E_T>50$ GeV [@pw]. Again it would be sufficient to measure the ratio ($Z$+gaps/$Z$ inclusive).
One problem is that even with the $E_T>50$ GeV cut, the QCD background arising from the QCD $b\bar{b}$ central exclusive production is comparable to the electroweak $qq \to Z+2$ jet signal. Therefore we should concentrate on the leptonic decay modes of the $Z$ boson, which results in a smaller event rate[^6].
With tagged protons
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So far we have discussed measurements which do not depend on observing the forward protons. However, when the forward proton detectors become operational we can do more. Both the longitudinal and transverse momentum of the forward protons can, in principle, be measured. Hence we can study the $k_t$ behaviour of the cross section section for $W$+gap events, and scan the proton opacity, as described in [@KMRphoton].
Generalized, unintegrated gluon distribution $f_g$
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The cross section for the central exclusive production of a system $A$ is calculated using $Q_t$ factorization, and essentially has the form [@KMR] $$\sigma(pp \to p+A+p) ~\simeq ~\frac{S^2}{b^2} \left|\frac{\pi}{8} \int\frac{dQ^2_t}{Q^4_t}\: f_g(x_1, x_1', Q_t^2, \mu^2)f_g(x_2,x_2',Q_t^2,\mu^2)~ \right| ^2~\hat{\sigma}(gg \to A).
\label{eq:M}$$ The first factor, $S^2 $, is the soft survival factor discussed in the previous section and the factor $1/b^2$ arises from the integration over the transverse momentum of the forward proton assuming the form $d\sigma /dp^2_t \propto {\rm exp}(-bp^2_t)$. Also, $f_g$ denotes the generalized, unintegrated gluon distribution in the limit of $p_t \to 0$. Below we give a more precise definition of $f_g$ and explain how it is determined.
The generalized gluon distribution has not yet been measured explicitly. However, for the case of interest, where the skewedness is small, it can be obtained from the conventional diagonal gluon distribution, $g$, known from the global parton analyses. The procedure is as follows. We consider the skewedness first in the transverse momenta, and then in the longitudinal momenta, carried by the two $t$-channel gluons in Fig. \[fig:parts\]. The transverse momentum, $p_t$, transferred through the rapidity gap in a central exclusive process is limited by the proton form factor. This is not a large transverse momentum: $p_t^2 \simeq -t \ll Q_t^2$. Typically, we have the hierarchy |t\_[min]{}| 10\^[-4]{} [GeV]{}\^2; p\_t\^2 \~0.2 [GeV]{}\^2; Q\_t\^2 \~4 \^2; which follows, respectively, from (\[eq:tmin\]) with the relevant $\xi \sim 0.01$ or less; from the proton form factor; and, finally, from the presence of the Sudakov-like form factor, $T$ of (\[eq:T\]), in the integrand of (\[eq:M\]) – the maximum of the integrand occurs at $Q_t^2 \sim 4~\GeV^2$, see [@KMRProsp; @KMRmm].
As a consequence, it is reasonable to assume the following factorization of the gluon distribution: $F(t)~f_g(x,x',Q^2_t,\mu^2)$. That is, we work in terms of the distribution $f_g(x,x',Q^2_t,\mu^2)$, which is skewed only in longitudinal momentum fractions, $x \ne x'$ in Fig. \[fig:parts\]; $Q_t$ is essentially the transverse momentum of each $t$-channel gluon. Finally, $\mu^2 \sim M_A^2/4$ is the factorization scale which separates the partonic distribution from the matrix element of the hard subprocess $gg \to A$. The precise scale, to be used, is given in (\[eq:Delta\]) below.
In the region of interest (x’ \~) (x \~) 1. The generalized distribution can be obtained using the Shuvaev transform [@shuv] which assumes that the Gegenbauer moments of the generalized parton distributions are equal to the Mellin moments of the diagonal distributions. The accuracy of this assumption is $O(x^2)$. As a result we obtain [@MR01][^7]. f\_g(x,x’,Q\_t\^2,\^2) = R\_g ( xg(x,Q\_t\^2)), \[eq:rat6\] where $R_g$ accounts for the $x \ne x'$ skewedness. Note that the double log Sudakov suppression $T(Q_t,\mu)$ is now included in the unintegrated gluon distribution $f_g$, since to provide $Q_t$ factorization we choose an axial/planar gauge (and not the Feynman gauge used to draw $T$ in Fig. \[fig:parts\]). The Sudakov form factor $T$ ensures that the gluon remains untouched in the evolution up to the hard scale $\mu$, so that the rapidity gaps survive. It results from resumming the virtual contributions in the DGLAP evolution, and is given by [@WMR] $$T(Q_t,\mu) = \exp\left(-\int_{Q_t^2}^{\mu^2}\frac{\alpha_S(k_t^2)}{2\pi}\frac{dk_t^2}{k_t^2}
\int_\Delta^{1-\Delta}\left[zP_{gg}(z) + \sum_q P_{qg}(z)\right]dz\right), \label{eq:T}$$ where $\Delta$ is specified in (\[eq:Delta\]) below.
The main uncertainty here comes from the lack of knowledge of the integrated gluon distribution $g(x,Q_t^2)$ at low $x$ and small scales. For example, taking $Q_t^2=4~\GeV^2$ we find that a variety of recent MRST [@mstw] and CTEQ [@cteq] global analyses give gluons which have a spread of xg = (3-3.8) [for]{} x=10\^[-2]{} [and]{} xg = (3.4-4.5) [for]{} x=10\^[-3]{}. These are big uncertainties bearing in mind that the cross section for central exclusive production depends on $(xg)^4$.
Exclusive $\Upsilon$ production as a probe of $f_g$
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\[t\]
![Exclusive $\Upsilon$ production via (a) photon exchange, and (b) via odderon exchange.[]{data-label="fig:upsilon"}](upsilon.eps){height="5cm"}
To reduce the uncertainty associated with $f_g$ we can measure exclusive $\Upsilon$ production. The process is shown in Fig. \[fig:upsilon\](a). The cross section for $\gamma p \to \Upsilon p$ [@mrt] is given in terms of exactly the same generalized, unintegrated gluon distribution $f_g$ that occurs in Fig. \[fig:parts\]. If the $\Upsilon$ is detected at the LHC in the rapidity interval $(|y|< 2.5)$ then it will sample the $x_{1,2}$ intervals ($10^{-4}-10^{-2}$). To probe even lower $x$ we can either detect more forward $\Upsilon$ production at ALICE or LHCb [@LHCb] or study exclusive $J/\psi$ production, see [@KMRphoton].
Of course, there may be competition between production via photon exchange, Fig. \[fig:upsilon\](a), and via odderon exchange, where the upper three $t$-channel gluons form the C $=-1$ odderon state[^8], see Fig. \[fig:upsilon\](b). To date, odderon exchange has not been observed. On the other hand, a lowest order perturbative QCD calculation indicates that the odderon process (b) may be comparable to the photon-initiated process (a), see, for example, [@bmsc]. If the upper proton is tagged, it will be straightforward to separate the two mechanisms since odderon production has no $1/q^2$ singularity characteristic of the photon. Photon exchange populates mainly the low $q_t$ region, that is at low $p_t$ of the outgoing proton, whereas odderon production occurs at relatively large $p_t$. Without a proton tagger, an odderon would be revealed by a measured cross section that is larger than that predicted for photon exchange. Exclusive $\Upsilon$ (or $J/\psi$) production could be the first hint of the odderon’s existence.
Returning to process (a), we note that the cross section is of the form \^[(a)]{} = N\_(p p) where the photon flux $N_\gamma$ is well known. For small $q_t$, we may neglect the proton form factor and use the leading log approximation N\_ = . The expression for $\sigma(\gamma p \to \Upsilon p) \propto f_g^2$ is given in [@mrt]. The cross section [@bmsc] .|\_[y=0]{} 50 [pb]{} at the LHC energy[^9]. The signal will be diluted by the $\Upsilon \to \mu\mu$ branching fraction of 0.025.
In order to use this process to constrain the gluon distribution of the proton it would be preferable to tag the lower proton. Otherwise there will be some admixture of proton excitations and to calculate precisely the gap survival factor $S^2$, as before, we will have to resolve the partonic content of the different diffractive eigenstates. On the other hand, for low values of $\xi$, the expected value $S^2$ is close to 1 (see the continuous curve in Fig. \[fig:W2\]), and it may be sufficient to use the existing HERA data for the ratio of the cross sections of diffractive $J/\psi$ photoproduction with and without the proton dissociation, to extract the contribution to the cross section from ‘elastic’ events in which the lower proton does not dissociate.
Improvements to the hard subprocess
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The hard matrix elements necessary to predict the cross section for a central exclusive process are calculated using perturbative QCD mainly at leading log accuracy [@KMRProsp]. Of course, there exist numerous NLO results, but as a rule these are applicable to inclusive processes which do not allow for the fact that the centrally produced system $A$ must be in a colour singlet state and obey the $J_z=0$ selection rule, that is the projection of the total angular momentum along the incoming proton directions should be zero. Moreover the relation between the generalized, unintegrated gluon distribution, $f_g$, and the conventional diagonal gluon, $g$, was also based on a LO calculation [@MR01].
Numerically, the largest effects may come from the next-to-leading-log (NLL) corrections to the double log Sudakov-like factor $T$ of (\[eq:T\]). To include these NLL corrections phenomenologically we choose the limits of the integration over $k^2_t$ in such a way to reproduce the result of an explicit first-loop calculation. It has been shown (see footnote 9 in [@KKMRext]) that this is achieved by choosing = [with]{} =0.62M\_A \[eq:Delta\] in (\[eq:T\]).
On the other hand, the lower limit $Q_t^2$ of the integration has not yet been validated at this level. Clearly the contribution from the very small $k_t$ region vanishes due to destructive interference between the emissions from the active gluon ($x$) and the screening gluon ($x'$). However, variation of the lower limit from $Q^2_t/2$ to $2Q^2_t$ may alter the prediction by up to an order of magnitude.
Fortunately, the contribution from the region of $k_t \sim Q_t$ can be calculated with better precision. We are seeking the next-to-leading log correction to the double logarithmic $T$ factor of (\[eq:T\]), that is, for the single log terms in $T$. Therefore, in the region $k_t \sim Q_t$ we need to keep only the LO BFKL-like term which contains the longitudinal logs. This term is given by the usual BFKL kernel [@bfkl] which sums all the leading $\alpha_s$log$M_A$ contributions. If we consider the $k_t^2$ integration in (\[eq:T\]) with a lower cut-off of $k_0^2$, then the above summation amounts to the replacement \_[k\_0\^2]{}... \_[k\_0\^2]{}(1-)... = \_[Q\_t\^2]{}... . \[eq:TTT\] First, we notice in the middle expression for the integral above that for $k_t \gg Q_t$ the last term in the brackets is negligible, while for $k_t \ll Q_t$ the whole expression in brackets goes to zero and removes the infrared divergence. Now we can put the infrared cut-off $k_0=0$. The result of an explicit calculation of the middle expression for the integral shows that it is equivalent to the integral in (\[eq:T\]) with the lower limit given by $Q^2_t$, as indicated in (\[eq:TTT\]).
Thus, in summary, both the upper and lower limits of the $k_t^2$ integral in (\[eq:T\]) are fixed so as to reproduce the one-loop contributions. The potentially large ambiguity in the value of $T$ has been removed.
To check experimentally the formalism used in the perturbative QCD calculations for the central exclusive matrix element we should study an exclusive process with the emission of one additional jet. Note that the physical origin of the Sudakov $T$ factor is that it expresses the probability ${\it not}$ to emit additional gluons. The formula for $T(Q_t,\mu)$ of (\[eq:T\]) can be written as $T={\rm exp}(-n)$ where $n$ is the mean number of gluons emitted in the interval $(Q_t,\mu)$. Thus, the observation of the explicit emission of additional gluon jets would provide a direct test of the formalism. Since the central system must be colour neutral, we need [*either*]{} consider the emission of two extra jets [*or*]{} to have the possibility to compensate the colour of one emitted gluon by rearrangement of the colour content of the system $A$. The optimum choice is to observe the emission of a third jet in the production of a pair of high $E_T$ jets, that is, in high $E_T$ dijet production.
Three-jet events as a probe of the Sudakov factor
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Traditionally, the search for the exclusive dijet signal at the Tevatron, $p\bar{p} \to p+jj+\bar{p}$, is performed [@CDFjj] by plotting the cross section in terms of the variable R\_[jj]{} = M\_[jj]{}/M\_A . \[eq:jj\] Ideally, we might expect exclusive dijet production to show up as a narrow peak centred at $R_{jj}=1$, since, for these events, the mass of the dijet system, $M_{jj}$, is equal to the mass, $M_A$, of the whole central system. Unfortunately, in practice, the $R_{jj}$ distribution is strongly smeared out by QCD bremsstrahlung, hadronization, the jet searching algorithm and other experimental effects [@CDFjj; @KMRrj].
![The rapidities of the three jets in the central system. Note that the rapidity $y_A$ of the whole central system does not necessarily occur at $y=0$. The rapidity interval containing the three jets is denoted by $\delta\eta$, outside of which there is no hadronic activity.\[fig:2\]](f2.eps){height="7cm"}
To weaken the role of this smearing it was proposed in Ref. [@KMRrj] that the observed dijet distribution be studied in terms of a new variable R\_j = 2E\_T ([cosh]{} \^\*)/M\_A , \[eq:j\] where only the transverse energy $E_T$ and the rapidity $\eta$ of the jet with the [*largest*]{} $E_T$ are used in the numerator. Here $\eta^* = \eta -y_A$ where $y_A$ is the rapidity of the whole central system[^10]. Clearly the jet with the largest $E_T$ is less affected by hadronization, final parton radiation etc. In particular, final state radiation at lowest order in $\alpha_S$ will not affect $R_j$ at all, since it does not change the kinematics of the highest $E_T$ jet used to evaluate (\[eq:j\]). Even with the emission of an extra jet during the final parton shower, we will still have $R_j=1$. Thus, to see the role of QCD radiation on the $R_j$ distribution, we only have to account explicitly for additional gluon radiation in the initial state. At leading order, it is sufficient to consider the emission of a third gluon jet, as shown in Fig. \[fig:2\], where we take all three jets to lie in a specified rapidity interval $\delta\eta$. The reason why it is sufficient to consider only one extra jet, is that the effect of the other jets, which, at LO, carry lower energy due to the strong ordering, is almost negligible in terms of the $R_j$ distribution.
![The $R_j$ distribution of exclusive two- and three-jet production at the LHC. Without smearing, exclusive two-jet production would be just a $\delta$-function at $R_j=1$. The distribution for three-jet production is shown for two choices of the rapidity interval, $\delta\eta$, containing the jets; these distributions are shown with and without smearing. Here, we have taken the highest $E_T$ jet to have $E_T>50$ GeV. To indicate the effect of jet smearing, we have assumed a Gaussian distribution with a typical resolution $\sigma=0.6/\sqrt{E_T~{\rm in~ GeV}}$. []{data-label="fig:Rj"}](Rj.eps){height="12cm"}
The cross section $d\sigma/dR_j$, as a function of $R_j$, for the exclusive production of a high $E_T$ dijet system accompanied by a third (lower $E_T$) jet was calculated and discussed in detail in [@KMRrj]. The mass of the central system, $M_A$, can be accurately measured by the missing mass reconstructed from the tagged protons, or, failing that, by accurately summing both the light-cone momentum fractions $\xi^+$ and $\xi^-$ of the hadrons observed in the calorimeter, see (\[eq:xi\]). As shown in [@clp], the usage of the $R_j$ variable is quite beneficial for the extraction of exclusive dijet events. In particular, the dependence on the jet selection criteria (for example, the cone radius and $R$-parameter) is less marked for the $R_j$ than for the conventional $R_{jj}$ variable. Further discussion can be found in [@KMRrj]. Moreover, note that studying the $R_j$ distribution, we do not need to select events with a rather large $E_T$ of the third jet. To calculate $R_j$ it is sufficient to measure only the $E_T$ of the largest $E_T$ jet, together with the rapidities of first and second jets, and the values of $M_A$ and $y_A$ as measured via $\xi^+$ and $\xi^-$.
In Fig. \[fig:Rj\] we show the $R_j$ distribution of both exclusive two- and three-jet production expected at the LHC. For three-jet production we show predictions for two choices of the rapidity interval $\delta\eta$ within which the third jet must lie. If we take the largest $E_T$ jet to have $E_T>50$ GeV at the LHC, we see that the cross section for exclusive three-jet production reaches a value of the order of 100 pb. Of course, if we enlarge the rapidity interval $\delta\eta$ where we allow emission of the third jet, then $d\sigma/dR_j$ will increase, see Fig. \[fig:Rj\]. Indeed, the measurement of the exclusive two- and three-jet cross sections [*as a function of $E_T$*]{} of the highest jet allow a check of the Sudakov factor; with much more information coming from the observation of the $\delta\eta$ dependence of three-jet production. Note that the background from double-Pomeron-exchange should be small for $R_j \gapproxeq 0.5$, and can be removed entirely by imposing an $E_T$ cut on the third jet, say $E_T>5$ GeV.
Another way to observe the effect of the Sudakov suppression is just to study the $E_T$ dependence of exclusive dijet production. On dimensional grounds we would expect $d\sigma/dE_T^2 \propto 1/E_T^4$. This behaviour is modified by the anomalous dimension of the gluon and by a stronger Sudakov suppression with increasing $E_T$. Already the existing CDF exclusive dijet data [@CDFjj] exclude predictions which omit the Sudakov effect. In the $E_T$ interval from 10 to 35 GeV the Tevatron observations show that the cross section falls by an order of magnitude faster than the prediction [@pes], based on [@pes2], which does not include the Sudakov suppression, see Fig. 20a of [@CDFjj].
It is clear that precise measurements of the $E_T$ behaviour of the exclusive dijet cross section at the LHC offer the possibility to probe the corrections to (\[eq:T\]). However the study of exclusive three-jet events will provide much more information and, indeed, allow a study of the integrand of (\[eq:T\]).
Soft-hard factorization: enhanced absorptive effects
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![(a) A typical enhanced diagram, where the shaded boxes symbolically denote $f_g$, and the soft rescattering is on an intermediate parton, giving rise to a gap survival factor $S_{\rm en}$; (b) and (c) are the Reggeon and QCD representations, respectively.[]{data-label="fig:enh"}](enh.eps){height="6cm"}
We mentioned in Section 3 that the soft-hard factorization implied by Fig. \[fig:parts\] may be already violated if the different diffractive eigenstates of the $pp$ interaction have different partonic distributions. To date, there is no unambiguous model to distribute the partons obtained in the global analyses between the different diffractive components. However, this is not a major uncertainty. It is seen in Fig. \[fig:W3\] that the difference between $W^+$ and $W^-$, which have a different valence quark contribution, is not large.
Another potential source of the violation of soft-hard factorization arises from the so-called enhanced Reggeon diagrams, which occur from the rescattering of an intermediate parton generated in the evolution of $f_g$. Such a diagram is shown in Fig. \[fig:enh\](a). Fig. \[fig:enh\](b) is the Pomeron representation of the diagram, where the rapidity of the intermediate parton fixes the position of the triple-Pomeron vertex. Clearly such diagrams will violate the soft-hard factorization.
The contribution of the first Pomeron loop diagram, Fig. \[fig:enh\](b) was calculated in perturbative QCD in Ref. [@bbkm]. A typical perturbative diagram is shown in Fig. \[fig:enh\](c). For LHC energies it was found that the probability of such rescattering may be numerically large[^11]. The reason is that the gluon density grows in the low $x$ region and, for low $k_t$ partons, approaches the saturation limit. The recent estimates of [@WZ] indicate that in the region of $x^- \sim 10^{-6}$ (which is relevant for the LHC kinematics) the value of the saturation scale, $Q_S$, exceeds 1 GeV. In other words, a parton with $k_t<Q_S$ will receive an important absorptive correction. However, this is true for the central region of impact parameter space. On the other hand, eikonal absorption, which is shown in Fig. \[fig:parts\] and discussed in Section 3, makes the centre of the disk almost black. This eikonal factor, $S^2$, already forces the central exclusive signal to occur at relatively large $b_t$, namely $b_t>0.5-0.6$ fm, see Fig. 22 of [@KMRnewsoft]. In this peripheral region of the proton, the value of $Q_S$ is rather low. In fact, $Q_S^2 \sim 0.3~ \GeV^2$ is found in Fig. 11 of [@WZ]. As a consequence, the enhanced diagram will affect only the very beginning of the QCD evolution – the region that cannot be described perturbatively and which, in our calculation of the cross sections for central exclusive processes, is already included phenomenologically – that is, described in terms of the multichannel eikonal framework.
This is probably one reason why semi-enhanced screening corrections are not seen in leading neutron production measured at HERA [@HERAln]. As the energy of the virtual photon increases, the available rapidity region grows, so the number of intermediate partons that may participate in the rescattering increases, and we expect that the fraction of leading neutrons should decrease. This is not seen in the data [@HERAln]. Further discussion is given in [@KKMR; @KMRln; @KMRneutr].
Pomeron-$p$ interactions as a probe of the enhanced effect
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Experimentally, it appears at first sight, that we may study the role of semi-enhanced absorption by observing the $W$+2 gaps process shown in Fig. \[fig:WZ\](a). To do this one may vary the transverse momentum of the accompanying quark jet, $q$, and the size of the rapidity gap $\Delta \eta_2$. For lower transverse momentum of the quark jet we expect a stronger absorptive effect, that should decrease with increasing $\Delta \eta_2$, since the number of partons in the rest of the rapidity interval increases. Unfortunately, the photon-initiated process of Fig. \[fig:WZ\](a) occurs at large impact parameter $b_t$ where the probability of rescattering is small. Moreover it will be very difficult to observe the quark jet at low $E_T$ in the main calorimeter $(-5<\eta<-3)$. Therefore we will discuss other processes which depend on rescattering on intermediate partons.
![Schematic diagrams for (a) the inclusive production of a system $A$, (b) and (c) for the diffractive production of $A$ without and with ‘enhanced’ soft rescattering on intermediate partons. The system $A$ is taken to be either a $W$ boson or an $\Upsilon$ or a pair of high $E_T$ jets.[]{data-label="fig:3en"}](3en.eps){height="7cm"}
The observations we have in mind are the measurements of ratio $R$ of diffractive (one-gap) events for $W$ (or $\Upsilon$ or dijet) production as compared to the number of events for the inclusive process (shown in Fig. \[fig:WZ\](b) for $W$ production). These processes are shown schematically in Fig. \[fig:3en\]. In other words, $R$ is the ratio of the process in diagram (c) to that in diagram (a). That is R = = S\^2S\^2\_[en]{}\_[[over]{} b\_t]{}, \[eq:Ren\] where $a^{\rm incl}$ and $a^{\rm diff}$ are the parton densities determined from the global analyses of inclusive and diffractive deep inelastic scattering data, respectively. The heavy central system $A$ is either $W$ or a pair of high $E_T$ jets or $\Upsilon$ or a Drell-Yan $\mu^+\mu^-$ pair. For $W$ or $\mu^+\mu^-$ pair production the parton densities $a$ are quark distributions, whereas for dijet or $\Upsilon$ production they are mainly gluon densities. The diffractive parton densities are known from analyses [@mrw2; @h1] of diffractive deep inelastic data. Thus measurements of the ratio $R$ will probe the gap survival factor averaged over the impact parameter $b_t$.
If we neglect the effect of enhanced rescattering on intermediate partons, that is set $S_{\rm en}^2=1$ then $R$ is the ratio of process (b) to (a). However this ratio will be reduced by the rescattering on the intermediate partons as sketched in Fig. \[fig:3en\](c)). The largest contribution to the extra suppression factor $S_{\rm en}^2$ comes from the interaction of the upper intermediate partons with the lower proton, since the partial energy corresponding to this interaction is larger, which leads to a larger absorptive cross section. Note that the enhanced effect is practically forbidden in a rapidity interval close to the rapidity of $A$ (shown by the bold lines in Fig. \[fig:3en\](c)), as the transverse momenta of partons in this region are quite large. The mean number of partons, $\langle N \rangle$, emitted in the evolution from 1 GeV up to the hard scale $\mu \sim M_A/2$ is given by the power of the exponent in the $T$ factor of (\[eq:T\]). We have $\langle N \rangle \sim 1$ or 2 for $M_A \simeq$ 10 or 100 GeV respectively, that is for exclusive $\Upsilon$ or $W$ production. These are the partons with $p_t>Q_S(b_t)$ which do not suffer the enhanced absorptive effect. As a rule, each parton occupies about one unit of rapidity. Therefore in the evaluation of the enhanced effect we exclude from our calculation an appropriate rapidity interval on either side of $A$, which is shown by the bold lines in Fig. \[fig:3en\](c). The presence of a “threshold” factor, that is the existence of a rapidity interval on either side of $A$ where the enhanced contribution is effectively forbidden, was emphasized in [@KKMR]. An analogous threshold strongly suppresses the enhanced absorption to exclusive Higgs production [@KMRln].
In the region of relatively large $b_t$ the value of $Q_S$ is low. Hence we evaluate the enhanced absorption in terms of Reggeon framework, using the phenomenological triple-Pomeron coupling extracted from diffractive $J/\psi$ production [@KMRj]. The absorptive corrections for $J/\psi$ production are low, and the value $g_{3\funp} \sim g_N/3$ should be close to the original bare triple-Pomeron coupling. Here, $g_N$ is the coupling of the Pomeron to the proton. The contribution of the one-loop Regge diagram of Fig. \[fig:enh\](b) can be quite large. Indeed, its strength, as compared to the original cross section, can be estimated by evaluating Fig. \[fig:enh\](b). We obtain r = d’, where $\sigma_{pp}(\eta')$ is the proton-proton cross section evaluated at the energy $\sqrt{s}=m_N\exp(\eta'/2)$. The factor 4 comes from the AGK cutting rules [@AGK] and $B \simeq 5~ \GeV^{-2}$ is the sum of the $t$ slopes of the form factors involved. The integral is taken over the appropriate rapidity intervals of the intermediate partons in Fig. \[fig:3en\](c). Since the result is sizeable, we allow for the summation of the higher order diagrams by replacing the absorptive factor $1-r$ by $S^2_{\rm en}={\rm exp}(-r)$.
![The predictions of the ratio $R$ of (\[eq:Ren\]) for $W$ and $\Upsilon$ production with (continuous curves) and without (dashed curves) enhanced soft rescattering on intermediate partons.[]{data-label="fig:upsw"}](upsw.eps){height="15cm"}
![The predictions of the ratio $R$ of (\[eq:Ren\]) for the production of a pair of high $E_T$ jets with (continuous curves) and without (dashed curves) enhanced soft rescattering on intermediate partons.[]{data-label="fig:upsd"}](upsd.eps){height="15cm"}
Experimentally, we can measure the rapidity $y_A$ of the central system ($A=W, ~\Upsilon$ or dijet) and also the momentum fraction carried in the gap direction by the soft hadrons, $\xi^- = \sum \xi_i^-$, recall the analogous sum of (\[eq:xi\]). Thus we know the value of $x_\funp=\xi^- +\xi^-_A$. That is, we can observe a double distribution $d^2 \sigma^{\rm diff}/dx_\funp dy_A$, and form the ratio $R$ using the inclusive cross section, $d\sigma^{\rm incl}/dy_A$. If we neglect the enhanced absorption, it is straightforward to calculate the ratio $R$ of (\[eq:Ren\]). The results are shown by the dashed curves in Figs. \[fig:upsw\] and \[fig:upsd\] as a function of the rapidity $y_A$ of the heavy system $A$. When we allow for the enhanced rescattering the ratios are reduced, and lead to steeper $y_A$ distributions, as shown by the continuous curves. These plots correspond to a fixed value of $\xi^-=10~\GeV/7$ TeV. That is, we assume that the soft hadrons observed in the central calorimeter carry a longitudinal momentum[^12] $p_z$=10 GeV in the direction of the rapidity gap associated with Pomeron exchange[^13].
The results shown in Figs. \[fig:upsw\] and \[fig:upsd\] should be regarded as an indication of the size of possible enhanced effects and not as quantitative predictions. First, the model used to estimate $S^2_{\rm en}$ is quite naive. Next, the results for large negative $y_A$ sample the diffractive gluon density at large $\beta$, where it is not well constrained. Finally, for $\Upsilon$ (and dijet production at the lower $E_T$ values) we need, for large positive $y_A$, the conventional gluon density at very small $x=\beta x_\funp \sim M_A{\rm exp}(-y_A)/\sqrt{s}\sim 10^{-4}$, where it is not well determined in the global analyses.
First, we discuss the results shown in Fig. \[fig:upsw\] for the ratio $R$ for $W$ production at positive $y_A$. Here, due to the large $W$ mass, we have no available rapidity interval for enhanced rescattering, that is $S^2_{\rm en}\simeq 1$. Moreover, this region corresponds to $x \sim 10^{-3}$ where the quark densities are well known. Therefore inclusive production acts as a good luminosity monitor and the ratio $R$ will yield information about the eikonal factor $S^2$. Note that we now probe the survival factor at much smaller $b_t$ than that corresponding to Figs. \[fig:W2\] and \[fig:W3\]. Here we expect the eikonal (non-enhanced) survival factor to be $S^2=0.08$.
On the other hand, diffractive production of a relatively light $\Upsilon$ is associated with larger rapidity intervals available for secondaries, and hence the possibility of more soft rescattering with intermediate partons, leads to more enhanced absorption. Indeed, the expected enhanced survival factor $S^2_{\rm en}\sim 0.2-0.3$. For $\Upsilon$ production the variation of $R$ with $y_A$ is weak. The smaller rescattering of intermediate partons with $y>y_A$ on the lower proton is compensated by stronger rescattering of the partons with $y<y_A$ on the upper proton.
Perhaps the most informative probe of $S^2_{\rm en}$ is to observe the ratio $R$ for dijet production in the region $E_T \sim 15-30$ GeV. For example for $E_T \sim$ 15 GeV we predict $S^2_{\rm en} \sim$ 0.25, 0.4 and 0.8 at $y_A=-2, ~0$ and $2$ respectively.
Conclusions and Outlook
=======================
Most of the diffractive measurements described above can be performed, without detecting the very forward protons, by taking advantage of the relatively low luminosity in the early LHC data runs. This allows the use of a veto trigger to select events with no hadronic activity in the region corresponding to the large rapidity gap(s). In this way we are able to study central exclusive diffractive processes, which should be experimentally accessible at the LHC, that probe the various individual components of the formalism used to predict their cross sections. The components are sketched in Fig. \[fig:parts\]. To summarize, the gap survival factor, $S^2$, caused by eikonal rescattering may be studied as indicated in Figs. \[fig:W2\], \[fig:W3\] and \[fig:upsw\], and the possible enhanced, $S^2_{\rm en}$, contributions as shown in Figs. \[fig:enh\], \[fig:upsw\] and \[fig:upsd\]. The relevant unintegrated gluon distribution, $f_g$, can be constrained by observing $\Upsilon$ production, see Fig. \[fig:upsilon\], and the QCD radiative effect, $T$, may be checked by observing exclusive two- and three-jet events, see Figs. \[fig:Rj\].
In the first LHC runs it may be difficult to measure the absolute values of the cross sections with sufficient accuracy. For instance, it will take time to determine the luminosity with a precision better than, say, 10$\%$. Thus the measurements of the ratios of the rate of events with and without rapidity gaps (such as ($W$+gaps/$W$ inclusive) and ($Z$+gaps/$Z$ inclusive) etc.) will be more reliable in the early data runs.
When the forward proton detectors are operating, even at moderate integrated luminosity[^14], much more can be done. First, it is possible to measure directly the cross section $d^2\sigma_{\rm SD}/dtdM^2_X$ for single diffractive dissociation, $pp \to p+X$, and also the cross section $d^2\sigma_{\rm DPE}/dy_1 dy_2$ for soft central diffractive production, $pp \to p+X+p$. These measurements will strongly constrain the models used to describe diffractive processes and the effects of soft rescattering. The predictions of a very recent detailed model can be found in Figs. 20 and 23 of [@KMRnewsoft]. It turns out that $d^2\sigma_{\rm DPE}/dy_1 dy_2$ is particularly sensitive to the detailed partonic content and sizes of the various diffractive eigenstates, see Fig. 20(d).
Next, a study of the transverse momentum distributions of both of the tagged protons, and the correlations between their momenta, $\vec{p}_{t1}$ and $\vec{p}_{t2}$, is able to scan the proton optical density (opacity) [@KMRtag; @KMRphoton] (see also [@Petrov; @krp]). In principle, there are additional smearing effects caused by the intrinsic transverse momentum spread of the proton beams, see for instance, [@KP]. Incorporation of these effects requires detailed studies, including, in particular, the detector resolution. Therefore the predictions based on measurements of the proton transverse momenta should allow for this smearing.
We emphasize that the selection of central exclusive dijet production in the kinematical region corresponding to the sought-after Higgs signal ($E_T \sim M_H/2$) provides an ideal “standard candle”. At leading log accuracy, this process includes all the components of the theoretical formalism used to predict the central exclusive Higgs signal; the same parton densities in the same kinematical region, the same gap survival factors $S^2$ (and $S^2_{\rm en}$) and the same QCD radiative effect $T$. The only small difference arises at next-to-leading log order. It is caused by the interference between the large-angle soft gluon radiation from the initial active gluons and the coloured composition of the dijet system. This effect should be evaluated allowing for the detector acceptance and the selection cuts used in the experiment.
Finally note, that when the statistics allow, a valuable check of the formalism of central exclusive diffractive production will come from the central diphoton production, see for instance, [@kmrs]. The first CDF results [@cdfdiphot] are quite encouraging.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Mike Albrow, Brian Cox, Albert De Roeck, Sasha Nikitenko, Andy Pilkington, Krzysztof Piotrzkowski and Risto Orava for encouraging us to write this article and for valuable advice. MGR thanks the IPPP at the University of Durham for Physics for hospitality. This work was supported by INTAS grant 05-103-7515, by grant RFBR 07-02-00023, by the Russian State grant RSGSS-5788.2006.02, and by the Russia-Israel grant 06-02-72041-204; 205; 210; 200.
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[^1]: CMS will have the CASTOR calorimeter operating with the coverage of $5.1< |\eta |<6.5$ and a Zero Degree Calorimeter (ZDC) with an acceptance for neutral particles with $\eta >8$. Both calorimeters have an electromagnetic and hadronic section. Moreover, CMS is expected to have integrated read out with TOTEM (see [@TOTEM]) allowing CMS to benefit from the TOTEM forward coverage and TOTEM, in turn, from the CMS central coverage [@CMS-TOTEM]. ATLAS plan a Cerenkov detector LUCID with an acceptance $5.4 < \eta < 6.1$ and a ZDC with an acceptance $8.3 < \eta < 9.2$. Diffractive studies are under discussion also at ALICE, see for instance, [@RS]. The ALICE detector has a central barrel covering the pseudorapidity range $-0.9 < \eta < 0.9$ and (on one side) a muon spectrometer covering the region of $2.4 < \eta < 4$ and a ZDC. Additional detectors for trigger purposes and for event classification are placed on both sides of the central barrel, such that the range $-3.7 < \eta < 5$ is covered. This configuration allows the possibility of a (double) rapidity gap trigger by requiring no activity in the event classification detectors, see [@RS].
[^2]: There are quite a few theoretical studies of the $S^2$ factor, starting from the publications [@DKS; @BJ] to more recent ones [@KMRsoft; @GLMrev] and references therein. Following Bjorken ([@BJ]) this quantity is often called the survival probability or soft survival factor.
[^3]: The CDF collaboration [@cdf] have demonstrated that this method provides an accurate determination of $\xi$.
[^4]: Even for the leptonic decay channels, the momentum of the $W$ boson can be reconstructed with knowledge of the missing transverse energy and $M_W$.
[^5]: Even so it may not be exactly the same $b_t$ region, since, in general, the impact parameter distributions of quarks and gluons may be different.
[^6]: Note that in the recent study [@nikit] it was demonstrated that the so-called Track Counting Veto (TCV) is robust for selection of the central rapidity gap events in vector-boson fusion $H \to \tau^+\tau^-$ searches at CMS. The idea of track counting is close in spirit to the hadron level gap selection of [@DKS; @pw] and, as compared to the (more) standard calorimeter jet veto technique, the TCV has various advantages. In particular, it does not involve calorimeter scale uncertainties, suffers less from pile-up contributions and so can be used at higher luminosities. Moreover, it is planned to apply this technique at CMS using data on central $Z \to \mu^+\mu^-$ events. Though the original motivation for this analysis [@nikit] is to study the detector conditions, such a measurement, in particular, presented in terms of the ratio ($Z$+gaps/$Z$ inclusive) may appear to be one of the first tests of models for soft survival factor.
[^7]: In the actual computations we use a more precise form as given by eq.(26) of Ref.[@MR01]
[^8]: Here the colour indices of the gluons are convoluted with the $d_{abc}$ tensor.
[^9]: The estimate assumes that the forward protons are not tagged, and so is enhanced by a factor of about 2 caused by the dissociation of the lower proton in Fig. \[fig:upsilon\](a).
[^10]: Note that the transverse momentum of the dijet system can be neglected, since it is very small compared to the $E_T$ resolution.
[^11]: Of course, the higher-order Reggeon diagrams will considerably reduce the size of the effect. Moreover, allowing for the more complicated multi-Pomeron vertices makes the suppression of the first Pomeron loop contribution even stronger. This effect has be seen in a particular model, see Fig. 17 of [@KMRnewsoft]. It is practically impossible to quantify the reduction within the framework of perturbative QCD.
[^12]: This value of $p_z$ corresponds to the realistic assumption that at the edge of the central calorimeter we will have about 2 particles with $p_t \sim$ 0.5 to 1 GeV.
[^13]: In this way we essentially fix the value of $x_\funp$, so that a possible interaction between the upper proton and the Pomeron placed in the gap interval (which causes an additional enhanced correction) will not affect the $y_A$ dependence of $R$.
[^14]: That is, before these detectors accumulate enough luminosity to probe for new physics signals.
|
---
abstract: 'We present measurements of the [near-side]{}of triggered di-hadron correlations using neutral strange baryons ([$\Lambda$]{}, [$\bar{\Lambda}$]{}) and mesons ([$K^0_S$]{}) at intermediate transverse momentum (3 $<$ [$p_T$]{} $<$ 6 [GeV/$c$]{}) to look for possible flavor and baryon/meson dependence. This study is performed in [$d$+Au]{}, [Cu+Cu]{}and [Au+Au]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}measured by the STAR experiment at RHIC. The [near-side]{}di-hadron correlation contains two structures, a peak which is narrow in azimuth and pseudorapidity consistent with correlations due to jet fragmentation, and a correlation in azimuth which is broad in pseudorapidity. The particle composition of the [jet-like correlation]{}is determined using identified associated particles. The dependence of the conditional yield of the [jet-like correlation]{}on the trigger particle momentum, associated particle momentum, and centrality for correlations with unidentified trigger particles are presented. The neutral strange particle composition in jet-like correlations with unidentified charged particle triggers is not well described by [PYTHIA]{}. However, the yield of unidentified particles in jet-like correlations with neutral strange particle triggers is described reasonably well by the same model.'
author:
- 'B. Abelev'
- 'L. Adamczyk'
- 'J. K. Adkins'
- 'G. Agakishiev'
- 'M. M. Aggarwal'
- 'Z. Ahammed'
- 'I. Alekseev'
- 'A. Aparin'
- 'D. Arkhipkin'
- 'E. C. Aschenauer'
- 'M. U. Ashraf'
- 'A. Attri'
- 'G. S. Averichev'
- 'X. Bai'
- 'V. Bairathi'
- 'L. S. Barnby'
- 'R. Bellwied'
- 'A. Bhasin'
- 'A. K. Bhati'
- 'P. Bhattarai'
- 'J. Bielcik'
- 'J. Bielcikova'
- 'L. C. Bland'
- 'M. Bombara'
- 'I. G. Bordyuzhin'
- 'J. Bouchet'
- 'J. D. Brandenburg'
- 'A. V. Brandin'
- 'I. Bunzarov'
- 'J. Butterworth'
- 'H. Caines'
- 'M. Calder[ó]{}n de la Barca S[á]{}nchez'
- 'J. M. Campbell'
- 'D. Cebra'
- 'I. Chakaberia'
- 'P. Chaloupka'
- 'Z. Chang'
- 'A. Chatterjee'
- 'S. Chattopadhyay'
- 'J. H. Chen'
- 'X. Chen'
- 'J. Cheng'
- 'M. Cherney'
- 'W. Christie'
- 'G. Contin'
- 'H. J. Crawford'
- 'S. Das'
- 'L. C. De Silva'
- 'R. R. Debbe'
- 'T. G. Dedovich'
- 'J. Deng'
- 'A. A. Derevschikov'
- 'B. di Ruzza'
- 'L. Didenko'
- 'C. Dilks'
- 'X. Dong'
- 'J. L. Drachenberg'
- 'J. E. Draper'
- 'C. M. Du'
- 'L. E. Dunkelberger'
- 'J. C. Dunlop'
- 'L. G. Efimov'
- 'J. Engelage'
- 'G. Eppley'
- 'R. Esha'
- 'O. Evdokimov'
- 'O. Eyser'
- 'R. Fatemi'
- 'S. Fazio'
- 'P. Federic'
- 'J. Fedorisin'
- 'Z. Feng'
- 'P. Filip'
- 'Y. Fisyak'
- 'C. E. Flores'
- 'L. Fulek'
- 'C. A. Gagliardi'
- 'L. Gaillard'
- 'D. Garand'
- 'F. Geurts'
- 'A. Gibson'
- 'M. Girard'
- 'L. Greiner'
- 'D. Grosnick'
- 'D. S. Gunarathne'
- 'Y. Guo'
- 'A. Gupta'
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title: 'Near-side azimuthal and pseudorapidity correlations using neutral strange baryons and mesons in [$d$+Au]{}, [Cu+Cu]{}and [Au+Au]{} collisions at $\sqrt{s_{NN}}$ = 200 GeV'
---
Introduction {#section-introduction}
============
Ultrarelativistic heavy-ion collisions create a unique environment for the investigation of nuclear matter at extreme temperatures and energy densities. Measurements of nuclear modification factors [@Adler:2003kg; @Adams:2003am; @CMS:2012aa; @Aamodt:2010jd; @Aad:2015wga] show that the nuclear medium created is nearly opaque to partons with large transverse momentum ([$p_T$]{}). Anisotropic flow measurements demonstrate that the medium exhibits partonic degrees of freedom and has properties close to those expected of a perfect fluid [@Adams:2003am; @Adams:2004bi; @Adare:2006ti; @Alver:2006wh].
Studies of jets in heavy ion collisions are possible through single particle measurements [@Adler:2003kg; @Adams:2003am; @CMS:2012aa; @Aamodt:2010jd], di-hadron correlations [@Agakishiev:2011st; @Adams:2005ph; @STARConical:2008nd; @Adler:2002tq; @Adams:2006yt; @RidgePaper:2009qa; @Abelev:2009jv; @Agakishiev:2014ada; @Aamodt:2011vg; @Chatrchyan:2011eka; @Chatrchyan:2012wg], and measurements of reconstructed jets [@Aad:2012vca; @Aad:2010bu; @CMS:2012aa; @Chatrchyan:2013kwa; @Abelev:2012ej] and their correlations with hadrons [@Adamczyk:2013jei; @Adam:2015doa]. Measurements of reconstructed jets provide direct evidence for partonic energy loss in the medium. Di-hadron and jet-hadron correlations enable studies at intermediate momenta, where the interplay between jets and the medium is important and direct jet reconstruction is challenging.
Properties of jets have been studied extensively using di-hadron correlations relative to a trigger particle with large transverse momentum at the Relativistic Heavy Ion Collider (RHIC) [@Agakishiev:2011st; @Adams:2005ph; @STARConical:2008nd; @Adler:2002tq; @Adams:2006yt; @RidgePaper:2009qa; @Abelev:2009jv; @Agakishiev:2014ada] and the Large Hadron Collider (LHC) [@Aamodt:2011vg; @Chatrchyan:2011eka; @Chatrchyan:2012wg]. Systematic studies of associated particle distributions on the opposite side of the trigger particle in azimuth ([$\Delta\phi$]{} $\approx$ 180$^{\circ}$) revealed significant modification, including the disappearance of the peak at intermediate transverse momentum, approximately 2–4 [GeV/$c$]{} [@Adler:2002tq; @Adare:2007vu] and its reappearance at high [$p_T$]{} [@Adams:2006yt; @Adare:2010ry]. The associated particle distribution on the near side of the trigger particle, the subject of this paper, is also significantly modified in central [Au+Au]{}collisions [@Adams:2005ph; @RidgePaper:2009qa; @Adler:2005ee]. In [$p$+$p$]{}and [$d$+Au]{}collisions, there is a peak that is narrow in azimuth and pseudorapidity ([$\Delta\eta$]{}) around the trigger particle, which we refer to as the [jet-like correlation]{}. In [Cu+Cu]{}and [Au+Au]{}collisions this peak is observed to be broader than that in [$d$+Au]{}collisions, although the yields are comparable [@Agakishiev:2011st]. Besides the shape modifications of jet-like correlations at intermediate transverse momenta, the production mechanism of hadrons may differ from simple fragmentation. In central [$A$+$A$]{}collisions baryon production is enhanced relative to that in [$p$+$p$]{}collisions [@Abelev:2006jr; @Agakishiev:2011ar; @Abelev:2013xaa]. The baryon to meson ratios measured in [Au+Au]{}collisions increase with increasing $p_T$ until reaching a maximum of approximately three times that observed in [$p$+$p$]{}collisions at $p_T\approx$ 3 GeV/$c$ in both the strange and non-strange quark sectors. A fall-off of the baryon to meson ratio is observed for $p_T>$ 3 GeV/$c$ and both the strange and non-strange baryon to meson ratios in [Au+Au]{}collisions approach the values measured in [$p$+$p$]{}collisions at $p_T\approx$ 6 GeV/$c$. Using statistical separation di-hadron correlation studies with pion and non-pion triggers [@Abdelwahab:2014cvd] showed that significant enhancement of near-side jet-like yields in central Au+Au collisions relative to [$d$+Au]{}collisions is present for pion triggered correlations. In contrast, for the non-pion triggered sample which consists mainly of protons and charged kaons no statistically significant difference is observed.
In this paper, studies of two-particle correlations on the near-side in [$d$+Au]{}, [Cu+Cu]{}and [Au+Au]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}measured by the STAR experiment are presented. Results from two-particle correlations in pseudorapidity and azimuth for neutral strange baryons ([$\Lambda$, $\bar{\Lambda}$]{}) and mesons ([$K^0_S$]{}) at intermediate [$p_T$]{}(3 $<$ [$p_T$]{}$~<$ 6 [GeV/$c$]{}) in the different collision systems are compared to unidentified charged particle correlations ([h-h]{}). Both identified strange trigger particles associated with unidentified charged particles ([$K^0_S$-h]{}, [$\Lambda$-h]{}) and unidentified charged trigger particles associated with identified strange particles ([h-$K^0_S$]{}, [h-$\Lambda$]{}) are studied. The [near-side]{}[jet-like]{}yield is studied as a function of centrality of the collision and transverse momentum of trigger and associated particles to look for possible flavor and baryon/meson dependence. The composition of the [jet-like correlation]{}is studied using identified associated particles to investigate possible medium effects on particle production. The results are compared to expectations from [PYTHIA]{} [@Sjostrand:2006za].
Experimental setup and particle reconstruction {#section-experiment}
==============================================
The Solenoidal Tracker At RHIC (STAR) experiment [@Ackermann:2002ad] is a multipurpose spectrometer with a full azimuthal coverage consisting of several detectors inside a large solenoidal magnet with a uniform magnetic field of 0.5 T applied parallel to the beam line. This analysis is based exclusively on charged particle tracks detected and reconstructed in the Time Projection Chamber (TPC) [@Anderson:2003ur] with a pseudorapidity acceptance $|\eta|<$ 1.5. The TPC has in total 45 pad rows in the radial direction allowing up to 45 independent spatial and energy loss ($dE/dx$) measurements for each charged particle track. Charged particle tracks used in this analysis were required to have at least 15 fit points in the TPC, a distance of closest approach to the primary vertex of less than 1 cm and a pseudorapidity $|\eta|<$ 1.0. These tracks are referred to as charged hadron tracks because the majority of them come from charged hadrons. The results presented in this paper are based on analysis of data from [$d$+Au]{}, [Cu+Cu]{}, and [Au+Au]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}taken by the STAR experiment in 2003, 2005, and 2004, respectively.
For [$d$+Au]{}collisions, the events analyzed were selected using a minimally biased (MB) trigger requiring at least one beam-rapidity neutron in the Zero Degree Calorimeter (ZDC), located 18 m from the nominal interaction point in the Au beam direction and accepting 95$\pm$3% of the hadronic cross section [@Adams:2003im]. For [Cu+Cu]{}collisions, the MB trigger was based on the combined signals from the Beam-Beam Counters (BBC) placed at forward pseudorapidity (3.3 $<|\eta|<$ 5.0) and a coincidence between the two ZDCs. The MB [Au+Au]{}events required a coincidence between the two ZDCs, a signal in both BBCs and a minimum charged particle multiplicity in an array of scintillator slats aligned parallel to the beam axis and arranged in a barrel, the Central Trigger Barrel (CTB), to reject non-hadronic interactions. An additional online trigger for central [Au+Au]{}collisions was used to sample the most central 12% of the total hadronic cross section. This trigger was based on the energy deposited in the ZDCs in combination with the multiplicity in the CTB. Centrality selection is based on the primary charged particle multiplicity $N_{ch}$ within the pseudorapidity range $|\eta|<$ 0.5, as in [@Ackermann:2000tr; @Adler:2002xw]. Calculation of the number of participating nucleons, [$N_{\mathrm{part}}$]{}, in each centrality class is done as in [@Adamczyk:2013tvk; @Abelev:2008ab; @Abelev:2008zk].
In order to achieve a more uniform detector acceptance in [Cu+Cu]{}and [Au+Au]{}data sets, only those events with a primary collision vertex position along the beam axis ($z$) within 30 cm of the center of the STAR detector were used for the analysis. For [$d$+Au]{}collisions this vertex position selection was extended to $|z|~<$ 50 cm. The number of events after the vertex cuts in individual data samples is summarized in [Tab. \[table-eventcount\]]{}.
System Centrality No. of events \[10$^6$\]
------------ ------------ --------------------------
[$d$+Au]{} 0-95% 3
[Cu+Cu]{} 0-60% 38
[Au+Au]{} 0-80% 28
[Au+Au]{} 0-12% 17
: Number of events after cuts (see text) in the data samples analyzed.[]{data-label="tabevents"}
\[table-eventcount\]
We identify weakly decaying neutral strange ([$V^{0}$]{}) particles [$\Lambda$]{}, [$\bar{\Lambda}$]{}and [$K^0_S$]{}by topological reconstruction of their decay vertices from their charged hadron daughters measured in the TPC as described in [@Adler:2002uv]: $$\begin{aligned}
\label{vzerodecays}
\Lambda &\rightarrow& p+\pi^-, BR=(63.9\pm0.5)\% \nonumber \\
\bar{\Lambda} &\rightarrow& \bar{p}+\pi^+, BR=(63.9\pm0.5)\% \\
K^{0}_{S} &\rightarrow& \pi^+ + \pi^-, BR=(68.95\pm0.14)\% \nonumber\end{aligned}$$ where $BR$ denotes the branching ratio. The [$V^{0}$]{}reconstruction software pairs oppositely charged particle tracks into [$V^{0}$]{}candidates. Reconstructed [$\Lambda$]{}and [$K^0_S$]{}particles are required to be within $|\eta|<$ 1.0. Topological cuts are optimized for each data set and chosen to have a signal-to-background ratio of at least 15:1. For the analyses presented here, no difference was observed between results with [$\Lambda$]{}and [$\bar{\Lambda}$]{}trigger particles. Therefore the correlations with [$\Lambda$]{}and [$\bar{\Lambda}$]{}trigger particles were combined to increase the statistical significance of the results. In the remainder of the discussion the combined particles are refered to simply as [$\Lambda$]{}baryons.
Method {#section-method}
======
Correlation technique
---------------------
The analysis in this paper follows the method in [@Agakishiev:2011st]. A [high-[$p_T$]{}]{}trigger particle was selected and the raw distribution of associated tracks relative to that trigger particle in pseudorapidity ([$\Delta\eta$]{}) and azimuth ([$\Delta\phi$]{}) is formed. This distribution, ${d^2N_{\mathrm{raw}}}/{d\Delta\phi\, d\Delta\eta}$, is normalized by the number of trigger particles, [$N_{\mathrm{trigger}}$]{}, and corrected for the efficiency and acceptance of associated tracks: $$\begin{aligned}
\label{eq:di-hadronDist}
{\frac{d^2N}{d\Delta\phi\, d\Delta\eta}(\Delta\phi,\Delta\eta)}& = & \frac{1}{N_{\mathrm{trigger}}} \frac{d^2N_{\mathrm{raw}}}{d\Delta\phi d\Delta\eta} \nonumber \\
& & \frac{1}{\varepsilon_{\mathrm{assoc}}(\phi,\eta)} \frac{1}{\varepsilon_{\mathrm{pair}}(\Delta\phi,\Delta\eta)}.\end{aligned}$$ The efficiency correction $\varepsilon_{\mathrm{assoc}}(\phi,\eta)$ is a correction for the single particle reconstruction efficiency in TPC and $\varepsilon_{\mathrm{pair}}(\Delta\phi,\Delta\eta)$ is a correction for the finite TPC track-pair acceptance in [$\Delta\phi$]{}and [$\Delta\eta$]{}, including track merging effects. Since the correlations are normalized by the number of trigger particles, the efficiency correction is only applied for the associated particle. The fully corrected correlation functions are averaged between positive and negative [$\Delta\phi$]{}and [$\Delta\eta$]{}regions and are reflected about [$\Delta\phi$]{} = 0 and [$\Delta\eta$]{} = 0 in the plots.
Single particle efficiency correction
-------------------------------------
For unidentified charged associated particles, the efficiency correction $\varepsilon_{\mathrm{assoc}}(\phi,\eta)$ is the correction for charged particles, identical to that applied in [@Agakishiev:2011st]. This single charged track reconstruction efficiency is determined as a function of [$p_T$]{}, $\eta$, and centrality by simulating the TPC response to a particle and embedding the simulated signals into a real event. The efficiency is found to be approximately constant for [$p_T$]{}$>$ 2 [GeV/$c$]{}and ranges from around 75% for central [Au+Au]{}events to around 85% for peripheral [Cu+Cu]{}events. The efficiency for reconstructing a track in [$d$+Au]{}events is 89%.
For identified associated strange particles, the reconstruction efficiency $\varepsilon_{\mathrm{assoc}}(\phi,\eta)$ is determined in a similar way, but forcing the simulated particle to decay through the channel in [Equation \[vzerodecays\]]{} and then correcting for the respective branching ratio. The efficiency for reconstructing [$\Lambda$]{}, [$\bar{\Lambda}$]{}, and [$K^0_S$]{}ranges from 8% to 15%, increasing with momentum and decreasing with system size [@Aggarwal:2010ig]. No correction for the reconstruction efficiency is applied for identified trigger particles because the reconstruction efficiency does not vary significantly within the [$p_T^{\mathrm{trigger}}$]{}bins used in this analysis and the correlation function is normalized by the number of trigger particles.
The systematic uncertainty associated with the efficiency correction for unidentified associated particles is 5% and is strongly correlated across centralities and [$p_T$]{}bins within each data set but not between data sets. For identified associated particle ratios the systematic uncertainties on the efficiency correction partially cancel out and are negligible compared to the statistical uncertainties.
For the inclusive spectra the feeddown correction due to secondary [$\Lambda$]{}baryons from $\Xi$ baryon decays is 15%, independent of [$p_T$]{} [@Agakishiev:2011ar]. For identified [$\Lambda$]{}trigger particles, we assume that feeddown lambdas do not change the correlation. Correlations with $\Xi$ triggers were performed to check this assumption. For identified associated particles, we assume the same correlation between primary and secondary [$\Lambda$]{}particles and correct the yield of [$\Lambda$]{}associated particles by reducing the yield by 15%.
Pair acceptance correction {#acceptanceCorrection}
--------------------------
The requirement that each track falls within $|\eta|<1.0$ in TPC results in a limited acceptance for track pairs. The geometric acceptance for a track pair is [$\approx$]{}100% for [$\Delta\eta$]{}[$\approx$]{} $0$ and close to 0% near [$\Delta\eta$]{}[$\approx$]{} $2$. The track pair acceptance is limited in azimuth by the 12 TPC sector boundaries, leading to dips in the acceptance of track pairs in [$\Delta\phi$]{}. To correct for the limited geometric acceptance, a mixed event analysis was performed using trigger particles from one event combined with associated particles from another event, as done in [@RidgePaper:2009qa]. The event vertices were required to be within 2 cm of each other along the beam axis and the events were required to have the same charged particle multiplicity within 10 particles. To increase statistics of the mixed event sample, each event with a trigger particle was mixed with ten other events.
Yield extraction {#yieldmethod}
----------------
An example of a 2D correlation function after the corrections described above is shown in [Fig. \[Fig:SampleCorr2D\]]{}. The notation and method used to extract the yield in this paper follow [@Agakishiev:2011st; @RidgePaper:2009qa]. The [jet-like correlation]{}is narrow in both [$\Delta\phi$]{}and [$\Delta\eta$]{}and is contained within $|\Delta\phi|<0.78$ and $|\Delta\eta|<0.78$ for the kinematic cuts in [$p_T^{\mathrm{trigger}}$]{}and [$p_T^{\mathrm{associated}}$]{}used in this analysis. The di-hadron correlation from [Equation \[eq:di-hadronDist\]]{} is projected onto the [$\Delta\eta$]{}axis:
$$\begin{aligned}
\label{eq:IDphi}
\left.{\frac{dN}{d\Delta\eta}}\right|_{\Delta\phi_1,\Delta\phi_2}
\equiv\int_{\Delta\phi_1}^{\Delta\phi_2}d\Dphi\frac{d^2N}{d\Dphi{d}\Deta}.\end{aligned}$$
All other correlations, including those from [$v_2$]{}, [$v_{3}$]{}, and higher order flow harmonics, are assumed to be independent of [$\Delta\eta$]{}within the $\eta$ acceptance of the analysis, consistent with [@Back:2004mh; @phobosFlow2; @RidgePaper:2009qa; @Alver:2009id]. We make the assumption that the $\eta$ dependence observed for [$v_{3}$]{}measured using the two particle cumulant method [@Adamczyk:2013waa] is entirely due to nonflow. With these assumptions, both correlated and uncorrelated backgrounds such as flow are constant in [$\Delta\eta$]{}. The [jet-like correlation]{}can then be determined by: $$\begin{aligned}
\label{eq:NJeta}
\frac{dN_{J}\left(\Deta\right)}{d\Deta}=
\left.{\frac{dN}{d\Delta\eta}}\right|_{\Delta\phi_1,\Delta\phi_2}-b_{\Delta\eta}\end{aligned}$$ where ${b_{\Delta\eta}}$ is a constant offset determined by fitting a constant background $b_{\Delta\eta}$ plus a Gaussian to $\frac{dN_{J}}{d\Deta}\left(\Deta\right)$. Variations in the method for extracting the constant background, such as fitting a constant at large [$\Delta\eta$]{}, lead to differences in the yield smaller than the statistical uncertainty due to the background alone. Nevertheless, a 2% systematic uncertainty is applied to account for this. This uncertainty is uncorrelated with the uncertainty on the efficiency for a total uncertainty of 5.5% on all yields. Examples of correlations are given in [Fig. \[Fig:SampleCorr\]]{}. Where the track merging effect discussed below is negligible the yield from the fit and from bin counting are consistent. When the dip due to track merging is negligible, the yield determined from fit is discarded to avoid any assumptions about the shape of the peak and instead we integrate [Equation \[eq:NJeta\]]{} over [$\Delta\eta$]{}using bin counting to determine the [jet-like yield]{}${Y^{\Delta\eta}_{J}}$: $$\begin{aligned}
{Y^{\Delta\eta}_{J}}= \int_{\Delta\eta_1}^{\Delta\eta_2} d{\ensuremath{\Delta\eta}}\; &
\frac{dN_{J}\left(\Deta\right)}{d\Deta}. \label{eq:Yjeta} \end{aligned}$$ The choice of $\Delta\phi_1$, $\Delta\phi_2$, $\Delta\eta_1$, and $\Delta\eta_2$ is arbitrary. For this analysis we choose $\Delta\phi_1=\Delta\eta_1$ = $-0.78$ and $\Delta\phi_2=\Delta\eta_2$ = 0.78 in order to be consistent with previous studies and in order to include the majority of the peak [@Agakishiev:2011st].
Track merging correction {#section-method}
------------------------
The track merging effect in unidentified particle (h) correlations discussed in [@Agakishiev:2011st] is also present for [$V^{0}$-h]{}and [h-$V^{0}$]{}correlations. This effect leads to a loss of tracks at small [$\Delta\phi$]{}and [$\Delta\eta$]{}due to overlap between the trigger and associated particle tracks and is manifested as a dip in the correlation function. When one of the particles is a [$V^{0}$]{}, this overlap is between one of the [$V^{0}$]{}daughter particles and the unidentified particle. The size of the dip due to track merging depends strongly on the relative momenta of the particle pair. The effect is larger when the momentum difference of the two overlapping tracks is smaller. For [$V^{0}$-h]{}correlations, the typical associated particle momentum is approximately 1.5 [GeV/$c$]{}. Since the [$K^0_S$]{}decay is symmetric, the track merging effect is greatest for [$K^0_S$-h]{}correlations with a trigger [$K^0_S$]{}momentum of approximately 3 [GeV/$c$]{}. In a [$\Lambda$]{}decay, the proton daughter carries more of the [$\Lambda$]{}momentum than the pion daughter. Therefore this effect is larger for [$\Lambda$]{}trigger particles with lower momenta. Because track merging affects both signal and background particles and the signal sits on top of a large combinatorial background, the effect is larger for collisions with a higher charged track multiplicity. Since the dip in [$V^{0}$-h]{}and [h-$V^{0}$]{}correlations is the result of a [$V^{0}$]{}daughter merged with an unidentified particle, the dip is wider in [$\Delta\phi$]{}and [$\Delta\eta$]{}than in unidentified particle correlations.
For identified [$V^{0}$]{}associated particles in the kinematic range studied in this paper, there was no evidence for track merging. A straightforward extension of the method in [@Agakishiev:2011st] to [$V^{0}$]{}trigger particles did not fully correct for track merging. The residual effect was dependent on the helicity of the associated particle, demonstrating that this was a detector effect. When the track merging dip is present, it is corrected by fitting a Gaussian to the peak, excluding the region impacted by track merging, and using the Gaussian fit to extract the yield. The event mixing procedure described in [@Agakishiev:2011st] was not applied to simplify the method since the yield would still need to be corrected using a fit to correct for the residual effect.
This correction is only necessary for the data points in [Fig. \[rjpttrig\]]{} specified below. To investigate the effect of using a fit where the peak is excluded from the fit region, we used a toy model where a Gaussian signal with a constant background was thrown with statistics comparable to the data with a residual track merging effect When the peak is excluded from the fit for samples with high statistics, the yield is determined correctly from the fit. For the low statistics samples comparable to the points with a residual track merging effect, the yield from the fit is usually within uncertainty of the true value but there is an average skew of about 13% in the extracted yield. A 13% systematic uncertainty is added in quadrature to the statistical uncertainty on the yield from the fit so that these points can be compared to the other points. When the residual track merging effect is corrected by a fit, the track merging correction applied by the fit is approximately the same size as the statistical uncertainty on the yield. We therefore conclude that when no dip is evident, the track merging effect is negligible compared to the statistical uncertainty on the yield.
Summary of systematic uncertainties {#sec-syserr}
-----------------------------------
Systematic uncertainties are summarized in [Tab. \[table-syserr\]]{}. All data points have a 5% systematic uncertainty due to the single track reconstruction efficiency and a 2% systematic uncertainty due to the yield extraction method. This is a total 5.5% systematic uncertainty. In addition, there is a 13% systematic uncertainty due to the yield extraction for data points with residual track merging. It is added in quadrature to the statistical uncertainty so that these data can be compared to data without residual track merging. This uncertainty is only in the yields in [Fig. \[rjpttrig\]]{} listed below.
source value (%)
---------------------------------------- -----------
$\varepsilon$ 5%
yield extraction 2%
yield with track merging (see caption) 13%
total 5.5%
: Summary of systematic uncertainties due to the efficiency $\varepsilon$, yield extraction for all points, and yield extraction in the presence of a residual track merging effect. The 13% systematic uncertainty due to the yield extraction for data points with residual track merging is added in quadrature to the statistical uncertainty, which is on the order of 20-30% for these data points. This uncertainty is only in the yields in [Fig. \[rjpttrig\]]{} listed below.[]{data-label="tabevents"}
\[table-syserr\]
Results {#section-results}
=======
Charged particle-$V^0$ correlations {#hV0Corr}
-----------------------------------
Previous studies demonstrated that the [jet-like correlation]{}in [h-h]{}correlations is nearly independent of collision system [@Agakishiev:2011st; @Abelev:2009ah; @RidgePaper:2009qa], with some indications of particle type dependence [@Abdelwahab:2014cvd], and that it is qualitatively described by PYTHIA [@Agakishiev:2011st] at intermediate momenta. This indicates that the [jet-like correlation]{}is dominantly produced by fragmentation, even at intermediate momenta (2 $<$ [$p_T$]{}$<$ 6 [GeV/$c$]{}) where recombination predicts significant modifications to hadronization. The composition of the [jet-like correlation]{}can be studied using correlations with identified associated particles. For the analysis presented here, the size of [$d$+Au]{}data sample was limited and the [Au+Au]{}data set was limited by the presence of residual track merging. Therefore it was only possible to determine the composition of the [jet-like correlation]{}in [Cu+Cu]{}collisions for a relatively large centrality range (0-60%).
These measurements are compared to inclusive baryon to meson ratios in [$p$+$p$]{}collisions from the STAR experiment [@Abelev:2006cs] and the ALICE experiment [@Aamodt:2011zza] and simulations of [$p$+$p$]{}collisions in PYTHIA [@Sjostrand:2006za] using the Perugia 2011 [@Skands:2010ak] tune and Tune A [@Field:2005sa] in [Fig. \[bmratio\]]{}. The ratio in the [jet-like correlation]{}in [Cu+Cu]{}collisions is consistent with the inclusive particle ratios from [$p$+$p$]{}. This further supports earlier observations that the jet-like correlation in heavy-ion collisions is dominantly produced by the fragmentation process, which also governs the production of particles in [$p$+$p$]{}collisions at these momenta. It also implies that production of strange particles through recombination is not significant in the [jet-like correlation]{}, even in [$A$+$A$]{}collisions, where the inclusive spectra show an enhancement of [$\Lambda$]{}production of up to a factor of three relative to the [$K^0_S$]{} [@Agakishiev:2011ar; @Abelev:2013xaa].
The experimentally measured particle ratios in [$p$+$p$]{}collisions at [$\sqrt{s}$]{}= 200 and 7000 GeV are consistent with each other. However, they are not described well by PYTHIA. PYTHIA is able to match the light quark meson ($\pi$ and $\omega$) production [@Aamodt:2010my; @ALICE:2011ad], but generally underestimates production of strange particles, especially strange baryons [@Aamodt:2010my; @ALICE:2011ad; @Abelev:2006cs; @Aamodt:2011zza]. Tune A has been adjusted to match low momentum h-h correlations [@Field:2005sa], while the Perugia 2011 tune has been tuned to match inclusive particle spectra better, including data from the LHC [@Skands:2010ak]. The most recent MONASH tune [@Skands:2014pea], which is a variation of Tune A, had some success in capturing the inclusive strange meson yield at the LHC, but the [$\Lambda$]{}yield is still underestimated by a factor of 2. The discrepancy grows with the strange quark content of the baryon. Since [h-$V^{0}$]{}correlations are dominated by gluon and light quark jet fragmentation, PYTHIA underestimates the generation of strange quarks in those jets. This effect is enhanced in strange baryon production since the formation of an additional di-quark is required in PYTHIA. The probability of such a combination is significantly suppressed in PYTHIA, whereas the data seem to suggest that di-quark formation is not necessary to form strange baryons. The discrepancy between PYTHIA and the data in [Fig. \[bmratio\]]{} can therefore be attributed exclusively to the problems of describing strange baryon production in PYTHIA. On the other hand, strange particle triggered correlations, such as [$K^0_S$-h]{}and [$\Lambda$-h]{}, originate predominantly from the fragmentation of strange quarks. It should be easier for PYTHIA to describe the production of strange particles from the fragmentation of strange quarks than light quarks and gluons. We therefore studied the [$V^{0}$-h]{}correlations in more detail.
Correlations with identified strange trigger particles
------------------------------------------------------
The [jet-like yield]{}as a function of [$p_T^{\mathrm{trigger}}$]{}is shown in [Fig. \[rjpttrig\]]{} for [$K^0_S$-h]{}and [$\Lambda$-h]{}correlations for [$d$+Au]{}, [Cu+Cu]{}, and [Au+Au]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}. The data are tabulated in [Tab. \[trigpttable\]]{}. Due to residual track merging effects discussed in [Section \[section-method\]]{}, fits are used for [$\Lambda$-h]{}correlations in some [$p_T^{\mathrm{trigger}}$]{}ranges: in [Cu+Cu]{}collisions, [2.0 $< p_T^{\mathrm{trigger}} <$ 3.0 GeV/$c$]{}; in 0-12% Au+Au collisions, [3.0 $< p_T^{\mathrm{trigger}} <$ 4.5 GeV/$c$]{}; and in 40-80% Au+Au collisions, [2.0 $< p_T^{\mathrm{trigger}} <$ 4.5 GeV/$c$]{}. There is no significant difference in the yields between the collision systems, however, the data are not sensitive enough to distinguish the 20% differences observed for identified pion triggers [@Abdelwahab:2014cvd]. No system dependence is observed for [h-h]{}correlations in [@Agakishiev:2011st; @Abdelwahab:2014cvd]. This includes no significant difference between results from [Au+Au]{}collisions in 40-80% and 0-12% central collisions. For this reason we only compare to [h-h]{}correlations from 40-80% [Au+Au]{}collisions.
Next the [jet-like yields]{}are studied as a function of collision centrality expressed in terms of number of participating nucleons ([$N_{\mathrm{part}}$]{}) calculated from the Glauber model [@Miller:2007ri]. The extracted [jet-like yield]{}as a function of [$N_{\mathrm{part}}$]{}is shown in [Fig. \[rjcentr\]]{} for [h-h]{} [@Agakishiev:2011st], [$K^0_S$-h]{}, and [$\Lambda$-h]{}correlations for [$d$+Au]{}, [Cu+Cu]{}, and [Au+Au]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}. All yields are determined using bin counting. While there is no centrality dependence in the [jet-like yield]{}of [h-h]{}correlations, there is a centrality dependence in the yields of the [$K^0_S$-h]{}correlations. These data are compared to PYTHIA [@Sjostrand:2006za] calculations from the Perugia 2011 [@Skands:2010ak] tune in [Fig. \[rjcentr\]]{}. There is a hint of a particle species ordering, with the [jet-like yield]{}from [$K^0_S$-h]{}correlations generally above that of the [jet-like yield]{}from [h-h]{}correlations and the [jet-like yield]{}from [$\Lambda$-h]{}generally below that of the [h-h]{}correlations. This is different from the particle type ordering observed in PYTHIA.
The [jet-like yield]{}as a function of [$p_T^{\mathrm{associated}}$]{}is shown in [Fig. \[rjptassoc\]]{} for [$K^0_S$-h]{}and [$\Lambda$-h]{}correlations for [$d$+Au]{}and [Cu+Cu]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}. All yields are determined using bin counting. The [$\Lambda$-h]{}and [$K^0_S$-h]{}correlations are only shown for [$d$+Au]{}and [Cu+Cu]{}collisions since residual track merging made measurements in [Au+Au]{}collisions difficult. Data are compared to the [jet-like yield]{}from [h-h]{}correlations [@Agakishiev:2011st]. The trend is similar for [h-h]{}, [$K^0_S$-h]{}, and [$\Lambda$-h]{}correlations, although the wide centrality bins required by low statistics may mask centrality dependencies such as those shown in [Fig. \[rjcentr\]]{}.
------------------- ------------------------------ ------------------- ---------------------
Collision system, [$p_T^{\mathrm{trigger}}$]{} [[$K^0_S$-h]{}]{} [[$\Lambda$-h]{}]{}
centrality (GeV/$c$) yield yield
[$d$+Au]{}, 3.0-5.0 0.162 $\pm$ 0.028 0.079 $\pm$ 0.018
0-95%
[Cu+Cu]{}, 2.0-2.5 0.036 $\pm$ 0.004 0.026 $\pm$ 0.005
0-60% 2.5-3.0 0.059 $\pm$ 0.006 0.071 $\pm$ 0.007
3.0-3.5 0.098 $\pm$ 0.009 0.084 $\pm$ 0.017
3.5-5.0 0.144 $\pm$ 0.011 0.142 $\pm$ 0.013
[Au+Au]{}, 2.0-2.5 0.063 $\pm$ 0.008 -
40-80% 2.5-3.0 0.084 $\pm$ 0.023 0.061 $\pm$ 0.010
3.0-3.5 0.139 $\pm$ 0.022 -
3.5-4.5 0.172 $\pm$ 0.021 0.096 $\pm$ 0.030
4.5-5.5 0.170 $\pm$ 0.037 0.184 $\pm$ 0.040
[Au+Au]{}, 3.0-3.5 0.105 $\pm$ 0.021
0-12% 3.5-4.5 0.160 $\pm$ 0.036 0.128 $\pm$ 0.022
4.5-5.5 0.240 $\pm$ 0.045 0.091 $\pm$ 0.033
------------------- ------------------------------ ------------------- ---------------------
: The jet-like yield in $\mid$[$\Delta\eta$]{}$\mid<$0.78 as a function of [$p_T^{\mathrm{trigger}}$]{}for [$K^0_S$-h]{}and [$\Lambda$-h]{}correlations for [$1.5$ GeV/$c$ $<$ $p_T^{\mathrm{associated}}$ $<$ $p_T^{\mathrm{trigger}}$]{}in minimum bias [$d$+Au]{}, 0-60% [Cu+Cu]{}, and [Au+Au]{}collisions at [$\sqrt{s_{{NN}}}$ = 200 GeV]{}, as shown in [Fig. \[rjpttrig\]]{}.[]{data-label="trigpttable"}
Conclusions {#Conclusions}
===========
Measurements of di-hadron correlations with identified strange associated particles demonstrated that the ratio of [$\Lambda$]{}to [$K^0_S$]{}for the [jet-like correlation]{}in [Cu+Cu]{}collisions is comparable to that observed in [$p$+$p$]{}collisions. This provides additional evidence that the [jet-like correlation]{}is dominantly produced by fragmentation. Measurements of di-hadron correlations with identified strange trigger particles show some centrality dependence, indicating that fragmentation functions or particle production mechanisms may be modified in heavy ion collisions. These studies provide hints of possible mass ordering, although the measurements are not conclusive due to the statistical precision of the data.
These measurements provide motivation for future studies of strangeness production in jets. Larger data sets and data from collisions at higher energies could provide more robust tests of the strangeness production mechanism. Studies in [$p$+$p$]{}would be essential in order to search for modifications of strangeness production in jets in heavy ion collisions.
We thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL, the KISTI Center in Korea, and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Office of Nuclear Physics within the U.S. DOE Office of Science, the U.S. NSF, the Ministry of Education and Science of the Russian Federation, NSFC, CAS, MoST and MoE of China, the National Research Foundation of Korea, NCKU (Taiwan), GA and MSMT of the Czech Republic, FIAS of Germany, DAE, DST, and UGC of India, the National Science Centre of Poland, National Research Foundation, the Ministry of Science, Education and Sports of the Republic of Croatia, and RosAtom of Russia.
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|
---
abstract: |
We propose that the Yang-Baxter deformation of the symmetric space $\sigma$-model parameterized by an $r$-matrix solving the homogeneous (classical) Yang-Baxter equation is equivalent to the non-abelian dual of the undeformed model with respect to a subgroup determined by the structure of the $r$-matrix. We explicitly demonstrate this on numerous examples in the case of the $AdS_5$ $\sigma$-model. The same should also be true for the full $AdS_5
\times S^5$ supercoset model, providing an explanation for and generalizing several recent observations relating homogeneous Yang-Baxter deformations based on non-abelian $r$-matrices to the undeformed $AdS_5 \times S^5$ model by a combination of T-dualities and non-linear coordinate redefinitions. This also includes the special case of deformations based on abelian $r$-matrices, which correspond to TsT transformations: they are equivalent to non-abelian duals of the original model with respect to a central extension of abelian subalgebras.
---
Title: Homogeneous Yang-Baxter deformations as non-abelian duals of the AdS\_5 sigma-model Authors: B. Hoare, A.A. Tseytlin Abstract: We propose that the Yang-Baxter deformation of the symmetric space sigma-model parameterized by an r-matrix solving the homogeneous (classical) Yang-Baxter equation is equivalent to the non-abelian dual of the undeformed model with respect to a subgroup determined by the structure of the r-matrix. We explicitly demonstrate this on numerous examples in the case of the AdS\_5 sigma-model. The same should also be true for the full AdS\_5 x S\^5 supercoset model, providing an explanation for and generalizing several recent observations relating homogeneous Yang-Baxter deformations based on non-abelian r-matrices to the undeformed AdS\_5 x S\^5 model by a combination of T-dualities and non-linear coordinate redefinitions. This also includes the special case of deformations based on abelian r-matrices, which correspond to TsT transformations: they are equivalent to non-abelian duals of the original model with respect to a central extension of abelian subalgebras. Comments: 28 pages; v2: comments and references added Report Number: Imperial-TP-AT-2016-03
[Imperial-TP-AT-2016-03]{}
[**Homogeneous Yang-Baxter deformations\
as non-abelian duals of the $\mathbf{AdS_5}$ $\mathbf{\sigma}$-model**]{}
[B. Hoare$^{a,}$[^1] and A.A. Tseytlin$^{b,}$[^2]]{}\
[*$^{a}$ Institut für Theoretische Physik, ETH Zürich,\
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland.*]{}
[*$^{b}$ The Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.*]{}
Introduction {#secint}
============
The class of integrable deformations of the superstring model based on the “Yang-Baxter” [@Klimcik:2002zj; @Klimcik:2008eq; @dmv] has recently been under active investigation. It generalizes the usual group-space or coset-space as $\mathcal{L}=\Tr ( J J) \to \Tr ( J \mathcal{O} J)$ where $J= g^{-1} d g$ and $\mathcal{O}$ depends on $g$ and a constant antisymmetric operator $R$ acting on the Lie algebra (referred to as the “$r$-matrix”) satisfying a (modified) classical Yang-Baxter equation (cYBE), i.e. $[RX,RY] -
R ( [RX,Y] + [X, RY]) = c [X,Y] $.
The two inequivalent cases are $c=1$ and $c=0$.[[^3]]{} The first case represents a non-trivial $q$-deformation of the symmetry algebra of the original symmetric space [@Delduc:2013fga; @dmv]. The second case (based on the homogeneous cYBE) studied in [@yabe1; @yabe2; @yjor; @ysum; @vtsum; @ysugra1; @Hoare:2016hwh; @ysugra2; @Borsato:2016ose; @Osten:2016dvf] appears to be simpler and more closely related to the original coset $\s$-model. Indeed, it was observed on particular examples [@yabe1; @yabe2; @yjor; @vtsum] and proved in general [@Osten:2016dvf] that for abelian $r$-matrices the resulting deformed model can be obtained from the original undeformed one by abelian T-dualities (more precisely, by a TsT transformation combining T-duality with a linear coordinate shift which is a special case of the $O(d,d)$ T-duality transformation).
Furthermore, for several examples defined by non-abelian $r$-matrices solving the homogeneous cYBE it was recently observed [@ysugra2; @Borsato:2016ose] that the resulting deformed can be related to the original supercoset model by a combination of T-dualities along [*non*]{}-commuting directions and [*non*]{}-linear coordinate transformations (required to be able to perform the T-dualities).
Here we will generalize (and provide an explanation for) these observations by demonstrating that the homogeneous YB deformations of a symmetric space are equivalent to non-abelian duality (NAD) transformations [@nadold; @os; @vene; @gr; @alv; @elit] of the original model with respect to various (in general, non-semi-simple) subgroups of the global symmetry group. We will focus on the bosonic $AdS_5$ but the same should be true in general and also for the supercoset model.
The subgroup in which we will dualize is determined by the structure of the $r$-matrix. Not all subgroups have a corresponding classical $r$-matrix and hence not all NAD transformations correspond to a homogeneous YB deformation. Indeed the corresponding subalgebra should be quasi-Frobenius (or a central extension thereof), which, in particular, implies it is solvable. Therefore, the correspondence is absent, for example, if the subgroup is non-abelian and simple (such as $SO(1,2) \subset SO(2,4)$ or $SO(3) \subset SO(6)$). In these cases one would not expect the NAD to be a deformation of the original $\sigma$-model. On the other hand, the case of abelian $r$-matrices is naturally included: as we shall explain below, the abelian TsT transformation may be viewed as a special case of non-abelian duality with respect to a central extension of an abelian symmetry group. It is an interesting open question what the criteria is for a subgroup to give NAD model that can be understood as a deformation of the original $\sigma$-model, when one needs to centrally extend the group, and if these cases are in correspondence with the homogeneous YB deformations.
As the non-abelian duality should preserve the classical integrability of a symmetric [@sfet], one may then, instead of studying homogeneous YB deformations, directly consider all possible NAD transforms of the with respect to all possible (centrally-extended) subgroups of $PSU(2,2|4)$. As mentioned above, there will be special NAD models that have a “deformation” interpretation (i.e. depend on free parameters that when taken to zero give back the original model) and other NAD models (corresponding to simple subgroups) that are close cousins of the original model (e.g. sharing the same first-order structure) but not reducing to it in a limit and not allowing one to reverse the NAD transform.[[^4]]{}
An advantage of NAD over the YB deformation is that it can be performed systematically as a path integral transformation (determining also the dilaton). This allows us, in particular, to answer the question of which NAD transforms of the will still be Weyl-invariant so that the corresponding backgrounds will be solutions of the standard type II supergravity equations (and will thus define critical string models).
In general, the NAD transform of a Weyl-invariant is not Weyl but only scale-invariant [@elit] (in the context of GS superstring this means that the corresponding dual background solves only the generalized supergravity equations of [@Arutyunov:2015mqj; @Wulff:2016tju]). The condition for NAD to preserve Weyl invariance is that the structure constants of the group which is dualized should be traceless, i.e. $n_a \equiv f^{c}_{\ c a} =0$ [@alv; @elit]. An equivalent “unimodularity” condition was found also for the homogeneous YB deformations of the as a condition on the corresponding non-abelian $r$-matrices [@Borsato:2016ose]. If $n_a\not=0$ the NAD-transformed background does not solve the Weyl-invariance or supergravity equations but it can still be mapped by a formal T-duality (along the $n_a$ direction, as discussed in section \[secnad\]) to a proper supergravity solution, in agreement with the general discussion in [@Arutyunov:2015mqj].
While abelian TsT transformations (and thus equivalent abelian YB deformations) of are related, via AdS/CFT, to non-commutative gauge theories [@Hashimoto:1999ut; @Maldacena:1999mh; @Lunin:2005jy; @Dhokarh:2008ki; @vtsum] the role of the non-abelian duals of (beyond a supergravity solution generating technique) is presently unclear. In the TsT case one is guided by the action of T-duality on open strings or D-branes but similar intuition is absent in the NAD case (cf. though [@Forste:1996hy]). Still, the quantum Weyl-invariant and integrable obtained by NAD from the model may provide new non-trivial examples of solvable string models.
Let us briefly comment on the case of the [*in*]{}homogeneous YB deformation of the [@dmv] that non-trivially depends on one “quantum” deformation parameter. The corresponding 2d theory is scale but not Weyl invariant, i.e. the associated background [@ABF; @HT2] only solves the generalized equations of [@Arutyunov:2015mqj; @Wulff:2016tju]. At the same time, it is classically related [@Vic; @HT1], by Poisson-Lie (PL) duality [@KS], to the “$\l$-model” of [@hms2; @hms1] (generalizing the bosonic model of [@sfet]) which is Weyl-invariant at the quantum level [@apphol; @BTW; @Lunin; @Borsato:2016ose]. This suggests that as for the non-abelian duality in the case of $n_a \not=0$ the PL duality (which should preserve quantum equivalence on a flat 2d background and thus scale invariance [@Alekseev:1995ym]) here should be “Weyl-anomalous” at the quantum level.
A special “undeformed” (level $k\to \infty$ or $q=e^{i\pi \ov k} \to 1$) limit of the $\l$-model is just the NAD of the supercoset model with respect to the full $PSU(2,2|4)$ which is still Weyl invariant. The $\l$-model can thus be interpreted as a Weyl-invariance preserving $q$-deformation of the full NAD of the superstring model. It would be of interest to see if one can construct similar non-trivial deformations of NADs of with respect to some subgroups of $PSU(2,2|4)$ (cf. [@sfth]).
Among other open questions, it would be useful to give a general proof of the equivalence between the homogeneous YB deformations of the supercoset model and its non-abelian duals with respect to the corresponding subalgebras. One may be able to establish this relation by identifying the underlying first-order actions which include auxiliary gauge fields.[[^5]]{}
The structure of this paper is as follows. We shall start in section 2 with a review of the non-abelian duality transformation of a bosonic $\s$-model, explaining also how one can interpret the abelian TsT transformation as a special case. In section 3, after reviewing the Yang-Baxter deformation and the NAD of the symmetric space we shall turn our attention to the $AdS_5$ $\sigma$-model. For a range of cases of different types of $r$-matrices (abelian and non-abelian, both unimodular and jordanian) we will explicitly demonstrate that the Yang-Baxter deformation is equivalent to the non-abelian dual of the $AdS_5$ with respect to a specific (centrally-extended) subalgebra that can be identified from the $r$-matrix. In the Appendix we will present a large number of additional jordanian examples.
Non-abelian duality {#secnad}
===================
In this section we shall discuss non-abelian duality (NAD) in bosonic string [@nadold; @os; @vene; @gr; @alv; @elit; @SF]. NAD relates one bosonic with a non-abelian global symmetry group $H$ (i.e. with $H$-invariant metric, $B$-field and dilaton $\p$) to another one that generically has a smaller symmetry group (and no symmetry in the case when $H$ is non-abelian and simple). The standard abelian (or “torus”) T-duality [@bu] is a special case when $H$ is abelian, i.e. $\mathbb{R}^d$ or $U(1)^d$. The $O(d,d)$ generalization of the T-duality (and, in particular, the special TsT case) can also be viewed, as we shall explain below, as a particular case of NAD when one considers a central extension of the abelian group.
If the original is classically integrable then NAD maps it, as for the standard T-duality case (see, e.g., [@Frolov:2005ty; @Hatsuda:2006ts; @Ricci:2007eq]), to a classically integrable model [@sfet]. If the original is quantum Weyl-invariant, its NAD counterpart is also Weyl-invariant unless $H$ is non-semisimple with the generators in the adjoint representation satisfying $\tr T_a\not=0$, i.e. $n_a\equiv f^c_{\
ca}\not=0$ [@alv; @elit]. In the latter case the NAD transformation preserves only the scale invariance of the $\s$-model.
It should be possible to generalize NAD to the case of the GS superstring and then for $n_a =0$ NAD should again preserve Weyl invariance, i.e. (viewed as a transformation on the target space couplings of the $\sigma$-model) will map a supergravity solution to a supergravity solution.[[^6]]{} For $n_a \not=0$ NAD may not preserve Weyl invariance,[[^7]]{} i.e. may transform a Weyl-invariant into a scale-invariant one. Then the original target space background that was a supergravity solution will be mapped by NAD to a solution of the generalized supergravity equations or superstring scale invariance conditions [@Arutyunov:2015mqj; @Wulff:2016tju].
As we shall see below, in agreement with the general expectation [@Arutyunov:2015mqj], in that case one can still apply a formal (classical, i.e. ignoring the dilaton transformation) T-duality to associate to the resulting generalized background a solution of the standard Weyl-invariance or supergravity equations that has a linear non-isometric term in the dilaton obstructing the reverse T-duality at the quantum level [@HT1; @HT2].
Non-abelian duality of bosonic in curved 2d background {#ssecnad}
------------------------------------------------------
Let us start with a bosonic string depending on group $H$ coordinates $h$ and “spectator” coordinates $x^r$ that is invariant under a global $H$-symmetry, $h \to h_0 h, \ h_0\in H$. Its action can be written in conformal gauge $g_{ij}= e^{2 \s} \eta_{ij}$, with $\eta = \diag(-1,1)$, as $$\begin{aligned}
I[h,x] & = {1\ov 4 \pi \a'} \int d^2 z \big[ E_{ab}(x)\, J_+^a J^b_- + L_{ar}(x)\, J_+^a j^r_ - + M_{sb}(x)\, j_+^s J^b_ - + K_{rs}(x)\, j_+^r j^s_ -
\no \\
& \hspace{75pt} - 2 \a' \p(x)\, \del_+ \del_- \s
\big] \ , \la{1} \\
J^a_i & = \tr (h^{-1} \del_i h \, \bar T^a ) \ , \qquad j^r_i = \del_i x^r \ , \qquad [T_a,T_b]=f^c_{\ ab}T_c \ , \qquad
\tr(T_a \bar{T}^b)= \delta_{a}^b \la{2} \ .\end{aligned}$$ Here $i=0,1$, $\del_\pm = \del_0 \pm \del_1$, $a,b= 1, 2, \ldots, d=\dim H$, $r,s= 1,\ldots, n$. $T_a$ is a basis for the algebra $\mathfrak{h}$ of $H$ (taken below to be given by matrices in the adjoint representation, $T_a = - f^c_{\ ab}$) and $\bar T^a$ is a dual algebra basis (assuming $H$ is embedded into a simple group).[[^8]]{} We have also included a possible $H$-invariant dilaton term ${1\ov 4 \pi} \int d^2 z
\sqrt g R^{(2)} \phi(x) = -{1\ov 2 \pi} \int d^2 z\, \p(x)\, \del^2 \s$ where $\s$ is the conformal factor of the 2d metric. The path integral over $h$ with the action [(\[1\])]{} can be obtained from the path integral over the 2d gauge field $A_\pm^a$ and “Lagrange-multiplier” field $v_a$ with the following first-order or “interpolating” action $$\begin{aligned}
\hat I[A,v,x] & = {1\ov 4 \pi \a'} \int d^2 z \big[ E_{ab}(x)\, A_+^a A^b_- + L_{ar}(x)\, A_+^a j^r_ - + M_{sb}(x)\, j_+^s A^b_ - + K_{rs}(x)\, j_+^r j^s_ - \no \\
& \hspace{75pt}
+\ v_a \ep^{ij} F^a_{ ij} + 2 \a' \s\, n_a \del^i A^a_i - 2 \a' \p(x)\, \del_+ \del_- \s \big] \la{3} \ , \\
& \la{4} F^a_{ij} \equiv \del_i A^a_j - \del_j A^a_i + f^a_{\ bc} A^b_i A^c_j \ ,\qquad \qquad n_a = \tr T_a= f^c_{\ c a} \ .\end{aligned}$$ Integrating over $v_a$ gives $F_{ij} =0$ or $A_i= A^a_i T_a = h^{-1} \del_i h $ and the first line in [(\[3\])]{} becomes equivalent to the first line of [(\[1\])]{} on a flat 2d background.
The $n_a$-dependent term [@elit] in the second line of [(\[3\])]{} [@elit], which is non-local in the 2d metric ($\sigma = - \ha \del^{-2} \sqrt g R^{(2)}
$), is important for the quantum equivalence of [(\[3\])]{} and [(\[1\])]{} on a curved 2d background. It comes from the Jacobian of the transformation from $A_+ =
h^{-1} \del_+ h, \ A_-=h'^{-1} \del_- h'$ to $h, h'$ on a curved 2d background and is related to the “mixed” anomaly [@alv] ($\del_i {\delta
\Gamma \ov \delta A_i} \sim R^{(2)}$) which is present only if $H$ is such that $n_a \not=0$.
The dual of the model [(\[1\])]{} is then found by integrating out the gauge field $A^a_i$ in [(\[3\])]{} $$\begin{aligned}
\td I[v,x] & = \int d^2 z \Big[\big(\del_+ v_a + M_{sa}(x)\, \del_+ x^s + \a' n_a \del_+ \s\big) N^{ab} \big(\del_- v_b - L_{br}(x)\, \del_- x^r - \a' n_b \del_- \s\big) \no \\
& \hspace{75pt} \la{5}
+ K_{rs}(x)\, \del_+x^r \del_- x^s - 2 \a' \big( \p(x) + \ha \ln \det N \big) \, \del_+ \del_- \s \Big] \ ,
\\
N^{ab} & = \big[ E_{ab}(x) - v_c f^c_{\ ab} \big]^{-1} \la{6} \ .\end{aligned}$$ If $n_a \not=0$ [(\[5\])]{} does not have the interpretation of a local action on a curved 2d background. If we formally ignore the $n_a$-dependent terms in [(\[5\])]{} the corresponding dual target space background (metric $\td G$, 2-form field $\td B$ and dilaton $\td \p$) will no longer represent a solution of the critical string Weyl-invariance conditions [@elit], explaining what was observed in [@vene] for a specific example. Considered on a flat 2d background the dual [(\[5\])]{} will still be scale invariant so that the dual metric and $B$-field will solve the scale-invariance conditions (i.e., in the superstring context, the generalized supergravity equations [@Arutyunov:2015mqj; @Wulff:2016tju]).
Let us note that the scale-invariant dual background $(\td G, \td B)$ has a remarkable property that it can still be naturally associated to a proper solution of the Weyl-invariance conditions by applying a formal (2d flat space) duality transformation in the direction of the isometric part of $v_a$ parallel to $n_a$. The key observation is that while for $n_a\not=0$ the model [(\[5\])]{} can not be interpreted as a local on a curved 2d background, applying a 2d duality transformation to one scalar field in [(\[5\])]{} restores such an interpretation.[[^9]]{} Let us split the Lagrange multiplier field $v_a$ in [(\[3\])]{} as $ v_a = u_a + n_a y $. Then $y$ will appear in [(\[3\])]{} and thus also in [(\[5\])]{} only through its derivatives, i.e. shifts of $y$ will be an abelian isometry of the dual background.[[^10]]{} Indeed, we will have $v_a F^a_{ ij} = u_a F^a_{ ij} + y (\del_i \A_j - \del_j \A_i)$ where $\A_i = n_a A^a_i = \tr A_i $ and we have used that $n_a f^a_{\ bc} = f^d_{\ da} f^a_{\ bc} =0$ as follows from the Jacobi identity. Then the $y$-dependent and related terms in [(\[3\])]{} will be (up to integration by parts) v\_a \^[ij]{} F\^a\_[ ij]{} +2 ’ n\_a \^i A\^a\_i -2 \^[ij]{} \_i y \_j +2 ’ \^i \_i . Applying T-duality in the $y$ direction amounts to replacing $\del_i y $ by $B_i$ adding at the same time the term $-2 \ep^{ij} \del_i \td y\, B_j$. Then integrating over $B_i$ gives $\A_i= \del_i \td y$ and thus the last term in [(\[7\])]{} becomes (after integrating by parts) $2 \a' \s\, \del^i \A_i
\to -2 \a' \td y\ \del_+ \del_- \s\ $. The latter is the standard $R^{(2)} \p$ dilaton term with $\p$ linear in the dual coordinate, $\p\sim \td y$. Integrating the rest of $A_i^a$ out will then give a local Weyl-invariant with a linear non-isometric dilaton term. The above steps can be carried out explicitly, e.g., on the examples of the models discussed in [@elit].
We conclude that the scale-invariant background found by applying the NAD transformation to the action [(\[1\])]{}, which, taking $n_a \neq 0$, does not solve the Weyl-invariance equations, is still related by formal T-duality to a Weyl-invariant background with a linear non-isometric dilaton. This background cannot then be T-dualized back to give a standard Weyl-invariant with local couplings to the worldsheet metric. Thus NAD provides a particular example of the general case discussed in [@Arutyunov:2015mqj].
TsT duality as special case of non-abelian duality {#sectst}
--------------------------------------------------
Let us now consider the particular case of $O(2,2)$ T-duality where one starts with a metric having two abelian isometries, $ds^2 = dx^2 + f_1(x) dy_1^2 + f_2 (x) dy_2^2$, applies T-duality $y_1 \to \td y_1$, then shifts $y_2 \to y_2 + \g \td y_1$ introducing the parameter $\g$, and finally T-dualizes back $\td y_1 \to {\td {\td y}}_1 \equiv y_1$. This generates the following non-trivial background depending on $\g$ $$\begin{split} \la{8}
ds^2 & = dx^2 + U(x) \big[ f_1(x) dy_1^2 + f_2 (x) dy_2^2\big] \ ,
\\ B_{y_1y_2} & = \g f_1 f_2 U \ , \qquad e^{2\p} = U(x) \equiv {1 \ov 1 + \g^2 f_1(x) f_2(x) } \ . \end{split}$$ Applications of such TsT transformations were discussed, e.g., in [@tst; @Hashimoto:1999ut; @Maldacena:1999mh; @Lunin:2005jy; @Frolov:2005ty].
Below we shall show that this transformation of the corresponding 2d may be viewed as a special case of NAD with the algebra of $H$ being the centrally-extended 2d translation algebra (or Heisenberg algebra) \[P\_r, P\_s\]= \_[rs]{} Z , =0 , r,s=1,2 . Let us start with $\mathcal{L}= f_1 ( \del_i y_1)^2 + f_2 (\del_i y_2)^2$. The “interpolating” action that corresponds to the first two steps of the above TsT transformation may be written as = f\_1 (A\_i)\^2 + f\_2 (\_i y\_2 + \_i y\_1)\^2 + 2 \^[ij]{} \_i y\_1 A\_j .T-dualizing again $\td y_1\to y_1$ by introducing another abelian gauge field $A'_i$ we find ’= f\_1 (A\_i)\^2 + f\_2 (\_i y\_2 + A’\_i )\^2 + 2 \^[ij]{} A’\_i A\_j + 2 \^[ij]{} \_i y\_1 A’\_j .Redefining $A_i \equiv A^{1}_i$, $\del_i y_2 + \g A'_i \equiv A^{2}_i$, and then sending $y_1 \to -\g v_2$, $y_2 \to \gamma v_1$ we arrive at ’= f\_1 (A\^[1]{}\_i)\^2 + f\_2 (A\^[2]{}\_i)\^2 +2 \^[ij]{} ( v\_1 \_i A\^[1]{}\_j + v\_2 \_i A\^[2]{}\_j ) - 2\^[-1]{} \^[ij]{} A\^[1]{}\_i A\^[2]{}\_j .If we first integrate over $v_r$ we have $A^{r}_i=\del_i y^r_i$ and thus go back to the original model (the last term in [(\[12\])]{} is then total derivative). Integrating instead over the two gauge fields in [(\[12\])]{} one ends up with the corresponding to the TsT background [(\[8\])]{} (with $v_r \to y_r$). Note that without the last $-2\g^{-1} \ep^{ij} A^{1}_i A^{2}_j $ term [(\[12\])]{} is the first-order action for T-dualizing on both $y_1$ and $y_2$.[[^11]]{}
Let us now derive the same Lagrangian [(\[12\])]{} starting instead with the NAD model [(\[3\])]{} with $\mathfrak{h}$ taken to be the centrally-extended algebra [(\[9\])]{} with $T_a=(P_1,P_2, Z)$ (for which only $f^3_{\ 12}$ is non-zero and thus $n_a=0$). Here v\_a \^[ij]{} F\_[ij]{}\^a = 2 \^[ij]{} , where $w\equiv v^3$ and $C_i\equiv A^3_i$ corresponds to the central generator $Z$ in [(\[9\])]{}. Note that as the original is assumed to be invariant under the abelian translations, the central generator $Z$ should act trivially on the coordinates. Gauging it is still possible as integrating out $v_r$ and $w$ brings us back to the original model: we get $A^r_i = \del_i y^r $ (and $dC $ expressed in terms of $y^r$), so that $ f_r (A^{r}_i)^2 \to f_r (\del_i
y^r)^2$.
Since the gauge field $C_i$ does not enter the rest of the action, we can readily integrate it out getting the condition $w=w^{(0)}=\const$. Then [(\[13\])]{} becomes the same as the last three terms in [(\[12\])]{} with $w^{(0)}\sim -\g^{-1}$. Thus considering the central extension of the abelian translation group allows one to introduce an extra free parameter $\g$ (absent in the standard first-order abelian T-duality action): the TsT parameter $\g$ acquires the interpretation of the background value of the dual coordinate corresponding to the central generator $Z$. Let us note also that in the example we considered above the origin of the $B$-field of the resulting background can be traced to the non-abelian nature of the Heisenberg algebra [(\[9\])]{} (which is also related to the non-commutativity of the dual gauge theory [@Hashimoto:1999ut; @Maldacena:1999mh; @Lunin:2005jy]).
One can obviously consider various generalizations. First, one can readily repeat the above discussion for generic model with two abelian isometries $\mathcal{L} = f_1 ( \del_i y_1)^2 + f_2 (\del_i y_2)^2
+ g_1 \partial^i y_1 \partial_i y_2 + g_2 \ep^{ij} \partial_i y_1 \partial_j y_2 + h^{i}_1 \partial_i y_1 +h^{i}_2 \partial_i y_2 $ where $f_{r}$, $g_{r}$ and $h^{i}_r$ are functions of the remaining fields, including fermionic degrees of freedom. If the rank of the abelian isometry algebra is greater than 2, constructing its central extensions and performing NAD will introduce several continuous parameters $w^{(0)}_m$ as expected in the general $O(d,d)$ T-duality case. Instead of an abelian isometry group one may also have a non-abelian one, $H$, that has an abelian (translational) subgroup. While the direct application of the NAD transformation with respect to $H$ may give a dual model with no free parameters, starting instead with a centrally-extended group and then applying NAD will lead to a model containing several free parameters. In general, the central extension of an algebra satisfying $f^{c}_{\ ca} = 0$ will also satisfy this property.
For example, one may start with the euclidean $AdS_3$ space $ds^2 =z^{-2} [
dz^2 + dy_1^2 + dy_2^2] $ and perform NAD with respect to the 2d Euclidean group $ISO(2) \subset SO(1,3)$ with the algebra $[J, P_r] = \ep_{rs} P_s$, $[P_r,P_s]=0$. Considering its central extension then allows one to introduce a free parameter. For rank four and higher algebras one may be able to introduce several parameters. Such examples will be discussed below for the $AdS_5$ $\sigma$-model.
Homogeneous Yang-Baxter deformations of the and non-abelian duality {#secgen}
===================================================================
We now turn to demonstrating the conjectured equivalence between homogeneous Yang-Baxter (YB) deformations of a coset and non-abelian duals (NADs) of that same model. We shall first make some general remarks and then focus on the bosonic $AdS_5$ model but similar considerations should apply also to the full supercoset model.
In general, the homogeneous YB deformation of the $F/G$ symmetric space is based on a solution to the classical Yang-Baxter equation (cYBE) for $\mathfrak{f} = Lie(F)$. These solutions are in correspondence with the quasi-Frobenius subalgebras of $\mathfrak{f}$.
The homogeneous YB deformations with $r$-matrices corresponding to abelian subgroups are equivalent to TsT transformations of the original coset model [@Osten:2016dvf]. As we have seen in section \[sectst\], the TsT transformation can, in fact, be reformulated as the NAD with respect to a centrally-extended abelian subgroup.
Furthermore, in [@Borsato:2016ose] it was shown that the YB deformation of the supercoset model [@yjor; @dmv] corresponds to a supergravity solution (i.e. is one-loop Weyl invariant when supplemented with an appropriate dilaton) if the $r$-matrix $r = r^{\a\b} e_\a \wedge e_\b$ is unimodular, i.e. satisfies $$\label{uni}
r^{\a\b}[e_\a,e_\b] = 0 \ .$$ Here the $e_\a$ generate the quasi-Frobenius subalgebra $\mathfrak{h}$ of $\mathfrak{f}$.[[^12]]{} The unimodularity condition , combined with the quasi-Frobenius property, implies that the structure constants on $\mathfrak{h}$ satisfy [@Borsato:2016ose] $$f^{\gamma}_{\ \gamma\alpha} = 0 \ .$$ As discussed in section \[secnad\], this is precisely the same as the requirement on $\mathfrak{h}$ for the NAD with respect to $\mathfrak{h}$ of an $H$-invariant to preserve the one-loop Weyl invariance [@alv; @elit].
Based on these observations we shall make a conjecture that the homogeneous YB deformation based on a classical $r$-matrix is always equivalent to the NAD transformation of the original coset model with respect to the corresponding (centrally-extended) quasi-Frobenius subalgebra. As already mentioned, for abelian $r$-matrices this follows from the results in section \[sectst\] and [@Osten:2016dvf]. For non-abelian $r$-matrices we will not prove this conjecture directly, but will provide a comprehensive range of examples (including those in [@Hoare:2016hwh; @ysugra2; @Borsato:2016ose]), explicitly demonstrating its validity.
As our primary focus is on deformations of the $AdS_5 \times S^5$ superstring and its lower-dimensional counterparts here we will restrict our attention to the bosonic $AdS_5$ $\s$-model. The isometry algebra $\mathfrak{so}(2,4)$ admits a number of different subgroups corresponding to non-abelian $r$-matrices of various types, both unimodular and jordanian.[[^13]]{} While it is unclear if it is a universal rule, we find that for unimodular $r$-matrices we need to consider the centrally-extended algebra, while for jordanian $r$-matrices this appears not to be required.
YB deformation and non-abelian duality for the symmetric space
---------------------------------------------------------------
Our starting point for both the homogeneous YB deformation and the NAD transformation will be the symmetric space $\s$-model. Here we will define them for a generic symmetric space $F/G$ with both $F$ and $G$ being semi-simple. In the case of $AdS_5$ we have $F = SO(2,4)$ and $G = SO(1,4)$.
For $\mathfrak{f} = Lie(F)$ and $\mathfrak{g} = Lie(G)$ we use the standard bilinear form on $\mathfrak{f}$ to define $\mathfrak{p}$ as the orthogonal complement of $\mathfrak{g}$ in $\mathfrak{f}$ so that for a symmetric space $$\mathfrak{f} = \mathfrak{g} \oplus \mathfrak{p} \ , \qquad \Tr[\mathfrak{g}\mathfrak{p}] = 0 \ , \qquad
[\mathfrak{g}, \mathfrak{g}] \subset \mathfrak{g} \ , \qquad
[\mathfrak{g}, \mathfrak{p}] \subset \mathfrak{p} \ , \qquad
[\mathfrak{p}, \mathfrak{p}] \subset \mathfrak{g} \ . \qquad$$ The Lagrangian for the symmetric space is then given by $$\label{ssslag}
\mathcal{L} = \Tr[J_+ P J_-] \ , \qquad \qquad J = f^{-1} df \ , \qquad f \in F \ ,$$ where $P$ is the projector onto $\mathfrak{p}$, $\Tr$ is appropriately normalized and $\pm$ are light-cone coordinates on the worldsheet. This action has a global $F$ symmetry $f \to f_0 f$ and a local $G$ gauge symmetry $f \to f g$.
The homogeneous YB deformation [@Klimcik:2002zj; @Klimcik:2008eq; @Delduc:2013fga; @yjor] of the symmetric space is defined as $$\label{yblag}
\mathcal{L} = \Tr[J_+ P \frac{1}{1-R_f P} J_-] \ , \qquad\qquad R_f = \Ad_f^{-1} R \Ad_f \ ,$$ where operator $R$ is an antisymmetric solution of the cYBE for the algebra $\mathfrak{f}$ $$[RX,RY] = R([RX,Y] + [X,RY]) \ , \qquad X,Y \in \mathfrak{f} \ .$$ As this equation is homogeneous in $R$ we can always multiply any solution by an overall constant $\eta$. Then in the limit $\eta\to 0$ we recover the Lagrangian of the symmetric space $\s$-model. The deformed action preserves the local $G$ gauge symmetry $f \to fg$; however, the global symmetry is broken to a subgroup of $F$ depending on the choice of $R$.
In general, we will write the operator $R$ in terms of an $r$-matrix taking values in $\mathfrak{f} \otimes \mathfrak{f}$ $$r = T_1 \wedge T_2 + T_3 \wedge T_4 + \ldots \ , \qquad \qquad T_r \wedge T_s = T_r \otimes T_s - T_s \otimes T_r \ ,$$ where $T_a$ is a basis of $\mathfrak{f}$ and by the use of the wedge product we enforce the antisymmetry of the operator. The operator $R$ is then defined using the bilinear form as $$R X = \Tr_2(r (1 \otimes X)) \ ,$$ where $\Tr_2$ denotes that the contraction is taken over the second space in the tensor product $\mathfrak{f} \otimes \mathfrak{f}$.
To find the non-abelian dual of the symmetric space with respect to a subgroup $H
\subset F$ we first write the group element $f$ as $$f = h f' \ , \qquad \qquad h \in H \ , \qquad f' \in F \ .$$ Substituting this into the Lagrangian and gauging the global $H$ symmetry, i.e. replacing $h^{-1} d h \to A$, where $A \in \mathfrak{h} = Lie(H)$, we find as in [(\[3\])]{} $$\label{nadlag}
\mathcal{L} = \Tr\big[(f'^{-1} A_+ f' + f'^{-1} \partial_+f')P(f'^{-1} A_- f' + f'^{-1} \partial_- f') +
v F_{-+} (A)\big] \ .$$ Here $v$ is a Lagrange multiplier imposing the flatness of the connection $A$, i.e. $F_{-+}(A) \equiv \partial_- A_+ - \partial_+ A_- + [A_-,A_+]=0$, and hence the equivalence of and . In general, the algebra $\mathfrak{h}$ need not be semi-simple. In this case $v$ should not be taken in the algebra $\mathfrak{h}$, but rather in its dual $\bar{\mathfrak{h}}$. Introducing $T_a$ as a basis for $\mathfrak{f}$ and $e_\a$ as a basis for $\mathfrak{h}$ we can define a basis of $\bar{\mathfrak{h}}$ to be $$\label{dualbasis}
\bar e_\a = \sum_{a=1}^{\dim F} \Tr[T_a e_\a] \, T_a \ ,$$ as $F$ is assumed to be semi-simple. The NAD model (defined on a flat 2d background, cf. [(\[3\])]{} and [(\[5\])]{}) is then found upon integrating out the gauge field $A$ in .
The Lagrangian still has a local $G$ gauge symmetry $f' \to f'
g$, but now it also possesses an additional local $H$ gauge symmetry: $f' \to h f'$, $A \to h A h^{-1} - dh h^{-1}$, $v \to h v h^{-1}$.[[^14]]{} In the case when the NAD is with respect to the full group $H = F$, the $H$ gauge symmetry may be used to fix $f' = 1$. In this gauge fixing the $G$ gauge symmetry then has a compensating action on $A$ and $v$ which may be used to fix $v$. When $H$ is a subgroup one may no longer be able to fix $f' = 1$. The gauge condition may, in general, involve both $f'$ and $v$. Our approach will be to gauge fix $f'$ as far as possible and use the remaining gauge symmetry to constrain $v$.
As suggested by the example of the TsT transformation discussed in section \[sectst\], in some cases it will not be enough to perform the NAD in a subgroup $H$ of $F$. Rather we will need to start with a central extension of $H$ (which, in general, need not be admissible in the full group). We shall define the NAD with respect to the centrally-extended group $H_{c.e.}$ as (cf. [(\[13\])]{})[[^15]]{} $$\begin{split}\label{nadlagce}
{\hat {\mathcal{L} }}& = \Tr\big[(f'^{-1} A_+ f' + f'^{-1} \partial_+f')P(f'^{-1} A_- f' + f'^{-1} \partial_- f') +
v F_{-+} (A)\big]
\\ & \qquad + w_m (\partial_- C^m_+ - \partial_+ C^m_-) + w_m [A_-,A_+]_{c.e.}^m \ .
\end{split}$$ Here the first line is identical to , i.e. no central extension is present and the Lie brackets remain those of $\mathfrak{f}$ and the subalgebra $\mathfrak{h}$. In the second line the index $m$ labels the central extensions of $\mathfrak{h}$, while $w_m$ and $C^m$ are the corresponding Lagrange multipliers and gauge fields respectively. The bracket $[\ ,\, ]_{c.e.}$ is then the Lie bracket on the centrally-extended algebra $\mathfrak{h}_{c.e.}$.
On integrating out the Lagrange multipliers $v$ and $w_m$ in we still recover the symmetric space , demonstrating that, like [(\[nadlag\])]{}, [(\[nadlagce\])]{} is still equivalent to [(\[ssslag\])]{}. However, integrating out the gauge fields in will now lead to a more general model (depending on extra parameters) than that found from . Indeed, following the discussion in section \[sectst\], integrating out $C^m$ implies that $w_m=w^{(0)}_m=\const$. Substituting this back into gives $$\label{nadfin}
{\hat {\mathcal{L} }} = \Tr\big[(f'^{-1} A_+ f' + f'^{-1} \partial_+f')P(f'^{-1} A_- f' + f'^{-1} \partial_- f')
+ v F_{-+} (A)\big] + w^{(0)}_m [A_-,A_+]^m_{c.e.} \ .$$ Then integrating out the gauge field $A_\pm$ defines the NAD model, which will now depend on arbitrary constant parameters $w_m^{(0)}$. As was shown in section \[sectst\], in the case when $\mathfrak{h}$ is abelian these constants can then be interpreted as the parameters of the TsT transformations.
and {#seccon}
-----
Let us now turn to the $F/G= SO(2,4)/SO(1,4)$ symmetric space for $AdS_5$ and first briefly state our conventions for the algebra $\mathfrak{so}(2,4)$. Introducing the $\g$-matrices (here $\s_{1,2,3}$ are the standard Pauli matrices and $\s_0$ is the $2 \times 2$ identity matrix) $$\g_0 = i \s_3 \otimes \sigma_0 \ , \quad \g_1 = \s_2 \otimes \s_2 \ , \quad \g_2 = - \s_2 \otimes \s_1 \ , \quad \g_3 = \s_1 \otimes \s_0 \ , \quad \g_4 = \s_2 \otimes \s_3 \ , $$ we define the following basis for $\mathfrak{so}(2,4)$ $$\textstyle T_{ij} = \frac14 [\g_i, \g_j] \ , \quad T_{i5} = -T_{5i} = \frac12 \g_i \ , \quad T_{55} = 0 \ , \qquad i,j = 0,\ldots 4 \ .$$ We then use the standard matrix trace as our bilinear form. It is this basis that we use to define bases for the duals of non semi-simple subalgebras of $\mathfrak{so}(2,4)$ as in .
The deformations based on non-abelian $r$-matrices that we consider are more naturally understood in the Poincaré patch of $AdS_5$. Therefore, we will use the corresponding basis of $\mathfrak{so}(2,4)$ $$D = T_{45} \ , \qquad P_\mu = T_{\mu 5} - T_{\mu 4} \ , \qquad K_\mu = T_{\mu 5} + T_{\mu 4} \ , \qquad M_{\mu \nu} = T_{\mu \nu} \ , \qquad \mu,\nu = 0,\ldots3 \ .$$ The symmetric space $AdS_5$ can be represented as the coset $SO(2,4)/SO(1,4)$. The subalgebra $\mathfrak{so}(1,4)$ corresponding to the gauge group is spanned by $T_{ij}$ where $i,j = 0,\ldots , 4$ and therefore the projector $P$ onto the coset part of the algebra is given by $$P(X) = - \Tr[X T_{05}] T_{05} + \sum_{i=1}^4 \Tr[X T_{i5}] T_{i5} \ .$$ Taking the gauge-fixed field $f$ to be $$\label{f}
f = \exp[- x_0 P_0 + x_1 P_1 + x_2 P_2 + x_3 P_3]\, \exp[\log z \, D] \ ,$$ and substituting it into the Lagrangian of the symmetric space we find that it takes the form of the with the target space metric being the $AdS_5$ metric in Poincaré patch $$ds^2 = \frac{- dx_0^2 + dx_1^2 + dx_2^2 + dx_3^2 + dz^2}{z^2} \ .$$
Abelian -matrices {#secabe}
-----------------
The first examples of the relation between homogeneous YB deformations and NAD that we will consider are based on abelian $r$-matrices. While the equivalence in these cases follows from the results of section \[sectst\] and [@Osten:2016dvf], it will be instructive to consider two examples explicitly. Prior to the general investigation of [@Osten:2016dvf], deformations based on abelian $r$-matrices and their relation to TsT transformations of have been studied extensively on a case by case basis [@yabe1; @yabe2; @ysum; @vtsum; @ysugra1; @Hoare:2016hwh; @ysugra2].
[**Example 1:**]{} The first case we shall consider corresponds to the rank 2 abelian $r$-matrix $$\la{321}
r = \eta \, P_2 \wedge P_3 \ ,$$ where $P_\mu$ are translation generators and $\eta$ is a free deformation parameter. This $r$-matrix and also the rank 4 one below were first discussed in [@yabe2] where it was shown that the metrics and $B$-fields of corresponding YB deformed models are those of the “non-commutative dual” backgrounds of [@Hashimoto:1999ut; @Maldacena:1999mh]. Using the gauge-fixed field $f$ , the Lagrangian of the corresponding YB deformed model with $r$-matrix given by [(\[321\])]{} is found to be that of the with the following target space metric and $B$-field (cf. [(\[8\])]{}) $$\begin{split}\label{abel2}
ds^2 & = \frac{-dx_0^2 + dx_1^2 + dz^2}{z^2} + \frac{z^2}{z^4 +\eta^2} (dx_2^2 + dx_3^2) \ ,
\\
B & = \frac{\eta}{z^4 + \eta^2} dx_2 \wedge dx_3 \ .
\end{split}$$ On the other hand, let us consider the NAD of the $AdS_5$ with respect to the central extension of the algebra $\mathfrak{h} = \{P_2,P_3\}$ (equivalent to [(\[9\])]{}). In this case the dual algebra is given by $\bar{\mathfrak{h}} = \{K_2,K_3\}$. We use the local $H$ gauge symmetry of the NAD model to fix $$f' = \exp[-x_0 P_0 + x_1 P_1] \exp[\log z \, D] \ ,$$ and also parametrize the gauge field and Lagrange multiplier as $$A_\pm = A_{1\pm} P_2 + A_{2\pm} P_3 \ , \qquad \qquad v = \frac{1}{2\eta}(x_2 K_2 + x_3 K_3) \ .$$ Substituting these expressions into the Lagrangian where we take the explicit form of the central extension term to be (cf. [(\[13\])]{}) $$\label{2ce}
w_m^{(0)} [A_-,A_+]_{c.e.}^m = \frac{1}{\eta}(A_{1+}A_{2-} - A_{2+}A_{1-}) \ ,$$ and integrating out the gauge field we find that the NAD of $AdS_5$ with respect to the central extension of $\mathfrak{h} = \{P_2,P_3\}$ again gives the based on the background metric and $B$-field (up to a total derivative term in the $B$-field).[[^16]]{}
[**Example 2:**]{} Our second example is defined by the rank 4 abelian $r$-matrix $$\label{rmatabel4}
r = \eta \, P_0 \wedge P_1 + \zeta \, P_2 \wedge P_3 \ ,$$ where $\eta$ and $\zeta$ are independent parameters. Using again the gauge-fixed field $f$ , the deformed metric and $B$-field corresponding to the YB deformed model are found to be $$\begin{split}\label{abel4}
ds^2 & = \frac{z^2}{z^4 - \eta^2} (- dx_0^2 + dx_1^2) + \frac{z^2}{z^4 + \zeta^2} ( dx_2^2 + dx_3^2) + \frac{dz^2}{z^2} \ ,
\\
B & = \frac{\eta}{z^4 - \eta^2} dx_0 \wedge dx_1 + \frac{\zeta}{z^4 + \zeta^2} dx_2 \wedge dx_3 \ .
\end{split}$$ Next, we construct the NAD of the $AdS_5$ with respect to central extension of the algebra $\mathfrak{h} = \{P_0, P_1; P_2,P_3 \}$ which is implied by the form of the $r$-matrix $$\label{ceabel4}
[P_0,P_1] = Z_1 \ , \qquad\qquad [P_2,P_3] = Z_2 \ .$$ The dual algebra is given by $\bar{\mathfrak{h}} = \{K_0,K_1;K_2,K_3\}$. We now use the local $H$ gauge symmetry of to fix $$f' = \exp[\log z \, D] \ ,$$ and parametrize the gauge field and the Lagrange multiplier in [(\[nadfin\])]{} as $$A_\pm = A_{1\pm} P_0 + A_{2\pm} P_1 + A_{3\pm} P_2 + A_{4\pm} P_3 \ , \qquad v = \frac{1}{2\eta}(x_1 K_0 - x_0 K_1) + \frac{1}{2\zeta}(x_2 K_2 + x_3 K_3) \ .$$ We then substitute these expressions into the Lagrangian where, according to the explicit form of the central extension term is now $$\label{4ce}
w_m^{(0)} [A_-,A_+]_{c.e.}^m = \frac{1}{\eta}(A_{1+}A_{2-} - A_{2+}A_{1-}) + \frac{1}{\zeta}(A_{3+}A_{4-} - A_{4+}A_{3-}) \ .$$ Integrating out the gauge field $A$ we conclude that the NAD transform of $AdS_5$ with respect to the central extension of $\mathfrak{h} = \{P_0,P_1;P_2,P_3\}$ gives the based again on the metric and $B$-field in . In addition, in both examples the NAD procedure determines also the dilaton field given by the standard T-duality expression [@bu] (cf. [(\[5\])]{},[(\[8\])]{}).
Unimodular non-abelian -matrices {#secuni}
--------------------------------
Let us now turn our attention to homogeneous YB deformations based on non-abelian $r$-matrices. We start with unimodular examples for which the YB deformation of the supercoset background [@yjor; @dmv] preserves the satisfaction of the supergravity equations or one-loop Weyl invariance (with an appropriate dilaton) [@Borsato:2016ose].
All abelian $r$-matrices, defining deformations which are equivalent to sequences of TsT transformations of [@Osten:2016dvf], are unimodular. Furthermore, at rank 2 all unimodular $r$-matrices are abelian. However, for the algebra $\mathfrak{so}(2,4)$ there are rank 4 and rank 6 non-abelian unimodular $r$-matrices. Those of rank 4 were classified in [@Borsato:2016ose] and fall into three classes characterized by the algebra of the corresponding generators. Defining the $r$-matrix as $$r = e_1 \wedge e_2 + e_3 \wedge e_4 \ ,$$ the non-vanishing commutation relations determining the three classes are[[^17]]{} $$\begin{aligned}
\nonumber
& \text{ class 1:} \quad [e_1, e_4] = e_2 \ ,
\\
& \text{ class 2:} \quad [e_1, e_4] = e_3 \ , \qquad [e_1, e_3] = - e_4 \ , \la{333}
\\ \nonumber
& \text{ class 3:} \quad [e_1, e_4] = e_2 \ , \qquad [e_1, e_3] = -e_4 \ .\end{aligned}$$ In [@Borsato:2016ose] the deformations corresponding to examples from the first two classes were observed to be equivalent to a sequence of two TsT transformations (with a non-linear coordinate redefinition in between) of the model; however, a similar result for the last class was not found.
Let us now consider one example from each class demonstrating that the corresponding YB deformation is equivalent to the NAD of the $AdS_5$ with respect to the following central extension of the algebra $\mathfrak{h} = \{e_1,e_2;e_3,e_4\}$[[^18]]{} $$[e_1,e_2] = Z_1 \ , \qquad \qquad [e_3,e_4] = Z_2 \ .$$ As mentioned in section \[sectst\], centrally extending a unimodular algebra preserves this property. At the end of this section we will also consider one rank 6 example.
[**Class 1 example:**]{} An $r$-matrix from class 1 that we shall consider is [@Borsato:2016ose] $$\label{class1rmat}
r = \eta \, M_{+3} \wedge P_+ + \zeta \, P_2 \wedge P_3 \ ,$$ where $\eta$ and $\zeta$ are two free parameters. Here we have introduced light-cone indices in the $\mathfrak{so}(1,3)$ Lorentz subalgebra spanned by $M_{\mu \nu}$ (not be confused with the light-cone coordinates on the worldsheet) defined as $\Lambda_\pm = \Lambda_0 \pm \Lambda_1 $, i.e. $M_{+3}= M_{03} + M_{13}$. Fixing the group-valued field $f$ as $$f = \exp[-\ha(x_- P_+ + x_+ P_-) + x_2 P_2 + x_3 P_3] \exp[\log z \, D] \ ,$$ we find the YB deformed model corresponds to following metric and $B$-field $$\begin{split}\label{class1back}
ds^2 & = \frac{-dx_- dx_+ + dz^2}{z^2} + \frac{- \eta^2 x_+^2 dx_+^2 - 2 \eta \zeta x_+ dx_+ dx_2 + z^4(dx_2^2 + dx_3^2) }{z^2(z^4 + \zeta^2)} \ ,
\\
B & = \frac{\eta x_+ dx_+ \wedge dx_3 + \zeta dx_2 \wedge dx_3} {z^4 +\z^2} \ .
\end{split}$$ Let us now compare this to the NAD of the $AdS_5$ with respect to the central extension of the algebra $\mathfrak{h} = \{M_{+3}, P_+; P_2, P_3\}$. The dual algebra is given by $\bar{\mathfrak{h}} = \{ M_{-3}, K_-; K_2, K_3\}$. We first partially use the local $H$ gauge symmetry of to fix $f'$ to be $$f' = \exp[-\ha x_+ P_-] \exp[\log z \, D] \ .$$ This leaves one gauge freedom corresponding to $M_{+3}$ that we cannot use to fix $f'$ further. We therefore use it to fix the $K_3$ component of the Lagrange multiplier $v$ to zero, so that it can be parametrized as $$\la{439}
v = -\big(\frac{x_-}{4\eta} + \frac{x_+x_2}{2\zeta}\big) M_{-3} + \frac12\sqrt{\frac{-x_2}{2\eta\zeta}} \, K_-
+ \big(\frac{x_3}{2\zeta} + \frac{\eta x_+}{\zeta} \sqrt{\frac{-x_2}{2\eta\zeta}}\big) K_2 \ .$$ The reason for this choice is to make manifest the comparison with [(\[class1back\])]{}. Parametrizing the gauge field as $$\label{aa1}
A_\pm = A_{1\pm} M_{+3} + A_{2\pm} P_+ + A_{3\pm} P_2 + A_{4\pm} P_3 \ ,$$ we again take the explicit form of the central extension term in to be given by . Integrating out the gauge field we finally find that the NAD of the $AdS_5$ with respect to the central extension of $\mathfrak{h} =
\{M_{+3},P_+;P_2,P_3\}$ gives exactly the same as defined by the metric and $B$-field in .
[**Class 2 example:**]{} An example of $r$-matrix from class 2 is [@Borsato:2016ose] $$\label{class2rmat}
r = \eta \, M_{23} \wedge P_1 + \zeta \, P_2 \wedge P_3 \ .$$ Fixing the group-valued field $f$ as in [(\[f\])]{} with $$\la{341}
x_2 = r \cos \theta \ , \qquad x_3 = r \sin \theta \ ,$$ we find the following metric and $B$-field of the YB deformed model $$\begin{split}\label{class2back}
ds^2 & = \frac{- dx_0^2 + dz^2}{z^2} + \frac{(z^4 + \z^2) dx_1^2 - 2 \eta \zeta r dx_1 dr + (z^4 + \eta^2 r^2) dr^2 + z^4 r^2 d \theta^2}{z^2(z^4 + \eta^2 r^2 + \zeta^2)} \ ,
\\
B & = \frac{\eta r^2 dx_1 \wedge d\theta + \zeta r dr \wedge d\theta} {z^4 + \eta^2 r^2 + \z^2} \ .
\end{split}$$ Next, we compare this to the NAD of $AdS_5$ with respect to the central extension of the algebra $\mathfrak{h} = \{M_{23}, P_1; P_2, P_3\}$. The dual algebra is given by $\bar{\mathfrak{h}} = \{ M_{23}, K_1; K_2, K_3\}$. We partially use the local $H$ gauge symmetry of to fix $$f' = \exp[-x_0 P_0] \exp[\log z \, D] \ .$$ This leaves one gauge freedom corresponding to $M_{23}$ that we can use to set the $K_3$ component of the Lagrange multiplier $v$ to zero, which we then parametrize as $$v = \big(-\frac{x_1}{\eta} + \frac{r^2}{2\zeta}\big) M_{23} + \frac{\theta}{2\eta} K_1 + \frac{r}{2\zeta} K_2 \ .$$ Parametrizing the gauge field as $$\label{aa2}
A_\pm = A_{1\pm} M_{23} + A_{2\pm} P_1 + A_{3\pm} P_2 + A_{4\pm} P_3 \ ,$$ we again take the explicit form of the central extension term in to be given by . Integrating out the gauge field we find that the NAD of the $AdS_5$ with respect to the central extension of $\mathfrak{h} =
\{M_{23},P_1;P_2,P_3\}$ gives the defined by the metric and $B$-field .
Let us note that one can also consider the $r$-matrix $$\label{class2rmatalt}
r = \eta \, M_{01} \wedge P_3 + \zeta \, P_0 \wedge P_1 \ ,$$ which can be understood as an analytic continuation of . The corresponding quasi-Frobenius algebra satisfies the commutation relations given in footnote \[footcomrel\]. By repeating the above discussion using the obvious analytic continuation it is clear that the corresponding YB deformed model is equivalent to the NAD of $AdS_5$ with respect to the central extension of $\mathfrak{h} = \{M_{01}, P_3 ; P_0, P_1\}$.
[**Class 3 example:**]{} Let us now consider the following $r$-matrix from class 3[[^19]]{} $$\label{class3rmat}
r = \eta \, M_{+3} \wedge P_+ + \zeta \, P_1 \wedge P_3 \ .$$ Fixing the group-valued field $f$ as in [(\[f\])]{}, i.e. $$f = \exp[-\ha(x_- P_+ + x_+ P_-) + x_2 P_2 + x_3 P_3 ] \exp[\log z \, D] \ ,$$ we find the following metric and $B$-field of the YB deformed model $$\begin{aligned}
\nonumber
ds^2 & = \frac{dx_2^2 + dz^2}{z^2} + \frac{-2(2z^4 +2\eta\zeta x_+ + \zeta^2)dx_-dx_+ -\zeta^2 dx_-^2 - (2\eta x_+ + \zeta)^2 dx_+^2 + 4z^4 dx_3^2}{4z^2(z^4 + 2\eta \zeta x_+ + \zeta^2)} \ ,
\\\label{class3back}
B & = \frac{-\zeta dx_- \wedge dx_3 + (2\eta x_+ + \zeta)dx_+ \wedge dx_3} {2(z^4 + 2\eta \zeta x_+ + \zeta^2)} \ .\end{aligned}$$ We are now to compare this to the NAD of $AdS_5$ with respect to the central extension of $\mathfrak{h} = \{M_{+3}, P_+; P_1, P_3\}$ with the dual algebra being $\bar{\mathfrak{h}} = \{ M_{-3}, K_-; K_1, K_3\}$. We partially use the local $H$ gauge symmetry of to fix $$f' = \exp[x_2 P_2] \exp[\log z \, D] \ .$$ This leaves one gauge freedom corresponding to $M_{+3}$ that we can use this to fix the $K_3$ component of the Lagrange multiplier $v$ to zero, which we then parametrize as $$v = -\big(\frac{x_- + x_+}{4\eta} + \frac{x_+^2}{4\zeta}\big) M_{-3} + \big(\frac{x_3}{2\zeta} + \sqrt{\frac{-x_+}{2\eta \zeta}} (1 + \frac{2 \eta x_+}{3\zeta}) \big) K_-
+ \big(\frac{x_3}{\zeta} + \sqrt{\frac{-x_+}{2\eta\zeta}} (1+ \frac{4\eta x_+}{3\zeta}) \big) K_1 \ .$$ Parametrizing the gauge field as $$\label{aa4}
A_\pm = A_{1\pm} M_{+3} + A_{2\pm} P_+ + A_{3\pm} P_1 + A_{4\pm} P_3 \ ,$$ choosing the central extension term in to be and integrating out the gauge field we finally conclude that the NAD of the $AdS_5$ with respect to the central extension of $\mathfrak{h} =
\{M_{+3},P_+,P_1,P_3\}$ is again equivalent to the YB deformed corresponding to the metric and $B$-field in .
[**Rank 6 example:**]{} Finally, we consider the following example of a rank 6 unimodular $r$-matrix [@Borsato:2016ose] $$\label{rank6rmat}
r = \eta \, M_{01} \wedge M_{23} + \zeta \, P_0 \wedge P_1 + \kappa \, P_2 \wedge P_3 \ ,$$ where $\eta$, $\zeta$ and $\kappa$ are free parameters. Fixing the group-valued field $f$ as in [(\[f\])]{} with $$\la{341a}
x_0 = t \cosh \chi \ , \qquad x_1 = t \sinh \chi \ , \qquad x_2 = r \cos \theta \ , \qquad x_3 = r \sin \theta \ ,$$ we find the following metric and $B$-field of the YB deformed model $$\begin{split}\label{rank6back}
ds^2 & = \frac{ dz^2}{z^2} + \frac{-z^2(z^4 + \eta^2 t^2 r^2 + \kappa^2) dt^2 + 2 \eta \zeta z^2 t r^2 dt d\theta + (z^4- \zeta^2)z^2 r^2 d\theta^2 }{z^8 + z^4(\eta^2 t^2 r^2 - \zeta^2 + \kappa^2) - \zeta^2 \kappa^2}
\\ & \qquad \qquad
+ \frac{z^2(z^4 + \eta^2 t^2 r^2 - \zeta^2) dr^2 - 2 \eta \kappa z^2 t^2 r dr d\chi + (z^4+\kappa^2)z^2 t^2 d\chi^2 }{z^8 + z^4(\eta^2 t^2 r^2 - \zeta^2 + \kappa^2) - \zeta^2 \kappa^2} \ ,
\\
B & = \frac{-\eta \zeta \kappa t r dt \wedge dr + \zeta (z^4 + \kappa^2) t dt \wedge d\chi - \kappa (z^4 - \zeta^2) r d\theta \wedge dr - \eta z^4 t^2 r^2 d\theta \wedge d\chi } {z^8 + z^4(\eta^2 t^2 r^2 - \zeta^2 + \kappa^2) - \zeta^2 \kappa^2} \ .
\end{split}$$ Now we compare this to the NAD of $AdS_5$ with respect to the central extension of the algebra $\mathfrak{h} = \{M_{01},M_{23}; P_0, P_1; P_2, P_3\}$.[[^20]]{} The dual algebra is given by $\bar{\mathfrak{h}} = \{M_{01}, M_{23}; K_0, K_1; K_2, K_3\}$. We partially use the local $H$ gauge symmetry of to fix $$f' = \exp[\log z \, D] \ .$$ This leaves two free gauge transformations corresponding to $M_{01}$ and $M_{23}$ that we can use to set the $K_0$[[^21]]{} and $K_2$ components of the Lagrange multiplier $v$ to zero, which we then parametrize as $$v = -\big(\frac{\theta}{\eta} - \frac{t^2}{2\zeta}\big) M_{01} - \big(\frac{\chi}{\eta} - \frac{r^2}{2\kappa}\big) M_{23}
+ \frac{t}{2\zeta} K_1 + \frac{r}{2\kappa} K_3 \ .$$ Parametrizing the gauge field as $$\label{aarank6}
A_\pm = A_{1\pm} M_{01} + A_{2\pm} M_{23} + A_{3\pm} P_0 + A_{4\pm} P_1 + A_{5\pm} P_2 + A_{6\pm} P_3 \ ,$$ we again take the explicit form of the central extension term in to be given by $$\label{6ce}
w_m^{(0)} [A_-,A_+]_{c.e.}^m = \frac{1}{\eta}(A_{1+}A_{2-} - A_{2+}A_{1-}) + \frac{1}{\zeta}(A_{3+}A_{4-} - A_{4+}A_{3-})
+\frac{1}{\kappa}(A_{5+}A_{6-} - A_{6+}A_{5-}) \ .$$ Integrating out the gauge field we find that the NAD of the $AdS_5$ with respect to the central extension of $\mathfrak{h} =
\{M_{01},M_{23};P_0,P_1;P_2,P_3\}$ gives the defined by metric and $B$-field .
Jordanian -matrices {#secjor}
-------------------
We now turn to our final group of examples, the jordanian $r$-matrices. Jordanian $r$-matrices have the form (see, e.g., [@jord] and references therein) $$r = T_1 \wedge T_2 + \ldots \ ,\qquad \qquad [T_1,T_2] = T_2 \ ,$$ and the corresponding deformations of $AdS_5$ have been extensively studied on a case by case basis in [@yjor; @ysum; @vtsum; @ysugra1; @Hoare:2016hwh; @ysugra2]. When built out of bosonic generators the jordanian $r$-matrices do not satisfy the unimodularity property [@Borsato:2016ose]. Indeed, the backgrounds corresponding to the jordanian deformations of the supercoset model [@yjor; @dmv] do not solve the supergravity equations [@ysugra1; @Hoare:2016hwh; @ysugra2], solving instead the generalized equations of [@Arutyunov:2015mqj; @Wulff:2016tju].
In the following we will compare the YB deformations arising from jordanian $r$-matrices to the NAD of $AdS_5$ with respect to the corresponding quasi-Frobenius subalgebra itself (i.e. without central extension). A possible reason why we do not need to consider central extensions is that, unlike for the unimodular $r$-matrices, the central extension of interest turns out to be trivial. Indeed, let us consider the simplest case of the $r$-matrix $D\wedge
P_0$. The two generators here have the commutation relation $[D,P_0] = P_0$. If we try to centrally extend this 2d algebra we get $[D,P_0] = P_0 + Z$, but now defining $P'_0 = P_0 + Z$ we see that this extension is trivial. For the extended $r$-matrix $D \wedge P_0 + M_{01} \wedge P_1 + M_{+2} \wedge P_2 +
M_{+3} \wedge P_3$ we consider the central extension $[D,P_0] = P_0 + Z$, $[M_{01},P_1] = P_0 + Z$, $[M_{+2},P_2] = P_+ + Z$ and $[M_{+3},P_3] = P_+ +
Z$. Here we only consider a single extension as there is only a single free parameter scaling the whole $r$-matrix. Again by shifting $P_0$ we see that this extension is trivial. Similar statements hold for the remaining examples that we consider.
As for the abelian $r$-matrices and unimodular non-abelian $r$-matrices in sections \[secabe\] and \[secuni\] we will again provide exhaustive evidence for the equivalence of the YB and NAD constructions. In this section we will discuss two special cases in detail, while in Appendix \[appjor\] we will summarize the key information for twenty additional examples.
[**Example 1:**]{} The first example is the case of the rank 2 jordanian $r$-matrix $$\label{jorp0rmat}
r = \eta \, D \wedge P_0 \ ,$$ where $D$ is the dilatation operator. Parametrizing the group-valued field $f$ as in [(\[f\])]{} with $$\la{355}
x_1 = r \cos \theta \ , \qquad x_2 = r \sin\theta \, \cos \phi \ , \qquad x_3 = r \sin\theta \, \sin \phi \ ,$$ we find the following metric and $B$-field of the corresponding YB deformed model $$\begin{split}\label{jorp0back}
ds^2 & = \frac{-z^4 dx_0^2 + z^2(z^2 - \eta^2) dr^2 + 2 \eta^2 z r dz dr + (z^4-\eta^2 r^2) dz^2}{z^2(z^4 -\eta^2 z^2 - \eta^2 r^2)} + \frac{r^2}{z^2}(d\theta^2 + \sin^2\theta \, d\phi^2) \ ,
\\
B & = \frac{\eta r dr \wedge dx_0 + \eta z dz \wedge dx_0} {z^4 - \eta^2 z^2 - \eta^2 r^2} \ .
\end{split}$$ In [@ysugra2] it was observed that this background can be also obtained from the $AdS_5$ via a generalized “TsT” transformation in which the shift is replaced with a more complicated non-linear field redefinition.[[^22]]{}
It turns out that this background can be derived in one step by applying the NAD transformation to the $AdS_5$ with respect to the non-semisimple subalgebra $\mathfrak{h} = \{D, P_0\}$ (with the dual algebra being given by $\bar{\mathfrak{h}} = \{D,K_0\}$). We first use the local $H$ gauge symmetry of to fix $f'$ as $$\label{para1}
f'= \exp\big[\frac{1}{z}(x_1 \, P_1 + x_2 \, P_2 + x_3 \, P_3) \big]\ ,$$ where $x_i$ are again given by [(\[355\])]{}. We also parametrize the Lagrange multiplier $v$ in terms of the coordinates $x_0$ and $z$ as $$\label{rr0}
v=\eta^{-1} \big[- x_0 D + \ha {z} K_0\big] \ .$$ Substituting this into and integrating out the gauge field, we find that the NAD of the $AdS_5$ with respect to $\mathfrak{h} = \{D,P_0\}$ gives the based on the metric and $B$-field .[[^23]]{}
The fact that the algebra $\mathfrak{h}$ is not unimodular (i.e. $n_a$ in [(\[4\])]{} is non-zero) and hence the NAD transformation does not preserve Weyl invariance is then recognized as the reason why the full YB deformed background found in [@ysugra2] solves only the generalized (scale-invariance) but not standard supergravity (Weyl-invariance) equations.
From the analysis in section \[ssecnad\] we expect that if we T-dualize the full YB deformed background in the direction of $v$ corresponding to $n_\a = f^{\gamma}_{\ \gamma \alpha}$ we find a supergravity solution with a linear dilaton in that direction. From the Lagrange multiplier this implies that the dilaton should be linear in $x_0$ (i.e. the coordinate associated to the dilatation generator $D$). This is in agreement with the results of [@ysugra2].
[**Example 2:**]{} As our final examples let us consider the following pair of extended jordanian $r$-matrices [@Hoare:2016hwh] $$\begin{aligned}
\label{jorext1rmat}
r & = \eta \, \big(D \wedge P_0 + M_{01} \wedge P_1 + M_{+2} \wedge P_2 + M_{+3} \wedge P_3\big) \ ,
\\ \label{jorext2rmat}
r & = \eta \, \big(D \wedge P_1 + M_{01} \wedge P_0 + M_{+2} \wedge P_2 + M_{+3} \wedge P_3\big) \ ,\end{aligned}$$ which are closely related by an analytic continuation. In both cases we fix the group-valued field $f$ as in [(\[f\])]{} with $x_2$ and $x_3$ written in terms of polar coordinates as in [(\[341\])]{}, i.e. $$f= \exp[-x_0 P_0 + x_1 P_1 + r (\cos \theta \, P_2 + \sin \theta \, P_3)] \exp[\log z \, D] \ .$$ Then the metrics and $B$-fields of the YB deformed model are found to be $$\begin{split}\label{jorext1back}
ds^2 & = \frac{-dx_0^2 + dz^2}{z^2 - \eta^2} + \frac{z^2(dx_1^2 + dr^2)}{z^4 + \eta^2 r^2} + \frac{r^2}{z^2}d\theta^2 \ , \qquad
B = -\frac{\eta dx_0 \wedge dz}{z(z^2 - \eta^2)} + \frac{\eta r dx_1 \wedge dr}{z^4 + \eta^2 r^2} \ ,
\end{split}$$ and $$\begin{split}\label{jorext2back}
ds^2 & = \frac{dx_1^2 + dz^2}{z^2 + \eta^2} + \frac{z^2( - dx_0^2 + dr^2)}{z^4 - \eta^2 r^2} + \frac{r^2}{z^2}d\theta^2 \ , \qquad
B = - \frac{\eta dx_1 \wedge dz}{z(z^2 + \eta^2)} + \frac{\eta r dx_0 \wedge dr}{z^4 - \eta^2 r^2} \ ,
\end{split}$$ respectively.
These two $r$-matrices and are built from the generators of the same algebra = {D,M\_[01]{},M\_[+2]{},M\_[+3]{},P\_0,P\_1,P\_2,P\_3} , with the dual algebra being $\bar{\mathfrak{h}} = \{D,M_{01},M_{-2},M_{-3},K_0,K_1,K_2,K_3\} .$ Let us then consider the NAD of the $AdS_5$ with respect to $\mathfrak{h}$. After partially using the local $H$ gauge symmetry to completely fix $f'$ in [(\[nadlag\])]{}, i.e. $$f' = 1 \ ,$$ there are still three remaining gauge symmetries, corresponding to the generators $M_{01}$, $M_{+2}$, $M_{+3}$. The Lagrange multiplier $v \in \bar{\mathfrak{h}}$ contains a piece $v_0 K_0 + v_1 K_1 + \ldots$. For $v_0^2 > v_1^2$ we can use the remaining gauge symmetries to fix $$\label{regime1}
v =\eta^{-1} \big[ - x_0 D + \ha {z} K_0 + {x_1} M_{01} +\ha r (\cos \theta \, M_{-2} + \sin \theta \, M_{-3})\big] \ ,$$ while for $v_1^2 > v_0^2$ we fix $$\label{regime2}
v =\eta^{-1} \big[ {x_1} D + \ha {z} K_1 - x_0 M_{01} +\ha r (\cos \theta \, M_{-2} + \sin \theta \, M_{-3})\big] \ .$$ Substituting $v$ in into and integrating out the gauge field we find the based on the metric and $B$-field , while using $v$ in we recover the based on . Thus the two YB deformations correspond to different “patches” of the NAD of $AdS_5$ model with respect to $\mathfrak{h}$.
To conclude, let us observe that for the jordanian examples the parameter $\eta$, which appears in the $r$-matrix as an overall factor, also enters the dual model in a similar way, with the Lagrange multiplier $v \sim \eta^{-1}$ (cf. , and ). This is consistent as using the automorphism $P_\m \to \lambda P_\m$, $K_\m \to \lambda^{-1} K_\m$ of $\mathfrak{so}(2,4)$ we can set $\eta = 1$ in the $r$-matrix, while the same can be done in the NAD model by rescaling $v$. Similar observations also hold for the examples in Appendix \[appjor\] where in some cases we also use the inner automorphism generated by the $SO(1,1)$ subgroup of the $SO(1,3)$ Lorentz algebra.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank R. Borsato, S. van Tongeren and L. Wulff for discussions of related questions and useful comments on the draft. We also thank the organizers of the conference and focus program on “Integrability in Gauge and String Theory” at Humboldt University Berlin 23/08-02/09/2016 for kind hospitality. The work of BH is partially supported by grant no. 615203 from the European Research Council under the FP7. The work of AAT was supported by the ERC Advanced grant no. 290456, the STFC Consolidated grant ST/L00044X/1 and by the Russian Science Foundation grant 14-42-00047.
Further examples corresponding to jordanian -matrices {#appjor}
=====================================================
In this Appendix we present a number of further examples of the relation between YB deformed models based on jordanian $r$-matrices and non-abelian duals of the $AdS_5$ $\s$-model. This list (including the cases discussed already in section \[secjor\]), while not a classification, covers the majority of the $r$-matrices considered in [@yjor; @ysum; @vtsum; @ysugra1; @Hoare:2016hwh; @ysugra2] up to certain automorphisms of $\mathfrak{so}(2,4)$ including those based on $P_\m
\leftrightarrow K_\m$ and $+ \leftrightarrow -$.
For each case we will give the $r$-matrix and the parametrization of the group element $f$ for the YB deformed model . We will then provide the corresponding data for the NAD model : the algebra $\mathfrak{h}$ in which we dualize, its dual $\bar{\mathfrak{h}}$ and the parametrizations of the group element $f'$ and of the Lagrange multiplier $v \in \bar{\mathfrak{h}}$. As was mentioned in section \[secjor\], for the jordanian $r$-matrices we do not need to consider central extensions of $\mathfrak{h}$.
For reasons of brevity we will not present the explicit forms of the metrics and $B$-fields, but just state that in all cases one finds the complete agreement between the YB deformed and NAD transformed $AdS_5$ model.
Note that the pair of examples 1 and 2 below follow a similar pattern as the extended jordanian $r$-matrices in example 2 in section \[secjor\]: the two YB deformed models both correspond to the same NAD model. The difference on the NAD side appears in the gauge fixing of the Lagrange multiplier $v$. The same is true also for the pair of examples 3 and 4.
[-0.57cm]{}[-1.43cm]{}
1\. r & = ( D P\_0 + M\_[01]{} P\_1 + M\_[+2]{} P\_2) , & f & = , &\
& = {D, P\_0, M\_[01]{}, P\_1, M\_[+2]{}, P\_2} , & f’ & = , &\
| & = {D, K\_0, M\_[01]{}, K\_1, M\_[-2]{}, K\_2} , & v & = -D + K\_0 + M\_[01]{} + M\_[-2]{} . &
2\. r & = ( D P\_1 + M\_[01]{} P\_0 + M\_[+2]{} P\_2) , & f & = , &\
& = {D, P\_1, M\_[01]{}, P\_0, M\_[+2]{}, P\_2} , & f’ & = , &\
| & = {D, K\_1, M\_[01]{}, K\_0, M\_[-2]{}, K\_2} , & v & = D + K\_1 - M\_[01]{} + M\_[-2]{} . &
3\. r & = ( D P\_0 + M\_[01]{} P\_1) , & f & = , &\
& = {D, P\_0, M\_[01]{}, P\_1} , & f’ & = , &\
| & = {D, K\_0, M\_[01]{}, K\_1} , & v & = - D + K\_0 + M\_[01]{} . &
4\. r & = ( D P\_1 + M\_[01]{} P\_0) , & f & = , &\
& = {D, P\_1, M\_[01]{}, P\_0 } , & f’ & = , &\
| & = {D, K\_1, M\_[01]{}, K\_0} , & v & = D + K\_1 - M\_[01]{} . &
5\. r & = D P\_1 , & f & = , &\
& = {D, P\_1 } , & f’ & = ,\
| & = {D, K\_1} , & v & = D + K\_1 . &
6\. r & = ( D P\_2 + M\_[23]{} P\_3) , & f & = , &\
& = {D, P\_2, M\_[23]{}, P\_3} , & f’ & = , &\
| & = {D, K\_2, M\_[23]{}, K\_3} , & v & = D + K\_2 + M\_[23]{} . &
7\. r & = ( M\_[01]{} M\_[+2]{} + M\_[23]{} M\_[+3]{}) , & f & = , &\
& = {M\_[01]{}, M\_[+2]{}, M\_[23]{}, M\_[+3]{}} , & f’ & = , &\
| & = {M\_[01]{}, M\_[-2]{}, M\_[23]{}, M\_[-3]{}} , & v & = M\_[01]{} + M\_[-2]{} + M\_[23]{} . &
8\. r & = ( (D + M\_[01]{}) P\_+ + 2M\_[+2]{} P\_2 + 2M\_[+3]{} P\_3) , & f & = , &\
& = {D+ M\_[01]{} ,P\_+ , M\_[+2]{}, P\_2 , M\_[+3]{}, P\_3} , & f’ & = , &\
| & = {D +M\_[01]{}, K\_-, M\_[-2]{}, K\_2, M\_[-3]{}, K\_3} , & v & = - (D + M\_[01]{}) + K\_- + ( M\_[-2]{} + M\_[-3]{}) . &
9\. r & = ( (D + M\_[01]{}) P\_+ + 2M\_[+2]{} P\_2) , & f & = , &\
& = {D+ M\_[01]{} ,P\_+ , M\_[+2]{}, P\_2 } , & f’ & = , &\
| & = {D +M\_[01]{}, K\_-, M\_[-2]{}, K\_2} , & v & = - (D + M\_[01]{}) + K\_- + M\_[-2]{} . &
10\. r & = (D + M\_[01]{}) P\_+ , & f & = , &\
& = {D+ M\_[01]{} ,P\_+ } , & f’ & = , &\
| & = {D +M\_[01]{}, K\_-} , & v & = - (D + M\_[01]{}) + K\_- . &
11\. r & = ( D P\_+ + M\_[+2]{} P\_2 + M\_[+3]{} P\_3) , & f & = , &\
& = {D ,P\_+ , M\_[+2]{}, P\_2 , M\_[+3]{}, P\_3} , & f’ & = , &\
| & = {D , K\_-, M\_[-2]{}, K\_2, M\_[-3]{}, K\_3} , & v & = - D + K\_- + ( M\_[-2]{} + M\_[-3]{}) . &
12\. r & = ( D P\_+ + M\_[+2]{} P\_2) , & f & = , &\
& = {D ,P\_+ , M\_[+2]{}, P\_2 } , & f’ & = , &\
| & = {D, K\_-, M\_[-2]{}, K\_2} , & v & = - D + K\_- + M\_[-2]{} . &
13\. r & = D P\_+ , & f & = , &\
& = {D ,P\_+ } , & f’ & = , &\
| & = {D , K\_-} , & v & = - D+ K\_- . &
14\. r & = ( M\_[01]{} P\_+ + M\_[+2]{} P\_2 + M\_[+3]{} P\_3) , & f & = , &\
& = {M\_[01]{} ,P\_+ , M\_[+2]{}, P\_2 , M\_[+3]{}, P\_3} , & f’ & = , &\
| & = {M\_[01]{} , K\_-, M\_[-2]{}, K\_2, M\_[-3]{}, K\_3} , & v & = - M\_[01]{}+ K\_- + ( M\_[-2]{} + M\_[-3]{}) . &
15\. r & = ( M\_[01]{} P\_+ + M\_[+2]{} P\_2) , & f & = , &\
& = {M\_[01]{} ,P\_+ , M\_[+2]{}, P\_2 } , & f’ & = , &\
| & = {M\_[01]{}, K\_-, M\_[-2]{}, K\_2} , & v & = - M\_[01]{} + K\_- + M\_[-2]{} . &
16\. r & = M\_[01]{} P\_+ , & f & = , &\
& = {M\_[01]{} ,P\_+ } , & f’ & = , &\
| & = {M\_[01]{} , K\_-} , & v & = - M\_[01]{}+ K\_- . &
17\. r & = M\_[01]{} (P\_+ + K\_+) , & f & = , &\
& = {M\_[01]{} ,P\_+ + K\_+ } , & f’ & = , &\
| & = {M\_[01]{} , P\_- + K\_-} , & v & = (P\_- + K\_-) . &
18\. r & = M\_[01]{} (P\_+ - K\_+) , & f & = , &\
& = {M\_[01]{} ,P\_+ - K\_+ } , & f’ & = , &\
| & = {M\_[01]{} , P\_- - K\_-} , & v & = - (P\_- - K\_-) . &
19\. r & = (D-P\_1) P\_0 , & f & = , &\
& = {D-P\_1,P\_0} , & f’ & = , &\
| & = {D - K\_1, K\_0} , & v & = - (D-K\_1) + K\_0 . &
20\. r & = (D + M\_[01]{} - M\_[23]{}) P\_+ , & f & = , &\
& = {D+ M\_[01]{} - M\_[23]{} ,P\_+ } , & f’ & = , &\
| & = {D +M\_[01]{} + M\_[23]{}, K\_-} , & v & = - (D + M\_[01]{} + M\_[23]{}) + K\_- . &
[30]{}
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[^1]: bhoare@ethz.ch
[^2]: Also at Lebedev Institute, Moscow. tseytlin@imperial.ac.uk
[^3]: One may also consider $c=-1$. However, no solutions exist for the real form of the superalgebra $\mathfrak{psu}(2,2|4)$, the symmetry algebra of the $AdS_5 \times S^5$ superstring.
[^4]: A potentially important feature of the NAD transform compared to the YB deformation is that it may be applied also to the less symmetric (and non-integrable) D3-brane background away from the near-horizon limit. As in the case of the TsT duality example in this may help to understand the dual gauge theory interpretation of the resulting model.
[^5]: For the bosonic YB deformed model such a first-order action was given in [@HT1].
[^6]: A prescription for the NAD transformation of the RR fields has already been proposed (by analogy with T-duality) in [@sfth2]. It has also been checked on examples that for simple groups $H$, like $SU(2)$, NAD indeed maps supergravity solutions to supergravity solutions and that the transformation is a symmetry of the supersymmetry variations of supergravity [@Lozano:2011kb].
[^7]: Here we are not considering the possibility of dualizing in fermionic directions.
[^8]: For example, the case of a coset space may be included by taking $E_{ab}$ to be degenerate.
[^9]: Equivalently, one may say that in order to preserve the local interpretation of the dual model and thus its Weyl invariance one should not dualize in the $n_a$-direction of the algebra in the first place.
[^10]: If $H$ is non-abelian and simple (such that $n_a=0$) then the dual background has no isometries. If $n_a \neq 0$ then the dual background will have at least one abelian isometry.
[^11]: Indeed, the $\g=\infty$ limit of [(\[8\])]{} is the double T-dual background. This last term in [(\[10\])]{} which is absent in this limit, is analogous to “current-current” deformation related to $O(2,2)$ duality (cf. [@Kiritsis:1993ju]).
[^12]: Compared to section \[secnad\] we now use Greek indices $\a,\b,\ldots$ for the subalgebra $\mathfrak{h}$ in which we will dualize and Latin indices $a,b,\ldots$ for the algebra $\mathfrak{f}$.
[^13]: Due to the compact nature of the isometry algebra $\mathfrak{so}(6)$ of $S^5$, it only admits abelian solutions of the cYBE.
[^14]: One may be concerned that the action of $H$ on $v$ does not preserve $v \in \bar{\mathfrak{h}}$ if $\mathfrak{h} \neq \bar{\mathfrak{h}}$. Indeed, this may be the case, but one can always write the new terms as the sum of a part valued in $\bar{\mathfrak{h}}$ and a part that drops out in the bilinear form. To show this we first write an algebra element $X \in \mathfrak{f}$ as $X = X^\a \bar
e_\a + Y^{I} t_{I}$ where $t_{I}$ is some extension of $\bar e_\a$ to a basis of $\mathfrak{f}$. Then if $\Tr[t_{I} e_{\a}] = c \neq 0$ for some $I$ and $\a$, it follows from that $\bar e_\a = c t_{I} + \ldots$. We can then define a new generator $\tilde t_I = t_I - c \bar e_\a / \sum_{a=1}^{\dim
F} \Tr[T_a e_\a]^2$, which does satisfy $\Tr[\tilde t_I e_\a] = 0$. Replacing $t_I \to \tilde t_I$ in the basis of $\mathfrak{f}$ still gives a basis of $\mathfrak{f}$. Therefore, applying this process iteratively, we can construct a basis $(\bar e_\a$, $\tilde t_I)$ with the desired property $\Tr[\tilde t_I
e_\a] = 0$.
[^15]: One may attempt to promote the Lagrange multiplier terms to a gauged WZW model as in [@sfet] in order to construct a $\lambda$-model type theory with an additional deformation parameter based on . It would be interesting to see if this preserves integrability, or if one also needs to modify the first term. There may also be subtleties for centrally-extended and non-semi-simple algebras.
[^16]: In what follows we will always ignore total derivative terms when comparing the $B$-fields.
[^17]: In [@Borsato:2016ose] the $r$-matrices are grouped into four classes, with the non-vanishing commutation relations $[e_1, e_3] = e_4$, $[e_1, e_4] = e_3$ determining the additional class. Here we note that these commutation relations are related to those of class 2 by analytic continuation and hence for our purposes do not need to be considered independently.\[footcomrel\]
[^18]: One can check that the Jacobi identity is satisfied and this central extension is consistent for all three classes.
[^19]: This is the fourth example in [@Borsato:2016ose] for which a TsT interpretation has thus far not been found.
[^20]: The central extension is given by $[M_{01},M_{23}] = Z_1$, $[P_0,P_1] = Z_2$, $[P_2,P_3] = Z_3$ as follows from the structure of the $r$-matrix . Again, one can check that the Jacobi identity is satisfied.
[^21]: Writing $v = v_0 K_0 + v_1 K_1 + \ldots$, this is a valid gauge fixing in the “patch” in which $v_1^2 > v_0^2$.
[^22]: The coordinate transformation is required is to make the dilatation symmetry a linear isometry.
[^23]: If we instead consider the centrally-extended algebra $[D,P_0] = P_0 + Z$, which, as discussed above, amounts to a trivial extension, (with $A_\pm = A_{1\pm}
D + A_{2\pm} P_0$ and the explicit form of the central extension term as in ) we again recover the background after taking $f'$ as in and $v = \eta^{-1} [-x_0 D + \ha (z-1) K_0] $. This simple shift in the Lagrange multiplier is directly correlated to the shift in $P_0$ required to reach the trivially-extended algebra. Similar statements should hold for the remaining examples.
|
---
abstract: 'If $M$ is an $R$-module, we study the submodules $K\leq M$ with the property that $K$ is invariant with respect to all monomorphisms $K\rightarrow M$. Such submodules are called *strictly invariant*. For the case of $\mathbb{Z}$-modules (i.e. Abelian groups) we prove that in many situations these submodules are invariant with respect to all homomorphisms $K\rightarrow M$, submodules which were called *strongly invariant*.'
author:
- 'Simion Breaz, Grigore Călugăreanu and Andrey Chekhlov [^1]'
title: Strictly invariant submodules
---
Introduction
============
Let $K$ be a submodule of a module $M$, and let $\mathcal{X}$ be a class of homomorphisms such that $f(K)$ makes sense for all $f\in \mathcal{X}$. We say that $K$ is invariant with respect to the class $\mathcal{X}$ if the inclusion $f(K)\leq K$ holds for all $f\in \mathcal{X}$. For instance, $K$ is fully invariant, injective invariant, respectively characteristic, if $K$ is invariant with respect to that class $\mathcal{X}$, where $\mathcal{X}$ is $\mathrm{End}(M)$, $\mathrm{Mon}(M)$ (i.e. the set of all monic endomorphisms of $M$), respectively $\mathrm{Aut}(M)$. In module theory there are important classes of modules which can be characterized by the invariance of some submodules with respect to some classes of homomorphisms. For instance, a module $M$ is quasi-injective (pseudo-injective) if and only if it is fully invariant (characteristic) as a submodule of the injective hull of $M$, cf. [@Fa64] (respectively [@ESS]). We refer to [As-et al]{} for some general statements about modules which are invariant with respect to classes of endomorphisms of injective hulls.
Injective invariant subgroups of Abelian groups were termed S-characteristic and left invariant, respectively, in [@bae] or [@gol]. These were used in [@bre] for the study of (co)hopfian modules.
The submodules $K$ which are invariant with respect to $\mathcal{X}=\mathrm{Hom}(K,M)$ are called *strongly invariant*, and these are studied in [@cal], with a special attention to the case of Abelian groups. We will say that the submodule $K$ of $M$ is *strictly invariant* if it is invariant with respect to the set $\mathcal{X}=\mathrm{Mon}(K,M)$ of all monomorphisms $K\rightarrow M$. Clearly, strongly invariant submodules are strictly invariant and strictly invariant submodules are characteristic.
For reader’s convenience we mention that the same notions are discussed in the case of non-Abelian groups in [@sw], where strongly invariant (normal) subgroups are termed *homomorph containing* and strictly invariant subgroups are termed *isomorph containing*.
In the next section we study general properties of strictly invariant submodules. Among these it is proved that the set of all strictly invariant submodules of a module is a complete lattice with respect to the inclusion relation, Proposition \[lattice\]. Moreover, if the additive group of the module has no elements of order $2$, then every strictly invariant submodule is invariant with respect to idempotent endomorphisms, Proposition \[w2\].
In the third section we study strictly invariant subgroups of Abelian groups. We mention that in Example \[non-strongly\] it is proved that there exist strictly invariant submodules which are not strongly invariant. However, we were not able to construct such an example for the case of Abelian groups. Therefore we are focussed on finding conditions (as general as possible) on the group and/or on the subgroup, which imply that the strictly invariant subgroups are strongly invariant, in order to argue the enunciation of the following conjecture: *every strictly invariant subgroup of an Abelian group is strongly invariant*. Very large classes of Abelian groups are shown to support this conjecture.
In this context we mention that in the case of Abelian groups, all pseudo-injective groups are quasi-injective, [@js67]. A similar situation occurred in [@bre]: denoting by $\mathcal{Q}(G)$, the family of all subgroups $N\leq G$ such that every homomorphism $N\longrightarrow G$ extends to an endomorphism of $G$ and by $\mathcal{P}(G)$, the family of all subgroups $N\leq G$ such that every injective homomorphism $N\longrightarrow
G$ extends to an endomorphism of $G$, though we strongly suspect that $\mathcal{Q}(G)=\mathcal{P}(G)$ for Abelian groups, the proof which shows that *finitely generated subgroups from* $\mathcal{P}(G)$* are also in* $\mathcal{Q}(G)$ was already very hard (and the general question is still open).
Notice that *for noncommutative groups* it is easy to give examples of strictly invariant subgroups which are not strongly invariant: the dihedral 2-groups of order at least 8 and the infinite dihedral group. The order 8 group $D_{8}$ has *a unique cyclic maximal subgroup* $H$ (of order 4) which clearly is strictly but not strongly invariant in $D_{8}$. Indeed, there are other two order 4 subgroups which are Klein, and all the other order 2 subgroups are (clearly) cyclic.
We finally mention that, starting from [@bel], Dikranian, Giordano Bruno, Goldsmith, Salce, Virili and Zanardo defined and studied fully inert subgroups of Abelian groups in [@dar] - [@dik2], [@gol1], [gol2]{}. Replacing fully invariant subgroups by strongly invariant subgroups, led the first and second authors to study the strongly inert subgroups of Abelian groups in [@bre1]. A natural continuation of all these (kindly suggested by the referee) would be to study the *strictly inert subgroups* and compare these with strongly inert subgroups. We postpone this to a forthcoming paper.
All modules we consider are over a unital ring denoted $R$. $\mathbb{F}_{2}$ denotes the field with two elements and $\mathbb{Z}(2)$ the Abelian group with two elements. For other notations for Abelian groups we refer to [fuc1]{} and [@fuc2].
General properties
==================
Using the above definitions we obtain the following chart $$\begin{array}{ccccc}
\text{\textrm{strongly-invariant}} & \mathrm{\overset{(\ast )}{\Longrightarrow }} & \text{\textrm{fully-invariant}} & & \\
\mathrm{\Downarrow (4)} & & \mathrm{\Downarrow (1)} & & \\
\text{\textrm{strictly-invariant}} & \mathrm{\overset{(1)}{\Longrightarrow }}
& \text{\textrm{injective-invariant}} & \mathrm{\overset{(3)}{\Longrightarrow }} & \text{\textrm{characteristic}}\end{array}$$
The following examples (the numbering corresponds to these) show that all reversed implications fail ((2) presents fully invariant subgroup which is not strictly invariant; as for (\*), such examples are given in [@cal]).
First, *an injective invariant subgroup which is not strictly invariant*.
[Let $G=\left\langle a_{1}\right\rangle \oplus \left\langle
a_{2}\right\rangle \oplus \left\langle a_{3}\right\rangle $ with $o(a_{i})=2^{i}$, and $H=\langle 2a_{2}\rangle \oplus \langle
a_{1}+2a_{3}\rangle $. Since $G$ is finite, characteristic and injective invariant subgroups coincide (because injective functions from $G$ to $G$are bijective). It is proved in [@fuc2 p. 9] that $H$ is characteristic, and it is easy to see that $H$ is not strictly invariant (e.g. take $2a_{2}\longmapsto a_{1}$ and $a_{1}+2a_{3}\longmapsto a_{2}$). Moreover, it is not fully invariant. ]{}
[If $p$ is a prime, the subgroup $p\mathbb{Z}$ of $\mathbb{Z}$ is not strictly invariant but it is fully invariant. ]{}
Other examples may be found in [@cal] or [@che1].
Next, a *characteristic subgroup which is not injective invariant.*
[By [@Ar Theorem 2.14], for every prime $p$ there exists a torsion free Abelian group $G$ of rank $2$ with endomorphism ring isomorphic to $R=\mathbb{Z}[\sqrt{-p}]$. Since the units of $R$ are $\pm 1$, it follows that all subgroups are characteristic. Moreover, $\mathbb{Q}\otimes R$ is a division ring, hence all non-zero endomorphisms of $G$ are injective. Let $x\in G$ be a non-zero element, and let $f$ and $g$ be two endomorphisms of $G$ which are $\mathbb{Q}$-independent in $\mathbb{Q}\otimes R$. Suppose $f(x)$ and $g(x)$ are not $\mathbb{Z}$-independent. Then there exist two non-zero integers $m$ and $n$ such that $mf(x)=ng(x)$ and so $mf-ng$ is not injective. Hence $mf=ng$, a contradiction. Thus for every non-zero element $x$ of $G$, the subgroup $Rx$ has to be of rank $2$, hence the subgroup $\langle x\rangle $ is not injective invariant. ]{}
Next, we present an example of *strictly invariant submodule which is not strongly invariant*.
\[non-strongly\] [Let $R$ be a ring such that there exist non-isomorphic simple modules $S_{1}$, $S_{2}$ and $T$ such that ]{}
**1.** [the endomorphism rings of these modules are isomorphic to $\mathbb{Z}_{2}$; ]{}
**2.** [there are non-splitting exact sequences $$0\rightarrow S_{1}\rightarrow K\rightarrow T\rightarrow 0,\text{ and }0\rightarrow S_{2}\rightarrow K\rightarrow T\rightarrow 0.$$By using the pullback diagram $$\xymatrix{ & & 0\ar[d] & 0\ar[d] & \\ & & S_2\ar[d]\ar@{=}[r] & S_2\ar[d] &
\\ 0\ar[r] & S_1\ar[r]\ar@{=}[d] & M\ar[r]^{\psi}\ar[d]^{\varphi} &
L\ar[r]\ar[d] & 0\\ 0\ar[r] & S_1\ar[r] & K\ar[r]\ar[d] & T\ar[r]\ar[d] &
0\\ & & 0 & 0 & }$$we construct a module $M$ such that the set of its submodules is $\{0,S_{1},S_{2},S_{1}\oplus S_{2},M\}$ and $M/S_{1}\oplus S_{2}\cong T$ (this module is also used in [@js Lemma 2]). ]{}
[Let $\varphi :M\rightarrow K$ be a non-zero homomorphism. Then $\varphi (S_{2})=0$. If $\varphi (S_{1})=0$ then $\varphi $ induces a non-zero homomorphism $T\rightarrow M$, which is impossible. We obtain that $\varphi (S_{1})\neq 0$, and it follows that $\varphi |_{S_{1}}$ is the inclusion map. Therefore, if $\varphi _{1},\varphi _{2}:M\rightarrow K$ are two non-zero homomorphisms then the restriction of these homomorphisms to $S_{1}\oplus S_{2}$ coincide. It follows that $(\varphi _{1}-\varphi
_{2})(S_{1}\oplus S_{2})=0$, hence $\varphi _{1}=\varphi _{2}$. This way $\mathrm{Hom}(M,K)=\{0,\varphi \}$ and in the same way we obtain $\mathrm{Hom}(M,L)=\{0,\mathrm{\psi }\}$. ]{}
[It is easy to see that if $\rho :M\rightarrow K\times L$ is the homomorphism induced by $\varphi $ and $\psi $ then $\rho $ is a monomorphism. Since $\mathrm{Hom}(M,K\times L)\cong \mathrm{Hom}(M,K)\times
\mathrm{Hom}(M,L)$, it follows that $\rho $ is the only monomorphism from $M$ into $K\times L$. We conclude that $\rho (M)$ is strictly invariant. Since there exists an epimorphism M]{}$\mathrm{\rightarrow }$ [K, and $K\times 0$ is not contained in $\rho (M)$, it follows that $\rho (M)$ is not strongly invariant. ]{}
For reader’s convenience we recall the concrete example described in [@As-et; @al Example 3.1].
Let $R=\left(
\begin{array}{ccc}
\mathbb{F}_{2} & \mathbb{F}_{2} & \mathbb{F}_{2} \\
0 & \mathbb{F}_{2} & 0 \\
0 & 0 & \mathbb{F}_{2}\end{array}\right) $. Then the right $R$-module $M=\left(
\begin{array}{ccc}
\mathbb{F}_{2} & \mathbb{F}_{2} & \mathbb{F}_{2} \\
0 & 0 & 0 \\
0 & 0 & 0\end{array}\right) $ satisfies the required conditions: the simple submodules are $S_{1}=\left(
\begin{array}{ccc}
0 & \mathbb{F}_{2} & 0 \\
0 & 0 & 0 \\
0 & 0 & 0\end{array}\right) $ and $S_{2}=\left(
\begin{array}{ccc}
0 & 0 & \mathbb{F}_{2} \\
0 & 0 & 0 \\
0 & 0 & 0\end{array}\right) ,$ and it is easy to see that the simple R-module $M/(S_{1}\oplus
S_{2})$ is not isomorphic to $S_{1}$ or $S_{2}$.
In what follows we study the basic properties of strictly invariant submodules. First observe that *strict invariance is not a transitive property.*
\[non-transitive\] [In order to show this, we observe that if $S$ and $T$ are non-isomorphic simple modules, $0\rightarrow S\rightarrow
K\rightarrow T\rightarrow 0$ is a non-splitting exact sequence and $M=K\oplus S$, then $S\oplus 0$ is strictly invariant in $K\oplus 0$, $K\oplus 0$ is strictly invariant in $M$, but $S\oplus 0$ is not strictly invariant in $M$. ]{}
[For the case of Abelian groups, consider $G=H\oplus L=\mathbb{Z}(2^{\infty })\oplus \mathbb{Z}(2)$ with $K=\mathbb{Z}(2)<H$. Then $K=S(H)$, the socle, is strongly and so strictly invariant in $H$. It is not strictly invariant in $G$, since the composition of the isomorphism $K\cong L$ with the injection $\iota _{L}:L\longrightarrow G$ does not map $K$ into $K$. Finally, $H$ is a fully invariant direct summand - as divisible part of $G$ - and so strongly and strictly invariant in $G$. ]{}
Next, *if* $H\leq L\leq M$* and* $H$* is strictly invariant in* $M$* then* $L$* might not be strictly invariant in* $M$.
[It suffices to take $K$ as in Example \[non-transitive\], $M=K\oplus K$, $H=S\oplus S$ and $K=K\oplus S$. ]{}
Let $M$ be a module and let $H\leq K$ be submodules of $M$. If $H$ is strictly invariant in $M$ and $K/H$ is strongly invariant in $M/H$, then $K$ is strictly invariant in $M$.
Let $f:K\longrightarrow M$ be an injective homomorphism. Since $H$ is strictly invariant in $M$, the map $\widetilde{f}:K/H\longrightarrow M/H$, $\widetilde{f}(k+H)=f(k)+H$ is well-defined and a homomorphism. Since $K/H$ is strongly invariant in $G/H$, $\widetilde{f}(K/H)\subseteq K/H$ which shows that $f(K)\subseteq K$.
Notice that we cannot weaken the hypothesis $K/H$ is strongly invariant in $M/H$ only to strictly invariant, as the example below shows .
*If* $H\leq K\leq M$* and* $K$* is strictly invariant in* $M $* then* $K/H$* might not be strictly invariant in* $M/H$*.*
[For instance, if $M=H\oplus K=\mathbb{Z}_{2}\oplus \mathbb{Z}_{4}$, the socle $H+2K$ is strictly invariant in $M$ but $(H+2K)/2K=\mathbb{Z}_{2}$ is not strictly invariant in $M/2K=\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$. ]{}
Further, *the intersection of a family of strictly invariant submodules is not (in general) strictly invariant.*
\[intersection\] [In order to prove this, we use the same module as in Example \[non-transitive\]. It is easy to see that the socle $S\oplus S$ of $M$ and $K$ are strictly invariant submodules of $M$, but $S\oplus 0=(S\oplus S)\cap K$ is not strictly invariant. ]{}
[For the case of Abelian groups we can consider $G=D\oplus R$, where $D$ is a divisible $p$-group and $R$ is a reduced $p$-group. Then $D$ and $G[p]=D[p]\oplus R[p]$ are strongly invariant subgroups; however, the subgroup $D\cap G[p]=D[p]$ is not strictly invariant in $G$ (this covers the missing example in [@cal], where an example of two strongly invariant subgroups with not strongly invariant intersection was not given). ]{}
[Intersections of strictly invariant subgroups may not be strictly invariant also in torsion-free groups. To see this we use Example 2 (p. 107, [@che1]). We recall some details about this example. ]{}
[Let $E_{1}$, $E_{2}$, $E_{3}$ and $E_{4}$ be torsion-free groups of rank 1, let $p$, $q$, $p_{2}$ and $p_{3}$ be distinct primes, let the types of the groups $E_{1}$, $E_{2}$, and $E_{3}$ be pairwise incomparable, and let $E_{1}\cong E_{4}$, $p_{2}E_{2}=E_{2}$, $p_{3}E_{3}=E_{3}$, $pE_{1}\neq
E_{1}$, $pE_{2}\neq E_{2}$, $pE_{3}\neq E_{3}$, $p_{2}E_{1}\neq E_{1}$, $p_{2}E_{3}\neq E_{3}$, $p_{3}E_{1}\neq E_{1}$, $p_{3}E_{2}\neq E_{2}$, $qE_{1}\neq E_{1}$, $qE_{2}\neq E_{2}$, $qE_{3}\neq E_{3}$. ]{}
[A group $G$ is constructed as subgroup of a divisible torsion-free group, using a vector space over the field of rational numbers. Write $A=\left\langle E_{1},E_{2},p^{-\infty }(e_{1}+e_{2})\right\rangle $, $B=\left\langle E_{3},E_{4},q^{-\infty }(e_{3}+e_{4})\right\rangle $ and $G=A\oplus B$ where $0\neq e_{i}\in E_{i}$, $i\in \{1,2,3,4\}$ and $p^{-\infty }a$ is the infinite set $p^{-1}a$, $p^{-2}a$,... If $\mathbf{t}(E_{i})$ denotes the type of $E_{i}$, it is shown that $A$ and $E_{1}\oplus
E_{4}=G(\mathbf{t}(E_{1}))$ are strongly invariant in $G$ but $E_{1}=A\cap G(\mathbf{t}(E_{1}))\cong E_{4}$ is not strictly invariant. ]{}
In the sequel we prove some basic properties of strictly invariant submodules. We denote by $\mathcal{T}(M)$ the set of all strictly invariant submodules of $M$.
\[lattice\] Let $M$ be an $R$-module. If $\{S_{i}\}_{i\in I}$ is a family of submodules from $\mathcal{T}(M)$ then $\sum_{i\in I}S_{i}\in
\mathcal{T}(M)$. Consequently, $(\mathcal{T}(M),\subseteq )$ is complete lattice.
Let $\{S_{i}\}_{i\in I}$ be a family of strictly invariant submodules of a module $M$ and let $f:\sum\limits_{i\in I}S_{i}\longrightarrow M$ be an injective homomorphism. Denoting by $\iota _{i}:S_{i}\longrightarrow
\sum\limits_{i\in I}S_{i}$ ($i\in I$) the inclusions, the compositions $f\circ \iota _{i}:S_{i}\longrightarrow G$ are also injective. By hypothesis, $(f\circ \iota _{i})(S_{i})\subseteq S_{i}$ and so $f(\sum\limits_{i\in
I}S_{i})\subseteq \sum\limits_{i\in I}S_{i}$, as required.
The existence of inf’s now follows because an ordered set $A$ is a complete lattice if and only if for every subset $B\subseteq A$, there exists $\sup B$.
Using Example \[intersection\], we observe that in general the infimum of a family $\{S_{i}\}_{i\in I}$ from $\mathcal{T}(M)$ is not the intersection of these submodules, that is, the complete lattice $(\mathcal{T}(M),\subseteq )$ above is not a complete sublattice of the lattice of all submodules of $M$.
Let $M$ be an $R$-module. If $K\leq M$, we denote by $\mathcal{M}_{M}(K)$ the sum of all submodules $f(K)$, where $f$ ranges all monomorphisms $f:K\rightarrow M$. We denote by $\mathcal{S}(M)$ the lattice of all submodules of $M$ and $\mathcal{M}_{M}(\mathcal{S}(M))=\{\mathcal{M}_{M}(K):K\leq M\}$.
Let $M$ be an $R$-module. Then $\mathcal{M}_{M}(-):\mathcal{S}(M)\rightarrow
\mathcal{S}(M)$ is an idempotent decreasing operator, and $\mathcal{M}_{M}(\mathcal{S}(M))=\mathcal{T}(M)$.
If $f\colon \mathcal{M}_{M}(K)\rightarrow M$ is an injective homomorphism, then for every monomorphism $g:K\rightarrow M$, since $g(K)\leq \mathcal{M}_{M}(K)\leq M$, we can consider $f\circ g:K\rightarrow M$, which is also a monomorphism. Then $(f\circ g)(K)\leq \mathcal{M}_{M}(K)$, and we conclude that $f(\mathcal{M}_{M}(K))\leq \mathcal{M}_{M}(K)$. Finally for every submodule $K$ of $M$ we have $\mathcal{M}_{M}(K)\in \mathcal{T}(M)$ and the surjectivity follows from the fact that $K\in \mathcal{T}(M)$ clearly implies $\mathcal{M}_{M}(K)=K$.
Let $M$ be an $R$-module and $K\leq M$.
1. If $f:K\rightarrow M$ is a homomorphism and $f(K)\nsubseteq \mathcal{M}_{M}(K)$ then for every $\alpha :K\rightarrow \mathcal{M}_{M}(K)$ there exists $x\in K$ such that $f(x)=\alpha (x)$.
2. $f(K)\cap \mathcal{M}_{M}(K)\neq 0$ for every $0\neq f\in \mathrm{Hom}\,(K,G)$.
3. If $H\leq M$ is a submodule of $M$ such that $H\cap \mathcal{M}_{M}(K)=0$, then $\mathrm{Hom}\,(H,K)=0$.
1\. The image of $f-\alpha $ is not contained in $\mathcal{M}_{M}(K)$. Then $f-\alpha $ is not a monomorphism.
For 2 and 3 we apply 1 taking for $\alpha $ the inclusion map.
\[ch\] Let $H$ be a strictly invariant submodule of $M$. Then:
1. $f(H)\subseteq H$ for every non-zero homomorphism $f:H\rightarrow G$ such that $f(f(H)\cap H)=0$.
2. $f(H)\cap H\neq 0$ for every $0\neq f\in \mathrm{Hom}\,(H,M)$.
3. $\mathrm{Hom}(H,L)=0$ for every $L\leq M$ such that $L\cap H=0$.
1\. Suppose there exists $f:H\rightarrow G$, $f\neq 0$, such that $f(f(H)\cap
H)=0$ and consider $\overline{f}:H\rightarrow G$, $\overline{f}(h)=h+f(h)$ for every $h\in H$. If $h+f(h)=0$ then $h\in f(H)\cap H$, hence $f(h)=0$ and so $h=0$. Therefore, $\overline{f}$ is a monomorphism and $f(H)\subseteq H$ by strictly invariance.
The statements 2 and 3 are consequences of 1.
\[direct-summands\] A direct summand is strictly invariant if and only if it is fully invariant.
For any pair $A$, $N$ of modules, denote by $S_{A}(N)=\sum_{f\in \mathrm{Hom}(N,A)}f(N)$ the $N$-*socle* of $A$, a submodule of $A$.
\[sd\] Let $M=A\oplus B$ be an $R$-module. If $K\leq A$ and $L\leq B$ are submodules such that $K\oplus L$ is strictly invariant in $M$ then
1. $K$ is strictly invariant in $A$.
2. $L$ is strictly invariant in $B$.
3. $S_{A}(L)\leq K$ and $S_{B}(K)\leq L$.
Let $f:K\rightarrow A$ be a monomorphism. Then $\overline{f}=f\oplus \iota
_{L}:K\oplus L\rightarrow A\oplus B$ is a monomorphism ($\iota
_{N}:L\rightarrow B$ denotes the inclusion map) and, since $K\oplus L$ is strictly invariant, the inclusion $f(K)\leq K$ follows.
Any homomorphism $f:L\rightarrow A$ induces a homomorphism $\overline{f}:K\oplus L\rightarrow A\oplus B$, $\overline{f}(k+\ell )=k+f(\ell )+\ell $ for every $k\in K$, $\ell \in L$. If $k+f(\ell )+\ell =0$ then $\ell =0$ and we also obtain $k=0$. Therefore, $\overline{f}$ is injective, and it is easy to see that $f(\ell )\in K$ for every $\ell \in L$.
In the following result, for any (finite or infinite) cardinal $k$, $M^{(k)}$ denotes the direct sum of $k$ copies of $M$.
\[+\] Let $H$ be a strictly invariant submodule of $M$. Then the following conditions are equivalent:
$1)$ $H^{2}$ is a strictly invariant submodule of $M^{2}$.
$2)$ $H^{(k)}$ is a strictly invariant submodule of $M^{(k)}$ for every cardinal number $k$.
$3)$ $H^{(k)}$ is a strongly invariant submodule of $M^{(k)}$ for every cardinal number $k$.
$4)$ $H$ is a strongly invariant submodule of $M$.
1\) $\Rightarrow $ 4) Follows from Proposition \[sd\].
4\) $\Rightarrow $ 3) Since $H$ is strongly invariant in $M$, by [@cal], $H^{(k)}$ is strongly invariant in $M^{(k)}$ for any finite $k$. For any infinite cardinal $k$, since every element of $M^{(k)}$ belongs to a direct summand isomorphic to $M^{(n)}$ for some finite $n$, $H^{(k)}$ is also strongly invariant in $M^{(k)}$.
3\) $\Rightarrow $ 2) Obvious.
2\) $\Rightarrow $ 1) Obvious.
\[w2\] Let $M=A\oplus B$ be a module such that the additive group $A$ has no elements of order $2$. If a submodule $H\leq M$ is strictly invariant then there exist $K\leq A$, $L\leq B$ such that
$1)$ $H=K\oplus L$.
$2)$ $K$ is strictly invariant in $A$ and $L$ is strictly invariant in $B$.
$3)$ $S_{A}(L)\leq K$ and $S_{B}(K)\leq L$.
By Proposition \[sd\], it suffices to prove that $H=K\oplus L$ with $K\leq
A$ and $L\leq B$.
Let $\pi _{A}:M\rightarrow A$ and $\pi _{B}:M\rightarrow B$ be the projections and suppose $H$ is strictly invariant in $M$. If $\pi \in \{\pi _{A},\pi _{B}\}$ and $\pi (H)\nleq H$, there is $0\neq
h\in H$ such that $\pi (h)\notin H$. Therefore $(\pi +1)(h)\notin H$ and so the restriction $(\pi +1)|_{H}$ is not injective (otherwise $H$ is not strictly invariant). Hence $\ker \,((\pi +1)|_{H})\neq 0$ and if $\pi (x)=-x$ for $0\neq x\in H$ then $\pi (x)=-\pi (x)$, i.e. $2\pi (x)=0$, a contradiction.
It follows that $\pi _{A}(H)\leq H$, and similarly $\pi _{B}(H)\leq H$. Therefore $H=M\oplus N$, where $M=\pi _{A}(H)\leq A$ and $N=\pi _{B}(H)\leq
B $.
This follows from the construction used in Example \[non-strongly\].
Strictly invariant subgroups
============================
As mentioned in the Introduction, for fairly large classes of groups, we show that our conjecture, strictly invariant subgroups of Abelian groups are strongly invariant, holds.
We start the investigation of strictly invariant subgroups of Abelian groups with a consequence of Proposition \[sd\] (in this section, unless otherwise stated, group means Abelian group). By $\mathbb{P}$ we denote the set of all prime numbers and for an Abelian group $G$ and a prime $p$, $G_{p}=\{x\in G:\exists n\in \mathbb{N},p^{n}x=0\}$ denotes the $p$-component of $G$. For an element $x\in G$, the $p$*-height of* $x$, denoted $h_{p}(x)$, is the smallest integer $n$ such that $x\in p^{n}H$. If $x\in p^{n}H$ for all positive integers $n$ then we say that $x$ is of infinite height.
If $G$ is an Abelian group, we denote by $D(G)$ its divisible part. Moreover, if $p$ is a prime then $D_{p}(G)$ denotes the $p$-component of $D(G)$.
\[divisible-part\] Let $H$ be a strictly invariant subgroup of a group $G $. Then
1. $D(H)=D(G)$ whenever $D(H)$ is not a torsion group.
2. $D_{p}(H)=D_{p}(G)$ whenever $p$ is a prime and $D_{p}(H)\neq 0$.
We chose a decomposition $H=H_{0}\oplus D(H)$. Using [fuc1]{}, we can find a direct decomposition $G=K\oplus D(H)$ such that $H_{0}\leq K$. By Proposition \[sd\], $S_{K}(D(H))\leq H_{0}$. Since $H_{0}$ is reduced and every image of a divisible group is divisible, it follows that $S_{K}(D(H))=0$.
1\. If $D(H)$ is not torsion then it has a direct summand isomorphic to $\mathbb{Q}$, hence for every non-reduced group $L$ we have non-zero homomorphisms $D(H)\rightarrow L$. It follows that $K$ is reduced, hence $D(H)=D(G)$.
2\. If $D_{p}(H)\neq 0$ then $D(H)$ has a direct summand isomorphic to $\mathbb{Z}(p^{\infty })$. From $S_{K}(D(H))=0$ it follows that $D_{p}(K)=0$, hence $D_{p}(H)=D_{p}(G)$.
\[str-inv-p\] Let $G$ be a group and let $H$ be a $p$-subgroup of $G$. Then $H$ is strictly invariant in $G$ if and only if it satisfies one of the following conditions:
1. $H=G_{p}$.
2. there exists a non-negative integer $n$ such that $H=G[p^{n}]$.
3. there exists a non-negative integer $n$ such that $H=G[p^{n}]+D_{p}(G)$.
Suppose $H$ is strictly invariant. Since $H$ is a $p$-group, we can suppose that $G$ is also a $p$-group.
As in the proof of Corollary \[divisible-part\] we can find direct decompositions $G=K\oplus D(H)$ and $H=H_{0}\oplus D(H)$ with $H_{0}\leq K$. Using Proposition \[sd\] it follows that $H_{0}$ is strictly invariant in $K$. Therefore, we can assume w.l.o.g. that $H$ is reduced.
*Case I: $H$ is not bounded.* We will prove that $H=G$. Let us fix an element $y\in G$. We can assume w.l.o.g that there exists $u$, the smallest positive integer such that $p^{u}y\in H$.
If $p^uy=0$, we chose $\langle x\rangle$ a direct summand of $H$ such that $\mathrm{ord}(x)\geq\mathrm{ord}(y)$. If $H=\langle x\rangle \oplus L$ then $\langle x+y\rangle +L=\langle x+y\rangle \oplus L$ and $\mathrm{ord}(x+y)=\mathrm{ord}(x)$. It follows that $\langle x+y\rangle +L\cong H$, and we obtain that $x+y\in H$, hence $y\in H$.
Suppose that $p^{u}y\neq 0$ and we can find non-zero elements in $\langle
p^{u}y\rangle $ whose heights computed in $H$ are infinite. Let $\oplus
_{n>0}B_{n}$ be a basic $p$-subgroup of $H$, where for every $n$ the group $B_{n}$ is isomorphic to a direct sum of cyclic groups of order $p^{n}$ (see [@fuc1 Theorem 32.4]). Since $H$ has an unbounded basic subgroup, there exists $n>0$ such that $p^{n}>\mathrm{ord}(y)$ and $B_{n}\neq 0$. Since all non-zero elements of $B_{n}$ are of finite height, it follows that $\langle
p^{u}y\rangle \cap B_{n}=0$, hence we can find a $B_{n}$-high subgroup $C\leq H$ such that $\langle p^{u}y\rangle \leq C$. By the proof of [@fuc1 Proposition 27.1] it follows that $H=B\oplus C$. Therefore, there exists a decomposition $H=\langle x\rangle \oplus L$ such that $\mathrm{ord}(x)\geq \mathrm{ord}(y)$ and $p^{u}y\in L$. Write $H=\langle x\rangle \oplus
L$, and consider the homomorphism $f:H\rightarrow G$, $f(mx+\ell
)=mx+my+\ell $, for all $m\in \mathbb{Z}$ and $\ell \in L$. If $kx+ky+\ell
=0 $, it follows that $ky\in H$, hence $p^{u}$ divides $k$. Therefore $ky+\ell \in L$, hence $kx=0$. Since $\mathrm{ord}(x)\geq \mathrm{ord}(y)$ we obtain $ky=0$, and it follows that $\ell =0$. Finally, $f$ is a monomorphism and it is easy to conclude that $y\in H$.
If $p^{u}y\neq 0$ and all non-zero elements of $\langle p^{u}y\rangle $ are of finite heights (computed in $H$) then by [@fuc1 Theorem 33.4], there exists a basic subgroup $B$ of $H$ such that $\langle p^{u}\rangle \leq B$. Since $B$ is unbounded, we can find a cyclic direct summand $\langle
x\rangle $ of $B$, hence of $G$, such that $\mathrm{ord}(x)\geq \mathrm{ord}(y)$, and $\langle x\rangle \cap \langle p^{u}y\rangle =0$. Then there exists a decomposition $H=\langle x\rangle \oplus L$ such that $p^{u}y\in L$, and we can repeat the proof used in the previous case to conclude that $y\in H$.
*Case II: $H$ is bounded.* If $p^{n}=\exp H$ then clearly $H\leq
G[p^{n}]$. Assume that $H<G[p^{n}]$ and let $x\in H$ be such that $\mathrm{ord}(x)=p^{n}$. Since $G[p^{n}]$ is generated by the elements of order $p^{n} $ in $G$, there exists $y\in G[p^{n}]$ such that $\mathrm{ord}(y)=p^{n} $ and $y\notin H$. By [@fuc1 Theorem 27.1], there exist $K,L\leq G$ such that $G[p^{n}]=\left\langle x\right\rangle \oplus
K=\left\langle y\right\rangle \oplus L$. By Dedekind’s law, $H=H\cap
G[p^{n}]=\left\langle x\right\rangle \oplus (H\cap K)$. Since $L\cong K$, there exists $L_{1}\leq L $ such that $L_{1}\cong H\cap K$ and it follows that $\left\langle y\right\rangle \oplus L_{1}$ and $\left\langle
x\right\rangle \oplus (H\cap K)=H$ are isomorphic. Since $y\notin H$ this contradicts the fact that $H$ is strictly invariant in $G$. Thus $H=G[p^{n}]$, as desired.
As for the converse, it is enough to observe that if $H$ verifies any of the conditions 1–3 then it is strongly invariant.
Namely, if $H$ has an unbounded basic subgroup, we write $G=D_{p}(G)\oplus R$ with reduced $R$ and $H=D_{p}(G)\oplus K$ with $K\leq R$. Let $k$ be a positive integer. There exists a cyclic direct summand $C\cong \mathbb{Z}(p^{n})$ of $R$ with $n\geq k$. Applying Proposition \[w2\] for the direct decomposition $G=(D_{p}(G)\oplus C)\oplus L$, it follows that $S_{L}(D_{p}(G)\oplus C)=S_{L}(C)\leq H$. It is easy to see that $G[p^{n}]\leq H$, hence $G[p^{k}]\leq H$ for all $k$ and the proof is complete.
From now on, starting with the next corollary, the results are all in the line of the conjecture, stating that every strictly invariant subgroup of an Abelian group is strongly invariant.
First we are able to show that
\[che\] Every torsion strictly invariant subgroup of any group is strongly invariant.
It is proved in [@cal] that a torsion subgroup is strongly invariant if and only if all its primary components are strongly invariant. Using Theorem \[str-inv-p\], the conclusion now follows. Indeed, all subgroups from the theorem, $G_{p}$, $G[p^{n}]$ and $G[p^{n}]+D_{p}(G)$ are strongly invariant.
In the following proposition, $r_{p}(K)$ denotes the $p$-rank of $K$. We will prove that if the divisible part of a group is large enough, then all strictly invariant subgroups are strongly invariant.
Let $G=D(G)\oplus R$ be a group and $r_{p}(D(G))\geq \max \{r_{p}(R),\aleph
_{0}\}$ for every $p\in \mathbb{P}\cup \{0\}$. Then every strictly invariant subgroup of $G$ is strongly invariant.
Let $H$ be a strictly invariant subgroup of $G$. Then $H=D(G)\oplus K$ (we use Corollary \[divisible-part\]) and we can suppose that $K\leq R$. It suffices to prove that $K$ is strongly invariant in $R$. In order to do this, let us fix a homomorphism $f:K\rightarrow R$.
By the rank hypotheses it follows that $K$ can be embedded in $D(G)$ and $D(G)\cong D(G)\oplus D(G)$. Therefore, there exists a monomorphism $\alpha
:H\rightarrow D(G)$. We consider the homomorphism $g:H\rightarrow G$, defined by $g(d+k)=\alpha (d+k)+f(k)$ for all $d\in D(G)$ and $k\in K$. It is easy to see that $g$ is a monomorphism. Therefore $f(k)\in K$ for all $k\in K$ and the proof is complete.
The following results refer to torsion-free subgroups or torsion-free groups.
Let $H$ be a strictly invariant torsion-free subgroup of a group $G$. If $H$ is of finite rank then it is strongly invariant.
Let $f:H\rightarrow G$ be a homomorphism. We claim that there exists a positive integer $k$ such that for all $x\in H$ we have $f(x)\neq kx$.
By contradiction suppose that the above claim fails. Then for every positive integer $k$ there exists $x_{k}\in H$ such that $f(x_{k})=kf(x_{k})$. We will prove by induction on the cardinality of $S$ that every non-empty finite subset $S\subseteq \{x_{k}:k\in \mathbb{N}^{\star }\}$ is linearly independent. Since for $|S|=1$ the property is obvious, suppose that all non-empty subsets $S\subseteq \{x_{k}:k\in \mathbb{N}^{\star }\}$ of cardinality at most $n$ are linearly independent. Let $\{x_{k_{1}},\dots
,x_{k_{n+1}}\}$ be a subset of cardinality $n+1$, and suppose that there exist integers $\alpha _{1},\dots ,\alpha _{n+1}$ such that $\sum_{i=1}^{n+1}\alpha _{i}x_{k_{i}}=0$. Applying $f$ we obtain $\sum_{i=1}^{n+1}k_{i}\alpha _{i}x_{k_{i}}=0$, and so $\sum_{i=1}^{n}(k_{i}-k_{n+1})\alpha _{i}x_{k_{i}}=0$. By the induction hypothesis it follows that $(k_{i}-k_{n+1})\alpha _{i}=0$ for all $i\in
\{1,\dots ,n\}$, and now it is easy to conclude that $\{x_{k_{1}},\dots
,x_{k_{n+1}}\}$ is linearly independent. Hence the rank of $H$ is infinite, a contradiction.
Let $k$ be a positive integer such that for all $x\in H$ we have $f(x)\neq
kx $. Then the map $g:H\rightarrow G$, $g(x)=kx+f(x)$ is a monomorphism. Using the strictly invariance of $H$, it follows that $g(H)\subseteq H$, hence $f(H)\subseteq H$, and the proof is complete.
If $G$ is torsion-free and all rank 2 pure subgroups of $G$ are indecomposable, then every strictly invariant subgroup of $G$ is strongly invariant.
Suppose there exists a non-injective homomorphism $f:H\rightarrow G$. Then there is a non-zero element $x\in H$ such that $f(x)=x$. Indeed, let $f\in
\mathrm{Hom}(H,G)$ and $f(H)\nleq H$. If $g$ is the embedding of $H$ in $G$ then $(f-g)H\nleq H$, so there exists a non-zero $x\in H$ such that $f(x)=x$. Take $y$ a non-zero element from the kernel of $f$, and let $L$ be the pure subgroup generated by $x$ and $y$. For every non-zero element $z\in L$, we have a relation $kz=mx+ny$ with $k\neq 0$. Then $kf(z)=mx$, and so $kf^{2}(z)=mx=kf(z)$. From the torsion-free hypothesis, we can view $f$ as an idempotent endomorphism of $L$ whose image has rank $1$. It follows that $L$ is not indecomposable, a contradiction.
Examples of such groups include the purely indecomposable groups determined by Griffith in the reduced case (see Theorem **88.5** [@fuc2]) and in particular the so-called cohesive groups considered by Dubois (see Exercise 17, § **88**, [@fuc2]).
Next, it is easy to see that the only strictly invariant subgroups of rank 1 torsion-free groups are the trivial ones (i.e. $0$ or the whole group). This is clear for $\mathbb{Z}$ and follows from Corollary \[divisible-part\] for $\mathbb{Q}$. By Theorem \[str-inv-p\], this also holds for $\mathbb{Z}(p)^{\mathbb{N}}$.
\[comp-dec\] A subgroup of a completely decomposable group is strictly invariant if and only if it is a fully invariant direct summand.
Let $G=\oplus _{i\in I}G_{i}$ be a completely decomposable group, where all groups $G_{i}$ are of rank $1$. If $H$ is a strictly invariant subgroup of $G $, then using Proposition \[w2\] it follows that for all $i\in I$ we have $\pi _{i}(H)\leq H$, where $\pi _{i}:G\rightarrow G_{i}$ denotes the projection. Then $H=\oplus _{i\in I}(H\cap G_{i})$, and, using Proposition \[sd\], $H\cap G_{i}$ is a strictly invariant subgroup of $G_{i}$ for every $i\in I$. By the preceding paragraph, $H=\oplus _{j\in J}G_{j}$, where $J$ is the set of all $j\in I$ such that $H\cap G_{j}=G_{j}$. The conclusion is now a consequence of Corollary \[direct-summands\].
Let $G$ be a separable torsion-free group and $H$ a nonzero strictly invariant subgroup. Then $H$ is strongly invariant.
Let $f:H\longrightarrow G$ be a homomorphism. If $x\in H$ then there exists a finite rank completely decomposable $G_{1}\oplus \dots \oplus G_{n}$ direct summand of $G$ such that $x,f(x)\in G_{1}\oplus \dots \oplus G_{n}$. Using Proposition \[w2\] it follows that $K=H\cap (G_{1}\oplus \dots
\oplus G_{n})$ is a strictly invariant subgroup of $G_{1}\oplus \dots \oplus
G_{n}$. Then by Proposition \[comp-dec\], $K$ is strongly invariant, whence $f(x)\in H$.
Finally, we show that the *groups, all whose subgroups are strictly invariant*, coincide with those all whose subgroups are strongly invariant.
\[all\]The only groups in which every subgroup is strictly invariant are the direct sums of cocyclic groups, at most one, for each prime number.
The proof in [@cal] holds verbatim with obvious changes for torsion groups. Indeed, a subgroup $H$ of a torsion group $G$ is strictly invariant if and only if the $p$-component $H_{p}$ is strictly invariant in $G_{p}$ for each prime $p$, in a $p$-group $G$ every subgroup is strictly invariant if and only if $G$ is cocyclic, and, in a torsion group every subgroup is strictly invariant if and only if each $p$-component has this property.
Further, there are no torsion-free nor (genuine) mixed groups with only strictly invariant subgroups. Indeed, using the multiplication with $\frac{1}{p}$ for a suitable prime $p$, it is easy to see that rank 1 torsion-free groups are strictly invariant simple (i.e. have only trivial strictly invariant subgroups). Therefore, no torsion-free groups have only strictly invariant subgroups.
As for (genuine) mixed groups $G$, the torsion part $T(G)$ must be as in the strongly invariant case and so is a direct summand (see [@cal]). If $G=T(G)\oplus F$, we continue with $F$ as above.
**Acknowledgement.** Thanks are due to the referee, for corrections and suggestions which improved our presentation.
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http://groupprops.subwiki.org/wiki/Homomorph-containing\_subgroup
S. Breaz, Department of Mathematics, Babeş-Bolyai University, 1 Kogălni-ceanu Street, Cluj-Napoca, Romania
E-mail: bodo@math.ubbcluj.ro
G. Călugăreanu, Department of Mathematics, Babeş-Bolyai University, 1 Kogălniceanu Street, Cluj-Napoca, Romania
E-mail: calu@math.ubbcluj.ro
A. Chekhlov, Faculty of Mechanics and Mathematics, Tomsk State University, Tomsk, Russia
E-mail: cheklov@math.tsu.ru
[^1]: 2010 AMS Subject Classification: 16D10, 16D80,20K27 Key words: strictly invariant submodule, strongly invariant submodule, Abelian group, strictly invariant subgroup
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abstract: 'This paper primarily focuses on figuring out the best array of cameras, or visual sensors, so that such a placement enables the maximum utilization of these visual sensors. Maximizing the utilization of these cameras can convert to another problem that is simpler for the formulation, that is maximizing the total coverage with these cameras. To solve the problem, the coverage problem is first defined subject to the capabilities and limits of cameras. Then, each camera’ pose is analyzed for the best arrangement.'
author:
- 'Lin Xu [^1]'
title: A Preliminary Study on Optimal Placement of Cameras
---
Cameras, coverage
Introduction
============
Cameras are ubiquitous in our lives, as we can almost see it everywhere we go. Streets, restaurants, offices, schools, areas where people go and stay, are under these “eyes.” Cameras, or in a broad way of definition, visual sensors, are widely utilized for video surveillance, recording, military uses, and so on. Given the area that the cameras need to watch, if all cameras are the same, the size of the area is proportional to the number of cameras. Therefore, the cost of setting cameras is positively related to the number of cameras employed. Sometimes, designers make coverage mistakes, ineffective setting visual sensors, as shown below. In this case, optimal placement of cameras is critical.
In 1973, mathematician Victor Klee proposed the art gallery problem, a real-world problem trying to find a minimum number of guards who together can observe the whole gallery. These guards are at fixed positions, and the gallery shapes like a polygon. Similarly, there is also the Floodlight illumination problem, coping with the illumination of planar regions by light sources. For more information, please refer to \[1\],\[2\]. However, these solutions contain unrealistic factors, such as unlimited field of view, the infinite depth of field, making these algorithms unsuitable for most real-world computer vision applications.
Problem Formulation
===================
In this paper, as shown in Figure 2, the problem of optimal camera placement for a given region is formulated. We focus on the static camera placement problem, where the goal is to determine optimal position and number of cameras for a region to be observed, given a set of task-specific constraints, and a set of possible cameras to use in the layout.
For simplicity, we first assume that the detection range of each visual sensor is unbounded. However, there are several constraints needed to consider: (1) the number of cameras that given, (2) the each camera’s limited field-of-view, (3) fixed position, (4) the room without obstacles. The optimal placement is defined as the cameras are placed so that they cover as much space as possible.
The field-of-view of a camera can be described by a triangle, as shown in Figure 3. There are three parameters, with $d$ representing the length of sight, $\varphi$ determining the pose of a camera, and $2\alpha$ defining the field-of-view angle.
Main Result
===========
For a cuboid 3D space, consider the camera working in 2D space such as 2D space from the side view and 2D space from the top view as shown in Figure 4. As for the coverage problem, we can claim that if a place is covered by both the side view and the top view of a camera, then it can be covered by the field-of-view angle of the camera in 3D space. With this result, we divide the coverage problem in 3D space into two subproblems, namely from the top view and side view.
From the top view
-----------------
From the top view, the best arrangement should be that all spaces are covered ideally, and the overlapped areas should be minimal. Due to such an analysis, the cameras should be placed in a staggered way such that these visual sensors would not monitor a space multiple times, shown in Figure 5.
**Proposition 1**. *As shown in Figure 6, given that cameras with the same field-of-view angle 2 and unbounded the depth of view, the staggered arrangement is optimal for a rectangular space.*
**Proof**. In order to get the maximum coverage, the pose of a camera should be changed such that all field-of-view angle is included in the given space, as shown in Figure 6. Six cameras are labelled by A to F, and their field-of-view angles also correspond to1 to 6 in Figure 6, respectively.
At that time, since all field-of-view angles are in the space, so$$\text{angle}1=\text{angle}2=\text{angle}3=\text{angle}4=\text{angle}5=\text{angle}6=2\varphi=2\alpha.$$ Therefore, AJ // CB // FG // HD// IE. We can see that no fields of view overlap until the fifth camera is in position, so they maximize the coverage while no space is counted twice and no space is out of sight. The dashed area is the fields of view overlap due to the insufficient length of the space, but it is indeed the best one because applying five cameras will not coverage all spaces. The dashed area can be more even if we slightly adjust the pose of the first camera at the corner, and the rest sensors still form sets of a rectangle. For such an arrangement, the length of a given space, $l$, and the vertex angle of the sensor, $\alpha$, do matter. The position of the cameras is determined by them if follows the optimal placements, for which the first camera should be at one corner of a space, and the others follow, as shown in Figure 7. Let us name them from top to bottom as number 1, number 2, number 3, etc.
Since Camera \#1 is at the corner, where we can see it as the origin (0, 0), camera \#2’s position could be calculated by camera \#1. Let us assume that $x$ refers to the horizontal position, and $y$ represents the vertical position of a camera. Then$$\begin{aligned}
x & =w\\
y & =-w\tan \alpha.\end{aligned}$$ So the position of camera \#2 is $\left( w,-w\tan \alpha \right) $, with $y$ position vertically shifts downward at the point where camera \#1’s sight edge intersects that of camera \#2. Similarly, we can calculate other cameras’ positions by each moving downward a distance of $-w\tan \alpha$ and horizontal positions switch between $0$ and $w$.
From the side view
------------------
For analyzing from the side view, it is relatively more straightforward, since all visual sensors are located on the top of the space. In order to achieve the maximum coverage, cameras have to be equally spaced so that they can spread area as much as they can, shown in Figure 8.
In this case, each camera is separated by a distance of one-fifth of the length horizontally. Combined the analysis of side view and top view, we can assume that both are the optimal placements that depend on real situations.
Conclusion and future work
==========================
We have presented an initial study for optimal visual sensor arrangement in a given space. We have simplified the three-dimension problem to be a two-dimension problem. However, the camera models are still ideal, and the space is somewhat simple, such as some variables like the pitch angles of cameras are not considered. Future works will include these variables and make it more useful to the practice.
[9]{}
E. Horster, and R. Lienhart. On the optimal placement of multiple visual sensors. Proceeding VSSN ’06 Proceedings of the 4th ACM international workshop on Video surveillance and sensor networks, Santa Barbara, California, USA, October 27, 2006, Pages 111-120.
T.C. Shermer. Recent Results in Art Galleries. Proceedings of the IEEE, vol. 80, no. 9, 1992, Pages 1384-1399.
[^1]: Lin Xu is with Beijing 101 Middle School, Beijing 100091, People’s Republic of China, Email: linxu\_2002@163.com
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abstract: 'Multitime differential games are related to the modeling and analysis of cooperation or conflict in the context of a multitime dynamical systems. Their theory involves either a curvilinear integral functional or a multiple integral functional and an $m$-flow as constraint. The aim of this paper is to give original results regarding multitime hybrid differential games with curvilinear integral functional constrained by an $m$-flow: fundamental properties of multitime upper and lower values, viscosity solutions of multitime (HJIU) PDEs, representation formula of viscosity solutions for multitime (HJ) PDEs, and max-min representations.'
author:
- 'Constantin Udrişte, Elena-Laura Otobîcu, Ionel Ţevy'
title: |
Multitime hybrid differential games with\
curvilinear integral functional
---
\[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Remark]{}
[**Mathematics Subject Classification 2010**]{}: 49L20, 91A23, 49L25, 35F21.
[**Key words**]{}: multitime hybrid differential games, curvilinear integral cost, multitime dynamic programming, multitime viscosity solutions.
Multitime hybrid differential game with\
curvilinear integral functional
========================================
Let $t=(t^\alpha)\in \Omega_{0T}\subset \mathbb{R}^m_+$, $\alpha =1,...,m,$ be an evolution multi-parameter, called multitime. Consider an arbitrary $C^1$ curve $\Gamma_{0T}$ joining the diagonal opposite points $0=(0,\ldots,0)$ and $T=(T^1,\ldots,T^m)$ in the $m$-dimensional parallelepiped $\Omega_{0T}=[0,T]$ (multitime interval) in $\mathbb{R}^m_+$ endowed with the product order, a $C^2$ state vector $x:\Omega_{0T}\rightarrow \mathbb{R}^n, x(t)=(x^i(t)),$ $i=1,...,n$, a $C^1$ control vector $u(t)=(u_\alpha(t)):\Omega_{0T}\rightarrow U\subset \mathbb{R}^{qm},$ for the first equip of $m$ players (who wants to maximize), a $C^1$ control vector $v(t)=(v_\alpha(t)):\Omega_{0T}\rightarrow V\subset \mathbb{R}^{qm},$ for the second equip of $m$ players (who wants to minimize), $u_\alpha(\cdot)=\Phi(\cdot,\eta_1(\cdot)), v_\alpha(\cdot)=\Psi(\cdot,\eta_2(\cdot)),$ a running cost $L_\alpha(t,x(t),u_\alpha(t),v_\alpha(t))dt^\alpha$ as a nonautonomous closed Lagrangian $1$-form (satisfies $D_\beta L_\alpha=D_\alpha L_\beta),$ a terminal cost $g(x(T))$ and the $C^1$ vector fields $X_\alpha=(X_\alpha^i)$ satisfying the complete integrability conditions (CIC) $D_{\beta}X_{\alpha}=D_{\alpha}X_{\beta}$ (m-flow type problem).
In our paper, a [*multitime hybrid differential game*]{} is given by a multitime dynamics, as a PDE system controlled by two controllers (first equip, second equip) and a target including a curvilinear integral functional. The approach we follow below is those in the paper [@[2]], but we must be more creative since our theory is multitemporal one (see also [@[8]]-[@[21]]). More precisely, we introduce and analyze a multitime differential game whose Bolza payoff is the sum between a path independent curvilinear integral (mechanical work) and a function of the final event (the terminal cost, penalty term), and whose evolution PDE is an m-flow: [*Find $$\min_{v(\cdot)\in V}\max_{u(\cdot)\in U} J(u(\cdot),v(\cdot))=\int_{\Gamma_{0T}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha+g(x(T)),$$ subject to the Cauchy problem $$\frac{\partial x^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x(s),u_\alpha(s),v_\alpha(s)),$$ $$x(0)=x_0, \,\,s\in \Omega_{0T}\subset \mathbb{R}_+^m, \,\,x\in \mathbb{R}^n.$$*]{}
Let $D_{\alpha}$ be the total derivative operator and $[X_{\alpha},X_{\beta}]$ be the bracket of vector fields. Suppose the piecewise complete integrability conditions (CIC) $$\left( \frac{\partial X_{\alpha}}{\partial u^a_\lambda}\delta^{\gamma}_{\beta} - \frac{\partial X_{\beta}}{\partial u^a_\lambda}\delta^{\gamma}_{\alpha}\right)\frac{\partial u^a_\lambda}{\partial s^{\gamma}}+\left( \frac{\partial X_{\alpha}}{\partial v^b_\lambda}\delta^{\gamma}_{\beta} - \frac{\partial X_{\beta}}{\partial v^b_\lambda}\delta^{\gamma}_{\alpha}\right)\frac{\partial v^b_\lambda}{\partial s^{\gamma}}=\left[ X_{\alpha},X_{\beta}\right] + \frac{\partial X_{\beta}}{\partial s^{\alpha}} - \frac{\partial X_{\alpha}}{\partial s^{\beta}},$$ where $a, b =1,...,q$, are satisfied throughout.
To simplify, suppose that the curve $\Gamma_{0T}$ is an increasing curve in the multitime interval $\Omega_{0T}$. If we vary the starting multitime and the initial point, then we obtain a larger family of similar multitime problems containing the functional $$J_{x,t}(u(\cdot),v(\cdot))=\int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha+g(x(T)),$$ and the evolution constraint $$\frac{\partial x^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x(s),u_\alpha(s),v_\alpha(s)),$$ $$x(t)=x, \,\,s\in \Omega_{tT}\subset \mathbb{R}_+^m,\,\, x\in \mathbb{R}^n.$$
We assume that each vector field $X_\alpha:\Omega_{0T}\times \mathbb{R}^n\times U\times V\rightarrow \mathbb{R}^n$ is uniformly continuous, satisfying $$\left\{\begin{array}{ll}
\Vert X_\alpha(t,x,u_\alpha,v_\alpha)\Vert\leqslant A_\alpha\\
\Vert X_\alpha(t,x,u_\alpha,v_\alpha) - X_\alpha(t,\hat{x},u_\alpha,v_\alpha)\Vert\leqslant A_\alpha\Vert x-\hat{x}\Vert,
\end{array}\right.$$ for some constant 1-form $A=(A_\alpha)$ and all $t\in \Omega_{0T}, \,\,x, \hat{x}\in \mathbb{R}^n, u\in U,v\in V.$
Suppose the functions $$g:\mathbb{R}^n\rightarrow \mathbb{R}, \quad L_\alpha:\Omega_{0T}\times \mathbb{R}^n\times U\times V\rightarrow \mathbb{R}$$ are uniformly continuous and satisfy the boundedness conditions $$\left\{\begin{array}{ll}
\vert g(x)\vert\leqslant B\\
\vert g(x)-g(\hat{x})\vert\leqslant B\Vert x-\hat{x}\Vert,
\end{array}\right.$$
$$\left\{\begin{array}{ll}
\vert L_\alpha(t,x,u_\alpha,v_\alpha)\vert\leqslant C_\alpha\\
\vert L_\alpha(t,x,u_\alpha,v_\alpha)-L_\alpha(t,\hat{x},u_\alpha,v_\alpha)\vert\leqslant C_\alpha\Vert x-\hat{x}\Vert,
\end{array}\right.$$ for constant $1$-form $C=(C_\alpha)$ and all $t\in \Omega_{0T},\,\, x, \hat{x}\in \mathbb{R}^n,\,\, u\in U,\,\,v\in V.$
\(i) The set $$\mathcal {U}(t)=\left\lbrace u_\alpha(\cdot):\mathbb{R}^m_+\rightarrow U \vert \ u_\alpha(\cdot) \mathrm{ \ is \ measurable \ and \ satisfies \ CIC}\right\rbrace$$ is called **the control set for the first equip of players**. (ii) The set $$\mathcal {V}(t)=\left\lbrace v_\alpha(\cdot):\mathbb{R}^m_+\rightarrow V \vert \ v_\alpha(\cdot) \mathrm{ \ is \ measurable \ and \ satisfies \ CIC}\right\rbrace$$ is called **the control set for the second equip of players**.
\(i) A map $\Phi:\mathcal {V}(t)\rightarrow \mathcal {U}(t)$ is called **a strategy for the first equip of players**, if the equality $v(\tau)=\widehat{v}(\tau), t\leq \tau \leq s \leq T$ implies $\Phi[v](\tau)=\Phi[\widehat{v}](\tau).$ (ii) A map $\Psi:\mathcal {U}(t)\rightarrow \mathcal {V}(t)$ is called **a strategy for the second equip of players**, if the equality $u(\tau)=\widehat{u}(\tau), t\leq \tau \leq s \leq T$ implies $\Psi[u](\tau)=\Psi[\widehat{u}](\tau).$
Let $ \mathcal{A}(t)$ be ** the set of strategies for the first equip of players** and $\mathcal{B}(t)$ be ** the set of strategies for the second equip of players**.
\(i) The function $$m(t,x)=\min_{\Psi\in \mathcal{V}} \max_{u(\cdot)\in U} J_{t,x}( u(\cdot),\Psi[u](\cdot))$$ is called **the multitime lower value function**. (ii) The function $$M(t,x)=\max_{\Phi\in \mathcal{U}} \min_{v(\cdot)\in V} J_{t,x}(\Phi[v](\cdot),v(\cdot))$$ is called **the multitime upper value function**.
The multitime lower value function $m(t,x)$ and the multitime upper value function $M(t,x)$ are piecewise continuously differentiable (see below, the boundedness and continuity of the values functions).
Properties of lower and upper values
====================================
**(multitime dynamic programming optimality conditions)** For each pair of strategies $(\Phi,\Psi),$ the lower and upper value functions can be written respectively in the form $$\begin{split}
m(t,x)\ & =\min_{\Psi\in \mathcal{B}(t)} \max_{u_\alpha\in \mathcal{U}(t)}\bigg\{ \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha \\& +m(t+h,x(t+h))\bigg\}
\end{split}$$ and $$\begin{split}
M(t,x)\ & =\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)}\bigg\{ \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha \\ & +M(t+h,x(t+h))\bigg\},
\end{split}$$ for all $(t,x) \in \Omega_{tT}\times \mathrm{R}^n$ and all $h\in \Omega_{0T-t}.$
First we recognize the Bellman principle (we write the value of a decision problem at a certain point in multitime in terms of the payoff from some initial choices and the value of the remaining decision problem that results from those initial choices).
To confirm the first statement, we shall use the function $$\begin{split}
w(t,x)\ &=\min_{\Psi\in \mathcal{B}(t)} \max_{u_\alpha\in \mathcal{U}(t)}\bigg\{ \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha \\ & +m(t+h,x(t+h))\bigg\}.
\end{split}$$
We will show that, for all $\varepsilon >0,$ the lower value function $m(t,x)$ will satisfies two inequalities, $m(t,x)\leq w(t,x)+2\varepsilon$ and $m(t,x)\geq w(t,x)-3\varepsilon.$ Since $\varepsilon>0$ is arbitrary, it follows $m(t,x)=w(t,x).$
i) For $\varepsilon >0,$ there exists a strategy $\Upsilon\in \mathcal{B}(t)$ such that
$$\begin{split}\label{eq.7}
w(t,x)\ &\geqslant \max_{u_\alpha\in \mathcal{U}(t)}\bigg\{ \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u_\alpha(s),\Upsilon[u_\alpha](s))ds^\alpha \\ & +m(t+h,x(t+h))\bigg\}-\varepsilon.
\end{split}$$
We shall use the state $x(\cdot)$ which solves the (PDE), with the initial condition $\overline{x}=x(t+h)$ (Cauchy problem) on the set $\Omega_{tT}\setminus \Omega_{tt+h},$ for each $\overline{x}\in \mathbb{R}^n$. We can write $$\begin{split}
m(t+h,\overline{x}) \ & =\min_{\Psi\in \mathcal{B}(t+h)} \max_{u_\alpha\in \mathcal{U}(t+h)}\bigg\{ \int_{\Gamma_{t+hT}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha \\ & +g(x(T))\bigg\}.
\end{split}$$
Thus there exists a strategy $\Upsilon_{\overline{x}}\in \mathcal{B}(t+h)$ for which
$$\begin{split}\label{eq.8}
m(t+h,\overline{x})\ & \geqslant \max_{u_\alpha\in \mathcal{U}(t+h)}\bigg\{ \int_{\Gamma_{t+hT}} L_\alpha (s,x(s),u_\alpha(s),\Upsilon_{\overline{x}}[u_\alpha](s))ds^\alpha \\& + g(x(T))\bigg\}-\varepsilon.
\end{split}$$
Define a new strategy $$\Psi\in\mathcal{B}(t), \Psi[u_\alpha](s)\equiv
\left\{\begin{array}{ll}
\Upsilon[u_\alpha](s) & s\in \Omega_{tt+h}\\
\Upsilon_{\overline{x}}[u_\alpha](s) & s\in \Omega_{tT}\setminus\Omega_{tt+h},
\end{array}\right.$$ for each control $u_\alpha\in \mathcal{U}(t).$ For any $u_\alpha\in \mathcal{U}(t)$, replacing the inequality $\eqref{eq.8}$ in the inequality $\eqref{eq.7}$, we obtain $$w(t,x)\geqslant \int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha+g(x(T))-2\varepsilon.$$ Consequently $$\max_{u_\alpha\in \mathcal{U}(t)}\left\lbrace \int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha+g(x(T))\right\rbrace\leq w(t,x)+2\varepsilon.$$
Hence $$m(t,x)\leq w(t,x)+2\varepsilon.$$
ii) On the other hand, there exists a strategy $\Psi\in\mathcal{B}(t)$ for which we can write the inequality
$$\label{eq.9}
m(t,x)\geqslant \max_{u_\alpha\in \mathcal{U}(t)}\left\lbrace \int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha+g(x(T))\right\rbrace-\varepsilon.$$
By the definition of $w(t,x),$ we have $$\begin{split}
w(t,x) \ & \leqslant \max_{u_\alpha\in U(t)}\bigg\{ \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha \\& +m(t+h,x(t+h))\bigg\}
\end{split}$$ and consequently there exists a control $u^1_\alpha\in \mathcal{U}(t)$ such that $$\begin{split}\label{eq.10}
w(t,x) \ & \leqslant \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u^1_\alpha(s),\Psi[u^1_\alpha](s))ds^\alpha \\& +m(t+h,x(t+h))+\varepsilon.
\end{split}$$
Define a new control $${u_\alpha^\star}\in \mathcal{U}(t), {u_\alpha^\star}(s)\equiv
\left\{\begin{array}{ll}
u^1_\alpha(s) & s\in \Omega_{tt+h}\\
u_\alpha(s) & s\in \Omega_{tT}\setminus\Omega_{tt+h},
\end{array}\right.$$ for each control $u_\alpha\in \mathcal{U}(t+h)$ and then define the strategy ${\Psi}^\star\in\mathcal{B}(t+h), \Psi^\star[u_\alpha](s)\equiv\Psi[{u_\alpha^\star}](s), s\in \Omega_{tT}\setminus\Omega_{tt+h}.$ We find the inequality $$\begin{split}
\ & m(t+h,x(t+h)) \\ & \leq\max_{u_\alpha\in \mathcal{U}(t+h)}\left\lbrace \int_{\Gamma_{tt+h}} L_\alpha(s,x(s),u_\alpha(s),\Psi^\star[u_\alpha](s))ds^\alpha+g(x(T))\right\rbrace
\end{split}$$ and so there exists the control $u^2_\alpha\in \mathcal{U}(t+h)$ for which $$\begin{split}\label{eq.11}
\ & m(t+h, x(t+h))\\& \leq \int_{\Gamma_{tT}\setminus \Gamma_{tt+h}} L_\alpha(s,x(s),u^2_\alpha(s),\Psi^\star[u^2_\alpha](s))ds^\alpha+g(x(T)) +\varepsilon.
\end{split}$$
Define a new control $$u_\alpha\in \mathcal{U}(t), u_\alpha(s)\equiv
\left\{\begin{array}{ll}
u^1_\alpha(s) & s\in \Omega_{tt+h}\\
u^2_\alpha(s) & s\in \Omega_{tT}\setminus\Omega_{tt+h}.
\end{array}\right.$$
Then the inequalities $\eqref{eq.10}$ and $\eqref{eq.11}$ yield $$w(t,x)\leq \int_{\Gamma_{tT}} L_\alpha(s,x(s),u_\alpha(s),\Psi[u_\alpha](s))ds^\alpha+g(x(T)) +2\varepsilon,$$ and so $\eqref{eq.9}$ implies the inequality $$w(t,x)\leq m(t,x)+3\varepsilon.$$
This inequality and $m(t,x)\leq w(x,t)+2\varepsilon$ complete the proof.
**(boundedness and continuity of the values functions)** The lower, upper value function $m(t,x)$, $M(t,x)$ satisfy the boundedness conditions $$\vert m(t,x)\vert, \vert M(t,x)\vert\leq D$$ $$\vert m(t,x)-m(\hat{t},\hat{x})\vert, \vert M(t,x)-M(\hat{t},\hat{x})\vert\leq E\,\, \ell (\Gamma_{\hat{t}\,t})+ F\,\Vert x-\hat{x}\Vert,$$ for some constants $D, E, F$ and for all $t, \hat{t}\in \Omega_{0T}, x, \hat{x} \in \mathbb{R}^n.$
We prove only the statements for upper value function $M(t,x).$
Since $\vert g(x)\vert\leqslant B, \vert L_\alpha(t,x,u_\alpha,v_\alpha)\vert\leqslant C_\alpha, \alpha=\overline{1,m}$, we find $$\begin{split}
\vert J_{t,x}(u(\cdot),v(\cdot))\vert \ & =\Big\vert \int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha+g(x(T))\Big\vert \\& \leq \Big\vert \int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha \Big\vert + \vert g(x(T))\vert \\& \leq \int_{\Gamma_{tT}} \Vert L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))\Vert \Vert ds^\alpha \Vert +\vert g(x(T))\vert \\& \leq \Vert C \Vert \int_{\Gamma_{tT}} ds + B= \Vert C \Vert l(\Gamma_{tT}) +B\leq \Vert C \Vert l(\Gamma_{0T}) +B=D \\ & \Longrightarrow \vert M(t,x)\vert\leq D,
\end{split}$$ for all $u_\alpha(\cdot)\in \mathcal{U}(t),v_\alpha(\cdot)\in \mathcal{V}(t).$
Let $x_1,x_2\in \mathbb{R}^n,\,\, t_1, t_2 \in \Omega_{0T}.$ For $\varepsilon>0$ and the strategy $\Phi\in \mathcal{A}(t_1),$ we have
$$\label{eq:1}
M(t_1,x_1)\leq \min_{v_\alpha \in \mathcal{V}(t_1)} J(\Phi[v_\alpha],v_\alpha)+\varepsilon.$$
Define the control $$\overline{v}_\alpha\in \mathcal{V}(t_1),\overline{v}_\alpha(s)\equiv
\left\{\begin{array}{ll}
{v}_\alpha^1(s) & s\in \Omega_{0t_2}\setminus\Omega_{0t_1}\\
{v}_\alpha(s) & s\in \Omega_{0T}\setminus\Omega_{0t_2},
\end{array}\right.$$ for any $v_\alpha \in \mathcal{V}(t_2)$ and some $v_\alpha^1 \in V$ and for each $v_\alpha \in \mathcal{V}(t_2), \underline{\Phi}\in \mathcal{A}(t_2)$ (the restriction of $\Phi$ over $\Omega_{0T}\setminus \Omega_{0t_1})$ by $\underline{\Phi}[v_\alpha]=\Phi[\overline{v}_\alpha], s\in \Omega_{0T}\setminus\Omega_{0t_2}.$
Choose the control $v_\alpha\in \mathcal{V}(t_2)$ so that $$\label{eq:2}
M(t_2,x_2)\geq J(\underline{\Phi}[v_\alpha],v_\alpha)-\varepsilon.$$
By the inequality $\eqref{eq:1},$ we have $$\label{eq:3}
M(t_1,x_1)\leq J(\Phi[\overline{v}_\alpha],\overline{v}_\alpha)+\varepsilon.$$
We know that the (unique, Lipschitz) solution $x(\cdot)$ of the Cauchy problem $$\left\{\begin{array}{ll}
\frac{\partial x^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x(s),u_\alpha(s),v_\alpha(s))\\
x(t)=x, \quad s\in \Omega_{tT}\subset \mathbb{R}_+^m, x\in \mathbb{R}^n, i=\overline{1,n}, \alpha =\overline{1,m},
\end{array}\right.$$ is the response to the controls $u_\alpha(\cdot), v_\alpha(\cdot)$ for $s\in \Omega_{0T}.$
We choose $x_1(\cdot)$ as solution of the Cauchy problem $$\left\{\begin{array}{ll}
\frac{\partial x^i_1}{\partial s^\alpha}(s)=X^i_\alpha(s,x_1(s),\Phi[\overline{v}_\alpha](s),\overline{v}_\alpha(s))\\
x_1(t_1)=x_1, \quad s\in \Omega_{0T}\setminus \Omega_{0t_1}.
\end{array}\right.$$ Equivalently, $x_1(\cdot)$ is solution of integral equation $$x_1(s)= x_1(t_1) + \int_{\Gamma_{t_1s}}X_\alpha(\sigma,x_1(\sigma),\Phi[\overline{v}_\alpha](\sigma),\overline{v}_\alpha(\sigma))d\sigma^\alpha.$$ Take $x_2(\cdot)$ as solution of the Cauchy problem $$\left\{\begin{array}{ll}
\frac{\partial x^i_2}{\partial s^\alpha}(s)=X^i_\alpha(s,x_2(s),\underline{\Phi}[v_\alpha](s),v_\alpha(s))\\
x_2(t_2)=x_2, \quad s\in \Omega_{0T}\setminus \Omega_{0t_2}.
\end{array}\right.$$ Equivalently, $x_2(\cdot)$ is solution of integral equation $$x_2(s)= x_2(t_2) + \int_{\Gamma_{t_2s}}X_\alpha(\sigma,x_2(\sigma),\underline{\Phi}[v_\alpha](\sigma),\overline{v}_\alpha(\sigma))d\sigma^\alpha.$$
It follows that $$\Vert x_1(t_2)-x_1 \Vert = \Vert x_1(t_2)-x_1(t_1)\Vert \leq \Vert A\Vert \,\ell(\Gamma_{t_1t_2}).$$
Since $v_\alpha=\overline{v}_\alpha$ and $\underline{\Phi}[v_\alpha]=\Phi[\overline{v}_\alpha],$ for $s\in \Omega_{0T}\setminus\Omega_{0t_2}$, we find the estimation $$\begin{split}
\Vert x_1(s)-x_2(s)\Vert \ & \leq \Vert x_1(t_1) - x_2(t_2)\Vert + \Vert \int_{\Gamma_{t_1t_2}}\cdots \Vert\\ & \leq
\Vert A \Vert \ell(\Gamma_{t_1t_2})+ \Vert x_1-x_2\Vert
,\,\, \hbox{on}\,\, t_2\leq s\leq T.
\end{split}$$
Thus the inequalities $\eqref{eq:2}$ and $\eqref{eq:3}$ imply $$M(t_1,x_1)-M(t_2,x_2) \leq J(\Phi[\overline{v}_\alpha],\overline{v}_\alpha])-J(\underline{\Phi}[v_\alpha],v_\alpha])+2\varepsilon$$ $$\leq \Big\vert \int_{\Gamma_{t_1t_2}} L_\alpha (s,x_1(s),\Phi[\overline{v}_\alpha](s),\overline{v}_\alpha(s))ds^\alpha$$ $$+\int_{\Gamma_{t_2T}} (L_\alpha (s,x_1(s),\underline{\Phi}[v_\alpha](s),v_\alpha(s)) -L_\alpha (s,x_2(s),\underline{\Phi}[v_\alpha](s),v_\alpha(s)))ds^\alpha$$ $$+g(x_1(T))-g(x_2(T))+2\varepsilon\Big\vert$$ $$\leq\int_{\Gamma_{t_1t_2}} \vert L_\alpha (s,x_1(s),\Phi[\overline{v}_\alpha](s),\overline{v}_\alpha(s))ds^\alpha\vert$$ $$+\int_{\Gamma_{t_2T}} \vert (L_\alpha (s,x_1(s),\underline{\Phi}[v_\alpha](s),v_\alpha(s)) -L_\alpha (s,x_2(s),\underline{\Phi}[v_\alpha](s),v_\alpha(s)))ds^\alpha\vert$$ $$+\vert g(x_1(T))-g(x_2(T))\vert +2\varepsilon$$ $$\leq \Vert C \Vert \ell(\Gamma_{t_1t_2}) +\Vert C \Vert \ell(\Gamma_{t_2T})\,(\Vert A \Vert \ell(\Gamma_{t_1t_2})+ \Vert x_1-x_2\Vert) +B\, \Vert x_1-x_2\Vert) +2\varepsilon$$ $$\leq \Vert C \Vert \ell(\Gamma_{t_1t_2}) +\Vert C \Vert \ell(\Gamma_{0T})\,(\Vert A \Vert \ell(\Gamma_{t_1t_2})+ \Vert x_1-x_2\Vert) +B\, \Vert x_1-x_2\Vert) +2\varepsilon.$$
Since $\varepsilon$ is arbitrary, we obtain the inequality $$\label{eq:7}
M(t_1,x_1)-M(t_2,x_2)\leq E\,\ell(\Gamma_{t_1t_2}) + F \,\Vert x_1-x_2\Vert.$$
Let $\varepsilon>0$ and choose the strategy $\Phi\in \mathcal{A}(t_2)$ such that $$\label{eq:4}
M(t_2,x_2)\leq \min_{v_\alpha \in \mathcal{V}(t_2)} J(\Phi[v_\alpha],v_\alpha)+\varepsilon.$$
For each control $v_\alpha \in \mathcal{V}(t_1)$ and $s\in \Omega_{0T}\setminus\Omega_{0t_2},$ define the control $\underline{v}_\alpha\in \mathcal{V}(t_2), \underline{v}_\alpha(s)=v_\alpha(s).$
For some $u^1_\alpha \in U,$ we define the strategy $\overline{\Phi}\in \mathcal{A}(t_1)$ (the restriction of $\Phi$ over $\Omega_{0T}\setminus\Omega_{0t_2}$) by $$\overline{\Phi}[{v}_\alpha]=
\left\{\begin{array}{ll}
u^1_\alpha & s\in \Omega_{0t_2}\setminus\Omega_{0t_1}\\
\Phi[\underline{v}_\alpha] & s\in \Omega_{0T}\setminus\Omega_{0t_2}.
\end{array}\right.$$
Now choose a control $v_\alpha\in \mathcal{V}(t_1)$ so that $$\label{eq:5}
M(t_1,x_1)\geq J(\overline{\Phi}[v_\alpha],v_\alpha)-\varepsilon.$$
By the inequality $\eqref{eq:4},$ we have $$\label{eq:6}
M(t_2,x_2)\leq J(\Phi[\underline{v}_\alpha],\underline{v}_\alpha)+\varepsilon.$$
We choose $x_1(\cdot)$ as solution of the Cauchy problem (PDE system + initial condition) $$\left\{\begin{array}{ll}
\frac{\partial x_1^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x_1(s),\overline{\Phi}[v_\alpha],v_\alpha(s)), s\in \Omega_{0T}\setminus\Omega_{0t_1} \\
x_1(t_1)=x_1, \quad s\in \Omega_{0T}\setminus \Omega_{0t_1}
\end{array}\right.$$ and $x_2(\cdot)$ as solution of the Cauchy problem (PDE system + initial condition) $$\left\{\begin{array}{ll}
\frac{\partial x_2^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x_2(s),\Phi[\underline{v}_\alpha],\underline{v}_\alpha(s)), s\in \Omega_{0T}\setminus\Omega_{0t_2}\\
x_2(t_2)=x_2, \quad s\in \Omega_{0T}\setminus \Omega_{0t_2}.
\end{array}\right.$$ Using the associated integral equations, it follows that $$\Vert x_1(t_2)-x_1 \Vert = \Vert x_1(t_2)-x_1(t_1)\Vert \leq \Vert A\Vert \,\ell(\Gamma_{t_1t_2}).$$ Also, for $s\in \Omega_{0T}\setminus\Omega_{0t_2}, v_\alpha=\underline{v}_\alpha$ and $\overline{\Phi}[v_\alpha]=\Phi[\underline{v}_\alpha],$ we find $$\begin{split}
\Vert x_1(s)-x_2(s)\Vert \ & \leq \Vert x_1(t_1) - x_2(t_2)\Vert + \Vert \int_{\Gamma_{t_1t_2}}\cdots \Vert\\ & \leq
\Vert A \Vert \ell(\Gamma_{t_1t_2})+ \Vert x_1-x_2\Vert,\,\, \hbox{on}\,\, t_2\leq s\leq T.
\end{split}$$
Thus, the relations $\eqref{eq:5}$ and $\eqref{eq:6}$ imply $$M(t_2,x_2)-M(t_1,x_1) = J(\overline{\Phi}[v_\alpha],v_\alpha])-J(\Phi[\underline{v}_\alpha],\underline{v}_\alpha])+2\varepsilon$$ $$= - \int_{\Gamma_{t_1t_2}} L_\alpha (s,x_1(s),\overline{\Phi}[{v}_\alpha](s),{v}_\alpha(s))ds^\alpha$$ $$+\int_{\Gamma_{t_2T}} (L_\alpha (s,x_1(s),{\Phi}[\underline{v}_\alpha](s),\underline{v}_\alpha(s)) - L_\alpha (s,x_2(s),{\Phi}[\underline{v}_\alpha](s),\underline{v}_\alpha(s)))ds^\alpha$$ $$+g(x_1(T))-g(x_2(T))+2\varepsilon$$ $$\leq \Vert C \Vert \ell(\Gamma_{t_1t_2}) +\Vert C \Vert \ell(\Gamma_{0T})\,(\Vert A \Vert \ell(\Gamma_{t_1t_2})+ \Vert x_1-x_2\Vert) +B\, \Vert x_1-x_2\Vert) +2\varepsilon.$$
Since $\varepsilon$ is arbitrary, we obtain the inequality $$\label{eq:8}
M(t_2,x_2)-M(t_1,x_1)\leq E\,\ell(\Gamma_{t_1t_2}) + F \,\Vert x_1-x_2\Vert.$$
By $2.17$ and $2.22$, we proved the continuity of the lower and upper value functions.
Viscosity solutions of\
multitime (HJIU) PDEs
=======================
**(PDEs for multitime upper value function, resp. multitime lower value function)**
The multitime upper value function $M(t,x)$ and the multitime lower value function $m(t,x)$ are the viscosity solutions of Hamilton-Jacobi-Isaacs-Udrişte (HJIU) PDEs:
- the multitime upper (HJIU) PDEs $$\frac{\partial M}{\partial t^\alpha}(t,x)+\min_{v_\alpha\in \mathcal{V}} \max_{u_\alpha\in \mathcal{U}} \left\lbrace \frac{\partial M}{\partial x^i}(t,x) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\right\rbrace =0,$$ with the terminal condition $M(T,x)=g(x),$
- the multitime lower (HJIU) PDEs $$\frac{\partial m}{\partial t^\alpha}(t,x)+\max_{u_\alpha \in \mathcal{U}} \min_{v_\alpha \in \mathcal{V}} \left\lbrace \frac{\partial m}{\partial x^i}(t,x) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\right\rbrace =0,$$ with the terminal condition $m(T,x)=g(x).$
If we introduce the so-called upper and lower Hamiltonian $1$-forms defined respectively by $$H^+_\alpha(t,x,p)=\min_{v_\alpha\in \mathcal{V}} \max_{u_\alpha \in \mathcal{U}}\lbrace p_i(t) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\rbrace,$$ $$H^-_\alpha(t,x,p)=\max_{u_\alpha\in \mathcal{U}} \min_{v_\alpha\in \mathcal{V}}\lbrace p_i(t) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\rbrace,$$ then the multitime (HJIU) PDE systems can be written in the form $$\frac{\partial M}{\partial t^\alpha}(t,x)+H^+_{\alpha}\left( t,x,\frac{\partial M}{\partial x}(t,x)\right) =0$$ and $$\frac{\partial m}{\partial t^\alpha}(t,x)+H^-_\alpha\left( t,x,\frac{\partial m}{\partial x}(t,x)\right) =0.$$
The proof will be given in another paper.
Representation formula of viscosity\
solutions for multitime (HJ) PDEs
====================================
In this section, we want to obtain a representation formula for the viscosity solution $M(t,x)$ of the multitime (HJ) PDEs system $$\frac{\partial M}{\partial t^\alpha}+H_\alpha\left( t,x,\frac{\partial M}{\partial x}(t,x)\right) =0, (t,x)\in \Omega_{0T}\times \mathbb{R}^n,\alpha=\overline{1,m},$$ $$M(0,x)=g(x), x\in \mathbb{R}^n \,(\hbox{initial\, condition}),$$ where the unique solution $M(t,x)$ satisfies the inequalities $$\label{eq:11}
\left\{\begin{array}{ll}
\vert M(t,x)\vert\leq D\\
\vert M(t,x)-M(\hat{t},\hat{x})\vert\leq E\,\,\ell(\Gamma_{t\hat t})+ F\,\,\Vert x-\hat{x}\Vert,
\end{array}\right.$$ for some constants $D, E, F$ (for $m=1,$ see also \[4\]).
Also, we assume that $g:\mathbb{R}^n \rightarrow \mathbb{R}, H_\alpha:\Omega_{0T} \times \mathbb{R}^n\times \mathbb{R}^p\rightarrow \mathbb{R},$ satisfy the inequalities $$\left\{\begin{array}{ll}
\vert g (x)\vert\leq B\\
\vert g(x)-g (\hat{x})\vert\leq B \Vert x-\hat{x}\Vert
\end{array}\right.$$ and $$\label{eq:8}
\left\{\begin{array}{ll}
\vert H_\alpha(t,x,0)\vert\leq K_\alpha\\
\vert H_\alpha (t,x,p)-H_\alpha(\hat{t},\hat{x},\hat{p})\vert\leq K_\alpha\,\, (\ell(\Gamma_{t\hat t})+\Vert x-\hat{x}\Vert +\Vert p-\hat{p}\Vert).
\end{array}\right.$$
**Max-min representation of a Lipschitz function as affine functions** (for $m=1,$ see also \[2\], \[3\]).
\[l-2\]
For each $\alpha$, let $$\label{eq:10}
\left\{\begin{array}{ll}
U=B(0,1)\subset \mathbb{R}^{n}\\
V=B(0,P)\subset \mathbb{R}^{n}\\
X_\alpha(u_\alpha)=K_\alpha u_\alpha,\, K_\alpha \in \mathbb{R}\\
L_\alpha(t,x,u_\alpha,v_\alpha)=H_\alpha (t,x,v_\alpha)-<K_\alpha u_\alpha,v_\alpha>.
\end{array}\right.$$ Let $H_\alpha$ be a Lipschitz 1-form. For some constant $P>0$ and for each $t\in \Omega_{0T},\,\, x \in \mathbb{R}^n,$ we have $$H_\alpha (t,x,{p})=\max_{v_\alpha \in V}\min_{u_\alpha \in U}\left\lbrace <X_\alpha(u_\alpha),{p}> + L_\alpha(t,x,u_\alpha,v_\alpha)\right\rbrace ,$$ if $\Vert {p}\Vert \leq P$.
In view of the assumption $H_\alpha (t,x,v_\alpha)-H_\alpha(t,x,{p})\leq K_\alpha \Vert {p}-v_\alpha\Vert,$ by the Cauchy-Schwarz formula, and by the condition $||u||\leq 1$, we have for any $x\in \mathbb{R}^n,$ $$\begin{split}
H_\alpha (t,x,{p})\ & =\max_{v_\alpha \in V} \left\lbrace H_\alpha(t,x,v_\alpha) - K_\alpha\Vert {p}-v_\alpha\Vert\right\rbrace \\ & =\max_{v_\alpha \in V}\min_{u_\alpha\in U}\left\lbrace H_\alpha(t,x,v_\alpha)\,+\, <K_\alpha u_\alpha,{p}-v_\alpha>\right\rbrace.
\end{split}$$
**Max-min representation of a Lipschitz function as positive homogeneous functions** (for m=1, see also \[2\],\[3\]).
Let $H_\alpha$ be a Lipschitz $1$-form which is homogeneous in $p,$ i.e., $$H_\alpha(t,x,\lambda p)=\lambda H_\alpha(t,x,p),\,\,\lambda \geq 0.$$ Then there exist compact sets $U, V\subset \mathbb{R}^{2n}$ and vector fields $$X_\alpha:[0,T]\times \mathbb{R}^n\times U\times V\rightarrow \mathbb{R}^n$$ satisfying $$\Vert X_\alpha(x)-X_\alpha(\hat{x})\Vert\leqslant A_\alpha\Vert x-\hat{x}\Vert$$ and such that, for each $\alpha$, $$H_\alpha (t,x,p)=\max_{v_\alpha \in V}\min_{u_\alpha \in U}\left\lbrace <X_\alpha(t,x,u_\alpha,v_\alpha),p> \right\rbrace ,$$ for all $t\in \Omega_{0T},x\in \mathbb{R}^n,p \in \mathbb{R}^n.$
Let $u_\alpha=(u^1_\alpha,u^2_\alpha),v_\alpha=(v^1_\alpha,v^2_\alpha)$ ($2n$-dimensional controls) and $$\label{eq:9}
\left\{\begin{array}{ll}
U=V=B(0,1)\times B(0,1)\subset \mathbb{R}^{2n}\\
L_\alpha(t,x,u^1_\alpha,v^1_\alpha)=H_\alpha (t,x,v_\alpha^1)-<K_\alpha u^1_\alpha,v_\alpha^1>\\
X_\alpha(t, x, u_\alpha, v_\alpha)=K_\alpha u^1_\alpha+ C_\alpha v^2_\alpha+ (L_\alpha(t,x,u^1_\alpha,v^1_\alpha)- C_\alpha)u^2_\alpha.\\
\end{array}\right.$$ According to Lemma $\eqref{l-2}$ and the assumptions $\eqref{eq:9},$ if $\Vert\eta\Vert =1,$ we have $$\begin{split}
H_\alpha (t,x,\eta)\ & =\max_{v^1_\alpha \in V^1}\min_{u^1_\alpha\in U^1}\left\lbrace <K_\alpha u^1_\alpha,\eta> + L_\alpha (t,x,u^1_\alpha, v^1_\alpha)\right\rbrace,
\end{split}$$ for $U^1=V^1=B(0,1)\in \mathbb{R}^n$.
For any $p\neq 0$, we can write $$\begin{split}
H_\alpha(t,x,p)\ & =\Vert p\Vert H_\alpha \left( t,x,\frac{p}{\Vert p\Vert}\right) \\ & =\max_{v^1_\alpha\in V^1}\min_{u^1_\alpha\in U^1}\left\lbrace <K_\alpha u^1_\alpha,p>+L_\alpha (t,x,u^1_\alpha,v^1_\alpha)\Vert p\Vert\right\rbrace .
\end{split}$$ Then, if we choose $C_\alpha>0$ such that $\vert L_\alpha\vert\leq C_\alpha,$ we find $$\begin{split}
H_\alpha (t,x,p)\ & =\max_{v^1_\alpha \in V^1}\min_{u^1_\alpha\in U^1}\bigg\{ <K_\alpha u^1_\alpha,p> +C_\alpha\Vert p\Vert +( L_\alpha (t,x,u^1_\alpha, v^1_\alpha)-C_\alpha)\Vert p\Vert\bigg\}\\ & =\max_{v^1_\alpha \in V^1}\min_{u^1_\alpha\in U^1}\max_{v^2_\alpha \in V^1}\min_{u^2_\alpha\in U^1}\bigg\{ <K_\alpha u^1_\alpha,p> +<C_\alpha v^2_\alpha, p>\\ & +( L_\alpha (t,x,u^1_\alpha, v^1_\alpha)-C_\alpha)< u^2_\alpha,p> \bigg\} \\ & =\max_{v_\alpha \in V}\min_{u_\alpha\in U}\bigg\{ <X_\alpha(t,x,u_\alpha, v_\alpha),p>\bigg\}.
\end{split}$$ Now, interchanging $\min_{u_\alpha^1\in U^1}$ and $\max_{v_\alpha^2\in V^1}$, the result in Lemma follows.
We are now in a position to give the main result of this section.
For each $t \in \Omega_{0T}$ and $x \in \mathbb{R}^n,$ the upper value function $M(t,x)$ verifies the equality $$\begin{split}
M(t,x)=\max_{\Phi\in \mathcal{U}(T-t)}\min_{v_\alpha\in V(T-t)}\bigg\{ \ & - \int_{\Gamma_{T-tT}} L_\alpha(T-s,x(s),\Phi[v_\alpha](s),v_\alpha(s))ds^\alpha \\ & +g(x(T))\bigg\} ,
\end{split}$$ where for each pair of controls $v_\alpha\in V(T-t)$, $u_\alpha=\Phi[v_\alpha]\in U(T-t),$ the state function $x(\cdot)$ solves the problem $$\left\{\begin{array}{ll}
\frac{\partial x^i}{\partial s^\alpha}(s)=-F^i_\alpha u_\alpha(s), s\in \Omega_{0T}\setminus \Omega_{0T-t}\\
x(T-t)=x.
\end{array}\right.$$
Let $$H^1_\alpha(t,x,p)=\max_{v_\alpha\in V}\min_{u_\alpha\in U}\left\lbrace <X_\alpha(u_\alpha),p>+L_\alpha(t,x,u_\alpha,v_\alpha)\right\rbrace,$$ $U=B(0,1)\subset \mathbb{R}^{pm}, V=B(0,P)\subset \mathbb{R}^{qm}$ and $X^i_\alpha, L_\alpha$ Lipschitz functions with the assumptions $\eqref{eq:10}.$
Then $H_\alpha(t,x,p)=H^1_\alpha(t,x,p)$ provided $\vert p\vert\leq P.$ Since $M(t,x)$ satisfies $\eqref{eq:11},$ it follows that $M(t,x)$ is also the unique viscosity solution of the multitime (HJ) PDEs system (for $m=1,$ see also \[4\]) $$\frac{\partial M}{\partial t^\alpha}+H^1_\alpha\left( t,x,\frac{\partial M}{\partial x}(t,x)\right) =0, \,\,(t,x)\in \Omega_{0T}\times \mathbb{R}^n,\alpha=\overline{1,m},$$ $$M(0,x)=g(x),\,\, x\in \mathbb{R}^n.$$ If we take $M^1(t,x)=M(T-t,x),$ one observes that $M^1(t,x)$ is a viscosity solution of this system (for $m=1,$ see also \[2\])
$$\frac{\partial M^1}{\partial t^\alpha}+H^+_\alpha\left( t,x,\frac{\partial M^1}{\partial x}(t,x)\right) =0,\,\, (t,x)\in \Omega_{0T}\times \mathbb{R}^n,\alpha=\overline{1,m},$$
$$M^1(T,x)=g(x),\,\, x\in \mathbb{R}^n$$
and $$H^+_\alpha(t,x,p)=\max_{v_\alpha\in V}\min_{u_\alpha\in U}\left\lbrace -<X_\alpha(u_\alpha),p>+L_\alpha(T-t,x,u_\alpha,v_\alpha)\right\rbrace.$$ Using the above developments, we obtain $$\begin{split}
M^1(t,x)=M(t,x)=\max_{\Phi\in \mathcal{U}(t)}\min_{v_\alpha\in V(t)}\bigg\{ \ & - \int_{\Gamma_{tT}} L_\alpha(T-s,x(s),\Phi[v_\alpha](s),v_\alpha(s))ds^\alpha \\ & +g(xT))\bigg\} ,
\end{split}$$ where $x(\cdot)$ is the solution of the Cauchy problem $$\left\{\begin{array}{ll}
\frac{\partial x^i}{\partial s^\alpha}(s)=-X^i_\alpha(u_\alpha(s))=-F^i_\alpha u_\alpha(s), s\in \Omega_{0T}\setminus \Omega_{0T-t}\\
x(t)=x,
\end{array}\right.$$ for the control $u_\alpha(\cdot)=\Phi[v_\alpha].$
[99]{} L. C. Evans, *An Introduction to Mathematical Optimal Control Theory*, Lectures Notes, University of California, Departament of Mathematics, Berkeley, (2005). L. C. Evans, P. E. Souganidis, *Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations*, Indiana University Mathematics Journal, 33, 5, (1984), 773-797. M. G. Crandall, L. C. Evans, P. L. Lions, *Some properties of viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc., 282, 2, (1984), 487-502. M. G. Crandall, P. L. Lions, *Viscosity Solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc., 277, (1983), 1-42. E. N. Barron, L. C. Evans, R. Jensen, *Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls*, Journal of Differential Equations, 53, (1984), 213-233. L. G' omez Esparza, G. Mendoza Torres, L. M. Saynes Torres, *A Brief Introduction to Differential Games*, International Journal of Physical and Mathematical Sciences, 4, 1, (2013). G. Jank, *Introduction to Non-cooperative Dynamical Game Theory*, Coimbra, (2001). P. E. Souganidis, *Existence of viscosity solutions of Hamilton-Jacobi equations*, Journal of Differential Equations, 56, (1985), 345-390. K. Margellos, J. Lygeros, *Hamilton-Jacobi formulation for reach-avoid differential games*, IEEE Trans. Automat. Contr., 56, 8, (2011), 1849-1861. C. Udrişte, I. Ţevy, *Multi-time Euler-Lagrange-Hamilton theory*, WSEAS Trans. Math., 6, 6, (2007), 701-709. C. Udrişte, *Multitime stochastic control theory*, in Selected Topics on Circuits, Systems, Electronics, Control and Signal Processing, Proc. of the 6-th WSEAS International Conference on Circuits, Systems, Electronics, Control an Signal Processing (CSECS’07), Cairo, Egypt, December 29-31, (2007), 171-176. C. Udrişte, *Multi-time controllability, observability and bang-bang principle*, J. Optim. Theory Appl., 138, 1 (2008), 141–157. C. Udriste, L. Matei, I. Duca, [*Multitime Hamilton-Jacobi Theory*]{}, Proceedings of the 8th WSEAS International Conference on Applied Computer and Applied Computational Science, 509-513, 2009. C. Udrişte, *Equivalence of multitime optimal control problems*, Balkan J. Geom. Appl. 15, 1, (2010), 155-162. C. Udrişte, *Simplified multitime maximum principle*, Balkan J. Geom. Appl. 14, 1, (2009), 102-119. C. Udrişte, I. Ţevy, *Multitime dynamic programming for curvilinear integral actions*, J. Optim. Theory and Appl., 146, (2010), 189-207. C. Udrişte, L. Matei, *Lagrange-Hamilton Theories* (in Romanian), Monographs and Textbooks 8, Geometry Balkan Press, Bucharest, (2008). C. Udrişte, A. Bejenaru, *Multitime optimal control with area integral costs on boundary*, Balkan J. Geom. Appl., 16, 2, (2011), 138-154 C. Udrişte, *Multitime maximum principle for curvilinear integral cost*, Balkan J. Geom. Appl., 16, 1, (2011), 128-149. C. Udrişte, I. Ţevy, *Multitime dynamic programming for multiple integral actions*, J. Glob. Optim., 51, 2, (2011), 345-360. A. W. Starr, *Nonzero-sum differential games: concepts and models*, Division of Engineering and Applied Physics Harvard University-Cambridge, Massachusetts, Technical Report, 590, (1969). A. Davini, M. Zavidovique, *On the (non) existence of viscosity solutions of multi-time Hamilton-Jacobi equations,* Preprint (2013), [http://www.math.jussieu.fr/ zavidovique/articles/NonCommutingNov2013.pdf]{}
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**Higher order potential expansion for the continuous limits of the Toda hierarchy** {#higher-order-potential-expansion-for-the-continuous-limits-of-the-toda-hierarchy .unnumbered}
======================================================================================
Introduction
============
The continuous limits of discrete systems are one of the remarkably important research areas in soliton theory [@Toda-73; @Case-74; @Kupershmidt-85; @Ablowitz-81]. In recent years, more attention was focused on the continuous limit relations between hierarchies of discrete systems and hierarchies of soliton equations [@Zeng-95a; @Zeng-95b; @Morosi-96; @Morosi-98a; @Morosi-98b]. The so-called recombination method, i.e., properly combining the objects (such as the vector fields) of discrete systems, was first proposed to study the continuous limit of the Ablowitz-Ladik hierarchy [@Zeng-95a] and the Kac-van Moerbeke hierarchy [@Zeng-95b]. Morosi and Pizzocchero also used the recombination method to study the continuous limits of some integrable lattices in their recent works [@Morosi-96; @Morosi-98a; @Morosi-98b]. Up to now, there has not been much work concerning the continuous limit relations between lattices and differential equations, which have different numbers of potentials. Furthermore, to the best of our knowledge, there is no work which successfully gives a way to introduce the higher order terms in potential expansion to study the continuous limit relations between hierarchies of lattices and hierarchies of soliton equations. Illumined by Gieseker’s conjecture [@Gieseker], we will propose a method for finding the higher order terms in potential expansion to study the continuous limit relation between the Toda hierarchy and the KdV hierarchy by the recombination method.
In 1996, Gieseker proposed a conjecture [@Gieseker]:
[**Conjecture.**]{}
*Denote $w(n,t)$ and $v(n,t)$, where $n\in\mathbb Z$ and $t\in\mathbb R$, are the two potentials of the Toda hierarchy, and let $f$ be a function of $x\in \mathbb R$ and $t\in \mathbb R$. There are $\Phi_i(f)$’s, which are the differential polynomials of $f$, so that if we define*
\[w-v-q\] $$w(n,t)=-2+ f(x,t) h^2+h^2\sum_{i=1}^L \Phi_i(f(x,t)) h^i,$$ $$v(n,t)=1+ f(x,t)h^2-h^2\sum_{i=1}^L \Phi_i(f(x,t)) h^i,$$
where $h$ is the small step of lattice and $x=nh$, then by taking suitable linear combinations of the equations of Toda hierachy under the definition (\[w-v-q\]), we can produce asymptotic series whose leading terms in $h$ are the KdV hierarchy if $L$ is large enough.
In [@Gieseker], Gieseker proposed a way to introduce $\Phi_i(f)$ by using the Toda lattice $$\label{toda}
w_t= v-Ev=v-v^{(1)}, \qquad v_t=v(E^{(-1)}w-w)=v(w^{(-1)}-w),$$ where the shift operator $E$ is defined by $$(E f) (n) = f(n+1), \quad
f^{(k)} (n) =E^{(k)} f (n)=f(n+k), \quad n, k\in {\mathbb Z}.$$ For instance, in order to find $\Phi_1(f)$, substituting the definition (\[w-v-q\]) into the equation (\[toda\]) and expanding the shift terms out by Taylor’s theorem
$$\label{Phi_1a}
\frac{d f}{dt} + \frac{d \Phi_1(f)}{dt} h =
-\frac{d f}{dx} h - \frac{d^2 f}{2dx^2} h^2 + \frac{d \Phi_1(f)}{dx} h^2 +O(h^3),$$
$$\frac{d f}{dt} - \frac{d \Phi_1(f)}{dt} h =
-\frac{d f}{dx} h + \frac{d^2 f}{2dx^2} h^2 - \frac{d \Phi_1(f)}{dx} h^2 +O(h^3).$$
Combining the above two equations we know $$\frac{d f}{dt} = -\frac{d f}{dx} h + O(h^3),$$ then by the chain rule we have $$\frac{d \Phi_1(f)}{dt} = -\frac{d \Phi_1(f)}{dx} h + O(h^2),$$ Notice the above equation and the equation (\[Phi\_1a\]) one can get $$\frac{d \Phi_1(f)}{dx} = \frac 1 4 \frac{d^2 f}{dx^2},$$ by integration it yields $$\Phi_1(f) = \frac 1 4 \frac{d f}{dx}.$$ We can see that the integration must be used in this process for finding $\Phi_i(f)$. As a consequence, there is a problem that whether this process can be continued indefinitely and the $\Phi_i(f)$’s, found in this process, are the differential polynomials of $f$.
The Gieseker’s conjecture were proved in the following three cases of (\[w-v-q\]) [@ZLC]: $$\mbox{(a)} \qquad L=0, \quad f(x,t)=\frac12 q(x,t);$$ $$\mbox{(b)} \qquad L=1, \quad f(x,t)=\frac12 q(x,t), \quad \Phi_1(f)=\frac{1}{8} q_x;$$ $$\mbox{(c)} \qquad L=2, \quad f(x,t)=\frac12 q(x,t), \quad \Phi_1(f)=\frac{1}{8} q_x,
\quad \Phi_2(q)=-\frac{1}{32} q^2.$$ It was found that the fewer equations in the Toda hierarchy are needed in the recombination method for the case (c) to give the KdV hierarchy than for the case (a).
In this paper, we will give a new method to introduce $\Phi_i(f)$ required in (\[w-v-q\]) instead of the Gieseker’s process in order that we can derive the continuous limit relation between the Toda hierarchy and the KdV hierarchy by the recombination method. Following our approach for finding $\Phi_i(f)$, one can easily see that the $\Phi_i(f)$’s are all differential polynomials of $f$. Compared with the previous work in [@ZLC], we will show that the fewer equations in the Toda hierarchy are needed in the recombination method for giving the KdV hierarchy if higher order terms are introduced in the potential expansion (\[w-v-q\]). We will also present that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the KdV hierarchy in continuous limit.
Basic notation and some known results
=====================================
For latter use, we list some notation and results in [@ZLC]. Let us consider the following discrete isospectral problem [@TUTD; @Newell], $$\label{TDEIGEN}
L y = (E + w + v E^{(-1)}) y =\lambda y,$$ where $w=w(n,t)$ and $v=v(n,t)$ depend on integer $n\in {\mathbb Z}$ and real variable $t \in {\mathbb R},$ and $\lambda$ is the spectral parameter.
The equation in the Toda hierarchy associated with (\[TDEIGEN\]) can be written as the following Hamiltonian equation [@TUTD] $$\label{TDHIERARCHY}
{\left( \begin{array}{c} w \\ v \end{array} \right)}_{t_m}
=J K_{m+1}=J \frac{\delta H_{m+1}}{\delta u}, \quad
m=0,1,...,$$ where $\frac{\delta}{\delta u}=
{(\frac{\delta}{\delta w}, \frac{\delta}{\delta v})}^{T},$ and the Poisson tensor $J$ and the Hamiltonians $H_i$ are defined by $$J\equiv \left( \begin{array}{cc}
0 & J_{12} \\ J_{21} & 0 \end{array} \right)
\equiv \left( \begin{array}{cc}
0 & (1-E) v \\ v(E^{(-1)} -1) & 0 \end{array} \right),$$ $$\label{Ki}
K_i\equiv \left(\begin{array}{c} K_{i,1} \\ K_{i,2} \end{array} \right)
=\frac{\delta H_{i}}{\delta u}=
\left( \begin{array}{c} -b_i^{(1)} \\ \frac{a_i}{v} \end{array}
\right), \quad i=0,1,...,$$ $$H_0=\frac{1}{2}\ln v, \quad H_i=-\frac{b_{i+1}}{i}, \quad
i=1,2,...,$$ with $a_0=\frac{1}{2},$ $b_0=0,$ and $$\label{ab}
b_{i+1}^{(1)}=w b_i^{(1)} -(a_i^{(1)}+a_i), \qquad
a_{i+1}^{(1)}-a_{i+1} = w(a_i^{(1)}-a_i)+vb_i-v^{(1)}b_i^{(2)},$$ for $i=0,1,....$ The Lax pairs for the $m$th equation of the Toda hierarchy (\[TDHIERARCHY\]) are given by (\[TDEIGEN\]) and $$y_{t_m} = A_m y = \sum_{i=0}^m (- v b_i^{(1)} E^{(-1)}- a_i )
(E + w + v E^{(-1)})^{m-i} y, \qquad m = 0,1,....$$ The equations (\[TDHIERARCHY\]) have the bi-Hamiltonian formulation $$\label{TDBIHS}
G K_{i-1} = J K_i, \quad i=1,2,...,$$ $$G \equiv \left( \begin{array}{cc}
vE^{(-1)}-v^{(1)}E & w(1-E)v \\
v(E^{(-1)}-1)w & v(E^{(-1)}-E)v \end{array} \right),$$ where $G$ is the second Poisson tensor. The Toda hierarchy also has a tri-Hamiltonian formulation and a Virasoro algebra of master symmetries [@Ma-JMP; @Ma-99]. The first four covariants $K_i$’s are $$K_0 = \left( \begin{array}{c}
0 \\ \frac{1}{2v} \end{array} \right), \quad
K_1 = \left( \begin{array}{c}
1 \\ 0 \end{array} \right), \quad
K_2 = \left( \begin{array}{c}
w \\ 1 \end{array} \right), \quad
K_3 = \left( \begin{array}{c}
v+v^{(1)}+w^2 \\ w+w^{(-1)} \end{array} \right).$$
The Schrödinger spectral problem is given by $$\label{KDVEIGEN}
\overline L \overline y = (\partial^2_x+q) \overline y
= -\overline\lambda \overline y.$$ which is associated with the KdV hierarchy [@Newell] $$\label{KDVHIERARCHY}
q_{t_m} = B_0 P_m=B_0 \frac{\delta \overline H_m}{\delta q},
\qquad m=0,1,...,$$ where the vector field possesses the bi-Hamiltonian formulation with two Poisson tensors $B_0$ and $B_1$ $$\label{KdVBIHS}
B_0 P_{k+1} = B_1 P_k, \quad k=0,1,...,$$ $$B_0 = \partial\equiv\partial_x, \quad
B_1 = \frac{1}{4} \partial^3+q \partial+
\frac{1}{2} q_x, \quad
\overline H_i = \frac{4 \bar b_{i+2}}{2i+1},\quad
i = 0,1,...,$$ with $\bar b_0=0,$ $\bar b_1=1,$ and $$\bar b_{i+1}=(\frac{1}{4}\partial^2+q-
\frac{1}{2}\partial^{-1}q_x)\bar b_i,
\quad i=0,1,...,$$ where $\partial^{-1}\partial=\partial\partial^{-1}=1$. The first three covariants $P_k$’s read as $$P_0 = 2, \quad P_1=q, \quad
P_2 = \frac{1}{4}(3{q}^2+q_{xx}).$$ The well-known KdV equation is the second one: $$\label{KDV}
q_{t_2}=\frac{1}{4}{(3 q^2+q_{xx})}_x.$$ The Lax pairs for the $m$th equation of the KdV hierarchy (\[KDVHIERARCHY\]) are given by (\[KDVEIGEN\]) and $$\label{KdV-Lax-t}
\overline y_{t_m} = \overline A_m \overline y
= \sum_{i=0}^m (- \frac{1}{2}\overline b_{i,x} + \overline b_i \partial)
(\partial^2+q)^{m-i} \overline y, \qquad m=0,1,....$$
Let us consider the Toda hierarchy on a lattice with a small step $h$. We interpolate the sequences $(w(n))$ and $(v(n))$ with two smooth functions of a continuous variable $x$, and relate $w(n)$ and $v(n)$ to $f(x)$ by using (\[w-v-q\]). Suppose $$E^{(k)} w(n) = -2+f(x+kh)h^2
+h^2\sum_{i=1}^L \Phi_i(f(x+kh)) h^i,$$ $$E^{(k)} v(n) = 1+f(x+kh)h^2
-h^2\sum_{i=1}^L \Phi_i(f(x+kh)) h^i,
\qquad k\in\mathbb Z.$$
In [@ZLC], we got the following result.
\[prop-spectral\] Under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$, the Lax operator of the Toda hierarchy goes to the Lax operator of the KdV hierarchy in continuous limit, i.e., we have $$L = \overline L h^2 + O(h^3),$$
\[lemma-Ki\] Under the relation (\[w-v-q\]), we have $$\label{KIEXPAND}
K_i = \left( \begin{array}{c} -b_i^{(1)} \\ \frac{a_i}{v} \end{array}
\right)
=\left( \begin{array}{c} \alpha_i \\ \gamma_i \end{array}
\right) + O(h), \qquad i=0,1,...,$$ where $\alpha_i$ and $\gamma_i$ are given by
\[w1\] $$\label{w1-0-1}
\alpha_0 = 0, \qquad \alpha_1 = 1, \qquad
\gamma_0=\frac{1}{2}, \qquad \gamma_1=0,$$ $$\alpha_i = (-1)^{(i-1)} C_{2i-2}^{i-1},
\qquad
\gamma_i = (-1)^i C_{2i-2}^{i},
\qquad i=2,3,....$$
Define $\widetilde J= \left( \begin{array}{cc} 0 &\widetilde J_{21}
\\\widetilde J_{12} & 0
\end{array} \right)$ by requiring that $J\widetilde J=I.$ Then the following lemma is true.
\[lemma-TKi\] Under the relation (\[w-v-q\]), we have $$\label{w2}
TK_i \equiv\widetilde JG K_i=K_{i+1}+\delta_{i+1} K_0, \quad
i = 0,1,...,$$ where $$\label{w3}
\delta_i = -2(\alpha_i+\gamma_i)
= (-1)^i \frac{2}{i} C_{2i-2}^{i-1}, \quad i=1,2,....$$
\[prop-Poisson\] Under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$, the Poisson tensors of the Toda hierarchy go to those of the KdV hierarchy in continuous limit, $$\label{JWEXPAND}
J = - B_0 \left( \begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array} \right) h + O(h^2), \quad
W_{ij}+W_{kl} = -B_1 h^3 +O(h^4),$$ where $W\equiv \frac{1}{4}G\widetilde JG+G=(W_{ij}),$ $1\leq i,j \leq 2,$ and $$(i,j,k,l)\in
\left\{ (1,1,1,2), (1,1,2,1), (1,2,2,2), (2,1,2,2) \right\}.$$
Higher order potential expansion and the continuous limits of the Toda hierarchy
================================================================================
Now, we give a new method to introduce $\Phi_i(f)$ required in (\[w-v-q\]) and derive the continuous limits of the Toda hierarchy under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$.
\[lemma-T\] Define the operator as $$\label{TDEF}
T\equiv \widetilde JG
= \left( \begin{array}{cc} T_{11} & T_{12} \\
T_{21} & T_{22}
\end{array} \right).$$ Then under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$, the operator $T$ has the following expansions for its entries: $$T_{11} = -2 + \frac{1}{2} h^2 q + O(h^3), \quad
T_{12} = 2 + h\partial +(\frac{1}{2}\partial^2+q)h^2+O(h^3),$$ $$T_{21}=2- h\partial +(\frac{1}{2}\partial^2-
\frac{1}{2}\partial^{-1}q_x)h^2
+O(h^3), \quad
T_{22} = -2+ \frac{1}{2}h^2\partial^{-1}q\partial +O(h^3).$$
[*Proof.* ]{} The result can be found in [@ZLC] (see the proof of Lemma 3 in [@ZLC]).
\[lemma-Ki-expand\] Under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$, we have the following expansions, $$K_i\equiv \left(\begin{array}{c}K_{i,1} \\ K_{i,2} \end{array}
\right)$$ $$\label{Ki-expand}
=\left(\begin{array}{c}
\alpha_{i}+\Psi_{i,1,0}(q)h^2
+h^2\sum\limits_{j=1}^L
h^j(\zeta_{i,1}\Phi_{j}+\Psi_{i,1,j}(q,\Phi_1,...,\Phi_{j-1})) \\
\gamma_{i}+\Psi_{i,2,0}(q)h^2
+h^2\sum\limits_{j=1}^L
h^j(\zeta_{i,2}\Phi_{j}+\Psi_{i,2,j}(q,\Phi_1,...,\Phi_{j-1}))
\end{array} \right)+O(h^{L+3}),$$ for $i=0,1,2,...,$ where $\alpha_{i}$ and $\gamma_{i}$ are given in Lemma \[lemma-Ki\], $$\zeta_{0,1}=0,\quad \zeta_{0,2}=\frac{1}{2},\quad
\zeta_{1,1}=0, \quad \zeta_{1,2}=0,$$ $$\label{zeta}
\zeta_{i+1,1}=-2\zeta_{i,1}+2\zeta_{i,2}+\alpha_i-2\gamma_i, \quad
\zeta_{i+1,2}=2\zeta_{i,1}-2\zeta_{i,2}+\alpha_i-\frac{1}{2}\delta_{i+1},
\quad i=0,1,...,$$ $\Psi_{i,1,j}(q,\Phi_1,...,\Phi_{j-1})$ stands for the term which is a differential polynomial of $q$, $\Phi_1$, ..., $\Phi_{j-1}$, and etc.
[*Proof.*]{} Define $c_i=-vb_i^{(1)},$ $i=0,1,....$ Using the identity [@TUTD] $$\sum_{i=0}^k(a_ia_{k-i}+b_ic_{k-i})=0, \qquad k=1,2,...,$$ we can show by the mathematical induction that $a_i$, $b_i$, $c_i$, $i=0,1,...,$ are polynomials of $w$, $v$, $w^{(-1)}$, $v^{(-1)}$, $w^{(1)}$, $v^{(1)}$, $...$. According to the definition of $K_i$ in (\[Ki\]), we conclude that $K_i$ has the expansion formula (\[Ki-expand\]). Notice Lemma \[lemma-Ki\] and Lemma \[lemma-TKi\], we can prove (\[zeta\]) by the mathematical induction.
\[lemma-beta\] Define the combination coefficients $\beta_{k,i}$, $0\leq i \leq k+1$, $k=0,1,...$, as follows $$\beta_{0,0}=2, \qquad \beta_{0,1}=1, \qquad
\beta_{1,0}=-2, \qquad \beta_{1,1}=2, \qquad \beta_{1,2}=1,$$ $$\label{beta}
\beta_{k+1,i}=\beta_{k,i-1}, \quad 1\leq i \leq k+2, \qquad
\beta_{k+1,0}=\sum_{i=0}^{k+1}\beta_{k,i} \delta_{i+1},$$ then we have $$\sum_{i=0}^{k+1} \beta_{k,i}\alpha_i=0,
\qquad \sum_{i=0}^{k+1} \beta_{k,i}\gamma_i=0,
\qquad k=1,2,....$$
[*Proof.*]{} It is easy to check the case when $k=1$. If the lemma is true for $k$, then $$\sum_{i=0}^{k+1} \beta_{k,i} K_i
= O(h) \left(\begin{array}{c} 1 \\ 1 \end{array} \right),$$ so according to Lemma \[lemma-TKi\], we have $$\sum_{i=0}^{k+2} \beta_{k+1,i} K_i
= \widetilde JG \sum_{i=0}^{k+1} \beta_{k,i} K_i
= O(h) \left(\begin{array}{c} 1 \\ 1 \end{array} \right),$$ which completes the proof.
Let $\beta_{k,i}$ be defined by (\[beta\]). Then we have \[lemma-beta-zeta\] $$\label{beta-zeta}
\sum_{i=0}^{k+1}\beta_{k,i}(\zeta_{i,2}-\zeta_{i,1})
=(-4)^{k}, \qquad k=1,2,....$$
[*Proof.*]{} It is easy to check the case when $k=1$. If the lemma is true for $k$, then we have (according to Lemma \[lemma-Ki\] and Lemma \[lemma-Ki-expand\]) $$\begin{aligned}
\sum_{i=0}^{k+2}\beta_{k+1,i}(\zeta_{i,2}-\zeta_{i,1})
& = & \frac{1}{2}\sum_{i=0}^{k+1}\beta_{k,i}\delta_{i+1}
+\sum_{i=1}^{k+2}\beta_{k,i-1}(\zeta_{i,2}-\zeta_{i,1}) \\
& = & \frac{1}{2}\sum_{i=0}^{k+1}\beta_{k,i}\delta_{i+1}
+\sum_{i=0}^{k+1}\beta_{k,i}(-4\zeta_{i,2}+4\zeta_{i,1}
-\frac{1}{2}\delta_{i+1}+2\gamma_i) \\
& = & -4\sum_{i=0}^{k+1}\beta_{k,i}(\zeta_{i,2}-\zeta_{i,1})
+ 2\sum_{i=0}^{k+1}\beta_{k,i}\gamma_i, \end{aligned}$$ Note Lemma \[lemma-beta\], and the proof is completed.
\[prop-Phi\] Given an integer $m>0$, let $\beta_{k,i}$ be defined by (\[beta\]), and set $$\Phi_{2k-1}= (-1)^k 2^{-2k-1}
\left[-\frac{1}{2}\partial P_k+2\sum\limits_{i=0}^{k+1}
\beta_{k,i}(\Psi_{i,1,2k-1}-\Psi_{i,2,2k-1})\right],$$ $$\begin{aligned}
\label{Phi}
\Phi_{2k}& = & (-1)^k 2^{-2k-1}
\left[\frac{1}{2}P_{k+1}-(\frac{1}{2}\partial^2
+\frac{3}{2}q)\frac{1}{2}P_k
-\partial\sum\limits_{i=0}^{k+1}
\beta_{k,i}(\zeta_{i,2}\Phi_{2k-1}+\Psi_{i,2,2k-1})\right.
\nonumber\\
& & \left. +2\sum\limits_{i=0}^{k+1}
\beta_{k,i}(\Psi_{i,1,2k}-\Psi_{i,2,2k})\right], \end{aligned}$$ for $k=1,2,...,m-1.$ Then under the relation (\[w-v-q\]) with $L=2m-2$, $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]) we have $$\label{VFP1}
\widetilde P_m \equiv \sum_{i=0}^{m+1} \beta_{m,i} K_i =
\frac{1}{2} P_m h^{2m}
\left( \begin{array}{c} 1 \\ 1 \end{array} \right)
+ O(h^{2m+1}),$$ and $$\label{EQNP1}
{\left( \begin{array}{c} w \\ v \end{array} \right)}_{t_m}
+\frac{1}{h^{2m-1}} J \widetilde P_m =
\frac{1}{2}( q_{t_m} - B_0 P_{m} ) h^2
\left( \begin{array}{c} 1 \\ 1 \end{array} \right)
+ O(h^3).$$
[*Proof.* ]{} It is easy to check the case when $m=1$, If the equation (\[VFP1\]) is valid for $m$, then we have (according to Lemma \[lemma-Ki-expand\]) $$\begin{aligned}
T \widetilde P_m & = &
\widetilde JG \sum_{i=0}^{m+1} \beta_{m,i} K_i \\
& = & \widetilde JG
\left[\frac{1}{2} P_m h^{2m}
\left( \begin{array}{c} 1 \\ 1 \end{array} \right)
+h^{2m+1}\sum_{i=0}^{m+1} \beta_{m,i}
\left( \begin{array}{c}
\zeta_{i,1}\Phi_{2m-1}+\Psi_{i,1,2m-1} \\
\zeta_{i,2}\Phi_{2m-1}+\Psi_{i,2,2m-1} \end{array} \right)\right. \nonumber\\
& & \left. +h^{2m+2}\sum_{i=0}^{m+1} \beta_{m,i}
\left( \begin{array}{c}
\zeta_{i,1}\Phi_{2m}+\Psi_{i,1,2m} \\
\zeta_{i,2}\Phi_{2m}+\Psi_{i,2,2m} \end{array} \right)
+ O(h^{2m+3})\right], \end{aligned}$$ note the definition of $\Phi_{2m-1}$ and $\Phi_{2m}$ in (\[Phi\]), we obtain (due to (\[beta-zeta\])) $$\label{2k+1}
-2 \sum_{i=0}^{m+1} \beta_{m,i}
(\zeta_{i,1}\Phi_{2m-1}+\Psi_{i,1,2m-1})
+ 2 \sum_{i=0}^{m+1} \beta_{m,i}
(\zeta_{i,2}\Phi_{2m-1}+\Psi_{i,2,2m-1}) + \frac{1}{2}\partial P_m = 0,$$ and $$(\frac{1}{2}\partial^2+\frac{3}{2}q)\frac{1}{2}P_m
+\partial \sum_{i=0}^{m+1} \beta_{m,i}(\zeta_{i,2}\Phi_{2m-1}+\Psi_{i,2,2m-1})$$ $$\label{2k+2a}
-2\sum_{i=0}^{m+1} \beta_{m,i}(\zeta_{i,1}\Phi_{2m}+\Psi_{i,1,2m})
+ 2 \sum_{i=0}^{m+1} \beta_{m,i}(\zeta_{i,2}\Phi_{2m}+\Psi_{i,2,2m})
= \frac{1}{2}P_{m+1}.$$ Combining the above two equations (\[2k+1\]) and (\[2k+2a\]), and noting the equation (\[KdVBIHS\]), we have $$(\frac{1}{2}\partial^2-\frac{1}{2}\partial^{-1}q_x
+\frac{1}{2}\partial^{-1}q\partial)\frac{1}{2}P_m
-\partial \sum_{i=0}^{m+1} \beta_{m,i}(\zeta_{i,1}\Phi_{2m-1}+\Psi_{i,1,2m-1})$$ $$\label{2k+2b}
+2\sum_{i=0}^{m+1} \beta_{m,i}(\zeta_{i,1}\Phi_{2m}+\Psi_{i,1,2m})
- 2 \sum_{i=0}^{m+1} \beta_{m,i}(\zeta_{i,2}\Phi_{2m}+\Psi_{i,2,2m})
= \frac{1}{2}P_{m+1}.$$ So we get $$T \widetilde P_m = \frac{1}{2} P_{m+1} h^{2m+2}
\left( \begin{array}{c} 1 \\ 1 \end{array} \right)
+ O(h^{2m+3}).$$ On the other hand (according to Lemma \[lemma-TKi\]), $$T \widetilde P_m = \widetilde JG \sum_{i=0}^{m+1} \beta_{m,i} K_i
= \sum_{i=0}^{m+1} \beta_{m,i}( K_{i+1} + \delta_{i+1} K_0)
= \widetilde P_{m+1}.$$ The equation (\[EQNP1\]) is the corollary of the equation (\[VFP1\]) and Proposition \[prop-Poisson\]. The proof is finished.
We give an example here. For $m=3$, using Proposition \[prop-Phi\], we can get $$\Phi_1 = \frac{1}{8} q_x, \qquad \Phi_2 = -\frac{1}{32}q^2,
\qquad \Phi_3=-\frac{1}{384}q_{xxx}, \qquad
\Phi_4=\frac{1}{254}(q^3+qq_{xx}+q_x^2),$$ then under the relation (\[w-v-q\]) with $L=4$, $f(x,t)=\frac 12 q(x,t)$ and the above $\Phi_i$’s we have $$-10 K_0 + 4 K_1 -2 K_2 + 2 K_3 + K_4 = \frac{1}{2} P_3 h^6
\left( \begin{array}{c} 1 \\ 1 \end{array} \right) + O(h^7).$$ In the previous work in [@ZLC], we must combine $K_0$, $K_1$, ..., $K_6$ for giving $P_3$ under the relation (\[w-v-q\]) with $L=0$. In general, $K_0$, $K_1$, ..., $K_{2m}$ are needed to be combined for giving $P_m$ under the relation (\[w-v-q\]) with $L=0$ [@ZLC]. Proposition \[prop-Phi\] shows us that almost only half of them, i.e., $K_0$, $K_1$, ..., $K_{m+1}$, are needed to give $P_m$ by introducing $\Phi_i(f)$ (\[Phi\]). Furthermore, according to the recursion formula for $\Phi_i(f)$ (\[Phi\]) it is easy to see that all the $\Phi_i(f)$’s, introduced by (\[Phi\]), are differential polynomials of $f$, and our process for finding $\Phi_i(f)$ can be continued indefinitly.
In what follows, we will derive the continuous limit relations between the Hamiltonians, the Lax pairs of the Toda hierarchy and those of the KdV hierarhcy, respectively.
\[lemma-tilde-w-q\] If there is a relation between $\widetilde w(n)$, $n\in\mathbb Z$, and $q(x)$, $x\in\mathbb R$ $$\label{tilde-w-q}
\widetilde w(n)=q^{(s_1)}(x)q^{(s_2)}(x)\cdots q^{(s_m)}(x) h^l,$$ where $h$ is the step of lattice, $x=nh$, $s_i$, $1\leq i\leq m$ and $l$ are nonnegtive integers, and denote $\widetilde S$ as the operator which stands for submitting the relation (\[tilde-w-q\]) into a polynomial of $\widetilde w$, $\widetilde w^{(-1)}$, $\widetilde w^{(1)}$, ..., and then expanding in Taylor series, then we have the formula $$\frac{\delta }{\delta q} \circ \widetilde S
= h^l \widetilde Z \circ \widetilde S\circ \frac{\delta}{\delta \widetilde w},$$ where $\widetilde Z$ stands for a differential operator.
The proof for Lemma \[lemma-tilde-w-q\] is given in Appendix A.
\[prop-Hk\] Given an integer $m>0$, set $$\widetilde H_m \equiv \sum_{i=0}^{m+1} \beta_{m,i} H_i
- \sum_{i=1}^{m+1} \beta_{m,i} \frac{\alpha_{i+1}}{i},$$ under the relation (\[w-v-q\]) with $L=2m-2$, $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]), we have $$\int S(\widetilde H_m) dx
= \frac{1}{2}h^{2m+2}\int \overline H_m dx +O(h^{2m+3}),$$ where $S$ is an operator which stands for submitting the relation (\[w-v-q\]) with $L=2m-2$, $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]) into a polynomial of $w$, $v(n)$, $w^{(-1)}$, $v^{(-1)}$, $w^{(1)}$, $v^{(1)}$, ..., and then expanding in Taylor series.
[*Proof.*]{} According to Lemma \[lemma-tilde-w-q\], under the relation (\[w-v-q\]) with $L=2m-2$, $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]), (since $\Phi_i$’s are differential polynomials of $q$), we have $$\begin{aligned}
\frac{\delta}{\delta q} \circ S
& = & \sum_{j=0}^\infty (-\partial)^j
\frac{\partial}{\partial q^{(j)}} \circ S \\
& = & \sum_{j=0}^\infty (-\partial)^j \sum_{k\in\mathbb Z}
\left[(\frac{\partial S(w^{(k)})}{q^{(j)}}) S\circ
\frac{\partial}{\partial w^{(k)}}
+ (\frac{\partial S(v^{(k)})}{q^{(j)}}) S\circ
\frac{\partial}{\partial v^{(k)}}\right] \\
& = & \frac{1}{2} h^2 \sum_{j=0}^\infty (-\partial)^j \sum_{k\in\mathbb Z}
\frac{(kh)^{j}}{j!} S \circ (\frac{\partial}{\partial w^{(k)}}+
\frac{\partial}{\partial v^{(k)}})
+ h^3 Z\circ S \circ (\frac{\delta}{\delta w}-\frac{\delta}{\delta v}) \\
& = & \frac{1}{2} h^2 S \circ (\frac{\delta}{\delta w}+\frac{\delta}{\delta v})
+ h^3 Z\circ S \circ (\frac{\delta}{\delta w}-\frac{\delta}{\delta v}),\end{aligned}$$ where $Z$ stands for a differential operator, and we do not care about its concrete form. Note Lemma \[lemma-Ki\] and the definition of $H_i$ in (\[Ki\]), we can have the expansion $$S(\widetilde H_m) = \sum_{i=2}^\infty \widetilde H_{m,i} h^i,$$ where $\left. \widetilde H_{m,i} \right|_{q=0} =0$, and according to Proposition \[prop-Phi\], we have $$\begin{aligned}
\frac{\delta}{\delta q} \circ S (\widetilde H_m)
& = &\sum_{i=2}^\infty h^i\frac{\delta \widetilde H_{m,i}}{\delta q} \\
& = & \left[ \frac{1}{2} h^2 S \circ (\frac{\delta}{\delta w}+
\frac{\delta}{\delta v})
+ h^3 Z \circ S \circ (\frac{\delta}{\delta w}-\frac{\delta}{\delta v}) \right]
\sum_{i=0}^{m+1} \beta_{m,i} H_m \\
& = & \frac{1}{2} h^{2m+2} \frac{\delta \overline H_m}{\delta q}
+ O(h^{2m+3}). \end{aligned}$$ Then one can get [@TUTD] $$\widetilde H_{m,i} \in {\rm Const.} + \mbox{Image}(\partial),
\qquad 2\leq i \leq 2m+1.$$ As we mentioned above, there is no constant item in each $\widetilde H_{m,i}$, $i\geq 2$, (i.e., $\left. \widetilde H_{m,i} \right|_{q=0} =0$), so $$\int \widetilde H_{m,i} dx = 0, \qquad 2\leq i \leq 2m+1.$$ Just using the same deduction, we conclude $$\int \widetilde H_{m,2m+2} dx = \frac{1}{2} \int \overline H_m dx,$$ which completes the proof.
\[lemma:A\_m-expand\] Under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$, we have $$\label{A_m-expand}
A_k = \alpha_k-\gamma_k+
\sum_{i=2}^\infty A_{k,i} h^i ,
\qquad k=0,1,...,$$ where $$A_{k,2i}|_{q=0} =0,\quad
A_{k,2i+1}|_{q=0}
=\xi_{k,2i+1} \partial^{2i+1},
\qquad i=1,2,...,$$ $\xi_{k,2i+1}$is a constant, and $\alpha_k$ and $\gamma_k$ are given in Lemma \[lemma-Ki\].
[*Proof.*]{} For $k=0$ and $k=1$, we have $$A_0|_{q=0}=-\frac{1}{2},
\qquad A_1|_{q=0}
=1+\sum_{j=0}^\infty \frac{1}{(2j+1)!}
h^{2j+1}(-\partial)^{2j+1}.$$ If the lemma is valid for $k-1$, note $\alpha_k=-2\alpha_{k-1}+2\gamma_{k-1}$ (see Lemma \[lemma-Ki\]), we have $$\begin{aligned}
A_k|_{q=0}
&=& \left. A_{k-1} (E + w + v E^{(-1)})
- vb_k^{(1)}E^{(-1)}
-a_k \right|_{q=0} \\
&=& \left[\alpha_{k-1}-\gamma_{k-1}+
\sum_{i=0}^\infty \xi_{k-1,2i+1}
h^{2i+1} \partial^{2i+1}\right]
\sum_{j=1}^\infty
\frac{2}{(2j)!}h^{2j}\partial^{2j} \\
& & +\alpha_k\sum_{j=0}^\infty
\frac{1}{j!}h^j(-\partial)^j -\gamma_k \\
&\equiv& \alpha_k-\gamma_k+
\sum_{i=0}^\infty \xi_{k,2i+1}
h^{2i+1} \partial^{2i+1}. \end{aligned}$$
\[lemma-tilde-A\_m\] Define $$\widetilde A_k \equiv
\sum_{i=1}^{k+1} \beta_{k,i} A_{i-1}
\qquad k=1,2,....$$ Then under the relation (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$, we have $$\widetilde A_k =\sum_{i=2}^\infty
\widetilde A_{k,i} h^{i},$$ where $$\widetilde A_{k,2i}|_{q=0}=0,\quad
\widetilde A_{k,2i+1}|_{q=0} =
\widetilde \xi_{k,2i+1} \partial^{2i+1},
\qquad i=1,2,...,$$ $\widetilde \xi_{k,2i+1}$is a constant.
[*Proof.*]{} According to Lemma \[lemma:A\_m-expand\], we only need to prove $$\sum_{i=1}^{k+1} \beta_{k,i}
(\alpha_{i-1}-\gamma_{i-1})=0.$$ It is easy to check the cases: $k=1$ and $k=2$, and for $k\geq 3$, note Lemma \[lemma-beta\], we have $$\sum_{i=1}^{k+1}\beta_{k,i}(\alpha_{i-1}-\gamma_{i-1})
=\sum_{i=1}^{k+1}\beta_{k-1,i-1} (\alpha_{i-1}-\gamma_{i-1})
=\sum_{i=0}^k \beta_{k-1,i}(\alpha_{i}-\gamma_{i})=0,$$ which completes the proof.
\[prop-Ak\] Given an integer $m>0$, under the relation (\[w-v-q\]) with $L=2m-2$, $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]), we have $$\widetilde A_m \equiv \sum_{i=1}^{m+1} \beta_{m,i} A_{i-1}
= - \overline A_m h^{2m-1}+O(h^{2m}).$$
[*Proof.*]{} It is valid for $m=1,2$. According to Proposition \[prop-Phi\], we have $$\begin{aligned}
\label{widetilde-A_k-L}
[ \widetilde A_m, L]
&=& \sum_{i=1}^{m+1} \beta_{m,i}\frac{d w}{d t_{i-1}}
+\sum_{i=1}^{m+1} \beta_{m,i}\frac{d v}{d t_{i-1}} E^{(-1)}
\nonumber \\
&=& J_{12}\sum_{i=1}^{m+1} \beta_{m,i} K_{i,2}+
J_{21}\sum_{i=1}^{m+1}\beta_{m,i} K_{i,1} E^{(-1)}
\nonumber \\
&=& - B_0 P_m h^{2m+1}+O(h^{2m+2}) \nonumber \\
&=& -[ \overline A_m, \overline L]h^{2m+1}+O(h^{2m+2}).\end{aligned}$$ Under the relation (\[w-v-q\]) with $L=2m-2$, $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]), Proposition \[prop-spectral\] and Lemma \[lemma-tilde-A\_m\] together imply $$L = \overline L h^2+\sum_{i=3}^\infty L_i h^i,
\quad\qquad
\widetilde A_m = \sum_{i=2}^\infty
\widetilde A_{m,i} h^i,$$ where $L_i$ and $\widetilde A_{m,i}$ are differential operators. Comparing the terms of $h^4$ in (\[widetilde-A\_k-L\]), we know $$[\widetilde A_{m,2}, \overline L]=0,$$ According to [@Drinfeld-Sokolov-85], $\widetilde A_{m,2}$ can be represented by $$\widetilde A_{m,2}=\sum_{j=0}^\infty
\eta_{m,2,j} (\overline L)^j,$$ where $\eta_{m,2,j}$ are constants. Note Lemma \[lemma-tilde-A\_m\], we have $$\widetilde A_{m,2}|_{q=0}=0
=\sum_{j=0}^\infty
\eta_{m,2,j} (\partial^2)^j.$$ Then one can get $\eta_{m,2,j}=0$ for all $j$, and $$\label{A_m_2}
\widetilde A_{m,2} =0.$$ Comparing the terms of $h^5$ in (\[widetilde-A\_k-L\]), we know $$[\widetilde A_{m,3}, \overline L]=0,$$ then $\widetilde A_{m,3}$ can be represented by [@Drinfeld-Sokolov-85] $$\widetilde A_{m,3}=\sum_{j=0}^\infty
\eta_{m,3,j} (\overline L)^j,$$ where $\eta_{m,3,j}$ are constants. Note Lemma \[lemma-tilde-A\_m\], and we have $$\widetilde A_{m,3}|_{q=0}
=\widetilde \xi_{m,3}\partial^3
=\sum_{j=0}^\infty
\eta_{m,3,j} (\partial^2)^j.$$ Then one can get $\eta_{m,3,j}=0$ for all $j$, and $$\widetilde A_{m,3} =0.$$ In the same way, we conclude $$\widetilde A_{m,i} =0, \qquad
i=2,...,2m-2.$$ Comparing the terms of $h^{2m+1}$ in (\[widetilde-A\_k-L\]), we know $$[\widetilde A_{m,2m-1}, \overline L]
=-[\overline A_m, \overline L],$$ then $\widetilde A_{m,2m-1}+\overline A_m$ can be represented by [@Drinfeld-Sokolov-85] $$\widetilde A_{m,2m-1}+\overline A_m
=\sum_{j=0}^\infty
\eta_{m,2m-1,j} (\overline L)^j,$$ where $\eta_{m,2m-1,j}$ are constants. Note Lemma \[lemma-tilde-A\_m\] and (\[KdV-Lax-t\]), we have $$\left. \left( \widetilde A_{m,2m-1}+\overline A_m
\right) \right|_{q=0}
=\widetilde \xi_{m,2m-1} \partial^{2m-1}
+\partial^{2m-1}
=\sum_{j=0}^\infty
\eta_{m,2m-1,j} (\partial^2)^j.$$ Then we get $\eta_{m,2m-1,j}=0$ for all $j$ and $$\widetilde A_m \equiv
\sum_{i=1}^{2m} \beta_{m,i} A_{i-1}
= - \overline A_m h^{2m-1} +O(h^{2m}).$$ Thus the proof is completed.
Conclusions and remarks
=======================
In this paper, by introducing the higher order terms in the potential expansion, we have proved that there is the continuous limit relation between the Toda hierarchy and the KdV hierarchy. Compared with the [@ZLC], the fewer members of the Toda hierarchy are needed to recover the KdV hierarchy by the recombination method. For example, Proposition \[prop-Phi\] shows that under the potential expansion (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$ and (\[Phi\]), we can combine $K_0$, $K_1$, $...$, $K_{m+1}$, to get $P_m$ in continuous limit. However, under the lower finite potential expansion, for example (\[w-v-q\]) with $f(x,t)=\frac 12 q(x,t)$ and $L=0$, we need $K_0$, $K_1$, $...$, $K_m$, $...$, $K_{2m}$, to recover $P_m$ through the continuous limit process [@ZLC].
Compared with the [@Gieseker], a new method for introducing $\Phi_i(f)$ in the potential expansion (\[w-v-q\]) was presented in this paper. Moreover, from the recursion formula for $\Phi_i(f)$ (\[Phi\]), it is easy to see that the $\Phi_i(f)$’s, introduced in our construction, are all differential polynomials of $f$, and our process for determining $\Phi_i(f)$ can be continued indefinitly. However, this can not be obtained in [@Gieseker], since the $\Phi_i(f)$’s are obtained by integration there.
It was also shown that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the KdV hierarchy in continuous limit
Acknowledgements {#acknowledgements .unnumbered}
================
This work was in part supported by the City University of Hong Kong (Project Nos.: 7001178, 7001041), the RGC of Hong Kong (Project No.: 9040466), and Chinese Basic Research Project “Nonlinear Science". One of the authors (R.L. Lin) acknowledges warm hospitality at the City University of Hong Kong.
Appendix A. Proof of Lemma \[lemma-tilde-w-q\] {#appendix-a.-proof-of-lemma-lemma-tilde-w-q .unnumbered}
==============================================
Denote $\widetilde w_i=q^{(s_1)}\cdots q^{(s_{i-1})}q^{(s_{i+1})}\cdots q^{(s_m)},$ for $i=1,...,m$, then we have $$\begin{aligned}
\frac{\delta }{\delta q} \circ \widetilde S
& = & \sum_{j=0}^\infty
(-\partial)^j \frac{\partial}{\partial q^{(j)}} \circ \widetilde S \\
& = & \sum_{j=0}^\infty (-\partial)^j
\sum_{k\in{\mathbb Z}}
\left( \frac{\partial \widetilde S(\widetilde w^{(k)})}{\partial q^{(j)}} \right)
\widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}} \\
& = & h^l \sum_{i=1}^m \sum_{j=s_i}^\infty (-\partial)^j
\sum_{k\in{\mathbb Z}} \frac{(kh)^{j-s_i}}{(j-s_i)!}
\left( e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}} \\
& = & h^l \sum_{i=1}^m \sum_{j=0}^\infty (-\partial)^{j+s_i}
\sum_{k\in{\mathbb Z}} \frac{(kh)^{j}}{j!}
\left( e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}} \\
& = & h^l \sum_{i=1}^m (-\partial)^{s_i}
\sum_{j=0}^\infty \sum_{k\in{\mathbb Z}}
\sum_{p=0}^j \frac{(-kh)^{j}}{p! (j-p)!}
\left( \partial^p e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\partial^{j-p}\circ \widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}} \\
& = & h^l \sum_{i=1}^m (-\partial)^{s_i} \sum_{p=0}^\infty
\sum_{k\in{\mathbb Z}} \sum_{j=p}^\infty
\frac{(-kh)^{j}}{p! (j-p)!}
\left( \partial^p e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\partial^{j-p}\circ \widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}}
\\
& = & h^l \sum_{i=1}^m (-\partial)^{s_i} \sum_{p=0}^\infty
\left( \partial^p e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\sum_{k\in{\mathbb Z}} \sum_{j=0}^\infty
\frac{(-kh)^{j+p}}{p! j!}
\partial^{j}\circ \widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}}
\\
& = & h^l \sum_{i=1}^m (-\partial)^{s_i} \sum_{p=0}^\infty
\frac{(-kh)^{p}}{p! }
\left( \partial^p e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\sum_{k\in{\mathbb Z}} \sum_{j=0}^\infty
\frac{(-kh)^{j}}{j!}
\partial^{j}\circ \widetilde S\circ\frac{\partial}{\partial \widetilde w^{(k)}} \\
& = & h^l \sum_{i=1}^m (-\partial)^{s_i} \sum_{p=0}^\infty
\frac{(-kh)^{p}}{p! }
\left( \partial^p e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\widetilde S\circ \sum_{k\in{\mathbb Z}} E^{(-k)}
\circ\frac{\partial}{\partial \widetilde w^{(k)}} \\
& = & h^l \sum_{i=1}^m (-\partial)^{s_i} \sum_{p=0}^\infty
\frac{(-kh)^{p}}{p! }
\left( \partial^p e^{kh\partial} \widetilde S(\widetilde w_i) \right)
\widetilde S \circ\frac{\delta}{\delta \widetilde w^{(k)}} \\
& \equiv & h^l \widetilde Z \circ \widetilde S\circ \frac{\delta}{\delta \widetilde w}.\end{aligned}$$ The proof for Lemma \[lemma-tilde-w-q\] is finished.
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|
---
abstract: 'A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulation provides unique solutions for all combinations of wavenumbers in the closed upper half-plane for which Maxwell’s equations have a unique solution. This includes the challenging combination of a real positive wavenumber in the outer region and an imaginary wavenumber inside the object. The formulation, or variants thereof, is particularly suitable for numerical field evaluations as confirmed by examples involving both smooth and non-smooth objects.'
author:
- 'Johan Helsing[^1] and Anders Karlsson[^2]'
date: 'February 15, 2020'
title: 'An extended charge-current formulation of the electromagnetic transmission problem'
---
Introduction {#sec:introduction}
============
This work is about transmission problems. A simply connected homogeneous isotropic object is located in a homogeneous isotropic exterior region. A time harmonic incident wave, generated in the exterior region, is scattered from the object. The aim is to evaluate the fields in the interior and exterior regions.
We present boundary integral equation (BIE) formulations for the solution of the scalar Helmholtz and the electromagnetic Maxwell transmission problems. We show that our integral equations have unique solutions for all wavenumbers $k_1$ of the exterior domain and $k_2$ of the object with $0\leq{{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2)<\pi$, and for which the partial differential equation (PDE) formulations of the two problems have unique solutions. As we understand it, there is no other BIE formulation of the electromagnetic problem known to the computational electromagnetics community that can guarantee unique solutions for the wavenumber combination $${{\rm{Arg}}}(k_1)=0\,, \quad {{\rm{Arg}}}(k_2)=\pi/2\,, \quad \mbox{and} \quad
k_2^2/k_1^2\ne -1\,.
\label{eq:plasmonic}$$
We refer to the combination as the [*plasmonic condition*]{} since it enables discrete quasi-electrostatic surface plasmons in smooth, infinitesimally small, objects [@TzarSihv18], continuous spectra of quasi-electrostatic surface plasmons in non-smooth objects [@HelsPerf18], and undamped surface plasmon waves along planar surfaces [@Raeth88 Appendix I]. Wavenumbers with ${{\rm{Arg}}}(k_1)=0$ and $\pi/4<{{\rm{Arg}}}(k_2)\leq\pi/2$ are of special interest in the areas of nano-optics and metamaterials because in this range weakly damped surface plasmons in subwavelength objects and weakly damped dynamic surface plasmon waves in objects of any size can occur. These phenomena become increasingly pronounced, and useful in applications, as ${{\rm{Arg}}}(k_2)$ approaches $\pi/2$ [@Homola08; @LuTsGuHo19]. It is important to have uniqueness under the plasmonic condition, despite that there are no known materials that satisfy this condition exactly, since non-uniqueness implies spurious resonances that deteriorate the accuracy of the numerical solution also for ${{\rm{Arg}}}(k_1)=0$, $\pi/4<{{\rm{Arg}}}(k_2)<\pi/2$.
It is relatively easy to find a BIE formulation of the scalar transmission problem since one has access to the fundamental solution to the scalar Helmholtz equation. It remains to make sure that the boundary conditions are satisfied and that the solution is unique. To find a BIE formulation of the electromagnetic transmission problem, based on the same fundamental solution, is harder. Apart from satisfying the boundary conditions and uniqueness one also has to make sure that the solution satisfies Maxwell’s equations. Otherwise the two problems are very similar.
Our BIE formulation of the scalar problem is a modification of the formulation in [@KleiMart88 Section 4.2]. While our formulation guarantees unique solutions under the plasmonic condition, provided that the object surface is smooth, the formulation in [@KleiMart88 Section 4.2] does not.
Our BIE formulation of the electromagnetic problem is a further development of the classic formulation by Müller, [@Muller69 Section 23]. In [@MautHarr77] it is shown that the Müller formulation has unique solutions for $0\leq
{{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2)<\pi/2$, but as shown in [@HelsKarl19], it may have spurious resonances under the plasmonic condition. The Müller formulation has four unknown scalar surface densities, related to the equivalent electric and magnetic surface current densities, and that leads to dense-mesh/low-frequency breakdown in field evaluations. Despite these shortcomings, the Müller formulation has been frequently used. Its advantages are emphasized in a recent paper [@LaiOneil19] on scattering from axisymmetric objects where accurate solutions are obtained away from the low-frequency limit.
One way to overcome low-frequency breakdown in the Müller formulation is to increase the number of unknown densities from four to six by adding the equivalent electric and magnetic surface charge densities [@HelsKarl17; @TaskOija06; @VicGreFer18]. The charge densities can be introduced in two ways, leading to two types of formulations. The first type is decoupled charge-current formulations, where the charge densities are introduced after the BIE has been solved. The other type is coupled charge-current formulations, where the charge densities are present from the start. Unfortunately, both types of formulations can give rise to new complications such as spurious resonances and near-resonances. Several formulations in the literature ignore these complications, but in [@VicGreFer18] a stable formulation is presented. In line with all other formulations in literature, uniqueness in [@VicGreFer18] is not guaranteed under the plasmonic condition.
The main result of the present work is our extended charge-current BIE formulation of the electromagnetic transmission problem where two additional surface densities, related to electric and magnetic volume charge densities, are introduced. The formulation is given by the representation (\[eq:repEH\]) and the system (\[eq:EHsys\]) below. The formulation is free from low-frequency breakdown and it provides unique solutions also under the plasmonic condition. Just like the Müller- and charge-current formulations it is a direct formulation, meaning that the surface densities are related to boundary limits of fields, or derivatives of fields. This is in contrast to indirect formulations [@EpsGreNei13; @EpsGreNei19; @KresRoac78; @VicGreFer18], where the surface densities lack immediate physical interpretation. Albeit somewhat more numerically expensive than competing formulations, our new formulation enables high achievable accuracy and since it is more robust this should outweigh the disadvantage of having eight densities. From a broader perspective one can say that our paper, and many other papers [@HelsKarl17; @LaiOneil19; @MautHarr77; @Muller69; @TaskOija06; @VicGreFer18], use integral representations of the electric and magnetic fields for modeling. It is also possible to start with representations of scalar and vector potentials and antipotentials [@EpsGreNei13; @EpsGreNei19; @LiFuShank18].
The paper is organized as follows: Section \[sec:pre\] introduces notation and definitions common to the scalar and the electromagnetic problems. The scalar problem and two closely related homogeneous problems, to be used in a uniqueness proof, are defined in Section \[sec:scalarprob\]. Scalar integral representations containing two surface densities are introduced in Section \[sec:intrepA\]. Section \[sec:inteqA\] proposes a system of BIEs for these densities. This system contains two free parameters and, as seen in Section \[sec:uniqueA\], unique solutions are guaranteed by giving them proper values. Section \[sec:eval\] concerns the evaluation of near fields. The procedure for finding BIEs for the scalar problem is then adapted to the electromagnetic problem, defined along with two auxiliary homogeneous problems in Section \[sec:EMprob\]. Integral representations of electric and magnetic fields in terms of eight scalar surface densities are given in Section \[sec:intrepC\] and a corresponding system of BIEs is proposed in Section \[sec:inteqC\]. This BIE system contains four free parameters and again, as shown in Section \[sec:uniqueC\], unique solutions are guaranteed by choosing them properly. Section \[sec:twoD\] presents reduced two-dimensional (2D) versions of the electromagnetic BIE system whose purpose is to facilitate initial tests and comparisons. Section \[sec:geomdisc\] reviews test domains and discretization techniques and Section \[sec:numex\] presents numerical examples, including what we believe is the first high-order accurate computation of a surface plasmon wave on a non-smooth three-dimensional (3D) object.
Appendix A presents boundary values of integral representations. Appendix B and C derive conditions for our representations of the electric and magnetic fields to satisfy Maxwell’s equations. In Appendix D a set of points $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is identified for which the electromagnetic problem has at most one solution.
![Geometry in $\mathbb{R}^3$. Inside $\Gamma$ the volume is $\Omega_2$ and the wavenumber $k_2$. Outside $\Gamma$ the volume is $\Omega_1$ and the wavenumber $k_1$. The outward unit normal is ${{\boldsymbol\nu}}$ at ${{\boldsymbolr}}$ and ${{\boldsymbol\nu}}'$ at ${{\boldsymbolr}}'$.[]{data-label="fig:amoeba0"}](hkfig1.pdf){height="50mm"}
Notation {#sec:pre}
========
Let $\Omega_2$ be a bounded volume in $\mathbb{R}^3$ with a smooth closed surface $\Gamma$ and simply connected unbounded exterior $\Omega_1$. The outward unit normal at position ${{\boldsymbolr}}$ on $\Gamma$ is ${{\boldsymbol\nu}}$. We consider time-harmonic fields with time dependence $e^{-{\rm i}t}$, where the angular frequency is scaled to one. The relation between time-dependent fields $F({{\boldsymbolr}},t)$ and complex fields $F({{\boldsymbolr}})$ is $$F({{\boldsymbolr}},t)=\Re{\rm e}\left\{F({{\boldsymbolr}})e^{-{\rm i}t}\right\}.
\label{eq:timedep}$$ The volumes $\Omega_1$ and $\Omega_2$ are homogeneous with wavenumbers $k_1$ and $k_2$. See Figure \[fig:amoeba0\], which depicts a non-smooth $\Gamma$ that is used later in numerical examples. An incident field is generated by a source somewhere in $\Omega_1$.
Layer potentials and boundary integral operators {#sec:laypot}
------------------------------------------------
The fundamental solution to the scalar Helmholtz equation is taken to be $$\Phi_k({{\boldsymbolr}},{{\boldsymbolr}}')=
\frac{e^{{\rm i}k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert}}
{4\pi\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert}\,,\quad
{{\boldsymbolr}},{{\boldsymbolr}}'\in\mathbb{R}^3\,.
\label{eq:fund}$$ Two scalar layer potentials are defined in terms of a general surface density $\sigma$ as $$\begin{split}
S_k\sigma({{\boldsymbolr}})&=2\int_{\Gamma}\Phi_{k}({{\boldsymbolr}},{{\boldsymbolr}}')
\sigma({{\boldsymbolr}}')\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,\\
K_k\sigma({{\boldsymbolr}})&=2\int_{\Gamma}(\partial_{\nu'}\Phi_{k})
({{\boldsymbolr}},{{\boldsymbolr}}')\sigma({{\boldsymbolr}}')\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\end{split}
\label{eq:STk}$$ where ${\rm d}\Gamma$ is an element of surface area, $\partial_{\nu'}={{\boldsymbol\nu}}'\cdot\nabla'$, and ${{\boldsymbol\nu}}'={{\boldsymbol\nu}}({{\boldsymbolr}}')$. We use (\[eq:STk\]) also for ${{\boldsymbolr}}\in\Gamma$, in which case $S_k$ and $K_k$ are viewed as boundary integral operators. Further, we need the operators $K_k^{\rm
A}$ and $T_k$, defined by $$\begin{split}
K^{\rm A}_k\sigma({{\boldsymbolr}})&=2\int_{\Gamma}
(\partial_\nu\Phi_{k})({{\boldsymbolr}},{{\boldsymbolr}}')\sigma({{\boldsymbolr}}')
\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Gamma\,,\\
T_k\sigma({{\boldsymbolr}})&=2\int_{\Gamma}
(\partial_\nu\partial_{\nu'}\Phi_{k})
({{\boldsymbolr}},{{\boldsymbolr}}')\sigma({{\boldsymbolr}}')\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Gamma,
\end{split}$$ and where $T_k\sigma$ is to be understood in the Hadamard finite-part sense. We also need the vector-valued layer potentials $$\begin{split}
{\cal S}_k{{\boldsymbol\sigma}}({{\boldsymbolr}})&=2\int_\Gamma
\Phi_{k}({{\boldsymbolr}},{{\boldsymbolr}}'){{\boldsymbol\sigma}}({{\boldsymbolr}}')\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,\\
\boldsymbol{\cal N}_k\sigma({{\boldsymbolr}})&=2\int_\Gamma
\nabla\Phi_{k}({{\boldsymbolr}},{{\boldsymbolr}}')\sigma({{\boldsymbolr}}')\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,\\
{\cal R}_k{{\boldsymbol\sigma}}({{\boldsymbolr}})&=2\int_{\Gamma}\nabla\Phi_{k}
({{\boldsymbolr}},{{\boldsymbolr}}')\times{{\boldsymbol\sigma}}({{\boldsymbolr}}')\,{\rm d}\Gamma'\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\end{split}
\label{eq:cSNR}$$ with corresponding operators ${\cal S}_k$, $\boldsymbol{\cal N}_k$, and ${\cal R}_k$ for ${{\boldsymbolr}}\in\Gamma$. The notation $$\tilde S_{k}={\rm i}k_1 S_k\,,\quad
\tilde{\cal S}_{k}={\rm i}k_1 {\cal S}_k\,,
\label{eq:barS}$$ will be used for brevity.
The fundamental solution $\Phi_k$ and the operators $S_k$, $K_k$, $K^{\rm A}_k$, and $T_k$ are identical to the corresponding constructs in [@ColtKres98 Eqs. (2.1) and (3.8)–(3.11)]. The potentials of correspond to the potentials in [@TaskOija06 Eqs. (3) and (9)], scaled with a factor of two.
Limits of layer potentials {#sec:limits}
--------------------------
It is convenient to introduce the notation $$\begin{split}
A^+({{\boldsymbolr}}^\circ)&=\lim_{\Omega_1\ni{{\boldsymbolr}}\to {{\boldsymbolr}}^\circ}A({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}^\circ\in\Gamma\,,\\
A^-({{\boldsymbolr}}^\circ)&=\lim_{\Omega_2\ni{{\boldsymbolr}}\to {{\boldsymbolr}}^\circ}A({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}^\circ\in\Gamma\,,
\end{split}$$ for limits of a function $A({{\boldsymbolr}})$ as $\Omega_1\cup\Omega_2\ni{{\boldsymbolr}}\to{{\boldsymbolr}}^\circ\in\Gamma$. For compositions of operators and functions, square brackets $[\cdot]$ indicate parts where limits are taken. In this notation, results from classical potential theory on limits of layer potentials include [@ColtKres98 Theorem 3.1] and [@ColtKres83 Theorem 2.23] $$\begin{split}
[S_k\sigma]^\pm({{\boldsymbolr}})&=S_k\sigma({{\boldsymbolr}})\,,\quad {{\boldsymbolr}}\in\Gamma\,,\\
[K_k\sigma]^\pm({{\boldsymbolr}})&=\pm\sigma({{\boldsymbolr}})+K_k\sigma({{\boldsymbolr}})\,,
\quad {{\boldsymbolr}}\in\Gamma\,,\\
{{\boldsymbol\nu}}\cdot[\nabla S_k\sigma]^\pm({{\boldsymbolr}})&=
\mp\sigma({{\boldsymbolr}})+K_k^{\rm A}\sigma({{\boldsymbolr}})\,,
\quad {{\boldsymbolr}}\in\Gamma\,,\\
{{\boldsymbol\nu}}\cdot[\nabla K_k\sigma]^\pm({{\boldsymbolr}})&=
T_k\sigma({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,.
\end{split}
\label{eq:SKATlim}$$ See also [@KirsHett15 Theorem 5.46] for statements on the second and fourth limit of (\[eq:SKATlim\]) in a more modern function-space setting.
The layer potentials of (\[eq:cSNR\]) have limits $$\begin{split}
[{\cal S}_k{{\boldsymbol\sigma}}]^\pm({{\boldsymbolr}})&=
{\cal S}_k{{\boldsymbol\sigma}}({{\boldsymbolr}})\,,\quad {{\boldsymbolr}}\in\Gamma\,,\\
{{\boldsymbol\nu}}\cdot[\boldsymbol{\cal N}_k\sigma]^\pm({{\boldsymbolr}})&=
\mp\sigma({{\boldsymbolr}})+{{\boldsymbol\nu}}\cdot\boldsymbol{\cal N}_k\sigma({{\boldsymbolr}})\,,
\quad {{\boldsymbolr}}\in\Gamma\,,\\
{{\boldsymbol\nu}}\times[{\cal R}_k{{\boldsymbol\sigma}}]^\pm({{\boldsymbolr}})&=
\pm{{\boldsymbol\sigma}}({{\boldsymbolr}})+{{\boldsymbol\nu}}\times{\cal R}_k{{\boldsymbol\sigma}}({{\boldsymbolr}})\,,
\quad {{\boldsymbolr}}\in\Gamma\,.
\end{split}$$
Scalar transmission problems {#sec:scalarprob}
============================
We present three scalar transmission problems called problem [A]{}, problem ${\sf A}_0$, and problem ${\sf B}_0$. Problem [A]{} is the problem of main interest. Problem ${\sf A}_0$ and ${\sf B}_0$ are needed in proofs.
Problem [A]{} and ${\sf A}_0$ {#sec:A0}
-----------------------------
The transmission problem [A]{} reads: Given an incident field $U^{\rm in}$, generated in $\Omega_1$, find the total field $U({{\boldsymbolr}})$, ${{\boldsymbolr}}\in\Omega_1\cup\Omega_2$, which, for a complex jump parameter $\kappa$ and for wavenumbers $k_1$ and $k_2$ such that $$0\leq{{\rm{Arg}}}(k_1), {{\rm{Arg}}}(k_2)<\pi\,,
\label{eq:k1k2}$$ solves $$\begin{split}
\Delta U({{\boldsymbolr}})+k_1^2U({{\boldsymbolr}})&=0\,,\quad{{\boldsymbolr}}\in\Omega_1\,,\\
\Delta U({{\boldsymbolr}})+k_2^2U({{\boldsymbolr}})&=0\,,\quad{{\boldsymbolr}}\in\Omega_2\,,
\end{split}
\label{eq:helmA12}$$ except possibly at an isolated point in $\Omega_1$ where the source of $U^{\rm in}$ is located, subject to the boundary conditions $$\begin{aligned}
U^+({{\boldsymbolr}})&=U^-({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,,
\label{eq:BCA1}\\
\kappa{{\boldsymbol\nu}} \cdot[\nabla U]^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\cdot[\nabla U]^-({{\boldsymbolr}})\,,\quad {{\boldsymbolr}}\in\Gamma\,,
\label{eq:BCA2}\\
\left(\partial_{\hat{{{\boldsymbolr}}}}-{\rm i}k_1\right)
U^{\rm sc}({{\boldsymbolr}})&=o\left(\lvert{{\boldsymbolr}}\rvert^{-1}\right),
\quad\lvert{{\boldsymbolr}}\rvert\rightarrow\infty\,.
\label{eq:BCA3}\end{aligned}$$ Here $\hat{{{\boldsymbolr}}}={{\boldsymbolr}}/|{{\boldsymbolr}}|$, the scattered field $U^{\rm
sc}$ is source free in $\Omega_1$ and given by $$U({{\boldsymbolr}})=U^{\rm in}({{\boldsymbolr}})+U^{\rm sc}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\,,$$ and the incident field satisfies $$\Delta U^{\rm in}({{\boldsymbolr}})+k_1^2U^{\rm in}({{\boldsymbolr}})=0\,,
\quad{{\boldsymbolr}}\in\mathbb{R}^3\,,$$ except at the possible isolated source point in $\Omega_1$.
The homogeneous version of problem [A]{}, that is problem [A]{} with $U^{\rm in}$=0, is referred to as problem ${\sf A}_0$.
Problem ${\sf B}_0$ {#sec:B0}
-------------------
The transmission problem ${\sf B}_0$ reads: Find $W({{\boldsymbolr}})$, ${{\boldsymbolr}}\in\Omega_1\cup\Omega_2$, which, for a complex jump parameter $\alpha$ and for wavenumbers $k_1$ and $k_2$ such that (\[eq:k1k2\]) holds, solves $$\begin{split}
\Delta W({{\boldsymbolr}})+k_2^2W({{\boldsymbolr}})&=0\,,\quad{{\boldsymbolr}}\in\Omega_1\,,\\
\Delta W({{\boldsymbolr}})+k_1^2W({{\boldsymbolr}})&=0\,,\quad{{\boldsymbolr}}\in\Omega_2\,,
\end{split}
\label{eq:helmB12}$$ subject to the boundary conditions $$\begin{aligned}
W^+({{\boldsymbolr}})&=W^-({{\boldsymbolr}})\,,
\quad {{\boldsymbolr}}\in\Gamma\,,
\label{eq:BCB1}\\
\alpha{{\boldsymbol\nu}}\cdot[\nabla W]^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\cdot[\nabla W]^-({{\boldsymbolr}})\,,
\quad {{\boldsymbolr}}\in\Gamma\,,
\label{eq:BCB2}\\
\left(\partial_{\hat{{{\boldsymbolr}}}}-{\rm i}k_2\right)
W({{\boldsymbolr}})&=o\left(\lvert{{\boldsymbolr}}\rvert^{-1}\right),
\quad\lvert{{\boldsymbolr}}\rvert\rightarrow\infty\,.
\label{eq:BCB3}\end{aligned}$$
Uniqueness and existence {#sec:uniqueexist}
------------------------
We now review uniqueness theorems by Kress and Roach [@KresRoac78] and Kleinman and Martin [@KleiMart88] for solutions to problem [A]{}, along with corollaries for problem ${\sf A}_0$ and ${\sf
B}_0$. Propositions and corollaries apply only under conditions on $k_1$, $k_2$, $\kappa$, and $\alpha$ that are more restrictive than those of (\[eq:k1k2\]). Conjugation of complex quantities is indicated with an overbar symbol.
Assume that $(\ref{eq:k1k2})$ holds. Let in addition $k_1$, $k_2$, $\kappa$, $\kappa^{-1}$ $\in \mathbb{C}\backslash 0$ be such that $$\begin{split}
{{\rm{Arg}}}(k_1^2\bar k_2^2\kappa)&=
\left\{
\begin{array}{lll}
0 & \text{if} & \Re{\rm e}\{k_1\}\Re{\rm e}\{k_2\}\ge 0\,,\\
\pi & \text{if} & \Re{\rm e}\{k_1\}\Re{\rm e}\{k_2\}<0 \,,
\end{array}
\right.
\\
{{\rm{Arg}}}(k_2)&\ne 0\quad \text{if}\quad {{\rm{Arg}}}(k_1)=\pi/2\,.
\end{split}
\label{eq:KRunique}$$ Then problem [A]{} has at most one solution. \[prop:KR\]
This is [@KresRoac78 Theorem 3.1], supplemented with a condition to compensate for a minor flaw in the proof. The original conditions in [@KresRoac78 Theorem 3.1] permit combinations of $k_1$, $k_2$, and $\kappa$ for which problem [A]{} has nontrivial homogeneous solutions. Examples can be found with ${{\rm{Arg}}}(k_1)=\pi/2$, ${{\rm{Arg}}}(k_2)=0$, and ${{\rm{Arg}}}(\kappa)=\pi$, using the example for the sphere in [@KresRoac78 p. 1434].
Assume that $(\ref{eq:k1k2})$ holds. Let in addition $k_1$, $k_2$, $\kappa$, $\kappa^{-1}$ $\in \mathbb{C}\backslash 0$ be such that $$0\leq{{\rm{Arg}}}(k_1\kappa)\leq \pi\quad \text{and}\quad
0\leq{{\rm{Arg}}}(\bar k_1 k_2^2\bar\kappa)\leq\pi\,.
\label{eq:crit2KL}$$ Then problem [A]{} has at most one solution. \[prop:KM\]
This is the uniqueness theorem in [@KleiMart88 p. 309].
The conditions (\[eq:KRunique\]) intersect with the conditions (\[eq:crit2KL\]). If any of these sets of conditions holds, then we say that *the conditions of Proposition \[prop:KR\] or \[prop:KM\] hold*. These conditions are sufficient for our purposes but, as pointed out in [@KresRoac78 p. 1434], uniqueness can be established for a wider range of conditions.
If the conditions of Proposition \[prop:KR\] or \[prop:KM\] hold, then problem ${\sf A}_0$ has only the trivial solution $U({{\boldsymbolr}})=0$. \[cor:uniqueA0\]
In Ref. [@KleiMart88], the condition (\[eq:k1k2\]) is not directly included in the formulation of what corresponds to our problem [A]{}. Instead, the condition $0\leq{{\rm{Arg}}}(k_1)<\pi$ is added for the problem to have at most one solution and $0\leq{{\rm{Arg}}}(k_2)<\pi$ is added for the existence of a unique solution.
Assume that $$\begin{split}
{{\rm{Arg}}}(\bar k_1^2 k_2^2\alpha)&=
\left\{
\begin{array}{lll}
0 & \text{if} & \Re{\rm e}\{k_1\}\Re{\rm e}\{k_2\}\ge 0\,,\\
\pi & \text{if} & \Re{\rm e}\{k_1\}\Re{\rm e}\{k_2\}<0 \,,
\end{array}
\right.
\label{eq:KRuniqueB0}
\\
{{\rm{Arg}}}(k_1)&\ne 0\quad \text{if}\quad {{\rm{Arg}}}(k_2)=\pi/2\,,
\end{split}$$ or $$0\leq{{\rm{Arg}}}(k_2\alpha)\leq\pi\quad \text{and}\quad
0\leq{{\rm{Arg}}}(k_1^2\bar k_2\bar\alpha)\leq\pi\,,
\label{eq:crit2KLB0}$$ holds. Then problem ${\sf B}_0$ has only the trivial solution $W({{\boldsymbolr}})=0$. \[prop:uniqueB0\]
Interchange $k_1$ and $k_2$ and replace $\kappa$ by $\alpha$ in Proposition \[prop:KR\] and \[prop:KM\]. Then use Corollary \[cor:uniqueA0\].
If any of the sets of conditions (\[eq:KRuniqueB0\]) or (\[eq:crit2KLB0\]) holds we say that *the conditions of Proposition \[prop:uniqueB0\] hold*.
![In each image, the gray region and the solid black lines constitute a set of points $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ for which, when $\kappa=k_2^2/k_1^2$, problem [A]{} has at most one solution and problem ${\sf A}_0$ only has the trivial solution. Dashed lines and circles are not included: (a) a set of points obtained using techniques from [@KleiMart88; @KresRoac78]; (b) the set of points discussed in the second paragraph of Section \[sec:uniqueA0kappa\].[]{data-label="fig:hexagon"}](hkfig2a.pdf "fig:"){height="50mm"} ![In each image, the gray region and the solid black lines constitute a set of points $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ for which, when $\kappa=k_2^2/k_1^2$, problem [A]{} has at most one solution and problem ${\sf A}_0$ only has the trivial solution. Dashed lines and circles are not included: (a) a set of points obtained using techniques from [@KleiMart88; @KresRoac78]; (b) the set of points discussed in the second paragraph of Section \[sec:uniqueA0kappa\].[]{data-label="fig:hexagon"}](hkfig2b.pdf "fig:"){height="50mm"}
Uniqueness and existence when $\kappa=k_2^2/k_1^2$ {#sec:uniqueA0kappa}
--------------------------------------------------
The parameter value $\kappa=k_2^2/k_1^2$ is relevant for the electromagnetic transmission problem. By using similar techniques as in [@KleiMart88; @KresRoac78] one can show that when $\kappa=k_2^2/k_1^2$ and $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a), then problem [A]{} has at most one solution and problem ${\sf A}_0$ has only the trivial solution $U({{\boldsymbolr}})=0$.
We also mention that stronger results, including existence results, are available for problem [A]{} with (\[eq:k1k2\]) extended to $0\leq{{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2)\leq\pi$. Using methods from [@Axels06], developed for the more general Dirac equations, one can prove that problem [A]{}, with $\kappa=k_2^2/k_1^2$, has at most one solution in finite energy norm for $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ in the set of points of Figure \[fig:hexagon\](b). Furthermore, such a solution exists in Lipschitz domains given that $\kappa\notin [-c_\Gamma,-1/c_\Gamma]$, where $c_\Gamma\ge 1$ is a geometry-dependent constant which assumes the value $c_\Gamma=1$ for smooth $\Gamma$ [@HelsRose20 Proposition 5.2].
Integral representations for problem [A]{} {#sec:intrepA}
==========================================
We introduce two fields $$\begin{aligned}
U_1({{\boldsymbolr}})&= \frac{1}{2}K_{k_1}\mu({{\boldsymbolr}})
-\frac{1}{2}S_{k_1}\varrho({{\boldsymbolr}})
+U^{\rm in}({{\boldsymbolr}})\,,\quad {{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\label{eq:U1}\\
U_2({{\boldsymbolr}})&=-\frac{1}{2}K_{k_2}\mu({{\boldsymbolr}})
+\frac{\kappa}{2}S_{k_2}\varrho({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\label{eq:U2}\end{aligned}$$ where $\mu$ and $\varrho$ are unknown surface densities. The relations in Section \[sec:limits\] give limits of $U_1({{\boldsymbolr}})$ and $U_2({{\boldsymbolr}})$ at ${{\boldsymbolr}}\in\Gamma$ $$\begin{aligned}
U_1^\pm({{\boldsymbolr}})&=\pm\frac{1}{2}\mu({{\boldsymbolr}})
+\frac{1}{2}K_{k_1}\mu({{\boldsymbolr}})
-\frac{1}{2}S_{k_1}\varrho({{\boldsymbolr}})
+U^{\rm in}({{\boldsymbolr}})\,,
\label{eq:U1pm}\\
U_2^\pm({{\boldsymbolr}})&=\mp\frac{1}{2}\mu({{\boldsymbolr}})
-\frac{1}{2}K_{k_2}\mu({{\boldsymbolr}})
+\frac{\kappa}{2}S_{k_2}\varrho({{\boldsymbolr}})\,.
\label{eq:U2pm}\end{aligned}$$ Limits for the normal derivatives of $U_1({{\boldsymbolr}})$ and $U_2({{\boldsymbolr}})$ at ${{\boldsymbolr}}\in\Gamma$ are $$\begin{aligned}
{{\boldsymbol\nu}}\cdot[\nabla U_1]^\pm({{\boldsymbolr}})&=\pm\frac{1}{2}\varrho({{\boldsymbolr}})
+\frac{1}{2}T_{k_1}\mu({{\boldsymbolr}})
-\frac{1}{2}K_{k_1}^{\rm A}\varrho({{\boldsymbolr}})
+{{\boldsymbol\nu}}\cdot\nabla U^{\rm in}({{\boldsymbolr}})\,,
\label{eq:gU1pm}\\
{{\boldsymbol\nu}}\cdot[\nabla U_2]^\pm({{\boldsymbolr}})&=\mp\frac{\kappa}{2}\varrho({{\boldsymbolr}})
-\frac{1}{2}T_{k_2}\mu({{\boldsymbolr}})
+\frac{\kappa}{2}K_{k_2}^{\rm A}\varrho({{\boldsymbolr}})\,.
\label{eq:gU2pm}\end{aligned}$$
We now construct the ansatz $$U({{\boldsymbolr}})=\left\{
\begin{array}{ll}
U_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_1\,,\\
U_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_2\,,
\end{array}
\right.
\label{eq:U}$$ for the solution to problem [A]{}. The fundamental solution makes $U$ of (\[eq:U\]) automatically satisfy the PDEs of (\[eq:helmA12\]) and the radiation condition . It remains to determine $\mu$ and $\varrho$ to ensure that the boundary conditions (\[eq:BCA1\]) and (\[eq:BCA2\]) are satisfied.
Integral equations for problem [A]{} {#sec:inteqA}
====================================
We propose the system of second-kind integral equations on $\Gamma$ $$\begin{bmatrix}I-\beta_1(K_{k_1}-c_1K_{k_2})&\beta_1(S_{k_1}-c_1\kappa S_{k_2})\\
-\beta_2(T_{k_1}-c_2\kappa^{-1}T_{k_2})&I+\beta_2(K_{k_1}^{\rm A}-c_2K_{k_2}^{\rm A})
\end{bmatrix}
\begin{bmatrix}
\mu\\
\varrho
\end{bmatrix}=
2\begin{bmatrix}
\beta_1U^{\rm in}\\
\beta_2\partial_\nu U^{\rm in}
\end{bmatrix}
\label{eq:kombi}$$ for the determination of $\mu$ and $\varrho$. Here $I$ is the identity and $$\beta_i=(1+c_i)^{-1},\quad i=1,2\,,$$ where $c_1$ and $c_2$ are two free parameters such that $$c_i\neq -1,0\,,\quad i=1,2\,.
\label{eq:ci}$$
We now prove that a solution $\{\mu,\varrho\}$ to (\[eq:kombi\]), under certain conditions and via $U$ of (\[eq:U\]), represents a solution to problem [A]{}. Since $U$ of (\[eq:U\]) satisfies (\[eq:helmA12\]) and (\[eq:BCA3\]) for any $\{\mu,\varrho\}$, it remains to show that $\{\mu,\varrho\}$ from (\[eq:kombi\]) makes $U$ satisfy (\[eq:BCA1\]) and (\[eq:BCA2\]). For this we need to show that, under certain conditions, $U_1$ of (\[eq:U1\]) is zero in $\Omega_2$ and $U_2$ of (\[eq:U2\]) is zero in $\Omega_1$. We introduce the auxiliary field $$W({{\boldsymbolr}})=\left\{
\begin{array}{ll}
U_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_1\,,\\
-c_1^{-1}U_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_2\,.
\end{array}
\right.
\label{eq:W}$$
The field $W$ of (\[eq:W\]), with $\{\mu,\varrho\}$ from (\[eq:kombi\]) and $U_1$ and $U_2$ from (\[eq:U1\]) and (\[eq:U2\]), is the unique solution to problem ${\sf B}_0$ with $\alpha=c_2/(c_1\kappa)$ provided that the conditions of Proposition \[prop:uniqueB0\] hold. This is so since $W$, by construction, satisfies (\[eq:helmB12\]) and (\[eq:BCB3\]). Furthermore, the boundary conditions (\[eq:BCB1\]) and (\[eq:BCB2\]) are satisfied. This can be checked by substituting $U_1^-$ of and $U_2^+$ of into , and ${{\boldsymbol\nu}}\cdot[\nabla U_1]^-$ of and ${{\boldsymbol\nu}}\cdot[\nabla U_2]^+$ of into , and using . As a consequence, according to Proposition \[prop:uniqueB0\], we have $$W({{\boldsymbolr}})=0\,,\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,.
\label{eq:W0}$$
Several useful results for ${{\boldsymbolr}}\in\Gamma$ follow from (\[eq:W\]) and (\[eq:W0\]) $$\begin{aligned}
U_1^-({{\boldsymbolr}})&=0\,,
\label{eq:01m}\\
U_2^+({{\boldsymbolr}})&=0\,,
\label{eq:02p}\\
{{\boldsymbol\nu}}\cdot[\nabla U_1]^-({{\boldsymbolr}})&=0\,,
\label{eq:g01m}\\
{{\boldsymbol\nu}}\cdot[\nabla U_2]^+({{\boldsymbolr}})&=0\,.
\label{eq:g02p}\end{aligned}$$ Now, from (\[eq:U1pm\]) and (\[eq:01m\]), and from (\[eq:U2pm\]) and (\[eq:02p\]) $$\begin{aligned}
U_1^+({{\boldsymbolr}})&=\mu({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,,
\label{eq:mu1}\\
U_2^-({{\boldsymbolr}})&=\mu({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,.
\label{eq:mu2}\end{aligned}$$ Similarly, from (\[eq:gU1pm\]) and (\[eq:g01m\]), and from (\[eq:gU2pm\]) and (\[eq:g02p\]) $$\begin{aligned}
{{\boldsymbol\nu}}\cdot[\nabla U_1]^+({{\boldsymbolr}})&=\varrho({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Gamma\,,
\label{eq:rho1}\\
\kappa^{-1}{{\boldsymbol\nu}}\cdot[\nabla U_2]^-({{\boldsymbolr}})&=
\varrho({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,.
\label{eq:rho2}\end{aligned}$$ It is now easy to see that (\[eq:BCA1\]) and (\[eq:BCA2\]) are satisfied and we conclude:
Assume that $\{k_1,k_2,\alpha=c_2/(c_1\kappa)\}$ is such that the conditions of Proposition \[prop:uniqueB0\] hold. Then a solution $\{\mu,\varrho\}$ to (\[eq:kombi\]) represents, via (\[eq:U\]), a solution also to problem [A]{}. Furthermore, (\[eq:U\]) and (\[eq:kombi\]) correspond to a direct integral equation formulation of problem [A]{} with $\mu$ and $\varrho$ linked to limits of $U$ and $\nabla U$ via (\[eq:mu1\])–(\[eq:rho2\]). \[thm:exA\]
Unique solution to problem [A]{} from (\[eq:kombi\]) {#sec:uniqueA}
====================================================
We use the Fredholm alternative to prove that, under certain conditions, the system (\[eq:kombi\]) has a unique solution $\{\mu,\varrho\}$ and that this solution represents, via (\[eq:U\]), the unique solution to problem [A]{}. Three conditions are referred to with roman numerals
- $c_2=\kappa$ and (\[eq:ci\]) holds.
- $k_1$, $k_2$, and $\kappa$ make the conditions of Proposition \[prop:KR\] or \[prop:KM\] hold or, if $\kappa=k_2^2/k_1^2$, $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a).
- $\{k_1,k_2,\alpha=c_2/(c_1\kappa)\}$ makes the conditions of Proposition \[prop:uniqueB0\] hold.
We start with the observation that (\[eq:kombi\]) is a Fredholm second-kind integral equation with compact (differences of) operators when condition (i) holds and $\Gamma$ is smooth. Then the Fredholm alternative can be applied to (\[eq:kombi\]). Let $\mu_0$ and $\varrho_0$ be solutions to the homogeneous version of (\[eq:kombi\]). Let $U_{10}$, $U_0$, and $W_0$ be the fields (\[eq:U1\]), (\[eq:U\]), and (\[eq:W\]) with $\mu=\mu_0$ and $\varrho=\varrho_0$. From Section \[sec:inteqA\] we know that $W_0=0$ if (iii) holds. We shall now prove that also $U_0=0$ and, from that, $\mu_0=0$ and $\varrho_0=0$.
It follows from Theorem \[thm:exA\], which requires (iii), that $\{\mu_0,\varrho_0\}$ represents a solution to problem ${\sf A}_0$. If (ii) holds, then $U_0=0$ according to Corollary \[cor:uniqueA0\]. It then follows that $U_{10}=0$ in $\Omega_1$ so that $U_{10}^+=0$ and $[\nabla U_{10}]^+=0$. Then $\mu_0=0$ and $\varrho_0=0$ from (\[eq:mu1\]) and (\[eq:rho1\]). Now, from the Fredholm alternative, the system (\[eq:kombi\]) has a unique solution $\{\mu,\varrho\}$. By Theorem \[thm:exA\] this solution represents a solution to problem [A]{}. If problem [A]{} has at most one solution, which requires (ii), this solution to problem [A]{} is unique and we conclude:
Assume that conditions (i), (ii), (iii) hold. Then the system (\[eq:kombi\]) has a unique solution $\{\mu,\varrho\}$ which represents the unique solution to problem [A]{}. \[thm:exunA\]
Note that, when (i) holds, $\alpha=1/c_1$ in (iii) and it is always possible to find a constant $c_1$ so that (\[eq:crit2KLB0\]) holds under the assumption (\[eq:k1k2\]). In this respect, condition (iii) in Theorem \[thm:exunA\] does not introduce any additional constraint to problem [A]{}. A simple rule that satisfies condition (iii) is $$c_1=\left\{
\begin{array}{lll}
e^{{\rm i}{{\rm{Arg}}}(k_2)} &\mbox{if}&\Re{\rm e}\{k_1\}\ge 0\,,\\
e^{{\rm i}({{\rm{Arg}}}(k_2)-\pi)}&\mbox{if}&\Re{\rm e}\{k_1\}< 0\,.
\end{array}\right.$$ This rule gives $c_1={\rm i}$ when $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))=(0,\pi/2)$. It is also possible to choose $c_1=-{\rm i}$ when $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))=(0,\pi/2)$.
Our results, so far, extend those of [@KleiMart88 Section 4.1], where a direct formulation of problem [A]{} is presented in [@KleiMart88 Eq. (4.10)]. To see this, note that [@KleiMart88 Eq. (4.10)] corresponds to with $c_2=\kappa$ and $c_1=1/\kappa$. Now (\[eq:kombi\]) with $c_2=\kappa$ and $c_1$ in agreement with (\[eq:crit2KLB0\]) provides unique solutions over a broader range of $k_1$, $k_2$, and $\kappa$ than does [@KleiMart88 Eq. (4.10)]. For example, if $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))=(0,\pi/2)$ and ${{\rm{Arg}}}(\kappa)=\pi$, then with $c_2=\kappa$ and $c_1=\pm{\rm i}$ is guaranteed to have a unique solution while [@KleiMart88 Eq. (4.10)] is not.
A weakly singular representation of $U$ {#sec:eval}
=======================================
Once the solution $\{\mu,\varrho\}$ has been obtained from (\[eq:kombi\]), the field $U({{\boldsymbolr}})$ can be evaluated via (\[eq:U\]). When ${{\boldsymbolr}}$ is close to $\Gamma$, this could be problematic due to the rapid variation with ${{\boldsymbolr}}'$ in the Cauchy-type singular kernels of $K_{k_1}$ and $K_{k_2}$ in (\[eq:U1\]) and (\[eq:U2\]). To alleviate this problem we introduce $$V({{\boldsymbolr}})=\left\{
\begin{array}{ll}
U_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_1\,,\\
U_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_2\,.
\end{array}
\right.
\label{eq:V}$$ From (\[eq:W\]) and (\[eq:W0\]) it follows that $V$ is a null-field such that $V=0$ in $\Omega_1\cup\Omega_2$, and hence $U=U+V$. The Cauchy-type kernel singularities in the representation of $U+V$ cancel out and we are left with better-behaved weakly singular kernels. In the numerical examples in Section \[sec:numex\] we exploit $U=U+V$ for near-field evaluation.
Electromagnetic transmission problems {#sec:EMprob}
=====================================
We present three electromagnetic transmission problems called problem [C]{}, problem ${\sf C}_0$, and problem ${\sf D}_0$. The main problem is [C]{}, whereas problems ${\sf C}_0$ and ${\sf D}_0$ are needed in proofs.
The prerequisites in Section \[sec:pre\] hold, with regions $\Omega_1$ and $\Omega_2$ that are dielectric and non-magnetic. The electric field is denoted ${{\boldsymbolE}}$ and the magnetic field ${{\boldsymbolH}}$. The electric field is scaled such that ${{\boldsymbolE}}=\eta_1^{-1}{{\boldsymbolE}}_{\rm unscaled}$, where $\eta_1=\sqrt{\mu_0/\varepsilon_1}$ is the wave impedance of $\Omega_1$ and $\varepsilon_1$ is the permittivity of $\Omega_1$. Furthermore, problems [C]{}, ${\sf C}_0$, and ${\sf
D}_0$ contain a complex parameter $\kappa$ which plays a somewhat similar role as the parameter $\kappa$ of Section \[sec:A0\] played in problem [A]{} and ${\sf A}_0$. This new $\kappa$ has the value $\kappa=\varepsilon_2/\varepsilon_1$, where $\varepsilon_2$ is the permittivity of $\Omega_2$. For non-magnetic materials, this is equivalent to $$\kappa=k_2^2/k_1^2.
\label{eq:kappa}$$
Problems [C]{} and ${\sf C}_0$ {#sec:C0}
------------------------------
The transmission problem [C]{} reads: Given an incident field ${{\boldsymbolH}}^{\rm in}$, generated in $\Omega_1$, find ${{\boldsymbolE}}({{\boldsymbolr}}),\,{{\boldsymbolH}}({{\boldsymbolr}})$, ${{\boldsymbolr}}\in\Omega_1\cup\Omega_2$, which, for wavenumbers $k_1$ and $k_2$ and with $\kappa$ from (\[eq:kappa\]) such that $$0\leq{{\rm{Arg}}}(k_1), {{\rm{Arg}}}(k_2)<\pi
\quad\mbox{and}\quad
\kappa\neq -1\,,
\label{eq:k1k2Maxkap}$$ solve Maxwell’s equations $$\begin{split}
\nabla\times{{\boldsymbolE}}({{\boldsymbolr}})&= {\rm i}k_1{{\boldsymbolH}}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,\\
\nabla\times{{\boldsymbolH}}({{\boldsymbolr}})&=-{\rm i}k_1{{\boldsymbolE}}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\,,\\
\nabla\times{{\boldsymbolH}}({{\boldsymbolr}})&=-{\rm i}k_1\kappa{{\boldsymbolE}}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_2\,,
\end{split}
\label{eq:Max123C}$$ except possibly at an isolated point in $\Omega_1$ where the source of ${{\boldsymbolH}}^{\rm in}$ is located, subject to the boundary conditions $$\begin{aligned}
{{\boldsymbol\nu}}\times{{\boldsymbolE}}^+({{\boldsymbolr}})&={{\boldsymbol\nu}}\times{{\boldsymbolE}}^-({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in \Gamma\,,\label{eq:rv2C}\\
{{\boldsymbol\nu}}\times{{\boldsymbolH}}^+({{\boldsymbolr}})&={{\boldsymbol\nu}}\times{{\boldsymbolH}}^-({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in \Gamma\,,\label{eq:rv1C}\\
\left(\partial_{\hat{{{\boldsymbolr}}}}-{\rm i}k_1\right){{\boldsymbolH}}^{\rm sc}
({{\boldsymbolr}})&=o\left(\vert {{\boldsymbolr}}\vert^{-1}\right),
\quad\lvert{{\boldsymbolr}}\rvert\rightarrow\infty\,.
\label{eq:radcondC}\end{aligned}$$ The scattered field ${{\boldsymbolH}}^{\rm sc}$ is source free in $\Omega_1$ and defined by $${{\boldsymbolH}}({{\boldsymbolr}})={{\boldsymbolH}}^{\rm in}({{\boldsymbolr}})+{{\boldsymbolH}}^{\rm sc}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in \Omega_1\,.
\label{eq:decomp}$$ The condition and decomposition also hold for ${{\boldsymbolE}}$. The incident field satisfies $$\begin{split}
\nabla\times{{\boldsymbolE}}^{\rm in}({{\boldsymbolr}})&= {\rm i}k_1{{\boldsymbolH}}^{\rm in}
({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\mathbb{R}^3\,,\\
\nabla\times{{\boldsymbolH}}^{\rm in}({{\boldsymbolr}})&=-{\rm i}k_1{{\boldsymbolE}}^{\rm in}
({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\mathbb{R}^3\,,
\end{split}$$ except at the possible isolated source point in $\Omega_1$.
The homogeneous problem ${\sf C}_0$ is problem [C]{} with ${{\boldsymbolE}}^{\rm in}={{\boldsymbolH}}^{\rm in}={{\boldsymbol0}}$.
Problem ${\sf D}_0$ {#sec:D0}
-------------------
The transmission problem ${\sf D}_0$ reads: find ${{\boldsymbolE}}_W({{\boldsymbolr}}),\,{{\boldsymbolH}}_W({{\boldsymbolr}})$, ${{\boldsymbolr}}\in\Omega_1\cup\Omega_2$, which, for a complex jump parameter $\lambda$ and for $k_1$, $k_2$, and $\kappa$ such that holds, solve $$\begin{split}
\nabla\times{{\boldsymbolE}}_W({{\boldsymbolr}})&= {\rm i}k_1{{\boldsymbolH}}_W({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,\\
\nabla\times{{\boldsymbolH}}_W({{\boldsymbolr}})&=-{\rm i}k_1\kappa{{\boldsymbolE}}_W({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\,,\\
\nabla\times{{\boldsymbolH}}_W({{\boldsymbolr}})&=-{\rm i}k_1{{\boldsymbolE}}_W({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_2\,,
\end{split}
\label{eq:D0123}$$ subject to the boundary conditions $$\begin{aligned}
\lambda\kappa{{\boldsymbol\nu}}\times {{\boldsymbolE}}_W^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\times{{\boldsymbolE}}_W^-({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,,
\label{eq:RV2D0}\\
{{\boldsymbol\nu}}\times{{\boldsymbolH}}_W^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\times{{\boldsymbolH}}_W^-({{\boldsymbolr}})\,,\quad{{\boldsymbolr}}\in\Gamma\,,
\label{eq:RV1D0}\\
\left(\partial_{\hat{{{\boldsymbolr}}}}-{\rm i}k_2\right){{\boldsymbolH}}_W({{\boldsymbolr}})&=o
\left(\vert{{\boldsymbolr}}\vert^{-1}\right),
\quad\lvert{{\boldsymbolr}}\rvert\rightarrow\infty\,.
\label{eq:radcondD0}\end{aligned}$$ The radiation condition also holds for ${{\boldsymbolE}}_W$.
Uniqueness and existence of solutions to problem ${\sf
C}$, ${\sf C}_0$, and ${\sf D}_0$ {#sec:uniqueex}
------------------------------------------------------
In Appendix D it is shown that when $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a), then problem [C]{} has at most one solution and problem ${\sf C}_0$ has only the trivial solution ${{\boldsymbolE}}={{\boldsymbolH}}={{\boldsymbol0}}$. It is also shown that when the conditions of Proposition \[prop:uniqueB0\] hold for $\{k_1,k_2,\alpha=\lambda\}$, then problem ${\sf D}_0$ has only the trivial solution ${{\boldsymbolE}}_W= {{\boldsymbolH}}_W={{\boldsymbol0}}$.
The stronger results for problem [A]{}, discussed in Section \[sec:uniqueA0kappa\], carry over to problem [C]{}. One can prove that there exists a unique solution in finite energy norm to problem [C]{} in Lipschitz domains when $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](b) and $\kappa$ is outside a certain interval on the real axis [@HelsRose20 Proposition 7.4].
Integral representations for problem [C]{} {#sec:intrepC}
==========================================
Let $\sigma_{\rm E}$, $\varrho_{\rm E}$, ${{\boldsymbolM}}_{\rm s}$, ${{\boldsymbolJ}}_{\rm s}$, $\varrho_{\rm M}$, and $\sigma_{\rm M}$ be six unknown, scalar- and vector-valued, surface densities and define the four fields $$\begin{aligned}
{{\boldsymbolE}}_1({{\boldsymbolr}})&=
-\frac{1}{2}\boldsymbol{\cal N}_{k_1}\varrho_{\rm E}({{\boldsymbolr}})
-\frac{1}{2}{\cal R}_{k_1}\left({{\boldsymbol\nu}}'\sigma_{\rm M}
+{{\boldsymbolM}}_{\rm s})({{\boldsymbolr}}\right)\nonumber\\
&\qquad
+\frac{1}{2}\tilde{\cal S}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})+ {{\boldsymbolE}}^{\rm in}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\label{eq:E1rep1}\\
{{\boldsymbolE}}_2({{\boldsymbolr}})&=
\frac{1}{2\kappa}\boldsymbol{\cal N}_{k_2}\varrho_{\rm E}({{\boldsymbolr}})
+\frac{1}{2\kappa}{\cal R}_{k_2}\left({{\boldsymbol\nu}}'\sigma_{\rm M}
+\kappa{{\boldsymbolM}}_{\rm s})({{\boldsymbolr}}\right)\nonumber\\
&\qquad
-\frac{1}{2}\tilde{\cal S}_{k_2}(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\label{eq:E2rep1}\end{aligned}$$ $$\begin{aligned}
{{\boldsymbolH}}_1({{\boldsymbolr}})&=
\frac{1}{2}\tilde{\cal S}_{k_1}\left({{\boldsymbol\nu}}'\sigma_{\rm M}
+{{\boldsymbolM}}_{\rm s})({{\boldsymbolr}}\right)
+\frac{1}{2}{\cal R}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})\nonumber\\
&\qquad
-\frac{1}{2}\boldsymbol{\cal N}_{k_1}\varrho_{\rm M}({{\boldsymbolr}})
+{{\boldsymbolH}}^{\rm in}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in \Omega_1\cup \Omega_2\,,
\label{eq:H1rep11}\\
{{\boldsymbolH}}_2({{\boldsymbolr}})&=
-\frac{1}{2}\tilde{\cal S}_{k_2}\left({{\boldsymbol\nu}}'\sigma_{\rm M}
+\kappa{{\boldsymbolM}}_{\rm s})({{\boldsymbolr}}\right)
-\frac{1}{2}{\cal R}_{k_2}(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})\nonumber\\
&\qquad
+\frac{1}{2}\boldsymbol{\cal N}_{k_2}\varrho_{\rm M}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,.
\label{eq:H2rep11}\end{aligned}$$ The introduction of $\sigma_{\rm E}$ and $\sigma_{\rm M}$ is inspired by the integral representations for the generalized Helmholtz transmission problem in [@Vico16; @VicGreFer18].
The integral representations of the fields ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ for problem [C]{} are $${{\boldsymbolE}}({{\boldsymbolr}})=\left\{
\begin{array}{ll}
{{\boldsymbolE}}_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_1\,,\\
{{\boldsymbolE}}_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_2\,,
\end{array}\right.
\quad
{{\boldsymbolH}}({{\boldsymbolr}})=\left\{
\begin{array}{ll}
{{\boldsymbolH}}_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in \Omega_1\,,\\
{{\boldsymbolH}}_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in \Omega_2\,.
\end{array}\right.
\label{eq:repEH}$$
Integral equations for problem [C]{} {#sec:inteqC}
====================================
For the determination of $\{\sigma_{\rm E},\varrho_{\rm E},{{\boldsymbolM}}_{\rm s},{{\boldsymbolJ}}_{\rm s},\varrho_{\rm M},\sigma_{\rm M}\}$ we propose the system of second-kind integral equations on $\Gamma$ $$\left(I+{\bf D}{\bf Q}\right){{\boldsymbol\mu}}=2{\bf D}{{\boldsymbolf}}\,.
\label{eq:EHsys}$$ Here ${{\boldsymbol\mu}}$ and ${{\boldsymbolf}}$ are column vectors with six entries each $$\begin{aligned}
{{\boldsymbol\mu}}&=\left[\sigma_{\rm E};\varrho_{\rm E};{{\boldsymbolM}}_{\rm s};
{{\boldsymbolJ}}_{\rm s};\varrho_{\rm M};\sigma_{\rm M}\right],\\
{{\boldsymbolf}}&=\left[0;{{\boldsymbol\nu}}\cdot{{\boldsymbolE}}^{\rm in};
-{{\boldsymbol\nu}}\times{{\boldsymbolE}}^{\rm in};{{\boldsymbol\nu}}\times{{\boldsymbolH}}^{\rm in};
{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}^{\rm in};0\right],\end{aligned}$$ ${\bf Q}$ is a $6\times 6$ matrix whose non-zero operator entries $Q_{ij}$ map scalar- or vector-valued densities to scalar or vector-valued functions $$\begin{aligned}
Q_{11}&=-K_{k_1}+c_3K_{k_2}\,,\quad
Q_{12} =-\tilde S_{k_1}+c_3\kappa\tilde S_{k_2}\,,\quad
Q_{14} =\nabla\cdot({\cal S}_{k_1}-c_3\kappa {\cal S}_{k_2})\,,\\
Q_{21}&=-{{\boldsymbol\nu}}\cdot(\tilde{\cal S}_{k_1}-c_4\tilde{\cal
S}_{k_2}){{\boldsymbol\nu}}'\,,\quad
Q_{22} =K_{k_1}^ {\rm A}-c_4K_{k_2}^{\rm A}\,,\\
Q_{23}&={{\boldsymbol\nu}}\cdot({\cal R}_{k_1}-c_4\kappa {\cal R}_{k_2})\,,\quad
Q_{24} =-{{\boldsymbol\nu}}\cdot(\tilde{\cal S}_{k_1}-c_4\kappa\tilde{\cal S}_{k_2})\,,\\
Q_{26}&={{\boldsymbol\nu}}\cdot({\cal R}_{k_1}-c_4{\cal R}_{k_2}){{\boldsymbol\nu}}'\,,\quad
Q_{31} ={{\boldsymbol\nu}}\times(\tilde{\cal S}_{k_1}-c_5\kappa^{-1}\tilde{\cal S}_{k_2}){{\boldsymbol\nu}}'\,,\\
Q_{32}&=-{{\boldsymbol\nu}}\times(\boldsymbol{\cal
N}_{k_1}-c_5\kappa^{-1}\boldsymbol{\cal N}_{k_2})\,,\quad
Q_{33} =-{{\boldsymbol\nu}}\times({\cal R}_{k_1}-c_5{\cal R}_{k_2})\,,\\
Q_{34}&={{\boldsymbol\nu}}\times(\tilde{\cal S}_{k_1}-c_5\tilde{\cal S}_{k_2})\,,\quad
Q_{36} =-{{\boldsymbol\nu}}\times({\cal R}_{k_1}-c_5\kappa^{-1}{\cal
R}_{k_2}){{\boldsymbol\nu}}'\,,\\
Q_{41}&=-{{\boldsymbol\nu}}\times({\cal R}_{k_1}-c_6\kappa^{-1}{\cal
R}_{k_2}){{\boldsymbol\nu}}'\,,\quad
Q_{43} =-{{\boldsymbol\nu}}\times(\tilde{\cal S}_{k_1}-c_6\kappa\tilde{\cal
S}_{k_2})\,,\\
Q_{44}&=-{{\boldsymbol\nu}}\times({\cal R}_{k_1}-c_6{\cal R}_{k_2})\,,\quad
Q_{45} ={{\boldsymbol\nu}}\times(\boldsymbol{\cal N}_{k_1}-c_6\boldsymbol{\cal
N}_{k_2})\,,\\
Q_{46}&=-{{\boldsymbol\nu}}\times(\tilde{\cal S}_{k_1}-c_6\tilde{\cal
S}_{k_2}){{\boldsymbol\nu}}'\,,\quad
Q_{51} =-{{\boldsymbol\nu}}\cdot({\cal R}_{k_1}-c_7\kappa^{-1}{\cal
R}_{k_2}){{\boldsymbol\nu}}'\,,\\
Q_{53}&=-{{\boldsymbol\nu}}\cdot(\tilde{\cal S}_{k_1}-c_7\kappa \tilde{\cal
S}_{k_2})\,,\quad
Q_{54} =-{{\boldsymbol\nu}}\cdot({\cal R}_{k_1}-c_7{\cal R}_{k_2})\,,\\
Q_{55}&=K_{k_1}^{\rm A}-c_7K_{k_2}^{\rm A}\,,\quad
Q_{56} =-{{\boldsymbol\nu}}\cdot(\tilde{\cal S}_{k_1}-c_7\tilde{\cal
S}_{k_2}){{\boldsymbol\nu}}'\,,\\
Q_{63}&=\nabla\cdot({\cal S}_{k_1}-c_8\kappa{\cal S}_{k_2})\,,\quad
Q_{65} =-\tilde S_{k_1}+c_8\kappa\tilde S_{k_2}\,,\quad
Q_{66} =-K_{k_1}+c_8K_{k_2}\,,\end{aligned}$$ ${\bf D}$ is a diagonal $6\times 6$ matrix of scalars with non-zero entries $$\begin{split}
&D_{ii}=(1+c_{i+2})^{-1}\,,\quad i=1,\ldots,6\,,\\
c_3=&\gamma_{\rm E}c\,,\quad c_4=c_6=c\,,\quad c_5=c_7=\lambda\kappa c\,,
\quad c_8=\gamma_{\rm M}c\,,
\end{split}$$ and $c$, $\lambda$, $\gamma_{\rm E}$, and $\gamma_{\rm M}$ are free parameters such that $$c_i \neq -1,0,\quad i=3,\ldots,8\,.
\label{eq:neq0}$$
Criteria for to represent a solution to problem [C]{}
-----------------------------------------------------
We now prove that a solution ${{\boldsymbol\mu}}$ to , under certain conditions and via , represents a solution to problem [C]{}.
The fundamental solution makes ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ of satisfy the radiation condition . It remains to prove that ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ satisfy Maxwell’s equations and the boundary conditions and . For this we first need to show that, under certain conditions, ${{\boldsymbolE}}_1$ and ${{\boldsymbolH}}_1$ of and are zero in $\Omega_2$ and ${{\boldsymbolE}}_2$ and ${{\boldsymbolH}}_2$ of and are zero in $\Omega_1$. We introduce the auxiliary fields $${{\boldsymbolE}}_W({{\boldsymbolr}})=\left\{
\begin{array}{ll}
{{\boldsymbolE}}_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_1\,,\\
-c^{-1}{{\boldsymbolE}}_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_2\,,
\end{array}\right.
\quad
{{\boldsymbolH}}_W({{\boldsymbolr}})=\left\{
\begin{array}{ll}
{{\boldsymbolH}}_2({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_1\,,\\
-c^{-1}{{\boldsymbolH}}_1({{\boldsymbolr}})\,, & {{\boldsymbolr}}\in\Omega_2\,.
\end{array}\right.
\label{eq:EHW1}$$
The fields ${{\boldsymbolE}}_W$ and ${{\boldsymbolH}}_W$, with ${{\boldsymbol\mu}}$ from , is the unique trivial solution to problem ${\sf
D}_0$ provided the sets $\{k_1,k_2,\alpha=\lambda\bar\gamma_{\rm
M}\}$, $\{k_1,k_2,\alpha=\bar\gamma_{\rm E}\bar\kappa\}$, and $\{k_1,k_2,\alpha=\lambda\}$ are such that the conditions of Proposition \[prop:uniqueB0\] hold. This statement is now shown in several steps. The fundamental solution makes ${{\boldsymbolE}}_W$ and ${{\boldsymbolH}}_W$ satisfy . Using Appendix A in combination with one can show that and are satisfied. Appendix B shows that if ${{\boldsymbol\mu}}$ is a solution to and if the conditions of Proposition \[prop:uniqueB0\] hold for $\{k_1,k_2,\alpha=\lambda\bar\gamma_{\rm M}\}$ and $\{k_1,k_2,\alpha=\bar\gamma_{\rm E}\bar\kappa\}$, then $$\begin{aligned}
\nabla\cdot{{\boldsymbolE}}_W({{\boldsymbolr}})&=0\,,
\quad{{\boldsymbolr}}\in \Omega_1\cup\Omega_2\,,\label{eq:divEW}\\
\nabla\cdot{{\boldsymbolH}}_W({{\boldsymbolr}})&=0\,,
\quad{{\boldsymbolr}}\in \Omega_1\cup\Omega_2\,.\label{eq:divHW}\end{aligned}$$ Appendix C shows that if and hold, then ${{\boldsymbolE}}_W$ and ${{\boldsymbolH}}_W$ satisfy . If the conditions of Proposition \[prop:uniqueB0\] also hold for $\{k_1,k_2,\alpha=\lambda\}$, then ${\sf D}_0$ only has the trivial solution ${{\boldsymbolE}}_W={{\boldsymbolH}}_W={{\boldsymbol0}}$, that is, $$\begin{split}
{{\boldsymbolE}}_2({{\boldsymbolr}})&={{\boldsymbolH}}_2({{\boldsymbolr}})=
{{\boldsymbol0}}\,,\quad{{\boldsymbolr}}\in \Omega_1\,,\\
{{\boldsymbolE}}_1({{\boldsymbolr}})&={{\boldsymbolH}}_1({{\boldsymbolr}})=
{{\boldsymbol0}}\,,\quad{{\boldsymbolr}}\in \Omega_2\,.
\end{split}
\label{eq:H1H2zero}$$ By that the statement is proven.
From and Appendix A we obtain the surface densities as boundary values of the full 3D electromagnetic fields $$\begin{aligned}
[\nabla\cdot {{\boldsymbolE}}_1]^+({{\boldsymbolr}})&=
\kappa[\nabla\cdot {{\boldsymbolE}}_2]^-({{\boldsymbolr}})=
-{\rm i}k_1\sigma_{\rm E}({{\boldsymbolr}})\,,\label{eq:bvalues1}\\
{{\boldsymbol\nu}}\cdot{{\boldsymbolE}}_1^+({{\boldsymbolr}})&=
\kappa{{\boldsymbol\nu}}\cdot{{\boldsymbolE}}_2^-({{\boldsymbolr}})=
\varrho_{\rm E}({{\boldsymbolr}})\,,\label{eq:bvalues2}\\
{{\boldsymbol\nu}}\times{{\boldsymbolE}}_1^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\times{{\boldsymbolE}}_2^-({{\boldsymbolr}})=
-{{\boldsymbolM}}_{\rm s}({{\boldsymbolr}})\,,\label{eq:bvalues3}\\
{{\boldsymbol\nu}}\times{{\boldsymbolH}}_1^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\times{{\boldsymbolH}}_2^-({{\boldsymbolr}})=
{{\boldsymbolJ}}_{\rm s}({{\boldsymbolr}})\,,\label{eq:bvalues4}\\
{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_1^+({{\boldsymbolr}})&=
{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_2^-({{\boldsymbolr}})=
\varrho_{\rm M}({{\boldsymbolr}})\,,\label{eq:bvalues5}\\
[\nabla\cdot {{\boldsymbolH}}_1]^+({{\boldsymbolr}})&=
[\nabla\cdot {{\boldsymbolH}}_2]^-({{\boldsymbolr}})=-{\rm i}k_1\sigma_{\rm M}({{\boldsymbolr}})\,.
\label{eq:bvalues6}\end{aligned}$$ Due to and , ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ of satisfy and . Appendix B shows that – imply $$\nabla\cdot{{\boldsymbolE}}({{\boldsymbolr}})=\nabla\cdot{{\boldsymbolH}}({{\boldsymbolr}})=0\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\label{eq:divH0E0}$$ when $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a). Finally, from the representations – and the divergence condition , Appendix C shows that is satisfied. We conclude:
Assume that $\{k_1,k_2,\alpha=\lambda\bar\gamma_{\rm M}\}$, $\{k_1,k_2,\alpha=\bar\gamma_{\rm E}\bar\kappa\}$, and $\{k_1,k_2,
\alpha=\lambda\}$ are such that the conditions of Proposition \[prop:uniqueB0\] hold. Then a solution ${{\boldsymbol\mu}}$ to represents, via , a solution also to problem [C]{}. Furthermore, and correspond to a direct integral equation formulation of problem [C]{} with ${{\boldsymbol\mu}}$ linked to limits of ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ via –. \[thm:exC\]
The surface densities in – can be given the following physical interpretations: $-{\rm
i}k_1\sigma_{\rm E}$ and $-{\rm i}k_1\sigma_{\rm M}$ are the electric and magnetic volume charge densities at $\Gamma^+$, $\varrho_{\rm E}$ and $\varrho_{\rm M}$ are the equivalent electric and magnetic surface charge densities on $\Gamma^+$, and ${{\boldsymbolM}}_{\rm s}$ and ${{\boldsymbolJ}}_{\rm s}$ are the equivalent magnetic and electric surface current densities on $\Gamma^+$.
Unique solution to problem [C]{} from {#sec:uniqueC}
======================================
We now prove that if there exists a solution to problem [C]{}, then, under certain conditions, there exists a solution ${{\boldsymbol\mu}}$ to and it represents the unique solution to problem [ C]{}. Three conditions are referred to
- The conditions in hold.
- $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a).
- $\{k_1,k_2,\alpha=\lambda\bar\gamma_{\rm M}\}$, $\{k_1,k_2,\alpha=\bar\gamma_{\rm E}\bar\kappa\}$, and $\{k_1, k_2,
\alpha=\lambda\}$ are such that the conditions of Proposition \[prop:uniqueB0\] hold.
Let ${{\boldsymbol\mu}}_{0}$ be a solution to the homogeneous version of and assume that (i), (ii), and (iii) hold. Since (iii) holds, ${{\boldsymbol\mu}}_{\rm 0}$ represents a solution to problem [ C]{}$_0$, according to Theorem \[thm:exC\]. Since (ii) holds, this solution is the trivial solution ${{\boldsymbolE}}={{\boldsymbolH}}={{\boldsymbol0}}$, according to Section \[sec:uniqueex\]. The limits of fields in – are then zero and hence ${{\boldsymbol\mu}}_0={{\boldsymbol0}}$. Then has at most one solution ${{\boldsymbol\mu}}$. Since ${{\boldsymbol\mu}}$ is linked to limits of ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ via – it follows that if problem [C]{} has a solution, then via – this solution gives a ${{\boldsymbol\mu}}$ that solves . We conclude:
Assume that there exists a solution to problem [C]{} and that condition (ii) holds. Then this solution is unique. If conditions (i) and (iii) also hold, then there exists a unique solution ${{\boldsymbol\mu}}$ to and this solution represents via the unique solution to problem [C]{}. \[thm:exunC\]
From , , and it is seen that $\sigma_{\rm E}$ and $\sigma_{\rm M}$ are zero. Despite this, $\sigma_{\rm E}$ and $\sigma_{\rm M}$ are needed in to guarantee uniqueness. Often, however, one can omit $\sigma_{\rm E}$ and $\sigma_{\rm M}$ from and still get the correct unique solution. \[rem:div\]
Determination of uniqueness parameters
--------------------------------------
The system contains the free parameters $\lambda$, $\gamma_{\rm E}$, $\gamma_{\rm M}$, and $c$. Unique solvability of requires that the conditions of Proposition \[prop:uniqueB0\] hold for the sets $\{k_1,k_2,\alpha=\lambda\bar\gamma_{\rm M}\}$, $\{k_1,k_2,\alpha=\bar\gamma_{\rm E}\bar\kappa\}$, and $\{k_1,k_2,\alpha=\lambda\}$ while the choice of $c$ is restricted by (\[eq:neq0\]). Because of their role in ensuring unique solvability of , we refer to $\{\lambda,\gamma_{\rm
E},\gamma_{\rm M},c\}$ as [*uniqueness parameters*]{}.
Generally, there are many parameter choices for which the conditions of Proposition \[prop:uniqueB0\] and (\[eq:neq0\]) hold for a given $\{k_1,k_2\}$ satisfying . A valid choice, which also works well numerically, when ${{\rm{Arg}}}(k_1)=0$ and $\pi/4\leq{{\rm{Arg}}}(k_2)\leq\pi/2$ is $$\lambda=e^{-{\rm i}{{\rm{Arg}}}(k_2)},\quad
\gamma_{\rm E}=\kappa^{-1}e^{{\rm i}({{\rm{Arg}}}(k_2)-\pi)},\quad
\gamma_{\rm M}=1\,,\quad
c=\lambda^{-1},
\label{eq:parameter2}$$ and when ${{\rm{Arg}}}(k_1)=0$ and $\pi/2\leq{{\rm{Arg}}}(k_2)\leq 3\pi/4$ $$\lambda=e^{{\rm i}(\pi-{{\rm{Arg}}}(k_2))},\quad
\gamma_{\rm E}=\kappa^{-1}e^{{\rm i}{{\rm{Arg}}}(k_2)},\quad
\gamma_{\rm M}=1\,,\quad
c=\lambda^{-1}.
\label{eq:parameter1}$$
2D limits {#sec:twoD}
=========
As a first numerical test of our formulations we consider, in Section \[sec:numex\], the 2D transverse magnetic (TM) transmission problem where an incident TM wave is scattered from an infinite cylinder. This problem is independent of the $z$-coordinate and we introduce the vector $r=(x,y)$, the unit tangent vector $\tau=(\tau_x,\tau_y)$, and the unit normal vector $\nu=(\nu_x,\nu_y)$, where $(\tau_x,\tau_y,0)=\hat{{{\boldsymbolz}}}\times(\nu_x,\nu_y,0)$ and $\hat{{{\boldsymbolz}}}$ is the unit vector in the $z$-direction. The incident wave has polarization ${{\boldsymbolH}}^{\rm
in}(r)=\hat{{{\boldsymbolz}}}H^{\rm in}(r)$, which implies ${{\boldsymbolM}}_{\rm
s}=\hat{{{\boldsymbolz}}}M$, ${{\boldsymbolJ}}_{\rm s}=\tau J$, $\varrho_{\rm M}=0$, and $\sigma_{\rm M}=0$.
The integral representations , , and –, as well as the systems and , are transferred to two dimensions by exchanging the fundamental solution for the 2D fundamental solution $$\Phi_k(r,r')=\frac{\rm i}{4}H_0^{(1)}(k|r-r'|)\,,\quad r,r'\in\mathbb{R}^2,
\label{eq:Phi}$$ where $H_0^{(1)}$ is the zeroth order Hankel function of the first kind.
Integral representations in two dimensions {#sec:intrep2D}
------------------------------------------
Since $\sigma_{\rm E}$ is zero, see Remark \[rem:div\], the 2D representation of the field ${{\boldsymbolH}}$ in , to be used in evaluation of the magnetic field, is $$H(r)=\left\{
\begin{array}{ll}
\dfrac{1}{2}\tilde S_{k_1}M(r)-\dfrac{1}{2} K_{k_1}J(r)+H^{\rm in}(r)\,,&
r\in\Omega_1\,,\\
-\dfrac{\kappa}{2}\tilde S_{k_2}M(r)+\dfrac{1}{2}K_{k_2}J(r)\,,&
r\in\Omega_2\,.
\end{array}\right.
\label{eq:H22D}$$ By letting $U=H$, $U^{\rm in}=H^{\rm in}$, $\mu=-J$, $\varrho=-{\rm
i}k_1M$, and $\kappa=k_2^2/k_1^2$ in the scalar representation it becomes identical to . According to Section \[sec:eval\] one may add null-fields to . That gives the representation $$H(r)=\frac{1}{2}(\tilde S_{k_1}-\kappa\tilde S_{k_2})M(r)
-\frac{1}{2}(K_{k_1}- K_{k_2})J(r)+H^{\rm in}(r)\,,
\quad r\in\Omega_1\cup \Omega_2,
\label{eq:Hnull}$$ which is to prefer for evaluations at points $r$ close to $\Gamma$.
Integral equations with four, three, and two densities
------------------------------------------------------
In the TM problem the system becomes $$\left(I+\tilde{\bf D}\tilde{\bf Q}\right)\tilde{{{\boldsymbol\mu}}}
=2\tilde{\bf D}\tilde{{{\boldsymbolf}}}\,.
\label{eq:EHsysTM}$$ Here $\tilde{{{\boldsymbol\mu}}}$ and $\tilde{{{\boldsymbolf}}}$ are column vectors with four entries each $$\begin{aligned}
\tilde{{{\boldsymbol\mu}}}&=\left[\sigma_{\rm E};\varrho_{\rm E};M;J\right],\\
\tilde{{{\boldsymbolf}}}&=\left[0;{\rm i}k_1^{-1}\partial_\tau H^{\rm in};
{\rm i}k_1^{-1}\partial_\nu H^{\rm in};-H^{\rm in}\right],\end{aligned}$$ $\tilde{\bf Q}$ is a $4\times 4$ matrix with non-zero scalar operator entries $$\begin{aligned}
\tilde Q_{11}&=-K_{k_1}+c_3K_{k_2}\,,\quad
\tilde Q_{12} =-\tilde S_{k_1}+c_3\kappa\tilde S_{k_2}\,,\quad
\tilde Q_{14} =-C_{k_1}+c_3\kappa C_{k_2}\,,\\
\tilde Q_{21}&=-(\tilde S_{k_1}-c_4\tilde S_{k_2})\nu\cdot\nu'\,,\quad
\tilde Q_{22} =K_{k_1}^{\rm A}-c_4K_{k_2}^{\rm A}\,,\\
\tilde Q_{23}&=C_{k_1}^{\rm A}-c_4\kappa C_{k_2}^{\rm A}\,,\quad
\tilde Q_{24} =-(\tilde S_{k_1}-c_4\kappa \tilde S_{k_2})\nu\cdot\tau'\,,\\
\tilde Q_{31}&=(\tilde S_{k_1}-c_5\kappa^{-1}\tilde S_{k_2})
\tau\cdot\nu'\,,\quad
\tilde Q_{32} =-C_{k_1}^{\rm A}+c_5\kappa^{-1}C_{k_2}^{\rm A}\,,\\
\tilde Q_{33}&=K_{k_1}^{\rm A}-c_5K_{k_2}^{\rm A}\,,\quad
\tilde Q_{34} =(\tilde S_{k_1}-c_5\tilde S_{k_2})\tau\cdot\tau'\,,\\
\tilde Q_{41}&=C_{k_1}-c_6\kappa^{-1}C_{k_2}\,,\quad
\tilde Q_{43} =\tilde S_{k_1}-c_6\kappa \tilde S_{k_2}\,,\quad
\tilde Q_{44} =-K_{k_1}+c_6 K_{k_2}\,,\end{aligned}$$ $\tilde{\bf D}$ is a diagonal $4\times 4$ matrix of scalars with non-zero entries $$\tilde D_{ii}=(1+c_{i+2})^{-1}\,,\quad i=1,2,3,4\,,$$ and $$\begin{split}
C_k\sigma(r)&=
2\int_\Gamma (\partial_{\tau'}\Phi_k)(r,r')\sigma(r')\,{\rm d}\Gamma',
\quad r\in\Gamma\,,\\
C_k^{\rm A}\sigma(r)&=
2\int_\Gamma (\partial_{\tau}\Phi_k)(r,r')\sigma(r')\,{\rm d}\Gamma',
\quad r\in\Gamma.
\end{split}$$
If we omit $\sigma_{\rm E}$, see Remark \[rem:div\], the system reduces to $$\left(I+\hat{\bf D}\hat{\bf Q}\right)\hat{{{\boldsymbol\mu}}}
=2\hat{\bf D}\hat{{{\boldsymbolf}}}\,.
\label{eq:threedens}$$ Here $\hat{\bf Q}$ and $\hat{\bf D}$ are $\tilde{\bf Q}$ and $\tilde{\bf D}$ with the first row and column deleted, $\hat{{{\boldsymbolf}}}$ is $\tilde{{{\boldsymbolf}}}$ with the first entry deleted, and $\hat{{{\boldsymbol\mu}}}$ contains the three densities $\{\varrho_{\rm
E},M,J\}$.
A third alternative is to only use the densities $M$ and $J$. The integral representation and system are now suitable, where the change of variables in Section \[sec:intrep2D\] makes equal to and equal to $$\begin{bmatrix}I+\beta_2(K_{k_1}^{\rm A}-c_2K_{k_2}^{\rm A})
&\beta_2{\rm i}k_1^{-1}(T_{k_1}-c_2\kappa^{-1}T_{k_2})\\
\beta_1(\tilde S_{k_1}-c_1\kappa\tilde S_{k_2})&I-\beta_1(K_{k_1}-c_1K_{k_2})\end{bmatrix}
\begin{bmatrix}
M\\
J
\end{bmatrix}=
2\begin{bmatrix}
\beta_2{\rm i}k_1^{-1}\partial_\nu H^{\rm in}\\
-\beta_1H^{\rm in}
\end{bmatrix}.
\label{eq:kombi2}$$ If the conditions in Theorem \[thm:exunA\] hold, then has a unique solution $\{M,J\}$. Via it represents the unique solution to the 2D TM problem.
Test domains and discretization {#sec:geomdisc}
===============================
This section reviews domains and discretization schemes that are used for numerical tests in the next section.
![Non-smooth test domains: (a) boundaries $\Gamma$ of 2D domains with corner opening angles $\alpha=\pi/2$ (solid blue) and $\alpha=31\pi/18$ (dashed orange); (b) cylindrical coordinates $(\rho,\theta,z)$ of a point ${{\boldsymbolr}}$ on the surface of an axisymmetric object with generating curve $\gamma$; (c) cross section of the object generated by $\gamma$ with conical point opening angle $\alpha=31\pi/18$.[]{data-label="fig:amoeba12"}](hkfig3a.pdf "fig:"){height="35mm"} ![Non-smooth test domains: (a) boundaries $\Gamma$ of 2D domains with corner opening angles $\alpha=\pi/2$ (solid blue) and $\alpha=31\pi/18$ (dashed orange); (b) cylindrical coordinates $(\rho,\theta,z)$ of a point ${{\boldsymbolr}}$ on the surface of an axisymmetric object with generating curve $\gamma$; (c) cross section of the object generated by $\gamma$ with conical point opening angle $\alpha=31\pi/18$.[]{data-label="fig:amoeba12"}](hkfig3b.pdf "fig:"){height="35mm"} ![Non-smooth test domains: (a) boundaries $\Gamma$ of 2D domains with corner opening angles $\alpha=\pi/2$ (solid blue) and $\alpha=31\pi/18$ (dashed orange); (b) cylindrical coordinates $(\rho,\theta,z)$ of a point ${{\boldsymbolr}}$ on the surface of an axisymmetric object with generating curve $\gamma$; (c) cross section of the object generated by $\gamma$ with conical point opening angle $\alpha=31\pi/18$.[]{data-label="fig:amoeba12"}](hkfig3c.pdf "fig:"){height="35mm"}
The 2D one-corner object and the 3D “tomato” {#sec:droptoma}
--------------------------------------------
Numerical tests in two dimensions involve a one-corner object whose boundary $\Gamma$ is parameterized as $$r(s)=\sin(\pi s)\left(\cos((s-0.5)\alpha),\sin((s-0.5)\alpha)\right)\,,
\quad s\in[0,1]\,,
\label{eq:gamma2D}$$ and where $\alpha$ is a corner opening angle. See Figure \[fig:amoeba12\](a) for illustrations.
Numerical tests in three dimensions involve an object whose surface $\Gamma$ is created by revolving the generating curve $\gamma$, parameterized as $${{\boldsymbolr}}(s)=
\sin(\pi s)\left(\sin((0.5-s)\alpha),0,\cos((0.5-s)\alpha)\right)\,,
\quad s\in[0,0.5]\,,
\label{eq:gamma3D}$$ around the $z$-axis. For $\alpha>\pi$ this object resembles a “tomato”. See Figure \[fig:amoeba0\] and Figure \[fig:amoeba12\](b,c) for illustrations with $\alpha=31\pi/18$.
The reason for testing integral equations in axisymmetric domains, rather than in general domains, is the availability of efficient high-order solvers. Use of axisymmetric domains and solvers as a robust test-bed for new integral equation reformulations of scattering problems is contemporary common practice [@EpsGreNei19; @LaiOneil19].
RCIP-accelerated Nyström discretization schemes {#sec:disc}
-----------------------------------------------
Nyström discretization, relying on composite Gauss–Legendre quadrature, is used for all our systems of integral equations. Large discretized linear systems are solved iteratively using GMRES. In the presence of singular boundary points which call for intense mesh refinement, the Nyström scheme is accelerated by recursively compressed inverse preconditioning (RCIP) [@Hels18]. The RCIP acts as a fully automated, geometry-independent, and fast direct local solver and boosts the performance of the original Nyström scheme to the point where problems on non-smooth $\Gamma$ are solved with the same ease as on smooth $\Gamma$. Accurate evaluations of layer potentials close to their sources on $\Gamma$ are accomplished using variants of the techniques first presented in [@Hels09].
The schemes used in the numerical examples are not entirely new. For 2D problems we use the scheme in [@HelsKarl19 Section 11.3], relying on 16-point composite quadrature. For 3D problems we use a modified unification of the schemes in [@HelsKarl17] and [@HelsPerf18], relying on 32-point composite quadrature. A key feature in the scheme of [@HelsKarl17] is an FFT-accelerated separation of variables, pioneered by [@YouHaoMar12] and used also in [@EpsGreNei19; @LaiOneil19].
An important technique in [@HelsKarl17] is the split of the numerator in $\Phi_k({{\boldsymbolr}},{{\boldsymbolr}}')$ of (\[eq:fund\]) into parts that are even and odd in $\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert$. Let $G(k,{{{\boldsymbolr}}},{{{\boldsymbolr}}}')$ be one of the $2\pi$-periodic kernels of Section \[sec:laypot\]. Azimuthal Fourier coefficients $$G_n=
\frac{1}{\sqrt{2\pi}}\int_{-\pi}^{\pi}e^{-{\rm i}n(\theta-\theta')}
G(k,{{{\boldsymbolr}}},{{{\boldsymbolr}}}')\,{\rm d}(\theta-\theta')\,,
\quad n=0,\pm 1,\pm 2,\ldots\,,
\label{eq:GF}$$ are, for ${{\boldsymbolr}}$ and ${{\boldsymbolr}}'$ close to each other, computed in different ways depending on the parity of these parts. When $\Im{\rm
m}\{k\}$ is small, the split $$e^{{\rm i}k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert}=
\cos(k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert)
+{\rm i}\sin(k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert)
\label{eq:split1}$$ is efficient for $\Phi_k({{\boldsymbolr}},{{\boldsymbolr}}')$. When $\Im{\rm m}\{k\}$ is large, the terms on the right hand side of (\[eq:split1\]) can be much larger in modulus than the function on the left hand side. Then numerical cancellation takes place. To fix this problem for large $\Im{\rm m}\{k\}$, not encountered in [@HelsKarl17], we introduce a bump-like function $$\chi(k,\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert)=
e^{-\left(\Im{\rm m}\{k\}{\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert/4.6}\right)^8}\,,$$ modify the split (\[eq:split1\]) to $$e^{{\rm i}k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert}=
(1-\chi)e^{{\rm i}k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert}
+\chi\cos(k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert)
+{\rm i}\chi\sin(k\lvert{{\boldsymbolr}}-{{\boldsymbolr}}'\rvert)\,,
\label{eq:split2}$$ and compute $G_n$ of (\[eq:GF\]) with techniques (direct transform or convolution) appropriate for parts of $G(k,{{{\boldsymbolr}}},{{{\boldsymbolr}}}')$ associated with each of the terms on the right hand side of (\[eq:split2\]).
Numerical examples {#sec:numex}
==================
The systems (\[eq:EHsys\]), (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]) and the representations (\[eq:repEH\]), (\[eq:H22D\]), and (\[eq:Hnull\]) are now put to the test. In all examples we take $k_1$ real and positive, $\varepsilon_1=1$, and $\varepsilon_2=-1.1838$. This parameter combination satisfies the plasmonic condition and has been used in previous work on 2D surface plasmon waves [@Annsop16; @HelsKarl18; @HelsKarl19]. In situations involving non-smooth surfaces, it may happen that solutions for $\varepsilon_2=-1.1838$ do not exist. We then compute limit solutions as $\varepsilon_2$ approaches $-1.1838$ from above in the complex plane. Such limit solutions, discussed in the context of Laplace transmission problems in [@HelsPerf18 Section 2.2], have boundary traces that may best be characterized as lying in fractional-order Sobolev spaces [@HelsRose20] and are given a downarrow superscript. For example, the limit of the field ${{\boldsymbolH}}$ is denoted ${{\boldsymbolH}}^\downarrow$. The uniqueness parameters $\{\lambda,\gamma_{\rm E},\gamma_{\rm M},c\}$, needed in (\[eq:EHsys\]), (\[eq:EHsysTM\]), and (\[eq:threedens\]), are chosen according to (\[eq:parameter1\]). The uniqueness parameters needed in (\[eq:kombi2\]) are chosen as $\{c_1,c_2\}=\{-{\rm
i},\kappa\}$.
Our codes are implemented in [Matlab]{}, release 2018b, and executed on a workstation equipped with an Intel Core i7-3930K CPU and 64 GB of RAM. When assessing the accuracy of computed field quantities we most often adopt a procedure where to each numerical solution we also compute an overresolved reference solution, using roughly 50% more points in the discretization of the system under study. The absolute difference between these two solutions is denoted the [*estimated absolute error*]{}. Throughout the examples, field quantities are computed at $10^6$ field points on a rectangular Cartesian grid in the computational domains shown in the figures.
Unique solvability on the unit circle {#sec:C2}
-------------------------------------
We compute condition numbers of the discretized system matrices in (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]). The boundary $\Gamma$ is the unit circle and $k_1$ is swept through the interval $[0,10]$. Recall that the systems (\[eq:EHsysTM\]) and (\[eq:kombi2\]) are guaranteed to be free from wavenumbers for which the solution is not unique (false eigenwavenumbers) while the system (\[eq:threedens\]) is not.
Condition number analysis of 2D limits of 3D systems on the unit circle is a revealing test for detecting if a given system of integral equations has false eigenwavenumbers when the plasmonic condition holds. For example, in [@HelsKarl19 Figure 9] it is shown that the original Müller system and the “${{\boldsymbolE}}$-system” of [@VicGreFer18] exhibit several false eigenwavenumbers in such a test.
![Condition numbers of system matrices on the unit circle, $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1\in[0,10]$: (a) the systems (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]); (b) the Müller system and a repeat of the bottom curve in (a).[]{data-label="fig:cond2"}](hkfig4a.pdf "fig:"){height="47mm"} ![Condition numbers of system matrices on the unit circle, $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1\in[0,10]$: (a) the systems (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]); (b) the Müller system and a repeat of the bottom curve in (a).[]{data-label="fig:cond2"}](hkfig4b.pdf "fig:"){height="47mm"}
Figure \[fig:cond2\](a) shows results obtained with (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]) using 768 discretizations points on $\Gamma$ and approximately $20,\!700$ values of $k_1\in[0,10]$. The regularly recurring high peaks correspond to true eigenwavenumbers just below the positive $k_1$-axis (weakly damped dynamic surface plasmons). One can see that neither the four-density system (\[eq:EHsysTM\]) nor the two-density system (\[eq:kombi2\]) exhibits any false eigenwavenumbers, as expected, and that (\[eq:kombi2\]) is the best conditioned system. Furthermore, which is more remarkable, the three-density system (\[eq:threedens\]) also appears to be free from false eigenwavenumbers. For comparison, Figure \[fig:cond2\](b) shows condition numbers of the original Müller system, corresponding to $\{c_1,c_2\}=\{1,\kappa\}$ in . Here one can see 13 false eigenwavenumbers. Some distance away from these wavenumbers the results from the Müller system and (\[eq:kombi2\]) with $\{c_1,c_2\}=\{-{\rm i},\kappa\}$ overlap.
Field accuracy for the 2D one-corner object {#sec:F2}
-------------------------------------------
An incident plane wave with ${{\boldsymbolH}}^{\rm in}(r)=\hat{{{\boldsymbolz}}}e^{{\rm i}k_1d\cdot r}$, $k_1=18$, and direction of propagation $d=\left(\cos(\pi/4),\sin(\pi/4)\right)$ is scattered against the 2D one-corner object of Section \[sec:droptoma\]. The corner opening angle is $\alpha=\pi/2$. A number of $800$ discretization points is placed on $\Gamma$ and the performance of the three systems (\[eq:EHsysTM\]), (\[eq:threedens\]), (\[eq:kombi2\]) are compared.
![The field $H^\downarrow(r,0)$ on the 2D one-corner object with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=18$: (a) the field $H^\downarrow(r,0)$ itself; (b,c,d) $\log_{10}$ of estimated absolute field error using the systems (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]), respectively.[]{data-label="fig:field2"}](hkfig5a.pdf "fig:"){height="46mm"} ![The field $H^\downarrow(r,0)$ on the 2D one-corner object with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=18$: (a) the field $H^\downarrow(r,0)$ itself; (b,c,d) $\log_{10}$ of estimated absolute field error using the systems (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]), respectively.[]{data-label="fig:field2"}](hkfig5b.pdf "fig:"){height="46mm"} ![The field $H^\downarrow(r,0)$ on the 2D one-corner object with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=18$: (a) the field $H^\downarrow(r,0)$ itself; (b,c,d) $\log_{10}$ of estimated absolute field error using the systems (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]), respectively.[]{data-label="fig:field2"}](hkfig5c.pdf "fig:"){height="46mm"} ![The field $H^\downarrow(r,0)$ on the 2D one-corner object with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=18$: (a) the field $H^\downarrow(r,0)$ itself; (b,c,d) $\log_{10}$ of estimated absolute field error using the systems (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]), respectively.[]{data-label="fig:field2"}](hkfig5d.pdf "fig:"){height="46mm"}
Figure \[fig:field2\](a) shows the total magnetic field $\Re{\rm
e}\{H^\downarrow(r)\}$, see (\[eq:timedep\]), and Figures \[fig:field2\](b,c,d) show $\log_{10}$ of the estimated absolute error obtained with (\[eq:EHsysTM\]), (\[eq:threedens\]), and (\[eq:kombi2\]), respectively. The number of GMRES iterations required to solve the discretized linear systems is 266 for (\[eq:EHsysTM\]), 154 for (\[eq:threedens\]), and 143 for (\[eq:kombi2\]). The absolute errors for the systems (\[eq:EHsysTM\]) and (\[eq:threedens\]) are estimated using the solution from (\[eq:kombi2\]) as reference.
It is interesting to observe, in Figure \[fig:field2\], that the field accuracy is high for all three systems. The number of digits lost is in agreement with what could be expected for computations on the unit circle, considering the condition numbers shown in Figure \[fig:cond2\] and assuming that $k_1$ is not close to a true eigenwavenumber. Note also that (\[eq:kombi2\]) is a system of Fredholm second-kind integral equations with operator differences that are compact on smooth $\Gamma$ – a property often sought for in integral equation modeling of PDEs. The system (\[eq:threedens\]), on the other hand, contains a Cauchy-type singular difference of operators. Still, the performance of the two systems is very similar.
Unique solvability on the unit sphere {#sec:C3}
-------------------------------------
We repeat the experiment of Section \[sec:C2\], but now on the unit sphere using the system (\[eq:EHsys\]). Inspired by the good performance of the system (\[eq:threedens\]), reported above and where $\sigma_{\rm E}$ is omitted, we omit both $\sigma_{\rm E}$ and $\sigma_{\rm M}$ from (\[eq:EHsys\]) to get a six-scalar-density system. Again, there is noo proof that this system has a unique solution, but every solution to the time harmonic Maxwell’s equations corresponds to a solution to this system.
![Condition numbers of system matrices on the unit sphere, $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1\in[0,10]$: (a) azimuthal modes $n=0,5,10$ of the system (\[eq:EHsys\]) with $\sigma_{\rm E}$ and $\sigma_{\rm M}$ omitted; (b) azimuthal mode $n=0$ of the pseudo-Müller system and a repeat of the top curve in (a).[]{data-label="fig:cond3"}](hkfig6a.pdf "fig:"){height="47mm"} ![Condition numbers of system matrices on the unit sphere, $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1\in[0,10]$: (a) azimuthal modes $n=0,5,10$ of the system (\[eq:EHsys\]) with $\sigma_{\rm E}$ and $\sigma_{\rm M}$ omitted; (b) azimuthal mode $n=0$ of the pseudo-Müller system and a repeat of the top curve in (a).[]{data-label="fig:cond3"}](hkfig6b.pdf "fig:"){height="47mm"}
The Fourier–Nyström scheme of [@HelsKarl17], see Section \[sec:disc\], decomposes the reduced system (\[eq:EHsys\]) into a sequence of smaller, modal, systems on the generating curve $\gamma$. Figure \[fig:cond3\](a) shows result for the azimuthal modes $n=0,5,10$, with 768 discretization points on $\gamma$, and with approximately $3,\!500$ values of $k_1\in[0,10]$. No false eigenwavenumbers can be seen. For comparison, Figure \[fig:cond3\](b) shows results for a six-scalar-density variant of the Müller system. The original four-scalar-density Müller system [@Muller69 p. 319] uses the surface current densities ${{\boldsymbolM}}_{\rm s}$ and ${{\boldsymbolJ}}_{\rm s}$ and contains compact differences of hypersingular operators. These operator differences are hard to implement in three dimensions, even though it definitely is possible on axisymmetric surfaces [@LaiOneil19]. Our variant of the Müller system is derived from the original Müller system via integration by parts and relating the surface divergence of ${{\boldsymbolM}}_{\rm s}$ and ${{\boldsymbolJ}}_{\rm s}$ to $\varrho_{\rm M}$ and $\varrho_{\rm E}$, see [@HelsKarl17 Eqs. (36) and (35)]. This corresponds to omitting both $\sigma_{\rm E}$ and $\sigma_{\rm M}$ from (\[eq:EHsys\]) and setting $c_4=c_6=1$, and $c_5=c_7=\kappa$. Figure \[fig:cond3\](b) shows that this pseudo-Müller system exhibits at least $32$ false eigenwavenumbers for $k_1\in[0,10]$.
Field accuracy for the 3D “tomato” {#sec:F3}
----------------------------------
An incident linearly polarized plane wave with ${{\boldsymbolE}}^{\rm
in}({{\boldsymbolr}})=\hat{{{\boldsymbolx}}}e^{{\rm i}k_1z}$ and $k_1=5$ is scattered against the 3D “tomato” of Section \[sec:droptoma\]. The conical point opening angle is $\alpha=31\pi/18$. The same six-scalar-density version of the system (\[eq:EHsys\]) is used as in Section \[sec:C3\]. Only two azimuthal modes, $n=-1$ and $n=1$, are present in this problem and the Fourier coefficients of the surface densities of these modes are either identical or have opposite signs. Therefore only one modal system needs to be solved numerically.
![Field images on a cross section of the 3D “tomato” subjected to an incident plane wave ${{\boldsymbolE}}^{\rm in}({{\boldsymbolr}})=\hat{{{\boldsymbolx}}} e^{{\rm i}k_1z}$ and with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=5$: (a) the field $E_\rho({{\boldsymbolr}},0)$ with colorbar range set to $[-4.55,4.55]$ ; (b) $\log_{10}$ of estimated absolute field error in $E_\rho({{\boldsymbolr}},0)$; (c) the field $H_\theta({{\boldsymbolr}},0)$; (d) $\log_{10}$ of estimated absolute field error in $H_\theta({{\boldsymbolr}},0)$.[]{data-label="fig:field3"}](hkfig7a.pdf "fig:"){height="51mm"} ![Field images on a cross section of the 3D “tomato” subjected to an incident plane wave ${{\boldsymbolE}}^{\rm in}({{\boldsymbolr}})=\hat{{{\boldsymbolx}}} e^{{\rm i}k_1z}$ and with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=5$: (a) the field $E_\rho({{\boldsymbolr}},0)$ with colorbar range set to $[-4.55,4.55]$ ; (b) $\log_{10}$ of estimated absolute field error in $E_\rho({{\boldsymbolr}},0)$; (c) the field $H_\theta({{\boldsymbolr}},0)$; (d) $\log_{10}$ of estimated absolute field error in $H_\theta({{\boldsymbolr}},0)$.[]{data-label="fig:field3"}](hkfig7b.pdf "fig:"){height="51mm"} ![Field images on a cross section of the 3D “tomato” subjected to an incident plane wave ${{\boldsymbolE}}^{\rm in}({{\boldsymbolr}})=\hat{{{\boldsymbolx}}} e^{{\rm i}k_1z}$ and with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=5$: (a) the field $E_\rho({{\boldsymbolr}},0)$ with colorbar range set to $[-4.55,4.55]$ ; (b) $\log_{10}$ of estimated absolute field error in $E_\rho({{\boldsymbolr}},0)$; (c) the field $H_\theta({{\boldsymbolr}},0)$; (d) $\log_{10}$ of estimated absolute field error in $H_\theta({{\boldsymbolr}},0)$.[]{data-label="fig:field3"}](hkfig7c.pdf "fig:"){height="51mm"} ![Field images on a cross section of the 3D “tomato” subjected to an incident plane wave ${{\boldsymbolE}}^{\rm in}({{\boldsymbolr}})=\hat{{{\boldsymbolx}}} e^{{\rm i}k_1z}$ and with $\varepsilon_1=1$, $\varepsilon_2=-1.1838$, and $k_1=5$: (a) the field $E_\rho({{\boldsymbolr}},0)$ with colorbar range set to $[-4.55,4.55]$ ; (b) $\log_{10}$ of estimated absolute field error in $E_\rho({{\boldsymbolr}},0)$; (c) the field $H_\theta({{\boldsymbolr}},0)$; (d) $\log_{10}$ of estimated absolute field error in $H_\theta({{\boldsymbolr}},0)$.[]{data-label="fig:field3"}](hkfig7d.pdf "fig:"){height="51mm"}
Figure \[fig:field3\] shows the electric field in the $\rho$-direction, $E_\rho({{\boldsymbolr}},0)$, and the magnetic field in the $\theta$-direction, $H_\theta({{\boldsymbolr}},0)$, on the cross section in Figure \[fig:amoeba12\](c). The results are obtained with $576$ discretization points on the generating curve $\gamma$ and with 242 GMRES iterations. Since the field $E_\rho({{\boldsymbolr}},0)$ is singular at the origin, the colorbar range in Figure \[fig:field3\](a) is restricted to the most extreme values of $E_\rho({{\boldsymbolr}},0)$ away from the origin. The precision shown in Figure \[fig:field3\](b,d) is consistent with the condition numbers of Figure \[fig:cond3\](a) in the sense discussed in Section \[sec:F2\]. We conclude by noting that Figure \[fig:field3\] clearly shows an accurately computed surface plasmon wave on a non-smooth 3D object in a setup with negative permittivity ratio. To simulate such surface waves is the ultimate goal of this work.
Conclusions
===========
A new system of Fredholm second-kind integral equations is presented for an electromagnetic transmission problem involving a single scattering object. Our work can be seen as an extension of the work by Kleinman and Martin [@KleiMart88] on direct methods for scalar transmission problems. Thanks to the introduction of certain uniqueness parameters, our new system gives unique solutions for a wider range of wavenumber combinations than do other systems of integral equations for Maxwell’s equations, for example the original Müller system. In particular, unique solutions are guaranteed for smooth scatterers under the plasmonic condition , which is of great interest in physical and engineering applications.
The favorable properties of our new system extend beyond what can be proven rigorously. In a numerical example, a reduced version of the system in combination with a high-order Fourier–Nyström discretization scheme is shown to produce accurate field images of a surface plasmon wave on a non-smooth axisymmetric scatterer.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank Andreas Rosén (formerly Andreas Axelsson) for many useful conversations. This work was supported by the Swedish Research Council under contract 621-2014-5159.
Appendix {#appendix .unnumbered}
========
A. Boundary limits of ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ {#a.-boundary-limits-of-boldsymbole-and-boldsymbolh .unnumbered}
---------------------------------------------------------------
The relations in Section \[sec:limits\] give the following limits at $\Gamma$ for the integral representations of ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ in –: $$\begin{aligned}
[\nabla\cdot{{\boldsymbolE}}_1]^\pm&=
\mp\frac{{\rm i}k_1}{2}\sigma_{\rm E}
-\frac{{\rm i}k_1}{2}\tilde{\cal S}_{k_1}\varrho_{\rm E}
+\frac{1}{2}\nabla\cdot\tilde{\cal S}_{k_1}
({{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})\,,\\
{{\boldsymbol\nu}}\cdot{{\boldsymbolE}}_1^\pm&=\pm\frac{1}{2}\varrho_{\rm
E}-\frac{1}{2}{{\boldsymbol\nu}}\cdot\boldsymbol{\cal
N}_{k_1}\varrho_{\rm E}-\frac{1}{2}{{\boldsymbol\nu}}\cdot {\cal
R}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm M}+{{\boldsymbolM}}_{\rm s})
\nonumber\\
&\qquad\qquad
+\frac{1}{2}{{\boldsymbol\nu}}\cdot \tilde{\cal S}_{k_1}
({{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})
+{{\boldsymbol\nu}}\cdot{{\boldsymbolE}}^{\rm in},\end{aligned}$$ $$\begin{aligned}
{{\boldsymbol\nu}}\times{{\boldsymbolE}}_1^\pm&=\mp\frac{1}{2}{{\boldsymbolM}}_{\rm
s}-\frac{1}{2}{{\boldsymbol\nu}}\times\boldsymbol{\cal
N}_{k_1}\varrho_{\rm E}-\frac{1}{2}{{\boldsymbol\nu}}\times{\cal
R}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm M}+{{\boldsymbolM}}_{\rm s})
\nonumber\\
&\qquad\qquad
+\frac{1}{2}{{\boldsymbol\nu}}\times\tilde{\cal S}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})+{{\boldsymbol\nu}}\times{{\boldsymbolE}}^{\rm in},\\
{{\boldsymbol\nu}}\times{{\boldsymbolH}}_1^\pm&=
\pm\frac{1}{2}{{\boldsymbolJ}}_{\rm s}
+\frac{1}{2}{{\boldsymbol\nu}}\times\tilde{\cal S}_{k_1}
({{\boldsymbol\nu}}'\sigma_{\rm M}+{{\boldsymbolM}}_{\rm s})
+\frac{1}{2}{{\boldsymbol\nu}}\times{\cal R}_{k_1}
({{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})\nonumber\\
&\qquad\qquad
-\frac{1}{2}{{\boldsymbol\nu}}\times\boldsymbol{\cal N}_{k_1}
\varrho_{\rm M}+{{\boldsymbol\nu}}\times {{\boldsymbolH}}^{\rm in},\\
{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_1^\pm&=\pm\frac{1}{2}\varrho_{\rm
M}+\frac{1}{2}{{\boldsymbol\nu}}\cdot\tilde{\cal
S}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm M}+{{\boldsymbolM}}_{\rm s})
+\frac{1}{2}{{\boldsymbol\nu}}\cdot{\cal R}_{k_1}
({{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})\nonumber\\
&\qquad\qquad
-\frac{1}{2}{{\boldsymbol\nu}}\cdot\boldsymbol{\cal N}_{k_1}
\varrho_{\rm M}+{{\boldsymbol\nu}}\cdot {{\boldsymbolH}}^{\rm in},\\
[\nabla\cdot{{\boldsymbolH}}_1]^\pm&=
\mp\frac{{\rm i}k_1}{2}\sigma_{\rm M}
+\frac{1}{2}\nabla\cdot\tilde{\cal S}_{k_1}
({{\boldsymbol\nu}}'\sigma_{\rm M}+{{\boldsymbolM}}_{\rm s})
-\frac{{\rm i}k_1}{2}\tilde{\cal S}_{k_1}\varrho_M\,,\\
[\nabla\cdot{{\boldsymbolE}}_2]^\pm&=
\pm\frac{{\rm i}k_1}{2\kappa}\sigma_{\rm E}
+\frac{{\rm i}k_1}{2}\tilde{\cal S}_{k_2}\varrho_{\rm E}
-\frac{1}{2}\nabla\cdot\tilde{\cal S}_{k_2}
(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})\,,\\
{{\boldsymbol\nu}}\cdot{{\boldsymbolE}}_2^\pm&=
\mp\frac{1}{2\kappa}\varrho_{\rm E}
+\frac{1}{2\kappa}{{\boldsymbol\nu}}\cdot\boldsymbol{\cal N}_{k_2}
\varrho_{\rm E}
+\frac{1}{2\kappa}{{\boldsymbol\nu}}\cdot{\cal R}_{k_2}
({{\boldsymbol\nu}}'\sigma_{\rm M}+\kappa{{\boldsymbolM}}_{\rm s})
\nonumber\\
&\qquad\qquad
-\frac{1}{2}{{\boldsymbol\nu}}\cdot\tilde{\cal S}_{k_2}
(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})\,,\\
{{\boldsymbol\nu}}\times{{\boldsymbolE}}_2^\pm&=
\pm\frac{1}{2}{{\boldsymbolM}}_{\rm s}
+\frac{1}{2\kappa}{{\boldsymbol\nu}}\times\boldsymbol{\cal N}_{k_2}
\varrho_{\rm E}+\frac{1}{2\kappa}{{\boldsymbol\nu}}\times{\cal R}_{k_2}
({{\boldsymbol\nu}}'\sigma_{\rm M}+\kappa{{\boldsymbolM}}_{\rm s})
\nonumber\\
&\qquad\qquad
-\frac{1}{2}{{\boldsymbol\nu}}\times\tilde{\cal S}_{k_2}
(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})\,,\\
{{\boldsymbol\nu}}\times{{\boldsymbolH}}_2^\pm&=
\mp\frac{1}{2}{{\boldsymbolJ}}_{\rm s}
-\frac{1}{2}{{\boldsymbol\nu}}\times\tilde{\cal S}_{k_2}
({{\boldsymbol\nu}}'\sigma_{\rm M}+\kappa{{\boldsymbolM}}_{\rm s})
\nonumber\\
&\qquad
-\frac{1}{2}{{\boldsymbol\nu}}\times{\cal R}_{k_2}
(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})
+\frac{1}{2}{{\boldsymbol\nu}}\times\boldsymbol{\cal N}_{k_2}
\varrho_{\rm M}\,,\\
{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_2^\pm&=
\mp\frac{1}{2}\varrho_{\rm M}
-\frac{1}{2}{{\boldsymbol\nu}}\cdot\tilde{\cal S}_{k_2}({{\boldsymbol\nu}}'\sigma_{\rm M}
+\kappa{{\boldsymbolM}}_{\rm s})
\nonumber\\
&\qquad
-\frac{1}{2}{{\boldsymbol\nu}}\cdot{\cal R}_{k_2}
(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s})
+\frac{1}{2}{{\boldsymbol\nu}}\cdot\boldsymbol{\cal N}_{k_2}
\varrho_{\rm M}\,,\\
[\nabla\cdot{{\boldsymbolH}}_2]^\pm&=
\pm\frac{{\rm i}k_1}{2}\sigma_{\rm M}
-\frac{1}{2}\nabla\cdot\tilde{\cal S}_{k_2}({{\boldsymbol\nu}}'\sigma_{\rm M}
+\kappa{{\boldsymbolM}}_{\rm s})
+\frac{{\rm i}k_1}{2}\kappa\tilde{\cal S}_{k_2}\varrho_M\,.\end{aligned}$$
B. Divergence conditions {#b.-divergence-conditions .unnumbered}
------------------------
The derivations of the conditions for , , and to hold are all very similar. For this reason we only present a detailed derivation of the condition for to hold.
The fields ${{\boldsymbolE}}_W$ and ${{\boldsymbolH}}_{W}$ are defined through , –, and the solution to . Appendix A and give the relations on $\Gamma$ $$\begin{aligned}
\lambda\kappa{{\boldsymbol\nu}}\times{{\boldsymbolE}}_W^+&=
{{\boldsymbol\nu}}\times{{\boldsymbolE}}_W^-\,,\label{eq:RVA1}\\
\lambda\kappa{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_W^+&=
{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_W^-\,,\label{eq:RVA2}\\
\gamma_{\rm M}[\nabla\cdot{{\boldsymbolH}}_W]^+&=
[\nabla\cdot{{\boldsymbolH}}_W]^-\,.\label{eq:RVA3}\end{aligned}$$ By combining the surface divergence of with we get $$\lambda\kappa({\rm i}k_1{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_2^+-{{\boldsymbol\nu}}\cdot[\nabla\times{{\boldsymbolE}}_2)]^+)=
{\rm i}k_1{{\boldsymbol\nu}}\cdot{{\boldsymbolH}}_1^-
-{{\boldsymbol\nu}}\cdot[\nabla\times {{\boldsymbolE}}_1)]^-\,,$$ where we have used ${{\boldsymbol\nu}}\cdot(\nabla\times{{\boldsymbol\nu}}\times({{\boldsymbol\nu}}\times
{{\boldsymbolE}}_i))=-{{\boldsymbol\nu}}\cdot(\nabla\times{{\boldsymbolE}}_i)$, $i=1,2$. By – and limits in Appendix A this leads to $$\begin{gathered}
\lambda\kappa\left(\kappa^{-1}{{\boldsymbol\nu}}\cdot
[\nabla(\nabla\cdot{\cal S}_{k_2})]^+({{\boldsymbol\nu}}'\sigma_{\rm M}
+\kappa{{\boldsymbolM}}_{\rm s})
-{\rm i}k_1{{\boldsymbol\nu}}\cdot\boldsymbol{\cal N}_{k_2}\varrho_{\rm M}
+{\rm i}k_1\varrho_{\rm M}\right)\\
=-{{\boldsymbol\nu}}\cdot[\nabla(\nabla\cdot {\cal S}_{k_1})]^-
({{\boldsymbol\nu}}'\sigma_{\rm M}+{{\boldsymbolM}}_{\rm s})
+{\rm i}k_1{{\boldsymbol\nu}}\cdot\boldsymbol{\cal N}_{k_1}\varrho_{\rm M}
+{\rm i}k_1\varrho_{\rm M}\,.
\label{eq:surfdiv}\end{gathered}$$ A comparison of with the limits ${{\boldsymbol\nu}}\cdot[\nabla(\nabla\cdot{{\boldsymbolH}}_1)]^-$ and ${{\boldsymbol\nu}}\cdot[\nabla(\nabla\cdot{{\boldsymbolH}}_2)]^+$ gives $$\lambda{{\boldsymbol\nu}}\cdot[\nabla(\nabla\cdot{{\boldsymbolH}}_2)]^+=
{{\boldsymbol\nu}}\cdot[\nabla(\nabla\cdot{{\boldsymbolH}}_1)]^-\,.
\label{eq:divcond2}$$ Let $\psi_W=\nabla\cdot {{\boldsymbolH}}_W$, with ${{\boldsymbolH}}_W$ from . The fundamental solution and the boundary conditions and make $\psi_W$ satisfy $$\left\{
\begin{array}{ll}
\Delta\psi_W({{\boldsymbolr}})+k_2^2\psi_W({{\boldsymbolr}})=0\,,& {{\boldsymbolr}}\in\Omega_1\,,\\
\Delta\psi_W({{\boldsymbolr}})+k_1^2\psi_W({{\boldsymbolr}})=0\,,& {{\boldsymbolr}}\in\Omega_2\,,\\
\gamma_{\rm M} \psi_W^+({{\boldsymbolr}})=\psi_W^-({{\boldsymbolr}})\,,& {{\boldsymbolr}}\in\Gamma\,,\\
\lambda{{\boldsymbol\nu}}\cdot[\nabla\psi_W]^+({{\boldsymbolr}})=
{{\boldsymbol\nu}}\cdot[\nabla\psi_W]^-({{\boldsymbolr}})\,,& {{\boldsymbolr}}\in\Gamma\,,\\
\left(\partial_{\hat{{{\boldsymbolr}}}}-{\rm i}k_2\right)\psi_W({{\boldsymbolr}})=
o\left(\vert{{\boldsymbolr}}\vert^{-1}\right)\,,& \vert{{\boldsymbolr}}\vert\to\infty\,.
\end{array}\right.
\label{eq:psiWeq}$$ By rescaling $\psi_W$ in $\Omega_1$, problem becomes identical to problem ${\sf B}_0$ with $\alpha=\lambda\bar{\gamma}_{\rm
M}/\vert{\gamma}_{\rm M}\vert^2$. Thus if $\{k_1,k_2,\alpha=\lambda\bar{\gamma}_{\rm M}\}$ is such that the conditions of Proposition \[prop:uniqueB0\] hold, then only has the trivial solution $\nabla\cdot{{\boldsymbolH}}_W=0$ for ${{\boldsymbolr}}\in\Omega_1\cup\Omega_2$.
The condition for $\nabla\cdot{{\boldsymbolE}}_W=0$ is that the set $\{k_1,k_2,\alpha=\bar\gamma_{\rm E}\bar\kappa\}$ is such that the conditions of Proposition \[prop:uniqueB0\] hold. The condition for to hold is that $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a).
C. Fulfillment of Maxwell’s equations {#c.-fulfillment-of-maxwells-equations .unnumbered}
-------------------------------------
We show that ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ of satisfy and that ${{\boldsymbolE}}_W$ and ${{\boldsymbolH}}_W$ of satisfy if $\nabla\cdot{{\boldsymbolE}}_i({{\boldsymbolr}})=\nabla\cdot{{\boldsymbolH}}_i({{\boldsymbolr}})=0$, $i=1,2$, and ${{\boldsymbolr}}\in\Omega_1\cup\Omega_2$.
The rotation of and can be written $$\begin{aligned}
\nabla\times{{\boldsymbolH}}_1({{\boldsymbolr}})&=
\frac{{\rm i}k_1}{2}{\cal R}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm M}
+{{\boldsymbolM}}_{\rm s})({{\boldsymbolr}})
-\frac{{\rm i}k_1}{2}\tilde{\cal S}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})
\nonumber\\
+\frac{1}{2}\nabla&(\nabla\cdot{\cal S}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s}))({{\boldsymbolr}})+\nabla\times {{\boldsymbolH}}^{\rm in}({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in \Omega_1\cup \Omega_2\,,
\label{eq:H1rep11NN}\\
\nabla\times{{\boldsymbolH}}_2({{\boldsymbolr}})&=
-\frac{{\rm i}k_1}{2}{\cal R}_{k_2}({{\boldsymbol\nu}}'\sigma_{\rm M}
+\kappa{{\boldsymbolM}}_{\rm s})({{\boldsymbolr}})
+\frac{{\rm i}k_1}{2}\tilde{\cal S}_{k_2}
({{\boldsymbol\nu}}'\sigma_{\rm E}+\kappa{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})
\nonumber\\
&-\frac{1}{2}\nabla(\nabla\cdot{\cal S}_{k_2}
(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}+{{\boldsymbolJ}}_{\rm s}))({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,.
\label{eq:H2rep11NN}\end{aligned}$$ If $\nabla\cdot{{\boldsymbolE}}_i=0$, $i=1,2$, it follows from and that $$\begin{aligned}
\tilde{\cal S}_{k_1}\varrho_{\rm E}({{\boldsymbolr}})
-\nabla\cdot{\cal S}_{k_1}({{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})&=0\,,
\quad{{\boldsymbolr}}\in \Omega_1\cup\Omega_2\,,\label{eq:E1rep1mod}\\
\tilde{\cal S}_{k_2}\varrho_{\rm E}({{\boldsymbolr}})
-\nabla\cdot{\cal S}_{k_2}(\kappa^{-1}{{\boldsymbol\nu}}'\sigma_{\rm E}
+{{\boldsymbolJ}}_{\rm s})({{\boldsymbolr}})&=0\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,.\label{eq:E2rep1mod}\end{aligned}$$ The Ampère law $$\begin{split}
\nabla\times{{\boldsymbolH}}_1({{\boldsymbolr}})&=-{\rm i}k_1{{\boldsymbolE}}_1({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,\\
\nabla\times{{\boldsymbolH}}_2({{\boldsymbolr}})&=-{\rm i}k_1\kappa{{\boldsymbolE}}_2({{\boldsymbolr}})\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\end{split}
\label{eq:Max24}$$ now follows by combining and with , and by combining and with . The Faraday law $$\nabla\times{{\boldsymbolE}}_i({{\boldsymbolr}})={\rm i}k_1{{\boldsymbolH}}_i({{\boldsymbolr}})\,,
\quad i=1,2\,,
\quad{{\boldsymbolr}}\in\Omega_1\cup\Omega_2\,,
\label{eq:Max13}$$ follows in the same manner from $\nabla\cdot{{\boldsymbolH}}_i=0$, $i=1,2$, and by combining the rotation of with and the rotation of with . From and it follows that ${{\boldsymbolE}}$ and ${{\boldsymbolH}}$ of satisfy and that ${{\boldsymbolE}}_W$ and ${{\boldsymbolH}}_W$ of satisfy .
D. Uniqueness for problems [C]{}, ${\sf C}_0$, and ${\sf D}_0$ {#d.-uniqueness-for-problems-c-sf-c_0-and-sf-d_0 .unnumbered}
--------------------------------------------------------------
We sketch a proof that problem ${\sf C}_0$ has only the trivial solution and that problem [C]{} has at most one solution by relating these problems to problem ${\sf A}_0$ and [A]{}. We also justify that the criteria for problem ${\sf D}_0$ to only have the trivial solution are the same as the criteria in Proposition \[prop:uniqueB0\] that make problem ${\sf B}_0$ only have the trivial solution.
Let $S_R$ be a sphere of radius $R$ with outward unit normal ${{\boldsymboln}}$. Assume that $S_R$ is sufficiently large to contain $\Gamma$ and let $\Omega_{1,R}=\{{{\boldsymbolr}}\in\Omega_1:\vert{{\boldsymbolr}}\vert < R\}$. From Gauss’ theorem we obtain energy relations for problem ${\sf A}_0$ and problem ${\sf C}_0$ $$\int_{S_R}(U\nabla\bar U)\cdot{{\boldsymboln}}\,{\rm d}S=
\int_{\Omega_{1,R}}\left(\vert\nabla U\vert^2
-\bar k_1^2\vert U\vert^2\right){\rm d}v+
\int_{\Omega_{2}}\left(\bar\kappa^{-1}\vert\nabla U\vert^2-\bar k_1^2\vert
U\vert^2\right){\rm d}v\,,
\label{eq:unablau}$$ $$\begin{gathered}
-{\rm i}\bar k_1\int_{S_R}(\bar{{{\boldsymbolE}}}\times{{{\boldsymbolH}}})\cdot{{\boldsymboln}}
\,{\rm d}S=
\int_{\Omega_{1,R}}\left(\vert k_1\vert^2\vert{{\boldsymbolE}}\vert^2-\bar k_1^2\vert
{{\boldsymbolH}}\vert^2\right){\rm d}v\\
+\int_{\Omega_{2}}\left(\vert k_1\kappa\vert^2\bar{\kappa}^{-1}
\vert{{\boldsymbolE}}\vert^2-\bar k_1^2\vert{{\boldsymbolH}}\vert^2\right){\rm d}v\,.
\label{eq:barEH}\end{gathered}$$ The right hand sides of and are equivalent. By using techniques similar to those in [@KleiMart88 pp. 309–310] and [@KresRoac78 p. 1434] it follows that when $({{\rm{Arg}}}(k_1),{{\rm{Arg}}}(k_2))$ is in the set of points of Figure \[fig:hexagon\](a), then $U=0$ and ${{\boldsymbolE}}={{\boldsymbolH}}={{\boldsymbol0}}$ in $\Omega_1\cup \Omega_2$. Standard arguments give that problem [C]{} has at most one solution when problem ${\sf C}_0$ only has the trivial solution.
In the same manner as above the energy relation for problem ${\sf
D}_0$ is shown to be equivalent to the energy relation for problem ${\sf B}_0$. We can again use [@KleiMart88 pp. 309–310] and [@KresRoac78 p. 1434] to find the criteria that lead to $W=0$ and ${{\boldsymbolH}}_W={{\boldsymbolE}}_W={{\boldsymbol0}}$. These are the criteria for the set $\{k_1,k_2,\alpha=\lambda\}$ in Proposition \[prop:uniqueB0\].
[00]{}
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[^1]: Centre for Mathematical Sciences, Lund University, Sweden
[^2]: Electrical and Information Technology, Lund University, Sweden
|
Amorphous metallic alloys have been the subject of extensive investigations over the last two decades because of their surprising transport, magnetic and superconducting properties. To explain the most unusual low-temperature physical properties of glassy materials the concept of two-level tunneling systems (TLS) was suggested (for a review see [@black]). According to a theoretical model [@VZ], the electron scattering processes on TLS result in the commonly occurring logarithmic temperature dependence of the resistivity. On the other hand the resistivities of some amorphous alloys ($Ni_xNb_{1-x}$, $Fe_{x}Au_{1-x}$) were claimed not to exhibit any logarithmic feature at low temperatures [@Harris] and therefore these alloys were regarded as a different class of metallic glasses.
Point contact (PC) spectroscopy [@YS] offers a very sensitive method to investigate scattering processes in conducting materials, since back scattering of electrons in a PC causes a noticeable change in the current through the PC and the R(V) characteristics give quantitative information about the energy dependence of the electron scattering processes on quasiparticles (TLS, phonons, magnons, etc.,). In particular, the strong electron-TLS coupling gives rise to a peak in the PC differential resistance around $V=0$. This phenomenon is usually referred to as the zero bias anomaly (ZBA). Experiments on metallic PCs containing nonequilibrium defects [@RB; @anomaly] as well as on metallic glass PCs [@Keijsers; @bal] demonstrated the existence of ZBA suggested by the theory [@VZ; @KK].
The present study was motivated by the expected absence of the logarithmic peak in $Ni_xNb_{1-x}$, and was aimed at investigating TLS scattering in these metallic alloys with the help of PC spectroscopy. The $Ni_xNb_{1-x}$ metallic glasses (MG) are also interesting from another point of view, since a superconducting ground state develops at high enough $Nb$ concentration [@Rapp]. We examined $Ni_{44}Nb_{56}$ and $Ni_{59}Nb_{41}$ by measurements on bulk samples investigating the temperature and magnetic field dependence of resistivity and by PC spectroscopy based on break junction technique [@muller]. This technique permits very stable PC in the resistance range 1-200 $\Omega$ to be made by breaking the sample in ultra high vacuum and then forming the contact between the freshly fractured atomically clean surfaces. Due to the relatively large resistivity of $Ni_xNb_{1-x}$ MG these contacts were basically in the Maxwell limit [@YS], where the contact resistance is calculated as $R_{PC}=\rho/d$ ($\rho$ being the electrical resistivity, $d$ the contact diameter).
Figure \[RT\] presents the results of bulk sample measurements. We found that $Ni_{44}Nb_{56}$ has a superconducting transition at 1.8 K, while the resistance of $Ni_{59}Nb_{41}$ starts to decrease only below 700 mK indicating a superconducting transition just outside of the temperature range of the measurements (300 mK). For $Ni_{59}Nb_{41}$ we observed logarithmic behavior between 700mK and 25K [@comment]. It is hard to confirm the logarithmic character for $Ni_{44}Nb_{56}$ because of the interfering presence of the superconducting fluctuations. In our experiments these fluctuations start at 8 K, where the $R(T)$ curve splits from that of the non-superconducting sample. This broad range of fluctuations is in good quantitative agreement with theoretical calculations [@Maki] and experiments on other amorphous superconductors [@SCinMG]. The inset in Fig. \[RT\] shows a similarly broad magnetic field region of superconducting fluctuations (1.5-2.6 T) at constant temperature (300 mK). After suppressing the superconductivity by applying a magnetic field of 12 T the temperature dependence of the resistivity in both samples shows a clear logarithmic behavior up to 25 K. In this logarithmic region a small but noticable magnetic resistance is observed. Above the temperature range shown in Fig. \[RT\] both samples exhibited decreasing resistivity with increasing temperature. The normal state resistivity at 50 K was $\rho \approx 2.5~\mu\Omega m$ for $Ni_{44}Nb_{56}$ and $\rho \approx
1.6~\mu\Omega m$ for $Ni_{59}Nb_{41}$.
The PC spectra measurements showed a common behavior of $dV/dI(V)$ for all contacts in the resistance range $R_{PC}=1-30~\Omega$ (corresponding to the contact diameter $d$ = 2000 - 60 nm) which means rather good reproducibility of results for different samples. For $R_{PC}>60~\Omega$ ($d<$ 30 nm) individual features start to prevail and the $dV/dI(V)$ curves for the samples of the same resistance may differ significantly. This sets the length scale of the material inhomogeneity to $\sim$30 nm. This value is in agreement with the small angle neutron scattering measurements performed on these metallic glasses [@Svab] where inhomogeneity was found on the length scale of $\sim$18 nm. The regime of electron flow in point contacts depends on the relationship between the contact diameter $d$ and the elastic $(l_{el})$ and the inelastic $(l_{in})$ mean free paths [@YS]. Due to the large resistivity of $Ni_xNb_{1-x}$ MG, the transition to the thermal limit ($l_{el},l_{in}\ll d $) for low ohmic contacts occurs at small voltage bias (because of the strong energy dependence of the inelastic mean free path). In the thermal regime the excess electron energy is dissipated inside the contact, which results in the increase of the point contact temperature with respect to the bias voltage [@YS]: $T^2_{PC}=T^2_{bath}+V^2/{4L}$, where L is the Lorenz number. This equation relates R(V) measurements done by PC technique to the temperature dependence of the resistivity.
Figure \[highbias\] shows the differential resistance of $Ni_{59}Nb_{41}$ junctions. The low ohmic contacts (curve 1) exhibit clear logarithmic peaks in the voltage region of 1-12 mV, as presented on the enlarged scale for a $4~\Omega$ junction. According to the above equation and calculating by the standard Lorenz number, the bias voltage 12 meV corresponds to $T_{PC}=38$ K in the thermal regime, which is somewhat higher than the border of the logarithmic region ($25$ K) in the bulk sample measurements of $R(T)$. This difference is due to the high resistivity of the material: the voltage drop in the vicinity of the contact is comparable with the voltage drop over the bulk part of the sample, which shifts the logarithmic region towards higher voltages. The decrease in the contact resistance between 0 and 50 mV is comparable to that for the bulk samples in the temperature range 5-160 K.
The $dV/dI(V)$ dependences for high ohmic $Ni_{59}Nb_{41}$ junctions show step-like singularities at high biases (curve 2, Fig. \[highbias\]), which can be repeatedly reproduced for the same contact but vary in amplitude and position for different samples. The origin of these high bias anomalies is the subject of ongoing investigations.
The PC characteristics of the superconducting $Ni_{44}Nb_{56}$ MG below the critical temperature are quite different from these in ordinary superconductors and can be understood only qualitatively.
Figure \[PCres\] shows the PC differential resistance and I-V curves of $Ni_{44}Nb_{56}$ at different contact resistances. The junctions with small normal-state resistance ($\lesssim 1.5\Omega$) present conventional $I-V$ curves (Fig. \[PCres\]b) of a current driven contact with a clear voltage jump above a certain critical current value and with excess current at high voltages. The evaluation of the excess current [@Bardas] for these low resistance junctions shows, that the normal resistance - excess current product is constant giving the close-to-BCS value of $3.2\pm 0.2$ for $2\Delta /k_BT_c$. For higher resistances the $I-V$ curves are smeared, $R_NI_{exc}$ vanishes and an increasing residual resistance is observed at zero bias. We found that this residual resistance increases rapidly for decreasing contact diameter and may differ significantly for contacts with the same $R_N$. The transition between the jump-like curves and the smeared ones is also sample dependent, varying between $1-2~\Omega$. These phenomena can be understood qualitatively in terms of percolation-like superconductivity. In large enough contacts the current can find continuous superconducting percolation paths between the two electrodes, but below a certain contact diameter no such paths exist any more, and the current must flow through normal regions as well. In this case the residual resistance is determined by the fraction of normal and SC regions along the current paths, which explains the strongly contact-dependent behavior. The characteristic width of percolation paths is most probably close to the material inhomogeneity scale of 18 nm [@Svab]. This size scale of percolation is in agreement with the value of coherence length ($\approx
10$ nm) calculated from $H_{c2}$.
As Fig. \[PC-field\] shows, the step-like $I-V$ curve is smeared under the influence of magnetic field as well, but the zero bias resistance and the excess current remains constant up to 1.2 T. We believe that this smearing follows from the vortex dynamics at high current densities in the contact area: the resistance caused by vortices is superimposed on the step-like zero-field $I-V$ curve. The $B=1.6$ T and $2$ T curves are already within the fluctuation region of $H_{c2}$ (see inset in Fig. \[RT\]), thus one obtains completely different $I-V$ curves with larger zero bias resistance and small excess current. Going above $H_{c2}$, at $B=5$ T we regain the positive logarithmic peak attributed to electron scattering on two level systems (see inset in Fig. \[PC-field\]).
Recording the differential resistance curves of $Ni_{44}Nb_{56}$ as the function of temperature, we found that the transition is broadening by decreasing contact diameter. Similar behavior was observed by Naidyuk [*et al.*]{} in superconducting heavy fermion point contacts [@Naidyuk].
The differential resistance of some contacts displays large fluctuations around zero bias (Fig. \[PC-jumps\]). The plot of this noise as the function of time shows that the resistance is switching between two (or more) discrete states on the time-scale of seconds. This slow two level fluctuation is not sensitive to magnetic fields up to 1.5 T and decreases rapidly at larger fields. In [@Keijsers] a similar fluctuation was superimposed on the logarithmic ZBA in $Fe_{80}B_{20}$ and $Fe_{32}Ni_{36}Cr_{14}P_{12}B_{6}$ metallic glasses. This fluctuation was explained as the effect of slowly moving defects influencing electron-TLS coupling. In $Ni_{44}Nb_{56}$ contacts the motion of such relatively large defects can result in shutting down one of the percolation paths and suppressing superconductivity in a sizeable part of the constriction. These fluctuations were only observed in relatively small contacts ($d\lesssim 200$ nm). In such small areas only a few percolation paths are present, which explains that shutting down one of them has an observable effect. It makes the superconducting characteristics an extremely sensitive detector for the slow relaxing TLS motion.
In conclusion we demonstrated that in contrast to earlier observations both the bulk resistivity and the PC differential resistance of amorphous $Ni_xNb_{1-x}$ alloys exhibit low-energy logarithmic behavior which is characteristic of electron scattering on the fast relaxing TLSs in full accordance with the Vladar-Zawadowski model [@VZ]. In $Ni_{59}Nb_{41}$ we found reproducible structures in the point contact spectra at high biases and higher ohmic contacts. We also studied the unusual features of superconducting $Ni_{44}Nb_{56}$ contacts which can be explained by a percolation type of superconductivity, heating effects in the normal phase with increasing bias and the influence of slow configurational changes close to the contact.
We acknowledge E. Sváb, A. Zawadowski and I.K. Yanson for useful discussions. This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), the Stichting Fundamenteel Onderzoek der Materie (FOM) and by the Hungarian National Science Foundation under grant No. T026327.
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---
abstract: |
While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.
nonlinear oscillations, periodic response, global analysis, harmonic balance, existence criterion
author:
- 'Thomas Breunung[^1] and George Haller'
title: 'When does a Periodic Response Exist in a Periodically Forced Multi-Degree-of-Freedom Mechanical System?'
---
Introduction
============
Nonlinear mechanical systems are generally assumed to approach a periodic orbit under external periodic forcing. While approximately periodic responses are commonly observed in numerical routines (eg. numerical time integration, numerical continuation or harmonic balance) and experiments, concluding the existence of periodic response rigorously in a nonlinear system is more delicate.
For nonlinear system close to a solvable limit, perturbation methods remain a powerful tool to compute periodic responses. Among these, the method of averaging requires slowly varying amplitude equations (cf. Sanders et al. [@Verhulst_Avg]), while the method of multiple scales (cf. Nayfeh [@Nayfeh_Perturb]) assumes evolution on different time scales generated by small parameters. The method of normal forms (cf. Murdock [@Murdock_NF]) introduces a series of smooth coordinate changes to approximate and simplify the essential dynamics in a Taylor series in a small enough neighborhood of an equilibrium. Due to the truncation of infinite series arising in these procedures, the approximate dynamics remains valid only for sufficiently small values of an underlying perturbation parameter. How small that parameter is required to be is generally unclear, and hence the relevance of the results obtained from perturbation procedures under physically relevant parameter values is a priori unknown.
Rigorous numerics (cf. van den Berg and Lessard [@Berg_RigorousNumerics]), estimating the ignored tail of Taylor expansions, is only applicable to specific numerical examples. In applications, therefore, one typically employs an additional numerical method to verify the predictions of perturbation methods. This is clearly not optimal.
Due to the broad availability of effective numerical packages, numerical time-integration is often used to compute periodic responses. Modern, lightly damped engineering structures, however, require long integration times to reach a steady state response. Furthermore, the observed periodic response depends on the initial condition and unstable branches of the full set of periodic responses cannot be located in this fashion.
More advanced numerical schemes, such as harmonic balance (cf. Mickens [@Mickens_HB]) and numerical collocation (cf. Ascher et al. [@Ascher_collo]), reformulate the underlying ordinary differential equations or boundary value problem into a set of algebraic equations by approximating the periodic response in terms of a set of finite basis functions (e.g., polynomial or Fourier basis). Coupled with numerical continuation schemes, this approach enables the computation of periodic responses even for high-amplitude oscillations (cf. Dankowicz and Schilder [@coco]). To justify this procedure a priori and estimate the error due to the truncation of the infinite-dimensional basis-function space, the existence of the periodic orbit would need to be established by an analytic criterion. While this can be guaranteed under small forcing or small nonlinearities by Poincaré map arguments, the problem remains unsolved for general, forced nonlinear mechanical systems.
Recently, Jain et al. [@NPO] have proposed iterative methods to efficiently compute periodic responses of periodically forced nonlinear mechanical systems without small parameter assumptions. Their existence criterion, derived via the Banach fixed-point theorem, however, fails for high forcing amplitudes and forcing frequencies in resonance with an eigenfrequency of the linearized system.
In the absence of small parameters, general fixed point theorems, such as Brouwer’s or Schauder’s fixed point theorem or the Leray-Schauder principle are powerful tools to prove the existence of periodic orbits (cf. Bobylev et al. [@Bobylev] or Precup [@Precup] for a summary). Lefschetz [@lefschetz1943existence], for example, proved the existence of a periodic orbit for a specific one-degree-of-freedom, forced nonlinear mechanical system. His result, however, requires the damping force to be of the same order as the geometric nonlinearities. Therefore, his argument does not apply to common mechanical systems, such as the classic Duffing oscillator, with linear damping.
This restriction on the damping was relaxed significantly by Lazer [@Lazer_Schauder]. Based on Schauder’s fixed point theorem, Lazer’s result allows for linear damping but requires a growth restriction on the nonlinearity. This result was further strengthened and extended to higher dimensions by Mawhin [@Mawhin_LazerExt], who required the damping simply to be differentiable. Both results, however, restrict the growth of the nonlinearties to be less than linear for sufficiently high displacements, i.e., exclude polynomial or even linear stiffness forces. As Martelli [@Martelli] noted, this restriction can be relaxed to a linear growth with sufficiently low slope, depending on the eigenfrequency of the system and the forcing frequency. Due to the growth restriction on the nonlinearities these results are inapplicable for simple polynomial nonlinearities.
The results of Mawhin and Lazer have been extended to nonsmooth systems by Chu et al. [@Chu_singular] and Torres [@torres2003existence] and to more complex differential operators (cf. Mawhin [@Mawhin2001]), relying on an extension of the generalized continuation theorems by Gaines and Mawhin [@Mahwin_Cidx] and Manásevich and Mawhin [@mawhin_pLaplacian]. Furthermore, Antman and Lacabonara [@Antman2009] give an existence criterion for periodic solutions based on the principle of guiding functions (cf. Krasnosel’skij [@Krasnos_guidingFunc]). This result, however, relies on the specific form of the nonlinearity and external forcing for shells.
A general existence criterion for periodic orbits in second-order differential equations with linear dissipation can be found in the work of Rouche and Mawhin [@Rouche_PO]. Their result implies the existence of a periodic response for dissipative nonlinear mechanical systems for arbitrary large forcing amplitudes. It appears, however, that the results in [@Rouche_PO] are not known in the mechanical vibrations literature.
In this paper, we refine the Rouche-Mawhin results to be directly relevant for mechanical systems. This gives a general sufficient criterion for the existence of periodic orbits in multi-degree-of-freedom forced-damped nonlinear mechanical systems, without any restriction on the magnitude of the forcing or vibration amplitude. We also give mechanically relevant examples of periodically forced systems violating our criterion in which a periodic response does not exist. These show that the assumptions in our results are indeed relevant for mechanical systems and cannot be individually omitted without loosing the conclusion. Further, we identify a broad family of nonlinear mechanical systems for which our theorem guarantees the existence of a periodic response. This result enables the rigorous computation of periodic orbits for a large class of strongly nonlinear mechanical systems.
Problem statement
=================
We consider a general $N$-degree-of-freedom mechanical system of the form $$\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{S}(\mathbf{q})= \mathbf{f}(t), \qquad \mathbf{f}(t+T)=\mathbf{f}(t) ,\qquad \mathbf{q}\in \mathbb{R}^N,
\label{eq:sys0}$$ where the mass matrix $\mathbf{M}\in \mathbb{R}^{N\times N}$ is positive definite and $\mathbf{C}\in \mathbb{R}^{N\times N}$ denotes the damping matrix. The geometric nonlinearities $\mathbf{S}(\mathbf{q})$ contain all position-dependent forces, including potential and non-potential forces, such as follower forces [@Bolotin]. We also assume that $\mathbf{S}(\mathbf{q})$ to be continuous in its arguments. The external forcing $\mathbf{f}$ is assumed to be $T$-periodic and continuous.
To emphasize the importance of rigorous existence criteria for periodic orbits of nonlinear mechanical systems, we next present two examples for which the popular harmonic balance procedure leads to wrong conclusions. First, addressing the common belief that periodic forcing always leads to a periodic response of system , we start with an example illustrating the contrary. Furthermore, we demonstrate numerically that the harmonic balance method yields false results on this example. Secondly, we demonstrate that for a linear system the harmonic balance procedure can predict an inaccurate periodic orbit.
Motivating examples {#sec:counter_ex1}
-------------------
In the harmonic balance procedure, the equation of motion of the dynamical system is evaluated along an assumed periodic orbit of the form $$\mathbf{q}^*(t)=\frac{\mathbf{c}_0}{2}+
\sum_{k=1}^{K}
\left[
\mathbf{s}_k\sin(k\Omega t)+\mathbf{c}_k\cos(k\Omega t)
\right],\qquad
\mathbf{s}_k,\mathbf{c}_k\in\mathbb{R}^N.
\label{eq:HB_sol}$$ Next, the forcing term $\mathbf{f}(t)$ and the nonlinearity evaluated along the postulated periodic orbit, $\mathbf{S}(\mathbf{q}^*(t))$, are projected on the first $K$ Fourier modes and higher modes are ignored. As a result, one obtains a finite set of nonlinear algebraic equations for the unknown constants $\mathbf{c}_k$ and $\mathbf{s}_k$. This set of equations can generally not be solved analytically, therefore iterative methods, such as a Newton-Raphson iteration, are employed to generate an approximate solution. For more details, we refer to Mickens [@Mickens_HB].
Besides the a priori assumption of the existence of a periodic orbit, the truncation of the periodic orbit at some finite order $K$ needs to be justified. Classic results (cf. Bobylev et al. [@Bobylev] or Leipholz [@leipholz1977direct]) show that this truncation can be justified for a sufficiently large $K$ when a periodic orbit of system actually exists. If the existence of a periodic orbit cannot be guaranteed a priori, conditions derived by Urabe [@urabe] or Stokes [@stokes] might guarantee the existence of a periodic orbit close to the harmonic balance approximation. These conditions, however, can only be evaluated a posteriori, as they rely on the harmonic balance solution itself. Notably, Kogelbauer et al. [@Kogelbauer_HB] strengthen the results of Urabe [@urabe] and Stokes [@stokes], and provide an explicitly verifiable condition for the existence of a periodic orbit. In practice, however, their conditions restrict the forcing and response amplitudes to small values.
To illustrate issues that can arise with harmonic balance, we consider the two degree-of-freedom oscillator $$\begin{bmatrix}
m_1 & 0\\
0 & m_2
\end{bmatrix}
\ddot{\mathbf{q}}
+
\begin{bmatrix}
c_1+c_2 & -c_2\\
-c_2 &c_1+c_2
\end{bmatrix}
\dot{\mathbf{q}}
+
\begin{bmatrix}
k_1+k_2 & -k_2\\
-k_2 &k_1+k_2
\end{bmatrix}
\mathbf{q}
+\kappa
\begin{bmatrix}
q_1^2+q_2^2 \\
q_1^2+q_2^2
\end{bmatrix}
=
\begin{bmatrix}
f_1 \\
f_2
\end{bmatrix}.
\label{eq:eqm_counter1}$$ The nonlinearities assumed in system may arise in a Taylor series approximation of a more complex nonlinear forcing vector, which is terminated at second order. Specifically, quadratic nonlinearities arise in the modeling of ship capsize (cf. Thompson [@thompson_cap]), ear drums (cf. Mickens [@Mickens_Intro_NL]) and shells (cf. Antman and Lacabonara [@Antman2009]). Touzé et al. [@touze_springsys_cons] study a spring-mass system in which quadratic nonlinearities arise naturally due to the geometry.
We assume forcing in the form of a triangular wave $$f_1(t)=-f_2(t)=\frac{2 f_m}{\pi}\int_0^t\operatorname{sign}(\cos(\Omega s))ds=f_m\frac{8}{\pi^2}\sum_{k=0}^{\infty}(-1)^k\frac{\sin((2k+1)\Omega t)}{(2k+1)^2},
\label{eq:trigwave}$$ where the parameter $f_m$ denotes the amplitude and $\Omega=2\pi/T$ the excitation frequency. For the remaining parameters, we assume the following non-dimensional numerical values $$m_1=m_2=1,\quad k_1=1,\quad k_2=4,\quad c_1=0.001,\quad c_2=0,\quad f_m=0.01178,\quad \Omega=1,\quad \kappa=1.
\label{eq:pars_counter1}$$ We apply the harmonic balance to system with the parameters and forcing . We solve the resulting algebraic system of equations with a Newton-Raphson iteration, whereby we evaluate the nonlinearity in the time domain and transform the time signal to the frequency domain using fast Fourier transforms (cf. Cameron and Griffin [@cameron_AFT]). Following common practice (cf. Cochelin and Vergez [@cochelin2009high]), we start with a low number of harmonics and successively increase the number $K$ of harmonics in ansatz until the resulting oscillation amplitude appears converged, i.e., does not change with increasing $K$. We depict the result of this harmonic balance procedure in Fig. \[fig:C1\].
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]{};
table\[row sep=crcr\][ 0 0.0949124991201851\
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1.88115875796349 0.0238017977398286\
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1.92986562080984 0.0524372735726866\
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1.960547109217 0.0670169814086803\
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1.99007804180888 0.0783688324820027\
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6.26132474668949 0.0941037314611455\
6.27513141647271 0.0947983293102519\
6.28318530717959 0.0949124991201851\
]{};
table\[row sep=crcr\][ 0 0.0949124839101785\
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1.56322183434438 -0.272574722313198\
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]{};
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3.99741442084678 0.0349916599297346\
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4.07450166046975 0.0723009395213206\
4.08907536746315 0.077349019724859\
4.10326555585146 0.0816232326318476\
4.11745574423976 0.0852562873745049\
4.13164593262807 0.0882408686607494\
4.14545260241129 0.0905166338833201\
4.15925927219451 0.0921684564714935\
4.17306594197773 0.0931931778455315\
4.18687261176094 0.0935888358512607\
4.20067928154416 0.0933546695913012\
4.21448595132738 0.0924911212222002\
4.2282926211106 0.0909998347071088\
4.24209929089382 0.0888836515901668\
4.25590596067704 0.0861466039095022\
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4.29847652584196 0.0738610193212255\
4.31305023283536 0.0683488651078772\
4.32800745843384 0.0620176846726412\
4.34296468403233 0.0550177445344335\
4.3583054282359 0.0471599001915814\
4.37402969104457 0.0384116428000363\
4.39013747245832 0.0287437369423298\
4.40662877247717 0.0181307457241004\
4.42388710970619 0.00628020199573509\
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4.45993785858459 -0.0207760660891561\
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4.4994402749088 -0.0536548860320956\
4.52091731679381 -0.0727802823855299\
4.54354491449408 -0.0937646989696814\
4.5680901052198 -0.117378453162655\
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4.62638493319339 -0.17623103608156\
4.6658873495176 -0.217582874088577\
4.78592867291058 -0.343869296893761\
4.81737719852791 -0.375678839235242\
4.84460701948926 -0.402363461089394\
4.86953572882007 -0.425941416512715\
4.89254684512543 -0.446871164983301\
4.91440740561553 -0.465924202571331\
4.93511741029036 -0.483157454786843\
4.95467685914992 -0.498646580037265\
4.9734692707993 -0.512761671071282\
4.99187816384359 -0.525820024654867\
5.0095200196777 -0.537586500168167\
5.02677835690672 -0.548359787965753\
5.04365317553066 -0.558163229231853\
5.0601444755495 -0.56702418016016\
5.07625225696325 -0.574973464389647\
5.09197651977192 -0.582044856188822\
5.1073172639755 -0.588274595459843\
5.12265800817907 -0.59383157408644\
5.13761523377756 -0.598591904070939\
5.15218894077096 -0.602597356713424\
5.16676264776435 -0.605971369630776\
5.18133635475775 -0.608708242271676\
5.19552654314606 -0.610756472557536\
5.20971673153436 -0.612192988878705\
5.22390691992267 -0.613015487201231\
5.23809710831098 -0.613222641091941\
5.25228729669929 -0.612814102557242\
5.2664774850876 -0.611790502185328\
5.2806676734759 -0.610153448662497\
5.29485786186421 -0.607905527652004\
5.30943156885761 -0.604964743275954\
5.324005275851 -0.601388310892813\
5.3385789828444 -0.597182233229801\
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5.58057922265581 -0.445381344812608\
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5.6534477576228 -0.37625370170084\
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5.95067467656708 -0.0721111627302831\
5.97253523705718 -0.0526072904421886\
5.993245241732 -0.0349489654582396\
6.01280469059156 -0.0190704149276746\
6.03159710224094 -0.00460122843882793\
6.04962247668014 0.00850629372666223\
6.06688081390917 0.0203095641708515\
6.0837556325331 0.0311108938035956\
6.10024693255194 0.0409309495347241\
6.1163547139657 0.0497950724306104\
6.13207897677436 0.0577327064752851\
6.14741972097794 0.0647768449734176\
6.16237694657643 0.0709634986264955\
6.17695065356982 0.076331188648207\
6.19152436056322 0.0810356868059063\
6.20571454895153 0.0849696654575887\
6.21990473733984 0.0882575865733672\
6.23371140712305 0.090830265567531\
6.24751807690627 0.0927802113510703\
6.26132474668949 0.0941037282720405\
6.27513141647271 0.0947983146057796\
6.28318530717959 0.0949124787463997\
]{};
From the amplitudes depicted in Fig. \[fig:C1\_apmls\], one would normally conclude the converge of the displacement amplitude of the first coordinate to a value of about $0.6$. Also, the time series for the choices of $K$, shown in Fig. \[fig:C1\_tseries\], are practically indistinguishable. There is, therefore, every indication that the harmonic balance method has correctly identified a periodic orbit for system .
The periodic orbit suggested by the numerical result in Fig. \[fig:C1\], however, does *not exist* in system for the parameters and the forcing . More generally, in Appendix \[app:C1\] we prove that if the amplitude $f_m$ of the forcing is above a certain threshold, no periodic orbit exists for system . Since the value of the forcing amplitude is above this threshold, the periodic orbit indicated by the harmonic balance procedure in Fig. \[fig:C1\] does not actually exists.
One might argue that due to the discontinuity of the forcing , the assumption of a twice differentiable solution is not justified. Indeed, due to the Lipschitz continuity of the forcing just the existence and uniqueness of a local solution can be guaranteed by Picard’s theorem (cf. Coddington and Levinson [@Coddington]). Our nonexistence proof, however, relies only on the fact that the amplitudes of the forcing is above a certain threshold. One can therefore easily replace the forcing by a smoother, even analytic alternative and obtain the same conclusion.
The unforced limit of system has two fixed points which are connected through a homoclinic orbit. As Thompson and Steward [@Thompson_NLC] observe, these features can give rise unbounded (escape) behavior for the forced system. For small enough forcing amplitudes, the existence of the periodic orbit is guaranteed by the general results of Haro and de Llave [@Haro_tori]. For larger forcing amplitudes, however, the result of Haro and de Llave [@Haro_tori] cannot guarantee the existence of a periodic orbit and exceeding the threshold rules out the possibility of any periodic motion.
As we will see shortly, the crucial reason for the nonexistence of a periodic orbit in the above example is the form of the nonlinearity. Indeed, for a simple system with a single quadratic nonlinearity, Thompson and Steward [@Thompson_NLC] were unable to continue a periodic orbit numerically for arbitrarily high forcing amplitudes. Difficulties in applying harmonics balance to systems with quadratic nonlinearities have also lead to the practical guidelines by Mickens [@mickens_HBguide], who heuristically restricts the harmonics balance procedure to systems with odd nonlinearities.
Next, we give an example for which the harmonic balance procedure yields an inaccurate periodic orbit, due its unavoidable truncation of the basis function space. We consider the linear forced-damped oscillator $$\ddot{q}+c\dot{q}+kq=f(t), \qquad k=400,~~ c=0.01,
\label{eq:lin_sys}$$ with a forcing shown in Fig. \[fig:C2\_forcing\]. This forcing is clearly dominated by a fundamental harmonic. The harmonic balance procedure for the choice of five harmonics yields a periodic orbit, which is dominated by the fundamental harmonic with an amplitude at about $0.0025$ (c.f. Fig. \[fig:C2\_tseries\]). Increasing the number of harmonics considered to ten and fifteen, we obtain the periodic orbits in Fig. \[fig:C2\_tseries\], which are practically indistinguishable from the periodic orbit obtained from five harmonics. In practice, one would practically terminate the harmonic balance procedure and conclude the convergence of method to a periodic orbit with the maximal position equal to $0.0025$ in magnitude.
[0.49]{}
table\[row sep=crcr\][ 0 0.000524999999999665\
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[0.49]{}
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]{};
The untruncated version of the periodic orbit depicted in Fig \[fig:C2\_tseries\], however, differs significantly from the approximation obtained from the harmonic balance procedure. The maximal position along the true periodic orbit is twice the value predicted with a low-order truncation. We note that the forcing $f(t)$ and the corresponding periodic orbit $q^*(t)$ of system are given by $$f(t)=\frac{1}{400}\left(399\sin(t)+0.01\cos(t)+0.2\cos(20t)\right),\qquad q^*(t)=0.0025(\sin(t)+\sin(20t)).
\label{eq:lin_sol}$$ The periodic orbit has a fundamental harmonic, which is identified correctly by the harmonic balance approximations (c.f. Fig \[fig:C2\_tseries\]). The higher-frequency component, however, is truncated in the harmonic balance procedure and is therefore not captured. Eventually, increasing the number of harmonics considered above twenty, we obtain the correct result from the harmonic balance, but there is no rigorous criterion that would indicate this in advance. Even in the case of infinite harmonics in ansatz , the approximate periodic orbit from the harmonic balance procedure can differ significantly from the actual periodic orbit, as we demonstrate in Appendix \[app:fejer\].
As we have illustrated, even in the case of an apparently convergent harmonic balance approximation the existence of a periodic orbit cannot be guaranteed. Rigorous criteria for the existence of periodic orbits, however, can exclude false positives.
Existence of a periodic response
================================
With the mean forcing defined as $$\bar{\mathbf{f}}=\frac{1}{T}\int_0^T\mathbf{f}(t)dt,
\label{eq:f_mean}$$ we will prove the following general result
Assume that the forcing $\mathbf{f}(t)$ in system is continuous and the following conditions hold:
1. The damping matrix $\mathbf{C}$ is definite, i.e., there exists a constant $C_0\!>\!0$ such that $$|\mathbf{x}^T\mathbf{C}\mathbf{x}|>C_0|\mathbf{x}|^2, \qquad \mathbf{x}\in\mathbb{R}^N.
\label{eq:damping_cond}$$ \[cond:damping\]
2. The geometric nonlinearities derive from a potential, i.e., there exists a continuously differentiable scalar function $V(\mathbf{q})$ such that $$\mathbf{S}(\mathbf{q})=\frac{\partial V(\mathbf{q})}{\partial \mathbf{q}}.
\label{eq:potential}$$ \[cond:potential\]
3. For each degree of freedom, the quantity $q_j(S_j(\mathbf{q})-\bar{f}_j)$ has a constant, nonzero sign far enough from the origin. Specifically, there exists a distance $r\!>\!0$ and an integer $1\leq n \leq N$ such that $$\begin{split}
q_j\left(S_j(\mathbf{q})-\bar{f}_j \right) &>0, \qquad |q_j|>r,~~j=1,...,n,
\\
q_j\left(S_j(\mathbf{q})-\bar{f}_j \right)&<0, \qquad |q_j|>r,~~j=n+1,...,N.
\end{split}
\label{eq:sign_cond}$$ \[cond:sign\_cond\]
Then system has a twice continuously differentiable periodic orbit. \[thm:Existence\]
We deduce this theorem from Theorem 6.3 by Rouche and Mawhin [@Rouche_PO] after the removal of an unnecessary zero-mean forcing assumption in its original version. We detail the proof in Appendix \[app:Exist\_proof\].
As a consequence of the proof of Theorem \[thm:Existence\], we obtain an upper bound on the amplitude of the existing periodic orbit. Specifically, with the squared $L_2$-norm $$C_f^2:=\int_0^T \mathbf{f}^T\mathbf{f} dt,
\label{eq:def_CF}$$ of the forcing an estimate for the maximal oscillation amplitude is given by $$\sup_{0\leq t\leq T} |\mathbf{q}| \leq \sqrt{N}\left(r+\sqrt{T}\frac{C_f}{C_0}\right),
\label{eq:bnd_ampls}$$ where the constant $C_0$ is defined in equation . We detail the derivation of this estimate in Appendix \[app:bound\_ampl\]. Our bound is stricter than that obtained by Rouche and Mawhin [@Rouche_PO], who have an additional summand of $\sqrt{T}C_f/C_0$ in equation . Further, the bound confirms the intuition arising from linear theory, that the maximal response amplitude is proportional to the quotient of forcing amplitude and minimum damping coefficient. The inequality confirms this intuition for the full nonlinear system without small-parameter assumptions.
For a single degree-of-freedom harmonic oscillator ($N\!=\!1$, $r=0$, damping coefficient $c$ and eigenfrequency $\omega_0$) and single harmonic forcing with amplitude $f$ at resonance, i.e., for a system $$\ddot{q}+c\dot{q}+\omega_0^2q=f\sin(\omega_0 t),
\label{eq:lin_osci}$$ the relationship between the bound and the exact solution $q_{lin}\!=\!fT/(2\pi c)$ is as follows: $$\sup_{0\leq t\leq T} |\mathbf{q}| \leq \sqrt{N}\left(r+\sqrt{T}\frac{C_f}{C_0}\right)
=\frac{fT}{2c}>\frac{fT}{2\pi c}=q_{lin}.
\label{eq:bnd_lin}$$ As expected, the bound is conservative, but only by a factor of $\pi$.
Condition \[cond:sign\_cond\] implies a sign change of the geometric nonlinearities minus the mean forcing component-wise inside the interval $[-r, r]$. In Fig. \[fig:c3\_illu\], we sketch graphs of three different geometric nonlinearities. If the value of the geometric nonlinearities $S_j(\mathbf{q})$ is greater than the mean forcing for $q_j\!<\!-r$ (i.e. lies in the upper left dotted square of Fig. \[fig:c3\_illu\]), then the quantity $(S_j(\mathbf{q})-\bar{f}_j)q_j$ evaluated for $q_j\!<\!-r$ is negative. Therefore, for condition \[cond:sign\_cond\] to hold, $(S_j(\mathbf{q})-\bar{f}_j)q_j$ for $q_j\!>\!r$ needs to be negative, which implies that the geometric nonlinearity $S_j(\mathbf{q})$ needs to be below $\bar{f}_j$. In Fig. \[fig:c3\_illu\], the graph of of $S_j(\mathbf{q})$ needs to end in the lower right dotted square. Such a nonlinearity is depicted in blue in Fig. \[fig:c3\_illu\]. Similarly, a geometric nonlinearity satisfying $(S_j(\mathbf{q})-\bar{f}_j)q_j\!>\!0$ for all $|q_j|\!>\!r$ needs to be contained in the two shaded regions of Fig. \[fig:c3\_illu\]. For the red curve, we have $(S_j(\mathbf{q})-\bar{f}_j)q_j\!<\!0$ for $q_j\!<\!-r$ and $(S_j(\mathbf{q})-\bar{f}_j)q_j\!>\!0$ for $q_j\!>\!r$ , i.e. condition \[cond:sign\_cond\] is not satisfied.
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plot \[smooth\] coordinates [ (-0.5,4.25) (1,2.25) (2,1.75) (3,1.5) (4.5,4.45)]{};
(4,2) node\[below\]; (4,2)– (4,1.8) ;
(-0.05,2) node\[below\]; (-0,2)– (-0,1.8); (-1,2) – (5,2) node\[above\]; (2,0.5) – (2,4.5)node\[left\][$S_j(\mathbf{q})$]{}; (-1,3) – (5,3) node\[above\];
\[thm:posdef\] If the geometric nonlinearities $\mathbf{S}(\mathbf{q})$ are differentiable, condition \[cond:sign\_cond\] holds if
1. The Hessian of $V(\mathbf{q})$ is definite for $|\mathbf{q}|\!>\!r^*$, i.e., for some constant $C_v\!>\!0$, $$|\mathbf{x}^T\frac{\partial^2 V(\mathbf{q})}{\partial ^2 \mathbf{q}} \mathbf{x}|> C_v|\mathbf{x}|^2,\qquad \mathbf{x}\in\mathbb{R}^N,~~ |\mathbf{q}|\geq r^*.$$ \[cond:conv\_cond\]
See Appendix \[app:convex\] for a proof.
Condition \[cond:conv\_cond\] is more restrictive than condition \[cond:sign\_cond\]. For example, consider the potential $$V(q_1,q_2)=k_1 q_1^2+k_2 q_2^2,
\label{eq:saddle_NL}$$ which satisfies \[cond:conv\_cond\] only if $k_1$ and $k_2$ have the same sign ($k_1k_2>0$), while it satisfies condition \[cond:sign\_cond\] for any non-zero $k_1$ and $k_2$. However, condition \[cond:conv\_cond\] is more intuitive as it restricts the global geometry to be cup-shaped sufficiently far from the origin. In addition, condition \[cond:conv\_cond\] is generally easier to verify, since positive or negative definiteness of the Hessian can be verified through direct eigenvalue computation, the leading minor criterion (i.e. Sylvester’s criterion) or the Cholesky decomposition (cf. Horn and Johnson [@Horn_matrix]).
Theorems \[thm:Existence\] and \[thm:posdef\] can guarantee the existence of periodic orbits for arbitrary large forcing and response amplitudes. These theorems, therefore, enable an a priori justification of the use of otherwise heuristic approaches, such as harmonic balance or numerical collocation. In the following, we demonstrate the use of these theorems on various mechanical systems.
Examples
========
First, we illustrate via mechanically relevant examples that conditions \[cond:damping\]-\[cond:sign\_cond\] of Theorem \[thm:Existence\] cannot be omitted without replacement by some other requirement. Next, we identify a large class of high-dimensional mechanical systems for which the existence of periodic orbits can be guaranteed by Theorems \[thm:Existence\] and \[thm:posdef\].
The importance of condition \[cond:damping\]-\[cond:sign\_cond\]
----------------------------------------------------------------
In the following we show that if one of the conditions \[cond:damping\]-\[cond:sign\_cond\] is violated, one can find mechanical systems with no periodic orbits that nevertheless satisfy the remaining conditions of Theorem \[thm:Existence\]. This underlines the importance of conditions \[cond:damping\]-\[cond:sign\_cond\].
All solutions in a one-degree-of-freedom, undamped linear oscillator grow unbounded when the oscillator is forced at resonance. As a consequence, no periodic orbits may exists in such a system. Indeed, any undamped linear oscillator violates condition \[cond:damping\], because damping matrix is identically zero and therefore neither positive nor negative definite.
We have seen in section \[sec:counter\_ex1\] that system has no orbit. Indeed, trying to apply Theorem \[thm:Existence\] to this problem, we find that condition \[cond:potential\] is not satisfied. To examine condition \[cond:sign\_cond\], we evaluate the quantity $S_j(\mathbf{q})-\bar{f}_j$ along $q_2=0$. This parabola opens upward and is positive outside a closed interval, i.e., $$((k_1+k_2)q_1+\kappa q_1^2-\bar{f}_1)>0, \qquad q_1>\sqrt{\frac{|\bar{f}_1|}{\kappa}},~~\text{or} ~~ q_1<-\frac{k_1+k_2}{\kappa}-\sqrt{\frac{|\bar{f}_1|}{\kappa}}.$$ Therefore, the quantity $q_1((k_1+k_2)q_1+\kappa q_1^2-\bar{f}_1)$ is positive for all $q_1\!>\!\sqrt{\frac{|\bar{f}_1|}{\kappa}}$ and negative for $q_1\!<\!-\frac{k_1+k_2}{\kappa}-\sqrt{\frac{|\bar{f}_1|}{\kappa}}$. This implies that no constant $r$ exists such that $q_1((k_1+k_2)q_1-k_2q_2+\kappa(q_1^2+q_2^2)-\bar{f}_j)$ has a constant sign for all $|q_i|>r$, i.e. condition \[cond:sign\_cond\] is violated.
We now consider a slight modification of system in the form of $$\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}
\ddot{\mathbf{q}}
+
\begin{bmatrix}
c_1 & 0\\
0 & c_2
\end{bmatrix}
\dot{\mathbf{q}}
+
\begin{bmatrix}
\omega_1^2 & 0\\
0 &\omega_2^2
\end{bmatrix}
\mathbf{q}
+\kappa
\begin{bmatrix}
q_1q_2^2 \\
0
\end{bmatrix}
=
\begin{bmatrix}
f_1(t) \\
a\sin(\Omega t)
\end{bmatrix},
\label{eq:eqm_counter3}$$ which satisfies conditions \[cond:sign\_cond\], for the choice $$\kappa>0,~~\bar{f}_1 \leq 0,\quad \Rightarrow q_1(q_1(\omega_1^2+\kappa q_2^2)-\bar{f}_1)>0,~~~~\text{for all } q_1,q_2\in \mathbb{R}.
\label{eq:conv_cond_eval}$$ Condition \[cond:damping\] is also satisfied for positive damping values $c_1,c_2>0$. Condition \[cond:potential\], however, is not satisfied as the geometric nonlinearities of system do not derive from a potential. In the following, we will show that system has no $T$-periodic orbits for an appropriately chosen set of parameters.
Assuming the contrary, we consider a periodic solution $\mathbf{q}^*(t)$ and solve the second equation in to obtain $$q_2^*(t)=A(\omega_2,c_2,a,\Omega)\sin(\Omega t-\psi(\omega_2,c_2,a,\Omega)),
\label{eq:c3_q2tilde}$$ where the amplification factor $A(\omega_2,c_2,a)$ and the phase shift $\psi(\omega_2,c_2,a)$ are constants depending on the damping coefficient and eigenfrequency, as well as the forcing amplitude and frequency, as indicated. The exact form of $A$ and $\psi$ can be determined from linear theory (see, e.g., Géradin and Rixen [@Rixen]). Substituting equation into the first equation in , we obtain $$\ddot{q}_1^*+c_1\dot{q}_1^*+q_1^*(\omega_1^2+\frac{\kappa A^2}{2}-\frac{\kappa A^2}{2} \cos(2\Omega t -2 \psi))=f_1(t),
\label{eq:T_vary_stiffness}$$ which is a modification of classic forced-damped Matthieu equation (cf., Guckenheimer and Holmes [@G+H] for the undamped-unforced limit, or Nayfeh and Mook [@Nayfeh_Mook] for the unforced limit). Compared to the standard Matthieu equation, an additional term $q_1^*\kappa A^2/2$ arises in equation . For the unforced Matthieu equation ($f_1\!=\!0$), a change of stability of the trivial solution is commonly observed for various values of the damping, stiffness and forcing frequency. Utilizing this observation, we can prove the nonexistence of a periodic orbit in system with the following fact:
\[thm:PO\_tvary\] If the trivial solution of the system $$\ddot{q}_1^*+c_1\dot{q}_1^*+q_1^*( k_1+\frac{\kappa A^2}{2}-\frac{\kappa A^2}{2} \cos(2\Omega t -2 \psi))=0,
\label{eq:mat_unforced}$$ is unstable for some parameter values $a$, $\Omega$, $c_2>0$, $c_1>0$, $\omega_1$, $\omega_2$ and $\kappa$, then we can find a $T$-periodic forcing $f_1(t)$ satisfying condition , such that system has no periodic orbit.
The proof relies on the fact that a $T$-periodic solution to system does not exist if a non-trivial $T$-periodic solution exists in the homogeneous system and additional orthogonality conditions between the external forcing and non-trivial $T$-periodic solutions are violated (cf. Farkas [@Farkas_PO]). In Appendix \[app:T\_vary\], we show the existence of non-trivial $T$-periodic solutions and show that the orthogonality conditions are generally violated for appropriately chosen $f_1$.
We can use the above fact to establish the nonexsistence of a periodic orbit for system . To this end, we have to find a set of parameters for which the trivial solution of system is unstable. For the non-dimensional parameters $$\omega_2=1,~~c_1=c_2=0.01,~~\kappa=1,~~\Omega=1,
\label{eq:pars}$$ we calculate the monodromy matrix for the equilibrium at the origin using numerical integration, covering a parameter range for the forcing amplitude $a$ and the eigenfrequency $\omega_1$. We depict the result of the Floquet analysis performed on the monodromy matrix in Fig. \[fig:Stab\_map\], where we indicate the system configurations with stable trivial solution in green, while red indicates a system configuration with an unstable trivial solution. The critical system configurations can be found on the stability boundary, which is highlighted in black in Fig. \[fig:Stab\_map\]. As we prove in Appendix \[app:T\_vary\], for the black configurations, we can find a continuous $T$-periodic forcing $f_1$ such that the system , and hence system , has no periodic orbit.
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2.31963927855711 0.0498997995991984\
2.31963927855711 0.0498997995991984\
2.31182364729459 0.05\
]{};
Condition \[cond:sign\_cond\] requires the sign of the quantities $q_j(S_j(\mathbf{q})-\bar{f}_j)$ to be constant and non-zero for $|q_j|\!>\!r$. If the geometric nonlinearities minus the mean forcing has a constant sign outside the region $|q_j|\!>\!r$ for some degree of freedom ($\operatorname{sign}(S_j(\mathbf{q}) - \bar{\mathbf{f}}_j)=const.$ for $|q_j|\!>\!r$), then the quantities $q_j(S_j(\mathbf{q})-\bar{f}_j)$ evaluated for $q_j\!>\!r$ and for $q_j\!<\!-r$ have opposite sign. Therefore, condition \[cond:sign\_cond\] is violated. This is certainly the case if the geometric nonlinearities have a global minimum value and the mean forcing of a single coordinate is below that minimum value, i.e., $$S_j(\mathbf{q})>S_{\min}>\bar{f}_l, \quad j=1,...,N, \quad 1\leq l\leq N, \qquad \mathbf{q}\in\mathbb{R}^N.
\label{eq:global_min}$$ Then $\left(S_l(\mathbf{q}) - \bar{\mathbf{f}}_l\right)$ is always positive and $q_l\left(S_l(\mathbf{q}) - \bar{\mathbf{f}}_l\right)$ changes sign. For system with geometric nonlinearities and mean forcing satisfying , we have the following fact:
\[thm:global\_min\] If the geometric nonlinearities and the mean forcing of system satisfy the conditions , then no periodic orbit exists for system .
We detail this proof in Appendix \[app:global\_extrm\].
The conclusion of Fact also holds in the case of geometric nonlinearities with global maxima $S_{max}$ and a mean forcing larger than $S_{max}$, i.e. for systems satisfying, $$S_j(\mathbf{q})<S_{\max}<\bar{f}_l, \quad j=1,...,N, \quad 1\leq l\leq N \qquad \mathbf{q}\in\mathbb{R}^N.
\label{eq:global_max}$$ hold.
An example for a nonlinearity satisfying and is the simple pendulum, whose geometric nonlinearity $S(q)\!=\!c_p\sin(q)$ has global maximum and minimum value. Therefore, the damped forced-pendulum $$\ddot{q}+c\dot{q}+c_p\sin(q)=f(t),\qquad |\bar{f}|>|c_p|,
\label{eq:pendulum}$$ has no $T$-periodic solution.
The previous example indicates that the mean value of the forcing plays a critical role in the existence of periodic orbits. One might wonder if a zero-mean restriction of the forcing, as in the theorem of Rouche and Mawhin [@Rouche_PO] (cf. Theorem \[thm:RM\] in Appendix \[app:Exist\_proof\]), allows relaxing some of our conditions, notably condition \[cond:sign\_cond\]. In the following example, we show that even for zero-mean forcing ($\bar{f}_j=0$) condition \[cond:sign\_cond\] cannot be relaxed.
We consider the nonlinear oscillator $$\ddot{q}+c\dot{q}+\omega^2q+\kappa q^2=f \cos(\Omega t),
\label{eq:eqm_x^2}$$ the simplest example with geometric nonlinearities violating condition \[cond:sign\_cond\]. As forcing, we choose simple single harmonic forcing with amplitude $f$. For system , we have then the following fact:
\[thm:xsprt\_NL\] If the forcing amplitude $f$ for mechanical system is above the threshold $$|f|>\frac{ \omega^2}{ 2|\kappa|} \left(|-\Omega^2+\mathrm{i}c\Omega+\omega^2|+ 2\omega^2 \right)+|\kappa|\frac{\omega^4}{4\kappa^2},
\label{eq:c4_f_thres}$$ then no $T$-periodic solution to system exist.
We detail the proof in Appendix \[app:xsprt\_NL\] .
Therefore, choosing any forcing amplitude exceeding the threshold will necessarily rule out the existence of a periodic orbit.
Examples with periodic orbits guaranteed by Theorem \[thm:Existence\].
----------------------------------------------------------------------
In the following, we give examples in which Theorem \[thm:Existence\] guarantees the existence of a periodic response. Since the damping condition \[cond:damping\] and the assumption \[cond:potential\] on the geometric nonlinearities derived from a potential are simple to verify, we focus on condition \[cond:sign\_cond\] and \[cond:conv\_cond\]. We start with the classic Duffing oscillator and proceed with higher-dimensional examples.
The forced-damped Duffing oscillator is simple harmonic oscillator with an additional cubic nonlinearity added, i.e. $$\ddot{q}+c\dot{q}+\omega^2q +\kappa q^3=f \cos(\Omega t),
\label{eq:duffing}$$ where we have chosen single harmonic forcing with amplitude $f$ and frequency $\Omega$.
For $\kappa\!\geq \!0$, the potential of eq. is positive definite for all $q$ (cf. Fig. \[fig:Duf\_pot\_pos\]). Therefore, condition \[cond:conv\_cond\] is trivially satisfied. Furthermore, condition is satisfied for arbitrarily small radius $r$, which can be used for the upper bound on the amplitudes . Therefore, both Theorems \[thm:Existence\] and \[thm:posdef\] apply and guarantee the existence of a periodic solution without any numerics. We compute periodic responses with the automated continuation package <span style="font-variant:small-caps;">coco</span> [@coco] and show the amplitudes in Fig. \[fig:Duf\_FRF\_pos\].
[0.49]{}
(-1,2) – (5,2) node\[below\][$q$]{}; (2,1) – (2,4.5);
plot ([2+]{}, [2+(0.1\*+0.1\*)]{} )node\[right\][$qS(q)$]{};
plot ([2+]{}, [2+0.1+0.1\*3\*]{} ) node\[right\];
plot ([2+]{}, [2+(0.1/2\*+0.1/4\*)]{} ) node\[right\];
[0.49]{}
table\[row sep=crcr\][ 0.690523210767485 0.952337939006846\
nan nan\
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0.740720652509768 0.971631990923166\
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0.681348297080721 0.94949717529313\
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6.63839898395265 7.74221443674985\
6.63849719810129 7.74223885939222\
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4.93476335413663 0.00428227050993613\
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1.07492200563165 0.446616666096186\
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1.03703758379492 0.193877711726034\
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1.27623059663951 0.0158937329391242\
1.35699051589863 0.0118762184436263\
1.58555501388757 0.00660264359273821\
2.07713987571579 0.00301665040320787\
3 0.00124994850400695\
]{};
For negative values of the coefficient $\kappa$, the Hessian of the potential is not globally positive (cf. Fig \[fig:Duf\_pot\_neg\] green curve). However, outside the ball of radius $r^*\!=\! \omega\sqrt{-1/(3\kappa)}$, the second derivative of the potential is negative. Therefore, the existence of a periodic orbit is guaranteed by Theorem \[thm:posdef\]. Furthermore, two nontrivial equilibria arise at $q_0=\omega\sqrt{-1/\kappa}$ (cf. Fig \[fig:Duf\_pot\_neg\], black curve). If we select this $q_0$ as the radius $r$ in Theorem \[thm:Existence\], then condition \[cond:sign\_cond\] is satisfied. Again, both Theorems \[thm:Existence\] and \[thm:posdef\] guarantee the existence of a periodic response.
We numerically continue the trivial and the two nontrivial periodic orbits for increasing forcing amplitude at the fixed forcing frequency $\Omega\!=\!1$ and show the amplitudes of the periodic response in Fig. \[fig:Duf\_bifi\_neg\]. For larger forcing amplitudes, numerical continuation becomes more challenging, yet our results continue to imply the existence of a periodic response rigorously.
[0.49]{}
(-1,2) – (5,2) node\[above\][$q$]{}; (2,1) – (2,4.5);
plot ([2+0.75\*]{}, [2+(1\*-0.1\*\^3)]{} ); (2+0.75\*2.5,2+2.2361+0.2) node\[above\][$qS(q)$]{};
plot ([2+0.75\*]{}, [2+1-0.1\*3\*]{} ) node\[below\];
plot ([2+0.75\*]{}, [2+(1/2\*-0.1/4\*)]{} )node\[right\];
(2+0.75\*3.1623,2) – (2+0.75\*3.1623,1.8)node\[below\][$r$]{}; (2-0.75\*3.1623,2) – (2-0.75\*3.1623,1.8)node\[below\][$-r$]{};
(2+0.75\*1.8257,2) – (2+0.75\*1.8257,1.8) node\[below\][$r^*$]{}; (2-0.75\*1.8257,2) – (2-0.75\*1.8257,1.8)node\[below\][$-r^*$]{};
[0.49]{}
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\[ex:osci\] We consider the $N$-dimensional oscillator chain depicted in Fig. \[fig:Osci\_chain\]. Two adjacent masses are connected via nonlinear springs and linear dampers. The first and last mass are suspended to the wall. The equation of motion of the $j$-th mass is given by $$m_j\ddot{q}_j-c_j(\dot{q}_{j-1}-\dot{q}_{j})+c_{j+1}(\dot{q}_{j}-\dot{q}_{j+1})-S_{j}(q_{j-1}-q_{j})+S_{j+1}(q_{j}-q_{j+1})=f_j(t), \qquad j=1,...,N,
\label{eq:eqm_chain}$$ where we set the coordinates $q_0$ and $q_{N+1}$ to zero. Variants of such systems have been investigated by Shaw and Pierre [@SHAW+Pierre], Breunung and Haller [@TB_Backbone] and Jain et al. [@NPO]. Physically, the system may represent, e.g., a discretized beam. As we detail in Appendix \[app:chain\_sys\], the following fact holds for the chain system :
\[thm:chain\_sys\] For positive masses, damping coefficients and hardening spring stiffnesses, i.e. $$m_j>0,\qquad c_j>0, \qquad \frac{\partial S_{j}(\delta)}{\partial \delta}>0 , \qquad j=1,...,N+1,
\label{eq:pars_chain}$$ system satisfies the conditions of Theorem \[thm:Existence\] and hence must have a steady-sate response.
=\[thick,decorate,decoration=[zigzag,pre length=0.3cm,post length=0.3cm,segment length=6]{}\] =\[thick,decoration=[markings, mark connection node=dmp, mark=at position 0.45 with [ (dmp) \[thick,inner sep=0pt,transform shape,rotate=-90,minimum width=8pt,minimum height=3pt,draw=none\] ; ($(dmp.north east)+(5pt,0)$) – (dmp.south east) – (dmp.south west) – ($(dmp.north west)+(5pt,0)$); ($(dmp.north)+(0,-3pt)$) – ($(dmp.north)+(0,3pt)$); ]{} ]{}, decorate\]
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($(M1.south) + (-0.5,-0.125)$) circle (0.125); ($(M1.south) + (0.5,-0.125)$) circle (0.125); ($(M1) - (0.5,0)$) – ($(M1) + (0.5,0)$) node\[midway,below\][$f_1$]{}; ($(M1.north) + (-0.5,0.1)$) – ($(M1.north) + (0.5,0.1)$) node\[midway,above\][$q_1$]{};
(LBs) at (2.25,-0.25); (LBd) at (2.45,0.25); plot \[smooth\] coordinates [(2.35,-0.6) (LBs) (LBd) (2.35,0.6)]{} ; ($(M1.east) - (0,0.25)$) – (LBs) node \[midway,below\] [ $S_{2}$]{}; (1.05,-0.4)– (1.85,-0.05) ;
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(RBs)–($(M2.west) - (0,0.25)$) node \[midway,below\] [$S_{N}$]{}; (2.95,-0.4)– (3.75,-0.05) ; (RBd) – ($(M2.west) + (0,0.25)$) node \[midway,above\] ;
($(M2.south) + (-0.5,-0.125)$) circle (0.125); ($(M2.south) + (0.5,-0.125)$) circle (0.125); ($(M2) - (0.5,0)$) – ($(M2) + (0.5,0)$) node\[midway,below\][$f_N$]{}; ($(M2.north) + (-0.5,0.1)$) – ($(M2.north) + (0.5,0.1)$) node\[midway,above\][$q_N$]{};
at (7.275,-0.125) (RW) \[minimum width=0.25cm, minimum height=1.75cm\] ;
($(M2.east) - (0,0.25)$) – ($(RW.west) - (0,0.125)$) node \[midway,below\] [ $S_{N+1}$]{}; (5.95,-0.4)– (6.75,-0.05) ; ($(M2.east) + (0,0.25)$) – ($(RW.west) + (0,0.375)$) node \[midway,above\] ;
(-2.25,0.75)–(-2.25,-0.75)–(7.15,-0.75)–(7.15,0.75);
at (2.45,-0.875) (BW) \[minimum width=9.4cm, minimum height=0.25cm\] ;
The systems investigated by Shaw and Pierre [@SHAW+Pierre], Breunung and Haller [@TB_Backbone], Jain et al. [@NPO] satisfy the conditions , as the stiffness forces are of the form $S_j(\delta)\!=\!k_j \delta+\kappa_j \delta^3$ with $k_j\!>\!0$ and $\kappa_j\!\geq\!0$. Therefore, we can guarantee the existence of the periodic response of these systems for arbitrary large forcing amplitudes.
In the derivations in Appendix \[app:chain\_sys\], we further detail that the assumptions on the parameters can be relaxed to include either the cases $c_{N+1}\!=\!0$ and $S_{N+1}(q_{N+1})\!=\!0$ or $c_1\!=\!0$ and $S_1(q_1)\!=\!0$. In both cases, the damping matrix $\mathbf{C}$ and the second derivative of the potential remain positive definite. The conditions on the first and the $N+1$-th damping coefficient and stiffness force cannot be relaxed simultaneously. In the case of $S_1(q_1)=S_{N+1}(q_N)\!=\!0$, the system is not connected to the walls and hence a non-periodic, free rigid body motion of the whole chain can be initiated with appropriate forcing. For $c_1\!=\!c_{N+1}\!=\!0$, this motion is undamped and hence if the springs are linear, then forcing at resonance cannot result in a periodic response.
Conclusions
===========
We have discussed the example of a specific mechanical system for which the application of the harmonic balance procedure leads to the wrong conclusion about the existence of a periodic response. This underlines the necessity of rigorous existence criteria for periodic orbits in damped-forced nonlinear mechanical systems. Such existence criteria can give a priori justification for the use of formal perturbation methods and numerical continuation, eliminating erroneous conclusions or wasted computational resources.
To obtain such an existence criterion, we have extend a theorem by Rouche and Mawhin [@Rouche_PO] to obtain generally applicable sufficient conditions for the existence of a periodic response in periodically forced, nonlinear mechanical systems. Roughly speaking, these conditions guarantee a periodic orbit under arbitrarily large forcing and response amplitudes, as long as the dissipation acts on all degrees of freedom, the spring forces are potential, and the potential function is strictly convex or strictly concave outside a neighborhood of the origin.
Since the conditions of our theorem are sufficient but not necessary, the question arises whether they can be relaxed. With mechanically relevant examples, we have illustrated that none of the conditions in our theorem can be individually omitted while keeping the others. Based on these results, we identify a large class of nonlinear mechanical systems for which numerical procedures, such as the harmonic balance and the collocation method, are a priori justified. This enables the reliable computation of periodic orbits for large forcing and oscillation amplitudes in this class of systems.
Theorem \[thm:Existence\] guarantees the existence of a periodic orbit but gives not immediate conclusion about the stability of the orbit. For positive definite damping, we do observe both stable and unstable periodic orbits numerically (c.f. Fig \[fig:Duf\_pot\_neg\]) when the conditions of Theorem \[thm:Existence\] hold.
We have limited our discussion to periodic forcing, for which extensive mathematical literature exists. Quasi-periodic forcing is also of interest in engineering applications; indeed, the harmonic balance method has been extended to compute quasi-periodic steady states response of nonlinear mechanical systems (cf. Chua and Ushida [@Chua_QPHB]). The extension of the present results to quasi-periodic forcing, however, is not immediately clear.
Our discussion is restricted to mechanical equations of motions with position depended nonlinearities, as it is customary in the structural vibrations literature. It is also of interest, however, to extend our conclusions to velocity-dependent nonlinearities.
#### Acknowledgements.
[We are thankful to Florian Kogelbauer and Walter Lacarbonara for fruitful discussion on this work. ]{}
#### Conflict of Interest.
[ The authors declare that they have no conflict of interest. ]{}
#### Funding.
[We recieved no funding for this study.]{}
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Proofs of the main theorems
============================
In the following, we prove the main Theorems \[thm:Existence\] and \[thm:posdef\] and derive an upper bound on the amplitudes of the steady-state responses.
Proof of Theorem \[thm:Existence\] {#app:Exist_proof}
----------------------------------
We base our proof of Theorem \[thm:Existence\] on a Theorem by Rouche and Mawhin [@Rouche_PO], who analyze systems of the following form $$\ddot{\mathbf{q}}+\bar{\mathbf{C}}\dot{\mathbf{q}}+\frac{\partial \bar{V}(\mathbf{q})}{\partial \mathbf{q}}=\mathbf{g}(t), \qquad \mathbf{g}(t)=\mathbf{g}(t+T),~~\bar{V}\in C^1.
\label{eq:RM_sys}$$
\[thm:RM\] Assume system satisfies the following conditions:
1. The damping matrix $\bar{\mathbf{C}}$ is positive or negative definite. \[cond:damping\_RM\]
2. There exists a distance $r\!>\!0$ and an integer $1\!\leq \!n\!\leq N$ such that $$\begin{split}
q_j\frac{\partial \bar{V}(\mathbf{q})}{\partial q_j} &>0, \qquad |q_j|>r,~~j=1,...,n,
\\
q_j\frac{\partial \bar{V}(\mathbf{q})}{\partial q_j}&<0, \qquad |q_j|>r,~~j=n+1,...,N.
\end{split}
\label{eq:sign_cond_RM}$$ \[cond:sign\_RM\]
3. The forcing $\mathbf{g}$ is continuous with zero mean value, i.e., $$\bar{\mathbf{g}}=\frac{1}{T}\int_{0}^{T}\mathbf{g}(t)dt=\mathbf{0} .$$ \[cond:forcing\_RM\]
Then system has at least one $T$-periodic solution.
The proof relies on a homotopy of equation to the equation $\ddot{\mathbf{q}}=0$. Conditions \[cond:damping\_RM\]-\[cond:forcing\_RM\] ensure a bound on the solution for all homotopy parameters. In addition, condition \[cond:sign\_RM\] ensures a non-zero Brouwer degree, i.e., the existence of at least one T-periodic solution during the homotopy. For a detailed proof, we refer to Rouche and Mawhin [@Rouche_PO].
We transform system such that it is in the form and then show that the conditions of Theorem \[thm:Existence\] imply that Theorem \[thm:RM\] applies. First, we absorb the mean forcing into the potential by setting $$\tilde{V}(\mathbf{q})=V(\mathbf{q})-\mathbf{q}^T\bar{\mathbf{f}}, \qquad \tilde{\mathbf{f}}=\mathbf{f}-\bar{\mathbf{f}}.
\label{eq:mod_pot}$$ The equation of motion with nonlinearity derived from the potential $\tilde{V}$ and forcing $\tilde{\mathbf{f}}$ is equivalent to system . Further, we right-multiply equation with the inverse of the mass matrix $\mathbf{M}$ and obtain $$\ddot{\mathbf{q}}+\mathbf{M}^{-1}\mathbf{C}\dot{\mathbf{q}}+\mathbf{M}^{-1}\frac{\partial \tilde{V}}{\partial \mathbf{q}}=\mathbf{M}^{-1}\mathbf{\mathbf{f}}(t).
\label{eq:eqm_RPO}$$ The potential for the geometric nonlinearities of system is given by $\bar{V}(\mathbf{q})\!:=\!\tilde{V}(\mathbf{M}^{-1}\mathbf{q})$. Therefore, system can be rewritten in the form . Since the mass matrix is positive definite by assumption, the product $\bar{\mathbf{C}}\!:=\!\mathbf{M}^{-1}\mathbf{C}$ is positive or negative definite. Therefore, condition \[cond:damping\] of Theorem \[thm:Existence\] implies that condition \[cond:damping\_RM\] of Theorem \[thm:RM\] is satisfied.
Rewriting condition \[cond:sign\_cond\] for the the potential $\bar{V}\!:=\!\tilde{V}(\mathbf{M}^{-1}\mathbf{q})$, one recovers the equivalent condition \[cond:sign\_RM\]. Since $\tilde{\mathbf{f}}$ has zero mean, condition \[cond:forcing\_RM\] holds. Therefore, the conditions in Theorem \[thm:Existence\] ensure that Theorem \[thm:RM\] applies, and hence the existence of a periodic orbit can be guaranteed.
Maximal amplitude of the periodic response {#app:bound_ampl}
------------------------------------------
Essential to the proof of Rouche and Mawhin [@Rouche_PO] is an upper bound on the periodic solution of equation . In the following, we show that this can be obtained for system in its original form. Therefore, a transformation to equation is not necessary. We derive an upper bound on the solutions of $$\mathbf{M} \ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{S}(\mathbf{q})= \mathbf{f}(t),~\qquad \mathbf{q}\in C^2(T),
\label{eq:sys0_lams}$$ in the $C^0$ norm defined by $$||\mathbf{q}||_{C^0}=\max_{0\leq t\leq T}\left|\mathbf{q}\right|.
\label{eq:def_norm}$$ First, we follow the derivation by Rouche and Mawhin [@Rouche_PO] by left-multiplying equation with $\dot{\mathbf{q}}^T$ and integrating over one period to obtain $$\int_0^T\dot{\mathbf{q}}^T\mathbf{M}\ddot{\mathbf{q}}~dt+
\int_0^T\dot{\mathbf{q}}^T\mathbf{C}\dot{\mathbf{q}}~dt
+
\int_0^T\dot{\mathbf{q}}^T\mathbf{S}(\mathbf{q})~dt
= \int_0^T\dot{\mathbf{q}}^T\mathbf{f}(t)~dt.
\label{eq:eqm_times_qdot}$$ Observing that $$\int_0^T\dot{\mathbf{q}}^T\mathbf{M}\ddot{\mathbf{q}}~dt=
\int_0^T \frac{d}{dt}\left(\frac{1}{2} \dot{\mathbf{q}}^T\mathbf{M}\dot{\mathbf{q}}\right) dt=0,$$ where we have used the symmetry of the mass matrix ($\mathbf{M}=\mathbf{M}^T$) and the periodicity of $\mathbf{q}$. Similarly $$\int_0^T\dot{\mathbf{q}}^T\mathbf{S}(\mathbf{q})~dt=
\int_0^T\frac{d}{dt}\left(V(\mathbf{q})\right)dt=0,$$ where we have used the fact that the geometric nonlinearities arise from a potential and again the periodicity of $\mathbf{q}$. Therefore, from equation , we obtain $$\left|\int_0^T \dot{\mathbf{q}}^T\mathbf{C}\dot{\mathbf{q}} ~dt\right|=\left|\int_0^T\dot{\mathbf{q}}^T \mathbf{f}~ dt \right|.
\label{eq:vel_est}$$ With the assumption of of a positive or negative definite $\mathbf{C}$ matrix (cf. equation ), we obtain a lower bound on the left hand side of equation to $$C_0\int_0^T |\dot{\mathbf{q}}|^2 dt\leq \left|\int_0^T \dot{\mathbf{q}}^T\mathbf{C}\dot{\mathbf{q}} dt\right|.
\label{eq:vel_est_LHS}$$ For the right hand side of equation , we obtain an upper bound by using the Cauchy-Schwartz inequality $$\left|\int_0^T\dot{\mathbf{q}} \mathbf{f} ~dt\right|\leq
\left(\int_0^T |\dot{\mathbf{q}} |^2 ~dt\right)^{1/2} \left(\int_0^T |\mathbf{f} |^2 ~dt\right)^{1/2}.
\label{eq:vel_est_RHS}$$ Using the definition of $C_f$ and combining the estimates and , we obtain from equation , that $$\left(\int_0^T |\dot{\mathbf{q}} |^2 ~dt\right)^{1/2}\leq \frac{C_f}{C_0}.
\label{eq:ydot_sqrt}$$ Equation is and upper bound on the $L_2$-norm of the velocity of the periodic orbit. Rouche and Mawhin [@Rouche_PO] derive the same bound. Now we depart from the derivations by Rouche and Mawhin [@Rouche_PO] and integrate system for one period, which yields $$\int_0^T \mathbf{S}(\mathbf{q})~dt=T\bar{\mathbf{f}},\quad \Leftrightarrow \quad \int_0^T \left(\mathbf{S}(\mathbf{q}) -\bar{\mathbf{f}}\right)dt=0,
\label{eq:eqm_intT}$$ where we used the definition of the mean forcing $\bar{\mathbf{f}}$. Applying the mean-value theorem to , we conclude that there exist $t_j$ such that $$S_j(\mathbf{q}(t_j))-\bar{f}_j=0,\qquad 0\leq t_j\leq T,\qquad j=1,...,N.
\label{eq:eqm_MVTHM}$$ From condition , we conclude that equation is only satisfied if $|q_j(t_j)|<r$. We conclude $$\begin{split}
q_j^2(t)&=\left(q_j(t_j)+\int_{t_j}^t\dot{q}_j(s)ds\right)^2=
q_j(t_j)^2+2q_j(t_j)\int_{t_j}^t\dot{q}_j(s)ds+\left(\int_{t_j}^t\dot{q}_j(s)ds\right)^2
\\
&\leq r^2+2r|t-t_j|^{\frac{1}{2}}\left(\int_{t_j}^t\dot{q}_j^2(s)ds\right)^{1/2}+
|t-t_j|\int_{t_j}^t\dot{q}_j^2(s)ds
\\&\leq r^2+2r\sqrt{T}\left(\int_{0}^T\dot{q}_j^2(s)ds\right)^{1/2}+
T\int_{0}^T\dot{q}_j^2(s)ds\leq \left(r+\sqrt{T}\frac{C_f}{C_0}\right)^2,
\label{eq:qj^2}
\end{split}$$ where we have used the upper bound . Therefore, for the $C^0$-norm of the positions, we obtain $$||\mathbf{q}||_{C^0} < \left(\sum_{j=1}^N \sup_{0\leq t \leq T} (q_j^2(t)) \right)^{1/2}\leq \sqrt{N}\left(r+\sqrt{T} \frac{C_f}{C_0} \right) ,
\label{eq:pos_C0_bound}$$ In contrast, Rouche and Mawhin [@Rouche_PO] use the bound to obtain an upper estimate on the oscillatory part of the position $\tilde{\mathbf{q}}:=\mathbf{q}-1/T\int_0^T\mathbf{q}\,dt$ to $||\tilde{\mathbf{q}}||_{C^0}\leq TC_f/C_0$. From equation , they directly obtain that each component of the mean $\bar{\mathbf{q}}:=1/T\int_0^T\mathbf{q}\,dt$ is bounded by $r+TC_f/C_0$. Adding the mean and oscillatory part, Rouche and Mawhin [@Rouche_PO] derive the bound $$||\mathbf{q}||_{C^0} \leq \sqrt{N}\left(r+\sqrt{T} \frac{C_f}{C_0} \right) +\frac{TC_f}{C_0},
\label{eq:pos_C0_RM}$$ which includes the additional summand $TC_f/C_0$ compared to our bound .
Proof of Theorem \[thm:posdef\] {#app:convex}
-------------------------------
In the following, we show that condition \[cond:conv\_cond\] implies that condition \[cond:sign\_cond\] is satisfied. We note that each continuous function $S_j(\mathbf{q})$ has a maximum and a minimum value in a ball of radius $r^*$, which we label with $S_{\max}^j$ and $S_{\min}^j$. Choosing the radius $$r_j=r^*+\max(0,\frac{\bar{f}_j-S^j_{min}}{C_v},\frac{S_{\max}^j-\bar{f}_j}{C_v}),
\label{eq:r_estimate}$$ ensures that the quantity $q_j(S_j(\mathbf{q})-\bar{f}_j)$ has a constant, non-zero sign for all $$\mathbf{q}\in\mathbb{Q}_j:=\{\mathbf{q}\in\mathbb{R}^N|~|q_j|> r_j \}.$$ First, we assume a positive definite Hessian outside a ball of radius $r^*$. Using a Taylor series expansion of the nonlinearity, we note that outside the $r^*$ ball the following holds: $$\mathbf{h}^T(\mathbf{S}(\mathbf{q}+\mathbf{h})-\mathbf{S}(\mathbf{q}))
=\int_0^1 \mathbf{h}^T \frac{\partial^2 V (\mathbf{q}+s\mathbf{h})}{\partial \mathbf{q}^2}\mathbf{h} ds>C_v|\mathbf{h}|^2,\quad \mathbf{q},\mathbf{h}\in \mathbb{R}^N,\quad 0\leq s \leq 1,\quad|\mathbf{q}+s\mathbf{h}|>r^*.
\label{eq:pos_def_V}$$ For every point $\mathbf{q}\!\in\!\mathbb{Q}_j$, we select $\mathbf{h}$ to be the vector pointing from the $q_j$-axis to $\mathbf{q}$ with minimal length. Denoting the $j$-th unit vector by $\mathbf{e}_j$, we set $\mathbf{h}\!=\!\mathbf{q}-q_j\mathbf{e}_j$ and $\mathbf{q}=q_j\mathbf{e}_j$. Since $|q_j|\!>\!r^*$, line connecting $q_j\mathbf{e}_j$ and $\mathbf{q}$ is in the region, where the potential $V(\mathbf{q})$ is positive definite. From equation we obtain $$(\mathbf{q}-q_j\mathbf{e}_j)^T(\mathbf{S}(\mathbf{q})-\mathbf{S}(q_j\mathbf{e}_j))=
\sum_{\begin{array}{c}
n=1
\\
n\neq j
\end{array}}^N
q_n\left[S_n(\mathbf{q})-S_n(q_j\mathbf{e}_j)\right]>0.
\label{eq:r_est_notej}$$ Further, we reduce the $j$-th coordinate until we reach $|q_j|=r^*$. The line connecting between the points $\operatorname{sign}(q_j)r^*\mathbf{e}_j$ and $q_j\mathbf{e}_j$ lies in the region with a positive definite Hessian. We evaluate for $\mathbf{q}=\operatorname{sign}(q_j)r^*\mathbf{e}_j$ and $\mathbf{h}\!=\!\mathbf{q}-\operatorname{sign}(q_j)r^*\mathbf{e}_j$ to obtain $$\begin{split}
&(\mathbf{q}-\operatorname{sign}(q_j)r^*\mathbf{e}_j)^T(\mathbf{S}(\mathbf{q})-\mathbf{S}(\operatorname{sign}(q_j)r^*\mathbf{e}_j)\\ =
&\sum_{\begin{array}{c}
n=1
\\
n\neq j
\end{array}}^N
q_n\left[S_n(\mathbf{q})-S_n(q_j\mathbf{e}_j)\right]+
(q_j-\operatorname{sign}(q_j)r^*)\left[S_j(\mathbf{q})-S_j(\operatorname{sign}(q_j)r^*\mathbf{e}_j)\right]
\\
&>(q_j-\operatorname{sign}(q_j)r^*)\left[S_j(\mathbf{q})-S_j(\operatorname{sign}(q_j)r^*\mathbf{e}_j)\right]
>C_v(q_j-\operatorname{sign}(q_j)r^*)^2,\qquad \mathbf{q}\in \mathbb{Q}_j.
\label{eq:r_estimate_ej}
\end{split}$$ For $q_j\!>\!0$, equation implies that $(q_j-\operatorname{sign}(q_j)r^*)$ is positive. Therefore, we obtain from equation that $$S_j(\mathbf{q})>C_v(q_j-r^*)+S_j(r^*\mathbf{e}_j)>
C_v(q_j-r^*)+S^j_{\min}>
C_v(r^*+\frac{\bar{f}_j-S_{\min}^j}{C_v}-r^*)+S_{\min}^j=\bar{f}_j,\qquad q_j>r_j.
\label{eq:r_estimat_qjpos}$$ Similarly, for $q_j\!<\!0$ the quantity $S(q_j-\operatorname{sign}(q_j)r^*)$ is negative, therefore equation implies $$S_j(\mathbf{q})<C_v(q_j+r^*)+S_j(-r^*\mathbf{e}_j)<C_v(q_j+r^*)+S_{\max}^j<C_v(-r^*-\frac{S_{\max}^j-\bar{f}_j}{C_v}+r^*)+S_{\max}^j=\bar{f}_j,\qquad q_j^*<-r_j.
\label{eq:r_estimat_qjneg}$$ Equations and together imply $$q_j(S_j(\mathbf{q})-\mathbf{f}_j)>0,\qquad \mathbf{q}\in\mathbb{Q}_j,
\label{eq:final_cond}$$ which is equivalent to the upper condition , if we set $r:=\max_j(r_j)$.
The same argument can be repeated for potentials having a negative definite Hessian. The sign in equation changes, therefore one obtains $$q_j(S_j(\mathbf{q})-\mathbf{f}_j)<0,\qquad \mathbf{q}\in\mathbb{Q}_j,
\label{eq:final_cond2}$$ which is equivalent to the lower condition , if we set $r:=\max_j(r_j)$.
Derivations for specific examples
=================================
Necessary bound on the forcing amplitude for system {#app:C1}
-----------------------------------------------------
In the following, we prove a necessary bound on the forcing amplitude for the existence of periodic solutions for system with parameters . Specifically, we assume the existence of a twice continuous differentiable periodic orbit $\mathbf{q}^*$. Transforming system to modal coordinates, we obtain
$$q_1^*=x_1+x_2,\qquad q_2^*=x_1-x_2,$$
$$\ddot{x}_1+c_1\dot{x}_1+k_1 x_1 +2\kappa x_1^2=-2\kappa x_2^2,
\label{eq:c1_quad_dof}$$
$$\ddot{x}_2+c_1\dot{x}_2+(k_1+2k_2) x_2 =f_1.
\label{eq:c1_lin_dof}$$
The equation of motion of the second modal degree-of-freedom is linear and therefore the assumed periodic response of the second degree of freedom $x_2$ can be obtained analytically: $$\begin{split}
x_2&=\frac{8f_m}{\pi^2}\sum_{k=0}^{\infty} \frac{(-1)^k\sin((2k+1)\Omega t-\varphi_k)}{(2k+1)^2\sqrt{((k_1+2k_2)-(2k+1)^2\Omega^2)^2+((2k+1)c_1\Omega)^2}}=\frac{8f_m}{\pi^2}\sum_{k=0}^{\infty} c_k\sin((2k+1)\Omega t-\varphi_k) ,
\\
\varphi_k&=\tan^{-1}\left( \frac{(2k+1)c_1\Omega}{(k_1+2k_2)-(2k+1)^2\Omega^2}\right),
\end{split}
\label{eq:x2_sol}$$ Here we have relabeled the amplitudes for notational convenience. Next, we integrate over one period and impose periodicity to obtain $$\int_0^T \left(k_1 x_1+2 \kappa x_1^2 \right) dt= -2 \kappa \int_0^T x_2^2 dt,
= - T \kappa \frac{64f_m^2}{\pi^4} \sum_{k=0}^{\infty}|c_k|^2.
\label{eq:c1_mean_val_thm}$$ The infinite sum converges to the limit $c_{\infty}$, since it can be majorized by $1/k^6$, i.e. $$\begin{split}
c_{\infty}:=\sum_{k=0}^{\infty}|c_k|^2&=
\sum_{k=0}^{\infty}\frac{1}{(2k+1)^4(((k_1+2k_2)-(2k+1)^2\Omega^2)^2+((2k+1)c_1\Omega)^2)}
\\
&\leq\frac{1}{c_1^2\Omega^2}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^6}\leq\frac{1}{c_1^2\Omega^2}\sum_{k=1}^{\infty}\frac{1}{k^6}.
\end{split}$$ For the parameters , we compute the value $c_{\infty}$ numerically and obtain $$c_{\infty}:= 1371.7577441>1371.757744027918.$$ By the mean-value theorem applied to equation , there must be a time instance $t^*\!\in \! \left[ 0,T\right]$ at which the integrand on the left-hand side multiplied by $T$ is equal to the infinite sum on the right hand side. Calculating the minimum of the parabola in that integrand and inserting the numerical parameter values yields $$-\frac{k_1^2}{8\kappa}T\leq \left(k_1\tilde{x}_1(t^*)+2 \kappa \tilde{x}_1^2(t^*) \right)T=-\kappa T\frac{64f_m^2}{\pi^4}c_{\infty}, \qquad 0\leq t^*<T.
\label{eq:bound_fm}$$ Solving for the forcing amplitude, we obtain $$|f_m|<\sqrt{\frac{k_1^2\pi^4}{512 \kappa^2 c_{\infty}}}=0.011777,\qquad \kappa>0.
\label{eq:f_thres}$$ Since the forcing amplitude is above the threshold , the periodic orbit indicated by the harmonic balance method does not exist.
Failure of the harmonic balance with infinite harmonics {#app:fejer}
-------------------------------------------------------
In the following we construct a forcing for the linear system , such that even for infinite number of harmonics in ansatz , the harmonic balance procedure yields a periodic orbit that differs from the actual periodic orbit significantly. Generally speaking, the computability of a finite number of terms in a Fourier series of a periodic solution does not guarantee the pointwise convergence of that series to the periodic orbit. We consider the function $$f_f=\sum_{k=1}^{K}\frac{2}{k^2}\sin(p_k t)\sum_{l=1}^{q_k}\frac{1}{l}\sin(l t),\qquad p_k=2^{k^3+1},\quad q_k=2^{k^3},
\label{eq:fejer_forcing}$$ which is a truncated version of a classic example due to Fejér (c.f. Edwards [@Edwards]). We note that the function is analytic and therefore the forcing $$f(t)=\ddot{f}_f+c\dot{f}_f+kf_f,
\label{eq:fejer_forcing2}$$ is well-defined. Applying this forcing in system , we obtain the periodic orbit in the form $q^*\!=\!f_f$. The harmonic balance procedure, therefore, produces a Fourier series of the function . As Edwards [@Edwards] details, the function $f_f$ can be bounded from above by a constant independent of $K$, while its Fourier series at $t\!=\!0$ is unbounded for $K\!\rightarrow \!\infty$. Therefore, for large enough $K$ the Fourier series of $f_f$ will deviate from the function at $t\!=\!0$. Choosing an appropriately large $K$ leads to a large deviation of the approximative periodic orbit obtained by the harmonic balance from the unique periodic orbit of system with forcing . Therefore, the harmonic balance fails to approximate the periodic orbit at $t\!=\!0$.
Proof of Fact \[thm:PO\_tvary\] {#app:T_vary}
--------------------------------
In the following, we show that no periodic orbit for system exists, for an appropriately chosen set of parameters. For these sets of parameters, one of the Floquet multipliers of the unforced limit of system equals to one in norm. This introduces the possibility of resonance between the external periodic forcing and the nontrivial solution of the homogeneous part , under which no periodic orbit for system can exist.
For further analysis, we introduce the matrices and vectors $$\mathbf{x}:=
\begin{bmatrix}
q_1^*\\
\tilde{q}^*
\end{bmatrix},\quad
\mathbf{A}(t):=
\begin{bmatrix}
0 & 1\\
-k_1-\frac{\kappa A^2}{2}+\frac{\kappa A^2}{2} \cos(2\Omega t -2 \psi) & -c_1
\end{bmatrix},\quad
\mathbf{g}(t):=
\begin{bmatrix}
0\\
f_1(t)
\end{bmatrix}.
\label{eq:At_sys_notation}$$ With the notation , we express system in first-order form $$\dot{\mathbf{x}}=\mathbf{A}(t)\mathbf{x}+\mathbf{g}(t),\qquad \mathbf{A}(t+T/2)=\mathbf{A}(t),\qquad T=2\pi/\Omega,
\label{eq:At_sys}$$ and denote its homogeneous part by $$\dot{\mathbf{x}}=\mathbf{A}(t)\mathbf{x}.
\label{eq:At_sys_hom}$$ Furthermore, we define the adjoint problem, $$\dot{\mathbf{y}}=-\mathbf{A}(t)^T\mathbf{y}.
\label{eq:At_adj}$$
To show the non-existence of a periodic orbit of system , we use the following Theorem:
\[thm:PO\_Farkas\] Assume that system has $k$ linearly independent, nontrivial $T$-periodic solutions and denote $k$ linearly independent$T$-periodic solutions to the adjoint system by $\tilde{\mathbf{y}}_1$, $\tilde{\mathbf{y}}_2$,..., $\tilde{\mathbf{y}}_k$. Then the non-autonomous system has a $T$-periodic solution if and only if the orthogonality conditions $$\int_0^T\tilde{\mathbf{y}}_j^T\mathbf{g}(t)dt=0,\qquad j=1,...,k,
\label{eq:ortho}$$ hold.
For a proof, we refer to Farkas [@Farkas_PO].
First, we note system is periodic with period $T/2$ (cf. equation ), where $T$ is determined by the external forcing $f_2$ (cf. equation ). We denote the complex conjugate Floquet multipliers of system by $\rho_1$ and $\rho_2$ and further obtain from Liouvilles theorem that $$\rho_1\rho_2=e^{\int_{t_0}^{t_0+T/2} \operatorname{Tr}\left[ \mathbf{A}(s)\right] ds}=e^{-\frac{c_1T}{2}}, \qquad \rho_1=\bar{\rho}_2.
\label{eq:Flo_mult}$$ Equation imply that the Floquet multipliers are located either on the circle with radius $e^{-c_1T/4}$ (red circle in Fig. \[fig:Flo\_mult\]) or on the real axis (blue line in Fig. \[fig:Flo\_mult\]) in the complex plane.
If the forcing $f_2$ is zero, then the parameter $A$ in system is zero and, due to the positive damping value $c_1$, the trivial solution of system stable. Therefore, the Floquet multipliers are located on the red circle in Fig. \[fig:Flo\_mult\]. If we observe instability of the trivial solution to eq. for some non-zero forcing ($A\!\neq\! 0$), then the Floquet multipliers must have crossed the unit circle in the complex plane. In this critical case, one of the Floquet multipliers is either one or negative one, which we mark with a black square in Fig. \[fig:Flo\_mult\].
(-1,2) – (5,2) node\[above\][$\mbox{Re}(\rho)$]{}; (2,-0.5) – (2,4.5)node\[left\][$\mbox{Im}(\rho)$]{}; (2,2) circle (2cm); (2,2) circle (2.25cm); (-0.75,2) – (1.95,2); (2.05,2) – (4.75,2) ;
(4.2,1.95) rectangle ++ (0.1,0.1); (4.25,2) node\[below\]; (4.25,2)– (4.25,1.8) ;
(-0.3,1.95) rectangle ++ (0.1,0.1); (-0.35,2) node\[below\]; (-0.25,2)– (-0.25,1.8);
(2,2) – (3.41421,3.41421) node \[midway, above, sloped\] [$e^{-\frac{c_1T}{4}}$]{};
If one of the multipliers, $\rho_1$, is one, there exists a non trivial $T/2$-periodic solution of the homogeneous part of system . In the case of a Floquet multiplier of negative one, a non-trivial $T$-periodic solution exists (cf. Farkas [@Farkas_PO]). As Farkas details further, in both cases, the adjoint system has a non-trivial $T/2$ or $T$-periodic solution, which we denote by $\tilde{\mathbf{y}}$. Analyzing equation , we conclude that a nontrivial $\tilde{\mathbf{y}}$ implies a non-constant value of both coordinates $\tilde{y}_1(t)$ and $\tilde{y}_2(t)$. We choose the forcing $$f_1(t)=
\begin{cases}
\tilde{y}_2(t),& \text{if } \int_0^T\tilde{y}_2(t) dt<0,
\\
-\tilde{y}_2(t) & \text{if } \int_0^T\tilde{y}_2(t) dt \geq 0,
\end{cases}
\label{eq:f1_forcing}$$ which satisfies the negative mean-forcing requirement . Further, note that the forcing is $T/2$ periodic for the case $\rho_1=1$ and $T$ periodic in the case $\rho_1=-1$. Then the orthogonality condition is $$\int_0^{T} \tilde{\mathbf{y}}\mathbf{g}(t)dt= \pm \int_0^{T} \tilde{y}_2^2dt\neq 0,
\label{eq:orto_cond}$$ where the sign depends on the choice of the forcing . Clearly, the orthogonality condition is not satisfied and therefore, by theorem \[thm:PO\_Farkas\] system , has no periodic solution.
Proof of Fact \[thm:global\_min\] {#app:global_extrm}
---------------------------------
In the following, we show that no periodic orbit for system exists if the geometric nonlinearities possess a global minimum, and the mean forcing is below this minimum value (i.e., equation is satisfied). To prove the nonexistence of a $T$-periodic orbit, we proceed as in Appendix \[app:C1\], assuming the existence of a twice differentiable periodic orbit $\mathbf{q}^*$ for system . Integrating equation for one period and imposing periodicity yields $$\int_0^T \mathbf{S}( \mathbf{q}^*(t)) dt=T\bar{\mathbf{f}},\qquad \Leftrightarrow \qquad \int_0^T \left( \mathbf{S}(\mathbf{q}^*(t)) -\bar{\mathbf{f}} \right)dt=0 .
\label{eq:c2_int}$$ By the mean-value theorem, there exist time instances $t^*_j$ within the period at which the integrand in equation is equal to zero, i.e., $$S_j(\mathbf{q}^*(t^*_j))-\bar{f}_j=0,\qquad j=1,...,N,\quad 0\leq t_j\leq T.
\label{eq:mvt_S}$$ However, due to the choice of the forcing , we obtain for $j\!=\!l$ that $$S_l(\mathbf{q}^*(t^*))-\bar{f}_l>0,$$ which contradicts . Therefore, the periodic orbit cannot exist.
Proof of Fact \[thm:xsprt\_NL\] {#app:xsprt_NL}
-------------------------------
In the following, we prove that if the forcing amplitude $f$ in the oscillator is above the threshold , then no periodic solution to system exists. Again, we assume the existence of a twice continuous differentiable periodic orbit $q^*$ and split the coordinate $q^*$ into a constant and a purely oscillatory part, i.e. $$\bar{q}:=\frac{1}{T}\int_0^Tq^*(t)dt,\qquad \tilde{q}(t)=q^*-\bar{q}.
\label{eq:q_split}$$ Substituting the definitions into the equation of motion , yields $$\ddot{\tilde{q}}+c\dot{q}+\omega_0^2(\bar{q}+\tilde{q})+\kappa (\bar{q}^2+2\bar{q}\tilde{q}+\tilde{q}^2) =f\cos(\Omega t).
\label{eq:eqm_qt}$$ Integrating equation over one period, we obtain $$\int_0^T \tilde{q}^2dt =-T\left(\frac{\omega^2}{\kappa}\bar{q}+\bar{q}^2\right)\leq\frac{T\omega^4}{4\kappa^2},
\label{eq:qt_L2_bnd}$$ where we have used that $\tilde{q}$ has zero mean (cf. definition ). Furthermore, we note that the left-hand side of is positive. Since the right-hand side of equation is a parabola which is concave downwards, it is positive on a closed interval. We thus obtain the upper bound on $\bar{q}$ in the form $$|\bar{q}|<\frac{\omega^2}{|\kappa|},
\label{eq:x0_bound}$$ which is independent of the sign of $\kappa$. Since $q^*$ is twice continuously differentiable, it can be expressed in a convergent Fourier series. We denote the Fourier coefficients of $\tilde{q}$ by $$\tilde{q}^k:=\frac{1}{T}\int_0^Tq_t e^{-\mathrm{i} k \Omega t} dt,~\qquad k\in\mathbb{Z}.
\label{eq:xt_Fseries}$$ Using Parseval’s identity and equation , we obtain an upper bound on the Fourier coefficients of the assumed periodic orbit as follows $$|\tilde{q}^k|\leq\left(\sum_{k\in \mathbb{Z}}|\tilde{q}^k|^2\right)^{1/2}=\left( \frac{1}{T} \int_0^T \tilde{q}^2dt\right)^{1/2}\leq \frac{ \omega^2}{2|\kappa|},\qquad k\in \mathbb{Z}.
\label{eq:xt_bound}$$ Multiplying equation with $e^{-\mathrm{i}\Omega t}$ and integrating over one period yields $$\int_0^T(\ddot{\tilde{q}}+c\dot{\tilde{q}}+\omega_0^2\tilde{q}+2\kappa\bar{q}\tilde{q}) e^{-i \Omega t} dt+\int_0^T\kappa \tilde{q}^2 e^{-i \Omega t} dt=\frac{f}{2}.
\label{eq:F_entw}$$ From equation , we obtain $$\begin{split}
\left|\frac{f}{2}\right|
&\leq\left|\int_0^T(\ddot{\tilde{q}}+c\dot{\tilde{q}}+\omega_0^2\tilde{q}+2\kappa\bar{q}\tilde{q}) e^{-i \Omega t} dt\right|+|\kappa|\int_0^T |\tilde{q}^2(t)||e^{-i\Omega t}| dt
\\
&\leq |(-\Omega^2+\mathrm{i}c\Omega+\omega^2+2\kappa\bar{q})\tilde{q}^1|+|\kappa|\frac{\omega^4}{4\kappa^2}
\leq
\frac{ \omega^2}{ 2|\kappa|} \left(|-\Omega^2+\mathrm{i}c\Omega+\omega^2|+ 2\omega^2 \right)+|\kappa|\frac{\omega^4}{4\kappa^2},
\end{split}
\label{eq:c4_contracition}$$ where we have used the upper bounds and . Equation gives an upper bound for the forcing amplitude $f$ of the oscillator . For forcing amplitudes exceeding this threshold, we obtain a contradiction and therefore no periodic orbit can exist for the oscillator .
Proof of Fact \[thm:chain\_sys\] {#app:chain_sys}
--------------------------------
We show that the chain system with the parameters satisfies the conditions of Theorem \[thm:Existence\] and hence a steady-state response exists. First, we show that the conditions \[cond:potential\] and \[cond:conv\_cond\] on the geometric nonlinearities are satisfied for the set of parameters . The definiteness of the damping matrix (i.e. condition \[cond:damping\]) can be shown in a fashion similar to the definiteness of the Hessian.
As for condition \[cond:potential\], the spring forces of system can be derived from the potential $$V(\mathbf{q})=\int_0^{q_1}S_{1}(-p) dp +\sum_{j=2}^{N} \int_0^{q_{j-1}-q_j} S_{j}(p) d p+\int_0^{q_N}S_{N+1}(p) dp.
\label{eq:pot_chain}$$ Since the spring forces in of system are continuous by assumption, the integrals in equation exist. With the notation $$S_{j,l}:=\frac{\partial }{\partial q_l}\left(S_j(q_{j-1}-q_j)\right),$$ the Hessian of the potential is given by $$\small
\mathbf{H}:=\frac{\partial^2 V(\mathbf{q})}{\partial \mathbf{q}^2}=
\begin{bmatrix}
-S_{1,1}+S_{2,1} & S_{2,2}& 0&& &
\\
-S_{2,1}& -S_{2,2}+S_{3,2} & S_{3,3}& 0 &&
\\
0& -S_{3,2}& -S_{3,3}+S_{4,3} &S_{4,4}& 0 &
\\
&0&\ddots &\ddots&\ddots &0
\\
& &0 &-S_{N\!-\!1,N\!-\!2} & -S_{N\!-\!1,N\!-\!1} + S_{N,N\!-\!1} &S_{N,N}
\\
& & &0& -S_{N,N\!-\!1} & -S_{N,N}+S_{N\!+\!1,N}
\end{bmatrix}.
\label{eq:chain_hessian}$$ Due to the choice of parameters , we have following identities $$S_{j,j}<0,\qquad S_{j+1,j}>0,\qquad S_{j,j}=-S_{j,j-1},$$ which implies that the main diagonal entries of the Hessian are positive and the off-diagonal elements negative. We define the matrices $$\mathbf{H}^j=\begin{bmatrix}
-S_{1,1}-S_{2,2} & S_{2,2}& 0&& &
\\
S_{2,2}& -S_{2,2}-S_{3,3} & S_{3,3}& 0 &&
\\
0& S_{3,3}& -S_{3,3}-S_{4,4} &S_{4,4}& 0 &
\\
&0&\ddots &\ddots&\ddots &S_{j,j}
\\
& & &0& S_{j,j} & -S_{j,j}
\end{bmatrix}\in\mathbb{R}^{j\times j},
\label{eq:Hj}$$ which are equivalent to the leading minors of the Hessian, except for the last term in the main diagonal where the term $-S_{j+1,j}$ is missing. Therefore, $\mathbf{H}^N$ is not equal to $\mathbf{H}$. The matrices $\mathbf{H}^j$ can be constructed recursively as follows $$\mathbf{H}^1=-S_{1,1},
\quad
\mathbf{H}^{j+1}=
\begin{bmatrix}
\mathbf{H}^j & \mathbf{0}
\\
\mathbf{0}&0
\end{bmatrix}
+
\begin{bmatrix}
\mathbf{0} & \mathbf{0} &\mathbf{0}
\\
\mathbf{0} & -S_{j,j} &S_{j,j}
\\
\mathbf{0} & S_{j,j} &-S_{j,j}
\end{bmatrix}.$$ We show that the matrices $\mathbf{H}^j$ are positive definite by induction. As a first step, we note that $\mathbf{H}^1$ is positive definite. Performing the induction step, we have $$\mathbf{x}^T\mathbf{H}^{j}\mathbf{x}=
\mathbf{x}^T \begin{bmatrix}
\mathbf{H}^{j-1} & \mathbf{0}
\\
\mathbf{0}&0
\end{bmatrix}
\mathbf{x}+
\mathbf{x}^T
\begin{bmatrix}
\mathbf{0} & \mathbf{0} &\mathbf{0}
\\
\mathbf{0} & -S_{j,j} &S_{j,j}
\\
\mathbf{0} & S_{j,j} &-S_{j,j}
\end{bmatrix}
\mathbf{x}.
\label{eq:induc_step}$$ Since the matrix $\mathbf{H}^{j-1}$ is positive definite, the first summand in is always positive unless $\mathbf{x}$ aligns with the $x_j$-axis, i.e. $x_1\!=\!x_2\!=\!...\!=\!x_{j-1}\!=\!0$. Along this axis the first quadratic form is zero, the second quadratic form, however, yields $-S_{j,j}x_j^2$ which is positive. For the case $x_j\!=\!0$ and $|\tilde{\mathbf{x}}|\!=\!|\left[x_1,\dots,x_j-1\right]^T|\!>\!0$, we obtain $$\tilde{\mathbf{x}}^T\mathbf{H}^{j-1} \tilde{\mathbf{x}}+\mathbf{x}^T
\begin{bmatrix}
\mathbf{0} & \mathbf{0} &\mathbf{0}
\\
\mathbf{0} & -S_{j,j} &S_{j,j}
\\
\mathbf{0} & S_{j,j} &-S_{j,j}
\end{bmatrix}
\mathbf{x}
\geq \tilde{\mathbf{x}}^T\mathbf{H}^{j-1} \tilde{\mathbf{x}},\quad |\tilde{\mathbf{x}}|>0,$$ where we have used the fact, that the matrix in the second quadratic form in equation is positive semi definite. Mering both cases $$\mathbf{x}^T\mathbf{H}^{j}\mathbf{x}\geq
\left\{
\begin{array}{l l l}
-S_{j,j} x_j^2 >0, & |\tilde{\mathbf{x}}|=0, & |x_j|>0,
\\[3mm]
\tilde{\mathbf{x}}^T\mathbf{H}^{j-1} \tilde{\mathbf{x}} >0, & |\tilde{\mathbf{x}}|>0, & x_j=0,
\end{array}
\right.$$ which implies positive definiteness of all matrices $\mathbf{H}^j$. Since the Hessian can be written as the sum of the positive definite matrix $\mathbf{H}^N$ and a positive semidefinite the matrix, i.e. $$\mathbf{H}=\mathbf{H}^N+
\begin{bmatrix}
\mathbf{0} & \mathbf{0}\\
\mathbf{0} & -S_{N+1,N}
\end{bmatrix},
\label{eq:add_last}$$ we conclude that the Hessian positive definite.
Since the damping matrix is in the form of the Hessian , the positive definiteness proof applies for the damping matrix as well. Therefore, we have verified the remaining condition \[cond:damping\] of Theorem \[thm:Existence\], and the existence of a periodic orbit is guaranteed by Theorem \[thm:Existence\].
We note that in the case of $S_{N+1,N}\!=\!0$, the Hessian $\mathbf{H}$ coincides with the matrix $\mathbf{H}^N$, which is positive definite. Therefore, the assumptions on the parameters can be relaxed to include this case.
[^1]: Corresponding author, E-mail: brethoma@ethz.ch, Phone: +41 44 633 83 56
|
[**Kolmogorov Complexity, String Information, Panspermia and the Fermi Paradox** ]{}
V.G.Gurzadyan
Dipartimento di Fisica, Università “La Sapienza", Rome, and Yerevan Physics Institute
[**Abstract**]{} - Bit strings rather than byte files can be a mode of transmission both for intelligent signals and for travels of extraterrestrial life. Kolmogorov complexity, i.e. the minimal length of a binary coded string completely defining a system, can then, due to its universality, become a key concept in the strategy of the search of extraterrestrials. Evaluating, for illustration, the Kolmogorov complexity of the human genome, one comes to an unexpected conclusion that a low complexity compressed string - analog of Noah’s ark - will enable the recovery of the totality of terrestrial life. The recognition of bit strings of various complexity up to incompressible Martin-Löf random sequences, will require a different strategy for the analysis of the cosmic signals. The Fermi paradox “Where is Everybody?” can be viewed under in the light of such information panspermia, i.e. a Universe full of traveling life streams.
Introduction
============
Diametrically opposing views on the existence of advanced civilizations are mainly due to the uncertainty, to what extent features of our civilization can be attributed to extraterrestrials. For example, why cannot a civilization be developed on the basis of elementary particles in entangled quantum states instead of atoms or molecules? Various strategies for the search of advanced civilizations range from detection of electromagnetic signals [@CM] to the search of physical artefacts in our vicinity [@RW].
My discussion is based on outlining the information aspect of the carriers of life. The basic idea is to send a file containing all the information on a system, up to such complex systems as human beings. The amount of information then, of course, can be quite big. The size of the package will decrease drastically if, instead of sending a file (byte sequences), one sends the program (bit strings) which is able to recover it. Then, the Kolmogorov complexity, which is the minimal length of the program defining a system [@K], due to its universality will act as the quantitative descriptor of the messages.
The recent progress in the deciphering of the human genome is used below to illustrate these ideas (Thus far, sequences of three mammalian genomes are studied with reasonable accuracy: the complete human genome sequence was published in 2003 (the draft in 2001), the draft genomes of the mouse and rat were published in 2002 and 2004, respectively [@gen]). Evaluating the Kolmogorov complexity of the human genome, I arrive at an unexpected conclusion that low complexity strings will enable the complete recovering of the terrestrial life. If true, the methods of analysis of bit strings in the radiation arrived from space have to become an important goal.
Kolmogorov complexity of the human genome
=========================================
Kolmogorov complexity, $K$, is defined as the minimal length of a binary coded program (in bits) which is required to describe the system $x$, i.e. will enable recovery of the initial system $x$ completely [@K]: $$K(\phi(p),x) =min_{p:\phi=x} l(p),$$ where $\phi(p,x)$ is a recursive i.e. algorithmically calculable function, $l(p)$ is the length of the program $p$.
The universality of the Kolmogorov complexity is due to the proof by Kolmogorov that the complexities defined by various Turing machines [@K] differ by no more than an additive constant $C$ $$|K(\phi(p),x) - K(x\mid y)|\leq C,$$ where the conditional complexity $K(x\mid y)$ of object $x$ when the complexity of another object $y$ is known, is $$K(x\mid y)=min\,\ l(p).$$ The amount of information of object $x$ with respect to an object $y$ is evaluated from the complexities $$I(y:x)=K(x)- K(x\mid y).$$ Complexities obtained by different algorithms differ from the asymptotic (minimal) one by another additive constant. In other words, a system can be recovered from a compressed string almost independently on the computer and the algorithm.
Obviously, a repeat or periodic string has a low complexity, and it can be compressed more compactly than the chaotic one with random sequences. The precise complexity is usually unreachable for physical complex systems; however a value not too different from it can be estimated, as for example, for the maps of Cosmic Microwave Background radiation [@G].
Let us estimate the complexity of the human genome. The human genome [@gen] contains 2.9 $10^9$ base pairs, those of the rat and mouse contain 2.75 $10^9$ and 2.6 $10^9$ pairs, respectively. The number of predicted genes is about 23.000 for the human and over 22.000 for the mouse genome. About 99% of mouse genes are similar to the human ones, and of these 96% have the same location for the mouse and human genome, 80% of mouse gene (orthologues) are also the best match for human gene. Therefore, the complexity of the human genome has to be $$K<10^{11},
\label{k}$$ with correspondingly smaller value for the code carrier part. Only less than 1% of mouse proteins are specific to mice, 99% per cent are shared with other mammals, and 98% are shared with the humans, while 27% are common to all animals and 23% to all species, including bacteria. Another feature of the mammalian genomes is the existence of repeat sequences. Namely, 46% of the human genome and 38% of the mouse genome consists of recognizable interspersed repeats while only 2% are the coding genes. The complexity of a string of length N is limited by $K(x)<N$, while the fraction of sequences with $$K<N-m$$ is small if $m$ is sufficiently large, as it is for the human genome. The chromosomes of the three studied organisms, 23, 21 and 20 pairs for human, rats and mice, respectively, are related to each other by 280 large regions of sequence similarity. So, the conditional complexity of terrestial species is small once the complexity of human genome is known. (The similarities in the genomes are not only the quantitative indications for the common ancestors but also for the time periods of the divergence from the common path of evolution.)
The energy, E, required to communicate $B$ bits of information is $E=BkT\, \ln2$, where $T$ is the temperature of the noise per bandwidth. A lower bound for the energy to transmit $B$ bits by an antenna can be evaluated by the formula [@RW] $$E=8\, \ln2\, BS (\frac{D}{A})^2\simeq 10^6 (B/10^{11})(d/1pc)^2(R/150m)^{-4}\, erg.
\label{E}$$ where $D$ is the communication distance, $d$, in units of the antenna’s aperture, $A$ is the antenna’s aperture, $R$, in units of the transmission wavelength, and $S$ is the noise spectral energy density. The Arecibo aperture, $R=150m$, and 3K antenna temperature is used for the normalization, so it is seen that larger antennae will enable the coverage of the Galaxy and even other galaxies within energy limits reasonable for our civilization in foreseen future.
Thus, the complexity of genomes of terrestrial organisms due to repeat sequences and common fractions is comparable to the human one and the resulting package can be transmitted to galactic distances.
Network of von Neumann automata
===============================
The self-reproduction of information carriers is an efficient strategy for spreading over the Universe. A simplified example of such strategy can include sending N self-reproducing von Neumann cellular automata [@vN], as suggested by Tipler [@T]. The automata would create duplicates of their own from the environment upon arrival at the destination, and send them in other N directions. At a speed $0.001$c, an automaton will arrive to the nearest star situated 1pc away in $\tau=3,000$ years, and the time of creation of automata network within the Galaxy will be $$10^4 \tau = 30\, 000\, 000\, years$$ At a speed $0.01c$, the Universe within the radius $10^{26}$ cm would be reached in the Hubble time.
Once the network of von Neumann automata is created within Galaxy in such cosmologically short time scale, the transmission of information packages i.e. packed travellers, can be as commonplace events as air travels on the Earth today. I do not discuss many traditionally debated issues, such as whether the civilizations would have other alternatives to the expansion in space, etc.
Conclusion
==========
I advance the idea of propagation of the life not via files containing the information on them, but the programs, i.e. coded strings defined by Kolmogorov complexity. Considering the Kolmogorov complexity of the human genome, I have shown that low-conditional-complexity strings are enough for the complete recovery of terrestrial life. The complexity of the coded strings, the existence of random sequences in the sense of Martin-Löf closely related to Hausdorff dimensionality, will require new methods in the studies of the cosmic signals, and can eventually approach the solution to the Fermi paradox "\`Where is Everybody?" [@W].
[99]{}
G.Cocconi, P.Morrison, Nature, [**184**]{}, 844,1959. C.Rose, G.Wright, Nature, [**431**]{}, 47, 2004. A.N.Kolmogorov, Probl.Information Transfer, [**1**]{}, 3, 1965; Li M., Vitanyi P., [*An Introduction to Kolmogorov Complexity and Its Applications*]{} (Springer, Heidelberg), 1997; A.Shen, Comp.J. [**42**]{}, 340, 1999. R.H.Waterstone et al, Nature [**420**]{}, 520, 2002; Nature, [**428**]{}, 493, 2004. V.G.Gurzadyan, Europhys.Lett., [**46**]{}, 114, 1999; in: [*The Physics of Communication*]{}, Proc.XXII Solvay Conference on Physics, Eds.I.Antoniou, V.A.Sadovnichy, H.Walther, World Sci., p.204, 2003. J. von Neumann, The General and Logical Theory of Automata, in: [it Cerebral Mechanisms in Behaviour]{} (The Hixon Symposium) (John Wiley, New York), 1951. F.Tipler, QJRAS, 21, 267, 1980. S.Webb [*Where Is Everybody?: Fifty Solutions to the Fermi Paradox and the Problem of Extraterrestrial Life*]{} (Springer, Heidelberg), 2002.
|
---
abstract: 'Recently, large-scale cascading failures in complex systems have garnered substantial attention. Such extreme events have been treated as an integral part of the self-organized criticality (SOC). Recent empirical work has suggested that some extreme events systematically deviate from the SOC paradigm, requiring a different theoretical framework. We shed additional theoretical light on this possibility by studying financial crisis. We build our model of financial crisis on the well-known forest fire model in scale-free networks. Our analysis shows a non-trivial scaling feature indicating supercritical behavior, which is independent of system size. Extreme events in the supercritical state result from bursting of a fat bubble, seeds of which are sown by a protracted period of a benign financial environment with few shocks. Our findings suggest that policymakers can control the magnitude of financial meltdowns by keeping the economy operating within reasonable duration of a benign environment.'
author:
- Deokjae Lee
- 'Jae-Young Kim'
- Jeho Lee
- 'B. Kahng'
title: Forest Fire Model as a Supercritical Dynamic Model in Financial Systems
---
Large-scale cascading failures have garnered attention in many complex systems, such as power grids and communication networks [@buldyrev_catastrophic_2010; @jacobson_congestion_1988; @hancock_cell_2010; @lee_branching_2012], because once they happen, their impact can be unexpectedly catastrophic. A case in point is the crippling blow to the world economy preceded by the failure of an investment bank, Lehman Brothers, and the subsequent financial meltdown with the evaporation of more than \$10 trillion from the global equity market [@stiglitz_freefall:_2010]. In the past, such an extreme event was treated as an integral part of the self-organized criticality [@bak_self-organized_1987; @scheinkman_self-organized_1994; @jensen_self-organized_1998], which is characterized by a power-law distribution. Partly due to the scarcity of extreme events, few suspected the possibility that some of them could systematically deviate from a power-law distribution. Recently, however, researchers have begun to consider extreme events as supercritical phenomena, characterizing extreme events as distinguishable by their sizes from the rest of the statistical population [@sornette_why_2004; @sornette_dragon-kings:_2012]. The objective of our work is to shed additional light on such supercritical behavior by studying financial meltdown.
We build our model of financial crisis on the existing forest fire (FF) model introduced by Drossel and Schwabl [@drossel_self-organized_1992; @malamud_forest_1998; @jagla_forest-fire_2013], because it captures two essential features in financial meltdown. First, its non-conservative ingredient naturally mimics financial meltdown, where asset prices tend not to be conserved. When an asset market collapses, traders have difficulty pricing assets, as was the case in the collapse of the mortgage-backed securities market on the eve of the 2008 Financial Crisis. The assets that were previously considered liquid become illiquid, causing chronic problems for banks with speculative bets on these assets. The upshot is that an important quantity, the value of assets, will not be conserved over time. Second, there exists a separation of two time scales. It takes a long time for banks to build up a fat bubble, which is represented by a percolation cluster consisting of counterparties of vulnerable banks that make speculative bets on risky assets. In contrast, the meltdown of this cluster takes place very quickly as trees burn up in a short time.
Here, we model the FF dynamics in scale-free networks, which are employed to capture entangled counterparty relationships among banks worldwide. For example, on the eve of the 2008 financial crisis, Lehman alone was counterparty to almost a million derivatives contracts and a huge borrower in the repo market, and its zillions of derivative and repo contracts connected the bank to numerous counterparties all over the world [@blinder_after_2013]. Our analysis shows a non-trivial scaling feature indicating supercritical behavior, which is independent of system size. Prior research on the FF model did not detect this supercritical behavior [@drossel_self-organized_1992; @schenk_finite-size_2000; @grassberger_critical_2002; @pruessner_broken_2002]. We are able to detect it because it becomes more pronounced and conspicuous in scale-free networks, where the percolation threshold vanishes when the degree exponent is between two and three.
{width=".9\textwidth"}
[*Model:*]{} Building on [@drossel_self-organized_1992], we model the contagion of financial crisis through an inter-banking network of size $N$, which is represented by a scale-free network with the degree distribution $P_d(k) \sim k^{-\gamma}$ [@lu_complex_2006; @cho_percolation_2009]. It is known to be ubiquitous, and empirical research suggests that an inter-banking network can be approximated by a scale-free network [@boss_network_2004; @soramaki_topology_2007]. In the inter-banking network, each node represents a bank or bank-like firm, whereas a link between two nodes represents a counterparty relationship. A bank may lend money to its counterpart bank or invest in its financial products or assets. When one bank defaults on some debt, this event can leave its counterpart creditors or investors dangerously short of funds. To shed some meaningful light on the dynamics of such a complex system, our model focuses only on cascading bank failures in the inter-banking network. Defaults of non-financial firms or individuals are treated as external shocks to the system.
The dynamics of the FF model in the inter-banking network is defined as follows: Each node can be in one of the two states: vulnerable or healthy, which corresponds to a tree-occupied state or an empty state. In the vulnerable state, the node has insufficient cash reserves and is susceptible to financial shock. In the healthy state, the bank has enough cash or liquid assets on hand to meet depositors’ (or creditors’) demands, and is resilient to financial shocks. Initially all nodes are healthy, and the following steps are repeated: i) a randomly chosen node becomes vulnerable; ii) a randomly chosen node experiences a shock with a probability of $1/\theta$. If the chosen node is vulnerable, the whole cluster of vulnerable nodes containing the chosen node fails, and all the failed nodes become healthy. This approach to modeling of financial contagion differs from typical epidemic models, where healthy individuals are susceptible to infection from infected individuals. Actually such contact processes are supposed to exist but are ignored in the FF model because their time scale is too short compared with that of growing trees. We call the number of nodes in the failed cluster the avalanche size. The probability distribution of avalanche sizes, which is denoted as $P_s(s)$, is our primary interest (Fig. \[Fig-SchematicIllustration\] A,B).
[*Implication to financial systems:*]{} The parameter $\theta$ controls the average duration between two successive external shocks (two successive instances of lightening in the context of forest fire), which may be interpreted as the availability of liquidity in a financial system. In the model, the extreme events result from bursting of bubbles, seeds of which are sown by economic expansion with few shocks for a long period, which corresponds to the case when $\theta$ is large. That is, as banks do not experience defaults on their loans, more and more banks get involved in transactions of risky assets with many other counterparties, building up an extremely fat bubble. Historically, the fragility of the financial system has been increased by long periods of easy access to money, during which defaults on loans were infrequent [@roubini_crisis_2011; @krugman_end_2013].
After the expansion with easy money, the moment arrives for a dramatic reversal of the expansion$-$this is now known as a Minsky moment in the financial community [@minsky_can_1984; @minsky_stabilizing_2008]. Usually an external shock, such as sudden increases in interest rates, acts as a wakeup call of a financial meltdown. Assets that were previously considered liquid become illiquid and values of risky assets are heavily devaluated. Banks with imprudent practices can no longer borrow money from the inter-banking money market at a reasonable cost and fail. The devaluation and the propagation of failures induce each other amplifying the meltdown [@minsky_can_1984; @minsky_stabilizing_2008; @krugman_return_2009; @roubini_crisis_2011]. On the other hand, in the FF model, there are no locally conserved “carriers” of vulnerability such as sand grains in the sandpile model [@bak_self-organized_1987]. The failure of a node simply causes failures of all vulnerable nodes connected to the failed node. This is a simplified version of the real situation in which vulnerability is amplified by the collapse of asset markets and the subsequent evaporation of liquidity. The non-conservative nature of the FF model is an essential ingredient for the supercritical behavior because conventional conservative avalanche models such as sandpile model do not exhibit supercriticality in regular lattices or scale-free networks [@goh_sandpile_2003].
In the FF model, the separation of two time scales, the periods of expansion and meltdown, seems to be reasonable for modeling financial crisis. After the Minsky moment, a failure of one vulnerable bank tends to trigger a financial meltdown. Since the timescale for such a meltdown in reality is much smaller than that for expansion [@stiglitz_freefall:_2010], we describe a financial meltdown as a series of bank failures occurring in one time step.
In the next time step, the failed nodes become healthy again. An interpretation of this rule is that the failed banks are refinanced through government bailouts or acquisitions by other actors. In reality, failed banks may also be dissolved, or new banks may enter the system, and the bank network evolves. However, after a transient period of the evolution which is rather short compared with the interval between two successive financial crises, the network should be still scale-free and have similar statistical properties with the previous one. Thus, statistical properties of the FF model on such dynamic networks can be obtained by repeated simulations in an ensemble of scale-free networks.
![(Color online) (A) Probability distributions of avalanche sizes for various $q$. The tails of the distributions systematically deviate from a power-law behavior. (B) Probability distributions of avalanche sizes for various degree exponent values $\gamma$. Simulations were run on scale-free networks with exponent $\gamma=2.5$ for (A), containing $N=10^7$ nodes and $L=10N$ links for (A) and (B). []{data-label="Fig-MainResult"}](Fig2.eps){width=".9\linewidth"}
![(Color online) (A) Plot of data collapse in small-size avalanche region and (B) intermediate-size avalanche region for different system sizes $N$. The parameter $q$ is fixed as $10^{-4}$. The underlying networks are scale-free networks with degree exponent $\gamma=2.5$. (A) indicates the crossover point $s_{c1}$ between the first and second region scales as $N^{0.53}$ and (B) indicates the other crossover point $s_{c2}$ between the second region and third region scales as $N$. []{data-label="Fig-Scaling"}](Fig3.eps){width=".92\linewidth"}
[*Avalanche size distribution:*]{} The simulation results of $P_s(s)$ for various $q \equiv \theta/N$ are shown in Fig. \[Fig-MainResult\]A. They show that a parameter $q$, the availability of liquidity relative to the system size, positively affects the magnitude of the large-scale financial meltdown. When $q$ is sufficiently small, the size distribution decays in a power law-like manner. In this case, external shocks are frequent, small-scale avalanches are more likely, and large-scale avalanches are less likely. In contrast, when $q$ is large, the distribution exhibits a supercritical behavior and can be characterized in three distinct regions: i) in the first region, the size distribution decays at a rate close to a power-law; ii) in the second region, a bump exists whose pattern can be described by an increasing power-law function, called the supercritical region; iii) in the third region, the distribution tails off sharply.
In the model, a protracted period without shock allows the development of bubbles, which are represented by a giant cluster of complex transactional relationships among vulnerable banks. This is equivalent to the giant cluster in a percolation theory [@cohen_percolation_2002] (Fig. \[Fig-SchematicIllustration\]C). The failure of one bank in a giant cluster causes the failure of the whole cluster. It is a large-scale financial meltdown in the model. The size distribution in the first region is due to the failures of banks in finite-sized clusters as well as giant clusters, whereas the distribution in the second region stems from the failures of banks in giant clusters only.
We examine the sensitivity of our key findings to a change in the degree exponent $\gamma$, which controls the degrees of mega banks. For all levels of $\gamma$ from $2.1$ to $5$, supercritical behavior is apparent (Fig. \[Fig-MainResult\]B). We also run simulations on regular lattics, in which mega banks are outright absent. Supercritical behavior also appears when $q$ is sufficiently large, which will be shown later. This result shows that the absence of mega banks does not eliminate the possibility of a supercritical financial meltdown completely if $q$ is large. Indeed, history suggests that large-scale financial meltdowns did occur in pre-modern eras prior to the evolution of modern mega banks [@roubini_crisis_2011; @mackay_memoirs_2009; @kindleberger_manias_2005].
[*Finite-size scaling of supercritical behavior*]{}: A bump in an avalanche size distribution is found in many systems. Such bumps are usually believed to be a finite-size effect that vanishes in the thermodynamic limit, manifesting critical behavior. However, the bump in our avalanche size distribution is qualitatively different in that it sustains in the thermodynamic limit, implying genuine supercritical behavior. Furthermore, the bump exhibits the increasing power-law behavior, which has not been observed in other avalanche dynamics to our knowledge. Here we systematically analyze these observations based on a finite-size scaling analysis of the avalanche size distribution.
We first denote the crossover point between the first and second regions as $s_{c1}$. In Fig. \[Fig-Scaling\], we show that the aforementioned behavior is observed in systems of different sizes, but the crossover point depends on the system size in a power-law manner (i.e., $s_{c1} \sim N^\mu$). Based on this result, we make the usual scaling ansatz: $$P_s^{(<)}(s) = c_1 G_<(s/s_{c1}).$$ The scaling function $G_<(x)$ behaves as $G_<(x) \sim x^{-\tau}$ for $x < 1$. To eliminate the size dependency, $c_1$ is determined as $c_1 \sim N^{-\mu \tau}$.
The crossover point between the second and the third regions is denoted as $s_{c2}$. To characterize the scaling behavior in the bump pattern for different system sizes, we introduce another scaling hypothesis: $$P_s^{(>)}(s) = c_2 G_>(s/s_{c2})$$ where $s_{c2} \sim N$, because $s_{c2}$ represents a massive-scale avalanche comparable to the system size in order of magnitude (Fig. \[Fig-Scaling\]B). Then the scaling function behaves as $G_>(x) \sim x^\zeta$ for $x < 1$ and sharply decays for $x > 1$. Because the two avalanche size distribution functions are continuous at $s_{c1}$, the coefficient $c_2$ must depend on the system size as $c_2 \sim N^{\zeta - \mu(\tau +\zeta)}$.
We numerically confirm the scaling behavior using the data collapse procedure. The scaling hypothesis implies that curves $N^{-\mu\tau} P_s^{(<)}(s/N^\mu)$ for different $N$ should collapse into the same curve in the first region, and that the collapsing part extends as $N$ grows. Similarly, the curves $N^{-\zeta + \mu(\tau + \zeta)} P_s^{(>)}(s/N)$ for different $N$ should collapse into the same curve in the second region. This data collapse is well established by the choice of $\tau=2.55$, $\mu=0.53$, and $\zeta=0.75$ for networks with $\gamma=2.5$ (Fig. \[Fig-Scaling\]). The chosen exponents do not depend on $q$, but on $\gamma$ (Table. \[Table-Exponents\]). Thus, the scaling behavior is independent of the parameter $q$. It depends only on the topology of the underlying network.
A scaling relation between the exponents $\tau$, $\mu$, and $\zeta$ is obtained by considering the average size of avalanches $\left< s \right>$. For a duration $\Delta t$, the average number of avalanches is given by $\rho \Delta t / q N$, where $\rho$ is the density of the vulnerable banks in the steady state. Then, the average number of failed banks in the duration is $\left< s \right> \rho \Delta t / q N$. This must be equal to the average number of banks $(1-\rho) \Delta t$ that become vulnerable in the duration because the number of vulnerable banks in the system is steady on average. Thus, we obtain $\left< s \right> = q N (1 - \rho) / \rho$ [@drossel_self-organized_1992]. On the other hand, we have $\left< s \right> = \int_1^{s_{c1}} s P_s^{(<)} (s) ds + \int_{s_{c1}}^{\infty} s P_s^{(>)}(s) ds \sim N^{\zeta - \mu(\tau + \zeta) + 2}$. Thus, the relation $\mu(\tau + \zeta) - \zeta = 1$ is obtained. Our measurement of the exponents in the simulations also fits the relation well. [*This relation implies that the second region is sustained in the thermodynamic limit because the probability of the avalanches in the region is constant as*]{} $$\int_{s_{c1}}^{s_{c2}} P_s^{(>)}(s) ds \sim N^{\zeta - \mu(\tau + \zeta) + 1} \sim {\rm const}.$$
![(Color online) Average density of vulnerable nodes as a function of $q$. The density is measured just before each shock. Degree exponent of the underlying network is taken as $\gamma=2.5$. The scaling behavior is evident in the range of $q$ where the plateau is observed in the average density. []{data-label="Fig-Density"}](Fig4.eps){width=".9\linewidth"}
![(Color online) Data collapse of the avalanche size distributions on two dimensional square lattices of different sizes for a fixed $q=0.7$. The linear size of a lattice is denoted as $L$. The collapse of the decreasing part indicates that the crossover point scales as $L^{1.9}$. The inset shows the cutoff point of the increasing part scales along the order of the system size $L^2$.[]{data-label="Fig-2DLattice"}](Fig5.eps){width=".9\linewidth"}
The range of $q$ in which the scaling behavior holds can be estimated by the average density of vulnerable nodes measured just before each shock. We find that there exists a range of $q$ in which the average density measured just before each shock is independent of $q$. The scaling behavior holds in the range, whereas deviation is observed outside the range (Fig. \[Fig-Density\]). A finite upper bound $q_0$ ($\approx 10^{-4}$) of the range exists, but the lower bound seems to vanish as the system size increases. For any $0 < q < q_0$, the scaling behavior holds if the system size is large enough. The vanishing lower bound is supported by the fact that the percolation threshold vanishes in scale-free networks with $2 < \gamma <3$ [@cohen_percolation_2002]. The density of vulnerable nodes is always higher than the percolation threshold, implying a supercritical giant cluster exists in the thermodynamic limit.
We have two divergent size scales $s_{c1}$ and $s_{c2}$ in the model. The scaling of $s_{c2}$ of the order of $N$ is clear, but scaling of the crossover point $s_{c1}$ is not trivial. To understand the characteristics of the exponent $\mu$, we simulate the model on the two dimensional square lattice. When $q$ is sufficiently large, a bump again appears in the tail of the distribution (Fig. \[Fig-2DLattice\]). We find that the crossover points scale as $s_{c1} \sim L^{1.9}$ and $s_{c2} \sim L^2$, where $L$ is the linear size of the system and $N=L^2$. The value $1.9$ of the exponent $s_{c1}$ is close to the fractal dimension of the percolating cluster around the percolation threshold on the lattice. The scaling is expected because $s_{c1}$ is the starting point of the bump and thus represents the typical size of small giant clusters. The existence of the bump and the cutoff of the order of $N$ are also expected if the density of vulnerable nodes is maintained to be larger than the percolation threshold by sufficiently large $q$. However, the bump is too narrow to exhibit any power-law behavior. Therefore, the arguments we made for the FF model on scale-free networks will not be directly applicable to the bump in the lattice. We remark that in a previous study [@schenk_finite-size_2000], a similar distribution was observed and was interpretted as a violation of simple finite-size scaling ansatz, but no scaling analysis for $s_{c1}$ was provided.
\[Table-Exponents\]
[*Discussion:*]{} We have studied the FF model on scale-free networks and derived a scaling relation for the avalanche sizes in the supercritical region, which implies that the supercritical dynamics can occur generically, independent of system size. In particular, the dynamics of our model generate not only small-scale bank failures but also extremely large-scale financial meltdowns. The dynamics are shaped by the formation of a bubble, which is represented by a cluster of counterparty relationships among vulnerable banks that make speculative bets on risky assets.
The size of a bubble is controlled by the duration in which the system is not exposed to external shocks (e.g., no lightening in the context of forest fire). When the duration is short, small-scale bank failures are more likely, but the possibility for an extreme financial meltdown is reduced. When the duration is long enough, however, small-scale bank failures become less frequent, as is usually the case in an era of easy access to money. History, however, suggests that a protracted era of easy money promotes imprudent banking practices and development of speculative bubbles [@roubini_crisis_2011; @krugman_end_2013; @shiller_irrational_2005]. In our model, the system evolves to a supercritical state in this munificent environment, increasing the likelihood of development of an unusually large cluster of counterparty relationships among vulnerable banks with speculative bets on risky assets. This cluster is equivalent to a supercritical percolation cluster in the context of forest fire. When one bank in this cluster fails, other counterparties in the cluster are affected, and cascading bank failures occur.
This is reminiscent of the financial meltdown triggered by the demise of Lehman Brothers, which was acting as the prime broker for many hedge funds in executing trades, holding collateral, receiving, and disbursing monies [@blinder_after_2013]. Lehman’s failure immediately wiped out plenty of hedge funds. The bankruptcy of Lehman’s European subsidiary alone froze \$40 billion in clients’ funds [@buchanan_forecast_2014]. Furthermore, its biggest counterparties, such as Bank of America, Citigroup, and Deutsche Bank, were critically affected and eventually bailed out by governments. Then, a credit crunch hammered banking systems globally, and the shutdown of some asset markets made it difficult to conserve the value of an asset. It became blatantly obvious that this non-conservative nature is one of the essential features of financial crisis. Our findings highlight the importance of policy interventions in keeping the economy operating within reasonable duration of easy money regime, which seems to be one of the root causes of large-scale financial meltdowns.
This work was supported by the SNU research grant in form of brain fusion project and the NRF in Korea with grant Nos. 2010-0015066 and 2014-069005.
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---
abstract: 'We show that the trace decategorification, or zeroth Hochschild homology, of the twisted Heisenberg category defined by Cautis and Sussan is isomorphic to a quotient of $W^-$, a subalgebra of $W_{1+\infty}$ defined by Kac, Wang, and Yan. Our result is a twisted analogue of that by Cautis, Lauda, Licata, and Sussan relating $W_{1+\infty}$ and the trace decategorification of the Heisenberg category.'
address:
- |
Department of Mathematics\
University of Southern California\
Los Angeles, CA
- |
Department of Mathematics\
University of Virginia\
Charlottesville, VA
author:
- Can Ozan Oğuz
- Michael Reeks
bibliography:
- 'references.bib'
nocite: '[@*]'
title: Trace of the Twisted Heisenberg Category
---
Introduction
============
Categorification is the process of enriching an algebraic object by increasing its categorical dimension by one, e.g. passing from a set to a category or from a 1-category to a 2-category. The original object can be recovered through the inverse process of decategorification. The most commonly used decategorification functor is the split Grothendieck group $K_0$, but it is natural to ask whether alternative decategorification functors may give additional insight into the categorified object. One such alternative, advocated in [@BGHL], is the trace decategorification, which often encodes more information than $K_0$.
The trace, or zeroth Hochschild homology, of a $\Bbbk$-linear additive category $\mathcal{C}$ is the $\Bbbk$-vector space given by $${\operatorname{Tr}}(\mathcal{C}) := \bigg(\bigoplus_{x\in \operatorname{Ob}(\mathcal{C})} {\operatorname{End}}_{\mathcal{C}}(X)\bigg)\bigg/ \operatorname{span}_{\Bbbk}\{fg-gf\},$$ where $f$ and $g$ run through all pairs of morphisms $f:x\rightarrow y$ and $g:y\rightarrow x$ with $x,y \in \operatorname{Ob}(\mathcal{C})$. If a $\Bbbk$-linear category $\mathcal{C}$ carries a monoidal structure, then $\operatorname{span}\{fg-gf\}$ is an ideal, and ${\operatorname{Tr}}(\mathcal{C})$ becomes an algebra where multiplication in the trace is induced from tensor product of $\mathcal{C}$. The trace has the advantage that it is, unlike $K_0$, invariant under passage to the Karoubi envelope, cf. [@BHLZ Proposition 3.2]. Since passing to the Karoubi envelope often prevents one from working with diagrams, trace seems to be a more suitable option to decategorify diagrammatic categories.
The traces of several interesting categories have been computed. in [@BHLZ] and [@BHLW], the trace of any categorified type ADE quantum group is shown to be isomorphic to a current algebra. In [@SVV], traces of quiver Hecke algebras are studied. In [@EL], the trace of the Hecke category is shown to be a semidirect product of the Weyl group and a polynomial algebra. A unifying approach to Heisenberg categorifications was given in [@Rosso2016] via Frobenius algebras; in [@Jack], the degree zero part of the trace of these categories are computed.
The trace ${\operatorname{Tr}}(\mathcal{C})$ is closely related to the $K_0({\mathcal{C}})$ through the Chern character map $$h_\mathcal{C}: K_0(\mathcal{C}) \rightarrow {\operatorname{Tr}}(\mathcal{C})$$ which sends the isomorphism class of an object to the class of its identity morphism in the trace. Interestingly, the map $h_\mathcal{C}$ is usually injective, but is often not surjective. Thus, the trace often contains additional structure which has no analogue in the Grothendieck group.
One interesting example in which $h_{\mathcal{C}}$ fails to be surjective is given by the Heisenberg category $\mathcal{H}$ defined in [@Khovanov]. It is a $\mathbb{C}$-linear additive monoidal category. Therefore ${\operatorname{Tr}}(\mathcal{H})$ carries an algebra structure. There is an injective algebra homomorphism from the Heisenberg algebra $\mathfrak{h}$ to $K_0(\mathcal{H})$ (they are conjecturally isomorphic). In [@CLLS], ${\operatorname{Tr}}(\mathcal{H})$ is shown to be isomorphic to a quotient of $W_{1+\infty}$, a filtered algebra which is important in conformal field theory. In particular, it properly contains $\mathfrak{h}$ in filtration degree zero. Hence ${\operatorname{Tr}}(\mathcal{H})$ likely contains more information than $K_0(\mathcal{H})$. This fits into a larger framework, studied in [@CLLSS], involving the elliptic Hall algebra.
We study a twisted version of Khovanov’s Heisenberg category. The twisted Heisenberg algebra $\mathfrak{h}_{tw}$ is a unital associative algebra generated by $h_{m/2}$, $m\in 2\mathbb{Z}+1$, subject to the relations $$\left[h_{\frac{n}{2}}, h_{\frac{m}{2}}\right] = \frac{n}{2} \delta_{n,-m}.$$ In [@CS], a twisted version of the Heisenberg category, denoted $\mathcal{H}_{tw}$, is introduced. It is also a $\mathbb{C}$-linear additive monoidal category, with an additional $\mathbb{Z}/2\mathbb{Z}$-grading. It is proved that $K_0({\mathcal{H}_{tw}})$ contains $\mathfrak{h}_{tw}$ (again, they are conjecturally isomorphic).
The goal of this paper is to study the trace ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$, and determine additional structure analogous to that in the untwisted version. We show that the even part of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ is isomorphic as an algebra to a quotient of a subalgebra of $W_{1+\infty}$ that we will denote by $W^-$. We give explicit descriptions of $W_{1+\infty}$ and $W^-$ in Section \[W-algebra\]. This confirms the expectation in [@CLLS] that there should be a relationship between $\mathcal{H}_{tw}$ and one of two subalgebras of $W_{1+\infty}$ defined in [@Wang].
\[main theorem\] There is an algebra isomorphism $${{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}} \longrightarrow W^-/\langle w_{0,0}, C-1\rangle.$$
Even though the isomorphism between $K_0(\mathcal{H}_{tw})$ and the twisted Heisenberg algebra $\mathfrak{h}_{tw}$ is still conjectural, we are able to completely characterize ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$.
To prove Theorem \[main theorem\], we first compute sets of algebra generators and relations for both $W^-$ and ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, adapting arguments used in [@CLLS] to accommodate the new supercommutative elements arising from the twisting (cf. Section \[trace\]). We then study actions of each algebra on its canonical level one Fock space representation. These Fock space representations are isomorphic, and so induce a linear map $\Phi:{{\operatorname{Tr}}(\mathcal{H}_{tw})}\rightarrow W^-$. We prove that $\Phi$ is an algebra homomorphism by studying the actions of both $W^-$ and ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ on their Heisenberg subalgebras. Finally, we check that the actions of the generators are identified under $\Phi$, and deduce that $\Phi$ is an algebra isomorphism.
An important tool in studying the connection between these algebras is the relationship between ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ and the degenerate affine Hecke-Clifford algebra ${\mathfrak{H}^C}_n$ of type $A_{n-1}$. The trace of ${\mathfrak{H}^C}_n$ as a vector space was computed by the second author in [@Mike]. The algebra ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ admits a triangular decomposition, where ${\operatorname{Tr}}({\mathfrak{H}^C}_n)$ is identified with the upper (respectively lower) half. This identification simplifies some of the computations and the calculation of the graded rank of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$.
The structure of the paper is as follows. In Section 2, we describe the $W$-algebra $W^-$ of interest, describe its gradings and a set of generators, and study its Fock space representation. In Section 3, we describe trace decategorification in more detail and present the twisted Heisenberg category studied in [@CS], as well as its gradings. We also identify a copy of the degenerate affine Hecke-Clifford algebra within the trace. In Section 4, we study a subalgebra of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ consisting of circular diagrams called bubbles, and describe how they interact with other elements of the trace. Section 5 contains a number of calculations of diagrammatic relations in the trace that are useful for computing a generating set of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$. Finally, in Section 6, we describe a triangular decomposition of the trace, and then establish a generating set. This allows us to prove the desired isomorphism by using the action of each algebra on its Fock space.
[**Acknowledgements**]{} The authors thank Andrea Appel, Victor Kleen, Aaron Lauda, Joshua Sussan, and Weiqiang Wang for helpful discussions and advice concerning the paper; Maxwell Siegel for his notation suggestions; and Joshua Sussan for suggesting this project. The first author was partially supported by the NSF grant DMS-1255334. The second author was partially supported by a GRA fellowship in Weiqiang Wang’s NSF grant and by a GAANN fellowship.
W-algebra
=========
In this section, we will recall the W-algebra we are interested in, its structure as a $\mathbb{Z}$-graded and $\mathbb{N}$-filtered algebra, and one of its subalgebras – the twisted Heisenberg algebra – as well as their Fock space representations.
Twisted Heisenberg algebra $\mathfrak{h}_{tw}$
----------------------------------------------
We recall the definition of the twisted Heisenberg algebra. The twisted Heisenberg algebra $\mathfrak{h}_{tw}$ is a unital associative algebra generated by $h_{n}$ for $n\in \mathbb{Z}+\frac{1}{2}$ subject to the relation that $[h_n,h_m]=n\delta_{n,-m}$.
W-algebra $W^-$ {#W-algebra}
---------------
Let $\mathcal{D}$ denote the Lie algebra of differential operators on the circle. The central extension $\hat{\mathcal{D}}$ of $\mathcal{D}$ is described in [@Wang]. It is generated by by $C$ and by $w_{k,l}=t^kD^l$ for $l\in\mathbb{Z}$ and $k\in\mathbb{Z}_{\geq 0}$ where $t$ is a variable over $\mathbb{C}$, and $D=t\frac{d}{dt}$, subject to relations that $C$ and $w_{0,0}$ are central, and: $$\label{WForm}
[t^rf(D),t^sg(D)]=t^{r+s}(f(D+s)g(D)-f(D)g(D+r))+\psi(t^rf(D),t^sg(D))C,$$ where $$\psi(t^rf(D),t^sg(D))= \begin{cases}
{\displaystyle}\sum_{-r\leq j\leq -1}f(j)g(j+r) & r=-s\geq 0 \\
0 & r+s\neq 0
\end{cases}$$ for $f,g$ polynomials.
The W-algebra $W_{1+\infty}$ is the universal envelopping algebra of $\hat{\mathcal{D}}$. It is shown in [@CLLS] that trace of Khovanov’s Heisenberg category is isomorphic to a quotient of $W_{1+\infty}$.
In this paper, we are interested in the universal enveloping algebra of a central extension of a Lie subalgebra of $\mathcal{D}$ fixed by a degree preserving anti-involution. Define the map:
$$\begin{aligned}
\sigma:&\mathcal{D}&\longrightarrow \mathcal{D} \\
&1&\mapsto \sigma(1) = -1\\
&t&\mapsto \sigma(t)=-t\\
&D&\mapsto \sigma(D)=-D.\end{aligned}$$
This is a degree preserving anti-involution of $\mathcal{D}$, and the Lie subalgebra fixed by $-\sigma$ is $$\mathcal{D}^-:=\{a\in\mathcal{D}|\sigma(a)=-a\}.$$ Let $\hat{\mathcal{D}}^-$ be a central extension of $\mathcal{D^-}$ where the $2$-cocycle is the restriction of the $2$-cocycle $\psi$ given above. Therefore $\hat{\mathcal{D}}^-$ is a Lie subalgebra of $\hat{\mathcal{D}}$.
More explicitly, $\hat{\mathcal{D}}^-$ is the Lie algebra over the vector space spanned by $\{C\}\cup \{ t^{2k-1} g(D+(2k-1)/2); \ g\text{ even}\} \cup \{ t^{2k} f(D+k) ;\ f\text{ odd}\}$ where $k\in\mathbb{Z}$ and even and odd refer to even and odd polynomial functions. Its Lie bracket is given by Equation .
Denote by $W^-$ the universal enveloping algebra of $\hat{\mathcal{D}}^-$. Our main result relates the trace of twisted Heisenberg category to a quotient of $W^-$.
$$\begin{tikzpicture}[scale=1.1]
\node[font=\Large] at (-0.5,1.5) {$\mathcal{D}$};
\draw[->] (0,1.5) to (3,1.5);
\node[font=\Large] at (3.4,1.55) {$\widehat{\mathcal{D}}$};
\node at (1.4,1.7) {central extension};
\draw[->] (3.9,1.5) to (6.7,1.5);
\node[font=\Large] at (7.5,1.5) {$W_{1+\infty}$};
\node at (5.3,1.7) {enveloping algebra};
\node[rotate=90,scale=1.5] at (-.5,.8) {$\subset$};
\node[rotate=90,scale=1.5] at (3.4,.8) {$\subset$};
\node[rotate=90,scale=1.5] at (7.4,.8) {$\subset$};
\node[text width=1cm] at (-1,.8) {fixed by -$\sigma$};
\node[font=\Large] at (-0.4,0.05) {$\mathcal{D}^-$};
\draw[->] (0,0) to (3,0);
\node[font=\Large] at (3.4,0.05) {$\widehat{\mathcal{D}^-}$};
\node at (1.4,0.2) {central extension};
\draw[->] (3.9,0) to (6.7,0);
\node[font=\Large] at (7.5,0) {$W^-$};
\node at (5.3,.2) {enveloping algebra};
\end{tikzpicture}$$ Note that not all $w_{k,\ell}$ are contained in $W^-$.
When $k-\ell$ is an even integer, $w_{k,\ell}\not\in W^-$. Moreover, the difference $k-\ell$ being odd is not sufficient. For example, $t^2D=w_{2,1}\not\in W^-$ since an element starting with $t^2$ should be followed by $f(D+1)$ where $f$ is an odd polynomial function. Hence $t^2D=w_{2,1}\not\in W^-$ but $t^2(D+1)=t^2D+t^2=w_{2,1}+ w_{2,0}\in W^-$ (and, indeed, $\sigma(t^2 (D+1)) = t^2(-D-1) = - t^2(D+1)$).
Gradings on $W^-$
-----------------
There is a natural $ \mathbb{Z}^{\geq 0}$ filtration of $W^-$ called the *differential filtration* with $w_{k, \ell}$ in degree $ \ell$; denote this filtration by $|\cdot|_{dot}$. It is convenient to define an additional filtration: the *difference filtration*, where $w_{k, \ell}$ is in degree $\ell-k$, denoted $|\cdot|_{diff}$. That this is a filtration follows from the fact that $W^-$ also carries a filtration with $w_{k, \ell}$ in degree $k$.
These filtrations are compatible, so we have a $(\mathbb{Z}\times \mathbb{Z}^{\geq 0})$-filtration with with an element $f=t^j g(D-j/2) \in W^-$ in bidegree $\leq(|f|_{diff}, |f|_{dot}) = (\operatorname{deg}(g)-j, \operatorname{deg}(g))$, where $\operatorname{deg}(g)$ is the polynomial degree of $g(w) \in \mathbb{C}[w]$. Define the following subalgebras of $W^-$: $$W^{-,>} = \mathbb{C}\langle t^j g(D-j/2) | \deg(g)-j\geq 1\rangle;$$ $$W^{-,<} = \mathbb{C}\langle t^j g(D-j/2) | \deg(g)-j\leq 1\rangle;$$ $$W^{-,0} = \mathbb{C}\langle g(D) |\deg(g) \text{ odd}\rangle.$$ Let $W^{-,{\omega}}[\leq r, \leq k]$ denote the set of elements in difference degree $\leq r$ and differential degree $\leq k$, with $\omega \in \{>,<,0\}$.
Denote by ${\operatorname{gr}}W^-$ the associated graded object with respect to this filtration. Hence ${\operatorname{gr}}W^-$ is $(\mathbb{Z}\times \mathbb{Z}^{\geq 0})$-graded with $|w_{k,\ell}| = (\ell-k, \ell)$. For $\omega \in \{>,<,0\}$, define a generating series for the graded dimension of ${\operatorname{gr}}(W^-)^{\omega}$ by $$P_{{\operatorname{gr}}(\mathcal{W^-})^\omega}(t,q) = \sum_{r\in \mathbb{Z}} \sum_{k\in \mathbb{Z}, k\geq 0} \dim {\operatorname{gr}}(W^-)^\omega[r,k] t^r q^k.$$
\[grDim Walg\] The graded dimensions of ${\operatorname{gr}}(W^-)^>$ and ${\operatorname{gr}}(W^-)^<$ are given by:
$$P_{{{\operatorname{gr}}(\mathcal{D}^-)}^>} = \prod_{r\geq 0} \prod_{k>0} \frac{1}{1-t^{2r+1} q^k};$$ $$P_{{{\operatorname{gr}}(\mathcal{D}^-)}^<} = \prod_{r\leq 0} \prod_{k>0} \frac{1}{1-t^{2r-1} q^k}.$$
The algebra $W^-$ is generated by elements of the form $t^j g(D-j/2)$, where $ \deg(g)-j$ is odd. Hence ${{\operatorname{gr}}(\mathcal{D}^-)}^>$ is freely generated by elements $w_{k, \ell}$ with $k-\ell$ odd; such elements have bidgegree $(k-\ell, \ell)$. The proposition follows.
Let $W^-_{r,s}$ denote the rank $r$, differential filtration $s$ part of $W^-$. It is easy to see that the differential filtration zero part of $W^-$, namely ${\displaystyle}\bigcup_{r\in\mathbb{Z}} W^-_{r,0}$, is spanned as a vector space by $\{C\}\cup\{t^{2n+1}\}_{n\in\mathbb{Z}}$. As an algebra, we have that $$[t^{2n+1},t^{2m+1}]=(2n+1)\delta_{n,-m}$$ Hence we have an isomorphism between the differential filtration zero part of $W^-$ and the twisted Heisenberg algebra $\mathfrak{h}_{tw}$ given by: $$\begin{aligned}
\phi:&\mathfrak{h}_{tw}&\longrightarrow {\displaystyle}\bigcup_{r\in\mathbb{Z}} W^-_{r,0}\\
&h_{\frac{2n+1}{2}}&\mapsto \frac{1}{\sqrt{2}} t^{2n+1}\end{aligned}$$ where $n\in\mathbb{Z}$.
Generators of the algebra $W^-$
-------------------------------
The following lemma describes a generating set for $W^-$ as an algebra.
\[GenSetW\] The algebra $W^-/\langle w_{0,0}, C\rangle$ is generated by $w_{1,0}$, $w_{0,3}$, and $w_{\pm 2,1} \pm w_{\pm 2,0}$.
Let $t^k g(D-{k/2})$ be an arbitrary element of $W^-$. Without loss of generality, we may assume $g$ is a monic monomial of the form $g(w) = w^\ell$ with $\ell-k$ odd, since lower terms in $g$ are just monomials of this form with lower degree, and thus can be generated separately. Therefore, we have $$\label{lowerTerms} t^k g(D-k/2) = \sum_{i=0}^\ell \binom{\ell}{i}(-1)^{\ell - i} (k/2)^{\ell - i} t^k D^i.$$ The leading term of this element with respect to differential degree is $t^k D^\ell.$ We will generate the leading term first, and address lower terms afterwards. There are two cases, depending on the parities of $k$ and $\ell$.
First, suppose that $k=2n$ is even and $\ell = 2m+1$ is odd (recall that $k$ and $\ell$ must have opposite parity in $W^-$). Hence, we must generate $w_{\pm 2n, 2m+1}$. The following calculations are easy, using Formula \[WForm\]: $$[w_{-2,1} - w_{-2,0}, w_{1,0}] = w_{-1,0},$$ $$[w_{1,0}, w_{0,3}] = -3(w_{1,2}+ w_{1,1})-w_{1,0},$$ $$\label{1,2n} [w_{1,2b},w_{0,3}] = -3w_{1, 2b+2} + O(w_{1,2b+1}),$$
where $O(\omega)$ refers to terms with lower differential degree than $\omega$. Hence, starting with $w_{1,2} - w_{1,1}$, we can use the Equation above to generate $w_{1,2b}$ for any $b$. Now we have:
$$\label{2n+1,0}[w_{\pm 2a, 1}, w_{1,0}] = w_{\pm 2a+1,0},$$
$$\label{2n,0} [w_{\pm 2a+1, 0}, w_{1,2} - w_{1,1}] = -(4a+2)w_{2a+2,1} - (2a+1)(2a+2)w_{2a+2,0}.$$
Thus, starting from $w_{2,1} + w_{2,0}$, we can generate $w_{2a,1}$ for any $a$. Finally, we have:
$$\label{0,2n}[w_{-1,0}, w_{1,2b}] = \sum_{i=0}^{2b-1} \binom{2b}{i} (-1)^{2b-i+1} w_{0,i} = w_{0,2b-1} + O(w_{0,2b-2}),$$
$$\begin{aligned}
&= -\sum_{i=0}^{2b-2} \binom{2b-1}{i} (\pm 1)^{2b-i}(2)^{2b-2-i} t^{2a}D^{i+1} \\&= w_{2a,2b-1} + O(w_{2a,2b-2}).\end{aligned}$$
So, we can generate a polynomial with leading term $w_{\pm 2n, 2m+1}$.
Next, suppose that $k = 2n+1$ is odd and positive and $\ell = 2m$ is even. Using Formula , we have:
$$\begin{aligned}
&= t^{2a+1} \sum_{i=0}^{2b} \binom{2b+1}{i} (2a+1)^{2b+1 -i} D^i \\&= w_{2a+1, 2b} + O(w_{2a+1, 2b-1}).\end{aligned}$$
Now Equations and give that we can generate $w_{2a+1,0}$ and $w_{0,2b+1}$. Hence we can generate a polynomial with leading term $w_{2a+1,2b}.$
Finally, assume that $k=-(2n+1)$ is odd and $n=2m$ is even. Using Formula , we have:
$$[w_{-2a,1}, w_{1,0}] = w_{1-2a,0}.$$
By Equation , we can therefore generate $w_{-(2a+1),0}$ for any $a$. Next, note that:
$$[w_{-1,0}, w_{1,2b}] = -\sum_{i=0}^{2b-1} \binom{2b-1}{i} (-1)^{2b-1-i} D^i = w_{0,2b-1} + O(w_{0,2b-2}).$$
By Equation , we can generator $w_{0, 2b+1}$ for any $b$. Finally, we have
$$\begin{aligned}
[w_{-(2a+1),0}, w_{0,2b-1}]& = t^{-(2a+1)} \sum_{i=0}^{2n-2} \binom{2n-1}{i} (-1)^{2n-i} (2a+1)^{2n-1-i} D^{i} \\&= w_{-(2a+1), 2b-2} + O(w_{-(2a+1), 2b-3}).\end{aligned}$$
Thus, we can generate a polynomial with leading term $w_{-(2n+1), 2m}.$
It remains to adjust the lower terms of these equations so that they match those in Equation . But note that each equation used above to generate the leading term results in lower terms which lie in different filtrations of $W^-$. Therefore we can adjust the coefficients of lower terms by scaling individual equations above. Since there is no dependency between these equations, we can choose constant coefficients for the generators so that our generated polynomial has the correct lower terms.
Fock space representation of $W^-$
----------------------------------
The algebra $W^-$ inherits a Fock space representation from $W_{1+\infty}$. Let $W^{-,\geq} = W^{-,0} \oplus W^{-,>}$. For parameters $c,d \in \mathbb{C}$, let $\mathbb{C}_{c,d}$ be a one-dimensional module for $W^{-,\geq}$ on which each $w_{k,\ell}$ with $(k,\ell) \not= (0,0)$ acts as zero, $C$ acts as $c$, and $w_{0,0}$ acts as $d$. Let $\mathcal{M}_{c,d} := {\operatorname{Ind}}_{W^{-,\geq}}^{W^-} \mathbb{C}_{c,d}$. This induced module possesses the following properties:
\[FockSpaceW\] [@AFMO; @FKRW] The $W^-$-module $\mathcal{M}_{c,d}$ has a unique irreducible quotient $\mathcal{V}_{c,d}$, which is isomorphic as a vector space to $\mathbb{C}[w_{-1,0}, w_{-2,0},w_{-3,0},\ldots ]$.
\[FockSpaceW faithful\] [@SV] The action of $W^-/(C-1,w_{0,0})$ is faithful on $\mathcal{V}_{1,0}$.
This follows immmediately from the argument in [@SV] for $W_{1+\infty}$ because $W^-$ is a subalgebra.
Proposition \[FockSpaceW\] allows us to the compute the action of the generators on $\mathcal{V}_{1,0}$, which we record for convenience below.
\[FockSpaceW action\] Let $k$ be a positive integer. The generators of $W^-$ act on $\mathcal{V}_{1,0}$ as follows: $$\begin{aligned}
[w_{1,0}, w_{-k,0}] &= \delta_{1,k},\\
[w_{-2,1}-w_{-2,0}, w_{-k,0}] &= (k+2) w_{-(k+2),0} ,\\
[w_{2,1} + w_{2,0}, w_{-k,0}] &= -(k+2)w_{2-k,0} ,\\
[w_{0,3}, w_{-k,0}] &= 3k w_{-k,2} - 3k^2 w_{-k,1} + k^3 w_{-k,0}.\end{aligned}$$
Twisted Heisenberg category {#twistedHeisenberg}
===========================
We will now describe the main object of interest in the paper, the twisted Heisenberg category $\mathcal{H}_{tw}$. After defining the category, we recall the trace decategorification functor and some of its properties. We then describe some filtrations of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$, identify a copy of the degenerate affine Hecke-Clifford algebra ${\mathfrak{H}^C}_n$, and describe the trace of ${\mathfrak{H}^C}_n$. Finally, we identify a set of distinguished elements in ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ which generate the nonzero filtration degrees of the algebra.
Definition of $\mathcal{H}_{tw}$
--------------------------------
The twisted Heisenberg category $\mathcal{H}^{t}$ is defined in [@CS] as the Karoubi envelope of a $\mathbb{C}$-linear $\mathbb{Z}/2\mathbb{Z}$-graded additive monoidal category, whose moprhisms are described diagrammatically. There is an injective algebra homomorpshim from $\mathfrak{h}_{tw}$ to the split Grothendieck group of the twisted Heisenberg caterogy $K_0({\mathcal{H}^{t}})$. As in the untwisted case, this map is conjecturally surjective.
The object of our main interest is the trace decategorification or zeroth Hochschild homology of $\mathcal{H}^{t}$. It is shown in [@BGHL Proposition 3.2] that trace of an additive category is isomorphic to the trace of its Karoubi envelope,. Therefore, we can work with the non-idempotent completed version of $\mathcal{H}^t$. We will denote it by $\mathcal{H}_{tw}$. Focusing our attention to $\mathcal{H}_{tw}$ allows us to work with the diagrammatics introduced in [@CS].
The category $\mathcal{H}_{tw}$ is the $\mathbb{C}$-linear, $\mathbb{Z}/2\mathbb{Z}$-graded monoidal additive category whose objects are generated by $P$ and $Q$. A generic object is a sequence of $P$’s and $Q$’s. The morphisms of $\mathcal{H}_{tw}$ are generated by oriented planar diagrams up to boundary fixing isotopies, with generators $$\hspace{0.9cm}
\begin{tikzpicture}
\draw[->](-2,-0.25) to (-2,0.75);
\draw (-1.99,0.25) circle [radius=2pt];
\draw[<-](-1,-0.25) to (-1,0.75);
\draw (-1,0.25) circle [radius=2pt];
\draw[->] (0,-0.25) to (1,0.75);
\draw[->] (1,-0.25) to (0,0.75);
\draw[->] (1.5,0.5) arc (180:360:5mm);
\draw[<-] (3,0.5) arc (180:360:5mm);
\draw[->] (4.5,0) arc (180:0:5mm);
\draw[<-] (6,0) arc (180:0:5mm);
\end{tikzpicture}$$ where the first diagram corresponds to a map $P\rightarrow P\{1\}$ and the second diagram corresponds to a map $Q\rightarrow Q\{1\}$, where $\{1\}$ denotes the $\mathbb{Z}/2\mathbb{Z}$-grading shift. The first two diagrams above have degree one, and the last five have degree zero. The identity morphisms of $P$ and $Q$ are indicated by an undecorated upward and downward pointing arrow, respectively. These generators satisfy the following relations:
$$\label{R2}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to [out=45,in=-45] (0,2);
\draw [->](0.5,0) to [out=135,in=-135] (0.5,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (1.5,0) to (1.5,2);
\draw[->] (2,0) to (2,2);
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to [out=45,in=-45] (0,2);
\draw [<-](0.5,0) to [out=135,in=-135] (0.5,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (1.5,0) to (1.5,2);
\draw[<-] (2,0) to (2,2);
\end{tikzpicture}
\hspace{1.5cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to (1.5,2);
\draw [->](1.5,0) to (0,2);
\draw[->](0.75,0) to [out=45,in=-45] (0.75,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to (1.5,2);
\draw [->](1.5,0) to (0,2);
\draw[->](0.75,0) to [out=135,in=225] (0.75,2);
\end{tikzpicture}$$
$$\label{R3}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [<-](0,0) to [out=45,in=-45] (0,2);
\draw [->](0.5,0) to [out=135,in=-135] (0.5,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (1.5,0) to (1.5,2);
\draw[->] (2,0) to (2,2);
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3,2) arc (180:360:5mm);
\draw[<-] (3,0) arc (180:0:5mm);
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3,2) arc (180:360:5mm);
\draw[<-] (3,0) arc (180:0:5mm);
\draw (3.05,1.8) circle [radius=3pt];
\draw (3.95,0.2) circle [radius=3pt];
\end{tikzpicture}$$
$$\label{d00=1}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3,2) arc (-180:180:5mm);
\end{tikzpicture}\hspace{6pt}
=1
\hspace{1.5cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\end{tikzpicture}\hspace{6pt}
=0$$
$$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (1,2);
\draw[->] (1,0) to (0,2);
\draw (0.25,0.5) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (1,2);
\draw[->] (1,0) to (0,2);
\draw (0.75,1.5) circle [radius=3pt];
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (1,2);
\draw[->] (1,0) to (0,2);
\draw (0.75,0.5) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (1,2);
\draw[->] (1,0) to (0,2);
\draw (0.25,1.5) circle [radius=3pt];
\end{tikzpicture}$$
$$\label{caps}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (180:0:5mm);
\draw (3.05,2.2) circle [radius=2.5pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (180:0:5mm);
\draw (3.95,2.2) circle [radius=2.5pt];
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (180:0:5mm);
\draw (3.05,2.2) circle [radius=2.5pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (180:0:5mm);
\draw (3.95,2.2) circle [radius=2.5pt];
\end{tikzpicture}$$
$$\label{cups}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (180:360:5mm);
\draw (3.05,1.8) circle [radius=2.5pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (180:360:5mm);
\draw (3.95,1.8) circle [radius=2.5pt];
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (180:360:5mm);
\draw (3.05,1.8) circle [radius=2.5pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (180:360:5mm);
\draw (3.95,1.8) circle [radius=2.5pt];
\end{tikzpicture}$$
$$\label{d01=0}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw (0,0.6) circle [radius=3pt];
\draw (0,1.2) circle [radius=3pt];
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0,0) to (0,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{1mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0,0) to (0,2);
\draw (0,0.6) circle [radius=3pt];
\draw (0,1.2) circle [radius=3pt];
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3,2) arc (-180:180:5mm);
\draw (3.95,2.2) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=\hspace{6pt}0$$
$$\label{cicj=-cjci}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw (0,1.6) circle [radius=3pt];
\draw[fill=] (0.2,1) circle [radius=0.3pt];
\draw[fill=] (0.4,1) circle [radius=0.3pt];
\draw[fill=] (0.6,1) circle [radius=0.3pt];
\draw[->] (0.8,0) to (0.8,2);
\draw (0.8,0.3) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=\hspace{6pt}-\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw (0,0.3) circle [radius=3pt];
\draw[fill=] (0.2,1) circle [radius=0.3pt];
\draw[fill=] (0.4,1) circle [radius=0.3pt];
\draw[fill=] (0.6,1) circle [radius=0.3pt];
\draw[->] (0.8,0) to (0.8,2);
\draw (0.8,1.6) circle [radius=3pt];
\end{tikzpicture}. \hspace{2cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0,0) to (0,2);
\draw (0,1.6) circle [radius=3pt];
\draw[fill=] (0.2,1) circle [radius=0.3pt];
\draw[fill=] (0.4,1) circle [radius=0.3pt];
\draw[fill=] (0.6,1) circle [radius=0.3pt];
\draw[<-] (0.8,0) to (0.8,2);
\draw (0.8,0.3) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=\hspace{6pt}-\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0,0) to (0,2);
\draw (0,0.3) circle [radius=3pt];
\draw[fill=] (0.2,1) circle [radius=0.3pt];
\draw[fill=] (0.4,1) circle [radius=0.3pt];
\draw[fill=] (0.6,1) circle [radius=0.3pt];
\draw[<-] (0.8,0) to (0.8,2);
\draw (0.8,1.6) circle [radius=3pt];
\end{tikzpicture}.$$
Also, if we let $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\end{tikzpicture}\hspace{6pt}
:=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[fill](0,1) circle[radius=3pt];
\end{tikzpicture}$$ we get the following relations:
$$\label{anticommute}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[fill](0,0.6) circle[radius=3pt];
\draw(0,1.2) circle[radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw(0,0.6) circle[radius=3pt];
\draw[fill](0,1.2) circle[radius=3pt];
\end{tikzpicture}$$
$$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw[fill] (0,1.6) circle [radius=3pt];
\draw[fill] (0.2,1) circle [radius=0.3pt];
\draw[fill] (0.4,1) circle [radius=0.3pt];
\draw[fill] (0.6,1) circle [radius=0.3pt];
\draw[->] (0.8,0) to (0.8,2);
\draw[fill] (0.8,0.3) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw[fill] (0,0.3) circle [radius=3pt];
\draw[fill] (0.2,1) circle [radius=0.3pt];
\draw[fill] (0.4,1) circle [radius=0.3pt];
\draw[fill] (0.6,1) circle [radius=0.3pt];
\draw[->] (0.8,0) to (0.8,2);
\draw[fill] (0.8,1.6) circle [radius=3pt];
\end{tikzpicture}
\hspace{1.5cm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw (0,1.6) circle [radius=3pt];
\draw[fill] (0.2,1) circle [radius=0.3pt];
\draw[fill] (0.4,1) circle [radius=0.3pt];
\draw[fill] (0.6,1) circle [radius=0.3pt];
\draw[->] (0.8,0) to (0.8,2);
\draw[fill] (0.8,0.3) circle [radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw (0,0.3) circle [radius=3pt];
\draw[fill] (0.2,1) circle [radius=0.3pt];
\draw[fill] (0.4,1) circle [radius=0.3pt];
\draw[fill] (0.6,1) circle [radius=0.3pt];
\draw[->] (0.8,0) to (0.8,2);
\draw[fill] (0.8,1.6) circle [radius=3pt];
\end{tikzpicture}$$
$$\label{dotSlide: bottomLeft}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,0.5) circle[radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.75,1.5) circle[radius=3pt];
\end{tikzpicture}\hspace{6pt}
+\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[->](0.5,0) to (0.5,2);
\end{tikzpicture}
\hspace{2mm}+\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[->](0.5,0) to (0.5,2);
\draw(0,1.2) circle[radius=3pt];
\draw(0.5,0.6) circle[radius=3pt];
\end{tikzpicture}$$
$$\label{dotSlide: topLeft}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.75,0.5) circle[radius=3pt];
\end{tikzpicture}\hspace{6pt}
+
\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[->](0.5,0) to (0.5,2);
\end{tikzpicture}\hspace{2mm}
-
\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[->](0.5,0) to (0.5,2);
\draw(0,1.2) circle[radius=3pt];
\draw(0.5,0.6) circle[radius=3pt];
\end{tikzpicture}$$
$$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (1,2);
\draw[<-](1,0) to (0,2);
\draw[fill](0.25,0.5) circle[radius=3pt];
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (1,2);
\draw[<-](1,0) to (0,2);
\draw[fill](0.75,1.5) circle[radius=3pt];
\end{tikzpicture}
+\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\draw[<-](0.5,0) to (0.5,2);
\end{tikzpicture}
\hspace{2mm}-\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\draw[<-](0.5,0) to (0.5,2);
\draw(0,1.2) circle[radius=3pt];
\draw(0.5,0.6) circle[radius=3pt];
\end{tikzpicture}$$
$$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (1,2);
\draw[<-](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (1,2);
\draw[<-](1,0) to (0,2);
\draw[fill](0.75,0.5) circle[radius=3pt];
\end{tikzpicture}
+\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\draw[<-](0.5,0) to (0.5,2);
\end{tikzpicture}
\hspace{2mm}+\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\draw[<-](0.5,0) to (0.5,2);
\draw(0,1.2) circle[radius=3pt];
\draw(0.5,0.6) circle[radius=3pt];
\end{tikzpicture}$$
.
If $x$ and $y$ are morphisms, the diagram corresponding to $x\otimes y$ is obtained by placing the diagram of $y$ to the right of the diagram of $x$. Since the relative positions of the hollow dots are important, we will work with the convention that the hollow dots in the diagram of $y$ will be placed below the height of hollow dots in the diagram of $x$.
Trace decategorification {#trace}
------------------------
In [@BGHL], the trace or zeroth Hochschild homology of a $\Bbbk$-linear additive category $\mathcal{C}$ is proposed as an alternative decategorification functor. Here we will recall its definition, and point out one subtlety occuring in our case due to the supercommutative nature of hollow dots and solid dots.
Let $\mathcal{C}$ be a $\Bbbk$-linear additive category. Then its trace decategorification, denoted ${\operatorname{Tr}}(\mathcal{C})$, is defined as follows:
$${\operatorname{Tr}}(\mathcal{C})\simeq \bigg( \bigoplus_{x\in \mathcal{O}b(\mathcal{C})} {\operatorname{End}}(x) \bigg) \big/\mathcal{I},$$
where $\mathcal{I}$ is the ideal generated by $\operatorname{span}_{\Bbbk}\{fg-gf\}$ for all $f:x\rightarrow y$ and $g:y\rightarrow x$ , $x,y\in \mathcal{O}b(\mathcal{C})$. Note that here we quotient out by an ideal, so ${\operatorname{Tr}}(\mathcal{C})$ has an algebra structure.
Trace decategorification has a nice diagrammatic interpretation, in which we consider our string diagrams to be drawn on an annulus instead of a plane. The annulus recaptures the trace relation $fg=gf$ diagrammatically since we can slide $f$ or $g$ around the annulus to change their composition order.
$$\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw (0,0.3) to (0,0.5);
\draw (-0.25,0.5) rectangle (0.25,0.9);
\draw (0,0.9) to (0,1.2);
\draw (-0.25,1.2) rectangle (0.25,1.6);
\draw[->] (0,1.6) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\node at (0,0.7){\small $f$};
\node at (0,1.4){\small $g$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw (0,0.3) to (0,0.5);
\draw (-0.25,0.5) rectangle (0.25,0.9);
\draw (0,0.9) to (0,1.2);
\draw (-0.25,1.2) rectangle (0.25,1.6);
\draw[->] (0,1.6) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\node at (0,0.7){\small $g$};
\node at (0,1.4){\small $f$};
\end{tikzpicture}$$
As described in Section \[twistedHeisenberg\], ${\mathcal{H}_{tw}}$ has a $\mathbb{Z}/2\mathbb{Z}$-grading where
(-2,-0.25) to (-2,0.75); (-2,0.25) circle \[radius=3pt\];
and
(-1,-0.25) to (-1,0.75); (-1,0.25) circle \[radius=3pt\];
have degree one, and other generating diagrams have degree zero. We also have supercommutativity relations (\[cicj=-cjci\]) and (\[anticommute\]) and supercyclicity relations (\[caps\]) and (\[cups\]). These relations have several interesting diagrammatic consequences.
Working with relation (\[anticommute\]), we have the following compuation: $$\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw[->] (0,0.3) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\draw (0,0.7) circle [radius=2pt];
\fill (0,1.4) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw[->] (0,0.3) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\fill (0,0.7) circle [radius=2pt];
\draw (0,1.4) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw[->] (0,0.3) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\draw (0,0.7) circle [radius=2pt];
\fill (0,1.4) circle [radius=2pt];
\end{tikzpicture}
=0.$$ Here the first equality is obtained by sending the solid dot around the annulus using trace relation, and the second equality is a consequence of relation (\[anticommute\]). Therefore the above diagram is equal to zero in the trace.
\[supertrace\] To demonstrate the subtlety with supercyclicity relations (\[caps\]) and (\[cups\]), consider the following situation: $$\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw[->] (0,0.3) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\draw (0,1.4) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=\hspace{6pt}-\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rounded corners]
\filldraw[blue,dashed,fill=blue!5!white] (0.5,1) circle [radius=40pt];
\filldraw[fill,blue,fill=white] (0.5,1) circle [radius=4pt];
\draw[->] (0,0.3) to (0,1.7);
\draw (0,1.7) arc(180:0:5mm);
\draw (1,1.7) to (1,0.3);
\draw (1,0.3) arc(360:180:5mm);
\draw (0,0.7) circle [radius=2pt];
\end{tikzpicture}$$ If we denote
(-2,-0.25) to (-2,0.75); (-2,0.25) circle \[radius=3pt\];
by $f$, with the usual trace relation, we would get $f\circ id=id\circ f$. However, in this case, we gain an extra negative sign from the supercyclicity relations. So, we must replace the usual trace relation $fg=gf$ with the supertrace relation $fg=(-1)^{|f||g|}gf$ in the ideal $\mathcal{I}$, where $|f|,|g|$ are the degrees of $f$ and $g$ with respect to the $\mathbb{Z}/2\mathbb{Z}$ grading. This example can be generalized to show that composition of an an odd morphism with a cycle of odd length is zero in the supertrace, since it will be equal to its negative when a hollow dot travels around the annulus and arrives to its original position.
We wish to restrict our study to the following subalgebra of the trace.
The *even trace* of ${\mathcal{H}_{tw}}$ is defined by $${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}\simeq \bigg( \bigoplus_{x\in \mathcal{O}b({\mathcal{H}_{tw}})} {\operatorname{End}}_{\overline{0}}(x) \bigg) \big/\mathcal{I}_{\overline{0}}$$ where ${\operatorname{End}}_{\overline{0}}(x)$ consists of even degree endomorphisms and $\mathcal{I}_{\overline{0}}$ is its ideal generated by $\operatorname{span}_{\mathbb{C}}\{fg-gf\}$ for all $f:x\rightarrow y$ and $g:y\rightarrow x$ , $x,y\in \mathcal{O}b({\mathcal{H}_{tw}})$.
This is the restriction of the trace to only the *even* part (with respect to the $\mathbb{Z}/2\mathbb{Z}$ grading induced by $\deg(c_i) = 1$). The odd part of the trace is not zero (it contains, e.g., $\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](-2,-0.25) to (-2,0.75);
\draw (-2,0.25) circle [radius=3pt];
\end{tikzpicture}$), but is not interesting from a representation theoretic viewpoint as explained above. The example of trace functions on the finite Hecke-Clifford algebra in [@WW Section 4.1] demonstrates the importance of the even trace.
Wan and Wang study the space of trace functions on the finite Hecke-Clifford algebra $\mathcal{H}_n$: linear functions $\phi: \mathcal{H}_n \rightarrow \mathbb{C}$ such that $\phi([h,h'])=0$ for all $h,h'\in \mathcal{H}_n$, and $\phi(h)=0$ for all $h\in (\mathcal{H}_n)_{\overline{1}}$. This latter requirement encodes the information that odd elements act with zero trace on any $\mathbb{Z}_2$-graded $\mathcal{H}_n$-module (because multiplication by an odd element results in a shift in degree). The space of such trace functions is clearly canonically isomorphic to the dual of the even cocenter, rather than of the full cocenter. The same observation holds for the trace of the affine Hecke-Clifford algebra, as studied in [@Mike].
We will see in Section 4 that the structure of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ is largely controlled by the even trace of the degenerate affine Hecke-Clifford algebra in type $A$; we therefore do not lose interesting representation-theoretic information by restricting to ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, and greatly simplify our calculations by doing so. For instance, the ambiguity in the supercyclicity relations identified in Example 3.\[supertrace\] does not interfere with calculations in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$.
Since $\mathcal{I}_{\overline{0}}$ is an ideal of $\bigoplus_{x\in \mathcal{O}b({\mathcal{H}_{tw}})}{\operatorname{End}}_{\overline{0}}(x)$, the compositions $fg$ and $gf$ must be even morphisms, even though individually $f$ and $g$ may be odd morphisms. This situation is analogous to the even cocenter of the degenerate affine Hecke-Clifford algebra studied in [@Mike], where Clifford generators $c_i$ do no appear individually (as they are odd generators), but still have an impact on the cocenter via the relation $c_i^2=-1$.
Diagrammatically, the above definition means that we will have an even number of hollow dots on our diagrams. In a diagram with $2n$ hollow dots, sliding one around the annulus from top to the bottom will multiply the diagram by $(-1)^{2n-1}(-1)=1$ where $(-1)^{2n-1}$ is a result of changing relative height with the remaining $2n-1$ hollow dots using relation (\[cicj=-cjci\]) and $(-1)$ is the result of sliding it through a clockwise cup using relation (\[cups\]).
For the sake of clarity, when working with diagrams in the even trace we will not draw them on an annulus, but will instead draw them inside square brackets, e.g. $ \left[ {
\xy (0,0)*{\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.5]
\hspace{1mm}
\draw[->] (0,0) to (1,2);
\draw[->] (1,0) to (0,2);
\end{tikzpicture}}; \endxy}\;\: \right]$. This notation refers to the equivalence class of the diagram in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$.
Our main theorem will relate ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ and $W^-$. In particular, we will establish that the correspondence in Table 1 gives an isomorphism between ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ and $W^-$. Recall that $w_{k,\ell} = t^{\ell} D^k \in W^-$.
[ |c|c|c|c| ]{} ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ & bidegree ($k$-$l$,$k$) & values of ($l$,$k$) & $W^-$\
(3,2) arc (-180:180:5mm); (3.95,2.2) circle \[radius=2pt\]; at (4.3,2.2) [$2a$]{};
& ($2a$+$1$,$2a$+$1$) & (0,$2a$+$1$) & $-2w_{0,2a+1}$\
$\left[\;\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,1);
\end{tikzpicture}\;\; \right]$ & ($1$,$0$) & ($-1$,$0$) & $\sqrt{2}w_{-1,0}$\
$\left[\;\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,1);
\fill (0,.5) circle[radius=2pt];
\node at (0.2,0.5) {\small $2$};
\end{tikzpicture} \right]$ & ($3$,$2$) & ($-1$,$2$) & $\sqrt{2}(w_{-1,2}-w_{-1,1})$\
$\left[\;\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0,0) to (0,1);
\end{tikzpicture}\;\; \right]$ & ($-1$,$0$) & ($1$,$0$) & $\sqrt{2}w_{1,0}$\
$\left[ \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.5]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\end{tikzpicture}\;\; \right]$ & ($n$,0) & ($-n$,0) & $\sqrt{2}w_{-n,0}$\
$\left[ \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.5]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (.05,1.6) circle (3pt);
\node at (-0.48,1.7) {\small $2a$};
\end{tikzpicture}\;\; \right]$ & ($n$+$a$,$a$) & ($-n$,$a$) & $\sqrt{2}(w_{-n,a}+O(w_{-n,a-1}))$\
$\left[\;\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,-0.25) to (1,0.75);
\draw[->] (1,-0.25) to (0,0.75);
\filldraw (.19, .55) circle[radius=2pt];
\end{tikzpicture}\;\; \right]$ + $\left[\;\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,-0.25) to (1,0.75);
\draw[->] (1,-0.25) to (0,0.75);
\filldraw (.8, .55) circle[radius=2pt];
\end{tikzpicture}\;\; \right]$ & ($3$,$1$) & ($-2$,$1$) & $2\sqrt{2}(w_{-2,1}-w_{-2,0})$\
\
Degenerate affine Hecke-Clifford algebra
----------------------------------------
We recall the definition of the degenerate affine Hecke-Clifford algebra of type $A_{n-1}$, denoted ${\mathfrak{H}^C}_n$, which was first studied in [@Naz]. Let $\mathcal{C}\ell_n$ be the Clifford algebra with generators $c_1, \ldots, c_n$, subject to the relations: $$c_i^2 = -1 \qquad \text{for } 1\leq i \leq n,$$ $$c_i c_j = - c_j c_i \qquad \text{if } i\not=j.$$ The symmetric group $S_n$ has a natural action on $\mathcal{C}\ell_n$ by permuting the generators.
Define the Sergeev algebra, or finite Hecke-Clifford algebra of type $A_{n-1}$, to be the semidirect product $$\mathbb{S} := \mathcal{C}\ell_n \rtimes \mathbb{C} S_n$$ corresponding to this action.
The degenerate affine Hecke algebra, ${\mathfrak{H}^C}_n$, is isomorphic as a vector space to $\mathbb{S} \otimes \mathbb{C}[x_1, \ldots, x_n]$. It is an associative unital algebra over $\mathbb{C}[u]$, where $u$ is a formal parameter usually set to 1, generated by $s_1,s_2,...,s_{n-1}$, $x_1,x_2,...,x_n$, and $c_1,c_2,...,c_n$ subject to relations making $\mathbb{C}[x_1, \ldots, x_n]$, $C\ell_V$, and $\mathbb{C} S_n$ subalgebras, along with the additional relations:
$$\begin{aligned}
x_i c_i &= -c_i x_i, \quad x_i c_j = c_j x_i \quad (i\not= j) ,\\
\sigma c_i & = c_{\sigma(i)} \sigma \quad (1\leq i \leq n, \sigma \in S_n),\\
x_{i+1}s_{i}-s_{i}x_{i} & =u(1-c_{i+1}c_{i}) , \\
x_{j}s_{i} & =s_{i}x_{j}\quad (j\neq i,i+1) .\end{aligned}$$
It also has a $\mathbb{Z}/{2\mathbb{Z}}$ grading via $\deg(s_i)=\deg(x_i)=0$ and $\deg(c_i)=1$. This algebra is also called the affine Sergeev algebra, and later on we will see that it controls the endomorphisms of up strands in $\mathcal{H}_{tw}$.
Trace of the degenerate affine Hecke-Clifford algebra
-----------------------------------------------------
The second author computes the trace (or zeroth Hochschild homology) of the even part of ${\mathfrak{H}^C}_n$ as a vector space in [@Mike], where he gives an explicit description of a vector space basis for types $A$, $B$ and $D$. Here we recall the result for type $A$.
Let $I$ be the standard root system of type $A_{n-1}$, and let $W= S_n$ be the Weyl group. For a partition $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k) \vdash n$, let $J_\lambda$ be the unique minimal subset of $I$ (up to conjugation by $W$) such that $W_{J_\lambda}$ contains an element of cycle type $\lambda.$ Let $w_{\lambda} \in W$ be the element $(1,\ldots, \lambda_1)(\lambda_1 +1, \ldots, \lambda_1 + \lambda_2)\ldots (n-\lambda_n +1, \ldots, n)$. Then $w_\lambda \in W_{J_\lambda}$.
Let $V$ be the standard representation of $\mathfrak{sl}_n$, with basis $\{x_1, \ldots, x_n\}$. Denote by $V^2$ the vector space with basis $\{x_1^2, \ldots, x_n^2\}$.
Finally, fix a basis $\{f_{J_{\lambda};i}\}$ of the vector space $S((V^2)^{W_{J_\lambda}})^{N_W(W_{J_\lambda})}$, where $S(U)$ denotes the symmetric algebra of the vector space $U$, and $N_W(W_{J_\lambda})$ denotes the normalizer of the parabolic subgroup $W_{J_\lambda}$ in $W$. We have the following description of a basis for ${\operatorname{Tr}}({\mathfrak{H}^C}_n)_{\overline{0}}$ in type $A$.
\[cocenter HC\][@Mike Theorem 5.4] The set $\{w_{\lambda}f_{J_{\lambda;i}}\}_{\lambda\in\mathcal{OP}_n}$ is a basis of ${\operatorname{Tr}}({\mathfrak{H}^C}_n)_{\overline{0}}$, where $\mathcal{OP}_n$ is the set of partitions of $n$ with all odd parts.
Let $n=3$. Then we have $\mathcal{OP}_3=\{(1,1,1),(3)\}$.
For $\lambda=(1,1,1)$, we have $w_\lambda = 1$ and $J_{\lambda} = \emptyset$, since ${W}_{J_\lambda} = \{1\}$. Thus $N_{W}({W}_{J_\lambda}) = {S_3}$. So, we choose a basis $\{f_{J_{\lambda};i}\}$ of the vector space $S((V^2)^{W_J})^{N_W(W_J)} = \mathbb{C}[x_1^2 , x_2^2, x_3^2]^{S_3}$, i.e. the symmetric polynomials in 3 variables. We can take $\{f_{J_{\lambda};i}\} = \{s_\nu\}$, the Schur polynomials in 3 variables.
For $\lambda=(3)$, we have $w_{\lambda}=(123)$, a $3$-cycle in $S_3$. Thus $J_\lambda = I$, ${W}_{J_\lambda}=W$ and $N_{W}(W_{J_\lambda})=W$. Therefore $f_{J_{\lambda},i}$ is a basis of $\mathbb{C}[x_1^2+x_2^2+x_3^2]$, polynomials in the variable $(x_1^2+x_2^2+x_3^2)$ (in this case, the $N_W(W_{J_\lambda})$-invariance is superfluous).
Therefore a basis of $Tr({\mathfrak{H}^C}_3)_{\overline{0}}$ is given by $\{ s_\nu\}\cup \{(123)(x_1^2+ x_2^2+ x_3^2)^{n}\}_{n\in \mathbb{N}}$ where $\{s_\nu\}$ are the Schur polynomials in 3 variables.
Note that this bases does not contain any classes indexed by partitions with even parts. Correspondingly, we will see that degree zero diagrams in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ containing even cycles are zero.
Distinguished elements $h_n$
----------------------------
Define the elements: $$\begin{aligned}
{\ifthenelse{\isempty{(x_1^{j_1} \cdots x_n^{j_n}) (c_1^{\epsilon_1} \cdots c_n^{\epsilon_n})}{}}{h_{n}}{h_{n}^{(x_1^{j_1} \cdots x_n^{j_n}) (c_1^{\epsilon_1} \cdots c_n^{\epsilon_n})}}} &:= \left[ \;{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (.05,1.6) circle (2pt);
\filldraw (.78,1.6) circle (2pt); \filldraw (1.58,1.6) circle (2pt);
\filldraw (3.18,1.6) circle (2pt);
\draw (.35,1.2) circle (2.5pt);
\draw (.62,1.2) circle (2.5pt);
\draw (1.425,1.2) circle (2.5pt);
\draw (3.025,1.2) circle (2.5pt);
\node at (-0.18,1.7) {${\scriptstyle}j_1$};
\node at (.55,1.7) {${\scriptstyle}j_2$};
\node at (1.35,1.7) {${\scriptstyle}j_3$};
\node at (3.45,1.7) {${\scriptstyle}j_n$};
\node at (.051,1.25) {$ \epsilon_1$};
\node at (.99,1.25) {$\epsilon_2$};
\node at (1.76,1.25) {$\epsilon_3$};
\node at (3.37,1.25) {$\epsilon_n$};
\end{tikzpicture}}; \endxy}
\; \right],
\\
{\ifthenelse{\isempty{(x_1^{j_1} \cdots x_n^{j_n}) (c_1^{\epsilon_1} \cdots c_n^{\epsilon_n})}{}}{h_{-n}}{h_{-n}^{(x_1^{j_1} \cdots x_n^{j_n}) (c_1^{\epsilon_1} \cdots c_n^{\epsilon_n})}}} &:= \left[ \;{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw[<-] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[<-] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[<-] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[<-] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (.05,1.6) circle (2pt);
\filldraw (.78,1.6) circle (2pt); \filldraw (1.58,1.6) circle (2pt);
\filldraw (3.18,1.6) circle (2pt);
\draw (.35,1.2) circle (2.5pt);
\draw (.62,1.2) circle (2.5pt);
\draw (1.425,1.2) circle (2.5pt);
\draw (3.025,1.2) circle (2.5pt);
\node at (-0.18,1.7) {${\scriptstyle}j_1$};
\node at (.55,1.7) {${\scriptstyle}j_2$};
\node at (1.35,1.7) {${\scriptstyle}j_3$};
\node at (3.45,1.7) {${\scriptstyle}j_n$};
\node at (.051,1.25) {$ \epsilon_1$};
\node at (.99,1.25) {$\epsilon_2$};
\node at (1.76,1.25) {$\epsilon_3$};
\node at (3.37,1.25) {$\epsilon_n$};
\end{tikzpicture}}; \endxy}
\; \right],\nonumber\end{aligned}$$
where $\epsilon_i \in \{0,1\}$. In both of these elements, we consider the hollow dots to be descending in height from left to right, so that the dot labeled $\epsilon_1$ is the highest.\
These elements are analogues to those denoted $h_{\pm n} \otimes (x_1^{j_1} \cdots x_n^{j_n}) $ in [@CLLS].
Additionally, set $${\ifthenelse{\isempty{\sum x_i^{j_i}}{}}{h_{n}}{h_{n}^{\sum x_i^{j_i}}}} = \sum {\ifthenelse{\isempty{x_i^{j_i}}{}}{h_{n}}{h_{n}^{x_i^{j_i}}}}.$$
\[basic hn facts\] For $n\geq 1$ and $1\leq i \leq n-1$ we have
1. ${\ifthenelse{\isempty{x_i}{}}{h_{\pm n}}{h_{\pm n}^{x_i}}} ={\ifthenelse{\isempty{x_{i+1}}{}}{h_{\pm n}}{h_{\pm n}^{x_{i+1}}}} \pm {\ifthenelse{\isempty{}{}}{h_{\pm i}}{h_{\pm i}^{}}} {\ifthenelse{\isempty{}{}}{h_{\pm (n-i)}}{h_{\pm (n-i)}^{}}} $.
2. ${\ifthenelse{\isempty{ x_i c_j}{}}{h_{\pm n}}{h_{\pm n}^{ x_i c_j}}} = - {\ifthenelse{\isempty{x_{i+1}c_{j+1}}{}}{h_{\pm n}}{h_{\pm n}^{x_{i+1}c_{j+1}}}} $.
Part (1) is just [@CLLS Lemma 14], except our solid dot sliding relation through crossing involves an extra term with hollow bubbles. But cycles with single hollow dot are zero since sending the hollow dot around the annulus gives us the same diagram with a negative sign. For the above calculations, our $n$-cycles split into smaller cycles with single hollow dot at least on one of them. The proof of part 2 depends on the relative position of $i$ and $j$, but is a straightforward computation.
Let $w\in S_n$, and define the elements: $$f_{w; j_1, \ldots, j_n; \epsilon_1, \ldots, \epsilon_n}=
{
\xy (0,0)*{\begin{tikzpicture}
\draw[very thick][->] (-.55,0) -- (-.55,1.5);
\draw[very thick][->] (.55,0) -- (.55,1.5);
\draw[fill=white!20,] (-.8,.4) rectangle (.8,.8);
\node () at (0,.55) {$w$};
\node () at (0,1.25) {$\cdots$};
\node () at (0,.25) {$\cdots$};
\filldraw (-.55,1.25) circle (2pt);
\filldraw (.55,1.25) circle (2pt);
\draw (-.55,.9) circle (2pt);
\draw (.55,.9) circle (2pt);
\draw (-.75,.9) node {$\epsilon_1$};
\draw (.8,.9) node {$\epsilon_n$};
\draw (-.75,1.25) node {$j_1$};
\draw (.8,1.25) node {$j_n$};
\end{tikzpicture}}; \endxy}$$ and $$f_{w; j_1, \ldots, j_n; \epsilon_1, \ldots, \epsilon_n}=
{
\xy (0,0)*{\begin{tikzpicture}
\draw[very thick][<-] (-.55,0) -- (-.55,1.5);
\draw[very thick][<-] (.55,0) -- (.55,1.5);
\draw[fill=white!20,] (-.8,.4) rectangle (.8,.8);
\node () at (0,.55) {$w$};
\node () at (0,1.25) {$\cdots$};
\node () at (0,.25) {$\cdots$};
\filldraw (-.55,1.25) circle (2pt);
\filldraw (.55,1.25) circle (2pt);
\draw (-.55,.9) circle (2pt);
\draw (.55,.9) circle (2pt);
\draw (-.75,.9) node {$\epsilon_1$};
\draw (.8,.9) node {$\epsilon_n$};
\draw (-.75,1.25) node {$j_1$};
\draw (.8,1.25) node {$j_n$};
\end{tikzpicture}}; \endxy}.$$
\[red to hn lm\] Let $w\in S_n$ and $(n_1, \ldots, n_r)$ be a composition of $n$. Then $$[f_{\pm w; j_1, \ldots, j_n; \epsilon_1, \ldots, \epsilon_n}] = \sum d_{n_1, \ldots, n_r} {\ifthenelse{\isempty{ p_{n_1} c_{n_1}}{}}{h_{n_1}}{h_{n_1}^{ p_{n_1} c_{n_1}}}} \ldots {\ifthenelse{\isempty{ p_{n_r} c_{n_r}}{}}{h_{n_r}}{h_{n_r}^{ p_{n_r} c_{n_r}}}}$$ for constants $d_{n_1, \ldots, n_r} \in \mathbb{C}$, polynomials $p_{n_i}$ in $i$ variables, and elements $c_{n_i}$ consisting of at most $i$ Clifford generators (e.g. $c_{n_3} = \{ c_1^{\epsilon_1} c_2^{\epsilon_2} c_3^{\epsilon_3} | \epsilon_i \in \{0,1\}\}$).
We proceed by induction on $\sum {\epsilon_i}$. The base case is $\sum \epsilon_i = 0$; then $$[f_{\pm w; j_1, \ldots, j_n; \epsilon_1, \ldots, \epsilon_n}] = [f_{\pm w; j_1, \ldots, j_n}]$$ and we apply [@CLLS Lemma 15].
Now assume the statement is true for $\sum \epsilon_i = k$ for all $k<m\leq n$. Take $(\epsilon_1, \ldots, \epsilon_n)$ so that $\sum \epsilon_i = m$. Choose $g\in S_n$ such that $gwg^{-1} = w_\lambda$, where $\lambda$ is the cycle type of $w$ (so $gwg^{-1} = (s_1 \ldots s_{n_1 -1})\ldots (s_{n_1 + \ldots + n_{r-1}} \ldots s_{n_1 +\ldots +n_r -1})$). Let $p= x_1^{j_1} \ldots x_n^{j_n}$ and $c= c_1^{\epsilon_1} \ldots c_n^{\epsilon_n}$. Then we have $$f_{\pm w; j_1, \ldots, j_n; \epsilon_1, \ldots, \epsilon_n} = pcw = (-1)^\epsilon cpw$$ where $$\epsilon = \sum_{\epsilon_i = 1} j_i.$$
Thus conjugating by $g$ gives that $$\begin{aligned}
gpcwg^{-1}& = (-1)^{\epsilon} gcpwg^{-1} \\ &=(-1)^\epsilon (g.c) gpwg^{-1} \\&= (-1)^{\epsilon} \left[ (g.c)(g.p)gwg^{-1} + (g.c) p_L wg^{-1}\right], \end{aligned}$$ where $p_L$ is a polynomial of degree less than $j_1 + \ldots + j_n$. Note that $gwg^{-1}$ is a product of cycles, so the first term in the above expression has the correct form. In the second term, we have $\{i | \epsilon_{g(i)} =1\} \leq m$ (strict inequality can occur if $g$ has fixed points). If $\{i | \epsilon_{g(i)} =1\} < m$, we are done by induction, so assume that we have equality.
Now repeat the process on the second term, choosing a $g' \in S_n$ such that $g'(wg^{-1}) (g')^{-1}$ is a product of cycles, and conjugating $(g.c) p_L wg^{-1}$. Each application of this process results in one term in which the symmetric group element is a product of cycles (which has the desired form), and one term with the degree of the polynomial part strictly lesser and the degree of the Clifford part weakly lesser.
If the degree of the Clifford part ever strictly decreases, we are done. If not, the conjugation will eventually reduce the degree of the polynomial part to 0, so we have an element of the form $c' \sigma$, $c' \in C\ell_n$ and $\sigma \in S_n$. Choose a $g''\in S_n$ such that $g'' \sigma (g'')^{-1}$ is a product of cycles; then $$g'' c \sigma (g'')^{-1} = (g''c) g'' \sigma (g'')^{-1}.$$ This now has the desired form.
\[red to hn\] Let $w\in S_n$ and $(n_1, \ldots, n_r)$ a composition of $n$. Then $$[f_{\pm w; j_1, \ldots, j_n; \epsilon_1, \ldots, \epsilon_n}] = \sum d_{n_1, \ldots, n_r} {\ifthenelse{\isempty{x_1^{\ell_1} c_1^{k_1}}{}}{h_{ \pm n_1}}{h_{ \pm n_1}^{x_1^{\ell_1} c_1^{k_1}}}}\ldots {\ifthenelse{\isempty{x_1^{\ell_r} c_{1}^{k_r}}{}}{h_{\pm n_r}}{h_{\pm n_r}^{x_1^{\ell_r} c_{1}^{k_r}}}}$$ where $d_{n_1, \ldots, n_r} \in \mathbb{C}$ and $\ell_1, \ldots, \ell_r, k_1, \ldots, k_r \in \mathbb{N}$.
This follows immediately from the preceding lemmas.
Proposition \[red to hn\] allows us to write any element in ${{\operatorname{Tr}}^>(\mathcal{H}_{tw})_{\overline{0}}}$ or ${{\operatorname{Tr}}^<(\mathcal{H}_{tw})_{\overline{0}}}$ as a linear combination of the elements ${\ifthenelse{\isempty{}{}}{h_{n}}{h_{n}^{}}}$. We will therefore direct our attention to these elements in future computations.
Gradings in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$
----------------------------------------------------------------------
The next lemma follows from diagrammatic computations in the next section. We record it here for convenience of terminology.
\[rank filtration\] The algebra ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ is $\mathbb{Z}$-filtered where $\deg\left({\ifthenelse{\isempty{x_1^{2a}}{}}{h_{n}}{h_{n}^{x_1^{2a}}}}\right) \leq n$ for any $a\geq 0$.
This is called the rank filtration. Denote by ${{\operatorname{Tr}}^>(\mathcal{H}_{tw})_{\overline{0}}}$ (resp. ${{\operatorname{Tr}}^<(\mathcal{H}_{tw})_{\overline{0}}}$) the subalgebra of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ generated by ${\ifthenelse{\isempty{x_1^{2a}}{}}{h_{n}}{h_{n}^{x_1^{2a}}}}$, $n\geq 1$ (resp. $n\leq 1$).
\[dot filtration\] The algebra ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ is $\mathbb{Z}^{\geq 0}$-filtered where $\deg\left({\ifthenelse{\isempty{ x_1^{2a}}{}}{h_{n}}{h_{n}^{ x_1^{2a}}}}\right)\leq a$ for any $a \geq 0$.
Dots can slide through crossings modulo a correction term containing fewer dots.
This is called the dot filtration, and corresponds to the differential filtration (given by $\deg(w_{\ell, k}) = k$) in $W^-$.
These filtrations are compatible, so ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ is $(\mathbb{Z}\times \mathbb{Z}^{\geq 0})$-filtered with ${\ifthenelse{\isempty{ x_1^{2a}}{}}{h_{n}}{h_{n}^{ x_1^{2a}}}}$ in bidegree $(n, a)$. For $\omega \in \{>,<,0\}$ denote the associated graded object by ${\operatorname{gr}}{{\operatorname{Tr}}^\omega(\mathcal{H}_{tw})_{\overline{0}}}$. Define a generating series for the graded dimension of ${\operatorname{gr}}{{\operatorname{Tr}}^\omega(\mathcal{H}_{tw})_{\overline{0}}}$ by $$P_{{\operatorname{gr}}{{\operatorname{Tr}}^\omega(\mathcal{H}_{tw})_{\overline{0}}}}(t,q) = \sum_{r\in \mathbb{Z}} \sum_{k\in \mathbb{Z}, k\geq 0 } \dim {\operatorname{gr}}{{\operatorname{Tr}}^\omega(\mathcal{H}_{tw})_{\overline{0}}}[r,k] t^r q^k.$$
The following is an easy calculation using Proposition \[triangularDecomposition\] and Proposition \[cocenter HC\]. They are not used in the proof of the main result, but we record them here for convenience.
\[grDim Tr\] The graded dimensions of ${\operatorname{gr}}{{\operatorname{Tr}}^>(\mathcal{H}_{tw})_{\overline{0}}}$ and ${\operatorname{gr}}{{\operatorname{Tr}}^<(\mathcal{H}_{tw})_{\overline{0}}}$ are given by: $$P_{{{\operatorname{gr}}(\mathcal{D}^-)}^>} = \prod_{r\geq 0} \prod_{k>0} \frac{1}{1-t^{2r+1} q^k};$$ $$P_{{{\operatorname{gr}}(\mathcal{D}^-)}^<} = \prod_{r\leq 0} \prod_{k>0} \frac{1}{1-t^{2r-1} q^k}.$$
Note that the rank grading and dot gradings are shifted by 1 for clockwise bubbles (so $d_2$ is in bidegree $(1,2)$ and $d_4$ is in bidegree $(1,3)$). This is a consequence of the decomposition formula in Lemma \[bubbleDecomp\].
Bubbles
=======
We investigate the endomorphisms of 1 in ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$, known as bubbles. We prove that all bubbles can be written in terms of clockwise bubbles, and deduce formulas for moving bubbles past strands in the trace.
Definition and basic properties
-------------------------------
Elements of ${\operatorname{End}}_{\mathcal{H}_{tw}}(1)$ are $\mathbb{C}$-linear combinations of possibly intersecting or nested closed diagrams, which may have dots. We can always separate the nested pieces, and resolve any crossing that occur between different closed diagrams using the defining relations and end up with non intersecting, not nested closed oriented diagrams. Each one can be deformed into an oriented circle, possibly with dots, via an isotopy. A single closed, oriented, non self intersecting diagram is called a bubble. They are the building blocks of endomorphisms of the identity object in $\mathcal{H}_{tw}$.
We define $$\bar{d}_{k,l}:=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\draw (3.95,1.8) circle [radius=2pt];
\node at (4.2,1.8) {$l$};
\node at (4.2,2.2) {$k$};
\end{tikzpicture}
\hspace{0.5cm}
\text{and}
\hspace{0.5cm}
d_{k,l}:=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\draw (3.95,1.8) circle [radius=2pt];
\node at (4.2,1.8) {$l$};
\node at (4.2,2.2) {$k$};
\end{tikzpicture}
\hspace{0.5cm}
\text{for}
\hspace{2mm} k,l\in\mathbb{Z}_{\geq 0}.$$
Given any closed diagram with any configuration of dots, it is possible to collect the hollow dots and the solid dots together, possibly after multiplying the diagram by $-1$, by using relation . Solid dots move freely along caps and cups, and hollow dots may capture a negative sign while moving along caps or cups, depending on the orientation. After regrouping, we may assume that the dots are placed on the right middle side of the diagram as above.
Moreover, using the left two equations in relation , we can erase a pair of hollow dots, possibly by changing the sign of the diagram.
Therefore the set $\{d_{k,l},\bar{d}_{k,l}|k\in\mathbb{Z}_{\geq0}, l\in\{0,1\}\}$ is a spanning set for ${\operatorname{End}}_{\mathcal{H}_{tw}}(1)$.
In our defining relations, we have that
$$\bar{d}_{0,0}=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\end{tikzpicture}\hspace{6pt}
=1
\hspace{1cm} \text{and}
\hspace{1cm}
\bar{d}_{0,1}=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\draw (4,2) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=0.$$
Further, we have the following.
\[hollow dots\] We have that $\bar{d}_{k,1}=0$ and $d_{k,1}=0$ for all non-negative integers $k$.
An example computation shows that $$\bar{d}_{1,1}=\hspace{6pt}\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\draw (3.95,2.2) circle [radius=2pt];
\fill (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\draw (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.75,1.6) circle [radius=2pt];
\draw (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\bar{d}_{1,1}=0,$$ where in the second equality, negative sign comes from relation (11), and the third equality comes from sliding the solid dot around.
More generally, if we have $k$ solid dots where $k$ is an even integer, then sliding the hollow dot around the circle and passing it through $k$ solid dots multiplies the diagram by $(-1)^{k+1}$, so the diagram is zero. If $k$ is an odd number, sliding a solid dot around the circle and passing it through a hollow dot catches a minus sign, so these diagrams are zero as well.
These arguments do not depend on the orientation of the bubble, hence the result follows.
From now on, we will assume that the second index in $\bar{d}_{k,l}$ and $d_{k,l}$ is always zero. We will omit it from our notation and write $d_k$ instead of $d_{k,0}$, and $\bar{d}_k$ instead of $\bar{d}_{k,0}$.
\[odd dots\] We have that $d_{2n+1}=\bar{d}_{2n+1}=0$ for all non-negative integers $n$.
Note that $$\bar{d}_1=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (4,2) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\draw (4,2) circle [radius=2pt];
\draw (3.95,2.2) circle [radius=2pt];
\fill (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\draw (4,2) circle [radius=2pt];
\draw (3.05,2.2) circle [radius=2pt];
\fill (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (4,2) circle [radius=2pt];
\draw (3.95,2.2) circle [radius=2pt];
\draw (3.05,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (4,2) circle [radius=2pt];
\draw (3.95,2.2) circle [radius=2pt];
\draw (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}$$ $$=
\hspace{6pt}-\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\draw (4,2) circle [radius=2pt];
\draw (3.95,2.2) circle [radius=2pt];
\fill (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}-\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (4,2) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
=\hspace{6pt}0.$$ The same arguments works for any odd number of solid dots and works for clockwise oriented bubbles.
\[bubbleDecomp\] We have that $${\displaystyle}\bar{d}_{2n}=\sum_{2a+2b=2n-2}\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (0,0) arc (0:180:5mm);
\draw (0,0) arc (360:180:5mm);
\draw[<-] (0.1,0) arc (-180:0:5mm);
\draw (0.1,0) arc (180:0:5mm);
\fill (-0.2,0.4) circle[radius=2pt];
\fill (0.9,0.4) circle[radius=2pt];
\node at (-0.04,0.6) {$2a$};
\node at (1.05,0.6) {$2b$};
\end{tikzpicture}
=\sum_{2a+2b=2n-2}\bar{d}_{2a}d_{2b}$$ for any integers $a,b$ and $n\geq1$.
For the $n=1$ case, we have the following computation:
$$\begin{aligned}
\bar{d}_2=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\fill (3.95,1.8) circle [radius=2pt];
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (0,0) arc (10:180:5mm);
\draw (0,-0.18) arc (350:180:5mm);
\draw (0.1,0) arc (170:-170:5mm);
\draw (-0.02,-0.2) to (0.1,0.01);
\draw (-0.01,0.02) to (0.11,-0.2);
\fill (-0.1,0.23) circle[radius=2pt];
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (0,0) arc (10:180:5mm);
\draw (-1,-0.08) arc (180:350:5mm);
\draw (0.1,0) arc (170:-170:5mm);
\draw (-0.02,-0.2) to (0.1,0.01);
\draw (-0.01,0.02) to (0.11,-0.2);
\fill (0.2,0.23) circle[radius=2pt];
\end{tikzpicture}
\hspace{6pt}
+\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (0,0) arc (0:180:5mm);
\draw (0,0) arc (360:180:5mm);
\draw[<-] (0.1,0) arc (-180:0:5mm);
\draw (0.1,0) arc (180:0:5mm);
\end{tikzpicture}
\hspace{6pt}
+\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (0,0) arc (0:180:5mm);
\draw (0,0) arc (360:180:5mm);
\draw[<-] (0.1,0) arc (-180:0:5mm);
\draw (0.1,0) arc (180:0:5mm);
\draw (-0.04,-0.15) circle[radius=2pt];
\draw (0.13,0.1) circle[radius=2pt];
\end{tikzpicture}\hspace{6pt}
=\hspace{6pt}d_0\end{aligned}$$
where the first diagram on right hand side is zero since it contains a left curl, the second term is $\bar{d}_0d_0=d_0$ and the last term is zero by Lemma \[hollow dots\].
For general $n$, if you replace one of the solid dots with a right-twist curl, and slide the remaining $2n-1$ dots through the crossings using relations \[dotSlide: bottomLeft\] and \[dotSlide: topLeft\] repeatedly, we will get many resolution terms, consisting of a sum of product of counterclockwise and clockwise bubbles, some with only solid dots, some with hollow dots as well. The terms with hollow dots are zero, and so are the terms with an odd number of solid dots. Also, the figure eight shape contains a left twist curl, so it is zero as well, which proves the statement.
Algebraic independence of bubbles
---------------------------------
A categorified Fock space action for $\mathcal{H}_{tw}$ is described in [@CS Section 6.3]. $\mathcal{H}_{tw}$ acts on the category $\mathfrak{S}$, whose objects are induction and restriction functors between $\mathbb{Z}/2\mathbb{Z}$-graded finite dimensional $\mathbb{S}_n$-modules, for all $n\geq1$. Morphisms of $\mathfrak{S}$ are natural transformations between the induction and restriction functors.
Following Khovanov’s approach from [@Khovanov], let $\mathfrak{S}_n$ be the subcategory of $\mathfrak{S}$, whose objects start with induction or restriction functors from $\mathbb{Z}/2\mathbb{Z}$-graded finite dimensional $\mathbb{S}_n$-modules. For every $n\in\mathbb{Z}_{\geq1}$, we have a functor $F_n:\mathcal{H}_{tw}\rightarrow\mathfrak{S}_n$ sending $P$ to ${\operatorname{Ind}}_n^{n+1}$ and sending $Q$ to ${\operatorname{Res}}_n^{n-1}$.
Note that $F_n$ sends ${\operatorname{End}}_{\mathcal{H}_{tw}}(1)$ to the center of $\mathbb{S}_n$, which is same as the center of $\mathbb{C} [S_n]$.
Explicit descriptions of the actions of a crossing, a cup and a cap are provided in [@CS]. We would like to study the action of clockwise bubbles to show their algebraic independence. Note that $d_{2k}$ is obtained as composition of a cup, $k$ copies of ${\ifthenelse{\isempty{x_1}{}}{h_{1}}{h_{1}^{x_1}}}$ and a cap. $$\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {$k$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (1,-3.5) arc (0:180:5mm);
\fill (1,-3.8) circle[radius=2pt];
\draw[->] (1,-4) to (1,-3.5);
\draw[->] (0,-3.5) to (0,-4);
\node at (0,-4.15) {\vdots};
\node at (1,-4.15) {\vdots};
\fill (1,-4.8) circle[radius=2pt];
\draw[->] (1,-5) to (1,-4.5);
\draw[->] (0,-4.5) to (0,-5);
\draw[->] (0,-5) arc (-180:0:5mm);
\draw [decorate,decoration={brace,amplitude=2pt},xshift=-4pt,yshift=0pt] (1.3,-3.5) -- (1.3,-5) node [black,midway,xshift=15pt] {\footnotesize $k$ dots};
\end{tikzpicture}$$
Therefore to study the action of $d_{2k}$, we need to know the action of ${\ifthenelse{\isempty{x_1}{}}{h_{1}}{h_{1}^{x_1}}}$ in addition to actions of cups and caps. Now ${\ifthenelse{\isempty{x_1}{}}{h_{1}}{h_{1}^{x_1}}}$ is defined as a combination of caps, cups and crossings: $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.15]
\draw[->] (0,0) to (0,1);
\fill (0,.5) circle[radius=2pt];
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,.5);
\draw[<-] (0.5,0.5) arc (-180:0:3mm);
\draw[->] (0,.5) to (.5,1);
\draw[->] (.5,.5) to (0,1);
\draw[->] (1.1,1) to (1.1,.5);
\draw[->] (0,1) to (0,1.5);
\draw[<-] (1.1,1) arc (0:180:3mm);
\end{tikzpicture}$$
Using the explicit description of Fock space representation of $\mathcal{H}$ in [@CS], we compute the required actions. These computations give that ${\ifthenelse{\isempty{x_1}{}}{h_{-1}}{h_{-1}^{x_1}}}$ acts by sending $$1 \mapsto J_{n+1}={\displaystyle}\sum_{i=1}^{n}(1-c_{n+1}c_i)(i,n+1).$$ This is the $(n+1)$-st even Jucys-Murphy element. Therefore ${\ifthenelse{\isempty{x_1^2}{}}{h_{1}}{h_{1}^{x_1^2}}}$ acts by multiplication by $J_{n+1}^2$. This is analogous to the untwisted case where the same element acts as multiplication by a Jucys-Murphy element.
Finally, the action of the bubble $d_{2k}$ is given by multiplication by $${\displaystyle}\sum_{i=1}^{n}(i\leftrightarrow n+1)J_{n+1}^{2k}(n+1\leftrightarrow i)-c_n(i\leftrightarrow n+1)J_{n+1}^{2k}(n+1\leftrightarrow i)c_1,$$ where $(i\leftrightarrow n)$ denotes the $n$-cycle $s_is_{i+1}....s_{n-1}$.
Here we can apply the filtration argument on the number of disturbances of permutations as done in [@Khovanov Section 4] to obtain the following.
\[bubbles\] The elements $\{d_{2k}\}_{k\geq0}$ are algebraically independent, i.e. there is an isomorphism $${\operatorname{End}}_{\mathcal{H}_{tw}}(1)\cong k[d_0,d_2,d_4,...].$$
Therefore the bubbles are algebraically independent, and they form of a copy of a polynomial ring in infinitely many variables.
Counter-clockwise bubble slide moves
------------------------------------
In order to describe ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ as a vector space, it would be convenient to have a standard form for our diagrams in the trace. In particular, we want to collect all the bubbles appearing in a diagram on the rightmost part of the diagram. In order to do so, we must describe how bubbles slide through upward and downward strands. Note that since we can work with local relations, the bubbles don’t have to interact with solid dots or crossings, they can simply slide through under a crossing or under a solid dot.
All calculations in this section take place in the trace, though we omit the brackets in some situations for readability.
\[bubbleslide to curl\] We have that ${\displaystyle}[\bar{d}_{2n},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]=2\sum_{k=1}^n \left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.8,0.8) {\small 2k-1};
\end{tikzpicture}\;\right]$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for any positive integer $n$.
The proof is a direct computation, given below: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\node at (2.5,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\fill (3.5,1) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$1$};
\node at (3.6,1) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.7,0.67) circle (1pt);
\draw (0.85,0.65) circle [radius=1pt];
\draw (0.98,0.57) circle (1pt);
\node at (0.65,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
\\ &=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\fill (3.5,1) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$1$};
\node at (3.6,1) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
\\ &=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\fill (3.5,1) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$2$};
\node at (3.7,1) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.65,0.8) {\small $2n$-$2$};
\fill (1,0.75) circle (1pt);
\node at (1.1,0.75) {};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.7,0.67) circle (1pt);
\draw (0.85,0.65) circle (1pt);
\draw (0.98,0.57) circle (1pt);
\node at (0.65,0.8) {\small $2n$-$2$};
\fill (1,0.75) circle (1pt);
\node at (1.1,0.75) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\fill (3.5,1) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$2$};
\node at (3.7,1) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\fill (3.5,1) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$4$};
\node at (3.7,1) {\small $4$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {\small $2n$-$3$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.60]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {\small $2n$-$1$};
\end{tikzpicture}\end{aligned}$$ Continuing to slide dots in the first term in this way, we obtain:
$$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3.5,1) circle (2pt);
\node at (3.8,1) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2\sum_{k=1}^n\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.8,0.8) {\small $2k$-$1$};
\end{tikzpicture}.\end{aligned}$$
\[curl split\] We have that $${\displaystyle}\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.75,0.8) {2n+1};
\end{tikzpicture}\;\right]
=
\sum_{a+b=n}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\fill (4,1) circle (2pt);
\node at (3,1.8) {2a};
\node at (4.3,1) {2b};
\end{tikzpicture}\;\right]$$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for any non-negative integer $n$.
This is an easy computation using induction on $n$. The base case is $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (1,0.27) circle (1pt);
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\draw (3.5,1) circle (2pt);
\draw (4,0.8) circle (2pt);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\end{tikzpicture}$$ where the first term after the first equality contains a left twist curl, and the last term is zero since a bubble with a hollow dot is zero.
For the induction step, suppose the statement holds for $n\geq 1$. Then $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$3$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$2$};
\fill (1,0.27) circle (1pt);
\node at (1.1,0.27) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$+$2$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\draw (3.5,1) circle (2pt);
\draw (4,0.8) circle (2pt);
\node at (3,1.8) {\small $2n$+$2$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$2$};
\fill (1,0.27) circle (1pt);
\node at (1.1,0.27) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$+$2$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$1$};
\fill (1,0.27) circle (1pt);
\node at (1.1,0.27) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\fill (4,1) circle (2pt);
\node at (3,1.8) {\small $2n$+$1$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\fill (4,1) circle (2pt);
\draw (3.5,1) circle (2pt);
\draw (4,0.8) circle (2pt);
\node at (3,1.8) {\small $2n$+$1$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$+$2$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$1$};
\fill (1,0.27) circle (1pt);
\node at (1.1,0.27) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$+$2$};
\node at (4.3,1) {};
\end{tikzpicture},\end{aligned}$$ where on the second line, we know that counter-clockwise bubbles with odd number of hollow dots are zero by Proposition \[odd dots\], and the terms with hollow dots are zero by Proposition \[hollow dots\].
Now we can apply our induction hypothesis to the upper part of $\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$1$};
\fill (1,0.27) circle (1pt);
\node at (1.1,0.27) {\small $2$};
\end{tikzpicture}$ to get that
$$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.75]
\draw (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw[->] (0.95,0.46) to [out=75, in=270] (1,1);
\fill (0.8,0.67) circle (1pt);
\node at (0.7,0.78) {\small $2n$+$3$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}\sum_{a+b=n}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\fill (4,1) circle (2pt);
\node at (3,1.8) {2a};
\node at (4.3,1) {2b};
\fill (4,0.5) circle (2pt);
\node at (4.2,0.5) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$+$2$};
\node at (4.3,1) {};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}\sum_{a+b=n+1}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\fill (4,1) circle (2pt);
\node at (3,1.8) {2a};
\node at (4.3,1) {2b};
\end{tikzpicture},\end{aligned}$$
as desired.
Obtaining an explicit formula for sliding counter-clockwise bubbles is difficult since we express their commutators in terms of left twist curls with some dots on the curl, whose resolution terms still leave us with counter-clockwise bubbles on the left side of ${\ifthenelse{\isempty{x_i^a}{}}{h_{1}}{h_{1}^{x_i^a}}}$. However, the situation is better with clockwise oriented bubbles.
Clockwise bubble slide moves
----------------------------
We can compute an explicit formula for clockwise bubble slides.
[\[clockwise bubbles 1\]]{} We have $${\displaystyle}[d_{2n},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]=2\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[->] (0,0) to (0,1);
\fill (0,0.75) circle (1pt);
\node at (0.2,0.75) {2n};
\end{tikzpicture} \;\right]
+2\sum_{a+b=2n-1}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.02,0.75) circle (1pt);
\node at (0.8,0.75) {a};
\fill (1.45,0.5) circle (1pt);
\node at (1.55,0.5) {b};
\end{tikzpicture} \;\right]$$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for all $n\geq 0$.
This is a direct computation, given below: $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\node at (2.5,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
+2\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\fill (0,1.6) circle (2pt);
\node at (0.34,1.6) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$1$};
\fill (3.5, 1) circle (2pt);
\node at (3.8,1) {};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$1$};
\draw (1,0.9) circle (1pt);
\draw (1.12,0.67) circle (1pt);
\end{tikzpicture}\hspace{6pt}
+2\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\fill (0,1.6) circle (2pt);
\node at (0.34,1.6) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$1$};
\fill (3.5, 1) circle (2pt);
\node at (3.8,1) {};
\end{tikzpicture}\hspace{6pt}
+2
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
+2\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\fill (0,1.6) circle (2pt);
\node at (0.34,1.6) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$2$};
\fill (3.5, 1) circle (2pt);
\node at (3.8,1) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$2$};
\fill (1.24,0.72) circle (1pt);
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$2$};
\fill (1.24,0.72) circle (1pt);
\draw (1,0.9) circle (1pt);
\draw (1.12,0.67) circle (1pt);
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
+2\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\fill (0,1.6) circle (2pt);
\node at (0.34,1.6) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (3.3,0) to (3.3,2);
\fill (3,1.5) circle (2pt);
\node at (2.7,1.8) {\small $2n$-$2$};
\fill (3.5, 1) circle (2pt);
\node at (3.8,1) {\small $2$};
\end{tikzpicture}\hspace{6pt}
+
2\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$2$};
\fill (1.24,0.72) circle (1pt);
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.8) circle (1pt);
\node at (0.7,0.8) {\small $2n$-$1$};
\end{tikzpicture}\hspace{6pt}
+
2\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\fill (0,1.6) circle (2pt);
\node at (0.34,1.6) {\small $2n$};
\end{tikzpicture}\end{aligned}$$ Continuing to slide dots in the first term in this way, we obtain: $$\begin{aligned}
&
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (4,0) to (4,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
+2\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\fill (0,1.6) circle (2pt);
\node at (0.34,1.6) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}2\sum_{a+b=2n-1}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.02,0.75) circle (1pt);
\node at (0.8,0.75) {\small $a$};
\fill (1.45,0.5) circle (1pt);
\node at (1.55,0.5) {\small $b$};
\end{tikzpicture}.\end{aligned}$$
In particular, we can refine this statement to obtain the following recursive formula for computing $[d_{2n},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]$.
[\[clockwise bubbles recursive\]]{} We have$$[d_{2n},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]=[d_{2n-2},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]\circ x_1^2 + 4\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.85) circle (1pt);
\node at (1.1,0.85) {\small $2$};
\fill (1.45,0.5) circle (1pt);
\node at (1.7,0.5) {\small $2n$-$3$};
\end{tikzpicture} \;\right]
-2\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0.5,1) arc (-180:180:5mm);
\draw[->] (0,0) to (0,2);
\fill (1.5,1) circle (2pt);
\node at (1.8,1) {\small $2n-2$};
\end{tikzpicture} \;\right]$$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for all $n\geq 0$.
This lemma follows from the observation that $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.02,0.75) circle (1pt);
\node at (0.8,0.75) {\small $a$};
\fill (1.45,0.5) circle (1pt);
\node at (1.65,0.5) {\small $2k$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.02,0.75) circle (1pt);
\node at (0.8,0.75) {\small $a$+$1$};
\fill (1.45,0.5) circle (1pt);
\node at (1.70,0.5) {\small $2k$-$1$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2k$-$1$};
\fill (2,1.4) circle (2pt);
\node at (1.8,1.4) {\small $a$};
\end{tikzpicture}\hspace{6pt}
+
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2k$-$1$};
\fill (2,1.4) circle (2pt);
\node at (1.8,1.4) {\small $a$};
\draw (2,1.2) circle (2pt);
\draw (2.5,1.1) circle (2pt);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.02,0.75) circle (1pt);
\node at (0.8,0.75) {\small $a$+$1$};
\fill (1.45,0.5) circle (1pt);
\node at (1.70,0.5) {\small $2k$-$1$};
\end{tikzpicture},$$ where the second term after the first equality is zero by Lemma \[odd dots\], and the third term is zero by Lemma \[hollow dots\]. Applying this result to the summands in the statement of Lemma \[clockwise bubbles 1\] yields the result.
Finally, we obtain an explicit formula for computing $[d_{2n},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]$.
\[clockwise bubbles explicit\] We have $${\displaystyle}[d_{2n},{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}]=(2+4n)\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[->] (0,0) to (0,1);
\fill (0,0.75) circle (1pt);
\node at (0.2,0.75) {2n};
\end{tikzpicture} \;\right]
-\sum_{a+b=n-1}(2+4a)
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0.5,1) arc (-180:180:5mm);
\draw[->] (0,0) to (0,2);
\fill (1.5,1) circle (2pt);
\fill (0,1.5) circle (2pt);
\node at (0.3,1.5) {2a};
\node at (1.8,1) {2b};
\end{tikzpicture} \;\right]$$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for all $n\geq 0$.
We claim that $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.85) circle (1pt);
\node at (1.1,0.85) {\small $2$};
\fill (1.45,0.5) circle (1pt);
\node at (1.7,0.5) {\small $2n$-$3$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[->] (0,0) to (0,1);
\fill (0,0.75) circle (1pt);
\node at (0.2,0.75) {$2n$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
{\displaystyle}\sum_{\substack{a+b=n-1\\a\neq0}}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2b$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2a$};
\end{tikzpicture}\hspace{6pt}$$ for $n\geq2$. We proceed via induction on $n$. The base case $n=2$ is a direct computation. Now suppose the formula holds for some $n\geq 2$. Then $$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.85) circle (1pt);
\node at (1.1,0.85) {\small $2$};
\fill (1.45,0.5) circle (1pt);
\node at (1.8,0.5) {\small $2n-3$};
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.85) circle (1pt);
\node at (1.1,0.85) {\small $3$};
\fill (1.45,0.5) circle (1pt);
\node at (1.8,0.5) {\small $2n-4$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$-$4$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.85) circle (1pt);
\node at (1.1,0.85) {\small $4$};
\fill (1.45,0.5) circle (1pt);
\node at (1.8,0.5) {\small $2n-5$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$-$4$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.02,0.75) circle (1pt);
\node at (1.12,0.75) {\small $2$};
\fill (1,0.92) circle (1pt);
\node at (1.1,0.92) {\small $2$};
\fill (1.45,0.5) circle (1pt);
\node at (1.8,0.5) {\small $2n-5$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$-$4$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2$};
\end{tikzpicture}.\end{aligned}$$ Now we can apply the induction hypothesis to the lower part of the first term in the last expression. This gives us:
$$\begin{aligned}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw (1,0) to [out=90, in=85](1.05,0.5);
\draw (1.05,0.5) arc (-175:175:2mm);
\draw[->] (1.05,0.46) to [out=95, in=270] (1,1);
\fill (1.01,0.85) circle (1pt);
\node at (1.1,0.85) {\small $2$};
\fill (1.45,0.5) circle (1pt);
\node at (1.8,0.5) {\small $2n$-$3$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\Bigg(\;\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[->] (0,0) to (0,1);
\fill (0,0.65) circle (1pt);
\node at (0.3,0.65) {\small $2n$-$2$};
\fill (0,0.85) circle (1pt);
\node at (0.15,0.85) {\small $2$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
{\displaystyle}\sum_{\substack{a+b=n-2\\a\neq0}}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2b$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2a$};
\fill (2,1.7) circle (2pt);
\node at (1.7,1.7) {\small $2$};
\end{tikzpicture}\hspace{6pt}
\Bigg)-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$-$4$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\Bigg(\;\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[->] (0,0) to (0,1);
\fill (0,0.65) circle (1pt);
\node at (0.2,0.65) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
{\displaystyle}\sum_{\substack{a+b=n-2\\a\neq0}}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2b$};
\fill (2,1.4) circle (2pt);
\node at (1.4,1.4) {\small $2a$+$2$};
\end{tikzpicture}\hspace{6pt}
\Bigg)-
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2n$-$4$};
\fill (2,1.4) circle (2pt);
\node at (1.7,1.4) {\small $2$};
\end{tikzpicture}\hspace{6pt}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[->] (0,0) to (0,1);
\fill (0,0.65) circle (1pt);
\node at (0.2,0.65) {\small $2n$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}
{\displaystyle}\sum_{\substack{a+b=n-1\\a\neq0}}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (2.5,1) arc (-180:180:5mm);
\draw[->] (2,0) to (2,2);
\fill (3,1.5) circle (2pt);
\node at (3,1.8) {\small $2b$};
\fill (2,1.4) circle (2pt);
\node at (1.4,1.4) {\small $2a$};
\end{tikzpicture}\end{aligned}$$
Applying this result to the recursive formula in Lemma \[clockwise bubbles recursive\] proves the statement.
Commutators of bubbles with downward strands are similar to those of bubbles with upward strands.
\[clockwise bubbles with down recursive\] We have $${\displaystyle}[d_{2n},{\ifthenelse{\isempty{}{}}{h_{-1}}{h_{-1}^{}}}]=-2\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[<-] (0,0) to (0,1);
\fill (0,0.75) circle (1pt);
\node at (0.2,0.75) {\small $2n$};
\end{tikzpicture} \;\right]
-2\sum_{a+b=2n-1}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[<-] (1,0) to [out=90, in=-75](0.95,0.5);
\draw (0.95,0.5) arc (5:355:2mm);
\draw (0.95,0.46) to [out=75, in=270] (1,1);
\fill (1.00,0.75) circle (1pt);
\node at (0.85,0.75) {\small $a$};
\fill (0.55,0.5) circle (1pt);
\node at (0.40,0.5) {\small $b$};
\end{tikzpicture} \;\right]$$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for all $n\geq 0$.
This follows from a computation similar to those in the proofs of Lemmas \[clockwise bubbles 1\] and \[clockwise bubbles recursive\].
Finally we have an explicit formula for commutators of clockwise oriented bubbles and a single downward strand.
\[clockwise bubbles with down explicit\] We have $${\displaystyle}[d_{2n},{\ifthenelse{\isempty{}{}}{h_{-1}}{h_{-1}^{}}}]=-(2+4n)\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=1.5]
\draw[<-] (0,0) to (0,1);
\fill (0,0.75) circle (1pt);
\node at (0.2,0.75) {\small $2n$};
\end{tikzpicture} \;\right]
+\sum_{a+b=n-1}(2+4a)\hspace{2mm}
\left[\; \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0.5,1) arc (-180:180:5mm);
\draw[<-] (2.3,0) to (2.3,2);
\fill (1.5,1) circle (2pt);
\fill (2.3,1.5) circle (2pt);
\node at (2.6,1.5) {\small $2a$};
\node at (1.8,1) {\small $2b$};
\end{tikzpicture}\;\right]$$ in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ for $n\geq0$.
This follows from Lemma \[clockwise bubbles with down recursive\], using a similar argument as in the proof of Proposition \[clockwise bubbles explicit\].
Note that in this formula, we are still left with clockwise bubbles on the left side of a downward strand, but with fewer dots on it. Hence the formula may be applied inductively in order to move all the bubbles to the rightmost part of the diagram.
Diagrammatic lemmas
===================
This section contains some technical computations to derive relations between diagrams consisting of up and down strands. These relations allow us to find a generating set of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$ in Section 6.
Differential degree zero part of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$
-------------------------------------------------------------------------------------------
The differential degree zero part of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ consists of elements $\{{\ifthenelse{\isempty{}{}}{h_{n}}{h_{n}^{}}}\}_{n\in \mathbb{Z}}$. First, we have the following basic fact.
[@Mike Proposition 3.9] \[evenCyclesZero\] We have $${\ifthenelse{\isempty{}{}}{h_{2n}}{h_{2n}^{}}} \cong 0$$ for any $n\in \mathbb{Z}$.
By Proposition \[triangularDecomposition\], the proof in the Hecke-Clifford algebra applies here, as well.
The elements of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ satisfy the following relations.
\[basic hn commutators\] The following commutators are zero for all non-negative integers $n,m$:
1. $[{\ifthenelse{\isempty{}{}}{h_{n}}{h_{n}^{}}},{\ifthenelse{\isempty{}{}}{h_{m}}{h_{m}^{}}}]=0$,
2. $[{\ifthenelse{\isempty{}{}}{h_{-n}}{h_{-n}^{}}},{\ifthenelse{\isempty{}{}}{h_{-m}}{h_{-m}^{}}}]=0$,
3. $[{\ifthenelse{\isempty{}{}}{h_{2n}}{h_{2n}^{}}},{\ifthenelse{\isempty{}{}}{h_{-2n}}{h_{-2n}^{}}}]=0$.
Parts (1) and (2) follow from the fact that similarly oriented strands can be split apart when they cross twice. Part (3) follows immediately from Proposition \[evenCyclesZero\].
To obtain a copy of the twisted Heisenberg algebra in the ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, we need to look at commutators between elements with odd numbers of oppositely oriented strands.
\[twistedHeisRels\] We have, for any $n,m \in \mathbb{Z}^{\geq 0}$, $$[{\ifthenelse{\isempty{}{}}{h_{2n+1}}{h_{2n+1}^{}}},{\ifthenelse{\isempty{}{}}{h_{-2m+1}}{h_{-2m+1}^{}}}]=(\delta_{n,-m})(-2(2n+1)).$$
First note that [@CLLS Lemma 19] and [@CLLS Lemma 20] holds in our twisted case with a small modification, since all the arguments in their proofs use the fact that the resolution terms contain left twist curls, hence are zero. There are extra resolution terms with hollow dots due to relation , but two hollow dots on a diagram containing a left twist curl still gives zero. The only modification comes in the case $m=n$ where we get two copies of counter-clockwise bubbles instead of one, since a two hollow dots on a counter-clockwise bubble end up canceling each other without changing the sign of the diagram. We immediately get that when $m\neq n$, our commutator is zero since we have no solid dots. Therefore we have
$$\begin{aligned}
h_{2n+1}h_{-2m+1}&=\left[ \;\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.8]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\end{tikzpicture}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.8]
\draw[<-] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[<-] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[<-] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[<-] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\end{tikzpicture}\; \right]
\\&=
\left[ \;\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.8]
\draw (3,0) .. controls ++(0,1.25) and ++(0,-1.75) .. (0,1.5);
\draw (0,0) .. controls ++(0,1) and ++(0,-.7) .. (.6,1.5);
\draw (.6,0) .. controls ++(0,1) and ++(0,-.7) .. (1.2,1.5);
\draw (1.8,0) .. controls ++(0,1) and ++(0,-.7) .. (2.4,1.5);
\draw (2.4,0) .. controls ++(0,1) and ++(0,-.7) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (6,0) .. controls ++(0,1.25) and ++(0,-1.75) .. (3.6,1.5) ;
\draw (3.6,0) .. controls ++(0,1) and ++(0,-.7) .. (4.2,1.5);
\draw (4.2,0) .. controls ++(0,1) and ++(0,-.7) .. (4.8,1.5);
\draw (5.4,0) .. controls ++(0,1) and ++(0,-.7) .. (6,1.5);
\node at (4.8,-.65) {$\dots$};
\node at (5.4,1.85) {$\dots$};
\draw[blue, dotted] (-0.4,0) -- (6.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (6.4,1.5);
\draw[->] (3,1.5) .. controls ++(0,0.5) and ++(0,-.5) .. (3.6,2.5);
\draw (3.6,1.5) .. controls ++(0,1) and ++(0,-1) .. (0,2.5);
\draw[->] (2.4,1.5) .. controls ++(0,0.5) and ++(0,-.5) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls ++(0,0.5) and ++(0,-.5) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls ++(0,0.5) and ++(0,-.5) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls ++(0,0.5) and ++(0,-.5) .. (.6,2.5);
\draw (4.2,1.5) -- (4.2,2.5);
\draw (4.8,1.5) -- (4.8,2.5);
\draw (6,1.5) -- (6,2.5);
\draw (2.4,0) .. controls ++(0,-.5) and ++(0,.5) .. (3,-1);
\draw (1.8,0) .. controls ++(0,-.5) and ++(0,.5) .. (2.4,-1);
\draw (.6,0) .. controls ++(0,-.5) and ++(0,.5) .. (1.2,-1);
\draw (0,0) .. controls ++(0,-.5) and ++(0,.5) .. (.6,-1);
\draw (3,0) .. controls ++(0,-.5) and ++(0,.5) .. (3.6,-1);
\draw[,->] (3.6,0) .. controls ++(0,-1) and ++(0,+1) .. (0,-1);
\draw[->] (4.2,0) -- (4.2,-1);
\draw[->] (5.4,0) -- (5.4,-1);
\draw[->] (6,0) -- (6,-1);
\end{tikzpicture}\;\right]
\\&=h_{-(2m+1)}h_{2n+1}(-2\bar{d}_0(2n+1)).\end{aligned}$$
Hence $[{\ifthenelse{\isempty{}{}}{h_{(2n+1)}}{h_{(2n+1)}^{}}},{\ifthenelse{\isempty{}{}}{h_{-(2m+1)}}{h_{-(2m+1)}^{}}}]=\delta_{n,-m}(-2(2n+1))$.
Therefore the subset $A=\{{\ifthenelse{\isempty{}{}}{h_{(2n+1)}}{h_{(2n+1)}^{}}}\}_{n\in\mathbb{Z}}$ of the filtration degree zero part of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ is isomorphic to the twisted Heisenberg algebra via
$$\begin{aligned}
\phi:&\mathfrak{h}_{tw}&\stackrel{\sim}{\longrightarrow} A \\
&{\ifthenelse{\isempty{}{}}{h_{\frac{2n+1}{2}}}{h_{\frac{2n+1}{2}}^{}}}&\mapsto \frac{1}{2}{\ifthenelse{\isempty{}{}}{h_{-(2n+1)}}{h_{-(2n+1)}^{}}}.\end{aligned}$$
In the $W$-algebra $W^-$, we have an isomophic copy of the twisted Heisenberg algebra as well, given by $B=\{\omega_{2n+1,0}\}_{n\in\mathbb{Z}}$, with the isomorphism given by
$$\begin{aligned}
\psi:&\mathfrak{h}_{tw}&\stackrel{\sim}{\longrightarrow} B \\
&{\ifthenelse{\isempty{}{}}{h_{\frac{2n+1}{2}}}{h_{\frac{2n+1}{2}}^{}}}&\mapsto \frac{1}{\sqrt{2}}\ \omega_{2n+1,0}.\end{aligned}$$
Therefore we have an isomorphism between the degree zero part of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ and the degree zero part of $W^-$:
$$\begin{aligned}
\psi\circ\phi^{-1}:&A&\stackrel{\sim}{\longrightarrow} B \\
&{\ifthenelse{\isempty{}{}}{h_{-(2n+1)}}{h_{-(2n+1)}^{}}}&\mapsto \sqrt{2}\ w_{2n+1,0}.\end{aligned}$$
Nonzero differential degree part of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$
----------------------------------------------------------------------------------------------
We have the following basic facts about diagrams in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}^{>}$, which we may copy from the corresponding facts in the trace of the affine Hecke-Clifford algebra because of the triangular decomposition of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ described in Proposition \[triangularDecomposition\].
\[odd with odd dots zero\][@Mike Propositions 3.9, 4.2] In $Tr({\mathfrak{H}^C})$ for any $m,n \in \mathbb{Z}$, we have $${\ifthenelse{\isempty{x_1^{2m+1}}{}}{h_{2n+1}}{h_{2n+1}^{x_1^{2m+1}}}} = 0,$$ $${\ifthenelse{\isempty{x_1^{2m}}{}}{h_{2n}}{h_{2n}^{x_1^{2m}}}} =0.$$
Hence any diagram containing an odd cycle with an odd number of dots or an even cycle with an even number of dots is zero. Therefore, the difference of the number of strands and number of solid dots must be odd. This agrees with the fact that in the $W$-algebra $W^-$, $l-k$ has to be an odd number for $w_{l,k}$.
The generators of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}^{>}$ satisfy the following relations.
$$\left[\hspace{2mm} \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\end{tikzpicture}
,\hspace{3mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\end{tikzpicture}\hspace{2mm}\right]
=-4
\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\end{tikzpicture}
\hspace{10mm}
\text{and}
\hspace{8mm}
\left[\hspace{2mm} \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\end{tikzpicture}
,\hspace{3mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\end{tikzpicture}\hspace{2mm}\right]
=2
\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\end{tikzpicture}$$
$$\left[\hspace{2mm} \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.75,0.5) circle[radius=3pt];
\end{tikzpicture}
,\hspace{3mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\end{tikzpicture}\hspace{2mm}\right]
=0
\hspace{10mm}
\text{and}
\hspace{10mm}
\left[\hspace{2mm} \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\end{tikzpicture}
,\hspace{3mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\end{tikzpicture}\hspace{2mm}\right]
=2
\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\end{tikzpicture}$$
$$\left[\hspace{2mm} \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[->](.5,0) to (.5,2);
\end{tikzpicture}
\hspace{2mm}
,\hspace{3mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-](0,0) to (0,2);
\end{tikzpicture}\hspace{2mm}\right]
=-4
\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\end{tikzpicture}
\hspace{10mm}
\text{and}
\hspace{8mm}
\left[\hspace{2mm} \begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\draw[->](.5,0) to (.5,2);
\end{tikzpicture}
\hspace{2mm}
,\hspace{3mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (0,2);
\end{tikzpicture}\hspace{2mm}\right]
=0$$
\[commutators\] For $m,n\in \mathbb{Z}$ with $mn>0$, we have
1. $[{\ifthenelse{\isempty{x_1}{}}{h_{2m }}{h_{2m }^{x_1}}}, {\ifthenelse{\isempty{ x_1}{}}{h_{2n}}{h_{2n}^{ x_1}}}] = 2(n-m) {\ifthenelse{\isempty{x_1}{}}{h_{2n+2m}}{h_{2n+2m}^{x_1}}}.$
2. $ [{\ifthenelse{\isempty{c_1}{}}{h_{m}}{h_{m}^{c_1}}}, {\ifthenelse{\isempty{c_1}{}}{h_{n}}{h_{n}^{c_1}}}] = -2{\ifthenelse{\isempty{c_1}{}}{h_{n}}{h_{n}^{c_1}}}.$
Part (1) is a slight modification of [@CLLS Lemma 23]. By Proposition \[odd with odd dots zero\], if at least one of the indices inside the commutator is odd, the commutator will be zero. Hence we will work with the case where both indices are even numbers. The modification we need in [@CLLS Lemma 23] is a result of us having two resolution terms in our relations and . As a consequence of having even number of strands in both of our elements, canceling the two empty dots in our resolution terms give rise to the same sign as the other resolution term, hence we have a coefficient of two in front of our result.
Part (2) follows easily the proof of [@CLLS Lemma 23] since moving an empty dot through a crossing is for free in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, and we get a negative sign from changing relative heights of hollow dots.
\[VirasorowithUp\] For $n\geq0$, we have $$[{\ifthenelse{\isempty{(x_1 + \ldots + x_{2n})}{}}{h_{\pm 2n}}{h_{\pm 2n}^{(x_1 + \ldots + x_{2n})}}}, {\ifthenelse{\isempty{}{}}{h_{ 1}}{h_{ 1}^{}}}] = \pm 4n {\ifthenelse{\isempty{}{}}{h_{\pm(2n+1)}}{h_{\pm(2n+1)}^{}}}.$$
First note that we have:
$$\begin{aligned}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\draw [->](1.5,0) to (1.5,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to (1.5,2);
\draw [->](1.5,0) to (0,2);
\draw [fill](.25,1.63) circle[radius=3pt];
\draw[->](0.75,0) to [out=135,in=225] (0.75,2);
\end{tikzpicture}\hspace{2mm}\right] \hspace{6pt}
\\&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to (1.5,2);
\draw [->](1.5,0) to (0,2);
\draw [fill](.65,1.2) circle[radius=3pt];
\draw[->](0.75,0) to [out=135,in=225] (0.75,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
+
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
-
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\draw (.08,1.5) circle[radius=3pt];
\draw (.67, 1.2) circle[radius=3pt];
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
\\&{\addtocounter{equation}{1}\tag{\theequation}}\label{zeroDotMove}
\hspace{6pt}=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](0.25,1.5) circle[radius=3pt];
\draw [->](-.5,0) to (-.5,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
+
\hspace{6pt}2
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\end{tikzpicture}\hspace{2mm}\right].\end{aligned}$$
Hence $[{\ifthenelse{\isempty{x_1}{}}{h_{2}}{h_{2}^{x_1}}}, {\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}] = 2 {\ifthenelse{\isempty{}{}}{h_{3}}{h_{3}^{}}}$.
Next, moving the solid dot in ${\ifthenelse{\isempty{x_2}{}}{h_{2}}{h_{2}^{x_2}}}$ around to the bottom of the crossing using the trace relation gives:
$$\begin{aligned}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](.75,.5) circle[radius=3pt];
\draw [->](1.5,0) to (1.5,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to (1.5,2);
\draw [->](1.5,0) to (0,2);
\draw[->](0.75,0) to [out=45,in=-45] (0.75,2);
\draw [fill](.9,.75) circle[radius=3pt];
\end{tikzpicture}\hspace{2mm}\right] \hspace{6pt}
\\&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [->](0,0) to (1.5,2);
\draw [->](1.5,0) to (0,2);
\draw[->](0.75,0) to [out=45,in=-45] (0.75,2);
\draw [fill](1.3,.25) circle[radius=3pt];
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
+
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
-
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\draw (1.77,.5) circle[radius=3pt];
\draw (1.05, .3) circle[radius=3pt];
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
\\&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->](0,0) to (1,2);
\draw[->](1,0) to (0,2);
\draw[fill](.75,.5) circle[radius=3pt];
\draw [->](-.5,0) to (-.5,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
+
\hspace{6pt}2\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (2,0) .. controls (2,1.25) and (0,.25) .. (0,2);
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (1,2);
\draw[->] (1,0) .. controls (1,1) and (1.8,.8) .. (2,2);
\end{tikzpicture}\hspace{2mm}\right].\end{aligned}$$
So, $[{\ifthenelse{\isempty{(x_1 + x_2)}{}}{h_{2}}{h_{2}^{(x_1 + x_2)}}}, {\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}} ] = 4{\ifthenelse{\isempty{}{}}{h_{3}}{h_{3}^{}}}.$
Next, we claim that $[{\ifthenelse{\isempty{x_{2n}}{}}{h_{2n}}{h_{2n}^{x_{2n}}}}, {\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}] = 2 {\ifthenelse{\isempty{}{}}{h_{2n+1}}{h_{2n+1}^{}}}$ for any $n$. Indeed, we have: $$\begin{aligned}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (3.18,.2) circle (3pt);
\draw[->](3.5,0) to (3.5,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (2.8,.55) circle (3pt);
\draw[->](3,0) ..controls (3.2,1) and (3,.8)..(2.8,2);
\end{tikzpicture}\hspace{2mm}\right] \hspace{6pt}
\\&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (3.18,.2) circle (3pt);
\draw[->](2.8,0) ..controls (3,1) and (2.8,.8)..(2.6,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
+
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3,0) .. controls (3,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\draw[->](3.4,0) ..controls (3.6,1) and (3,.8)..(2.8,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
\\&-
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3,0) .. controls (3,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\draw (2.85,.4) circle (3pt);
\draw (3.43, .2) circle (3pt);
\draw[->](3.4,0) ..controls (3.6,1) and (3,.8)..(2.8,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
\\&=
\hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (3.18,.2) circle (3pt);
\draw[->](-.3,0) to (-.3,2);
\end{tikzpicture}\hspace{2mm}\right]\hspace{6pt}
+
\hspace{6pt}2
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\draw[->] (2.8,0) .. controls (2.8,1) and (3.6,.8) .. (3.6,2);
\end{tikzpicture}\hspace{2mm}\right],\end{aligned}$$ where the last equality is obtained by pushing the crossings at the bottom of the diagrams without dots to the top. Indeed, diagrammatic calculations similar to the above give that $$[{\ifthenelse{\isempty{ x_{a}}{}}{h_{2n}}{h_{2n}^{ x_{a}}}}, {\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}] = 2{\ifthenelse{\isempty{}{}}{h_{2n+1}}{h_{2n+1}^{}}}$$ for any $1 < a \leq 2n$.
Finally, note that $${\ifthenelse{\isempty{x_1}{}}{h_{2n}}{h_{2n}^{x_1}}} {\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}\hspace{6pt} = \hspace{6pt}
\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (.1,1.55) circle (3pt);
\draw[->](.4,0) ..controls (.4,1) and (.2,.8)..(.4,2);
\end{tikzpicture}\hspace{2mm}\right].$$ The dot will slide over the top-leftmost crossing in the same manner as in Equation , meaning the correction terms will cancel out. Hence, we have the desired result.
\[upVirasorowithUp\] Let $m$ be an odd integer. We have $$[ {\ifthenelse{\isempty{(x_1+ x_2)}{}}{h_{2}}{h_{2}^{(x_1+ x_2)}}}, {\ifthenelse{\isempty{}{}}{h_{m}}{h_{m}^{}}}] = 4m{\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}}.$$
We compute directly:
$$\begin{aligned}
{\ifthenelse{\isempty{}{}}{h_{m}}{h_{m}^{}}}{\ifthenelse{\isempty{x_1}{}}{h_{2}}{h_{2}^{x_1}}}
\hspace{6pt} &= \hspace{6pt} \left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw[->] (3,0) .. controls ++(0,1.25) and ++(0,-1.75) .. (0,2);
\draw[->] (0,0) .. controls ++(0,1) and ++(0,-1.2) .. (.6,2);
\draw[->] (.6,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.2,2);
\draw[->] (1.8,0) .. controls ++(0,1) and ++(0,-1.2) .. (2.4,2);
\draw[->] (2.4,0) .. controls ++(0,1) and ++(0,-1.2) .. (3,2);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.65) {$\dots$};
\draw[->] (3.4,0)--(4.5,2);
\draw[->] (4.5,0)--(3.4,2);
\filldraw (3.7,1.4) circle (2pt);
\end{tikzpicture}}; \endxy}
\; \;\;\right]\hspace{6pt}
\\&=
\hspace{6pt}
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3,0) .. controls (3,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (.6,.8) .. (.6,1.5);
\draw (.6,0) .. controls (.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\draw (2.4,0) .. controls (2.4,1) and (3,.8) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (4.2,0) .. controls (4.2,1.25) and (3.6,.25) .. (3.6,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (3.6,0) .. controls (3.6,1) and (4.2,.8) .. (4.2,1.5);
\draw[blue, dotted] (-0.4,0) -- (4.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (4.4,1.5);
\draw[->] (3,1.5) .. controls (3,2) and (3.6,2) .. (3.6,2.5);
\draw[->] (3.6,1.5) .. controls (3.6,2.5) and (0,1.5) .. (0,2.5);
\draw[->] (2.4,1.5) .. controls (2.4,2) and (3,2) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls (1.2,2) and (1.8,2) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls (.6,2) and (1.2,2) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls (0,2) and (.6,2) .. (.6,2.5);
\draw[->] (4.2,1.5) -- (4.2,2.5);
\filldraw (3.6,1.5) circle (2pt);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\draw (1.8,0) .. controls (1.8,-.5) and (2.4,-.5) .. (2.4,-1);
\draw (.6,0) .. controls (.6,-.5) and (1.2,-.5) .. (1.2,-1);
\draw (0,0) .. controls (0,-.5) and (.6,-.5) .. (.6,-1);
\draw (3,0) .. controls (3,-.5) and (3.6,-.5) .. (3.6,-1);
\draw (3.6,0) .. controls (3.6,-1) and (0,0) .. (0,-1);
\draw (4.2,0) -- (4.2,-1);
\end{tikzpicture}}; \endxy}
\right]
\hspace{6pt}
\\&=
\hspace{6pt}
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3,0) .. controls (3,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (.6,.8) .. (.6,1.5);
\draw (.6,0) .. controls (.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\draw (2.4,0) .. controls (2.4,1) and (3,.8) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (4.2,0) .. controls (4.2,1.25) and (3.6,.25) .. (3.6,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (3.6,0) .. controls (3.6,1) and (4.2,.8) .. (4.2,1.5);
\draw[blue, dotted] (-0.4,0) -- (4.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (4.4,1.5);
\draw[->] (3,1.5) .. controls (3,2) and (3.6,2) .. (3.6,2.5);
\draw[->] (3.6,1.5) .. controls (3.6,2.5) and (0,1.5) .. (0,2.5);
\draw[->] (2.4,1.5) .. controls (2.4,2) and (3,2) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls (1.2,2) and (1.8,2) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls (.6,2) and (1.2,2) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls (0,2) and (.6,2) .. (.6,2.5);
\draw[->] (4.2,1.5) -- (4.2,2.5);
\filldraw (3.0,1.95) circle (2pt);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\draw (1.8,0) .. controls (1.8,-.5) and (2.4,-.5) .. (2.4,-1);
\draw (.6,0) .. controls (.6,-.5) and (1.2,-.5) .. (1.2,-1);
\draw (0,0) .. controls (0,-.5) and (.6,-.5) .. (.6,-1);
\draw (3,0) .. controls (3,-.5) and (3.6,-.5) .. (3.6,-1);
\draw (3.6,0) .. controls (3.6,-1) and (0,0) .. (0,-1);
\draw (4.2,0) -- (4.2,-1);
\end{tikzpicture}}; \endxy}
\right]
\hspace{6pt}
-
\hspace{6pt}
\left[ \;\;\;{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3,0) .. controls (3,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (.6,.8) .. (.6,1.5);
\draw (.6,0) .. controls (.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\draw (2.4,0) .. controls (2.4,1) and (3,.8) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (4.2,0) .. controls (4.2,1.25) and (3.6,.25) .. (3.6,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (3.6,0) .. controls (3.6,1) and (4.2,.8) .. (4.2,1.5);
\draw[blue, dotted] (-0.4,0) -- (4.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (4.4,1.5);
\draw[->] (3,1.5) .. controls (3,2) and (0,2) .. (0,2.5);
\draw[->] (3.6,1.5)--(3.6,2.5);
\draw[->] (2.4,1.5) .. controls (2.4,2) and (3,2) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls (1.2,2) and (1.8,2) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls (.6,2) and (1.2,2) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls (0,2) and (.6,2) .. (.6,2.5);
\draw[->] (4.2,1.5) -- (4.2,2.5);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\draw (1.8,0) .. controls (1.8,-.5) and (2.4,-.5) .. (2.4,-1);
\draw (.6,0) .. controls (.6,-.5) and (1.2,-.5) .. (1.2,-1);
\draw (0,0) .. controls (0,-.5) and (.6,-.5) .. (.6,-1);
\draw (3,0) .. controls (3,-.5) and (3.6,-.5) .. (3.6,-1);
\draw (3.6,0) .. controls (3.6,-1) and (0,0) .. (0,-1);
\draw (4.2,0) -- (4.2,-1);
\end{tikzpicture}}; \endxy}
\right]\hspace{6pt}
\\&+
\hspace{6pt}
\left[ \;\;\;{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3,0) .. controls (3,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (.6,.8) .. (.6,1.5);
\draw (.6,0) .. controls (.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\draw (2.4,0) .. controls (2.4,1) and (3,.8) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (4.2,0) .. controls (4.2,1.25) and (3.6,.25) .. (3.6,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (3.6,0) .. controls (3.6,1) and (4.2,.8) .. (4.2,1.5);
\draw (3.6, 1.2) circle (3pt);
\draw[blue, dotted] (-0.4,0) -- (4.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (4.4,1.5);
\draw[->] (3,1.5) .. controls (3,2) and (0,2) .. (0,2.5);
\draw[->] (3.6,1.5)--(3.6,2.5);
\draw[->] (2.4,1.5) .. controls (2.4,2) and (3,2) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls (1.2,2) and (1.8,2) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls (.6,2) and (1.2,2) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls (0,2) and (.6,2) .. (.6,2.5);
\draw[->] (4.2,1.5) -- (4.2,2.5);
\draw (2.8,1.7) circle (3pt);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\draw (1.8,0) .. controls (1.8,-.5) and (2.4,-.5) .. (2.4,-1);
\draw (.6,0) .. controls (.6,-.5) and (1.2,-.5) .. (1.2,-1);
\draw (0,0) .. controls (0,-.5) and (.6,-.5) .. (.6,-1);
\draw (3,0) .. controls (3,-.5) and (3.6,-.5) .. (3.6,-1);
\draw (3.6,0) .. controls (3.6,-1) and (0,0) .. (0,-1);
\draw (4.2,0) -- (4.2,-1);
\end{tikzpicture}}; \endxy}
\right].\end{aligned}$$
Cancelling the empty dots in the last term results in a change in sign, and both of the latter diagrams are $(m+2)$-cycles. Hence we have: $$=\hspace{6pt}\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3,0) .. controls (3,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (.6,.8) .. (.6,1.5);
\draw (.6,0) .. controls (.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\draw (2.4,0) .. controls (2.4,1) and (3,.8) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (4.2,0) .. controls (4.2,1.25) and (3.6,.25) .. (3.6,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (3.6,0) .. controls (3.6,1) and (4.2,.8) .. (4.2,1.5);
\draw[blue, dotted] (-0.4,0) -- (4.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (4.4,1.5);
\draw[->] (3,1.5) .. controls (3,2) and (3.6,2) .. (3.6,2.5);
\draw[->] (3.6,1.5) .. controls (3.6,2.5) and (0,1.5) .. (0,2.5);
\draw[->] (2.4,1.5) .. controls (2.4,2) and (3,2) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls (1.2,2) and (1.8,2) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls (.6,2) and (1.2,2) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls (0,2) and (.6,2) .. (.6,2.5);
\draw[->] (4.2,1.5) -- (4.2,2.5);
\filldraw (3.0,1.95) circle (2pt);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\draw (1.8,0) .. controls (1.8,-.5) and (2.4,-.5) .. (2.4,-1);
\draw (.6,0) .. controls (.6,-.5) and (1.2,-.5) .. (1.2,-1);
\draw (0,0) .. controls (0,-.5) and (.6,-.5) .. (.6,-1);
\draw (3,0) .. controls (3,-.5) and (3.6,-.5) .. (3.6,-1);
\draw (3.6,0) .. controls (3.6,-1) and (0,0) .. (0,-1);
\draw (4.2,0) -- (4.2,-1);
\end{tikzpicture}}; \endxy}
\right]
\hspace{6pt} -\hspace{6pt} 2{\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}}$$
Sliding the solid dot in the first diagram all the way to the left results in $m$ total crossing resolutions, each of which yieds a term of $-2 {\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}}$. So, $$\begin{aligned}
&=\hspace{6pt}\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3,0) .. controls (3,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (.6,.8) .. (.6,1.5);
\draw (.6,0) .. controls (.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\draw (2.4,0) .. controls (2.4,1) and (3,.8) .. (3,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.15) {$\dots$};
\draw (4.2,0) .. controls (4.2,1.25) and (3.6,.25) .. (3.6,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (3.6,0) .. controls (3.6,1) and (4.2,.8) .. (4.2,1.5);
\draw[blue, dotted] (-0.4,0) -- (4.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (4.4,1.5);
\draw[->] (3,1.5) .. controls (3,2) and (3.6,2) .. (3.6,2.5);
\draw[->] (3.6,1.5) .. controls (3.6,2.5) and (0,1.5) .. (0,2.5);
\draw[->] (2.4,1.5) .. controls (2.4,2) and (3,2) .. (3,2.5);
\draw[->] (1.2,1.5) .. controls (1.2,2) and (1.8,2) .. (1.8,2.5);
\draw[->] (.6,1.5) .. controls (.6,2) and (1.2,2) .. (1.2,2.5);
\draw[->] (0,1.5) .. controls (0,2) and (.6,2) .. (.6,2.5);
\draw[->] (4.2,1.5) -- (4.2,2.5);
\filldraw (.15,2.2) circle (2pt);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\draw (1.8,0) .. controls (1.8,-.5) and (2.4,-.5) .. (2.4,-1);
\draw (.6,0) .. controls (.6,-.5) and (1.2,-.5) .. (1.2,-1);
\draw (0,0) .. controls (0,-.5) and (.6,-.5) .. (.6,-1);
\draw (3,0) .. controls (3,-.5) and (3.6,-.5) .. (3.6,-1);
\draw (3.6,0) .. controls (3.6,-1) and (0,0) .. (0,-1);
\draw (4.2,0) -- (4.2,-1);
\end{tikzpicture}}; \endxy}
\right]
\;\; \;\; - \hspace{6pt}2m{\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}}
\\&
=\;\; \left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[baseline=(current bounding box.center),scale=.8]
\draw[->] (0,0)--(1.2,2);
\draw[->] (1.2,0)--(0,2);
\filldraw (.35,1.4) circle (2pt);
\end{tikzpicture}
\;\;\;
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.8]
\draw[->] (3,0) .. controls ++(0,1.25) and ++(0,-1.75) .. (0,2);
\draw[->] (0,0) .. controls ++(0,1) and ++(0,-1.2) .. (.6,2);
\draw[->] (.6,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.2,2);
\draw[->] (1.8,0) .. controls ++(0,1) and ++(0,-1.2) .. (2.4,2);
\draw[->] (2.4,0) .. controls ++(0,1) and ++(0,-1.2) .. (3,2);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.65) {$\dots$};
\end{tikzpicture}}; \endxy}\;\;\;
\right]
\;\; - \hspace{6pt}2m{\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}}\end{aligned}$$ Hence we have $$[{\ifthenelse{\isempty{x_1}{}}{h_{2}}{h_{2}^{x_1}}}, {\ifthenelse{\isempty{}{}}{h_{m}}{h_{m}^{}}}] = 2m{\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}}.$$ A similar computation gives that $$[{\ifthenelse{\isempty{x_2}{}}{h_{2}}{h_{2}^{x_2}}}, {\ifthenelse{\isempty{}{}}{h_{m}}{h_{m}^{}}}] = 2m{\ifthenelse{\isempty{}{}}{h_{m+2}}{h_{m+2}^{}}},$$ giving the desired result.
\[upVirasorowithDown\] We have $$[{\ifthenelse{\isempty{(x_1+x_2 +\ldots+ x_{2n})}{}}{h_{2n}}{h_{2n}^{(x_1+x_2 +\ldots+ x_{2n})}}}, {\ifthenelse{\isempty{}{}}{h_{-(2m+1)}}{h_{-(2m+1)}^{}}}] = \left\{\begin{array}{lr} -4(2m+1) {\ifthenelse{\isempty{}{}}{h_{2n-2m-1}}{h_{2n-2m-1}^{}}} & \qquad \text{if } n>m\geq 1 \\ 0 & \qquad \text{if } n=m\geq 1 \\ -2(2m+1) {\ifthenelse{\isempty{}{}}{h_{2n-2m-1}}{h_{2n-2m-1}^{}}}& \qquad \text{if } 1\leq n < m.\end{array}\right.$$
We follow the methods of [@CLLS Lemma 26], substituting our new relations as necessary.
As in that case, let $\beta_n = {\ifthenelse{\isempty{x_1}{}}{h_{2n}}{h_{2n}^{x_1}}}$ and $\alpha_m = {\ifthenelse{\isempty{x_1}{}}{h_{2m+1}}{h_{2m+1}^{x_1}}}$, and proceed by induction on $m$. When $m=1$, we can compute directly: $$\begin{aligned}
\label{Lm26 setup}
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw[->] (3,0) .. controls ++(0,1.25) and (0,.25) .. (0,2);
\draw[->] (0.0,0) .. controls ++(0,1) and ++(0,-1.2) .. (0.6,2);
\draw[->] (0.6,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.2,2);
\draw[->] (1.2,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.8,2);
\draw[->] (2.4,0) .. controls ++(0,1) and ++(0,-1.2) .. (3,2);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\draw[<-] (-0.6,0) -- (-0.6,2);
\filldraw (.05,1.6) circle (2pt);
\end{tikzpicture}}; \endxy}
\;\;\right]
\;\; &{\xy {\ar@{=}^{\eqref{R2}}
(-1,0)*{};(1,0)*{}};
\endxy} \;\;
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3.0,0) .. controls ++(0,1.25) and ++(0,-1.1) .. (0,1.5);
\draw (0.0,0) .. controls ++(0,.5) and ++(0,-.6) .. (0.6,1.5);
\draw (0.6,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.2,1.5);
\draw (1.2,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.8,1.5);
\draw (2.4,0) .. controls ++(0,.5) and ++(0,-.6) .. (3,1.5);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.35) {$\dots$};
\draw[<-] (-0.6,0) -- (-0.6,1.5);
\filldraw (.1,1.2) circle (2pt);
\draw[blue, dotted] (-1,0) -- (3.4,0);
\draw[blue, dotted] (-1,1.5) -- (3.4,1.5);
\draw[->] (0.0,1.5) .. controls ++(0,.45) and ++(0,-.6) .. (-0.6,2.5);
\draw (-0.6,1.5) .. controls ++(0,.45) and ++(0,-.6) .. (0.0,2.5);
\draw[->] (0.6,1.5) -- (0.6,2.5);
\draw[->] (1.2,1.5) -- (1.2,2.5);
\draw[->] (1.8,1.5) -- (1.8,2.5);
\draw[->] (3,1.5) -- (3,2.5);
\draw[<-] (0.0,0) .. controls ++(0,-.45) and ++(0,.6) .. (-0.6,-1);
\draw[->] (-0.6,0) .. controls ++(0,-.45) and ++(0,.6) .. (0.0,-1);
\draw[<-] (0.6,0) -- (0.6,-1);
\draw[<-] (1.2,0) -- (1.2,-1);
\draw[<-] (2.4,0) -- (2.4,-1);
\draw[<-] (3.0,0) -- (3,-1);
\end{tikzpicture}}; \endxy}
\;\;\right]
\\ &+\hspace{6pt}2 \;\;
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3.0,0) .. controls ++(0,1.25) and ++(0,-1.1) .. (0,1.5);
\draw (0.0,0) .. controls ++(0,.5) and ++(0,-.6) .. (0.6,1.5);
\draw (0.6,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.2,1.5);
\draw (1.2,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.8,1.5);
\draw (2.4,0) .. controls ++(0,.5) and ++(0,-.6) .. (3,1.5);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.35) {$\dots$};
\draw (-0.6,0) -- (-0.6,1.5);
\filldraw (.1,1.2) circle (2pt);
\draw[blue, dotted] (-1,0) -- (3.4,0);
\draw[blue, dotted] (-1,1.5) -- (3.4,1.5);
\draw (-0.6,1.5) .. controls ++(0,.35) and ++(0,.35) .. (0.0,1.5);
\draw[->] (0.6,1.5) -- (0.6,2.5);
\draw[->] (1.2,1.5) -- (1.2,2.5);
\draw[->] (1.8,1.5) -- (1.8,2.5);
\draw[->] (3,1.5) -- (3,2.5);
\draw[->] (-0.6,0) .. controls ++(0,-.35) and ++(0,-.35) .. (0.0,0);
\draw[<-] (0.6,0) -- (0.6,-1);
\draw[<-] (1.2,0) -- (1.2,-1);
\draw[<-] (2.4,0) -- (2.4,-1);
\draw[<-] (3.0,0) -- (3,-1);
\end{tikzpicture}}; \endxy}
\;\;\right]\end{aligned}$$ where the trailing terms arising from relation have the same sign after cancelling the empty dots, and thus add together. We claim that the diagram in the second term is ${\ifthenelse{\isempty{}{}}{h_{2n-1}}{h_{2n-1}^{}}}$. Indeed, sliding the dot gives: $$\begin{aligned}
\;\;
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3.0,0) .. controls ++(0,1.25) and ++(0,-1.1) .. (0,1.5);
\draw (0.0,0) .. controls ++(0,.5) and ++(0,-.6) .. (0.6,1.5);
\draw (0.6,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.2,1.5);
\draw (1.2,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.8,1.5);
\draw (2.4,0) .. controls ++(0,.5) and ++(0,-.6) .. (3,1.5);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.35) {$\dots$};
\draw (-0.6,0) -- (-0.6,1.5);
\filldraw (.3,.65) circle (2pt);
\draw[blue, dotted] (-1,0) -- (3.4,0);
\draw[blue, dotted] (-1,1.5) -- (3.4,1.5);
\draw(-0.6,1.5) .. controls ++(0,.35) and ++(0,.35) .. (0.0,1.5);
\draw[->] (0.6,1.5) -- (0.6,2.5);
\draw[->] (1.2,1.5) -- (1.2,2.5);
\draw[->] (1.8,1.5) -- (1.8,2.5);
\draw[->] (3,1.5) -- (3,2.5);
\draw[->] (-0.6,0) .. controls ++(0,-.35) and ++(0,-.35) .. (0.0,0);
\draw[<-] (0.6,0) -- (0.6,-1);
\draw[<-] (1.2,0) -- (1.2,-1);
\draw[<-] (2.4,0) -- (2.4,-1);
\draw[<-] (3.0,0) -- (3,-1);
\end{tikzpicture}}; \endxy}
\;\;\right] \;\; &{\xy {\ar@{=}^{\eqref{dotSlide: bottomLeft}}
(-1,0)*{};(1,0)*{}};
\endxy} \;\;
\overline{d}_{0,0}{\ifthenelse{\isempty{}{}}{h_{2n-1}}{h_{2n-1}^{}}} + \overline{d}_{0,1}{\ifthenelse{\isempty{}{}}{h_{2n-1}}{h_{2n-1}^{}}}
= {\ifthenelse{\isempty{}{}}{h_{2n-1}}{h_{2n-1}^{}}}\end{aligned}$$ by relations and .
Now, sliding the solid dot over the crossing on the right hand side of Equation gives:
$$\;\; {\xy {\ar@{=}^{\eqref{dotSlide: bottomLeft}}
(-1,0)*{};(1,0)*{}};
\endxy} \;\;
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3.0,0) .. controls ++(0,1.25) and ++(0,-1.1) .. (0,1.5);
\draw (0.0,0) .. controls ++(0,.5) and ++(0,-.6) .. (0.6,1.5);
\draw (0.6,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.2,1.5);
\draw (1.2,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.8,1.5);
\draw (2.4,0) .. controls ++(0,.5) and ++(0,-.6) .. (3,1.5);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.35) {$\dots$};
\draw[<-] (-0.6,0) -- (-0.6,1.5);
\filldraw (-0.55,2.2) circle (2pt);
\draw[blue, dotted] (-1,0) -- (3.4,0);
\draw[blue, dotted] (-1,1.5) -- (3.4,1.5);
\draw[->] (0.0,1.5) .. controls ++(0,.45) and ++(0,-.6) .. (-0.6,2.5);
\draw (-0.6,1.5) .. controls ++(0,.45) and ++(0,-.6) .. (0.0,2.5);
\draw[->] (0.6,1.5) -- (0.6,2.5);
\draw[->] (1.2,1.5) -- (1.2,2.5);
\draw[->] (1.8,1.5) -- (1.8,2.5);
\draw[->] (3,1.5) -- (3,2.5);
\draw[<-] (0.0,0) .. controls ++(0,-.45) and ++(0,.6) .. (-0.6,-1);
\draw[->] (-0.6,0) .. controls ++(0,-.45) and ++(0,.6) .. (0.0,-1);
\draw[<-] (0.6,0) -- (0.6,-1);
\draw[<-] (1.2,0) -- (1.2,-1);
\draw[<-] (2.4,0) -- (2.4,-1);
\draw[<-] (3.0,0) -- (3,-1);
\end{tikzpicture}}; \endxy}
\;\;\right]
+\hspace{6pt}2 \;\;
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (3.0,0) .. controls ++(0,1.25) and ++(0,-1.1) .. (0,1.5);
\draw (0.0,0) .. controls ++(0,.5) and ++(0,-.6) .. (0.6,1.5);
\draw (0.6,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.2,1.5);
\draw (1.2,0) .. controls ++(0,.5) and ++(0,-.6) .. (1.8,1.5);
\draw (2.4,0) .. controls ++(0,.5) and ++(0,-.6) .. (3,1.5);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.35) {$\dots$};
\draw[<-] (-0.6,0) -- (-0.6,1.5);
\draw[blue, dotted] (-1,0) -- (3.4,0);
\draw[blue, dotted] (-1,1.5) -- (3.4,1.5);
\draw[->] (0.0,2.5) arc (360:180:3mm);
\draw[->] (0.0,1.5) arc (0:180:3mm);
\draw[->] (0.6,1.5) -- (0.6,2.5);
\draw[->] (1.2,1.5) -- (1.2,2.5);
\draw[->] (1.8,1.5) -- (1.8,2.5);
\draw[->] (3,1.5) -- (3,2.5);
\draw[<-] (0.0,0) .. controls ++(0,-.45) and ++(0,.6) .. (-0.6,-1);
\draw[->] (-0.6,0) .. controls ++(0,-.45) and ++(0,.6) .. (0.0,-1);
\draw[<-] (0.6,0) -- (0.6,-1);
\draw[<-] (1.2,0) -- (1.2,-1);
\draw[<-] (2.4,0) -- (2.4,-1);
\draw[<-] (3.0,0) -- (3,-1);
\end{tikzpicture}}; \endxy}
\;\;\right]
\;\;$$ where the trailing terms arising from relation have the same sign after canceling the empty dots, and thus add together. We can use the trace relation to slide the top cup in the second term to the bottom; after simplication, this term is therefore equal to ${\ifthenelse{\isempty{}{}}{h_{n-1}}{h_{n-1}^{}}}$. The first term is equal to $\beta_n\alpha_{-1}$ as in [@CLLS Lemma 26]. Thus,
$ \left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw[->] (3,0) .. controls ++(0,1.25) and (0,.25) .. (0,2);
\draw[->] (0.0,0) .. controls ++(0,1) and ++(0,-1.2) .. (0.6,2);
\draw[->] (0.6,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.2,2);
\draw[->] (1.2,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.8,2);
\draw[->] (2.4,0) .. controls ++(0,1) and ++(0,-1.2) .. (3,2);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\draw[<-] (-0.6,0) -- (-0.6,2);
\filldraw (.05,1.6) circle (2pt);
\end{tikzpicture}}; \endxy}
\;\;\right]
\;\;
=
\;\;
\left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw[->] (3,0) .. controls ++(0,1.25) and (0,.25) .. (0,2);
\draw[->] (0.0,0) .. controls ++(0,1) and ++(0,-1.2) .. (0.6,2);
\draw[->] (0.6,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.2,2);
\draw[->] (1.2,0) .. controls ++(0,1) and ++(0,-1.2) .. (1.8,2);
\draw[->] (2.4,0) .. controls ++(0,1) and ++(0,-1.2) .. (3,2);
\node at (1.8,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\draw[<-] (3.6,0) -- (3.6,2);
\filldraw (.05,1.6) circle (2pt);
\end{tikzpicture}}; \endxy}
\;\;\;\right] +\hspace{6pt}4 {\ifthenelse{\isempty{}{}}{h_{2n-1}}{h_{2n-1}^{}}}$
as desired. The base case of the induction is proved. The induction step follows from examination of the Jacobi identity, exactly as in [@CLLS Lemma 26], using our Lemma \[VirasorowithUp\] in place of [@CLLS Lemma 24].
\[VirasoroGen\] Let $n \in \mathbb{Z}$. We have $$[{\ifthenelse{\isempty{x_1^2}{}}{h_{1}}{h_{1}^{x_1^2}}}, {\ifthenelse{\isempty{}{}}{h_{2n-1}}{h_{2n-1}^{}}}] = 2{\ifthenelse{\isempty{x_1 + \ldots + x_{2n}}{}}{h_{2n}}{h_{2n}^{x_1 + \ldots + x_{2n}}}} +2{\ifthenelse{\isempty{x_2+\ldots+x_{2n-1}}{}}{h_{2n}}{h_{2n}^{x_2+\ldots+x_{2n-1}}}}.$$
This is a straightforward diagrammatic calculation similar to Lemmas \[upVirasorowithUp\] and \[upVirasorowithDown\]. We have $$\left[\hspace{2mm}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (3.2,0) .. controls (3.2,1.25) and (0,.25) .. (0,2)
node[pos=0.85, shape=coordinate](X){};
\draw[->] (0,0) .. controls (0,1) and (.8,.8) .. (.8,2);
\draw[->] (.8,0) .. controls (.8,1) and (1.6,.8) .. (1.6,2);
\draw[->] (2.4,0) .. controls (2.4,1) and (3.2,.8) .. (3.2,2);
\node at (1.6,.35) {$\dots$};
\node at (2.4,1.65) {$\dots$};
\filldraw (-.3,1.5) circle (3pt);
\node at (-.7,1.5) {$2n$};
\draw[->](-.3,0) to (-.3,2);
\end{tikzpicture}\hspace{2mm}\right]
\hspace{6pt}
=
\hspace{6pt} \left[ \;\;\;
{
\xy (0,0)*{
\begin{tikzpicture}[scale=0.8]
\draw (2.4,0) .. controls (2.4,1.25) and (0,.25) .. (0,1.5)
node[pos=0.85, shape=coordinate](X){};
\draw (0,0) .. controls (0,1) and (0.6,.8) .. (0.6,1.5);
\draw (0.6,0) .. controls (0.6,1) and (1.2,.8) .. (1.2,1.5);
\draw (1.8,0) .. controls (1.8,1) and (2.4,.8) .. (2.4,1.5);
\node at (1.2,.35) {$\dots$};
\node at (1.8,1.65) {$\dots$};
\draw (3,0) -- (3,1.5);
\filldraw (0.05,2.25) circle (2pt);
\node at (-.4,2.25) {$2n$};
\draw[blue, dotted] (-0.4,0) -- (3.4,0);
\draw[blue, dotted] (-0.4,1.5) -- (3.4,1.5);
\draw[->] (3,1.5) .. controls ++(0,0.95) and ++(0,-1.15) .. (0,2.5);
\draw[->] (2.4,1.5) .. controls ++(0,0.5) and ++(0,-0.5) .. (3,2.5);
\draw[->] (0,1.5) .. controls ++(0,0.5) and ++(0,-0.5) .. (0.6,2.5);
\draw[->] (0.6,1.5) .. controls ++(0,0.5) and ++(0,-0.5) .. (1.2,2.5);
\draw[->] (1.2,1.5) .. controls ++(0,0.5) and ++(0,-0.5) .. (1.8,2.5);
\draw (3,0) .. controls ++(0,-0.95) and ++(0,+0.95) .. (0,-1);
\draw (0,0) .. controls ++(0,-0.5) and ++(0,+0.5) .. (0.6,-1);
\draw (0.6,0) .. controls ++(0,-0.5) and ++(0,+0.5) .. (1.2,-1);
\draw (1.8,0) .. controls ++(0,-0.5) and ++(0,+0.5) .. (2.4,-1);
\draw (2.4,0) .. controls (2.4,-.5) and (3,-.5) .. (3,-1);
\end{tikzpicture}}; \endxy}
\; \;\;\right]$$
Sliding the dots all the way to the right side of the diagram results in $2(2n-1)$ resolution terms. Each of these resolution terms contains a $2n$-cycle and a single solid dot - there are 2 resolution terms containing a solid dot on the first strand and 2 containing a solid dot on the last strand, and 4 resolution terms with a dot on each other strand. All empty dots cancel in such a way that no resolution terms cancel with each other. The result follows.
The following lemmas will allow us to generate bubbles with arbitrary numbers of dots using just ${\ifthenelse{\isempty{x_1^2}{}}{h_{\pm 1}}{h_{\pm 1}^{x_1^2}}}$.
\[figure 8 split\] We have $${\displaystyle}\sum_{a+b=2n-1}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.1,1.3) {\small $a$};
\node at (0.1,-.4) {\small $b$};}
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\sum_{i+j=n-1}(1+2j)\hspace{1mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {\small $2i$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {\small $2j$};
\end{tikzpicture}$$
We compute: $${\displaystyle}\sum_{a+b=2n-1}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.1,1.3) {\small $a$};
\node at (0.1,-.4) {\small $b$};}
\end{tikzpicture}\hspace{6pt}$$ $$=
\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$1$};}
\end{tikzpicture}\hspace{4pt}
+
\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$2$};}
\end{tikzpicture}\hspace{4pt}
+
\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$3$};
\node at (0.2,-.4) {\small $2$};}
\end{tikzpicture}\hspace{4pt}
+
\hspace{4pt}
\dots
\hspace{4pt}
+
\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,-.4) {\small $2n$-$2$};}
\end{tikzpicture}$$ $$=
\hspace{4pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$1$};}
\end{tikzpicture}\hspace{4pt}
+
\hspace{4pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$3$};
\node at (0.2,-.4) {\small $2$};}
\end{tikzpicture}\hspace{4pt}
+
\hspace{4pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$5$};
\node at (0.2,-.4) {\small $4$};}
\end{tikzpicture}\hspace{4pt}
+
\hspace{4pt}
\dots
\hspace{4pt}
+
\hspace{4pt}2
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,-.4) {\small $2n$-$2$};}
\end{tikzpicture}$$ because we have $$\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$2i$};
\node at (0.2,-.4) {\small $2i$-$1$};}
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$2i$-$1$};
\node at (0.2,-.4) {\small $2i$};}
\end{tikzpicture}.$$ Moreover, we can decompose these figure eights into a linear combination of products of two bubbles using dot slide relations $\ref{dotSlide: bottomLeft}$ and $\ref{dotSlide: topLeft}$ as follows: $$\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.2,1.3) {\small $2n$-$2a$-$1$};
\node at (0.2,-.4) {\small $2a$};}
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
{\displaystyle}\sum_{\substack{i+j=n-1\\j\geq a}}\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {\small $2i$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {\small $2j$};
\end{tikzpicture}.$$ Combining these results, we get that $${\displaystyle}\sum_{a+b=2n-1}
\begin{tikzpicture}[baseline=(current bounding box.center),rotate=90]
\raisebox{3mm}{
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.1,1.3) {\small $a$};
\node at (0.1,-.4) {\small $b$};}
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\sum_{i+j=n-1}(1+2j)\hspace{1mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {\small $2i$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[<-] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.2,2.2) {\small $2j$};
\end{tikzpicture}.$$
\[updots with downdots\] We have $${\displaystyle}[{\ifthenelse{\isempty{x_1^{2a}}{}}{h_{1}}{h_{1}^{x_1^{2a}}}},{\ifthenelse{\isempty{x_1^{2b}}{}}{h_{-1}}{h_{-1}^{x_1^{2b}}}}]=-2\bar{d}_{2(a+b)}-\sum_{i+j=2(a+b)-1}(2+4j)\bar{d}_{2i}d_{2j}$$ for $a,b\in \mathbb{Z}_{\geq0}.$
We compute: $$\begin{aligned}
{\displaystyle}\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (0,0) to (0,2);
\draw[<-] (0.75,0) to (0.75,2);
\fill (0,1.2) circle (2pt);
\fill (0.75,1.2) circle (2pt);
\node at (-0.3,1.3) {\small $2a$};
\node at (1.1,1.3) {\small $2b$};
\end{tikzpicture}\hspace{6pt}
&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,2);
\draw[->] (0.0,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,2);
\fill (-0.55,1.7) circle (2pt);
\fill (-0.1,1.7) circle (2pt);
\node at (-.95,1.8) {\small $2a$};
\node at (0.35,1.8) {\small $2b$};
\end{tikzpicture}
=\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,2);
\draw[->] (0.0,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,2);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.1,1.7) circle (2pt);
\node at (0.3,1.3) {\small $2a$};
\node at (0.35,1.8) {\small $2b$};
\end{tikzpicture}\hspace{6pt}
-\hspace{6pt}2
\sum_{j=0}^{2a-1}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,2) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,2);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (0,1.8) circle (2pt);
\node at (0.2,1.2) {\small $j$};
\node at (1.2,1.8) {\small $2(a$+$b)$-$1$-$j$};
\end{tikzpicture}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,2);
\draw[->] (0.0,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,2);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (-0.55,1.2) circle (2pt);
\node at (0.3,1.3) {\small $2a$};
\node at (-0.9,1.3) {\small $2b$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}2
\sum_{i=0}^{2b-1}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,2) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,2);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (0,1.8) circle (2pt);
\node at (0.66,1.2) {\small $2a$+$i$};
\node at (0.66,1.8) {\small $2b$-$1$-$i$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}2
\sum_{j=0}^{2a-1}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,2) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,2);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (0,1.8) circle (2pt);
\node at (0.2,1.2) {\small $j$};
\node at (1.2,1.8) {\small $2(a$+$b)$-$1$-$j$};
\end{tikzpicture}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[->] (-0.6,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,2);
\draw[<-] (0.0,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,2);
\fill (-0.55,1.7) circle (2pt);
\fill (-0.1,1.7) circle (2pt);
\node at (-.95,1.8) {\small $2b$};
\node at (0.35,1.8) {\small $2a$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}2
\sum_{j=0}^{2(a+b)-1}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,2) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,2);
\filldraw (-0.05,1.2) circle (2pt);
\filldraw (0,1.8) circle (2pt);
\node at (0.2,1.2) {\small $j$};
\node at (1.2,1.8) {\small $2(a$+$b)$-$1$-$j$};
\end{tikzpicture}
\\&=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[->] (-0.6,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,2);
\draw[<-] (0.0,1) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,2);
\fill (-0.55,1.7) circle (2pt);
\fill (-0.1,1.7) circle (2pt);
\node at (-.95,1.8) {\small $2b$};
\node at (0.35,1.8) {\small $2a$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}2
\sum_{j=0}^{2(a+b)-1}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (0,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.3,1.3) {$j$};
\node at (1.2,-.2) {$2(a$+$b)$-$1$-$j$};
\end{tikzpicture}
\\&=
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (0,0) to (0,2);
\draw[->] (0.75,0) to (0.75,2);
\fill (0,1.2) circle (2pt);
\fill (0.75,1.2) circle (2pt);
\node at (-0.3,1.3) {\small $2b$};
\node at (1.1,1.3) {\small $2a$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}2\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[->] (3,2) arc (-180:180:5mm);
\fill (3.95,2.2) circle [radius=2pt];
\node at (4.5,2.2) {\small $2(a$+$b)$};
\end{tikzpicture}\hspace{6pt}
-
\hspace{6pt}2
\sum_{j=0}^{2(a+b)-1}\hspace{2mm}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[->] (-0.6,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.0,1);
\draw[<-] (0.0,0) .. controls ++(0,.4) and ++(0,-.5) .. (-0.6,1);
\draw[<-] (-0.6,1) .. controls ++(0,.4) and ++(0, .4) .. (-0.0,1);
\draw[->] (0.0,0) .. controls ++(0,-.5) and ++(0,-.5) .. (-0.6,0);
\filldraw (0,1.2) circle (2pt);
\filldraw (-0.1,-.3) circle (2pt);
\node at (0.3,1.3) {$j$};
\node at (1.2,-.2) {$2(a$+$b)$-$1$-$j$};
\end{tikzpicture}.\end{aligned}$$
Therefore ${\displaystyle}[{\ifthenelse{\isempty{x_1^{2a}}{}}{h_{1}}{h_{1}^{x_1^{2a}}}},{\ifthenelse{\isempty{x_1^{2b}}{}}{h_{-1}}{h_{-1}^{x_1^{2b}}}}]=-2\bar{d}_{2(a+b)}-\sum_{i+j=2(a+b)-1}(2+4j)\bar{d}_{2i}d_{2j}$.
Algebra isomorphism
===================
In this section, we will study the structure of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, first as a vector space and then as an algebra. We show that ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ has a triangular decomposition into two copies of the trace of ${\mathfrak{H}^C}_n$ and a polynomial algebra. We then describe a generating set for ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, which allows us to define the algebra homomorphism to $W^-$. Finally, we prove that this homomorphism is an isomorphism.
Trace of $\mathcal{H}_{tw}$ as a vector space
---------------------------------------------
Let $m,n \geq 0$ and define $J_{m,n}$ to be the 2-sided ideal in ${\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n)$ generated by diagrams which contain at least one arc connecting a pair of upper points.
\[ses\] There exists a split short exact sequence $$0 \rightarrow J_{m,n} \rightarrow {\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n) \rightarrow ({\mathfrak{H}^C})^{op}_m \otimes {\mathfrak{H}^C}_n \otimes \mathbb{C}[d_0,d_2,d_4....] \rightarrow 0.$$
In ${\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n)$, due to the middle diagram in relation , we can assume our diagrams have no crossing between opposite oriented strands. Taking the quotient ${\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n)/J_{m,n}$ kills diagrams with cups connecting two upper points, and those with caps connecting two lower points. Therefore we are left with diagrams, possibly with bubbles, which have no caps or cups and have crossings only among like-oriented strands. Note that in the quotient ${\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n)/J_{m,n}$, the diagram in relation simplifies to $$\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw [<-](0,0) to [out=45,in=-45] (0,2);
\draw [->](0.5,0) to [out=135,in=-135] (0.5,2);
\end{tikzpicture}\hspace{6pt}
=
\hspace{6pt}
\begin{tikzpicture}[baseline=(current bounding box.center),scale=0.75]
\draw[<-] (1.5,0) to (1.5,2);
\draw[->] (2,0) to (2,2);
\end{tikzpicture}$$ and therefore we can move the bubbles to the rightmost part of our diagrams for free. This gives us a short exact sequence $$0 \rightarrow J_{m,n} \rightarrow {\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n) \rightarrow {\operatorname{End}}_{\mathcal{H}_{tw}}(P^m) \otimes {\operatorname{End}}_{\mathcal{H}_{tw}}(Q^n) \otimes {\operatorname{End}}_{\mathcal{H}_{tw}}(1) \rightarrow 0.$$ By [@CS Proposition 7.1], we have that ${\operatorname{End}}_{\mathcal{H}_{tw}}(P^m)$ is isomorphic to $({\mathfrak{H}^C})^{op}_m$ and that ${\operatorname{End}}_{\mathcal{H}_{tw}}(Q^n)$ is isomoprhic to ${\mathfrak{H}^C}_n$. By Proposition \[bubbles\], it follows that ${\operatorname{End}}_{\mathcal{H}_{tw}}(1)$ is isomorphic to $\mathbb{C}[d_0,d_2,d_4....]$. Hence the result follows.
\[fg\] If $f,g\in {\mathfrak{H}^C}_n$ such that $fg=1$, then $f,g\in {\mathcal{C}\ell_n}\rtimes \mathbb{C}[S_n]\subset {\mathfrak{H}^C}_n$.
There is an $\mathbb{N}$-filtration on ${\mathfrak{H}^C}_n$ given by $\deg(x_i)=1$ for $i\in \{1,...,n\}$ and other generators have degree zero. Under this filtration, the degree zero part of ${\mathfrak{H}^C}_n$ is the semidirect product $Cl_n\rtimes \mathbb{C}[S_n]$. Therefore, in the associated graded object, we see that if $fg=1$, $\deg(gr(f)gr(g))=\deg(gr(f))+\deg(gr(g))=\deg(1)=0$, hence $gr(f),gr(g)$ are in degree zero part. Therefore $f,g\in {\mathcal{C}\ell_n}\rtimes \mathbb{C}[S_n]$.
\[indec objects\] The indecomposable objects of $\mathcal{H}_{tw}$ are of the form $P^mQ^n$ for $m,n\in \mathbb{Z}_{\geq 0}$.
First, note that if $QP$ appears in an object, that object can be decomposed into more components using the diagram in relation . Hence all indecomposable objects must be of the form $P^mQ^n$.
On the other hand, to see that every sequence of the form $P^mQ^n$ is an indecomposable object, we will show that any idempotent in ${\operatorname{End}}(P^mQ^n)$ has to be the identity.
Let $f,g$ be two maps as mentioned in Lemma \[fg\]. Note that $gf$ is an idempotent since $(gf)(gf)=g(fg)f=gf$. Since we had the splitting short exact sequence $0 \rightarrow J_{m,n} \rightarrow {\operatorname{End}}_{\mathcal{H}_{tw}}(P^mQ^n) \rightarrow {\operatorname{End}}(P^m) \otimes {\operatorname{End}}(Q^n) \otimes {\operatorname{End}}(id) \rightarrow 0$ in Lemma \[ses\], we know that the maps $f$ and $g$ will decompose into $(f_1,f_2)$ and $(g_1,g_2)$ where $f_1,g_1:P^m\rightarrow P^m$ and $(f_2,g_2):Q^n\rightarrow Q^n$. Now $g_1f_1$ is the identity map in ${\operatorname{End}}(P^m)$, and by the above lemma $g_1,f_1\in \mathcal{C}\ell_n\rtimes \mathbb{C}[S_n]$. Similarly, $f_2, g_2\in \mathcal{C}\ell_n\rtimes \mathbb{C}[S_n]$.
But in $\mathcal{C}\ell_n\rtimes \mathbb{C}[S_n]$, $g_1f_1=1$ implies that $f_1g_1=1$ as well. To see this, consider the diagrams corresponding to $g_1$ and $f_1$ which consist of a permutation and some hollow dots on top. After composing these diagrams, we can collect all the hollow dots on the top since hollow dots can pass through crossing for free, possibly gaining a sign. Furthermore, each strand has an even number of hollow dots, since this composition is the identity map. So, the hollow dots cancel with each other. This shows that the corresponding permutations of $f_1$ and $g_1$ are inverses of each other, and in particular they commute. Therefore $f_1g_1=1$. Similarly, $f_2g_2=1$. Thus we have that $fg=1$.
\[triangularDecomposition\] We have the triangular decomposition of ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$: $${{\operatorname{Tr}}(\mathcal{H}_{tw})}\cong \bigoplus_{m,n\in \mathbb{Z}_{\geq0}} {\operatorname{Tr}}(({\mathfrak{H}^C}_m)^{op} \otimes {\mathfrak{H}^C}_n \otimes \mathbb{C}[d_0,d_2,d_4....]).$$
As shown in [@BGHL], to find ${{\operatorname{Tr}}(\mathcal{H}_{tw})}$, it is enough to consider the direct sum over indecomposable objects of endomorphism spaces of objects of $H_{tw}$. Let $I=\operatorname{span}_\mathbb{C}\{fg-gf\}$ where $f:x\rightarrow y$ and $g:y\rightarrow x$ for $x,y$ objects of a $\mathbb{C}$-linear category. Therefore by Lemma \[indec objects\] we have $${{\operatorname{Tr}}(\mathcal{H}_{tw})}\cong \bigg(\bigoplus_{m,n\in \mathbb{Z}_{\geq0}} {\operatorname{End}}_{{\mathcal{H}_{tw}}}(P^mQ^n)\bigg)\big/I.$$
By Lemma \[ses\], this gives us $${{\operatorname{Tr}}(\mathcal{H}_{tw})}\cong \bigg(\bigoplus_{m,n\in \mathbb{Z}_{\geq0}} (({\mathfrak{H}^C}_m)^{op} \otimes {\mathfrak{H}^C}_n \otimes \mathbb{C}[d_0,d_2,d_4....])\oplus J_{m,n}\bigg)\big/I.$$ Recall that the ideal $J_{m,n}$ is generated by diagrams containing at least one cup connecting two upper points. Therefore, the diagrams in $J_{m,n}$ must also contain caps, since they are dealing with endomorphisms. Using the trace relation and the relations in $\mathcal{H}_{tw}$, we can express the elements of $J_{m,n}$ as direct sum of endomorphisms of $P^{m'}Q^{n'}$ for $m'\leq m$ and $n'\leq n$. Hence we have $$\begin{aligned}
{{\operatorname{Tr}}(\mathcal{H}_{tw})}&\cong \bigoplus_{m,n\in \mathbb{Z}_{\geq0}} {\operatorname{Tr}}(({\mathfrak{H}^C}_m)^{op} \otimes {\mathfrak{H}^C}_n \otimes \mathbb{C}[d_0,d_2,d_4....])
\\&\cong \bigg(\bigoplus_{m,n\in \mathbb{Z}_{\geq0}} {\operatorname{Tr}}(({\mathfrak{H}^C}_m)^{op} \otimes {\mathfrak{H}^C}_n)\bigg) \otimes \mathbb{C}[d_0,d_2,d_4....].\end{aligned}$$
Generators of the algebra ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$
------------------------------------------------------------------------------------
The following gives a generating set for ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ as an algebra.
\[GenSetTr\] The algebra ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ is generated by ${\ifthenelse{\isempty{}{}}{h_{-1}}{h_{-1}^{}}}$, ${\ifthenelse{\isempty{(x_1 + x_2)}{}}{h_{\pm 2}}{h_{\pm 2}^{(x_1 + x_2)}}}$, and $d_0 + d_2$.
First, Proposition \[clockwise bubbles with down explicit\] implies that ${\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}$ and $(d_0+d_2)$ allow us generate a differential degree two element ${\ifthenelse{\isempty{x_1^2}{}}{h_{1}}{h_{1}^{x_1^2}}}$; since all relations in $\mathcal{H}_{tw}$ are local, we can evaluate the commutator $[{\ifthenelse{\isempty{x_1^2}{}}{h_{1}}{h_{1}^{x_1^2}}}, (d_0+d_2)]$ by moving the dot to the bottom of the upward strand and sliding the bubbles over the upper portion. We can therefore apply Lemma \[clockwise bubbles with down explicit\] repeatedly to show that $\operatorname{ad}(d_0+d_2)^n{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}$ has a leading term of ${\ifthenelse{\isempty{x_1^{2n}}{}}{h_{1}}{h_{1}^{x_1^{2n}}}}$.
By Lemma \[upVirasorowithUp\], the elements ${\ifthenelse{\isempty{}{}}{h_{-1}}{h_{-1}^{}}}$ and ${\ifthenelse{\isempty{x_1+x_2}{}}{h_{2}}{h_{2}^{x_1+x_2}}}$ are sufficient to generate ${\ifthenelse{\isempty{}{}}{h_{2m+1}}{h_{2m+1}^{}}}$ for all integers $m>0$. Then we can generate ${\ifthenelse{\isempty{x_1+\ldots+x_n}{}}{h_{2n}}{h_{2n}^{x_1+\ldots+x_n}}}$ from ${\ifthenelse{\isempty{x_1^2}{}}{h_{1}}{h_{1}^{x_1^2}}}$ and ${\ifthenelse{\isempty{}{}}{h_{2m+1}}{h_{2m+1}^{}}}$ by using Lemma \[VirasoroGen\]. Lemma \[upVirasorowithDown\], ${\ifthenelse{\isempty{}{}}{h_{-1}}{h_{-1}^{}}}$ and ${\ifthenelse{\isempty{x_1+x_2+\ldots + x_n}{}}{h_{2n}}{h_{2n}^{x_1+x_2+\ldots + x_n}}}$ allow us to generate ${\ifthenelse{\isempty{}{}}{h_{2r+1}}{h_{2r+1}^{}}}$ for all integers $r$.
Proposition \[red to hn\] implies that all elements with nonzero rank degree can be written as a sum of elements of the form ${\ifthenelse{\isempty{x_1^{\ell} c_1^{k}}{}}{h_{ \pm n}}{h_{ \pm n}^{x_1^{\ell} c_1^{k}}}}$. By Propositions \[evenCyclesZero\] and \[odd with odd dots zero\], all elements of this form except for the ones generated in the preceding paragraphs are 0 in ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$, so we have generated all of ${{\operatorname{Tr}}^>(\mathcal{H}_{tw})_{\overline{0}}}$ and ${{\operatorname{Tr}}^<(\mathcal{H}_{tw})_{\overline{0}}}$.
Finally, Lemma \[updots with downdots\] allows us to generate $d_{2n}$, applying Lemma \[bubbleDecomp\] to split up the $\overline{d}_{2n}$ terms.
The isomorphism
---------------
There is an obvious isomorphism of vector spaces between the Fock space representations of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ and $W^-$:
$$\label{VSIsom}
\phi: V= \mathbb{C}[{\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}, {\ifthenelse{\isempty{}{}}{h_{2}}{h_{2}^{}}}, \ldots] \rightarrow \mathbb{C}[w_{-1,0}, w_{-2,0}, \ldots] = \mathcal{V}_{1,0}.$$
Recall that each algebra acts faithfully on its Fock space representation.
\[Heis commutes\] The map $\phi$ in Equation commutes with the action of the twisted Heisenberg subalgebras in $V$ and $\mathcal{V}_{1,0}$, i.e.: $$\phi({\ifthenelse{\isempty{}{}}{h_{r}}{h_{r}^{}}}v) =\sqrt{2}w_{-r,0}\phi(v).$$
The vector space realizations of $V$ and $\mathcal{V}_{1,0}$ in Equation imply that the action of ${\ifthenelse{\isempty{}{}}{h_{r}}{h_{r}^{}}}$ on $V$ is simply the adjoint action of ${\ifthenelse{\isempty{}{}}{h_{r}}{h_{r}^{}}}$ on the subalgebra ${{\operatorname{Tr}}^>(\mathcal{H}_{tw})_{\overline{0}}}$, and the action of $w_{-r,0}$ on $\phi(v)$ is the adjoint action of $w_{-r,0}$ on $(W^-)^-$. The Lemma follows from our computation of these twisted Heisenberg relations in Propositions \[FockSpaceW action\] and \[twistedHeisRels\].
\[bubbles commutes\] For any $v\in V$ we have $\phi((d_0+d_2)v) = -2w_{0,3} \phi(v)$.
Propositions \[FockSpaceW action\] and \[clockwise bubbles explicit\] give that $w_{0,3}$ maps $w_{-1,0}$ to an element with leading term $w_{-1,2}$, and $(d_0+d_2)$ maps ${\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}}$ to an element with leading term ${\ifthenelse{\isempty{x_1^2}{}}{h_{1}}{h_{1}^{x_1^2}}}$. Comparision of the actions of these terms on the twisted Heisenberg subalgebras on either side gives that that their images in the endomorphisms of the Fock space are identical.
\[Virasoro commutes\] For any $v \in V$ we have $\phi({\ifthenelse{\isempty{(x_1 + x_2)}{}}{h_{\pm 2}}{h_{\pm 2}^{(x_1 + x_2)}}} v) = 2\sqrt{2}(w_{\mp 2, 1} + w_{\mp 2, 0}) \phi(v)$.
This follows from comparision of Lemma \[upVirasorowithUp\] and Proposition \[FockSpaceW action\].
Now extend $\phi$ to a map $$\Phi: {{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}\longrightarrow W^-/\langle w_{0,0}, C-1 \rangle$$ by mapping $${\ifthenelse{\isempty{}{}}{h_{1}}{h_{1}^{}}} \mapsto \sqrt{2}w_{-1,0} \qquad {\ifthenelse{\isempty{(x_1 + x_2)}{}}{h_{\pm 2}}{h_{\pm 2}^{(x_1 + x_2)}}} \rightarrow 2\sqrt{2}w_{\mp 2, 1} + w_{\mp 2, 0} \qquad d_2+d_0 \mapsto -2w_{0,3}$$ and extending algebraically, i.e. $$\Phi(a_1 \ldots a_k) = \Phi(a_1) \ldots \Phi(a_k)$$ for generators $a_1, \ldots, a_k$ of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$.
\[well defined\] The map $\Phi$ above is well defined.
Suppose $A\in {{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ has two representations in terms of generators, $A= a_{i_1} \ldots a_{i_k} = a_{j_1} \ldots a_{j_{\ell}}$. Then $a_{i_1} \ldots a_{i_k}. V = a_{j_1}\ldots a_{j_{\ell}}.V$, so applying $\Phi$ gives $\Phi(a_{i_1} \ldots a_{i_k}).\mathcal{V}_{1,0}=\Phi(a_{j_1} \ldots a_{j_\ell}).\mathcal{V}_{1,0}$. Hence $\Phi(a_{i_1} \ldots a_{i_k}) = \Phi(a_{j_1} \ldots a_{j_\ell})$ by the faithfulness of the Fock space representation for $W^-$.
\[main thm\] The map $\Phi$ is an isomorphism of algebras.
We immediately have that $\Phi$ is surjective, because it maps generators to generators. Thus, it remains to show that $\Phi$ is injective. Let $A := a_{i_1} \ldots a_{i_k}\in {{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$ and assume that $\Phi(A).\mathcal{V}_{1,0} = 0$. Then $\Phi(A) = 0$ by the faithfulness of the representation. But then $\Phi(a_{i_1})\ldots \Phi(a_{i_k}) .\mathcal{V}_{1,0}=0$. Then, by Lemmas \[Heis commutes\], \[bubbles commutes\], and \[Virasoro commutes\], we have $\Phi(a_{i_1})\ldots \Phi(a_{i_k} ). \mathcal{V}_{1,0} = \phi(a_{i_1}\ldots a_{i_k}. V) = \phi(A.v)=0$. But $\phi$ is an isomorphism, so this implies that $A.V=0$. Hence $A = 0$ by the faithfulness of the Fock space representation of ${{{\operatorname{Tr}}(\mathcal{H}_{tw})}_{\overline{0}}}$.
|
---
abstract: 'Style transfer deals with the algorithms to transfer the stylistic properties of a piece of text into that of another while ensuring that the core content is preserved. There has been a lot of interest in the field of text style transfer due to its wide application to tailored text generation. Existing works evaluate the style transfer models based on content preservation and transfer strength. In this work, we propose a reinforcement learning based framework that directly rewards the framework on these target metrics yielding a better transfer of the target style. We show the improved performance of our proposed framework based on automatic and human evaluation on three independent tasks: wherein we transfer the style of text from formal to informal, high excitement to low excitement, modern English to Shakespearean English, and vice-versa in all the three cases. Improved performance of the proposed framework over existing state-of-the-art frameworks indicates the viability of the approach.'
author:
- 'Abhilasha Sancheti[^1]'
- Kundan Krishna
- Balaji Vasan Srinivasan
- Anandhavelu Natarajan
bibliography:
- 'mybibliography.bib'
title: Reinforced Rewards Framework for Text Style Transfer
---
Introduction {#sec: introduction}
============
Text style transfer deals with transforming a given piece of text in such a way that the stylistic properties change to that of the target text while preserving the core content of the given text. This is an active area of research because of its wide applicability in the field of content creation including news rewriting, generating messages with a particular style to maintain the personality of a brand, etc. The stylistic properties may denote various linguistic phenomenon, from syntactic changes [@W17-4902; @xu2012paraphrasing] to sentiment modifications [@shen2017style; @li2018delete; @fu2017style] or extent of formality in a sentence [@rao2018dear].
Most of the existing works in this area either use copy-enriched sequence-to-sequence models [@W17-4902] or employ an adversarial [@shen2017style; @fu2017style; @P18-1080] or much simpler generative approaches [@li2018delete] based on the disentanglement of style and content in text. On the other hand, more recent works like [@subramanian2018multiple] and [@dai2019style] perform the task of style transfer without disentangling style and content, as practically this condition cannot always be met. However, all of these works use word-level objective function (eg. cross-entropy) while training which is inconsistent with the desired metrics (content preservation and transfer strength) to be optimized in style transfer tasks. These metrics are generally calculated at a sentence-level and use of word level objective functions is not sufficient. Moreover, discreteness of these metrics makes it even harder to directly optimize the model over these metrics.
Recent advancements in Reinforcement Learning and its effectiveness in various NLP tasks like sequence modelling [@keneshloo2018deep], abstractive summarization [@paulus2017deep], and a related one machine translation [@D18-1397] have motivated us to leverage reinforcement learning approaches in style transfer tasks.
In this paper, we propose a reinforcement learning (RL) based framework which adopts to optimize sequence-level objectives to perform text style transfer. Our reinforced rewards framework is based on a sequence-to-sequence model with attention [@bahdanau2014neural; @luong2015effective] and copy-mechanism [@W17-4902] to perform the task of text style transfer. The sentence generated by this model along with the ground truth sentence is passed to a content module and a style classifier which calculates the metric scores to finally obtain the reward values. These rewards are then propagated back to the sequence-to-sequence model in the form of loss terms.
The rest of our paper is organized as follows: we discuss related work on text style transfer in Section \[sec: related\]. The proposed reinforced rewards framework is introduced in Section \[sec: model\]. We evaluate our framework and report the results on formality transfer task in Section \[sec: experiments-formality\], on affective dimension like excitement in Section \[sec: experiments-beyond\] and on Shakespearean-Modern English corpus in Section \[sec: experiments-beyond-affect\]. In Section \[sec: discussion\], we discuss few qualitative sample outputs. Finally, we conclude the paper in Section \[sec: conclusion\].
Related Work {#sec: related}
============
Style transfer approaches can be broadly categorized as style transfer with parallel corpus and style transfer with non-parallel corpus.
Parallel corpus consists of input-output sentence pairs with mapping. Since such corpora are not readily available and difficult to curate, efforts here are limited. [@xu2012paraphrasing] introduced a parallel corpus of $30$K sentence pairs to transfer Shakespearean English to modern English and benchmark various phrase-based machine translation methods for this task. [@W17-4902] use a copy-enriched sequence-to-sequence approach for Shakespearizing modern English and show that it outperforms the previous benchmarks by [@xu2012paraphrasing]. Recently, [@rao2018dear] introduced a parallel corpus of formal and informal sentences and benchmark various neural frameworks to transfer sentences across different formality levels. Our approach contributes in this field of parallel style transfer and extends the work by [@W17-4902] by directly optimizing the metrics used for evaluating the style transfer tasks.
Another class of explorations are in the area of non-parallel text style transfer [@shen2017style; @fu2017style; @li2018delete; @P18-1080] which does not require mapping between the input and output sentences. [@fu2017style] compose a non-parallel dataset for paper-news titles and propose models to learn separate representations for style and content using adversarial frameworks. [@shen2017style] assume a shared latent content distribution across a given corpora and propose a method that leverages refined alignment of latent representations to perform style transfer. [@li2018delete] define style in terms of attributes (such as, sentiment) localized to parts of the sentence and learn to disentangle style from content in an unsupervised setting. Although these approaches perform well on the transfer task, content preservation is generally observed to be low due to the non-parallel nature of the data. Along this line, parallel style transfer approaches have shown better performance in benchmarks despite the data curation challenges [@rao2018dear].
Style transfer models are primarily evaluated on **content preservation** and **transfer strength**. But the existing approaches do not optimize on these metrics and rather teach the model to generate sentences to match the ground truth. This is partly because of the reliance on a differentiable training objective and discreteness of these metrics makes it challenging to differentiate the objective. Leveraging recent advancements in reinforcement learning approaches, we propose a reinforcement learning based text style transfer framework which directly optimizes the model on the desired evaluation metrics. Though there exists some prior work on reinforcement learning for machine translation [@D18-1397], sequence modelling [@keneshloo2018deep] and abstractive summarization [@paulus2017deep] dealing model optimization for qualitative metrics like Rouge [@lin2004rouge], they do not consider style aspects which is one of the main requirements of style transfer tasks. More recently, efforts [@xu2018unpaired; @gong2019reinforcement] have been made to incorporate RL in style transfer tasks in a non-parallel setup. However, our work is in the field of parallel text style transfer which is not much explored.
Our work is different from these related works in the sense that we take care of content preservation and transfer strength with the use of a content module (to ensure content preservation) and cooperative style discriminator (style classifier) without explicitly separating content and style. We illustrate the improvement in the performance of the framework on the task of transferring text between different levels of formality [@rao2018dear]. Furthermore, we present the generalizability of the proposed approach by evaluating it on a self-curated excitement corpus as well as modern English to Shakespearean corpus [@W17-4902].
Reinforced Rewards Framework {#sec: model}
============================
The proposed approach takes an input sentence $x= x_{1}\ldots x_{l}$ from source style $s_{1}$ and translates it to sentence $y= y_{1}\ldots y_{m}$ with style $s_{2}$, where *x* and *y* are represented as a sequence of words. If $x$ is given by ($c_{1},s_{1}$) where $c_{1}$ represents the content and $s_{1}$ the style of the source, our objective is to generate $y=(c_{1},s_{2})$ which has same content as the source but with the target style.
Our approach is based on a copy-enriched sequence-to-sequence framework [@W17-4902] which allows the model to retain factual parts of the text while changing the style specific text using an attention mechanism. At the time of training, the framework takes in the source style and the target style sentence as input to the attention based sequence-to-sequence encoder-decoder model. The words in the input sentence are mapped into an embedding space and the sentence is encoded into a latent space by the LSTM encoder. The network learns to pay attention to the words in the source sentence and creates a context vector based on the attention. The decoder model is a mixture of RNN and pointer (PTR) network where the RNN predicts the probability distribution over the vocabulary and the pointer network predicts the probability over the words in the input sentence based on the context vector. A weighted average of the two probabilities yields the final probability distribution at time step t given by, $$\textstyle
P_{t}(w) = \delta P_{t}^{RNN}(w) + (1-\delta) P_{t}^{PTR}(w),$$ where $\delta$ is computed based on encoder outputs and previous decoder hidden states. The decoder generates the transferred sentence by selecting the most probable word at each time step. This model is trained to minimize cross entropy loss given by $$\textstyle
L_{ml} = - \sum_{t=1}^{m}{\log(p(P_{t}(y_{t}^{*})))},$$ where *m* is the maximum length of the output sentence and $y_{t}^{*}$ is the ground truth word at time *t* in the transferred sentence.
![Model overview[]{data-label="fig: overview"}](overview.pdf){width="7.5cm" height="2.5cm"}
While this framework optimizes for generating sentences close to the ground truth, it does not explicitly teach the network to preserve the content and generate sentences in target style. To achieve this, we introduce a style classifier and a content module which takes in the generated sentence from the sequence-to-sequence model along with the ground truth target sentence to provide reward to the sentence, as shown in Figure \[fig: overview\]. We leverage BLEU [@papineni2002bleu] score to measure the reward for preserving content and because of the lack of any formal score for transfer strength, we use a cooperative discriminator to provide score to the generated sentence. This score from the discriminator is used as a measure to reward for transfer strength. These rewards are then back propagated as explicit loss terms to penalize the network for incorrect generation.
Content Module: Rewarding Content Preservation
----------------------------------------------
To preserve the content while transferring the style, we leverage Self-Critic Sequence Training (SCST) [@rennie2017self] approach and optimize the framework with BLEU scores as the reward. SCST is a policy gradient method for reinforcement learning and is used to train end-to-end models directly on non-differentiable metrics. We use BLEU score as reward for content preservation because it measures the overlap between the ground truth and the generated sentences. Teaching the network to favor this would result in high overlap with the ground truth and subsequently preserve the content of the source sentence since ground truth ensures this preservation.
We produce two output sentences $y^s$ and $y^{'}$, where $y^s$ is sampled from the distribution $p(y_{t}^{s}|y_{1:t-1}^{s},x)$ at each decoding time step and $y^{'}$ (baseline output) is obtained by greedily maximizing the output distribution at each time step. The BLEU score between the sampled and greedy sequences is computed as the reward and the corresponding content-preservation loss is given by, $$\textstyle
L_{cp} = (r(y^{'})-r(y^{s}))\sum_{t=1}^{m}{\log(p(y_{t}^{s}| y_{1:t-1}^s,x))},$$ where the log term is the log likelihood on sampled sequence and the difference term is the difference between the reward (BLEU score) for the greedily sampled $y^{'}$ and multinomially sampled $y^{s}$ sentences. Note that our formulation is flexible and does not require the metric to be differentiable because rewards are used as weights to the log-likelihood loss. Minimizing $L_{cp}$ is equivalent to encouraging the model to generate sentences which have higher reward as compared to the baseline $y^{'}$ and thus increasing the reward expectation of the model. The framework can now be trained end to end by using this loss function along with the cross entropy loss to preserve the content of the source sentence in the transferred sentence.
Style Classifier: Rewarding Transfer Strength {#subsec: ts}
---------------------------------------------
To optimize the model to generate sentences which belong to the target style, it is possible to use a similar loss function as above and use it with the SCST framework [@rennie2017self]. However, that will require a formal measure for the target style aspect. Here, we present an alternate framework where such a formal measure is not readily available. We train a convolutional neural network based style classifier as proposed by [@kim2014convolutional] on the training dataset. This style classifier predicts the likelihood that an input sentence is in the target style, and the likelihood is taken as a proxy to the reward for style of a sentence and appended to a discriminator-based loss function extended from [@P18-1152]. Based on the transfer direction, we add the following term to the cross-entropy loss, $$L_{ts}=
\begin{cases}
- \log(1-s(y^{'})),& \text{high to low level} \\
- \log(s(y^{'})),& \text{low to high level}
\end{cases}$$ In this formulation, $y^{'}$ is the greedily generated output from the decoder and s($y^{'}$) is the likelihood score predicted by the classifier for $y^{'}$. When transfer is done from high to low level of style, minimization of $L_{ts}$ will encourage generation of sentences such that the classifier score is as low as possible. When the sentences are transferred from low to high level of style then the formulation ensures that the generated sentences have a score as high as possible. The framework is trained end-to-end using this loss function to generate the sentences which belong to the target style.
Training and Inference
----------------------
The overall loss function thus can be written as a combination of the $3$ loss functions, $$Loss= \alpha L_{ml}+ \beta L_{cp}+ \gamma L_{ts}$$ We train various models using this loss function and different training methodologies (setting $\alpha=1.0$, $\beta=0.125$, $\gamma=1.0$ after hyper-parameter tuning) as described in the next section. During the inference phase, the model predicts a probability distribution over the vocabulary based on the sentence generated so far and the word having the highest probability is chosen as the next word till the maximum length of the output sentence is reached. Note that unlike training phase in which case both the input and ground truth transferred sentences are available to the model, only the input sentence is made available to the model.
Experiments: Reinforcing Formality (GYAFC Dataset) {#sec: experiments-formality}
==================================================
We evaluate the proposed approach on the GYAFC [@rao2018dear] dataset which is a parallel corpus for formal-informal text. We present the transfer task results in both the directions - formal to informal and vice-versa. This dataset (from Entertainment and Music domain) consists of $\sim$56K informal-formal sentence pairs: $\sim$52K in train, $\sim$1.5K in test and $\sim$2.5K in validation split.
We use both human and automatic evaluation measures for content preservation and transfer strength to illustrate the performance of the proposed approach.
**Content preservation** measures the degree to which the target style model outputs have the same meaning as the input style sentence. Following [@rao2018dear], we measure preservation of content using BLEU [@papineni2002bleu] score between the ground truth and the generated sentence since the ground truth ensures that content of the source style sentence is preserved in it. For human evaluation, we presented $50$ randomly selected model outputs to the Mechanical turk annotators and requested them to rate the outputs on a Likert [@bertram2007likert] scale of 6 as described in [@rao2018dear].
**Transfer strength** measures the degree to which style transfer was carried out. We reuse the classifiers that we built to provide rewards to the generated sentences (Section \[subsec: ts\]). A score above $0.5$ from the classifier represents that the generated sentence belongs to the target style and to the source style otherwise. We define accuracy as the fraction of generated sentences which are classified to be in the target style. The higher the accuracy, higher is the transfer strength. For human evaluation, we ask the Mechanical turk annotators to rate the generated sentence on a Likert scale of $5$ as described in [@rao2018dear].
Following [@fu2017style] who illustrate the trade-off between the two metrics - content preservation and transfer strength, we combine the two evaluation measures and present an **overall score** for the transfer task since both the measures are central to different aspects of text style transfer task. The trade-off arises because the best content preservation can be achieved by simply copying the source sentence. However, the transfer strength in such scenario will be the worst. We compute overall score in the following way $$\rm{Overall} = \frac{\rm{BLEU} \times \rm{Accuracy}}{\rm{BLEU} + \rm{Accuracy}}$$ which is similar to F1-score since content preservation can be considered as measuring recall of the amount of source content retained in the target style sentence and transfer strength acts as a measure of precision with which the transfer task is carried out. In the above formulation, both BLEU and accuracy scores are normalized to be between $0$ and $1$.
We first ran an **ablation study** to demonstrate the improvement in performance of the model with introduction of the two loss terms in the various settings differing in the way training is being carried out. Below we provide details about each of the settings.\
**CopyNMT:** Trained with $L_{ml}$ **TS:** Trained with $L_{ml}$ followed by $\alpha L_{ml}+ \gamma L_{ts}$ **CP:** Trained with $L_{ml}$ followed by $\alpha L_{ml}+ \beta L_{cp}$ **TS+CP:** Trained with $L_{ml}$ followed by $\alpha L_{ml}+\beta L_{cp}+\gamma L_{ts}$ **TS**$\rightarrow$**CP:** Trained with $L_{ml}$ followed by $\alpha L_{ml}+ \gamma L_{ts}$ and finally with $\alpha L_{ml}+ \beta L_{cp}$ **CP**$\rightarrow$**TS:** Trained with $L_{ml}$ followed by $\alpha L_{ml}+ \beta L_{cp}$ and finally with $\alpha L_{ml}+ \gamma L_{ts}$\
\
Training with $L_{ml}$ alone in all the above settings is done for $10$ epochs with all the hyper-parameters set as default in the off-the-shelf implementation of [@W17-4902]. Each of the iterative model training is done using the model with the best performance on validation set for $5$ more epochs.
------------------- ---------------- -------------------- ------------------- ---------------- -------------------- -------------------
**Models** BLEU$\uparrow$ Accuracy$\uparrow$ Overall$\uparrow$ BLEU$\uparrow$ Accuracy$\uparrow$ Overall$\uparrow$
CopyNMT 0.263 0.774 0.196 0.280 0.503 0.180
TS 0.240 0.801 0.184 0.271 0.527 0.179
CP 0.272 0.749 0.199 0.281 0.487 0.178
TS+CP 0.259 0.772 0.194 0.271 0.527 0.179
CP$\rightarrow$TS 0.227 **0.817** 0.178 0.259 **0.5441** 0.175
TS$\rightarrow$CP **0.286** 0.723 **0.205** **0.298** 0.516 **0.189**
------------------- ---------------- -------------------- ------------------- ---------------- -------------------- -------------------
: Ablation study to demonstrate the improvement of the addition of the loss terms on formality transfer task. []{data-label="tab: ablation-table"}
We can observe from Table \[tab: ablation-table\] that $L_{ts}$ and $L_{cp}$ helps in improving the accuracy which measures transfer strength (TS) and BLEU score which measures content preservation (CP) respectively as compared to CopyNMT. When all the three loss terms are used simultaneously (TS+CP) the resulting performance lies between TS and CP, indicating that there is a trade-off between the two metrics and improvement in one metric is at the cost of another as observed by [@fu2017style]. This phenomenon is evident from the results of TS$\rightarrow$CP and CP$\rightarrow$TS where the network gets a bit biased towards the latter optimization. Moreover, improvement in CP$\rightarrow$TS and TS$\rightarrow$CP as compared to TS and CP respectively suggests that incremental training better helps in teaching the framework. Since the performance on both transfer strength and content preservation metrics plays an important role in text style transfer task, we chose TS$\rightarrow$CP, which has the maximum overall score, over the other models for further analysis.
**Baselines:** We compare the proposed approach TS$\rightarrow$CP against the state-of-the-art cross-aligned autoencoder style transfer approach (Cross-Aligned) by [@shen2017style][^2], parallel style transfer approach (CopyNMT) by [@W17-4902][^3] and neural encoder-decoder based transformer model [@vaswani2017attention][^4]
------------------------------------- -------------------- ------------------------ ----------------------- -------------------- ------------------------ -----------------------
**Models** **BLEU$\uparrow$** **Accuracy$\uparrow$** **Overall$\uparrow$** **BLEU$\uparrow$** **Accuracy$\uparrow$** **Overall$\uparrow$**
Transformer [@vaswani2017attention] 0.125 **0.933** 0.110 0.099 **0.894** 0.089
Cross-Aligned [@shen2017style] 0.116 0.670 0.098 0.117 0.766 0.101
CopyNMT [@W17-4902] 0.263 0.774 0.196 0.280 0.503 0.180
TS$\rightarrow$CP (Proposed) **0.286** 0.723 **0.205** **0.298** 0.516 **0.189**
Transformer [@vaswani2017attention] 0.077 **0.922** 0.071 0.069 0.605 0.062
Cross-Aligned [@shen2017style] 0.059 0.818 0.055 0.061 0.547 0.054
CopyNMT [@W17-4902] 0.143 0.919 0.124 0.071 **0.813** 0.065
TS$\rightarrow$CP (Proposed) **0.153** **0.922** **0.131** **0.088** 0.744 **0.078**
Transformer [@vaswani2017attention] 0.027 **0.736** 0.026 0.046 **0.915** 0.043
Cross-Aligned [@shen2017style] 0.044 0.614 0.041 0.049 0.537 0.044
CopyNMT [@W17-4902] 0.104 0.495 0.085 0.111 0.596 0.093
TS$\rightarrow$CP (Proposed) **0.127** 0.489 **0.100** **0.137** 0.567 **0.110**
------------------------------------- -------------------- ------------------------ ----------------------- -------------------- ------------------------ -----------------------
: Comparison of TS$\rightarrow$CP with the baselines on the three transfer tasks in both the directions. All the scores are normalized to be between 0 and 1.[]{data-label="tab: result-table"}
**Results:** It can be seen from Table \[tab: result-table\] that even though the transformer model has the best accuracy, it fails in preserving the content. Closer look at the outputs (formal to informal transfer task in Table \[tab: sampleformality\]) reveal that it generates sentences in target style but the sentences do not preserve the meaning of the input and sometimes are out of context (discussed in the Section \[sec: discussion\]). Cross-Aligned performs the worst in informal to formal transfer task among all the other approaches because it is generating a lot of unknowns and is not able to preserve content. TS$\rightarrow$CP, on the other hand, has the highest overall score and performs the best in preserving the content. We also observed that the dataset had many sentences containing proper nouns like name of the songs, person or artists. In such cases, copy mechanism helps in retaining the proper nouns whereas other models are not able to do so. This is evident from the higher BLEU scores for our proposed model.
**Task**
---------- ------- ------- ------- ------- ------- -------
R$>$C R$>$T R$>$S R$>$C R$>$T R$>$S
I-F 88.67 81.34 70.00 70.00 72.67 83.67
F-I 73.34 88.67 61.22 59.34 79.34 91.80
E-NE 64.00 79.34 68.00 60.67 71.34 71.73
NE-E 76.67 70.67 68.00 69.34 74.00 70.00
: Human evaluation results of 50 randomly selected model outputs. The values represent the % of times annotators rated model outputs from TS$\rightarrow$CP (R) as better than the baseline CopyNMT (C), Transformer (T) and Cross-Aligned (S) over the metrics. I-F (E-NE) refers to informal to formal (exciting to non-exciting) task.[]{data-label="tab: human-evaluation"}
Table \[tab: human-evaluation\] presents the human evaluation results aggregated over three annotators per sample. It can be seen that in at least 70% of the cases, annotators rated model outputs from TS$\rightarrow$CP as better than the three baselines on both the evaluated metrics except for the content preservation as compared to CopyNMT in formal to informal task wherein, both the models perform equally good. One reason behind this is that both the models use copy-mechanism.
Experiments: Beyond Formality (Excitement Dataset) {#sec: experiments-beyond}
==================================================
In order to demonstrate the generalizability of our approach on an affective style dimension like excitement (the feeling of enthusiasm and eagerness), we curated our own dataset using reviews from Yelp dataset[^5] which is a subset of Yelp’s businesses, reviews, and user data. We request human annotators to provide rewrites for given exciting sentences such that they sound as non-exciting/boring as possible. Reviews with rating greater than or equal to $3$ were filtered out and considered as exciting to get the non-exciting/boring rewrites. We also asked the annotators to rate the given and transferred sentences on a Likert scale of $1$ (No Excitement at all) to $5$ (Very high Excitement). The dataset thus curated was split into train ($\sim$36K), test (1K) and validation (2K) sets. We evaluate the transfer quality on content preservation and transfer strength metrics as defined in Section \[sec: experiments-formality\].
For measuring the transfer strength we train a classifier as described in Section \[subsec: ts\]. We use the annotations provided by the human annotators on these sentences to get the labels for the two styles. Sentences with a rating greater than or equal to $3$ were considered as exciting and non-exciting otherwise.
**Results:** The transfer task in this case is to convert the input sentence with high excitement (exciting) to a sentence with low excitement (non-exciting) and vice-versa. We can observe from Table \[tab: result-table\] that model performance in the case of excitement transfer task is similar to what we observed in the formality transfer task. However, CopyNMT performs the best in transferring style in case of non-exciting to exciting transfer task because the model has picked up on expressive words (‘awesome’, ‘great’, and ’amazing’) which helps in boosting the transfer strength. TS$\rightarrow$CP (with highest overall score) consistently outperforms Cross-Aligned in all the metrics and both the directions. Table \[tab: human-evaluation\] presents the human evaluation results on this transfer task. We notice that humans preferred outputs from our proposed model at least 60% of the times on both the measures as compared to the other three baselines. This provides an evidence that the proposed RL-based framework indeed helps in improving generation of more content preserving sentences which align with the target style.
Experiments: Beyond Affective Elements (English Dataset) {#sec: experiments-beyond-affect}
========================================================
Besides affective style dimensions, our approach can also be extended to other style transfer tasks like converting modern English to Shakespearean English. To illustrate the performance of our model on this task we experimented with the corpus used in [@W17-4902]. The dataset consists of $\sim$21K modern-Shakespearean English sentence pairs with $\sim$18K in train, $\sim$1.5K in test and $\sim$1.2K in validation split. We use the same evaluation measures as in the previous two tasks for illustrating the model performance and generalizability of the approach. For this task we present only the automatic evaluation results because manual evaluation of this task is not easy since it requires an understanding of Shakespearean English and finding such population is a difficult task due to limited availability. **Results:** We can observe from Table \[tab: result-table\] that model performance in the case of this transfer task is also similar to what we have observed in the earlier two transfer tasks. Although Cross-Aligned has better accuracy than TS$\rightarrow$CP, it fails to preserve the content (sample 3 of Table \[tab: sampleenglish\]). Similar is the case with transformer which outperforms others in accuracy but is not able to retain the content (sample 1 of Table \[tab: sampleenglish\]). TS$\rightarrow$CP outperforms the three baselines in preserving the content with the highest overall score. This establishes the viability of our approach to various types of text style transfer tasks.
These experiments further indicate that our proposed reinforcement learning framework improves the transfer strength and content preservation of parallel style transfer frameworks and is also generalizable across various stylistic expression.
Discussion {#sec: discussion}
==========
In this section, we provide few qualitative samples from the baselines and the proposed reinforcement learning based model. We can observe from the transformer model output for Input $1$ and $2$ in formal to informal column of Table \[tab: sampleformality\] that it generates sentences with correct target style but does not preserve the content. It either adds random content or deletes the required content (‘band’ instead of ‘better’ in $1$ and ‘hot’ instead of ‘talented’ in $2$). As mentioned earlier, in sample output $3$ of Table \[tab: sampleformality\], Cross-Aligned is unable to retain the content and tend to generate unknown tokens. CopyNMT, even though is able to preserve content, tend to generate repeated token like ’please’ in sample input $2$ (informal to formal task) which results in lower BLEU score than our proposed approach. Transformer model outputs for exciting to non-exciting task in samples $1$ and $2$ of Table \[tab: sample\], miss specific content words like ‘environment’ and ‘alisha’ respectively. However, it is able to generate the sentences in target style. Similary, Cross-Aligned and CopyNMT are also not able to retain the name of the server in sample $2$ of Table \[tab: sample\]. Sample $2$ of Shakespearean to Modern English and $1$ of Modern to Shakespearean English task in Table \[tab: sampleenglish\] provide evidence for high accuracy and lower BLEU scores for transformer model. From sample $2$ of Shakespearean to modern English transfer task, we can observe that Cross-Aligned although can generate the sentence in the target style is not able to preserve the entities like ’father’ and ’child’. On the other hand, TS$\rightarrow$CP can not only generate the sentences in the target style but is also able to retain the entities. There are few cases when CopyNMT is better in preserving the content as compared to other models, for instance, sample $1$ of formal to informal transfer task and sample $3$ of non-exciting to exciting transfer task since it leverages copy-mechanism.
Another point to notice is the lexical level changes made to reflect the target style. For example, the use of ‘would’, ‘don’t’ and ‘inform’ instead of ‘want’, ‘dono’ and ‘let me know’ respectively for transforming informal sentences into formal ones. Use of colloquial words like ‘u’, ‘gonna’ and ‘mama’ for converting the formal sentences to informal can be observed from the sample outputs. Not only lexical level changes but structural transformations can also be observed as in ’Please inform me if you find out’. In case of excitement transfer task, use of strong expressive words like ‘amazing’ and ‘great’ makes the sentence sound more exciting while less expressive words such as ‘okay’ and ‘good’ makes the sentence less exciting. Use of ‘thou’ for you and ‘hither’ for here are more frequently used in Shakespearean English than in modern English. These sample outputs indeed provide an evidence that our model is able to learn these lexical or structural level differences in various transfer tasks, be it formality, beyond formality or beyond affective dimensions.
Conclusion and Future Work {#sec: conclusion}
==========================
The primary contribution of this work is a reinforce rewards based sequence-to-sequence model which explicitly optimizes over content preservation and transfer strength metrics for style transfer with parallel corpus. Initial results are promising and generalize to other stylistic characteristics as illustrated in our experimental sections. Leveraging this approach for simultaneously changing multiple stylistic properties (for e.g. high excitement and low formality) is a subject of further research.
[^1]: This work was done while the authors were working at Adobe Research, Bangalore, India.
[^2]: We use the off-the-shelf implementation provided by the authors at\
<https://github.com/shentianxiao/language-style-transfer>
[^3]: <https://github.com/harsh19/Shakespearizing-Modern-English>
[^4]: <https://github.com/pytorch/fairseq> We also tried using the model proposed by [@gong2019reinforcement] to compare against out proposed approach but we couldn’t get stable performance on our datasets.
[^5]: https://www.yelp.com/dataset
|
---
abstract: 'Berry phases mix states of positive and negative energy in the propagation of fermions and bosons in external gravitational and electromagnetic fields and generate Zitterbewegung oscillations. The results are valid in any reference frame and to any order of approximation in the metric deviation.'
address:
- 'Department of Physics and Prairie Particle Physics Institute, University of Regina, Regina, Sask S4S 0A2, Canada'
- 'International Institute for Advanced Scientific Studies, 89019 Vietri sul Mare (SA), Italy.'
author:
- Giorgio Papini
bibliography:
- '<your-bib-database>.bib'
title: Zitterbewegung and gravitational Berry phase
---
Zitterbewegung, gravitational Berry phase, covariant wave equations. PACS No.: 04.62.+v, 11.30.Fs, 95.30.Sf
Introduction {#1}
============
The contribution of external gravitational fields to the solution of covariant wave equations is contained in a Berry phase [@caipap1]. This should be expected because in general relativity the space of parameters of Berry’s theory coincides with space-time. The wave equations for fermions and bosons [@caipap2], [@caipap3], [@papsc], [@pap1] have been solved exactly to first order in the metric deviation $\gamma_{\mu\nu}=g_{\mu\nu}- \eta_{\mu\nu}$ for any metric and the solutions give the correct Einstein deflection when applied to geometrical optics and can be used in interferometry, gyroscopy, in the study of neutrino helicity and flavour oscillations [@pap2] and of spin-gravity coupling [@pap3]. They also reproduce a variety of known effects like those discussed in [@COW], [@PW], [@BW],[@MASH].
It is shown below that the gravitational Berry phase gives rise to a field-dependent Zitterbewegung (ZB) in the propagation of particles in a gravitational background.
Dirac and Klein-Gordon equations {#2}
================================
Consider first the covariant Dirac equation $$\label{CDE}
[i\gamma^{\mu}(x){\cal D}_\mu-m]\Psi(x)=0\,.$$ The notations are those of [@pap2]. The first order solutions of (\[CDE\]) are of the form $$\label{E}
\Psi(x) = {\hat T}(x) \psi(x)\,,$$ where $\psi(x)$ is a solution of the flat space-time Dirac equation $$\label{DE}
\left(i\gamma^{\hat{\mu}}\partial_{\mu}-m\right)\psi(x)=0\,,$$ here a plane wave of four-momentum $k^{\alpha}$ satisfying the relation $k_{\alpha}k^{\alpha}=m^{2}$, and $\gamma^{\hat{\mu}}$ are the usual constant Dirac matrices. The operator $\hat{T}$ is given by [@pap2] $$\label{T}
\hat{T}=
-\frac{1}{2m}\left(-i\gamma^{\mu}(x)\mathcal{D}_{\mu}-m\right)e^{-i\Phi_{T}}\,,$$ $$\label{PHIS}
\Phi_{T}=\Phi_{S}+\Phi_{G}\,,\qquad
\Phi_{S}(x)=\int_{P}^{x}dz^{\lambda}\Gamma_{\lambda}(z)\,,$$ and $$\begin{aligned}
\label{PH}
\Phi_{G}(x) = -\frac{1}{4}\int_P^xdz^\lambda\left[\gamma_{\alpha\lambda,
\beta}(z)-\gamma_{\beta\lambda, \alpha}(z)\right]\left[\left(x^{\alpha}-
z^{\alpha}\right)k^{\beta}-\left(x^{\beta}-z^{\beta}\right)k^{\alpha}\right]+
\\ \nonumber
\frac{1}{2}\int_P^xdz^\lambda\gamma_{\alpha\lambda}(z)k^{\alpha}\,,\end{aligned}$$ where $\Gamma_{\lambda}$ represents the spin connection. The solutions $\psi(x)$ of (\[DE\]) can include wave packets, if so desired. In this case the ZB decays in time [@Lock], which is not an essential point in what follows. In (\[PHIS\]) and (\[PH\]), the path integrals are taken along the classical world line of the particle starting from a reference point $P$.
In most applications $\psi(x)$ is represented by a positive energy solution $\psi(x)=u(\vec{k})e^{-ik_{\mu}x^{\mu}}$. However the influence of negative energy solutions $\psi^{(1)}(x)=v(\vec{k})e^{ik_{\mu}x^{\mu}}$ can not be neglected because the wave functions $\psi(x)$ by themselves do not form a complete set. A relationship between $\Psi(x)$ and $\Psi^{(1)}(x) = \hat{T}_{1}\psi^{(1)}(x)$ must therefore be found. The spin-up ($\uparrow$) and spin-down ($\downarrow$) components of the spinors $u$ and $v$ obey the well-known equations $$\label{spin}
u_{\downarrow}=\gamma^{5}v_{\uparrow}\,,\qquad v_{\downarrow}=\gamma^{5}u_{\uparrow}\,.$$ The required relation between $\Psi(x)$ and $\Psi^{(1)}(x)$ follows from (\[spin\]), or simply from the replacement of $\psi(x)$ with $\gamma^{5}\psi(x)$ in (\[DE\]). If, in fact, $\Psi(x)=e^{-ik_{\mu}x^{\mu}}\hat{T}u$ is a solution of (\[CDE\]), it then follows from (\[spin\]), the relations $\left\{\gamma^{5},\gamma^{\hat{\mu}}\right\}
=0$, $\sigma^{{\hat \alpha}{\hat \beta}}=\frac{i}{2}[\gamma^{\hat
\alpha}, \gamma^{\hat \beta}]$, $ \gamma^\mu(x)=e^\mu_{\hat \alpha}(x) \gamma^{\hat
\alpha}\,,
\Gamma_\mu(x)=-\frac{1}{4} \sigma^{{\hat \alpha}{\hat \beta}} e^\nu_{\hat \alpha}e_{\nu\hat{\beta};\, \mu}\,,$ and $ [\gamma^{5},\Gamma^{\mu}]=0$ that $\Psi^{(1)}(x)=e^{ik_{\mu}x^{\mu}}\hat{T}_{1}v$ also is a solution of (\[CDE\]) and $\hat{T}_{1}=\gamma^{5}\hat{T}\gamma^{5}$. It is useful to further isolate the gravitational contribution in the vierbein components by writing $e^{\mu}_{\hat{\alpha}}\simeq\delta^{\mu}_{\hat{\alpha}}+h^{\mu}_{\hat{\alpha}}$, which leads to $$\begin{aligned}
\label{T}
{\hat
T}=\frac{1}{2m}\left\{\left(1-i\Phi_{G}\right)\left(m+\gamma^{\hat{\alpha}}
k_{\alpha}\right)-i
\left(m+\gamma^{\hat{\alpha}}k_{\alpha}\right)\Phi_{S}+
\left(k_{\beta}h^{\beta}_{\hat{\alpha}}+
\Phi_{G,\alpha}\right)\gamma^{\hat{\alpha}}\right\}\equiv
\\ \nonumber\hat{T}_{0}+\hat{T}_{G}\,,\end{aligned}$$ where $\hat{T}_{0}\equiv\frac{1}{2m}\left(m+\gamma^{\hat{\alpha}}k_{\alpha}\right)$ and $\hat{T}_{G}$ contains the gravitational corrections. The operator ${\hat T}_1$ can be immediately calculated from (\[T\]).
The gravitational field mixes the positive and negative energy solutions of (\[DE\]). In fact the eigenstates $U^{\pm}=
1/\sqrt{2}(u \pm v)$ of $\gamma^{5}$ and the eigenstates $u$ and $v$ of $\hat{T}_{0}$ are not the same and $\hat{T},\,\hat{T}_{1}$ mix $u$ and $v$. The mixing is effected by $\hat{T}_{G}$ and $\hat{T}_{1G}$ which are entirely due to Berry phase.
Mixing manifests itself as follows.
The state of a fermion in a gravitational field can be written in the form $$\label{PS}
|\Phi(t) \rangle =
\alpha(t)|\psi(t)\rangle+\beta(t)|\psi^{(1)}(t)\rangle=\alpha_{0}\hat{T}(t)|\psi(t)\rangle+\beta_{0}\hat{T}_{1}(t)|\psi^{(1)}(t)\rangle\,,$$ where $|\alpha_{0}|^2+|\beta_{0}|^2=1$, from which one obtains $$\begin{aligned}
\label{al}
\alpha(t)=
\langle \psi|\Phi(t)\rangle=
\alpha_{0}\langle\psi|\hat{T}|\psi\rangle +\beta_{0}\langle
\psi|\hat{T}_{1}|\psi^{(1)}\rangle\,; \qquad
\\ \nonumber\beta(t)=\langle
\psi^{(1)}|\Phi(t)\rangle =\alpha_{0}\langle
\psi^{(1)}|\hat{T}|\psi \rangle +\beta_{0}\langle
\psi^{(1)}|\hat{T}_{1}|\psi^{(1)}\rangle \,. \qquad\end{aligned}$$ If at $t=0$ the gravitational field is not present, then $\hat{T}_{G}=0\,,\hat{T}_{1G}=0$ and $\alpha(0)\equiv\alpha_{0}=\,,\beta(0)\equiv\beta_{0}$. It follows from (\[al\]) that as the system propagates in a gravitational field, shifts from $|\psi(t)\rangle$ to $|\psi^{(1)}(t)\rangle$ produce oscillations. Thus the geometrical structure of space-time, represented by gravity, affects Hilbert space by producing oscillations between the positive and negative energy states.
The presence of an electromagnetic field [@dinesh] can be accommodated by adding the term $ qA_{\alpha}$, where $q$ is the charge of the particle, to $\Phi_{G,\alpha}$ in $\hat{T}$ and $\hat{T}_{1}$. The relationship between external electromagnetic fields and ZB has been investigated extensively by Feschbach and Villars [@FV] for both Dirac and Klein-Gordon equations.
In order to obtain the transition probabilities $|\alpha(t)|^{2}\,,|\beta(t)|^{2} $ from (\[al\]) in a concrete case, one can choose for simplicity $$\label{psi0}
\psi(x)=f_{0,R}e^{-ik_\alpha x^\alpha}=\sqrt{\frac{E+m}{2m}}
\left(\begin{array}{c}
f_{R} \\
\frac{\sigma^{3} k}{E+m}\, f_{R} \end{array}\right)
\,e^{-ik_\alpha x^\alpha}\,,$$ where $ f_{R}$ is the positive helicity eigenvector. The normalizations are $\langle \psi|\psi\rangle =1$, where $\langle
\psi|= \langle\psi^{\dag}|\gamma^{\hat{0}}$, $\langle
\psi^{(1)}|\psi^{(1)}\rangle =-1$ and $\langle
\psi|\psi^{(1)}\rangle = \langle \psi^{(1)}|\psi\rangle =0$. In addition, one needs explicit expressions of the metric components for the purpose of calculating $\hat{T}$ and $\hat{T}_{1}$. The choice of the metric $$\label{LTmetric}
\gamma_{00}=2\phi\,, \quad \gamma_{ij}=2\phi\delta_{ij}\,, \quad$$ where $\phi=-\frac{GM}{r}$, and *M*, *R*, are mass and radius of the source, is again dictated by simplicity. The vierbein components to order $\mathcal{O}(\gamma_{\mu\nu})$ are given by $$\label{3.5}
e^0_{\hat{i}}=0\,{,}\quad
e^0_{\hat{0}}=1-\phi\,{,}\quad
e^i_{\hat{0}}=0 \,{,}\quad
e^l_{\hat{k}}=\left(1+\phi\right)\delta^l_k\,.$$ Without loss of generality, one may consider particles starting from $z=-\infty$, and propagating along $x=b\geq R\,,y=0$ in the field of the gravitational source and set $k^{3}\equiv k$ and $k^{0}\equiv E$.
Returning to (\[al\]), if originally the system is in a positive energy state, then $ \alpha_{0}=1\,, \beta_{0}=0$, $
|\Phi(t)\rangle = \hat{T} |\psi\rangle $ and from (\[PS\]) and $\Phi_{G,3}=(E^{2}/k +k)\phi$ one gets $$\label{al2}
\beta(t)=
\frac{e^{-2iq_{\alpha}x^{\alpha}}}{2m}\left\{-\langle \psi^{(1)}|
\left[Eh_{\hat{0}}^{0}\gamma^{\hat{0}}+\left(-kh_{\hat{3}}^{3}+\left(\frac{E^2}{k}+k\right)
\phi(z)\right)\gamma^{\hat{3}}\right]|\psi\rangle\right\}\,,$$ where $q_{0}\equiv E$ because the field does not depend on time, hence energy is conserved, and $q_{i}\equiv k_{i}^{(i)}-k_{i}^{(f)}$. The first two terms in (\[al2\]) are due to $\Gamma_{\mu}$ and refer to $\Phi_{S}$. The remaining two terms come from $\Phi_{G,3}$ and are also Berry phase contributions. Thus, according to (\[al\]) and (\[al2\]), the propagation of the particle has two overlapping components: one in which the state of the particle does not change, the other in which oscillations take place from and to energy states of opposite sign with a frequency $2E$, or in ordinary units $2E/\hbar$. This is at least as large as the ZB frequency $2m$. The particle therefore behaves as if it were trying to conserve energy-momentum and angular momentum during its propagation. The presence of the gravitational Berry phase translates into a ZB that vanishes when there is no gravity acting on the particle and is therefore due to a real force, as pointed out in [@FV] for the case of an external electromagnetic field. Because the approach is covariant, the result holds true in any frame of reference. Moreover, the non-local potential $K_{\lambda}(x,x_{0})=\Phi_{G,\lambda}(x, x_{0})+\Gamma_{\lambda}(x)$ can be calculated to any order, meaning that a ZB also exists at any order.
The transition amplitude $\langle\psi|\hat{T}|\psi\rangle$ can be better calculated using the relation $\langle\psi|\hat{T}|\psi\rangle=\int_{\lambda_{0}}^{\lambda}\langle\psi|\dot{x}^{\mu}\partial_{\mu}\hat{T}|\psi\rangle d\lambda$, where $\dot{x}^{\mu}=k^{\mu}/m$ and $\lambda$ is an affine parameter along the particle world line. The calculation is outlined in [@pap2].
The probability of the transition $\psi \rightarrow \psi^{(1)}$ follows from (\[al2\]) and is $$\label{BL}
P_{\psi\rightarrow \psi^{(1)}}=|\beta(t)|^{2} =\left[\frac{1}{2m^2} (k^2 - \frac{E^3}{k})\right]^2 \phi^2(z) \,.$$
If $ \alpha_{0}=0, \beta_{0}=1$, then $|\alpha(t)|^2$ represents the probability for the inverse process $\psi^{(1)}\rightarrow \psi$. One finds $$\begin{aligned}
\label{BE}
P_{\psi^{(1)}\rightarrow \psi}=|\alpha(t)|^2=|\langle \psi|\hat{T}|\psi^{(1)}\rangle|^{2}=
|\langle \psi|\gamma^{5}\hat{T_{1}}\gamma^{5}|\psi^{(1)}\rangle|^{2}= \qquad
\\ \nonumber|\langle \psi^{(1)}|\hat{T}_{1}|\psi\rangle|^{2}=P_{\psi\rightarrow \psi^{(1)}}\,.\end{aligned}$$ According to (\[BL\]) and (\[BE\]), the transitions proceed in both directions with the same probability, as expected.
As mentioned above, an external electromagnetic field can be introduced by simply adding the corresponding Berry phase to (\[al2\]). The additional term in curly brackets is therefore $\langle \psi^{(1)}|-q A_{\mu}\hat{\gamma}^{\mu}|\psi\rangle$. If the addition corresponds to an electromagnetic wave of amplitude $f$ and frequency $\omega$, in vanishing gravity, there is a resonance at $\omega =2E$ that leads to $|\beta(t)|^2=(qkf/m^2 \omega)^2 \cos^2(2Ex_{0})$. If gravity is also present, the resonance condition becomes $\omega=2E$, $C=qkf/m^2 \omega\equiv A$, with $C$ represented by the terms of (\[al2\]) in curly brackets, and $|\beta(t)|^2=(A/m)^2 \sin^2(Et)$. The prospects of achieving resonance in laboratory conditions in the near future do not appear favourable.
ZB appears to be universal in condensed matter physics and is the subject of recent, intense research [@ZR]. It is in this area that lie the best opportunities to observe ZB.
Entirely similar conclusions can be reached for the covariant Klein-Gordon equation $$\label{CKG}
\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}+m^2\right)\Phi(x)=0\equiv \hat{\mathcal{T}}\Phi(x)\,,$$ which has the first order solution $$\label{SCKG}
\Phi(x) = e^{-i\Phi_{G}}\varphi(x)\,,$$ where $\varphi(x)$ satisfies the Klein-Gordon equation in flat space-time $$\label{KG}
\left(\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}+m^2\right) \varphi(x)=0\equiv\hat{\mathcal{T}}_{0}\varphi(x)\,.$$ Following the procedure of [@FV] one can write the plane wave solutions of (\[KG\]) as $$\label{FV}
\varphi^{+}(x)=e^{-ip_{\mu}x^{\mu}}\chi^{+}(p)\,\,,\,\,
\varphi^{-}(x)=e^{ip_{\mu}x^{\mu}}\chi^{-}(p)\,,$$ where $\chi^{\pm}(p)$ are known functions of $p$. Representing a generic state of the system $\Lambda(x)$ in terms of the free-field solutions $\varphi^{\pm}$ one sees immediately that $\Lambda(x)$ is not an eigenstate of $\hat{\mathcal{T}}_{0}$ and that, therefore, the gravitational part $\hat{\mathcal{T}}-\hat{\mathcal{T}}_{0}$ due to $\Phi_{G}$ mixes the states of positive and negative energy.
Similar results can be obtained for all known relativistic wave equations.
Summary and discussion {#3}
======================
It was shown in [@FV] that static electric and magnetic fields in flat space-time excite a field-dependent ZB. This result has been extended, in this work, not only to electromagnetic fields of any type in curved space-time, but also to any gravitational fields of weak to intermediate strength. The extension is based on the notion of Berry phase. Since the approach is covariant, the result holds true in any reference frame. Moreover, the gauge potential $K_{\lambda}(x, x_{0})$ exists to any order, hence the results remain valid to any order of approximation in $\gamma_{\mu\nu}$ for both fermions and bosons.
Particle propagation is affected by gravitational and electromagnetic Berry phases. They imply gauge structures that mix the field-free states giving rise to oscillations of frequency at least as high as $2m$. This action can be interpreted, in the gravitational case, as an example of how the curvature of space-time can affect Hilbert space by determining transitions between states of positive and negative energy. The transitions involve $\hbar$. Though resonance conditions between ZB and the external fields exist in principle, their realization for particles in vacuum seems unlikely at present. The significance of the results is related to the role played by Berry phase and the related potential $K_{\lambda}(x, x_{0})$ in the mixing of positive and negative energy states that are necessarily contained in the eigenfunctions of relativistic particles. ZB oscillations appear as the particles strive to conserve energy-momentum and angular momentum along their world lines.
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abstract: 'Let $(R,{{{\mathfrak}{m}}},k)$ be a local Cohen–Macaulay (CM) ring of dimension one. It is known that $R$ has finite CM type if and only if $R$ is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer-Thrall theorem for excellent rings.'
address:
- |
Department of Mathematics\
University of Kansas\
Lawrence, KS 66045
- |
Department of Mathematics and Statistics\
University of Nebraska–Lincoln\
Lincoln, NE 68588-0323
author:
- 'Graham J. Leuschke'
- Roger Wiegand
date: 'April 15, 2003 (filename: BCMTrevised1.tex)'
title: 'Local Rings of Bounded Cohen–Macaulay Type'
---
[^1]
Let $R$ be a Cohen–Macaulay (CM for short) local ring. We say that $R$ has finite, respectively bounded, CM type provided there are only finitely many indecomposable maximal Cohen–Macaulay (MCM) $R$-modules up to isomorphism, respectively, there is a bound on the multiplicities of the indecomposable MCM $R$-modules. The one-dimensional CM local rings of finite CM type have been completely classified [@Drozd-Roiter], [@Green-Reiner], [@Cimen:thesis], [@Cimen:paper], [@CWW]. One consequence of the characterization is that a one-dimensional CM local ring $R$ has finite CM type if and only if the completion ${\widehat}R$ has bounded CM type and is reduced.
Here we study the one-dimensional CM local rings of bounded but infinite CM type. In [@Leuschke-Wiegand:hyperbrt] we classified the complete equicharacteristic hypersurface singularities of dimension one having bounded but infinite CM type: Up to isomorphism, the only ones are $k[[X,Y]]/(Y^2)$ and $k[[X,Y]]/(XY^2)$. In §1 of this paper we consider complete equicharacteristic rings that are not hypersurfaces, and we show that only one additional isomorphism type arises (Theorem \[classifydim1\]). In the second section we prove that bounded CM type ascends to and descends from the completion. Our main results in dimension one are summarized in Theorem \[summary\]. In the third section of the paper we study ascent and descent in higher dimensions.
One-dimensional complete local rings
====================================
We begin by quoting two results we will need from [@Leuschke-Wiegand:hyperbrt]. Recall that an $R$-module $M$ is said to [*have (constant) rank $r$*]{} [@Scheja-Storch] provided $K\otimes_RM \cong K^r$, where $K$ is the total quotient ring of $R$ (obtained by inverting all nonzerodivisors). We denote by $\nu_R(M)$ the number of generators required for $M$ as an $R$-module.
[@Leuschke-Wiegand:hyperbrt Theorem 3.1]\[hyperclassify\] Let $R$ be a complete, equicharacteristic hypersurface of dimension one (i.e., $R\cong k[[X,Y]]/(f)$ for some field $k$ and some non-zero non-unit $f$ in the formal power series ring $k[[X,Y]]$). Then $R$ has bounded but infinite CM type if and only if either $R \cong k[[X,Y]]/(Y^2)$ or $R \cong k[[X,Y]]/(XY^2)$. Further, if $R$ has unbounded CM type, then $R$ has, for each positive integer $r$, an indecomposable MCM module of constant rank $r$.
\[bigranks\] Let $(R,{{{\mathfrak}{m}}})$ be a one-dimensional local CM ring. Assume that either
1. $R$ has multiplicity at least $4$, or
2. $R$ has a birational extension $S$ such that $\nu_R(S) = 3$ and $\nu_R(\frac{{{{\mathfrak}{m}}}S}{{{{\mathfrak}{m}}}}) > 1$.
Then $R$ has, for each positive integer $r$, an indecomposable MCM module of constant rank $r$.
While Proposition \[bigranks\] is not stated explicitly in [@Leuschke-Wiegand:hyperbrt], it follows immediately from Lemma 2.2 and Theorem 2.3 there. Also, one needs (0.5) and (2.1) of [@CWW] to deduce the “Further" statement in Proposition \[hyperclassify\]. Before stating our main result (Theorem 1.5), we state two lemmas that will be useful here and in the next section. Part (1) of the first lemma is due to Bass [@Bass:ubiquity (7.2)]. We include the proof here, since the context in [@Bass:ubiquity] is a bit different from ours.
\[endoring\] Let $(R,{{{\mathfrak}{m}}})$ be a Gorenstein local ring of dimension one with total quotient ring $K$. Let $E = {\operatorname{End{}}}_R({{{\mathfrak}{m}}}) = \{\lambda \in K\ |\ \lambda {{{\mathfrak}{m}}}\subseteq {{{\mathfrak}{m}}}\}$. Assume that $E$ is local.
1. If $M$ is an indecomposable MCM $R$-module and $M\not\cong R$, then $M$ is an indecomposable MCM $E$-module ([*naturally*]{}, that is, in such a way that the $R$-module structure induced by the natural map $R\hookrightarrow E$ agrees with the original structure).
2. Every indecomposable MCM $E$-module is an indecomposable MCM $R$-module.
In particular, $R$ has finite (respectively, bounded) CM type if and only if $E$ has finite (respectively, bounded) CM type.
\(1) (Bass, [@Bass:ubiquity]) Since there is no surjection from $M$ to $R$, we have $M^* = {\operatorname{Hom{}}}_R(M,{{{\mathfrak}{m}}})$, which is an $E$-module. Therefore $M\cong M^{**}$ is an $E$-module as well. Clearly $_EM$ is indecomposable and (since $E$ is local) maximal Cohen–Macaulay.
\(2) Since $R\hookrightarrow E$ is module-finite, any MCM $E$-module $N$ is MCM as an $R$-module. Since $R\hookrightarrow E$ is birational and $_RN$ is torsion-free, any $R$-endomorphism of $N$ is automatically $E$-linear. It follows that $N$ is indecomposable as an $R$-module.
The final statement is immediate from (1) and (2).
We are indebted to Tom Marley (private communication) for showing us the following lemma, and to the anonymous referee for suggesting an improvement to the proof. Let ${\operatorname{e}}(R)$ denote the multiplicity of the local ring $R$.
\[tom\] Let $(R,{{{\mathfrak}{m}}},k)$ be a one-dimensional local CM ring with $k$ infinite, and suppose ${\operatorname{e}}(R) = \nu_R({{{\mathfrak}{m}}}) = 3$. Let $N$ be the nilradical of $R$. Then:
1. $N^2 = 0$.
2. $\nu_R(N) \le 2$.
3. If $\nu_R(N) =2$, then ${{{\mathfrak}{m}}}$ is generated by three elements $x,y,z$ such that ${{{\mathfrak}{m}}}^2={{{\mathfrak}{m}}}x$ and $N=Ry+Rz$.
4. If $\nu_R(N) = 1$, then ${{{\mathfrak}{m}}}$ is generated by three elements $x,y,z$ such that ${{{\mathfrak}{m}}}^2={{{\mathfrak}{m}}}x$, $N = Rz$, and $yz = z^2 = 0$.
We note that $R$ has minimal multiplicity [@Abhyankar] and hence ${{{\mathfrak}{m}}}$ has reduction number 1. Since the residue field is infinite, then, there is an element $x\in {{{\mathfrak}{m}}}$ such that ${{{\mathfrak}{m}}}^2 = x{{{\mathfrak}{m}}}$. Since $R$ is CM, $x$ is a nonzerodivisor. We recall the formula [@Sally:1978 (1.1)] $$\label{Sallyineq}
\nu_R(J)\le {\operatorname{e}}(R) - {\operatorname{e}}(R/J)$$ for an ideal $J$ of height $0$ in a one-dimensional CM local ring $R$. Now the image of $x$ is a reduction element for $R/N^2$, so the right-hand side of (1.4.1), with $J = N^2$, is ${\operatorname{e}}(R/(x)) - {\operatorname{e}}(R/(Rx + N^2)$. But $N^2 \subseteq {{{\mathfrak}{m}}}^2 \subset Rx$, so this expression is $0$. We conclude from (\[Sallyineq\]) that $N^2 = 0$. Similarly, putting $J = N$, we see than $\nu_R(N) \le 2$.
For (3) and (4), we observe that $N/{{{\mathfrak}{m}}}N \subseteq {{{\mathfrak}{m}}}/{{{\mathfrak}{m}}}^2,$ that is, minimal generators of $N$ are also minimal generators of ${{{\mathfrak}{m}}}/x{{{\mathfrak}{m}}}$ and of ${{{\mathfrak}{m}}}$. Indeed, since $x$ is a nonzerodivisor mod $N$, $N \cap x{{{\mathfrak}{m}}}=xN$; it follows that $N\cap {{{\mathfrak}{m}}}^2 = {{{\mathfrak}{m}}}N$, and the map $N/{{{\mathfrak}{m}}}N {{\longrightarrow}}{{{\mathfrak}{m}}}/{{{\mathfrak}{m}}}^2$ is injective. This proves (3).
To complete the proof of (4), let $z$ generate $N$. Choose any $w$ such that ${{{\mathfrak}{m}}}= Rx+Rw+Rz$. Now $zw = xg$ for some $g$, and (as $x$ is a nonzerodivisor) $g = bz$ for some $b\in R$. Then $z(w-bx) = 0$, and we may take $y = w-bx$.
\[classifydim1\] Let $k$ be an infinite field. The following is a complete list, up to $k$-isomorphism, of the one-dimensional, complete, equicharacteristic, CM local rings with bounded but infinite CM type and with residue field $k$:
1. $k[[X,Y]]/(Y^2)$;
2. $T:= k[[X,Y]]/(XY^2)$;
3. $E:={\operatorname{End{}}}_T({{{\mathfrak}{m}}}_T)$, where ${{{\mathfrak}{m}}}_T$ is the maximal ideal of $T$. We have a presentation $E\cong k[[X,Y,Z]]/(XY,YZ,Z^2)$.
Moreover, if $(R,{{{\mathfrak}{m}}},k)$ is a one-dimensional, complete, equicharacteristic CM local ring and $R$ does [*not*]{} have bounded CM type, then $R$ has, for each positive integer $r$, an indecomposable MCM module of constant rank $r$.
The rings in (1) and (2) have bounded but infinite CM type by Proposition \[hyperclassify\]. To show that $E$ has bounded CM type, it suffices, by Lemma \[endoring\], to check that $E$ is local, a fact that will emerge in the next paragraph, where we verify the presentation of $E$ given in (3). We routinely use decapitalization to denote specialization of variables.
The element $x+y$ is a nonzerodivisor of $T$, and the fraction $z := \frac{y^2}{x+y}$ is easily checked to be in $E:={\operatorname{End{}}}_T({{{\mathfrak}{m}}}_T)$ but not in $T$. Now $E = {\operatorname{Hom{}}}_T({{{\mathfrak}{m}}}_T,T)$ since ${{{\mathfrak}{m}}}_T$ does not have $T$ as a direct summand, and it follows by duality over the Gorenstein ring $T$ that $E/T \cong {\operatorname{Ext{}}}_R^1(T/{{{\mathfrak}{m}}}_T,T)\cong k$. Therefore $E = T[z].$ Since $z^2 = \frac{y^2(x+y)^2}{(x+y)^2} \in {{{\mathfrak}{m}}}_T$, $E$ is local. One verifies the relations $xz =0, yz = y^2 = z^2$ in $E$. Thus the map $S:=k[[X',Y',Z']] \twoheadrightarrow E$ (sending $X'\mapsto x, Y'\mapsto y,$ and $Z'\mapsto z$) induces a surjection from $A:= S/(X'Z', {Y'}^2-Y'Z', Y'Z'-{Z'}^2)$ to $E$. We will build an inverse map, but first we note that the change of variables $X' = X, Y' = Y+Z, Z' = Y$ transforms $A$ to the ring $B:= k[[X,Y,Z]]/(XY,YZ,Z^2)$. Since $B/(z)$ is one-dimensional and $z^2=0$, $B$ and $A$ are one-dimensional.
To build the inverse map, we note that we have $x'y'^2=0$ in $A$, so we have, at least, a map from $T$ to $A$ taking $x$ to $x'$ and $y$ to $y'$. Next, we note that the defining ideal $I$ of $A$ is the ideal of $2 \times 2$ minors of the matrix ${\varphi}= \left[\smallmatrix X' & Y' \\ Y'-Z' & 0 \\ 0 & Z' \endsmallmatrix\right].$ By (the converse of) the Hilbert–Burch theorem [@BH (1.4.16)], $I$ has a free resolution $$\CD 0 {{\longrightarrow}}S^2 @>{\varphi}>> S^3 {{\longrightarrow}}I {{\longrightarrow}}0.\endCD$$ Therefore $A$ has projective dimension $2$ over $S$ and hence depth $1$. Therefore $A$ is Cohen-Macaulay, and, since ${{{\mathfrak}{m}}}_A^2 = (x'+y'){{{\mathfrak}{m}}}$, it follows that $x'+y'$ is a non-zerodivisor of $A$. Now our map $T {{\longrightarrow}}A$ extends to a map $T[\frac{1}{x+y}] {{\longrightarrow}}A[\frac{1}{x'+y'}]$, and the restriction of this map provides the desired map from $E$ to $A$.
We now know that each of the rings on our list has bounded but infinite CM type. To show that the list is complete and to prove the “Moreover" statement, assume now that $(R,{{{\mathfrak}{m}}},k)$ is a one-dimensional, complete, equicharacteristic CM local ring with with $k$ infinite and having infinite CM type. Suppose, moreover, that $R$ does [*not*]{} have indecomposable MCM modules of arbitrarily large (constant) rank. We will show that $R$ is isomorphic to one of the rings on the list.
If $R$ is a hypersurface, Proposition \[hyperclassify\] tells us that $R$ is isomorphic to either $k[[X,Y]]/(Y^2)$ or $k[[X,Y]]/(XY^2)$. Thus we assume that $\nu_R({{{\mathfrak}{m}}}) \ge 3$. But ${\operatorname{e}}(R) \le 3$ by Proposition \[bigranks\]. Therefore we may assume that ${\operatorname{e}}(R) = \nu_R({{{\mathfrak}{m}}}) = 3$. Thus we are in the situation of Lemma \[tom\]. Moreover, the nilradical $N$ of $R$ is non-trivial, by [@CWW (0.5), (1.2)].
We claim that $N$ is principal. If not, then by (3) of Lemma \[tom\], we can find elements $x,y,z$ in $R$ such that $$\label{relationsN2gen}
{{{\mathfrak}{m}}}= Rx+Ry+Rz, \ {{{\mathfrak}{m}}}^2={{{\mathfrak}{m}}}x, \ {\text{and}} \ \ N=Ry+Rz$$ Put $S := R[\frac{y}{x^2},\frac{z}{x^2}] = R + R \frac{y}{x^2} + R\frac{z}{x^2}$, and note that ${{{\mathfrak}{m}}}S = {{{\mathfrak}{m}}}+ R\frac{y}{x} + R\frac{z}{x}$. It is easy to verify (by clearing denominators) that $\{1,\frac{y}{x^2},\frac{z}{x^2}\}$ is a minimal generating set for $S$ as an $R$-module, and that the images of $\frac{y}{x}$ and $\frac{z}{x}$ form a minimal generating set for $\frac {{{{\mathfrak}{m}}}S}{{{{\mathfrak}{m}}}}$. Thus we are in case (2) of Proposition \[bigranks\], and our basic assumption is violated. This proves our claim that $N$ is principal.
Using Lemma \[tom\](4), we find elements $x,y,z$ in $R$ such that $$\label{tumrelations}
{{{\mathfrak}{m}}}= Rx+Ry+Rz, \ {{{\mathfrak}{m}}}^2={{{\mathfrak}{m}}}x, \ N = Rz, \ {\text{and}} \ \ yz = z^2 = 0.$$ Since $y^2 \in {{{\mathfrak}{m}}}x \subset Rx$, we see that $R/Rx$ is a three-dimensional $k$-algebra. Further, since $\cap_n (Rx^n) = 0$, it follows that $R$ is finitely generated (and free) as a module over the discrete valuation ring $V := k[[x]]$.
We claim that $R = V + Vy + Vz$ (and therefore $\{1,y,z\}$ is a basis for $R$ as a $V$-module). To see this, we note that $R = V[[y,z]] = V[[y]] + Vz$, since $yz = z^2 = 0$. Let $h \in V[[y]]$, say, $h = v_0 +v_1y + v_2y^2 + \dots$, with $v_i \in V$. For each $n \ge 1$ write $y^{n+1} = xh_n$, with $h_n \in {{{\mathfrak}{m}}}^n$. (This is possible since ${{{\mathfrak}{m}}}^{n+1} = x{{{\mathfrak}{m}}}^n$.) Then $v_2y^2 + v_3y^3 + \dots = x(v_2h_1 + v_3h_2 + \dots) \in xR$. Therefore $h\in V+ Vy + xR$, and it follows that $R = V + Vy + Vz + xR$. Our claim now follows from Nakayama’s Lemma.
In order to understand the structure of $R$ we must analyze the equation that puts $y^2$ into $x{{{\mathfrak}{m}}}$. Thus we write $y^2 = x^rq$, where $r \ge 1$ and $q\in {{{\mathfrak}{m}}}-{{{\mathfrak}{m}}}^2$. Write $q = \alpha x + \beta y + \gamma z$, with $\alpha,\beta, \gamma \in V$. Since $x$ is a non-zerodivisor and $yz = z^2 = 0$, we see immediately that $\alpha = 0$. Thus we have $$y^2 = x^r(\beta y + \gamma z)$$ with $\beta,\gamma\in V$; moreover, at least one of $\beta,\gamma$ must be a unit of $V$ (since $q\notin {{{\mathfrak}{m}}}^2$).
We claim that $r = 1$. For suppose $ r\ge 2$. Put $v: = \frac{y}{x^2}$ and $w:= \frac{z}{x^2}$. The relations $vw = w^2 = 0$ and $v^2 = x^{r-2} (\beta v + \gamma w)$ show that $S:= R[v,w]$ is a module-finite birational extension of $R$. Moreover, $\nu_R(S) = 3$ and it is easy to see that $\frac{{{{\mathfrak}{m}}}S}{{{{\mathfrak}{m}}}}$ is minimally generated by the images of $\frac{y}{x}$ and $\frac{z}{x}$. The desired contradiction now follows from Proposition 1.2.
Thus we have $$\label{y-square}
y^2 = x(\beta y + \gamma z)$$ with $\beta,\gamma\in V$, and at least one of $\beta,\gamma$ is a unit of $V$. We will produce a hypersurface subring $A:=V[[g]]$ of $R$ such that $R = {\operatorname{End{}}}_A({{{\mathfrak}{m}}}_A)$. We will then show that $A \cong k[[X,Y]]/(XY^2)$, and the proof will be complete.
[**Case 1**]{}: $\beta$ is a unit. Consider the subring $A:=V[y]\subset R$. From (\[y-square\]) we see that $xz\in A$, and it follows easily that $z \in {\operatorname{End{}}}_A({{{\mathfrak}{m}}}_A)$, so $R \subset {\operatorname{End{}}}_A({{{\mathfrak}{m}}}_A)$. As before (since $A$ is Gorenstein) $R = {\operatorname{End{}}}_A({{{\mathfrak}{m}}}_A)$.
[**Case 2**]{} $\beta$ is not a unit (whence $\gamma$ is a unit). This time we put $A:=V[y+z]\subset
R$. The equation $$xy(1-\beta\gamma^{-1}) = x(y+z) - \gamma^{-1}(y+z)^2$$ shows that $xy \in A$. Therefore $xz\in A$ as well, and as before we conclude that $R={\operatorname{End{}}}_A({{{\mathfrak}{m}}}_A)$.
By Lemma \[endoring\], $A$ has infinite CM type but does not have indecomposable MCM modules of arbitrarily large constant rank. Moreover, $A$ cannot have multiplicity $2$, since it has a module-finite birational extension of multiplicity greater than $2$. By Proposition \[hyperclassify\], $A \cong k[[X,Y]]/(XY^2)$, as desired.
Ascent and descent in dimension one
===================================
In this section we show that bounded CM type passes to and from the ${{{\mathfrak}{m}}}$-adic completion of an equicharacteristic one-dimensional CM local ring $(R,{{{\mathfrak}{m}}},k)$ with $k$ infinite. Contrary to the situation in higher dimension (see Theorem \[ascentgen\] below), we need not assume that $R$ is excellent with an isolated singularity. Indeed, in dimension one this assumption would make ${\widehat}R$ reduced, in which case finite and bounded CM type are equivalent [@CWW]. We do, however, insist that $k$ be infinite, in order to use the crucial fact from §1 that failure of bounded CM type implies the existence of MCM modules of unbounded [*constant*]{} rank and also to use the explicit equations worked out in [@BGS] for the indecomposable MCM modules over $T:=k[[X,Y]]/(XY^2)$.
Given a local ring $R$ with completion ${\widehat}R$ and a finitely generated module $M$ over ${\widehat}R$, we say $M$ is [*extended*]{} (from $R$) provided there is a finitely generated $R$-module $A$ such that ${\widehat}A \cong M$. The following proposition appears, in a narrower context, in the 2002 University of Nebraska Ph.D. thesis of M. Arnavut [@Arnavut:thesis]. The argument is adapted from [@Weston:1986 (1.5)].
\[coextended\] Let $R$ be a one-dimensional CM local ring with completion ${\widehat}R$, and let $K$ be the total quotient ring of ${\widehat}R$. Let $M$ and $N$ be finitely generated ${\widehat}R$-modules such that $K\otimes_{{\widehat}R}M \cong K\otimes_{{\widehat}R}N$. Then $M$ is extended if and only if $N$ is extended.
Assume $N$ is extended, say, $N\cong {\widehat}A$, where $_RA$ is finitely generated. Choose an $R$-module homomorphism ${\varphi}:M {{\longrightarrow}}N$ such that ${\varphi}\otimes 1_K$ is an isomorphism. We obtain an exact sequence $$\CD
0 {{\longrightarrow}}V {{\longrightarrow}}M @>{\varphi}>> N @>\pi>> W {{\longrightarrow}}0,
\endCD$$ in which both $V$ and $W$ are torsion modules (therefore of finite length). Let $L = {\varphi}(M)={\operatorname{Ker}}(\pi)$. Now ${\operatorname{Hom{}}}_{{\widehat}R}(N, W) =
{\operatorname{Hom{}}}_{{\widehat}R}({\widehat}A, {\widehat}W) = ({\operatorname{Hom{}}}_R(A,W)){\widehat}{} =
{\operatorname{Hom{}}}_R(A,W)$, since $W$ and ${\operatorname{Hom{}}}_R(A,W)$ have finite length. Therefore there is a homomorphism $\rho \in {\operatorname{Hom{}}}_R(A,W)$ such that ${\widehat}\rho = \pi$. Letting $B = {\operatorname{Ker}}(\rho)$, we see that ${\widehat}B
\cong L$.
Next, we consider the short exact sequence $$\label{quartz}
0 {{\longrightarrow}}V {{\longrightarrow}}M {{\longrightarrow}}L {{\longrightarrow}}0,$$ viewed as an element of ${\operatorname{Ext{}}}_{{\widehat}R}^1(L,V) = {\operatorname{Ext{}}}_R^1(B,V)$ (again, because $V$ and ${\operatorname{Ext{}}}_R^1(B,V)$ have finite length). Therefore (\[quartz\]) is the completion of a short exact sequence $$0 {{\longrightarrow}}V {{\longrightarrow}}C {{\longrightarrow}}B {{\longrightarrow}}0.$$ Then ${\widehat}C \cong M$, as desired.
A finitely generated module $M$ over a Noetherian ring $R$ is said to be [*generically free*]{} provided $M_P$ is $R_P$-free for each $P\in {\operatorname{Ass}}(R)$. For a generically free $R$-module $M$, we let ${\operatorname{rank{}}}_P(M)$ denote the rank of the free $R_P$-module $M_P$, for $P\in {\operatorname{Ass}}(R)$.
\[rankfibers\] Let $(R,{{{\mathfrak}{m}}})$ be a one-dimensional CM local ring with completion ${\widehat}R$, and let $M$ be a generically free ${\widehat}R$-module. Then $M$ is extended from $R$ if and only if ${\operatorname{rank{}}}_P(M) = {\operatorname{rank{}}}_Q(M)$ whenever $P$ and $Q$ are minimal primes of ${\widehat}R$ lying over the same prime of $R$. In particular, every ${\widehat}R$-module of constant rank is extended from an $R$-module (necessarily of the same constant rank).
Suppose $M \cong {\widehat}W$, and let $P$ and $Q$ be primes of ${\widehat}R$ lying over $p\in
{\operatorname{Spec}}(R)$. Let $r = {\operatorname{rank{}}}_P(M)$. We have a flat local homomorphism $R_p {{\longrightarrow}}{\widehat}R_P$. It follows from faithfully flat descent [@EGA4.2 (2.5.8)] that $W_p$ is $R_p$-free of rank $r$. From the change of rings $R_p {{\longrightarrow}}{\widehat}R_Q$ we see that ${\operatorname{rank{}}}_Q(M) = r$. This proves the “only if” implication and the parenthetical remark in the last sentence of the statement.
For the converse, let $\{p_1,\dots, p_s\}$ be the minimal primes of $R$, and let $r_i =
{\operatorname{rank{}}}_P(M_P)$ for $P$ in the fiber over $p_i$. Let $J_1\cap\dots\cap J_s$ be a primary decomposition of $(0)$ in $R$, with $\sqrt{J_i} = p_i$. (Since $R$ is CM, ${{{\mathfrak}{m}}}\notin {\operatorname{Ass}}(R)$.) Put $W =
\bigoplus_{i=1}^s(R/J_i)^{r_i}$. Then $K\otimes_{{\widehat}R}{\widehat}W \cong K\otimes_R M$. By Proposition \[coextended\], $M$ is extended.
Here is our main result of this section.
\[updowndim1\] Let $(R,{{{\mathfrak}{m}}},k)$ be a one-dimensional equicharacteristic CM local ring with completion ${\widehat}R$. Assume that $k$ is infinite. Then $R$ has bounded CM type if and only if ${\widehat}R$ has bounded CM type. If $R$ has unbounded CM type, then $R$ has, for each $r$, an indecomposable MCM module of constant rank $r$.
Assume that ${\widehat}R$ does not have bounded CM type. Fix a positive integer $r$. By Theorem \[classifydim1\] we know that ${\widehat}R$ has an indecomposable MCM module $M$ of constant rank $r$. By Corollary \[rankfibers\] there is a finitely generated $R$-module $N$, necessarily MCM and with constant rank $r$, such that ${\widehat}N \cong M$. Obviously $N$ too must be indecomposable.
Assume from now on that ${\widehat}R$ has bounded CM type. If ${\widehat}R$ has [*finite*]{} CM type, the same holds for $R$, [@CWW]. Therefore we assume that ${\widehat}R$ has infinite CM type. Then ${\widehat}R$ is isomorphic to one of the rings of Theorem \[classifydim1\]: $k[[X,Y]]/(Y^2), T :=
k[[X,Y]]/(XY^2),$ or $E:= {\operatorname{End{}}}_T({{{\mathfrak}{m}}}_T).$ If ${\widehat}R \cong
k[[X,Y]]/(Y^2)$, then ${\operatorname{e}}(R) = 2$, and $R$ has bounded CM type by [@Leuschke-Wiegand:hyperbrt (2.1)]. Suppose for the moment that we have verified bounded CM type for any local ring $S$ whose completion is isomorphic to $E$. If, now, ${\widehat}R \cong T$, put $S
:= {\operatorname{End{}}}_R({{{\mathfrak}{m}}})$. Then ${\widehat}S \cong E$, whence $S$ has bounded CM type. Therefore so has $R$, by Lemma \[endoring\].
Therefore we may assume that ${\widehat}R = E$. Our plan is to examine each of the indecomposable non-free $E$-modules and then use Proposition \[coextended\] and Corollary \[rankfibers\] to determine exactly which MCM $E$-modules are extended from $R$. From now on we use the presentation for $E={\widehat}R$ given in the proof of Theorem \[classifydim1\]: $E \cong k[[X,Y,Z]]/(XZ, Y^2-YZ, YZ-Z^2)$. By Lemma \[endoring\] the indecomposable non-free MCM $E$-modules are exactly the indecomposable non-free MCM $T$-modules, namely, the cokernels of the following matrices over $T$ (see [@BGS (4.2)]): $$\label{Tmods1}
[y];\ \ [xy];\ \ [x];\ \ [y^2]$$ $$\label{Tmods2}
\alpha:=\left[\begin{matrix} y & x^k\\ 0 & -y \end{matrix}\right];\ \
\beta:=\left[\begin{matrix} xy & x^{k+1}\\ 0 & -xy \end{matrix}\right];\ \
\gamma:=\left[\begin{matrix} xy & x^k\\ 0 & -y \end{matrix}\right];\ \
\delta:=\left[\begin{matrix} y & x^{k+1}\\ 0 & -xy \end{matrix}\right].$$ Let $P:=(x)$ and $Q:=(y)$ be the two minimal prime ideals of $T$. Note that $T_P \cong k((Y))$ and $T_Q \cong k((X))[Y]/(Y^2)$. With the exception of $U:= {\operatorname{Coker}}[y]$ and $V:={\operatorname{Coker}}[xy]$, each of the modules in (\[Tmods1\]) and (\[Tmods2\]) is generically free, with ${\varphi},
({\operatorname{rank{}}}_P{\operatorname{Coker}}({\varphi}),{\operatorname{rank{}}}_Q{\operatorname{Coker}}({\varphi}))$ given in the following list: $$\label{Tmods-ranks}
[x],(1,0);\ \ [y^2],(0,1);\ \ \alpha,(0,1);\ \
\beta,(2,1);\ \ \gamma,(1,1);\ \ \delta,(1,1).$$ Let $M$ be a MCM ${\widehat}R$-module, and write $$\label{Emod}
M \cong (\oplus_{i=1}^aA_i)\oplus (\oplus_{j=1}^bB_j)\oplus(\oplus_{k=1}^cC_k)\oplus
(\oplus_{l=1}^dD_l)\oplus U^e \oplus V^f,$$ where the $A_i,B_j,C_k,D_l$ are indecomposable generically free modules of ranks $(1,0),$ $(0,1),$ $(1,1),$ $(2,1)$ (and, again, $U = {\operatorname{Coker}}([y])$ and $V = {\operatorname{Coker}}([xy]))$.
Suppose first that $R$ is a domain. Then $M$ is extended if and only if $b = a+d$ and $e = f= 0$. Now the indecomposable MCM $R$-modules are those whose completions have $(a,b,c,d,e,f)$ minimal and non-trivial with respect to these relations. (We are using implicitly the fact (see [@Wiegand:2001 (1.2)] for example) that for two finitely generated $R$-modules $N_1$ and $N_2$, $N_1$ is isomorphic to a direct summand of $N_2$ if and only if ${\widehat}N_1$ is isomorphic to a direct summand of ${\widehat}N_2$.) The only possibilities are $(0,0,1,0,0,0)$, $(1,1,0,0,0,0)$ and $(0,1,0,1,0,0)$, and we conclude that the indecomposable $R$-modules have rank $1$ or $2$.
Next, suppose that $R$ is reduced but not a domain. Then $R$ has exactly two minimal prime ideals, and we see from Corollary \[rankfibers\] that every generically free ${\widehat}R$-module is extended from $R$; however, neither $U$ nor $V$ can be a direct summand of an extended module. In this case, the indecomposable MCM $R$-modules are generically free, with ranks $(1,0), (0,1), (1,1)$ and $(2,1)$ at the minimal prime ideals.
Finally, we assume that $R$ is not reduced. We must now consider the two modules $U$ and $V$ that are not generically free. We will see that $U:={\operatorname{Coker}}[y]$ is always extended and that $V:={\operatorname{Coker}}[xy]$ is extended if and only if $R$ has two minimal prime ideals. Note that $U \cong Txy = Exy$ (the nilradical of $E = {\widehat}R$), and $V \cong Ty = Ey$.
The nilradical $N$ of $R$ is of course contained in the nilradical $Exy$ of ${\widehat}R$. Moreover, since $Exy \cong E/(0:_E xy) = E/(y,z)$ is a faithful cyclic module over $E/(y,z) \cong k[[x]]$, every non-zero submodule of $Exy$ is isomorphic to $Exy$. In particular, $N{\widehat}R \cong Exy$. This shows that $U$ is extended.
Next we deal with $V$. The kernel of the map $Ey \twoheadrightarrow Exy$ (multiplication by $x$) is $Ey^2$. Thus we have a short exact sequence $$\label{2.3.4}
0{{\longrightarrow}}Ey^2{{\longrightarrow}}V {{\longrightarrow}}U{{\longrightarrow}}0.$$ Now $Ey^2 = Ty^2 = {\operatorname{Coker}}([x])$ is generically free of rank $(1,0)$, and since the total quotient ring $K$ (of both $T$ and ${\widehat}R$) is Gorenstein we see that $$KV \cong Ky^2\oplus KU.$$
If, now, $R$ has two minimal primes, every generically free ${\widehat}R$-module is extended, by Corollary \[rankfibers\]. In particular, $Ey^2$ is extended, and by Proposition \[coextended\] so is $V$. Thus every indecomposable MCM ${\widehat}R$-module is extended, and $R$ has bounded CM type.
If, on the other hand, $R$ has just one minimal prime ideal, then the module $M$ in (\[Emod\]) is extended if and only if $b = a+d+f$. The ${\widehat}R$-modules corresponding to indecomposable MCM $R$-modules are therefore $U$, $V\oplus W$, where $W$ is some generically free module of rank $(0,1)$, and the modules of constant rank $1$ and $2$ described above.
We conclude this section with a summary of the main results of §§1 and 2.
\[summary\] Let $(R,{{{\mathfrak}{m}}},k)$ be an equicharacteristic one-dimensional local CM ring with $k$ infinite. Then $R$ has bounded but infinite CM type if and only if the completion ${\widehat}R$ is isomorphic to one of the following:
1. $k[[X,Y]]/(Y^2)$;
2. $T:= k[[X,Y]]/(XY^2)$;
3. $E:={\operatorname{End{}}}_T({{{\mathfrak}{m}}}_T)$, where ${{{\mathfrak}{m}}}_T$ is the maximal ideal of $T$.
Moreover, if $R$ does [*not*]{} have bounded CM type, then $R$ has, for each positive integer $r$, an indecomposable MCM module of constant rank $r$.
Ascent, descent, and Brauer–Thrall in higher dimensions
=======================================================
In this section we study ascent and descent of bounded CM type to and from the completion in dimension greater than one. We prove that bounded CM type ascends to the completion of an excellent CM local ring with an isolated singularity. An easy corollary of this result is a generalization of the Brauer–Thrall theorem of Yoshino and Dieterich [@Yoshino:book (6.4)]. See Theorem \[BrauerThrall\].
For descent, we have a less complete picture. We show that bounded CM type descends from the completion of a Henselian local ring, and we investigate the case of a two-dimensional normal local domain such that the completion is also a normal domain. Example \[nobound\] indicates why descent is less tractable than ascent.
For Henselian rings, ascent and descent are easy:
\[updownRh\] Let $R$ be a Henselian local ring with completion ${\widehat}R$. If ${\widehat}R$ has bounded CM type, then so has $R$. Conversely, if $R$ has bounded CM type and ${\widehat}R$ has at most an isolated singularity, then ${\widehat}R$ has bounded CM type.
Assume ${\widehat}R$ has bounded CM type, and let $M$ be an indecomposable MCM $R$-module. Since $R$ is Henselian, the endomorphism ring $E:= {\operatorname{End{}}}_R(M)$ is local, meaning $E/J$ is a division ring (where $J$ is the Jacobson radical of $E$). Passing to ${\widehat}R$, we observe that ${\operatorname{End{}}}_{{\widehat}R}({\widehat}M) =
{\widehat}R \otimes_R {\operatorname{End{}}}_R(M)$. Since ${\widehat}E/{\widehat}J = (E/J){\widehat}{} = E/J$, we see that ${\operatorname{End{}}}_{{\widehat}R}({\widehat}M)$ is local as well, so ${\widehat}M$ is indecomposable. Since $M$ was arbitrary, it follows that $R$ has bounded CM type.
For the converse we use Elkik’s theorem [@Elkik Thèoréme 3] on extensions of vector bundles over Henselian pairs. Since ${\widehat}R$ has an isolated singularity, every MCM ${\widehat}R$-module $M$ is locally free on the punctured spectrum of ${\widehat}R$, and so by Elkik’s theorem is isomorphic to ${\widehat}N$ for some (necessarily MCM) $R$-module $N$. It follows immediately that bounded CM type extends to ${\widehat}R$.
For ascent to the Henselization we recycle an argument from [@Wiegand:1998] and [@Leuschke-Wiegand:2000]. Recall [@Demeyer-Ingraham] that the extension $R {{\longrightarrow}}{R^{\text h}}$ is [*separable*]{}, meaning that the sequence $$\label{sep}
\CD
0 @>>> J @>>> {R^{\text h}}\otimes_R {R^{\text h}}@>\mu>> {R^{\text h}}@>>> 0,
\endCD$$ where $\mu(u\otimes v) = uv$, is split exact as a sequence of ${R^{\text h}}\otimes_R {R^{\text h}}$-modules. Tensoring (\[sep\]) with an arbitrary finitely generated ${R^{\text h}}$-module $N$ shows that $N$ is a direct summand of the extended module ${R^{\text h}}\otimes_R N$, where the action of ${R^{\text h}}$ on ${R^{\text h}}\otimes_R N$ is by change of rings. Write $_RN$ as a directed union of finitely generated $R$-modules $A_\alpha$. Then, since $N$ is a finitely generated ${R^{\text h}}$-module, $N$ is a direct summand of ${R^{\text h}}\otimes_R A_\alpha$ for some $\alpha$. Thus any finitely generated ${R^{\text h}}$-module $N$ is a direct summand of an extended module.
\[recycle\] Let $R$ be a CM local ring with Henselization ${R^{\text h}}$. Assume that ${R^{\text h}}$ is Gorenstein on the punctured spectrum. If $R$ has bounded CM type, then ${R^{\text h}}$ does as well.
Let $M$ be an indecomposable MCM ${R^{\text h}}$-module. Put $d = \dim(R)$. Since ${R^{\text h}}$ is Gorenstein on the punctured spectrum, $M$ is a $d^{\text th}$ syzygy of some finitely generated ${R^{\text h}}$-module $N$, [@EG 3.8]. By the argument above, $N$ is a direct summand of ${R^{\text h}}\otimes_R B$ for some finitely generated $R$-module $B$. Letting $A$ be a $d^{\text th}$ syzygy of $_RB$, we see (using the Krull–Schmidt Theorem over ${R^{\text h}}$) that $M$ is a direct summand of ${R^{\text h}}\otimes_RA$. Since $_RA$ is MCM, we can write $A$ as a direct sum of modules $C_i$ of low multiplicity. Using Krull–Schmidt again, we deduce that $M$ is a direct summand of some ${R^{\text h}}\otimes_RC_i$, thereby getting a bound on the multiplicity of $M$.
\[ascentgen\] Let $R$ be an excellent CM local ring with at most an isolated singularity. If $R$ has bounded CM type, then the completion ${\widehat}R$ also has bounded CM type.
As $R$ has geometrically regular formal fibres, ${\widehat}R$ and the Henselization ${R^{\text h}}$ both also have isolated singularities [@EGA4.2 (6.5.3)]. By Proposition \[updownRh\] and Corollary \[recycle\], then, bounded CM type ascends to ${R^{\text h}}$ and thence to ${\widehat}R$.
Theorem \[ascentgen\] allows us to verify a version of the Brauer–Thrall conjecture. The complete case of this theorem is due to Yoshino and Dieterich [@Yoshino:book (6.4)].
\[BrauerThrall\] Let $(R, {{{\mathfrak}{m}}}, k)$ be an excellent equicharacteristic CM local ring with perfect residue field $k$. Then $R$ has finite CM type if and only if $R$ has bounded CM type and $R$ has at most an isolated singularity.
If $R$ has finite CM type, then $R$ has at most an isolated singularity by [@Huneke-Leuschke], and of course $R$ has bounded CM type. Suppose now that $R$ has bounded CM type and at most an isolated singularity. According to Theorem \[ascentgen\], ${\widehat}R$ also has bounded CM type. By the Brauer–Thrall theorem of Yoshino and Dieterich, ${\widehat}R$ has finite CM type. This descends to $R$ by [@Wiegand:1998 (1.4)], and we are done.
One cannot remove the hypothesis of excellence. For example, let $S$ be any one-dimensional analytically ramified local domain. It is known [@Matlis pp. 138–139] that there is a one-dimensional local domain $R$ between $S$ and its quotient field such that ${\operatorname{e}}(R) = 2$ and ${\widehat}R$ is not reduced. Then $R$ has bounded but infinite CM type by [@Leuschke-Wiegand:hyperbrt (2.1), (0.1)], and of course $R$ has an isolated singularity.
Proving descent of bounded CM type in general seems quite difficult. Part of the difficulty lies in the fact that, in general, there is no bound on the number of indecomposable MCM ${\widehat}R$-modules required to decompose the completion of an indecomposable MCM $R$-module. Here is an example to illustrate. Recall [@RWW Prop. 3] that when $R$ and ${\widehat}R$ are two-dimensional normal domains, a torsion-free ${\widehat}R$-module $M$ is extended from $R$ if and only if $[M]$ is in the image of the natural map on divisor class groups ${\operatorname{Cl}}(R) {{\longrightarrow}}{\operatorname{Cl}}({\widehat}R)$.
\[nobound\] Let $A$ be a complete local two-dimensional normal domain containing a field, and assume that the divisor class group ${\operatorname{Cl}}(A)$ has an element $\alpha$ of infinite order. (See, for example, [@Wiegand:2001 (3.4)].) By Heitmann’s theorem [@Heitmann], there is a unique factorization domain $R$ contained in $A$ such that ${\widehat}R = A$. Choose, for each integer $n$, a divisorial ideal $I_n$ corresponding to $n\alpha \in
{\operatorname{Cl}}({\widehat}R)$. For each $n \ge 1$, let $M_n:= I_n \oplus N_n$, where $N_n$ is the direct sum of $n$ copies of $I_{-1}$. Then $M_n$ has trivial divisor class and therefore is extended from $R$ by [@RWW Prop. 3]. However, no non-trivial proper direct summand of $M_n$ has trivial divisor class, and it follows that $M_n$ (a direct sum of $n+1$ indecomposable ${\widehat}R$-modules) is extended from an indecomposable MCM $R$-module.
It is important to note that the example above does not give a counterexample to descent of bounded CM type, but merely points out one difficulty in studying descent.
We finish with a positive result. Recall [@Geroldinger-Schneider] that the [*Davenport constant*]{} ${\operatorname{D}}(G)$ of a finite abelian group $G$ is the least positive integer $d$ such that every sequence of $d$ (not necessarily distinct) elements of $G$ has a non-empty subsequence whose sum is $0$. It is easy to see that ${\operatorname{D}}(G) \le |G|$, with equality if $G$ is cyclic.
Let $R$ be a local normal domain of dimension two such that ${\widehat}R$ is also a normal domain. Assume that the cokernel $G$ of the natural map ${\operatorname{Cl}}(R) {{\longrightarrow}}{\operatorname{Cl}}({\widehat}R)$ is finite. If the ranks of the indecomposable MCM ${\widehat}R$-modules are bounded by $n$, then the ranks of the indecomposable MCM $R$-modules are bounded by $m:=n{\operatorname{D}}(G)$. In particular, if ${\widehat}R$ has bounded CM type, so has $R$.
Let $M$ be an indecomposable MCM $R$-module, and put $N:= {\widehat}M$. Write $$N \cong Z_1^{a_1} \oplus \dots \oplus Z_k^{ a_k},$$ where the $Z_i$ are indecomposable MCM modules, each of rank at most $n$. It will suffice to show that $a_1+\dots+a_k \le {\operatorname{D}}(G)$.
Let $\pi:{\operatorname{Cl}}({\widehat}R) {{\longrightarrow}}G$ be the natural map, let $\zeta_i$ be the divisor class of $Z_i$, and let $\gamma_i = \pi(\zeta_i) \in
G$. If $a_1+\dots+a_k > {\operatorname{D}}(G)$, there is a sequence $(b_1,\dots,b_k)\in {\mathbb{N}}^k$ such that
1. $b_i \le a_i$ for all $i$;
2. $b_j < a_j$ for some $j$;
3. $b_k > 0$ for some $k$;
4. $b_1\gamma_1+ \dots + b_k\gamma_k = 0$.
Put $N' := Z_1^{b_1} \oplus \dots \oplus Z_k^{ b_k}$. By (4), the divisor class of the module $N'$ is in the image of the map ${\operatorname{Cl}}(R) {{\longrightarrow}}{\operatorname{Cl}}({\widehat}R)$. Therefore $N'$ is extended from $R$, by [@RWW Prop. 3], say $N' \cong {\widehat}M'$. Also, (1), (2) and (3) say that $N'$ is a non-trivial proper direct summand of $N$. By [@Wiegand:2001 (1.2)], $M'$ is a non-trivial proper direct summand of $M$, contradiction.
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[^1]: Leuschke’s research was supported by an NSF Postdoctoral Fellowship, and Wiegand’s was supported by grants from the NSA and the NSF
|
---
abstract: 'The techniques for optical calibration of Jefferson Lab’s large-acceptance magnetic hadron spectrometer, BigBite, have been examined. The most consistent and stable results were obtained by using a method based on singular value decomposition. In spite of the complexity of the optics, the particles’ positions and momenta at the target have been precisely reconstructed from the coordinates measured in the detectors by means of a single back-tracing matrix. The technique is applicable to any similar magnetic spectrometer and any particle type. For $0.55\,\mathrm{GeV}/c$ protons, we have established a vertex resolution of $1.2\,\mathrm{cm}$, angular resolutions of $7\,\mathrm{mrad}$ and $13\,\mathrm{mrad}$ (in-plane and out-of-plane, respectively), and a relative momentum resolution of $1.6\,\mathrm{\%}$.'
address:
- 'Jožef Stefan Institute, 1000 Ljubljana, Slovenia'
- 'University of Ljubljana, 1000 Ljubljana, Slovenia'
- 'Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA'
- 'Kent State University, Kent, OH, 44242, USA'
- 'Glasgow University, Glasgow, G12 8QQ, Scotland, United Kingdom'
- 'College of William and Mary, Williamsburg, VA, 23187, USA'
- 'University of Virginia, Charlottesville, VA, 22908, USA'
- 'Massachusetts Institute of Technology, Cambridge, MA, 02139, USA'
- 'Los Alamos National Laboratory, Los Alamos, NM, 87545, USA'
- 'University of Kentucky, Lexington, KY, 40506, USA'
- 'Hampton University , Hampton, VA, 23668, USA'
- 'Tel Aviv University, Tel Aviv 69978, Israel'
- 'Duke University, Durham, NC, 27708, USA'
- 'Hebrew University of Jerusalem, Jerusalem, Israel'
- 'GSI, 64291 Darmstadt, Germany'
- 'Yerevan Physics Institute, Yerevan, Armenia'
- 'George Washington University, Washington, D.C., 20052, USA'
- 'Rutgers University, New Brunswick, NJ, 08901, USA'
author:
- 'M. Mihovilovič'
- 'S. Širca'
- 'K. Allada'
- 'B. D. Anderson'
- 'J. R. M. Annand'
- 'T. Averett'
- 'A. Camsonne'
- 'R. W. Chan'
- 'J.-P. Chen'
- 'K. Chirapatpimol'
- 'C. W. de Jager'
- 'S. Gilad'
- 'D. J. Hamilton'
- 'J.-O. Hansen'
- 'D. W. Higinbotham'
- 'J. Huang'
- 'X. Jiang'
- 'G. Jin'
- 'W. Korsch'
- 'J. J. LeRose'
- 'R. A. Lindgren'
- 'N. Liyanage'
- 'E. Long'
- 'R. Michaels'
- 'B. Moffit'
- 'P. Monaghan'
- 'V. Nelyubin'
- 'B. E. Norum'
- 'E. Piasetzky'
- 'X. Qian'
- 'Y. Qiang'
- 'S. Riordan'
- 'G. Ron'
- 'G. Rosner'
- 'B. Sawatzky'
- 'M. Shabestari'
- 'A. Shahinyan'
- 'R. Shneor'
- 'R. Subedi'
- 'V. Sulkosky'
- 'J. W. Watson'
- 'Y.-W. Zhang'
title: Methods for optical calibration of the BigBite hadron spectrometer
---
optical calibration ,magnetic spectrometers ,BigBite ,track reconstruction 29.30.Aj ,29.85.Fj ,25.30.-c
Introduction
============
One of the recent acquisitions in experimental Hall A of the Thomas Jefferson National Accelerator Facility (TJNAF) is the BigBite spectrometer. It was previously used at the NIKHEF facility for the detection of electrons [@lange-general; @lange-optics]. At Jefferson Lab, BigBite has been re-implemented as a versatile spectrometer that can be instrumented with various detector packages optimized for the particular requirements of the experiments. BigBite complements the High-Resolution Spectrometers, which are part of the standard equipment of Hall A [@alcorn]. Adding BigBite allows one to devise more flexible experimental setups involving double- and even triple-coincidence measurements.
In 2005, the BigBite spectrometer was first used in Hall A as the hadron arm in the E01-015 experiment, which investigated nucleon-nucleon short-range correlations [@subedi; @shneor]. In 2006, it was instrumented as the electron arm for the measurement of the neutron electric form factor (experiment E02-013 [@riordan]). In 2008 and 2009, it has been used in two large groups of experiments spanning a broad range of physics topics. We studied near-threshold neutral pion production on protons (experiment E04-007 [@e04007]) and measured single-spin asymmetries in semi-inclusive pion electro-production on polarized $^3\mathrm{He}$ (experiments E06-010 and E06-011 [@qian11; @huang11]). In the same period, we also measured parallel and perpendicular asymmetries on polarized $^3\mathrm{He}$ in order to extract the $g_2^\mathrm{n}$ polarized structure function in the deep-inelastic regime (experiment E06-014 [@e06014]), and measured double-polarization asymmetries in the quasi-elastic processes ${}^3\vec{\mathrm{He}}(\vec \mathrm{e},\mathrm{e}'\mathrm{d})$, ${}^3\vec{\mathrm{He}}(\vec \mathrm{e},\mathrm{e}'\mathrm{p})$, and ${}^3\vec{\mathrm{He}}(\vec \mathrm{e},\mathrm{e}'\mathrm{n})$ (experiments E05-102 and E08-005 [@e05102; @e08005]). In 2011, the investigation of short-range correlations has been continued in the E07-006 experiment [@e07006] exploring the repulsive part of the nucleon-nucleon interaction.
BigBite is a non-focusing spectrometer consisting of a single dipole with large momentum and angular acceptances (the details are presented in Section \[sec:BB\]). The magnetic optics of such spectrometers tend to become complicated towards the edges of their acceptances, especially for the momentum and the dispersive angle. It was not clear from the outset that particle momentum and interaction vertex reconstruction could be accomplished by using a single procedure for all momenta.
The calibration presented in this paper allows for a full description of BigBite optics by means of a single reconstruction matrix. The method was developed and successfully used with the data obtained in the E05-102 experiment with the detector package configured for hadrons (Section \[sec:exp\]), but it is applicable to any magnetic spectrometer with a similar optical configuration and any particle type. Various calibration procedures are discussed in Section \[sec:cal\].
The BigBite spectrometer {#sec:BB}
========================
The BigBite spectrometer [@lange-general] consists of a single room-temperature dipole magnet, shown in Fig. \[BBSpectrometer\]. Energizing the magnet with a current of $518\,\mathrm{A}$ results in a mean field density of $0.92\,\mathrm{T}$, corresponding to a central momentum of $p_\mathrm{c}=0.5\,\mathrm{GeV}/c$ and a bending angle of $25^\circ$. The magnet is coupled to a hadron detector package consisting of two multi-wire drift chambers (MWDC) [@nilanga; @ChanMSc] for particle tracking and two planes of scintillation detectors (denoted by dE and E) [@ShneorMSc] for triggering, particle identification, and energy determination.
![The BigBite spectrometer on its support frame. BigBite consists of a dipole magnet, followed by the detector package assembled from a pair of multi-wire drift chambers (MWDC) and two scintillator planes (dE and E). The directions of the incoming electron beam and the scattered particles, the target cell, and the Helmholtz coil (holding field) assembly are also shown.\[BBSpectrometer\]](eed_Simon_annotated.eps){width="13cm"}
Each MWDC consists of six planes of wires. The wires in the first two planes are oriented at an angle of $60^\circ$ with respect to the dispersive direction, while the wires in the third and fourth plane are aligned horizontally. The wires of the last two planes are oriented at $-60^\circ$. Each wire plane in the first and the second MWDC contains $141$ and $200$ wires, respectively. The spacing between the wires in all planes is $1\,\mathrm{cm}$. The intrinsic spatial resolution of the MWDCs is about $100\,\mu\mathrm{m}$ and $200\,\mu\mathrm{m}$ for the dispersive and non-dispersive coordinates, respectively, and about $0.15\,\mathrm{mrad}$ and $0.35\,\mathrm{mrad}$ for the dispersive and non-dispersive angles, respectively.
The dE- and E-planes (also called the trigger planes) each consist of $24$ plastic scintillator bars. The bars are $50\,\mathrm{cm}$ long and $8.6\,\mathrm{cm}$ wide. For the dE-plane, thinner bars ($0.3\,\mathrm{cm}$) were used to detect low-energy particles, while for the E-plane, a thickness of $3\,\mathrm{cm}$ was chosen to allow for the detection of more energetic particles. The light pulses in each bar were detected by photomultiplier tubes mounted at each end of the bar. To double the spatial and momentum resolution, the bars in the E-plane are offset from those in the dE-plane by one half of the bar width ($4.3\,\mathrm{cm}$).
Experimental details and data {#sec:exp}
=============================
The E05-102 experiment was performed in Hall A [@alcorn] at Jefferson Lab. In the experiment, a polarized ${}^3\mathrm{He}$ target was used in conjunction with the polarized continuous-wave electron beam. Scattered electrons were detected by the left High Resolution Spectrometer (HRS) in coincidence with protons and deuterons that were detected by BigBite. A variety of kinematic settings were employed (Table \[table\_kinematics\]), with the momentum-transfer vector $\vec{q}$ pointing towards BigBite. This ensured that the protons and deuterons from elastic and quasi-elastic scattering were always within its acceptance.
---------------- ---------------------- -------------------- -------------------
Setting $E_{\mathrm{beam}}$
\[-6pt\] label \[$\mathrm{GeV}/c$\] HRS \[${}^\circ$\] BB \[${}^\circ$\]
$1$-pass $1.245$ $17.0$ $-74.0$
$2$-pass $2.425$ $12.5$ $-75.0$
\[-6pt\] $14.5$ $-82.0$
$3$-pass $3.606$ $12.5$ $-75.0$
\[-6pt\] $17.0$ $-74.0$
---------------- ---------------------- -------------------- -------------------
: Kinematics settings of the E05-102 experiment (the incoming electron energy $E_{\mathrm{beam}}$ and the angles of the HRS and BigBite spectrometers with respect to the beam direction).\[table\_kinematics\]
The core component of the polarized ${}^3\mathrm{He}$ target was a pressurized cylindrical glass cell with a length of $40\,\mathrm{cm}$ and a diameter of $1.9\,\mathrm{cm}$ (see Fig. \[TGSYSTEM\]). The thickness of the glass cylinder was $1.7\,\mathrm{mm}$, while the thickness of the end windows was $140\,\mu\mathrm{m}$. The gas in the cell was polarized to approximately $60\,\%$ by hybrid spin-exchange optical pumping [@he3a; @he3b] driven by an infra-red laser system. The direction of the nuclear polarization was maintained by three pairs of Helmholtz coils surrounding the cell.
In addition to the ${}^3\mathrm{He}$ helium target, a $40\,\mathrm{cm}$-long multi-foil carbon target was used for calibration, as described below. It consists of seven $0.252\,\mathrm{mm}$-thick carbon foils mounted to a plastic frame (Fig. \[TGSYSTEM\]) which are preceded by a single slanted BeO foil for beam positioning. Below the multi-foil target, a dummy (reference) cell was installed that could be either evacuated or filled with hydrogen, deuterium, unpolarized helium-3 or nitrogen.
![The target system including the polarized ${}^3\mathrm{He}$ cell at the top, the multi-foil carbon optics target, and the reference cell at the bottom. The slanted BeO foil is used for visual inspection of the beam impact point.\[TGSYSTEM\]](./He3Targetb.eps){width="85.00000%"}
![\[Left\] Schematics of the BigBite sieve-slit collimator. \[Center, Right\] Sieve pattern reconstruction by using the simplex method and the SVD, respectively (see subsubsection \[sub:nmsvd\]). The SVD technique resolves more holes and yields a much clearer pattern. The holes at the left edge are missing due to geometrical obstacles between the target and BigBite.\[BBSieve\]](BigBiteSieve3.eps "fig:"){height="8cm"} ![\[Left\] Schematics of the BigBite sieve-slit collimator. \[Center, Right\] Sieve pattern reconstruction by using the simplex method and the SVD, respectively (see subsubsection \[sub:nmsvd\]). The SVD technique resolves more holes and yields a much clearer pattern. The holes at the left edge are missing due to geometrical obstacles between the target and BigBite.\[BBSieve\]](SievePlot_NM_Color.eps "fig:"){height="8cm"} ![\[Left\] Schematics of the BigBite sieve-slit collimator. \[Center, Right\] Sieve pattern reconstruction by using the simplex method and the SVD, respectively (see subsubsection \[sub:nmsvd\]). The SVD technique resolves more holes and yields a much clearer pattern. The holes at the left edge are missing due to geometrical obstacles between the target and BigBite.\[BBSieve\]](SievePlotColor.eps "fig:"){height="8cm"}
For the optics calibration of BigBite, a special set of measurements was performed with a $4\,\mathrm{cm}$-thick lead sieve-slit collimator positioned at the entrance to the BigBite magnet (see Fig. \[BBSpectrometer\]). The sieve-slit collimator has $82$ circular holes that are almost uniformly positioned over the whole acceptance of the spectrometer, Fig. \[BBSieve\] (left). The collimator also contains four elongated holes used to remove ambiguities in horizontal and vertical orientations and to allow for easier identification of the hole projections at the detector package.
Prior to any optics analysis, a series of cuts were applied to the collected calibration data to eliminate backgrounds. A HRS-BigBite coincidence trigger system was used to acquire electron-proton and electron-deuteron coincidences, at typical rates between $700\,\mathrm{Hz}$ and $1\,\mathrm{kHz}$. True coincidences were selected by applying a cut on the raw coincidence time. The backgrounds were further reduced by PID and HRS acceptance cuts. Finally, only those events that produce consistent hits in all BigBite detectors, and could consequently be joined to form single particle tracks, were selected.
Methods of optical calibration {#sec:cal}
==============================
The purpose of optical calibration is to establish the mapping between the detector variables that are measured directly, and the target variables corresponding to the actual physical quantities describing the particle at the reaction vertex. In the MWDCs, two position coordinates ($x_{\mathrm{Det}}$ and $y_{\mathrm{Det}}$) and two angles ($\theta_{\mathrm{Det}}$ and $\phi_{\mathrm{Det}}$) are measured. From this information, we wish to reconstruct the location of the interaction vertex ($y_{\mathrm{Tg}}$), the in-plane and out-of-plane scattering angles ($\phi_{\mathrm{Tg}}$ and $\theta_{\mathrm{Tg}}$), and the particle momentum relative to the central momentum ($\delta_{\mathrm{Tg}}=(p_\mathrm{Tg}-p_\mathrm{c})/p_\mathrm{c}$). This can be done in many ways. We have considered an analytical model as well as a more sophisticated approach based on transport-matrix formalism, with several means to estimate the reliability of the results and the stability of the algorithms.
Quasi-elastic protons from scattering on the multi-foil carbon target were used to calibrate $y_\mathrm{Tg}$; the same target was also used to calibrate $\theta_\mathrm{Tg}$ and $\phi_\mathrm{Tg}$ when the sieve-slit collimator was in place. In turn, elastic protons and deuterons (from hydrogen and deuterium targets) were used to calibrate $\theta_\mathrm{Tg}$, $\phi_\mathrm{Tg}$, and $\delta_{\mathrm{Tg}}$. The $\delta_{\mathrm{Tg}}$ matrix elements could also be determined by quasi-elastic events from $^3\mathrm{He}$ under the assumption that the energy losses are well understood.
![The schematic of the dispersive (top) and non-dispersive (bottom) planes of the BigBite spectrometer. Small angular deflections in the non-dispersive plane occur if the particle trajectory is not perpendicular to the effective field boundary [@lange-optics; @penner; @brown]. At the entrance to the magnet, they are at most $18\,\mathrm{mrad}$ (close to the acceptance boundaries in the dispersive direction). At the exit field boundary, the effect acts in the opposite sense and partially cancels the deflection at the entrance.\[BBScheme\]](BigBite9.eps){width="15cm"}
The analytical model
--------------------
The magnetic field of the BigBite magnet is oriented in the $y_{\mathrm{Tg}}$ direction (see Fig. \[BBScheme\]). Field mapping has shown [@lange-general] that the field density is almost constant inside the magnet, with fringe fields that decrease exponentially outside of the magnet. In the analytical model, the true field was approximated by a constant field within the effective field boundaries, while edge effects were neglected. Under these assumptions all target coordinates were calculated by applying a circular-arc approximation [@ShneorPhD] of the track inside the field: the particle transport was divided into free motion (drift) in the $(y,z)$ plane and circular motion in the $(x,z)$ plane (see Fig. \[BBScheme\]), described by the Lorentz equation $$p_{y} = \mathrm{const} \>, \qquad p_{xz}= e R B_y \>.$$ To determine the momentum, the radius $R$ of the trajectory needs to be calculated first. This can be done by using the track information obtained from the detector package, combined with the geometrical properties of BigBite. A few reference points are needed, as shown in Fig. \[BBScheme\]; the point $\mathrm{Tg}$ represents the position of the particle at the target, and $D$ corresponds to the point where the particle hits the detector package. The point $B$ at which the particle exits the magnet is the intersection between the extrapolated particle track through the detector package and the effective exit face of the magnet. Similarly, the point $A$ lies at the intersection of the effective entrance face of the magnet and the particle track from the target. The point $O$ is the center of the circular trajectory. In order for all these points to correspond to a single particle track through the spectrometer, the conditions $$\begin{aligned}
\overline{A\mathrm{{Tg}}} \perp \overline{AO} \>, \qquad
\overline{OB} \perp \overline{BD} \>, \qquad
|\overline{AO}| = |\overline{BO}| = R \>, \nonumber\end{aligned}$$ must be satisfied. In the target coordinate system, this becomes $$\begin{aligned}
x_O &=& -\frac{d_{\mathrm{Tg}}}{(x_A-x_{\mathrm{Tg}})}(z_O-z_A) + x_A
= -\frac{z_D - z_B}{x_D - x_B}(z_O - z_B)
+ x_B\>, \label{analitic.eq1}\\
R^2 &=& (z_O - z_A)^2\left[1 + \left(\frac{d_{\mathrm{Tg}}}{x_A-x_{\mathrm{Tg}}} \right)^2 \right]
= (z_B - z_O)^2\left[1
+ \left(\frac{z_D - z_B}{x_D - x_B} \right)^2 \right]\,.\label{analitic.eq2}\end{aligned}$$ The coordinates $x_B$ and $z_B$ of $B$, and the coordinates $x_D$ and $z_D$ of $D$ can be directly calculated from the information obtained by the detector package. The position of the target $(x_{\mathrm{Tg}},z_{\mathrm{Tg}})$ is known. Since only thin targets are employed, $x_{\mathrm{Tg}}$ is set to zero. The coordinate $z_A$ of $A$ corresponds to the known distance $d_{\mathrm{Tg}}$ between the target center and the effective field boundary at the entrance to the magnet. By expressing $z_O$ from Eq. (\[analitic.eq1\]) and inserting it into Eq. (\[analitic.eq2\]), an equation for $x_A$ is obtained which has three complex solutions in general. The physically meaningful result for $x_A$ should be real and lie within the effective field boundaries. Two additional physical constraints are applied. The particle track should always represent the shortest possible arc of the circle (the arc between $A$ and $B$ in Fig. \[BBScheme\]). Moreover, the track should bend according to the polarity of the particle and orientation of the magnetic field. From $x_A$, the radius $R$ and the momentum $p_{xz}$ can be calculated. The particle flight path $l_{xz}$ in the $(x,z)$ plane can also be calculated by using the cosine formula for the angle $\beta = \measuredangle AOB$, $$\begin{aligned}
l_{xz} &=& \sqrt{x_A^2 + d_{\mathrm{Tg}}^2} + R\beta + \sqrt{(x_D-x_B)^2
+ (z_D - z_B)^2}\>, \nonumber \\
\cos \beta &=& \frac{(x_A-x_O)(x_B-x_O)
+ (z_A-z_O)(z_B-z_O)}{R^2} \> . \nonumber\end{aligned}$$ By using this information, all target coordinates can be expressed as $$\begin{aligned}
\phi_{\mathrm{Tg}} &=& \phi_{\mathrm{Det}} \>,\nonumber \\
\theta_{\mathrm{Tg}} &=& \arctan\left(\frac{x_A}{d_{\mathrm{Tg}}}\right) \>,\nonumber \\
y_{\mathrm{Tg}} &=& y_{\mathrm{Det}}
- l_{xz}\tan\phi_{\mathrm{Det}}\>, \nonumber \\
\delta_{\mathrm{Tg}} &=& \frac{p_{xz}}{p_\mathrm{c}}\frac{\sqrt{1
+\tan^2\phi_{\mathrm{Tg}} + \tan^2\theta_{\mathrm{Tg}}}}{\sqrt{1
+ \tan^2\theta_{\mathrm{Tg}}}} - 1\>,\nonumber \\
L &=&l_{xz}\sqrt{1 + \tan^2\phi_{\mathrm{Tg}}}\>, \nonumber \end{aligned}$$ where $p_\mathrm{c}$ is the central momentum and $L$ is the total flight-path of the particle.
With the analytical approximation, resolutions of a few percent can be achieved, but they deteriorate when moving towards the edges of the acceptance where the fringe fields begin to affect the optics. This is particularly true for $\phi_{\mathrm{Tg}}$. Figure \[MissingMassPlot\] (left) shows the reconstructed mass of the neutron from the process $\mathrm{{}^2H(e,e'p)n}$, obtained by using the analytical model. The relative resolution is $0.35\,\mathrm{\%}$.
![\[Left\] The reconstructed mass of the undetected neutron (missing mass) from the process $\mathrm{{}^2H(e,e'p)n}$ by using the analytical model and the matrix-formalism (SVD) approach. The width (sigma) of the peak determined with the analytical model is $3.3\,\mathrm{MeV}/c^2$ (corresponding to $0.35\,\mathrm{\%}$ relative resolution). The width of the peak reconstructed by the SVD method is $4\,\mathrm{MeV}/c^2$. \[Right\] The absolute calibration of $\delta_{\mathrm{Tg}}$ as a function of the particle momentum measured by BigBite. The relative resolution of $\delta_{\mathrm{Tg}}$ is better in the analytical model than in the matrix method, but the absolute momentum calibration is inferior to the matrix approach, except in the narrow region around $p \approx 0.55\,\mathrm{GeV}/c$. \[MissingMassPlot\]](MissingMassPlotNew2.eps "fig:"){width="49.00000%"} ![\[Left\] The reconstructed mass of the undetected neutron (missing mass) from the process $\mathrm{{}^2H(e,e'p)n}$ by using the analytical model and the matrix-formalism (SVD) approach. The width (sigma) of the peak determined with the analytical model is $3.3\,\mathrm{MeV}/c^2$ (corresponding to $0.35\,\mathrm{\%}$ relative resolution). The width of the peak reconstructed by the SVD method is $4\,\mathrm{MeV}/c^2$. \[Right\] The absolute calibration of $\delta_{\mathrm{Tg}}$ as a function of the particle momentum measured by BigBite. The relative resolution of $\delta_{\mathrm{Tg}}$ is better in the analytical model than in the matrix method, but the absolute momentum calibration is inferior to the matrix approach, except in the narrow region around $p \approx 0.55\,\mathrm{GeV}/c$. \[MissingMassPlot\]](figure_TargetDeltaAbsCalMomDepNew.eps "fig:"){width="49.00000%"}
The analytical method requires just a few geometry parameters, but these need to be known quite accurately. Had no survey been performed, the sizes of spectrometer components and the distances between them could be obtained, in principle, by calibrating with elastic events. However, the solution is not unique. Different combinations of parameters have been shown to yield almost identical results for the target variables, while only one combination is correct.
The matrix formalism
--------------------
In spite of its shortcomings, the analytical model is a good starting point. Due to its simplicity, it can be implemented and tested quickly, and lends itself well to online estimation of the experimental data. For the off-line analysis, a more sophisticated approach based on the transport matrix formalism is needed. In this approach, a prescription is obtained that transforms the detector variables directly to the target variables. Various parameterizations of this transformation are possible. We have adopted a polynomial expansion of the form [@bertozzi; @nilangaTN] $$\begin{aligned}
\Omega_{\mathrm{Tg}} = \sum_{i,j,k,l}
a_{ijkl}^{\Omega_{\mathrm{Tg}}} \,\,
x_\mathrm{Det}^i \,
\theta_{\mathrm{Det}}^j \,
y_{\mathrm{Det}}^k \,
\phi_{\mathrm{Det}}^l \>, \qquad
\Omega_{\mathrm{Tg}} \in \left\{
\delta_{\mathrm{Tg}}, \theta_{\mathrm{Tg}},
\phi_{\mathrm{Tg}}, y_{\mathrm{Tg}} \right\} \>. \label{eq1}\end{aligned}$$ Knowing the optics of a spectrometer is equivalent to determining the expansion coefficients $a_{ijkl}^{\Omega_{\mathrm{Tg}}}$ (the so-called optical “matrix”) and establishing the limitations of such a parameterization.
Ideally, one would like to obtain a single optical matrix with full reconstruction functionality for all particle species and momenta, with as few high-order terms as possible. In a large-acceptance spectrometer like BigBite, this represents a considerable challenge. In particular, one must clearly understand the contributions of the high-order elements. Uncontrolled inclusion of these terms typically causes oscillations of the reconstructed variables at the edges of the acceptance. In the following sections, we discuss the procedure of constructing the optical matrix in which special attention is devoted to checking the convergence of the method and estimating the robustness of the matrix elements.
### Decoupled description {#sub:direct}
The determination of the optical matrix starts with a low-order analysis in order to estimate the dominant matrix elements. As in the analytical model, the BigBite magnet is assumed to be an ideal dipole. This assumption decouples the in-plane and out-of-plane variables, resulting in the simplification that $\delta_{\mathrm{Tg}}$ and $\theta_{\mathrm{Tg}}$ depend only on $x_{\mathrm{Det}}$ and $\theta_{\mathrm{Det}}$, while $y_{\mathrm{Tg}}$ and $\phi_{\mathrm{Tg}}$ depend only on $y_{\mathrm{Det}}$ and $\phi_{\mathrm{Det}}$.
Since each target coordinate depends only on two detector coordinates, the matrix elements were estimated by examining two-dimensional histograms of target coordinates (as given by the HRS) versus BigBite detector variables, using various detector-variable cuts. Since BigBite in this approximation does not bend horizontally, only first-order polynomials were utilized to fit the data for $y_{\mathrm{Tg}}$ and $\phi_{\mathrm{Tg}}$, while expansions up to third-order were applied for $\delta_{\mathrm{Tg}}$ and $\theta_{\mathrm{Tg}}$: $$\begin{aligned}
\delta_{\mathrm{Tg}}(x, \theta) &=& \left[a_{0000}^{\delta_{\mathrm{Tg}}}
+a_{1000}^{\delta_{\mathrm{Tg}}}x +
a_{2000}^{\delta_{\mathrm{Tg}}}x^2 \right]
+\left[a_{0100}^{\delta_{\mathrm{Tg}}} +
a_{1100}^{\delta_{\mathrm{Tg}}}x +
a_{2100}^{\delta_{\mathrm{Tg}}}x^2 \right]\theta \nonumber\\
&+& \left[a_{0200}^{\delta_{\mathrm{Tg}}}
+ a_{1200}^{\delta_{\mathrm{Tg}}}x \right] \theta^2
+ \left[a_{0300}^{\delta_{\mathrm{Tg}}}
+ a_{1300}^{\delta_{\mathrm{Tg}}}x \right ]\theta^3\>, \nonumber \\
\theta_{\mathrm{Tg}}(x, \theta) &=& \left[a_{0000}^{\theta_{\mathrm{Tg}}}
+ a_{1000}^{\theta_{\mathrm{Tg}}}x + a_{2000}^{\theta_{\mathrm{Tg}}}x^2 \right]
+ \left[a_{0100}^{\theta_{\mathrm{Tg}}} + a_{1100}^{\theta_{\mathrm{Tg}}}x
+ a_{2100}^{\theta_{\mathrm{Tg}}}x^2 \right] \theta\>, \nonumber \\
\phi_{\mathrm{Tg}}(y,\phi) &=& a_{0000}^{\phi_{\mathrm{Tg}}}
+ a_{0001}^{\phi_{\mathrm{Tg}}}\phi\>, \nonumber \\
y_{\mathrm{Tg}}(y, \phi) &=& \left[a_{0001}^{y_{\mathrm{Tg}}}
+ a_{0011}^{y_{\mathrm{Tg}}}y \right]\phi + \left[ a_{0000}^{y_{\mathrm{Tg}}}
+ a_{0010}^{y_{\mathrm{Tg}}}y\right]\,. \nonumber \end{aligned}$$ The calculated matrix elements are shown in the second column of Table \[table1\]. The $a_{0001}^{\phi_{\mathrm{Tg}}}$ matrix element was set to $1$ since there is no in-plane bending. This approximation could not be used for further physics analysis because higher-order corrections are needed. However, the low-order terms are very robust and do not change much when more sophisticated models with higher-order terms are considered. The results obtained by using this method serve as a benchmark for more advanced methods, in particular as a check whether the matrix elements computed by automated numerical algorithms converge to reasonable values.
----------------------------------------------------- ------------- ---------- ----------
Matrix Decoupled N&M SVD
\[-6pt\] element description
$a_{0010}^{y_\mathrm{Tg}}\,[\mathrm{m/m}]$ $0.998$ $1.024$ $0.917$
$a_{0001}^{y_\mathrm{Tg}}\,[\mathrm{m/rad}]$ $-2.801$ $-2.839$ $-2.766$
$a_{0001}^{\phi_\mathrm{Tg}}\,[\mathrm{rad/rad}]$ $1.000$ $1.052$ $0.9517$
$a_{1000}^{\theta_\mathrm{Tg}}\,[\mathrm{rad/m}]$ $0.497$ $0.549$ $0.551$
$a_{0100}^{\theta_\mathrm{Tg}}\,[\mathrm{rad/rad}]$ $-0.491$ $-0.490$ $-0.484$
$a_{1000}^{\delta_\mathrm{Tg}}\,[\mathrm{1/m}]$ $-0.754$ $-0.716$ $-0.676$
$a_{0100}^{\delta_\mathrm{Tg}}\,[\mathrm{1/rad}]$ $2.811$ $2.881$ $2.802$
----------------------------------------------------- ------------- ---------- ----------
: The dominant matrix elements of the BigBite optics model (Eq. (\[eq1\])) determined by a decoupled description (subsubsection \[sub:direct\]), by simplex minimization (N&M), and by singular value decomposition (SVD, subsubsection \[sub:nmsvd\]). \[table1\]
### Higher order matrix formalism {#sub:nmsvd}
For the determination of the optics matrix a numerical method was developed in which matrix elements up to fourth order were retained. Their values were calculated by using a $\chi^2$-minimization scheme, wherein the target variables calculated by Eq. (\[eq1\]) were compared to the directly measured values, $$\begin{aligned}
\chi^2\left(a_i^{\Omega_{\mathrm{Tg}}}\right) =
\sqrt{\left( \Omega_{\mathrm{Tg}}^{\mathrm{Measured}} -
\Omega_{\mathrm{Tg}}^{\mathrm{Optics}}\left(x_{\mathrm{Det}},
y_{\mathrm{Det}}, \theta_{\mathrm{Det}}, \phi_{\mathrm{Det}};
a_i^{\Omega_{\mathrm{Tg}}}\right) \right)^2} \>, \qquad
i = 1,2,\,\ldots\>,M\,. \label{eq2}\end{aligned}$$ The use of $M$ matrix elements for each target variable means that a global minimum in $M$-dimensional space must be found. Numerically this is a very complex problem; two techniques were considered for its solution.
Our first choice was the downhill simplex method developed by Nelder and Mead [@nelder; @nrc]. The method tries to minimize a scalar non-linear function of $M$ parameters by using only function evaluations (no derivatives). It is widely used for non-linear unconstrained optimization, but it is inefficient and its convergence properties are poorly understood, especially in multi-dimensional minimizations. The method may stop in one of the local minima instead of the global minimum [@lagarias; @mckinnon], so an additional examination of the robustness of the method was required.
The set of functions $\Omega_{\mathrm{Tg}}$ is linear in the parameters $a_i^{\Omega_{\mathrm{Tg}}}$. Therefore, Eq. (\[eq2\]) can be written as $$\begin{aligned}
\chi^2 = \sqrt{ \, \left| A \, \vec{a} - \vec{b} \, \right |^2}\>,\label{eq3}\end{aligned}$$ where the $M$-dimensional vector $\vec{a}$ contains the matrix elements $a_i^{\Omega_{\mathrm{Tg}}}$, and the $N$-dimensional vector $\vec{b}$ contains the measured values of the target variable being considered. The elements of the $N\times M$ matrix $A$ are various products of detector variables ($x_\mathrm{Det}^i \theta_{\mathrm{Det}}^j
y_{\mathrm{Det}}^k \phi_{\mathrm{Det}}^l$) for each measured event. The system $A\,\vec{a} = \vec{b}$ in Eq. (\[eq3\]) is overdetermined ($N>M$), thus the vector $\vec{a}$ that minimizes the $\chi^2$ can be computed by singular value decomposition (SVD). It is given by $A = UWV^\mathrm{T}$, where $U$ is a $N\times M$ column-orthogonal matrix, $W$ is a $M\times M$ diagonal matrix with non-negative singular values $w_i$ on its diagonal, and $V$ is a $M\times M$ orthogonal matrix [@golub; @nrc]. The solution has the form $$\vec{a} = \sum_{i=1}^M\left(\frac{ \vec{U}_i\cdot\vec{b}}{w_i} \right)
\vec{V}_i \>.$$
The SVD was adopted as an alternative to simplex minimization since it produces the best solution in the least-square sense, obviating the need for robustness tests. Another great advantage of SVD is that it can not fail; the method always returns a solution, but its meaningfulness depends on the quality of the input data. The most important leading-order matrix elements computed by using both techniques are compared in Table \[table1\].
Calibration results
===================
Vertex position
---------------
The matrix for the vertex position variable $y_{\mathrm{Tg}}$ was obtained by analyzing the protons from quasi-elastic scattering of electrons on the multi-foil carbon target. The positions of the foils were measured by a geodetic survey to sub-millimeter accuracy, allowing for a very precise calibration of $y_{\mathrm{Tg}}$. The vertex information from the HRS was used to locate the foil in which the particle detected by BigBite originated. This allowed us to directly correlate the detector variables for each coincidence event to the interaction vertex. When Eq. (\[eq1\]) is written for $y_{\mathrm{Tg}}$, a linear equation for each event can be formed: $$\begin{aligned}
{y_\mathrm{Tg}}_{(n)}^{\mathrm{Measured}} =
{y_{\mathrm{Tg}}}_{(n)}^{\mathrm{Optics}}
&:=& a_{0000}^{y} + a_{0001}^{y}\phi_{(n)} +
a_{0002}^{y}\phi_{(n)}^2 + a_{0003}^{y}\phi_{(n)}^3 +
\cdots\ \nonumber\\
&+& a_{0010}^{y}y_{(n)} + a_{0020}^{y}y_{(n)}^2 +
a_{0030}^{y}y_{(n)}^3 + a_{0040}^{y}y_{(n)}^4 +
\cdots\ \nonumber \\
&+& a_{0100}^{y}\theta_{(n)} + a_{0200}^{y}\theta_{(n)}^2+
a_{0300}^{y}\theta_{(n)}^3 + a_{0400}^{y}\theta_{(n)}^4 +
\cdots \nonumber \\
&+& a_{1000}^{y}x_{(n)} + a_{2000}^{y}x_{(n)}^2 +
a_{3000}^{y}x_{(n)}^3 + a_{4000}^{y}x_{(n)}^4 +
\cdots\ \nonumber \\
&+& a_{1111}^{y}x_{(n)}\theta_{(n)} y_{(n)} \phi_{(n)}\>,
\label{TgYAnsatz}\end{aligned}$$ where $n = 1,2,\ldots,N$, and $N$ is the number of coincidence events used in the analysis. The overdetermined set of Eqs. (\[TgYAnsatz\]) represents a direct comparison of the reconstructed vertex position $y_{\mathrm{Tg}}^{\mathrm{Optics}}$ to the measured value $y_{\mathrm{Tg}}^{\mathrm{Measured}}$. Initially a consistent polynomial expansion to fourth degree ($i+j+k+l \leq 4$) was considered, which depends on $70$ matrix elements $a_{ijkl}^{y}$. Using this ansatz in Eq. (\[eq2\]) defines a $\chi^2$-minimization function, which serves as an input to the simplex method. To be certain that the minimization did not converge to one of the local minima, the robustness of this method was examined by checking the convergence of the minimization algorithm for a large number of randomly chosen initial sets of parameters (see Fig. \[TargetYConvergence\]).
![\[Left\] Robustness checks of the simplex minimization method for select matrix elements $a_{ijkl}^{\Omega_{\mathrm{Tg}}}$. The analysis was done for a large set of randomly chosen initial conditions for each target coordinate. The fact that the vast majority of the initial conditions converge to a single value is an indication of the robustness of the method. \[Right\] The values of the $\chi^2$-function before and after simplex minimization for all four target coordinates. The method converges to a single $\chi^2$ value for a wide range of initial conditions (note the log scales). The solution with the smallest $\chi^2$ represents the result used in the optics-matrix. \[TargetYConvergence\] ](figure_Simplex_JoinedMatrixElementPlot.eps "fig:"){width="49.00000%"} ![\[Left\] Robustness checks of the simplex minimization method for select matrix elements $a_{ijkl}^{\Omega_{\mathrm{Tg}}}$. The analysis was done for a large set of randomly chosen initial conditions for each target coordinate. The fact that the vast majority of the initial conditions converge to a single value is an indication of the robustness of the method. \[Right\] The values of the $\chi^2$-function before and after simplex minimization for all four target coordinates. The method converges to a single $\chi^2$ value for a wide range of initial conditions (note the log scales). The solution with the smallest $\chi^2$ represents the result used in the optics-matrix. \[TargetYConvergence\] ](figure_Chi2Values.eps "fig:"){width="49.00000%"}
The results were considered to be stable if the $\chi^2$ defined by Eq. (\[eq2\]) converged to the same value for the majority of initial conditions. Small variations in $\chi^2$ were allowed: they are caused by small matrix elements which are irrelevant for $y_{\mathrm{Tg}}$, but have been set to non-zero values in order to additionally minimize $\chi^2$ in a particular minimization process. These matrix elements could be easily identified and excluded during the robustness checks because they are unstable and converge to a different value in each minimization. Ultimately only $25$ matrix elements that had the smallest fluctuations were kept for the $y_{\mathrm{Tg}}$ matrix.
The SVD method was used next. To compute the matrix elements for $y_{\mathrm{Tg}}$, the linear set of Eqs. (\[TgYAnsatz\]) first needs to be rewritten in the form $A\,\vec{a} = \vec{b}$ used in Eq. (\[eq3\]): $$\begin{aligned}
\left(
\begin{array}{cccc}
1 & \phi_{(1)} & \cdots & x_{(1)}\theta_{(1)}y_{(1)}\phi_{(1)} \\
1 & \phi_{(2)} & \cdots & x_{(2)}\theta_{(2)}y_{(2)}\phi_{(2)} \\
1 & \phi_{(3)} & \cdots & x_{(3)}\theta_{(3)}y_{(3)}\phi_{(3)} \\
\vdots & \vdots & \ddots & \vdots \\
1 & \phi_{(N-2)} & \cdots & x_{(N-2)}\theta_{(N-2)}y_{(N-2)}\phi_{(N-2)} \\
1 & \phi_{(N-1)} & \cdots & x_{(N-1)}\theta_{(N-1)}y_{(N-1)}\phi_{(N-1)} \\
1 & \phi_{(N)} & \cdots & x_{(N)}\theta_{(N)}y_{(N)}\phi_{(N)} \\
\end{array}
\right)
\left(
\begin{array}{c}
a_{0000} \\
a_{0001} \\
\vdots \\
a_{1111}
\end{array}
\right) =
\left(
\begin{array}{c}
{y_{\mathrm{Tg}}}_{(1)} \\
{y_{\mathrm{Tg}}}_{(2)} \\
{y_{\mathrm{Tg}}}_{(3)} \\
\vdots \\
{y_{\mathrm{Tg}}}_{(N-2)} \\
{y_{\mathrm{Tg}}}_{(N-1)} \\
{y_{\mathrm{Tg}}}_{(N)}
\end{array}
\right)\>, \nonumber\end{aligned}$$ where $\vec{a}$ contains $M$ unknown matrix elements $a_{ijkl}^{y}$ to be determined by the SVD, $\vec{b}$ contains $N$ measured values of $y_{\mathrm{Tg}}$, and $A$ is filled with the products of detector variables accompanying the matrix elements in the polynomial expansion of Eq. (\[TgYAnsatz\]) for each event.
The SVD analysis also began with $70$ matrix elements, but was not applied to one combined data set as in the simplex method in order to extract the most relevant ones. Rather, it was used on each set of data separately. From the comparison of the matrix elements obtained with different calibration data sets, only the elements fluctuating by less than $100\,\%$ were selected. Although this choice appears to be arbitrary, the results do not change much by modifying this criterion, for example, by including elements with as much as $\pm 1000\, \mathrm{\%}$ fluctuation. The final set of matrix elements contained only $37$ of the best entries. With these elements, the entire analysis was repeated in order to calculate their final values. The most relevant elements are listed in Table \[table1\]. The result of the calibration of $y_\mathrm{Tg}$ is shown in Fig. \[TgYResult\].
![The reconstructed vertex position (reaction point) for the multi-foil carbon target and the empty cell of the production target, by using the SVD technique. The vertical dashed lines indicate the actual positions of the carbon foils and the empty-cell glass windows. The small shoulder to the right of the reconstructed empty-cell entry window is due to the jet of $^4\mathrm{He}$ gas used to cool the window at the beam impact point. \[TgYResult\]](figure_Carbon_Target_VeryNew.eps){width="60.00000%"}
Angular coordinates
-------------------
For the calibration of the angular variables $\theta_{\mathrm{Tg}}$ and $\phi_{\mathrm{Tg}}$, a set of quasi-elastic data on carbon and deuterium targets taken with the sieve-slit collimator was analyzed. The particles that pass through different holes can be well separated and localized at the detector plane.
By knowing the detector coordinates and the accurate position of the corresponding hole in the sieve, the target variables can be calculated. From the reaction point at the target (see Fig. \[BBSieveSlitDiagram\]), $\theta_{\mathrm{Tg}}$ and $\phi_{\mathrm{Tg}}$ can be calculated: $$\begin{aligned}
\tan \phi_{\mathrm{Tg}} = \frac{y_{\mathrm{Sieve}} -
y_{\mathrm{Tg}}}{z_{\mathrm{Sieve}} - z_{\mathrm{Tg}}} \>,
\qquad\tan \theta_{\mathrm{Tg}} =
\frac{x_{\mathrm{Sieve}} -
x_{\mathrm{Tg}}}{z_{\mathrm{Sieve}} - z_{\mathrm{Tg}}} \>. \nonumber\end{aligned}$$
By using the values of the target variables, a set of linear equations has been written for all measured events, and matrix elements determined by using both numerical approaches. In the simplex method, $30$ matrix elements for $\theta_{\mathrm{Tg}}$ and $68$ elements for $\phi_{\mathrm{Tg}}$ were retained. Robustness checks for both angular variables were repeated to ensure that the global minimum had been reached.
![Position of the sieve-slit collimator relative to the target. The vector of the particle track through a particular hole in the sieve is the difference of the position vector at the hole and the reaction-point vector. BigBite is positioned at $-75^\circ$ with respect to the beam direction. Other settings are listed in Table \[table\_kinematics\]. \[BBSieveSlitDiagram\]](ScatteringDiagram3.eps){width="65.00000%"}
The SVD analysis also started with $70$ matrix elements, which were ultimately reduced to $37$ for $\theta_{\mathrm{Tg}}$ and $51$ for $\phi_{\mathrm{Tg}}$, again taking into account only those elements that fluctuated by less than $100\,\mathrm{\%}$. Figure \[BBSieve\] (right) shows the reconstructed sieve pattern. The majority of the holes are reconstructed, except those obscured by parts of the experimental apparatus due to specific geometric constraints during the experiment. In order to demonstrate the effect of gradually excluding redundant matrix elements, Fig. \[TgPhME\] shows the reconstructed top row of the sieve-slit collimator holes when the elements with up to $\pm 1000\,\%$, $\pm 100\,\%$, and $\pm 20\,\%$ fluctuations are retained. There is virtually no difference in the reconstructed pattern when all elements exceeding the $\pm 100\,\%$ fluctuations are dropped, while errors start to appear when those fluctuating by less than $\pm 100\,\%$ are dropped.
![The reconstructed positions of the holes in the top row of the sieve-slit collimator, computed from $\phi_\mathrm{Tg}$. The quality of the reconstruction depends on the number of included matrix elements. There is almost no difference when the elements fluctuating by up to $\pm 1000\,\%$ are retained ($70$ elements, dashed lines) or only those that fluctuate by up to $\pm 100\,\%$ ($51$ elements, full line). The quality deteriorates if too many elements are dropped (i.e. keeping $18$ elements fluctuating by less than $\pm 20\,\%$, dotted lines). \[TgPhME\]](TgPhMEComparison.eps){width="60.00000%"}
The quality of the sieve-pattern reconstruction was examined by comparing the centers of the reconstructed holes with their true positions. Figure \[ThPhCenterPositionError\] shows that, with the exception of a few holes near the acceptance edges, these deviations are smaller than $2\,\mathrm{mm}$ in the vertical, and smaller than $4\,\mathrm{mm}$ in the horizontal direction. This is much less than the hole diameter, which is $19.1\,\mathrm{mm}$.
![Distribution of vertical ($\Delta X$) and horizontal ($\Delta Y$) deviation of the center of each reconstructed sieve-slit hole from its true position. Observed deviations are much smaller than the diameter of a hole, which is $19.1\,\mathrm{mm}$. The horizontal and vertical histograms (top and right axis labels, respectively) represent the distributions in the horizontal and vertical directions. \[ThPhCenterPositionError\]](SieveErrorPlot.eps){width="48.00000%"}
Once the sieve pattern was reconstructed, an absolute calibration had to be performed to correct for any BigBite misalignment and mispointing. For that purpose hydrogen and deuterium elastic data were used. By comparing the direction of the momentum transfer vector from the HRS to the calculated values of $\theta_{\mathrm{Tg}}$ and $\phi_{\mathrm{Tg}}$, the zero-order matrix elements could be properly determined and the offsets corrected. In addition, the precise distance between the target and the sieve-slit collimator was obtained, which we were not able to measure precisely due to physical obstacles between the target and BigBite. From this analysis, the sieve slit was determined to be positioned $1.13\,\mathrm{m}$ away from the target.
Momentum
--------
The matrix elements for the $\delta_{\mathrm{Tg}}$ variable were obtained by using data from elastic scattering of electrons on hydrogen and deuterium for which the particle momentum in BigBite should be exactly the same as the momentum transfer $\vec{q}$ given by the HRS. We assumed that $\delta_{\mathrm{Tg}}$ depends only on $x_{\mathrm{Det}}$ and $\theta_{\mathrm{Det}}$, while the dependencies involving $y_{\mathrm{Det}}$ and $\phi_{\mathrm{Det}}$ were neglected. Furthermore, the use of in-plane coordinates in the analysis for $\delta_{\mathrm{Tg}}$ could result in an erroneous matrix due to the strong $\phi_{\mathrm{Tg}}$ dependence inherent to elastic scattering (events strongly concentrated at one edge of the acceptance). Considering only $x_{\mathrm{Det}}$ and $\theta_{\mathrm{Det}}$ matrix elements, $\delta_{\mathrm{Tg}}$ can be expressed as $$\begin{aligned}
\delta_{\mathrm{Tg}} = \frac{q_{\mathrm{HRS}} - \Delta_{\mathrm{Loss}}}
{p_\mathrm{c}}-1 =
a_{0000}^\delta + a_{1000}^\delta x_{\mathrm{Det}} +
a_{0100}^\delta\theta_{\mathrm{Det}}+\cdots \,.
\label{momentumeq}\end{aligned}$$ In order to obtain the optics matrix applicable to all types of particles, energy losses $\Delta_{\mathrm{Loss}}$ for particle transport through the target enclosure and materials within the BigBite spectrometer were studied carefully. The energy losses were estimated by the Bethe-Bloch formula [@leo], but since the losses were significant, the formula had to be integrated over the complete particle track for each particle type and each initial momentum. The two largest contributions to the total momentum loss came from the target cell walls and from the air between the target and the detectors. (The latter losses could be alleviated by using a helium bag between the target and the detectors, but its benefits were considered to be smaller than the technical problems involved.) The resulting corrections that were taken into account in Eq. (\[momentumeq\]) are shown in Fig. \[figure\_MomentumLosses\] (left).
![\[Left\] Momentum losses of protons and deuterons inside the target and the total momentum losses up to the MWDCs. \[Right\] Quality of reconstructed momentum for elastic protons and deuterons. If energy losses are not taken into account, two peaks are visible (center and right histograms summed to the full curve). With proper inclusion of energy losses both peaks merge into one (left histogram), resulting in better momentum resolution. \[figure\_MomentumLosses\]](MomentumLossesPlot.eps "fig:"){width="49.00000%"} ![\[Left\] Momentum losses of protons and deuterons inside the target and the total momentum losses up to the MWDCs. \[Right\] Quality of reconstructed momentum for elastic protons and deuterons. If energy losses are not taken into account, two peaks are visible (center and right histograms summed to the full curve). With proper inclusion of energy losses both peaks merge into one (left histogram), resulting in better momentum resolution. \[figure\_MomentumLosses\]](figure_MomentumDifference.eps "fig:"){width="49.00000%"}
The elastic data available for calibration (momentum range approximately $0.45\,\mathrm{GeV}/c$ to $0.7\,\mathrm{GeV}/c$) covered only about half of the BigBite momentum acceptance. To calibrate the low-momentum region from $0.2\,\mathrm{GeV}/c$ to $0.45\,\mathrm{GeV}/c$, we used protons from quasi-elastic scattering on ${}^3\mathrm{He}$ by exploiting the information from the scintillator dE- and E-planes; the deposited particle energy in each plane was directly mapped to the particle momentum, based on known properties of the scintillator material. The punch-through point, corresponding to the particular momentum at which the particle has just enough energy to penetrate through the scintillators, served as a reference.
Beside the proton punch-through point, two other points with exactly known energy deposits in the dE- and E-planes were identified, as illustrated in Fig. \[figure\_EdE\]. With the additional information from these points, a complete momentum calibration was possible. To compute the $\delta_{\mathrm{Tg}}$ matrix elements, both numerical approaches described above were used. Since the available data were rather sparse, the search for the most stable matrix elements was not performed and a complete expansion to fifth order was considered in both techniques. Since only a two-variable dependency was assumed, a complete description was achieved by using only $21$ matrix elements.
The comparison of the most relevant matrix elements obtained from both numerical approaches is again shown in Table \[table1\]. Figure \[figure\_EdE\] (right) shows that the $\delta_{\mathrm{Tg}}$ matrix is well under control. The reconstructed momentum agrees with the simulation of energy losses inside the scintillation planes for the complete momentum acceptance of BigBite, for both protons and deuterons. Figure \[MissingMassPlot\] shows the missing-mass peak for the $\mathrm{{}^2H( e, e'p)n}$ process. The resolution of the reconstructed neutron mass is approximately $4\,\mathrm{MeV}/c^2$.
![\[Left\] The energy losses in the thin ($3\,\mathrm{mm}$) scintillator dE-plane versus the energy losses in the thicker ($3\,\mathrm{cm}$) E-plane. The punch-through points, at which the protons and deuterons have just enough energy to penetrate both scintillation planes, are clearly visible. The black boxes show sections of events with precisely determined momenta that were used in the $\delta_{\mathrm{Tg}}$ calibration. \[Right\] Particle momentum as a function of energy losses in the E-plane for ${}^3\mathrm{He}$ data. The deuterons can be clearly distinguished from the protons. The measurements agree well with the simulation (dot-dashed line).\[figure\_EdE\]](figure_TargetDeltaCalibrationResultsWithSim_dE_vs_E_No4.eps "fig:"){width="49.00000%"} ![\[Left\] The energy losses in the thin ($3\,\mathrm{mm}$) scintillator dE-plane versus the energy losses in the thicker ($3\,\mathrm{cm}$) E-plane. The punch-through points, at which the protons and deuterons have just enough energy to penetrate both scintillation planes, are clearly visible. The black boxes show sections of events with precisely determined momenta that were used in the $\delta_{\mathrm{Tg}}$ calibration. \[Right\] Particle momentum as a function of energy losses in the E-plane for ${}^3\mathrm{He}$ data. The deuterons can be clearly distinguished from the protons. The measurements agree well with the simulation (dot-dashed line).\[figure\_EdE\]](figure_TargetDeltaCalibrationResultsWithSim_E_vs_p_No4.eps "fig:"){width="49.00000%"}
Resolution
----------
The quality of the BigBite optics was also studied. The resolution of the vertex position was estimated from the difference between the reconstructed $y_{\mathrm{Tg}}$ and the true position at the target by taking the width (sigma) of the obtained distribution. This part of the analysis was done by using $2$-pass ($2.425\,\mathrm{GeV}$ beam) quasi-elastic carbon data. The extracted values for the resolution of $y_\mathrm{Tg}$ in different momentum bins can be parameterized as $$\sigma_{y_\mathrm{Tg}} \approx 0.01\left(1 + \frac{0.02}{p^{4}} \right)\>,$$ where the particle momentum is in $\mathrm{GeV}/c$ and the result is in meters. It is best at the upper limit of the accepted momentum range (about $p=0.7\,\mathrm{GeV}/c$) where it amounts to $\sigma_{y_{\mathrm{Tg}}}=1.1\,\mathrm{cm}$. The deterioration of the resolution at lower momenta is due to multiple scattering [@leo] in the air between the scattering chamber and the MWDCs.
![The absolute resolution of $\phi_{\mathrm{Tg}}$ and the relative momentum resolution as functions of the momentum measured by BigBite, obtained by the SVD method. Irreducible multiple-scattering contributions, mostly due to the air between the scattering chamber and MWDCs, are shown by full and dashed lines for deuterons and protons, respectively. \[ResolutionPlots\]](figure_TargetPhMomentumDependence.eps "fig:"){width="49.00000%"} ![The absolute resolution of $\phi_{\mathrm{Tg}}$ and the relative momentum resolution as functions of the momentum measured by BigBite, obtained by the SVD method. Irreducible multiple-scattering contributions, mostly due to the air between the scattering chamber and MWDCs, are shown by full and dashed lines for deuterons and protons, respectively. \[ResolutionPlots\]](figure_TargetDeltaMomentumDependence.eps "fig:"){width="49.00000%"}
The resolutions of $\theta_{\mathrm{Tg}}$ and $\phi_{\mathrm{Tg}}$ were estimated by comparing them to the corresponding angles as determined from the momentum transfer $\vec{q}$ in elastic scattering on hydrogen and deuterium. The direction of $\vec{q}$ is given by the electron kinematics and determined by the HRS spectrometer. The corresponding HRS resolutions have been studied in [@GeJinPhD]. Based on these values, the resolution of the reconstructed $\vec{q}$ was estimated to be $6\,\mathrm{mrad}$ and $0.3\,\mathrm{mrad}$ for the vertical and horizontal angles, respectively. These contributions were subtracted in quadrature from the calculated peak widths, yielding the final resolutions attributable to BigBite. The result for $\phi_{\mathrm{Tg}}$ is shown in Fig. \[ResolutionPlots\] (left). The strong momentum dependence of the resolution is again caused by multiple scattering in the target and the spectrometer. Different resolutions for deuterons and protons occur because the peak broadening in multiple scattering strongly depends on the particle mass (at a given momentum). As before, the biggest contributions come from the air. In a typical kinematics of the E05-102 experiment, the resolutions of $\phi_{\mathrm{Tg}}$ and $\theta_{\mathrm{Tg}}$ are $\sigma_{\phi_\mathrm{Tg}} \approx 7\,\mathrm{mrad}$ and $\sigma_{\theta_\mathrm{Tg}} \approx 13\,\mathrm{mrad}$ for $0.55\,\mathrm{GeV}/c$ protons, and approximately $\sigma_{\phi_\mathrm{Tg}} \approx 11\,\mathrm{mrad}$ and $\sigma_{\theta_\mathrm{Tg}} \approx 13\,\mathrm{mrad}$ for $0.6\,\mathrm{GeV}/c$ deuterons. (Due to multiple scattering, these resolutions are clearly much larger than the intrinsic MWDC resolutions mentioned in Section \[sec:BB\].)
The resolution of $\delta_{\mathrm{Tg}} = (p-p_\mathrm{c})/p_\mathrm{c}$ was also determined from elastic data by comparing the magnitude of $\vec{q}$ to the momentum reconstructed by BigBite. The analysis was done separately for the hydrogen and deuterium data sets. Figure \[ResolutionPlots\] (right) shows the relative momentum resolution $\sigma_{p}/p$ as a function of momentum. The relative momentum resolution is approximately $1.6\,\%$ for $0.55\,\mathrm{GeV}/c$ protons, and $2\,\%$ for $0.6\,\mathrm{GeV}/c$ deuterons. Figure \[MissingMassPlot\] (right) shows the absolute resolution of $\delta_{\mathrm{Tg}}$.
Summary
=======
We have described the optics calibration of the BigBite spectrometer that was used to detect hadrons in the E05-102 experiment at Jefferson Lab. While the methods have been developed and applied to one spectrometer under very specific physical conditions, the same procedures can be applied to any spectrometer with a similar magnetic configuration and acceptance.
Two different approaches were considered: an analytical model that treats BigBite as an ideal dipole and a matrix formalism. The former approach results only in modest resolutions; still, resolutions of a few percent can be achieved by a suitable choice of parameters. The latter approach allows for a more precise calibration. Two numerical methods were used to determine the matrix elements, but the one based on singular value decomposition delivered better and more reliable results.
The vertex resolution for protons was found to be $1.2\,\mathrm{cm}$ at $0.55\,\mathrm{GeV}/c$ along the whole $40\,\mathrm{cm}$ target length. The resolution deteriorates significantly at lower momenta due to multiple scattering in the target, air, and detector material. The corresponding angular resolution is $7\,\mathrm{mrad}$ for the in-plane angle $\phi_{\mathrm{Tg}}$ and $13\,\mathrm{mrad}$ for the out-of-plane angle $\theta_{\mathrm{Tg}}$. The angular resolution worsens at lower momenta due to multiple scattering, with the effect more pronounced for deuterons. The relative momentum resolution for $0.55\,\mathrm{GeV}/c$ protons (best case) has been estimated to be $1.6\,\%$.
For $0.6\,\mathrm{GeV}/c$ deuterons (best case), we obtained the resolutions of $2\,\%$ (momentum), $11\,\mathrm{mrad}$ ($\phi_{\mathrm{Tg}}$), and $13\,\mathrm{mrad}$ ($\theta_{\mathrm{Tg}}$).
Acknowledgments
===============
The Authors wish to thank Bogdan Wojtsekhowski for fruitful discussions and valuable advice on the manuscript.
This work was supported in part by the U.S. Department of Energy and the U. S. National Science Foundation. It is supported by DOE contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, operates the Thomas Jefferson National Accelerator Facility. The UK collaborators acknowledge the funding by the UK Engineering and Physical Sciences Council and the UK Science and Technology Facilities Council.
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abstract: 'We consider a discrete-time Markov decision process with Borel state and action spaces. The performance criterion is to maximize a total expected [utility determined by unbounded return function.]{} It is shown the existence of optimal strategies under general conditions allowing the reward function to be unbounded both from above and below and the action sets available at each step to the decision maker to be not necessarily compact. To deal with unbounded reward functions, a new characterization for the weak convergence of probability measures is derived. Our results are illustrated by examples.'
author:
- |
F. Dufour\
Institut Polytechnique de Bordeaux\
INRIA Bordeaux Sud Ouest, Team: CQFD\
IMB, Institut de Mathématiques de Bordeaux, Université de Bordeaux, France\
e-mail: francois.dufour@math.u-bordeaux.fr
- |
A. Genadot\
IMB, Institut de Mathématiques de Bordeaux, Université de Bordeaux, France\
INRIA Bordeaux Sud Ouest, Team: CQFD\
e-mail: alexandre.genadot@math.u-bordeaux.fr
title: On the expected total reward with unbounded returns for Markov decision processes
---
\[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Remark]{} \[theorem\][Conjecture]{} \[theorem\][Assumption]{}
**Keywords:** Markov decision processes.
**AMS 2010 Subject Classification:** 90C40, 60J05.
Introduction {#sec-1}
============
In this paper, our objective is to provide sufficient conditions for the existence of optimal strategies in dynamic programming decision models under the expected total reward criterion. The model under consideration is rather general since the reward function may be unbounded both from above and below and the action sets available at each step to the decision maker may not be necessarily compact.
Typically, here is an example of model we are able to handle. Roughly speaking, the state space is given by $\mathbf{X}=[0,1]$, the action space is [$\mathbf{A}=\{1,2,\ldots\}$]{} and the reward function $r$ satisfies $$r(x,a)=\begin{cases}
0 & \text{ if $(x,a) \in \{0\}\times {\mathbf{A}}$} \\
\frac{1}{a^{2}} \frac{1}{\sqrt{x}}I_{]0,1/2]}(x)-a \frac{1}{\sqrt{1-x}}I_{]1/2,1[}(x) & \text{ if $ (x,a) \in ]0,1[\times {\mathbf{A}} $} \\
-\infty, & \text{ if $(x,a) \in \{1\} \times {\mathbf{A}}$.}
\end{cases}$$ Observe that this reward function is not upper semicontinuous nor bounded from above. Moreover, it can actually takes the $-\infty$ value. This example will be described in details in Section \[sec-examples\]. As far as we know, such example does not satisfy the standard conditions of the literature see, for instance, the references [@balder89; @balder92; @nowak88; @schal75; @schal79].
[Several approaches have been proposed in the literature to study the existence of optimal strategies for discrete-time Markov decision processes under the total expected utility criterion with unbounded reward function. A possible method is based on the analysis of the so-called dynamic programming equation, see for example [@bertsekas78; @hinderer70; @schal75] for general results in this direction. In addition, the total reward criterion can be seen as a special case of the so-called expected utility criteria as studied for example in [@kertz79].]{} In our paper, we use a different approach following the line of the works developed in [@balder89; @balder92; @nowak88; @schal75; @schal79] that, roughly speaking, consists in finding a suitable topology on the set of strategic probability measures [to ensure that this set is compact and the expected reward functional is semicontinuous.]{} This technique is quite classical in the literature. In the context of compact action sets, it has been studied by many authors, see *e.g.* [@balder89; @nowak88; @schal75; @schal79]. In [@balder92], this method has been generalized to the case of possible non-compact action sets by introducing the so-called strong coercivity condition on the reward function. A key condition used in all the aforementioned works [@balder89; @balder92; @nowak88; @schal75; @schal79] is to consider the reward function being bounded from above (or equivalently, bounded from below for a cost function). However, it has been emphasized in [@jaskiewicz14; @jaskiewicz11; @matkowski11] (and the references therein) that many real models do not satisfy such property, in particular in economy where the utility function may be logarithmic and may take the value $-\infty$ for some states (see example 2 in [@jaskiewicz11]). Recently, new sets of conditions have been studied in [@jaskiewicz14; @jaskiewicz11; @matkowski11] to deal with Markov decision processes [and in [@jaskiewicz11b] for min-max games]{} generalizing the so-called weighted norm approach due to Wessels [@wessels77]. Roughly speaking the authors provide existence results based on the analysis of the Bellman optimality equation. In this paper, *red*[we used another approach]{} and follow the line developed in [@balder92] to show the existence of optimal strategies in the general framework of unbounded reward function and so generalizing the results of [@balder92]. Moreover, it will be shown that the assumptions proposed in [@jaskiewicz11] satisfy our new set of hypotheses.
In order to deal with unbounded reward functions, we have derived a new characterization for the weak convergence of probability measures. We show that the determining class of test functions for weak convergence, usually the set of continuous and bounded functionals, can be relaxed to the set of not everywhere bounded and upper semicontinuous functions. Such a relaxation on the boundedness condition have also been considered in [@zapala08] but in keeping a continuity hypotheses.
The rest of the paper is organized as follows. In Section \[sec-model\], we define the control model. Section \[sec-portmanteau\] studies weak convergence of probability measures with not everywhere bounded and semicontinuous test functions. In Section \[sec-results\] we state our assumptions and establish the existence of optimal policies. Finally, Section \[sec-examples\] is dedicated to the presentation of examples illustrating our results.
Description of the control problem {#sec-model}
==================================
Basic notations
---------------
First of all, we introduce the following notations and terminology.
We write $\NN$ for the set of integers, that is, $\NN=\{0,1,2,\ldots\}$, $\NN^*=\NN\setminus\{0\}$ for the set of positive integers, $\RR$ for the set of real numbers, ${\overline{\RR}}=\RR\cup\{-\infty,+\infty\}$ for the set of extended real numbers.
Given a topological space $Z$, we say that a function $f:Z\rightarrow{\overline{\RR}}$ is upper semicontinuous (respectively, upper semicompact) on $Z$ if $\{z\in Z: f(z)\geq \beta\}$ is a closed (respectively, relatively compact) set of $Z$ for every $\beta\in\RR$. By ${\overline{\boldsymbol{\mathcal{C}}}}(Z)$ we will denote the family of real-valued functions on $Z$ which are bounded and continuous.
Recall from [@balder92] that, for two metric spaces $Y$ and $Z$ and a subset $B$ of $Y\times Z$, a function $u : B \to [-\infty,+\infty)$ is said to be strongly coercive on $B$ if for every sequence $\{(y_k,z_k)\}_{k\in \NN}$ in $B$, such that $y_k\to y^{*}$ for some $y^{*}\in Y$ and $\beta =\limsup_{k} u(y_k,z_k)>-\infty$, there exists a subsequence $\{z_{k_j}\}$ of $\{z_k\}$ and $z^*\in Z$ such that $z_{k_j}\to z^*$, $(y^{*},z^{*})\in B$ and $u(y^{*},z^*)\geq\beta$. As a direct consequence of the definition, strongly coercive functions are upper semicontinuous. However, it is not hard to exhibit upper semicontinuous fonctions which are not strongly coercives. This definition of strongly coercivity is consistent with the notion of coercivity in the literature. Indeed, if $u : Y\times Z\to [-\infty,+\infty)$ is strongly coercive with $(Z,\|\cdot\|)$ a normed vector space, then, for any $y\in Y$, we have $\ds \lim_{\|z\|\to\infty} u(y,z)=-\infty$.
We use the symbol $f^{+}$ (respectively $f^{-}$) to denote the positive part (respectively, negative part) of a function $f:Z\rightarrow{\overline{\RR}}$.
The Borel $\sigma$-algebra of $Z$ is denoted by $\mathfrak{B}(Z)$. The set of probability measures on $(Z,\mathfrak{B}(Z))$ is denoted by $\boldsymbol{\mathcal{P}}(Z)$. In this work, $\boldsymbol{\mathcal{P}}(Z)$ will be considered as a topological space equipped with the weak topology. If $\mu$ is a measure on $Z_1\times Z_2$ then $\mu_{| Z_{1}}$ denotes the marginal of the measure $\mu$ on $Z_1$.
A Borel subset of a complete and separable metric space is called Borel space. If $Z_1$ and $Z_2$ are two Borel spaces and $Q$ is a stochastic kernel on $Z_2$ given $Z_1$, then, for a function $v:Z_2\rightarrow{\overline{\RR}}$, we define $Qv:Z_1\rightarrow{\overline{\RR}}$ as $$Qv(z_1):=\int_{Z_2}v^{+}(z_2)Q(dz_2|z_1)-\int_{Z_2}v^{-}(z_2)Q(dz_2|z_1),$$ provided that [one of the two terms in the right member of the previous equation is finite.]{} For a measure $\mu$ on $Z_1$, we denote by $\mu Q$ the measure $\ds \int_{Z_1} Q(\cdot|z_1)\mu(dz_1)$ on $Z_2$.
The control model.
------------------
Let us consider the standard discrete-time non-stationary control model: $$\begin{aligned}
\big(\{\mathbf{X}_{t}\}_{t\in \NN^{*}},\{\mathbf{A}_{t}\}_{t\in \NN^{*}},\{\Psi_{t}\}_{t\in \NN^{*}},\{Q_{t}\}_{t\in \NN^{*}},\{r_{t}\}_{t\in \NN^{*}},\nu\big) \label{5tuple}\end{aligned}$$ consisting of:
1. A sequence of Borel spaces $\{\mathbf{X}_{t}\}_{t\in \NN^{*}}$ where $\mathbf{X}_{t}$ is the state space at time $t\in\NN^{*}$.
2. A sequence of Borel spaces $\{\mathbf{A}_{t}\}_{t\in \NN^{*}}$ where $\mathbf{A}_{t}$ represents the control or action set at time $t$. For notational convenience, we introduce recursively the set $\mathbf{H}_{t}$ of histories up to time $t$ by defining $\mathbf{H}_{1}=\mathbf{X}_{1}$ and $\mathbf{H}_{t+1}=\mathbf{H}_{t}\times \mathbf{A}_{t}\times \mathbf{X}_{t+1}$ for $t\geq 1$. The set of histories will be denoted by $\mathbf{H}_{\infty}=\prod_{t\in \NN^{*}} \mathbf{X}_{t}\times \mathbf{A}_{t}$.
3. A sequence of multifunctions $\{\Psi_{t}\}_{t\in \NN^{*}}$ defined recursively by setting $\Psi_{1} \colon \mathbf{X}_{1} \to 2^{\mathbf{A}_{1}} \backslash \{\emptyset\}$ and $\Psi_{t} : \boldsymbol{\mathcal{K}}_{t-1} \times \mathbf{X}_{t} \rightarrow 2^{\mathbf{A}_{t}} \backslash \{\emptyset\}$ for $t\geq 2$ where $\boldsymbol{\mathcal{K}}_{t}\subset\mathbf{H}_{t}\times \mathbf{A_{t}}$ denotes the graph of $\Psi_{t}$. It is assumed that $\boldsymbol{\mathcal{K}}_{t}\in\mathfrak{B}(\mathbf{H}_{t}\times \mathbf{A}_{t})$. The set $\Psi_{t}(h_{t})$ for any $h_{t}\in \boldsymbol{\mathcal{K}}_{t-1} \times \mathbf{X}_{t}$ (by a slight abuse of notation $\boldsymbol{\mathcal{K}}_{0}\times \mathbf{X}_{1}$ means $\mathbf{X}_{1}$) is the set of available actions the decision maker can choose knowing the admissible history $h_{t}$ up to time $t$. By $\boldsymbol{\mathcal{H}}_{t}$ we denote the set of admissible histories up to time $t$, that is $\boldsymbol{\mathcal{H}}_{t}=\boldsymbol{\mathcal{K}}_{t-1} \times \mathbf{X}_{t}$ for $t\in \NN^{*}$.
4. A sequence of stochastic kernels $\{Q_{t}\}_{t\in \NN^{*}}$ on $\mathbf{X}_{t+1}$ given $\boldsymbol{\mathcal{K}}_{t}$, which stands for the transition probability function from time $t$ to time $t+1$.
5. The reward function $r_{t}:\boldsymbol{\mathcal{K}}_{t}\rightarrow [-\infty,+\infty[$ at step $t$ for $t\in \NN^{*}$.
6. Finally, a probability measure $\nu$ on $(\mathbf{X}_{1},\mathfrak{B}(\mathbf{X}_{1}))$ describing the initial distribution of the process.
A control policy (a policy, for short) is a sequence $\pi=\{\pi_{t}\}_{t\in\NN}$ of stochastic kernels $\pi_{t}$ on $\mathbf{A}_{t}$ given $\boldsymbol{\mathcal{H}}_{t}$ such that $\pi_{t}(\Psi_{t}(h_{t})|h_{t})=1$ for any $h_{t}\in \boldsymbol{\mathcal{H}}_{t}$. Let $\Pi$ be the set of all policies.
To state the optimal control problem we are concerned with, we introduce the canonical space $(\Omega,\mathcal{F})$ consisting of the set of sample paths $\Omega=\mathbf{H}_{\infty}$ and the associated product $\sigma$-algebra $\mathcal{F}$. The projection from $\Omega$ to the $t$-th state space and the $t$-th action space are denoted by $X_{t}$ and $A_{t}$. That is, for $$\omega=(y_{1},b_{1},\ldots,y_{t},b_{t}\ldots)\in \Omega \quad\hbox{we have}\quad X_{t}(\omega)=y_{t}\;\;\hbox{and}\;\; A_{t}(\omega)=b_{t}$$ for $t\in \NN^{*}$. Consequently, $\{X_{t}\}_{t\in \NN^{*}}$ is the state process and $\{A_{t}\}_{t\in \NN^{*}}$ is the control process. For notational convenience, we will write $\mathcal{H}_{t}=\sigma\{X_{1},A_{1},\ldots,X_{t-1},A_{t-1},X_{t}\}$ for $t\geq 2$ and $\mathcal{H}_{1}=\sigma\{X_{1}\}$. It is a well known result that for every policy $\pi \in \Pi$ and any initial probability measure $\nu$ on $(\mathbf{X}_1,\mathfrak{B}(\mathbf{X}_1))$ there exists a unique probability measure $\mathbb{P}^{\pi}$ on $(\Omega,\mathcal{F})$ such that the marginal of $\mathbb{P}^{\pi}$ on $\boldsymbol{\mathcal{H}}_{t}\times\mathbf{A}_{t}$ denoted by $\mathbb{P}^{\pi}_{| \boldsymbol{\mathcal{H}}_{t}\times \mathbf{A}_{t}}$ satisfies $\mathbb{P}^{\pi}_{| \boldsymbol{\mathcal{H}}_{t}\times \mathbf{A}_{t}}(\boldsymbol{\mathcal{K}}_{t})=1$, for any $t\in \NN^{*}$ and $$\mathbb{P}^{\pi}(X_{1}\in B)=\nu(B), \quad \text{ for } B\in \mathfrak{B}(\mathbf{X}_{1}),$$ $$\mathbb{P}^{\pi}(X_{t+1}\in C|\mathcal{H}_{t}\vee\sigma\{A_{t}\})=Q_{t}(C|X_{1},A_{1},\ldots,X_{t-1},A_{t-1},X_{t},A_{t}) \quad \text{ for } C\in \mathfrak{B}(\mathbf{X}_{t+1}),$$ $$\mathbb{P}^{\pi}(A_{t}\in D| \mathcal{H}_{t})=\pi_{t}(D|X_{1},A_{1},\ldots,X_{t-1},A_{t-1},X_{t}) \quad \text{ for } D\in \mathfrak{B}(\mathbf{A}_{t}),$$ $\mathbb{P}^{\pi}-a.s.$, for any $t\in\NN^{*}$.
We refer to $\mathbb{P}^{\pi}$ as the *strategic probability measure* generated by the policy $\pi$. Observe that we have chosen to drop in the notation of $\mathbb{P}^{\pi}$ its dependence with respect to the initial distribution $\nu\in \boldsymbol{\mathcal{P}}(\mathbf{X}_1)$. The expectation with respect to $\mathbb{P}^{\pi}$ is denoted by $\mathbb{E}^{\pi}$. Let $\boldsymbol{\mathcal{S}}$ be the family of strategic probability measures, that is, $$\label{eq-def-P-nu}
\boldsymbol{\mathcal{S}}=\{\mathbb{P}^\pi:\pi\in\Pi\}\subseteq\boldsymbol{\mathcal{P}}(\Omega).$$
#### Statement of the control problem.
The expected reward functional $\mathcal{J} \colon \boldsymbol{\mathcal{S}} \to [-\infty,+\infty[$ is defined by [$$\begin{aligned}
\mathcal{J}(\mathbb{P}^\pi)=\sum_{t=1}^{\infty} \int_{\boldsymbol{\mathcal{K}}_{t}} r_{t}^{+}(H_{t},A_{t}) d\mathbb{P}^\pi_{|\mathbf{H}_{t}\times \mathbf{A}_{t}} - \sum_{t=1}^{\infty} \int_{\boldsymbol{\mathcal{K}}_{t}} r_{t}^{-}(H_{t},A_{t}) d\mathbb{P}^\pi_{|\mathbf{H}_{t}\times \mathbf{A}_{t}}\end{aligned}$$ where $\mathbb{P}^\pi\in \boldsymbol{\mathcal{S}}$]{} and by convention $(+\infty)-(+\infty)=-\infty$.
The optimal control problem we consider consists in maximizing the expected reward $\mathcal{J}$ over the set of strategic probability measures $\boldsymbol{\mathcal{S}}$.
Weak convergence with not everywhere bounded and semicontinuous functions {#sec-portmanteau}
=========================================================================
By definition, a sequence of probability measures $(\mu_n)$ on a metric space $Y$ converges weakly towards a measure $\mu$ if for any $u\in \bar{\boldsymbol{\mathcal{C}}}(X)$, $$\lim_{n\to\infty} \int_Y u d\mu_n=\int_Yud\mu.$$ It is well known that by Alexandroff’s theorem, also known as Portmanteau’s theorem, see [@bogachev07 Corollary 8.2.5], the condition of continuity for the class of [test functions $u$]{} can be relaxed to semicontinuity. Weak convergence of $\mu_n$ towards $\mu$ is thus equivalent to the fact that $$\limsup_{n\to\infty} \int_Y u d\mu_n\leq\int_Yud\mu$$ for any upper semicontinuous [function $u$]{} which is bounded from above. The boundedness condition can itself be relaxed to some kind of uniform integrability for the test functions, see *e.g.* [@zapala08]. For instance, let us rewrite [@zapala08 Theorem 2] in our setting.
\[thm:zapala\] A sequence $(\mu_{n})$ of probability measures on a metric space $Y$ converges weakly towards a probability measure $\mu$ if and only if, for any continuous [function $u$]{} which is asymptotically uniformly integrable, that is $$\forall\epsilon>0,~\exists n_\epsilon\in\NN,~C_\epsilon>0,~\forall n\geq n_\epsilon,\quad\int_{\{|u|\geq C_\epsilon\}} |u| d\mu_n <\epsilon,$$ we have $$\lim_{n\to\infty} \int_Y u d\mu_n=\int_Y ud\mu.$$
We propose here an approach allowing to relax both everywhere semicontinuity and boundedness conditions.
\[thm:newhanger\] A sequence $(\mu_{n})$ of probability measures on a metric space $Y$ converges weakly towards a probability measure $\mu$ if and only if, for any function $u:Y\rightarrow [-\infty,+\infty[$ satisfying the following conditions:
- $u^+$ is integrable with respect to $\mu$: $$\begin{aligned}
\label{Hyp-lem-tight-lsc2}
\int_{Y} u^{+} d\mu <+\infty;\end{aligned}$$
- for any $\epsilon>0$, there exists a closed subset $Y_{\epsilon}$ of $Y$ satisfying $$\begin{aligned}
\label{Hyp-lem-tight-lsc1}
\sup_{n} \int_{Y\setminus Y_{\epsilon}} [u^{+}\vee 1] d\mu_{n} <\epsilon;\end{aligned}$$
- the restriction of $u$ on $Y_{\epsilon}$ is upper semicontinuous and bounded above;
we have, $$\begin{aligned}
\limsup_{n\rightarrow \infty} \int_{Y} u d\mu_{n} \leq \int_{Y} u d\mu.\end{aligned}$$
**Proof:** The *only if* part is obvious from Alexandroff’s Theorem. Let us consider the *if* part. From (\[Hyp-lem-tight-lsc1\]), we have $\ds \sup_{n}\int_{Y} u^{+} d\mu_{n} < + \infty $ and so, $\ds \int_{Y} u d\mu_{n}$ is well defined for any $n\in \NN$. For $\epsilon>0$, consider $Y_{\epsilon}$ satisfying the hypotheses. Write $u_{\epsilon}(y) = u(y)\wedge M_{\epsilon}$ where $M_{\epsilon}=\sup_{Y_{\epsilon}} u^{+}$. We have $$\begin{aligned}
\int_{Y} u d\mu_{n} = \int_{\{u^{+}\leq M_{\epsilon} \}} u_{\epsilon} d\mu_{n} + \int_{\{u^{+} > M_{\epsilon} \}} u^{+} d\mu_{n}.\end{aligned}$$ Observe now that $\{u^{+} > M_{\epsilon} \} \subset Y\setminus Y_{\epsilon}$ and $u_{\epsilon}\equiv M_{\epsilon}$ on $\{u^{+} > M_{\epsilon} \}$ showing $$\begin{aligned}
\int_{Y} u d\mu_{n} \leq \int_{Y} u_{\epsilon} d\mu_{n} + \int_{Y\setminus Y_{\epsilon}} u^{+} d\mu_{n}.\end{aligned}$$ Therefore, equation (\[Hyp-lem-tight-lsc1\]) implies $$\begin{aligned}
\int_{Y} u d\mu_{n} \leq \int_{Y} u_{\epsilon} d\mu_{n} + \epsilon\end{aligned}$$ for any $\epsilon>0$. Clearly, by using the hypotheses, [for any $\eta>0$]{} there exists a closed subset $Z_{\eta}$ of $Y$ such that the restriction of $u_{\epsilon}$ on $Z_{\eta}$ is upper semicontinuous and $\sup_{n} \mu_{n}(Y\setminus Z_{\eta}) <\eta$. Moreover, $u_{\epsilon}$ is clearly bounded above on $Y$ and so $u_{\epsilon}$ satisfies the conditions of Lemma 2.4 in [@balder89] whose proof is detailed in Appendix \[app:balder\], implying $\ds \limsup_{n\rightarrow \infty} \int_{Y} u_{\epsilon} d\mu_{n}\leq \int_{Y} u_{\epsilon} d\mu$. Therefore, $$\ds \limsup_{n\rightarrow \infty} \int_{Y} u d\mu_{n} \leq \int_{Y} u_{\epsilon} d\mu + \epsilon.$$ Now, it follows from (\[Hyp-lem-tight-lsc2\]) that $\ds \int_{Y} u d\mu$ is well defined and since $u_{\epsilon}\leq u$ we get $$\begin{aligned}
\limsup_{n\rightarrow \infty} \int_{Y} u d\mu_{n} \leq \int_{Y} u d\mu + \epsilon\end{aligned}$$ for any $\epsilon>0$, showing the result. $\Box$
Clearly, the condition (\[Hyp-lem-tight-lsc1\]) may be relaxed to the existence, for any $\epsilon>0$, of a closed subset $Y_\epsilon$ and an integer $n_\epsilon$ such that $$\sup_{n\geq n_\epsilon} \int_{Y\setminus Y_{\epsilon}} [u^{+}\vee 1] d\mu_{n} <\epsilon.$$ Then, it is not hard to see that the latter condition plus the fact that $u$ is upper semicontinuous and bounded from above on $Y_\epsilon$ implies the asymptotic uniform integrability of $u$, and the latter condition is thus somehow stronger in this sense than asymptotic uniform integrability. But in our case, $u$ is semicontinuous only on a subset $Y_\epsilon$, making, in this sense, the set of conditions of Theorem \[thm:newhanger\] weaker than the conditions in Theorem \[thm:zapala\].
The following result is a direct consequence of Theorem \[thm:newhanger\].
\[lem-tight-usc\] Let $Y$ be a metric space. Consider a subset $\cal P$ of $\boldsymbol{\mathcal{P}}(Y)$ and a function $u$ on $Y$ satisfying
- for any $\epsilon>0$, there exists a closed subset $Y_\epsilon$ of $Y$ such that $$\sup_{\mathbb{P}\in\cal P}\int_{Y\setminus Y_\epsilon} (u^{+}\vee 1) d \mathbb{P} <\epsilon,$$
- the restriction of $u$ to $Y_\epsilon$ is upper semicontinuous and bounded above.
Then, the function $\ds \mathbb{P} \to \int_Y u d\mathbb{P}$ defined on $\cal P$ is upper semicontinuous and bounded above.
Existence result under general conditions {#sec-results}
=========================================
We start this section with the introduction and a discussion of the assumptions under consideration in this work. Then, we will prove the existence of an optimal control strategy for the model presented in section \[sec-model\].
\[Regular-Hypotheses\]
\[Condition-C\] The following condition holds: $$\lim_{n\rightarrow\infty} \sup_{p\geq n} \sup_{\pi\in\Pi} \bigg[ \sum_{t=n}^{p} \mathbb{E}^\pi \big[ r_t(H_{t},A_{t})\big] \bigg]^{+}= 0.$$
\[Continuity-transition-kernel\] For any $t\in \NN^{*}$, $g\in {\overline{\boldsymbol{\mathcal{C}}}}(\mathbf{X}_{t+1})$ and $\epsilon>0$ there exists $C_{\epsilon}$ a closed subset of $\boldsymbol{\mathcal{K}}_{t}$ satisfying $$\begin{aligned}
\sup_{\pi\in\Pi} \mathbb{E}^\pi\big[ \mathbf{I}_{\boldsymbol{\mathcal{K}}_{t}\setminus C_{\epsilon}}(H_{t},A_{t}) \big]<\epsilon\end{aligned}$$ and such that the real-valued mapping defined on $C_{\epsilon}$ by $\ds (x_{1},a_{1},\ldots,x_{t},a_{t})\rightarrow Q_tg(x_{1},a_{1},\ldots,x_{t},a_{t})$ is continuous.
\[Reward+Multifunction\]
For any $t\in \NN^{*}$, $\epsilon>0$ there exists $K_{\epsilon}$, a closed subset of $\boldsymbol{\mathcal{H}}_{t}$, satisfying $$\begin{aligned}
\label{w-tight-condition}
\sup_{\pi\in\Pi} \mathbb{E}^\pi\Big[ \mathbf{I}_{\boldsymbol{\mathcal{H}}_{t}\setminus K_{\epsilon}}(H_{t}) \big[1\vee r_t^{+}(H_{t},A_{t}) \big] \Big]<\epsilon\end{aligned}$$ and such that the restriction of $r_{t}$ to $[K_{\epsilon} \times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ is strongly coercive and bounded above.
\[Hyp-discussion\] Discussion of the hypotheses.
(a) \[Schal-condition\] Assumption \[Condition-C\] is the so-called Condition (C) in Schäl’s papers [@schal75b; @schal75]. It is slightly weaker than Condition (A3) in Balder’s result [@balder92], [see also the discussion in [@kertz79 Remark 6.11]]{}.
(b) \[Balder-condition\] [Assumption \[Continuity-transition-kernel\] is a standard condition, see for example Condition (C2) in [@balder89] and also Condition (A1) in [@balder92]. Specific conditions that can be expressed in terms of the primitive data of the model and implying Assumption \[Continuity-transition-kernel\] are presented in [@balder89 Section 3].]{}
(c) \[New-condition-1\] The condition (\[w-tight-condition\]) in Assumption \[Reward+Multifunction\] is new and generalizes condition (A2) in [@balder92] to the case where the reward function may not be bounded above. Observe that in [@balder89; @balder92; @schal75b; @schal75] the reward functions are bounded above. In [@jaskiewicz11], the authors studied a discounted Markov decision process on general state and action spaces with possibly unbounded reward function with application to economic models. We will show in Section \[example-Nowak\] that the approach presented in [@jaskiewicz11] can be easily embedded into our framework. Moreover, our condition (\[w-tight-condition\]) incorporates the case of history-dependent action spaces contrary to the framework discussed in [@balder92].
(d) \[New-condition-1-sufficiency\]
It can be shown easily that the following set of conditions implies that Assumption \[Reward+Multifunction\] holds. There are written explicitly in terms of the parameters of the model which makes them easier to check than Assumption \[Reward+Multifunction\], as outlined in Section \[example-general\] through an example:
(i) For any $t\in \NN^{*}$, there exists a $\RR_{+}\union\{+\infty\}$-valued measurable mapping $\Phi_{t}$ defined on $\boldsymbol{\mathcal{H}}_{t}$ satisfying $$\begin{aligned}
\sup_{a_{t}\in \Psi_{t}(h_{t})} \big[1\vee r_{t}^{+}(h_{t},a_{t}) \big] \leq \Phi_{t}(h_{t}).\end{aligned}$$
(ii) For any $\epsilon>0$, there exists $K_{\epsilon}$ a closed subset of $\mathbf{X}_{1}$, satisfying $$\begin{aligned}
\label{sufficient-tight-1-condition}
\int_{\mathbf{X}_{1}\setminus K_{\epsilon}} \Phi_{1}(x_{1}) \nu(dx_{1}) < \epsilon\end{aligned}$$ and such that the restriction of $r_{1}$ to $[K_{\epsilon} \times \mathbf{A}_{1}]\cap\boldsymbol{\mathcal{K}}_{1}$ is strongly coercive and bounded above.
(iii) For any $t\in \NN^{*}$ and $\epsilon>0$, there exists $K_{\epsilon}$ a closed subset of $\boldsymbol{\mathcal{H}}_{t+1}$, satisfying $$\begin{aligned}
\label{sufficient-tight-2-condition}
\sup_{(h_{t},a_{t})\in \boldsymbol{\mathcal{K}}_{t}} \int_{\mathbf{X}_{t+1}} \mathbf{I}_{\boldsymbol{\mathcal{H}}_{t+1}\setminus K_{\epsilon}} (h_{t},a_{t},x_{t+1})
\Phi_{t+1}(h_{t},a_{t},x_{t+1})
Q_{t}(dx_{t+1}|h_{t},a_{t}) <\epsilon\end{aligned}$$ and such that the restriction of $r_{t+1}$ to $[K_{\epsilon} \times \mathbf{A}_{t+1}]\cap\boldsymbol{\mathcal{K}}_{t+1}$ is strongly coercive and bounded above.
(e) [Observe that Assumption \[Continuity-transition-kernel\] is related to the family of transition kernels $\{Q_{t}\}_{t\in \NN^{*}}$ and states roughly speaking that $Q_{t}$ is weakly continuous on a closed subset of $\boldsymbol{\mathcal{K}}_{t}$ for $t\in \NN^{*}$ while Assumption \[Reward+Multifunction\] is associated with the reward functions $\{r_{t}\}_{t\in \NN^{*}}$ and imposes that $r_{t}$ is strongly coercive on a closed subset of $\boldsymbol{\mathcal{H}}_{t}$ for $t\in \NN^{*}$. We would like to emphasize that the closed sets involved in Assumption \[Continuity-transition-kernel\] and \[Reward+Multifunction\] are different by definition since in Assumption \[Continuity-transition-kernel\] it is a closed subset of $\boldsymbol{\mathcal{K}}_{t}$ while in Assumption \[Reward+Multifunction\] it is a closed subset of $\boldsymbol{\mathcal{H}}_{t}$.]{}
(f) \[lower-bound-reward\] From Assumption \[Reward+Multifunction\], it follows that for any $t\in \NN^{*}$, $$\ds \sup_{\mathbb{P}\in \boldsymbol{\mathcal{S}}} \int_{\boldsymbol{\mathcal{K}}_{t}} r_{t}^{+}(H_{t},A_{t}) d\mathbb{P}_{|\mathbf{H}_{t}\times \mathbf{A}_{t}} \leq \sup_{\mathbb{P}\in \boldsymbol{\mathcal{S}}} \int_{[K_{\epsilon} \times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}} r_{t}^{+}(H_{t},A_{t})
d\mathbb{P}_{|\mathbf{H}_{t}\times \mathbf{A}_{t}} +\epsilon$$ for some $\epsilon>0$ and a closed set $K_{\epsilon}$ in $\boldsymbol{\mathcal{H}}_{t}$. However, $r_{t}$ is bounded above on $[K_{\epsilon}\times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ and so, $\ds \sup_{\mathbb{P}\in \boldsymbol{\mathcal{S}}} \int_{\boldsymbol{\mathcal{K}}_{t}} r_{t}^{+}(H_{t},A_{t}) d\mathbb{P}_{|\mathbf{H}_{t}\times \mathbf{A}_{t}} <\infty$.
\[Reward-upper-semicompact\] Suppose Assumptions \[Regular-Hypotheses\] and \[Reward+Multifunction\] hold. The mapping $\mathcal{J}$ is upper semicompact.
**Proof:** We have to show that for any $\beta \in \RR$, the set $\{\mathbb{P}\in \boldsymbol{\mathcal{S}} : \mathcal{J}(P)\geq \beta\}$ is relatively compact in $\boldsymbol{\mathcal{S}}$ for the weak topology. We will proceed along the line described in [@balder92 Lemma 4.2] to show the result. The main difference is that in our case, the reward function is not necessarily bounded above and that the action sets are history-dependent leading to the introduction of a new set of conditions given by Assumption \[Reward+Multifunction\].
There exist compact sets $\widehat{\mathbf{X}}_{t}$ and $\widehat{\mathbf{A}}_{t}$ such that $\mathbf{X}_{t}\in\mathfrak{B}(\widehat{\mathbf{X}}_{t})$ and $\mathbf{A}_{t}\in\mathfrak{B}(\widehat{\mathbf{A}}_{t})$ for $t\in \NN^{*}$. Equipped with the weak topology, $\boldsymbol{\mathcal{P}}(\mathbf{H}_{\infty})$ and $\boldsymbol{\mathcal{S}}$ are topological subspaces of the compact space $\boldsymbol{\mathcal{P}}(\widehat{\mathbf{H}}_{\infty})$ where $\widehat{\mathbf{H}}_{\infty}=\prod_{t\in \NN^{*}} \mathbf{X}_{t} \times \mathbf{A}_{t}$. Consider $\{\mathbb{P}_{j}\}_{j\in \NN}$ a sequence in $\{\mathbb{P}\in \boldsymbol{\mathcal{S}} : \mathcal{J}(\mathbb{P})\geq \beta\} \subset \boldsymbol{\mathcal{P}}(\widehat{\mathbf{H}}_{\infty})$. Since $\boldsymbol{\mathcal{P}}(\widehat{\mathbf{H}}_{\infty})$ is compact, there exists a subsequence of $\{\mathbb{P}_{j}\}_{j\in \NN}$ (still denote by $\{\mathbb{P}_{j}\}_{j\in \NN}$) that converges to $\mathbb{P}_{\infty}\in \boldsymbol{\mathcal{P}}(\widehat{\mathbf{H}}_{\infty})$. In order to get the the result, it is sufficient to show that $\mathbb{P}_{\infty}\in \boldsymbol{\mathcal{S}}$ or equivalently that $\mathbb{P}_{\infty | \widehat{\mathbf{X}}_{1}}=\nu$ and for any $k\in \NN^{*}$ $$\begin{aligned}
\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{k}\times \widehat{\mathbf{A}}_{k}}(\boldsymbol{\mathcal{K}}_{k})=1,
\label{Const-1} \\
\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{k+1}}=P_{\infty | \widehat{\mathbf{H}}_{k}\times \widehat{\mathbf{A}}_{k}}Q_{k}.
\label{Const-2}\end{aligned}$$ Clearly, we have $\mathbb{P}_{\infty | \widehat{\mathbf{X}}_{1}}=\nu$. The other two equalities will be shown by induction. Let us assume that (\[Const-1\]) and (\[Const-2\]) hold for $k\in \{1,\ldots,t-1\}$.
Let us first show that $\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t} \times \widehat{\mathbf{A}}_{t}}(\boldsymbol{\mathcal{K}}_{t})=1$. To do so, it is sufficient to prove, for the function $v \colon \boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t} \to [-\infty,+\infty[$ defined by $v=r_{t}\wedge 0$ on $\boldsymbol{\mathcal{K}}_{t}$ and $v=-\infty$ otherwise, that $$\ds \int_{\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}} v d\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}} > -\infty.$$ Indeed, in such a case, the probability measure $\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}$ is necessarily supported by $\boldsymbol{\mathcal{K}}_{t}$.
For $\epsilon>0$, let us denote by $K_{\epsilon}$ the closed subset of $\boldsymbol{\mathcal{H}}_{t}$ satisfying Assumption \[Reward+Multifunction\] and write $Z_{\epsilon}=[K_{\epsilon}\times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$. By hypothesis, $r_{t | Z_{\epsilon}}$ is strongly coercive and so is $\big[r_{t | Z_{\epsilon}}\big]_{K_{\epsilon} \times \widehat{\mathbf{A}}_{t}}$ where $$\big[r_{t | Z_{\epsilon}}\big]_{K_{\epsilon} \times\widehat{\mathbf{A}}_{t}}=\left\{
\begin{array}{rl}
r_{t } &\text{on } Z_{\epsilon},\\
-\infty&\text{on } (K_{\epsilon}\times \widehat{\mathbf{A}}_{t})\setminus Z_{\epsilon}.
\end{array}
\right.$$ Now, observe that $v_{| K_{\epsilon} \times \widehat{\mathbf{A}}_{t}}=\big[r_{t | Z_{\epsilon}}\big]_{K_{\epsilon} \times \widehat{\mathbf{A}}_{t}} \wedge 0$. Thus, by items (i) and (iii) of Proposition 2.2 in [@balder92], it follows that $v_{| K_{\epsilon} \times \widehat{\mathbf{A}}_{t}}$ is strongly coercive and so, upper semicontinuous.
By the induction hypothesis, we easily obtain that $\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t} \times \widehat{\mathbf{A}}_{t}}(\boldsymbol{\mathcal{H}}_{t} \times \widehat{\mathbf{A}}_{t})=1$. Recalling that for any $j\in \NN^{*}$, $\mathbb{P}_{j | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}(\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t})=1$, we get that the restriction of $\mathbb{P}_{j | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}$ to $\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}$ converges weakly to the restriction of $\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}$ to $\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}$. Now, combining the Portmanteau theorem and , it follows that $\ds \mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}\big((\boldsymbol{\mathcal{H}}_{t}\setminus K_{\epsilon})\times \widehat{\mathbf{A}}_{t}\big)
\leq \liminf_{j\rightarrow\infty} \mathbb{P}_{j | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}\big((\boldsymbol{\mathcal{H}}_{t}\setminus K_{\epsilon})\times \widehat{\mathbf{A}}_{t}\big) <\epsilon$ since $K_{\epsilon}$ is a closed subset of $\boldsymbol{\mathcal{H}}_{t}$. Therefore, we can apply Corollary \[lem-tight-usc\] to the function $v$ and the set of probability measures given by the restriction of $\mathbb{P}_{j | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}$ to $\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}$ with $j\in \NN\cup\{\infty\}$. We obtain $\ds \limsup_{j\rightarrow \infty} \int_{\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}} v d \mathbb{P}_{j | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}
\leq \int_{\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}} v d\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}$. However, recalling that $\mathcal{J}(\mathbb{P}_{j})\geq \beta$ we get with Assumption \[Condition-C\] that $\beta\leq\mathcal{J}(\mathbb{P}_{j})\leq \mathcal{J}_{t}(\mathbb{P}_{j})+1$. From Remark \[Hyp-discussion\].(\[lower-bound-reward\]), it implies that $\ds \inf_{j\in \NN} \int_{\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}} (r_{t}\wedge 0) d\mathbb{P}_{j | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}} >-\infty$. Therefore, $\ds \int_{\boldsymbol{\mathcal{H}}_{t}\times \widehat{\mathbf{A}}_{t}} v d\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times\widehat{\mathbf{A}}_{t}} > -\infty$ showing that $\mathbb{P}_{\infty | \widehat{\mathbf{H}}_{t}\times \widehat{\mathbf{A}}_{t}}(\boldsymbol{\mathcal{K}}_{t})=1$ by definition of $v$, as required.
Now, following exactly the same arguments as in the second part of the proof of Theorem 2.1 in [@balder89] and using Assumption \[Continuity-transition-kernel\], we obtain that is valid for $k=t$, giving the last part of the result. $\Box$
\[Reward-upper-semicontinuous\] Suppose Assumptions \[Condition-C\] and \[Reward+Multifunction\] hold. The mapping $\mathcal{J}$ is upper semicontinuous.
**Proof:** Consider $t\in \NN^{*}$ and $\epsilon>0$. According to Assumption \[Reward+Multifunction\] there exists $K_{\epsilon}$, a closed subset of $\boldsymbol{\mathcal{H}}_{t}$, such that the restriction of $r_{t}$ to $Z_{\epsilon}=[K_{\epsilon}\times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ is strongly coercive and bounded above and $\ds \sup_{\pi\in\Pi} \int_{\boldsymbol{\mathcal{K}}_{t}\setminus Z_{\epsilon}} \big[1\vee r_t^{+}(H_{t},A_{t}) \big] d\mathbb{P}^{\pi}_{| \mathbf{H}_{t} \times \mathbf{A}_{t}}<\epsilon$. By item (i) of Proposition 2.2 in [@balder92], the restriction of $r_{t}$ to $Z_{\epsilon}$ is upper semicontinuous. Observe that $Z_{\epsilon}$ is a closed subset of $\boldsymbol{\mathcal{K}}_{t}$. Therefore, from Corollary \[lem-tight-usc\] we obtain that the mapping defined on $\boldsymbol{\mathcal{S}}$ by $\ds \mathbb{P}\to \int_{\boldsymbol{\mathcal{K}}_{t}} r_{t}(H_{t},A_{t}) d\mathbb{P}^{\pi}_{| \mathbf{H}_{t} \times \mathbf{A}_{t}}$ is upper semicontinuous and bounded above. Now, taking into account Assumption \[Condition-C\], an application of Proposition 10.1 in [@schal75b] gives the result. $\Box$\
We are now able to state our main result.
\[main-result\] Suppose that Assumptions \[Regular-Hypotheses\] and \[Reward+Multifunction\] hold and that there exists a strategic probability measure $P_m\in\mathbb{P}$ such that $\mathcal{J}(P_m)>-\infty$. Then there exists a policy $\pi^\ast\in \Pi$ such that $$\sup_{P\in \mathbb{P}}\mathcal{J}(P)=\mathcal{J}(P^{\pi^\ast}).$$
**Proof:** From Proposition \[Reward-upper-semicompact\], $\mathcal{J}$ is upper semicompact and so, the set $\{\mathcal{J}\geq \mathcal{J}(P_m)\}$ is relatively compact. The map $\mathcal{J}$ being also upper semicontinuous according to Proposition \[Reward-upper-semicontinuous\], it admits a maximum on the compact set given by the closure of $\{\mathcal{J}\geq \mathcal{J}(P_m)\}$ and the result follows. $\Box$
Examples {#sec-examples}
========
This section provides examples illustrating our results. The first example described a controlled model for which the reward function is unbounded and not strongly coercive on its domain of definition and takes the value $-\infty$. It is shown that this model satisfies our assumptions.
A set of hypotheses has been introduced in [@jaskiewicz11] to ensure in particular the existence of an optimal policy for Makov decision processes with unbounded rewards. It is shown in the second example that our conditions are satisfied [in such a setting. Moreover, we would like to emphasize that this set of conditions is satisfied for a large class of economical models as described in [@jaskiewicz11 Section 5]. For the sake of completeness we describe one of such model at the end of the Section \[example-Nowak\].]{}
An example with unbounded and non strongly coercive reward function {#example-general}
-------------------------------------------------------------------
We consider a model with state space $\mathbf{X}_{t}= [0,1]$ and action space $\mathbf{A}_{t}=\mathbb{N}^\ast$. Let us introduce $$\mathbf{A}(x)=\begin{cases} \{1,\ldots,p\} & \text{ if } x\in [0,1/2]\\
\mathbb{N}^\ast & \text{ if } x\in ]1/2,1], \end{cases}$$ for some $p\in\mathbb{N}^\ast$. The sequence of multifunctions $\{\Psi_{t}\}_{t\in \NN^{*}}$ are defined recursively by setting $\Psi_{1} \colon \mathbf{X}_{1} \to 2^{\mathbf{A_{1}}} \backslash \{\emptyset\}$ with $\Psi_{1}(x)=\mathbf{A}(x)$ and $\Psi_{t} : \boldsymbol{\mathcal{K}}_{t-1} \times \mathbf{X}_{t} \rightarrow 2^{\mathbf{A_{t}}} \backslash \{\emptyset\}$ for $t\geq 2$ given by $\Psi_{t}(x_{1},a_{1},\ldots,x_{t-1},a_{t-1},x_{t})=\mathbf{A}(x_{t})$ where $\boldsymbol{\mathcal{K}}_{t}\subset\mathbf{H}_{t}\times \mathbf{A}_{t}$ denotes the graph of $\Psi_{t}$. The transition probability function from time $t$ to time $t+1$ is $$Q_t( dy | x_{1},a_{1},\ldots,x_{t-1},a_{t-1},x_{t},a_{t})=\beta(t+x_{t}/a_{t},5/2)(dy)$$ where $\beta(\alpha_{1},\alpha_{2})$ denotes the beta probability distribution on $[0,1]$ with parameters $(\alpha_{1},\alpha_{2})\in \RR_{+}^{2}$. The reward functions we consider are given by $$r_{t}(x_{1},a_{1},\ldots,x_{t-1},a_{t-1},x_{t},a_{t})= \begin{cases}
0 & \text{ if $(x_{t},a_{t}) \in \{0\}\times \NN^{*}$} \\
\frac{1}{a_{t}^{2}} \frac{1}{\sqrt{x_{t}}}I_{]0,1/2]}(x_{t})-a_{t} \frac{1}{\sqrt{1-x_{t}}}I_{]1/2,1[}(x_{t}) & \text{ if $(x_{t},a_{t}) \in ]0,1[\times \NN^{*}$} \\
-\infty, & \text{ if $(x_{t},a_{t}) \in \{1\} \times \NN^{*}$.}
\end{cases}$$ The initial probability measure $\nu$ is $\beta(1,2)$.
Observe that the reward function is unbounded, takes the value $-\infty$ and is not upper semicontinuous at point $(x_{1},a_{1},\ldots,x_{t-1},a_{t-1},0,a_{t})\in \boldsymbol{\mathcal{H}}_{t}$ and therefore not strongly coercive. Moreover, the action sets available at each step to the decision maker are not compact. [Let us show that Assumptions \[Regular-Hypotheses\] and \[Reward+Multifunction\] are satisfied. To check that Assumtpion \[Reward+Multifunction\] holds, we will use the approach developed in Remark \[Hyp-discussion\](\[New-condition-1-sufficiency\]). For any positive integer $t\geq1$ and $h_t=(x_{1},a_{1},\ldots,x_{t})\in \boldsymbol{\mathcal{H}}_{t}$, we set $$\label{eq:ex:maj}
\Phi_t(h_t)=I_{\{0\}}(x_t)+\frac{1}{\sqrt{x_t}}I_{]0,1/2]}(x_t)+I_{]1/2,1]}(x_t).$$ This measurable function satisfies, for any $h_t\in \boldsymbol{\mathcal{H}}_{t}$, $$\begin{aligned}
\sup_{a_{t}\in \Psi_{t}(h_{t})} \big[1\vee r_{t}^{+}(h_{t},a_{t}) \big] \leq \Phi_{t}(h_{t}).\end{aligned}$$ We have that for any positive integer $t\geq2$, $\epsilon\in [0,1/2]$, $\gamma\in [0,1/2]$ and any $(h_{t-1},a_{t-1})\in \boldsymbol{\mathcal{K}}_{t-1}$, $$\begin{aligned}
&\int_{\mathbf{X}_{t}}\mathbf{I}_{\boldsymbol{\mathcal{K}}_{t-1}\times([0,\epsilon[\union]1/2,1/2+\gamma[)}(h_{t}) \Phi_{t}(h_{t}) Q_{t}(dx_{t}|h_{t-1},a_{t-1})\nonumber \\
& \leq \int_{]0,\epsilon[} \frac{1}{\sqrt{y}} Q_{t}(dx_{t}|h_{t-1},a_{t-1})
+ Q_{t}(]1/2,1/2+\gamma [|h_{t-1},a_{t-1}) \nonumber\\
~&\leq \frac{1}{\mathsf{B}(t+x_{t-1}/a_{t-1},5/2)}
\bigg[ \int_{]0,\epsilon[} \frac{1}{\sqrt{y}} y^{t-1} dy + \int_{]1/2,1/2+\gamma [} y^{t-1} dy \bigg] \nonumber\\
~&\leq \frac{1}{\mathsf{B}(t+1,5/2)} \Big[ \frac{1}{t-1/2} \epsilon^{t-1/2} + \frac{1}{t} \big[ (1/2+\gamma)^{t}-(1/2)^{t} \big] \Big] \nonumber\\
& \leq \frac{Kt^{7/2}}{t-1/2} \Big[ \epsilon^{t-1/2}+\big[ (1/2+\gamma)^{t}-(1/2)^{t} \big] \Big],
\label{Ineq-exemple2}\end{aligned}$$ for some positive constant $K$ and where $\mathsf{B}(\alpha_{1},\alpha_{2})$ denotes the beta function with parameters $(\alpha_{1},\alpha_{2})\in \RR_{+}^{2}$. For $t=1$, similar calculations lead to $$\int_{\mathbf{X}_{1}}\mathbf{I}_{[0,\epsilon[\union]1/2,1/2+\gamma[}(x_1) \Phi_{1}(x_1)\nu(dx_1)\leq \nu(]1/2,1/2+\gamma [)+\frac{2}{\mathsf{B}(1,2)}\sqrt{\epsilon}\big(1-\epsilon/3\big).\label{Ineq-exemple2bis}$$ ]{} Using equation and choosing $\epsilon=1/2$ and $\gamma=0$ in equation , we can show that for $t\geq2$ and any policy $\pi\in \Pi$, $$\begin{aligned}
\mathbb{E}^\pi \big[ r^+_{t}(H_{t},A_t)\big] \leq \mathbb{E}^\pi \Big[ \mathbf{I}_{[0,\frac{1}{2}]}(X_t) \big[1\vee r^+_{t}(H_{t},A_t)\big]\Big]\leq \frac{Kt^{7/2}}{t-1/2} (1/2)^{t-1/2}\leq c t^{5/2} 2^{-t}\end{aligned}$$ for some positive constant $c$ and so, Assumption \[Condition-C\] is satisfied since the series with main term $t^{5/2}2^{-t}$ is convergent. Clearly, Assumption \[Continuity-transition-kernel\] is satisfied.
Now, regarding Assumption \[Reward+Multifunction\], let us consider $t\in\NN^*$ and $K_{\epsilon}$ defined by $\boldsymbol{\mathcal{K}}_{t-1} \times ([\epsilon,1/2]\cup[1/2+\epsilon,1])$ (with the slight abuse of notation $\boldsymbol{\mathcal{K}}_{0}\times ([\epsilon,1/2]\cup[1/2+\epsilon,1])$ means $[\epsilon,1/2]\cup[1/2+\epsilon,1]$) a closed subset of $\boldsymbol{\mathcal{H}}_{t}$ for any $\epsilon\in ]0,1/4]$. By using equations and , the set $K_{\epsilon}$ satisfies, for $t\geq2$, $$\lim_{\epsilon\rightarrow 0} \sup_{(h_{t-1},a_{t-1})\in \boldsymbol{\mathcal{K}}_{t-1}}\int_{\mathbf{X}_{t}}\mathbf{I}_{\boldsymbol{\mathcal{H}}_{t}\setminus K_{\epsilon}}(h_{t}) \Phi_{t}(h_{t}) Q_{t}(dx_{t}|h_{t-1},a_{t-1})=0,$$ and for $t=1$, using equation $$\lim_{\epsilon\rightarrow 0} \int_{\mathbf{X}_{1}}\mathbf{I}_{[0,\epsilon[\union]1/2,1/2+\epsilon[}(x_1) \Phi_{1}(x_1)\nu(dx_1)=0.$$ [This shows the existence of a closed subset of $\mathbf{X}_{1}$ (respectively, $\boldsymbol{\mathcal{H}}_{t+1}$) satisfying condition (respectively, condition ). Now it remains to prove that the restriction of $r_{t}$ to $[K_{\epsilon} \times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ is bounded above and strongly coercive to get the sufficient conditions proposed for Assumption \[Reward+Multifunction\], in Remark \[Hyp-discussion\](\[New-condition-1-sufficiency\]).]{} Clearly, the restriction of $r_{t}$ to $[K_{\epsilon} \times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ is bounded above. Let us show that it is strongly coercive. Indeed, for $t\in\mathbb{N}^\ast$, let $\{g^{k},x^{k},a^{k}\}_{k\in \NN}$ be a sequence in $[\boldsymbol{\mathcal{K}}_{t-1} \times([\epsilon,1/2]\cup[1/2+\epsilon,1]) \times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ such that $\{(g^{k},x^{k})\}_{k\in \NN}$ converges to $(g^\ast,x^{\ast})\in \boldsymbol{\mathcal{K}}_{t-1} \times ([\epsilon,1/2]\cup[1/2+\epsilon,1])$ as $k$ tends to infinity and $\ds \limsup_{k\rightarrow\infty} r_{t}(g^{k},x^{k},a^{k})>-\infty$. Let us show that there necessarily exits a converging subsequence $\{a^{\phi(k)}\}_{k\in \NN}$ of $\{a^{k}\}_{k\in \NN}$ to $a^{*}$ such $a^{*} \in \mathbf{A}(x^{*})$ and $\ds \limsup_{k\rightarrow\infty} r_{t}(g^{k},x^{k},a^{k})\leq r_{t}(g^{*},x^{*},a^{*})$.
We first prove that there exists a subsequence $\{a^{\phi(k)}\}_{k\in \NN}$ of $\{a^k\}_{k\in \NN}$ such that $\{a^{\phi(k)}\}_{k\in \NN}$ converges towards some $a^\ast\in \mathbf{A}(x^\ast)=\mathbb{N}^\ast$. Assume that $x^\ast$ is in $[\epsilon,1/2]$. Since $\{x^k\}$ is valued in $[\epsilon,1/2]\cup[1/2+\epsilon,1]$, this means that there is some $k_0$ such that for any $k\geq k_0$ we have $x_k\in[\epsilon,1/2]$ and thus $$\mathbf{A}(x^k)=\{1,\ldots,p\}.$$ Therefore, $a_k$ is in the compact set $\{1,\ldots,p\}$ for all $k\geq k_0$ and has a convergent subsequences to some $a^\ast\in\{1,\ldots,p\}=\mathbf{A}(x^\ast)$.\
Now, assume that $x^\ast$ is in $[1/2+\epsilon,1]$. Since $\{x^k\}_{k\in \NN}$ is valued in $[\epsilon,1/2]\cup[1/2+\epsilon,1]$, this means that there is some $k_0$ such that for any $k\geq k_0$ we have $x^k\in[1/2+\epsilon,1]$ and thus $$\mathbf{A}(x^k)=\mathbb{N}^\ast.$$ We proceed now by contradiction. Assume that for any subsequences $\{a^{\phi(k)}\}_{k\in \NN}$ of $\{a^k\}_{k\in \NN}$, either $\{a^{\phi(k)}\}_{k\in \NN}$ diverges or it converges to some $a^\ast\notin \mathbf{A}(x^\ast)=\mathbb{N}^\ast$. Of course, the latter claim can not happen: if the integer valued $\{a^{\phi(k)}\}_{k\in \NN}$ converges to some $a^\ast$, then, necessarily, $a^\ast\in\mathbb{N}^\ast$. Thus, assume that any subsequences $\{a^{\phi(k)}\}_{k\in \NN}$ of $\{a^k\}_{k\in \NN}$ diverges. This implies that the original sequence goes to infinity. Then, since for any $k\geq k_0$ we have $x_k\in[1/2+\epsilon,1]$, $$r_t(g^k,x^k,a^k)=-a^k\frac{1}{\sqrt{1-x^k}}.$$ Therefore, $\ds \limsup_{k\rightarrow\infty} r(g^k,x^k,a^k)=-\infty$, showing a contradiction. Consequently, there exists a subsequence $\{a^{\phi(k)}\}_{k\in \NN}$ of $\{a^k\}_{k\in \NN}$ such that $\{a^{\phi(k)}\}_{k\in \NN}$ converges towards some $a^\ast\in \mathbf{A}(x^\ast)=\mathbb{N}^\ast$.
Now, in both cases, $x^\ast \in [\epsilon,1/2]$ or $x^\ast \in [1/2+\epsilon,1]$, the upper semi-continuity of $r_t$ on $[\boldsymbol{\mathcal{K}}_{t-1} \times([\epsilon,1/2]\cup[1/2+\epsilon,1]) \times \mathbf{A}_{t}$ implies that $\ds \limsup_{k\rightarrow\infty} r_{t}(g^{k},x^{k},a^{k})\leq r_{t}(g^{*},x^{*},a^{*})$. Consequently, the restriction of $r_{t}$ to $[K_{\epsilon} \times \mathbf{A}_{t}]\cap\boldsymbol{\mathcal{K}}_{t}$ is strongly coercive showing that Assumption \[Reward+Multifunction\] is satisfied.
A model by A. Jaśkiewicz and A. Nowak {#example-Nowak}
-------------------------------------
The objective of this example is to show that our set of conditions are more general than those introduced in [@jaskiewicz11] to study discounted Markov decision processes with unbounded (from above and below) reward function. In [@jaskiewicz11], the authors consider the following discounted model with unbounded rewards where ${\cal X}$ (respectively, ${\cal A}$) is the Borel state (respectively, action) space. The decision maker can choose the action in a compact set ${\cal A}(x)\subset {\cal A}$ when the state process is in $x\in {\cal X}$. The set valued mapping $\psi(x)={\cal A}(x)$ defined on ${\cal X}$ is assumed to be upper semicontinuous. The transition probability denoted by $q$ on ${\cal X}$ given the graph $\boldsymbol{\mathcal{K}}$ of $\psi$ is weakly continuous. The reward function is given by $\beta^{t}u$ where the discount factor $\beta\in (0,1)$ and the reward function $u$ is upper semicontinuous on $\boldsymbol{\mathcal{K}}$. In addition to these classical hypotheses, it is assumed that there exists a sequence $\{{\cal X}_{j}\}_{j\in \NN}$ of non-empty Borel subsets of ${\cal X}$ satisfying ${\cal X}=\cup_{j\in \NN} \mathring{{\cal X}}_{j}$ (where $\mathring{{\cal X}}_{j}$ denotes the interior of ${\cal X}_{j}$) and from the set ${\cal X}_{j}$ only transitions to ${\cal X}_{j+1}$ are allowed, in other words the transition kernel $q$ satisfies $q({\cal X}_{j+1} | x,a)=1$ for any $x\in {\cal X}_{j}$ and $a\in {\cal A}(x)$. Moreover, the reward function $u$ is piecewise bounded on ${\cal X}$, that is $\sup_{x\in {\cal X}_{j}}\sup_{a\in {\cal A}(x)} u^{+}(x,a)=m_{j}<\infty$ with $\sum_{t\in \NN} \beta^{t}m_{j+t}<\infty$ for any $j\in \NN$. By using the dynamic programming approach, it is proved in Theorem 1 in [@jaskiewicz11], the existence of an optimal policy which is stationary maximizing $\mathbb{E}^{\pi}\big[\sum_{t\in \NN} \beta^{t}u(X_{t},A_{t})\big]$ over $\Pi$ when the initial distribution is degenerated to the point $x\in {\cal X}$. It can be easily shown that the previous model can be embedded to our framework by choosing a set ${\cal X}_{j}$ satisfying $x\in {\cal X}_{j}$ and defining $\mathbf{X}_{t}={\cal X}_{j+t-1}$ for $t\in \NN^{*}$, $\mathbf{A}_{t}={\cal A}$ and $\Psi_{1} \colon \mathbf{X}_{1} \to 2^{\mathbf{A}_{1}} \backslash \{\emptyset\}$ with $\Psi_{1}(x)={\cal A}(x)$ for any $x\in \mathbf{X}_{1}$ and for $t\geq 2$, $\Psi_{t} : \boldsymbol{\mathcal{K}}_{t-1} \times \mathbf{X}_{t} \rightarrow 2^{\mathbf{A}_{t}}$ with $\Psi_{t}(x_{1},a_{1},\ldots,x_{t-1},a_{t-1},x_{t})={\cal A}(x_{t})$ for any $(x_{1},a_{1},\ldots,x_{t-1},a_{t-1},x_{t})\in \boldsymbol{\mathcal{K}}_{t-1} \times \mathbf{X}_{t}$ and where $\boldsymbol{\mathcal{K}}_{t}\subset\mathbf{H}_{t} \times \mathbf{A}_{t}$ denotes the graph of $\Psi_{t}$. In our context, the stochastic kernel $Q_{t}$ on $\mathbf{X}_{t+1}$ given $\boldsymbol{\mathcal{K}}_{t}$ is defined by $Q_{t}(\cdot | x_{1},a_{1},\ldots,x_{t},a_{t}) =q(\cdot |x_{t},a_{t})$ and the reward function is given by $r_{t}:\boldsymbol{\mathcal{K}}_{t}\rightarrow [-\infty,+\infty[$ with $r_{t}(x_{1},a_{1},\ldots,x_{t},a_{t})=\beta^{t}u(x_{t},a_{t})$ for any $(x_{1},a_{1},\ldots,x_{t},a_{t})\in \boldsymbol{\mathcal{K}}_{t}$. Let us show that this model satisfies Assumptions \[Regular-Hypotheses\] and \[Reward+Multifunction\]. Clearly, we have $$\sup_{p\geq n} \sup_{\pi\in\Pi} \bigg[ \sum_{t=n}^{p} \mathbb{E}^\pi \big[ r_t(X_{t},A_{t})\big] \bigg]^{+}\leq\sum_{t=n}^{\infty} \beta^{t}m_{j+t}$$ and so Assumption \[Condition-C\] is satisfied since $\sum_{t\in \NN} \beta^{t}m_{j+t}<\infty$. Moreover, Hypothesis \[Continuity-transition-kernel\] holds since the stochastic kernel $q$ is weakly continuous. Finally, it is easy to show that Assumption \[Reward+Multifunction\] is satisfied. Indeed, $r_{t}$ is upper semicontinuous implying that $r_{t}$ is strongly continuous on the graph of $\Psi_{t}$ since $\Psi_{t}$ is upper semicontinuous with compact values. Now, by construction $r_{t}$ is bounded above on $\boldsymbol{\mathcal{K}}_{t}$ showing the claim.
[To conclude this section, we present now an economic model borrowed from [@jaskiewicz11 Section 5, Example 1]. We will show that Assumptions \[Regular-Hypotheses\] and \[Reward+Multifunction\] hold for this example. Let ${\cal X}=[0,\infty)$ be the set of all possible capital stocks. The variable $X_t$ represents a capital stock at the beginning of the period $t$, during which a portion $A_t\in {\cal A}(X_t)=[0,X_t]$ of the capital is consumed. The evolution of the capital stock is given, for $t\in\mathbb{N}$, by $$\label{eq-ex2-ev}
X_{t+1}=(1+\rho)(X_t-A_t)+\Xi_t,$$ with initial condition $X_1$. In the evolution equation , $\rho>0$ is a constant rate of growth and $\Xi_t$ is a random income received in period $t$. The random variables $\{\Xi_t\}_{t\in\mathbb{N}^\ast}$ are assumed to be i.i.d. with probability distribution $\mu$ supported by $[0,z]$ for some $z\geq1$. The objective is to maximize the expected total utility of consumption, given by $r_t(X_t,A_t)=(A_t)^\sigma$ with $\sigma\in(0,1)$. For a positive real $d>0$ and $t\in\mathbb{N}^\ast$, introduce $${\cal X}_t=[0,k_t],\quad {\cal A}={\cal X}.$$ with $k_t=(1+\rho)^td+z(1+\rho)[(1+\rho)^{t-1}-1]/\rho$ and $$Q_t(C|x,a)=\int_0^z I_B((1+\rho)(x-a)+\xi)\mu(d\xi),$$ where $C\in\mathfrak{B}({\cal X}_{t+1})$, $x\in {\cal X}_t$ and $a\in{\cal A}(x)$. Observe that the reward function $r_{t}$ is not bounded from above on ${\cal X}\times {\cal A}$. However, it has been shown in [@jaskiewicz11 Section 5, Example 1] that the set of hypotheses introduced in [@jaskiewicz11] are satisfied provided that $m_t=k^\sigma_t$ and $\beta(1+\rho)^\sigma<1$. Therefore, by applying the arguments developed in the previous paragraph, we can claim that Assumptions \[Regular-Hypotheses\] and \[Reward+Multifunction\] are satisfied. ]{}
Appendix : Balder’s lemma {#app:balder}
=========================
We detail here the proof of [@balder89 Lemma 2.4], for the sake of clarity.
\[lem-balder\] Let $(\mu_{n})$ be a sequence of probability measures on a metric space $Y$ converging weakly to a probability measure $\mu$. Consider a function $u:Y\rightarrow [-\infty,+\infty[$, bounded from above, satisfying the following conditions: for any $\epsilon>0$, there exists a closed subset $Y_{\epsilon}$ of $Y$ such that $$\begin{aligned}
\label{Hyp-lem-balder}
\sup_{n\in\mathbb{N}}\mu_{n}(Y\setminus Y_{\epsilon}) <\epsilon\end{aligned}$$ and the restriction of $u$ on $Y_{\epsilon}$ is upper semicontinuous. Then, $$\begin{aligned}
\limsup_{n\rightarrow \infty} \int_{Y} u d\mu_{n} \leq \int_{Y} u d\mu.\end{aligned}$$
**Proof:** Let us assume in a first step that $u$ is also bounded from below. Let $\|u\|=\sup_{Y}|u|$ and define, for any $\epsilon>0$, the function $$u_\epsilon=u I_{Y_\epsilon} - \|u\|I_{Y\setminus Y_\epsilon}.$$ Let us show that $u_\epsilon$ is upper semicontinuous on $Y$. For $\beta\in \mathbb{R}$, consider the level set $$A_\beta=\{ x\in Y : u_\epsilon(x) < \beta\}=\{x\in Y_\epsilon : u(x) < \beta\}\cup \{x\in Y\setminus Y_\epsilon : -\|u\| < \beta\}.$$ Our aim is to show that $A_\beta$ is open. If $\beta\leq -\|u\|$, we clearly have $A_\beta=\emptyset$ which is an open set. Otherwise, we can write $$A_\beta=\{x\in Y_\epsilon : u_\epsilon(x)<\beta\}\cup (Y\setminus Y_\epsilon).$$ Since $u$ is upper semicontinuous on $Y_\epsilon$, the level set $\{x\in Y_\epsilon : u_\epsilon(x) < \beta\}$ is open in $Y_\epsilon$, and so there exists an open set $O$ of $Y$ such that $$\{x\in Y_\epsilon : u_\epsilon(x)<\beta\}=Y_\epsilon\cap O.$$ Thus $A_\beta=(Y_\epsilon\cap O)\cup (Y\setminus Y_\epsilon)$. Let $x\in A_\beta$. If $x\in Y\setminus Y_\epsilon$, by the closedness of $Y_\epsilon$, we can find $\eta>0$ such that $B(x,\eta)\subset Y\setminus Y_\epsilon\subset A_\beta$. Otherwise, $x\in Y_\epsilon\cap O$. In this case, since $x\in O$ which is an open set, we can find $\eta'>0$ such that $B(x,\eta')\subset O$. Then $$B(x,\eta')\cap A_\beta= [B(x,\eta')\cap Y_\epsilon \cap O] \cup [ B(x,\eta')\cap (Y\setminus Y_\epsilon)]=[B(x,\eta')\cap Y_\epsilon] \cup [ B(x,\eta')\cap Y\setminus Y_\epsilon]=B(x,\eta').$$ Thus $B(x,\eta')\subset A_\beta$ showing that $A_\beta$ is open. This implies that $u_\epsilon$ is upper semicontinuous on $Y$.\
Remark that $$\sup_{n\in\mathbb{N}}\left|\int_Y u d\mu_n -\int_Y u_\epsilon d\mu_n\right|\leq 2\epsilon \|u\|.$$ Now, using the fact that $u_\epsilon$ is upper semicontinuous and bounded on the whole space $Y$, $$\begin{aligned}
\limsup_{n\rightarrow \infty} \int_{Y} u d\mu_{n} \leq \limsup_{n\rightarrow \infty} \int_{Y} u_\epsilon d\mu_{n}+2\|u\|\epsilon
\leq \int_{Y} u_\epsilon d\mu+2\|u\|\epsilon \leq \int_{Y} u d\mu+2\|u\|\epsilon,\end{aligned}$$ showing the result.\
In the case where $u$ is no longer bounded from below, we introduce $u_m=u\vee (-m)$ for which the previous step holds. Then, we apply the monotone convergence theorem to obtain the result. $\Box$
[10]{}
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abstract: 'I show that the so-called causality paradox of time-dependent density functional theory arises from an incorrect formulation of the variational principle for the time evolution of the density. The correct formulation not only resolves the paradox in real time, but also leads to a new expression for the causal exchange-correlation kernel in terms of Berry curvature. Furthermore, I show that all the results that were previously derived from symmetries of the action functional remain valid in the present formulation. Finally, I develop a model functional theory which explicitly demonstrates the workings of the new formulation.'
author:
- Giovanni Vignale
title: 'On the “Causality Paradox" of Time-Dependent Density Functional Theory'
---
Introduction
============
Time-dependent density functional theory (TDDFT) [@TDFTBook; @Runge84; @Gross90] is becoming a standard tool for the computation of time-dependent phenomena in condensed matter physics and quantum chemistry. Naturally the growing number of application has generated a new interest in the foundations of the theory (see, for example the recent critique by Schirmer and Drew [@Schirmer07], and the rebuttal by Maitra, Burke, and van Leeuwen [@Maitra07]). In this paper I address the so-called “causality paradox", a problem that has troubled TDDFT for many years [@Gross96], and has been the object of many discussions and technically sophisticated resolutions [@Rajagopal96; @vanLeeuwen98; @vanLeeuwen01; @Mukamel05; @TDFTBook]. I do not disagree with those resolutions, but I wish to propose a new one, which I find technically simpler, more direct, and closer to the spirit of the original formulation of TDDFT.
As an introduction to the problem, let us recall that the formal basis of TDDFT is the Runge-Gross (RG) theorem[@Runge84], which establishes a biunivocal correspondence between the time dependent particle density $n({\bf r},t)$ of a many-body system and the potential $v({\bf r}, t)$ that gives rise to that density starting from an assigned quantum state $|\psi_0\rangle$ at the initial time $t=0$. According to the RG theorem the potential that gives rise to $n({\bf r},t)$ starting from $|\psi_0\rangle$ is determined by $n({\bf r},t)$ and $|\psi_0\rangle$ up to a time-dependent constant. Similarly, the time-dependent quantum state $|\psi(t)\rangle$ is determined by $n({\bf r},t)$ and $|\psi_0\rangle$ up to a time-dependent phase factor. In this sense, both $v({\bf r},t)$ and $|\psi(t) \rangle$ are [*functionals*]{} of $n({\bf r},t)$ and $|\psi_0\rangle$ over a time interval $0 \leq t \leq T$. They should be denoted by $v[n,|\psi_0\rangle;{\bf r},t]$ and $|\psi[n,|\psi_0\rangle]\rangle$ respectively. From now on, however, the dependence on the initial state will not be explicitly noted and we will write simply $v[n;{\bf r},t]$ and $|\psi[n]\rangle$. Admittedly, there is no proof that every reasonable density can be produced by some local potential: but it is generally assumed that the densities for which the theorem holds are dense enough in the space of densities to provide an arbitrarily good approximation to the physical densities one might encounter in real life. In this paper I will assume [*tout court*]{} that all the time-dependent densities can be produced by some local potential: i.e., all time-dependent densities are [*v-representable*]{}.
Another theorem, proved by van Leeuwen in 1999,[@vanLeeuwen99] extends and strengthens the RG theorem. According to van Leeuwen’s theorem, the density $n({\bf r},t)$ which evolves in an interacting system under the action of an external potential $v({\bf r},t)$ starting from an initial state $|\psi_0\rangle$, can be reproduced in a noninteracting system evolving under the action of an appropriate and uniquely determined potential $v_s({\bf r},t)$, starting from any initial state $|\psi_{s0}\rangle$ that has the same density and divergence of the current density as $|\psi_0\rangle$. This theorem provides the basis for the extremely useful Kohn-Sham method of calculating the density. The effective potential $v_s({\bf r},t)$ – a functional of $n({\bf r},t)$, $|\psi_0\rangle$, and $|\psi_{s0}\rangle$ – is known as the Kohn-Sham potential. The difference $v({\bf r},t) - v_s({\bf r},t)-v_H({\bf r},t)$, where $v_H({\bf r},t)$ is the Hartree potential, is known as the exchange-correlation (xc) potential, denoted by $v_{xc}[n;{\bf r},t]$ – also a functional of $n({\bf r},t)$, $|\psi_0\rangle$, and $|\psi_{s0}\rangle$.
In general the potentials $v({\bf r},t)$, $v_s({\bf r},t)$ and $v_{xc}({\bf r},t)$ depend on the density $n({\bf r}',t')$ at different positions and earlier times $t'<t$, but cannot be affected by changes in the density at later times $t'>t$. This obvious causality requirement implies that the functional derivatives of these potentials with respect to $n({\bf r},t')$ and, in particular, the [*exchange-correlation kernel*]{} $f_{xc}({\bf r},t;{\bf r}',t') \equiv \delta v_{xc}[n;{\bf r},t]/\delta n({\bf r}',t')$ vanish for $t<t'$.
An interesting question is whether the potentials $v({\bf r},t)$, $v_s({\bf r},t)$ and $v_{xc}({\bf r},t)$ can be generated from functional derivatives of an action functional $A[n,|\psi_0\rangle]$ (denoted from now on simply as $A[n]$) with respect to the density, in close analogy with static DFT, where the potentials are functional derivatives of energy functionals with respect to the density. The existence of such a representation was suggested by RG in their original paper [@Runge84], and was subsequently used by this author [@Vignale95; @Vignale96] to derive several theorems in TDFT. In the mid-nineties, however, it became clear that the representation was problematic to say the least [@Gross96]. If the potential could be written as a functional derivative of an action functional, $$\label{RGV}
v[n;{\bf r},t]\equiv\frac{\delta A[n]}{\delta n({\bf r},t)}~,$$ then we should also have $$\frac{\delta v[n;{\bf r},t]}{\delta n({\bf r}',t')} = \frac{\delta^2 A[n]}{\delta n({\bf r},t)\delta n({\bf r}',t')}~.$$ But this equation is patently false, because the left hand side is different from zero only for $t>t'$ (by the causality requirement), while the right hand side is symmetric under interchange of $t$ and $t'$.
This startling observation became quickly known as [*the causality paradox*]{} and prompted several sophisticated resolutions [@Rajagopal96; @vanLeeuwen98; @vanLeeuwen01; @Mukamel05]. The best known is the Van Leeuwen’s construction of a “Keldysh action" in pseudotime [@vanLeeuwen98; @vanLeeuwen01]. More recently Mukamel [@Mukamel05] has shown how to construct causal response functions from symmetrical functional derivatives corresponding to “Liouville space pathways". The gist of these resolutions is that causality is not violated, but one must use a more abstract mathematical apparatus (Keldysh formalism, or the Liouville superoperator method) in order to connect functional derivatives of the action to causal response functions.
In this paper I re-examine the “paradox" from a more elementary point of view. I show that the variational principle for the time-evolution of the wave function, when properly implemented as a variational principle for the density, yields an expression for the potential as the sum of two terms: (1) the functional derivative of the RG action and (2) a correction term, which cannot be expressed as a functional derivative, but is still simple enough to be included in all the formal proofs. So the gist of the present resolution is that we learn to write the potential as functional derivative of an action [*plus*]{} a boundary term. Among other benefits, this approach explains why theorems that were originally proved under the incorrect assumption (\[RGV\])[@Vignale95; @Vignale96], turned out to be true after all. Furthermore, it leads to interesting expressions for the inverse of the density-density response function and the xc kernel in terms of “Berry curvature".
This paper is organized as follows. In the next section I discuss in detail the failure of the stationary action principle for the density, and show how the correct causal expressions for the xc potential and the xc kernel are derived from a modified variational principle. In section \[Theorems\] I explain why in many cases one can still pretend that the xc potential is the functional derivative of the xc action and get correct results. Appendix \[ETS\] clears up a technical point about the equal-time singularities of causal response functions. Finally, Appendix \[Model\] presents a pedagogical “time-dependent position density functional theory", which is conceptually equivalent to the full-fledged TDDFT but can be solved exactly, illustrating the workings of the new formulation.
Variational principle for the density {#section2}
=====================================
The starting point is the time-dependent quantum variational principle,[@Saraceno81] according to which the time-dependent Schrödinger equation is equivalent to the requirement that the action $$\label{Frenkel}
A_V[|\psi\rangle] = \int_0^T \langle \psi(t)|i \partial_t - \hat H_V|\psi(t)\rangle dt$$ ($\hbar =1$) be stationary ($\delta A_V=0$) with respect to a arbitrary variations of the wave function which vanish at the ends of the time interval $0\leq t\leq T$, i.e. $|\delta\psi(0)\rangle=|\delta \psi(T)\rangle=0$. Here $$\label{def-HV}
\hat H_V=\hat H_0+\int V({\bf r},t) \hat n({\bf r}) d{\bf r}$$ is the sum of the internal hamiltonian $\hat H_0$ (kinetic + potential) and the interaction with an external time-dependent potential field $V({\bf r},t)$.
The proof is straightforward. The variation of $A_V$ induced by a variation $\delta|\psi\rangle \equiv |\delta\psi \rangle$ is $$\begin{aligned}
\label{variation1}
\delta A_V[|\psi\rangle] &=& \int_0^T \langle \delta \psi(t)|i \partial_t - \hat H_V|\psi(t)\rangle dt \nonumber\\ &+& \int_0^T \langle \psi(t)|i \partial_t - \hat H_V|\delta\psi(t)\rangle dt~,\end{aligned}$$ and the second term on the right hand side can be integrated by parts to yield $$\begin{aligned}
\label{variation2}
\delta A_V[|\psi\rangle] &=& \int_0^T \langle \delta \psi(t)|i \partial_t - \hat H_V|\psi(t)\rangle dt \nonumber\\ &+& \int_0^T \langle (i \partial_t - \hat H_V)\psi(t)|\delta\psi(t)\rangle dt \nonumber\\
&+&i\left.\langle \psi(t)|\delta\psi(t)\rangle\right\vert_{t=0}^{t=T}~.\end{aligned}$$ The last term on the right hand side vanishes by virtue of the boundary conditions on $|\delta \psi(t)\rangle$, and the vanishing of the first two terms is equivalent to the time-dependent Schrödinger equation $(i\partial_t - \hat H_V)|\psi(t)\rangle=0$.
Since $|\psi(t)\rangle$ is, by virtue of the RG theorem, a functional of $n$ and $|\psi_0\rangle$, Runge and Gross suggested that a stationary action principle for the density could be formulated in terms of the functional $$\begin{aligned}
\label{RGVAction}
A_V[n] &=& \int_0^T \langle \psi[n]|i \partial_t - \hat H|\psi[n]\rangle dt \nonumber\\
&=&A_0 [n] - \int_0^T V({\bf r},t)n({\bf r},t) d{\bf r} dt~,
\nonumber\\\end{aligned}$$ where the “internal action" $$\label{RGAction}
A_0 [n]\equiv \int_0^T \langle \psi[n]|i \partial_t - \hat H_0|\psi[n]\rangle dt$$ is a universal functional of the density and the initial state. Then, setting $\delta A_V = 0$ for arbitrary variations of the density we easily find $$V({\bf r},t)=\frac{\delta A_0[n]}{\delta n({\bf r},t)},$$ when $n({\bf r},t)$ is the density corresponding to $V({\bf r},t)$. This implies that the external potential, [*viewed as a functional of the density*]{}, is the functional derivative of the internal action with respect to the density: $$\label{v-functional}
v[n;{\bf r},t]\equiv\frac{\delta A_0[n]}{\delta n({\bf r},t)},$$ and the time evolution of the density is determined by requiring $v[n;{\bf r},t]=V({\bf r},t)$, where $V({\bf r},t)$ is the actual external potential. The only problem with Eq. (\[v-functional\]), which would otherwise be very useful, is that it plainly contradicts causality, as discussed in the introduction. What went wrong?
The problem arises from the fact that the Frenkel variational principle $\delta A_V=0$ is valid only for variations of $|\psi\rangle$ that vanish at the endpoints of the time interval under consideration, i.e. at $t=0$ and $t=T$. But a variation of the density at any time $t<T$ inevitably causes a change in the quantum state at time $T$. Therefore we can only set $|\delta\psi(0)\rangle =0$, but have no right to set $|\delta \psi(T)\rangle =0$. Taking this into account, and going back to Eq. (\[variation2\]) we see that the correct formulation of the variational principle for the density is not $\delta A_V=0$ but $$\label{pre-main}
\delta A_V[n] = i\langle \psi_T[n]|\delta \psi_T[n]\rangle~.$$ Here $|\psi_T[n]\rangle \equiv|\psi[n;T]\rangle$ is the quantum state at time $T$ regarded as a functional of the density (and of course of the initial state). So we see that the action functional is not stationary, but its variation must be equal to another functional of the density, which is given on the right hand side of Eq. (\[pre-main\]).
Taking the functional derivative of Eq. (\[pre-main\]) with respect to $n({\bf r},t)$ and making use of Eqs. (\[RGVAction\]) and (\[v-functional\]) we get $$\label{mainresult}
v[n;{\bf r},t] = \frac{\delta A_0[n]}{\delta n({\bf r},t)}-i\left \langle \psi_T[n] \left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right.~,$$ where $\left.\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right.$ is a compact representation for the functional derivative of $|\psi_T[n]\rangle$ with respect to density.
This is the main result of this paper, since it shows that the external potential (and hence also the Kohn-Sham potential and the xc potential) is not merely a functional derivative of the Runge-Gross action $A_0[n]$. Notice that the additional “boundary term" is real, in spite of the $i$, because the quantum state $|\psi_T[n]\rangle$ is normalized to $1$ independent of density, implying that $$\left \langle \psi_T[n] \left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right. =
-\left.\left \langle \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\vert \psi_T[n]\right\rangle$$ is a purely imaginary quantity.
At first sight, however, Eq. (\[mainresult\]) is still problematic because it appears to depend on the arbitrary upper limit of the time interval ($T$) and therefore also on the density at times $t'>t$. However, this is only appearance. The point is that both the functional derivative of $A_0[n]$ and the boundary term, considered separately, have a non-causal dependence on the density, but the dependence on $n(t')$ with $t'>t$ cancels out exactly when the two terms are combined!
Let us show this in detail. Consider, for example, an increment of the upper limit of the time interval from $T$ to $T+\Delta T$. The density $n({\bf r},t)$ must be smoothly continued to the larger time interval $[0,T+\Delta T]$, and the quantum state $|\psi[n]\rangle$ satisfies in this time interval the Schrödinger equation $$\label{SE1}
(i\partial _t - \hat H_0 - \hat v[n;t])|\psi[n;t]\rangle=0~,$$ where $$\hat v[n;t] \equiv \int v[n;{\bf r}',t]\hat n({\bf r}') d{\bf r}'~,$$ $\hat n({\bf r})$ is the density [*operator*]{}, and $v[n;{\bf r},t]$ is the potential that yields $n({\bf r},t)$. Now in view of Eq. (\[SE1\]) the internal action over the time-domain $[0,T]$ can be written as $$A_0[n]=\int_0^{T} \langle \psi[n;t']|\hat v[n;t'])|\psi[n;t']\rangle dt'~,$$ and its variation, due to the extension of the upper limit from $T$ to $T+\Delta T$ is, to first order in $\Delta T$ given by $$\Delta A_0[n]=\Delta T \langle \psi_T[n]|\hat v[n;T]|\psi_T[n]\rangle~.$$ Taking the functional derivative with respect to $n({\bf r},t)$, with $t$ within the interval $[0,T]$, we see that $$\label{Proof-step1}
\Delta \frac{\delta A_0[n]}{\delta n({\bf r},t)}=\Delta T \langle \psi_T[n]\left \vert \frac{\delta \hat v[n;T]}{\delta n({\bf r},t)}\right\vert\psi_T[n]\rangle~.$$ The reason why we could take the functional derivative inside the expectation value on the right hand side of this equation is that $\langle \psi_T[n]|\hat v[n;T]|\psi_T[n]\rangle = \int v[n;{\bf r}',T] n({\bf r}',T) d{\bf r}'$ depends on $n({\bf r},t)$ only through the potential functional $v[n;{\bf r}',T]$: the density $n({\bf r}',T)$ is, by definition, unaffected by a variation of the density at the earlier time $t$.[^1]
Consider now the variation of the boundary term of Eq. (\[mainresult\]) again due to the change of the upper limit of the time interval from $T$ to $T+\Delta T$. To first order in $\Delta T$ we have $$|\psi_{T+\Delta T}[n]\rangle -|\psi_{T}[n]\rangle = - i \Delta T \hat H[n;t]|\psi_{T}[n]\rangle~,$$ where $\hat H[n,t]= \hat H_0 +\hat v[n;T]$ is the full time-dependent hamiltonian regarded as a functional of the density. Substituting this in the variation of the boundary term we get $$\begin{aligned}
&&-i \Delta \left \langle \psi_T[n] \left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right. =\nonumber\\ && \Delta T
\left\langle \psi_T[n] \left \vert \hat H[n;T]\frac{\delta}{\delta n({\bf r},t)} - \frac{\delta}{\delta n({\bf r},t)}\hat H[n;T] \right\vert \psi_T[n]\right \rangle.\nonumber\\\end{aligned}$$
This seemingly complicated expression contains the commutator between the Hamiltonian $\hat H[n;T]$ and the functional derivative with respect to $n({\bf r},t)$. This commutator is simply $- \frac{\delta \hat H[n;T]}{\delta n({\bf r},t)}$, which is evidently equal to $ - \frac{\delta \hat v[n;T]}{\delta n({\bf r},t)}$. Thus the variation of the boundary term is given by $$\label{Proof-step2}
-i \Delta \left \langle \psi_T[n] \left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right. = -\Delta T
\left\langle \psi_T[n] \left\vert \frac{\delta \hat v[n;T]}{\delta n({\bf r},t)} \right\vert \psi_T[n]\right \rangle.$$ Combining the two equations (\[Proof-step1\]) and (\[Proof-step2\]) we see that net variation of the potential $v[n;{\bf r},t]$ is exactly zero. This means that we have the freedom to change at will the upper limit $T$ of the time interval: the value of $v[n;{\bf r},t]$ will not change, as long as $T$ remains larger than or equal to $t$. But this means that we can always choose $T=t$, and this proves that the potential at time $t$ does not depend on what happens to the density at times later than $t$. QED.
Now that we have the correct expression for the potential as a functional of the density it is easy to construct the other two potentials of interest, namely the Kohn-Sham potential and the xc potential. For the Kohn-Sham potential we simply have $$\label{mainresultKS}
v_s[n;{\bf r},t] = \frac{\delta A_{0s}[n]}{\delta n({\bf r},t)}-i\left \langle \psi_{sT}[n] \left\vert \frac{\delta \psi_{sT}[n]}{\delta n({\bf r},t)}\right\rangle \right.~,$$ where $A_{0s}$ is the internal action for a noninteracting system and $|\psi_{sT}[n]\rangle$ is the non-interacting version of $|\psi_T[n]\rangle$ (starting from an initial state $|\psi_{s0}\rangle$). For the exchange-correlation potential we get $$\begin{aligned}
\label{mainresultxc}
&&v_{xc}[n;{\bf r},t] = \frac{\delta A_{xc}[n]}{\delta n({\bf r},t)}\nonumber\\&&-i\left \langle \psi_T[n] \left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right.+i\left \langle \psi_{sT}[n] \left\vert \frac{\delta \psi_{sT}[n]}{\delta n({\bf r},t)}\right\rangle \right., \nonumber\\\end{aligned}$$ where $A_{xc}[n]$ is the usual xc action defined in the Runge-Gross paper as the difference $A_0[n]-A_{0s}[n]-A_H[n]$, where the last term is the Hartree action. Notice that $v_{xc}$ depends not only on the density, but also on the two initial states $|\psi_0\rangle$ and $|\psi_{s0}\rangle$.
The above formulas allow us to obtain an elegant expression for the functional derivative of the potential with respect to the density. Consider, for example, the functional derivative of $v(n;{\bf r},t)$. From Eq. (\[mainresult\]) we see that this is given by
$$\begin{aligned}
\label{result1}
\frac{\delta v[n;{\bf r},t]}{\delta n({\bf r}',t')}= \frac{\delta^2 A_0[n]}{\delta n({\bf r},t)\delta n({\bf r}',t')}-i\left \langle \psi_T[n]\left\vert\frac{\delta^2 \psi_T[n]}{\delta n({\bf r},t)\delta n({\bf r}',t')}\right\rangle \right.
-i\left \langle \frac{\delta\psi_T[n]}{\delta n ({\bf r}',t')}\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle\right.~.\end{aligned}$$
The first two terms on the right hand side are symmetric under interchange of ${\bf r},t$ and ${\bf r}',t'$, while the last term, which has the structure of a [*Berry curvature*]{}, is antisymmetric under the same interchange. Let us then subtract from Eq. (\[result1\]) the same equation with ${\bf r},t$ and ${\bf r}',t'$ interchanged. When $t>t'$ then $\frac{\delta v[n;{\bf r}',t']}{\delta n({\bf r},t)}$ vanishes because of causality, so on the left hand side only $\frac{\delta v[n;{\bf r},t]}{\delta n({\bf r}',t')}$ is left. And on the right hand side the symmetric terms cancel out, leaving only the Berry curvature term. The result is $$\begin{aligned}
\label{result2}
\frac{\delta v[n;{\bf r},t]}{\delta n({\bf r}',t')} &=& -i\left[\left \langle \frac{\delta\psi_T[n]}{\delta n ({\bf r}',t')}\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right.
-\left \langle \frac{\delta\psi_T[n]}{\delta n ({\bf r},t)}\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r}',t')}\right\rangle \right. \right] \nonumber\\
&=&2 \Im m \left \langle \frac{\delta\psi_T[n]}{\delta n ({\bf r}',t')}\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right.~~~~~~(t>t').\end{aligned}$$ Notice that the right hand side of this formula is independent of $T$ (as long as $T$ is larger that $t$ and $t'$) since, as we have shown, $v[n;{\bf r},t]$ satisfies the causality requirements.
The above argument determines $\frac{\delta v[n;{\bf r},t]}{\delta n({\bf r}',t')}$ for $t>t'$, but leaves open the possibility of singular contributions at $t=t'$. In Appendix \[ETS\] I show that indeed the functional derivative $\delta v[n;{\bf r},t]/\delta n({\bf r}',t')$ contains an equal-time singularity of the form $$C_0 \delta(t-t')+C_1\dot\delta(t-t')+C_2 \ddot\delta(t-t'),$$ where $C_0$, $C_1$, and $C_2$ are functionals of the density at time $t$ ($\equiv n_t$) and functions of ${\bf r}$ and ${\bf r}'$. $\dot \delta$ and $\ddot \delta$ denote, respectively, the first and the second derivative of the $\delta$-function with respect to its own argument. Thus the complete expression for $\delta v/\delta n$ has the following form: $$\begin{aligned}
\label{result3}
\frac{\delta v[n;{\bf r},t]}{\delta n({\bf r}',t')} = 2 \theta(t-t') \Im m \left \langle \frac{\delta\psi_T[n]}{\delta n ({\bf r}',t')}\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right. + \hat S_\infty[n_t;{\bf r},{\bf r}']\delta(t-t')~,\end{aligned}$$ where $\hat S_\infty[n_t;{\bf r},{\bf r}']$ is the differential operator $C_0+C_1\frac{d}{dt}+C_2\frac{d^2}{dt^2}$. The coefficient of the leading term, $C_2$, is independent of interactions, and the coefficient of the linear term, $C_1$, vanishes in the linear response limit. An explicit demonstration of the equal-time singularities is provided in Eq. (\[f-derivative-singular\]) of Appendix \[Model\].
Equal-time singularities also enter the expression of the exchange-correlation kernel. Taking into account the fact that $C_2$ is independent of interactions we find $$\begin{aligned}
\label{result3-xc}
f_{xc}[n;{\bf r},t,{\bf r}',t'] &\equiv& \frac{\delta v_{xc}[n;{\bf r},t]}{\delta n({\bf r}',t')} \nonumber\\&=& 2 \theta(t-t') \left\{\Im m \left \langle \frac{\delta\psi_T[n]}{\delta n ({\bf r}',t')}\left\vert \frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle \right. - \Im m \left \langle \frac{\delta\psi_{sT}[n]}{\delta n ({\bf r}',t')}\left\vert \frac{\delta \psi_{sT}[n]}{\delta n({\bf r},t)}\right\rangle \right.\right\}\nonumber\\&+&\Delta C_1[n_t;{\bf r},{\bf r}'] \dot \delta (t-t')+f_{xc,\infty}[n_t;{\bf r},{\bf r}']\delta(t-t'),\nonumber\\\end{aligned}$$ where $\Delta C_1$ is the difference between the coefficients $C_1$ in the interacting and non-interacting systems and $f_{xc,\infty}$ denotes the difference between the coefficients $C_0$ in the interacting and non-interacting system. In the linear response regime, i.e. when the time-dependent potential is a weak perturbation to the ground-state, $\Delta C_1$ vanishes and $f_{xc,\infty}$ reduces to the well-known infinite-frequency xc kernel of linear response theory.[@TheBook] This behavior is demonstrated in Appendix \[Model\] for our exactly solved model – see Eq. (\[fxc-derivative-singular\]).
Finally, I note that the adiabatic approximation to the xc kernel is given by $$f_{xc}^{ad}({\bf r},t,{\bf r}',t') \simeq f_{xc,0}({\bf r},{\bf r}',t) \delta(t-t')~,$$ where $$f_{xc,0}({\bf r},{\bf r}',t) =f_{xc,\infty}[n_t;{\bf r},{\bf r}']+\int_0^t f_{xc}[n;{\bf r},t,{\bf r}',t']dt'$$ is the integral of the exchange-correlation kernel over all times $t'$ earlier than $t$. The implicit assumption here is that the retardation range of the $xc$ kernel is much shorter than the time scale of variation of the density, so that the $xc$ kernel can effectively be approximated as a $\delta$-function on that time scale. The first term on the right hand side of this expression is the contribution of the “true" $\delta$-function terms of Eq. (\[result3-xc\]). The time integral in the second term is restricted to times strictly less than $t$.
Why did “$v_{xc}[n;{\bf r},t]=\frac{\delta A_{xc}[n]}{\delta n({\bf r},t)}$" work? {#Theorems}
==================================================================================
The incorrect representation of the xc potential as a functional derivative of the xc action played a significant role in the early development of TDDFT, particularly in the proof of theorems that depend on symmetries of the action functional. Consider, for example, the “zero-force theorem" [@Vignale95; @Vignale96], according to which the net force exerted by the xc potential on the system is zero. This theorem was originally derived from the apparent invariance of the $xc$ action under a homogeneous time-dependent translation of the density: $$\label{Axc-invariance}
A_{xc}[n']=A_{xc}[n]$$ where $n'({\bf r},t) = n({\bf r}+ {\bf x}(t),t)$, and ${\bf x}(t)$ is an arbitrary time-dependent displacement that vanishes at $t=0$. The invariance of the action under this transformation implies $$\label{Axc2}
\int \frac{\delta A_{xc}[n]}{\delta n({\bf r},t)} \vec \nabla_{{\bf r}} n({\bf r}) d{\bf r} =0~,$$ and an integration by parts leads to $$\int n({\bf r}) \vec \nabla_{{\bf r}} \frac{\delta A_{xc}[n]}{\delta n({\bf r},t)} d{\bf r} =0~.$$ This would be the zero-force theorem if we could identify $\frac{\delta A_{xc}[n]}{\delta n({\bf r},t)}$ with $v_{xc}[n;{\bf r},t]$ which, of course, is incorrect.
Fortunately, the resolution of the puzzle is now at hand. The point is that we are making [*two*]{} errors, which are luckily compensating each other, leaving us with the correct result. The first error is in Eq. (\[Axc-invariance\]): it is not true that the xc action remains invariant under the transformation $n \to n'$. The invariance of $A_{xc}$ was “derived" in Ref. ([@Vignale95]) by showing that the change of the internal action under this transformation depends only the density, not on the wave function, and therefore cancels out in the difference $A_0 - A_{0s}$. However, we failed to include the boundary term $i \langle \psi_T[n]|\delta \psi_T[n]\rangle$, which [*does*]{} depend on the wave function and therefore does not cancel out, causing the xc action to vary, to first order in $n'-n$, by $$\delta A_{xc} = i \langle \psi_T[n]|\delta \psi_T[n]\rangle - i \langle \psi_{sT}[n]|\delta \psi_{sT}[n]\rangle~,$$ where $|\delta \psi_T[n]\rangle \equiv |\psi_T[n']\rangle - |\psi_T[n]\rangle$ and $|\delta \psi_{sT}[n]\rangle \equiv |\psi_{sT}[n']\rangle - |\psi_{sT}[n]\rangle$. Therefore, Eq. (\[Axc-invariance\]) must be amended as follows: $$\begin{aligned}
\label{Axc2bis}
A_{xc}[n']=A_{xc}[n]+ i \langle \psi_T[n]|\delta \psi_T[n]\rangle - i \langle \psi_{sT}[n]|\delta \psi_{sT},[n]\rangle~.\end{aligned}$$ and Eq. (\[Axc2\]) is replaced by: $$\begin{aligned}
\label{Axc2bis}
\int \left\{\frac{\delta A_{xc}[n]}{\delta n({\bf r},t)}\right. - i \left.\left \langle \psi_T[n]\left\vert\frac{\delta \psi_T[n]}{\delta n({\bf r},t)}\right\rangle\right.+i \left \langle \psi_{sT}[n]\left\vert\frac{\delta \psi_{sT}[n]}{\delta n({\bf r},t)}\right\rangle\right.\right\} \vec \nabla_{{\bf r}} n({\bf r}) d{\bf r} =0~.\end{aligned}$$ Integrating by parts, and using the correct formula for $v_{xc}$, Eq. (\[mainresultxc\]) we do indeed recover the zero force theorem $\int n({\bf r}) \vec \nabla_{{\bf r}} v_{xc}[n;{\bf r},t] d{\bf r} = 0$.
The lesson is quite general: we are allowed to pretend that the $xc$ potential is the functional derivative of the action, provided we calculate that functional derivative incorrectly, i.e. ignoring the boundary contribution. This is exactly what we did (unwittingly) in our earlier papers.
Conclusion
==========
I believe that the foregoing analysis provides a straightforward and pedagogically transparent resolution of the causality paradox in TDDFT. Compared to the resolutions proposed in Refs. [@vanLeeuwen98; @Mukamel05] the present approach is obviously much closer to the spirit of the original RG paper. Furthermore, our approach allows us to understand why in many cases we can get correct results from an incorrect representation of the xc potential.
In closing I wish to emphasize that what we have derived here is a [*variational principle*]{} for the time-dependent density. A variational principle is not as strong as a minimum principle, yet it is strong enough to formulate a dynamical theory. While the absolute numerical value of the RG action has no physical meaning (because a multiplication of the wave function by an arbitrary phase factor changes its value by an arbitrary constant), it must be borne in mind that the action determines the dynamics through its variations, and those variations are independent of the arbitrary additive constant (a similar situation occurs in classical mechanics, since the Lagrangian is defined up to an arbitrary total derivative with respect to time).
Acknowledgements
================
I am very grateful to Ilya Tokatly for a critical reading of the manuscript and for suggesting the analysis of equal-time singularities in Appendix \[ETS\], and to Carsten Ullrich for pressing a discussion of the adiabatic limit. This work has been supported by DOE under Grant No. DE-FG02-05ER46203.
Equal time singularities in $\delta v[n;{\bf r},t]/\delta n({\bf r}',t')$ {#ETS}
=========================================================================
Following Tokatly [@Tokatly05; @Tokatly07] we write the exact local conservation laws for particle number and momentum: $$\partial_t n+\nabla \cdot {\bf j}=0$$ and $$m\partial_t j_i +\partial_j\left(mnu_i u_j+P_{ij}\right)+n\partial_iv=0,$$ where ${\bf j}$ is the current density, $j_i$ is its $i$-th cartesian component, $u_i=j_i/n$ is the velocity field, $\partial_j$ denotes the derivative with respect to $r_j$ (with implied summation over repeated indices), and finally $P_{ij}$ is the stress tensor. These equations are valid both for interacting and non-interacting systems and together define the time-dependent potential $v$ as a functional of the density, provided the velocity field and the stress tensor are regarded as functionals of the density. Interaction effects enter implicitly throughf the form of these functionals.
Taking the divergence of the second equation and making use of the first, we recast the system in the more explicit form $$\partial_i (n \partial_i v) = m\partial_t^2 n +m\partial_i(u_i\partial_t n) -m\partial_i({\bf j} \cdot \nabla u_i) -\partial_i\partial_jP_{ij}.$$ This equation can be formally solved, yielding $$\label{vofn}
v = \hat G \left[m\partial_t^2 n +m\partial_i(u_i\partial_t n) -m\partial_i({\bf j} \cdot \nabla u_i) -\partial_i\partial_jP_{ij}\right],$$ where $\hat G$ is the inverse of the operator $\partial_i n \partial_i$.
For the limited purpose of identifying the equal-time singularities in $\delta v/\delta n$ we can ignore any retardation in the functional dependence of $P_{ij}$ and $u_i$ on the density. Then the right hand side of Eq. (\[vofn\]) depends on the density and its first two derivatives $\partial_t n$ and $\partial_t^2 n$ at time $t$. No higher derivatives are involved. Then taking the functional derivative with respect to $n({\bf r}',t')$ we get a singularity proportional to $\ddot \delta(t-t')$ from $\partial_t^2 n ({\bf r},t)$, a singularity proportional to $\dot \delta(t-t')$ from $\partial_t n ({\bf r},t)$, and, of course, a singularity proportional to $\delta(t-t')$ from the terms that do not contain time-derivatives of the density.
We can furthermore say that the coefficient of $\ddot \delta(t-t')$ is completely free of interaction effects, since the interactions enter only in the functional $P_{ij}$ and $u_i$.[^2] And we observe that the $\dot \delta$ singularity vanishes in the linear response regime (small perturbations around the ground-state) because the current vanishes in the ground-state.
Very little can be said in general about the explicit form of the $\delta$-function singularity. In the linear response regime, the dependence of $P_{ij}$ on density has been extensively studied, but only in local or semi-local approximations.[@Tao07] A fully nonlinear, but still local approximation to $P_{ij}$, known as nonlinear elastic local deformation approximation, has been formulated by Tokatly [@Tokatly05] and studied by Ullrich and Tokatly [@Ullrich06] in a model calculation. An accessible review of this theory can be found in Chapter 8 of Ref. [@TDFTBook].
Time-dependent position functional theory {#Model}
=========================================
The evolution of electronic systems subjected to time-dependent potentials is in general too complicated to allow us to construct the functionals $v[n;{\bf r},t]$, $|\psi_T[n]\rangle$ etc.., even in the simplest non-trivial case of a two-electron system. However, a simpler “position-functional theory" can be easily formulated, which is conceptually equivalent to the full-fledged theory and allows us to demonstrate explicitly all the main points of the theory.
Our model is based on a two-particle system in one-dimension, with a time-dependent hamiltonian of the form $$\label{def-HF}
\hat H_F(t) = \frac{1}{2}\left[\hat p_1^2+\hat p_2^2+\hat x_1^2+\hat x_2^2\right]+\frac{k}{2}(\hat x_1-\hat x_2)^2 -F(t) \hat x_1,$$ where $\hat x_1$, $\hat x_2$ are the position operators of the two particles $\hat p_1$ and $\hat p_2$ the canonical momentum operators, and $F(t)$ is a time-dependent force, which acts only on particle $1$. The two particles are subjected to a parabolic potential well, and interact with each other with a harmonic force with “elastic contant" $k>0$. The idea is that $\hat x_1$ plays the role of the density operator; its expectation value $x_1(t)$ is the time-dependent density, $-F(t)$ is the external potential. As in TDDFT, one can show that the time-dependent position $x_1(t)$ and the initial state of the system at $t=0$ uniquely determine the force $f(t)$ that produces it; but in this case the functional $f[x_1;t]$ can be explicitly constructed.
For definiteness, we start from an initial state described by the wave function $$\label{def-psi0}
\psi_0(x_1,x_2)=Ce^{-X^2}e^{-\sqrt{1+2k}x^2/4}$$ where $X\equiv(x_1+x_2)/2$, $x\equiv x_1-x_2$, and $C=\frac {(1+2k)^{1/8}}{\pi^{1/2}}$ is the normalization constant. This is the ground-state of the hamiltonian for $F=0$. The time evolution of this state under the full time-dependent hamiltonian is $$\begin{aligned}
\label{wavefunction}
\psi(x_1,x_2,t)&=&Ce^{i\phi(t)}e^{-[X-X_c(t)]^2}e^{2i [X-X_c(t)]{\dot X}_c(t)}\nonumber\\ &\times& e^{-\sqrt{1+2k}[x-x_c(t)]^2/4}e^{i[x-x_c(t)]{\dot x}_c(t)/2]}~,\nonumber\\\end{aligned}$$ where $X_c(t)$ and $x_c(t)$ are the solutions of the classical equations of motion $$\begin{aligned}
&&\ddot X_c(t)+X_c(t) = F(t)/2\nonumber\\
&&\ddot x_c(t)+(1+2k)x_c(t) = F(t)\end{aligned}$$ with initial conditions $X_c(0)=\dot X_c(0) =0$ and $x_c(0)=\dot x_c(0) =0$. The phase factor $\phi(t)$ is the classical action (including the zero-point energy): $$\label{def-phi}
\phi(t)=-\frac{1+\sqrt{1+2k}}{2} t+\int_0^t L(t')dt'~,$$ where $$\begin{aligned}
\label{Lagrangian}
L =\dot X_c^2-X_c^2+FX_c
+\frac{\dot x_c^2}{4}- \left(\frac{1+2k}{4}\right)x_c^2+\frac{F}{2}x_c\nonumber\\\end{aligned}$$ is the classical Lagrangian.
The solution of the equations of motion is $$\begin{aligned}
\label{classicalsolutions}
X_c(t) &=& \frac{1}{2}\int_0^t\sin(t-t')F(t')dt'\nonumber\\
x_c(t) &=& \int_0^t\frac{\sin[\sqrt{1+2k}(t-t')]}{\sqrt{1+2k}} F(t')dt'~.\end{aligned}$$
The solution of the quantum mechanical problem is obtained by substituting Eqs. (\[classicalsolutions\]) into Eqs. (\[wavefunction\]), (\[def-phi\]), and (\[Lagrangian\]). It is immediately evident that $x_c(t)$ and $X_c(t)$ are the expectation values of the quantum mechanical center of mass operator $\hat X = (\hat x_1+\hat x_2)/2$ and relative position $\hat x = \hat x_1 - \hat x_2$ respectively. The expectation values of $\hat x_1$ and $\hat x_2$ are given by $$\begin{aligned}
\langle \psi(t)|\hat x_1|\psi(t)\rangle \equiv x_1(t) &=& X_c(t)+\frac{x_c(t)}{2}\nonumber\\
\langle \psi(t)|\hat x_2|\psi(t)\rangle \equiv x_2(t) &=& X_c(t)-\frac{x_c(t)}{2}~.\end{aligned}$$
Our task is now to express the external force and the wave function as functionals of $x_1(t)$ – the “density" of our model. To do this, we observe that $x_2(t)$ is related to $x_1(t)$ by the classical equation of motion $$\ddot x_2(t)+x_2(t)=-k[x_2(t)-x_1(t)]$$ with initial condition $x_2(0)=\dot x_2(0)=0$. The solution of this equation, for given $x_1(t)$, is $$\label{def-x2}
x_2(t) = k\int_0^t \frac{\sin[\sqrt{1+k} (t-t')]}{\sqrt{1+k}} x_1(t') dt'~.$$ From this we can express both $X_c(t)$ and $x_c(t)$ as functionals of $x_1(t)$, and hence the whole time-dependent wave function $\psi(x_1,x_2,t)$ as a functional of $x_1(t)$. Furthermore, the force $f$, which produces the evolution $x_1(t)$ is given by $f(t)=\ddot x_1(t)+x_1(t)+k[x_1(t)-x_2(t)]$. Upon substituting the functional dependence of $x_2(t)$ on $x_1(t')$ in the expression for $f(t)$ we obtain the force functional $$\begin{aligned}
\label{f-functional}
f[x_1;t]&=&\ddot x_1(t)+(1+k) x_1(t)\nonumber\\
&-& k ^2\int_0^t \frac{\sin[\sqrt{1+k} (t-t')]}{\sqrt{1+k}} x_1(t') dt'~.\nonumber\\\end{aligned}$$ Observe how the force is uniquely and causally determined by $x_1(t)$. Knowing the force we can construct the phase $\phi(t)$ (Eq. (\[def-phi\])) as a functional of $x_1$, by substituting $F=f[x_1]$ in the Lagrangian (\[Lagrangian\]). Finally, we construct the internal action functional $$\begin{aligned}
A_0[x_1]&=&\int_0^T \langle \psi[x_1;t]|i\partial_t-\hat H_0|\psi[x_1,t]\rangle dt\nonumber\\
&=&-\int_0^T f[x_1;t']x_1(t') dt'~,\end{aligned}$$ where $f[x_1,t]$ is given by Eq. (\[f-functional\]).
We are now in a position to demonstrate explicitly the connection between the force and the functional derivative of the action. Namely, we can prove that $$\label{check1}
-f[x_1,t]=\frac{\delta A_0[x_1]}{\delta x_1(t)}- i \left\langle \psi_T[x_1] \left\vert \frac{\delta \psi_T[x_1]}{\delta x_1(t)}\right\rangle\right.~.$$ where the state $|\psi_T[x_1]\rangle$ is described by the wave function (\[wavefunction\]), evaluated at time $T$ and expressed as a functional of $x_1[t]$ (the negative sign on the left hand side comes from the fact that the force enters the hamiltonian $\hat H_F$ (Eq. \[def-HF\]) with a sign opposite to that of the potential in Eq. (\[def-HV\]).)
The calculation is greatly simplified by the following two observations: (i) All the terms that involve an expectation value of $\hat X-X_c$ or $\hat x - x_c$ are obviously zero, and (ii) because $x_c(t)$ and $X_c(t)$ are solutions of the classical equation of motion, the variation of the phase $\phi(T)$ comes only from the variation of the force (regarded as a functional of of $x_1$), and from the variation of $x_2(t)$ at the upper limit of integration. In this way we easily arrive at $$-i\left\langle \psi_T [x_1]\left\vert \frac{\delta \psi_T[x_1]}{\delta x_1(t)}\right\rangle\right.
=\int_0^T \frac{\delta f[x_1;t']}{\delta x_1(t)}x_1(t') dt' ~,$$ from which Eq. (\[check1\]) follows at once.
Similarly, we can show that the “Berry curvature" $2\Im m \left\langle \frac {\delta \psi_T[x_1]}{\delta x_1(t')} \left \vert \frac {\delta \psi_T[x_1]}{\delta x_1(t)} \right\rangle \right.$ is given by $$\frac{\delta x_2(T)}{\delta x_1(t')} \frac{\delta \dot x_2(T)}{\delta x_1(t)}-\frac{\delta x_2(T)}{\delta x_1(t)} \frac{\delta \dot x_2(T)}{\delta x_1(t')},$$ so making use of Eq. (\[def-x2\]) we obtain $$2\Im m \left\langle \frac {\delta \psi_T[x_1]}{\delta x_1(t')} \left \vert \frac {\delta \psi_T[x_1]}{\delta x_1(t)} \right\rangle \right.= k^2 \frac{\sin[\sqrt{1+k}(t-t')]} {\sqrt{1+k}}~,$$ from which the arbitrary time $T$ has disappeared! Armed with this result it is an easy matter to verify that $$\label{f-derivative}
-\frac{\delta f[x_1;t]}{\delta x_1(t')} = 2\Im m \left\langle \frac {\delta \psi_T[x_1]}{\delta x_1(t')} \left \vert \frac {\delta \psi_T[x_1]}{\delta x_1(t)} \right\rangle \right.~,$$ for $t>t'$, in agreement with Eq. (\[result2\]). We can also verify the presence of the singularity at $t=t'$ discussed in section \[section2\] after Eq. (\[result3\]). Indeed, the functional derivative of the first two terms in the expression of our force functional, Eq. (\[f-functional\]), gives $$\label{f-derivative-singular}
\left. \frac{\delta f[x_1;t]}{\delta x_1(t')}\right\vert_{sing} = \ddot \delta (t-t')+(1+k)\delta(t-t')~.$$ Notice that there is no term proportional to $\dot \delta$ in this simple model.
Finally, we observe that the analogue of the xc potential – an xc force in this case – is $F_{xc}[x_1;t] = f_s[x_1,t]-f[x_1,t]$, where the non-interacting force functional $f_s[x_1,t]$ is obtained from Eq. (\[f-functional\]) simply by putting $k=0$, so that $$\label{xcforce-s}
f_s[x_1,t] = \ddot x_1(t)+x_1(t)$$ and $$\begin{aligned}
\label{xcforce}
F_{xc}[x_1;t] &=& -kx_1(t)+k^2\int_0^t \frac{\sin [\sqrt{1+k}(t-t')]}{\sqrt{1+k}} x_1(t')\nonumber\\
&=&-k\{x_1(t)-x_2[x_1;t]\}.\end{aligned}$$ Of course, $F_{xc}$ is nothing but the force exerted by the second particle on the first, expressed as a functional of the basic variable $x_1(t)$. It is worth noting that the singular term $\ddot x_1(t)$ has cancelled out in $F_{xc}$. The singular part of the functional derivative of $F_{xc}$ is simply a $\delta$-function $$\label{fxc-derivative-singular}
\left. \frac{\delta F_{xc}[x_1;t]}{\delta x_1(t')}\right\vert_{sing} = -k\delta(t-t')~,$$ in agreement with the discussion following Eq. (\[result3-xc\]).
Let us now demonstrate explicitly how, given the knowledge of the exact functional $F_{xc}[x_1;t]$ (Eq. (\[xcforce\])), one can calculate the evolution of $x_1(t)$ within a “Kohn-Sham scheme". First of all, we introduce the “Kohn-Sham hamiltonian" $$\hat H_s(t) = \frac{1}{2}\left[\hat p_1^2+\hat p_2^2+\hat x_1^2+\hat x_2^2\right] -F(t) \hat x_1 -F_{xc}[x_1;t] \hat x_1(t)~,$$ which describes two [non-interacting]{} particles, with an effective force $F+F_{xc}$ acting only on particle “1". Then we solve the time-dependent Schrödinger equation $[i\partial_t - \hat H_s(t)] |\psi_s(t) \rangle =0$, starting with the non-interacting ground-state $$\psi_{s0}(x_1,x_2) = \frac{1}{\sqrt{\pi}} e^{-x_1^2/2}e^{-x_2^2/2}~,$$ which clearly has the same expectation values of $\hat x_1$ and $\hat p_1$ as its interacting counterpart (\[def-psi0\]). The solution is the time-dependent “Kohn-Sham wave function" $$\begin{aligned}
\label{KSwavefunction}
\psi_s(x_1,x_2,t)&=&\frac{e^{i\phi_1(t)}}{\sqrt{\pi}} e^{-[x_1-x_{1c}(t)]^2/2}e^{i [x_1-x_{1c}(t)]{\dot x}_{1c}(t)}\nonumber\\ &\times& e^{-x_2^2/2}~,\nonumber\\\end{aligned}$$ where $x_{1c}(t)$ is the solution of the equation of motion $$\label{eom-x1c}
\ddot x_{1c}(t)+x_1(t)= F(t)+F_{xc}[x_1;t]$$ with initial conditions $x_{1c}(0)=\dot x_{1c}(0)=0$, and $$\phi_1(t)=-\frac{t}{2}+\int_0^t L_1(t') dt'~,$$ with $$L_1(t) = \frac{\dot x_{1c}^2}{2}-\frac{x_{1c}^2}{2} + (F(t)+F_{xc}[x_{1c};t])x_{1c}(t)~.$$
Finally, we make use of the Kohn-Sham wave function to calculate the expectation value of $\hat x_1$: $$\langle \psi_{s}(t)|\hat x_1|\psi_s(t)\rangle=x_{1c}(t)~.$$
The equation of motion (\[eom-x1c\]) for $x_{1c}(t)$ can be rewritten explicitly as an integro-differential equation $$\begin{aligned}
\ddot x_{1c}(t)+(1+k)x_{1c}(t) &&= F(t) \nonumber\\
&&+k^2\int_0^t \frac{\sin [\sqrt{1+k}(t-t')]}{\sqrt{1+k}} x_{1c}(t')~,\nonumber\\\end{aligned}$$ with initial conditions $x_{1c}(0)=\dot x_{1c}(0)=0$. This equation can be solved by Laplace transformation. Denoting by $x_{1c}(s)$ the Laplace transform of $x_{1c}(t)$ we get $$(s^2+1+k)x_{1c}(s) = F(s)+ \frac{k^2}{s^2+1+k}x_{1c}(s),$$ and finally $$\begin{aligned}
x_{1c}(s) &=& \frac{s^2+1+k}{(s^2+1)(s^2+1+2k)}F(s)\nonumber\\
&=& \left[\frac{1}{s^2+1}+\frac{1}{s^2+1+2k}\right]\frac{F(s)}{2}~.\end{aligned}$$ Going back to the time domain we finally obtain $$\begin{aligned}
x_{1c}(t)&=& \frac{1}{2}\int_0^t \sin(t-t')F(t')\nonumber\\
&+&\frac{1}{2}\int_0^t \frac{\sin [\sqrt{1+2k}(t-t')]}{\sqrt{1+2k}}F(t')~,\end{aligned}$$ which of course agrees with the exact solution $x_1(t)=X_c(t)+x_c(t)/2$ obtained from Eqs. (\[classicalsolutions\]).
In TDDFT we do not have the luxury of knowing the exact xc force functional. But this example shows that, if we knew it, we could use it to predict the exact evolution of the density. So there is every reason to believe that a good approximation to the exact xc potential, obtained by whatever means, would enable us to make good predictions for the time evolution of the density.
[100]{} , edited by M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke and E. K. U. Gross, (Springer, Berlin 2006). E. Runge and E.K.U. Gross, Phys. Rev. Lett. [**52**]{}, 997 (1984). E. K. U. Gross and W. Kohn, Adv. Quant. Chem., [**21**]{}, 255 (1990). J. Schirmer and A. Dreuw, Phys. Rev. A [**75**]{}, 022513 (2007). Neepa T. Maitra, Kieron Burke, and Robert van Leeuwen, arXiv:0710.0018. E. K. U. Gross, J. F. Dobson, and M. Petersilka, in [*Density Functional Theory II*]{}, edited by R.F. Nalewajski, Vol. 181 of Topics in Current Chemistry (Springer, Berlin, 1996), p. 81. A. K. Rajagopal, Phys. Rev. A [**54**]{}, 3916 (1996). R. van Leeuwen, Phys. Rev. Lett. [**80**]{}, 1280 (1998). R. van Leeuwen, Int. J. Mod. Phys. A [**15**]{}, 1969 (2001). S. Mukamel, Phys. Rev. A [**71**]{}, 024503 (2005). R. van Leeuwen, Phys. Rev. Lett. [**82**]{}, 3863 (1999). G. Vignale, Phys. Rev. Lett. [**74**]{}, 3233 (1995). G. Vignale and W. Kohn, [*Electronic Density Functional Theory: Recent Progress and New Directions*]{}, edited by J. F. Dobson, G. Vignale, and M. K. Das (Plenum Press, New York, 1996), p. 199. P. Kramer and M. Saraceno, [*Geometry of the Time-Dependent Variational Principle in Quantum Mechanics*]{}, Lecture Notes in Physics Vol.140, Springer-Verlag (Berlin) 1981. G.F. Giuliani and G. Vignale, [*Quantum Theory of the Electron Liquid*]{} (Cambridge University Press, Cambridge, 2005), Section 5.6 and Appendix 11. C. A. Ullrich, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett. [**74**]{}, 872 (1995). I. V. Tokatly, Phys. Rev. B [**71**]{}, 165104 (2005); [*ibidem*]{} 165105 (2005). I. V. Tokatly, Phys. Rev. B [**75**]{}, 125105 (2007). J. Tao, G. Vignale, and I. V. Tokatly, Phys. Rev. B [**76**]{}, 195126 (2007). C. A. Ullrich and I. V. Tokatly, Phys. Rev. B [**73**]{}, 235102 (2006).
[^1]: More formally, $\frac{\delta n({\bf r},T)}{\delta n({\bf r}',t)} = \delta ({\bf r} - {\bf r}') \delta(T-t)$, which vanishes if $T>t$. It should be noted that the nature of our variational principle for the density is such that we are allowed to impose the condition $\delta n({\bf r},t) =0$ at $t=0$ and $t=T$, just as in the formulation of the variational principle for the wave function we can assume that the variations of the wave function vanish at $t=0$ and $t=T$. This makes the derivation of Eq. (\[Proof-step1\]) even simpler.
[^2]: It is worth noting that in a time-dependent [*current*]{} density functional theory, $u_i = j_i/n$ would not be a functional but a basic variable, so interaction effects would enter only through $P_{ij}$.
|
---
abstract: 'In this paper, The author introduces the concepts of the GA-$s$-convex functions in the first sense and second sense and establishes some integral inequalities of Hermite-Hadamard type related to the GA-$s$-convex functions.'
address: |
Department of Mathematics, Faculty of Arts and Sciences,\
Giresun University, 28100, Giresun, Turkey.
author:
- İmdat İşcan
title: 'Hermite-Hadamard type inequalities for GA-$s$-convex functions '
---
Introduction
============
In this section, we firstly list several definitions and some known results.
The following concept was introduced by Orlicz in [@O61]:
Let $0<s\leq 1$. A function $f:I\subseteq
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\mathbb{R}
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_{+}\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ where $%
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\mathbb{R}
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_{+}=\left[ 0,\infty \right) $, is said to be $s$-convex in the first sense if$$f\left( \alpha x+\beta y\right) \leq \alpha ^{s}f(x)+\beta ^{s}f(y)$$for all $x,y\in I$ and $\alpha ,\beta \geq 0$ with $\alpha ^{s}+\beta ^{s}=1$. We denote this class of real functions by $K_{s}^{1}.$
In [@HM94], Hudzik and Maligranda considered the following class of functions:
A function $f:I\subseteq
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\mathbb{R}
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_{+}\rightarrow
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%BeginExpansion
\mathbb{R}
%EndExpansion
$ where $%
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\mathbb{R}
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_{+}=\left[ 0,\infty \right) $, is said to be $s$-convex in the second sense if$$f\left( \alpha x+\beta y\right) \leq \alpha ^{s}f(x)+\beta ^{s}f(y)$$for all $x,y\in I$ and $\alpha ,\beta \geq 0$ with $\alpha +\beta =1$ and $s$ fixed in $\left( 0,1\right] $. They denoted this by $K_{s}^{2}.$
It can be easily seen that for $s=1$, $s$-convexity reduces to ordinary convexity of functions defined on $[0,\infty )$.
In [@DF99], Dragomir and Fitzpatrick proved a variant of Hermite-Hadamard inequality which holds for the $s$-convex functions.
Suppose that $f:%
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\mathbb{R}
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_{+}\mathbb{\rightarrow }%
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\mathbb{R}
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_{+}$ is an $s$-convex function in the second sense, where $s\in \lbrack
0,1) $ and let $a,b\in \lbrack 0,\infty )$, $a<b$. If $f\in L\left[ a,b%
\right] $, then the following inequalities hold $$2^{s-1}f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\dint%
\limits_{a}^{b}f(x)dx\leq \frac{f(a)+f(b)}{s+1}\text{.} \label{1-1}$$the constant $k=\frac{1}{s+1}$ is the best possible in the second inequality in (\[1-1\]).
The above inequalities are sharp. For recent results and generalizations concerning $s$-convex functions see [@ADK11; @DF99; @HBI09; @I13b; @KBO07]
A function $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R}$ is said to be a GA-convex function on $I$ if$$f(x^{t}y^{1-t})\leq tf(x)+(1-t)f(y)$$holds for all $x,y\in I$ and $t\in \left[ 0,1\right] $, where $x^{t}y^{1-t}$ and $tf(x)+(1-t)f(y)$ are respectively called the weighted geometric mean of two positive numbers $x$ and $y$ and the weighted arithmetic mean of $f(x)$ and $f(y)$.
For $b>a>0$, let $G\left( a,b\right) =\sqrt{ab}$, $L\left( a,b\right)
=\left( b-a\right) /\left( \ln b-\ln a\right) $, $I\left( a,b\right) =\left(
1/e\right) \left( b^{b}/a^{a}\right) ^{1/(b-a)}$, $A\left( a,b\right) =\frac{%
a+b}{2}$, and $L_{p}\left( a,b\right) =\left( \frac{b^{p+1}-a^{p+1}}{%
(p+1)(b-a)}\right) ^{\frac{1}{p}}$,$\ p\in
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\mathbb{R}
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\backslash \left\{ -1,0\right\} $, be the geometric, logarithmic, identric, arithmetic and p-logarithmic means of $a$ and $b$, respectively. Then $$\min \left\{ a,b\right\} <G\left( a,b\right) <L\left( a,b\right) <I\left(
a,b\right) <A\left( a,b\right) <\max \left\{ a,b\right\} .$$
In [@ZJQ13], Zhang et al. established some Hermite-Hadamard type integral inequalities for GA-convex functions and applied these inequalities to construct several inequalities for special means and they used the following lemma to prove their results:
Let $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R}$ be a differentiable mapping on $I^{\circ }$, and $a,b\in I^{\circ }$,with $a<b$. If $f^{\prime
}\in L\left[ a,b\right] $, then$$bf(b)-af(a)-\dint\limits_{a}^{b}f(x)dx=\left( \ln b-\ln a\right)
\dint\limits_{0}^{1}b^{2t}a^{2\left( 1-t\right) }f^{\prime }\left(
b^{t}a^{1-t}\right) dt.$$
Also, the main inequalities in [@ZJQ13] are pointed out as follows:
Let $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I$ with $a<b$ and $f^{\prime }\in L\left[ a,b%
\right] .$ If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-convex on $%
\left[ a,b\right] $ for $q\geq 1$, then$$\left\vert bf(b)-af(a)-\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{%
\left[ \left( b-a\right) A\left( a,b\right) \right] ^{1-1/q}}{2^{1/q}}$$$$\times \left\{ \left[ L(a^{2},b^{2})-a^{2}\right] \left\vert f^{\prime
}(a)\right\vert ^{q}+\left[ b^{2}-L(a^{2},b^{2})\right] \left\vert f^{\prime
}(b)\right\vert ^{q}\right\} ^{1/q}.$$
Let $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I$ with $a<b$ and $f^{\prime }\in L\left[ a,b%
\right] .$ If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-convex on $%
\left[ a,b\right] $ for $q>1$, then$$\left\vert bf(b)-af(a)-\dint\limits_{a}^{b}f(x)dx\right\vert \leq \left( \ln
b-\ln a\right)$$$$\times \left[ L(a^{2q/(q-1)},b^{2q/(q-1)})-a^{2q/(q-1)}\right] ^{1-1/q}\left[
A\left( \left\vert f^{\prime }(a)\right\vert ^{q},\left\vert f^{\prime
}(b)\right\vert ^{q}\right) \right] ^{1/q}.$$
Let $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I$ with $a<b$ and $f^{\prime }\in L\left[ a,b%
\right] .$ If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-convex on $%
\left[ a,b\right] $ for $q>1$ and $2q>p>0$, then$$\left\vert bf(b)-af(a)-\dint\limits_{a}^{b}f(x)dx\right\vert \leq \frac{%
\left( \ln b-\ln a\right) ^{1-1/q}}{p^{1/q}}$$$$\begin{aligned}
&&\times \left[ L(a^{(2q-p)/(q-1)},b^{(2q-p)/(q-1)})\right] ^{1-1/q} \\
&&\times \left\{ \left[ L(a^{p},b^{p})-a^{p}\right] \left\vert f^{\prime
}(a)\right\vert ^{q}+\left[ b^{p}-L(a^{p},b^{p})\right] \left\vert f^{\prime
}(b)\right\vert ^{q}\right\} ^{1/q}.\end{aligned}$$
In [@ZCZ10], Zhang et al. established the following Hermite-Hadamard type inequality for GA-convex (concave) functions:
If $b>a>0$ and $f:\left[ a,b\right] \rightarrow
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\mathbb{R}
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$ is a differentiable GA-convex (concave) function then$$f\left( I(a,b)\right) \leq (\geq )\frac{1}{b-a}\dint\limits_{a}^{b}f(x)dx%
\leq (\geq )\frac{b-L(a,b)}{b-a}f(b)+\frac{L(a,b)-a}{b-a}f(a).$$
In [@I13], the author proved the following identity and established some new Hermite-Hadamard-like type inequalities for the geometrically convex functions.
\[1.1\]Let $f:I\subseteq \left( 0,\infty \right) \mathbb{\rightarrow }%
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\mathbb{R}
%EndExpansion
$ be a differentiable mapping on $I^{\circ }$, and $a,b\in I$,with $a<b$. If $f^{\prime }\in L\left[ a,b\right] $, then$$f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}\frac{f(x)%
}{x}dx$$$$=\frac{\left( \ln b-\ln a\right) }{4}\left[ a\dint\limits_{0}^{1}t\left(
\frac{b}{a}\right) ^{\frac{t}{2}}f^{\prime }\left( a^{1-t}\left( ab\right) ^{%
\frac{t}{2}}\right) dt-b\dint\limits_{0}^{1}t\left( \frac{a}{b}\right) ^{%
\frac{t}{2}}f^{\prime }\left( b^{1-t}\left( ab\right) ^{\frac{t}{2}}\right)
dt\right] ,$$$$\frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}\frac{f(x)}{x}dx$$$$=\frac{\left( \ln b-\ln a\right) }{2}\left[ a\dint\limits_{0}^{1}t\left(
\frac{b}{a}\right) ^{t}f^{\prime }\left( a^{1-t}b^{t}\right)
dt-b\dint\limits_{0}^{1}t\left( \frac{a}{b}\right) ^{t}f^{\prime }\left(
b^{1-t}a^{t}\right) dt\right]$$
In this paper, we will give concepts $s$-GA-convex functions in the first and second sense and establish some new integral inequalities of Hermite-Hadamard-like type for these classes of functions by using Lemma [1.1]{}.
Definitions of GA-$s$-convex functions in the first and second sense
====================================================================
Now it is time to introduce two concepts, GA-$s$-convex functions in the first and second sense.
Let $0<s\leq 1$. A function $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R%
}$ is said to be a GA-$s$-convex (concave) function in the first sense on $I$ if$$f(x^{t}y^{1-t})\leq (\geq )\ t^{s}f(x)+(1-t^{s})f(y)$$holds for all $x,y\in I$ and $t\in \left[ 0,1\right] $.
Let $0<s\leq 1$. A function $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R%
}$ is said to be a GA-$s$-convex (concave) function in the second sense on $I
$ if$$f(x^{t}y^{1-t})\leq (\geq )\ t^{s}f(x)+(1-t)^{s}f(y)$$holds for all $x,y\in I$ and $t\in \left[ 0,1\right] $.
It is clear that when $s=1$, GA-$s$-convex functions in the first and second sense become GA-convex functions.
Inequalities for GA-$s$-convex functions in the first and second sense
======================================================================
Now we are in a position to establish some inequalities of Hermite–Hadamard type for GA-$s$-convex functions in the first and second sense
\[3.1\] Let $0<s\leq 1$. Suppose that $f:I\subseteq \left( 0,\infty
\right) \mathbb{\rightarrow }%
%TCIMACRO{\U{211d} }%
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\mathbb{R}
%EndExpansion
$ is GA-$s$-convex function in the first sense and $a,b\in I$ with $a<b$. If $f\in L\left[ a,b\right] $, then one has the inequalities:$$f\left( \sqrt{ab}\right) \leq \frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}\frac{%
f(x)}{x}dx\leq \frac{f(a)+sf(b)}{s+1} \label{3-1}$$
As $f$ is GA-$s$-convex function in the first sense, we have, for all $%
x,y\in I$$$f\left( \sqrt{xy}\right) \leq \frac{1}{2^{s}}f(x)+\left( 1-\frac{1}{2^{s}}%
\right) f(y). \label{3-1a}$$Now, let $x=a^{1-t}b^{t}$ and $y=a^{t}b^{1-t}$ with $t\in \left[ 0,1\right] $. Then we get by (\[3-1a\]) that:$$f\left( \sqrt{ab}\right) \leq \frac{1}{2^{s}}f(a^{1-t}b^{t})+\left( 1-\frac{1%
}{2^{s}}\right) f(a^{t}b^{1-t})$$for all $t\in \left[ 0,1\right] $. Integrating this inequality on $\left[ 0,1%
\right] $, we deduce the first part of (\[3-1\]).
Secondly, we observe that for all $t\in \left[ 0,1\right] $$$f(a^{t}b^{1-t})\leq t^{s}f(a)+(1-t^{s})f(b).$$Integrating this inequality on $\left[ 0,1\right] $, we get$$\dint\limits_{0}^{1}f(a^{t}b^{1-t})dt\leq \frac{f(a)+sf(b)}{s+1}.$$As the change of variable $x=a^{t}b^{1-t}$ gives us that$$\dint\limits_{0}^{1}f(a^{t}b^{1-t})dt=\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx,$$the second inequality in (\[3-1\]) is proved.
Similarly to Theorem \[3.1\], we will give the following theorem for GA-$s$-convex function in the second sense:
Suppose that $f:I\subseteq \left( 0,\infty \right) \mathbb{\rightarrow }%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is GA-$s$-convex function in the second sense and $a,b\in I$ with $a<b$. If $f\in L\left[ a,b\right] $, then one has the inequalities:$$2^{s-1}f\left( \sqrt{ab}\right) \leq \frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\leq \frac{f(a)+f(b)}{s+1} \label{3-2}$$
As $f$ is GA-$s$-convex function in the second sense, we have, for all $%
x,y\in I$$$f\left( \sqrt{xy}\right) \leq \frac{f(x)+f(y)}{2^{s}}. \label{3-2a}$$Now, let $x=a^{1-t}b^{t}$ and $y=a^{t}b^{1-t}$ with $t\in \left[ 0,1\right] $. Then we get by (\[3-2a\]) that:$$f\left( \sqrt{ab}\right) \leq \frac{f(a^{1-t}b^{t})+f(a^{t}b^{1-t})}{2^{s}}$$for all $t\in \left[ 0,1\right] $. Integrating this inequality on $\left[ 0,1%
\right] $, we deduce the first part of (\[3-2\]).
Secondly, we observe that for all $t\in \left[ 0,1\right] $$$f(a^{t}b^{1-t})\leq t^{s}f(a)+(1-t)^{s}f(b).$$Integrating this inequality on $\left[ 0,1\right] $, we get$$\dint\limits_{0}^{1}f(a^{t}b^{1-t})dt\leq \frac{f(a)+f(b)}{s+1}.$$As the change of variable $x=a^{t}b^{1-t}$ gives us that$$\dint\limits_{0}^{1}f(a^{t}b^{1-t})dt=\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx,$$the second inequality in (\[3-2\]) is proved.
The constant $k=1/(s+1)$ for $s\in \left( 0,1\right] $ is the best possible in the second inequality in (\[3-2\]). Indeed, as the mapping $f:\left[ a,b%
\right] \rightarrow \left[ a,b\right] $ given $f(x)=s+1$, $0<a<b$, is GA-$s$-convex in the second sense and $$\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}\frac{f(x)}{x}dx=s+1=\frac{f(a)+f(b)%
}{s+1}$$
\[2.1\]Let $f:I\subseteq \left( 0,\infty \right) \mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I^{\circ }$ with $a<b$ and $%
f^{\prime }\in L\left[ a,b\right] .$
a\) If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the second sense on $\left[ a,b\right] $ for $q\geq 1$ and $s\in \left( 0,1%
\right] ,$ then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \label{3-3}$$$$\begin{aligned}
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{q%
}}\left[ a\left\{ c_{1}(s,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(s,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{3}(s,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{4}(s,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \label{3-4}$$$$\begin{aligned}
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{3-\frac{1}{q%
}}\left[ a\left\{ c_{1}(s,q/2)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{3}(s,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{4}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$where $$\begin{aligned}
c_{1}(s,q) &=&\dint\limits_{0}^{1}t\left( 1-t\right) ^{s}\left( \frac{b}{a}%
\right) ^{qt}dt,\ c_{2}(s,q)=\dint\limits_{0}^{1}t^{s+1}\left( \frac{b}{a}%
\right) ^{qt}dt, \label{3-41} \\
c_{3}(s,q) &=&\dint\limits_{0}^{1}t\left( 1-t\right) ^{s}\left( \frac{a}{b}%
\right) ^{qt}dt,\ c_{4}(s,q)=\dint\limits_{0}^{1}t^{s+1}\left( \frac{a}{b}%
\right) ^{qt}dt, \notag\end{aligned}$$b) If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $ for $q\geq 1$ and $s\in \left( 0,1%
\right] ,$ then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \label{3-5}$$$$\begin{aligned}
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{q%
}}\left[ a\left\{ c_{5}(s,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(s,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{6}(s,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{4}(s,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \label{3-6}$$$$\begin{aligned}
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{3-\frac{1}{q%
}}\left[ a\left\{ c_{5}(s,q/2)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{6}(s,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{4}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] ,\end{aligned}$$where$$c_{5}(s,q)=\dint\limits_{0}^{1}t\left( 1-t^{s}\right) \left( \frac{b}{a}%
\right) ^{qt}dt,\ c_{6}(s,q)=\dint\limits_{0}^{1}t\left( 1-t^{s}\right)
\left( \frac{a}{b}\right) ^{qt}dt. \label{3-61}$$
a\) (1) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the second sense on $\left[ a,b\right] $, from lemma \[1.1\] and power mean inequality, we have$$\begin{aligned}
&&\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \\
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left[ a\dint\limits_{0}^{1}t%
\left( \frac{b}{a}\right) ^{t}\left\vert f^{\prime }\left(
a^{1-t}b^{t}\right) \right\vert dt+b\dint\limits_{0}^{1}t\left( \frac{a}{b}%
\right) ^{t}\left\vert f^{\prime }\left( b^{1-t}a^{t}\right) \right\vert dt%
\right] \end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \left( \frac{b}{a}\right) }{2}\left(
\dint\limits_{0}^{1}tdt\right) ^{1-\frac{1}{q}}\left(
\dint\limits_{0}^{1}t\left( \frac{b}{a}\right) ^{qt}\left\vert f^{\prime
}\left( a^{1-t}b^{t}\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \notag
\\
&&+\frac{b\ln \left( \frac{b}{a}\right) }{2}\left(
\dint\limits_{0}^{1}tdt\right) ^{1-\frac{1}{q}}\left(
\dint\limits_{0}^{1}t\left( \frac{a}{b}\right) ^{qt}\left\vert f^{\prime
}\left( b^{1-t}a^{t}\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}}
\label{3-3a}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \left( \frac{b}{a}\right) }{2}\left( \frac{1}{2}\right)
^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}t\left( \frac{b}{a}\right)
^{qt}\left( \left( 1-t\right) ^{s}\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}} \\
&&+\frac{b\ln \left( \frac{b}{a}\right) }{2}\left( \frac{1}{2}\right) ^{1-%
\frac{1}{q}}\left( \dint\limits_{0}^{1}t\left( \frac{a}{b}\right)
^{qt}\left( \left( 1-t\right) ^{s}\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}}\end{aligned}$$$$\begin{aligned}
&\leq &a\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{%
q}}\left\{ c_{1}(s,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}+c_{2}(s,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right\} ^{\frac{1}{q}} \\
&&+b\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{q}%
}\left\{ c_{3}(s,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}+c_{4}(s,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}.\end{aligned}$$
\(2) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the second sense on $\left[ a,b\right] $, from lemma \[2.1\] and power mean inequality, we have$$\begin{aligned}
&&\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \\
&\leq &\frac{\ln \frac{b}{a}}{4}\left[ a\dint\limits_{0}^{1}t\left( \frac{b}{%
a}\right) ^{\frac{t}{2}}\left\vert f^{\prime }\left( a^{1-t}\left( ab\right)
^{\frac{t}{2}}\right) \right\vert dt+b\dint\limits_{0}^{1}t\left( \frac{a}{b}%
\right) ^{\frac{t}{2}}\left\vert f^{\prime }\left( b^{1-t}\left( ab\right) ^{%
\frac{t}{2}}\right) \right\vert dt\right] \end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \frac{b}{a}}{4}\left( \dint\limits_{0}^{1}tdt\right) ^{1-%
\frac{1}{q}}\left( \dint\limits_{0}^{1}t\left( \frac{b}{a}\right) ^{\frac{qt%
}{2}}\left\vert f^{\prime }\left( a^{1-t}\left( ab\right) ^{\frac{t}{2}%
}\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \notag \\
&&+\frac{b\ln \frac{b}{a}}{4}\left( \dint\limits_{0}^{1}tdt\right) ^{1-\frac{%
1}{q}}\left( \dint\limits_{0}^{1}t\left( \frac{a}{b}\right) ^{\frac{qt}{2}%
}\left\vert f^{\prime }\left( b^{1-t}\left( ab\right) ^{\frac{t}{2}}\right)
\right\vert ^{q}dt\right) ^{\frac{1}{q}} \label{3-4a}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \frac{b}{a}}{4}\left( \frac{1}{2}\right) ^{1-\frac{1}{q}%
}\left( \dint\limits_{0}^{1}t\left( \frac{b}{a}\right) ^{\frac{qt}{2}}\left(
\left( 1-t\right) ^{s}\left\vert f^{\prime }\left( a\right) \right\vert
^{q}+t^{s}\left\vert f^{\prime }\left( \sqrt{ab}\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}} \\
&&+\frac{b\ln \frac{b}{a}}{4}\left( \frac{1}{2}\right) ^{1-\frac{1}{q}%
}\left( \dint\limits_{0}^{1}t\left( \frac{a}{b}\right) ^{\frac{qt}{2}}\left(
\left( 1-t\right) ^{s}\left\vert f^{\prime }\left( b\right) \right\vert
^{q}+t^{s}\left\vert f^{\prime }\left( \sqrt{ab}\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}}\end{aligned}$$$$\begin{aligned}
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{3-\frac{1}{q%
}}\left[ a\left\{ c_{1}(s,q/2)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{5}(s,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{6}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] ,\end{aligned}$$b) (1) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $, from the inequality (\[3-3a\]), we have$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert$$$$\begin{aligned}
&\leq &\frac{a\ln \left( \frac{b}{a}\right) }{2}\left( \frac{1}{2}\right)
^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}t\left( \frac{b}{a}\right)
^{qt}\left( \left( 1-t^{s}\right) \left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}} \\
&&+\frac{b\ln \left( \frac{b}{a}\right) }{2}\left( \frac{1}{2}\right) ^{1-%
\frac{1}{q}}\left( \dint\limits_{0}^{1}t\left( \frac{a}{b}\right)
^{qt}\left( \left( 1-t^{s}\right) \left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}}\end{aligned}$$$$\begin{aligned}
&\leq &a\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{%
q}}\left\{ c_{5}(s,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}+c_{2}(s,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right\} ^{\frac{1}{q}} \\
&&+b\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{q}%
}\left\{ c_{6}(s,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}+c_{4}(s,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}.\end{aligned}$$
\(2) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $, the inequality (\[3-6\]) is easily obtained by using the inequality (\[3-4a\]).
If taking $s=1$ in Theorem \[2.1\], we can derive the following inequalities for GA-convex.
\[2.1a\]Let $f:I\subseteq \mathbb{R}_{+}\mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I^{\circ }$ with $a<b$ and $%
f^{\prime }\in L\left[ a,b\right] .$ If $\left\vert f^{\prime }\right\vert
^{q}$ is GA-convex on $\left[ a,b\right] $ for $q\geq 1$, then$$\begin{aligned}
&&\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \label{3-7a} \\
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{2-\frac{1}{q%
}}\left[ a\left\{ c_{1}(1,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(1,q)\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}\right. \notag \\
&&\left. +b\left\{ c_{3}(1,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{4}(1,q)\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right\} ^{\frac{1}{q}}\right] , \notag\end{aligned}$$$$\begin{aligned}
&&\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \label{3-7b} \\
&\leq &\ln \left( \frac{b}{a}\right) \left( \frac{1}{2}\right) ^{3-\frac{1}{q%
}}\left[ a\left\{ c_{1}(1,q/2)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+c_{2}(1,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \notag \\
&&\left. +b\left\{ c_{3}(1,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{4}(1,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] \notag\end{aligned}$$
If taking $q=1$ in Theorem \[2.1\], we can derive the following corollary.
Let $f:I\subseteq \left( 0,\infty \right) \mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I^{\circ }$ with $a<b$ and $%
f^{\prime }\in L\left[ a,b\right] .$
a\) If $\left\vert f^{\prime }\right\vert $ is GA-$s$-convex function in the second sense on $\left[ a,b\right] $, $s\in \left( 0,1\right] $, then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left[ \left(
ac_{1}(s,1)+bc_{4}(s,1)\right) \left\vert f^{\prime }\left( a\right)
\right\vert \right. \\
&&\left. +\left( bc_{3}(s,1)+ac_{2}(s,1)\right) \left\vert f^{\prime }\left(
b\right) \right\vert \right] \end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{4}\left[ ac_{1}(s,1/2)\left%
\vert f^{\prime }\left( a\right) \right\vert +bc_{3}(s,1/2)\left\vert
f^{\prime }\left( b\right) \right\vert \right. \\
&&\left. +\left( ac_{2}(s,1/2)+bc_{4}(s,1/2)\right) \left\vert f^{\prime
}\left( \sqrt{ab}\right) \right\vert \right] \end{aligned}$$where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ are defined by (\[3-41\]).
b\) If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $, $s\in \left( 0,1\right] $, then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left[ \left(
ac_{5}(s,1)+bc_{4}(s,1)\right) \left\vert f^{\prime }\left( a\right)
\right\vert \right. \\
&&\left. +\left( ac_{2}(s,1)+bc_{6}(s,1)\right) \left\vert f^{\prime }\left(
b\right) \right\vert \right] ,\end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{4}\left[ ac_{5}(s,1/2)\left%
\vert f^{\prime }\left( a\right) \right\vert +bc_{6}(s,1/2)\left\vert
f^{\prime }\left( b\right) \right\vert \right. \\
&&\left. +\left( ac_{2}(s,1/2)+bc_{4}(s,1/2)\right) \left\vert f^{\prime
}\left( \sqrt{ab}\right) \right\vert \right] ,\end{aligned}$$where $c_{2}$, $c_{4}$, $c_{5}$, $c_{6}$ are defined by (\[3-41\]) and (\[3-61\]).
\[2.2\]Let $f:I\subseteq \left( 0,\infty \right) \mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I^{\circ }$ with $a<b$ and $%
f^{\prime }\in L\left[ a,b\right] .$
a\) If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the second sense on $\left[ a,b\right] $ for $q>1$ and $s\in \left( 0,1%
\right] ,$ then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \label{3-8}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{7}(s,q)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q)\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{9}(s,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \label{3-9}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{4}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{7}(s,q/2)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q/2)\left\vert f^{\prime }\left(
\sqrt{ab}\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{9}(s,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$where $$\begin{aligned}
c_{7}(s,q) &=&\dint\limits_{0}^{1}\left( 1-t\right) ^{s}\left( \frac{b}{a}%
\right) ^{qt}dt,\ c_{8}(s,q)=\dint\limits_{0}^{1}t^{s}\left( \frac{b}{a}%
\right) ^{qt}dt, \label{3-91} \\
c_{9}(s,q) &=&\dint\limits_{0}^{1}\left( 1-t\right) ^{s}\left( \frac{a}{b}%
\right) ^{qt}dt,\ c_{10}(s,q)=\dint\limits_{0}^{1}t^{s}\left( \frac{a}{b}%
\right) ^{qt}dt, \notag\end{aligned}$$b) If $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $ for $q>1$ and $s\in \left( 0,1\right]
,$ then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \label{3-10}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{11}(s,q)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q)\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{12}(s,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \label{3-11}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{4}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{11}(s,q/2)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q/2)\left\vert f^{\prime }\left(
\sqrt{ab}\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{12}(s,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] ,\end{aligned}$$where$$c_{11}(s,q)=\dint\limits_{0}^{1}\left( 1-t^{s}\right) \left( \frac{b}{a}%
\right) ^{qt}dt,\ c_{12}(s,q)=\dint\limits_{0}^{1}\left( 1-t^{s}\right)
\left( \frac{a}{b}\right) ^{qt}dt. \label{3-101}$$
a\) (1) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the second sense on $\left[ a,b\right] $, from lemma \[1.1\] and Hölder inequality, we have$$\begin{aligned}
&&\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \\
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left[ a\dint\limits_{0}^{1}t%
\left( \frac{b}{a}\right) ^{t}\left\vert f^{\prime }\left(
a^{1-t}b^{t}\right) \right\vert dt+b\dint\limits_{0}^{1}t\left( \frac{a}{b}%
\right) ^{t}\left\vert f^{\prime }\left( b^{1-t}a^{t}\right) \right\vert dt%
\right] \end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \left( \frac{b}{a}\right) }{2}\left(
\dint\limits_{0}^{1}t^{\frac{q}{q-1}}dt\right) ^{1-\frac{1}{q}}\left(
\dint\limits_{0}^{1}\left( \frac{b}{a}\right) ^{qt}\left\vert f^{\prime
}\left( a^{1-t}b^{t}\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}} \notag
\\
&&+\frac{b}{2}\ln \left( \frac{b}{a}\right) \left( \dint\limits_{0}^{1}t^{%
\frac{q}{q-1}}dt\right) ^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}\left(
\frac{a}{b}\right) ^{qt}\left\vert f^{\prime }\left( b^{1-t}a^{t}\right)
\right\vert ^{q}dt\right) ^{\frac{1}{q}} \label{3-8a}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}\left( \frac{b}{a}\right)
^{qt}\left( \left( 1-t\right) ^{s}\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}} \\
&&+\frac{b}{2}\ln \left( \frac{b}{a}\right) \left( \frac{q-1}{2q-1}\right)
^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}\left( \frac{a}{b}\right)
^{qt}\left( \left( 1-t\right) ^{s}\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{7}(s,q)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q)\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{9}(s,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] .\end{aligned}$$
(2)Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $, from lemma \[1.1\] and Hölder inequality, we have$$\begin{aligned}
&&\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \notag \\
&\leq &\frac{\ln \frac{b}{a}}{4}\left[ a\dint\limits_{0}^{1}t\left( \frac{b}{%
a}\right) ^{\frac{t}{2}}\left\vert f^{\prime }\left( a^{1-t}\left( ab\right)
^{\frac{t}{2}}\right) \right\vert dt+b\dint\limits_{0}^{1}t\left( \frac{a}{b}%
\right) ^{\frac{t}{2}}\left\vert f^{\prime }\left( b^{1-t}\left( ab\right) ^{%
\frac{t}{2}}\right) \right\vert dt\right] \label{3-9a}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{a\ln \frac{b}{a}}{4}\left( \frac{q-1}{2q-1}\right) ^{1-\frac{1}{%
q}}\left( \dint\limits_{0}^{1}\left( \frac{b}{a}\right) ^{\frac{qt}{2}%
}\left( \left( 1-t\right) ^{s}\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right) dt\right) ^{\frac{1}{q}} \\
&&+\frac{b\ln \frac{b}{a}}{4}\left( \frac{q-1}{2q-1}\right) ^{1-\frac{1}{q}%
}\left( \dint\limits_{0}^{1}\left( \frac{a}{b}\right) ^{\frac{qt}{2}}\left(
\left( 1-t\right) ^{s}\left\vert f^{\prime }\left( b\right) \right\vert
^{q}+t^{s}\left\vert f^{\prime }\left( \sqrt{ab}\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{4}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{7}(s,q/2)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q/2)\left\vert f^{\prime }\left(
\sqrt{ab}\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{9}(s,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] .\end{aligned}$$b) (1) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $, from the inequality (\[3-8a\]), we have$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert$$$$\begin{aligned}
&\leq &\frac{a\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}\left( \frac{b}{a}\right)
^{qt}\left( \left( 1-t^{s}\right) \left\vert f^{\prime }\left( a\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( b\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}} \\
&&+\frac{b\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}\right)
^{1-\frac{1}{q}}\left( \dint\limits_{0}^{1}\left( \frac{a}{b}\right)
^{qt}\left( \left( 1-t^{s}\right) \left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+t^{s}\left\vert f^{\prime }\left( a\right) \right\vert
^{q}\right) dt\right) ^{\frac{1}{q}}\end{aligned}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{11}(s,q)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(s,q)\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{12}(s,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(s,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] \end{aligned}$$
\(2) Since $\left\vert f^{\prime }\right\vert ^{q}$ is GA-$s$-convex function in the first sense on $\left[ a,b\right] $, the inequality (\[3-6\]) is easily obtained by using the inequality (\[3-9a\]).
If taking $s=1$ in Theorem \[2.2\], we can derive the following inequalities for GA-convex.
\[2.2a\]Let $f:I\subseteq \left( 0,\infty \right) \mathbb{\rightarrow R}$ be differentiable on $I^{\circ }$, and $a,b\in I^{\circ }$ with $a<b$ and $%
f^{\prime }\in L\left[ a,b\right] .$ If $\left\vert f^{\prime }\right\vert
^{q}$ is GA-convex function in the second sense on $\left[ a,b\right] $ for $%
q>1$, then$$\left\vert \frac{f(a)+f(b)}{2}-\frac{1}{\ln b-\ln a}\dint\limits_{a}^{b}%
\frac{f(x)}{x}dx\right\vert \label{3-12a}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{2}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{7}(1,q)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(1,q)\left\vert f^{\prime }\left(
b\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{9}(1,q)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(1,q)\left\vert f^{\prime }\left( a\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] ,\end{aligned}$$$$\left\vert f\left( \sqrt{ab}\right) -\frac{1}{\ln b-\ln a}%
\dint\limits_{a}^{b}\frac{f(x)}{x}dx\right\vert \label{3-12b}$$$$\begin{aligned}
&\leq &\frac{\ln \left( \frac{b}{a}\right) }{4}\left( \frac{q-1}{2q-1}%
\right) ^{1-\frac{1}{q}}\left[ a\left\{ c_{7}(1,q/2)\left\vert f^{\prime
}\left( a\right) \right\vert ^{q}+c_{8}(1,q/2)\left\vert f^{\prime }\left(
\sqrt{ab}\right) \right\vert ^{q}\right\} ^{\frac{1}{q}}\right. \\
&&\left. +b\left\{ c_{9}(1,q/2)\left\vert f^{\prime }\left( b\right)
\right\vert ^{q}+c_{10}(1,q/2)\left\vert f^{\prime }\left( \sqrt{ab}\right)
\right\vert ^{q}\right\} ^{\frac{1}{q}}\right] ,\end{aligned}$$where $c_{7}$, $c_{8}$, $c_{9}$, $c_{10}$ are defined by (\[3-91\]) and (\[3-101\]).
Application to Special Means
============================
Let $0<a<b,$and $q\geq 1.$ Then $$\begin{aligned}
&&\left\vert A\left( a,b\right) -L\left( a,b\right) \right\vert \leq \left[
\ln \left( \frac{b}{a}\right) \right] ^{1-\frac{1}{q}}\left( \frac{1}{2}%
\right) ^{2-\frac{1}{q}}\left( \frac{1}{q}\right) ^{\frac{1}{q}} \\
&&\times \left[ \left\{ b^{q}-L(a^{q},b^{q})\right\} ^{\frac{1}{q}}+\left\{
L(a^{q},b^{q})-a^{q}\right\} ^{\frac{1}{q}}\right]\end{aligned}$$$$\begin{aligned}
&&\left\vert G\left( a,b\right) -L\left( a,b\right) \right\vert \leq \left[
\ln \left( \frac{b}{a}\right) \right] ^{1-\frac{1}{q}}\left( \frac{1}{2}%
\right) ^{3-\frac{1}{q}}\left( \frac{2}{q}\right) ^{\frac{1}{q}} \\
&&\times \left[ \sqrt{a}\left\{ b^{q/2}-L(a^{q/2},b^{q/2})\right\} ^{\frac{1%
}{q}}+\sqrt{b}\left\{ L(a^{q/2},b^{q/2})-a^{q/2}\right\} ^{\frac{1}{q}}%
\right] .\end{aligned}$$
The assertion follows from the inequalities (\[3-7a\]) and (\[3-7b\]) in Corollary \[2.1a\] for $f(x)=x,\ x>0$.
Let $0<a<b\leq 1,$and $q>1.$ Then $$\left\vert A\left( a,b\right) -L\left( a,b\right) \right\vert \leq \ln
\left( \frac{b}{a}\right) \left( \frac{q-1}{2q-1}\right) ^{1-\frac{1}{q}}L^{%
\frac{1}{q}}(a^{q},b^{q})$$$$\left\vert G\left( a,b\right) -L\left( a,b\right) \right\vert \leq \frac{1}{2%
}\ln \left( \frac{b}{a}\right) \left( \frac{q-1}{2q-1}\right) ^{1-\frac{1}{q}%
}L^{\frac{1}{q}}(a^{q},b^{q})A\left( \sqrt{a},\sqrt{b}\right) .$$
The assertion follows from the inequalities (\[3-12a\]) and (\[3-12b\]) in Corollary \[2.2a\] for $f(x)=x,\ x>0$.
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---
abstract: 'Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension $1$. We import ideas from geometric group theory to study groups of birational transformations, and prove that a group of birational transformations that satisfies a fixed point property on $(0)$ cubical complexes is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.'
address:
- 'Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes'
- 'CNRS and Univ Lyon, Univ Claude Bernard Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne'
author:
- Serge Cantat and Yves de Cornulier
bibliography:
- 'references-fw.bib'
date: 'November 9, 2017'
title: 'Commensurating actions of birational groups and groups of pseudo-automorphisms'
---
Introduction
============
Birational transformations and pseudo-automorphisms
---------------------------------------------------
Let $X$ be a quasi-projective variety, over an algebraically closed field $\bfk$. Denote by $\Bir(X)$ the group of birational transformations of $X$ and by $\Aut(X)$ the subgroup of (regular) automorphisms of $X$. For the affine space of dimension $n$, automorphisms are invertible transformations $f\colon \A^n_\bfk \to \A^n_\bfk$ such that both $f$ and $f^{-1}$ are defined by polynomial formulas in affine coordinates: $$f(x_1, \ldots, x_n)=(f_1, \ldots, f_n), \; \; f^{-1}(x_1, \ldots, x_n)=(g_1, \ldots, g_n)$$ with $f_i$, $g_i \in \bfk[x_1, \ldots, x_n]$. Similarly, birational transformations of $\A^n_\bfk$ are given by rational formulas, i.e. $f_i$, $g_i\in \bfk(x_1, \ldots, x_n)$.
Birational transformations may contract hypersurfaces. Roughly speaking, [**[pseudo-automorphisms]{}**]{} are birational transformations that act as automorphisms in codimension $1$. Precisely, a birational transformation $f\colon X \dasharrow X$ is a pseudo-automorphism if there exist Zariski-open subsets $\U$ and $\V$ in $X$ such that $X\smallsetminus \U$ and $X\smallsetminus \V$ have codimension $\geq 2$ and $f$ induces an isomorphism from $\U$ to $\V$. The pseudo-automorphisms of $X$ form a group, which we denote by $\Psaut(X)$. For instance, all birational transformations of Calabi-Yau manifolds are pseudo-automorphisms; and there are examples of such manifolds for which $\Psaut(X)$ is infinite while $\Aut(X)$ is trivial (see [@Cantat-Oguiso:2015]). Pseudo-automorphisms are studied in Section \[par:pseudo-automorphisms\].
\[d\_psr\] Let $\Gamma\subset\Bir(X)$ be a group of birational transformations of an irreducible projective variety $X$. We say that $\Gamma$ is [**pseudo-regularizable**]{} if there exists a triple $(Y,\U,\varphi)$ where
1. $Y$ is a projective variety and $\varphi\colon Y\dasharrow X$ is a birational map;
2. $\U$ is a dense Zariski open subset of $Y$;
3. $\varphi^{-1}\circ \Gamma \circ \varphi$ yields an action of $\Gamma$ by pseudo-automorphisms on $\U$.
More generally if $\alpha:\Gamma\to\Bir(X)$ is a homomorphism, we say that it is pseudo-regularizable if $\alpha(\Gamma)$ is pseudo-regularizable.
One goal of this article is to use rigidity properties of commensurating actions, a purely group-theoretic concept, to show that many group actions are pseudo-regularizable. In particular, we exhibit a class of groups for which all actions by birational transformations on projective varieties are pseudo-regularizable.
Property [[(FW)]{}]{} {#par:Intro-FW}
---------------------
The class of groups we shall be mainly interested in is characterized by a fixed point property appearing in several related situations, for instance for actions on $(0)$ cubical complexes. Here, we adopt the viewpoint of commensurated subsets. Let $\Gamma$ be a group, and $\Gamma \times S\to S$ an action of $\Gamma$ on a set $S$. Let $A$ be a subset of $S$. One says that $\Gamma$ [**[commensurates]{}**]{} $A$ if the symmetric difference $$\gamma(A)\triangle A= \left( \gamma(A)\smallsetminus A \right) \, \cup\, \left( A\smallsetminus \gamma(A) \right)$$ is finite for every element $\gamma$ of $\Gamma$. One says that $\Gamma$ [**[transfixes]{}**]{} $A$ if there is a subset $B$ of $S$ such that $A\triangle B$ is finite and $B$ is $\Gamma$-invariant: $\gamma(B)=B,$ for every $\gamma$ in $\Gamma$.
A group $\Gamma$ has [**[Property [[(FW)]{}]{}]{}**]{} if, given any action of $\Gamma$ on any set $S$, all commensurated subsets of $S$ are automatically transfixed. For instance, $\SL_2(\Z[\sqrt{5}])$ and $\SL_3(\Z)$ have Property [[(FW)]{}]{}, but non-trivial free groups do not share this property. Property [[(FW)]{}]{} is discussed in Section \[par:FW\].
Let us mention that among various characterizations of Property [[(FW)]{}]{} (see [@Cornulier:Survey-FW]), one is: every combinatorial action of $\Gamma$ on a (0) cube complex fixes some cube. Another, for $\Gamma$ finitely generated, is that all its infinite Schreier graphs are one-ended.
Pseudo-regularizations
----------------------
Let $X$ be a projective variety. The group $\Bir(X)$ does not really act on $X$, because there are indeterminacy points; it does not act on the set of hypersurfaces either, because some of them may be contracted. As we shall explain, one can introduce the set $\Hyp(X)$ of all irreducible and reduced hypersurfaces in all birational models $X'\dasharrow X$ (up to a natural identification). Then there is a natural action of the group $\Bir(X)$ on this set, given by strict transforms. The rigorous construction of this action follows from a general categorical framework, which is developed in Section \[S:cat\]. Moreover, this action commensurates the subset $\Hy(X)$ of hypersurfaces of $X$. This construction leads to the following result.
[*[Let $X$ be a projective variety over an algebraically closed field. Let $\Gamma$ be a subgroup of $\Bir(X)$. If $\Gamma$ has Property [[(FW)]{}]{}, then $\Gamma$ is pseudo-regularizable.]{}*]{}
There is also a relative version of Property [[(FW)]{}]{} for pairs of groups $\Lambda \leq \Gamma$, which leads to a similar pseudo-regularization theorem for the subgroup $\Lambda$: this is discussed in Section \[par:distorsion\], with applications to distorted birational transformations.
\[rem:intro-extreme-cases\] Theorem A provides a triple $(Y,\U,\varphi)$ such that $\varphi$ conjugates $\Gamma$ to a group of pseudo-automorphisms on the open subset $\U\subset Y$. There are two extreme cases for the pair $(Y,\U)$ depending on the size of the boundary $Y\smallsetminus \U$. If this boundary is empty, $\Gamma$ acts by pseudo-automorphisms on a projective variety $Y$. If the boundary is ample, its complement $\U$ is an affine variety, and then $\Gamma$ actually acts by regular automorphisms on $\U$ (see Section \[par:affine\]). Thus, in the study of groups of birational transformations, [*[pseudo-automorphisms of projective varieties and regular automorphisms of affine varieties deserve specific attention]{}*]{}.
Classification in dimension $2$
-------------------------------
In dimension $2$, pseudo-automorphisms do not differ much from automorphisms; for instance, $\Psaut(X)$ coincides with $\Aut(X)$ if $X$ is a smooth projective surface. Thus, for groups with Property [[(FW)]{}]{}, Theorem A can be used to reduce the study of birational transformations to the study of automorphisms of quasi-projective surfaces. Combining results of Danilov and Gizatullin on automorphisms of affine surfaces with a theorem of Farley on groups of piecewise affine transformations of the circle, we will be able to prove the following theorem.
**
Let $X$ be a smooth, projective, and irreducible surface, over an algebraically closed field. Let $\Gamma$ be an infinite subgroup of $\Bir(X)$. If $\Gamma$ has Property [[(FW)]{}]{}, there is a birational map $\varphi\colon Y\dasharrow X$ such that
1. $Y$ is the projective plane $\P^2$, a Hirzebruch surface $\Hirz_m$ with $m\geq 1$, or the product of a curve $C$ by the projective line $\P^1$. If the characteristic of the field is positive, $Y$ is the projective plane $\P^2_\bfk$.
2. $\varphi^{-1}\circ \Gamma\circ \varphi$ is contained in $\Aut(Y)$.
The group $\Aut(Y)$ has finitely many connected components for all surfaces $Y$ listed in Assertion (1) of Theorem B. Thus, changing $\Gamma$ into a finite index subgroup $\Gamma_0$, one gets a subgroup of $\Aut(Y)^0$. Here $\Aut(Y)^0$ denotes the connected component of the identity of $\Aut(Y)$; this is an algebraic group, acting algebraically on $Y$.
\[applx\] Groups with Kazhdan Property (T) satisfy Property [[(FW)]{}]{}. Thus, Theorem B extends Theorem A of [@Cantat:Annals] and the present article offers a new proof of that result.
Theorem B can also be applied to the group $\SL_2(\Z[\sqrt{d}])$, where $d\ge 2$ is a non-square positive integer. Thus, every action of this group on a projective surface by birational transformations is conjugate to an action by regular automorphisms on $\P^2_\bfk$, the product of a curve $C$ by the projective line $\P^1_\bfk$, or a Hirzebruch surface. Moreover, in this case, Margulis’ superrigidity theorem can be combined with Theorem B to get a more precise result, see §\[scorcorSL2\].
In general, for a variety $X$ one can ask whether $\Bir(X)$ transfixes $\Hy(X)$, or equivalently is pseudo-regularizable. For a surface $X$, this holds precisely when $X$ is not birationally equivalent to the product of the projective line with a curve. See §\[surf\_birt\] for more precise results.
Acknowledgement
---------------
This work benefited from interesting discussions with Jérémy Blanc, Vincent Guirardel, Vaughan Jones, Christian Urech, and Junyi Xie.
Pseudo-automorphisms {#par:pseudo-automorphisms}
====================
This preliminary section introduces useful notation for birational transformations and pseudo-automorphisms, and presents a few basic results.
Birational transformations
--------------------------
Let $X$ and $Y$ be two irreducible and reduced algebraic varieties over an algebraically closed field $\bfk$. Let $f\colon X\dasharrow Y$ be a birational map. Choose dense Zariski open subsets $U\subset X$ and $V\subset Y$ such that $f$ induces an isomorphism $f_{U,V}\colon U\to V$. Then the graph $\mathfrak{G}_f$ of $f$ is defined as the Zariski closure of $\{(x,f_{U,V}(x)):x\in U\}$ in $X\times Y$; it does not depend on the choice of $U$ and $V$. The graph $\mathfrak{G}_f$ is an irreducible variety; both projections $u\colon \mathfrak{G}_f \to X$ and $v\colon \mathfrak{G}_f \to Y$ are birational morphisms and $f=v\circ u^{-1}$.
We shall denote by $\Ind(f)$ the indeterminacy set of the birational map $f$.
\[normal2\] Let $f:X\dashrightarrow Y$ be a rational map, with $X$ a normal variety and $Y$ a projective variety. Then the indeterminacy set of $f$ has codimension $\ge 2$.
The transformation of the affine plane $(x,y)\mapsto (x,y/x)$ is birational, and its indeterminacy locus is the line $\{x=0\}$: this set of co-dimension $1$ is mapped “to infinity”. If the affine plane is compactified by the projective plane, the transformation becomes $[x:y:z]\mapsto [x^2:yz:xz]$, with two indeterminacy points.
Assume that $X$ is normal; in particular, it is smooth in codimension $1$. The [**[jacobian determinant]{}**]{} $\Jac(f)(x)$ is defined in local coordinates, on the smooth locus of $X$, as the determinant of the differential $df_x$; $\Jac(f)$ depends on the coordinates, but its zero locus does not. The zeroes of $\Jac(f)$ form a hypersurface of the smooth part of $X$; the zero locus of $\Jac(f)$ will be defined as the Zariski closure of this hypersurface in $X$. The [**[exceptional set]{}**]{} of $f$ is the subset of $X$ along which $f$ is not a local isomorphism onto its image; by a corollary of Zariski’s main theorem, it coincides with the union of $\Ind(f)$, the zero locus of $\Jac(f)$, and additional parts which are contained in the singular locus of $X$ and have therefore codimension $\geq 2$. Its complement is the largest open subset on which $f$ is a local isomorphism (see [@Milne:BookAG; @Vakil], for instance).
The [**[total transform]{}**]{} of a subset $Z\subset X$ is denoted by $f_*(Z)$. If $Z$ is not contained in $\Ind(f)$, we denote by $f_\circ(Z)$ its [**[strict transform]{}**]{}, defined as the Zariski closure of $f(Z\smallsetminus \Ind(f))$. We say that a hypersurface $W\subset Z$ is contracted if it is not contained in the indeterminacy set and the codimension of its strict transform is larger than $1$.
Pseudo-isomorphisms
-------------------
A birational map $f\colon X \dasharrow Y$ is a [**[pseudo-isomorphism]{}**]{} if one can find Zariski open subsets $\U\subset X$ and $\V \subset Y$ such that
- $f$ realizes a regular isomorphism from $\U$ to $\V$ and
- $X\smallsetminus \U$ and $Y\smallsetminus \V$ have codimension $\geq 2$.
Pseudo-isomorphisms from $X$ to itself are called [**[pseudo-automorphisms]{}**]{} (see § \[par:Intro-FW\]). [*[The set of pseudo-automorphisms of $X$ is a subgroup $\Psaut(X)$ of $\Bir(X)$]{}*]{}.
Start with the standard birational involution $\sigma_n\colon \P^n_\bfk\dasharrow \P^n_\bfk$ which is defined in homogeneous coordinates by $\sigma_n[x_0:\ldots : x_n]=[x_0^{-1}:\ldots : x_n^{-1}].$ Blow-up the $(n+1)$ vertices of the simplex $\Delta_n=\{[x_0:\ldots : x_n ]; \; \prod x_i=0\}$; this provides a smooth rational variety $X_n$ together with a birational morphism $\pi\colon X_n\to \P^n_\bfk$. Then, $\pi^{-1}\circ \sigma_n\circ \pi$ is a pseudo-automorphism of $X_n$, and is an automorphism if $n\leq 2$.
\[pro:pseudo-isomorphism\] Let $f\colon X\dasharrow Y$ be a birational map between two (irreducible, reduced) normal algebraic varieties. Assume that the codimension of the indeterminacy sets of $f$ and $f^{-1}$ is at least $2$. Then, the following properties are equivalent:
1. The birational maps $f$ and $f^{-1}$ do not contract any hypersurface.
2. The jacobian determinants of $f$ and $f^{-1}$ do not vanish on the regular loci of $X\smallsetminus \Ind(f)$ and $Y\smallsetminus \Ind(f^{-1})$ respectively.
3. For every smooth point $q\in X\smallsetminus \Ind(f)$, $f$ is a local isomorphism from a neighborhood of $q$ to a neighborhood of $f(q)$, and the same holds for $f^{-1}$.
4. The birational map $f$ is a pseudo-isomorphism from $X$ to $Y$.
Denote by $g$ be the inverse of $f$. If the Jacobian determinant of $f$ vanishes at some (smooth) point of $X\smallsetminus \Ind(f)$, then it vanishes along a hypersurface $V\subset X$. If (1) is satisfied, the image of $V$ is a hypersurface $W$ in $Y$, and we can find a point $p\in V\smallsetminus\Ind(f)$ such that $f(p)$ is not an indeterminacy point of $g$. Since the product of the jacobian determinant of $f$ at $p$ and of $g$ at $f(p)$ must be equal to $1$, we get a contradiction. Thus (1) implies (2), and (2) is equivalent to (1). Now, assume that (2) is satisfied. Then $f$ does not contract any positive dimensional subset of $X^{reg}\smallsetminus\Ind(f)$: $f$ is a quasi-finite map from $X^{reg}\smallsetminus \Ind(f)$ to its image, and so is $g$. Zariski’s main theorem implies that $f$ realizes an isomorphism from $X^{reg}\smallsetminus \Ind(f)$ to $Y\smallsetminus \Ind(g)$ (see [@Milne:BookAG], Prop. 8.57). Thus, (2) implies (4) and (3). By assumption, $\Ind(f)$ and $\Ind(g)$ have codimension $\geq 2$; thus, (3) implies (2). Since (4) implies (1), this concludes the proof.
Let $X$ be a smooth projective variety with trivial canonical bundle $K_X$. Let $\Omega$ be a non-vanishing section of $K_X$, and let $f$ be a birational transformation of $X$. Then, $f^*\Omega$ extends from $X\smallsetminus \Ind(f)$ to $X$ and determines a new section of $K_X$; this section does not vanish identically because $f$ is dominant, hence it does not vanish at all because $K_X$ is trivial. As a consequence, $\Jac(f)$ does not vanish, $f$ is a pseudo-automorphism of $X$, and $\Bir(X)=\Psaut(X)$. We refer to [@Cantat-Oguiso:2015; @Fryers:preprint] for families of Calabi-Yau varieties with an infinite group of pseudo-automorphisms.
Projective varieties
--------------------
Let $f\colon X\dasharrow Y$ be a pseudo-isomorphism between two normal projective varieties. Then
1. the total transform of $\Ind(f)$ by $f$ is equal to $\Ind(f^{-1})$;
2. $f$ has no isolated indeterminacy point;
3. if $\dim(X)=2$, then $f$ is a regular isomorphism.
Let $p\in X$ be an indeterminacy point of the pseudo-isomorphism $f\colon X\dasharrow Y$. Then $f^{-1}$ contracts a subset $C\subset Y$ of positive dimension on $p$. Since $f$ and $f^{-1}$ are local isomorphisms on the complement of their indeterminacy sets, $C$ is contained in $\Ind(f^{-1})$. The total transform of a point $q\in C$ by $f^{-1}$ is a connected subset of $X$ that contains $p$ and has dimension $\geq 1$. This set $D_q$ is contained in $\Ind(f)$ because $f$ is a local isomorphism on the complement of $\Ind(f)$; since $p\in D_q\subset \Ind(f)$, $p$ is not an isolated indeterminacy point. This proves Assertions (1) and (2). The third assertion follows from the second one because indeterminacy sets of birational transformations of projective surfaces are finite sets.
Let $W$ be a hypersurface of $X$, and let $f\colon X \dasharrow Y$ be a pseudo-isomorphism. The divisorial part of the total transform $f_*(W)$ coincides with the strict transform $f_\circ(W)$. Indeed, $f_*(W)$ and $f_\circ(W)$ coincide on the open subset of $Y$ on which $f^{-1}$ is a local isomorphism, and this open subset has codimension $\geq 2$.
Recall that the Néron-Severi group $\NS(X)$ is the free abelian group of codimension $1$ cycles modulo cycles which are numerically equivalent to $0$. Its rank is finite and is called the Picard number of $X$.
\[thm:pseudo-automorphisms-neron-severi\] The action of pseudo-isomorphisms on Néron-Severi groups is functorial: $(g\circ f)_*=g_*\circ f_*$ for all pairs of pseudo-isomorphisms $f\colon X\dasharrow Y$ and $g\colon Y\dasharrow Z$. If $X$ is a normal projective variety, the group $\Psaut(X)$ acts linearly on the Néron-Severi group $\NS(X)$; this provides a morphism $$\Psaut(X)\to \GL(\NS(X)).$$ The kernel of this morphism is contained in $\Aut(X)$ and contains $\Aut(X)^0$ as a finite index subgroup.
As a consequence, if $X$ is projective the group $\Psaut(X)$ is an extension of a discrete linear subgroup of $\GL(\NS(X))$ by an algebraic group.
The first statement follows from the equality $f_*=f_\circ$ on divisors. The second follows from the first. To study the kernel $K$ of the linear representation $\Psaut(X)\to \GL(\NS(X))$, fix an embedding $\varphi\colon X\to \P^m_\bfk$ and denote by $H$ the polarization given by hyperplane sections in $\P^m_\bfk$. For every $f$ in $K$, $f_*(H)$ is an ample divisor, because its class in $\NS(X)$ coincides with the class of $H$. Now, a theorem of Matsusaka and Mumford implies that $f$ is an automorphism of $X$ (see [@KSC:book] exercise 5.6, and [@Matsusaka-Mumford:1964]). To conclude, note that $\Aut(X)^0$ has finite index in the kernel of the action of $\Aut(X)$ on $\NS(X)$ (see [@Matsusaka:1958; @Lieberman:1978]).
Affine varieties {#par:affine}
----------------
The group $\Psaut(\A^n_\bfk)$ coincides with the group $\Aut(\A^n_\bfk)$ of polynomial automorphisms of the affine space $\A^n_\bfk$: this is a special case of the following proposition.
\[pro:pseudo-automorphisms-affine\] Let $Z$ be an affine variety. If $Z$ is factorial, the group $\Psaut(Z)$ coincides with the group $\Aut(Z)$.
Fix an embedding $Z\to \A^m_\bfk$. Rational functions on $Z$ are restrictions of rational functions on $\A^m_\bfk$. Thus, every birational transformation $f\colon Z\to Z$ is given by rational formulas $
f (x_1, \ldots, x_m)= (f_1, \ldots, f_m)
$ where each $f_i$ is a rational function $$f_i=\frac{p_i}{q_i} \in \bfk(x_1, \ldots, x_m);$$ here, $p_i$ and $q_i$ are relatively prime polynomial functions. Since the local rings ${\mathcal{O}}_{Z,x}$ are unique factorization domains, we may assume that the hypersurfaces $W_Z(p_i)=\{x \in Z; \; p_i(z)=0\}$ and $W_Z(q_i)=\{x \in Z; \; q_i(z)=0\}$ have no common components. Then, the generic point of $W_Z(q_i)$ is mapped to infinity by $f$. Since $f$ is a pseudo-isomorphism, $W_Z(q_i)$ is in fact empty; but if $q_i$ does not vanish on $Z$, $f$ is a regular map.
Groups with Property [[(FW)]{}]{} {#par:FW}
=================================
Commensurated subsets and cardinal definite length functions (see [@Cornulier:Survey-FW])
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Let $G$ be a group, and $G \times S\to S$ an action of $G$ on a set $S$. Let $A$ be a subset of $S$. As in the Introduction, one says that $G$ [**[commensurates]{}**]{} $A$ if the symmetric difference $A\triangle gA$ is finite for every element $g\in G$. One says that $G$ [**[transfixes]{}**]{} $A$ if there is a subset $B$ of $S$ such that $A\triangle B$ is finite and $B$ is $G$-invariant: $gB=B$ for every $g$ in $G$. If $A$ is transfixed, then it is commensurated. Actually, $A$ is transfixed if and only if the function $g\mapsto\#(A\triangle gA)$ is bounded on $G$.
A group $G$ has [**[Property [[(FW)]{}]{} ]{}**]{} if, given any action of $G$ on a set $S$, all commensurated subsets of $S$ are automatically transfixed. More generally, if $H$ is a subgroup of $G$, then $(G,H)$ has [**[relative Property [[(FW)]{}]{}]{}**]{} if every commensurating action of $G$ is transfixing in restriction to $H$. This means that, if $G$ acts on a set $S$ and commensurates a subset $A$, then $H$ transfixes automatically $A$. The case $H=G$ is Property [[(FW)]{}]{} for $G$.
We refer to [@Cornulier:Survey-FW] for a detailed study of Property [[(FW)]{}]{}. The next paragraphs present the two main sources of examples for groups with Property [[(FW)]{}]{} or its relative version, namely Property (T) and distorted subgroups.
Property [[(FW)]{}]{} should be thought of as a rigidity property. To illustrate this idea, consider a group $K$ with Property ; by definition, this means that $K$ admits a commensurating action on a set $S$, with a commensurating subset $C$ such that the function $g\mapsto\#(C\triangle gC)$ has finite fibers. If $G$ is a group with Property [[(FW)]{}]{}, then, every homomorphism $G\to K$ has finite image.
Property [[(FW)]{}]{} and Property (T)
--------------------------------------
One can rephrase Property [[(FW)]{}]{} as follows: $G$ has Property [[(FW)]{}]{} if and only if every isometric action on an “integral Hilbert space” $\ell^2(X,\Z)$ has bounded orbits, where $X$ is any discrete set.
A group has Property if all its isometric actions on Hilbert spaces have fixed points. More generally, a pair $(G,H)$ of a group $G$ and a subgroup $H\subset G$ has relative Property if every isometric $G$-action on a Hilbert space has an $H$-fixed point. Thus, the relative Property implies the relative Property [[(FW)]{}]{}.
By a theorem of Delorme and Guichardet, Property is equivalent to Kazhdan’s Property (T) for countable groups (see [@delaHarpe-Valette:Ast]). Thus, Property (T) implies Property [[(FW)]{}]{}.
Kazhdan’s Property (T) is satisfied by lattices in semisimple Lie groups all of whose simple factors have Property (T), for instance if all simple factors have real rank $\ge 2$. For example, $\SL_3(\Z)$ satisfies Property (T).
Property [[(FW)]{}]{} is actually conjectured to hold for all irreducible lattices in semi-simple Lie groups of real rank $\ge 2$, such as $\SL_2(\mathbf{R})^k$ for $k\ge 2$. (here, irreducible means that the projection of the lattice [*modulo*]{} every simple factor is dense.) This is known in the case of a semisimple Lie group admitting at least one noncompact simple factor with Kazhdan’s Property (T), for instance in $\SO(2,3)\times\SO(1,4)$, which admits irreducible lattices (see [@Cornulier:MathZ]).
Distortion
----------
Let $G$ be a group. An element $g$ of $G$ is [**distorted**]{} in $G$ if there exists a finite subset $\Sigma$ of $G$ generating a subgroup $\langle\Sigma\rangle$ containing $g$, such that $\lim_{n\to\infty}\frac{1}{n}|g^n|_\Sigma=0$; here, $\vert g \vert_\Sigma$ is the length of $g$ with respect to the set $\Sigma$. If $G$ is finitely generated, this condition holds for some $\Sigma$ if and only if it holds for every finite generating subset of $G$. For example, every finite order element is distorted.
Let $K$ be a field. The distorted elements of $\SL_n(K)$ are exactly the virtually unipotent elements, that is, those elements whose eigenvalues are all roots of unity; in positive characteristic, these are elements of finite order. By results of Lubotzky, Mozes, and Raghunathan (see [@lubotzky1993cyclic; @Lubotzky-Mozes-Raghunathan:2001]), the same characterization holds in the group $\SL_n(\mathbf{Z})$, as soon as $n\ge 3$; it also holds in $\SL_n(\mathbf{Z}[\sqrt{d}])$ when $n\geq 2$ and $d\ge 2$ is not a perfect square. In contrast, in $\SL_2(\mathbf{Z})$, every element of infinite order is undistorted.
Let $G$ be a group, and $H$ a finitely generated abelian subgroup of $G$ consisting of distorted elements. Then, the pair $(G,H)$ has relative Property [[(FW)]{}]{}.
This lemma provides many examples. For instance, if $G$ is any finitely generated nilpotent group and $G'$ is its derived subgroup, then $(G,G')$ has relative Property ; this result is due to Houghton, in a more general formulation encompassing polycyclic groups (see [@Cornulier:Survey-FW]). Bounded generation by distorted unipotent elements can also be used to obtain nontrivial examples of groups with Property [[(FW)]{}]{}, including the above examples $\SL_{n}(\mathbf{Z})$ for $n\geq 3$, and $\SL_{n}(\mathbf{Z}[\sqrt{d}])$. The case of $\SL_{2}(\mathbf{Z}[\sqrt{d}])$ is particularly interesting because it does not have Property (T).
Subgroups of $\PGL_2(\bfk)$ with Property [[(FW)]{}]{} {#par:Bass}
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If a group $G$ acts on a tree $T$ by graph automorphisms, then $G$ acts on the set $E$ of directed edges of $T$ ($T$ is non-oriented, so each edge gives rise to a pair of opposite directed edges). Let $E_v$ be the set of directed edges pointing towards a vertex $v$. Then $E_v\triangle E_w$ is the set of directed edges lying in the segment between $v$ and $w$; it is finite of cardinality $2d(v,w)$, where $d$ is the graph distance. The group $G$ commensurates the subset $E_v$ for every $v$, and $\#(E_v\triangle gE_v)=2d(v,gv)$. As a consequence, if $G$ has Property [[(FW)]{}]{}, then it has Property (FA) in the sense that every action of $G$ on a tree has bounded orbits. This argument can be combined with Proposition 5.B.1 of [@Cornulier:Survey-FW] to obtain the following lemma.
Let $G$ be a group with Property [[(FW)]{}]{}, then all finite index subgroups of $G$ have Property [[(FW)]{}]{}, and hence have Property (FA). Conversely, if a finite index subgroup of $G$ has Property [[(FW)]{}]{}, then so does $G$.
On the other hand, Property (FA) is not stable by taking finite index subgroups.
\[lem:pgl2-fw\] Let $\bfk$ be an algebraically closed field and $\Lambda$ be a subgroup of $\GL_2(\bfk)$.
1. $\Lambda$ has a finite orbit on the projective line if and only if it is virtually solvable, if and only if its Zariski closure does not contain $\SL_2$.
2. Assume that all finite index subgroups of $\Lambda$ have Property [(FA)]{} (e.g., $\Lambda$ has Property FW). If the action of $\Lambda$ on the projective line preserves a non-empty, finite set, then $\Lambda$ is finite.
The proof of the first assertion is standard and omitted. The second assertion follows directly from the first one.
In what follows, we denote by $\oZ\subset \overline{\Q}$ the ring of algebraic integers (in some fixed algebraic closure $\overline{\Q}$ of $\Q$).
\[thm:Bass\] Let $\bfk$ be an algebraically closed field.
1. If $\bfk$ has positive characteristic, then $\GL_2(\bfk)$ has no infinite subgroup with Property (FA).
2. Suppose that $\bfk$ has characteristic zero and that $\Gamma\subset\GL_2(\bfk)$ is a countable subgroup with Property (FA), and is not virtually abelian. Then $\Gamma$ acts irreducibly on $\bfk^2$, and is conjugate to a subgroup of $\GL_2(\oZ)$. If moreover $\Gamma\subset\GL_2(K)$ for some subfield $K\subset\bfk$ containing $\overline{\Q}$, then we can choose the conjugating matrix to belong to $\GL_2(K)$.
The original statement [@Bass:Pacific Theorem 6.5] yields this statement, except the last fact, and assumes that $\Gamma$ is contained in $\GL_2(M)$ with $M$ a finitely generated field. The latter condition is actually automatic: indeed, being a countable group with Property (FA), $\Gamma$ is finitely generated [@Serre:AASL2 §6, Th. 15], and one can choose $K$ to be the field generated by entries of a finite generating subset.
For the last assertion, we have $\Gamma\cup B\Gamma B^{-1}\subset \GL_2(K)$ for some $B\in\GL_2(\bfk)$ such that $B\Gamma B^{-1}\subset
\GL_2(\oZ)$; we claim that this implies that $B\in \bfk^*\GL_2(K)$. First, since $\Gamma$ is absolutely irreducible, this implies that $B\mathcal{M}_2(K)B^{-1}\subset \mathcal{M}_2(K)$. The conclusion follows from Lemma \[algl\] below, which can be of independent interest.
\[algl\] Let $K\subset L$ be fields. Then the normalizer $\{B\in\GL_2(L):B\mathcal{M}_2(K)B^{-1}\subset \mathcal{M}_2(K)\}$ is reduced to $L^*\GL_2(K)=\{\lambda A:\lambda\in L^*,A\in\GL_2(K)\}$.
Write $$B=\begin{pmatrix}b_1 & b_2\\ b_3 & b_4\end{pmatrix}.$$ Since $BAB^{-1}\in \mathcal{M}_2(K)$ for the three elementary matrices $A\in\{E_{11},E_{12},E_{21}\}$, we deduce by a plain computation that $b_ib_j/b_kb_\ell\in K$ for all $1\le i$, $j$, $k$, $\ell\le 4$ such that $b_kb_\ell\neq 0$. In particular, for all indices $i$ and $j$ such that $b_i$ and $b_j$ are nonzero, the quotient $b_i/b_j=b_ib_j/b_j^2$ belongs to $K$. It follows that $B\in L^*\GL_2(K)$.
\[coro:Bass-k(C)\] Let $\bfk$ be an algebraically closed field. Let $C$ be a projective curve over $\bfk$, and let $\bfk(C)$ be the field of rational functions on the curve $C$. Let $\Gamma$ be an infinite subgroup of $\PGL_2(\bfk(C))$. If $\Gamma$ has Property (FA), then
1. the field $\bfk$ has characteristic $0$;
2. there is an element of $\PGL_2(\bfk(C))$ that conjugates $\Gamma$ to a subgroup of $\PGL_2(\oZ)\subset \PGL_2(\bfk(C))$.
A categoral limit construction {#S:cat}
==============================
The purpose of this section is to describe a general categorical construction, which can be used to construct various actions of groups of birational transformations, such as Manin’s construction of the Picard-Manin space (see [@Manin:cubic-forms; @Cantat:Annals]), as well as the commensurating action which is the main construction of this paper. A closely related construction is performed by V. Jones in [@jones2016no] to construct representations of Thompson’s groups, although it does not directly apply here.
Categories of projective varieties
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Here, in a category $\CC$, arrows between any two objects $X$ and $Y$ are assumed to form a set $\Hom_{\CC}(X,Y)$. Recall that a category is small if its objects form a set, and is essentially small if it is equivalent to a small category, or equivalently if there is a subset of the collection of objects meeting every isomorphism class. A subcategory $\CC$ of a category $\DD$ is full if all arrows of $\DD$ between objects of $\CC$ are also arrows of $\CC$.
\[eg:category-bira-1\] Our main example will be the following. Fix an algebraically closed field $\bfk$. Let $\cx=\cx_\bfk$ be the category whose objects are irreducible (reduced) projective $\bfk$-varieties and whose arrows are birational morphisms. Let $\cx^\sh$ be the category with the same objects, but whose arrows are birational maps. Similarly, one can consider the category $\cxn$ of irreducible (reduced) normal projective varieties, with arrows given by birational morphisms, and the category $\cxn^{\sh}$ with the same objects but whose arrows are all birational maps. By construction, $\cxn$ is a full subcategory of $\cx$, which is a subcategory of $\cx^\sh$.
Relative thinness and well-cofiltered categories {#s:rt}
------------------------------------------------
Given a category $\CC$ and an object $X\in\Ob(\CC)$, let us define the category $\CC_X$ whose objects are pairs $(Y,f)$ with $Y\in\Ob(\CC)$ and $f\in\Hom_\CC(Y,X)$, and whose arrows $(Y,f)\to(Z,g)$ are given by arrows $u\in\Hom_{\CC}(Y,Z)$ such that $g\circ u=f$. A category is [**thin**]{} if there is at most one arrow between any two objects. Let us say that a category is [**[relatively thin]{}**]{} if the category $\CC_X$ is thin for all $X\in\Ob(X)$.
\[varthin\] A category in which every arrow is invertible is relatively thin, and so are all its subcategories. This applies to the categories of Example \[eg:category-bira-1\]: the category $\cx^\sh_\bfk$ of birational maps between irreducible projective varieties, and to its subcategory $\cx_\bfk$, and similarly to $\cxn^\sh_\bfk$ and its subcategory $\cxn_\bfk$.
Recall that a category is [**cofiltered**]{} if it satisfies the following two properties (a) and (b):
- for any pair of objects $X_1$, $X_2$, there exists an object $Y$ with arrows $X_1\leftarrow Y\to X_2$;
- for any pair of objects $X$, $Y$ and arrows $u_1$, $u_2:X\to Y$, there exists an object $W$ and an arrow $w:W\to X$ such that $u_1\circ w=u_2\circ w$.
Note that (b) is automatically satisfied when the category is thin. We say that a category $\CC$ is [**[well-cofiltered]{}**]{} if it is relatively thin and for every object $X\in\Ob(\CC)$, the category $\CC_X$ is cofiltered (note that we do not require $\CC$ to be cofiltered).
\[eg:category-bira-2\] Coming again to the categories of Example \[eg:category-bira-1\], the category $\cx_\bfk$ is essentially small and well-cofiltered. It is relatively thin, as mentioned in Example \[varthin\]. To show that $(\cx_\bfk)_X$ is cofiltered, consider two birational morphisms $f_1\colon X_1 \to X$ and $f_2\colon X_2\to X$, and denote by $h$ the composition $f_2^{-1} \circ f_1$. The graph ${\mathfrak{G}}_h$ is a projective subvariety of $X_1\times X_2$. One can compose the projection of ${\mathfrak{G}}_h$ onto $X_1$ with $f_1$ (resp. onto $X_2$ with $f_2$) to get a birational morphism ${\mathfrak{G}}_h\to X$; this birational morphism is an object in $(\cx_\bfk)_X$ that dominates $f_1\colon X_1 \to X$ and $f_2\colon X_2\to X$, as in property (a).
The full subcategory $\cxn_\bfk$ of $\cx_\bfk$ enjoys the same properties. When $\bfk$ has characteristic zero, the resolution of indeterminacies implies that its full subcategory of non-singular varieties (and birational morphisms) is also well-cofiltered.
Filtering inductive limits {#catpre}
--------------------------
###
We shall say that a category $\EE$ [**[admits filtering inductive limits]{}**]{} if for every small, thin and cofiltered category $\DD$ and every contravariant functor $F:\DD\to\EE$, the colimit of $F$ exists (and then it also exists when “small” is replaced with “essentially small”). For example, the category of sets and the category of groups admit filtering inductive limits (see [@Vakil], § 1.4, for colimits).
###
Let us consider an essentially small category $\CC$, a category $\EE$ admitting filtering inductive limits, and a contravariant functor $F:\CC\to\EE$; we denote the functor $F$ by $X\mapsto F_X$ on objects and $u\mapsto F_u$ on arrows. Assume that $\CC$ is well-cofiltered. Then, for every object $X\in\Ob(\CC)$, we can restrict the functor $F$ to $\CC_X$ and take the colimit $\Fha_X$ of this restriction $F:\CC_X\to\EE$. Roughly speaking, $\Fha_X$ is the inductive limit in $\EE$ of all $F_Y$ for $Y\in\CC_X$. So, for every arrow $u:Y\to X$ in $\CC$, there is an arrow in $\EE$, $\phi_u:F_Y\to\Fha_X$ in $\EE$; and for every arrow $v:Z\to Y$ in $\CC_X$, the following diagram commutes $$\xymatrix{ F_Z \ar@{->}[rd]^{\phi_{u\circ v}} \ar@{<-}[d]_{F_v}& \\
F_Y\ar@{->}[r]^{\phi_u} & \Fha_X.}$$ The colimits $\Fha_X$ satisfy a universal property. To describe it, consider an object $E\in\Ob(\EE)$, together with arrows $\psi_Y:F_Y\to E$ for all $Y\in\CC_X$, and assume that for every arrow $v:Z\to Y$ in $\CC_X$ we have the relation $\psi_Z\circ F_v=\psi_Y$. Then, there exists a unique arrow $\psi:\Fha_X\to E$ in $\EE$ such that for every $(Y,u)\in\Ob(\CC_X)$ the following diagram commutes: $$\xymatrix{ \Fha_X \ar@{->}[r]^{\psi} \ar@{<-}[d]_{F_u}& E \\
F_Y.\ar@{->}[ru]_{\psi_Y} & }$$ This construction provides a bijection $\Phi_X$ from the inductive limit $\underleftarrow{\lim}_{Y\in\CC_X}\Hom_\EE(F_Y,E)$ to $\Hom(\Fha_X,E)$ whose reciprocal bijection maps an element $\psi\in\Hom(\Fha_X,E)$ to the family of arrows $(\psi\circ F_u)_{(Y,u)\in\CC_X}$.
###
We can now define the covariant functor $\alpha$ associated to $F$. At the level of objects, $\alpha$ maps $X\in \Ob(\CC)$ to the limit $\Fha_X$. Let us now describe $\alpha$ at the level of arrows. If we fix $(Y,u)\in\Ob(\CC_X)$, the family of arrows $(\phi_{u\circ v}:F_Z\to\Fha_X)_{(Z,v)\in\Ob(\CC_Y)}$ corresponds under $\Phi_Y$ to an arrow $\alpha_u:\Fha_Y\to\Fha_X$. For every $(Z,v)\in\Ob(\CC_Y)$, the following diagram commutes $$\label{eq:def-covariant-alpha}
\xymatrix{
\Fha_Y \ar@{->}[r]^{\alpha_u} & \Fha_X\\
F_Z,\ar@{->}[u]^{\phi_v}\ar@{->}[ru]_{\phi_{u\circ v}} &
}$$ and this characterizes the arrow $\alpha_u$. If $(W,w)\in\CC_Y$, the uniqueness readily proves that $\alpha_{u\circ w}=\alpha_u\circ\alpha_w$, that is,[*[ $X\mapsto\Fha_X$, $u\mapsto\alpha_u$ is a covariant functor $\CC\to\EE$, denoted $\alpha$, and called the [**[relative colimit functor]{}**]{} associated to $F$]{}*]{}.
###
Note that the previous diagram can be essentially rewritten in the form of the commutative square on the left of the next equation. The commutative diagram on the right only refers to $u$; it is obtained by composing the left diagram with the map $F_u$ and by using the equalities $\phi_v\circ F_v=\phi_{\mathrm{id}_{Y}}$ and $\phi_u\circ F_u=\phi_{\mathrm{id}_{X}}$. $$\label{eq:ddd}
\xymatrix{
\Fha_Y \ar@{->}[r]^{\alpha_u} & \Fha_X\\
F_Z\ar@{->}[u]^{\phi_v}\ar@{<-}[r]^{F_v} & F_Y\ar@{->}[u]^{\phi_u}
}
\quad\quad\quad\quad\quad \xymatrix{
\Fha_Y \ar@{->}[r]^{\alpha_u} & \Fha_X\\
F_Y\ar@{->}[u]^{\phi_{\mathrm{id}_{Y}}}\ar@{<-}[r]^{F_u} & F_X.\ar@{->}[u]_{\phi_{\mathrm{id}_{X}}}
}$$
\[alphaiso\] Suppose that $\CC$ is well-cofiltered. Then $\alpha$ maps arrows of $\CC$ to invertible arrows (i.e. isomorphisms) of $\EE$.
Fix $(Y,u)\in\CC_X$. The proof consists in constructing a map, and then show that it is the inverse map of $\alpha_u$.
Consider $(S,s)\in\Ob(\CC_X)$. By assumption, in $\CC$ we can find a commutative diagram as the one on the left of the following equation; hence, in $\EE$ we obtain the diagram on the right, where $g=\phi_x\circ F_w$ by definition. $$\xymatrix{ Y \ar@{<-}[r]^{x} \ar@{->}[d]_{u}& D\ar@{->}[d]^w \\
X\ar@{<-}[r]^{s} & S, }
\quad \quad \quad \quad \xymatrix{ F_Y \ar@{->}[r]^{F_x} \ar@{<-}[d]_{F_u}& F_D\ar@{<-}[d]_{F_w} \ar@{->}[r]^{\phi_x}& \Fha_Y\\
F_X\ar@{->}[r]^{F_s} & F_S. \ar@{.>}[ru]_{g} &}$$ A priori $g$ depends on the choice of $(D,x,w)$; let us show that it only depends on $(u,s)$ and, for that purpose, let us denote $g$ temporarily by $g=g_D$ ($x,w$ being implicit). First consider the case of a commutative diagram as the one on the left in the next equation; in $\EE$, this diagram induces the diagram depicted on the right, where everything not involving $g_D$ or $g_{D'}$ is commutative.
$$\xymatrix{ Y \ar@{<-}[rr]^{x}\ar@{<-}[rd]^{x'} \ar@{->}[dd]_{u}&& D\ar@{->}[dd]^w \\ & D'\ar@{<-}[ru]^{q}\ar@{->}[rd]^{w'}&\\
X\ar@{<-}[rr]^{v} && S, }
\quad \quad \quad \xymatrix{ F_Y \ar@{->}[rr]^{F_x}\ar@{->}[rd]^{F_{x'}} \ar@{<-}[dd]_{F_u}&& F_D\ar@{<-}[dd]^{F_w} \ar@{->}[rr]^{\phi_x}&& \Fha_Y\\ & F_{D'}\ar@{->}[ru]^{F_q}\ar@{<-}[rd]^{F_{w'}}\ar@{->}[rrru]_{\phi_{x'}}&&&\\
F_X\ar@{->}[rr]^{F_s} && F_S,\ar@{=>}[rruu]_{g_D,g_{D'}}&& }$$ Thus, by definition $g_{D'}=\phi_{x'}\circ F_{w'}=\phi_x\circ F_q\circ F_{w'}=\phi_x\circ F_w=g_D.$ Now consider, more generally two objects $D'$ and $D''$ and four arrows forming a diagram in $\CC_X$: $$\xymatrix{ Y \ar@{<-}[r] \ar@{<-}[d]& D'\ar@{->}[d] \\
D''\ar@{->}[r] & S; }$$ we have to show that $g_{D'}=g_{D''}$. Since $\CC$ is well-cofiltered, $\CC_X$ is thin and cofiltered, and we can complete the previous diagram into the one on the left of the following equation. Since this diagram is in $\CC_X$ which is a thin category, it is commutative; which means that if we complete it with both composite arrows $D\to Y$ and both composite arrows $D\to S$, the resulting arrows coincide; the resulting diagram, on the right of the equation, is a commutative one.
$$\xymatrix{ & D' &\\
Y \ar@{<-}[ru]\ar@{<-}[rd] & D\ar@{->}[u]\ar@{->}[d] & S
\ar@{<-}[lu]\ar@{<-}[ld]\\
& D''; &
}
\quad\quad\quad\quad \xymatrix{ & D' &\\
Y \ar@{<-}[ru]\ar@{<-}[rd]\ar@{<-}[r] & D\ar@{->}[r]\ar@{->}[u]\ar@{->}[d] & S
\ar@{<-}[lu]\ar@{<-}[ld]\\
& D''; &
}$$ Using the previous case, we deduce $g_{D'}=g_D=g_{D''}$. Thus, we have seen that $g_D$ does not depend on the choice of $D$; we now write it as $g_{u,s}$. In particular, when $S\in(\CC_X)_Y$, we can choose $D=S$ (and $w$ the identity); we thus deduce that $g_{u,s}=\phi_x$ (where $s=u\circ x$).
Consider $(T,t)\in\CC_S$ and choose $D\in\Ob(\CC_X)$ with $Y\stackrel{x}\leftarrow D\stackrel{w}\to T$ in $\CC_X$. Then we have the diagram in $\EE$ $$\xymatrix{
F_Y \ar@{->}[r]^{F_x}\ar@{<-}[dd] & F_D\ar@{->}[rd]^{\phi_x}&\\
& F_T\ar@{->}[u]^{F_w}\ar@{->}[r]^{g_{u,s\circ t}}& \Fha_Y\\
F_X\ar@{->}[r]^{F_s} & F_S,\ar@{->}[u]^{F_t}\ar@{->}[ru]_{g_{u,s}}&
}$$ where the left rectangle is commutative as well as the upper right triangle; since by definition $g_{u,s}=\phi_x\circ F_{t\circ w}=\phi_x\circ F_w\circ F_t=g_{u,s\circ t}\circ F_t$, the lower right triangle is also commutative. So the family $(g_s:F_S\to\Fha_Y)_{(S,s)\in\Ob(\CC_X)}$ defines an element $g_u:\Fha_X\to\Fha_Y$. Namely, for every $(S,s)\in\Ob(\CC_X)$ the following diagram commutes $$\xymatrix{
\Fha_X \ar@{->}[r]^{g_u} & \Fha_Y\\
F_S,\ar@{->}[u]^{\phi_s}\ar@{->}[ru]_{g_{u,s}} &
}$$ and $g_u$ is characterized by this property.
We now combine this with the map $\alpha_u$, and make use of the same notation as the one in Equations and . When $S=Z$ (so $s=u\circ v$, $g_{u,s}=\phi_v$), we obtain the commutative diagram. $$\xymatrix{
\Fha_Y\ar@{->}[r]^{\alpha_u}\ar@{<-}[rd]_{\phi_v} &\Fha_X \ar@{->}[r]^{g_u} & \Fha_Y\\
&F_Z,\ar@{->}[u]^{\phi_s}\ar@{->}[ru]_{\phi_{v}} &
}$$ Since this holds for all $(Z,v)\in(\CC_X)_Y$, the universal property of $\Fha_Y$ implies that $g_u\circ\alpha_u$ is the identity of $\Fha_Y$.
On the other hand, turning back to the notation of the beginning of the proof, both triangles in the following diagram are commutative $$\xymatrix{
\Fha_Y\ar@{->}[r]^{\alpha_u} & \Fha_X\\
F_D\ar@{->}[u]^{\phi_x}\ar@{<-}[r]_{F_w}\ar@{->}[ru]_{\phi_{u\circ x}} & F_S;\ar@{->}[u]_{\phi_s}
}$$ since $g_{u,s}=\phi_x\circ F_w$, this implies that the right triangle of the following diagram is commutative, the left-hand triangle from above also being commutative $$\xymatrix{
\Fha_X\ar@{->}[r]^{g_u}\ar@{<-}[rd]_{\phi_s}&\Fha_Y\ar@{->}[r]^{\alpha_u} & \Fha_X\\
&F_S.\ar@{->}[u]^{g_{u,s}}\ar@{->}[ru]_{\phi_{s}} &
}$$ Since this holds for all $(S,s)\in\CC_X$, by the universal property of $\Fha_X$, we obtain that $\alpha_u\circ g_u$ is the identity of $\Fha_X$. This ends the proof that $\alpha_u$ is invertible.
Good right-localization and extensions {#catpre2}
--------------------------------------
Given a category $\DD$ with a subcategory $\CC$ with the same objects, we say that $(\mathcal{C},\mathcal{D})$ is a [**[good right-localization]{}**]{} if (i) every arrow $u\colon X\to Y$ in $\mathcal{D}$ admits an inverse $u^{-1}\colon Y \to X$ and (ii) every arrow in $\DD$ can be decomposed as $g\circ f^{-1}$ where $f$ and $g$ are arrows of $\mathcal{C}$.
\[goodcofi\] Let $(\mathcal{C},\mathcal{D})$ be a good right-localization. Then $\CC$ is well-cofiltered.
Clearly any category in which all arrows are invertible is relatively thin. It follows that $\DD$ and its subcategory $\CC$ are relatively thin. Now consider a pair of objects $(Y,u)$ and $(Z,v)$ of $\CC_X$. Then $v^{-1}\circ u$ is an arrow of $\DD$ (because all arrows are invertible in $\DD)$, and it can be decomposed as $Y\stackrel{s}\leftarrow W\stackrel{t}\to Z$, with $s$ and $t$ arrows of $\CC$. By definition $v\circ t=u\circ s$ determines an arrow $W\to X$; endowing $W$ with the resulting composite arrow to $X$, the arrows $s$ and $t$ become arrows in $\CC_X$.
\[extcat\] Let $(\mathcal{C},\mathcal{D})$ be a good right-localization, and let $\mathcal{E}$ be another category. Consider a (covariant) functor $\beta$ from $\mathcal{C}$ to the category $\mathcal{E}$ mapping every arrow to an invertible arrow of $\EE$. Then $\beta$ has a unique extension to a functor from the category $\mathcal{D}$ to the category $\mathcal{E}$.
The uniqueness is clear. For the existence, consider an arrow $u$ in $\mathcal{D}$. We wish to map $u$ to $\beta(g)\circ \beta(f)^{-1}$, where $u=g\circ f^{-1}$. We have to prove that this does not depend on the choice of $(f,g)$. Thus write $u=g_1\circ f_1^{-1}=g_2\circ f_2^{-1}$. Since $\mathcal{C}_X$ is well-cofiltered (Lemma \[goodcofi\]), we can produce a diagram as follows in $\mathcal{C}$, where the left “square” and the whole square are commutative:
$$\xymatrix{ & & Y_1\ar@{->}[rd]^{g_1}\ar@{->}[ld]_{f_1} & \\
& X \ar@{<-}[rd]_{f_2}& Z\ar@{->}[u]_{h_1}\ar@{->}[d]^{h_2} & Y_2\\
& & X_2\ar@{->}[ru]_{g_2} & &
}$$ Then the right square is also commutative: indeed $g_1\circ h_1=(g_1\circ f_1^{-1})\circ (f_1\circ h_1)=(g_2\circ f_2^{-1})\circ (f_2\circ h_2)=g_2\circ h_2.$ Then $$\beta(g_1)\circ \beta(f_1)^{-1}=\beta(g_1)\circ \beta(h_1)\circ \beta(h_1)^{-1}\circ \beta(f_1)^{-1}=\beta(g_1\circ h_1)\circ \beta(f_1\circ h_1)^{-1}$$ $$=\beta(g_2\circ h_2)\circ \beta(f_2\circ h_2)^{-1}=\beta(g_2)\circ \beta(h_2)\circ \beta(h_2)^{-1}\circ \beta(f_2)^{-1}=\beta(g_2)\circ \beta(f_2)^{-1};$$ hence we can define without ambiguity $\beta(u)=\beta(g)\circ \beta(f)^{-1}$.
We have to prove $\beta(v)\circ \beta(u)=\beta(v\circ u)$ for any arrows $u$, $v$ of $\mathcal{D}$. This already holds for $u$, $v$ arrows of $\mathcal{C}$. Write $u=g\circ f^{-1}$, $v=j\circ h^{-1}$ with $f$, $g$, $h$, and $j$ arrows of $\mathcal{C}$. Write $h^{-1}\circ g=t\circ s^{-1}$ with $s$, $t$ arrows of $\mathcal{C}$. Then $g\circ s=h\circ t$, so $\beta(g)\circ \beta(s)=\beta(g\circ s)=\beta(h\circ t)=\beta(h)\circ \beta(t)$, which can be rewritten $\beta(h)^{-1}\circ \beta(g)=\beta(t)\circ \beta(s)^{-1}$. In turn, we get $$\beta(v\circ u)=\beta(j\circ h^{-1}\circ g\circ f^{-1})=\beta(j\circ t\circ s^{-1}\circ f^{-1})=\beta(j\circ t)\circ \beta(f\circ s)^{-1}$$ $$=\beta(j)\circ \beta(t)\circ \beta(s)^{-1}\circ \beta(f)^{-1}=\beta(j)\circ \beta(h)^{-1}\circ \beta(g)\circ \beta(f)^{-1}=\beta(v)\circ \beta(u).\qedhere$$
Combining Lemmas \[alphaiso\], \[goodcofi\] and \[extcat\], we deduce:
\[extencombi\] Consider a good right-localization $(\mathcal{C},\mathcal{D})$ and a category $\mathcal{E}$ admitting filtering inductive limits. Let $F$ be a contravariant functor from $\mathcal{C}$ to $\mathcal{E}$. Then the relative colimit functor $\alpha$, defined by $$X\mapsto\Fha_X=\underrightarrow{\lim}_{Y\to X}F(Y), \quad \quad u\mapsto\alpha_u,$$ has a unique extension to a covariant functor from $\mathcal{D}$ to $\mathcal{E}$.
Irreducible hypersurfaces {#catbir}
=========================
Let $X$ be a normal projective variety. In this chapter, we make use of the categorical construction of Section \[catpre\] to define an action of $\Bir(X)$ on the set of all irreducible hypersurfaces in all “models” $Y\to X$ of $X$. We also sketch an application to the construction of Picard-Manin spaces in Section \[par:Manin-Construction\].
Localizations in categories of projective varieties
---------------------------------------------------
Consider the categories $\cx$, $\cx^\sh$, $\cxn$, and $\cxn^\sh$ from examples \[eg:category-bira-1\] and \[eg:category-bira-2\].
Let $\bfk$ be an algebraically closed field.
1. The categories $\cx$ and $\cxn$ are well-cofiltered (as defined in §\[s:rt\]).
2. The pairs $(\cx,\cx^{\sh})$ $(\cxn,\cxn^{\sh})$ are good right-localizations of categories (as defined in §\[catpre2\]).
Property (1) follows from Example \[eg:category-bira-2\]. Let us now prove the second property. Clearly, every arrow in $\cx^{\sh}$ is invertible. Let $f:X\dasharrow X'$ be a birational map. The graph $\mathfrak{G}_f$ is an irreducible variety, both projections $X\stackrel{g}\leftarrow\mathfrak{G}_f\stackrel{g'}\to X'$ are birational morphisms and $f=g'\circ g^{-1}$. Since $f=g'\circ g^{-1}$ with $X\stackrel{g}\leftarrow\mathfrak{G}_f\stackrel{g'}\to X'$, we deduce that $(\cx,\cx^{\sh})$ is a good right-localization.
For $\cxn$, we only need to compose with the normalization map $Y\to\mathfrak{G}_f$ to get the result (see [@Vakil], §9.7).
To spare the reader from going into too much category theory, let us state explicitly Proposition \[extencombi\] in this case:
\[corlimf\] Let $\EE$ be a category admitting filtering inductive limits. Consider a contravariant functor $F$ from $\cxn$ to the category $\mathcal{E}$. Then the covariant functor $\alpha$, $X\mapsto\Fha_X=\underrightarrow{\lim}_{Y\to X}F(Y)$, $u\mapsto\alpha_u$ has a unique extension to a functor from $\cxn^\sharp$ to $\mathcal{E}$.
The functor of irreducible hypersurfaces
----------------------------------------
Let us construct the functor to which we will apply Corollary \[corlimf\]. For $X\in\cx$, define $\Hy(X)$ as the set of irreducible and reduced hypersurfaces of $X$.
\[hypermo\] Let $f:Y\to X$ be a birational morphism between two irreducible projective varieties (an arrow in $\cx$).
1. \[ihm2\] The number of $T\in\Hy(Y)$ such that $f(T)$ is not a hypersurface is finite (these $T$ are precisely the hypersurfaces contracted by $f$).
2. \[ihm1\] For every $S\in\Hy(X)$, the number of $S'\in\Hy(Y)$ such that $f(S')=S$ is positive and finite.
3. \[ihm3\] If $X$ is normal, this number is equal to 1, and the unique preimage $S'$ of $S$ is the strict transform $f^\circ(S)=(f^{-1})_\circ (S)$. In particular, $S\mapsto S'=f^\circ(S)$ is injective and its image has finite complement: this complement $\Hy(Y)\smallsetminus f^\circ(\Hy(X))$ is the set of hypersurfaces $T$ that are contracted by $f$.
Let $U\subset Y$ be a Zariski-dense open subset on which $f$ induces an isomorphism onto its image. Let $F$ be the complement of $U$.
(\[ihm2\]) If $f(T)$ is not a hypersurface, then $T\subset F$. So $T$ is one of the irreducible components of $F$, which leaves finitely many possibilities.
(\[ihm1\]) Since $Y$ is projective, $f$ is surjective, and hence $f(T)=S$, where $T=f^{-1}(S)$; $T$ is a proper subvariety. Then at least one irreducible component $T'$ of $T$ satisfies $f(T')=S$, and conversely, every $S'\in\Hy(Y)$ such that $f(S')=S$ has to be an irreducible component of $T$, hence there are finitely many of them.
(\[ihm3\]) We now use Theorem \[normal2\]: since $X$ is normal and $Y$ is projective, the indeterminacy set $\Ind(f^{-1})\subset X$ has codimension $\ge 2$. Hence the strict transform $(f^{-1})_\circ (S)$ of $S$ is well-defined and is equal to $\overline{f^{-1}(X\smallsetminus \Ind(f^{-1})}$. The total transform of $S$ by $f^{-1}$ may contain additional components of codimension $1$, but all of them are contracted into $\Ind(f^{-1})$, which has codimension $2$ (hence are not equal to $S$). This proves that $S'=(f^{-1})_\circ (S)$. Since $f(S')=S$, the map $f^\circ:=(f^{-1})_\circ$ is injective. Moreover, by construction, its image is made of hypersurfaces which are not contained in $f^{-1}(\Ind(f))$. Since every element $T\in \Hy(Y)$ which is not contracted coincides with $f^\circ(f(T))$, the image of $f^\circ$ is in fact equal to the complement of the set of contracted hypersurfaces.
From Proposition \[hypermo\], the map $X\mapsto\Hy(X)$ defines a contravariant functor from $\cxn$ to the category of sets, mapping an arrow $f:Y\to X$ to the (injective) map $$f^\circ:\Hy(X)\to\Hy(Y).$$ If $F$ denoted the functor $\Hy$ as in Section \[catpre\], then we would have $F_X=\Hy(X)$ and $F_f=f^\circ$. For $X$ in the category $\cxn$, define $\Hyp(X)$ as the filtering inductive limit $$\Hyp(X)=\underrightarrow{\lim}_{Y\to X}\Hy(Y).$$ By construction, $X\mapsto\Hyp(X)$ is a covariant functor from $\cxn$ to the category of sets. By Corollary \[corlimf\], it has a unique extension to a functor from $\cxn^\sh$ to the category of sets. The image of an object $Y\in \cxn^\sh$ is denoted $\Hyp(Y)$, and the image of an arrow $f\colon Y\dasharrow Y'$, that is of a birational map between two normal projective varieties $Y$ and $Y'$, is denoted by $f_\bullet$. By construction, $f_\bullet$ is a bijection from $\Hyp(Y)$ to $\Hyp(Y')$.
For an arrow $u$ in $\cxn$, i.e. a birational morphism $u:Y\to X$ between two normal projective varieties, we rewrite the commutative square on the right of Equation as $$\xymatrix{
\Hyp(Y) \ar@{->}[r]^{u_\bullet}_\sim & \Hyp(X)\\
\Hy(Y)\ar@{^{(}->}[u]^{i_Y}\ar@{<-^{)}}[r]^{u^\circ} & \Hy(X).\ar@{^{(}->}[u]_{i_X}
}$$ The two injections $i_X$ and $i_Y$ will simply be viewed as inclusions in what follows. So the bijection $u_\bullet^{-1}$ extends the injection which is given by the strict transform $u^\circ$. Since the image of $u^\circ$ has finite complement, the symmetric difference $\Hy(Y)\triangle u_\bullet^{-1}(\Hy(X))$ is finite. This latter property passes to inverses and compositions; hence, [*[for every birational map $v:X\dasharrow X'$ between normal irreducible projective varieties, the symmetric difference $\Hy(X)\triangle v_\bullet^{-1}(\Hy(X'))$ is finite.]{}*]{} To give a more precise statement, let us introduce the following notation: given a birational map $v:X\dasharrow X'$ between normal irreducible projective varieties, define $\exc(v)$ by $$\exc(v)=\# \left\{ S\in \Hy(X);\;\; v\; {\text{ contracts }}\; S \right\}.$$ This is the [**[number of contracted hypersurfaces]{}**]{} $S\in \Hy(X)$ by $v$.
\[pro:contraction-hyp-estimate\] Let $v:X\dasharrow X'$ be a birational transformation between normal irreducible projective varieties. Let $S$ be an element of $\Hy(X)$.
1. \[ipascontra\] If $S\in (v^{-1})_\circ\Hy(X')$, then $v_\bullet(S)=v_\circ(S)\in\Hy(X')$.
2. \[icontra\] If $S\notin (v^{-1})_\circ \Hy(X')$, then $v_\circ(S)$ has codimension $\ge 2$ (i.e. $v$ contracts $S$), and $v_\bullet(S)$ is an element of $\Hyp(X')\smallsetminus \Hy(X')$.
3. \[iconclusion\] The symmetric difference $v_\bullet(\Hy(X))\triangle \Hy(X')$ contains $\exc(v)+\exc(v^{-1})$ elements.
Let $U$ be the complement of $\Ind(v)$ in $X'$. Since, by Theorem \[normal2\], $\Ind(v)$ has codimension $\ge 2$, no $S\in\Hy(X)$ is contained in $\Ind(v)$.
Let us prove (\[ipascontra\]). When $v$ is a birational morphism the assertion follows from Proposition \[hypermo\]. To deal with the general case, write $v=g\circ f^{-1}$ where $f\colon Y\to X$ and $g\colon Y\to X'$ are birational morphisms from a normal variety $Y$. Since $f$ is a birational morphism, $f^\bullet(S)=f^\circ(S)\subset\Hy(Y)$; since $S$ is not contracted by $v$, $g_\bullet (f^\circ(S)) = g_\circ(f^\circ(S)) \in \Hy(X')$. Thus, $v_\bullet(S)=g_\bullet (f^\bullet(S))$ coincides with the strict transform $v_\circ(S)\in\Hy(X')$.
Now let us prove (\[icontra\]), assuming thus that $S\notin (v^{-1})_\circ\Hy(X')$. Let $S''\in\Hy(Y)$ be the hypersurface $(f^{-1})_\bullet(S)=(f^{-1})_\circ(S)$. Then $f(S'')=S$. If $g_\circ (S'')$ is a hypersurface $S'$, then $(v^{-1})_\circ (S')=S$, contradicting $S\notin (v^{-1})_\circ\Hy(X')$. Thus, $g$ contracts $S''$ onto a subset $S'\subset X'$ of codimension $\geq 2$. Since $S'=v_\circ(S)$, assertion (\[icontra\]) is proved.
Assertion (\[iconclusion\]) follows from the previous two assertions.
Let $g$ be a birational transformation of $\P^n_\bfk$ of degree $d$, meaning that $g^*(H)\simeq dH$ where $H$ denotes a hyperplane of $\P^n_\bfk$, or equivalently that $g$ is defined by $n+1$ homogeneous polynomials of the same degree $d$ without common factor of positive degree. The exceptional set of $g$ has degree $(n+1)(d-1)$; as a consequence, $\exc_{\P^n_\bfk}(g)\leq (n+1)(d-1)$. More generally, if $H$ is a polarization of $X$, then $\exc_X(g)$ is bounded from above by a function that depends only on the degree $\deg_H(g):=(g^*H)\cdot H^{\dim(X)-1}$.
Action of $\Bir(X)$ on $\Hyp(X)$
--------------------------------
Let us now restrict the functor $f\mapsto f_\bullet$ to the elements of $\Bir(X)$. The existence of this functor and Proposition \[pro:contraction-hyp-estimate\] give the following theorem.
\[thm:def-action-hypX\] Let $X$ be a normal projective variety. The group $\Bir(X)$ acts faithfully by permutations on the set $\Hyp(X)$ via the homomorphism $$\left\{ \begin{array}{ccc} \Bir(X)& \to & \Bij(\Hyp(X))\\
g & \mapsto & g_\bullet
\end{array}
\right.$$ This action commensurates the subset $\Hy(X)$ of $\Hyp(X)$: for every $g\in\Bir(X)$, $$\vert g_\bullet(\Hy(X))\triangle \Hy(X)\vert = \exc(g)+\exc(g^{-1}).$$
The only thing that has not been proven yet is the fact that the homomorphism $f\in \Bir(X)\mapsto \Bij(\Hyp(X))$ is injective. But the kernel of this homomorphism is made of birational transformations $f$ such that $f_\circ (W)=W$ for every irreducible hypersurface $W$ of $X$. Since $X$ is projective, one can embed $X$ in some projective space $\P^m_\bfk$; then, every point of $X(\bfk)$ is the intersection of finitely many irreducible hyperplane sections of $X$: since all these sections are fixed by $f$, every point is fixed by $f$, and $f$ is the identity.
Products of varieties
---------------------
Let $X,Y$ be irreducible, normal projective varieties. We consider the natural embedding of $\Bir(X)$ into $\Bir(X\times Y)$, given by the birational action $f\cdot(x,y)=(f(x),y)$, $f\in\Bir(X)$. There is a natural injection $j_Y$ of $\Hy(X)$ into $\Hy(X\times Y)$, given by $S\mapsto S\times Y$, which naturally extends to an injection of $\Hyp(X)$ into $\Hyp(X\times Y)$; this inclusion is $\Bir(X)$-equivariant. The following proposition will be applied to Corollary \[nondistab\].
\[pro:produ\] Let a group $\Gamma$ act on $X$ by birational transformations. Then $\Gamma$ transfixes $\Hy(X)$ in $\Hyp(X)$ if and only if it transfixes $\Hy(X\times Y)$ in $\Hyp(X\times Y)$. More precisely, the subset $\Hy(X\times Y)\smallsetminus j_Y(\Hy(X))$ is $\Bir(X)$-invariant.
The reverse implication is immediate, since any restriction of a transfixing action is transfixing. The direct implication follows from the latter statement, which we now prove. Consider $S\in\Hy(X\times Y)\smallsetminus j_Y(\Hy(X))$. This means that $S$ is an irreducible hypersurface of $X\times Y$ whose projection to $X$ is surjective. Now, for $\gamma\in\Bir(X)$, $\gamma$ induces an isomorphism between open dense subset $U,V$ of $X$, and hence between $U\times Y$ and $V\times Y$; in particular, $\gamma$ does not contract $S$. This shows that $\gamma$ stabilizes $\Hy(X\times Y)\smallsetminus j_Y(\Hy(X))$.
Manin’s construction {#par:Manin-Construction}
--------------------
Instead of looking at the functor $X\mapsto \Hy(X)$ from the category of normal projective varieties to the category of sets, one can consider the Néron-Severi functor $X\mapsto \NS(X)$ from the category of smooth projective varieties to the category of abelian groups. In characteristic zero, or for surfaces in arbitrary characteristic, the resolution of singularities shows that smooth projective varieties, together with birational morphisms, form a good right localization of the category of smooth projective varieties with birational maps between them. Thus, one can construct a functor, the relative colimit of Néron-Severi groups, $X\mapsto {\tilde{\NS}}(X)$ that maps birational maps $X\dasharrow Y$ to group isomorphisms ${\tilde{\NS}}(X)\to {\tilde{\NS}}(Y)$. In dimension $2$, this construction is known as the Picard-Manin space (see [@Cantat:Annals; @Manin:cubic-forms]). One may also replace $\NS(X)$ by other cohomology groups if they behave contravariantly with respect to birational morphisms (see [@Boucksom-Favre-Jonsson:PRIMS] for instance).
Pseudo-regularization of birational transformations
===================================================
In this section, we make use of the action of $\Bir(X)$ on $\Hyp(X)$ to characterize and study groups of birational transformations that are pseudo-regularizable, in the sense of Definition \[d\_psr\]. As before, $\bfk$ is an algebraically closed field.
An example {#par:example-intro6}
----------
Consider the birational transformation $f(x,y)=(x+1,xy)$ of $\P^1_\bfk\times \P^1_\bfk$. The vertical curves $C_i=\{x=-i\}$, $i\in \Z$, are exceptional curves for the cyclic group $\Gamma=\langle f \rangle$: each of these curves is contracted by an element of $\Gamma$ onto a point, namely $f^{i+1}_\circ(C_i)=(1,0)$.
Let $\varphi\colon Y\dasharrow \P^1_\bfk\times \P^1_\bfk$ be a birational map, and let $\U$ be a non-empty open subset of $Y$. Consider the subgroup $\Gamma_Y:=\varphi^{-1}\circ \Gamma\circ \varphi$ of $\Bir(Y)$. If $i$ is large enough, $\varphi^{-1}_\circ(C_i)$ is an irreducible curve $C'_i\subset Y$, and these curves $C'_i$ are pairwise distinct, so that most of them intersect $\U$. For positive integers $m$, $f^{i+m}$ maps $C_i$ onto $(m,0)$, and $(m,0)$ is not an indeterminacy point of $\varphi^{-1}$ if $m$ is large. Thus, $\varphi^{-1}\circ f^m\circ \varphi$ contracts $C'_i$, and $\varphi^{-1}\circ f^m\circ \varphi$ is not a pseudo-automorphism of $\U$. This argument proves the following lemma.
\[lem:counter-example\] Let $X$ be the surface $\P^1_\bfk\times \P^1_\bfk$. Let $f\colon X\dasharrow X$ be defined by $f(x,y)=(x+1,xy)$, and let $\Gamma$ be the subgroup generated by $f^\ell$, for some $\ell \geq 1$. Then the cyclic group $\Gamma$ is not pseudo-regularizable.
This shows that Theorem A requires an assumption on the group $\Gamma$. More generally, consider a subgroup $\Gamma\subset \Bir(X)$ such that $\Gamma$
1. contracts a family of hypersurfaces $W_i\subset X$ whose union is Zariski dense
2. the union of the family of strict transforms $f_\circ(W_i)$, for $f\in \Gamma$ contracting $W_i$, form a subset of $X$ whose Zariski closure has codimension at most $1$.
Then, $\Gamma$ cannot be pseudo-regularized.
Characterization of pseudo-Isomorphisms
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Recall that $f_\bullet$ denotes the bijection $\Hyp(X)\to \Hyp(X')$ which is induced by a birational map $f\colon X\dasharrow X'$. Also, for any nonempty open subset $U\subset X$, we define $\Hy(U)=\{H\in\Hy(X):H\cap U\neq\emptyset\}$; it has finite complement in $\Hy(X)$.
\[pro:charact-pseudo-isom\] Let $f:X\dasharrow X'$ be a birational map between normal projective varieties. Let $U\subset X$ and $U'\subset X'$ be two dense open subsets. Then, $f$ induces a pseudo-isomorphism $U\to U'$ if and only if $f_\bullet(\Hy(U))=\Hy(U')$.
If $f$ restricts to a pseudo-isomorphism $U\to U'$, then $f$ maps every hypersurface of $U$ to a hypersurface of $U'$ by strict transform. And $(f^{-1})_\circ$ is an inverse for $f_\circ \colon \Hy(U)\to \Hy(U')$. Thus, $f_\bullet(\Hy(U))=f_\circ(\Hy(U)=\Hy(U')$.
Let us now assume that $f_\bullet(\Hy(U))=\Hy(U')$. Since $X$ and $X'$ are normal, $\Ind(f)$ and $\Ind(f^{-1})$ have codimension $\ge 2$ (Theorem \[normal2\]).
Let $f_{U,U'}$ be the birational map from $U$ to $U'$ which is induced by $f$. The indeterminacy set of $f_{U,U'}$ is contained in the union of the set $\Ind(f)\cap U$ and the set of points $x\in U\smallsetminus \Ind(f)$ which are mapped by $f$ in the complement of $U'$; this second part of $\Ind(f_{U,U'})$ has codimension $2$, because otherwise there would be an irreducible hypersurface $W$ in $U$ which would be mapped in $X'\smallsetminus U'$, contradicting the equality $f_\bullet (\Hy(U))=\Hy(U')$. Thus, the indeterminacy set of $f_{U,U'}$ has codimension $\geq 2$. Changing $f$ in its inverse $f^{-1}$, we see that the indeterminacy set of $f^{-1}_{U',U}\colon U'\dasharrow U'$ has codimension $\geq 2$ too.
If $f_{U,U'}$ contracted an irreducible hypersurface $W\subset U$ onto a subset of $U'$ of codimension $\geq 2$, then $f_\bullet(W)$ would not be contained in $\Hy(U')$ (it would correspond to an element of $\Hyp(X')\smallsetminus \Hy(X')$ by Proposition \[pro:contraction-hyp-estimate\]). Thus, $f_{U,U'}$ satisfies the first property of Proposition \[pro:pseudo-isomorphism\] and, therefore, is a pseudo-isomorphism.
Characterization of pseudo-regularization
-----------------------------------------
Let $X$ be a (irreducible, reduced) normal projective variety. Let $\Gamma$ be a subgroup of $\Bir(X)$. Assume that the action of $\Gamma$ on $\Hyp(X)$ fixes (globally) a subset $A\subset \Hyp(X)$ such that $$\vert A\triangle \Hy(X)\vert < +\infty.$$ In other words, $A$ is obtained from $\Hy(X)$ by removing finitely many hypersurfaces $W_i \in \Hy(X)$ and adding finitely many hypersurfaces $W'_j\in \Hyp(X)\smallsetminus\Hy(X)$. Each $W'_j$ comes from an irreducible hypersurface in some model $\pi_j\colon X_j\to X$, and there is a model $\pi\colon Y\to X$ that covers all of them (i.e. $\pi\circ \pi_j^{-1}$ is a morphism from $Y$ to $X_j$ for every $j$). Then, $\pi^\circ(A)$ is a subset of $\Hy(Y)$. Changing $X$ into $Y$, $A$ into $\pi^\circ(A)$, and $\Gamma$ into $\pi^{-1}\circ \Gamma\circ \pi$, we may assume that
1. $A=\Hy(X)\smallsetminus\{E_1, \ldots, E_\ell\}$ where the $E_i$ are $\ell$ distinct irreducible hypersurfaces of $X$,
2. the action of $\Gamma$ on $\Hyp(X)$ fixes the set $A$.
In what follows, we denote by $\U$ the non-empty Zariski open subset $X\smallsetminus \cup_i E_i$ and by $\partial X$ the boundary $X\smallsetminus \U= E_1\cup \cdots \cup E_\ell$; $\partial X$ is considered as the boundary of the compactification $X$ of $\U$.
\[lem:pseudo-on-U\] The group $\Gamma$ acts by pseudo-automorphisms on the open subset $\U$. If $\U$ is smooth and there is an ample divisor $D$ whose support coincides with $\partial X$, then $\Gamma$ acts by automorphisms on $\U$.
In this statement, we say that the support of a divisor $D$ coincides with $\partial X$ if $D=\sum_{i} a_i E_i$ with $a_i>0$ for every $1\leq i\leq \ell$.
Since $A=\Hy(\U)$ is $\Gamma$-invariant, Proposition \[pro:charact-pseudo-isom\] shows that $\Gamma$ acts by pseudo-automorphisms on $\U$.
Since $D$ is an ample divisor, some positive multiple $mD$ is very ample, and the complete linear system $\vert mD\vert$ provides an embedding of $X$ in a projective space. The divisor $mD$ corresponds to a hyperplane section of $X$ in this embedding, and the open subset $\U$ is an affine variety because the support of $D$ is equal to $\partial X$. Proposition \[pro:pseudo-automorphisms-affine\] concludes the proof of the lemma.
By Theorem \[thm:def-action-hypX\], every subgroup of $\Bir(X)$ acts on $\Hyp(X)$ and commensurates $\Hy(X)$. If $\Gamma$ transfixes $\Hy(X)$, there is an invariant subset $A$ of $\Hyp(X)$ for which $A\triangle \Hy(X)$ is finite. Thus, one gets the following characterization of pseudo-regularizability (the converse being immediate).
\[thm:FW-pseudo-regularization\] Let $X$ be a normal projective variety over an algebraically closed field $\bfk$. Let $\Gamma$ be a subgroup of $\Bir(X)$. Then $\Gamma$ transfixes the subset $\Hy(X)$ of $\Hyp(X)$ if and only if $\Gamma$ is pseudo-regularizable.
Of course, this theorem applies directly when $\Gamma\subset \Bir(X)$ has property [[(FW)]{}]{} because Theorem \[thm:def-action-hypX\] shows that $\Gamma$ commensurates $\Hy(X)$.
Assuming ${\mathrm{char}}(\bfk)=0$, we may work in the category of smooth varieties (see Example \[eg:category-bira-2\] and § \[par:Manin-Construction\]). As explained in Remark \[rem:intro-extreme-cases\] and Lemma \[lem:pseudo-on-U\], there are two extreme cases, corresponding to an empty or an ample boundary $B=\cup_i E_i$.
If $\U=Y$, $\Gamma$ acts by pseudo-automorphisms on the projective variety $Y$. As explained in Theorem \[thm:pseudo-automorphisms-neron-severi\], $\Gamma$ is an extension of a subgroup of $\GL(\NS(Y))$ by an algebraic group (which is almost contained in $\Aut(Y)^0$).
If $\U$ is affine, $\Gamma$ acts by automorphisms on $\U$. The group $\Aut(\U)$ may be huge (for instance if $\U$ is the affine space), but there are techniques to study groups of automorphisms that are not available for birational transformations. For instance $\Gamma$ is residually finite and virtually torsion free if $\Gamma$ is a group of automorphisms generated by finitely many elements (see [@Bass-Lubotzky:1983]).
Distorted elements {#par:distorsion}
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Theorem \[thm:FW-pseudo-regularization\] may be applied when $\Gamma$ has Property [[(FW)]{}]{}, or for pairs $(\Lambda,\Gamma)$ with relative Property [[(FW)]{}]{}. Here is one application:
Let $X$ be an irreducible projective variety. Let $\Gamma$ be a distorted cyclic subgroup of $\Bir(X)$. Then $\Gamma$ is pseudo-regularizable.
The contraposition is useful to show that some elements of $\Bir(X)$ are undistorted. Let us state it in a strong “stable” way.
\[nondistab\] Let $X$ be a normal irreducible projective variety and let $f$ be an element of $\Bir(X)$ such that the cyclic group $\langle f\rangle$ does not transfix $\Hy(X)$ (i.e., $f$ is not pseudo-regularizable). Then the cyclic subgroup $\langle f\rangle$ is undistorted in $\Bir(X)$, and more generally for every irreducible projective variety, the cyclic subgroup $\langle f\times\mathrm{Id}_Y\rangle$ is undistorted in $\Bir(X\times Y)$.
The latter consequence indeed follows from Proposition \[pro:produ\]. This can be applied to various examples, such as those in Example \[gatransfix\].
Illustrating results
====================
Surfaces whose birational group is transfixing {#surf_birt}
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If $X$ is a projective curve, $\Bir(X)$ always transfixes $\Hy(X)$, since it acts by automorphisms on a smooth model of $X$. We now consider the same problem for surfaces, starting with the following result, which holds in arbitrary dimension.
\[ruled\] Let $X$ be a normal irreducible variety of positive dimension over an algebraically closed field $\bfk$. Then $\Bir(X\times\mathbb{P}^1)$ does not transfix $\Hy(X\times\mathbb{P}^1)$.
We can suppose that $X$ is affine and work in the model $X\times\mathbb{A}^1$. For $\varphi$ a nonzero regular function on $X$, define a regular self-map $f$ of $X\times\mathbb{A}^1$ by $f(x,t)=(x,\varphi(x)t)$. Denoting by $Z(\varphi)$ the zero set of $\varphi$, we remark that $f$ induces an automorphism of the open subset $(X\smallsetminus Z(\varphi))\times\mathbb{A}^1$. In particular, it induces a permutation of $\Hy((X\smallsetminus Z(\varphi))\times\mathbb{A}^1)$. Moreover, since $f$ contracts the complement $Z(\varphi)\times\mathbb{A}^1$ to the subset $Z(\varphi)\times\{0\}$, which has codimension $\ge 2$, its action on $\Hyp(X\times\mathbb{A}^1)$ maps the set of codimension $1$ components of $Z(\varphi)\times\mathbb{A}^1$ outside $M=\Hy(X\times\mathbb{A}^1)$. Therefore $M\smallsetminus f^{-1}(M)$ is the set of irreducible components of $Z(\varphi)\times\mathbb{A}^1$. Its cardinal is equal to the number of irreducible components of $Z(\varphi)$. When $\varphi$ varies, this number is unbounded; hence, $\Bir(X\times\mathbb{A}^1)$ does not transfix $\Hy(X\times\mathbb{A}^1)$.
Varieties that are birational to the product of a variety and the projective line are said to be [**ruled**]{}. Proposition \[ruled\] states that for any ruled irreducible projective variety $Y$ of dimension $\ge 2$, $\Bir(Y)$ does not transfix $\Hy(Y)$. The converse holds for surfaces, by the following theorem.
\[birtran\] Let $\bfk$ be an algebraically closed field. Let $X$ be an irreducible normal projective surface over $\bfk$. The following are equivalent:
1. \[notransfix\] $\Bir(X)$ does not transfix $\Hy(X)$;
2. \[kod\] the Kodaira dimension of $X$ is $-\infty$;
3. \[kod2\] $X$ is ruled;
4. \[kod3\] there is no irreducible projective surface $Y$ that is birationally equivalent to $X$, and such that $\Bir(Y)=\Aut(Y)$.
The equivalence between (\[kod\]) and (\[kod2\]) is classical (see [@BPVDVH]). The group $\Aut(Y)$ fixes $\Hy(Y)\subset \Hyp(Y)$, hence (\[notransfix\]) implies (\[kod3\]). If the Kodaira dimension of $X$ is $\geq 0$, then $X$ has a unique minimal model $X_0$, and $\Bir(X_0)=\Aut(X_0)$. Thus, (\[kod3\]) implies (\[kod\]). Finally, Proposition \[ruled\] shows that (\[kod2\]) implies (\[notransfix\]).
\[thm:fin-gen-transfix\] Let $X$ be an irreducible projective surface over an algebraically closed field $\bfk$. The following are equivalent:
1. \[fgnt\] some finitely generated subgroup of $\Bir(X)$ does not transfix $\Hy(X)$;
2. \[cynt\] some cyclic subgroup of $\Bir(X)$ does not transfix $\Hy(X)$;
3. \[kod21\]
- $\bfk$ has characteristic $0$, and $X$ is birationally equivalent to the product of the projective line with a curve of genus 0 or 1, or
- $\bfk$ has positive characteristic, and $X$ is a rational surface.
\[exp2facile\] Let $\bfk$ be an algebraically closed field that is not algebraic over a finite field. Let $t$ be an element of infinite order in the multiplicative group $\bfk^*$. Then the birational transformation $g$ of $\P^2_\bfk$ given, in affine coordinates, by $(x,y)\mapsto (tx+1,xy)$ does not transfix $\Hy(\P^2_\bfk)$. Indeed, it is easy to show that the hypersurface $C=\{x=0\}$ satisfies, for $n\in\Z$, $f^n(C)\in\Hy(\P^2_\bfk)$ if and only if $n\le 0$.
\[p2compli\] The example of non-transfixing element in Example \[exp2facile\] works under a small restriction on $\bfk$. Here is an example over an arbitrary algebraically closed field $\bfk$. Let $L$ and $L'$ be two lines in $\P^2_\bfk$ intersecting transversally at a point $q$. Let $f$ be a birational transformation of $\P^2_\bfk$ that contracts $L'$ onto $q$ and fixes the line $L$. For instance, in affine coordinates, the monomial map $(x,y)\mapsto (x,xy)$ contracts the $y$-axis onto the origin, and fixes the $x$-axis. Assume that there is an open neighborhood $\U$ of $q$ such that $f$ does not contract any curve in $\U$ except the line $L'$. Let $C$ be an irreducible curve that intersects $L$ and $L'$ transversally at $q$. Then, for every $n\geq 1$, the strict transform $f^n_\circ(C)$ is an irreducible curve, and the order of tangency of this curve with $L$ goes to infinity with $n$. Thus, the degree of $f^n_\circ(C)$ goes to infinity too and the $f^n_\circ(C)$ form an infinite sequence in $\Hy(\P^2_\bfk)$.
Now, assume that $C$ is contracted by $f^{-1}$ onto a point $p$, $p\notin \Ind(f)$, and $p$ is fixed by $f^{-1}$. Then, for every $m\geq 1$, $f^{-m}_\bullet(C)$ is not in $\Hy(\P^2_\bfk)$. This shows that the orbit of $C$ under the action of $f_\bullet$ intersects $\Hy(\P^2_\bfk)$ and its complement $\Hyp(\P^2_\bfk)\smallsetminus \Hy(\P^2_\bfk)$ on the infinite sets $\{f^n_\circ(C)\, ; \, n\geq 1\}$ and $\{f^{-m}_\bullet(C)\, ; \, m\geq 1\}$. In particular, $f$ does not transfix $\Hy(\P^2_\bfk)$.
Since such maps exist over every algebraically closed field $\bfk$, this example shows that property (\[cynt\]) of Theorem \[thm:fin-gen-transfix\] is satisfied for every rational surface $X$.
Trivially (\[cynt\]) implies (\[fgnt\]).
Suppose that (\[kod21\]) holds and let us prove (\[cynt\]). The case $X=\P^1\times\P^1$ is already covered by Lemma \[lem:counter-example\] in characteristic zero, and by the previous example in positive characteristic. The case $X=C\times\P^1$ in characteristic zero, where $C$ is an elliptic curve, is similar. To see it, fix a point $t_0\in C$ and a rational function $\varphi$ on $C$ that vanishes at $t_0$. Then, since $\bfk$ has characteristic zero, one can find a translation $s$ of $C$ of infinite order such that the orbit $\{s^n(t_0):n\in\Z\}$ does not contain any other zero or pole of $\varphi$ (here we use that the characteristic of $\bfk$ is $0$). Consider the birational transformation $f\in \Bir(X)$ given by $f(t,x)=(s(t),\varphi(t)x)$. Let $H$ be the hypersurface $\{t_0\}\times C$. Then for $n\in\Z$, we have $(f_\bullet)^nH\in\Hy(X)$ if and only if $n\le 0$. Hence the action of the cyclic group $\langle f\rangle$ does not transfix $\Hy(X)$.
Let us now prove that (\[fgnt\]) implies (\[kod21\]). Applying Theorem \[birtran\], and changing $X$ to a birationally equivalent surface if necessary, we assume that $X=C\times\mathbb{P}^1$ for some (smooth irreducible) curve $C$. We may now assume that the genus of $C$ is $\ge 2$, or $\ge 1$ in positive characteristic, and we have to show that every finitely generated group $\Gamma$ of $\Bir(X)$ transfixes $\Hy(X)$. Since the genus of $C$ is $\geq 1$, the group $\Bir(X)$ preserves the fibration $X\to C$; this gives a surjective homomorphism $\Bir(X)\to\Aut(C)$. Now let us fully use the assumption on $C$: if its genus is $\ge 2$, then $\Aut(C)$ is finite; if its genus is 1 and $\bfk$ has positive characteristic, then $\Aut(C)$ is locally finite[^1], and in particular the projection of $\Gamma$ on $\Aut(C)$ has a finite image. Thus the kernel of this homomorphism intersects $\Gamma$ in a finite index subgroup $\Gamma_0$. It now suffices to show that $\Gamma_0$ transfixes $\Hy(X)$. Every $f\in\Gamma_0$ has the form $f(t,x)= (t,\varphi_t(x))$ for some rational map $t\mapsto\varphi_t$ from $C$ to $\PGL_2$; define $U_f\subset C$ as the open and dense subset on which $\varphi_\gamma$ is regular: by definition, $f$ restricts to an automorphism of $U_f\times \P^1$. Let $S$ be a finite generating subset of $\Gamma_0$, and let $U_S$ be the intersection of the open subsets $U_g$, for $g\in S$. Then $\Gamma_0$ acts by automorphisms on $U_S\times \P^1$ and its action on $\Hy(X)$ fixes the subset $\Hy(U_S)$. Hence $\Gamma$ transfixes $\Hy(X)$.
It would be interesting to obtain characterizations of the same properties in dimension 3 (see Question \[ruled3\]).
Transfixing Jonquières twists {#par:AppendixII}
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Let $X$ be an irreducible normal projective surface and $\pi$ a morphism onto a smooth projective curve $C$ with rational connected fibers. Let $\Bir_\pi(X)$ be the subgroup of $\Bir(X)$ permuting the fibers of $\pi$. Since $C$ is a smooth projective curve, the group $\Bir(C)$ coincides with $\Aut(C)$ and we get a canonical homomorphism ${\mathrm{r}}_C\colon \Bir_\pi(X)\to\Aut(C)$.
The main examples to keep in mind are provided by $\P^1\times \P^1$, Hirzebruch surfaces, and $C\times \P^1$ for some genus $1$ curve $C$, $\pi$ being the first projection.
Let $\Hy_\pi(X)$ denote the set of irreducible curves which are contained in fibers of $\pi$, and define $\Hyp_\pi(X)=\Hy_\pi(X)\sqcup (\Hyp(X)\smallsetminus\Hy(X))$, so that $\Hyp(X)=\Hyp_\pi(X)\sqcup (\Hy(X)\smallsetminus\Hy_\pi(X)).$ An irreducible curve $H\subset X$ is an element of $\Hy(X)\smallsetminus\Hy_\pi(X)$ if and only if its projection $\pi(H)$ coincides with $C$; this curves are said to be transverse to $\pi$.
\[decomfi\] The decomposition $\Hyp(X)=\Hyp_\pi(X)\sqcup (\Hy(X)\smallsetminus\Hy_\pi(X))$ is $\Bir_\pi(X)$-invariant.
Let $H\subset X$ be an irreducible curve which is transverse to $\pi$. Since $\Bir_\pi(X)$ acts by automorphisms on $C$, $H$ can not be contracted by any element of $\Bir_\pi(X)$; more precisely, for every $g\in \Bir_\pi(X)$, $g_\bullet(H)$ is an element of $\Hy(X)$ which is transverse to $\pi$. Thus the set of transverse curves is $\Bir_\pi(X)$-invariant.
This proposition and the proof of Theorem \[thm:fin-gen-transfix\] lead to the following corollary.
\[cortran\] Let $G$ be a subgroup of $\Bir_\pi(X)$. If $\pi$ maps the set of indeterminacy points of the elements of $G$ into a finite subset of $C$, then $G$ transfixes $\Hy(X)$.
In the case of cyclic subgroups, we establish a converse under the mild assumption of algebraic stability. Recall that a birational transformation $f$ of a smooth projective surface is [**[algebraically stable]{}**]{} if the forward orbit of $\Ind(f^{-1})$ does not intersect $\Ind(f)$. By [@Diller-Favre], given any birational transformation $f$ of a surface $X$, there is a birational morphism $u\colon Y\to X$, with $Y$ a smooth projective surface, such that $f_Y:=u^{-1}\circ f\circ u$ is algebraically stable. If $\pi\colon X\to C$ is a fibration, as above, and $f$ is in $\Bir_\pi(X)$, then $f_Y$ preserves the fibration $\pi\circ u$. Thus, we may always assume that $X$ is smooth and $f$ is algebraically stable after a birational conjugacy.
\[protrahy\] Let $X$ be a smooth projective surface, and $\pi\colon X\to C$ a rational fibration. If $f\in \Bir_\pi(X)$ is algebraically stable, then $f$ transfixes $\Hy(X)$ if, and only if the orbit of $\pi(\Ind(f))$ under the action of ${\mathrm{r}}_C(f)$ is finite.
For $X=\mathbb{P}^1\times\mathbb{P}^1$, the reader can check (e.g., conjugating a suitable automorphism) that the proposition fails without the algebraic stability assumption.
Denote by $A\subset \Aut(C)$ the subgroup generated by ${\mathrm{r}}_C(f)$. Consider a fiber $F\simeq \P^1$ which is contracted to a point $q$ by $f$. Then, there is a unique indeterminacy point $p$ of $f$ on $F$. If the orbit of $\pi(q)$ under the action of $A$ is infinite, the orbit of $q$ under the action of $f$ is infinite too. Set $q_n=f^{n-1}(q)$ for $n\geq 1$ (so that $q_1=q$); this sequence of points is well defined because $f$ is algebraically stable: for every $n\geq 1$, $f$ is a local isomorphism from a neighborhood of $q_n$ to a neighborhood of $q_{n+1}$. Then, the image of $F$ in $\Hyp(X)$ under the action of $f^n$ is an element of $\Hyp(X)\smallsetminus \Hy(X)$: it is obtained by a finite number of blow-ups above $q_n$. Since the points $q_n$ form an infinite set, the images of $F$ form an infinite subset of $\Hyp(X)\smallsetminus \Hy(X)$. Together with the previous corollary, this argument proves the proposition.
\[gatransfix\]Consider $X=\P^1\times\P^1$, with $\pi(x,y)=x$ (using affine coordinates). Start with $f_a(x,y)=(ax,xy)$, for some non-zero parameter $a\in \bfk$. The action of ${\mathrm{r}}_C(f_a)$ on $C=\P^1$ fixes the images $0$ and $\infty$ of the indeterminacy points of $f_a$. Thus, $f_a$ transfixes $\Hyp(X)$ by Corollary \[cortran\]. Now, consider $g_a(x,y)=(a x, (x+1)y)$. Then, the orbit of $-1$ under multiplication by $a$ is finite if and only if $a$ is a root of unity; thus, if $a$ is not a root of unity, $g_a$ does not transfix $\Hy(X)$. Section \[par:example-intro6\] provides more examples of that kind.
Birational transformations of surfaces I
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From now on, we work in dimension $2$: $X$, $Y$, and $Z$ will be smooth projective surfaces over the algebraically closed field $\bfk$. (In dimension $2$, the resolution of singularities is available in all characteristics, so that we can always assume the varieties to be smooth.)
Regularization
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In this section, we refine Theorem \[thm:FW-pseudo-regularization\], in order to apply results of Danilov and Gizatullin. Recall that a curve $C$ in a smooth surface $Y$ has [**normal crossings**]{} if each of its singularities is a simple node with two transverse tangents. In the complex case, this means that $C$ is locally analytically equivalent to $\{xy=0\}$ (two branches intersecting transversally) in an analytic neighborhood of each of its singularities.
\[thm:FW-regularization-surfaces\] Let $X$ be a smooth projective surface, defined over an algebraically closed field $\bfk$. Let $\Gamma$ be a subgroup of $\Bir(X)$ that transfixes the subset $\Hy(X)$ of $\Hyp(X)$. There exists a smooth projective surface $Z$, a birational map $\varphi\colon Z\dasharrow X$ and a dense open subset $\U\subset Z$ such that, writing the boundary $\partial Z:= Z\smallsetminus \U$ as a finite union of irreducible components $E_i\subset Z$, $1\leq i\leq \ell$, the following properties hold:
1. \[mc2\] The boundary $\partial Z$ is a curve with normal crossings.
2. \[mc3\] The subgroup $\Gamma_Z:=\varphi^{-1}\circ \Gamma\circ \varphi\subset\Bir(Z)$ acts by automorphisms on the open subset $\U$.
3. \[mc4\] For all $i\in\{1,\dots,\ell\}$ and $g\in \Gamma_Z$, the strict transform of $E_i$ under the action of $g$ on $Z$ is contained in $\partial Z$: either $g_\circ(E_i)$ is a point of $\partial Z$ or $g_\circ(E_i)$ is an irreducible component $E_j$ of $\partial Z$.
4. \[mc5\] For all $i\in\{1,\dots,\ell\}$, there exists an element $g\in \Gamma_Z$ that contracts $E_i$ to a point $g_\circ(E_i)\in \partial Z$. In particular, $E_i$ is a rational curve.
5. \[mc6\] The pair $(Z, \U)$ is minimal for the previous properties, in the following sense: if one contracts a smooth curve of self-intersection $-1$ in $\partial Z$, then the boundary stops to be a normal crossing divisor.
Before starting the proof, note that the boundary $\partial Z$ may a priori contain an irreducible rational curve $E$ with a node.
We apply Theorem \[thm:FW-pseudo-regularization\] (which works in positive characteristic too, because $X$ is a surface), and get a birational morphism $\varphi_0\colon Y_0\to X$ and an open subset $\U_0$ of $Y_0$ that satisfy properties (1) and (3), except that we only know that the action of $\Gamma_0:=\varphi_0^{-1}\circ \Gamma\circ \varphi_0$ on $\U_0$ is by pseudo-automorphisms (not yet by automorphisms). We shall progressively modify the triple $(Y_0, \U_0, \varphi_0)$ to obtain a surface $Z$ with properties (\[mc2\]) to (\[mc6\]).
First, we blow-up the singularities of the curve $\partial Y_0= Y_0\smallsetminus \U_0$ to get a boundary that is a normal crossing divisor. This replaces the surface $Y_0$ by a new one, still denoted $Y_0$. This modification adds new components to the boundary $\partial Y_0$ but does not change the fact that $\Gamma_0$ acts by pseudo-automorphisms on $\U_0$. Let $\ell_0$ be the number of irreducible components of $Y_0\smallsetminus\U_0$.
Consider a point $q$ in $\U_0$, and assume that there is a curve $E_i$ of $\partial Y_0$ that is contracted to $q$ by an element $g\in \Gamma_0$; fix such a $g$, and denote by $D$ the union of the curves $E_j$ such that $g_\circ(E_j)=q$. By construction, $g$ is a pseudo-automorphism of $\U_0$. The curve $D$ does not intersect the indeterminacy set of $g$, since otherwise there would be a curve $C$ containing $q$ that is contracted by $g^{-1}$. And $D$ is a connected component of $\partial Y_0$, because otherwise $g$ maps one of the $E_j$ to a curve that intersects $\U_0$. Thus, there are small neighborhoods $\W$ of $D$ and $\W'$ of $q$ such that $\W\cap \partial Y_0=D$ and $g$ realizes an isomorphism from $\W\smallsetminus D$ to $\W'\smallsetminus \{ q\}$, contracting $D$ onto the smooth point $q\in Y_0$. As a consequence, there is a birational morphism $\pi_1\colon Y_0\to Y_1$ such that
1. $Y_1$ is smooth
2. $\pi_1$ contracts $D$ onto a point $q_1\in Y_1$
3. $\pi_1$ is an isomorphism from $Y_0\smallsetminus D$ to $Y_1\smallsetminus \{q_1\}$.
In particular, $\pi_1(\U_0)$ is an open subset of $Y_1$ and $\U_1=\pi_1(\U_0) \cup \{q_1\}$ is an open neighborhood of $q_1$ in $Y_1$.
Then, $\Gamma_1:=\pi_1\circ \Gamma_0\circ \pi_1^{-1}$ acts birationally on $Y_1$, and by pseudo-automorphisms on $\U_1$. The boundary $\partial Y_1=Y_1\smallsetminus \U_1$ contains $\ell_1$ irreducible components, with $\ell_1< \ell_0$ (the difference is the number of components of $D$), and is a normal crossing divisor because $D$ is a connected component of $\partial Y_0$.
Repeating this process, we construct a sequence of surfaces $\pi_k\colon Y_{k-1}\to Y_{k}$ and open subsets $\pi_k(\U_{k-1})\subset \U_k\subset Y_k$ such that the number of irreducible components of $\partial Y_k=Y_k\smallsetminus \U_k$ decreases. After a finite number of steps (at most $\ell_0$), we may assume that $\Gamma_k\subset\Bir(Y_k)$ does not contract any boundary curve onto a point of the open subset $\U_k$. On such a model, $\Gamma_k$ acts by automorphisms on $\U_k$.
We fix such a model, which we denote by the letters $Y$, $\U$, $\partial Y$, $\varphi$. The new birational map $\varphi\colon Y\dasharrow X$ is the composition of $\varphi_0$ with the inverse of the morphism $Y_0\to Y_k$. On such a model, properties (\[mc2\]) and (\[mc3\]) are satisfied. Moreover, (\[mc4\]) follows from (\[mc3\]). We now modify $Y$ further to get property (\[mc5\]).
Assume that the curve $E_i\subset Y\smallsetminus \U$ is not contracted by $\Gamma$. Let $F$ be the orbit of $E_i$: $F=\cup_{g\in\Gamma} g_\circ(E_i)$; this curve is contained in the boundary $\partial Y$ of the open subset $\U$. Changing $\U$ into $$\U'= \U \cup (F\smallsetminus \overline{\partial Y\smallsetminus F}),$$ the group $\Gamma$ also acts by pseudo-automorphisms on $\U'$. This operation decreases the number $\ell$ of irreducible components of the boundary. Thus, combining steps 2 and 3 finitely many times, we reach a model that satisfies Properties (\[mc2\]) to (\[mc5\]). We continue to denote it by $Y$.
If the boundary $\partial Y $ contains a smooth (rational) curve $E_i$ of self-intersection $-1$, it can be blown down to a smooth point $q$ by a birational morphism $\pi\colon Y\to Y'$; the open subset $\U$ is not affected, but the boundary $\partial Y'$ has one component less. If $E_i$ was a connected component of $\partial Y$, then $\U'=\pi(\U)\cup \{q\}$ is a neighborhood of $q$ and one replaces $\U$ by $\U'$, as in step 2. Now, two cases may happen. If the boundary $\partial Y'$ ceases to be a normal crossing divisor, we come back to $Y$ and do not apply this surgery. If $\partial Y'$ has normal crossings, we replace $Y$ by this new model. In a finite number of steps, looking successively at all $(-1)$-curves and iterating the process, we reach a new surface $Z$ on which all five properties are satisfied.
One may also remove property (\[mc6\]) and replace property (\[mc2\]) by
- The $E_i$ are rational curves, and none of them is a smooth rational curve with self-intersection $-1$.
But doing so, we may lose the normal crossing property. To get property (1’), apply the theorem and argue as in step 4.
Constraints on the boundary {#par:constraints-boundary}
---------------------------
We now work on the new surface $Z$ given by Theorem \[thm:FW-regularization-surfaces\]. Thus, $Z$ is the surface, $\Gamma$ the subgroup of $\Bir(Z)$, $\U$ the open subset on which $\Gamma$ acts by automorphisms, and $\partial Z$ the boundary of $\U$.
\[pro:Gizatullin-boundary\] There are four possibilities for the geometry of the boundary $\partial Z=Z\smallsetminus \U$.
1. \[giz1\] $\partial Z$ is empty.
2. \[giz2\] $\partial Z$ is a cycle of rational curves.
3. \[giz3\] $\partial Z$ is a chain of rational curves.
4. \[giz4\] $\partial Z$ is not connected; it is the disjoint union of finitely many smooth rational curves of self-intersection $0$.
Moreover, in cases (\[giz2\]) and (\[giz3\]), the open subset $\U$ is the blow-up of an affine surface.
Thus, there are four possibilities for $\partial Z$, which we study successively. We shall start with (1) and (4) in sections \[par:Projective-Surfaces\] and \[par:invariant-fibrations\]. Then case (3) is dealt with in Section \[par:completion-zigzag\]. Case (2) is slightly more involved: it is treated in Section \[par:Cycles-Thompson\].
Before that, let us explain how Proposition \[pro:Gizatullin-boundary\] follows from Section 5 of [@Gizatullin:1971]. First, we describe the precise meaning of the statement, and then we explain how the original results of [@Gizatullin:1971] apply to our situation.
– Consider the dual graph $\G_Z$ of the boundary $\partial Z$. The vertices of $\G_Z$ are in one to one correspondence with the irreducible components $E_i$ of $\partial Z$. The edges correspond to singularities of $\partial Z$: each singular point $q$ gives rise to an edge connecting the components $E_i$ that determine the two local branches of $\partial Z$ at $q$. When the two branches correspond to the same irreducible component, one gets a loop of the graph $\G_Z$.
We say that $\partial Z$ is a [**[chain]{}**]{} of rational curves if the dual graph is of type $A_\ell$: $\ell$ is the number of components, and the graph is linear, with $\ell$ vertices. Chains are also called [**[zigzags]{}**]{} by Gizatullin and Danilov.
We say that $\partial Z$ is a [**[cycle]{}**]{} if the dual graph is isomorphic to a regular polygon with $\ell$ vertices. There are two special cases: when $\partial Z$ is reduced to one component, this curve is a rational curve with one singular point and the dual graph is a loop (one vertex, one edge); when $\partial Z$ is made of two components, these components intersect in two distinct points, and the dual graph is made of two vertices with two edges between them. For $\ell= 3, 4, \ldots$, the graph is a triangle, a square, etc.
– To describe Gizatullin’s article, let us introduce some useful vocabulary. Let $S$ be a projective surface, and $C\subset S$ be a curve; $C$ is a union of irreducible components, which may have singularities. Assume that $S$ is smooth in a neighborhood of $C$. Let $S_0$ be the complement of $C$ in $S$, and let $\iota \colon S_0\to S$ be the natural embedding of $S_0$ in $S$. Then, $S$ is a [**[completion]{}**]{} of $S_0$: this completion is marked by the embedding $\iota \colon S_0\to S$, and its boundary is the curve $C$. Following [@Gizatullin:1971] and [@Danilov-Gizatullin:I; @Danilov-Gizatullin:II], we only consider completions of $S_0$ by curves (i.e. $S\smallsetminus \iota(S_0)$ is of pure dimension $1$), and we always assume $S$ to be smooth in a neighborhood of the boundary. Such a completion is
- [**[simple]{}**]{} if the boundary $C$ has normal crossings;
- [**[minimal]{}**]{} if it is simple and minimal for this property: if $C_i\subset C$ is an exceptional divisor of the first kind then, contracting $C_i$, the image of $C$ is not a normal crossing divisor anymore. Equivalently, $C_i$ intersects at least three other components of $C$. Equivalently, if $\iota'\colon S_0\to S'$ is another simple completion, and $\pi\colon S\to S'$ is a birational morphism such that $\pi\circ \iota=\iota'$, then $\pi$ is an isomorphism.
If $S$ is a completion of $S_0$, one can blow-up boundary points to obtain a simple completion, and then blow-down some of the boundary components $C_i$ to reach a minimal completion.
Now, consider the group of automorphisms of the open surface $S_0$. This group $\Aut(S_0)$ acts by birational transformations on $S$. An irreducible component $E_i$ of the boundary $C$ is [**[contracted]{}**]{} if there is an element $g$ of $\Aut(S_0)$ that contracts $E_i$: $g_\circ(E_i)$ is a point of $C$. Let $E$ be the union of the contracted components. In [@Gizatullin:1971], Gizatullin proves that $E$ satisfies one of the four properties stated in Proposition \[pro:Gizatullin-boundary\]; moreover, in cases (2) and (3), $E$ contains an irreducible component $E_i$ with $E_i^2>0$ (see Corollary 4, Section 5 of [@Gizatullin:1971]).
Thus, Proposition \[pro:Gizatullin-boundary\] follows from the properties of the pair $(Z,\U,\Gamma)$: the open subset $\U$ plays the role of $S_0$, and $Z$ is the completion $S$; the boundary $\partial Z$ is the curve $C$: it is a normal crossing divisor, and it is minimal by construction. Since every component of $\partial Z$ is contracted by at least one element of $\Gamma\subset \Aut(\U)$, $\partial Z$ coincides with Gizatullin’s curve $E$. The only thing we have to prove is the last sentence of Proposition \[pro:Gizatullin-boundary\], concerning the structure of the open subset $\U$.
First, let us show that $E=\partial Z$ supports an effective divisor $D$ such that $D^2>0$ and $D\cdot F \geq 0$ for every irreducible curve. To do so, fix an irreducible component $E_0$ of $\partial Z$ with positive self-intersection. Assume that $\partial Z$ is a cycle, and list cyclically the other irreducible components: $E_1$, $E_2$, ..., up to $E_m$, with $E_1$ and $E_m$ intersecting $E_0$. First, one defines $a_1=1$. Then, one chooses $a_2>0$ such that $a_1E_1+a_2E_2$ intersects positively $E_1$, then $a_3>0$ such that $a_1E_1+a_2E_2+a_3E_3$ intersects positively $E_1$ and $E_2$, ..., up to $\sum_{i=1}^m a_iE_i$ that intersects all components $E_i$, $1\leq i\leq m-1$ positively. Since $E_0^2>0$ and $E_0$ intersects $E_m$, one can find a coefficient $a_0$ for which the divisor $$D=\sum_{i=0}^ma_i E_i$$ satisfies $D^2>0$ and $D\cdot E_i>0$ for all $E_i$, $0\leq i\leq m$. This implies that $D$ intersects every irreducible curve $F$ non-negatively. Thus, $D$ is big and nef (see [@Lazarsfeld], Section 2.2). A similar proof applies when $\partial Z$ is a zigzag.
Let $W$ be the subspace of $\NS(X)$ spanned by classes of curves $F$ with $D\cdot F=0$. Since $D^2>0$, Hodge index theorem implies that the intersection form is negative definite on $W$. Thus, Mumford-Grauert contraction theorem provides a birational morphism $\tau\colon Z\to Z'$ that contracts simultaneously all curves $F$ with class $[F]\in W$ and is an isomorphism on $Z\smallsetminus F$; in particular, $\tau$ is an isomorphism from a neighborhood $\V$ of $\partial Z$ onto its image $\tau(\V)\subset Z'$. The modification $\tau$ may contract curves that are contained in $\U$, and may create singularities for the new open subset $\U'=\tau(\U)$, but does not modify $Z$ near the boundary $\partial Z$. Now, on $Z'$, the divisor $D'=\tau_*(D)$ intersects every effecitve curve positively and satisfies $(D')^2>0$. Nakai-Moishezon criterion shows that $D'$ is ample (see [@Lazarsfeld], Section 1.2.B); consequently, there is an embedding of $Z'$ into a projective space and a hyperplane section $H$ of $Z'$ for which $Z'\smallsetminus H$ coincides with $\U'$. This proves that $\U$ is a blow-up of the affine (singular) surface $\U'$.
Projective surfaces and automorphisms {#par:Projective-Surfaces}
-------------------------------------
In this section, we (almost always) assume that $\Gamma$ acts by regular automorphisms on a projective surface $X$. This corresponds to case (1) in Proposition \[pro:Gizatullin-boundary\]. Our goal is the special case of Theorem B which is stated below as Theorem \[thm:classification-virtually-autom\]. We shall assume that $\Gamma$ has property [[(FW)]{}]{} in some of the statements (this was not a hypothesis in Theorem thm:FW-regularization-surfaces).
We may, and shall, assume that $X$ is smooth. We refer to [@BPVDVH; @Beauville:Surfaces; @Hartshorne:book] for the classification of surfaces and the main notions attached to them.
### Action on the Néron-Severi group
The intersection form is a non-degenerate quadratic form $q_X$ on the Néron-Severi group $\NS(X)$, and Hodge index theorem asserts that its signature is $(1, \rho(X)-1)$, where $\rho(X)$ denotes the Picard number, i.e. the rank of the lattice $\NS(X)\simeq \Z^\rho$.
The action of $\Aut(X)$ on the Néron-Severi group $\NS(X)$ provides a linear representation preserving the intersection form $q_X$. This gives a morphism $$\Aut(X)\to \O(\NS(X); q_X).$$ Fix an ample class $a$ in $\NS(X)$ and consider the hyperboloid $${\mathbb{H}}_X=\{ u \in \NS(X)\otimes_\Z\R; \; q_X(u,u)=1 \; \, {\text{and}}\, \; q_X(u,a)>0 \}.$$ This set is one of the two connected components of $\{u; q_X(u,u)=1\}$. With the riemannian metric induced by $(-q_X)$, it is a copy of the hyperbolic space of dimension $\rho(X)-1$; the group $\Aut(X)$ acts by isometries on this space (see [@Cantat:Milnor]).
\[pro:BHW-Automorphisms\] Let $X$ be a smooth projective surface. Let $\Gamma$ be a subgroup of $\Aut(X)$. If $\Gamma$ has Property [[(FW)]{}]{}, its action on $\NS(X)$ fixes a very ample class, the image of $\Gamma$ in $\O(\NS(X); q_X)$ is finite, and a finite index subgroup of $\Gamma$ is contained in $\Aut(X)^0$.
The image $\Gamma^*$ of $\Gamma$ is contained in the arithmetic group $\O(\NS(X); q_X)$. The Néron-Severi group $\NS(X)$ is a lattice $\Z^\rho$ and $q_X$ is defined over $\Z$. Thus, $\O(\NS(X);\penalty-10000 q_X)$ is a standard arithmetic group in the sense of [@Bergeron-Haglund-Wise:2011], § 1.1. The main results of [@Bergeron-Haglund-Wise:2011] imply that the action of $\Gamma^*$ on the hyperbolic space ${\mathbb{H}}_X$ has a fixed point. Let $u$ be such a fixed point. Since $q_X$ is negative definite on the orthogonal complement $u^\perp$ of $u$ in $\NS(X)$, and $\Gamma^*$ is a discrete group acting by isometries on it, we deduce that $\Gamma^*$ is finite. If $a$ is a very ample class, the sum $\sum_{\gamma\in \Gamma^*}\gamma^*(a)$ is an invariant, very ample class.
The kernel $K\subset \Aut(X)$ of the action on $\NS(X)$ contains $\Aut(X)^0$ as a finite index subgroup. Thus, if $\Gamma$ has Property [[(FW)]{}]{}, it contains a finite index subgroup that is contained in $\Aut(X)^0$.
### Non-rational surfaces {#par:non-rational-surfaces}
In this paragraph, we assume that the surface $X$ is not rational. The following proposition classifies subgroups of $\Bir(X)$ with Property [[(FW)]{}]{}; in particular, such a group is finite if the Kodaira dimension of $X$ is non-negative (resp. if the characteristic of $\bfk$ is positive). Recall that we denote by $\overline{\Z}\subset \overline{\Q}$ the ring of algebraic integers.
Let $X$ be a smooth, projective, and non-rational surface, over the algebraically closed field $\bfk$. Let $\Gamma$ be an infinite subgroup of $\Bir(X)$ with Property [[(FW)]{}]{}. Then $\bfk$ has characteristic $0$, and there is a birational map $\varphi\colon X\dasharrow C\times \P^1_\bfk$ that conjugates $\Gamma$ to a subgroup of $\Aut(C\times \P^1_\bfk)$. Moreover, there is a finite index subgroup $\Gamma_0$ of $\Gamma$ such that $\varphi\circ \Gamma_0\circ \varphi^{-1}$, is a subgroup of $\PGL_2(\oZ)$, acting on $C\times \P^1_\bfk$ by linear projective transformations on the second factor.
Assume, first, that the Kodaira dimension of $X$ is non-negative. Let $\pi\colon X\to X_0$ be the projection of $X$ on its (unique) minimal model (see [@Hartshorne:book], Thm. V.5.8). The group $\Bir(X_0)$ coincides with $\Aut(X_0)$; thus, after conjugacy by $\pi$, $\Gamma$ becomes a subgroup of $\Aut(X_0)$, and Proposition \[pro:BHW-Automorphisms\] provides a finite index subgroup $\Gamma_0\leq \Gamma$ that is contained in $\Aut(X_0)^0$. Note that $\Gamma_0$ inherits Property [[(FW)]{}]{} from $\Gamma$.
If the Kodaira dimension of $X$ is equal to $2$, the group $\Aut(X_0)^0$ is trivial; hence $\Gamma_0=\{{{\rm Id}}_{X_0}\}$ and $\Gamma$ is finite. If the Kodaira dimension is equal to $1$, $\Aut(X_0)^0$ is either trivial, or isomorphic to an elliptic curve, acting by translations on the fibers of the Kodaira-Iitaka fibration of $X_0$ (this occurs, for instance, when $X_0$ is the product of an elliptic curve with a curve of higher genus). If the Kodaira dimension is $0$, then $\Aut(X_0)^0$ is also an abelian group (either trivial, or isomorphic to an abelian surface). Since abelian groups with Property [[(FW)]{}]{} are finite, the group $\Gamma_0$ is finite, and so is $\Gamma$.
We may now assume that the Kodaira dimension ${\sf{kod}}(X)$ is negative. Since $X$ is not rational, then $X$ is birationally equivalent to a product $S=C\times \P^1_\bfk$, where $C$ is a curve of genus ${\mathrm{g}}(C)\geq 1$. Denote by $\bfk(C)$ the field of rational functions on the curve$C$. We fix a local coordinate $x$ on $C$ and denote the elements of $\bfk(C)$ as functions $a(x)$ of $x$. The semi-direct product $\Aut(C)\ltimes \PGL_2(\bfk(C))$ acts on $S$ by birational transformations of the form $$(x,y)\in C\times \P^1_\bfk \mapsto \left(f(x), \frac{a(x)y+b(x)}{c(x)y+d(x)}\right),$$ and $\Bir(S)$ coincides with this group $\Aut(C)\ltimes \PGL_2(\bfk(C))$; indeed, the first projection $\pi\colon S \to C$ is equivariant under the action of $\Bir(S)$ because every rational map $\P^1_\bfk\to C$ is constant.
Since ${\mathrm{g}}(C)\geq 1$, $\Aut(C)$ is virtually abelian. Property [[(FW)]{}]{} implies that there is a finite index, normal subgroup $\Gamma_0\leq \Gamma$ that is contained in $\PGL_2(\bfk(C))$. By Corollary \[coro:Bass-k(C)\], every subgroup of $\PGL_2(\bfk(C))$ with Property [[(FW)]{}]{} is conjugate to a subgroup of $\PGL_2(\oZ)$ or a finite group if the characteristic of the field $\bfk$ is positive.
We may assume now that the characteristic of $\bfk$ is $0$ and that $\Gamma_0\subset \PGL_2(\oZ)$ is infinite. Consider an element $g$ of $\Gamma$; it acts as a birational transformation on the surface $S=C\times \P^1_\bfk$, and it normalizes $\Gamma_0$: $$g\circ \Gamma_0=\Gamma_0 \circ g.$$ Since $\Gamma_0$ acts by automorphisms on $S$, the finite set $\Ind(g)$ is $\Gamma_0$-invariant. But a subgroup of $\PGL_2(\bfk)$ with Property [[(FW)]{}]{} preserving a non-empty, finite subset of $\P^1(\bfk)$ is a finite group. Thus, $\Ind(g)$ must be empty. This shows that $\Gamma$ is contained in $\Aut(S)$.
### Rational surfaces {#par:auto-rational-surfaces}
We now assume that $X$ is a smooth rational surface, that $\Gamma\leq \Bir(X)$ is an infinite subgroup with Property [[(FW)]{}]{}, and that $\Gamma$ contains a finite index, normal subgroup $\Gamma_0$ that is contained in $\Aut(X)^0$. Recall that a smooth surface $Y$ is minimal if it does not contain any smooth rational curve of the first kind, i.e. with self-intersection $-1$. Every exceptional curve of the first kind $E\subset X$ is determined by its class in $\NS(X)$ and is therefore invariant under the action of $\Aut(X)^0$. Contracting such $(-1)$-curves one by one, we obtain the following lemma.
There is a birational morphism $\pi\colon X\to Y$ onto a minimal rational surface $Y$ that is equivariant under the action of $\Gamma_0$; $Y$ does not contain any exceptional curve of the first kind and $\Gamma_0$ becomes a subgroup of $\Aut(Y)^0$.
Let us recall the classification of minimal rational surfaces and describe their groups of automorphisms. First, we have the projective plane $\P^2_\bfk$, with $\Aut(\P^2_\bfk)= \PGL_3(\bfk)$ acting by linear projective transformations. Then comes the quadric $\P^1_\bfk\times \P^1_\bfk$, with $\Aut(\P^1_\bfk\times \P^1_\bfk)^0=\PGL_2(\bfk)\times \PGL_2(\bfk)$ acting by linear projective transformations on each factor; the group of automorphisms of the quadric is the semi-direct product of $\PGL_2(\bfk)\times \PGL_2(\bfk)$ with the group of order $2$ generated by the permutation of the two factors, $\eta(x,y)=(y,x)$. Then, for each integer $m\geq 1$, the Hirzebruch surface $\Hirz_m$ is the projectivization of the rank $2$ bundle ${\mathcal{O}}\oplus {\mathcal{O}}(m)$ over $\P^1_\bfk$; it may be characterized as the unique ruled surface $Z\to \P^1_\bfk$ with a section $C$ of self-intersection $-m$. Its group of automorphisms is connected and preserves the ruling. This provides a homomorphism $\Aut(\Hirz_m)\to \PGL_2(\bfk)$ that describes the action on the base of the ruling, and it turns out that this homomorphism is surjective. If we choose coordinates for which the section $C$ intersects each fiber at infinity, the kernel $J_m$ of this homomorphism acts by transformations of type $$(x,y)\mapsto (x,\alpha y+\beta(x))$$ where $\beta(x)$ is a polynomial function of degree $\leq m$. In particular, $J_m$ is solvable. In other words, $\Aut(\Hirz_m)$ is isomorphic to the group $$\left( \GL_2(\bfk)/\mu_m\right)\ltimes W_m$$ where $W_m$ is the linear representation of $\GL_2(\bfk)$ on homogeneous polynomials of degree $m$ in two variables, and $\mu_m$ is the kernel of this representation: it is the subgroup of $\GL_2(\bfk)$ given by scalar multiplications by roots of unity of order dividing $m$.
Given the above conjugacy $\pi\colon X\to Y$, the subgroup $\pi\circ\Gamma\circ\pi^{-1}$ of $\Bir(Y)$ is contained in $\Aut(Y)$.
Assume that the surface $Y$ is the quadric $\P^1_\bfk\times \P^1_\bfk$. Then, according to Theorem \[thm:Bass\], $\Gamma_0$ is conjugate to a subgroup of $\PGL_2(\oZ)\times \PGL_2(\oZ)$. If $g$ is an element of $\Gamma$, its indeterminacy locus is a finite subset $\Ind(g)$ of $\P^1_\bfk\times \P^1_\bfk$ that is invariant under the action of $\Gamma_0$, because $g$ normalizes $\Gamma_0$. Since $\Gamma_0$ is infinite and has Property [[(FW)]{}]{}, this set $\Ind(g)$ is empty (Lemma \[lem:pgl2-fw\]). Thus, $\Gamma$ is contained in $\Aut(\P^1_\bfk\times \P^1_\bfk)$.
The same argument applies for Hirzebruch surfaces. Indeed, $\Gamma_0$ is an infinite subgroup of $\Aut(\Hirz_m)$ with Property [[(FW)]{}]{}. Thus, up to conjugacy, its projection in $\PGL_2(\bfk)$ is contained in $\PGL_2(\oZ)$. If it were finite, a finite index subgroup of $\Gamma_0$ would be contained in the solvable group $J_m$, and would therefore be finite too by Property [[(FW)]{}]{}; this would contradict $\vert \Gamma_0\vert =\infty$. Thus, the projection of $\Gamma_0$ in $\PGL(\oZ)$ is infinite. If $g$ is an element of $\Gamma$, $\Ind(g)$ is a finite, $\Gamma_0$-invariant subset, and by looking at the projection of this set in $\P^1_\bfk$ one deduces that it is empty (Lemma \[lem:pgl2-fw\]). This proves that $\Gamma$ is contained in $\Aut(\Hirz_m)$.
Let us now assume that $Y$ is the projective plane. Fix an element $g$ of $\Gamma$, and assume that $g$ is not an automorphism of $Y=\P^2$; the indeterminacy and exceptional sets of $g$ are $\Gamma_0$ invariant. Consider an irreducible curve $C$ in the exceptional set of $g$, together with an indeterminacy point $q$ of $g$ on $C$. Changing $\Gamma_0$ in a finite index subgroup, we may assume that $\Gamma_0$ fixes $C$ and $q$; in particular, $\Gamma_0$ fixes $q$, and permutes the tangent lines of $C$ through $q$. But the algebraic subgroup of $\PGL_3(\bfk)$ preserving a point $q$ and a line through $q$ does not contain any infinite group with Property [[(FW)]{}]{} (Lemma \[lem:pgl2-fw\]). Thus, again, $\Gamma$ is contained in $\Aut(\P^2_\bfk)$.
### Conclusion, in Case (1)
Putting everything together, we obtain the following particular case of Theorem B.
\[thm:classification-virtually-autom\] Let $X$ be a smooth projective surface over an algebraically closed field $\bfk$. Let $\Gamma$ be an infinite subgroup of $\Bir(X)$ with Property [[(FW)]{}]{}. If a finite index subgroup of $\Gamma$ is contained in $\Aut(X)$, there is a birational morphism $\varphi\colon X\to Y$ that conjugates $\Gamma$ to a subgroup $\Gamma_Y$ of $\Aut(Y)$, with $Y$ in the following list:
1. $Y$ is the product of a curve $C$ by $\P^1_\bfk$, the field $\bfk$ has characteristic $0$, and a finite index subgroup $\Gamma'_Y$ of $\Gamma_Y$ is contained in $\PGL_2(\oZ)$, acting by linear projective transformations on the second factor;
2. $Y$ is $\P^1_\bfk\times \P^1_\bfk$, the field $\bfk$ has characteristic $0$, and $\Gamma_Y$ is contained in $\PGL_2(\oZ)\times \PGL_2(\oZ)$;
3. $Y$ is a Hirzebruch surface $\Hirz_m$ and $\bfk$ has characteristic $0$;
4. $Y$ is the projective plane $\P^2_\bfk$.
In particular, $Y=\P^2_\bfk$ if the characteristic of $\bfk$ is positive.
Denote by $\varphi \colon X\to Y$ the birational morphism given by the theorem. Changing $\Gamma$ in a finite index subgroup, we may assume that it acts by automorphisms on both $X$ and $Y$.
If $Y=C\times \P^1$, then $\varphi$ is in fact an isomorphism. To prove this fact, denote by $\psi$ the inverse of $\varphi$. The indeterminacy set $\Ind(\psi)$ is $\Gamma_Y$ invariant because both $\Gamma$ and $\Gamma_Y$ act by automorphisms. From Lemma \[lem:pgl2-fw\], applied to $\Gamma'_Y\subset \PGL_2(\bfk)$, we deduce that $\Ind(\psi)$ is empty and $\psi$ is an isomorphism. The same argument implies that the conjugacy is an isomorphism if $Y=\P^1_\bfk\times \P^1_\bfk$ or a Hirzebruch surface $\Hirz_m$, $m\geq 1$.
Now, if $Y$ is $\P^2_\bfk$, $\varphi$ is not always an isomorphism. For instance, $\SL_2(\C)$ acts on $\P^2_\bfk$ with a fixed point, and one may blow up this point to get a new surface with an action of groups with Property [[(FW)]{}]{}. But this is the only possible example, [*[i.e.]{}*]{} $X$ is either $\P^2_\bfk$, or a single blow-up of $\P^2_\bfk$ (because $\Gamma\subset \PGL_3(\C)$ can not preserve more than one base point for $\varphi^{-1}$ without loosing Property [[(FW)]{}]{}).
Invariant fibrations {#par:invariant-fibrations}
--------------------
We now assume that $\Gamma$ has Property [[(FW)]{}]{} and acts by automorphisms on $\U\subset X$, and that the boundary $\partial X=X\smallsetminus \U$ is the union of $\ell \geq 2$ pairwise disjoint rational curves $E_i$; each of them has self-intersection $E_i^2=0$ and is contracted by at least one element of $\Gamma$. This corresponds to the fourth possibility in Gizatullin’s Proposition \[pro:Gizatullin-boundary\]. Since $E_i\cdot E_j=0$, the Hodge index theorem implies that the classes $e_i=[E_i]$ span a unique line in $\NS(X)$, and that $[E_i]$ intersects non-negatively every curve.
From Section \[par:non-rational-surfaces\], we may, and do assume that $X$ is a rational surface. In particular, the Euler characteristic of the structural sheaf is equal to $1$: $\chi({\mathcal{O}}_X)=1$, and Riemann-Roch formula gives $$h^0(X, E_1)-h^1(X,E_1)+h^2(X,E_1)= \frac{E_1^2-K_X\cdot E_1}{2}+1.$$ The genus formula implies $K_X\cdot E_1=-2$, and Serre duality shows that $h^2(X, E_1)=h^0(X, K_X-E_1)=0$ because otherwise $-2=(K_X-E_1)\cdot E_1$ would be non-negative (because $E_1$ intersects non-negatively every curve). From this, we obtain $$h^0(X,E_1)= h^1(X,E_1)+2\geq 2.$$ Since $E_1^2=0$, we conclude that the space $H^0(X,E_1)$ has dimension $2$ and determines a fibration $\pi\colon X\to \P^1_\bfk$; the curve $E_1$, as well as the $E_i$ for $i\geq 2$, are fibers of $\pi$.
If $f$ is an automorphism of $\U$ and $F\subset \U$ is a fiber of $\pi$, then $f(F)$ is a (complete) rational curve. Its projection $\pi(f(F))$ is contained in the affine curve $\P^1_\bfk\smallsetminus \cup_i\pi(E_i)$ and must therefore be reduced to a point. Thus, $f(F)$ is a fiber of $\pi$ and $f$ preserves the fibration. This proves the following lemma.
There is a fibration $\pi\colon X\to \P^1_\bfk$ such that
1. every component $E_i$ of $\partial X$ is a fiber of $\pi$, and $\U=\pi^{-1}(\V)$ for an open subset $\V\subset \P^1_\bfk$;
2. the generic component of $\pi$ is a smooth rational curve;
3. $\Gamma$ permutes the fibers of $\pi$: there is a morphism $\rho\colon \Gamma\to \PGL_2(\bfk)$ such that $\pi\circ f=\rho(f)\circ \pi$ for every $f\in \Gamma$.
The open subset $\V\subsetneq \P^1_\bfk$ is invariant under the action of $\rho(\Gamma)$; hence $\rho(\Gamma)$ is finite by Property [[(FW)]{}]{} and Lemma \[lem:pgl2-fw\]. Let $\Gamma_0$ be the kernel of this morphism. Let $\varphi\colon X\dasharrow \P^1_\bfk\times \P^1_\bfk$ be a birational map that conjugates the fibration $\pi$ to the first projection $\tau\colon \P^1_\bfk\times \P^1_\bfk\to \P^1_\bfk$. Then, $\Gamma_0$ is conjugate to a subgroup of $\PGL_2(\bfk(x))$ acting on $\P^1_\bfk\times \P^1_\bfk$ by linear projective transformations of the fibers of $\tau$. From Corollary \[coro:Bass-k(C)\], a new conjugacy by an element of $\PGL_2(\bfk(x))$ changes $\Gamma_0$ in an infinite subgroup of $\PGL_2(\oZ)$. Then, as in Sections \[par:non-rational-surfaces\] and \[par:auto-rational-surfaces\] we conclude that $\Gamma$ becomes a subgroup of $\PGL_2(\oZ)\times \PGL_2(\oZ)$, with a finite projection on the first factor.
Let $\Gamma$ be an infinite group with Property [[(FW)]{}]{}, with $\Gamma\subset \Aut(\U)$, and $\U\subset Z$ as in case (4) of Proposition \[pro:Gizatullin-boundary\]. There exists a birational map $\psi\colon Z\dasharrow \P^1_\bfk\times \P^1_\bfk$ that conjugates $\Gamma$ to a subgroup of $\PGL_2(\overline{\Z})\times \PGL_2(\oZ)$, with a finite projection on the first factor.
Completions by zigzags {#par:completion-zigzag}
----------------------
Two cases remain to be studied: $\partial Z$ can be a chain of rational curves (a zigzag in Gizatullin’s terminology) or a cycle of rational curves (a loop in Gizatullin’s terminology). Cycles are considered in Section \[par:Cycles-Thompson\]. In this section, we rely on difficult results of Danilov and Gizatullin to treat the case of chains of rational curves (i.e. case (3) in Proposition \[pro:Gizatullin-boundary\]). Thus, in this section
1. $\partial X$ is a chain of smooth rational curves $E_i$
2. $\U=X\smallsetminus \partial X$ is an affine surface (singularities are allowed)
3. every irreducible component $E_i$ is contracted to a point of $\partial X$ by at least one element of $\Gamma\subset \Aut(\U) \subset \Bir(X)$.
In [@Danilov-Gizatullin:I; @Danilov-Gizatullin:II], Danilov and Gizatullin introduce a set of “standard completions” of the affine surface $\U$. As in Section \[par:constraints-boundary\], a completion (or more precisely a “marked completion”) is an embedding $\iota\colon \U\to Y$ into a complete surface such that $\partial Y=Y\smallsetminus \iota(\U)$ is a curve (this boundary curve may be reducible). Danilov and Gizatullin only consider completions for which $\partial Y$ is a chain of smooth rational curves and $Y$ is smooth in a neighborhood of $\partial Y$; the surface $X$ provides such a completion. Two completions $\iota\colon \U\to Y$ and $\iota'\colon \U\to Y'$ are isomorphic if the birational map $\iota'\circ \iota^{-1}\colon Y\to Y'$ is an isomorphism; in particular, the boundary curves are identified by this isomorphism. The group $\Aut(\U)$ acts by pre-composition on the set of isomorphism classes of (marked) completions.
Among all possible completions, Danilov and Gizatullin distinguish a class of “standard (marked) completions”, for which we refer to [@Danilov-Gizatullin:I] for a definition. There are elementary links (corresponding to certain birational mappings $Y\dasharrow Y'$) between standard completions, and one can construct a graph $\Delta_\U$ whose vertices are standard completions; there is an edge between two completions if one can pass from one to the other by an elementary link.
A completion is $m$-standard, for some $m\in \Z$, if the boundary curve $\partial Y$ is a chain of $n+1$ consecutive rational curves $E_0$, $E_1$, $\ldots$, $E_n$ ($n\geq 1$) such that $$E_0^2=0, \;\; E_1^2=-m, \; {\text{ and }} \; E_i^2=-2\;\; \text{if}\;\; i\geq 2.$$ Blowing-up the intersection point $q=E_0\cap E_1$, one creates a new chain starting by $E_0'$ with $(E_0')^2=-1$; blowing down $E_0'$, one creates a new $(m+1)$-standard completion. This is one of the elementary links.
Standard completions are defined by constraints on the self-intersections of the components $E_i$. Thus, the action of $\Aut(\U)$ on completions permutes the standard completions; this action determines a morphism from $\Aut(\U)$ to the group of isometries (or automorphisms) of the graph $\Delta_\U$ (see [@Danilov-Gizatullin:I]): $$\Aut(\U)\to \Iso(\Delta_\U).$$
The graph $\Delta_\U$ of all isomorphism classes of standard completions of $\U$ is a tree. The group $\Aut(\U)$ acts by isometries of this tree. The stabilizer of a vertex $\iota\colon \U\to Y$ is the subgroup $G(\iota)$ of automorphisms of the complete surface $Y$ that fix the curve $\partial Y$. This group is an algebraic subgroup of $\Aut(Y)$.
The last property means that $G(\iota)$ is an algebraic group that acts algebraically on $Y$. It coincides with the subgroup of $\Aut(Y)$ fixing the boundary $\partial Y$; the fact that it is algebraic follows from the existence of a $G(\iota)$-invariant, big and nef divisor which is supported on $\partial Y$ (see the last sentence of Proposition \[pro:Gizatullin-boundary\]).
The crucial assertion in this theorem is that $\Delta_\U$ is a simplicial tree (typically, infinitely many edges emanate from each vertex). There are sufficiently many links to assure connectedness, but not too many in order to prevent the existence of cycles in the graph $\Delta_\U$.
If $\Gamma$ is a subgroup of $\Aut(\U)$ that has the fixed point property on trees, then $\Gamma$ is contained in $G(\iota)\subset \Aut(Y)$ for some completion $\iota\colon \U\to Y$.
If $\Gamma$ has Property [[(FW)]{}]{}, it has Property [(FA)]{} (see Section \[par:Bass\]). Thus, if it acts by automorphisms on $\U$, $\Gamma$ is conjugate to the subgroup $G(\iota)$ of $\Aut(Y)$, for some zigzag-completion $\iota\colon \U\to Y$. Theorem \[thm:classification-virtually-autom\] of Section \[par:auto-rational-surfaces\] implies that the action of $\Gamma$ on the initial surface $X$ is conjugate to a regular action on $\P^2_\bfk$, $\P^1_\bfk\times \P^1_\bfk$ or $\Hirz_m$, for some Hirzebruch surface $\Hirz_m$. This action preserves a curve, namely the image of the zigzag into the surface $Y$. The following examples list all possibilities, and conclude the proof of Theorem B in the case of zigzags (i.e. case (3) in Proposition \[pro:Gizatullin-boundary\]).
Consider the projective plane $\P^2_\bfk$, together with an infinite subgroup $\Gamma\subset \Aut(\P^2_\bfk)$ that preserves a curve $C$ and has Property [[(FW)]{}]{}. Then, $C$ must be a smooth rational curve: either a line, or a smooth conic. If $C$ is the line “at infinity”, then $\Gamma$ acts by affine transformations on the affine plane $\P^2_\bfk\smallsetminus C$. If the curve is the conic $x^2+y^2+z^2=0$, $\Gamma$ becomes a subgroup of ${\sf{PO}}_3(\bfk)$.
When $\Gamma$ is a subgroup of $\Aut(\P^1_\bfk\times \P^1_\bfk)$ that preserves a curve $C$ and has Property [[(FW)]{}]{}, then $C$ must be a smooth curve because $\Gamma$ has no finite orbit (Lemma \[lem:pgl2-fw\]). Similarly, the two projections $C\to \P^1_\bfk$ being equivariant with respect to the morphisms $\Gamma\to \PGL_2(\bfk)$, they have no ramification points. Thus, $C$ is a smooth rational curve, and its projections onto each factor are isomorphisms. In particular, the action of $\Gamma$ on $C$ and on each factor are conjugate. From these conjugacies, one deduces that the action of $\Gamma$ on $\P^1_\bfk\times \P^1_\bfk$ is conjugate to a diagonal embedding $$\gamma\in \Gamma\; \mapsto \; (\rho(\gamma),\rho(\gamma)) \in \PGL_2(\bfk)\times \PGL_2(\bfk)$$ preserving the diagonal.
Similarly, the group $\SL_2(\bfk)$ acts on the Hirzebruch surface $\Hirz_m$, preserving the zero section of the fibration $\pi\colon \Hirz_m \to \P^1_\bfk$. This gives examples of groups with Property [[(FW)]{}]{} acting on $\Hirz_m$ and preserving a big and nef curve $C$.
Starting with one of the above examples, one can blow-up points on the invariant curve $C$, and then contract $C$, to get examples of zigzag completions $Y$ on which $\Gamma$ acts and contracts the boundary $\partial Y$.
Birational transformations of surfaces II {#par:Cycles-Thompson}
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In this section, $\U$ is a (normal, singular) affine surface with a completion $X$ by a cycle of $\ell$ rational curves. Every irreducible component $E_i$ of the boundary $\partial X=X\smallsetminus \U$ is contracted by at least one automorphism of $\U$. Our goal is to classify subgroups $\Gamma$ of $\Aut(\U)\subset \Bir(X)$ that are infinite and have Property [[(FW)]{}]{}: in fact, we shall show that no such group exists. This ends the proof of Theorem B since all the other possibilities of Proposition \[pro:Gizatullin-boundary\] have been dealt with in the previous section.
Let $(\A_\bfk^1)^*$ denote the complement of the origin in the affine line $\A^1_\bfk$; it is isomorphic to the multiplicative group ${\mathbb{G}}_m$ over $\bfk$. The surface $(\A_\bfk^1)^*\times (\A_\bfk^1)^*$ is an open subset in $\P^2_\bfk$ whose boundary is the triangle of coordinate lines $\{[x:y:z];\; xyz=0\}$. Thus, the boundary is a cycle of length $\ell=3$. The group of automorphisms of $(\A_\bfk^1)^*\times (\A_\bfk^1)^*$ is the semi-direct product $$\GL_2(\Z)\ltimes ({\mathbb{G}}_m(\bfk)\times {\mathbb{G}}_m(\bfk));$$ it does not contain any infinite group with Property [[(FW)]{}]{}.
Resolution of indeterminacies
-----------------------------
Let us order cyclically the irreducible components $E_i$ of $\partial X$, so that $E_i\cap E_j\neq \emptyset$ if and only if $i-j=\pm 1 (\mathrm{mod}\,\ell)$. Blowing up finitely many singularities of $\partial X$, we may assume that $\ell=2^m$ for some integer $m\geq 1$; in particular, every curve $E_i$ is smooth. (With such a modification, one may a priori create irreducible components of $\partial X$ that are not contracted by the group $\Gamma$.)
\[lem:Indet-Cyclic\] Let $f$ be an automorphism of $\U$ and let $f_X$ be the birational extension of $f$ to the surface $X$. Then
1. Every indeterminacy point of $f_X$ is a singular point of $\partial X$, i.e. one of the intersection points $E_i\cap E_{i+1}$.
2. Indeterminacies of $f_X$ are resolved by inserting chains of rational curves.
Property (2) means that there exists a resolution of the indeterminacies of $f_X$, given by two birational morphisms $\epsilon\colon Y \to X$ and $\pi\colon Y\to X$ with $f\circ \epsilon = \pi$, such that $\pi^{-1}(\partial X)=\epsilon^{-1}(X)$ is a cycle of rational curves. Some of the singularities of $\partial X$ have been blown-up into chains of rational curves to construct $Y$.
Consider a minimal resolution of the indeterminacies of $f_X$. It is given by a finite sequence of blow-ups of the base points of $f_X$, producing a surface $Y$ and two birational morphisms $\epsilon\colon Y \to X$ and $\pi\colon Y\to X$ such that $f_X=\pi\circ \epsilon^{-1}$. Since the indeterminacy points of $f_X$ are contained in $\partial X$, all necessary blow-ups are centered on $\partial X$.
The total transform $F=\epsilon^*(\partial X)$ is a union of rational curves: it is made of a cycle, together with branches emanating from it. One of the assertions (1) and (2) fails if and only if $F$ is not a cycle; in that case, there is at least one branch.
Each branch is a tree of smooth rational curves, which may be blown-down onto a smooth point; indeed, these branches come from smooth points of the main cycle that have been blown-up finitely many times. Thus, there is a birational morphism $\eta\colon Y\to Y_0$ onto a smooth surface $Y_0$ that contracts the branches (and nothing more).
The morphism $\pi$ maps $F$ onto the cycle $\partial X$, so that all branches of $F$ are contracted by $\pi$. Thus, both $\epsilon$ and $\pi$ induce (regular) birational morphisms $\epsilon_0\colon Y_0\to X$ and $\pi_0\colon Y_0\to X$. This contradicts the minimality of the resolution.
Let us introduce a family of surfaces $$\pi_k\colon X_k\to X.$$ First, $X_1=X$ and $\pi_1$ is the identity map. Then, $X_2$ is obtained by blowing-up the $\ell$ singularities of $\partial X_1$; $X_2$ is a compactification of $\U$ by a cycle $\partial X_2$ of $2\ell=2^{m+1}$ smooth rational curves. Then, $X_3$ is obtained by blowing up the singularities of $\partial X_2$, and so on. In particular, $\partial X_k$ is a cycle of $2^{k-1}\ell = 2^{m+k-1}$ curves.
Denote by $\Dual_k$ the [**[dual graph]{}**]{} of $\partial X_k$: vertices of $\Dual_k$ correspond to irreducible components $E_i$ of $\partial X_k$ and edges to intersection points $E_i\cap E_j$. A simple blow-up (of a singular point) modifies both $\partial X_k$ and $\Dual_k$ locally as follows
The group $\Aut(\U)$ acts on $\Hyp(X)$ and Lemma \[lem:Indet-Cyclic\] shows that its action stabilizes the subset $\B$ of $\Hyp(X)$ defined as $$\B=\left\{C\in \Hyp(X): \; \exists k\geq 1, C\; \, \text{is an irreducible component of } \; \partial X_k \right\}.$$ In what follows, we shall parametrize $\B$ in two distinct ways by rational numbers.
Farey and dyadic parametrizations {#fareydy}
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Consider an edge of the graph $\Dual_1$, and identify this edge with the unit interval $[0,1]$. Its endpoints correspond to two adjacent components $E_i$ and $E_{i+1}$ of $\partial X_1$, and the edge corresponds to their intersection $q$. Blowing-up $q$ creates a new vertex (see Figure \[fig:2\]). The edge is replaced by two adjacent edges of $\Dual_2$ with a common vertex corresponding to the exceptional divisor and the other vertices corresponding to (the strict transforms of) $E_i$ and $E_{i+1}$; we may identify this part of $\Dual_2$ with the segment $[0,1]$, the three vertices with $\{0, 1/2, 1\}$, and the two edges with $[0,1/2]$ and $[1/2,1]$.
Subsequent blow-ups may be organized in two different ways by using either a dyadic or a Farey algorithm (see Figure \[fig:3\]).
In the dyadic algorithm, the vertices are labelled by dyadic numbers $m/2^k$. The vertices of $\Dual_{k+1}$ coming from an initial edge $[0,1]$ of $\Dual_1$ are the points $\{n/2^{k}; \; 0\leq n \leq 2^k\}$ of the segment $[0,1]$. We denote by $\Dya(k)$ the set of dyadic numbers $n/2^k\in [0,1]$; thus, $\Dya(k)\subset \Dya(k+1)$. We shall say that an interval $[a,b]$ is a [**[standard dyadic]{}**]{} interval if $a$ and $b$ are two consecutive numbers in $\Dya(k)$ for some $k$.
In the Farey algorithm, the vertices correspond to rational numbers $p/q$. Adjacent vertices of $\Dual_k$ coming from the initial segment $[0,1]$ correspond to pairs of rational numbers $(p/q,r/s)$ with $ps-qr=\pm 1$; two adjacent vertices of $\Dual_k$ give birth to a new, middle vertex in $\Dual_{k+1}$: this middle vertex is $(p+r)/(q+s)$ (in the dyadic algorithm, the middle vertex is the “usual” euclidean middle). We shall say that an interval $[a,b]$ is a [**[standard Farey]{}**]{} interval if $a=p/q$ and $b=r/s$ with $ps-qr=-1$. We denote by $\Far(k)$ the finite set of rational numbers $p/q\in [0,1]$ that is given by the $k$-th step of Farey algorithm; thus, $\Far(1)=\{0,1\}$ and $\Far(k)$ is a set of $2^{k+1}$ rational numbers $p/q$ with $0\leq p\leq q$. (One can check that $1\leq q\leq \Fib(k)$, with $\Fib(k)$ the $k$-th Fibonacci number.)
By construction, the graph $\Dual_1$ has $\ell=2^m$ edges. The edges of $\Dual_1$ are in one to one correspondance with the singularities $q_j$ of $\partial X_1$. Each edge determines a subset $\B_j$ of $\B$; the elements of $\B_j$ are the curves $C\subset \partial X_k$ ($k\geq 1$) such that $\pi_k(C)$ contains the singularity $q_j$ determined by the edge. Using the dyadic algorithm (resp. Farey algorithm), the elements of $\B_j$ are in one-to-one correspondence with dyadic (resp. rational) numbers in $[0,1]$. Gluing these segments cyclically together one gets a circle $\SS^1$, together with a nested sequence of subdivisions in $\ell$, $2\ell$, $\ldots$, $2^{k-1}\ell$, $\ldots$ intervals; each interval is a standard dyadic interval (resp. standard Farey interval) of one of the initial edges .
Since there are $\ell=2^m$ initial edges, we may identify the graph $\Dual_1$ with the circle $\SS^1=\R/\Z=[0,1]/_{0\simeq 1}$ and the initial vertices with the dyadic numbers in $\Dya(m)$ modulo $1$ (resp. with the elements of $\Far(m)$ modulo $1$). Doing this, the vertices of $\Dual_k$ are in one to one correspondence with the dyadic numbers in $\Dya(k+m-1)$ (resp. in $\Far(k+m-1)$).
\[rem:Farey-Thompson\] [(a).–]{} By construction, the interval $[p/q,r/s]\subset [0,1]$ is a standard Farey interval if and only if $ps-qr=-1$, iff it is delimited by two adjacent elements of $\Far(m)$ for some $m$.
[(b).–]{} If $h\colon [x,y]\to [x',y']$ is a homeomorphism between two standard Farey intervals mapping rational numbers to rational numbers and standard Farey intervals to standard Farey intervals, then $h$ is the restriction to $[x,y]$ of a unique linear projective transformation with integer coefficients: $$h(t)=\frac{at+b}{ct+d}, \; \text{ for some element }\; \left( \begin{array}{cc} a & b \\ c & d \end{array}\right) \; \text{ of }\; \PGL_2(\Z).$$
[(c).–]{} Similarly, if $h$ is a homeomorphism mapping standard dyadic intervals to intervals of the same type, then $h$ is the restriction of an affine dyadic map $$h(t)=2^mt+\frac{u}{2^n}, \; {\text{ with }} m, n \in \Z.$$
In what follows, we denote by $\Tho$ the group of self-homeomorphisms of $\SS^1=\R/\Z$ that are piecewise $\PGL_2(\Z)$ mapping with respect to a finite decomposition of the circle in standard Farey intervals $[p/q,r/s]$. In other words, if $f$ is an element of $\Tho$, there are two partitions of the circle into consecutive intervals $I_i$ and $J_i$ such that the $I_i$ are intervals with rational endpoints, $h$ maps $I_i$ to $J_i$, and the restriction $f\colon I_i\to J_i$ is the restriction of an element of $\PGL_2(\Z)$ (see [@Navas:Book], §1.5.1).
Let $\U$ be an affine surface with a compactification $\U \subset X$ such that $\partial X:=X\smallsetminus \U$ is a cycle of smooth rational curves. In the Farey parametrization of the set $\B\subset \Hyp(X)$ of boundary curves, the group $\Aut(\U)$ acts on $\B$ as a subgroup of $\Tho$.
There is a unique orientation preserving self-homeomorphism of the circle that maps $\Dya(k)$ to $\Far(k)$ for every $k$. This self-homeomorphism conjugates $\Tho$ to the group $\Thom$ of self-homeomorphisms of the circle that are piecewise affine with respect to a dyadic decomposition of the circle, with slopes in $\pm 2^\Z$, and with translation parts in $\Z[1/2]$. Using the parametrization of $\B$ by dyadic numbers, the image of $\Aut(\U)$ becomes a subgroup of $\Thom$.
The reason why we keep in parallel the dyadic and Farey viewpoints is the following: the Farey viewpoint is more natural for algebraic geometers (this is related to toric –i.e. monomial– maps and appears clearly in [@Hubbard-Papadopol]), while the dyadic viewpoint is more natural to geometric group theorists, because this is the classical setting used in the study of Thompson groups (see [@Navas:Book], §1.5).
Lemma \[lem:Indet-Cyclic\] is the main ingredient. Consider the action of the group $\Aut(\U)$ on the set $\B$. Let $f$ be an element of $\Aut(\U)\subset \Bir(X)$. Consider an irreducible curve $E\in \B$, and denote by $F$ its image: $F=f_\bullet(E)$ is an element of $\B$ by Lemma \[lem:Indet-Cyclic\]. There are integers $k$ and $l$ such that $E\subset \partial X_k$ and $F\subset \partial X_l$. Replacing $X_k$ by a higher blow-up $X_m\to X$, we may assume that $f_{lm} :=\pi_l^{-1}\circ f\circ \pi_m$ is regular on a neighborhood of the curve $E$ (Lemma \[lem:Indet-Cyclic\]). Let $q_k$ be one of the two singularities of $\partial X_m$ that are contained in $E$, and let $E'$ be the second irreducible component of $\partial X_m$ containing $q$. If $E'$ is blown down by $f_{lm}$, its image is one of the two singularities of $\partial X_l$ contained in $F$ (by Lemma \[lem:Indet-Cyclic\]). Consider the smallest integer $n\geq l$ such that $\partial X_n$ contains the strict transform $F'=f_\bullet(E')$; in $X_n$, the curve $F'$ is adjacent to the strict transform of $F$ (still denoted $F$), and $f$ is a local isomorphism from a neighborhood of $q$ in $X_m$ to a neighborhood of $q':=F\cap F'$ in $X_n$.
Now, if one blows-up $q$, the exceptional divisor $D$ is mapped by $f_\bullet$ to the exceptional divisor $D'$ obtained by a simple blow-up of $q$: $f$ lifts to a local isomorphism from a neighborhood of $D$ to a neighborhood of $D'$, the action from $D$ to $D'$ being given by the differential $df_q$. The curve $D$ contains two singularities of $\partial X_{m+1}$, which can be blown-up too: again, $f$ lifts to a local isomorphism if one blow-ups the singularities of $\partial X_{n+1}\cap D'$. We can repeat this process indefinitely. Let us now phrase this remark differently. The point $q$ determines an edge of $\Dual_m$, hence a standard Farey interval $I(q)$. The point $q'$ determines an edge of $\Dual_n$, hence another standard Farey interval $I(q')$. Then, the points of $\B$ that are parametrized by rational numbers in $I(q)$ are mapped by $f_\bullet$ to rational numbers in $I(q')$ and this map respects the Farey order: if we identify $I(q)$ and $I(q')$ to $[0,1]$, $f_\bullet$ is the restriction of a monotone map that sends $\Far(k)$ to $\Far(k)$ for every $k$. Thus, on $I(q)$, $f_\bullet$ is the restriction of a linear projective transformation with integer coefficients (see Remark \[rem:Farey-Thompson\]-(b)). This shows that $f_\bullet$ is an element of $\Tho$.
Conclusion
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Consider the group $\Thom^*$ of self-homeomorphisms of the circle $\SS^1=\R/\Z$ that are piecewise affine with respect to a finite partition of $\R/\Z$ into dyadic intervals $[x_i,x_{i+1}[$ with $x_i$ in $\Z[1/2]/\Z$ for every $i$, and satisfy $$h(t)=2^{m_i}t+a_i$$ with $m_i\in \Z$ and $a_i\in \Z[1/2]$ for every $i$. This group is known as the Thompson group of the circle, and is isomorphic to the group $\Tho^*$ of orientation-preserving self-homeomorphisms in $\Tho$ (defined in §\[fareydy\]).
\[thm:Farley-Navas\] Every subgroup of $\Thom^*$ (and hence of $\Tho$) with Property [[(FW)]{}]{} is a finite cyclic group.
Indeed fixing a gap in an earlier construction of Farley [@Farley:2003][^2], Hughes proved [@hughesfarley] that $\Tho$ has Property PW, in the sense that it admits a commensurating action whose associated length function is a proper map (see also Navas’ book [@Navas:Book]). This implies the conclusion, because every finite group of orientation-preserving self-homeomorphisms of the circle is cyclic.
Thus, if $\Gamma$ is a subgroup of $\Aut(\U)$ with Property [[(FW)]{}]{}, it contains a finite index subgroup $\Gamma_0$ that acts trivially on the set $\B\subset \Hyp(X)$. This means that $\Gamma_0$ extends as a group of automorphisms of $X$ fixing the boundary $\partial X$. Since $\partial X$ supports a big and nef divisor, $\Gamma_0$ contains a finite index subgroup $\Gamma_1$ that is contained in $\Aut(X)^0$.
Note that $\Gamma_1$ has Property [[(FW)]{}]{} because it is a finite index subgroup of $\Gamma$. It preserves every irreducible component of the boundary curve $\partial X$, as well as its singularities. As such, it must act trivially on $\partial X$. When we apply Theorem \[thm:classification-virtually-autom\] to $\Gamma_1$, the conjugacy $\varphi\colon X\to Y$ can not contract $\partial X$, because the boundary supports an ample divisor. Thus, $\Gamma_1$ is conjugate to a subgroup of $\Aut(Y)$ that fixes a curve pointwise. This is not possible if $\Gamma_1$ is infinite (see Theorem \[thm:classification-virtually-autom\] and the remarks following it).
We conclude that $\Gamma$ is finite in case (2) of Proposition \[pro:Gizatullin-boundary\].
Birational actions of $\SL_2(\Z[\sqrt{d}])$ {#scorcorSL2}
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We develop here Example \[applx\]. If $\bfk$ is an algebraically closed field of characteristic $0$, therefore containing ${\overline{\Q}}$, we denote by $\sigma_1$ and $\sigma_2$ the distinct embeddings of $\Q(\sqrt{d})$ into $\bfk$. Let $j_1$ and $j_2$ be the resulting embeddings of $\SL_2(\Z[\sqrt{d}])$ into $\SL_2(\bfk)$, and $j=j_1\times j_2$ the resulting embedding into $$\mathsf{G}=\SL_2(\bfk)\times\SL_2(\bfk).$$
\[tSL2\] Let $\Gamma$ be a finite index subgroup of $\SL_2(\Z[\sqrt{d}])$. Let $X$ be an irreducible projective surface over an algebraically closed field $\bfk$. Let $\alpha:\Gamma\to\Bir(X)$ be a homomorphism with infinite image. Then $\bfk$ has characteristic zero, and there exist a finite index subgroup $\Gamma_0$ of $\Gamma$ and a birational map $\varphi:Y\dasharrow X$ such that
1. $Y$ is the projective plane $\P^2$, a Hirzebruch surface $\F_m$, or $C\times\P^1$ for some curve $C$;
2. $\varphi^{-1}\alpha(\Gamma)\varphi\subset\Aut(Y)$;
3. there is a unique algebraic homomorphism $\beta:\mathsf{G}\to\Aut(Y)$ such that $$\varphi^{-1}\alpha(\gamma)\varphi=\beta(j(\gamma))$$ for every $\gamma\in\Gamma_0$.
Theorem B ensures that the characteristic of $\bfk$ is $0$ and that (1) and (2) are satisfied. If $Y$ is $\P^2$ or a Hirzebruch surface $\F_m$, then $\Aut(Y)$ is a linear algebraic group. If $Y$ is a product $C\times \P^1$, a finite index subgroup of $\Gamma$ preserves the projection onto $\P^1$, so that it acts via an embedding into the linear algebraic group $\Aut(\P^1)=\PGL_2(\bfk)$.
When $\bfk$ has positive characteristic, $Y$ is the projective plane, and the $\Gamma$-action is given by a homomorphism $\Gamma\to\PGL_3(\bfk)$. Then we use the fact that for any $n$, every homomorphism $f:\Gamma\to\GL_n(\bfk)$ has finite image. Indeed, it is well-known that $\GL_n(\bfk)$ has no infinite order distorted elements: elements of infinite order have some transcendental eigenvalue and the conclusion easily follows. Since $\Gamma$ has an exponentially distorted cyclic subgroup, $f$ has infinite kernel, and infinite normal subgroups of $\Gamma$ have finite index (Margulis normal subgroup theorem).
On the other hand, in characteristic zero we conclude the proof of Theorem \[tSL2\] with the following lemma.
Let $\bfk$ be any field extension of $\Q(\sqrt{d})$. Consider the embedding $j$ of $\SL_2(\Z[\sqrt{d}])$ into $G=\SL_2(\bfk)\times\SL_2(\bfk)$ given by the standard embedding into the left-hand $\SL_2$ and its Galois conjugate in the right-hand $\SL_2$. Then for every linear algebraic group $H$ and homomorphism $f:\SL_2(\Z[\sqrt{d}])\to H(\bfk)$, there exists a unique homomorphism $\bar{f}:G\to H$ of $\bfk$-algebraic groups such that the homomorphisms $f$ and $\tilde{f}\circ j$ coincide on some finite index subgroup of $\Gamma$.
The uniqueness is a consequence of Zariski density of the image of $j$. Let us prove the existence. Zariski density allows to reduce to the case when $H=\SL_n$. First, the case $\bfk=\R$ is given by Margulis’ superrigidity, along with the fact that every continuous real representation of $\SL_n(\R)$ is algebraic. The case of fields containing $\R$ immediately follows, and in turn it follows for subfields of overfields of $\R$ (as soon as they contain $\Q(\sqrt{d})$).
Open problems
=============
Regularization and Calabi-Yau varieties
---------------------------------------
Let $\Gamma$ be a group with Property [[(FW)]{}]{}. Is every birational action of $\Gamma$ regularizable ? Here regularizable is defined in the same way as pseudo-regularizable, but assuming that the action on $\U$ is by automorphisms (instead of pseudo-automorphisms).
A particular case is given by Calabi-Yau varieties, in the strict sense of a simply connected complex projective manifold $X$ with trivial canonical bundle and $h^{k,0}(X)=0$ for $0< k<\dim(X)$. For such a variety the group $\Bir(X)$ coincides with $\Psaut(X)$. One can then ask (1) whether every infinite subgroup $\Gamma$ of $\Psaut(X)$ with property [[(FW)]{}]{} is regularizable on some birational model $Y$ of $X$ (without restricting the action to a dense open subset), and (2) what are the possibilities for such a group $\Gamma$.
Transfixing birational groups
-----------------------------
\[ruled3\] For which irreducible projective varieties $X$
1. \[bi1\] $\Bir(X)$ does not transfix $\Hy(X)$?
2. \[bi2\] some finitely generated subgroup of $\Bir(X)$ does not transfix $\Hy(X)$?
3. \[bi3\] some cyclic subgroup of $\Bir(X)$ does not transfix $\Hy(X)$.
We have the implications: $X$ is ruled $\Rightarrow$ (\[bi3\]) $\Rightarrow$ (\[bi2\]) $\Rightarrow$ (\[bi1\]). In dimension $2$, we have: ruled $\Leftrightarrow$ (\[bi1\]) $\Rightarrow\!\!\!\!\!\!\!\!\!/$ (\[bi2\]) $\Leftrightarrow$ (\[bi3\]) (see §\[surf\_birt\]). It would be interesting to find counterexamples to these equivalences in higher dimension, and settle each of the problems raised in Question \[ruled3\] in dimension $3$.
The affine space
----------------
The group of affine transformations of $\A^3_\C$ contains $\SL_3(\C)$, and this group contains many subgroups with Property [[(FW)]{}]{}. In the case of surfaces, Theorem B shows that groups of birational transformations with Property [[(FW)]{}]{} are contained in algebraic groups, up to conjugacy. The following question asks whether this type of theorem may hold for $\Aut(\A^3_\C)$.
Does there exist an infinite subgroup of $\Aut(\A^3_\C)$ with Property [[(FW)]{}]{} that is not conjugate to a group of affine transformations of $\A^3_\C$ ?
Length functions
----------------
Recall that a length function $\ell$ on a group $G$ is quasi-geodesic if there exists $M>0$ such that for every $n\ge 1$ and every $g\in G$ with $\ell(g)\le n$, there exist $1=g_0$, $g_1,\dots,$ $g_n=g$ in $G$ such that $\ell(g_{i-1}^{-1}g_i)\le M$ for all $i$. Equivalently $G$, endowed with the distance $(g,h)\mapsto\ell(g^{-1}h)$, is quasi-isometric to a connected graph.
Given an irreducible variety $X$, is the length function $$g\in \Bir(X) \mapsto |\Hy(X)\triangle g\Hy(X)|$$ quasi-geodesic? In particular, what about $X=\P^2$ and the Cremona group $\Bir(\P^2)$?
[^1]: Every finitely generated subgroup is finite.
[^2]: The gap in Farley’s argument lies in Prop. 2.3 and Thm. 2.4 of [@Farley:2003].
|
---
abstract: 'The holographic principle and its realisation as the AdS/CFT correspondence leads to the existence of the so called precursor operators. These are boundary operators that carry non-local information regarding events occurring deep inside the bulk and which cannot be causally connected to the boundary. Such non-local operators can distinguish non-vacuum-like excitations within the bulk that cannot be observed by any local gauge invariant operators in the boundary. The boundary precursors are expected to become increasingly non-local the further the bulk process is from the boundary. Such phenomena are expected to be related to the extended nature of the strings. Standard gauge invariance in the boundary theory equates to quantum error correction which furthermore establishes localisation of bulk information. I show that when double field theory quantum error correction prescriptions are considered in the bulk, gauge invariance in the boundary manifests residual effects associated to stringy winding modes. Also, an effect of double field theory quantum error correction is the appearance of positive cosmological constant. The emergence of spacetime from the entanglement structure of a dual quantum field theory appears in this context to generalise for de-Sitter spacetimes as well.'
address: 'ELI-NP, Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering, 30 Reactorului St, Bucharest-Magurele, 077125, Romania'
author:
- 'Andrei T. Patrascu'
title: Cosmological constant as quantum error correction from generalised gauge invariance in double field theory
---
Introduction
============
The AdS/CFT correspondence and its underlying holographic principle lead to a new way of regarding quantum field theories and their observables. Moreover, spacetime together with its geometry is seen as an emergent property of the entanglement structure of the boundary conformal field theory. It has been noticed that a local bulk observable is not in a one to one relation to boundary operators. One bulk observable can be holographically represented in many different forms by operators of the boundary conformal field theory. If one considers any point $x$ in the bulk space and one takes a point $Y$ on the boundary, then holography, as implemented by the AdS/CFT correspondence can map an associated bulk local field $\phi(x)$ into many different possible CFT operators. One can consider $\mathcal{O}[\phi(x)]$ which has no support in an open set containing $Y$ and holographically map the local bulk field to it. As a consequence, any local field of the CFT supported near $Y$ will commute with it. As we considered $Y$ to be an arbitrary point, if the CFT operator corresponding to it were to be unique we would arrive at the conclusion that $\mathcal{O}$ commutes with all local fields in the CFT and hence must be a multiple of the identity as the local field algebra is irreducible. However, one does not expect that any local bulk field to be mapped into the identity operator on the boundary. Indeed, one can avoid this conclusion if one realises that the bulk-boundary correspondence is not one-to-one or unique. Considering $Y$ and $Z$ two distinct boundary points, then the bulk operator $\phi(x)$ may be mapped either to the operator $\mathcal{O}$ on the boundary which commutes with the CFT local fields supported near $Y$, or to $\mathcal{O}'$ which commutes with the CFT local fields supported near $Z$. $\mathcal{O}$ and $\mathcal{O}'$ are inequivalent operators defined within the boundary CFT although they can be used interchangeably for the description of bulk physics. If we consider a fixed time subregion $A$ of the the CFT we can define a subregion in the bulk $\mathcal{C}[A]$ such that for any point $x\in \mathcal{C}[A]$, the bulk quantum field theory ensures that any bulk local operator $\phi(x)$ can be represented in the CFT as some non-local operator on $A$. While locality is not present in the bulk, one may imagine that it emerges by some mechanism. Bulk non-locality may be associated to a string theoretical effect and hence, to better understand its emergence one has to consider an intermediate situation between bulk non-locality and bulk locality. It is argued here that such an intermediate situation may be described by means of double field theory in the bulk. This theory would be an effective field theory that retains certain aspects of string theory related especially to their extended nature. Together with momentum modes, double field theory also incorporates winding modes which rely on the extended nature of strings and which may have effects that vanish when only supergravity type effective field theories are being considered. How to properly express double field theory in the context of the AdS/CFT correspondence however is highly non-trivial. If one considers a given bulk point $x$, one notices that it can lie within distinct causal wedges corresponding to different boundary regions, and hence the bulk operator $\phi(x)$ can have different representations in the boundary CFT with different spatial support. Reference \[7\] associates the non-uniqueness of the CFT operators corresponding to the operator $\phi(x)$ in the bulk with the possibility that the bulk field represents a form of logical operator which preserves a code subspace of the CFT Hilbert space. With such an interpretation it results that bulk field protects the CFT code subspace against erasures of parts of the boundary. Let the boundary operator corresponding to $\phi(x)$ act on a subsystem of the CFT which is protected against erasure of the boundary region denoted $A^{c}$. Such an operator can be represented in the boundary CFT as having a support on $A$, which is the complement of the erased region. The AdS-Rindler reconstruction of $\phi(x)$ on the boundary region $A$ can then be considered to act as a correction for the erasure of $A^{c}$. For various choices of the portion we wish to erase, we obtain different reconstructions of the bulk field $\phi(x)$. The deeper inside the bulk the operators are, the better protected against erasures they become, meaning that a larger region needs to be erased to prevent their reconstruction while operators near the boundary can be erased more easily by removing a smaller part of the boundary. As showed in \[17\] the code subspace can be seen as the low energy sector of the conformal field theory which corresponds to a smooth dual classical geometry. All the boundary CFT operators are physical and also have a bulk interpretation. The logical operators are special ones which map low energy states to other low energy states. Identical logical actions can be realised by distinct CFT operators, because those operators act on the high energy CFT states from outside the code subspace differently, although they act on low energy states in the same way \[17\]. The AdS/CFT holographic duality is known to map a D+1 dimensional conformal field theory on a flat spacetime into a D+2 dimensional quantum gravity theory on an AdS space background. It is well known that the large-N limit of the conformal field theory corresponds to the classical limit of the corresponding gravity theory. This correspondence is related to the existence of an SO(D,2) conformal symmetry group associated to the quantum field theory on the boundary which is identical to the isometry group of the AdS space. Different, more general quantum field theories, without conformal symmetry may be dual to bulk gravity theories with different spacetime manifolds. It is interesting to observe that the bulk theory becomes in the high energy limit a theory of quantum gravity. String theory, as such a theory, should therefore play an essential role in the holographic interpretation of various processes in the bulk. However, while it is sufficiently clear that string theory is holographical \[18\] there is no clear representation of the holographic duality in terms of string theoretical effects within the bulk. The recent understanding of the AdS/CFT correspondence in terms of error correction codes led to speculations on how various error correction properties may be understood in terms of the quantum field theory on the boundary. One idea was that any gauge invariant state already possesses some form of non-local entanglement originating in the initial requirement that it satisfies certain gauge constraints \[1\]. The connection between gauge invariance in the boundary theory and the emergence of the bulk spacetime appears in the natural error correction code that imposes localisation of the bulk information in different regions. It appears that such a connection is more general than the standard one based on error correction schemes. However, this connection has not been expressed in its most general form, as it did not take into account properties that may appear due to winding modes in the bulk. Such modes are not included in standard interpretations, yet they alter the gauge invariance transformations in a significant way. If we could re-interpret those generalised gauge transformations in terms of certain quantum information properties, our understanding both of quantum gravity and of quantum computing would significantly increase. If gauge invariance implies a form of entanglement for the boundary quantum field theory, how can that be generalised following the extension of the standard gauge invariance transformations to those associated to double field theory?
Double Field Theory
===================
The introduction of gauge symmetries in quantum information theory has been recently explored in \[1\] and \[2\]. Quantum error correction codes were required because the classical error correction based on analysing copies of the same information for discrepancies cannot be applied in quantum computing due to the no-cloning theorem. A way of providing additional robustness and the ability of correcting potential errors in quantum codes was brought by the fact that entanglement encodes the required information globally. A similar situation occurred when analysing precursor operators on the boundary of a holographic theory. Events occurring deep within the bulk could not be causally related to the boundary, yet, due to holography, they were encoded within the boundary by means of non-local operators. These non-local operators are what we call precursors and they exploit the fact that local information within the bulk can be represented non-locally on the boundary. Moreover, it has been showed that error correction in the boundary is linked to the gauge invariance of the boundary theory. Another situation in which non-local phenomena manifest themselves is double field theory. As this theory incorporates the string theoretical T-duality \[3\] which connects different length scales and even different topologies, understanding quantum error correction in the context of double field theory in the bulk will be of particular relevance, increasing the theoretical resilience of quantum codes on the boundary even further. As non-trivial topology has been shown to be equivalent to entanglement at least in the bipartite case \[4\], \[5\], understanding the effect of T-duality on quantum error corrections could provide new insights on the sets of observables that can come together and provide information about the boundary (when local) and about the causally disconnected events in the bulk (when non-local). However, in double field theory, gauge invariance is extended and the generalised metric is used. The theory we try to write in a T-duality invariant fashion is the NS-NS sector of supergravity. The degrees of freedom of this theory are contained in the $D$-dimensional metric tensor $g_{ij}$, $i,j=1,...,D$, the $D$-dimensional $2$-form $b_{ij}$ (Kalb-Ramond field) and the dilaton $\phi$, all depending on the spacetime coordinates $x^{i}$. Physical phenomena will not change under a pair of local gauge transformations. The first such symmetry is the diffeomorphism parametrised by the infinitesimal vectors $\lambda$ and encoded by the Lie derivative acting on arbitrary vectors $V^{i}$ like $$L_{\lambda}V^{i}=\lambda^{j}\partial_{j}V^{i}-V^{j}\partial_{j}\lambda^{i}=[\lambda,V]^{i}$$ where the last term is a Lie bracket which is antisymmetric and satisfies the Jacobi identity. The second gauge symmetry transformation of the $2$-form parametrised by the infinitesimal $1$-form parameter $\tilde{\lambda}_{i}$ is $$b_{ij}\rightarrow b_{ij}+\partial_{i}\tilde{\lambda}_{j}-\partial_{j}\tilde{\lambda}_{i}$$ The supergravity action takes the well known form $$S=\int d^{D}x\sqrt{g}e^{-2\phi}[R+4(\partial \phi)^{2}-\frac{1}{12}H^{ijk}H_{ijk}]$$ where the three-form $H_{ijk}=3\partial_{[i}b_{jk]}$ satisfies the Bianchi identity $\partial_{[i}H_{jkl]}=0$ and $R$ is the Ricci scalar constructed from $g_{ij}$. In double field theory, these gauge symmetries are extended with the explicit addition of the T-duality symmetry which relates the fields $g_{ij}$, $b_{ij}$, and $\phi$ mentioned above. To see how supergravity degrees of freedom can be put in a T-duality invariant formulation let us arrange all the objects in T-duality representations, hence having well defined transformation properties with respect to T-duality \[6\]. The metric and the two-form field can be combined into the symmetric generalised metric $$\mathcal{H}=
\left({\begin{array}{cc}
g^{ij} & -g^{ik}b_{kj} \\
b_{ik}g^{kj} & g_{ij}-b_{ik}g^{kl}b_{lj} \\
\end{array} } \right)$$ This metric is an $O(D,D)$ group element satisfying the property that its inverse is obtained by acting with the Minkowski metric on it $$\eta_{MN}=
\left({\begin{array}{cc}
0 & \delta^{i}_{\;\; j} \\
\delta_{i}^{\;\;j} & 0 \\
\end{array} } \right)$$ $$\mathcal{H}^{MN}=\eta^{MP}\mathcal{H}_{PQ}\eta^{QP}$$ where the uppercase indexes go from $1$ to $2D$ and refer to the doubled space. One may employ the notation $\sqrt{g}e^{-2\phi}\rightarrow e^{-2\phi}$ transforming the dilaton term which now becomes an $O(D,D)$ scalar. Using the standard notation for doubled coordinates $X^{M}=(\tilde{x}_{i},x^{i})$ we have the new coordinates $\tilde{x}_{i}$ representing the coordinates associated to the winding modes of the strings. In the context of supergravity these coordinates have no meaning and hence we need some form of constraint that while being T-duality invariant must restrict the dependence on such coordinates. There are several such constraints, one of the most common being the so called section (or strong) constraint $\eta^{MN}\partial_{M}\partial_{N}(...)=0$. This will be assumed in what follows. Following the notation of \[6\] this will be rewritten as $Y^{M\;\;N}_{\;\;\;\; P\;\;\; Q}\partial_{M}\partial_{N}(...)=0$ where $Y^{M\;\;N}_{\;\;\;\; P\;\;\; Q}=\eta^{MN}\eta_{PQ}$. The metric $g_{ij}$ and the Kalb-Ramond field transform under diffeomorphisms and the Kalb-Ramond field also transforms under gauge symmetry. In double field theory the standard diffeomorphisms can be unified with the gauge transformation leading to a generalised diffeomorphism implemented by a generalised Lie derivative and encoding the generalised gauge transformations of the two entities. The gauge parameter can be written in the double field theory as $$\xi^{M}=(\tilde{\lambda}_{i},\lambda^{i})$$ and the resulting generalised Lie derivative acting on a tensorial density $V^{M}$ with weight $\omega(V)$ will have the form $$\mathcal{L}_{\xi}V^{M}=\xi^{P}\partial_{P}V^{M}+(\partial^{M}\xi_{P}-\partial_{P}\xi^{M})V^{P}+\omega(V)\partial_{P}\xi^{P}V^{M}$$ Note that in this case $\omega(e^{-2\phi})=1$ and $\omega(\mathcal{H})=0$. The closure of these generalised diffeomorphisms imposes certain differential constraints on the theory. The closure of the group law implies that two successive gauge transformations parametrised by $\xi_{1}$ and $\xi_{2}$ acting on a given field $\xi_{3}$ generate a new transformation of the same group parametrised by $\xi_{12}(\xi_{1},\xi_{2})$
$$([\mathcal{L}_{\xi_{1}},\mathcal{L}_{\xi_{2}}]-\mathcal{L}_{\xi_{12}})\xi_{3}^{M}=0$$
i.e. the generalised Lie derivative must send tensors to tensors. The parameter resulting from this is $\xi_{12}=\mathcal{L}_{\xi_{1}}\xi_{2}$ with the constraint $$Y^{M\;\;N}_{\;\;\;\; P\;\;\; Q}(2\partial_{P}\xi^{R}_{[1}\partial_{Q}\xi_{2]}^{M}\xi_{3}^{S}-\partial_{P}\xi_{1}^{R}\xi_{2}^{S}\partial_{Q}\xi_{3}^{M})=0$$ The parameter $\xi_{12}$ is known as the D-bracket and its antisymmetric part (the C-bracket) is
$$\xi_{[12]}^{M}=\frac{1}{2}(\mathcal{L}_{\xi_{1}}\xi_{2}^{M}-\mathcal{L}_{\xi_{2}}\xi_{1}^{M})=[\xi_{1},\xi_{2}]^{M}+Y^{M\;\;N}_{\;\;\;\; P\;\;\; Q}\cdot\xi_{[1}^{Q}\partial_{P}\xi_{2]}^{N}$$
This corresponds to the extension of the usual Lie bracket, due to the correction depending on $Y$. This factor measures the departure from the conventional Riemannian geometry.
As can be seen, the explicit inclusion of T-duality gives rise to an extended form of gauge transformation which leads not only to a generalised Lie derivative and bracket but also to a new way of interpreting the quantum error correction and quantum secret sharing procedures in quantum gravitational contexts. Indeed, up to now quantum error correction was dominated by the fact that entanglement on the boundary encoded the bulk information non-locally with respect to boundary coordinates. Here, due to the introduction of coordinates related to the winding modes of the string, quantum error correction can make use of the additional $\tilde{x}$ coordinates as well. They are a stringy feature that allows us to use analogues of left and right rotation projectors in the boundary theory. Moreover, the patching of the boundary space will be somewhat unusual, as the patching function will now be related to the symmetry transformation defined by T-duality. This will result in a non-geometric structure on the boundary. It has been noted in \[1\] that while a local bulk operator is dual to several boundary precursor operators, those operators are all equivalent when acting on gauge invariant states. Given a bulk operator, the demand for bulk locality implies that the boundary precursor commutes with all spacelike separated boundary operators. Moreover, a bulk operator can be represented either as boundary precursors written in the form of bilocal operators distributed over the whole boundary or by representing bulk operators in the right bulk Rindler wedge as precursors smeared over the entire right half of the boundary \[1\]. This last representation allows one to eliminate bilocals connecting the two halves of the boundary or those who stretch only inside the left half of the boundary without any physical consequences. Such freedom arises because precursors may only act on gauge invariant states. The non-local nature of precursors relates to the idea of entanglement by the observation that given three patches $A$, $B$, and $C$ which cover the boundary conformal field theory, the precursors can only be reconstructed by combining at least two of these patches, i.e. $AB$, $BC$, or $CD$ but not from each $A$, $B$, or $C$ alone. This non-local storage of bulk information reminds us of entanglement and quantum error correction codes \[7-9\]. When double field theory is considered within the bulk we must remember that the symmetry made manifest by it is T-duality, which is specific to string theory. In the bulk this means we have to consider specific projector operators which single out left and right rotational states, a feature resulting from the closed string origins of this analysis. In the boundary limit this extends the way in which patches can be combined to provide useful information about the bulk states leading to information about the bulk state being encoded in combination of patches related via T-duality symmetry and resulting into non-geometric structures. Indeed a string can wrap around non-trivial cycles of the background leading to so-called winding states. Such states are created by vertex operators which depend on both coordinates associated with momentum excitations and T-dual coordinates associated with the winding excitations. Excitations of the vacuum by these operators may lead to non-geometric backgrounds. These backgrounds correspond to field theories with interactions depending on both types of coordinates. Given closed string theory in D-dimensional space with $d$ compactified directions, $\mathbb{R}^{n-1,1}\times T^{d}$ where $n+d=D$ and the coordinates $x^{i}=(x^{\mu},x^{a})$, $i=0,...,D-1$ where $a$ refers to the $d$-torus, the states are labelled by the momentum $p_{i}=(k_{\mu},p_{a})$ and the string windings $w^{a}$. States within the bulk will then be written as $${\ensuremath{\left|\Phi\right\rangle}}=\sum_{I}\int dk\sum_{p_{a},w^{a}}\phi_{I}(k_{\mu},p_{a},w^{a})\mathcal{O}^{I}{\ensuremath{\left|k_{\mu},p_{a},w^{a}\right\rangle}}$$ By means of a Fourier transform the dependence on the momenta is transformed into the spacetime dependence $x^{\mu}$ and $x^{a}$ while the winding mode $w^{a}$ becomes a new periodic coordinate which we called $\tilde{x}_{a}$. Physical strings will be annihilated by $$L_{0}-\bar{L}=N-\bar{N}-p_{a}w^{a}=0$$ Fields in the double field theory can be extended to the boundary and in this limit we can express them in terms of boundary operators depending on both normal and winding coordinates. Keeping the generalised gauge invariance and writing the states in the bulk within the context of double field theory we may connect bulk operators to boundary precursors by first using the bulk equations of motion for ${\ensuremath{\left|\Phi\right\rangle}}$, express it in terms of the boundary fields, and then use the boundary equations of motion to evolve this to a single time operator \[1\]. Here, the gauge invariance of the bulk-space modes is larger and involves both double coordinates. Demanding closure of the group law does not necessarily restrict us to the normal supergravity bulk. Instead we may obtain non-geometric structures unreachable from supergravity alone. Indeed, gauge invariance constraints on the boundary are translated into quantum error correction prescriptions as predicted by \[1\] but the gauge invariance here incorporates naturally diffeomorphisms and gauge transformations of the generalised metric and 2-form field
$$\begin{array}{c}
\mathcal{L}_{\xi}e^{-2\phi}=\partial_{M}(\xi^{M}e^{-2\phi})\\
\\
\mathcal{L}_{\xi}\mathcal{H}_{MN}=L_{\xi}\mathcal{H}+Y^{R\;\;M}_{\;\;\;\; P\;\;\; Q}\partial^{Q}\xi_{P}\mathcal{H}_{RN}+Y^{R\;\;N}_{\;\;\; P\;\;\; Q}\partial^{Q}\xi_{P}\mathcal{H}_{MR}
\end{array}$$
where $L_{\xi}$ is the usual Lie derivative in $2D$ dimensions. The term $Y$ incorporates non-Riemannian effects. The requirement of gauge invariance to this type of transformations can be interpreted as a quantum error correction code built naturally into the precursor operators only that now the extension towards the double field theory bulk space gauge transformations not only encode that information is non-locally spread over the boundary but also that they have a non-geometric component. The closure requirement for such transformations when analysed in a doubled bulk spacetime allows non-geometric effects to participate to the error correction mechanism. Non-geometric effects represent a departure from strict Riemannian geometry and are fundamentally invisible from the perspective of local standard quantum field theory not involving winding coordinates. It has been shown in \[10\] and \[11\] that in double field theory the Riemannian tensor is not fully determined in terms of physical fields. Moreover, the components of the Riemannian tensor that do not contain undetermined connections are zero. We can however define a set of projectors $$\begin{array}{cc}
P_{M}^{\;\;\;N}=\frac{1}{2}(\delta_{M}^{\;\;\;N}-\mathcal{H}_{M}^{\;\;\;N}), & \bar{P}_{M}^{\;\;\;N}=\frac{1}{2}(\delta_{M}^{\;\;\;N}+\mathcal{H}_{M}^{\;\;\;N})
\end{array}$$ which allow us to project onto the left-handed and right-handed subspaces. We use the simplifying notation for their action on indices as in \[10\] $$\begin{array}{cc}
W_{\underline{M}}=P_{M}^{\;\;\;N}W_{N}, & W_{\bar{M}}=\bar{P}_{M}^{\;\;\;N}W_{N}
\end{array}$$ The indeterminacy of the Riemann tensor can be traced back to the connection which can be decomposed into a determined and an undetermined part $$\Gamma_{MNK}=\hat{\Gamma}_{MNK}+\Sigma_{MNK}$$ where the hat denotes the part of the connection determined by the physical fields while the undetermined part can again be decomposed as $$\Sigma_{MNK}=\tilde{\Gamma}_{\underline{M}\;\underline{N}\;\underline{K}}+\tilde{\Gamma}_{\bar{M}\bar{N}\bar{K}}$$ To be sure that we work with a meaningful Riemannian curvature we have to define it in terms of projected indexes when working in double field theory, namely $R_{\underline{M}\;\underline{N}\;\underline{P}\;\underline{K}}$ and the scalar curvature then becomes $$R=R^{\underline{M}\;\underline{N}}_{\;\;\;\;\;\;\;\;\underline{M}\;\underline{N}}$$ When this method is used the undetermined connections drop out of $R$ \[10\]. However, there always exists the additional freedom given by the indeterminacy of the connection. In terms of quantum error correction codes, this allows a spread of quantum information on non-geometric structures that in the boundary limit becomes not only inaccessible to any local observables but also inaccessible to observers who ignore the stringy structure given by the winding coordinates in double field theory. The cosmological constant term in the doubled approach is $$\int dy^{2D}e^{-2\phi}\Lambda$$ with $\Lambda=\frac{4}{\alpha}$ as computed in \[12\] is required to match the DFT action with the effective action from string theory. This fact is particularly representative as normal effective actions do not encode non-geometric backgrounds. DFT is the first theory capable of detecting non-geometry. As has been shown before, extending gauge invariance to the generalised gauge invariance induced by double field theory we obtained additional tools for implementing quantum error correction, involving not only non-locality on Riemannian geometry but also spreading of the precursor operators in the boundary on manifolds defined by patching functions involving T-duality symmetry. This leads to non-Riemannian effects and to non-geometry. Non-localisation in non-geometry must take into account the way in which patches of the boundary manifold connect and T-duality plays a major role in this. Entanglement entropy has been associated with the cosmological constant in \[13\] yet bringing together the role of entanglement in quantum error correction codes has only recently been done in \[14\]. Connecting quantum error correction to gauge invariance has been done in \[1\] while finally in this work I relate the generalised gauge invariance of double field theory with an extended way of interpreting natural quantum error correction involving string-geometry phenomena. This generalised form of entanglement may be the source of the entanglement entropy part which plays a role in the cosmological constant.
Quantum error correction and stabiliser codes
=============================================
Recently, holography and its most direct manifestation, the AdS/CFT duality, have been interpreted in terms of quantum error correction protocols. In this sense, the quantum field theoretical description on the boundary, with its far larger number of degrees of freedom is regarded as the physical encoding of a logical quantum state manifested within the geometry of the bulk. To proper understand this interpretation and to relate T-duality and double field theory in the bulk with new forms of quantum error correction codes it is essential to understand the quantum error correction prescriptions in more detail. In general an arbitrary state of an individual qubit can be expressed as $${\ensuremath{\left|\phi\right\rangle}}=\alpha{\ensuremath{\left|0\right\rangle}}+\beta{\ensuremath{\left|1\right\rangle}}$$ with the two orthonormal basis states ${\ensuremath{\left|0\right\rangle}}$ and ${\ensuremath{\left|1\right\rangle}}$ and the coefficients satisfying $|\alpha |^{2}+|\beta |^{2}=1$. In quantum computation, the gate operations are represented by unitary operators acting on the Hilbert space of a collection of qubits. All operations must be reversible and hence unitary. The dynamical operation of a gate on a qubit is a member of the unitary group $U(2)$, $G$ which is a unitary matrix of dimension two such that $G^{\dagger}=G^{-1}$. Ignoring a global and unphysical phase factor, any gate operation on a qubit may be expressed as a linear combination of generators of the group $SU(2)$ in the form $$G=c_{I}\sigma_{I}+c_{x}\sigma_{x}+c_{y}\sigma_{y}+c_{z}\sigma_{z}$$ where $\sigma_{i}$ are the Pauli matrices including the identity. The main difference between classical and quantum error correction lies in the fact that we cannot duplicate quantum states (no-cloning theorem) and we cannot directly measure a single quantum state without destroying its quantum nature. Therefore, error correction protocols must be adapted in order to detect and correct errors without being forced to acquire any information about the state itself. Qubits employed in quantum information are susceptible to the classical bit errors like bit switching, but also to phase errors. Hence quantum error correction must take into account both. Errors in quantum mechanics are inherently continuous, as qubits experience angular shifts of the qubit state by any possible angle. What quantum error correction has in common with classical error correction is its reliance on redundancy in the encoding prescription. Such redundancy implies that a single quantum state is encoded over a larger Hilbert space, extending the domain of representation of, say, a qubit, beyond what would be required for a single qubit. Extending the space of states in order to obtain auxiliary symmetries that could simplify certain computations has been used in ref. \[2\] and \[19\]. There, the extension was based on the Batalin-Vilkovisky quantisation of gauge theories with non-closing gauge algebras and the extensions in the form of field-anti-field formalisms \[20\], \[21\]. Here, the extension will at first play a different role, as the focus will be on quantum error corrections. However, it has been noted in \[1\] that gauge invariance in the boundary field theory may be related to quantum error correction. As there are various ways to implement gauge invariance and to construct meaningful quantum gauge theories, even in the case of non-closing algebras, it is interesting to see how this may relate to the construction of more efficient quantum error correction codes. Ultimately, the gauge invariance of double field theory, with its manifest T-duality symmetry imposes a set of relations valid for all scales and fundamentally non-local. It will be seen in this article that such non-local relations, connecting even distinct topologies, may be obtained from holographic quantum codes by allowing certain extensions with respect to the requirements of \[17\]. Before we discuss those connections and start constructing more advanced holographic quantum error correction codes, let me first describe what types of quantum errors are to be expected in any general quantum code. Surely, errors existing in any quantum system depend on the specific physical mechanisms controlling the system. In general however we can identify three types of errors: coherent quantum errors, due to incorrect application of quantum gates, environmental decoherence errors due to the interaction of the quantum system with the environment, and loss, or quantum leakage. In our situation the focus will be on coherent quantum errors and errors due to qubit erasure (or loss). Their correction relies on multiple qubit encoding of a single qubit quantum information and on correction of individual errors. A first simple example is the so called 3-qubit code, which, while not capable of simultaneously correcting both bit and phase flips, is one of the first repetition codes used finally by Shor \[8\] to construct the 9-qubit code capable of simultaneous bit and phase flip error corrections. The main idea of the 3-qubit code is to encode a single logical qubit into three physical qubits such that any single $\sigma_{x}$ bit flip error will be corrected. There will be two logical basis states defined in terms of three physical qubits: ${\ensuremath{\left|0\right\rangle}}_{L}={\ensuremath{\left|000\right\rangle}}$ and ${\ensuremath{\left|1\right\rangle}}_{L}={\ensuremath{\left|111\right\rangle}}$. In general an arbitrary qubit state can be reformulated as $$\begin{array}{c}
\alpha{\ensuremath{\left|0\right\rangle}}+\beta{\ensuremath{\left|1\right\rangle}}\rightarrow \alpha{\ensuremath{\left|0\right\rangle}}_{L}+\beta{\ensuremath{\left|1\right\rangle}}_{L}=\\
=\alpha{\ensuremath{\left|000\right\rangle}}+\beta{\ensuremath{\left|111\right\rangle}}={\ensuremath{\left|\psi\right\rangle}}_{L}\\
\end{array}$$ A quantum circuit that would encode such a state with three qubits will start with three quantum states, the first encoding the original qubit state, and another two ancilla qubits initialised to ${\ensuremath{\left|0\right\rangle}}$. Two CNOT gates will couple the first qubit state to the second ${\ensuremath{\left|0\right\rangle}}$ state and the second ${\ensuremath{\left|0\right\rangle}}$ state to the third such that, in the end, the logical qubit will be encoded on three qubits. This code features a binary distance between the two codeword states and hence is capable of correcting for a single bit flip error. It is necessary to have three physical bit flips in order to transform the logical state from ${\ensuremath{\left|0\right\rangle}}_{L}$ to ${\ensuremath{\left|1\right\rangle}}_{L}$. Therefore if we assume ${\ensuremath{\left|\psi\right\rangle}}={\ensuremath{\left|0\right\rangle}}_{L}$, then with one single bit flip we will obtain a final state that still remains closer to ${\ensuremath{\left|0\right\rangle}}_{L}$. The distance between two codeword states, $d$, is related to the number of errors that can be corrected, $t$, by means of the relation $$t=[\frac{d-1}{2}]$$ The error correction prescription on the other side will need some additional ancilla qubits, because we cannot directly measure the logical state without destroying it. Those ancilla qubits are used to extract the syndrome information related to possible errors without discriminating the state of any qubit. The error correction connects the physical qubits to the new ancilla qubits by means of CNOT gates which check the parity of the three-qubit data block. In any case, there is either no error, or a single bit-flip error and in both cases the ancilla qubits are flipped to one unique state based on the parity of the data block. These qubits are then measured and provide the syndrome of the error. This will then allow us to apply the correction gate in a meaningful way. In order to correct for both bit and phase flip, the nine-qubits code may be employed. Other generalisations are possible but the simple discussion up to this point suffices for the matter at hand. Describing error correction codes from the perspective of the quantum state is often cumbersome and inefficient as the state representations and the circuits themselves will differ from code to code. The error correction prescription however can be described in a unified way by means of the so called stabiliser formalism \[22\], \[23\]. The basic idea is to describe quantum states in terms of operators. Given a state ${\ensuremath{\left|\psi\right\rangle}}$, one can say it is being stabilised by some operator $K$ if that state is an $+1$ eigenstate of $K$ namely $K{\ensuremath{\left|\psi\right\rangle}}={\ensuremath{\left|\psi\right\rangle}}$. A multi-qubit state will be described in an operatorial sense by analysing the group properties of the multi-qubit operators acting as stabilisers. Given the Pauli group for $N$-qubits $\mathcal{P}_{N}$, an $N$-qubit stabiliser state is defined by the $N$ generators of an Abelian subgroup $\mathcal{G}$ of the $N$-qubit Pauli group that satisfies $$\mathcal{G}=\{K^{i}\,|\,K^{i}{\ensuremath{\left|\psi\right\rangle}}={\ensuremath{\left|\psi\right\rangle}},\; [K^{i},K^{j}]=0,\; \forall (i,j)\}\subset \mathcal{P}_{N}$$ A given state ${\ensuremath{\left|\psi\right\rangle}}_{N}$ can be defined by specifying the generators of the stabiliser group. Each stabiliser operation squares to the identity. The use of stabiliser operators to describe quantum error correction codes allows us to see what logical operations can be applied directly to the encoded data. The preparation of logical state is based on the fact that valid codeword states are defined as simultaneous $+1$ eigenstates for each of the generators of the stabiliser group. Therefore it will be required to project our qubits into eigenstates of each of these operators. With the arbitrary input state ${\ensuremath{\left|\psi\right\rangle}}_{I}$ given, an ancilla initialised in the ${\ensuremath{\left|0\right\rangle}}$ state is used as a control qubit for the unitary and Hermitian operation $U$ performed on ${\ensuremath{\left|\psi\right\rangle}}_{I}$. A Hadamard gate is applied on the ancilla state and then it is coupled by means of the operation $U$ to our state ${\ensuremath{\left|\psi\right\rangle}}_{I}$. After inserting another Hadamard gate for the ancilla qubit, the state of the system will be $${\ensuremath{\left|\psi\right\rangle}}_{F}=\frac{1}{2}({\ensuremath{\left|\psi\right\rangle}}_{I}+U{\ensuremath{\left|\psi\right\rangle}}_{I}){\ensuremath{\left|0\right\rangle}}+\frac{1}{2}({\ensuremath{\left|\psi\right\rangle}}_{I}-U{\ensuremath{\left|\psi\right\rangle}}_{I}){\ensuremath{\left|1\right\rangle}}$$ We now measure the ancilla qubit in the computational basis. If the result is ${\ensuremath{\left|0\right\rangle}}$ then the input state becomes $${\ensuremath{\left|\psi\right\rangle}}_{F}={\ensuremath{\left|\psi\right\rangle}}_{I}+U{\ensuremath{\left|\psi\right\rangle}}_{I}$$ while if the measured outcome of the ancilla is ${\ensuremath{\left|1\right\rangle}}$ then the input state becomes $${\ensuremath{\left|\psi\right\rangle}}_{F}={\ensuremath{\left|\psi\right\rangle}}_{I}-U{\ensuremath{\left|\psi\right\rangle}}_{I}$$ Therefore this circuit projects onto the $\pm 1$ eigenstates of $U$. In order to project onto the positive eigenstate we read the measurement and decide whether to apply a gate that will project on the positive eigenstate, call it $Z$. To generalise this circuit for the situation in which we have several stabiliser operators we simply connect each stabiliser gate to the corresponding ancilla and measure the outcomes of all ancillas before projecting if necessary. In this way we will have projections onto the common eigenstates of the stabiliser operators. Quantum codes can be characterised by means of the number of physical qubits $(n)$ encoding a certain number of logical qubit $(k)$ with the associated distance between basis states $(d)$ as $[[n,k,d]]$. Then we may consider for the sake of example the quantum code $[[7,1,3]]$ which can correct $t=1$ error. The code defines one single logical qubit and hence must contain two meaningful logical code states ${\ensuremath{\left|0\right\rangle}}$ and ${\ensuremath{\left|1\right\rangle}}$ which are basis states for the code and can be written in a state vector notation for physical qubits as
$$\begin{array}{c}
{\ensuremath{\left|0\right\rangle}}_{L}=\frac{1}{8}({\ensuremath{\left|0000000\right\rangle}}+{\ensuremath{\left|1010101\right\rangle}}+{\ensuremath{\left|0110011\right\rangle}}+{\ensuremath{\left|1100110\right\rangle}}+{\ensuremath{\left|0001111\right\rangle}}+{\ensuremath{\left|1011010\right\rangle}}+{\ensuremath{\left|0111100\right\rangle}}+{\ensuremath{\left|1101001\right\rangle}})\\
\\
{\ensuremath{\left|1\right\rangle}}_{L}=\frac{1}{8}({\ensuremath{\left|1111111\right\rangle}}+{\ensuremath{\left|0101010\right\rangle}}+{\ensuremath{\left|1001100\right\rangle}}+{\ensuremath{\left|0011001\right\rangle}}+{\ensuremath{\left|1110000\right\rangle}}+{\ensuremath{\left|0100101\right\rangle}}+{\ensuremath{\left|1000011\right\rangle}}+{\ensuremath{\left|0010110\right\rangle}})\\
\end{array}$$
But in the case in which we work on a logical state encoded as a 7-qubits physical state, the total dimension of the Hilbert space must be $2^{7}$. However, the logically encoded state will only require a 2-dimensional subspace spanned by the states above. Stabiliser groups and their operators make such a reduction visible. For a $7$ qubits code we have six stabiliser operators. These will reduce the dimension of the code subspace as expected to $2^{7-6}=2^{1}=2$ which is the dimension of the logical qubit. Error correction for stabiliser codes is an extension of the state preparation prescription. Assume that on the encoded state $$\alpha{\ensuremath{\left|0\right\rangle}}+\beta{\ensuremath{\left|1\right\rangle}}\rightarrow\alpha{\ensuremath{\left|0\right\rangle}}_{L}+\beta{\ensuremath{\left|1\right\rangle}}_{L}={\ensuremath{\left|\psi\right\rangle}}_{L}$$ an error occurs at the level of an encoding qubit. This error is described by the operator $E$ acting over the $N$ physical qubits of the logical state. The erred state will then be $$K^{i}E{\ensuremath{\left|\psi\right\rangle}}_{L}=(-1)^{m}EK^{i}{\ensuremath{\left|\psi\right\rangle}}_{L}=(-1)^{m}E{\ensuremath{\left|\psi\right\rangle}}_{L}$$ The parameter $m$ is equal to zero if the error and the stabiliser commute and is equal to $1$ if they anti-commute. The error procedure implies a sequential measurement of each of the code stabilisers. If the error operator commutes with the stabiliser the state remains a $+1$ eigenstate of $K^{i}$, while if the error operator anti-commutes with the stabiliser then the logical state is flipped to a $-1$ eigenstate of $K^{i}$. The procedure of error correction is equivalent with that of state preparation. Since an error free state is already a $+1$ eigenstate of the stabilisers, the anti-commuting errors with any of the stabilisers will flip the relevant eigenstate and therefore when we measure the parity of these stabilisers we will obtain ${\ensuremath{\left|1\right\rangle}}$. For the $[[7,1,3]]$ code if the error operator is $E=X_{i}$ with $i=1,...,7$ encoding a bit flip on any one single qubit of the $7$ physical qubits, then, no matter where such bit flip would occur, $E$ will anti-commute with a unique combination of $K^{4}$, $K^{5}$, and $K^{6}$. After measuring these three operators we will obtain information about whether and where the $X_{i}$ error occurred. If $E=Z_{i}$ the error operator will anti-commute with a unique combination of $K^{1}$, $K^{2}$, and $K^{3}$ and will give us information about the $Z$ error. This example, based on the $[[7,1,3]]$ code, while certainly limited, is useful in understanding how the main idea of this article will be developed in the case of holographic quantum error correction codes and their topological properties. Indeed, stabiliser operators may be used to generally specify error correction codes and to reduce the dimensionality of the physical Hilbert space down to the subspace that encodes our logical states. Another source of errors is the actual loss of physical qubits. The loss of, say, a photon is assumed to be equivalent to measuring the photon in a basis, say $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ without knowing the answer. Such ignorance results in a possible logical bit-flip error on the encoded state, and hence the problem will be how to protect against logical bit flip errors. We already saw that the 3-qubit code allows us to obtain this type of correction. The important part is to encode the states into a redundancy code where an arbitrary logical state ${\ensuremath{\left|\psi\right\rangle}}_{L}$ is now given by $${\ensuremath{\left|\psi\right\rangle}}_{L}=\alpha{\ensuremath{\left|0\right\rangle}}_{1}^{N}{\ensuremath{\left|0\right\rangle}}_{2}^{N}...{\ensuremath{\left|0\right\rangle}}_{q}^{N}+\beta {\ensuremath{\left|1\right\rangle}}_{1}^{N}{\ensuremath{\left|1\right\rangle}}_{2}^{N}...{\ensuremath{\left|1\right\rangle}}_{q}^{N}$$ where ${\ensuremath{\left|0\right\rangle}}^{N}$ and ${\ensuremath{\left|1\right\rangle}}^{N}$ are the so called parity encoded states. The general parity encoding for a logical qubit is an N-photon GHZ state in the conjugate basis \[25\] $$\begin{array}{c}
{\ensuremath{\left|0\right\rangle}}_{L}^{N}=\frac{1}{\sqrt{2}}({\ensuremath{\left|+\right\rangle}}^{\otimes N}+{\ensuremath{\left|-\right\rangle}}^{\otimes N})\\
\\
{\ensuremath{\left|1\right\rangle}}_{L}^{N}=\frac{1}{\sqrt{2}}({\ensuremath{\left|+\right\rangle}}^{\otimes N}-{\ensuremath{\left|-\right\rangle}}^{\otimes N})\\
\end{array}$$ where ${\ensuremath{\left|\pm\right\rangle}}=({\ensuremath{\left|0\right\rangle}}\pm {\ensuremath{\left|1\right\rangle}})/2$. This type of encoding is useful because measuring any qubit in the $\{{\ensuremath{\left|0\right\rangle}},{\ensuremath{\left|1\right\rangle}}\}$ basis removes it from the state, with the result state being reduced $$\begin{array}{c}
P_{0,N}{\ensuremath{\left|0\right\rangle}}_{L}^{N}=(I_{N}+Z_{N}){\ensuremath{\left|0\right\rangle}}_{L}^{N}=\\
\\
=\frac{1}{\sqrt{2}}({\ensuremath{\left|+\right\rangle}}^{N-1}+{\ensuremath{\left|-\right\rangle}}^{N-1}){\ensuremath{\left|0\right\rangle}}_{N}={\ensuremath{\left|0\right\rangle}}_{L}^{N-1}{\ensuremath{\left|0\right\rangle}}_{N}\\
\\
P_{1,N}{\ensuremath{\left|0\right\rangle}}_{L}^{N}=(I_{N}-Z_{N}){\ensuremath{\left|0\right\rangle}}_{L}^{N}=\\
\\
=\frac{1}{\sqrt{2}}({\ensuremath{\left|+\right\rangle}}^{N-1}-{\ensuremath{\left|-\right\rangle}}^{N-1}){\ensuremath{\left|1\right\rangle}}_{N}={\ensuremath{\left|1\right\rangle}}_{L}^{N-1}{\ensuremath{\left|1\right\rangle}}_{N}\\
\end{array}$$ where $P_{i,N}$ are the projectors corresponding to the measurement in the ${\ensuremath{\left|0\right\rangle}}{\ensuremath{\left|1\right\rangle}}$ basis of the $N$-th qubit. Such encoding protects against the loss of qubits because it first encodes the system into a code structure that allows for the removal of qubits without eliminating the computational state and then it protects against logical errors induced by loss events. The basic idea is that this prescription maps errors un-correctable by standard error correction codes to errors that are correctable \[25\].
The extension of degrees of freedom in double field theory is required in order to encode global information regarding the phenomena that depart from the point like interpretation of elementary particles. Winding modes being strictly stringy objects will have to be represented through the doubling of the coordinates in the bulk space and their introduction will require a modification in the way logical states may be represented. The gauge invariance (a.k.a. redundancy) of the doubled field theory will incorporate additional transformations which may be interpreted in terms of quantum error correction codes. Their impact will be made clear later on in this article.
Holography as error correction
==============================
In order to exploit holography with double field theory in the bulk as a quantum error correction code, we need to better understand how holography may be interpreted as an error correction code to begin with. Double field theory adds additional information in this context as it tries to incorporate string theoretical phenomena in effective field theories at lower energies. Looking at the holographic principle from the perspective of quantum error correction codes helps us better understand non-localities in the bulk. We expect them to exist due to the extended nature of strings, however, they are usually not manifest as the bulk boundary duality is best understood in the context where bulk physics is described by classical gravity. Introducing double field theory in the bulk makes T-duality manifest and T-duality relates not only distinct geometries, but also distinct topologies. Moreover, the correspondence between non-trivial topology and entanglement \[4\] shows that T-duality may play the role of a change of the factorisation of the algebra describing the total quantum state. It is known that depending on the factorisation considered, a quantum state may appear either as entangled or as separable. For pure states we can switch between separability and entanglement in a unitary fashion. For mixed states however, we need some minimal amount of mixedness \[15\]. Incorporation of T-duality in the bulk is therefore crucial to the interpretation of the bulk-boundary duality as a quantum error correction code, as such codes rely on the existence of entanglement. Without a clear understanding of the topology changing phenomena occurring in the bulk, the quantum error correction code interpretation is not complete. The emergence of bulk locality and its relation to quantum error correction has been mentioned in \[7\] where it has been shown that all the bulk notions such as the Bogoliubov transformation, the localisation in the radial direction, and even the holographic entropy bound have natural boundary conformal field explanations in terms of quantum error corrections. Therefore, it is worthwhile exploring the interpretation of holography in terms of quantum error correction codes before we go further to understanding how T-duality and its associated topological uncertainty may impact such an interpretation. As mentioned in \[7\] but already well known to the holographic community, it is still a mystery how bulk locality emerges, even in an approximate way. Near the boundary it is quite clear that the relation $$\lim_{r\rightarrow\infty}r^{\Delta}\phi(r,x)=\mathcal{O}(x)$$ remains valid, where we have considered the limiting values of a bulk field $\phi$ and a conformal field theory operator $\mathcal{O}$. A dictionary based on this relation will manifestly respect locality in the $x$ direction simply because the conformal field theory does so too. Moving in the radial direction, such an approximate locality is less obvious. A local operator in the centre of the bulk is expected to commute with every local operator at the boundary given a fixed time slice containing that particular bulk operator. However, it is known that any operator that commutes with all local operators at a fixed time must be proportional to the identity. Because of this, bulk locality cannot be respected within the conformal field theory at the level of the algebra of operators \[7\]. Of course, together with the authors of \[7\], we may ask in what sense it is respected? The answer of \[7\] is to analyse the problem of bulk locality in terms of the stability of the bulk phenomena to errors in the boundary theory. The deeper into the bulk a process occurs the more resilient it will be to local errors. The radial direction in the bulk is seen from the perspective of the CFT as a measure of how well the CFT representations of the phenomena occurring within the bulk are protected from local erasures. The holographic principle appears as an upper bound on the amount of information that can be protected from erasures. It is important to understand that most error protection or correction codes add supplemental qubits into the description therefore increasing the total number of information and the required entanglement. While usual quantum field theories in the bulk would regain locality at least in some approximate way, the natural result, incorporating T-duality, will be manifestly non-local and hence using double field theory in the bulk will give a better insight into the nature of such non-localities. The idea that truncated subalgebras of bulk observables are relevant in the analysis has been explored both in \[7\] and in \[16\]. Such factorisation can be obtained in the context of double field theory as in this case, the strong constraint leads to the restoration of the non-stringy degrees of freedom. It must however be underlined that even when the strong constraint is employed, the stringy nature still remains manifest at least through the fact that the resulting theory may be defined on non-geometric backgrounds that couldn’t be obtained without T-duality. Given the AdS space and a metric having the asymptotic form $$ds^{2}\sim -(r^{2}+1)dt^{2}+\frac{dr^{2}}{r^{2}+1}+r^{2}d\Omega_{d-1}^{2}$$ we can identify the conformal field theory that is holographically dual to this system as living on the $S^{d-1}\times \mathbb{R}$. The time direction is given by $\mathbb{R}$. The Hilbert space is given by the configurations of the fields on the $d-1$ dimensional sphere $S^{d-1}$. In the usual context of a field theory in the bulk, we can construct CFT operators for the boundary which obey the bulk equations of motion. Following reference \[7\] we assume that interactions in the bulk are suppressed as powers of $\frac{1}{N}$. The bulk field $\phi(x)$ will be represented as $$\phi(x)=\int_{S^{d-1}\times \mathbb{R}}dY K(x;Y)\mathcal{O}(Y)$$ This integral is performed over the conformal boundary and $K(x;Y)$ is a smearing function which obeys the bulk equations of motion for the $x$ index and as $x$ approaches the boundary it yields the boundary limit equation. One does not naturally expect that such operators have the desired commutation relations in the bulk \[7\]. The expected commutations are being recovered in the perturbative domain within low point correlation functions \[23\] but it is expected that they would break down in states with enough excitations. When we have a representation of the bulk field $\phi(x)$ as in the above expression, it is possible to use the conformal field theoretical hamiltonian in order to express all operators $\mathcal{O}(Y)$ in terms of Heisenberg picture fields on a single Cauchy surface in the CFT \[7\]. If we then take $x$ to be near the boundary but not yet on it, the single time CFT representation of the bulk field still involves operators with support all over the single Cauchy surface in the CFT, $\Sigma$. A representation has been found whose boundary support becomes smaller as the operator approaches the boundary. This is exactly the AdS-Rindler representation discussed in \[7\]. In the AdS-Rindler construction the same bulk field operator $\phi(x)$ lies in multiple causal wedges. Its representation then can exist on different distinct regions of $\Sigma$. Given any bulk field operator $\phi(x)$ and any CFT local operator $\mathcal{O}(Y)$ chosen such that $x$ and $Y$ are spacelike separated, it is possible to choose a causal wedge $\mathcal{W}_{C}[A]$ such that $\mathcal{O}(Y)$ lies in the complement of $A$ in the surface $\Sigma$. But we assume locality in the boundary CFT. Therefore $\mathcal{O}(Y)$ must commute with the representation of $\phi(x)$ in that wedge. But no non-trivial operator in the boundary conformal field theory can commute with all local CFT operators on $\Sigma$. Therefore, the representations of the bulk field in the various wedges cannot be the same operator on the CFT Hilbert space. To better understand this apparent contradiction, the theory of quantum error corrections has been invoked by \[7\].
Holographic error correction codes with tensor networks
=======================================================
It is well known that the holographic principle implies that a theory of gravity in a bulk space is dual to a quantum field theory on a boundary. In the AdS/CFT context this is translated into a duality between a weakly coupled gravity in the bulk and a strongly coupled conformal field theory on the boundary. In order for this duality to be meaningful we need to relate the bulk operators to boundary operators. This mapping has however some surprising aspects. Of course, while introducing the bulk space we identify a new radial dimension from the boundary towards the bulk. It has been shown in \[27\], \[28\] that such a radial dimension can be seen as a renormalisation group scale. We can see the radial coordinate of a spacetime with asymptotically AdS geometry as a flow parameter of the boundary field theory. Recent research has completed the idea that geometry, seen as an emergent property, is related to quantum entanglement in the sense that the geometry in the bulk can be expressed in terms of the entanglement structure of the boundary quantum field theory. Describing quantum field theory by non-geometric means is highly complicated, therefore its connection to geometry offered a new understanding of various quantum field theoretical phenomena \[29\], \[30\]. The connection to geometry has been made clear already by the introduction of the Ryu-Takayanagi formula \[31\] and its covariant counterpart \[32\]. This formula is known to acquire corrections by various local and non-local terms. Such terms, derived also in \[33\] can be seen by means of so called holographic quantum correction codes. Such codes not only demonstrate the idea that entanglement is a source from where geometry emerges, but also allows us to better understand various prescriptions of the AdS/CFT dictionary. Such a construction is based on a tensor network which is expressed in terms of polygons that are uniformly tiling the bulk space. The terminology here will become that of quantum information theory, and hence we will have physical quantum information units encoding the information of logical quantum states. The physical variables associated to the quantum code will be on the boundary while the logical operators reside in the bulk. Holographic codes allow us to explicitly compute the mapping between boundary and bulk and hence to derive the dictionary of AdS/CFT. In essence, local operators in the bulk theory are being mapped into non-local operators of the bulk. This allows us to connect bulk geometry to the entanglement structure of the boundary quantum field theory. The bulk Hilbert space or the code space is a proper subspace of the boundary Hilbert space preserved by the bulk operators. The idea of reconstructing bulk operators on the boundary is based on the AdS-Rindler reconstruction described in the introduction. The ambiguity of the reconstruction prescription has been resolved by making use of some form of redundancy. Either we considered the highly different boundary operators as being different physical representations of the same type of action on the code subspace, or we considered the distinction in the boundary theory as given by the redundant description given by gauge invariance. However, the usual gauge invariance on the boundary seemed problematic. Extending it to the gauge invariance of double field theory however may have certain benefits. The fact that operators residing in the causal wedge of a certain boundary region $A$ can be reconstructed on the boundary is well known. However, if the the boundary region $A$ is a union of two or more disconnected components, then the domain of the bulk from where operators can be reconstructed on the boundary region $A$ increases. This introduces the so called entanglement wedge $E(A)$ which may extend further into the bulk and from which bulk operators may be reconstructed on the disconnected region $A$. Moreover, entangled pairs in the bulk with one of the members inside the entanglement wedge of the region $A$ and the other outside, will contribute to the entropy of the region $A$ and hence to the entanglement shared by $A$ and its complement $\bar{A}$. This means that there should be operators in $A$ capable of detecting the member of the pair inside the entanglement wedge $E(A)$. Given the general entropy formula $$S_{A}=\frac{A}{4G_{N}}+S_{bulk}(\rho_{E(A)})+\frac{\delta A}{4 G_{N}}+...$$ the first term is the leading Ryu-Takayanagi term, which is local on the entangling surface and state independent, the second term is the bulk entropy in the entanglement wedge $E(A)$ defined by the Ryu-Takayanagi minimal surface and therefore generally non-local and non-linearly dependent on the bulk state, and the third term is an additional quantum correction to the Ryu-Takayanagi area which is both local on the minimal surface and linear in the bulk state. The last term appears to also originate from a quantum error correction code which is based on the operators $\mathcal{O}$ associated precisely to the boundary between the entanglement wedge of the boundary area $A$ and that of its complement $\bar{A}$ \[34\]. Once the form of this correction was known, it has been noticed that it can be derived from holographic quantum error correction codes as well \[26\]. The operators $\mathcal{O}$ must be reconstructible from both $A$ and $\bar{A}$ and hence the operator itself must lie in the centre of either reconstructed algebra. Terms like the first and the third in the entropy formula are related to aspects of the code derived from the values of the operators in this centre \[26\]. The minimal area computation in the Ryu Takayanagi formalism is translated into the calculation of the so called “greedy” surface \[26\] in the context of holographic codes. MERA tensor networks also realise a hyperbolic geometry and entropy bounds as those found in holographic discussions. The description of such tensor networks rely on the so called perfect tensors which arise in the expansion of a pure state describing $2n$ $v$-dimensional spins in a suitable basis $${\ensuremath{\left|\psi\right\rangle}}=\sum_{a_{1},...,a_{2n}}T_{a_{1}...a_{2n}}{\ensuremath{\left|a_{1}a_{2}...a_{2n}\right\rangle}}$$ We call the tensor $T$ perfect if the state is maximally entangled across any bipartition cut of the set of $2n$ spins into two sets of $n$ spins. This expansion can be rewritten (by the maps of bra into ket states) in various forms and for the 6 spins code we have the operators $$\begin{array}{c}
\sum_{a_{1},...,a_{6}}T_{a_{1}...a_{6}}{\ensuremath{\left|a_{4}a_{5}a_{6}\right\rangle}}{\ensuremath{\left\langlea_{1}a_{2}a_{3}\right|}}\\
\\
\sum_{a_{1},...,a_{6}}T_{a_{1}...a_{6}}{\ensuremath{\left|a_{3}a_{4}a_{5}a_{6}\right\rangle}}{\ensuremath{\left\langlea_{1}a_{2}\right|}}\\
\\
\sum_{a_{1},...,a_{6}}T_{a_{1}...a_{6}}{\ensuremath{\left|a_{2}a_{3}a_{4}a_{5}a_{6}\right\rangle}}{\ensuremath{\left\langlea_{1}\right|}}\\
\end{array}$$ These expansions form isometric encoding maps which will encode a certain number of logical qubits into the emerging physical qubits. Holographic quantum error correction codes are then implemented by contracting perfect tensors taking into account the geometry of the bulk space (in this case hyperbolic) and its tiling by corresponding polygons. The uncontracted indices are particularly important in the description of gauge fields inside the bulk and hence will become important for the extension towards double field theory. In a holographic code there are two types of uncontracted indices, namely the bulk indices and the boundary indices. All other indices are contracted between tensors arising on different layers of tilings. The bulk and the boundary indices are however not separated. Because the code is essentially an isometric embedding of the bulk Hilbert space into the boundary Hilbert space, the two indices are related. Each polygon provides an isometry from incoming and bulk indices to outgoing indices.
Bulk gauge fields in the holographic tensor networks
====================================================
The implementation of bulk fields in the holographic tensor network has been discussed in \[26\]. There, a generalisation of the holographic quantum error correction code for bulk gauge fields is presented. As most of the novel aspects of double field theory are revealed in the generalised gauge transformations they introduce, understanding briefly how gauge fields and gauge invariance are implemented in a holographic quantum error correction code seems essential. New degrees of freedom on the links of the holographic tensor network are being introduced to that end, and additional connections to further copies of the holographic code are implemented by suitable isometries. In the case of double field theory such degrees of freedom on the links must be extended even further, considering the topological properties of T-duality. In the non-doubled case boundary regions allow the reconstruction of bulk algebras with central elements in the interior edges of the entanglement edge. In the case of double field theory bulk algebras are further extended leading to new error correction codes, previously unavailable. A tensor network has an upper bound for the amount of entanglement the state described by it can have, and this is based on the minimal cut dividing the network. This upper bound is saturated for connected regions in certain classes of holographic states. The case of planar graphs with non-positive curvature has been described in \[17\]. In order for a circuit interpretation of a network of perfect tensors to be valid, according to \[17\] it must satisfy three criteria. The first is the covering criterium, namely that to each edge (contracted or uncontracted index) is assigned a directionality. This condition is required in order to interpret the direction in which each tensor in the network processes information and hence to meaningfully define the input and the output indices. The second condition is the so called flow condition, which implies that each tensor has an equal number of incoming and outgoing indices. This is required for the interpretation that every tensor is a unitary gate. The last condition, namely the acyclicity condition is however rather special. It is already noted in \[17\] that this condition is non-local and is demanded so that the order of the application of the operations in the network to be consistent. Inconsistencies in this interpretation would be the presence of a closed time-like curve in the circuit picture. The assumptions made in \[17\] require for the graph to be, first, a planar embedding, namely the tensor network to be laid out in a planar form, the boundary of the network being a simple boundary of the embedding. Second, the tensors are required to be perfect, having an even number of legs and being unitary along any balanced distribution of legs. Finally, the network was expected to represent an AdS bulk and hence corresponding to a network equivalent of the AdS negative curvature. This implies that the distance function between two nodes of the network has no local maxima away from the boundary. Thinking in terms of the acyclic condition, which is a non-local property, it has been shown in \[17\] that the presence of a cycle implies the existence of an interior local maximum for the labelling. The proof goes, according to \[17\] as follows. Let there be a cycle $C$ in the construction of the tensor network. The node label values immediately in the interior of the loop will be larger or smaller (depending on the orientation of $C$) than those immediate to the exteriors. In the case of $C$ counterclockwise we may chose a node in the interior of $C$ with the lowest possible label. In the case of this note, the label is smaller than those of all its neighbours including those in the exterior of $C$, which means it contradicts the assumption that it is defined based on the graph distance function and its properties. In the clockwise case, we can chose a node in the interior of $C$ with the largest possible label. In this case it represents an interior maximum for the distance function and hence the surface homeomorphic to the disc cannot be negatively curved. Fascinatingly enough, precisely this acyclicity condition cannot uphold in the case of T-duality in the bulk. But it is well known from \[35\] and \[36\] that the T-dual of the AdS spacetime is the de-Sitter spacetime and hence abandoning the acyclic condition introduces into our network the cosmologically relevant de-Sitter space. In order for this to become clear let me first follow the results of ref. \[26\] regarding the inclusion of gauge fields in the tensor network representation, so that in the next chapter I can bring plausibility arguments for the statement above. In the case of lattice gauge theories, there appears the requirement of additional degrees of freedom on the links of the discrete graph model in order to describe the associated gauge fields. The various holonomies arise as paths through the lattice and the Gauss constraint provide a valid gauge interpretation. In the context of the holographic code in order to treat bulk gauge theories we need to introduce additional degrees of freedom on the links of the tensor network corresponding to the pentagon code. The tensor associated to the pentagon tiling has a total of six indices, five of them being associated to the network and away from the boundary they are connected to nearby tensors. Every such tensor also has an uncontracted index associated with local bulk degree of freedom. When $T$ is a perfect tensor, it describes an isometry from any three legs to the others, then an operator $\mathcal{O}$ acting on any bulk input may be transported along three of the output legs to the three neighbouring tensors. This procedure together with the negative curvature assumption allows us to transport local bulk operators up to the boundary because each tensor has at least three legs pointing towards the boundary. The additional degrees of freedom modelling bulk gauge fields can be introduced by adding a three index tensor $G_{ijk}$ To keep the connection to the bulk, one adds the new tensor to the bulk index and connects it to another bulk index leaving the third index for additional input. In this way one merges two bulk indices into one single index. This implies contracting the new tensor $G$ with a pair of neighbouring bulk inputs as described in \[26\].
Double field theory in the bulk and holographic tensor networks
===============================================================
The AdS/CFT holographic correspondence appears to be the best tool of understanding non-perturbative quantum gravity. Its connection to the quantum error correction codes has been recently made manifest in \[7\]. The quantum error correction interpretation of holography allows us to further expand both our understanding of quantum information processing and of holography. However, AdS/CFT is somehow restricted in its applicability, as we only now start to understand how it may generalise to different quantum field theories on the boundary and new types of geometry in the bulk. In its modern interpretation, the holographic duality states that the entanglement structure (and the quantum error correcting properties) of a quantum field theory is to be interpreted as defining the geometric properties of the bulk spacetime. As far as it is known now, the correspondence connects local operators supported deep inside the bulk to highly non-local operators on the boundary. This dictionary, interpreted as the encoding map of a quantum error correcting code allows us to see bulk local operators as the logical counterparts of the physical boundary theory. The logical operators map the code subspace $\mathcal{H}_{C}$ to itself, insuring the protection of the logical information against erasures of portions of the boundary. This idea allows us to finally see the relationship between the emergence of bulk geometry and the structure of the entanglement on the boundary theory. It was shown in \[38\] that codes can have holographic properties even when the underlying bulk geometry does not have negative curvature. Holographic codes have been analysed in the hope of providing new insights for quantum computing architectures \[37\] and their description in terms of operator algebra quantum error correction has been established \[39\], \[40\]. As noted also in \[37\] the understanding of the bulk both in terms of physics and geometry is insufficient. While AdS/CFT is very useful, understanding the bulk in more general contexts will reveal several aspects of quantum cosmology, quantum field theory, and quantum information. Also, \[37\] noticed that holographic codes are locally correctable provided that the bulk geometry is negatively curved in the asymptotic limit, but is not locally correctable for asymptotic flat or positive curvature. This may result in non-local physics on the boundary in the flat and positively curved cases. When approaching the boundary, it is interesting to consider therefore certain string theoretical effects which originate from their finite size. T-duality can be associated with topology changing effects and the field theoretical interpretation that makes T-duality manifest, namely double field theory, features properties that cannot be encoded in strictly local quantum field theories. Indeed string theory has access to both globally and locally non-geometric backgrounds where non-associative phenomena may occur. Obviously, the entanglement structure of such objects will be particularly complicated and I do not intend to explore it here directly. Instead of doing this, I will formulate a strategy for relating gauge symmetry in double field theories to particular features of holographic quantum error correction codes. Geometrical properties inside the bulk are related to the structure of quantum entanglement in the boundary, but then, to what can non-geometrical properties associated to double field theory and string theoretical T-duality in the bulk be related on the boundary? Can we establish a new form of quantum error correction on the boundary by thinking in terms of non-geometric properties in the bulk? It appears that the answer to both questions is in the positive. As is known, local operators in the bulk can be seen as logical operators acting on the code subspace. For holographic codes, the fact that a subsystem of the physical Hilbert space $\mathcal{H}$ is correctable with respect to a logical sub-algebra can be interpreted according to \[37\] in terms of a question about the bulk geometry. This can be seen from the perspective of the entanglement wedge hypothesis which provides us with the largest bulk region with a logical sub-algebra that can be represented on a given boundary region. In the case of double field theory and string geometry, several non Riemannian phenomena can occur in the bulk, leading to different ways in which manifolds can be patched together. If T-duality is being employed as a transition function, we find a new class of backgrounds which present a non-trivial dependence on the dual coordinates that are conjugate to the string winding number. As has been shown in the previous section, to introduce the degrees of freedom associated to gauge fields on links, the bulk indices of the code tensors must be linked with a tensor called $G_{ijk}$ whose role in the network is to add the degrees of freedom required. To keep the input from the bulk space into the network tensors $T$, this tensor has been extended to a six-fold tensor product $$T\rightarrow \bigotimes_{m=1}^{6}T$$ resulting that these units must be connected as in a pentagonal tiling of the hyperbolic disk. Each factor of the resulting tensor product was called a copy of the network. The first copy was considered to be the original network and the other copies were contracted with the three-legged tensor $G$. In this way the network will have six bulk input legs at each vertex. Five of these can be turned into inputs for the edges. Take one edge in the interior of the disk and one input leg from each of the two vertices it connects. These legs will be contracted with two of the three legs of the tensor $G$. These two legs of $G$ will be considered output legs and the remaining one will be an input leg associated with the current edge under consideration. This uses up five of the bulk legs at each vertex leaving the last one as a normal bulk input leg at each vertex. This prescription adds one $G$ for every two bulk legs and implicitly for two tensors $T$. This construction is as the one described in \[26\]. In order to introduce T-duality and the effects related to double field theory, certain adaptations must be included. The most intuitive modification that allows the doubling of the fields in the bulk is to increase the number of degrees of freedom associated to the tensor $G$, for example by adding a new index. This alters the connectivity of the bulk in the sense that now, two output legs of $G$ will be connected to the input legs of the vertices while another two legs of the tensor $G$ will play the role of input legs. This will connect the various layers of the bulk tensor network and will provide us with new topological structures not available before. In particular, the acyclicity condition must be altered in order to take into account effects that are related to T-duality. In particular, being a non-local property, it can be interpreted as a topological structure of the bulk spacetime. Topologically non-trivial bulk surfaces are not unexpected, particularly situations manifesting entanglement over spacelike separated regions, this being the main idea behind the ER-EPR duality \[41\]. The effect of this extension however has another interesting consequence. As shown previously, the condition of acyclicity not only reassures us that no closed timelike curves are possible, but also keeps the distance function from having a maximum away from the boundary. As doubling the number of bulk degrees of freedom alters this property, it becomes obvious that such a property can no longer be maintained, leading us to conclude that we cannot assume that the negative curvature assumed in the beginning can be preserved. Therefore, it appears that double field theory in the bulk, by its ability of manifestly implementing T-duality leads to positive curvature and hence to the emergence of a de-Sitter spacetime. The connections of such an observation with cosmological data remains to be discussed in a future article. Until then, following ref. \[37\], it will be relevant to move to the operator algebra quantum error correction code interpretation in order to see the way in which gauge freedom manifests itself in the bulk and how the extended gauge freedom of double field theory affects the standard holographic interpretation. In the simplest case, let there be $H$ a finite dimensional Hilbert space and an associated complex vector space of linear operators acting on this Hilbert space. This complex vector space forms an algebra $\mathcal{A}$. We can identify a subspace of this Hilbert space which can be written as a product of two tensor factors $$H\supseteq \bigoplus_{\alpha}H_{\alpha}\otimes H_{\bar{\alpha}}$$ The algebra then can be written as $$\mathcal{A}=\bigoplus_{\alpha}\mathcal{M}_{\alpha}\otimes I_{\bar{\alpha}}$$ We define the commutant of $\mathcal{A}$ as the algebra $\mathcal{A}'$ which contains all operators acting on the Hilbert space $H$ which commute with all the operators of the algebra $\mathcal{A}$. Then the commutant algebra can be written as $$\mathcal{A}'=\bigoplus_{\alpha}I_{\alpha}\otimes \mathcal{M}_{\bar{\alpha}}$$ The original algebra and its commutant share the same centre $Z(\mathcal{A})$ which contains elements of the form $$\bigoplus_{\alpha}m_{\alpha}I_{\alpha}\otimes I_{\bar{\alpha}}$$ where $I$ represents the identity matrix on the respective subspace. The above algebras describe both the classical and quantum aspects of the information related to a system. The operators in the common centre describe the classical data, like the area operator associated to the minimal area in the Ryu-Takayanagi formula, while $\mathcal{M}_{\alpha}$ describes the quantum data. In order to analyse error correction properties of holographic codes it is useful to consider the code subspace of the Hilbert space associated to a low energy domain of the boundary field theory. Here, the algebras $\mathcal{A}$ and $\mathcal{A}'$ contain logical operators that preserve the code subspace. In the case of a single summand in the decomposition of the Hilbert space, with $\mathcal{M}_{\bar{\alpha}}$ non-trivial, the algebra $\mathcal{A}$ represents the algebra of logical operators in a subsystem code, while the code subspace can be decomposed in two sides, $$H_{C}=H_{\alpha}\otimes H_{\bar{\alpha}}$$ the first representing the protected tensor part, $H_{\alpha}$, while the other represents a gauge part, $H_{\bar{\alpha}}$. The algebra $\mathcal{A}$ acts only on the protected part, while by means of the holographic duality, the gravitational bulk system has an emergent gauge symmetry. In double field theory this gauge symmetry is extended in a non-trivial way, by introducing additional components related to the winding modes. Basically, as stated in the second chapter, the doubled gauge parameter is extended as $\xi^{M}=(\tilde{\lambda}_{i}, \lambda^{i})$. The doubling has as effect the extension of the bulk algebra to one with non-trivial commutation relations linking the extended winding modes and the associated coordinates. The basis of the doubled space are the so called left and right moving modes that now can be formed given the non-local effects provided by T-duality. The idea of forming the left-invariant and right-invariant forms for both left and right moving modes implies the doubling of the associated algebra leading to elements belonging to the direct product of these. As we have these, let us consider the algebra generated by the operations $G_{I}$ obeying commutation relations of the form $$\begin{array}{ccc}
[G_{I},G_{J}]=i\cdot f_{IJ}^{\;\;\;\;K}G_{K}, & Tr(G_{I}G_{J})=\eta_{IJ}, & \det(\eta_{IJ})\neq 0\\
\end{array}$$ where $\eta_{IJ}$ is a non-degenerate metric doubled metric. In our context of the $AdS$ bulk spacetime, we need a doubling of this space into left and right components. Representing this in the form of groups, the doubled $AdS$ group will be $SO(D,D+1)$ according to \[42\]. The doubled $AdS$ algebra is generated by the doubled momenta, and the doubled Lorentz generators, and it results in the emergence of a left / right mixed index. We observe \[42\] that the left moving mode will exist in the $AdS$ space while the right moving one in the de-Sitter space. The details of the group representation of the doubled bulk space as well as the emergence of the de-Sitter components have been presented in \[42\] and I will not insist upon them here. The relevant aspect however is first, that the explicit introduction of T-duality in the bulk by means of field doubling leads to modes belonging to the de-Sitter component, and second, that such an extension can be encoded by extending the gauge component in the bulk corresponding to a boundary term that originated in a particular map of the boundary code subspace. To see how the boundary theory implements such a doubling in terms of quantum error code correction, let me follow again \[37\] to see how the error correction map can be extended. Of course, a more comprehensive discussion will have to take into account an actual string theory in the bulk, but this lies beyond the scope of the present article. Given an noise channel which can be written as acting on an operator $X$ as $$\mathcal{N}(X)=\sum_{a}N_{a}^{\dagger}XN_{a}$$ with $N_{a}$ being the Krauss operators \[37\], and considering the Hilbert space $H$ then quantum error correction would be a process that could reverse the effects of $\mathcal{N}$. Unless $\mathcal{N}$ is unitary, such a reversion would not be possible over the entire $H$ but we still hope to be able to reverse a subset $H_{C}\subset H$ of the original Hilbert space. Indeed considering the algebra of logical operators that act on the code space we denote the set of those linear operators that map $H$ into $H$ as $\mathcal{L}(H)$ and given the projector $P$ from $H$ to $H_{C}$, the operator $X\in \mathcal{L}(H)$ is called logical if $[X,P]=0$ and therefore $X$ maps the code space to itself $$X\cdot H_{C}=XP\cdot H=PX\cdot H\subseteq H_{C}$$ We say that the noise $\mathcal{N}$ is correctable on the code space $H_{C}$ with respect to the operator $X\in\mathcal{L}(H)$ if there exists such a recovery channel $\mathcal{R}$ that the property $$P(\mathcal{R}\cdot \mathcal{N})^{\dagger}(X)P=PXP$$ is satisfied. In the bulk however, correctability is represented in terms of the so called entanglement wedge hypothesis which states basically that if a bulk region is included in the entanglement wedge of a boundary region, then the complementary boundary region is correctable with respect to the logical bulk subalgebra associated to the above mentioned bulk region. In the case in which the bulk space is doubled and mixing indices appear connecting anti-de-Sitter and de-Sitter algebras, the standard representation of the bulk algebra as a tensor product over bulk states $$\mathcal{A}=\bigotimes_{x}\mathcal{A}_{x}$$ fails, in the sense that there can be no such simple decomposition as the group associated to the bulk space will include also a de-Sitter component. However, it is still possible to reconstruct a boundary theory by carefully restricting the doubled coordinates in the bulk region close to the boundary. There are several options by which such a restriction can be performed \[43\], \[44\], \[45\], each coming with advantages and disadvantages. The most important aspect is to keep the desirable effects of T-duality in the limit where the doubled coordinates become irrelevant. It is currently not clear what precisely the boundary encoding map associated to the mixing terms found in the bulk due to the extended gauge symmetry. Heuristically speaking, it is possible to imagine that stringy modes encoded in the bulk double field theory may have the effect of violating associativity of the operators in the boundary. Also, effects associated to the extended nature of the strings, from where double field theory extracts its stranger features, seem to be related to mixing of operators in the boundary and to a left-right symmetry which should not otherwise be present \[46\].
Conclusions
===========
This article explores the connection between quantum code correctability and the geometry of the bulk in the unexplored context in which the bulk coordinates are doubled in order for string theoretical duality to become manifest. It seems plausible nowadays that for holographic codes, the correctability of a subsystem of a Hilbert space can be expressed in terms of the bulk geometry by means of the so called entanglement wedge hypothesis. Such an observation basically equates the emergence of spacetime geometry with the structure of entanglement in a non-gravitational quantum field theory. However, such an analysis does not directly take into account gauge invariance, which may play an important role in establishing the connection between quantum error code correction and geometry. Even more so, stringy effects are not directly considered and they may be the key towards even more advanced quantum codes. This article is a heuristic attempt towards the exploration of the holography / quantum error correction duality in the context in which the stringy T-duality becomes manifest. The main observation is that T-duality plays an important role in the emergence of de-Sitter geometry in the bulk which may lead to further cosmological discussions. Moreover, T-duality appears to negate the requirement for acyclicity in the bulk, a phenomenon that may lead to new interpretations of the dualities between topologically distinct manifolds in the context of the ER-EPR duality. This article wishes to point out these new directions leaving for future work, the task of going from a heuristic approach to one based on more specific examples.
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abstract: '[We perform an analysis of the Renormalization Group evolution of the couplings in an extension to the Standard Model which contains a real triplet in the Higgs sector. Insisting that the model remain valid up to 1 TeV allow us to map out the region of allowed mass for the Higgs bosons. We conclude that it is possible for there to be no light Higgs bosons without any otherwise dramatic deviation from the physics of the Standard Model.]{}'
---
plus 2mm minus 2mm 23.0cm 17.0cm -1.0in -65pt
MC-TH-2002/14\
Cavendish-HEP-2003/01\
February 2003\
\
J.R. Forshaw$^1$, A. Sabio Vera$^2$ and B.E. White$^1$
$^1$[*Department of Physics & Astronomy*]{},\
[*University of Manchester*]{},\
[*Oxford Road*]{},\
[*Manchester M13 9PL, U.K.*]{}\
$^2$ [*Cavendish Laboratory*]{},\
[*University of Cambridge*]{},\
[*Madingley Road*]{},\
[*Cambridge CB3 OHE, U.K.*]{}\
Introduction
============
In a previous paper, we studied an extension of the Standard Model in which a real scalar $SU(2)$ triplet with zero hypercharge is added to the usual scalar $SU(2)$ doublet [@Forshaw:2001xq]. We showed that such an extension is allowed by the precision data and that the mass of the lightest Higgs boson can be as big as 500 GeV.
To recap, the Lagrangian of the model in terms of the usual Standard Model Higgs, $\Phi_1$, and the new triplet, $\Phi_2$, reads $$\begin{aligned}
{\cal L} &=& (D_\mu \Phi_1)^{\dagger}~ D^\mu \Phi_1
+ \frac{1}{2} (D_\mu \Phi_2)^{\dagger} ~D^\mu \Phi_2
- V_0(\Phi_1 , \Phi_2),
\label{eqn:lag}\end{aligned}$$ with a scalar potential $$\begin{aligned}
V_0(\Phi_1 , \Phi_2) &=&\mu_1^2 ~|\Phi_1|^2 + \frac{\mu_2^2}{2}
~|\Phi_2|^2 + \lambda_1 ~|\Phi_1|^4+ \frac{\lambda_2}{4} ~|\Phi_2|^4
+ \frac{\lambda_3}{2} ~|\Phi_1|^2 ~|\Phi_2|^2 \nonumber\\
&+& \lambda_4 \, {\Phi_1}^\dagger \sigma^\alpha \Phi_1 ~ {\Phi_2}_\alpha.\end{aligned}$$ $\sigma^\alpha$ are the Pauli matrices. The expansion of the field components is $$\begin{aligned}
\Phi_1 = \left(\begin{array}{c}\phi^+ \\
\frac{1}{\sqrt{2}}\left(h_c^0 + h^0 + i \phi^0\right)\end{array}\right)_
{Y=1},
~~\Phi_2 = \left(\begin{array}{c}\eta_1 \\
\eta_2 \\ \eta_c^0 + \eta^0 \end{array}\right)_{Y=0}\end{aligned}$$ where $\eta^\pm = ( \eta_1 \mp i \eta_2) / \sqrt{2}$ and $\phi^0$ is the Goldstone boson which is eaten by the $Z^0$.
The model violates custodial symmetry at tree level giving a prediction for the $\rho$-parameter of $$\begin{aligned}
\rho ~=~ 1 + 4 \left(\frac{\eta^0_c}{h^0_c}\right)^2.\end{aligned}$$ As discussed in [@Forshaw:2001xq], it is precisely this violation of custodial symmetry which allows the lightest Higgs to be much heavier than in the Standard Model. By giving the triplet a non-zero vacuum expectation value, one is in effect making a positive tree-level contribution to the $T$-parameter, and this is enough to allow a heavier Higgs.
In the neutral Higgs sector we have two CP-even states which mix with angle $\gamma$. The mass eigenstates $\{H^0, N^0\}$ are defined by $$\begin{aligned}
\left(\begin{array}{c}H^0\\N^0\end{array}\right) &=&
\left(\begin{array}{cc}\cos{\gamma}&-\sin{\gamma}\\
\sin{\gamma}&\cos{\gamma}\end{array}\right)
\left(\begin{array}{c}h^0\\\eta^0\end{array}\right).\end{aligned}$$ There is also mixing in the charged Higgs sector. We define the mass eigenstates $\{g^\pm, h^\pm\}$ by $$\begin{aligned}
\left(\begin{array}{c}g^\pm\\h^\pm\end{array}\right) &=&
\left(\begin{array}{cc}\cos{\beta}&-\sin{\beta}\\
\sin{\beta}&\cos{\beta}\end{array}\right)
\left(\begin{array}{c}\phi^\pm\\\eta^\pm\end{array}\right).\end{aligned}$$ The $g^\pm$ are the Goldstone bosons corresponding to $W^\pm$ and, at tree level, the mixing angle is $$\begin{aligned}
\tan{\beta} &=& 2 \frac{\eta^0_c}{h^0_c}.\end{aligned}$$ The precision electroweak data constrain $\beta$ to be smaller than about $4^\circ$ [@Forshaw:2001xq].
In this paper we wish to examine the renormalization group flow of the couplings and hence establish bounds on the scalar masses under the assumption that the triplet model remain valid up to some scale $\Lambda$. We take $\Lambda = 1$ TeV and make no statements about physics at higher scales. For the Lagrangian of (\[eqn:lag\]) to remain appropriate up to $\Lambda$, we demand that the scalar couplings $\lambda_i$ remain perturbative and that the vacuum remain stable (i.e. is a local minimum) up to $\Lambda$. We begin in the next section with the calculation of the beta-functions. In Section 3 we present our results away from the decoupling limit of the model and in Section 4 we discuss decoupling.
The one-loop effective potential and the beta-functions
=======================================================
The effective potential [@Coleman:jx; @Jackiw:cv; @Iliopoulos:ur; @Kastening:1991; @Ford:mv; @Quiros:dr] has the following one-loop expansion in the $\overline{\rm MS}$ renormalization scheme and ’t Hooft-Landau gauge: $$\begin{aligned}
\hspace{-2cm}V &=& V_0 + V_{\rm CT} + V_1 \nonumber\\
&=&
\frac{1}{2}{{{\mu }_1}}^2\,{h^0_c}^2 +
\frac{1}{2}{{{\mu }_2}}^2\,{\eta^0_c}^2 +
\frac{1}{4}{{\lambda }_1}\,{h^0_c}^4 +
\frac{1}{4}{{\lambda }_2}\,{\eta^0_c}^4+
\frac{1}{4}{{\lambda }_3}\, {h^0_c}^2 \, {\eta^0_c}^2-
\frac{1}{2}{\lambda_4} \,{h^0_c}^2\,{\eta^0_c} \nonumber\\
&+& \delta \Omega
-\frac{1}{2} \delta{{{\mu }_1}}^2\,{h^0_c}^2
- \frac{1}{2} \delta{{{\mu }_2}}^2\,{\eta^0_c}^2
+ \frac{1}{4}\delta{{\lambda }_1}\,{h^0_c}^4 +
\frac{1}{4}\delta{{\lambda }_2}\,{\eta^0_c}^4
+ \frac{1}{4}\delta{{\lambda }_3}\,{h^0_c}^2\,{\eta^0_c}^2
- \frac{1}{2}\delta\lambda_4 \,{h^0_c}^2\,{\eta^0_c} \nonumber\\
&+& \frac{1}{16 \pi^2} \left\{
\frac{3}{4} {m}_Z^4 \left(\log{\frac{{m}_Z^2}{\mu^2}}
-\frac{5}{6}\right)
+\frac{3}{2} {m}_W^4 \left(\log{\frac{{m}_W^2}{\mu^2}}
-\frac{5}{6}\right)
-3~ {m}_t^4 \left(\log{\frac{{m}_t^2}{\mu^2}}
-\frac{3}{2}\right)\right.\nonumber\\
&+&\frac{1}{4} {m}_{\phi^0}^4
\left(\log{\frac{{m}_{\phi^0}^2}{\mu^2}}-\frac{3}{2}\right)
+\frac{1}{2} {m}_{g^\pm}^4
\left(\log{\frac{{m}_{g^\pm}^2}{\mu^2}}
-\frac{3}{2}\right)+\frac{1}{2} {m}_{h^\pm}^4
\left(\log{\frac{{m}_{h^\pm}^2}{\mu^2}}-\frac{3}{2}\right)\nonumber\\
&+&\left.\frac{1}{4} {m}_{H^0}^4 \left(\log{
\frac{{m}_{H^0}^2}{\mu^2}}
-\frac{3}{2}\right)+\frac{1}{4} {m}_{N^0}^4
\left(\log{\frac{{m}_{N^0}^2}{\mu^2}}-\frac{3}{2}\right)\right\} \nonumber\\
&-& \frac{C_{\rm UV}}{64 \pi^2} \left\{
3 ~ {m}_Z^4 +6 ~{m}_W^4 - 12 ~ {m}_t^4 + {m}_{\phi^0}^4
+ 2 ~{m}_{g^\pm}^4 + 2 ~{m}_{h^\pm}^4 + {m}_{H^0}^4 + {m}_{N^0}^4 \right\}.\end{aligned}$$ $\mu$ is the renormalization scale and $C_{\rm UV} = \frac{2}{4-D}-\gamma_E + \log{4 \pi}$. We have included the contributions from all the relevant physical states including the heaviest fermion, the top quark. The terms with $\delta$ correspond to the counterterms of the theory and the tree-level masses are $$\begin{aligned}
m_Z^2 &=&\frac{1}{4}{h^0_c}^2\,\left( g^2 + {g'}^2 \right), \\
m_W^2 &=& \frac{1}{4}g^2\,{h^0_c}^2 + g^2\,{\eta^0_c}^2, \\
m_t^2 &=& \frac{1}{2}{h_t}^2\,{h^0_c}^2, \\
m_{\phi^0}^2 &=& {{{\mu }_1}}^2 + {{\lambda }_1}\,{h^0_c}^2 +
\frac{1}{2} \, \lambda_3 \, {\eta^0_c}^2 - \lambda_4 \, \eta^0_c,\\
m_{g^\pm}^2 &=& \mu_1^2 + \lambda_1 \, {h^0_c}^2 + \lambda_4 \, \eta^0_c
+ \frac{1}{2}\,\lambda_3\,{\eta^0_c}^2 - \lambda_4 \, h^0_c \, \tan{\beta},\\
m_{h^\pm}^2 &=& \mu_2^2 + \lambda_2 \, {\eta^0_c}^2 + \lambda_4 \, h^0_c
\, \tan{\beta} + \frac{1}{2}\,\lambda_3\,{h^0_c}^2,\\
m_{H^0}^2 &=& {{{\mu }_1}}^2 + 3\,{{\lambda }_1}\,{h^0_c}^2 +
\frac{1}{2}\,\lambda_3 \,{\eta^0_c}^2 - \lambda_4 \, \eta^0_c
+ \lambda_4 \, h^0_c \, \tan{\gamma}
- \lambda_3 \, h^0_c \, \eta^0_c \, \tan{\gamma}, \\
m_{N^0}^2 &=& \mu_2^2 + 3\, \lambda_2 \, {\eta^0_c}^2 - \lambda_4 \, h^0_c \,
\tan{\gamma} + \frac{1}{2}\, \lambda_3 \, h^0_c \left(h^0_c + 2 \,
\eta^0_c \, \tan{\gamma}\right).\end{aligned}$$ It is understood that we should substitute explicitly for the mixing angles, which are solutions to the equations $$\begin{aligned}
&&\hspace{-1cm}\lambda_4 h^0_c + \tan{\beta}\left(\mu_1^2 - \mu_2^2 +
\lambda_1 {h^0_c}^2
- \frac{1}{2} \lambda_3 {h^0_c}^2 + \lambda_4 \eta^0_c - \lambda_2
{\eta^0_c}^2 + \frac{1}{2}\lambda_3 {\eta^0_c}^2 - \lambda_4 h^0_c
\tan{\beta}\right) = 0,\label{eq:beta} \\
&&\hspace{-1cm}-\lambda_4 h^0_c + \lambda_3 h^0_c \eta^0_c +
\tan{\gamma} \left(\mu_1^2 - \mu_2^2 + 3 \lambda_1 {h^0_c}^2
-\frac{1}{2}\lambda_3 {h^0_c}^2 - \lambda_4 \eta^0_c \right. \nonumber\\
&&\left. - 3 \lambda_2
{\eta^0_c}^2 + \frac{1}{2} \lambda_3 {\eta^0_c}^2 + \lambda_4 h^0_c
\tan{\gamma} - \lambda_3 h^0_c \eta^0_c \tan{\gamma}\right) = 0.
\label{eq:gamma}\end{aligned}$$
The expressions for the counterterms are thus $$\begin{aligned}
\delta \Omega &=& \frac{C_{\rm UV}}{64 \pi^2}\,
\left(4\,\mu_1^4 + 3\,\mu_2^2\right),\\
\delta \mu_1^2 &=& -\frac{C_{\rm UV}}{32 \pi^2}\,\left(12\,\lambda_1\,\mu_1^2
+ 3 \lambda_3 \, \mu_2^2
+ 6 \, \lambda_4^2 \right),\\
\delta \mu_2^2 &=& -\frac{C_{\rm UV}}{32 \pi^2}\,\left(10\,\lambda_2\,\mu_2^2
+ 4 \, \lambda_3 \mu_1^2
+ 4 \, \lambda_4^2\right),\\
\delta \lambda_1 &=& \frac{C_{\rm UV}}{16 \pi^2}\,\left(\frac{9}{16}\,g^4
- 3\, h_t^4 + 12\,\lambda_1^2
+\frac{3}{4}\,\lambda_3^2 + \frac{3}{8}\,g^2\,{g'}^2+\frac{3}{16}\,{g'}^4
\right),\\
\delta \lambda_2 &=& \frac{C_{\rm UV}}{16 \pi^2}\,\left(6\,g^4 + 11\,
\lambda_2^2 + \lambda_3^2\right),\\
\delta \lambda_3 &=& \frac{C_{\rm UV}}{16 \pi^2}\,
\left(3\,g^4 + 6\,\lambda_1 \, \lambda_3
+5 \, \lambda_2 \, \lambda_3 + 2 \, \lambda_3^2\right),\\
\delta \lambda_4 &=& \frac{C_{\rm UV}}{8 \pi^2}\,\lambda_4 \,
\left(\lambda_1 + \lambda_3 \right),\end{aligned}$$ where $\delta \Omega$ is the counterterm for the vacuum energy.
The fact that the theory should be independent of the unphysical mass $\mu$ implies that the couplings and masses acquire a $\mu$ dependence governed by the Renormalization Group (RG) equation for the one-loop effective potential, i.e. $$\begin{aligned}
&&\left(\beta_{\mu_1} \frac{\partial}{\partial \mu_1^2}+
\beta_{\mu_2} \frac{\partial}{\partial \mu_2^2}+
\beta_{\lambda_1} \frac{\partial}{\partial \lambda_1}+
\beta_{\lambda_2} \frac{\partial}{\partial \lambda_2}+
\beta_{\lambda_3} \frac{\partial}{\partial \lambda_3}+
\beta_{\lambda_4} \frac{\partial}{\partial \lambda_4}\right. \nonumber\\
&&\hspace{3cm}\left.
-\gamma_{h^0} \, h^0_c \, \frac{\partial}{\partial h^0_c}-
\gamma_{\eta^0} \, \eta^0_c \, \frac{\partial}{\partial \eta^0_c}\right)
V_0(h^0_c, \eta^0_c) = - 2 \frac{\partial}{\partial \log{\mu^2}} V_1(h^0_c,
\eta^0_c).\end{aligned}$$ In terms of the tree level masses this equation is equivalent to $$\begin{aligned}
&&\hspace{-1cm}
\left(2\,\beta_{\mu_1} - 4\,\gamma_{h^0}\, \mu_1^2\right) \, {h^0_c}^2 +
\left(2\,\beta_{\mu_2} - 4\,\gamma_{\eta^0}\, \mu_2^2\right) \, {\eta^0_c}^2 +
\left(\beta_{\lambda_1} - 4\,\gamma_{h^0}\, \lambda_1\right) \, {h^0_c}^4
+
\left(\beta_{\lambda_2} - 4\,\gamma_{\eta^0}\, \lambda_2\right)\,{\eta^0_c}^4
\nonumber\\
&&+
\left(\beta_{\lambda_3} - 2\,\left(\gamma_{h^0}+\gamma_{\eta^0}\right)\,
\lambda_3 \right)\,{h^0_c}^2\,{\eta^0_c}^2
- 2 \, \left(\beta_{\lambda_4}-\left(2\,\gamma_{h^0} + \gamma_{\eta^0}\right)
\,\lambda_4 \right)\,{h^0_c}^2\,{\eta^0_c}\nonumber\\
&&=\frac{1}{8 \pi^2} \left(3\,m_Z^4 + 6\, m^4_W
-12\,m_t^4 + m^4_{\phi^0}+ 2\,m^4_{g^\pm}+
2\,m^4_{h^\pm} + m^4_{H^0} + m^4_{N^0} \right),\end{aligned}$$ and, matching powers of fields, we can derive the beta functions: $$\begin{aligned}
\beta_{\mu_1} &=& -\frac{2}{C_{\rm UV}} \delta {\mu_1^2} + 2 \gamma_{h^0}
\mu_1^2,\\
\beta_{\mu_2} &=& -\frac{2}{C_{\rm UV}} \delta {\mu_2^2} + 2 \gamma_{\eta^0}
\mu_2^2,\\
\beta_{\lambda_1} &=& \frac{2}{C_{\rm UV}} \delta {\lambda_1} + 4
\gamma_{h^0} \lambda_1,\\
\beta_{\lambda_2} &=& \frac{2}{C_{\rm UV}} \delta {\lambda_2} + 4
\gamma_{\eta^0} \lambda_2,\\
\beta_{\lambda_3} &=& \frac{2}{C_{\rm UV}} \delta {\lambda_3} + 2
\left(\gamma_{h^0}+\gamma_{\eta^0}\right) \lambda_3,\\
\beta_{\lambda_4} &=& \frac{2}{C_{\rm UV}} \delta {\lambda_4} +
\left(2\,\gamma_{h^0}+\gamma_{\eta^0}\right) \lambda_4.\end{aligned}$$ We can now make use of the anomalous dimensions for the two neutral Higgs fields $$\begin{aligned}
\gamma_{h^0} &=& \frac{1}{16 \pi^2}\,\left(3\,h_t^2
-\frac{9}{4}\,g^2 - \frac{3}{4}\,{g'}^2\right),\\
\gamma_{\eta^0} &=& -\frac{3}{8 \pi^2}\,g^2,\end{aligned}$$ to write down our final expressions for the one-loop beta functions: $$\begin{aligned}
\beta_{\mu_1} &=&
\frac{1}{16 \pi^2}\,\left(6 \, \lambda_4^2 + 12 \, \lambda_1 \mu_1^2
+ 3 \, \lambda_3 \, \mu_2^2\right)
+ \frac{1}{8 \pi^2}\,\left(3\,h_t^2
-\frac{9}{4}\,g^2 - \frac{3}{4}\,{g'}^2\right)\,\mu_1^2,\\
\beta_{\mu_2} &=&
\frac{1}{16 \pi^2}\,\left(4\,\lambda_4^2 + 4\, \lambda_3 \, \mu_1^2 + 10 \,
\lambda_2 \, \mu_2^2 \right)
-\frac{3}{4 \pi^2}\,g^2 \, \mu_2^2 ,\\
\beta_{\lambda_1} &=& \frac{1}{8 \pi^2}\left(\frac{9}{16}\,g^4 - 3\,{{h_t}}^4
+12 \, \lambda_1^2 + \frac{3}{4}\,\lambda_3^2 + \frac{3}{8}\,g^2\,{g'}^2
+ \frac{3}{16}\,{g'}^4\right) \nonumber\\
&+&\frac{1}{4 \pi^2}\,\left(3\,h_t^2
-\frac{9}{4}\,g^2 - \frac{3}{4}\,{g'}^2\right)\,\lambda_1,\\
\beta_{\lambda_2} &=& \frac{1}{8\pi^2}
\left(6\,g^4 + 11\,\lambda_2^2 + \lambda_3^2 \right)
-\frac{3}{2 \pi^2}\,g^2 \, \lambda_2,\\
\beta_{\lambda_3} &=& \frac{1}{8\pi^2}
\left(3 \, g^4 + 6\,\lambda_1 \, \lambda_3 + 5 \, \lambda_2 \, \lambda_3
+ 2 \, \lambda_3^2\right)
+\frac{1}{8 \pi^2}\,\left(3\,h_t^2
-\frac{33}{4}\,g^2 - \frac{3}{4}\,{g'}^2\right)\,\lambda_3,\\
\beta_{\lambda_4}&=& \frac{1}{4\pi^2}\,\lambda_4 \,
\left(\lambda_1 + \lambda_3 \right)+\frac{3}{32 \pi^2}\,\left(4\,h_t^2
- 7\,g^2 - \,{g'}^2\right)\,\lambda_4.\end{aligned}$$ In the gauge and top quark sector the beta functions for the $U(1)$, $SU(3)$ and Yukawa couplings are the same as in the Standard Model, i.e. $$\begin{aligned}
\beta_{g'} &=& \frac{41}{96 \pi^2} \, {g'}^3, \\
\beta_{g_S} &=& - \frac{7}{16 \pi^2} \, {g_S}^3,\\
\beta_{h_t} &=& \frac{1}{16 \pi^2} \,
\left\{\frac{9}{2}\,{h_t^2}-8\,{g_S}^2 -\frac{9}{4}\,g^2-
\frac{17}{12}\,{g'}^2\right\}\,h_t.\end{aligned}$$ The $SU(2)$ coupling is modified due to the extra Higgs triplet in the adjoint representation, i.e. $$\begin{aligned}
\beta_g &=& - \frac{5}{32 \pi^2}\,g^3.\end{aligned}$$ Working with the tree-level effective potential with couplings evolved using the one-loop $\beta$ and $\gamma$ functions we are able to resum the leading logarithms to all orders in the effective potential. It would be possible to include the next-to-leading logarithmic contributions by using the two-loop $\beta$ and $\gamma$ functions and including the one-loop part of the effective potential, see [@Kastening:1991; @Ford:mv; @Bando:1992wz].
Let us now turn to the RG analysis. We first introduce the parameter $t$, related to the scale $\mu$ through $\mu (t) = m_Z \exp{(t)}$. We shall perform evolution starting at $t=0$. The RG equations are coupled differential equations in the set $$\begin{aligned}
\left\{g_s, ~g, ~g',~h_t,~\mu_1,~\mu_2,~\lambda_1,~\lambda_2,~\lambda_3
,~\lambda_4\right\}.\end{aligned}$$ We choose rather to use the following set to define the input to the RG equations: $$\begin{aligned}
\left\{\alpha_s,~m_Z,~\sin^2{\theta_W},~m_t,~m_{h^\pm},~m_{H^0},~m_{N^0},~v,~
\tan{\beta},~\tan{\gamma} \right\}.\end{aligned}$$ Within the accuracy to which we are working, the values of the couplings at $t=0$ can be obtained from the input set using the appropriate tree-level expressions.
The vacuum conditions, $$\begin{aligned}
{h^0_c} \, \mu_1^2 + \lambda_1 \, {h^0_c}^3 - \lambda_4 \, {h^0_c}\,
\eta_c^0 + \frac{1}{2}\,\lambda_3 \, {h^0_c} \, {\eta^0_c}^2 = 0, \\
{\eta^0_c} \, \mu_2^2 + \frac{1}{2}\,\lambda_3 \,{h^0_c}^2\,{\eta^0_c}
-\frac{1}{2}\,\lambda_4\,{h^0_c}^2 + \lambda_2 \, {\eta^0_c}^3 = 0,\end{aligned}$$ allow us to write (defining $h^0_c \equiv v$ and $\eta^0_c ~\equiv~ \frac{v}{2}\,\tan{\beta}$) $$\begin{aligned}
m_Z^2 &=&\frac{1}{4}\,v^2\,\left( g^2 + {g'}^2 \right), \\
m_W^2 &=& \frac{1}{4}\,g^2\,{v}^2\,\left(1+\tan^2{\beta}\right), \\
m_t^2 &=& \frac{1}{2}{h_t}^2\,{v}^2, \\
m_{\phi^0}^2 &=& m_{g^\pm}^2 ~=~ 0,\\
m_{h^\pm}^2 &=& v\,\lambda_4 \,\left(\cot{\beta}+\tan{\beta}\right),\\
m_{H^0}^2 &=& v\,\left\{2\,v\,\lambda_1 +
\left(\lambda_4 - \frac{1}{2}\,v \,\lambda_3\, \tan{\beta}\right)
\,\tan{\gamma}\right\}, \\
m_{N^0}^2 &=& v\, \lambda_4\,\left(\cot{\beta}-\tan{\gamma}\right)
+\frac{1}{2}\,v^2\,\tan{\beta}\left(\lambda_2\,\tan{\beta}+
\lambda_3\, \tan{\gamma}\right), \\
\tan{(2\,\gamma)} &=& \frac{2\,\tan{\beta}\,\left(-2\,\lambda_4 +
v\,\lambda_3\,\tan{\beta}\right)}{2\,\lambda_4 - 4 \, v \, \lambda_1\,
\tan{\beta}+v\,\lambda_2\,\tan^3{\beta}}.\end{aligned}$$ Inverting these relations we can thus fix the $t=0$ boundary conditions for the subsequent evolution: $$\begin{aligned}
g_s &\equiv& \sqrt{4 \pi \alpha_s (m_Z)} ~\simeq~ 1.22, \\
v &\equiv& \frac{1}{2^{1/4}\sqrt{G_{\rm Fermi}}} ~\simeq~ 246 {\rm ~GeV},\\
g' &\equiv& g\,\tan{\theta_W} ~\simeq~ 0.35,\\
g &\equiv& 2\,\frac{m_Z}{v}\,\cos{\theta_W} ~\simeq~ 0.65,\\
h_t &\equiv& \sqrt{2}\,\frac{m_t}{v} ~\simeq~ 1.01,\\
\lambda_1 &=& \frac{1}{2\,v^2}\,\left(m_{H^0}^2\,{\cos^2{\gamma}} +
m_{N^0}^2\,{\sin^2{\gamma}} \right), \label{lam1} \\
\lambda_2 &=& -\frac{1}{v^2}\,\left\{m_{h^\pm}^2 - m_{H^0}^2 -
m_{N^0}^2 + m_{h^\pm}^2\,\cos (2\,\beta )
+ \left( m_{H^0}^2 - m_{N^0}^2 \right)\,\cos (2\,\gamma )
\right\} \,{\cot^2{\beta}},\\
\lambda_3 &=&
\frac{1 }{v^2}\,\cot{\beta}\,\left\{ m_{h^\pm}^2\,\sin(2\,\beta ) +
\left(-m_{H^0}^2 + m_{N^0}^2 \right) \,
\sin (2\,\gamma ) \right\}, \label{lam3} \\
\lambda_4 &=& \frac{1}{v}\,m_{h^\pm}^2\,\cos{\beta}\,\sin{\beta},\\
\mu_1^2 &=& \frac{1}{8} \, \left\{-4\,m_{H^0}^2\,{\cos^2{\gamma}} +
2\,m_{h^\pm}^2\,{\sin^2{\beta}} -
4\,m_{N^0}^2\,{\sin^2{\gamma}} \right. \nonumber\\
&+&\left.\left( m_{H^0}^2 - m_{N^0}^2 \right) \,
\sin (2\,\gamma )\,\tan{\beta}\right\},\\
\mu_2^2 &=&
\frac{1}{4}\,\left\{m_{h^\pm}^2 - m_{H^0}^2 - m_{N^0}^2 +
m_{h^\pm}^2\,\cos (2\,\beta ) \right. \nonumber\\
&+& \left. \left( m_{H^0}^2 - m_{N^0}^2 \right) \,
\left( \cos (2\,\gamma ) + 2\,\cot{\beta}\,\sin (2\,\gamma ) \right)
\right\}. \end{aligned}$$
To ensure that the system remains in a local minimum we impose the condition that the squared masses should remain positive, i.e. $$\begin{aligned}
&&\lambda_4 > 0,\\
&&2\,v\,\lambda_1 +
\left(\lambda_4 - \frac{1}{2}\,v \,\lambda_3\, \tan{\beta}\right)
\,\tan{\gamma} > 0, \\
&&\lambda_4\,\left(\cot{\beta}-\tan{\gamma}\right)
+\frac{1}{2}\,v\,\tan{\beta}\left(\lambda_2\,\tan{\beta}+
\lambda_3\, \tan{\gamma}\right) > 0.\end{aligned}$$ We impose the further requirement that the couplings remain perturbative. In particular we insist that $|\lambda_i (t)| < 4 \pi$ for $i={1,2,3}$ and $|\lambda_4| < 4 \pi v$. We run the evolution from $t = 0$ to $t_{\rm max}= \log{(\Lambda/m_Z)}$, with $\Lambda = 1$ TeV.
Results in the non-decoupling regime
====================================
In this section we present our results of the Higgs mass bounds in the regime where the triplet Higgs cannot be arbitrarily heavy. As we shall see in the next section, decoupling of the triplet occurs when both mixing angles and their sum $(\beta+\gamma)$ tend to zero and in this case, the triplet decouples from the doublet and can be arbitrarily heavy.
We are free to choose the 3 scalar masses and the 2 mixing angles at $t=0$. In Figure \[B004G0\] we show the range of Higgs masses allowed when there is no mixing in the neutral Higgs sector, $\gamma = 0$, for a value of $\beta = 0.04$. Such a value is towards the upper end of the range allowed by the precision data and is interesting because it allows a rather heavy lightest Higgs (e.g. for $\beta = 0.04,\; m_{H^0}>150$ GeV and for $\beta = 0.05, \; m_{H^0}>300$ GeV) [@Forshaw:2001xq]. The strong correlation between the $h^\pm$ and $N^0$ masses arises in order that $\lambda_2$ remain perturbative ($\Delta m \sim \beta^2 v$ for masses $\sim v$). The upper bound on the triplet Higgs masses ($\approx 550$ GeV) comes about from the perturbativity of $\lambda_3$ whilst that on $H^0$ ($\approx 520$ GeV) comes from the perturbativity of $\lambda_1$. These latter two bounds can be estimated crudely by ignoring the evolution of the couplings directly from equations (\[lam1\]) and (\[lam3\]). Evolution tightens the bounds due to the positivity of the beta functions, especially for the $H^0$ since $8 \pi^2 \beta_{\lambda_1} \approx 12 \lambda_1^2$. The hole at low masses is due to vacuum stability.
In Figure \[B004G01\] we show the allowed regions for $\gamma = 0.1$. The correlation of the mainly triplet Higgses is as in Figure \[B004G0\]. For large $m_{H^0}$ ($>450$ GeV), the upper limit on the triplet Higgs mass arises because $\lambda_1$ becomes too large (in this region $\lambda_1
\sim \lambda_3)$. For smaller $m_{H^0}$, $\lambda_1$ is much smaller than $\lambda_3$ and the upper bound comes from the largeness of $\lambda_3$ with the tree-level estimate being $m_{h^{\pm}}^2 < 2 \pi v^2 (\beta/\gamma)$. The upper limit on $m_{H^0}$ is again a consequence of the perturbativity of $\lambda_1$, except at low $h^\pm$ masses, where it is due to the negativity of $\lambda_3$ driving the vacuum unstable. For very low $m_{h^\pm}$, $\lambda_2$ becoming too large is the problem.
In Figure \[B004GPi4\] we show the allowed regions for $\gamma = \pi/4$. In this maximal mixing scenario one loses the distinction between doublet and triplet Higgses and the bounds are correspondingly more democratic. The largeness of $\tan(2 \gamma)$ can be arranged either by tuning $2 v\lambda_1
\approx \lambda_4/\beta$ or by having small enough $\lambda_1$ and $\lambda_4$. In the former case, all masses are approximately degenerate, as can be seen in the plot. In the latter case, which corresponds to light masses, the degeneracy is lifted. The bounds for $\gamma > \pi/4$ are very similar to those for $(\pi/2 - \gamma)$ on interchanging the neutral Higgses $N^0$ and $H^0$.
For $\beta < 0.04$ and small $\gamma$ (but still away from the decoupling regime) the allowed regions are very similar to those for $\beta=0.04$, i.e. as in Figure \[B004G0\]. For larger $\gamma$, the mass bounds are again as for larger $\beta$ but with the correlation between the neutral and charged Higgs masses becoming even stronger than for larger $\beta$.
We should stress that all of the previous discussion is valid for strictly non-zero $\beta$. The situation is quite different for $\beta = 0$. If the neutral mixing is not zero (which is required if we are to avoid decoupling) then the vacuum conditions dictate that $\mu_1^2 = -\lambda_1 v^2$ and $\lambda_4 = 0$ and this renders equation (\[eq:beta\]) redundant. Equation (\[eq:gamma\]) then yields $\mu_2^2 = 2 \lambda_1 v^2 - \frac{1}{2}
\lambda_3 v^2$ and we have complete degeneracy, i.e. $m^2_{H^0} = m^2_{N^0} = m^2_{h^\pm} = 2 \lambda_1 v^2$.
The decoupling limit
====================
So far we have worked in a regime where the triplet does not decouple from the doublet. Clearly for $\beta = \gamma = 0$ there is no mixing between the doublet and triplet and there is no bound on the triplet mass. This is a special case of the more general decoupling scenario, which occurs when $|\beta + \gamma| \ll \beta$, which we now discuss.
For small mixing angles, the (mainly) triplet Higgs has mass squared $\sim \lambda_4 v / \beta$. One possible solution to the mixing angle equations (\[eq:beta\]) and (\[eq:gamma\]) is that $\lambda_4 \sim
\beta v$ and any $\gamma$. In this case the triplet Higgs has mass $\sim v$. This is the regime of the previous section. However, it is also possible to solve the mixing angle equations with $\lambda_4 \sim v$ by keeping $\mu_2^2$ large, i.e. (\[eq:beta\]) gives $\lambda_4 v = \beta \mu_2^2 \sim v^2$. In this case, equation (\[eq:gamma\]) forces $\beta+\gamma \approx 0$. This is the decoupling limit in which the triplet mass lies far above the mass of the doublet and the low energy model looks identical to the Standard Model.
Tree level arguments on the perturbativity of $\lambda_3$ allow us to quantify the approach to decoupling from the point of view of the triplet Higgs mass. In particular (\[lam3\]) dictates that, for small $\beta$ and $\gamma$, $$m_{h^{\pm}}^2 \approx m_{N^0}^2 <
\frac{2 \pi v^2 \beta + \gamma \; m_{H^0}^2}{\beta+\gamma}.$$ By virtue of the smallness of $\beta_{\lambda_3}$ this relation picks up relatively small loop corrections. This bound clearly demonstrates decoupling. It also re-iterates the results of the previous section, i.e. for small $\beta \gg \gamma$ the limit is as in Figure 1 and for small $\beta \ll \gamma$ the limit is as in Figure 2.
We remark that the pseudo-decoupling regime, where $\beta$ is not too small, is of particular interest in that it again allows one to relax the mass bound on the lightest Higgs coming from the precision data without otherwise changing the physics of the Standard Model [@Forshaw:2001xq].
Conclusions
===========
We have computed the one-loop beta functions for the scalar couplings in an extension to the Standard Model which contains an additional real triplet Higgs. Through considerations of perturbativity of the couplings and vacuum stability we have been able to identify the allowed masses of the Higgs bosons in the non-decoupling regime. In the decoupling regime, the model tends to resemble the Standard Model.
We note that the theoretical mass bounds presented here will of course be tightened after considering the precision electroweak and direct search data. Such a study requires that the impact of the quantum corrections (to the $T$ parameter) for non-zero $\gamma$ be computed (they were not explored in [@Forshaw:2001xq]).
As a final remark, we wish to emphasise that the near degeneracy of the triplet Higgs masses (the mass splitting is naturally $\sim \beta^2 v$) ensures that, at least for small $\gamma$, the quantum corrections to the $T$ parameter are negligible (the $S$ parameter vanishing since the triplet has zero hypercharge) [@Forshaw:2001xq]. As shown in [@Forshaw:2001xq], this means that the lightest Higgs boson can be heavy as a result of the compensation arising from the explicit tree-level violation of custodial symmetry which the real triplet induces. Thus it is quite possible to be in a regime where all the Higgs bosons are heavy without any dramatic deviation from the physics of the Standard Model.
[**Acknowledgements**]{}: We would like to thank Ben Allanach, Arthur Hebecker, Apostolos Pilaftsis and Douglas Ross for discussions. ASV acknowledges the support of PPARC (Postdoctoral Fellowship: PPA/P/S/1999/00446).
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M. Bando, T. Kugo, N. Maekawa and H. Nakano, Phys. Lett. B [**301**]{} (1993) 83.
|
---
abstract: |
The effect of the strong intersite Coulomb correlations on the formation of the electron structure of the $t-V$–model has been studied. A qualitatively new result has been obtained which consists in the occurrence of a split-off band of the Fermi states. The spectral intensity of this band increases with the enhancement of a doping level and is determined by the mean-square fluctuation of the occupation numbers. This leads to the qualitative change in the structure of the electron density of states.
PACS numbers
: 71.10.Fd, 71.18.+y, 71.27.+a, 71.28.+d, 71.70.-d
author:
- 'Valery V. Val’kov'
- 'Maxim M. Korovushkin'
title: Hubbard fermions band splitting at the strong intersite Coulomb interaction
---
The key point of the theory of the strongly correlated electron systems is the statement on a principal role of the one-site Coulomb repulsion of two electrons with the opposite spin projections, e.g., the Hubbard correlations [@H63], in the formation of the ground state and the elementary excitation spectrum [@Anderson]. One of the brightest manifestations of the Hubbard correlations is splitting of the initial band of the energy spectrum into two Hubbard subbands when one-site energy of the Coulomb interaction $U$ exceeds the bandwidth $W$.
As was noted previously [@Rogdai1; @Rogdai2], when hole concentration in a system described by the Hubbard model is small and value $U$ is large the intersite Coulomb interaction starts playing an important role due to its relatively weak screening. In this case at distances close to interatomic the order of magnitude of characteristic value $V$ of the Coulomb interaction between electrons can be comparable with that of value $U$. Manifestation of the intersite correlations in the physical properties of systems has been considered in many works (see, for instance, [@Rogdai1; @Rogdai2; @Falicov; @Khomskii; @Matsukawa; @Littlewood; @Fedro; @Li; @Larsen; @Onishi; @Neudert; @Barreteau; @Miyake]). However, until recent time no attention has been paid to the fact that the strong intersite correlations (SIC) can cause splitting energy bands of the Fermi states into subbands [@VVMK].
Qualitatively, the physical origin of this phenomenon is similar to that of the occurrence of the Hubbard subbands due to the strong one-site correlations and is related to the fact that when the intersite Coulomb interactions are taken into account the energy of the electron located on site $f$ becomes dependent of the valence configuration of its nearest neighborhood. For example in the case of a lower Hubbard subband, if there is one electron on each of the $z$ nearest neighboring ions, the “setting” energy of an electron is determined as a sum of one-site energy $\varepsilon$ and energy $zV$ of the interaction. If a hole appears in the nearest neighborhood, the “setting” energy is $\varepsilon+(z-1)V$, i.e., its value is less than the previous one by $V$.
These simple arguments show that when configuration neighborhood deviates from nominal one, one should expect the occurrence of the states with the energies different by $V$ in the Fermi excitation spectrum. For the strongly correlated systems with a relatively narrow energy band of the Fermi states, the situation becomes possible when hopping parameter $t$ is commensurable with or less than $V$. Under these conditions splitting the initial band of the Fermi states is expected. Obviously, the more probable the deviation of the electron configuration of the neighborhood from the nominal one the higher is the spectral intensity of the split-off band. A quantitative measure of this deviation is a mean-square fluctuation of the occupation numbers. For this reason the split-off band was named the band of the fluctuation states (BFS) [@VVMK]. For undoped Mott-Hubbard insulators the electron configuration of the neighborhood corresponds to the nominal one. Upon doping the deviation occurs and in the energy spectrum of the Fermi states the BFS appears whose spectral intensity increases with doping level. The aim of this Letter is to prove the presented qualitative interpretation.
In order to demonstrate the effect as brightly as possible we will consider the Hubbard model in the regime of the strong electron correlations ($U=\infty$) at electron concentrations $n\ll1$. In this case the electron properties of the model are determined by the lower Hubbard subband. Considering the arguments presented in [@Rogdai1; @Rogdai2] we will take into account the Coulomb interaction between the electrons located on the neighboring sites. The obtained system of the Hubbard fermions in the atomic representation will be described by the Hamiltonian of the $t-V$–model: $$\label{Ham} \hat{H}=\sum\limits_{f\sigma}\varepsilon_0
X_f^{\sigma\sigma}+\sum\limits_{fm\sigma}t_{fm}X_f^{\sigma0}X_m^{0\sigma}+\frac
{V}{2}\sum\limits_{f\delta}\hat n_f\hat n_{f+\delta}.$$ Here the first term reflects an ensemble of noninteracting electrons in the Wannier representation. The occurrence a fermion with spin projection $\sigma$ at site $f$ increases the energy of the system by value $\varepsilon_0$, $X^{pq}_f=|f,p\rangle\langle
f,q|$ are the Hubbard operators [@H65; @SGOVV] describing the transition from the one-site state $|q\rangle$ to the state $|p\rangle$. The second term corresponds to the kinetic energy of the Hubbard fermions where matrix element $t_{fm}$ determines the intensity of electron hopping from site $f$ to site $m$. The last term of the Hamiltonian takes into account the Coulomb interaction between the electrons located on neighboring sites $f$ and $f+\delta$ with intensity $V$. The operator of the number of electrons on site $f$ is $\hat n_f=\displaystyle\sum_{\sigma}
X_f^{\sigma\sigma}$.
Below we limit our consideration to the case when the number of holes $h=(1/N)\sum_f\langle X_f^{00}\rangle$ in the system is small; i. e., the condition $h=1-n\ll1$ is satisfied. In this regime it is reasonable to extract in explicit form the mean-field effects caused by the intersite interactions. Using the condition of completeness of the diagonal $X$-operators in the reduced Hilbert space $X^{00}_f+X^{\uparrow\uparrow}_f+X^{\downarrow\downarrow}_f=1$ we express Hamiltonian (\[Ham\]) as $$\begin{aligned}
\label{Cor_VHam}
\hat H &=&-E_0+
\sum\limits_{f\sigma}\bigl(\varepsilon-4Vh\bigr)X_f^{\sigma\sigma}
+\sum\limits_{fm\sigma}t_{fm}X_f^{\sigma0}X_m^{0\sigma}
\nonumber\\
&&+\frac12\sum_{f\delta}V\bigl(X^{00}_f-h\bigr)\bigl(X^{00}_{f+\delta}-h\bigr),\end{aligned}$$ where $E_0=2NV(1-h)^2$ in the mean-field approximation determines the energy of the Coulomb interaction of the system containing $h$ holes per site. At $h=0$ value $E_0$ equals to the exact value of the energy of the ground state of the system (with disregard for $\varepsilon_0$), since in this case hoppings make no contribution. The renormalized value of a one-electron level $\varepsilon=\varepsilon_0+4V$ is determined by the fact that when there is one electron on each neighboring site the excitation energy increases by $4V$. The shift $\varepsilon-4hV$ is related to the decrease in the Coulomb repulsion energy when the average number of holes in the system is nonzero. Note that such mean-field renormalizations of the one-site energies of electrons were used previously, for example, in the Falicov–Kimball model during investigation of the transitions with the change in valence. Extraction of the obvious mean-field effects is needed for representation of the intersite interaction in the form containing the correlation effects only. One can see that the last term of the Hamiltonian (\[Cor\_VHam\]) will contribute only at the presence of noticeable fluctuations of the occupation numbers, i. e., when the SICs are relatively large.
The method used below for the description of the strong intersite interactions (SII) consists in generalization of the Hubbard idea [@H63] for consideration of the intersite interactions [@VVMK]. It follows from the exact equation of motion for operator $X_f^{0\sigma}$ $$\begin{aligned}
\label{eq1}
i\frac{d}{dt}X^{0\sigma}_f&&=(\varepsilon-4Vh)X^{0\sigma}_f-V
\sum_{\delta}X_f^{0\sigma}(X^{00}_{f+\delta}-h)\nonumber\\
&&+\sum_m t_{fm}\bigl((X_f^{00}+X_f^{\sigma\sigma})X_m^{0\sigma}+
X_f^{\bar{\sigma}\sigma}X_m^{0\bar{\sigma}}\bigr),\end{aligned}$$ that at large $V$ one should correctly consider the contributions related to the second term in the right part of equation (\[eq1\]). One of the ways of solving such a problem is to extend the basis set of operators by means of inclusion of the operators $$\label{phi}
\hat{\Phi}_f^{\sigma}=\sum_{\delta}X_f^{0\sigma}(X^{00}_{f+\delta}-h),$$ in which uncoupling is forbidden. This operator describes the correlated process of annihilation of an electron on site $f$, since the result of the action of operator $X_f^{0\sigma}$ depends on the nearest configuration neighborhood of site $f$. Recall for comparison that in the fundamental Hubbard’s work [@H63] the basis was expanded by means of addition of a set of one-site operators $a_{f\sigma}\hat{n}_{f\bar{\sigma}}$.
Inclusion of new basis operators requires the equations of motion: $$\begin{aligned}
\label{eq2}
&&i\frac{d}{dt}\hat \Phi^{\sigma}_f=
(\varepsilon-2Vh-V)\hat\Phi^{\sigma}_f-4h(1-h)VX_f^{0\sigma}
\nonumber\\
&&+\sum_{m\delta} t_{fm}
\Bigl((X_f^{00}+X_f^{\sigma\sigma})X_{m}^{0\sigma}+
X_f^{\bar{\sigma}\sigma}X_{m}^{0\bar{\sigma}}\Bigr)
(X_{f+\delta}^{00}-h)\nonumber\\
&&+\sum_{m\delta\sigma'}t_{f+\delta,m}X^{0\sigma}_f
(X^{\sigma'0}_mX^{0\sigma'}_{f+\delta}-X^{\sigma'0}_{f+\delta}X^{0\sigma'}_m)
\nonumber\\
&&-V\sum_{\delta(\delta\neq\delta_1)}
X_f^{0\sigma}(X_{f+\delta}^{00}-h)(X_{f+\delta_1}^{00}-h).\end{aligned}$$ One can see that among the operators containing large parameter $V$ a three-site operator has occurred, which reflects the correlation effects related to the presence of two holes in the first coordination sphere of site $f$. If hole concentration in the system is low, then the contribution of this term can be ignored. Calculation of the spectrum with regard of this three-center operator confirmed validity this approximation.
After the basis set of operators has been specified, the set of equations of motion is closed using the Zwanzig-Mori projection technique [@Zwanzig; @Mori]. In the main approximation one can neglect the kinetic correlators occurring after projection and limit the consideration to spatially homogeneous solutions. Then, for the Fourier transforms of the Green functions we obtain the closed system of two equations: $$\begin{aligned}
\label{system}
(\omega-\varepsilon_{\bf k})\langle\langle X_{{\bf
k}\sigma}|X_{{\bf k}\sigma}^\dagger\rangle\rangle &=&
\frac{1+h}{2}-\gamma_{\bf k}\langle\langle \Phi_{{\bf
k}\sigma}|X_{{\bf
k}\sigma}^\dagger\rangle\rangle,\\
(\omega-\xi_{\bf k})\langle\langle \Phi_{{\bf k}\sigma}|X_{{\bf
k}\sigma}^\dagger\rangle\rangle &=& -4h(1-h)\gamma_{\bf
k}\langle\langle X_{{\bf k}\sigma}|X_{{\bf
k}\sigma}^\dagger\rangle\rangle.\nonumber\end{aligned}$$ Here we made the following notation: $$\begin{aligned}
&&\varepsilon_{{\bf
k}}=(\varepsilon-4Vh)+\biggl(\frac{1+h}{2}\biggr)t_{\bf k},
\quad\gamma_{\bf k}=V-\frac{t_{1{\bf k}}}{8},\nonumber\\
&&\xi_{\bf k}=(\varepsilon-4Vh)-V(1-2h)+\biggl(\frac{2+h}{8}\biggr)t_{1{\bf k}},\nonumber\\
&&t_{\bf k}=t_{1{\bf k}}+4t'cos\,k_xcos\,k_y+
2t''(cos\,2k_x+cos\,2k_y),\nonumber\\
&&t_{1{\bf k}}=2t(cos\,k_x+cos\,k_y).\end{aligned}$$ The use of the spectral theorem allows obtaining the equation which determines the dependence of the chemical potential on doping: $$\begin{aligned}
\label{selfcons}
\frac{4h}{1+h}&=&\frac{1}{N} \sum_{\bf k\rm}
\left(1+\frac{\varepsilon_{\bf k}-\xi_{\bf k}}{2\nu_{\bf
k}}\right) \left(1-\emph{f}\;(E^+_{\bf k})\right),\end{aligned}$$ where $f(x)=(\exp(\frac{x-\mu}{T})+1)^{-1}$ is the Fermi-Dirac function, $\mu$ is the chemical potential of the system, and the two-band Fermi excitation spectrum is determined as $$\begin{aligned}
\label{spectrum}
&&E^{\mp}_{\bf k}=\frac{\varepsilon_{\bf k}+\xi_{\bf k}}{2}\mp
\nu_{\bf k},\\
&&\nu_{\bf k}= \sqrt{\biggl(\frac{\varepsilon_{\bf k}-\xi_{\bf
k}}{2}\biggr)^2+ 4h(1-h)\gamma^2_{\bf k}}.\nonumber\end{aligned}$$
Fig.\[bands\] (on the right) demonstrates a band picture of the energy spectrum of the $t-V$–model obtained by solving equations (\[system\]). The values of hopping parameters were chosen so that evolution of the Fermi contour upon hole doping would qualitatively correspond to that observed experimentally. The left part of the figure shows the energy spectrum of the same model calculated with disregard of the SICs. Note that here the mean-field contribution of the SIIs is taken into account by the above-mentioned renormalizations. Comparison of the two presented spectra shows that the correct account for the SICs yields the qualitative difference, specifically, the occurrence of an additional band (BFS) in the band structure of the $t-V$–model.
It is seen from (\[spectrum\]) that the resulting energy spectrum forms by hybridization of states from an ordinary Hubbard band with energies $\varepsilon_{\bf k}$ and the states induced by the fluctuations of configuration neighborhood with energies $\xi_{\bf k}$. Note that unlike $\varepsilon_{\bf k}$ the dependence of function $\xi_{\bf k}$ on the quasimomentum is determined only by the integral of hopping between the nearest neighbors $t_{1{\bf k}}$. At $V\gg|t_{\bf k}|$ the values of this function form an energy band located lower by $V$. The intensity of hybridization is determined by the value proportional to the one-site mean-square fluctuation of the occupation numbers $$\overline{(\Delta
n)^2}=\langle(X_f^{00}-h)(X_f^{00}-h)\rangle=h(1-h).$$ As a result the spectral intensity of the BFS in the regime $V\gg|t_{\bf k}|$ acquires the simple form: $$A^-({\bf k},\omega)\simeq \frac{4\left(1+h\right)\overline{(\Delta
n)^2}}{1+16\overline{(\Delta n)^2}+\sqrt{1+16\overline{(\Delta
n)^2}}}\delta(\omega-E^{-}_{\bf k}).$$ It follows from this formula that with an increase in $h$ the spectral weight of the BFS rapidly grows and at $h=0.2$ the relative contribution of the BFS reaches 30%.
![Band picture of the $t-V$–model without account for the SICs (left) and with account for the SICs (right) for the set of parameters $t'=-0.1,\, t''=-0.5,\, V=2.5$ (in terms of $|t|$) and holes concentration $h=0.15$. Dashed lines show the chemical potential.[]{data-label="bands"}](Bands.eps){width="41.00000%"}
![Density of states of the $t-V$–model calculated for $h=0.25$ without account for the SICs $V=1$ (top), with account for the SICs at $V=1$ (middle) an at $V=2.5$ (bottom). Here $t'=-0.1,\, t''=-0.5$ (all in terms of $|t|$). Dashed lines show the chemical potential.[]{data-label="DOS"}](DOS.eps){width="31.00000%"}
{width="53.00000%"}
The occurrence of the BFS qualitatively changes the electron density of states of the Hubbard fermions. The upper graph in Fig.\[DOS\] shows the density of states of the $t-V$–model with disregard of the SICs. It is seen that ignoring these correlations leads to trivial displacement of the band position.
The energy dependence of the sum electron density of states of the $t-V$–model for the same set of parameters, but with regard of the SICs, is shown by a bold solid line in the middle graph of Fig.\[DOS\]. Comparison with the upper graph evidences that the account for the SICs yields noticeable qualitative changes in the density of states: the latter acquires a three-peak structure instead of the two-peak one. The occurrence of the additional peak corresponds to the formation of the BFS. For clarity, BFS density $g^-(E)$ is reflected in the graph by the line which bounds the shaded area. The BFS fraction is 36% of the whole number of states of the system, whereas the fraction of the ground band with density of states $g^+(E)$ is 64%. Summation of densities $g^+(E)$ and $g^-(E)$ gives total density $g(E)$ with the three peaks.
More substantial changes in the energy structure of the $t-V$–model take place at high values of the intersite Coulomb interaction. It is seen from the lower graph in Fig.\[DOS\] which presents the same dependences as in the two upper graphs but calculated for $V=2.5$ (the rest parameters are not changed). In this case, the SICs lead to splitting the BFS off and the formation of a gap in the energy spectrum. The graphs presented in Fig.\[DOS\] are calculated at $h=0.25$. In the case of $h=0$ the SIC contribution becomes zero and the density of states of the system will be such as shown in the upper graph in Fig.\[DOS\]. This suggests the presence of a qualitatively new effect related to the account for the SIC: upon doping not only the chemical potential shifts but the density of states rearranges.
The growth of the BFS spectral intensity with an increase in doping level leads to renormalization of the dependence of the chemical potential on hole concentration in the system. As a result in the area of optimal doping $h\simeq0.2$ the square bounded by the Fermi contour noticeably grows. The Fermi contour calculated at $h=0.25$ with disregard of the SICs is shown by dotted lines in Fig.\[FS\]. If the correlations are taken into account then the Fermi contour noticeably grows and acquires the form shown by solid lines. In this case the change in the square bounded by the Fermi contour reaches $16\%$ (right graph). This increase is important for description of the Lifshitz quantum phase transitions occurring upon doping [@SGO] and for interpretation of the experimental data on measuring magnetic oscillations in the de Haas–van Alphen effect. Recently, such measurements have been performed by many researchers due to substantial improvement of quality of materials and novel techniques with the use of the strong magnetic fields.
Note in summary that the hybridization character of the energy spectrum with regard of the SICs is related to the fact that electron hoppings between the nearest neighbors lead to the transitions between the one-site energy levels which differ by $V$. Therefore, the intensity of such hybridization processes is proportional to both hopping parameter $t$ and the above-mentioned mean-square fluctuation of the occupation numbers. Note also that the considered modification of the energy structure due to the SICs is general and not limited merely by the $t-V$–model. One should expect that the effects discovered in this study will be especially important for the systems with variable valence.
This study was supported by the program “Quantum physics of condensed matter” of the Presidium of the Russian Academy of Sciences (RAS); the Russian Foundation for Basic Research (project No. 07-02-00226); the Siberian Division of RAS (Interdisciplinary Integration project No. 53). One of authors (M.K.) would like to acknowledge the support of the Dynasty Foundation.
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abstract: 'For $p\in(0,1),$ let $Q_p$ spaces be the space of all analytic functions on the unit disk $\mathbb{D}$ such that $\vert f''(z) \vert ^2 (1-\vert z\vert ^2)^p dA(z)$ is a $p$ - Carleson measure. In this paper, we prove that the Wolff’s Ideal Theorem on $H^\infty{(\mathbb{D})}$ can be extended to the Banach algebra $H^{\infty}(\mathbb{D})\cap Q_{p}$, and also to the multiplier algebra on $Q_p$ spaces.'
address: |
Department of Mathematics and Statistics\
Coastal Carolina University\
Box 261954\
Conway, SC 29528-6054\
(843)349-6569
author:
- 'Debendra P. Banjade'
title: 'Wolff’s Ideal Theorem on $Q_p$ Spaces'
---
Introduction
============
For $p\geq{0}$, let $Q_p$ be the space of all analytic functions on the unit disk $\mathbb{D}$ with the norm\
$$\left\vert\left\vert f \right\vert\right\vert_{Q_{p}}^2:=\int _{-\pi}^\pi \vert f\vert^2 d\sigma+\underset{a\in \mathbb{D}}{\sup}\int_\mathbb{D}\vert f'(z)\vert^2(1-\vert \varphi_{a}(z)\vert^2)^p dA(z)\ < \infty,$$ where $\varphi_{a}(z)=\frac{a-z}{1-\bar{a}z}$ is a M$\ddot{o}$bius map. It is well known that $Q_{0}=\mathcal{D}$, is the classical Dirichlet space and $Q_{1}=BMOA=BMO( \mathbb{T})\cap H^{2}(\mathbb{D}).$\
The case we are interested in is $p\in (0,1)$.\
Let $\mathcal{M}(Q_{p})$ be the multiplier algebra of $Q_p$ spaces which we define, $$\mathcal{M}(Q_{p}):=\left\{ \phi\in Q_{p} :\, M_{\phi}(f)=\phi f\in Q_p\; \text{ for all } f\in Q_{p}\right\}.$$ We know from\[X1\] that $\mathcal{M}(Q_{p})\subseteq H^{\infty}(\mathbb{D}) \cap Q_p$.\
In 1962, Carleson \[C\] proved his famous Corona theorem" characterizing when a finitely generated ideal in $H^{\infty}(\mathbb{D})$ is actually all of $H^{\infty}(\mathbb{D})$. Independently, Rosenblum \[R\], Tolokonnikov \[To\], and Uchiyama gave an infinite version of Carleson’s work on $H^{\infty}(\mathbb{D})$. In 1997, Nicolau and Xiao \[NX\] proved that the corona theorem holds for the Banach algebra $H^{\infty}(\mathbb{D})\cap Q_{p}$ and later Xiao \[X1\] gave a necessary and sufficient condition for the solvability of the corona theorem on $Q_p$ spaces whereas a similar result on $\mathcal{M}(Q_p)$ was established by Pau \[P\].\
In light of the corona theorem it is natural to ask whether the corona kind of result still holds if we replace the uniform lower bound by any $H^\infty(\mathbb{D})$ function. Namely, let $f_1, f_2, ...,f_n$ be $H^\infty$ functions, and suppose $g\in\text{H}^\infty$ satisfies
$$\vert g(z)\vert\leq \vert f_1(z)\vert+...\vert f_n(z)\vert \;\; \text{ for all } \; z\in \mathbb{D}.$$
\
Then the question is whether (1) always implies $g\in\mathcal{I}(f_1,...f_n)$, the ideal generated by $\{f_{j}\}_{j=1}^n$. Unfortunately, the answer is no (see \[G, p. 369\] for an example given by Rao).\
However, T. Wolff in \[G\] has proved the following version to show that (1) implies $g^3\in\mathcal{I}(f_1,...f_n).$
If $$\begin{aligned}
\{f_j\}_{j=1}^n \subset H^{\infty}(\mathbb{D}), g \in H^{\infty}(\mathbb{D}) \quad \text{and}\notag\\
|g(z)| \le \left( \sum_{j=1}^n \, |f_j(z)|^2\right)^{\frac 1{2}} \;\; \text{for all } \, z \in \mathbb{D},\end{aligned}$$ then $$g^3 \in \mathcal{I} ( \{ f_j\}_{j=1}^n),$$
the ideal generated by $\{f_j\}_{j=1}^n$ in $H^{\infty}(\mathbb{D})$.
It is also known that (1) is not sufficient for $g^2$ to be in $\mathcal{I} ( \{ f_j\}_{j=1}^n)$ (see Treil \[T1\]).
It was proved by authors in \[BT1\] and \[BT2\] that Wolff’s theorem can be extended to the multiplier algebras of Dirichlet and weighted Dirichlet spaces. This paper is devoted to the extension of Wolff’s theorem to the Banach algebra $H^{\infty}(\mathbb{D})\cap Q_{p}$ and also to the multiplier algebra of $Q_p$ spaces. In Dirichlet spaces, using complete Nevanlinna pick kernels, the authors used Hilbert space version directly and then applied abstract operator theory result to establish the theorem. But, for $Q_p$ spaces, we are unable to use those Hilbert space techniques because these are only Banach spaces for $p\in(0,1)$. To overcome this difficulty, we will apply $\bar{\partial}$ - method and some $p$-Carleson measures for $Q_p$ spaces.\
Let $g$, $\{f_j\}_{j=1}^n \subset H^{\infty}(\mathbb{D}) \cap Q_p$. Assume that $$|g(z)| \le \sqrt{\sum_{j=1}^{n}\vert f_j(z)\vert ^2} \;\; \; \text{for all} \; \; {\text{z}}\in\mathbb{D}.$$ Then there exist $\{g_j\}_{j=1}^{n} \subset H^{\infty}(\mathbb{D}) \cap Q_p$ such that $$\sum_{j=1}^{n}{f_j(z)g_j(z)}=g^3(z) \;\;\; \text{for all} \;\; z\in \mathbb{D}.$$
If $p\geq 1$, the Banach algebra $H^{\infty}(\mathbb{D}) \cap Q_p$ is just $H^{\infty}(\mathbb{D})$. Then Theorem 1 definitely holds true, which was the result proved by Wolff in Theorem A.\
Before stating our next theorem, it’s worthwhile to note that for $0<p<1$, a Blaschke product B is in $\mathcal{M}({Q_p})$ if and only if it is a finite Blaschke product (see \[P\]). It simply suggests us that the same Rao’s example serves for the counter example in $\mathcal{M}({Q_p})$ as in $H^\infty(\mathbb{D})$. So (1) is not sufficient for $g\in\mathcal{I}(f_1 ,...f_n)$ in $\mathcal{M}(Q_p)$.
Let $g$, $\{f_j\}_{j=1}^n \subset \mathcal{M}(Q_p)$. Assume that $$|g(z)| \le \sqrt{\sum_{j=1}^{n}\vert f_j(z)\vert ^2} \;\; \; \text{for all} \; \; {\text{z}}\in\mathbb{D}.$$
Then there exist $\{g_j\}_{j=1}^{n} \subset \mathcal{M}(Q_p)$ such that $$\sum_{j=1}^{n}{f_j(z)g_j(z)}=g^3(z) \;\;\; \text{for all} \;\; z\in \mathbb{D}.$$
We remark that the cases $p=0$ and $\varphi_{a}(z)=z$ of Theorem 2 were proved in \[BT1\] and \[BT2\] for infinite number of generators.\
The paper is organized as follows: In section 2, we collect some results of $Q_p$ and $Q_p (\mathbb{T})$. We prove our Theorems 1 and 2 in section 3 and we point out some interesting open problems in section 4 . We use the notation $A \lesssim B$ to indicate that there is a constant $c>0$ with $A\leq cB$ and the notation $A \approx B$ to indicate $A\lesssim B$ and $B \lesssim A$.
Basic properties of $Q_p$ spaces and $Q_p (\mathbb{T})$
=======================================================
The p-Carleson measures
-----------------------
Given an arc $\mathit{I}$ of the unit circle $\mathbb{T}$ with normalized length $\vert I \vert \leq 1$, let $$S(\mathit{I}):=\left\{ r{e}^{it}\in\mathbb{D}:1-\vert \mathit{I}\vert<r<1, e^{it}\in \mathit{I}\right\}.$$ $S(\mathit{I})$ is called the Carleson square or a sector based on $\mathit{I}.$ For $0<p<\infty$, we say that a positive Borel measure $\mu$ on $\mathbb{D}$ is a $p$-Carleson measure on $\mathbb{D}$ if $$\vert\vert \mu \vert \vert_{p}=\sup_{\mathit{I}\subset{\mathbb{T}}} \frac{\mu(S(\mathit{I})}{\vert \mathit{I}\vert^p}<\infty,$$ where the supremum is taken over all subarcs $\mathit{I}$ of $\mathbb{T}$.\
Equivalently, $\mu$ is a $p$-Carleson measure if and only if there is a constant $C>0$ such that $\mu(S(\mathit{I}))\leq C\vert \mathit{I}\vert^p$ for any subarcs $\mathit{I}$ of $\mathbb{T}$. Also, p-Carleson measures can be described in terms of conformal invariants of those positive measures $\mu$ for which $$\sup_{a\in\mathbb{D}} \int_{\mathbb{D}}\left(\frac{1-\vert a\vert^2}{\vert1-{\bar{a}}z\vert^2}\right)^p d\mu(z)<\infty,$$ and this quantity is equivalent to $\vert\vert\mu\vert\vert_p$ (see \[X2\]).
Let, $0<p\leq 1$. An analytic function f is in $Q_p$ if and only if the measure $\vert f'(z)\vert ^2 \left(1-\vert z\vert^2\right)^p dA(z)$ is a $p$-Carleson measure.
Let $f\in{Q_p}$, then by definition of $Q_p$ norm, we have that $$\underset{a\in \mathbb{D}}{sup}\int_\mathbb{D}\vert f'(z)\vert^2(1-\vert \varphi_{a}(z)\vert^2)^p dA(z)\ < \infty.$$ This implies that $$\begin{aligned}
&\underset{a\in \mathbb{D}}{\sup}\int_\mathbb{D}\vert f'(z)\vert^2 \frac{\left(1-\vert z\vert^2\right)^p \left(1-\vert a \vert^2\right)^p}{\vert 1-\bar{a}z\vert^{2p}} dA(z) \\
&=\underset{a\in \mathbb{D}}{\sup}\int_\mathbb{D} \left(\frac{ 1-\vert a \vert^2}{\vert 1-\bar{a}z\vert^{2}}\right)^p \vert f'(z)\vert^2 \left(1-\vert z\vert^2\right)^p dA(z)\ < \infty.\end{aligned}$$
Hence, by definition of $p$-Carleson measure, $\vert f'(z)\vert ^2 \left(1-\vert z\vert^2\right)^p dA(z)$ is an $p$-Carleson measure. Converse can be easily obtained just by reversing the above argument.
We need the following series of lemmas whose proofs are excluded here. One can refer to \[P\], \[NX\], \[X1\] and \[X2\] for complete proofs.
Let $z\in {\mathbb{D}}, t<-1$ and $c>0$. Then $$\int_\mathbb{D}\frac{(1-\vert w\vert ^2)^{t}}{\vert 1-\bar{w}z\vert^{2+t+c}} \approx (1-\vert z\vert ^2)^{-c}$$
Let $0<p<1$. If $\vert g(z)\vert^2 \left(1-\vert z\vert^2\right)^p dA(z)$ is a $p$-carleson measure, then $\vert g(z)\vert dA(z)$ is a Carleson measure.
One can obtain this result using Cauchy-Schwarz inequalty and Lemma 2 (see, for example, \[P, Lemma 2.2\]).
We also need the next result, which can be found in \[P\] ( see also Theorem 7.4.2 of \[X2\]).
Let $p>0$. Then $g\in\mathcal{M}(Q_p)$ if and only if $g\in H^{\infty}(\mathbb{D})$, and for all $f\in {Q_p}$, the measure $\vert f(z)\vert^2 \vert g'(z)\vert ^2 \left(1-\vert z\vert^2\right)^p dA(z)$ is a $p$-Carleson measure.
Boundary Values of $Q_p$ Spaces
-------------------------------
Let $0<p<1$. A function $f\in L^2(\mathbb {T})$ is said to be in $Q_p(\mathbb{T})$ if $$\vert \vert f\vert\vert _{Q_p (\mathbb{T})}^2:=\int_{-\pi}^\pi \vert f \vert ^2 d\sigma + \sup_{I\subset\mathbb{T}}\frac{1}{\vert I\vert^p} \int_I \int_I \frac{\vert f(\xi)-f(\eta)\vert ^2}{\vert \xi -\eta \vert ^{2-p}} \vert d\xi\vert \vert d\eta \vert <\infty,$$ where the supremum is taken over all arcs $I\subset\mathbb{T}.$\
Since $Q_p \subset H^2$, where $H^2$ is the classical Hardy space, any function $f\in Q_p$ has a non-tangential radial limit almost everywhere on $\mathbb{T}$. It is also true that, for $p\in (0,1), Q_{p}=Q_{p}(\mathbb{T})\cap{H}^{2}(\mathbb{D})$. The following Lemma from \[X2\] proves that an analytic function $f$ on $\mathbb{D}$ is in $Q_p$ if and only if its boundary values lie on $Q_p(\mathbb{T}).$
Let $p\in (0,1)$ and let $f\in H^2$. Then $f\in Q_p$ if and only if $$\Vert f\Vert_{Q_p(\mathbb{T})}^2=\int_{-\pi}^\pi \vert f \vert ^2 d\sigma +\sup_{I\subset T} \vert I\vert^{-p} \int_I \int_I \frac{\vert f(\zeta)-f(\eta)\vert^2}{\vert \zeta -\eta \vert ^{2-p}} \vert d\zeta \vert \vert d \eta \vert <\infty.$$
We can see in the proof that $$\Vert f\Vert_{Q_p}\approx \Vert f\Vert _{Q_p(\mathbb{T})}.$$ The next Lemma, proved in \[NX\] (see also \[X2, Corollary 7.1.1\]), is also an important tool for us to check that a function belongs to $Q_p(\mathbb{T}).$
Let $\; 0<p<1$ and $f\in L^2(\mathbb{T})$, and let $ F\in C^{1}(\mathbb{D})$ such that $\lim_{r\rightarrow 1^{-}} F(re^{it})=f(e^{it})$ for a.e. $e^{it}\in \mathbb{T}$. If $\vert\nabla F(z)\vert ^2 \left(1-\vert z\vert ^2\right)^p dA(z)$ is a $p$-Carleson measure, then $f\in Q_p(\mathbb{T}).$
With the help of this lemma, we can easily see that a function $f\in L^2(\mathbb{T})$ belongs to $Q_p(\mathbb{T})$ if and only if $\vert\nabla \tilde{f}(z)\vert ^2 \left(1-\vert z\vert ^2\right)^p dA(z)$ is a $p$-Carleson measure, where $\tilde{f}$ denotes the Poisson integral of $f$(for example, see \[NX\]).
Let $\mathcal{M}(Q_{p}(\mathbb{T}))$ be the space of multipliers on $Q_p(\mathbb{T}),$ that is $$\mathcal{M}(Q_{p}(\mathbb{T})):=\left\{ \phi\in Q_{p}(\mathbb{T}) :\, M_{\phi}(f)=\phi f\in Q_{p}(\mathbb{T})\; \text{ for all } f\in Q_{p}(\mathbb{T})\right\}.$$ As in $Q_p$, it’s clear that $\mathcal{M}(Q_{p}(\mathbb{T}))\subseteq L^\infty \cap Q_p(\mathbb{T})$. One of the important parts of the proofs of our Theorems 1 and 2 is establishing the solvability of the $\bar{\partial}$ - equation, which turns out to be simpler because of the following Lemma.
Let $p\in (0,1).$ If $d\lambda(z)=\vert g(z)\vert ^2\left(1-\vert z\vert ^2\right)^{p}dA(z)$ is a $p$-Carleson measure, then $\bar{\partial}u=g$ has a solution $v\in Q_{p}(\mathbb{T})\cap L^{\infty}(\mathbb{T})$ such that $$\begin{aligned}
(i) &\;\; \vert\vert v \vert \vert_{Q_p(\mathbb{T})}+\vert \vert v \vert \vert _{L^{\infty}(\mathbb{T})}\leq C \vert \vert \lambda \vert \vert _{p}^{\frac{1}{2}}, \; \text {where C is an absolute constant}.\\
(ii) & \;\; vf\in Q_p(\mathbb{T})\;\; \text{for all}\;\; f\in Q_p.\end{aligned}$$
Reader can find the exact proof of (i) in \[NX\]. The solution $v$ taken there was $$\begin{aligned}
v(z)=&\frac{i}{\pi} \int_D \frac{1-\vert \zeta \vert ^2}{\vert 1-\bar{\zeta} z\vert ^2}.\\
& \exp \left(\int_{ \vert w\vert \geq \vert \zeta\vert} \left( \frac{1+\bar{w} \zeta}{1-\bar{w} \zeta}-\frac{1+\bar{w} z}{1-\bar{w}z}\right)\vert g(w)\vert dA(w)\right) \vert g(\zeta)\vert dA(\zeta),\end{aligned}$$ that has the same boundary values of $zu(z)$.\
For (ii), we need to show that $vf\in Q_p(\mathbb{T})\;\; \text{for all}\;\; f\in Q_p$. To prove that $vf\in Q_p(\mathbb{T})$, by Lemma 6, it’s enough to show that $$\vert \nabla (vf)(z)\vert ^2 (1-\vert z\vert ^2)^p dA(z)$$ is a $p$- Carleson measure.\
Since $f\in Q_p$ and $v\in L^{\infty}(\mathbb{D})$, we have that $$\vert v(z)\vert^2 \vert \nabla f(z)\vert ^2 (1-\vert z\vert ^2)^p dA(z)$$ is a $p$- Carleson measure. So it remains to show that $$\vert f(z)\vert^2 \vert \nabla v(z)\vert ^2 (1-\vert z\vert ^2)^p dA(z)$$ is a $p$-Carleson measure. Since $\vert g(z)\vert ^2\left(1-\vert z\vert ^2\right)^{p}dA(z)$ is a $p$-Carleson measure, we can see in the proof of Theorem 1 in \[P\] that $$\vert f(z)\vert^2 \vert \nabla v(z)\vert ^2 (1-\vert z\vert ^2)^p dA(z)$$ is a $p$- Carleson measure.\
This completes the proof of Lemma 7.
A similar result for $1$-Carleson measure was proved in \[G, P. 320-322\].\
Proof of Theorems
==================
First, by using the normal families, we will assume that the given family of functions are analytic in some neighborhood of $\overline{\mathbb{D}}$ and then reduce our theorems to the problem of solving certain inhomogoneous Cauchy-Riemann equations. Doing that will allow us to find smooth solutions for both (4) and (6). Then we will convert our obtained smooth solutions into $H^{\infty}(\mathbb{D})\cap Q_p$ and $\mathcal{M}(Q_p)$ - solutions, using some correction functions and applying the size conditions (3) and (5), respectively.
Proof of Theorem 1
------------------
Let, $ f_1, ..., f_n, g\in H^{\infty}(\mathbb{D})\cap Q_p$ such that they satisfy (3). Also, suppose that $g, f_1,...f_n $ are analytic on $\overline{\mathbb{D}}$. Moreover, we assume $\vert \vert f_{j}\vert \vert \leq 1, \vert \vert g\vert \vert \leq 1.$
Set: $$\psi_{j}=\frac{g\; \overline{f_j}}{\sum_{l=1}^n |f_l|^2}, \;\; j=1,2,...,n.$$
Then, using (3), $\vert \psi_{j}\vert \leq 1$, and $C^{\infty}$ on $\overline{\mathbb{D}}$ and $$\psi_{1}f_1+...+\psi_{n}f_n=g.$$ Suppose we can solve $$\frac{\partial{b_{j,k}}}{\partial { \overline{z}}}=g\psi_{j}\frac{\partial{\psi_{k}}}{\partial { \overline{z}}}=g^{3}G_{j,k}(z),\;\; 1\leq j ,k \leq n,$$ with $$b_{j,k} \in L^{\infty}(\mathbb{T}) \cap Q_{p} (\mathbb{T}) .$$ The difficulty, of course, is that $\psi_j(z)$ may not be analytic on $\mathbb{D}$. To rectify that, we write
$$g_j(z)=g^{2}(z)\psi_j(z)+\sum_{k=1}^n\left( b_{j,k}(z)-b_{k,j}(z) \right)f_k(z),$$ then we get $$\sum_{j=1}^{n}g_{j}f_{j}=g^{2}\sum_{j=1}^{n} \psi_{j}f_{j}=g^3$$ and also
$$\frac{\partial g_j}{\partial \bar{z}}=0.$$ Provided the solution of (7) satisfying (8), the functions $g_{j}$’s are bounded analytic solutions for (4). Now, we will try to show that $g_{j}\in Q_p$.
Since, $\frac{\partial \overline{f_l}}{\partial \bar{z}}=\overline{f'_l},$
$$\frac{\partial \psi_{j}}{\partial \bar{z}} =\frac{g\bar{f'_j}}{\sum_{l=1}^n \, |f_l|^2}-\frac{g\bar{f_{j}}\sum_{l=1}^{n}{f_{l}\bar{f'_{l}}}}{\left(\sum_{l=1}^n \,\vert{ f_l}\vert^2 \right)^2}
=\frac{g \sum_{l=1}^n f_{l}\left(\bar{f_l}\bar{f'_j}-\bar{f_k}\bar{f'_l}\right)}{\left(\sum_{l=1}^n \,\vert{ f_l}\vert^2 \right)^2}$$ Thus, $$\begin{aligned}
\left\vert \frac{\partial \psi_{j}}{\partial \bar{z}} \right\vert^2 \lesssim \frac { 2 \vert g\vert^2 \left(\sum \vert f_l \vert ^2\right)^2 \sum \vert f'_l \vert ^2}{\left(\sum_{l=1}^n \,\vert{ f_l}\vert^2 \right)^4}=&\frac { 2 \vert g\vert^2 \sum \vert f'_l \vert ^2}{\left(\sum_{l=1}^n \,\vert{ f_l}\vert^2 \right)^2}\\
& \leq \frac{2\sum \vert f'_l \vert ^2}{\sum_{l=1}^n \,\vert{ f_l}\vert^2} \;\; \left(\texttt{using (3)}\right).\end{aligned}$$
Similarly, $$\frac{\partial \psi_{j}}{\partial{z}}=\frac{g' \bar{f_j}}{\sum_{l=1}^n \, |f_l|^2}-\frac{g\bar{f_{j}}\sum_{l=1}^{n}{f'_{l}\bar{f_{l}}}}{\left(\sum_{l=1}^n \,\vert{ f_l}\vert^2 \right)^2}$$ $\therefore$ $$\begin{aligned}
\left\vert \frac{\partial \psi_{j}}{\partial{z}}\right\vert^2& \lesssim \frac{\vert g'\vert^{2}\sum \vert f_{l} \vert^2}{\left(\sum_{l=1}^n \, \vert f_l\vert\right)^2 }+\frac{\vert g\vert^2 \left(\sum \vert f_l\vert^2 \right)^2 \sum \vert f'_l \vert ^2}{\left(\sum_{l=1}^n \,\vert{ f_l}\vert^2 \right)^4}\\
&\lesssim \frac{\vert g'\vert^2 }{\sum_{l=1}^n \, |f_l|^2}+\frac{\sum \vert f'_l \vert ^2}{\sum_{l=1}^n \,\vert{ f_l}\vert^2 }\end{aligned}$$ Hence, $$\begin{aligned}
\left \vert \nabla \psi_j \right\vert^2&=2\left\vert \frac{\partial \psi_{j}}{\partial{z}}\right\vert^2+2\left\vert \frac{\partial \psi_{j}}{\partial \bar{z}}\right\vert^2\\
&\lesssim \frac{\vert g'\vert^2 }{\sum_{l=1}^n \, |f_l|^2}+\frac{\sum \vert f'_l \vert ^2}{\sum_{l=1}^n \,\vert{ f_l}\vert^2}\end{aligned}$$ Applying the size condition (3), we get that $$\left \vert g^2 \;\; \nabla \psi_{j} \right\vert^2 \lesssim \vert g'\vert^2 \, +\sum {\vert f'_l \vert ^2 }$$ Also, since $\psi_{j}$’s are $C^{\infty}$ on $\overline{\mathbb{D}}$, by Lemma 6, to see that $g^{2}\psi_{j} \in Q_{p}(\mathbb{T})$, it is enough to show that $\left\vert \nabla (g^{2} \psi_{j} )(z)\right\vert^2 \left(1-\vert z \vert ^{2}\right)^{p}$ is a $p$ - Carleson measure.\
Using the fact $g, f_{k} \in Q_p$ , we get that for any Carleson box $S(I)$ on any interval $I$ of $\mathbb{T}$, $$\begin{aligned}
\int_{S(I)}\vert \psi _{j}(z)\nabla (g^2(z))\vert^2\left(1-\vert z\vert ^2\right)^{p}dA(z)& \lesssim \int _{S(I)} \vert g'(z)\vert^2 \left(1-\vert z\vert^2\right)^{p} dA(z)\\
& \lesssim\vert I\vert^{p}.\end{aligned}$$ And $$\begin{aligned}
\int_{S(I)}\vert g^2(z) \nabla \psi _{j}(z)\vert^2\left(1-\vert z\vert ^2\right)^{p}dA(z)& \lesssim \int _{S(I)} \vert g'(z)\vert^2 \left(1-\vert z\vert^2\right)^{p} dA(z)\\ & +\sum_{l=1}^n \int _{S(I)} \vert f'_{j}(z)\vert^2 \left(1-\vert z\vert^2\right)^{p}dA(z)\\
& \lesssim\vert I\vert^{p}.\end{aligned}$$ Thus, $g^{2} \psi_{j}\in Q_{p}\left(\mathbb{T}\right).$ Also since $ f_1,...,f_n\in H^{\infty} \cap Q_p$, by (8) we find that $$\sum_{k=1}^n \left(b_{j,k}-b_{k,j}\right)f_{k}\in Q_{p}\left(\mathbb{T}\right).$$ This implies that $g_{j}f\in Q_{p}(\mathbb{T})$.\
Therefore, the functions $g_j$’s are bounded analytic whose boundary values lies on $Q_{p}(\mathbb{T})$. Hence, $g_j$’s are the required solutions of (4).
Looking back at the above proof, we will be done if we can find a solution of $\bar{\partial}\;b_{j,k}=g\psi_{j}\bar{\partial}{\psi_k}$ satisfying (8). For this, it is enough to deal with an equation $\bar{\partial}(u)=G$, where $G=g\;\psi_j \overline\partial{\psi_k}$.\
We have, $$\begin{aligned}
\vert G\vert^2& \leq \vert g \vert^2 \; \vert \psi_{j} \vert^2 \; \vert \overline\partial{\psi_k}\vert^2\\
& \lesssim \frac{\vert g\vert^2\;\sum \vert f'_l \vert ^2}{\sum_{l=1}^n \,\vert{ f_l}\vert^2}\leq \sum \vert f'_l \vert ^2\end{aligned}$$\
Since $f_l\in Q_p$, by Lemma 1, $\vert f'_l(z)\vert ^2 \left(1-\vert z\vert^2\right)^p dA(z)$ is a $p$-Carleson measure. Therefore, $\vert G(z)\vert ^2\left(1-\vert z\vert^2\right)^p dA(z)$ is a $p$ - Carleson measure.
Hence, using Lemma 7, we obtain a solution $v\in Q_{p}(\mathbb{T})\cap L^{\infty}(\mathbb{T})$ of $\bar{\partial}u=G$. This completes the proof of Theorem 1.\
To prove Theorem 2, we use arguments similar to those we used in Theorem 1, but the difference is finding the solutions in $\mathcal{M}(Q_p)$ for the given data on $\mathcal{M}(Q_p)$.
Proof of Theorem 2
------------------
Let, $ f_1, ..., f_n, g\in \mathcal{M}(Q_p)$ such that they satisfy (5) and are analytic on $\overline{\mathbb{D}}$. Moreover, we assume $\vert \vert f_{j}\vert \vert \leq 1, \vert \vert g\vert \vert \leq 1.$ In this case, taking the $\psi_{j}$ as in Theorem 1, we will suppose that we can solve $$\frac{\partial{b_{j,k}}}{\partial { \overline{z}}}=g\psi_{j}\frac{\partial{\psi_{k}}}{\partial { \overline{z}}}=g^{3}G_{j,k}(z),\;\; 1\leq j ,k \leq n,$$ satisfying $$b_{j,k} \in L^{\infty}(\mathbb{T}) \cap Q_{p} (\mathbb{T})\;\; \text{with}\;\;b_{j,k}f \in Q_{p} (\mathbb{T}) \;\; \text{for all}\;\; f\in Q_p.$$\
Hence, $$g_j(z)=g^{2}(z)\psi_j(z)+\sum_{k=1}^n\left( b_{j,k}(z)-b_{k,j}(z) \right)f_k(z)$$ are bounded analytic on $\mathbb{D}$ and also $$\sum_{j=1}^{n}g_{j}f_{j}=g^{2}\sum_{j=1}^{n} \psi_{j}f_{j}=g^3$$ So the functions $g_{j}$’s are analytic solutions for (6). Now, we will try to show that $g_{j}\in \mathcal{M}\left(Q_p\right)$. For this, our aim is to show that ${g_{j}f}\in Q_{p} $ for all $f\in Q_{p}.$\
But, $Q_{p}=Q_{p}(\mathbb{T})\cap{H}^{2}(\mathbb{D})$, it’s sufficient to show that $g_{j} f \in Q_{p}(\mathbb{T})$ for all $f\in Q_{p}$ .
Also, since $\psi_{j}$’s are $C^{\infty}$ on $\overline{\mathbb{D}}$, by Lemma 6, to see that $g^{2}\psi_{j} f \in Q_{p}(\mathbb{T})$, it is enough to show that $\left\vert \nabla (g^{2} \psi_{j}f )(z)\right\vert^2 \left(1-\vert z \vert ^{2}\right)^{p}$ is a $p$ - Carleson measure.\
Using the facts that $g, f_{k} \in \mathcal{M}(Q_p)$,
$$\left \vert \nabla \psi_j \right\vert^2
\lesssim \frac{\vert g'\vert^2 }{\sum_{l=1}^n \, |f_l|^2}+\frac{\sum \vert f'_l \vert ^2}{\sum_{l=1}^n \,\vert{ f_l}\vert^2}$$ and the size condition (5), for any Carleson box $S(I)$ based on any interval $I$ of $\mathbb{T}$, we have that $$\begin{aligned}
\int_{S(I)}\vert {g^2(z)} \psi _{j}(z)\nabla f(z)\vert^2\left(1-\vert z\vert ^2\right)^{p}dA(z)& \lesssim \int _{S(I)} \vert f'(z)\vert^2 \left(1-\vert z\vert^2\right)^{p} dA(z)\\
& \lesssim\vert I\vert^{p}.\end{aligned}$$
$$\begin{aligned}
\int_{S(I)}\vert {f(z)} \psi _{j}(z)\nabla g^2(z)\vert^2\left(1-\vert z\vert ^2\right)^{p}dA(z)& \lesssim \int _{S(I)} \vert f(z)\vert^2 \vert g'(z)\vert^2 \left(1-\vert z\vert^2\right)^{p} dA(z)\\
& \lesssim\vert I\vert^{p}.\end{aligned}$$
and $$\begin{aligned}
\int_{S(I)}\vert {f(z)} \nabla \psi _{j}(z) g^2(z)\vert^2\left(1-\vert z\vert ^2\right)^{p}dA(z)& \lesssim \int _{S(I)} \vert f(z)\vert^2 \vert g'(z)\vert^2 \left(1-\vert z\vert^2\right)^{p} dA(z)\\ & +\sum_{l=1}^n \int _{S(I)} \vert f(z)\vert^2 \vert f'_{j}(z)\vert^2 \left(1-\vert z\vert^2\right)^{p}dA(z)\\
& \lesssim\vert I\vert^{p}.\end{aligned}$$ Therefore, $(g^{2} \psi_{j})f\in Q_{p}\left(\mathbb{T}\right)$.\
Also, since $ f_1,...f_n\in \mathcal{M}\left(Q_p\right)$, by (10) we find that $$\sum_{k=1}^n \left(b_{j,k}-b_{k,j}\right)f_{k}f\in Q_{p}\left(\mathbb{T}\right).$$ This implies that $g_{j}f\in Q_{p}$. Therefore the functions $g_j$’s are in $\mathcal{M}\left(Q_p\right)$. Now, it remains to show that we can find a solution of (9) satisfying (10). For this, we have that $$G(z)=g\;\psi_j \overline\partial{\psi_k}$$ which satisfies $$\begin{aligned}
\vert G\vert^2& \leq \vert g \vert^2 \; \vert \psi_{j} \vert^2 \; \vert \overline\partial{\psi_k}\vert^2\\
& \lesssim \frac{\vert g\vert^2\;\sum \vert f'_l \vert ^2}{\sum_{l=1}^n \,\vert{ f_l}\vert^2}\leq \sum \vert f'_l \vert ^2\end{aligned}$$ Hence, $\vert G(z)\vert ^2 (1-\vert z\vert^2 )^p$ is a $p$ - Carleson measure. Now, applying the second part of Lemma 7, we get the required solution of (9).\
This completes the proof of Theorem 2.
Remarks and Questions
=====================
Wolff’s Theorem for Infinite Number of Generators
-------------------------------------------------
Since we took only the finite number of generators, we were able to show that $\vert G(z)\vert^2 \left(1-\vert z\vert^2\right)^p dA(z)$ is a $p$ - Carleson measure . But,we do not know upto this point whether this can be done taking infinite number of generators, because the p-Carleson constant may depend on n. It would be interesting to see if one can generalize both Theorems 1 and 2 for any number of generators.
Generalized Ideal Problem on $Q_p$ Spaces
-----------------------------------------
We proved that the size conditions (3) and (5) imply $g^3 \in \mathcal{I}(f_1,...,f_n)$ in $ H^{\infty}(\mathbb{D})\cap Q_p$ and in $\mathcal{M}(Q_p)$. Several authors (for example, Treil\[T2\], Trent\[Tr1\],..) have given sufficient conditions for $g\in \mathcal{I}(f_1,...,f_n)$ in $H^{\infty}(\mathbb{D})$. It opens up the problem of whether a single sufficient condition can generalize the ideal problem.
$\mathit{Acknowledgements}$. The author would like to thank Dr. T. Trent for suggesting the problem and also would like to thank Dr. J. Xiao, Dr. J. Pau, Dr. A. Nicolau, and Dr. Z. Wu for having nice conversations on different properties of $Q_p$ spaces.
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D. Banjade and T. Trent, *Wolff’s ideal problem in the multiplier algebra on weighted Dirichlet space*, Houston Journal of Mathematics, to appear.
L. Carleson, *Interpolation by bounded analytic functions and the problem*, Annals of Math. **76** (1962), 547-559.
J.B. Garnett, *Bounded Analytic Functions*, Academic Press, New York, 1981.
A. Nicolau and J. Xiao, *Bounded functions in mobius invariant Dirichlet spaces*, J. Funct. Anal. **150** (1997), 383-425.
J. Pau, *Multipliers of $Q_p$ spaces and the corona theorem*, Bull. London Math.Soc **40** (2008), 327-336.
J. Pau, J. A. Peláez, *Multipliers of Möbius invariant $Q_S$ spaces*, Math. Z. **261** (2009), no. 3, 545-555.
M. Rosenblum, *A corona theorem for countably many functions*, Int. Equ. Op. Theory **3** (1980), 125–137
V.A. Tolokonnikov, *Estimate in Carleson’s corona theorem and infinitely generated ideals in the algebras $H^{\infty}$*, Functional Anal, Prilozhen **14** (1980), 85-86, in Russian.
S.R. Treil, *Estimates in the corona theorem and ideal of $H^{\infty}$: A problem of T. Wolff*, J. Anal. Math. **87** (2002), 481-495.
S.R. Treil, *The problem of ideals of $H^\infty$:Beyond the exponent 3/2*, J. Fun. Anal. **253** (2007), 220-240.
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|
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abstract: |
Since the calculation of $BR(B^- \to \eta^{\prime} K^-)$ in the framework of QCD improved factorization method, developed by Beneke [*et al.*]{}, leads to numerical values much below the experimental data, we include two different contributions, in an alternative way. First, we find out that the spectator hard-scattering mechanism increases the $BR$ value with almost $50
\%$, but the predictions depend on the combined singularities in the amplitude convolution. Secondly, by adding SUSY contributions to the Wilson coefficients, we come to a $BR$ depending on three parameters, whose values are constrained by the experimental data.
author:
- |
Marina-Aura Dariescu and Ciprian Dariescu\
Dept. of Theoretical Physics\
[*Al. I. Cuza*]{} University\
Bd. Carol I no. 11, 6600 Iaşi, Romania\
email: marina@uaic.ro
title: 'Alternative Approach to $B^- \to \eta^{\prime} K^-$ Branching Ratio Calculation'
---
Short title:\
Alternative Approach to $B^- \to \eta^{\prime} K^-$ BR Calculation
1.6em
Introduction
============
As a first evidence of a strong penguin, the $B^- \to
\eta^{\prime} K^-$ decay has become of a real interest after CLEO announced its large numerical value $BR(B^- \to \eta^{\prime}K^- )
= ( 6.5^{+1.5}_{-1.4} \pm 0.9 ) \times 10^{-5}$ \[1\], which could not be explained by the existent theoretical models. As improved measurements followed, providing even larger values, $(80^{+10}_{-9} \pm 7) \times 10^{-6}$ (CLEO \[2\]), $(76.9 \pm 3.5
\pm 4.4) \times 10^{-6}$ (BaBar \[3\]) and $(79^{+12}_{-11} \pm 9)
\times 10^{-6}$ (Belle \[4\]), the inclusion of new contributions for accommodating these data has quickly become a real theoretical challenge. In this respect, perturbative QCD mechanisms \[5\], with different $\eta^{\prime} g^* g^*$ vertex function \[5, 6\], have been considered as main candidates for significantly increasing the $BR(B^- \to \eta^{\prime} K^-)$ value. On the other hand, while searching for physics beyond the Standard Model (SM), supersymmetry has been employed in processes like $B \to J/\psi
K^{*}$ \[7\], $B \to \phi K$ \[8\], $B \to \pi K$ \[9, 10\], $B \to X_s
\gamma$ \[11\], and deviations from the SM predictions for the values of branching ratios and $CP$ asymmetries have been the main targets.
The present paper is organized as follows: in Section 2, we compute the $BR(B^- \to \eta^{\prime} K^-)$ in the improved factorization approach developed by Beneke [*et al.*]{} \[12\]. Since we get a $BR$ much below the experimental values, we incorporate two alternative contributions. The first one, presented in Section 3, comes from the so-called spectator hard scattering mechanism. Following a similar approach as in \[13\], we give a detailed calculation of the gluonic transition form factor which plays an important role in the evaluation of this contribution. Although it has been concluded that this mechanism could provide large $BR$ values \[13\], we show that the presence of combined singularities in the amplitude convolution is a source of large uncertainties. In Section 4, we employ a supersymmetric approach and include exchanges of gluino and squark with left-right squark mixing. Working in the mass insertion approximation \[14\], the values of the Wilson coefficients $c_{8g}$ and $c_{7 \gamma}$ can be significantly increased, by adding the SUSY contributions, and this has a strong numerical impact in the branching ratio estimation. Finally, one may use the experimental data to impose constraints on the flavor changing SUSY parameter $\delta_{LR}^{bs}$.
Improved QCD Factorisation
==========================
The relevant decay amplitude for $ B^- \to \eta^{\prime} K^-$, in the improved QCD factorization approach \[12\], is given by \[5, 15\] $$\begin{aligned}
A(B^- \to \eta^{\prime} K^-)
&=& - \, i {G_F \over \sqrt{2}}
(m^2_B - m^2_{\eta^{\prime}})
F^{B\to \eta^{\prime}}_0(m^2_K)
f_K [ V_{ub} V_{us}^* a_1(X)
\nonumber \\*
&& + \, V_{pb}V_{ps}^* \left( a^p_4(X)+ a^p_{10}(X)
+ r^K_\chi (a_6^p(X) + a_8^p(X)) \right) ]
\nonumber \\*
&-& i {G_F \over \sqrt{2}} (m^2_B - m^2_K)
F^{B\to K}_0(m^2_{\eta^{\prime}}) f^u_{\eta^{\prime}}
\left[ V_{ub}V_{us}^* a_2(Y) \right.
\nonumber \\*
&&
+ V_{pb}V_{ps}^* [ \left( a_3(Y)-a_5(Y) \right)
( 2 + \sigma )
\nonumber \\*
& & +
\left[ a_4^p (Y) - \frac{1}{2} a_{10}^p (Y)
+ r_{\chi}^{\prime} \left(
a_6^p(Y) - \frac{1}{2} a_8^p (Y) \right) \right]
\sigma
\nonumber \\*
& &
\left. + \frac{1}{2}
\left( a_9(Y) - a_7(Y) \right) (1 - \sigma ) \right] ,\end{aligned}$$ where $X = \eta^{\prime} K$ and $Y = K \eta^{\prime}$, $p$ is summed over $u$ and $c$, $r^{\prime}_\chi =
2m^2_{\eta^{\prime}}/(m_b-m_s)(2m_s)$, $r^K_\chi = 2m^2_K/m_b(m_u + m_s)$, $\sigma = f^s_{\eta^{\prime}}/f^u_{\eta^{\prime}}$, and \[12\] $$\begin{aligned}
a_1(M_1 M_2) & = & c_1 + {c_2 \over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi}
(V_{M_2} + H) \right],
\nonumber\\
a_2(M_1 M_2) & = & c_2 + {c_1 \over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi}
(V_{M_2} + H) \right],
\nonumber\\
a_3(M_1 M_2) & = & c_3 + {c_4 \over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi}
(V_{M_2} + H) \right],
\nonumber\\
a^p_4(M_1 M_2) & = &
c_4 + {c_3\over N_c} \left[ 1 + {C_F \alpha_s \over 4 \pi}
(V_{M_2} + H) \right] +
{C_F \alpha_s \over 4\pi N_c} P^p_{M_2,2},
\nonumber\\
a_5(M_1 M_2) & = & c_5 + {c_6 \over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi}
(-12-V_{M_2} - H) \right],
\nonumber\\
a^p_6(M_1 M_2) & = & c_6 + {c_5\over N_c}
\left( 1 - 6 {C_F \alpha_s \over 4
\pi} \right) + {C_F\alpha_s \over 4\pi N_c}
P^{p}_{M_2, 3},
\nonumber\\
a_7(M_1 M_2) & = & c_7 + {c_8 \over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi}
(-12-V_{M_2} - H) \right],
\nonumber\\
a^p_{8}(M_1 M_2) & = & c_8 + {c_7 \over N_c}
\left( 1 - 6{C_F \alpha_s\over 4
\pi} \right) + {\alpha\over 9\pi N_c} P^{p,EW}_{M_2, 3},
\nonumber\\
a_9(M_1 M_2) & = & c_9 + {c_{10} \over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi}
(V_{M_2} + H) \right],
\nonumber\\
a^p_{10}(M_1 M_2) & = & c_{10} + {c_9\over N_c}
\left[ 1 + {C_F \alpha_s \over 4 \pi} (V_{M_2} +
H) \right] +
{\alpha \over 9 \pi N_c} P^{p,EW}_{M_2, 2},\end{aligned}$$ where $C_F = (N^2_c-1)/2N_c$ and $N_c = 3$ is the number of colors. The vertex, the hard gluon exchange with the spectator and the penguin contributions, at $\mu =m_b$, are: $$\begin{aligned}
V_M &=& - 18 + \int^1_0 dx g(x)
\phi_M(x),
\nonumber\\
P^p_{M,2} &=& c_1 \left[
\frac{2}{3} + G_M(s_p) \right]
+ c_3 \left[
\frac{4}{3} + G_M(0) + G_M(1) \right]
\nonumber\\*
& & + \, (c_4+c_6) \left[
(n_f-2) G_M(0) + G_M (s_c) + G_M(1) \right]
\nonumber \\*
& & - \, 2 c_{8g}^{eff} \int^1_0 {dx\over 1-x}\phi_M(x),
\nonumber\\
P^{p,EW}_{M,2}& =& (c_1 + N_c c_2) \left[
\frac{2}{3} + G_M(s_p) \right] -
3 c_{7\gamma}^{eff} \int^1_0 {dx\over 1-x} \phi_M(x),
\nonumber\\
P^p_{M,3} &=& c_1 \left[ \frac{2}{3} + \hat{G}_M(s_p)
\right] + c_3 \left[ \frac{4}{3} + \hat{G}_M(0) +
\hat{G}_M(1) \right]
\nonumber\\*
& & + \, (c_4+c_6) \left[ (n_f-2) \hat{G}_M(0) +
\hat{G}_M(s_c) + \hat{G}_M(1) \right]
- 2 c_{8g}^{eff},
\nonumber\\
P^{p,EW}_{M,3} &=& (c_1 + N_c c_2)
\left[ \frac{2}{3} + \hat{G}_M(s_p) \right] -
3 c_{7\gamma}^{eff},
\nonumber\\
H &=& \frac{4 \pi^2}{N_c} \frac{f_B f_{M_1}}{m_B^2
F^{B\to M_1}_0(0)}
\nonumber \\*
& & \times
\int^1_0 {d \xi \over \xi} \phi_B(\xi)
\int^1_0 {dx\over \bar{x}} \phi_{M_2}(x)
\int^1_0 {dy \over \bar{y}} \left[
\phi_{M_1}(y) + \frac{2 \mu_{M_1}}{m_b}
\frac{\bar{x}}{x} \phi^p_{M_1}(y) \right] ,
\; \; \; \; \; \;\end{aligned}$$ where $\bar{x}= 1-x$, $\bar{y} = 1-y$ and the parameter $2\mu_M/m_b$ coincides with $r_\chi$. The functions $g(x)$, $G_M(x)$ and $\hat G_M(x)$ are given by $$\begin{aligned}
g(x) & = & 3 \left( {1-2x\over 1-x}\ln x - i\pi \right)
\nonumber \\*
& & +
\left[ 2 {\rm Li}_2(x) -\ln^2x + {2\ln x\over 1-x} -
(3+2i\pi)\ln x -(x \to \bar{x} ) \right],
\nonumber\\
G(s,x) & = & 4 \int^1_0 du \, u
\bar{u} \ln[s-u \bar{u} x]
\nonumber \\*
& = & - \frac{10}{9} + \frac{2}{3} \ln s - \frac{8s}{3x}
+ \frac{4}{3} \left( 1 + \frac{2s}{x} \right)
\sqrt{\frac{4s}{x} -1} \arctan \frac{1}{\sqrt{\frac{4s}{x}
-1}} \, ,
\nonumber \\
G_M(s) & = & \int^1_0 dx \, G(s-i\epsilon,
\bar{x}) \phi_M(x),
\nonumber\\
\hat G_M(s) & = & \int^1_0 dx \,
G(s-i\epsilon, \bar{x}) \phi^p_M(x),\end{aligned}$$ where $s_i = m_i^2/m_b^2$ are the mass ratios for the quarks involved in the penguin diagrams, namely $s_u = s_d = s_s =0$ and $s_c = (1.3/4.2)^2$.
As it can be noticed, except for the hard contribution where the wave functions for both $M_1$ and $M_2$ are involved, the coefficients $a_i$ are different for the $X$ and $Y$ final states, since they depend on the twist-2 and twist-3 wave functions of the $M_2$ meson. Thus, the twist-2 distribution amplitude $\phi_K(x)$ has the following expansion in Gegenbauer polynomials \[12, 16\] $$\phi_K(x) = 6x(1-x)[ 1+ \alpha_1^K C^{(3/2)}_1(2x-1) +
\alpha_2^K C^{3/2}_2(2x-1) + ...],$$ with $C^{3/2}_1(u) = 3 u$, $C^{3/2}_2(u) = (3/2)(5u^2-1)$, $\alpha_1^K = 0.3 \pm 0.3$, and $\alpha_2^K = 0.1 \pm 0.3$. The corresponding twist-3 amplitude, $\phi^p_K$, is 1.
The physical states $\eta$ and $\eta^{\prime}$ are mixtures of SU(3)-singlet and octet components $\eta_0$ and $\eta_8$ and therefore the corresponding decay constants, in the two-angle mixing formalism, are given by $$\begin{aligned}
f^u_{\eta^{\prime}} & = &
\frac{f_8}{\sqrt{6}} \sin \theta_8 +
\frac{f_0}{\sqrt{3}} \cos \theta_0 \; ,
\nonumber \\
f^s_{\eta^{\prime}} & = &
- \, 2 \frac{f_8}{\sqrt{6}} \sin \theta_8 +
\frac{f_0}{\sqrt{3}} \cos \theta_0 \; ,\end{aligned}$$ with $\theta_8 = -22.2^o$, $\theta_0 = -9.1^o$, $f_8 = 168$ MeV, and $f_0 =157$ MeV \[17\]. These lead to $f^u_{\eta^{\prime}} = 63.5$ MeV, $f^s_{\eta^{\prime}} = 141$ MeV and to the relevant form factor for the $B \to \eta^{\prime}$ transition $$F_0^{B \to \eta^{\prime}} =
F_0^{\pi} \left( \frac{\sin \theta_8}{\sqrt{6}}
+ \frac{\cos \theta_0}{\sqrt{3}} \right) = 0.137$$
Even though the $\eta^{\prime}$ flavor singlet meson has a gluonic content which could bring a contribution to the wave function, this is supposed to be small \[18\] and therefore we employ, in the calculation of $V_{\eta^{\prime}}$, $P^p_{\eta^{\prime},2}$ and $P^{p,EW}_{\eta^{\prime},2}$ in $a_i(Y)$, only the leading twist-2 distribution amplitude $$\phi_{\eta^{\prime}} = 6 x \bar{x} \, .$$ Also, since the twist-3 quark-antiquark distribution amplitude do not contribute, due to the chirality conservation, the penguin parts in $a_6^p(Y)$ and $a_8^p(Y)$ are missing. As for the $B$ meson wave function, we shall work with a strongly peaked one, around $z_0 = \lambda_B/m_B \approx 0.066 \pm 0.029$, for $\lambda_B =0.35 \pm 0.15$ GeV.
Putting everything together, we get, within the SM improved factorization approach \[12\], the numerical value $BR_{SM}(B \to \eta^{\prime} K ) =
3.65 \times 10^{-5}$, which although is in accordance with other theoretical estimations \[5, 15, 17\], yet lay below the experimental data \[1-4\]. Hence, in spite of the \`\`conservative” prediction that the conventional mechanism should be the dominant one, it has been getting clear that new contributions are needed in order to account for the existent data.
Spectator Hard-Scattering Mechanism
===================================
It has been considered that the spectator hard-scattering mechanism (SHSM), depicted in Figure 1, is a reliable framework for this process, which significantly increases the value of $BR(B
\to \eta^{\prime} K)$ \[5, 13\]. Following this idea, let us write down the corresponding di-gluon exchange amplitude for the $b$ quark decaying into an $s$ quark and a hard gluon $$\begin{aligned}
A_{hs} & = & - \, i \, C_F \,
g_s^3 \frac{f_B}{2 \sqrt{6}}
\frac{f_K}{2 \sqrt{6}}
\int dz \, dy \, \phi_B(z) \phi_K(y)
\nonumber \\*
& & \times \, {\rm Tr} \left[ \gamma_5 \rlap{/}{P_k} \Gamma_{\mu}
( \rlap{/}{P_B} + m_B) \gamma_5 \gamma_{\nu} \right]
\frac{\varepsilon^{\mu \nu \alpha \beta} Q_{1 \alpha}
Q_{2 \beta}}{Q_1^2 Q_2^2} \,
F_{\eta^{\prime} g^* g^*}
(Q_1^2, Q_2^2, m_{\eta^{\prime}}^2) \; \; \; \;\end{aligned}$$ in terms of the effective $b \to sg$ vertex \[19\] $$\Gamma_{\mu}^a = \frac{G_F}{\sqrt{2}} \,
\frac{g_s}{4 \pi^2} V^*_{ps} V_{pb} \, t^a
\left[ F_1^p \left( Q_1^2 \gamma_{\mu}
- Q_{1 \mu} Q_1 \right) L
- F_2^p i \sigma_{\mu \nu} Q_1^{\nu} m_b R \right]$$ and the transition form factor \[6\] $$< g_a^* g_b^* | \eta^{\prime}> = - \, i \,
\delta_{ab} \varepsilon^{\mu \nu \alpha \beta}
\varepsilon^{a*}_{\mu} \varepsilon^{b*}_{\nu}
Q_{1 \alpha} Q_{2 \beta} F_{\eta^{\prime} g^* g^*}
(Q_1^2, Q_2^2, m_{\eta^{\prime}}^2)$$ The quark contribution to the $\eta^{\prime} g^* g^*$ vertex $$F_{\eta^{\prime} g^* g^*}
(Q_1^2, Q_2^2, m_{\eta^{\prime}}^2) =
4 \pi \alpha_s \frac{1}{2N_c} \sum_{q=u,d,s}
f^q_{\eta^{\prime}} \, F(y,a) \, ,$$ with $$F(y,a) =
\int_0^1 dx \; \frac{\phi_{\eta^{\prime}}(x)}{\bar{x}
Q_1^2 + x Q_2^2 -x \bar{x} m_{\eta^{\prime}}^2 + i
\varepsilon} + ( x \leftrightarrow \bar{x} )
\; , \; \; a^2=m_{\eta^{\prime}}^2/m_B^2 \, ,$$ will play an important role in the evaluation of the amplitude $A_{hs}$. Performing the calculations in (9), we come to the following expression of the hard scattering amplitude: $$\begin{aligned}
A_{hs} & = &
- \, 2 \, i \, \frac{G_F}{\sqrt{2}} V_{ps}^*
V_{pb} \frac{\alpha_s^2}{N_c^3} f_B f_K
(2f^u_{\eta^{\prime}} + f^s_{\eta^{\prime}})
\int_0^1 dz \, \phi_B(z) \int_0^1 dy \phi_K(y)
\nonumber \\*
& &
\times \left[ F_1^p Q_1^2 \left( (P_B \cdot Q_1)
(P_K \cdot Q_2) - (P_K \cdot Q_1)(P_B \cdot Q_2)
\right) + \right.
\nonumber \\*
& & + \, \left.
F_2^p m_B m_b \left( (P_K \cdot Q_2)
Q_1^2 - (P_K \cdot Q_1)(Q_1 \cdot Q_2) \right)
\right] \frac{F(y,a)}{Q_1^2 Q_2^2}\end{aligned}$$ With the gluon momenta $$Q_1=\bar{z}
P_B - \bar{y}P_K \; , \; \; Q_2 = zP_B -yP_K \; ,$$ and neglecting, for the moment, both $m_{\eta^{\prime}}^2$ and $m_K^2$, the amplitude (14) becomes $$\begin{aligned}
A_{hs} & = &
i \, \frac{G_F}{\sqrt{2}} \, V_{ps}^* V_{pb}
\, \frac{\alpha_s^2}{2 N_c^3}
f_B f_K (2f^u_{\eta^{\prime}} + f^s_{\eta^{\prime}})
\, \frac{1}{z_0}
\nonumber \\*
& & \times \,
\int_0^1 \phi_K(y) \left[ m_B^2 F_1^p +
m_B m_b \frac{F_2^p}{y-z_0} \right] F(y,a)\end{aligned}$$ where, for the dominant contribution coming from the insertion of the $O_1^{u,c}$ and the magnetic-penguin $O_{8g}$ operators, one has \[13\] $$F_1^p = c_1 \left[ \frac{2}{3} + G[s_p,
(1-z_0)(y-z_0)] \right] \, , \; \;
F_2^p = - 2 c_{8g}$$ In what it concerns the $F(y,a)$ function, which is an essential input in the calculations, it can be first written as $$F(y,a) \, = \,
4 \int_0^1 dx \, \frac{6x \bar{x}
\; (Q_1^2+Q_2^2-2 x \bar{x}
m_{\eta^{\prime}}^2)}{\left[ Q_1^2+Q_2^2-2x
\bar{x} m_{\eta^{\prime}}^2
\right]^2 - \left[
(x - \bar{x}) (Q_1^2 - Q_2^2) \right]^2}$$ and it comes, after algebraic computations, to the following form $$\begin{aligned}
F(y,a) & = &
- \, \frac{12}{m_{\eta^{\prime}}^2} \left[
1 \, - \, \frac{Q_1^2-Q_2^2}{2 m_{\eta^{\prime}}^2}
\log \left| \frac{Q_1^2}{Q_2^2} \right| +
\, \frac{(Q_1^2-Q_2^2)^2 -
m_{\eta^{\prime}}^2 ( Q_1^2+Q_2^2)}{2
m_{\eta^{\prime}}^2 \sqrt{p^4-4 Q_1^2 Q_2^2}}
\right. \nonumber \\*
& & \times \left. \log \left|
1 + 2 \, \frac{\sqrt{p^4-4Q_1^2Q_2^2}}{p^2
- \sqrt{p^4-4Q_1^2Q_2^2}} \right|
\right],\end{aligned}$$ where we have introduced the notation $p^2 = Q_1^2 + Q_2^2 -m_{\eta^{\prime}}^2$. The logarithmic nature of the $F(y,a)$ function makes it very sensitive to the values of $Q_1^2$, $Q_2^2$, $m_{\eta^{\prime}}^2$. We recommend \[6\] for a detailed discussion of the $\eta^{\prime} g^* g^*$ vertex in the case of arbitrary gluon virtualities in the time-like, $Q_1^2>0$, $Q_2^2 >0$, $p^4-4Q_1^2Q_2^2 >0$, and space-like, $Q_1^2<0$, $Q_2^2 <0$, $p^4-4Q_1^2Q_2^2 <0$, regions.
Now, using $$Q_1^2 \approx \bar{z} \left[(y-z)m_B^2+\bar{y}
m_{\eta^{\prime}}^2 \right] \, , \; \;
Q_2^2 \approx z \left[ -(y-z)m_B^2+ y
m_{\eta^{\prime}}^2 \right] ,$$ where we have neglected $m_K^2$, the dominant term in (19) is: $$\begin{aligned}
F(y,a) & \approx &
- \, \frac{12}{m_{\eta^{\prime}}^2} \left[
1 \, - \, \frac{1}{2} \left[ \frac{y-z}{a^2}
+ (1-y-z) \right] \log \left| \frac{a^2+y-z}{z(z-y)}
\right| \right.
\nonumber \\*
& & + \, \left.
\frac{(y-z)a^2+(y-z)^2}{2a^2 \; |y-z|} \log
\left| \frac{y(1-a^2)-z+|y-z|}{y(1-a^2)
-z-|y-z|} \right| \right]\end{aligned}$$
On the other hand, by comparing the expressions in (20), it clearly results that we are in the limit where $|Q_1^2| \gg
|Q_2^2|$. So, the function $F(y,a)$ can be computed in this approximation and it simply yields $$F (y,a) \, = \,
- \, \frac{12}{m_{\eta^{\prime}}^2}
\left[ 1 + \left( \frac{y-z_0}{a^2} + \bar{y} \right)
\log \left| 1-
\frac{1}{\frac{y-z_0}{a^2}+ \bar{y}} \right| \right]$$ As it can be seen from (20), the term $(y-z_0)/a^2 + \bar{y} = Q_1^2/m_{\eta^{\prime}}^2$ takes a whole range of values, from $-0.87$ to $26.5$, as $Q_1^2$ goes from the space-like to the time-like regions. Consequently, a logarithmic singularity develops as $y \to z_0/(1-a^2)$, i.e. for $Q_1^2 \to m_{\eta^{\prime}}^2$. Inspecting (16), we also notice the pole at $y = z_0$ in the $F_2^p$ contribution. In addition, while $G[s_p,(1-z_0)(y-z_0)]$ is divergence free for all $s>0$, the $G[0,(1-z_0)(y-z_0)]$ gets a logarithimic singularity at $y=z_0$. Hence, in the course of numerically evaluating the scattering contribution, one must be careful about dealing with these combined singularities in the convolution (16).
As in the case of other hard-scattering theoretical estimations \[5, 13\], the amplitude of this contribution contains, as main uncertainty, the peaking position, $z_0$, in the $B$ meson distribution function and accordingly, the branching ratio is extremely sensitive to it. For $z_0 \in [0.063 , \, 0.068]$ and the average value $\alpha_s(Q_1^2)=0.28$, the total branching ratio, including besides the improved factorization approach, the spectator hard-scattering mechanism with the vertex function (22), is in the range from $BR(B \to \eta^{\prime} K) = 6.58 \times
10^{-5}$, for $z_0 =0.063$, to $BR(B \to \eta^{\prime} K) = 5.8
\times 10^{-5}$, for $z_0 =0.068$.
Comparing these results with the experimental data \[$1-3$\], we notice that they are still below the lowest limit. An alternative way which increases the $BR$ and avoids the uncertainties coming from the combined singularities in the convolution (16), would presumably look more reliable.
SUSY Gluonic Dipole Contribution
================================
Employing the Minimal Supersymmetric Standard Model (MSSM), we shall add to the effective SM Hamiltonian (1), the SUSY contribution $$H^{SUSY} \, = \, - \, i \, {G_F \over \sqrt{2}}
(V_{ub} V_{us}^* +V_{cb} V_{cs}^*)
\left( c_{8g}^{SUSY} O_{8g} + c_{7 \gamma}^{SUSY}
O_{7 \gamma} \right) ,$$ expressed in terms of the usual gluon and photon operators: $$\begin{aligned}
O_{8g} & = & \frac{g_s}{8 \pi^2} \, m_b
\bar{s} \sigma_{\mu \nu} (1+\gamma_5 ) G^{\mu \nu} b \, ,
\nonumber \\
O_{7 \gamma} & = & \frac{e}{8 \pi^2} \, m_b \bar{s}
\sigma_{\mu \nu} (1+\gamma_5 ) F^{\mu \nu} b \, ,\end{aligned}$$ and of the Wilson coefficients \[10, 20\] $$\begin{aligned}
c_{8g}^{SUSY} (M_{SUSY}) & = & - \, \frac{\sqrt{2} \pi
\alpha_s}{G_F (V_{ub} V_{us}^* +
V_{cb} V_{cs}^*) m_{\tilde{g}}^2} \, \delta^{bs}_{LR} \,
\frac{m_{\tilde{g}}}{m_b} \, G_0(x) \, ,
\nonumber \\
c_{7 \gamma}^{SUSY} (M_{SUSY}) & = & - \,
\frac{\sqrt{2} \pi \alpha_s}{G_F (V_{ub} V_{us}^* +
V_{cb} V_{cs}^*) m_{\tilde{g}}^2} \, \delta^{bs}_{LR} \,
\frac{m_{\tilde{g}}}{m_b} \, F_0(x) \, ,\end{aligned}$$ where $$\begin{aligned}
G_0 (x) & = &
\frac{x}{3(1-x)^4} \,
\left[ 22-20x-2x^2+16 x \ln(x) -x^2 \ln (x)
+ 9 \ln (x) \right] ,
\nonumber \\
F_0 (x) & = & - \;
\frac{4x}{9(1-x)^4} \,
\left[ 1+4x-5x^2+4 x \ln(x) + 2 x^2 \ln (x) \right]\end{aligned}$$ In the above expressions, $x=m_{\tilde{g}}^2 / m_{\tilde{q}}^2$, with $m_{\tilde{g}}$ being the gluino mass and $m_{\tilde{q}}$ an average squark mass, while the factor $\delta^{bs} = \Delta^{bs}/
m_{\tilde{q}}^2$, where $\Delta^{bs}$ are the off-diagonal terms in the sfermion mass matrices, comes from the expansion of the squark propagator in terms of $\delta$, for $\Delta \ll
m_{\tilde{q}}^2$. In principle, the dimensionless quantities $\delta^{bs}$, measuring the size of flavor changing interaction for the $\tilde{s} \tilde{b}$ mixing, are present in all the SUSY corrections to the Wilson coefficients in (1) and they are of four types, depending on the $L$ or $R$ helicity of the fermionic partners. In the followings, we focus on the $\delta^{bs}_{LR}$ insertions because only the SUSY Wilson coefficients (25), being proportional to the large factor $m_{\tilde{g}} /m_b$, are going to make an important contribution, even for small values of $\delta$.
In (3), we replace the Wilson coefficients $c_{8g}^{eff}$ and $c_{7 \gamma}^{eff}$, by the total quantities $$c_{8g}^{total} [x, \delta] \, = \,
c_{8g}^{eff} + c_{8g}^{SUSY} (m_b) \; ,
\; \;
c_{7 \gamma}^{total} [x, \delta] \, = \,
c_{7 \gamma}^{eff} + c_{7 \gamma}^{SUSY} (m_b) \; ,$$ where $c^{SUSY} (m_b)$ have been evolved from $M_{SUSY} = m_{\tilde{g}}$ down to the $\mu =m_b$ scale, using the relations \[10, 19\] $$\begin{aligned}
c_{8g}^{SUSY} (m_b) & = &
\eta c_{8g}^{SUSY}(m_{\tilde{g}} ) \, ,
\nonumber \\
c_{7 \gamma}^{SUSY} (m_b) & = & \eta^2
c_{7 \gamma }^{SUSY}(m_{\tilde{g}} ) + \frac{8}{3}
(\eta - \eta^2) c_{8g}^{SUSY}(m_{\tilde{g}} ) \, ,\end{aligned}$$ with $$\eta \, = \,
\left( \alpha_s(m_{\tilde{g}})/
\alpha_s(m_t) \right)^{2/21}
\left( \alpha_s(m_t)/
\alpha_s(m_b) \right)^{2/23}$$
We choose for $m_{\tilde{q}}$ the value $m_{\tilde{q}} = 500$ GeV and write $m_{\tilde{g}}$ as $m_{\tilde{g}} = \sqrt{x} \,
m_{\tilde{q}}$ and $\delta^{bs}_{LR} \equiv \rho e^{i \varphi}$. As the total branching ratio can be expressed in terms of three free parameters: $x , \, \rho , \, \varphi$, one is able to plot the $BR^{total}$, in units of $10^{-5}$, as a function of $(\rho , \varphi)$, for different values of $x$. By inspecting the 3D plots displayed in Figure 2, for $x=0.3$ (the upper) and $x=1$ (the lower surface), we notice that the SUSY contributions (25) to the Wilson coefficients have significantly increased the SM value, $BR_{SM} =3.65 \times 10^{-5}$, represented by the horizontal plane. Using the experimental data, one is able now to determine the $\delta^{bs}_{LR}$ complex values, for each $x$.
Let us take, for example, $x=1$, pointing out that the same discussion can be performed for any $x$-value. For $\rho =0.005$, the $BR^{total}$ is increasing from $5.1 \times 10^{-5}$, for $\varphi \approx \pm \pi /3$, to the maximum value $BR^{total}=
6.24 \times 10^{-5}$, for $\varphi =0$. As $\rho$ goes to bigger values, we find a better agreement with the large experimental data. For $\rho =0.01$, the data can be accommodated for $\varphi
\approx -\pi /4$, while, for $\rho=0.02$, one has to impose $\varphi \approx - \, 8 \pi /15$.
Concluding Remarks
==================
At first, we have analyzed the $B^- \to \eta^{\prime} K$ decay and computed its branching ratio using the improved factorization method developed by Beneke [*al.*]{} \[12\]. Since the obtained result, $BR_{SM} = 3.65 \times 10^{-5}$, is much below the experimental data, \[1-4\], we have have added new contributions.
In this respect, the so-called spectator hard-scattering mechanism, which is depicted in Figure 1, has allowed us to compute the amplitude in terms of the effective $b \to sg$ vertex and the transition form factor (11) which contains the quark contribution to the $\eta^{\prime} g^* g^*$, (21), as an essential input. The total $BR$ has, as a main uncertainty, the peaking position in the $B$ meson wave function, $z_0 = \lambda_B/m_B$, with $\lambda_B = 0.35 \pm 0.15$ GeV. Even the results are closer to the experimental data, we point out the combined singularities in the amplitude convolution (16) which must be treated carefully.
Secondly, we extend the SM to the MSSM and add SUSY contributions to the Wilson coefficients $c^{eff}_{8g}$ and $c^{eff}_{7
\gamma}$. The total $BR$ is expressed in terms of the parameters $x=m_{\tilde{g}}^2/m_{\tilde{q}}^2$, and $\delta^{bs}_{LR} = \rho
e^{i \varphi}$ whose contribution turns out to be important, even for very small values of $\rho$. Finally, by inspecting the 3D-graphics (see Figure 2), representing the $BR^{total}$ for $x=0.3$ (the upper surface) and $x=1$ (the lower surface), one is able to find numerical values for $\rho$ and $\varphi$ that can account for the experimental data or other theoretical predictions \[21\].
[**Acknowledgments**]{}
The authors wish to acknowledge the kind hospitality and fertile environment of the University of Oregon where this work has been carried out. Professor N.G. Deshpande’s inciting suggestions and constant support are highly regarded. M.A.D. thanks the U.S. Department of State, the Council for International Exchange of Scholars (C.I.E.S.) and the Romanian-U.S. Fulbright Commission for sponsoring her participation in the Exchange Visitor Program no. G-1-0005.
[99]{} CLEO Collaboration, B.H. Behrens et al., Phys. Rev. Lett. [**80**]{}, 3710 (1998). CLEO Collaboration, S.J. Richichi et al., Phys. Rev. Lett. [**85**]{}, 520 (2000). BaBar Collaboration, B. Aubert et al., Phys. Rev. Lett. [**91**]{}, 161801 (2003). Belle Collaboration, K. Abe et al., Phys. Lett. B [**517**]{}, 309 (2001). D.S. Du, D.S. Yang, G.H. Zhu, Phys. Rev. D [**59**]{}, 014007 (1999); D.S. Du, C.S. Kim, Y. Yang, Phys. Lett. B [**426**]{}, 133 (1998); A. Ali, A.Y. Parkhomenko, Phys. Rev. D [**65**]{}, 074020 (2002). D. London, N. Sinha, R. Sinha, hep-ph/0207007. R. Barbieri, A. Strumia, Nucl. Phys. [**B508**]{}, 3 (1997); A.K. Giri, R. Mohanta, hep-ph/0306041; C. Dariescu, M.A. Dariescu, N.G. Deshpande, D.K. Ghosh, hep-ph/0308305; J.F. Cheng, C.S. Huang, X.H. Wu, Phys. Lett. B [**585**]{}, 287 (2004); J.F. Cheng, C.S. Huang, X.H. Wu, hep-ph/0404055; Y. Wang, Phys. Rev. D [**69**]{}, 054001 (2004); S. Mishima, A.I. Sanda, Phys. Rev. D [**69**]{}, 054005 (2004). Y. Grossman, M. Neubert, A.L. Kagan, JHEP [**9910**]{}, 029 (1999). X.G. He, J.Y. Leou, J.Q. Shi, Phys. Rev. D [**64**]{}, 094018 (2001). T. Besmer, C. Greub, T. Hurth, Nucl. Phys. [**B609**]{}, 359 (2001). M. Beneke, et al., Nucl. Phys. [**B606**]{}, 245 (2001). M.Z. Yang, Y.D. Yang, Nucl. Phys. [**B609**]{}, 469 (2001). F. Gabbiani, et al., Nucl. Phys. [**B477**]{}, 321 (1996). M. Beneke, M. Neubert, Nucl. Phys. [**B651**]{}, 225, (2003). P. Ball, JHEP [**09**]{}, 005 (1998); P. Ball, V.M. Braun, hep-ph/9808229. A. Ali, G. Kramer, C.D. Lu, Phys. Rev. D [**58**]{}, 094009 (1998). T. Muta, M.Z. Yang, Phys. Rev. D [**61**]{}, 054007 (2000).
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[**Figure Captions**]{}
Fig.1. Feynman diagrams of the hard scattering mechanism for $B^- \to \eta^{\prime} K^-$. The gluons are represented by the dashed lines.\
Fig.2. Total branching ratios (SM+SUSY) for $B^- \to \eta^{\prime}
K^-$, in units of $10^{-5}$, as functions of $(\rho , \varphi)$, for $x=0.3$ (the upper plot) and $x=1$ (the lower plot), compared to the SM estimation represented by the horizontal plane.
|
---
address:
- |
Department of Mathematics\
Massachusetts Institute of Technology\
77 Massachusetts Avenue\
Cambridge, MA 02139
- |
Department of Mathematics\
Ben-Gurion University of the Negev\
P.O.B. 653\
Be’er-Sheva 84105\
Israel
author:
- 'Jennifer S. Balakrishnan'
- Amnon Besser
bibliography:
- 'biblio.bib'
title: 'Computing local $p$-adic height pairings on hyperelliptic curves'
---
intro.tex coleman-gross.tex hyperelliptic.tex coleman-integration.tex algorithm.tex implementation.tex
|
---
abstract: 'Owing to the significance of combinatorial search strategies both for academia and industry, the introduction of new techniques is a fast growing research field these days. These strategies have really taken different forms ranging from simple to complex strategies in order to solve all forms of combinatorial problems. Nonetheless, despite the kind of problem these approaches solve, they are prone to heavy computation with the number of combinations and growing search space dimensions. This paper presents a new approach to speed up the generation and search processes using a combination of stack and hash table data structures. This approach could be put to practice for the combinatorial approaches to speed up the generation of combinations and search process in the search space. Furthermore, this new approach proved its performance in diverse stages better than other known strategies.'
author:
-
-
-
bibliography:
- 'references.bib'
title: A New Approach to Speed up Combinatorial Search Strategies Using Stack and Hash Table
---
Combinatorial search; Covering array; Combinatorial interaction testing; Combinatorial optimisation.
Introduction
============
Combinatorial strategies have received lots of interest lately as a result of their diverse applications in areas of research, particularly in software engineering. In its simple form, a combinatorial strategy can reduce the several input parameters of a system to a small set of these parameters base on their interaction (combination) [@ref1]. This idea developed more recently to include the constraints and seeding among these input parameters also [@ref2; @ref3]. Similarly, these parameters could be the features or configurations to be set for a system, or they might simply be the values to be entered while the system is in operation. The rationale behind the reduction is that it is impossible to take all possibilities of these input parameters. Therefore, the reduction must be done in a systematic way by considering the combinations of these parameters. As it is impossible to consider all likely combination for a system to test, there is a need to generate an optimised set of combinations that have the effectiveness of all possible combinations.
Combinatorial strategies establish their effectiveness for different applications including software engineering, chemistry, biology, communication and many other fields [@ref4]. To this end, optimisation methods have been used to generate this set. Regardless of the optimisation technique used in the implemented strategies, evidence revealed that the serious issues in the development of these strategies are: how to construct the combinations and how to search for them later. To optimise, the strategy first has to generate all the possible combinations. Then, the optimisation algorithm attempts to cover these combinations with the smallest set. The complexity of this process is proportional with the number of input parameters. Hence, there is a need to speed up this process to enable the optimisation algorithms inside the combinatorial strategies work faster and efficiently.
This paper proposes an approach to construct the combinations and also search for them efficiently. The approach includes new algorithm and programming techniques to construct, store and search for combinations. The remaining part of this paper is organised as follows: Section 2 gives a brief review of the combinatorial interaction strategies; Section 3 and 4 introduces the terminologies used all through this paper and similarly formulate the problem; Section 5 contains the methodology as well as the algorithms for the proposed strategy; Section 6 illustrates the experimental framework; Section 7 shows the experimental results and finally, Section 8 presents our conclusions.
Combinatorial Interaction Strategies
====================================
In the last decade, various studies on combinatorial interaction approaches have gained a lot of awareness in such a way that several test generation approaches were developed. With the approaches generally dedicated to solving different problems, a few of them solve the generation of optimised set of input parameters by taking combinations into account. Though some others are also dedicated to generating those sets with constraints or seeding among the input factors, they still require particular configuration. Other research groups have started to examine (instead of software engineering alone) the application of these approaches in other research fields like biology, chemistry and electrical engineering to solve real life problems.
Evidence revealed that the use of meta-heuristic algorithms could achieve optimum sets of final combinatorial set covering every interaction among the input parameters. Most recently, different meta-heuristic algorithms have been adapted to solve this problem such as Simulated Annealing (SA) [@ref5], Genetic Algorithms (GA) [@ref6], Cuckoo Search (CS) [@ref7] and many other algorithms. Despite the wide range of approaches and algorithms used in generating the combinatorial interaction set, we still cannot find a “universal” strategy that can generate optimised sets for all the configuration since this problem is an NP-complete problem [@ref8]. Hence, each strategy could be useful for specific kind of configuration and application.
Although different strategies have been developed, the problem of search space complexity is still the same. As mentioned earlier, the main aim of the combinatorial strategies is to cover the entire interactions of input parameters by the smallest set. Hence, the strategy needs to search for a combination that can cover much of those interactions. To determine the number of interactions covered, the strategy must search for them among a large number of interactions which will definitely consume the program time as well as the resources of the computer. It will likewise cause the program to take more iteration for searching within the meta-heuristic algorithm.
In addition to the aforementioned issue, the problem of generating input parameter combination represents another serious problem apart from consuming time and resources. This problem appears worsen as the input parameter continues to grow in size since most of the algorithms’ complexities are growing with the number of parameters. To overcome this problem, a special algorithm is needed to be combined with efficient data structures in order to speed up the generation and sorting process. This paper aims to provide new approaches and algorithms that will solve these problems and at the same time, speed up the combinatorial interaction search strategies in general.
Preliminaries
=============
Combinatorial interaction strategies relies on Covering Array (CA) a well-known mathematical model to represent the combinatorial interaction set. The CA notation assures that all the interactions represented within one array. This mathematical object originates essentially from another object called orthogonal array (OA) [@ref9]. An $OA_\lambda (N; t, k, v)$ is an $N \times k$ array, where for every $N \times t$ sub-array, each $t-tuple$ occurs exactly $\lambda$ times, where $\lambda = N/v^t$; $t$ is the combination strength; $k$ is the number of input factors ($k \geq t$); and $v$ is the number of symbols or levels associated with each input factor. In covering all the combinations, each $t-tuple$ must occur at least once in the final test suite [@ref10]. When each $t-tuple$ occurs exactly once, $\lambda=1$, and it can be unmentioned in the mathematical syntax, that is, $OA (N; t, k, v)$. As an example, $OA( 9; 2, 4, 3)$ contains three levels of value ($v$) with a combination degree ($t$) equal to two, and four factors ($k$) can be generated by nine rows.
CA is another mathematical notation that is more flexible in representing test suites with larger sizes of different parameters and values. In general, CA uses the mathematical expression $CA_\lambda (N; t, k, v)$ [@ref11]. A $CA_\lambda(N ; t, k, v)$ is an $N \times k$ array over $(0, . . . , v-1)$ such that every $B={b_0, ..., b_{t-1}}$ $\ni $ is $\lambda$-covered and every $N\times t$ sub-array contains all ordered subsets from $v$ values of size d at least $\lambda$ times [@ref12], where the set of column $B=b_0, ..., b_{t-1} \supseteq {0, ..., k-1}$.To ensure optimality, we normally want $t-tuples$ to occur at least once. Thus, we consider the value of $\epsilon=1$, which is often omitted. The notation becomes $CA(N;t,k,v)$ [@ref13]. Based on this notation and since the strategy is mainly depending on the interaction degree ($t$), the combinatorial strategies are sometimes termed $t-way$ strategies.
Generation of n-combinations
============================
Different algorithms have been employed in the literature to generate the combinations of input parameters. The most common among them all is the n-bit enumerator. As the name implies, the algorithm starts by enumerating from $0$ to $2^{n – 1}$ ($n$ is the number of input parameters) thereafter, it filters the number base on the specified combination strength. For example, when $t = 4$, the algorithm will only choose those numbers with four true cases and neglect the rest. $N-bit$ enumerator has been used in different combinatorial search strategies in different ways as well as in other research fields (ex. [@ref14]). Often, working perfectly when the number of input parameters is small, however, owing to the complexity of the algorithm $O(n)$, $n-bit$ enumerator becomes complex and the generation process tends to get slower as the number of input parameters increases.
Nowadays, software systems are prone to complexity in different ways due to many configurations and feature that may be present in any software to ensure its functions are properly managed. For example, Software Product Lines (SPL) need several parameters to be adjusted with different configurations due to different products that must be tested. To this end, other algorithms have also been implemented in the literature to overcome this issue. So far, backtracking, counting, and subsets algorithms have been used in the literature to solve this problem by speeding up the construction of t-combinations. Though each of the algorithms has its own approach to constructing the combinations, they also have different limitations for the input parameters and performance. Hence, this paper proposes another approach not only to speed up the construction but to search for them in an efficient way.
Methodology
===========
The Generation of Parameter Combination
----------------------------------------
The algorithm used the CA notation as the base of input. As shown in Algorithm 1, the algorithm took $k$ input parameters and produced t-combination of them each time adding the combinations to a final array containing all $t$-combinations of $k$. Instead of enumerating all $n-bits$, a stack data structure was used to hold the parameters permanently by “push” them into the stack and then “pop” them when needed during the iterations. Additionally, a temporary array was created with index i to help the generated combinations in each iteration (Steps 1-2). A stack data structure ($S$) was created and the first parameter (0) was pushed inside (Steps 3-4). The algorithm continued to iterate until the stack became empty (Step 5). The index number $i$ of the Comb array was set to length of $S -1$ and the value $v$ of this index $i$ was set to the top value in the stack (i.e. pop) until the $v$ was less than $k$ (Steps 6-9). Furthermore, the algorithm continued to increment the $i$ and $v$ then pus the value of $v$ into $S$ until the index number was equal to the length of the required interaction strength t (Steps 9-15). The pseudo code is shown clearly in Algorithm 1. Additionally, for better understanding of the algorithm, a running example is illustrated in Figure 1.\
Let Comb be an array of length $t$ Let $i$ be the index of Comb array Create a stack $S$ $S \leftarrow 0$
Figure 1 shows a running example to illustrate how the combinations of input parameters were generated using three input parameters \[0, 1, and 2\]. With the first parameter pushed into the stack at start, the algorithm iterated and the stack pop its last value to the $i+1$ index of the Comb array. In the next iteration, the stack was pushed by $v+1$ value. The algorithm stopped when the stack became empty. The final array then contained all the interaction of input parameters which are, \[(0:1), (0:2), (1:2)\].
\[h!\]
{width="\linewidth"}
As could be seen in Figure 1, the algorithm kept the previous value of $v$ for the next iteration unless it became greater than the t value. For example, $v = 0$ in the first iteration and in the next iteration, it became $v+1$ which equals to 1. Then it was incremented and pushed into the stack again.
Searching for the Interaction Coverage
--------------------------------------
The combinatorial search strategies need to generate all the possible interaction elements between the input parameters. This step is vital so as to verify how many of these elements can be covered by the suggested solution. Most of the time, this will be the fitness function of the meta-heuristic used in the strategy. It is not clear in most of the implemented strategies which data structure and searching mechanism they used since they are close sourced. However, for the known strategies, there are different mechanisms to store and search for the interaction elements.
The elements could be saved in a database and search later. The searching process could be enhanced by using a kind of indexing mechanism when storing them. However, these will potentially slowdown the search as there could be another outside system that may need to be interfaced with. Thus, another direction is to store the elements in the same program in an array and then search for them.
\[h!\] {width="\linewidth"}
Since there will be a huge amount of elements, the time for searching will increase dramatically with the increase of the parameter and value numbers. To overcome this issue, an indexing mechanism was used to store the elements base on the interaction in a sorting array and then search for a specific element in its corresponding combination. This could speed up the search processes effectively but, the time required for finding a specific element will increase when the number of values increases (since there will be several elements with equal parameter combination). Hence, there is a need to find a new approach to store and search for the interaction elements efficiently.
The proposed approach in this paper uses the HashTable data structure to store the interaction elements. As shown in Figure 2, the data structure is composed of <Key, Value> pairs, and the elements are stored based on the key with each key holding different value. When the program needs to know the number of interaction elements that could be covered by the possible solution, it will send it to the search function. The search function then searches for the interaction element in the exact <key, value> pair. Hence, without the need to search the entire element set, the function knows the location of the specified element.
Experimental Framework
======================
To evaluate the proposed approach, two sets of experiments were performed. The first set of experiment was to evaluate the performances of the parameter combination generation algorithm. This was carried out by running the algorithm under different conditions of input parameters and interaction degrees. In addition, other algorithms were implemented within the same environment to compare the results with them.
In the second set of experiment, the performances of the search mechanism were evaluated and different sets of benchmark were considered. The benchmarks are varied in the number of input parameters and in individual value. The performance is defined by the time it takes the algorithm to find the set of interaction elements for a specific solution. For the purpose of comparison, two other mechanisms were used in the experiments. The first mechanism stored all interaction elements in an array and then search for the elements while the second mechanism, stored all elements in an indexed array and then search for the elements.
All experiments are conducted within an environment of desktop computer with windows 10 installed, CPU 2.9 GHz Intel Core i5, 8 GB 1867 MHz DDR3 RAM, 512 MB of flash HDD. The algorithms are implemented in .Net environment.
Experimental Results and Discussion
===================================
As mentioned earlier, the experiments were performed within two phases. The parameter combination generation algorithm with different parameter size interaction strength was evaluated. The parameters were varied from 20 to 400 input parameters. It is worth to mention here that the algorithm can take more than 1000 parameters as input. However, there was no evidence in the literature showing the use of more than this amount of parameter. In addition, the interaction strength was varied from 2 to 6 since this was the range of interactions used in the research so far. Figure 3 shows the comparison of these results, with the x-axis showing the parameter sizes and the y-axis showing the time in milliseconds in logarithmic scale.
The results showed several important points about the algorithm, and it could be noted that the algorithm performed very well for the generation. Also, it could generate the combination of 400 parameters when $t = 2$ with less than 5 milliseconds. The performance dropped when the interaction strength became higher as could be seen in the figure. However, it still performed well. For example, it could generate the combination of 100 parameters when $t = 6$ with less than 60 seconds. The drop in performance was due to the stack capacity and the several parameters pushed into the stack as the interaction strength increases. It could also be noted from the algorithm that when the interaction strength becomes higher, for example ($t = 6$), 6 parameters should be push and pop each time. This will slow down the algorithm.
\[h!\] {width="\linewidth"}
The second set of experiment was conducted to compare the algorithm with the existing available algorithms. The n-bit enumerator is implemented within our environment for comparison since it is the fundamental algorithm in this direction. When the algorithm was executed, we observed that the algorithm could not generate combinations more than 30 parameters within our environment due to enough memory exception. The limitation of the algorithm was due to the compiler and memory limitations since they could not perform large variable when the parameter ‘values became higher. However, it should be mentioned that the algorithm takes 546 ms for generating combinations of 20 parameters.
The last set of experiment was the search time in the search space. The search time for the relevant interaction elements was measured. This time indicated the maximum time taken by the algorithm to discover the relevant interaction elements for a specific solution. The maximum time was taken because the time may vary and decrease as the algorithm iterates since some of the interaction elements will be deleted. Hence, the maximum time gave a good indication about the time taken by the algorithm when the search space was full. Figures 5 and 6 show this time when $t = 2$ and 3 respectively for two different benchmarks.
As could be noted from the figures, two configurations were taken in the experiments for a covering array generation. The configurations were $CA (N; 2, 10 10)$ and $CA (N; 2, 10 20)$ in which the interaction strength $t = 2$ then $CA (N; 3, 10 10)$ and $CA (N; 2, 10 20)$ where the interaction strength $t = 3$. The configurations represent perfect benchmarks for this experiment since they have many parameters and many values for of the parameters. This will make the search space more complicated with many interaction elements.
{width="\linewidth"}
{width="\linewidth"}
As could be noticed from Figures 4 and 5, the maximum search time for the interaction elements was compared for three searching mechanism, “Hash Algo”, which is our mechanism with “Full-Search” and “indexing” mechanism. As mentioned earlier, the indexing mechanism saved the interaction elements in a sorted array and stored the indexing of each group of elements, however, “Full-Search” mechanism sort all the elements in an array without indexing and hence search exhaustively every time in Full-Search for all elements.
Both Figure 4 and 5 showed that our mechanism reduced the search time dramatically and hence can improve the total generation time of the solution. The figures also show that the “Full-Search” mechanism took more time to find the related interaction elements for a solution. The indexing mechanism showed better performance as compared to “Full-Search.” The searching time for the indexing mechanism was low when the number of parameters and values were low. However, as they were getting higher or the interaction strength is getting higher, the performance dropped due to the many interaction elements that must be searched for in one group of indexing. Our mechanism performed better in a dramatic way as compared to other mechanisms.
For example in Figure 4, when P=10 and t=2, the search time for our mechanism was less than 1(ms), while the indexing took more than 2 (ms). When P=20, our mechanism took less than 5(ms) for search while indexing took more than 8(ms). This improvement in performance could be seen clearly in case of t=3. When P=10, our mechanism took less than 8 (ms) for search and indexing took more than 22 (ms) whereas the “Full-Search” took more than 43 (ms).
The performance of other mechanisms continues to drop in case of t=3 when P=20 in which the indexing search time became 38 (ms) and our mechanism was 17 (ms). It should be mentioned here that this performance of the search affected the totally performance cumulatively since this search process is the most consuming computation in the combinatorial search strategies.
Conclusion
==========
In this paper, we have presented our proposed approach to generate and search for the interaction elements of the input parameters of the combinatorial search strategies. Based on our experience with these strategies, the generation of input parameters’ combinations and search for the interaction elements for the fitness function will slow down the generation process of the final test suite of the interaction. This paper serves as a guide and framework for future implementation of combinatorial strategies. The implemented approach proved its performance for generation input parameters faster than other algorithms for difference sizes and also its performance is faster than other algorithms when searching for the interaction elements. This paper is part of an existing research on combinatorial interaction testing for generating effective test cases for different applications.
Acknowledgment {#acknowledgment .unnumbered}
==============
The first author of the paper would like to thanks IDSIA institute and Swiss Excellence Scholarship for hosting and supporting this research.
|
---
abstract: 'Within a gauge approach to the t-J model, we propose a new, non-BCS mechanism of superconductivity for underdoped cuprates. The gluing force of the superconducting mechanism is an attraction between spin vortices on two different Néel sublattices, centered around the empty sites described in terms of fermionic holons. The spin fluctuations are described by bosonic spinons with a gap generated by the spin vortices. Due to the no-double occupation constraint, there is a gauge attraction between holon and spinon binding them into a physical hole. Through gauge interaction the spin vortex attraction induces the formation of spin-singlet (RVB) spin pairs with a lowering of the spinon gap. Lowering the temperature the approach exhibits two crossover temperatures: at the higher crossover a finite density of incoherent holon pairs are formed leading to a reduction of the hole spectral weight, at the lower crossover also a finite density of incoherent spinon RVB pairs are formed, giving rise to a gas of incoherent preformed hole pairs, and magnetic vortices appear in the plasma phase. Finally, at a even lower temperature the hole pairs become coherent, the magnetic vortices become dilute and superconductivity appears. The superconducting mechanism is not of BCS-type since it involves a gain in kinetic energy (for spinons) coming from the spin interactions.'
address:
- 'Dipartimento di Fisica “G. Galilei” and INFN, I-35131 Padova, Italy '
- 'College of Material Science and Optoelectronics Technology, Graduate University of Chinese Academy of Science, Beijing 100049, China'
- 'Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China'
- 'Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China'
author:
- 'P.A. Marchetti'
- 'F. Ye'
- 'Z.B. Su'
- 'L. Yu'
title: 'Non-BCS superconductivity for underdoped cuprates by spin-vortex attraction'
---
71.10.Hf ,11.15.-q ,71.27.+a
Introduction
============
We propose a new, non-BCS mechanism of superconductivity (SC) for hole-underdoped cuprates relying in essential way upon a “compositeness” [@masy] of the low-energy hole excitation appearing in the spin–charge gauge approach [@jcmp] to the 2D [ *t-J*]{} model, used to describe the CuO planes. This “composite” structure involves a gapful bosonic constituent carrying spin 1/2 (spinon $z_\alpha$) and a gapless spinless fermionic constituent carrying charge (holon $h$), supported on the empty sites. An attractive interaction mediated by an emergent slave-particle gauge field ($A_\mu,
\mu=0,1,2$) binds them into a hole resonance. In terms of this “composite” structure we interpret two crossovers appearing in the normal state of cuprates that are view as “precursors” of superconductivity and the recovery of full coherence of the hole at the superconducting transition.
“Normal” state
==============
To give the key ingredients of the proposed SC mechanism we start shortly reviewing some basic features of the spin-charge gauge approach to the normal state. In the underdoped region of the model the disturbance of hole doping on the antiferromagnetic (AF) background originates spin vortices dressing the holons, with opposite chirality in the two Néel sublattices (see Fig.1).
Propagating in this gas of slowly moving vortices the AF spinons, originally gapless in the undoped Heisenberg model, acquire a finite gap, leading to a short range AF order with inverse correlation length $$\label{ms}
m_s \approx (\delta |\log \delta|)^{1/2}.$$ In eq.(\[ms\]) $\delta$ is the doping concentration and the logarithmic correction is due to the long-range tail of the spin vortices. Eq. (\[ms\]) agrees with experimental data in [@ke]. From the no-double occupation constraint of the $t$-$J$ model emerges the slave-particle gauge field $A_\mu$. It is minimally coupled to holon and spinon and it takes care of the redundant $U(1)$ degrees of freedom coming from the spin-charge decomposition of the hole ($c_\alpha$) of the $t$-$J$ model into spinon and holon. The dynamics of the transverse mode of the gauge field is dominated by the contribution of the gapless holons. Their Fermi surface produces an anomalous skin effect, with momementum scale $$Q \approx (T k_F^2)^{1/3},$$ the Reizer momentum, where $k_F$ is the holon Fermi momentum. As a consequence of the $T$-dependence of the Reizer momentum, the hole (holon-spinon) and the magnon (spinon-antispinon) resonances formed by the gauge attraction have a strongly $T$-dependentent life-time leading to a behaviour of these excitations less coherent than in a standard Fermi-liquid. For the appearance of Reizer skin effect the presence of a gap for spinons is crucial, because gapless spinons would condense at low $T$ thus gapping the gauge field through the Anderson-Higgs mechanism and destroying the $T$-dependent skin effect that decreases the coherence of hole and magnon.
![Pictorial representation of the spin vortices dressing the holons represented by white circles at their center.[]{data-label="Fig.1"}](vortices.eps){width="6cm"}
Superconductivity mechanism
===========================
The gluing force of the proposed superconductivity mechanism is a long-range attraction between spin vortices centered on holons in two different Néel sublattices. Therefore its origin is magnetic, but it is not due to exchange of AF spin fluctuations as e.g. in the proposal of [@pi], [@sus] . Explicitely the relevant term in the effective Hamiltonian is: $$\label{zh}
J (1-2 \delta) \langle z^* z \rangle \sum_{i,j} (-1)^{|i|+|j|} \Delta^{-1}
(i - j) h^*_ih_i h^*_jh_j,$$ where $\Delta$ is the 2D lattice laplacian and $$\langle z^* z \rangle \sim \int d^2q (\vec q^2
+m_s^2)^{-1} \sim (\Lambda^2+m_s^2)^{1/2}-m_s ,$$ with $\Lambda \approx 1$ as a UV cutoff. We propose that, lowering the temperature, superconductivity is reached with a three-step process: at a higher crossover a finite density of incoherent holon pairs are formed, at a lower crossover a finite density of incoherent spinon RVB pairs are formed, giving rise to a gas of incoherent preformed hole pairs and a gas of magnetic vortices appears in the plasma phase, at a even lower temperature both the holon pairs and the RVB pairs, hence also the hole pairs, become coherent and the gas of magnetic vortices becomes dilute. This last temperature identifies the superconducting transition. Clearly this mechanism relies heavily on the “composite” structure of the hole appearing in the “normal” state. Let us analyze in a little more detail these three steps.
Holon pairing
=============
At the highest crossover temperature, denoted as $$T_{ph} \approx J (1-2 \delta) \langle z^* z \rangle ,$$ a finite density of incoherent holon pairs appears, as consequence of the attraction of spin vortices with opposite chirality. We propose to identify this temperature with the experimentally observed (upper) pseudogap (PG) temperature, where the in-plane resistivity deviates downward from the linear behavior. The formation of holon pairs, in fact, induces a reduction of the spectral weight of the hole, starting from the antinodal region [@mg]. The mechanism of holon pair formation is BCS-like in the sense of gaining potential energy from attraction and losing kinetic energy, as shown by the reduction of the spectral weight. As natural due to its magnetic origin, its energy scale is however related to $J$ and not $t$, since the attraction originates from the $J$-term of the $t$-$J$ model. We denote the BCS-like holon-pair field by $\Delta^h$.
Spinon pairing and incoherent hole pairs
========================================
The holon pairing alone is not enough for the appearence of superconductivity, since its occurence needs the formation and condensation of [*hole*]{} pairs. In the previous step instead we have only the formation of holon-pairs. One then firsty needs the formation also of spinon-pairs. It is the gauge attraction between holon and spinon, that, roughly speaking, using the holon-pairs as sources of attraction induces in turn the formation of short-range spin-singlet (RVB) spinon pairs (see Fig.2).
![Pictorial representation of hole pairs, holons are represented by white circles surrounded by vortices, spinons by black circles with spin (arrow); the black line represents spin-vortex attraction, the dashed line the gauge attraction[]{data-label="Fig.2"}](pairs.eps){width="6cm"}
This phenomenon occurs, however, only when the density of holon-pairs is sufficiently high, since this attraction has to overcome the original AF-repulsion of spinons caused by the Heisenberg $J$-term which is positive in our approach, in contrast with the more standard RVB [@RVB] and slave-boson [@lee] approaches. Summarizing, at a intermediate crossover temperature, denoted as $T_{ps}$, lower than $T_{ph}$ in agreement with previous remarks, a finite density of incoherent spinon RVB pairs are formed, which, combined with the holon pairs, gives rise to a gas of incoherent preformed hole pairs. We denote the RVB spinon-pair field by $\Delta^s$. It turns out that for a finite density of spinon pairs there are two (positive energy) excitations, with different energies, but the same spin and momenta. They are given, [*e.g.*]{}, by creating a spinon up and destructing a spinon down in one of the RVB pairs. The corresponding dipersion relation, thus exhibits two (positive) branches (see Fig.3): $$\label{sd}
\omega (\vec k) = 2t \sqrt{(m_s^2 - |\Delta^s|^2) + (|\vec k| \pm
|\Delta^s|)^2.}$$ The lower branch exhibits a minimum with an energy lower than $m_s$, analogous to the one appearing in a plasma of relativistic fermions [@wel]; it implies a backflow of the gas of spinon-pairs dressing the “bare” spinon.
![The positive branches of the spinon dispersion relation in presence of finite density of RVB spinon pairs[]{data-label="Fig.3"}](spind.eps){width="6cm"}
Hence RVB condensation lower the spinon kinetic energy, but, as explained above, its occurrence needs the gauge interaction to overcome the spinon Heisenberg repulsion. The two-branches dispersion of the spinon (\[sd\]) is reminiscent of the hourglass shape of the neutron resonance found in the superconducting region and slighily above in temperature for underdoped samples [@hour]. If a suitable attraction mechanism for spinon and antispinon works, one can show that a similar dispersion is induced for the magnon resonance [@mg], directly comparable with experimental data. As soon as we have a finite density of hole pairs the RVB-singlet hole-pair field $\Delta^c
\approx \Delta^s /\Delta^h$ is non-vanishing and the gradient of its phase describes magnetic vortices. Hence below $T_{ps}$ a gas of magnetic vortice (vortex-loops in space-time) appears, in the plasma phase, because the incoherence of the hole pairs leads to a vanishing expectation value of the phase of $\Delta^c$. Therefore, we propose to identify $T_{ps}$ with the experimental crossover corresponding to the appearance of the diamagnetic and (vortex) Nernst signal [@ong]. This interpretation is reinforced by the computation of the contour-plot of the spinon pair density in the $\delta-T$ plane [@mysy], ressembling the contour-plot of the diamagnetic signal. The presence of holon pairs is required in advance to have RVB pairs, and two factors contribute to the density of holon pairs: the density of holons and the strength of the attraction, $\approx J (1-m_s)$ from (\[zh\]). These two effects act in opposite way increasing doping, this yields a finite range of doping for a non-vanishing expectation value of $|\Delta^s|$, starting from a non-zero doping concentration, producing a “dome” shape of $T_{ps}$ and of the contour-plot. The RVB spinon pair formation is clearly not BCS-like, the energy gain coming from the kinetic energy of spinons as discussed above; its energy scale is again related to $J$.
Hole-pair coherence and superconductivity
=========================================
Finally, at a even lower temperature, the superconducting transition temperature $T_c$, both holon pairs and RVB pairs, hence also the hole pairs, become coherent and a d-wave hole condensate $$\langle \sum_{\alpha,\beta} \epsilon_{\alpha \beta} c_{i \alpha} c_{j \beta} \rangle$$ appears, corresponding to a non-vanishing expectation value of the hole-pair field $\Delta^c$. As soon as the holon and RVB pairs condense the slave-particle global gauge symmetry is broken from $U(1)$ to ${\bf
Z}_2$. The Anderson-Higgs mechanism then implies a gap for the gauge field $A_\mu$ increasing with the density of RVB pairs. In this “Higgs”-phase the magnetic vortices become dilute. Therefore the SC transition appears as a 3D XY-type transition for magnetic vortices in presence of a dynamical gauge field, similar in this respect to the transition in the “phase-fluctuation” scenario proposed in [@ek]. One can prove that in our model a gapless gauge field is inconsistent with the coherence of holon pairs, i.e. coherent holon pairs cannot coexist with incoherent spinon pairs; hence the condensation of both occurs simultaneously. Since the gauge field binding spinon and holon into a hole resonance becomes massive in the superconducting state, one expects that the life-time of such resonance becomes $T$-independent, because the $T$-dependent anomalous skin effect appearing in the normal state is suppressed by the mass. Therefore the hole become coherent at the superconducting transition.
The appearance of two temperatures, one for pair formation and a lower one for pair condensation, is typical of a BEC-BCS crossover regime for a fermion system with attractive interaction [@BEC]. In this sense the incoherent hole pairs discussed in previous section play a role analogous to that of the “preformed pairs” considered e,g, in [@ue]. In our approach, however, the gas of hole pairs appears only at finite doping, implying a fortiory a “dome” shape for the superconductivity region starting from a non-vanishing critical doping concentration at $T=0$. This result is in agreement with experimental data, but at odds with standard fermionic BEC-BCS attractive systems, where the condensation usually persists in the extreme BEC limit [@BEC], and with the original Mean Field version of the slave boson approach [@lee], where holon condensation occur for arbitrary small holon density at $T=0$.
The non-vanishing critical doping for the “dome” exhibited in our approach appears also in [@tes], where, however, a nodal structure is argued to be present for the hole already in the region where the magnetic vortices are not dilute. On the contrary in that region our approach preserves a finite FS, as consequence of the incoherence of the holon pairs, and nodes for holes appear only below the superconducting transition. The superconducting transition being of XY-type is “kinetic energy” driven, but again related to the $J$- scale since the dynamics of vortices is triggered by the mass of the spinon and of the gauge field, both originated from the Heisenberg term. The “kinetic energy” driven character of the superconducting transition appears consistent with some experimental data for underdoped and optimally doped samples [@bon].
Final comments
==============
Our approach exhibits another crossover [@jcmp], $$T^* \approx t/8 \pi |\log \delta|,$$ intersecting $T_{ps}$. Such crossover is not directly related to superconductivity. It corresponds to a change in the holon dispersion. It is characterized by the emergence of a “small” holon FS around the momenta $\pm \pi/2, \pm \pi/2$, with complete suppression of the spectral weight for holes in the antinodal region and partial suppression outside of the magnetic Brillouin zone. Induced physical effects are observable experimentally both in transport and thermodynamics [@ma]. This crossover appears only in two-dimensional bipartite lattices. Below $T^*$ the effect of short-range AF fluctuations become stronger and the transport physics of the corresponding normal state region (“pseudogap”) is dominated by the interplay between the short-range AF of spinons and the thermal diffusion induced by the gauge fluctuations triggered by the Reizer momentum, producing in turn the metal-insulator crossover [@jcmp]. We identify $T^*$ in experimental data with the inflection point of in-plane resistivity and the broad peak in the specific heat coefficient [@ma].
The above presentation is just a sketch of the approach, many details have been already worked out explicitely, others remain admittedly conjectural , but the mechanism of SC proposed here is rather complete in its main structure and qualitatively consistent with many experimental features, as partially discussed above. A more complete presentation of our approach to superconductivity will appear in [@mysy].
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abstract: 'The spectral distribution of parametrically excited dipole-exchange magnons in an in-plane magnetized epitaxial film of yttrium-iron garnet was studied by means of frequency- and wavevector-resolved Brillouin light scattering spectroscopy. The experiment was performed in a parallel pumping geometry where an exciting microwave magnetic field was parallel to the magnetizing field. It was found that for both dipolar and exchange spectral areas parallel pumping excites the lowest volume magnon modes propagating in the film plane perpendicularly to the magnetization direction. In order to interpret the experimental observations, we used a microscopic Heisenberg model that includes exchange as well as dipole-dipole interactions to calculate the magnon spectrum and construct the eigenstates. As proven in our calculations, the observed magnons are characterized by having the highest possible ellipticity of precession which suggests the lowest threshold of parametric generation. Applying different pumping powers we observe modifications in the magnon spectrum that are described theoretically by a softening of the spin stiffness.'
author:
- 'A. A. Serga'
- 'C. W. Sandweg'
- 'V. I. Vasyuchka'
- 'M. B. Jungfleisch'
- 'B. Hillebrands'
- 'A. Kreisel'
- 'P. Kopietz'
- 'M. P. Kostylev'
title: 'Brillouin light scattering spectroscopy of parametrically excited dipole-exchange magnons'
---
Introduction
============
Parallel pumping, the parametric excitation of spin waves by means of a linearly polarized microwave magnetic field parallel to an external bias magnetic field, was first described in 1960.[@Morgenthaler60; @Schloemann60] The technique has been used for the last five decades to investigate a number of interesting phenomena such as the excitation of magnetoelastic waves in ferromagnets,[@Schloemann69] spin-wave parametric instability,[@Schloemann60; @Wiese94] high density magnon gases and condensates,[@Demokritov06; @Demidov08; @Kloss10; @Neumann08] parametric amplification of spin-wave solitons and bullets,[@Kostylev_soliton; @Patton_soliton; @Serha_bullet] spin pumping and the inverse spin Hall effect.[@Sandweg11; @KurebayashiNatureMat11; @KurebayashiAPL11; @Ando11] While the main principles of parametric pumping have been revealed in experiments with bulk ferrimagnetic samples the more recent research is focused on the application of this technique to ferro- and ferrimagnetic films. Thus, the understanding of the peculiarities of parametric excitation in the application to magnetic films is of paramount importance for the correct interpretation of many interesting effects (see, for example, Refs. focused on microwave driven ferrite-platinum bi-layers and nano-contacts).
Parallel pumping has two essential advantages in comparison with the direct excitation of spin waves. Firstly, when spin waves are directly driven by a microstrip antenna using an alternating Oersted field, the wavenumber of the effectively excited spin waves is rather small, in the range of $10^{3}$ rad/cm due to the finite width of the antenna. Therefore, direct excitation of large-wavenumber spin waves can hardly be realized. Secondly, parallel pumping is a threshold process, i.e. the driving field amplitude $h$ should exceed a certain threshold value $h_\mathrm{crit}$, which is dependent on the magnetic losses in the material, in order to start the parametric generation. Once this threshold has been overcome, the amplitude of the parametric spin waves grows exponentially in time until higher order nonlinear processes limit this amplitude to a certain level.[@Lvov; @Gurevich_Melkov] With this method it is possible to achieve much higher spin-wave intensities in comparison to the direct linear excitation of spin waves with a microstrip antenna.
During the process of parallel parametric amplification, spin waves are excited at half of the initial microwave pumping frequency $\omega_\mathrm{p}$. In terms of energy quanta, a microwave photon splits into two magnons which are the quasiparticles of the dynamic magnetization. From the laws of the conservation of energy and momentum and from the fact that for the case of weakly localized pumping the photon wavenumber is negligibly small, [@Melkov_JETP] it then follows that the pumping process creates pairs of magnons with the same frequency ${\omega_{p}}/{2}$ and with opposite wavevectors $\bf k$ and $-\bf k$, as shown in Fig. \[Fig1\] (a). Different magnon groups with the same frequency ${\omega_{p}}/ {2}$ are parametrically driven at the same time, and only the one with the lowest damping survives.[@Lvov; @restoration1; @restoration2] The determination of the spectral position of this dominant group is of crucial importance not only for the investigation of the primarily pumped magnons but for all the magnon groups which acquire their energy in subsequent scattering processes from the dominant group.
Here we present a direct observation of the dominant group for different pumping frequencies and powers in a wide wavevector range up to $1.6\times 10^{5}$ rad/cm using Brillouin light scattering (BLS) spectroscopy. The experiments are performed on a single-crystal epitaxial film of yttrium-iron garnet (YIG).[@SagaYIG; @YIG_Magnonics] We show experimentally that the primarily excited spin waves of the dominant group are always located at the 90$^\circ$ branch of the backward volume magnetostatic wave (BVMSW) having the lowest energy. Previous works on this topic were carried out by Kabos *et al.*,[@Kabos97; @Wettling83] but only for dipolar-dominated spin waves with a wavenumber up to $4\times 10^{4}$ rad/cm. In our present work we complete the picture by obtaining more details for the dipolar-dominated waves as well as by exploring the parametric processes involving the exchange-dominated BVMSW. By changing the pumping frequency $\omega_\mathrm{p}$ in sufficiently small step sizes, as indicated in Fig. \[Fig1\] (a), it is possible to move along the dispersion branch of the BVMSW and to determine the spectral position of the dominant group. In addition, calculations of the magnon dispersion based on an effective spin model for YIG have been performed which are in excellent agreement with the experimentally obtained results.
This paper is organized as follows: Sec. \[sec:ppa\] provides the reader with basic information about the technique of parallel parametric amplification. Calculations of the relevant magnon modes are performed in Sec. \[sec:theory\]. In Sec. \[sec:setup\] details of the combined BLS/microwave experiment are given, especially the wide wavevector selectivity. Sec. \[sec:expdat\] shows the experimental results which clarify the position of the dominant group for different pumping frequencies and powers. These experimental data are supported by theoretical investigations of the spin-wave dispersion. Finally, in Sec. \[sec:conclusion\], we summarize our results and present our conclusions.
Parallel parametric amplification {#sec:ppa}
=================================
A key condition for the parametric amplification is ellipticity of magnetization precession. The circular precession implies that the tip of the magnetization vector goes around a circle lying in the plane perpendicular to the applied field. In this case the component of the magnetization along the external bias magnetic field is constant: $$M_\mathrm{z}=\mathrm{const}.$$ The parametric amplification can be considered as a response of the spin system to the application of a weak alternating magnetic field parallel to the bias magnetic field ${\bf H}_\mathrm{0}$, which is oscillating with a precession frequency $2\omega_\mathrm{0}$ which is twice the eigenfrequency of the spin wave $\omega_\mathrm{0}$ we intend to excite parametrically: $$h(t)=h_\mathrm{0}\cos(2\omega_\mathrm{0}t).$$ Therefore, for the purely circular precession, the average energy pumped into the system vanishes: $$\left\langle \Delta E\right\rangle \propto \left\langle \textbf{h}\cdot\textbf{M} \right\rangle =
\left\langle h \right\rangle M_\mathrm{z} = h_\mathrm{0}M_\mathrm{z}
\int^{T_\mathrm{0}}_{0}\frac{dt}{T_0}\cos(2\omega_\mathrm{0}t)\equiv 0,$$ where $\bf{M}$ denotes the magnetization and $T_\mathrm{0}$ denotes the time for one cycle of precession.
An elliptical precession implies that this circle is squeezed in some direction in the plane which is perpendicular to the $z$-axis. As shown in Fig. \[Fig1\] (b), in this case the $M_\mathrm{z}$ component of magnetization oscillates with double frequency: $$M_\mathrm{z}=\mathrm{const}.+m_\mathrm{z,0} \cos(2\omega_\mathrm{0}t).$$ This allows absorption of microwave energy by the spin system: $$\left\langle \Delta E\right\rangle \propto
\int^{T_\mathrm{0}}_{0}\frac{dt}{T_0}\cos^{2}(2\omega_\mathrm{0}t)\neq 0.$$
For a bulk magnetic material the dynamic demagnetizing field along the direction of propagation of the plane spin wave gives rise to the ellipticity of the magnetization precession. In this case, the threshold for the parallel pumping is given by a simple analytical formula:[@Gurevich_Melkov] $$h_\mathrm{crit}=\min \left\{ \frac{\omega_\mathrm{p} \Delta
H_{\bf k}}{\omega_\mathrm{M}\sin^{2}\vartheta_{\bf k}} \right\},$$ where $\Delta H_{\bf k} = 2 \Gamma_{\bf k}/ \gamma$, $\omega_\mathrm{M}=\gamma 4 \pi M_\mathrm{s}$, and $\vartheta_{\bf k}$ is the angle between the propagation direction of the parametrically amplified spin waves with wavevector $\bf k$ and the external bias magnetic field. $\Gamma_{\bf k}$ denotes the relaxation frequency of this magnon mode, $\gamma$ is the gyromagnetic ratio, and $M_\mathrm{s}$ is the saturation magnetization. Importantly, from this expression it follows that the threshold field $h_\mathrm{crit}$ is inversely proportional to the ellipticity $\epsilon_{\bf k}$, which is defined by $$\epsilon_{\bf k} = 1 - \frac{m_{x_{\bf k}}^2}{m_{y_{\bf k}}^2} ,
\label{eq:ellipticity}$$ where $m_x$ and $m_y$ are the amplitudes of the dynamic components of the total magnetization vector ${\bf M}=({\bf m}_x, {\bf m}_y, {\bf M}_z)$, and ${\bf y} \bot {\bf k}$. Consequently, $h_\mathrm{crit}$ reaches the smallest possible value at $\vartheta_{\bf k}={\pi}/{2}$ when the ellipticity reaches its maximum.[@Gurevich_Melkov; @Wiese94]
In the confined geometry of thin films the picture of the dynamic demagnetizing field is more complicated. The out-of-plane component of the dynamic demagnetizing field, caused by the presence of the two film surfaces, “compresses” the precession cone in the direction of the film plane. As a result, the value of $h_\mathrm{crit}$ becomes dependent not only on the direction (as for the bulk materials), but also on the value of the in-plane wavenumber. In addition, the finite film thickness results in quantization of the total magnon wavenumber in the direction perpendicular to the film surface. A simple analytical expression can easily be obtained for the mode of the uniform precession (the wave with the vanishing value of the total wavenumber) $h_\mathrm{crit} \propto 1 / \eta_{{\bf k},n}$, where $$\eta_{{\bf k},n} = \left(1 - \frac{m_{x_{\bf k},n}^2}{m_{y_{\bf k},n}^2}\right)\left(\frac{m_{y_{\bf k},n}}{m_{x_{\bf k},n}}\right) .$$ In this work we extend the validity of this formula for the entire range of the wavevectors parametrically excited in the films. Under this assumption, and with the calculation shown in the next section, we find that in the case of intermediate film thicknesses (when $m_{x_{\bf k},n} \lesssim m_{y_{\bf k},n}$) no qualitative difference between $\eta$ and $\epsilon$ exists. Therefore, in the following, we restrict ourself to the classical definition of ellipticity given by Eq. \[eq:ellipticity\]. However, in contrast to bulk materials[@Gurevich_Melkov], in this work the discrete character of the magnon spectrum is also taken into account.
Another particular type of parametric amplification is perpendicular pumping. It is characterized by the application of an alternating pumping field in the plane perpendicular to the direction of the equilibrium magnetization, i.e, the alternating magnetization induced by the pumping field is always perpendicular to the external bias magnetic field.[@Suhl57] In this case, the amplification is caused by the parametric decay of the uniform precession mode, driven out-of-resonance (i.e. at $2 \omega_0$). In our work we use the microwave Oersted field of a microstrip transducer for the parametric pumping. The Oersted field of such a transducer has two components: one in-plane and one out-of-plane. The in-plane component is parallel to the bias field and thus is responsible for the parallel pumping process. Similarly, the out-of-plane component of the alternating Oersted field is responsible for a possible perpendicular pumping process[@NeumannAPL09] which we want to avoid in our work. Fortunately, the in-plane component of the Oersted field is maximized below the longitudinal axis of the microstrip line and, the out-of-plane component is mostly localized near the microstrip edges. Therefore, by using a sufficiently small laser spot size and probing the area near the longitudinal axis one can exclude the contribution of the perpendicular pumping mechanism from the parametric process.
Theoretical investigations of the magnon spectrum {#sec:theory}
=================================================
In order to find the spectral positions of the parametrically injected magnons in a film sample, we used a recently developed microscopic approach [@KreiselEPJ] for calculating the magnon energies as well as the ellipticity of the precession. Our starting point is a microscopic Hamiltonian that describes the properties of the relevant magnon modes in YIG. The model for our magnetic films of a finite thickness $d$ contains both exchange and dipole-dipole interactions and is completed by a Zeeman term that takes into account the external bias magnetic field ${\bf H}_0$:
$$\begin{aligned}
{\cal{H}}&=-\frac 12 \sum_{ij} J_{ij} {\bf S}_i \cdot {\bf S}_j -\mu {\bf H}_\mathrm{0}\cdot \sum_i{\bf S}_i\notag\\
&\hspace{-.5cm}\phantom{=}-\frac 12 \sum_{ij,i\neq j}\frac{\mu^2}{|{\bf R}_{ij}|^3} \left[3 ({\bf S}_i\cdot\hat{\bf R }_{ij})({\bf S}_j\cdot\hat{\bf R }_{ij}) -{\bf S}_i \cdot {\bf S}_j\right].
\label{eq:hamiltonian}\end{aligned}$$
The spin operators ${\bf S}_i$ are normalized such that ${\bf S}_i^2=S(S+1)$ with an effective total spin $S$ per lattice site. The sums run over the sites ${\bf R}_i$ of a cubic lattice with spacing $a=12.376\,\text{\AA}$, and $\hat{\bf R}_{ij}={\bf R}_{ij}/|{\bf R}_{ij}|$ are unit vectors in the direction of ${\bf R}_{ij}={\bf R}_i-{\bf R}_j$. The relevant parameters are the exchange interactions $J_{ ij}=J$ of neighboring spins and the magnetic moment $\mu=g\mu_B$, where $g$ is the effective $g$-factor and $\mu_B$ is the Bohr magneton. In order to proceed using a compact notation we introduce the dipolar tensor $D_{ij}^{\alpha \beta}=D^{\alpha\beta}({\bf R}_i-{\bf R}_j)$, $$\begin{aligned}
D_{ij}^{\alpha\beta}&=(1-\delta_{ij})\frac{\mu^2}{|{\bf R}_{ij}|^3}\left[3\hat{R}_{ij}^\alpha\hat{R}_{ij}^\beta-\delta^{\alpha\beta}\right]
\label{eq:defdip}\end{aligned}$$ and rewrite our effective Hamiltonian (\[eq:hamiltonian\]) as $${\cal{H}}=-\frac 12 \sum_{ij}\sum_{\alpha\beta} \left[J_{ij}\delta^{\alpha\beta}+D_{ij}^{\alpha\beta}\right] S_i^\alpha S_j^\beta -h\sum_i S_i^z ,$$ with the z-axis of the frame of reference pointing along the direction of the external magnetic field ${\bf H}_0=h/\mu {\bf e}_z$. We assume that the continuous film is magnetically saturated by the magnetic field ${\bf H}_0$ in its plane and the ferromagnetic magnetic order is perfect. This allows us to expand the Hamiltonian in terms of bosonic operators describing fluctuations around the classical ground state. Inserting the Holstein–Primakoff transformation[@Holstein40] we obtain a bosonized spin Hamiltonian of the form[@KreiselEPJ] $${\cal{H}} = {\cal{H}}_{0} + \sum_{n=2}^{\infty} {\cal{H}}_n \; .$$ Considering the large effective spin $S=M_\mathrm{s} a^3/\mu \approx 14.2$, it is sufficient to retain only terms up to $n=2$ in the formal $1/S$ expansion in order to calculate the magnetic excitation spectrum. The quadratic part of the Hamiltonian reads $${\cal{H}}_2 = \sum_{ij} \left[ A_{ij} b^{\dagger}_i b^{\phantom{\dagger}}_j + \frac{B_{ij}}{2}
\left( b_i b_j + b^{\dagger}_i b^{\dagger}_j \right) \right]\; ,
\label{eq:H2}$$ with
$$\begin{aligned}
A_{ij} & = \delta_{ij} h + S ( \delta_{ij} \sum_{n} J_{ in} - J_{ ij} )
\nonumber
\\
& \phantom{=}+ S \left[ \delta_{ij} \sum_{n} D_{in}^{zz}-
\frac{D_{ij}^{xx} + D_{ij}^{yy}}{2} \right],
\\
B_{ij} & = - \frac{S}{2} \left[ D_{ij}^{xx} - 2 i D_{ij}^{xy} - D_{ij}^{yy} \right].
\end{aligned}$$
A thin film of thickness $d=Na$ is obviously not translational invariant in all spatial directions, such that a full Fourier transformation cannot be used. Instead we set $ {\bf R}_i = ( x_i , {\bf r}_i )$ with ${\bf r}_i = ( y_i , z_i )$ and use the property of the discrete translational invariance in the $y$ and $z$ directions to partially diagonalize ${\cal{H}}_2$ via a partial Fourier transformation. We expand the bosonic operator $$b_i = \frac{1}{\sqrt{N_{y} N_z}} \sum_{ {\bf k}} e^{ i {\bf k} \cdot {\bf r}_i }
b_{\bf k} ( x_i )\;,$$ where $N_y$ and $N_z$ is the number of lattice sites in the $y$ and $z$ direction. The Hamiltonian (\[eq:H2\]) then reads $$\begin{aligned}
{\cal{H}}_{2} & = \sum_{\bf k} \sum_{ x_i, x_j}
\Bigl \{ [ \mathbf{A}_{ \bf k} ]_{ ij } b^{\dagger}_{ \bf k} ( x_i ) b^{\phantom{\dagger}}_{ \bf k} ( x_j )
\nonumber
\\
&\hspace{-.5cm} +
\frac{ [ \mathbf{B}_{ \bf k} ]_{ ij } }{2} b_{ \bf k} (x_i ) b_{ -\bf k} ( x_j )
%\nonumber \\ &
+ \frac{ [ \mathbf{B}_{ \bf k} ]_{ ij }^{ \ast } }{2} b^{\dagger}_{ \bf k} ( x_i )
b^{\dagger}_{ - \bf k} ( x_j )
\Bigr\},
\label{eq:H2Gauss}\end{aligned}$$ with the $N\times N$ matrices $\mathbf{A}_{ \bf k}$ and $\mathbf{B}_{ \bf k}$ given by [@Costa00]
$$\begin{aligned}
[ \mathbf{A}_{ \bf k} ]_{ ij } & = \sum_{ \bf r}
e^{ - i {\bf k} \cdot {\bf r}} A ( x_i - x_j , {\bf r} )\; ,
\nonumber\\
&=SJ_{\bf k}(x_{ij})+\delta_{ij}\bigl[ h+S\sum_{n}D_0^{zz}(x_{in})\bigr]\nonumber\\
&\phantom{=}-\frac S2 \bigl[D_{\bf k}^{xx}(x_{ij})+D_{\bf k}^{yy}(x_{ij})\bigr ] , \\
[ \mathbf{B}_{ \bf k} ]_{ ij } & = \sum_{ \bf r}
e^{ - i {\bf k} \cdot {\bf r} } B ( x_i - x_j , {\bf r} )\nonumber\\
&\hspace{-.5cm}=-\frac S2 \bigl[D_{\bf k}^{xx}(x_{ij})-2i D_{\bf k}^{xy}(x_{ij})-D_{\bf k}^{yy}(x_{ij})\bigr].
\end{aligned}$$
\[eq:ab\]
The exchange matrix is given by $$\begin{aligned}
J_{\bf k}(x_{ij})&=J \bigl[\delta_{ij}\bigl\{6-\delta_{j1}-\delta_{jN}\notag\\
&\hspace{-.5cm}-2(\cos (k_ya)+\cos(k_za))\bigr\}-\delta_{i j+1}-\delta_{i j-1}\bigr] ,
\label{eq:exmatrix}\end{aligned}$$ and the dipolar tensor reads $$\begin{aligned}
D_{\bf k}^{\alpha \beta} (x_{ij}) &= {\sum_{{\bf r}_{ij}}}^\prime e^{-i {\bf k}\cdot {\bf r}_{ij}} D_{ij}^{\alpha \beta} ,
\label{eq:dipsum}\end{aligned}$$ where $\Sigma^\prime$ excludes the term $y_{ij}=z_{ij}= 0$ when $x_{ij}=0$. We can rewrite the quadratic Hamiltonian in matrix notation $${\cal{H}}_2=\frac 12 \sum_{\bf k} (\vec b_{\bf k}^\dagger,\vec b_{-\bf k}^{\phantom{\dagger}}) {\mathcal D}_{\bf k}\left(\begin{array}{c}
\vec b_{\bf k}\\
\vec b_{-\bf k}^\dagger
\end{array}\right)
\label{eq:H2vec}$$ with the grand dynamic matrix $${\mathcal D}_{\bf k}=\left(\begin{array}{cc}
\mathbf{A}_{ \bf k}&\mathbf{B}_{ \bf k} \\
\mathbf{B}_{ \bf k}^T&\mathbf{A}_{ \bf k}^T \\
\end{array}\right)$$ and the row vector $$\vec b_{\bf k}^\dagger =(b_{\bf k}^\dagger(x_1),\ldots,b_{\bf k}^\dagger(x_N))$$ Observing the low symmetry of the matrices
$$\begin{aligned}
\mathbf{A}_{ \bf k} = \mathbf{A}_{ \bf -k}^T\neq \mathbf{A}_{ \bf k}^T\\
\mathbf{B}_{ \bf k} = \mathbf{B}_{ \bf -k}^T\neq \mathbf{B}_{ \bf k}^T\end{aligned}$$
we have to use a full $2N$ square transformation $${\mathcal J}_{\bf k}=\left(\begin{array}{cc} {\mathbf U}_{\bf k}^\dagger&- {\mathbf V}_{\bf k}^\dagger\\
- {\mathbf W}_{\bf k}^\dagger& {\mathbf X}_{\bf k}^\dagger\end{array}\right)$$ to diagonalize the Hamiltonian (\[eq:H2vec\]). This transformation connects the vectors of the true magnon operators $\vec{\gamma}_{\bf k}^\dagger=(\gamma_{{\bf k},1 }^\dagger,\ldots,\gamma_{{\bf k},N}^\dagger)$ to those in the Holstein–Primakoff basis via $$\left(\begin{array}{c}
\vec{\gamma}_{\bf k}\\
\vec{\gamma}_{-\bf k}^\dagger
\end{array}\right)={\mathcal J}_{\bf k}\left(\begin{array}{c}
\vec{b}_{\bf k}\\
\vec{b}_{-\bf k}^\dagger
\end{array}\right)=\left(\begin{array}{c}
{\mathbf U}_{\bf k}^\dagger \vec{b}_{\bf k}^{\phantom{\dagger}}-{\mathbf V}_{\bf k}^\dagger \vec{b}_{-\bf k}^\dagger\\
-{\mathbf W}_{\bf k}^\dagger \vec{b}_{\bf k}+{\mathbf X}_{\bf k}^\dagger \vec{b}_{-\bf k}^\dagger
\end{array}\right)\label{eq:trans}$$ and transforms to the diagonal Hamiltonian $${\cal{H}}_2=\frac 12 \sum_{\bf k}(\vec \gamma_{\bf k}^\dagger,\vec \gamma_{-\bf k}^{\phantom{\dagger}}) {\mathcal E}_{\bf k}\left(\begin{array}{c}
\vec \gamma_{\bf k}\\
\vec \gamma_{-\bf k}^\dagger
\end{array}\right)+E_0^{(2)}$$ with the matrix $$\begin{aligned}
{\mathcal E}_{\bf k}&=\bigl(\mathcal J_{\bf k}^\dagger\bigr)^{-1} {\mathcal D}_{\bf k}^{\phantom{-1}}{\mathcal J}^{-1}_{\bf k}\notag\\
&=\mathop{\mathrm{diag}}(E_{{\bf k},1},\ldots,E_{{\bf k},N},E_{{\bf k},1},\ldots,E_{{\bf k},N})\end{aligned}$$ and the correction $$E_0^{(2)}=\frac 12\sum_{\bf k}\mathop{\mathrm{Tr}} {\mathcal E}_{\bf k}$$ to the groundstate energy.\
In order to numerically calculate the eigenenergies $E_{{\bf k},n}$, $n=1,\ldots,N$ of all magnon modes and the transformation matrix for a given in-plane wavevector $\bf k$ we carry out the slowly converging sum in (\[eq:dipsum\]) using Ewald summation technique to set up the grand canonical matrix ${\mathcal D}_{\bf k}$. Following the algorithm of Ref. , we first calculate the Cholesky decomposition $\mathcal K_{\bf k}$ of $D_{\bf k}$, which has the property $$\mathcal D_{\bf k}={\mathcal K}_{\bf k}^\dagger {\mathcal K}_{\bf k}^{\phantom{\dagger}}.$$ Then we diagonalize the matrix ${\mathcal M}_{\bf k}$ given by $${\mathcal M}_{\bf k}={\mathcal K}_{\bf k} I_p {\mathcal K}_{\bf k}\;,\quad I_p=\left(\begin{array}{cc}
1_N&0\\
0&-1_N
\end{array}\right)$$ using a unitary transformation ${\mathcal U}_{\bf k}$ to obtain the diagonal matrix ${\mathcal L}_{\bf k}$ which is connected to the eigenenergies via $${\mathcal E}_{\bf k}=I_p {\mathcal L}_{\bf k}.$$ The inverse transformation $${\mathcal J}^{-1}_{\bf k}=\left(\begin{array}{cc}
{\mathbf U}_{\bf k}& {\mathbf W}_{\bf k}\\
{\mathbf V}_{\bf k}& {\mathbf X}_{\bf k}
\end{array}\right)$$ is now obtained from the equation $${\mathcal J}^{-1}_{\bf k}={\mathcal K}_{\bf k}^{-1} {\mathcal U}_{\bf k}^{\phantom{-1}} {\mathcal E}_{\bf k}^{1/2}.$$ Having constructed the full transformation, we then obtain the amplitude structure of the modes across the film. It is worth noting that in our approach the anti-nodal points of the magnon modes are located on the film surfaces which is consistent with the unpinned surface spin configuration.[@SW_and_pinning]
In line with the discussion in the previous section we also calculate the ellipticity of the modes $\epsilon_{{\bf k},n}$. The ellipticity can be calculated from the expectation values of the spin operators in the magnon eigenstates,[@Majlis] $$\epsilon_{{\bf k},n}=1-\frac{\langle (S^x)^2\rangle_{{\bf k},n}}{\langle (S^y)^2\rangle_{{\bf k},n}}.$$ $S^\alpha$ denotes the $\alpha$ component of the total spin operator and the expectation value has to be taken in the magnon eigenstate $|{\bf k},n\rangle=\gamma_{{\bf k},n}^\dagger | 0\rangle$ of the $n$-th magnon mode. According to a calculation in lowest order $1/S$ we use the expansion of the Holstein–Primakoff transformation to express the spin operators in terms of boson operators to obtain
$$\begin{aligned}
\langle \bigl(S^{{x}}\bigr)^2\rangle_{{\bf k},n}&=\frac S4 \bigl[\langle \vec b_{-\bf k}\cdot \vec b_{\bf k}\rangle_{{\bf k},n}+ \langle \vec b_{-\bf k}^{\phantom{\dagger}}\cdot \vec b_{-\bf k}^\dagger\rangle_{{\bf k},n}\notag\\
&\phantom{=\frac S4 \bigl(}+\langle \vec b_{\bf k}^\dagger\cdot \vec b_{-\bf k}^\dagger\rangle_{{\bf k},n}+ \langle \vec b_{\bf k }^\dagger\cdot \vec b^{\phantom{\dagger}}_{\bf k }\rangle_{{\bf k},n}\bigr],\\
\langle \bigl(S^{{y}}\bigr)^2\rangle_{{\bf k},n}&=\frac S4 \bigl[\langle \vec b_{-\bf k}\cdot \vec b_{\bf k}\rangle_{{\bf k},n}- \langle \vec b_{-\bf k}^{\phantom{\dagger}}\cdot \vec b_{-\bf k}^\dagger\rangle_{{\bf k},n}\notag\\
&\phantom{=\frac S4 \bigl(}+\langle \vec b_{\bf k}^\dagger\cdot \vec b_{-\bf k}^\dagger\rangle_{{\bf k},n}- \langle \vec b_{\bf k }^\dagger\cdot \vec b^{\phantom{\dagger}}_{\bf k }\rangle_{{\bf k},n}\bigr].\end{aligned}$$
An inversion of the transformation (\[eq:trans\]) allows the calculation of the expectation values,
$$\begin{aligned}
\langle \vec b_{-\bf k }^{\phantom{\dagger}} \cdot \vec b_{-\bf k}^\dagger\rangle_{{\bf k},n}&=\sum_j\bigl([U_{\bf k}^\dagger]_{nj}[U_{\bf k}^{\phantom{\dagger}}]_{jn}+[W_{\bf k}^\dagger]_{nj}[W_{\bf k}^{\phantom{\dagger}}]_{jn}\bigr),\\
\langle \vec b_{\bf k }^\dagger \cdot \vec b_{\bf k }^{\phantom{\dagger}}\rangle_{{\bf k},n}&=\sum_j\bigl([V_{\bf k}^\dagger]_{nj}[V_{\bf k}^{\phantom{\dagger}}]_{jn}+[X_{\bf k}^\dagger]_{nj}[X_{\bf k}^{\phantom{\dagger}}]_{jn}\bigr),\\
\langle \vec b_{\bf k}^\dagger \cdot \vec b_{-\bf k}^\dagger\rangle_{{\bf k},n}&=\sum_j\bigl([U_{\bf k}^\dagger]_{nj}[V_{\bf k}^{\phantom{\dagger}}]_{jn}+[W_{\bf k}^\dagger]_{nj}[X_{\bf k}^{\phantom{\dagger}}]_{jn}\bigr),\\
\langle \vec b_{-\bf k} \cdot \vec b_{\bf k}\rangle_{{\bf k},n}&=\sum_j\bigl([V_{\bf k}^\dagger]_{nj}[U_{\bf k}^{\phantom{\dagger}}]_{jn}+[X_{\bf k}^\dagger]_{nj}[W_{\bf k}^{\phantom{\dagger}}]_{jn}\bigr),
\end{aligned}$$
such that the ellipticity is given by $$\epsilon_{{\bf k},n}=\frac{2B_{{\bf k},n}}{A_{{\bf k},n}+B_{{\bf k},n}},
\label{eq:quant_ellipticity}$$ with the abbreviations
$$\begin{aligned}
A_{{\bf k},n}&=\sum_j\bigl(|[V_{{\bf k}}]_{jn}|^2+|[X_{{\bf k}}]_{jn}|^2 \notag\\
&\phantom{= \sum_j\bigl(} +|[U_{{\bf k}}]_{jn}|^2+|[W_{{\bf k}}]_{jn}|^2\bigr),\\
B_{{\bf k},n}&=2\mathop{\mathrm{Re}}\sum_j \bigl([V_{{\bf k}}]_{jn}[U_{{\bf k}}^\ast]_{jn}+[X_{{\bf k}}]_{jn}[W_{{\bf k}}^\ast]_{jn}\bigr).\end{aligned}$$
In the following we only analyze the characteristics of magnons with $\vartheta_{\bf{k}}={\pi}/{2}$ as these magnons have the highest ellipticity and consequently one may expect the strongest parametric coupling to this type of magnons. Furthermore, in our experiment these magnons propagate along the length of the microstrip transducer. This length is much greater than the free propagation path for the magnons. Therefore, one may expect that, in contrast to all other magnons which have a non-vanishing component of the in-plane wavevector perpendicular to the longitudinal axis of the transducer, the threshold of the parallel pumping excitation for them is not affected by the energy loss due to magnons escaping the pumped area.[@NeumannAPL09; @Sholom] Figure \[figek\] shows the calculated magnon dispersion along with the ellipticity of the precession as a function of the magnon wavenumber $k$. The spectrum comprises the finite set (limited by the number of crystallographic elementary cells of $12.376$ [Å]{} on the film thickness) of the quantized volume magnon modes (BVMSW) and a steep branch corresponding to the surface Damon-Eshbach magnon mode.
Setup {#sec:setup}
=====
The measurements were performed using combined microwave and optical facilities where the magnon system was parametrically pumped by the microwave circuit, and the response of the magnon system is analyzed by means of Brillouin light scattering spectroscopy. The microwave circuit comprises a microwave source, a switch, and a microwave amplifier connected to a probe section (see Fig. \[Fig3\] (a)). In contrast to the conventional approach where the pumping frequency is held constant and the applied bias magnetic field $H_0$ is swept we examined the magnon spectrum at different pumping frequencies holding the applied field and consequently the ground state for small amplitude-excitations constant. This implies that the spectral characteristics shown in Fig. 2 were preserved during the experiment. The pumping microwave Oersted field was created by a 50 $\mu$m wide short-circuited microstrip line. The regular microstrip line was utilized instead of the commonly used microwave resonator [@KurebayashiNatureMat11; @Neumann08; @NeumannAPL09] as it enables us to change the pumping frequency without the complicated realignment of the probe section. The investigated sample, a 15 mm long and 3 mm wide YIG film with a thickness of 5 $\mu$m, was placed on top of the microstrip line. An external static field $H_{0}$ of 1750 Oe was in the film plane and perpendicular to the longitudinal axis of the microstrip. In order to excite dipolar as well as exchange-dominated magnons, the frequency of the microwave source was varied from 13.6 GHz up to 14.6 GHz in steps of 20 MHz, which corresponds to change ${\omega_\mathrm{p}}/{2}$ from 6.8 GHz to 7.3 GHz in steps of 10 MHz. Thus, according to the magnon spectrum, a wide range of wavenumbers from zero up to $1.6\cdot 10^{5}$ rad/cm could be investigated by means of BLS spectroscopy.
The experiments were performed at different microwave powers from 100 mW to 10 W. In order to reduce any possible thermal effects which might have potentially influenced the magnon spectrum at high pumping powers by modifying the saturation magnetization $M_\mathrm{s}$, anisotropy fields, and the exchange stiffness constant the pumping was applied in 2 $\mu$s microwave pulses separated by 20 $\mu$s time intervals. For the same reason, a metallized aluminum nitride substrate, which is known for its high thermal conductivity, was chosen as a base plate for the probe section.
In order to probe the parametrically pumped magnons, the BLS measurements were performed in the back scattering geometry, where a single-mode solid state laser with a wavelength of 532 nm was focused onto the sample by using an objective with a high numerical aperture. The focal point of 20 $\mu$m in diameter was centered on the longitudinal axis of the microstrip line where the microwave Oersted field is parallel to the bias magnetic field $H_{0}$, and thus the condition for parallel parametric pumping is satisfied. The backscattered light was collected with the same objective and sent to a tandem Fabry-Pérot interferometer for frequency and intensity analysis. In a classical description, Brillouin light scattering can be interpreted as the diffraction of the probing light from a moving Bragg grating produced by a magnon mode, see Fig. \[Fig3\] (b). As a result, some portion of the scattered light is shifted in frequency by the frequency of this mode (in our case ${\pm\omega_\mathrm{p}}/{2}$). The intensity of the inelastically scattered light is directly proportional to the number of magnons in the mode. In addition, the diffraction from the grating leads to a transfer of momentum during this process. By changing the angle $\Theta_\mathrm{B}$ between the sample and the incident light beam, the wavevector selection with a resolution of up to $4500$ rad/cm in a film plane can be implemented as shown in Ref. . The incident angle $\Theta_\mathrm{B}$ of the probing light determines the selected magnon wavenumber $k=2k_\mathrm{sc}\sin(\Theta_\mathrm{B})$, where the wavenumber of the scattered light $k_\mathrm{sc}$ is equal to the wavenumber of the incident light $k_\mathrm{in}$.
Experimental results {#sec:expdat}
====================
Figure \[Fig4\] shows the measured BLS intensity of parametrically injected magnons as a function of the frequency and the wavevector for different pumping powers. To increase the signal-to-noise ratio the intensity was integrated across the entire width of the inelastically scattered peak for ${\omega_\mathrm{p}}/{2}$. The color-coded intensity maps, where blue (dark) corresponds to low intensities and orange (bright) to higher intensities, were recorded by changing the pumping frequency for the given incident angle $\Theta_\mathrm{B}$ of the probing light (see Fig. \[Fig1\]). Once the frequency measurement cycle had been completed, the incident angle $\Theta_\mathrm{B}$ was changed and the measurement were repeated in this way for wavenumbers ranging from 0 to $1.6 \times 10^{5}$ rad/cm in steps of $8.2 \times 10^{3}$ rad/cm.
In Fig. \[Fig4\] (a), the color coded BLS-intensity map measured at the pumping power $P_{1}=0.1$ W, which is slightly (approximately 1 dB) above the threshold power of the parametric generation, is compared with the theoretical calculations of the relevant modes of the magnon spectrum. It is clearly visible that the detected magnons are located along the dispersion curve representing the lowest magnon mode $n=1$ while other regions of the spectrum show practically no increase in the BLS intensity. This behavior can be attributed to the fact that from all magnon groups pumped at the same time only the dominant group having the lowest damping and the highest coupling to the pumping field is significantly populated.
The lowest mode corresponds to the backward volume magnetostatic wave having one node along the thickness of the YIG film. Modes with higher frequencies belong to higher order standing-wave modes, quantized perpendicular to the film plane. As we have no reason to assume different damping for different modes spread across the relatively narrow frequency and wavenumber ranges, our attention must be focused on the coupling efficiency. Since the elliptical precession of the magnetization allows one to excite the corresponding magnon modes (compare Fig. \[Fig1\]), as discussed above, we assume that the threshold of parametric generation is proportional to the inverse ellipticity, $h_\mathrm{crit} \propto 1/\epsilon_{{\bf k},n}$. Observing Fig. \[Fig5\], one notices that the lowest magnon mode, $n=1$, has the largest ellipticity, and correspondingly the lowest threshold, in the entire pumping frequency range. Therefore, one would expect parametric amplification of this mode as the dominant magnon mode.
While the bias magnetic field is kept fixed during the measurement, the saturation magnetization $M_\mathrm{s}$ is influenced by the pumping of magnons. Further, single magnons carry a total spin of 1 and every magnon reduces the magnetization by one Bohr magneton $\mu_\mathrm{B}$. In order to account for that in the theoretical calculation of the spectrum we fix the exchange coupling $J$ at zero pumping and set the relevant spin stiffness to $$D=\frac{J M_\mathrm{s} a^5}{\mu^2}\;.$$ Since the absolute magnon densities, composed by thermal magnons and pumped magnons, are unknown at this point, we use the saturation magnetization as a fit parameter. An adjustment of $M_\mathrm{s}$ then changes the position of the ferromagnetic resonance and the slope of the spectrum that originates from the effective exchange couplings. However, the slope of the spectrum also depends on the saturation magnetization and therefore needs to be adjusted to the experimental data. The solid curves in Fig. \[Fig4\](b)-(c) correspond to the transversal BVMSW mode having the lowest energy. To fit the experimental data we varied the value of the saturation magnetization. We saw that with increasing pumping power, $M_\mathrm{s}$ decreased significantly, in accordance with the theory, from 1750 G for the undisturbed sample to 1676 G at pumping level of $P_{3}=10$ W. At the same time, one can see that the precision of the fit decreases for the largest pumping power. It can be connected to a spread of magnon population beyond the dominant magnon mode and to corresponding blurring of the BLS intensity over the wide frequency and $k$-number ranges. This spread is caused by a number of physical phenomena. Firstly, at high pumping levels a number of frequency-degenerated magnon groups with damping higher than one for the dominant group can be excited.[@Lvov] Secondly, two-magnon scattering processes from crystal defects, dopants, and other static magnetic non-uniformities breaks the law of conservation of momentum, and thus lead to a spreading of the parametrically excited magnon population along the isofrequency lines in the ${\bf k}$-space. Thirdly, interactions between magnons and lattice vibrations will cause a redistribution of the magnons towards states with lower energy. [@comment1] Finally, non-elastic four-magnon scattering processes are responsible for occupation of energy levels around the initial magnon state at half of the pumping frequency, and thus for consequent thermalization of the parametrically injected magnons across the entire spin-wave spectrum. [@Cherepanov86]
In addition, we want also to comment on the role of the intermodal concurrency in parametric generation.[@Kozhus_experiment] We have observed no fine structure in the occupation of quantized standing modes at the beginning of the magnon spectrum as it was predicted by the theory in Refs. and reported in Ref. . This might be due to the fact that under our experimental conditions the inverse ellipticity curve of the lowest BVMSW mode lies below the corresponding curves of the highest standing modes and, as a result, this mode is solely amplified across the entire range of the experimentally probed frequencies (Fig. \[Fig5\]). Moreover, in accordance with our estimation based on Eq. \[eq:quant\_ellipticity\], the inverse ellipticity curve of the lowest BVMSW mode intersects first with the 65th standing mode at 9.6 GHz. The in-plane wavenumber of the lowest mode at this frequency is $k = 4\times 10^{5}$ rad/cm. The wavenumber of the 65th mode, calculated across the film, is $n \pi/d = 4.08\times 10^{5}$ rad/cm. Both of these values are significantly larger than the highest optically accessible wavenumber value of $2.6\times 10^{5}$ rad/cm. As a result no intermodal jumps in the magnon generation, and consequently no discrete structure in the BLS intensity map are expected to be observed in our experiment.
Conclusion {#sec:conclusion}
==========
We have observed the generation of parametric magnons under parallel pumping using the wavevector resolved BLS technique. In the theoretical part we constructed the magnon eigenstates for a thin ferromagnetic film in the framework of a microscopic Heisenberg model in linear spin-wave theory to be able to calculate such physical characteristics as eigenfrequencies, ellipticity, and spatial distribution of magnon modes. A good agreement between the experimentally determined spectral position of photon-coupled magnon pairs in a tangentially magnetized YIG film and the lowest-frequency magnon mode propagating in a film plane perpendicularly to the magnetization direction was obtained. Combining the theoretical results with the experimental observations, the dominant parametric excitation of the lowest mode is understood as a result of its highest ellipticity in the range of accessible wavenumbers. A fine structure which was previously reported in the threshold of parametric generation in YIG films was not detected in our experiment. This is possibly caused by the structure of the magnon spectra of the magnetic films in our case: the ellipticity of the lowest-frequency transversal magnon mode may be maximal across the entire range of the optically detectable $k$-numbers. In addition, a significant broadening of the populated spectral area is observed at high pumping powers. This phenomenon is associated with a variety of non-linear and linear scattering processes leading to thermalization of the parametrically injected magnons.
Financial support from the Deutsche Forschungsgemeinschaft within the SFB/TRR 49, from the Australian Research Council, and from the Australian-Indian Strategic Research Fund is gratefully acknowledged.
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It is worth to note that below the lowest transversal magnon mode, $n=1$, there are plenty of magnon states with a nonzero $k$-vector component along the magnetization direction.
|
---
abstract: 'We show that the number of entire maximal graphs with finitely many singular points that are conformally equivalent is a universal constant that depends only on the number of singularities, namely $2^n$ for graphs with $n+1$ singularities. We also give an explicit description of the family of entire maximal graphs with a finite number of singularities all of them lying on a plane orthogonal to the limit normal vector at infinity.'
author:
- 'Isabel Fernández [^1]'
title: The number of conformally equivalent maximal graphs
---
Introduction {#sec:intro}
============
The present paper is devoted to the study of maximal graphs in the Lorentz-Minkowski space $\L^3=(\R^3,\langle\cdot,\cdot\rangle),$ where $\langle(x_1,x_2,x_3),(y_1,y_2,y_3)\rangle=x_1y_1+x_2y_2-x_3y_3$. Maximal graphs appear in a natural way when considering variational problems. If $u:\Omega\subset\R^2\equiv\{x_3=0\}\to\R$ is a smooth function defining a spacelike graph in $\L^3$ (that is, a graph with Riemannian induced metric), then its area is given by the expression $$A(u)=\int_\Omega\sqrt{1-|\nabla u|^2},$$ (recall that $|\nabla u|<1$ since the graph is spacelike). The corresponding equation for the critical points of the area functional in $\L^3$ is $$\label{eq:maximal}
\mbox{Div}\frac{\nabla u}{\sqrt{1-|\nabla u|^2}} =0.$$ Spacelike graphs satisfying this (elliptic) differential equation are called [*maximal graphs*]{}, since they represent local maxima for the area functional. Geometrically, this condition is equivalent to the fact that the mean curvature of the surface in $\L^3$ vanishes identically. Besides of their mathematical interest, these surfaces, and more generally those having constant mean curvature, have a significant importance in physics [@mardsen].\
{width=".8\textwidth"}\[fig:catrie\]
From a global point of view, it is known by Calabi’s theorem [@calabi] that the only everywhere regular complete maximal surface is the plane. In particular, there are no entire maximal graphs besides the trivial one. This motivates to allow the existence of singularities, i.e., points of the surface where the metric degenerates. We will focus here our attention to the case where the singular set is the [*smallest*]{} possible, that is, a finite number of points. The first and most known example is the Lorentzian catenoid (Figure \[fig:catrie\], left), an entire maximal graph with one singular point, and actually the only one as proved in [@ecker], but there are examples with any arbitrary number of singularities. Among them it is worth mentioning the Riemann type maximal graphs (Figure \[fig:catrie\], right) obtained in [@lopez-lopez-souam], with two singular points and characterized by the property of being foliated by circles and lines. Other highly symmetric examples with arbitrary number of singularities (even infinitely many) were constructed in [@praga] (Figure \[fig:3y4picos\]). Actually there is a huge amount of such graphs. Indeed, in [@conelike] the authors study the moduli space $\mathcal{G}_n$ of entire maximal graphs with $n+1$ singularities, proving that it is an analytic manifold of dimension $3n+4$. A global system of coordinates in this space is given by the position of the singular points in $\L^3$ and a real number called [*the logarithmic growth*]{} that controls the asymptotic behavior.\
![Entire maximal graphs with isolated singularities.[]{data-label="fig:3y4picos"}](3y4picos){width="\textwidth"}
If $u:\Omega\to\R$ defines a maximal graph, singular points appear where $|\nabla u|=1$. At a singular point, the PDE stops being elliptic. Moreover, the tangent plane of the surface becomes lightlike, the normal vector has no well defined limit, and the surface is asymptotic to a half of the light cone of the singular point. For this reason they are called [*conelike singularities*]{}. It should be pointed out that a maximal surface with isolated conelike singularities is an entire graph if and only if it is complete (that is, divergent curves have infinite length), as proved in [@conelike].\
If $S$ is a maximal surface with singular set $F\subset S$, its regular part $S\setminus F$ has a natural conformal structure associated to its Riemannian metric. The conformal type of a maximal surface has been widely studied, for example in [@paralore; @alias] parabolicity criteria for maximal surfaces are given, but there also exist hyperbolic examples, [@alarcon1; @alarcon2; @muy].
In the case of entire graphs with $n+1$ singularities, it turns out that $S\setminus F$ is conformally equivalent to a $n$-connected circular domain of the complex plane, that is, the plane with $n+1$ discs removed. Each one of these boundary circles corresponds to a singular point of the graph. Our aim in this paper is to study the space of entire maximal graphs with the same conformal structure, that is
> [**Problem.**]{} [*Given a $n$-connected circular domain $\Omega$ of the complex plane, how many entire maximal graphs with $n+1$ singularities are there whose conformal structure is biholomorphic to $\Omega$?*]{}
We will answer this question by proving that the number of (non congruent) maximal graphs supported by a fixed circular domain is finite and does not depend on the circular domain, but only on the number of connected component of the boundary, that is, the number of singularities. This will be the aim of Section \[sec:first\]. Thus, our problem reduces to compute the number of graphs for a fixed conformal structure. In Section \[sec:count\] we will fix an specific $n$-connected circular domain (Definition \[def:omega\]) and we will find out how many entire graphs are there with this conformal structure, obtaining that there are exactly $2^n$ non-congruent surfaces. Moreover, the graphs constructed in Section \[sec:count\] can be characterized by the property of having all their singularities in a plane orthogonal to the limit normal vector at infinity (Theorem \[th:alineados\]).
Let us point out that our main result contrast with the analogous problem in the related theory of solutions to the Monge-Ampère equation $$\label{eq:hess} \mbox{Hess}(u) =1.$$ Specifically, in [@GMM] it is proved that any solution to globally defined on $\R^2$ with finitely many isolated singularities is uniquely determined by its associated conformal structure, which is also a circular domain of the complex plane.
Preliminaries {#sec:prelim}
=============
Maximal surfaces {#sec:maximal}
----------------
A differentiable immersion $X:M\to\l^3$ from a surface $M$ to $\l^3$ is said to be spacelike if the tangent plane at any point is spacelike, that is to say, the induced metric on $M$ is Riemannian. The Gauss map of a spacelike surface in $\L^3$ takes values in the sphere of radius $-1$, $\h^2=\{p\in\L^3\;:\;\langle p,p\rangle=-1\}$. Since $\h^2$ has two connected components, $\h^2_+=\h^2\cap\{x_3>0\}$ and $\h^2_-=\h^2\cap\{x_3<0\}$, spacelike surfaces are always orientable.\
A maximal immersion is a spacelike immersion whose mean curvature vanishes. A remarkable property of maximal surfaces in $\l^3$ is the existence of a Weierstrass-type representation for maximal surfaces, similar to the one of minimal surfaces. Roughly speaking, the Weierstrass representation of a conformal maximal immersion $X:M\to\l^3$ is a pair $(g,\phi_3)$ of a meromorphic function and a holomorphic $1$-form defined on $M$ such that, up to translation, the immersion can be recovered as $$\label{eq:repr}
X(p):=\mbox{Real} \int_{p_0}^p \big(
\frac{i}{2}(\frac{1}{g}-g)\phi_3,
\frac{-1}{2}(\frac{1}{g}+g)\phi_3,
\phi_3
\big) ,$$ where $p_0\in M$ is an arbitrary point. It is worth mentioning that $g$ agrees with the stereographic projection of the Gauss map of the surface. We refer to [@kobayashi; @ecker] and Theorem \[th:representation\] below for more details.
We will focus our attention to entire maximal graphs, that is, maximal graphs defined on the whole plane $\{x_3=0\}$. As we explained in Section \[sec:intro\], the only everywhere regular example is the plane [@calabi], and so singularities (i.e., points where the induced metric converges to zero) appear in a natural way in this setting. The following theorem condense the information regarding the global structure of entire maximal graphs with isolated singularities (also called [*conelike singularities*]{}).
\[pro:global\] Let $S$ be a surface with isolated singularities in $\L^3$. Then the following two statements are equivalents:
(i) $S$ is a complete embedded maximal surface,
(ii) $S$ is an entire graph over any spacelike plane.
In this case $S$ is asymptotic at infinity to either a half-catenoid or a plane. If we label $F\subset S$ as the singular set, $S\setminus F$ is conformally equivalent to $\Omega_0:=\c\setminus\cup_{p\in F} D_p,$ where $D_p$ are pairwise disjoint closed discs. Moreover, the associated conformal reparameterization $X:\Omega_0\to\L^3$ extends analytically to $\Omega:=\c\setminus\cup_{p\in F}\mbox{Int}(D_p)$ by putting $X(\partial(D_p))=X(p).$ The point $p_\infty=\infty$ is called the [*end*]{} of the surface.
Double surface and representation theorem {#sec:repr}
-----------------------------------------
As showed in the previous section, the underlying conformal structure of an entire maximal graph with an isolated set of singularities is conformally equivalent to a circular domain in the complex plane. We now go into this aspect in depth to obtain a representation theorem for entire maximal graphs with a finite number of singularities that will be crucial in our study.
For any finitely connected circular domain $\Omega=\c\setminus\cup_{j=1}^k
\mbox{Int}(D_j)$, let $\Omega^*$ be its mirror surface and $\Nb$ the double surface obtained by gluing $\Omega$ and $\Omega^*$ along their common boundaries as in Figure \[fig:mirror\] (see [@farkas] for an explicit description of this construction). It is clear that $\Nb$ is a compact Riemann surface of genus $k-1$ minus two points. We denote by ${\overline}{\Nb}$ the compactification of $\Nb$ by adding these two points.
Finally, we label $J:\Nb\to\Nb$ as the mirror involution mapping a point in $\Omega$ into its mirror image and viceversa. Notice that $J$ extends to an antiholomorphic involution on ${\overline}{\Nb}$, and its fixed point set of $J$ coincides with $\partial\Omega\equiv\partial\Omega^*.$
![The double surface associated to a maximal surface with singularities[]{data-label="fig:mirror"}](mirror){width="16cm"}
This double surface is used in [@conelike] to give a characterization of complete maximal surfaces with a finite number of singularities in terms of their Weierstrass data:
\[th:representation\] Let $X:\Omega\to\l^3$ be a conformal immersion of an entire maximal graph with $n+1$ conelike singularities, where $\Omega=\c\setminus\cup_{j=1}^{n+1}\mbox{Int}(D_j),$ $D_j$ pairwise disjoint closed discs. Label ${\overline}{\Nb}$ as the compactification of the double surface of $\Omega.$ Then the Weierstrass data of $X,$ $(g,\phi_3)$, satisfy:
(i) $g$ is a meromorphic function on $\overline{\Nb}$ of degree $n+1,$ $|g|<1$ on $\Omega$, and $g \circ J=\frac{1}{\overline{g}}$,
(ii) $\phi_3$ is a holomorphic 1-form on $\overline{\Nb}\setminus\{p_\infty,J(p_\infty)\},$ where $p_\infty=\infty
\in \Omega,$ with poles of order at most two at $p_\infty$ and $J(p_\infty),$ and satisfying $J^*(\phi_3)=-\overline{\phi_3},$
(iii) the zeros of $\phi_3$ in $\overline{\Nb}\setminus\{p_\infty,J(p_\infty)\}$ coincide (with the same multiplicity) with the zeros and poles of $g.$
Conversely, let ${\overline}{\Nb}$ be a compact genus $n$ Riemann surface. Suppose that there exists an antiholomorphic involution $J:\overline{\Nb} \rightarrow \overline{\Nb}$ such that the fixed point set of $J$ consists of $n+1$ pairwise disjoint analytic Jordan curves $\gamma_j,$ $j=0,1,\ldots,n,$ and that $\overline{\Nb}\setminus\bigcup_{j=0}^n \gamma_j={\Omega_0} \cup
J({\Omega_0}),$ where $\overline{\Omega_0}$ is topologically equivalent (and so conformally) to ${\overline}{\c}$ minus a finite number of pairwise disjoint open discs.
Then, for any $(g,\phi_3)$ satisfying $(i),$ $(ii)$ and $(iii)$ the map $ X:\overline{\Omega_0}\setminus\{p_\infty\} \rightarrow \l^3$ given by Equation is well defined and $S=X(\overline{\Omega_0}\setminus\{p_\infty\})$ is an entire maximal graph with conelike singularities corresponding to the points $q_j:=X(\gamma_j),$ $j=0,$ $1,\ldots,n.$
Divisors on a Riemann surface. {#sub:riemann}
------------------------------
An important part of our work in this paper deals with classical properties of divisors on compact Riemann surfaces. We recall here the notation and basics results that will be used in the sequel (see [@farkas] for more details).
Let $\Sigma$ be a Riemann surface. A (multiplicative) divisor on $\Sigma$ is a formal symbol $\Db=p_1^{k_1}\cdot\ldots p_h^{k_h},$ where $p_{k_j}\in\Sigma$ and $k_j\in\z.$ We can also write the divisor $\Db$ as $$\Db=\prod_{p\in\Sigma} p^{k_p},$$ where $k_p\neq 0$ only for finitely many. We call $\div(\Sigma)$ to the multiplicative group of divisors on $\Sigma$. We can define an order in $\div(\Sigma)$, indeed, given $\Db_1=\prod_{p\in\Sigma}p^{k_p^1}$ and $\Db_2=\prod_{p\in\Sigma}
p^{k_p^2} \in \div(\Sigma)$, we say that $\Db_1\geq \Db_2$ if $k_p^1\geq k_p^2$ for all $p\in\Sigma.$
The degree of the divisor $\Db$ is defined as the integer $\deg(\Db)=\sum_{p\in\Sigma}k_{p}.$ $\Db\in \div(\Sigma)$ is an [*integral*]{} divisor if $k_p\geq 0$ for any $p\in\Sigma.$ We denote by $\div_k(\Sigma)$ the set of integral divisors of degree $k.$\
Let $f$ be a meromorphic function on $\Sigma.$ The associated divisor of $f$ is defined as $(f)=\prod_{p\in\Sigma} p^{k_p},$ where for any zero (resp. pole) $p$ of $f$ of order $\alpha$ we have $k_p=\alpha >0$ (resp. $k_p=-\alpha <0$), and $k_p=0$ in other case. Likewise we define the associated divisor of a meromorphic 1-form. Classical theory of Riemann surfaces give that both functions and $1$-forms are determined by their divisors up to a multiplying constant. Moreover, the degree of a meromorphic function on a compact Riemann surface is $0$, whereas the associated divisor of a $1$-form has degree $2n-2$, where $n$ is the genus of the surface.
A first approach to the problem {#sec:first}
===============================
Let $G$ be an entire maximal graph with $n+1$ conelike singularities. When $n=0$, Ecker [@ecker] characterized the Lorentzian catenoid (Figure \[fig:catrie\], left) as the unique entire maximal graph with $1$ singular point, so we will assume from now on that $n\geq 1$.\
As showed in Section \[sec:maximal\], the underlying conformal structure of a maximal graph is conformally equivalent to a circular domain $\Omega\subset\C$ with $n+1$ boundary components. Moreover, if we rotate the surface so that the end is horizontal, as a consequence of Theorem \[th:representation\] the divisors of the Weierstrass data $(g,\phi_3)$ of $G$ must be of the form $$\label{eq:div}
(g)=\frac{D\cdot p_\infty}{D^\ast\cdot p_\infty^\ast},\qquad (\phi_3)=\frac{D\cdot D^\ast}{p_\infty\cdot p_\infty^\ast},$$ where $p_\infty=\infty\in\overline\Omega$ is the end of the surface, $D\in\div_n(\overline\Omega)$, and $\,^\ast$ denotes the mirror involution. Notice that the divisor $D$ determines uniquely the Weierstrass data $(g,\phi_3)$ up to replacing by $(e^{i\theta}\,g,A\,\phi_3)$, for any $\theta,A\in\R$.
Conversely, for any integral divisor $D$ of degree $n$ on $\overline\Omega$ such that there exist a meromorphic function $g$ and $1$-form $\phi_3$ satisfying , it is immediate to check that $(g,\phi_3)$ fulfill conditions $(i)$ to $(iii)$ in Theorem \[th:representation\]. Thus by means of Equation we can obtain an entire maximal graph with $n+1$ conelike singularities, horizontal end, and conformal structure $\Omega$. Moreover, this graph is unique up to homotheties and vertical rotations.
The problem of finding out whether exists a pair $(g,\phi_3)$ satisfying for a given divisor $D$ is closely related with the Abel-Jacobi map of the corresponding compact Riemann surface $\overline{\Nb}$, $ \varphi: \div(\overline{\Nb}) \to \mathcal{J}(\overline{\Nb}) $, where $\mathcal{J}(\overline{\Nb})$ denotes the Jacobian bundle of $\overline{\Nb}$ (see [@farkas] for its definition). Abel Theorem states that $\Db\in\div(\overline{\Nb})$ is the divisor associated to a meromorphic function (resp. 1-form) on $\overline{\Nb}$ if and only if $\varphi(\Db)=0$ (resp. $\varphi(\Db)=T$, where $T\in \mathcal{J}(\overline{\Nb})$ is a fixed element in the Jacobian bundle). Thus, in our case the divisors $D$ coming from Weierstrass data are precisely those satisfying: $$\varphi(D)+\varphi(p_\infty) - \varphi(D^\ast) - \varphi(p_\infty^\ast)=0, \quad \varphi(D) + \varphi(D^\ast) - \varphi(p_\infty) - \varphi(p_\infty^\ast) = T.$$
This set of divisors is deeply studied in [@conelike], proving that the previous two equations are equivalent to $$\label{eq:spin}
2\varphi(D) - 2\varphi(p_\infty^\ast)=T.$$ Before going into the properties of this set, let us fix some notation. Let $\Omega$ be a $n$-connected circular domain and write $\partial\Omega=\cup_{j=0}^{n} \gamma_{c_j}(r_j)$, with $\gamma_{c_j}(r_j)=\{z\in\C\;,\;|z-c_j|=r_j\}$. Up to a Möbius transformations we can assume that $c_0=0,$ $r_0=1$ and $c_1\in\r^+$. Thus, we can parameterize the space $\Tb_n$ of marked (i.e., with an ordering in the boundary components) $n$-connected circular domains (up to biholomorphisms) by their corresponding uplas $v=(c_1,r_1,\ldots,c_n,r_1,\ldots,r_n)\in\R^+\times\C^{n-1}\times(\R^+)^n,$ of centers and radii, with the convention $c_0=0$ and $r_0=1$. By this identification, $\Tb_n$ can be considered as an open subset of $\r^+\times\C^{n-1}\times(\R^+)^n$, and therefore it inherits a natural analytic structure of manifold of dimension $3n-1$. We label as $\Omega(v)$ the circular domain defined by $v\in\Tb_n$. Now define the [*spinorial bundle*]{} $$\Sb_n=\{ (v,D)\;:\; v\in\Tb_n,\; 2 \varphi_v(D)- 2 \varphi_v(p_\infty^\ast) = T_v \},$$ where the subscript $v$ refers to the double surface of $\Omega(v)$, then
The spinorial bundle $\Sb_n$ defined above is an analytical manifold of dimension $3n-1$ . Moreover, the map $$\nu:\Sb_n\to\Tb_n$$ $$\nu(v,D)=v$$ is a finitely sheeted covering.
Thus, the number of divisors $D\in\div_n(\overline{\Omega(v)})$ satisfying Equation is a universal constant that depends not on the conformal structure $\Omega(v)$, but only on the number of boundary components (equivalently, the number of singularities of the maximal graph). As explained above, each divisor corresponds to a unique congruence class of entire maximal graphs with $n+1$ singularities and conformal structure $\Omega(v)$. Thus we have the following
\[co:rec\] For each $n\in\N$ there exists a constant $C(n)\in\N$ such that, for any $n$-connected circular domain $\Omega$, the number of non-congruent entire maximal graph with conformal structure biholomorphic to $\Omega$ is exactly $C(n)$.
\[re:comp\] Since the space $\Tb_n$ is simply-connected, it follows from Corollary \[co:rec\] that the number of connected components of $\Sb_n$ is $C(n)$. In particular, the number of connected components of the moduli space of entire maximal graphs with $n+1$ singularities is also $C(n)$. Indeed, label ${\mathcal{G}}_n$ as the space of [*marked*]{} entire maximal graph with horizontal end and $n+1$ singularities, where a mark means an ordering $m=(q_0,\ldots,q_n)$ of the singular points of the graph. As we commented in Section \[sec:intro\], ${\mathcal{G}}_n$ can be endowed with a differentiable structure of manifold of dimension $3n+4$ with coordinates given by $(G,m)\mapsto (m,c)$, being $c$ the logarithmic growth at the end. On the other hand, we can consider the map $$\epsilon: {\mathcal{G}}_n\to\Sb_n\times\L^3\times \S^1\times\R$$ $$\epsilon((G,m))=((v,D),q_0,g(1),h(1))$$ where, if $(g,\phi_3)$ denote the Weierstrass data of the graph, then
- $(v,D)\in\Sb_n$ is given by the conformal structure of $G$ (with the order in $v\in\Tb_n$ given by the order in $m$), and the divisor $D$ defined as in Equation ,
- $q_0$ is the first singular point in $m$,
- $h:=\frac{\phi_3}{dz}$ (here $z$ means the natural conformal parameter in $\Omega(v)\subset\C$, recall that $1\in\partial\Omega(v)$ for all $v\in\Tb_n$).
Then, it is clear from the above explanation that $\epsilon$ is bijective. Moreover, the induced topology in ${\mathcal{G}}_n$ by $\epsilon$ agree with the one given by its before mentioned differentiable structure, as proved in [@conelike]. Thus, the number of connected components of $\mathcal{G}_n$ is $C(n)$.
Counting maximal graphs on a given circular domain {#sec:count}
==================================================
As it was showed in the previous section, the number of maximal graphs that share the same underlying conformal structure only depends on the number of boundary components of the conformal support. Thus, in this section we will fix an specific circular domain and we will find out how many non-congruent maximal graphs are defined on that surface.\
Let $n\in \n,$ and $a_1<a_2<\ldots <a_{2n+2}\in\r.$ Throughout this section, ${\overline}{\Nb}_0$ will denote the (hyperelliptic) compact genus $n$ Riemann surface associated to the function $\sqrt{\prod_{j=1}^{2n+2}(z-a_j)},$ that is, $${\overline}{\Nb}_0:=\{(z,w)\in{\overline}{\c}^2\;:\;
w^2=\prod_{j=1}^{2n+2}(z-a_j)\}.$$ And we will also define $\Nb_0={\overline}{\Nb}_0\setminus\{z^{-1}(\infty)\}.$
The surface ${\overline}{\Nb}_0$ can be realized as a two sheeted covering of the Riemann sphere. Indeed, consider two copies of ${\overline}{\c}.$ Following [@farkas], we label these copies as sheet $I$ and sheet $II.$ We “cut” each copy along curves joining $a_{2j+1}$ with $a_{2j+2},$ for any $j=1,\ldots,n.$ We assume that these cuts does not intersect each others (see Figure \[fig:2copias\]). Each cut has two banks: a N-bank and a S-bank. We recover the surface ${\overline}\Nb_0$ by identifying the N-bank (resp. S-bank) of a cut in the sheet $I$ with the corresponding S-bank (resp. N-bank) in the sheet $II.$
{width=".5\textwidth"}\[fig:2copias\]
We denote by $z,w:{\overline}\nb_0\to{\overline}{\c}$ the two canonical projections, whose associated divisors are $$(w)=\frac{a_1\cdot\ldots\cdot
a_{2n+2}}{(p_\infty)^{n+1}\cdot(p_\infty^*)^{n+1}}
\qquad\mbox{and}\qquad
(dz)=\frac{a_1\cdot\ldots\cdot a_{2n+2}}{(p_\infty)^{2}\cdot
(p_\infty^*)^{2}},$$ where $a_j\equiv(a_j,0)$ and $\{p_\infty,p_\infty^*\}=z^{-1}(\{\infty\}).$ We will label $p_\infty$ as the one where the coefficient of degree $-(n+1)$ of the Laurent series of $w$ is $-1$.\
Finally we define $J_0:{\overline}{\Nb}_0\to{\overline}{\Nb}_0$ as the antiholomorphic involution given by $J_0(z,w)=({\overline}{z},-{\overline}{w}).$ The fixed points of $J_0$ are the Jordan curves $\gamma_j=\{(z,w)\in{\overline}{\Nb}_0\;:\;z\in[a_{2j-1},a_{2j}]\},$ $j=1,\ldots,n+1.$ Moreover, $\Nb_0\setminus\cup_{j=1}^{n+1}\gamma_j$ has two connected components, each one of them corresponding to a single-valued branch of $w$, and biholomorphic to a $n$-connected circular domain.
\[def:omega\] Let $n\in \n,$ and $a_1<a_2<\ldots <a_{2n+2}\in\r.$ Consider the above defined compact Riemann surface $${\overline}{\Nb}_0:=\{(z,w)\in{\overline}{\c}^2\;:\;
w^2=\prod_{j=1}^{2n+2}(z-a_j)\},$$ with the antiholomorphic involution $J_0(z,w)=({\overline}{z},-{\overline}{w})$. Label $\Delta$ as the set of fixed points of $J_0$. We will define $\bar\Omega_0$ as the closure of the connected component of ${\overline}\Nb_0\setminus\Delta$ containing $p_\infty$, and $\Omega_0$ will denote the circular domain $\Omega_0:=\bar\Omega_0\setminus\{p_\infty\}$.
\[pro:caract\] Let $(g,\phi_3)$ be Weierstrass data on $\Omega_0$ of an entire maximal graph with $n+1$ singularities and horizontal end. Then there exists $n+1$ distinct points $\{b_1,\ldots,b_{n+1}\}\subset\{a_1,\ldots,a_{2n+2}\}$, such that
$$\label{eq:datos}
g=e^{i\theta}\frac{w+P(z)}{w-P(z)} \quad \mbox{and} \quad
\phi_3=A \big( \frac{w}{P(z)} - \frac{P(z)}{w} \big) dz,$$
where $P(z)=\prod_{j=1}^{n+1}(z-b_j)$, $\theta\in\r$, and $A\in\r^\ast$.
By Theorem \[th:representation\], the associated divisors to $(g,\phi_3)$ are given by $$\label{eq:divisor}
(g)=\frac{D\cdot p_\infty}{J(D)\cdot
p_\infty^\ast}\quad\mbox{and}\quad (\phi_3)=\frac{D\cdot
J(D)}{p_\infty\cdot p_\infty^\ast}$$ where $D\in Div_n(\overline{\Omega}_0).$ Here, $p_\infty$ denotes the point in ${\overline}{\Omega}_0\cap z^{-1}(\infty)$, and $p_\infty^\ast=J(p_\infty)$.
We will denote by $F:{\overline}{\Nb}_0\to{\overline}{\Nb}_0$ the holomorphic involution given by $F(z,w)=({z},-{w})$.
\[claim1\] In the above conditions there exist $n+1$ distint points $\{b_1,\ldots,b_{n+1}\}\subset\{a_1,\ldots,a_{2n+2}\}$, such that $g=\frac{G_1}{G_2}$ for two meromorphic functions $G_1$, $G_2$ on $\overline\Nb_0$ satisfying
a) $(G_1)\geq\displaystyle{\frac{p_\infty^*}{b_1\cdot\ldots\cdot b_{n+1}}}$
b) $(G_2)\geq\displaystyle{\frac{p_\infty}{b_1\cdot\ldots\cdot b_{n+1}}}$
Since $g$ has degree $n+1$ and $\overline\Nb_0$ is hyperelliptic, the two meromorphic functions $g$ and $z$ satisfy a relation $P(g,z)=0,$ where $P$ is a polynomial in two variables with algebraic degree two in the first one and $n+1$ in the second (see [@farkas]). We can rewrite this relation as $P_2(z)g^2+P_1(z)g+P_0(z)=0,$ with $P_i$ polynomials whose maximum algebraic degree is $n+1.$ Solving this equation we obtain $$g=\frac{-P_1\pm \sqrt{P_1^2-4P_0P_2}}{2P_2}.$$
Consider the meromorphic function $f=\sqrt{P_1^2-4P_0P_2}=\pm(2gP_2+P_1).$ Let us check that $f=c
w,$ for some constant $c\in\r^*.$ Indeed, any meromorphic function on the hyperelliptic surface ${\overline}{\Nb}_0$ can be expressed as $f=R_1(z)+R_2(z)w,$ with $R_i$ rational functions (see [@farkas]). In our case, $f^2$ is a polynomial function in $z$, and so it follows that either $R_1=0$ or $R_2=0.$ The last case would imply that $g$ is a rational function of $z,$ which is impossible from Equation (\[eq:divisor\]) so $f=R_2(z)w.$ Now observe that $f$ has poles only at $p_\infty$ and $p_\infty^\ast$ with order at most $n+1$, which implies that $f/w$ is a holomorphic function on ${\overline}{\Nb}_0,$ and therefore constant. Thus, $f=cw$ for some $c\in\r^\ast$. Up to replace $P_i$ by $\pm cP_i,$ $i=1,2,$ we can suppose that $$g=\frac{ P_1+w }{2 P_2}.$$ We will also assume that the leading coefficient of $P_1$ is one. Since $P_1$ and $P_2$ are meromorphic functions of degree $\leq 2(n+1)$ that only depend on $z$, it is not hard to realize that implies that $$(P_1+w)=\frac{D\cdot
E}{(p_\infty)^{n-1}\cdot(p_\infty^*)^{n+1}}
\quad\mbox{and}\quad (P_2)=\frac{J(D)\cdot
E}{(p_\infty)^n\cdot(p_\infty^*)^n},$$ where $E:=F(J(D)\in Div_n(\overline\Omega_0)$. Thus, the meromorphic function $$h=\frac{P_2(P_1+w)}{w\prod_{e\in E}
(z-z(e))}\frac{dz}{\phi_3}$$ satisfies that $(h)=\frac{E\cdot p_\infty}{F(E)\cdot
p_\infty^*}=(\frac{1}{h}\circ F),$ and therefore up to a multiplying constant $h\circ F=1/h.$ On the other hand, $\mbox{deg}(h)=n+1$, and reasoning as before we can deduce that $h=(\hat{P}_1(z)+w)/\hat{P}_2(z),$ for some $\hat{P}_i(z)$ polynomial functions in $z$ with algebraic degree less than or equal to $n+1.$ Since $h\circ F=1/{h},$ we infer that $w^2=\hat{P}_1^2-\hat{P}_2^2$ and so, setting $S=-\hat{P}_1-\hat{P}_2$ we can write $h=(S-w)/(S+w).$
Looking at the divisor of $h$ is immediate to realize that there exists an integral divisor $B$ with $\deg B=n+1$ such that: $$(S-w)=\frac{E\cdot B}{p_\infty^{n} \cdot(p_\infty^*)^{n+1}}
\quad \textrm{and} \quad (S+w)=\frac{F(E)\cdot
B}{p_\infty^{n+1}\cdot (p_\infty^*)^{n}}.$$ Since points in $B$ are zeros of both $S+w$ and $S-w$, they must be $n+1$ distinct (recall that $w$ only has simple zeroes) points of $\{a_1\dots
a_{2n+2}\}.$ Setting $G_1=\displaystyle{\frac{{P}_1+w}{S-w}}$ and $G_2=\displaystyle{\frac{2{P}_2}{S-w}}$ the claim is proved.\
\[claim2\] Up to multiplicative constants, the functions $G_1$ and $G_2$ in Claim \[claim1\] are given by $G_1= \displaystyle{\frac{w}{P(z)}+1} $ [and]{} $G_2=\displaystyle{ \frac{w}{P(z)}-1},$ being $P(z)=\prod_{j=1}^n(z-b_j)$.
Call $B$ to the integral divisor given by $B=b_1\cdot\ldots\cdot
b_{n+1}$. By Riemann-Roch Theorem, the dimension of the linear space of meromorphic functions on $\overline{\Nb}_0$ satisfying condition $a)$ (resp. $b)$) in Claim \[claim1\] is $1+d$ where $d$ is the dimension of the linear space of meromorphic 1-forms $\nu$ on $\overline{\Nb}_0$ satisfying $(\nu)\geq \frac{B}{p_\infty^*}$ (resp. $(\nu)\geq
\frac{B}{p_\infty}$). Let us see that $d=0.$
Indeed, observe first that by the residues theorem, both spaces agree with the space $L(B)$ of holomorphic $1$-forms $\nu$ with $(\nu)\geq B$. But since $\{ \displaystyle{\frac{dz}{w}}, z\displaystyle{\frac{dz}{w}},\ldots,
z^{n-1}\displaystyle{\frac{dz}{w}}\}$ is a basis for the space of holomorphic 1-forms on $\overline{\Nb}_0,$ any $\nu\in L(B)$ must be of the form $\nu=P(z)\frac{dz}{w},$ where $P$ is a polynomial with algebraic degree less than $n.$ Thus, if a Weierstrass point $a_{j_0}$ is a zero of $\nu$ then its order is at least two. It follows that the number of zeroes of the holomorphic 1-form $\nu$ is at least $2(n+1)$ which is impossible because ${\overline}{\Nb}_0$ has genus $n.$
Therefore the dimension of the linear space of meromorphic functions satisfying condition $ a)$ (resp. $ b)$) in the Claim \[claim1\] is $1.$ It is easy to show that the function $\frac{w}{\prod_{j=1}^{n}(z-b_j)}+1$ (resp. $\frac{w}{\prod_{j=1}^{n}(z-b_j)}-1$) is a basis for this space, so Claim \[claim2\] is proved.
As a consequence of the previous claims, we can write: $$g= \frac{G_1}{G_2}=
c\,\frac{w+P(z)}{w-P(z)},$$ for a suitable constant $c\in\c^\ast$. As $g\circ J=1/\overline{g}$ we infer that $c=e^{i\theta}$ for some $\theta\in\R$.
To finish observe that the divisor of $\phi_3$ coincides with the divisor for the 1-form $ \big( \frac{w}{P(z)} - \frac{P(z)}{w}
\big) dz,$ and as a consequence $$\phi_3=A\, \big(
\frac{w}{P(z)} - \frac{P(z)}{w} \big) dz,$$ since $J^*(\phi_3)=-\overline{\phi_3}$ we get $A\in\r.$ This concludes the proof.
To finish the classification of the entire maximal graphs on the given circular domain $\Omega_0$ we need to find out when the pair given by are actually Weierstrass data. This is done in the following proposition. Figure \[fig:alineados\] shows two examples of the surfaces given by these Weierstrass representation.
\[pro:b\] Choose $b_1<b_2<\ldots< b_{n+1}$ points in $\{a_1,\ldots,a_{2n+2}\}$, and define $P(z)=\prod_{j=1}^{n+1}(z-b_j)$.
Then the pair $(g,\phi_3)$ given by Equation are Weierstrass data on $\Omega_0$ of an entire maximal graph with $n+1$ singularities if and only if $b_j\in\{a_{2j-1},a_{2j}\}$ for all $j=1,\ldots,n+1$.
We just have to check the conditions stated in Theorem \[th:representation\]. Recall that $J(z,w)=(\bar{z},-\bar{w})$, and define $Q(z)=w^2/P(z)=\prod_{j=1}^{n+1}(z-c_j)$. For simplicity, we will assume that $\theta=0$ and $A=1$.
Conditions $(ii)$ and $(iii)$ are straightforward for all the possible values of $b_1,\ldots,b_{n+1}$. Let us show when $(i)$ is accomplished.
First, notice that $g^{-1}(1)=\{b_1,\ldots,b_{n+1}\}$. In particular, $\deg(g)=n+1$. In particular, in order to be $g$ the Gauss map of a maximal surface with conelike singularities, any connected component in $\partial\Omega_0$ must have exactly one point with $g=1$, and so $b_j\in\{a_{2j-1},a_{2j}\}$ for every $j=1,\ldots, n+1$.
Conversely, assume that $b_j\in\{a_{2j-1},a_{2j}\}$, $j=1,\ldots, n+1$, and let us show that $g$ has no critical points on $\partial\Omega_0\equiv\cup_{j=1}^{n+1}[a_{2j-1},a_{2j}]$. After some computations one easily gets that $$dg=\frac{QdP-PdQ}{w(Q+P-2w)}.$$ Thus for critical points in $\Nb_0={\overline}{\Nb}_0\setminus\{z^{-1}(\infty)\}$ we have $QdP=PdQ$, or equivalently, $$\sum_{j=1}^{n+1} \frac{1}{z-b_j} =
\sum_{j=1}^{n+1} \frac{1}{z-c_j}.$$
If we assume that $b_j\in\{a_{2j-1},a_{2j}\}$ for all $j=1,\ldots,n+1,$ and we have a point $p_0\in
[a_{2j_0-1},a_{2j_0}]\subset \partial\Omega_0$, with $a_{2j_0-1}=b_{j_0}$ and $a_{2j_0}=c_{j_0}$ (the case $a_{2j_0-1}=c_{j_0}$ and $a_{2j_0}=b_{j_0}$ is similar) then we have that $$\frac{1}{z(p_0)-c_j} < \frac{1}{z(p_0)-b_{j+1}}<0,
\quad j=1,\ldots n+1,$$ (here we use the convention $b_{n+2}=b_1$), and this gives that $p_0$ cannot be a critical point of $g$.
To finish just notice that $g\circ J =1/\bar{g}$ and therefore $|g|=1$ on the $n+1$ connected components of $\partial\Omega_0$. Since $g$ is injective on each one of these curves, and $\deg(g)=n+1$, then $|g|\neq 1$ on $\Nb_0\setminus\partial\Omega_0$. Taking into account that $g(p_\infty)=0$ we have that $|g|<1$ on $\Omega_0$.
\[def:wei\] Let $\Omega_0$ the circular domain given in Definition \[def:omega\] for some real numbers $a_1<\ldots<a_{2n+2}$. For each subset $\tau=\{b_1,\ldots,b_{n+1}\}\subset\{a_1,\ldots,a_{2n+2}\}$ with $b_j\in\{a_{2j-1},a_{2j}\}$, $j=1,\ldots,n+1$, we will define the $G_\tau$ as the entire maximal graph with $n+1$ singularities with Weierstrass data $(g_\tau,\phi_3^\tau)$ on $\Omega_0$ given by $$g_\tau= = \frac{w+P(z)}{w-P(z)} \quad \mbox{and} \quad
\phi_3^\tau= \big( \frac{w}{P(z)} - \frac{P(z)}{w} \big) dz,$$ where $P(z)=\prod_{j=1}^{n+1}(z-b_j)$.
{width=".8\textwidth"}\[fig:alineados\]
Let $\Omega_0$ be the $n$-connected circular domain given in Definition \[def:omega\]. Then the number of non-congruent entire maximal graphs whose underlying conformal structure is $\Omega_0$ is exactly $2^{n}.$
From Propositions \[pro:caract\] and \[pro:b\] we know that any maximal graph $G$ with horizontal end defined on $\Omega_0$ have Weierstrass data $(g=e^{i\theta} g_\tau,\phi_3=A\phi_3^\tau),$ where $\theta\in\R$, $A\in\R^\ast$ and $(g_\tau,\phi_3^\tau)$ are given by Definition \[def:wei\].
Observe that replacing the set $\tau$ by its complementary $\{a_1,\ldots,a_{2n+2}\}\setminus \tau$ gives congruent surfaces (more specifically, $(g,\phi_3)$ are transform into $(-g,-\phi_3)$). So, we can assume without loss of generality that $b_1=a_1$. To avoid congruences, we will also normalize so that $g(a_1)=h(a_1)=1$, where $h=\displaystyle{\frac{\phi_3}{\frac{dz}{w}}}$. Looking at the expressions for $g$ and $\phi_3$ this means that $\theta=0,$ $A=1$. Thus, the number of non-congruent maximal graphs defined on $\Omega_0$ is the number of possible choices of $b_j\in\{a_{2j-1},a_{2j+1}\},$ $j=2,\ldots, n+1$, which is $2^n$.
Taking into account our previous discussion in Section \[sec:first\], we can conclude that:
\[th:main\] The number of non-congruent entire maximal graphs with the same conformal structure is $2^n$, where $n+1$ is the number of (conelike) singularities.
Equivalently, the number of connected components of the space ${\mathcal{G}}_n$ of entire marked maximal graphs with $n+1$ singularities and horizontal end is $2^n$.
Maximal graphs with coplanar singularities {#sec:cons}
==========================================
We will prove now that the surfaces constructed in the previous section are characterized by the property of having all its singularities on a plane orthogonal to the limit normal vector at infinity. In particular, for $n=1$, surfaces obtained in Section \[sec:count\] describe the whole moduli space of the entire maximal graphs with two singular points.
\[th:alineados\] Let $G\subset\l^3$ be an entire maximal graph with $n+1$ conelike singularities. Then $G$ has all its singularities lying on a timelike plane in $\L^3$ orthogonal (in the Lorentzian sense) to the normal vector at the end if and only if $G$ is congruent to one of the examples given in Definition \[def:wei\].
Assume that $G$ has all its singularities in an orthogonal plane to the normal vector at the end. Up to a rigid motion in $\L^3$ we can assume that the end is horizontal and the singularities lie in the plane $\{x_1=0\}$. Let $X:\Omega\to G\subset\L^3$ a conformal reparameterization of $G$. By the uniqueness result in [@klyachin] (see also [@conelike] Remark 2.5), the surface is symmetric with respect to the plane $\{x_1=0\}.$ This symmetry induces an antiholomorphic involution $T:\overline{\Omega}\to \overline{\Omega}$ leaving $\partial\Omega$ globally fixed. It follows that $T$ extends to an antiholomorphic involution $T:\overline\Nb\to \overline\Nb$, where $\Nb$ is the mirror surface, by putting $T\circ
J=J\circ T$ ($J$ is the mirror involution). Moreover, if $(g,\phi_3)$ are the Weierstrass data of the immersion, $g\circ T=\overline{g}$ and $T^*(\phi_3)={\overline}{\phi_3}.$ It is straightforward that $T$ must have exactly two fixed points on every connected component of the circular domain $\partial \overline{\Omega}.$ We call these points $p_1,\ldots,p_{2n+2}.$ Observe that the end $p_{\infty}\in\overline\Omega$ is also fixed by $T.$
Consider the holomorphic involution $F=J\circ T,$ whose fixed points are exactly $p_1,\ldots, p_{2n+2}.$ Therefore, ${\overline}\Nb$ is a compact genus $n$ Riemann surface with $2n+2$ fixed points, this means that ${\overline}\Nb$ is hyperelliptic with Weiersrtass points $p_1,\ldots,p_{2n+2}$ (see [@farkas]), $$\overline\Nb \equiv
\{(z,w)\in\overline{\c}^2\;:\;w^2=\prod_{i=1}^{2n+2}(z-a_j)\},$$ where $(a_j,0)$ corresponds to $p_j$ for any $j$ (and so $a_j\neq
a_k$ for $k\neq j$). With this identification we have $F(z,w)=(z,-w).$ Up to a Möbius transformation we can suppose that $z(p_{2n+1})=1,$ $z(p_{2n+2})=-1,$ and $z(p_\infty)=\infty.$
In what follows we will identify $a_j=(a_j,0)\in \overline\nb.$ To prove $a_j\in\r$ notice that the divisor associated to the meromorphic 1-form $d(\overline{z\circ J})$ coincides with the one of $dz$ and therefore $\overline{z\circ J}=k\,z+\lambda,$ for some $k,\lambda\in\r.$ Since $a_{2n+1}=1$ and $a_{2n+2}=-1$ are fixed by $J$ it follows that $\overline{z\circ J}=z$ which implies that $a_j\in\r.$ Moreover, since $w^2\circ J=\overline{w}^2,$ then $w\circ J=\pm
\overline{w}.$ Taking into account that $J$ interchanges the two points with $z=\infty,$ namely $p_\infty$ and $p_\infty^*=J(p_\infty),$ then $w\circ J=- \overline{w}.$ Therefore $J(z,w)=({\overline}{z},-{\overline}{w})$ and $T(z,w)=({\overline}{z},{\overline}{w}).$ In particular, $\Omega$ agrees with the circular domain $\Omega_0$ defined in Definition \[def:omega\] and by Propositions \[pro:caract\] and \[pro:b\] we are done.\
Conversely, let $G_\tau$ one of the graphs defined in Definition \[def:wei\]. Consider the involution $T(z,w) = (\bar{z},\bar{w})$ on ${\overline}\nb_0$ that fix globally any component of $\partial\Omega_0$. Moreover, $g_\tau\circ T=\bar{g}_\tau$ and $T^\ast(\phi_3^\tau)=\bar{\phi}_3^\tau$, thus, $T$ induces an isometry on the resulting surface, namely $I(x_1, x_2, x_3) = (-x_1, x_2, x_3).$ Since $\{a_1,\ldots,a_{2n+2}\}$ are fixed by $T$ it follows that all the singularities lie in the plane $\{x_1 = 0\}$.
[99]{}
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[^1]: Research partially supported by Spanish MEC-FEDER Grant MTM2007-64504, and Regional J. Andalucía Grants P06-FQM-01642 and FQM325.2000 Mathematics Subject Classification. Primary 53C50; Secondary 53A10, 53A30. Key words and phrases: maximal surfaces, conelike singularities, conformal structures.
|
---
abstract: |
The packing chromatic number $\chi_\rho(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set $V(G)$ can be partitioned into disjoint classes $X_1, \ldots, X_k$, where vertices in $X_i$ have pairwise distance greater than $i$.
For the 2-dimensional square lattice ${\mathbb{Z}^2}$ it is proved that $\chi_\rho({\mathbb{Z}^2}) \geq 12$, which improves the previously known lower bound 10.
author:
- Jan Ekstein $^a$
- Jiří Fiala $^b$
- Přemysl Holub $^a$
- Bernard Lidický $^b$
title: 'The packing chromatic number of the square lattice is at least 12 [^1] '
---
$^a$ Department of Mathematics and Inst. for Theoretical Computer Science (ITI), University of West Bohemia, Univerzitní 22, 306 14 Pilsen, Czech Republic\
E-mail: [{ekstein,holubpre}@kma.zcu.cz]{}
$^b$ Department of Applied Mathematics and Inst. for Theoretical Computer Science (ITI),\
Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic\
E-mail: [{fiala,bernard}@kam.mff.cuni.cz]{}
[**Keywords:**]{} Packing chromatic number; Square lattice;
[**ACM 1998 classification** ]{}: G.2.2 Graph theory
Introduction
============
The concept of packing coloring comes from the area of frequency planning in wireless networks. This model emphasizes the fact that some frequencies have higher throughput and hence they are used more sparely to avoid an interference.
In our model, the first frequency cannot be assigned to neighbouring nodes. The second frequency cannot be assigned to nodes in distance at most two and so on. In graph theory language we ask for a partition of the vertex set of a graph $G$ into disjoint color classes $X_1,\ldots,X_k$ according to the following constraints. Each color class $X_i$ should be an *$i$-packing*, that is, a set of vertices with the property that any distinct pair $u,v\in X_i$ satisfies ${{\rm dist}}(u,v)>i$. Here ${{\rm dist}}(u,v)$ denotes the shortest path distance between $u$ and $v$. Such partition is called a *packing $k$-coloring*, even though it is allowed that some sets $X_i$ may be empty. The smallest integer $k$ for which there exists a packing $k$-coloring of $G$ is called the *packing chromatic number* of $G$ and it is denoted by $\chi_\rho(G)$.
This concept, under the notion *broadcast chromatic number*, was introduced by Goddard et al. [@gohe-08]. The notion packing chromatic was proposed by Brešar et al. [@brklra-07].
Topic of this work is the packing chromatic number of the infinite square lattice ${\mathbb{Z}^2}$. The question of determining $\chi_\rho({\mathbb{Z}^2})$ was posed in [@gohe-08]. Also a lower bound 9 and an upper bound 23 were given there. The upper bound was improved by Schwenk [@schwenk-02] to 22 and later by Holub and Soukal [@holub-09] to 17. The lower bound was improved to 10 by Fiala et al. [@fili-09].
We further improve the lower bound from 10 to 12.
\[thm:main\] The packing chromatic number for the square lattice is at least 12.
The proof relies on computer. In the next section we describe the main idea of the algorithm, which proves the theorem. All necessary code for running the computation is available at <http://kam.mff.cuni.cz/~bernard/packing>.
The Result
==========
The algorithm for proving Theorem \[thm:main\] is a brute force search through all possible configurations on lattice $15\times9$. It is too time consuming to simply check everything. Hence we use the following observation to speed up the computation by avoiding several configurations.
If there exists a coloring of the square lattice with 11 colors then it is possible to color lattice $15\times9$ where color 9 is at position $[5,5]$.
Any other color at any other position could be fixed instead of 9 at $[5,5]$. Color 9 at $[5,5]$ just sufficiently reduces the number of configurations to check. We do not claim that it is the optimal choice.
If there exists a coloring we simply find any vertex of color 9 and take a piece of the lattice in its neighborhood.
So in the search through the configurations we assume that at position $[5,5]$ is precolored by 9. The coloring procedure gets a matrix and tries to color the vertices row by row. A pseudocode is given here in this note.
function boolean try_color(lattice, [x,y])
begin
for color := 1 to 11 do
if can use color on lattice at [x,y] then
lattice[x,y] := color
if [x,y] is the last point return true
else if try_color(lattice, next([x,y]) then return true
endif
endfor
return false
end
We have two implementations of this procedure. One is in the language C++ and the other is in Pascal. The first one is available online at <http://kam.mff.cuni.cz/~bernard/packing>. We include the full source code as well as descriptions of inputs and outputs. We were checking the outputs of both programs during the computation and we verified that they match. The total number of checked configurations was 43112312093324. The computation took about 120 days of computing time on a single core workstation in year 2009.
The procedure fails to color the matrix $15 \times 9$ with 9 at position $[5,5]$. Hence we conclude that the packing chromatic number for the square lattice is at least 12.
[99]{}
B. Brešar, S. Klavžar and D. F. Rall: [*On the packing chromatic number of Cartesian products, hexagonal lattice, and trees*]{}, Discrete Appl. Math. [**155**]{} (2007) 2303–2311.
J. Fiala, S. Klavžar and B. Lidický: [*The packing chromatic number of infinite product graphs*]{}, European J. of Combin. [**30**]{} (2009) Pages 1101–1113.
W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, J. M. Harris and D. F. Rall: [*Broadcast chromatic numbers of graphs*]{}, Ars Combin. [**86**]{} (2008) 33–49.
P. Holub and R. Soukal: [*A note on packing chromatic number of the square lattice*]{}, Electronic Journal of Combinatorics, to appear.
A. Schwenk: [*personal communication with W. Goddard et al.*]{}, (2002).
[^1]: This work was supported by the Ministry of Education of the Czech Republic as projects 1M0021620808, 1M0545 and GACR 201/09/0197.
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