aainstitutetext:  Center for Theoretical Physics and College of Physics, Jilin
University,
Changchun 130012, People’s Republic of China bbinstitutetext:  Max Planck
Institute for Gravitational Physics (Albert Einstein Institute),
Am Mühlenberg 1, 14476 Golm, Germany

# Holographic Euclidean thermal correlator

Song He a and Yi Li<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we compute holographic Euclidean thermal correlators of the
stress tensor and $U(1)$ current from the AdS planar black hole. To this end,
we set up perturbative boundary value problems for Einstein’s gravity and
Maxwell theory in the spirit of Gubser-Klebanov-Polyakov-Witten, with
appropriate gauge fixing and regularity boundary conditions at the horizon of
the black hole. The linearized Einstein equation and Maxwell equation in the
black hole background are related to the Heun equation of degenerate local
monodromy. Leveraging the connection relation of local solutions of the Heun
equation, we partly solve the boundary value problem and obtain exact two-
point thermal correlators for $U(1)$ current and stress tensor in the scalar
and shear channels.

††arxiv: 2308.13518

##  1 Introduction

As an embodiment of the holographic principle tHooft:1993dmi ; Susskind:1994vu
, the Anti-de Sitter gravity/conformal field theory (AdS/CFT) correspondence
Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj  establishes a connection
between a quantum gravity theory in AdS space and a conformal field theory on
the boundary. This equivalence is encapsulated in the Gubser-Klebanov-
Polyakov-Witten (GKPW) relation, where the partition function of the conformal
field theory with operator sources equals the gravity partition function with
prescribed boundary conditions

$\displaystyle\langle e^{\int\phi_{0}O}\rangle_{CFT}=Z_{\mathrm{G}}[\phi_{0}]$
(1)

In the most helpful limit to exploit this correspondence, the classical
gravity on-shell action becomes the generating functional of connected
correlators of the strongly-coupled CFT

$\displaystyle I_{CFT}[\phi_{0}]=I_{\mathrm{G,on-shell}}[\phi_{0}]$ (2)

Correlators are computed by functional differentiation of the generating
functional, which amounts to solving the perturbative boundary value problem
for the bulk fields’ equation of motion. This involves varying the boundary
value of the bulk fields and solving for the corresponding variation of the
on-shell configuration in the bulk. The near-boundary behavior is well-
established, allowing the extraction of holographic correlators
Fefferman:1985cfm ; deHaro:2000vlm ; Bianchi:2001kw ; Fefferman:2007rka .
However, solving the global boundary value problem is generally intricate,
exemplified in cases like pure gravity Anderson:2004yi .

Although the prescription is clear, explicit computation of holographic
Euclidean correlators in the GKPW approach has been limited to pure AdS space
and its quotient spaces, such as thermal AdS where the method of images can be
applied (e.g., see Alday:2020eua  for thermal bootstrap emphasis). In our
prior work He:2023hoj , we computed holographic torus correlators of the
stress tensor. This study focuses on Euclidean thermal two-point correlators
of the stress tensor and $U(1)$ current in four-dimensional CFTs. Beyond the
Hawking-Page transition Hawking:1982dh , the thermal state holographically
corresponds to a five-dimensional Euclidean AdS planar black hole
Witten:1998zw . Correlators are derived by solving perturbative boundary value
problems in Einstein’s gravity and Maxwell theory for the $U(1)$ gauge field
in the black hole background. Two important steps are involved. The first is
to appropriately fix the gauge and impose regularity boundary conditions at
the horizon, ensuring a unique solution. The second step identifies the
equations of motion as the Heun equation Heun:1888zt , and solves the boundary
value problems with the connection relation of local solutions. The general
connection relation was established in Bonelli:2022ten , and it has been
applied to exact thermal correlators in Minkowski signature Dodelson:2022yvn ;
Bhatta:2022wga , and employed in various black hole perturbation problems
Aminov:2020yma ; Bonelli:2021uvf ; Consoli:2022eey ; Giusto:2023awo ;
Aminov:2023jve ; Lei:2023mqx . In our case, the Heun equations feature
degenerate local monodromy, with characteristic exponents differing by an
integer. We compute the connection relation by taking a limit of the generic
case. Ultimately, we obtain exact two-point correlators for the $U(1)$ current
and stress tensor in the scalar and shear channels (as defined in
Kovtun:2005ev ).

Thermal two-point correlators, also known as thermal spectral functions, have
many important applications and have been studied in Kovtun:2005ev  using
gauge invariants in each channel. In the final discussion section, we comment
on our approach to holographic computation and relevant applications of
thermal two-point correlators.

##  2 Holographic setup

We start by reviewing the basics of holographic computation independent of the
bulk background geometry. For Einstein’s gravity, it’s customary to work in
the Fefferman-Graham gauge Fefferman:1985cfm ; Graham:1991jqw  near the
conformal boundary

$\displaystyle
ds^{2}=\frac{dr^{2}}{r^{2}}+\frac{1}{r^{2}}\mathbf{g}_{ij}(r,x)dx^{i}dx^{j}$
(3)

and in dimension four, we have the series expansion

$\displaystyle\mathbf{g}_{ij}=\mathbf{g}^{(0)}_{ij}+r^{2}\mathbf{g}^{(2)}_{ij}+r^{4}\mathbf{g}^{(4)}_{ij}+r^{4}\log
r\mathbf{h}^{(4)}_{ij}+\ldots$ (4)

The background metric of the holographic field theory $\gamma_{ij}$
corresponds to $\mathbf{g}^{(0)}_{ij}$, and the one-point correlator of the
stress tensor111Our convention for the stress tensor is, for classical field
theories, $\delta_{\gamma}S=\frac{1}{2}\int dVT_{ij}\delta\gamma^{ij}$. For
quantum theories $\langle T_{ij}\rangle=2\frac{\delta I}{\delta\gamma^{ij}}$.
The integration measure is assumed to be included in the definition of the
functional derivative., with appropriate renormalization, is given by
Emparan:1999pm ; Kraus:1999di ; deHaro:2000vlm

$\displaystyle\langle T_{ij}\rangle=\frac{4}{16\pi
G}\big{[}\mathbf{g}^{(4)}_{ij}-\frac{1}{8}\mathbf{g}^{(0)}_{ij}(\mathbf{P}^{(0)2}-\mathbf{P}^{(0)}_{kl}\mathbf{P}^{(0)kl})-\frac{1}{2}\mathbf{P}^{(0)}_{ik}\mathbf{P}^{(0)k}_{j}+\frac{1}{4}\mathbf{P}^{(0)}\mathbf{P}^{(0)}_{ij}\big{]}$
(5)

where $\mathbf{P}^{(0)}_{ij}$ is the Schouten tensor of
$\mathbf{g}^{(0)}_{ij}$. In our case, the holographic field theory lives on a
flat background, so the terms of Schouten tensor do not contribute. We have

$\displaystyle\langle T_{ij}\rangle=\frac{1}{4\pi G}\mathbf{g}^{(4)}_{ij}$ (6)

The Einstein equation near the conformal boundary determines the series (4) in
terms of $\mathbf{g}^{(0)}_{ij}$ and $\mathbf{g}^{(4)}_{ij}$ or equivalently
the one point correlator $\langle T_{ij}\rangle$, and imposes holographic Ward
identities of conservation and the Weyl anomaly on $\langle T_{ij}\rangle$.
Near boundary solutions of the Einstein equation are in one-to-one
correspondence to the pair $(\gamma_{ij},\langle T_{ij}\rangle)$. The global
geometry of the bulk spacetime fully determines the one-point correlator as a
functional of the boundary metric, from which we can compute multi-point
correlators by functional differentiation. In particular for two-point
correlator we have

$\displaystyle\langle T_{ij}(x)T_{kl}(y)\rangle=-2\frac{\delta\langle
T_{ij}(x)\rangle}{\delta\gamma^{kl}(y)}$ (7)

Similarly, for the $U(1)$ gauge field, we can put it in the radial gauge near
the conformal boundary using the Fefferman-Graham coordinates of the bulk
metric

$\displaystyle A=\mathbf{A}_{i}(r,x)dx^{i}$ (8)

with the series expansion

$\displaystyle\mathbf{A}_{i}=\mathbf{A}^{(0)}_{i}+r^{2}\mathbf{A}^{(2)}_{i}+r^{2}\log
r\mathbf{B}^{(2)}_{i}+\ldots$ (9)

The one-point correlator with appropriate renormalization is given by

$\displaystyle\langle J_{i}\rangle=-2\mathbf{A}^{(2)}_{i}$ (10)

The Maxwell equation near the conformal boundary determines the series (9) in
terms of $\mathbf{A}^{(0)}_{i}$ and $\mathbf{A}^{(2)}_{i}$ or equivalently the
one point correlator $\langle J_{i}\rangle$, and imposes the holographic Ward
identity of conservation on $\langle J_{i}\rangle$. The global geometry of the
bulk spacetime fully determines $\langle J_{i}\rangle$. Multi-point
correlators are computed by taking functional derivative with respect to the
source $\mathcal{A}$, for two point correlator we have

$\displaystyle\langle J_{i}(x)J_{j}(y)\rangle=\frac{\delta\langle
J_{i}(x)\rangle}{\delta\mathcal{A}^{j}(y)}$ (11)

Now we specialize in the holographic correlators from the five-dimensional AdS
planar black hole. The black hole is a solid cylinder
$\mathbb{B}^{2}\times\mathbb{R}^{3}$ with the metric

$\displaystyle
ds^{2}=\frac{1}{\rho^{2}}[(1-\frac{\rho^{4}}{\rho_{0}^{4}})^{-1}d\rho^{2}+(1-\frac{\rho^{4}}{\rho_{0}^{4}})dt^{2}+d\vec{x}^{2}]$
(12)

The period of Euclidean time $t$, namely the inverse temperature, is
$\beta=\pi\rho_{0}$. The conformal boundary is at $\rho=0$, and the horizon is
at $\rho=\rho_{0}$, being the $\mathbb{R}^{3}$ axis of the cylinder. The
standard Fefferman-Graham radial coordinate $r$ is related to $\rho$ by

$\displaystyle\rho=\frac{r}{\sqrt{1+\frac{r^{4}}{4\rho_{0}^{4}}}}$ (13)

For simplicity, we set $\rho_{0}=1$ in the metric, effectively working in the
unit of $\rho_{0}$, and we will recover $\rho_{0}$ dependence when final
results are obtained. As a convention, we label bulk spacetime coordinate
indices by Greek alphabets $\mu,\nu,\rho,\ldots$, the boundary spacetime
coordinate indices by Roman alphabets $i,j,k,\ldots$ and the boundary space
indices by $a,b,c,\ldots$.

##  3 $U(1)$ current

Now, we work on the boundary value problem of the $U(1)$ gauge field,
beginning with gauge-fixing. We can put it in the radial gauge $A_{\rho}=0$ in
the region $0\leq\rho<1$ (excluding the horizon) by a $U(1)$ gauge
transformation. For a global solution, its restriction to the region
$0\leq\rho<1$ must have a regular limit going to the horizon $\rho=1$.
Therefore, we formulate the boundary value problem in the radial gauge, with
the boundary condition that the solution has a regular limit as $\rho\to 1$
after a gauge transformation. To work out the explicit form of the boundary
condition, we introduce the “cylindrical radial coordinate” $\mathfrak{s}$

$\displaystyle\cosh 2\mathfrak{s}=\frac{1}{\rho^{2}}$ (14)

Near the horizon $\rho\to 1$ or $\mathfrak{s}\to 0$, the metric takes the form
of Euclidean metric in “cylindrical coordinates”

$\displaystyle ds^{2}\sim
d\mathfrak{s}^{2}+\mathfrak{s}^{2}d(2t)^{2}+d\vec{x}^{2}$ (15)

and the horizon is properly covered by the “Cartesian coordinates”

$\displaystyle X$ $\displaystyle=\mathfrak{s}\cos 2t$ $\displaystyle Y$
$\displaystyle=\mathfrak{s}\sin 2t$ $\displaystyle\vec{x}$
$\displaystyle=\vec{x}$ (16)

The gauge field is regular at the horizon if and only if its components in a
coordinate chart that properly covers the horizon, for example the “Cartesian
coordinates”, are regular. That is, we have

$\displaystyle A=\mathbf{A}_{i}dx^{i}$ (17)

and there exists a $U(1)$ gauge transformation $\Lambda$, such that

$\displaystyle\lim_{\mathfrak{s}\to
0}A+d\Lambda=A^{*}_{X}(\vec{x})dX+A^{*}_{Y}(\vec{x})dY+A^{*}_{a}(\vec{x})dx^{a}$
(18)

The components on the right-hand side can only depend on $\vec{x}$ because the
$t$-circle shrinks to a point as $\mathfrak{s}\to 0$. We find

$\displaystyle\lim_{\mathfrak{s}\to
0}\partial_{\mathfrak{s}}\Lambda=A^{*}_{X}(\vec{x})\cos
2t+A^{*}_{Y}(\vec{x})\sin 2t$ (19) $\displaystyle\lim_{\mathfrak{s}\to
0}\frac{\mathbf{A}_{t}+\partial_{t}\Lambda}{\mathfrak{s}}=-2A^{*}_{X}(\vec{x})\sin
2t+2A^{*}_{Y}(\vec{x})\cos 2t$ (20) $\displaystyle\lim_{\mathfrak{s}\to
0}\mathbf{A}_{a}+\partial_{a}\Lambda=A^{*}_{a}(\vec{x})$ (21)

From (19) we see $\Lambda$ can be approximated as a linear function of
$\mathfrak{s}$ as $\mathfrak{s}\to 0$ (or $\rho\to 1$), then from (21) we know

$\displaystyle\lim_{\rho\to 1}\mathbf{A}_{a}\;\mathrm{exists}$ (22)

In addition, by integrating (20) over $t$ we find

$\displaystyle\int_{0}^{\pi}dt\mathbf{A}_{t}|_{\rho=1}=0$ (23)

This gauge fixing and regularity boundary conditions at the horizon, together
with the boundary value

$\displaystyle{\mathbf{A}_{i}}|_{\rho=0}$ $\displaystyle=\mathcal{A}_{i}$ (24)

as a turned-on source on the CFT side, determine a unique solution to the
Maxwell equation as we will see.

We utilize the translational symmetry in $t,\vec{x}$ direction and work with
Fourier modes $\tilde{\mathbf{A}}_{i}$ with Matsubara frequency
$\omega=2m,m\in\mathbb{Z}$ and spatial momentum $\vec{p}$. For simplicity, we
also rotate the spatial momentum to the $x^{1}$ direction. The Maxwell
equation

$\displaystyle d\ast F=0$ (25)

then decouples to the transverse channel for
$\tilde{\mathbf{A}}_{2},\tilde{\mathbf{A}}_{3}$ and the longitudinal channel
for $\tilde{\mathbf{A}}_{t},\tilde{\mathbf{A}}_{1}$. For the transverse
component $\tilde{\mathbf{A}}_{2}$ (and the same for $\tilde{\mathbf{A}}_{3}$)
we have

$\displaystyle(\partial_{z}^{2}-\frac{2z}{1-z^{2}}\partial_{z}-\frac{4m^{2}+p^{2}(1-z^{2})}{4z(1-z^{2})^{2}})\tilde{\mathbf{A}}_{2}=0$
(26)

where we used the convenient coordinate $z=\rho^{2}$. This is an ordinary
differential equation with four regular singularities $z=0,1,-1,\infty$. By
the substitution $\tilde{\mathbf{A}}_{2}(z)=(1-z^{2})^{-\frac{1}{2}}w(z)$, we
get a Heun equation in the normal form for $w(z)$

$\displaystyle(\partial_{z}^{2}+\frac{\frac{1}{4}-(\frac{1}{2})^{2}}{z^{2}}+\frac{\frac{1}{4}-(\frac{m}{2})^{2}}{(z-1)^{2}}+\frac{\frac{1}{4}-(\frac{m}{2}i)^{2}}{(z+1)^{2}}+\frac{p^{2}+4m^{2}-2}{8z(z-1)}-\frac{p^{2}+4m^{2}+2}{8z(z+1)})w(z)=0$
(27)

with the Heun equation parameters

$\displaystyle
t=-1,a_{0}=\frac{1}{2},a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=\frac{1}{2},u=-\frac{p^{2}+4m^{2}+2}{8}$
(28)

We refer the readers to Appendix A for a brief review of Fuchsian differential
equations, the Heun equation, its connection problem, and notational
conventions. By the boundary condition (22), $\tilde{\mathbf{A}}_{2}$ is
regular at $z=1$, so it must be proportional to the solution of exponent
$\frac{|m|}{2}$ at $z=1$. The constant of proportionality is determined by the
boundary condition $\tilde{\mathbf{A}}_{2}|_{z=0}=\tilde{\mathcal{A}}_{2}$ and
the connection relation (A). We find

$\displaystyle\tilde{\mathbf{A}}_{2}(\omega=2m,p,z)=\tilde{\mathcal{A}}_{2}(\omega,p)(1-z^{2})^{-\frac{1}{2}}\big{[}w^{(0)}_{-}+\frac{p^{2}+4m^{2}}{4}(-2\psi(1)-1$
$\displaystyle+\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\theta\frac{m}{2}+\sigma
a)-\frac{1}{2}\partial_{a_{0}}^{2}F|_{a_{0}=\frac{1}{2}}-\frac{2}{p^{2}+4m^{2}}(1+2\partial_{t}\partial_{a_{0}}F|_{a_{0}=\frac{1}{2},t=-1}))w^{(0)}_{+}\big{]}$
(29)

For the longitudinal components
$\tilde{\mathbf{A}}_{t},\tilde{\mathbf{A}}_{1}$, we have

$\displaystyle\partial_{z}^{2}\tilde{\mathbf{A}}_{t}-\frac{p^{2}}{4z(1-z^{2})}\tilde{\mathbf{A}}_{t}+\frac{2mp}{4z(1-z^{2})}\tilde{\mathbf{A}}_{1}=0$
(30)
$\displaystyle\partial_{z}^{2}\tilde{\mathbf{A}}_{1}-\frac{2z}{1-z^{2}}\partial_{z}\tilde{\mathbf{A}}_{1}-\frac{4m^{2}}{4z(1-z^{2})^{2}}\tilde{\mathbf{A}}_{1}+\frac{2mp}{4z(1-z^{2})^{2}}\tilde{\mathbf{A}}_{t}=0$
(31)
$\displaystyle\frac{2m}{1-z^{2}}\partial_{z}\tilde{\mathbf{A}}_{t}+p\partial_{z}\tilde{\mathbf{A}}_{1}=0$
(32)

Plugging (32) into $\partial_{z}\big{(}z(1-z^{2})$(30)$\big{)}$, we obtain

$\displaystyle(\partial_{z}^{2}+\frac{1-3z^{2}}{z(1-z^{2})}\partial_{z}-\frac{p^{2}(1-z^{2})+4m^{2}}{4z(1-z^{2})^{2}})\partial_{z}\tilde{\mathbf{A}}_{t}=0$
(33)

When $m\neq 0$, the solution to this third-order differential equation is
determined by the three boundary conditions

$\displaystyle\tilde{\mathbf{A}}_{1}|_{z=1}\;\mathrm{regular}$
$\displaystyle\tilde{\mathbf{A}}_{1}|_{z=0}=\tilde{\mathcal{A}}_{1},\;\tilde{\mathbf{A}}_{t}|_{z=0}=\tilde{\mathcal{A}}_{t}$
(34)

By the substitution
$\partial_{z}\tilde{\mathbf{A}}_{t}=z^{-\frac{1}{2}}(1-z^{2})^{-\frac{1}{2}}w(z)$,
(33) can be transformed to the normal Heun equation

$\displaystyle\big{(}\partial_{z}^{2}+\frac{\frac{1}{4}-0^{2}}{z^{2}}+\frac{\frac{1}{4}-(\frac{m}{2})^{2}}{(z-1)^{2}}+\frac{\frac{1}{4}-(\frac{m}{2}i)^{2}}{(z+1)^{2}}+\frac{p^{2}+4m^{2}-6}{8z(z-1)}-\frac{p^{2}+4m^{2}+6}{8z(z+1)}\big{)}w(z)=0$
(35)

with

$\displaystyle
t=-1,a_{0}=0,a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=1,u=-\frac{p^{2}+4m^{2}+6}{8}$
(36)

By (32) the solution must be proportional to $w^{(1)}_{+}$ for
$\tilde{\mathbf{A}}_{1}$ to be regular at $z=1$. The constant of
proportionality can be further determined by using the connection relation (A)
and evaluating (30) at $z=0$. We find

$\displaystyle
z^{\frac{1}{2}}\sqrt{1-z^{2}}\partial_{z}\tilde{\mathbf{A}}_{t}=$
$\displaystyle\frac{2mp\tilde{\mathcal{A}}_{1}-p^{2}\tilde{\mathcal{A}}_{t}}{4}\big{[}-w^{(0)}_{-}(z)$
$\displaystyle+(2\psi(1)-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\frac{1}{2}+\theta\frac{m}{2}+\sigma
a)+\frac{1}{2}\partial_{a_{0}}^{2}F)w^{(0)}_{+}\big{]}$ (37)

Then we integrate to obtain $\tilde{\mathbf{A}}_{t}$ with the constant of
integration given by the boundary value $\tilde{\mathcal{A}}_{t}$

$\displaystyle\tilde{\mathbf{A}}_{t}=$
$\displaystyle\tilde{\mathcal{A}}_{t}+\frac{2mp\tilde{\mathcal{A}}_{1}-p^{2}\tilde{\mathcal{A}}_{t}}{4}\big{[}-(z\log
z+\ldots)$
$\displaystyle+(2\psi(1)+1-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\frac{1}{2}+\theta\frac{m}{2}+\sigma
a)+\frac{1}{2}\partial_{a_{0}}^{2}F)(z+\ldots))\big{]}$ (38)

We get $\tilde{\mathbf{A}}_{1}$ by plugging $\tilde{\mathbf{A}}_{t}$ back to
(32)

$\displaystyle\tilde{\mathbf{A}}_{1}=$
$\displaystyle\tilde{\mathcal{A}}_{1}(1+\ldots)+\frac{2m(p\tilde{\mathcal{A}}_{t}-2m\tilde{\mathcal{A}}_{1})}{4}$
$\displaystyle\times\big{(}2\psi(1)+1-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\frac{1}{2}+\theta\frac{m}{2}+\sigma
a)+\frac{1}{2}\partial_{a_{0}}^{2}F\big{)}(z+\ldots)$ (39)

When $m=0$, we get $\tilde{\mathbf{A}}_{1}=\tilde{\mathcal{A}}_{1}$ from (32).
We still solve for
$z^{\frac{1}{2}}\sqrt{1-z^{2}}\partial_{z}\tilde{\mathbf{A}}_{t}$ from (33),
which is a linear combination of $w^{(1)}_{+}=\sqrt{1-z}(1+\ldots)$ and
$w^{(1)}_{-}=\sqrt{1-z}(\log(1-z)+\ldots)$. Then we plug it into (30) and
evaluate at $z=1$. We have ${\tilde{\mathbf{A}}_{t}}(m=0)|_{z=1}=0$ from the
boundary condition (23), and we find
$z^{\frac{1}{2}}\sqrt{1-z^{2}}\partial_{z}\tilde{\mathbf{A}}_{t}$ must be
proportional to $w^{(1)}_{+}$, the same as the previous case when $m\neq 0$.
So, we can carry over the results for $m\neq 0$ and set $m=0$ in the
expression.

To obtain the holographic correlators, we recover the dependence on $\rho_{0}$
or the inverse temperature $\beta=\pi\rho_{0}$, and read off
$\mathbf{A}^{(2)}_{i}$ from the bulk gauge field $\mathbf{A}_{i}$ (in our case
the coefficient of $z^{1}$)

$\displaystyle\tilde{\mathbf{A}}_{2}^{(2)}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle=-\frac{p^{2}+\omega^{2}}{4}\tilde{\mathcal{A}}_{2}(\omega,p)\mathcal{C}_{1}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle\tilde{\mathbf{A}}_{t}^{(2)}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle=\frac{\omega
p\tilde{\mathcal{A}}_{1}(\omega,p)-p^{2}\tilde{\mathcal{A}}_{t}(\omega,p)}{4}\mathcal{C}_{2}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle\tilde{\mathbf{A}}_{1}^{(2)}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle=\frac{\omega
p\tilde{\mathcal{A}}_{t}(\omega,p)-\omega^{2}\tilde{\mathcal{A}}_{1}(\omega,p)}{4}\mathcal{C}_{2}(\omega=\frac{2m}{\rho_{0}},p)$
(40)

where

$\displaystyle\mathcal{C}_{1}(\omega=\frac{2m}{\rho_{0}},p)=(2\psi(1)+1-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\theta\frac{m}{2}+\sigma
a)$
$\displaystyle+\frac{1}{2}\partial_{a_{0}}^{2}F+\frac{2}{\rho_{0}^{2}p^{2}+4m^{2}}(1+\partial_{t}\partial_{a_{0}}F))\big{|}_{t=-1,a_{0}=\frac{1}{2},a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=\frac{1}{2},u=-\frac{\rho_{0}^{2}p^{2}+4m^{2}+2}{8}}$
$\displaystyle\mathcal{C}_{2}(\omega=\frac{2m}{\rho_{0}},p)=(2\psi(1)+1-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\frac{1}{2}+\theta\frac{m}{2}+\sigma
a)$
$\displaystyle+\frac{1}{2}\partial_{a_{0}}^{2}F)\big{|}_{t=-1,a_{0}=0,a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=1,u=-\frac{\rho_{0}^{2}p^{2}+4m^{2}+6}{8}}$
(41)

We compute two-point correlators by the formula for renormalized one-point
correlators (10). Rotating the spatial momentum to a general direction, we
find

$\displaystyle\langle\tilde{J}_{t}(\omega,p)\tilde{J}_{t}(-\omega,-p)\rangle$
$\displaystyle=\frac{p^{2}}{2}\mathcal{C}_{2}(\omega,p)$
$\displaystyle\langle\tilde{J}_{t}(\omega,p)\tilde{J}_{b}(-\omega,-p)\rangle$
$\displaystyle=-\frac{\omega}{2}\mathcal{C}_{2}(\omega,p)p_{b}$
$\displaystyle\langle\tilde{J}_{a}(\omega,p)\tilde{J}_{b}(-\omega,-p)\rangle$
$\displaystyle=\frac{p^{2}+\omega^{2}}{2}\mathcal{C}_{1}(\omega,p)(\delta_{ab}-\frac{p_{a}p_{b}}{p^{2}})+\frac{\omega^{2}}{2}\mathcal{C}_{2}(\omega,p)\frac{p_{a}p_{b}}{p^{2}}$
(42)

##  4 Stress tensor

Gauge fixing and regularity boundary conditions at the horizon for Einstein’s
gravity follow the same line as the Maxwell theory. We can make the solid
cylinder coordinates $\rho,t,\vec{x}$ the Fefferman-Graham coordinates of the
perturbed bulk metric in the region $0\leq\rho<1$ by a diffeomorphism. Then,
the boundary value problem is formulated in this gauge with the boundary
condition that the metric has a regular limit as $\rho\to 1$ after a
diffeomorphism. For a first-order perturbation of the bulk metric, we have

$\displaystyle\delta ds^{2}=\delta\mathbf{g}_{ij}dx^{i}dx^{j}$ (43)

And to the first order, the diffeomorphism is characterized by a vector $V$,
then the regularity boundary condition at the horizon is the variation of the
bulk metric

$\displaystyle\mathcal{L}_{V}(ds^{2})+\delta ds^{2}$ (44)

has a regular limit as $\rho\to 1$ (or $\mathfrak{s}\to 0$), that is, its
components in the ”Cartesian coordinates” (3) are regular. We find

$\displaystyle\lim_{\mathfrak{s}\to
0}2\partial_{\mathfrak{s}}V^{\mathfrak{s}}=\cos^{2}2t\delta g^{*}_{XX}+2\cos
2t\sin 2t\delta g^{*}_{XY}+\sin^{2}2t\delta g^{*}_{YY}$ (45)
$\displaystyle\lim_{\mathfrak{s}\to
0}\frac{\partial_{t}V^{\mathfrak{s}}+\frac{\sinh^{2}2\mathfrak{s}}{\cosh
2\mathfrak{s}}\partial_{\mathfrak{s}}V^{t}}{2\mathfrak{s}}$
$\displaystyle=-\cos 2t\sin 2t\delta g^{*}_{XX}+(\cos^{2}2t-\sin^{2}2t)\delta
g^{*}_{XY}+\cos 2t\sin 2t\delta g^{*}_{YY}$ (46)
$\displaystyle\lim_{\mathfrak{s}\to 0}\partial_{a}V^{\mathfrak{s}}+\cosh
2\mathfrak{s}\partial_{\mathfrak{s}}V^{x}=\cos 2t\delta g^{*}_{Xa}+\sin
2t\delta g^{*}_{Ya}$ (47) $\displaystyle\lim_{\mathfrak{s}\to
0}\frac{\delta\mathbf{g}_{tt}+\frac{\sinh
2\mathfrak{s}}{\cosh^{2}2\mathfrak{s}}((3+4\cosh
4\mathfrak{s})V^{\mathfrak{s}}+\sinh
4\mathfrak{s}\partial_{t}V^{t})}{4\mathfrak{s}^{2}}$
$\displaystyle=\sin^{2}2t\delta g^{*}_{XX}-2\cos 2t\sin 2t\delta
g^{*}_{XY}+\cos^{2}2t\delta g^{*}_{YY}$ (48)
$\displaystyle\lim_{\mathfrak{s}\to
0}\delta\mathbf{g}_{ta}+\frac{\sinh^{2}2\mathfrak{s}}{\cosh
2\mathfrak{s}}\partial_{a}V^{t}+\cosh
2\mathfrak{s}\partial_{t}V^{a}=-2\mathfrak{s}\sin 2t\delta
g^{*}_{Xa}+2\mathfrak{s}\cos 2t\delta g^{*}_{Ya}$ (49)
$\displaystyle\lim_{\mathfrak{s}\to 0}\delta\mathbf{g}_{ab}+\cosh
2\mathfrak{s}(\partial_{a}V^{b}+\partial_{b}V^{a})+2\sinh
2\mathfrak{s}V^{\mathfrak{s}}\delta_{ab}=\delta g^{*}_{ab}$ (50)

(45) and (47) show that $V^{\mathfrak{s}}$ and $V^{a}$ can be approximated by
linear function in $\mathfrak{s}$ as $\mathfrak{s}\to 0$ (or $\rho\to 1$).
Plugging into (50), we see

$\displaystyle\lim_{\rho\to 1}\delta\mathbf{g}_{ab}\;\mathrm{exists}$ (51)

By (46) we know

$\displaystyle V^{t}=O(\frac{1}{\mathfrak{s}})$ (52)

as $\mathfrak{s}\to 0$. Then by integrating (49) over $t$ we find

$\displaystyle\int_{0}^{\pi}dt\delta\mathbf{g}_{ta}|_{\rho=1}=0$ (53)

Similar to the case of $U(1)$ gauge field, we work in Fourier modes and rotate
the spatial momentum to the $x^{1}$ direction. And for simplicity, we use the
variable $\mathbf{h}_{ij}=\rho^{2}\delta\mathbf{g}_{ij}$ which on the
conformal boundary equals the variation of the CFT background metric

$\displaystyle{\mathbf{h}_{ij}}|_{\rho=0}=\delta\gamma_{ij}$ (54)

The linearized Einstein equation

$\displaystyle\frac{1}{2}(\nabla^{\lambda}\nabla_{\mu}\delta
g_{\lambda\nu}+\nabla^{\lambda}\nabla_{\nu}\delta
g_{\lambda\mu}-\nabla^{\lambda}\nabla_{\lambda}\delta
g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\delta g^{\lambda}_{\lambda})+4\delta
g_{\mu\nu}=0$ (55)

decouples to the scalar channel of $\tilde{\mathbf{h}}_{23}$ and
$\tilde{\mathbf{h}}_{22}-\tilde{\mathbf{h}}_{33}$, the shear channel of
$\tilde{\mathbf{h}}_{t2},\tilde{\mathbf{h}}_{12}$ and
$\tilde{\mathbf{h}}_{t3},\tilde{\mathbf{h}}_{13}$, and the sound channel of
$\tilde{\mathbf{h}}_{tt},\tilde{\mathbf{h}}_{11},\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33},\tilde{\mathbf{h}}_{t1}$.
In the scalar channel, we have

$\displaystyle\partial_{z}^{2}\tilde{\mathbf{h}}_{23}-\frac{1+z^{2}}{z(1-z^{2})}\partial_{z}\tilde{\mathbf{h}}_{23}-\frac{p^{2}(1-z^{2})+\omega^{2}}{4z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{23}=0$
(56)

and in the shear channel, we have

$\displaystyle\partial_{z}^{2}\tilde{\mathbf{h}}_{t2}-\frac{1}{z}\partial_{z}\tilde{\mathbf{h}}_{t2}-\frac{p^{2}}{4z(1-z^{2})}\tilde{\mathbf{h}}_{t2}+\frac{2mp}{4z(1-z^{2})}\tilde{\mathbf{h}}_{12}=0$
(57)
$\displaystyle\partial_{z}^{2}\tilde{\mathbf{h}}_{12}-\frac{1+z^{2}}{z(1-z^{2})}\partial_{z}\tilde{\mathbf{h}}_{12}-\frac{4m^{2}}{4z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{12}+\frac{2mp}{4z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{t2}=0$
(58)
$\displaystyle\frac{2m}{1-z^{2}}\partial_{z}\tilde{\mathbf{h}}_{t2}+p\partial_{z}\tilde{\mathbf{h}}_{12}=0$
(59)

The computation in these two channels is very similar to that of the
transverse channel and longitudinal channel of the $U(1)$ gauge field in the
previous section, so we will be brief.

For the scalar channel, by the substitution
$\tilde{\mathbf{h}}_{23}(z)=z^{\frac{1}{2}}(1-z^{2})^{-\frac{1}{2}}w(z)$ we
obtain Heun equation in the normal form

$\displaystyle\big{(}\partial_{z}^{2}+\frac{\frac{1}{4}-1^{2}}{z^{2}}+\frac{\frac{1}{4}-(\frac{m}{2})^{2}}{(z-1)^{2}}+\frac{\frac{1}{4}-(\frac{m}{2}i)^{2}}{(z+1)^{2}}+\frac{4m^{2}+p^{2}+2}{8z(z-1)}+\frac{-4m^{2}-p^{2}+2}{8z(z+1)}\big{)}w(z)=0$
(60)

with

$\displaystyle
t=-1,a_{0}=1,a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=0,u=-\frac{p^{2}+4m^{2}-2}{8}$
(61)

With the boundary conditions and the connection relation (A) we find

$\displaystyle\tilde{\mathbf{h}}_{23}=\widetilde{\delta\gamma}_{23}\Big{[}(1+\ldots)+\frac{(p^{2}+4m^{2})^{2}}{64}\big{(}4\psi(1)+5-\sum_{\sigma=\pm,\sigma^{{}^{\prime}}=\pm}\psi(-\frac{1}{2}+\sigma\frac{m}{2}+\sigma^{{}^{\prime}}a)$
$\displaystyle+\partial_{a_{0}}^{2}F-\frac{32}{p^{2}+4m^{2}}(4a^{2}-2a^{2}m^{2}+\frac{1}{4}m^{4}+4(\partial_{t}F)^{2}+(-8a^{2}+2m^{2})\partial_{t}F-4\partial_{t}F\partial_{t}\partial_{a_{0}}F$
$\displaystyle+(-2+4a^{2}-m^{2})\partial_{t}\partial_{a_{0}}F)\big{)}z^{2}(1+\ldots)\Big{]}$
(62)

For the shear channel, plugging (59) into $\partial_{z}$(57), we obtain

$\displaystyle\big{(}\partial_{z}^{2}-\frac{2z}{1-z^{2}}\partial_{z}-\frac{4m^{2}+p^{2}(1-z^{2})-8z(1-z^{2})}{4z(1-z^{2})^{2}}\big{)}\partial_{z}\tilde{\mathbf{h}}_{t2}=0$
(63)

By the substitution
$\partial_{z}\tilde{\mathbf{h}}_{t2}=(1-z^{2})^{-\frac{1}{2}}w(z)$, the
equation is transformed to Heun equation of normal form

$\displaystyle\big{(}\partial_{z}^{2}+\frac{\frac{1}{4}-(\frac{1}{2})^{2}}{z^{2}}+\frac{\frac{1}{4}-(\frac{m}{2})^{2}}{(z-1)^{2}}+\frac{\frac{1}{4}-(\frac{m}{2}i)^{2}}{(z+1)^{2}}+\frac{4m^{2}+p^{2}-10}{8z(z-1)}-\frac{4m^{2}+p^{2}+10}{8z(z+1)}\big{)}w(z)=0$
(64)

with

$\displaystyle
t=-1,a_{0}=\frac{1}{2},a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=\frac{3}{2},u=-\frac{p^{2}+4m^{2}+10}{8}$
(65)

With boundary conditions and the connection relation (A) we find

$\displaystyle\partial_{z}\tilde{\mathbf{h}}_{t2}=\frac{2mp\widetilde{\delta\gamma}_{12}-p^{2}\widetilde{\delta\gamma}_{t2}}{4}(1-z^{2})^{-\frac{1}{2}}\big{[}w^{(0)}_{\widetilde{-}}$
$\displaystyle+\frac{p^{2}+4m^{2}}{4}\big{(}-2\psi(1)-1+\frac{1}{2}\psi(1+a_{1}+a)+\frac{1}{2}\psi(1+a_{1}-a)+\frac{1}{2}\psi(a_{1}+a)+\frac{1}{2}\psi(a_{1}-a)$
$\displaystyle-\frac{1}{2}\partial_{a_{0}}^{2}F-\frac{2}{p^{2}+4m^{2}}(1+2\partial_{t}\partial_{a_{0}}F)\big{)}w^{(0)}_{\widetilde{+}}\big{]}$
(66)

and furthermore

$\displaystyle\tilde{\mathbf{h}}_{t2}=\widetilde{\delta\gamma}_{t2}(1+\ldots)+\frac{p^{2}+4m^{2}}{32}(2mp\widetilde{\delta\gamma}_{12}-p^{2}\widetilde{\delta\gamma}_{t2})\big{(}-2\psi(1)-1+\frac{1}{2}\psi(1+a_{1}+a)+\frac{1}{2}\psi(1+a_{1}-a)$
$\displaystyle+\frac{1}{2}\psi(a_{1}+a)+\frac{1}{2}\psi(a_{1}-a)-\frac{1}{2}\partial_{a_{0}}^{2}F-\frac{2}{p^{2}+4m^{2}}(1+2\partial_{t}\partial_{a_{0}}F)\big{)}(z^{2}+\ldots)$
$\displaystyle\tilde{\mathbf{h}}_{12}=\widetilde{\delta\gamma}_{12}(1+\ldots)+\frac{p^{2}+4m^{2}}{32}(2mp\widetilde{\delta\gamma}_{t2}-4m^{2}\widetilde{\delta\gamma}_{12})\big{(}-2\psi(1)-1+\frac{1}{2}\psi(1+a_{1}+a)+\frac{1}{2}\psi(1+a_{1}-a)$
$\displaystyle+\frac{1}{2}\psi(a_{1}+a)+\frac{1}{2}\psi(a_{1}-a)-\frac{1}{2}\partial_{a_{0}}^{2}F-\frac{2}{p^{2}+4m^{2}}(1+2\partial_{t}\partial_{a_{0}}F)\big{)}(z^{2}+\ldots)$
(67)

The coefficient of $z^{2}$ in the solution of perturbed bulk metric
corresponds to the variation of $\mathbf{g}^{(4)}_{ij}$ and hence the
variation of the one-point correlator by (6). Then we can read off the two-
point correlators

$\displaystyle\langle\tilde{T}_{23}(\omega=\frac{2m}{\rho_{0}},p)\tilde{T}_{23}(-\omega,-p)\rangle=\frac{1}{2\pi
G}\frac{(p^{2}+\omega^{2})^{2}}{32}\mathcal{C}_{3}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle\langle\tilde{T}_{t2}(\omega=\frac{2m}{\rho_{0}},p)\tilde{T}_{t2}(-\omega,-p)\rangle=\frac{1}{2\pi
G}\frac{p^{2}+\omega^{2}}{32}p^{2}\mathcal{C}_{4}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle\langle\tilde{T}_{t2}(\omega=\frac{2m}{\rho_{0}},p)\tilde{T}_{12}(-\omega,-p)\rangle=-\frac{1}{2\pi
G}\frac{p^{2}+\omega^{2}}{32}\omega
p\mathcal{C}_{4}(\omega=\frac{2m}{\rho_{0}},p)$
$\displaystyle\langle\tilde{T}_{12}(\omega=\frac{2m}{\rho_{0}},p)\tilde{T}_{12}(-\omega,-p)\rangle=\frac{1}{2\pi
G}\frac{p^{2}+\omega^{2}}{32}\omega^{2}\mathcal{C}_{4}(\omega=\frac{2m}{\rho_{0}},p)$
(68)

with

$\displaystyle\mathcal{C}_{3}(\omega=\frac{2m}{\rho_{0}},p)=\big{[}2\psi(1)+\frac{5}{2}-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(-\frac{1}{2}+\theta\frac{m}{2}+\sigma
a)$
$\displaystyle+\frac{1}{2}\partial_{a_{0}}^{2}F-\frac{16}{(\rho_{0}^{2}p^{2}+4m^{2})^{2}}\big{(}4a^{2}-2a^{2}m^{2}+\frac{1}{4}m^{4}+4(\partial_{t}F)^{2}+(-8a^{2}+2m^{2})\partial_{t}F$
$\displaystyle-4\partial_{t}F\partial_{t}\partial_{a_{0}}F+(-2+4a^{2}-m^{2})\partial_{t}\partial_{a_{0}}F\big{)}\big{]}\big{|}_{t=-1,a_{0}=1,a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=0,u=-\frac{\rho_{0}^{2}p^{2}+4m^{2}-2}{8}}$
$\displaystyle\mathcal{C}_{4}(\omega=\frac{2m}{\rho_{0}},p)=\big{(}2\psi(1)+1-\frac{1}{2}\sum_{\theta,\sigma=\pm}\psi(\theta\frac{m}{2}+\sigma
a)$
$\displaystyle+\frac{1}{2}\partial_{a_{0}}^{2}F+\frac{2}{\rho_{0}^{2}p^{2}+4m^{2}}(1+2\partial_{t}\partial_{a_{0}}F)\big{)}\big{|}_{t=-1,a_{0}=\frac{1}{2},a_{1}=\frac{|m|}{2},a_{t}=\frac{m}{2}i,a_{\infty}=\frac{3}{2},u=-\frac{\rho_{0}^{2}p^{2}+4m^{2}+10}{8}}$
(69)

The unsolved part for the stress tensor is the sound channel. We have

$\displaystyle\partial_{z}^{2}\tilde{\mathbf{h}}_{tt}-\frac{3-5z^{2}}{2z(1-z^{2})}\partial_{z}\tilde{\mathbf{h}}_{tt}-\frac{1+z^{2}}{2z}\partial_{z}(\tilde{\mathbf{h}}_{11}+\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})+\frac{-4z+12z^{3}-p^{2}(1-z^{2})}{4z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{tt}$
$\displaystyle-\frac{4m^{2}}{4z(1-z^{2})}(\tilde{\mathbf{h}}_{11}+\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})+\frac{2mp}{2z(1-z^{2})}\tilde{\mathbf{h}}_{t1}=0$
(70)
$\displaystyle\partial_{z}^{2}\tilde{\mathbf{h}}_{11}-\frac{3+z^{2}}{2z(1-z^{2})}\partial_{z}\tilde{\mathbf{h}}_{11}-\frac{1}{2z(1-z^{2})}\partial_{z}\tilde{\mathbf{h}}_{tt}-\frac{1}{2z}\partial_{z}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})$
$\displaystyle-\frac{4m^{2}}{4z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{11}-\frac{p^{2}+4z}{4z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{tt}-\frac{p^{2}}{4z(1-z^{2})}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})+\frac{2mp}{2z(1-z^{2})^{2}}\tilde{\mathbf{h}}_{t1}=0$
(71)
$\displaystyle\partial_{z}^{2}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})-\frac{2}{z(1-z^{2})}\partial_{z}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})-\frac{1}{z(1-z^{2})}\partial_{z}\tilde{\mathbf{h}}_{tt}-\frac{1}{z}\partial_{z}\tilde{\mathbf{h}}_{11}$
$\displaystyle-\frac{4m^{2}+p^{2}(1-z^{2})}{4z(1-z^{2})^{2}}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})-\frac{2}{(1-z^{2})^{2}}\tilde{\mathbf{h}}_{tt}=0$
(72)
$\displaystyle\partial_{z}^{2}\tilde{\mathbf{h}}_{t1}-\frac{1}{z}\partial_{z}\tilde{\mathbf{h}}_{t1}-\frac{2mp}{4z(1-z^{2})}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})=0$
(73)
$\displaystyle\partial_{z}^{2}(\tilde{\mathbf{h}}_{11}+\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})+\frac{1}{1-z^{2}}\partial_{z}^{2}\tilde{\mathbf{h}}_{tt}-\frac{z}{1-z^{2}}\partial_{z}(\tilde{\mathbf{h}}_{11}+\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})$
$\displaystyle+\frac{z}{(1-z^{2})^{2}}\partial_{z}\tilde{\mathbf{h}}_{tt}+\frac{2}{(1-z^{2})^{3}}\tilde{\mathbf{h}}_{tt}=0$
(74) $\displaystyle
2m\partial_{z}(\tilde{\mathbf{h}}_{11}+\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})+\frac{2mz}{1-z^{2}}\partial_{z}(\tilde{\mathbf{h}}_{11}+\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})-p\partial_{z}\tilde{\mathbf{h}}_{t1}-\frac{2pz}{1-z^{2}}\tilde{\mathbf{h}}_{t1}=0$
(75) $\displaystyle
p\partial_{z}(\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33})+\frac{p}{1-z^{2}}\partial_{z}\tilde{\mathbf{h}}_{tt}-\frac{2m}{1-z^{2}}\partial_{z}\tilde{\mathbf{h}}_{t1}+\frac{pz}{(1-z^{2})^{2}}\tilde{\mathbf{h}}_{tt}=0$
(76)

We don’t know how to analytically solve the boundary value problem here. For
future reference, we can reduce the sound channel to a five-dimensional first-
order equation of variables
$\tilde{\mathbf{h}}_{tt},\tilde{\mathbf{h}}_{11},\frac{\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33}}{2},\tilde{\mathbf{h}}_{t1},\partial_{z}\tilde{\mathbf{h}}_{t1}$
(a similar equation can be found in Kovtun:2005ev ), and by the substitution

$\displaystyle\begin{pmatrix}\tilde{\mathbf{h}}_{tt}\\\
\tilde{\mathbf{h}}_{11}\\\
\frac{\tilde{\mathbf{h}}_{22}+\tilde{\mathbf{h}}_{33}}{2}\\\
\tilde{\mathbf{h}}_{t1}\\\
\partial_{z}\tilde{\mathbf{h}}_{t1}\end{pmatrix}=\begin{pmatrix}0&-\frac{1}{3}(1-z^{2})^{2}&\frac{2}{3}z(1-z^{2})&0&0\\\
-z^{2}&1-z^{2}&\frac{2}{3}z&0&0\\\ \frac{1}{2}z^{2}&0&-\frac{1}{3}z&0&0\\\
0&0&0&1-z^{2}&0\\\ 0&0&0&0&z\end{pmatrix}H$ (77)

we can transform the equation into a Fuchsian system of normal form 222The
connection relation of local solutions of Fuchsian systems has been studied in
the Mathematics literature (for example, see haraoka2015linear  for very brief
introduction), and knowledge of the connection relation in our system will
enable us to solve the boundary problem. The main progress we can find in the
literature is techniques of “addition” and “middle convolution”
oshima2011fractional  that can transform a Fuchsian system to a simpler one,
and most importantly, if we know the connection relation for one system we
know it for the other. We are however not able to simplify our Fuchsian system
by these transformations.

$\displaystyle\partial_{z}H=(\frac{M_{0}}{z}+\frac{M_{1}}{z-1}+\frac{M_{-1}}{z+1})H$
(78)

with

$\displaystyle M_{0}$
$\displaystyle=\begin{pmatrix}-2&-\frac{2}{3}&0&\frac{p}{3m}&\frac{12m^{2}+p^{2}}{6mp}\\\
0&0&0&0&0\\\ 0&-m^{2}+\frac{p^{2}}{12}&-1&mp&0\\\ 0&0&0&0&0\\\
0&0&-\frac{mp}{3}&0&2\end{pmatrix}$ $\displaystyle M_{1}$
$\displaystyle=\begin{pmatrix}0&0&-\frac{1}{3}&0&-\frac{m}{p}\\\
0&-\frac{1}{2}&0&-\frac{p}{2m}&-\frac{p}{4m}\\\
\frac{p^{2}}{8}&\frac{-1+m^{2}}{2}&2&\frac{p(1-m^{2})}{2m}&\frac{p}{4m}\\\
0&0&0&-1&-\frac{1}{2}\\\ -\frac{mp}{4}&0&\frac{mp}{6}&0&0\end{pmatrix}$
$\displaystyle M_{-1}$
$\displaystyle=\begin{pmatrix}0&0&\frac{1}{3}&0&-\frac{m}{p}\\\
0&-\frac{1}{2}&0&-\frac{p}{2m}&-\frac{p}{4m}\\\
\frac{p^{2}}{8}&\frac{1+m^{2}}{2}&2&-\frac{p(1+m^{2})}{2m}&-\frac{p}{4m}\\\
0&0&0&-1&-\frac{1}{2}\\\ \frac{mp}{4}&0&\frac{mp}{6}&0&0\end{pmatrix}$ (79)

##  5 Summary and discussion

In our study, we calculated holographic Euclidean thermal correlators of the
$U(1)$ current and stress tensor for four-dimensional CFTs using the
$\mathrm{AdS}_{5}$ planar black hole, following the approach of GKPW. By
utilizing the connection relation of local solutions of the Heun equation, we
obtained exact correlators for the $U(1)$ current and stress tensor in the
scalar and shear channels.

Extensive research has focused on thermal two-point correlators (thermal
spectral functions). Notably, Kovtun:2005ev  demonstrated the presence of
gauge invariants in each channel that diagonalize coupled differential
equations. These invariants and their derivatives render the on-shell action
quadratic. Thermal two-point correlators have been computed using this
formalism numerically or analytically by approximations Kovtun:2006pf ;
Teaney:2006nc . For example in the longitudinal channel the gauge invariant is
$E_{L}=p\tilde{\mathbf{A}}_{t}-\omega\tilde{\mathbf{A}}_{1}$ and we have

$\displaystyle\partial_{z}^{2}E_{L}-\frac{2\omega^{2}z}{(1-z^{2})(\omega^{2}+p^{2}(1-z^{2}))}\partial_{z}E_{L}-\frac{\omega^{2}+p^{2}(1-z^{2})}{4z(1-z^{2})^{2}}E_{L}=0$
(80)

This is a Fuchsian differential equation with six singularities. The two
singularities $z=\pm\sqrt{1+\frac{\omega^{2}}{p^{2}}}$ are apparent
singularities since they don’t appear in the equation of the fields. One can
verify that these apparent singularities cannot be transformed away by a
substitution $E_{L}(z)=P(z)f(z)$ where $P(z)$ is a meromorphic function that
does not introduce new singularities. In essence, these apparent singularities
remain inherent to the equation. We don’t know how to relate this equation to
the Heun equation and obtain the exact holographic correlators. From the
technical standpoint, we want to work with equations of fields, and in the
Euclidean signature, the boundary conditions of fields with
gauge/diffeomorphism symmetry are clearly specified. This is the technical
reason for our approach of holographic computation, in addition to giving an
illustrative example of Euclidean boundary value problems.

Thermal two-point correlators find diverse applications. They encode the
information in operator product expansion (OPE) of holographic CFTs. For
instance, Fitzpatrick:2019zqz ; Li:2019tpf ; Karlsson:2022osn  computed
holographic correlators in the OPE limit via near-boundary analysis,
extracting OPE coefficients for multi-stress tensors. For integer operator
dimension with operator mixing, exact two-point correlators are necessary for
complete OPE coefficient extraction. Moreover, if we can analytically continue
to the Lorentzian signature, we will get better understanding of the linear
response to perturbations in thermal equilibrium, and compute transport
coefficients such as shear viscosity, thermal conductivity, and electric
conductivity Policastro:2001yc ; Hartnoll:2016apf , and higher order transport
coefficients (see Kovtun:2018dvd ; Grieninger:2021rxd  for formula of second
order coefficients in terms of two-point correlators and holographic
computation). In addition, we can probe the chaotic dynamics by studying the
pole-skipping of the correlators Grozdanov:2017ajz ; Blake:2017ris ;
Blake:2018leo . In addition, we consider spherical thermal correlators (scalar
case solved in Dodelson:2022yvn ) and stress tensor correlators from a higher
derivative gravity theory as interesting generalizations of our work.

###### Acknowledgements.

We would like to thank Alba Grassi, Cristoforo Iossa, Yun-Ze Li, Hongfei Shu,
Ashish Shukla and Yunda Zhang for their helpful discussion. S.H. would
appreciate the financial support from the Fundamental Research Funds for the
Central Universities and Max Planck Partner Group and the Natural Science
Foundation of China (NSFC) Grants No. 12075101 and No. 12235016.

##  Appendix A Fuchsian ODE, the Heun equation and connection problem

In this appendix, we briefly review Fuchsian differential equations, the Heun
equation, and its connection relation we used in the computation in the main
text.

An ordinary differential equation (ODE) is called Fuchsian if the coefficients
are rational functions and all singularities are regular. Eigenvectors of
local monodromy constitute a natural basis of local solutions around
singularities. When eigenvalues of the local monodromy are all distinct,
eigenvectors span the space of local solutions, and they take the form of a
series

$\displaystyle
w^{(z_{0})}_{k}=(z-z_{0})^{\rho_{k}}\sum_{i=0}^{\infty}c_{i}(z-z_{0})^{i}$
(81)

where $z_{0}$ is the singularity, $k$ labels the local solution and the
prefactor $(z-z_{0})^{\rho_{k}}$ captures the local monodromy. The
characteristic exponents $\rho_{k}$ are computed as the roots of the indicial
equation. We usually adopt the normalization that $c_{0}=1$. When we have
repeated eigenvalues of the local monodromy, that is some characteristic
exponents differ by integers, we may need generalized eigenvectors to span the
space of local solutions, and they are expressed as series with logarithms.
For a second order ODE, we label the two characteristic exponents as
$\rho_{+},\rho_{-}$, with $\mathrm{Re}\rho_{+}\geq\mathrm{Re}\rho_{-}$. There
is always a series solution without logarithm $w^{(z_{0})}_{+}$ with the
exponent $\rho_{+}$333One can show that the recursion relation of the
coefficients of the series with exponent $\rho_{+}$ is always non-degenerate..
If two exponents differ by an integer, the other solution $w^{(z_{0})}_{-}$ to
form a basis may contain logarithm. There is also no canonical choice of
$w^{(z_{0})}_{-}$ since we can add any constant multiple of $w^{(z_{0})}_{+}$
to $w^{(z_{0})}_{-}$. For computational convenience, we choose the convention
that the coefficient of the power $(z-z_{0})^{\rho_{+}}$ is zero in
$w^{(z_{0})}_{-}$.

The Heun equation is the second-order Fuchsian ODE with four regular
singularities. By Möbius transformation and substitutions, we can bring it to
the normal form

$\displaystyle\big{(}$
$\displaystyle\partial_{z}^{2}+\frac{\frac{1}{4}-a_{0}^{2}}{z^{2}}+\frac{\frac{1}{4}-a_{1}^{2}}{(z-1)^{2}}+\frac{\frac{1}{4}-a_{t}^{2}}{(z-t)^{2}}-\frac{\frac{1}{2}-a_{1}^{2}-a_{t}^{2}-a_{0}^{2}+a_{\infty}^{2}+u}{z(z-1)}+\frac{u}{z(z-t)}\big{)}w(z)=0$
(82)

The four singularities with exponents at these points are

$\displaystyle z=0,\rho=\frac{1}{2}\pm a_{0}$ $\displaystyle
z=1,\rho=\frac{1}{2}\pm a_{1}$ $\displaystyle z=t,\rho=\frac{1}{2}\pm a_{t}$
$\displaystyle z=\infty,\rho=-\frac{1}{2}\pm a_{\infty}$ (83)

We adopt the convention that $\mathrm{Re}a_{0}\geq 0$ etc., so the exponents
with plus sign will be the exponent with greater real part $\rho^{+}$. The
connection relation of the local solutions in the generic case (that is,
characteristic exponents do not differ by an integer) was studied in
Bonelli:2022ten  by relating the Heun equation to the Belavin-Polyakov-
Zamolodchikov (BPZ) equation Belavin:1984vu  satisfied by conformal blocks
with degenerate insertion444One can also refer to the relevant studies offered
by Cao:2016hvd ; He:2017lrg . in the Liouville field theory in the
semiclassical limit. By the Alday-Gaiotto-Tachikawa (AGT) correspondence, the
Liouville correlators can be exactly computed by localization in
supersymmetric gauge theories Alday:2009aq ; Nekrasov:2002qd ; Nekrasov:2003rj
; Alday:2009fs ; Nekrasov:2009rc . Without losing generality, let $z=0$ and
$z=1$ be two adjacent singularities, the connection relation between local
solutions around these two points is

$\displaystyle
w^{(1)}_{\theta}(z)=\sum_{\theta^{{}^{\prime}}=\pm}\mathcal{M}_{\theta\theta^{{}^{\prime}}}(a_{1},a_{0};a)e^{(\frac{\theta}{2}\partial_{a_{1}}-\frac{\theta^{{}^{\prime}}}{2}\partial_{a_{0}})F({a_{t}\atop
a_{\infty}}a{a_{1}\atop a_{0}};\frac{1}{t})}w^{(0)}_{\theta^{{}^{\prime}}}(z)$
(84)

where

$\displaystyle\mathcal{M}_{\theta\theta^{{}^{\prime}}}(a_{1},a_{0};a)=\frac{\Gamma(-2\theta^{{}^{\prime}}a_{0})\Gamma(1+2\theta
a_{1})}{\Gamma(\frac{1}{2}+\theta
a_{1}-\theta^{{}^{\prime}}a_{0}+a)\Gamma(\frac{1}{2}+\theta
a_{1}-\theta^{{}^{\prime}}a_{0}-a)}$ (85)

and $F$ is the Nekrasov-Shatashvili function, defined as power series in
$\frac{1}{t}$ with combinatorially defined rational functions of other
parameters as the coefficients, see Appendix B for details. The exchange
momentum $a$ is to be implicitly determined from the relation

$\displaystyle u=-\frac{1}{4}-a^{2}+a_{t}^{2}+a_{0}^{2}+t\partial_{t}F$ (86)

In our computation, the masslessness of the bulk fields leads to a degenerate
local monodromy of the Heun equation at $z=0$ (the conformal boundary), that
is, two exponents differ by an integer ($a_{0}$ becomes a half-integer). This
degenerate scenario can be derived as a limit of the generic case, as a
specific solution to the Heun equation continuously depends on the parameters.
The emergence of logarithm and the discontinuity of the local monodromy basis
reflect a qualitative change of the local monodromy, rather than a specific
solution. The solution $w^{(1)}_{+}$ remains well-defined and continuously
depends on parameters including $a_{0}$, even when $a_{1}$ approaches half-
integers 555Meanwhile $w^{(1)}_{-}$ is not continuous when $a_{1}$ approaches
half-integers. When both $a_{0}$ and $a_{1}$ are half-integers, the complete
connection relation is computed by solving two linear equations obtained from
the limits of $w^{(0)}_{+}$ and $w^{(1)}_{+}$.. We proceed to take the limit
$a_{0}\to\frac{N}{2},N\in\mathbb{N}$ while keeping other parameters, such as
$t,a_{1},a_{t},a_{\infty},a$, fixed 666Another curve in the parameter space
can also be chosen to approach the limit, such as fixing $u$, an explicit
parameter in the Heun equation, instead of $a$. However, as the connection
coefficients explicitly depend on $a$, fixing $a$ yields a relatively simple
expression for the limit.. For $a_{0}=0$ we have

$\displaystyle w^{(1)}_{+}=\lim_{a_{0}\to 0}\frac{1}{2a_{0}}\big{[}$
$\displaystyle\frac{\Gamma(1+2a_{1})\Gamma(1+2a_{0})}{\Gamma(\frac{1}{2}+a_{1}+a_{0}+a)\Gamma(\frac{1}{2}+a_{1}+a_{0}-a)}e^{(\frac{1}{2}\partial_{a_{1}}+\frac{1}{2}\partial_{a_{0}})F}z^{\frac{1}{2}-a_{0}}(1+\ldots)$
$\displaystyle-\frac{\Gamma(1+2a_{1})\Gamma(1-2a_{0})}{\Gamma(\frac{1}{2}+a_{1}-a_{0}+a)\Gamma(\frac{1}{2}+a_{1}-a_{0}-a)}e^{(\frac{1}{2}\partial_{a_{1}}-\frac{1}{2}\partial_{a_{0}})F}z^{\frac{1}{2}+a_{0}}(1+\ldots)\big{]}$
(87)

The quantity in the square bracket must vanish when $a_{0}=0$ for the limit to
exist. It indeed vanish because $\partial_{a_{0}}F|_{a_{0}=0}=0$ with $F$
being an even function of $a_{0}$. Then the limit becomes the derivative with
respect to $a_{0}$, and we get

$\displaystyle w^{(1)}_{+}=$
$\displaystyle\frac{\Gamma(1+2a_{1})}{\Gamma(\frac{1}{2}+a_{1}+a)\Gamma(\frac{1}{2}+a_{1}-a)}e^{\frac{1}{2}\partial_{a_{1}}F}z^{\frac{1}{2}}$
$\displaystyle(2\psi(1)-\psi(\frac{1}{2}+a_{1}+a)-\psi(\frac{1}{2}+a_{1}-a)+\frac{1}{2}\partial_{a_{0}}^{2}F-\log
z+\ldots)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(1+2a_{1})}{\Gamma(\frac{1}{2}+a_{1}+a)\Gamma(\frac{1}{2}+a_{1}-a)}e^{\frac{1}{2}\partial_{a_{1}}F}$
$\displaystyle\big{[}-w^{(0)}_{-}+\big{(}2\psi(1)-\psi(\frac{1}{2}+a_{1}+a)-\psi(\frac{1}{2}+a_{1}-a)+\frac{1}{2}\partial_{a_{0}}^{2}F\big{)}w^{(0)}_{+}\big{]}$
(88)

where $\psi$ denotes the digamma function. For $a_{0}=\frac{1}{2}$ we find

$\displaystyle w^{(1)}_{+}=$
$\displaystyle\lim_{a_{0}\to\frac{1}{2}}\big{[}\frac{\Gamma(1+2a_{1})\Gamma(2a_{0})}{\Gamma(\frac{1}{2}+a_{1}+a_{0}+a)\Gamma(\frac{1}{2}+a_{1}+a_{0}-a)}e^{(\frac{1}{2}\partial_{a_{1}}+\frac{1}{2}\partial_{a_{0}})F}$
$\displaystyle\times
z^{\frac{1}{2}-a_{0}}(1+\frac{-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u}{(1-2a_{0})t}z+\ldots)$
$\displaystyle+\frac{\Gamma(1+2a_{1})\Gamma(-2a_{0})}{\Gamma(\frac{1}{2}+a_{1}-a_{0}+a)\Gamma(\frac{1}{2}+a_{1}-a_{0}-a)}e^{(\frac{1}{2}\partial_{a_{1}}-\frac{1}{2}\partial_{a_{0}})F}z^{\frac{1}{2}+a_{0}}(1+\ldots)\big{]}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(1+2a_{1})}{\Gamma(1+a_{1}+a)\Gamma(1+a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}+\frac{1}{2}\partial_{a_{0}})F}+\Gamma(1+2a_{1})e^{\frac{1}{2}\partial_{a_{1}}F}z$
$\displaystyle\times\lim_{a_{0}\to\frac{1}{2}}\frac{1}{1-2a_{0}}\big{[}\frac{\Gamma(2a_{0})}{\Gamma(\frac{1}{2}+a_{1}+a_{0}+a)\Gamma(\frac{1}{2}+a_{1}+a_{0}-a)}e^{\frac{1}{2}\partial_{a_{0}}F}$
$\displaystyle\times\frac{-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u}{t}z^{\frac{1}{2}-a_{0}}$
$\displaystyle-\frac{\Gamma(2-2a_{0})}{2a_{0}\Gamma(\frac{1}{2}+a_{1}-a_{0}+a)\Gamma(\frac{1}{2}+a_{1}-a_{0}-a)}e^{-\frac{1}{2}\partial_{a_{0}}F}z^{-\frac{1}{2}+a_{0}}+\ldots\big{]}$
(89)

The quantity in the square bracket must vanish when $a_{0}=\frac{1}{2}$ for
the limit to exist, that is, we must have

$\displaystyle
e^{\partial_{a_{0}}F}\frac{-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u}{t}\Big{|}_{a_{0}=\frac{1}{2}}$
$\displaystyle=e^{\partial_{a_{0}}F}\frac{-\frac{1+t}{4}+ta_{0}^{2}+ta_{1}^{2}+(1-t)a^{2}-a_{\infty}^{2}-(1-t)t\partial_{t}F}{t}\Big{|}_{a_{0}=\frac{1}{2}}$
$\displaystyle=a_{1}^{2}-a^{2}$ (90)

By the expansion of $F$

$\displaystyle F$
$\displaystyle=\frac{(\frac{1}{4}-a^{2}-a_{t}^{2}+a_{\infty}^{2})(\frac{1}{4}-a^{2}-a_{1}^{2}+a_{0}^{2})}{\frac{1}{2}-2a^{2}}\frac{1}{t}+O(\frac{1}{t^{2}})$
(91)

one can verify (A) holds to the order of expansion. Again, the limit becomes
the derivative with respect to $a_{0}$ and we find

$\displaystyle w^{(1)}_{+}=$
$\displaystyle\frac{\Gamma(1+2a_{1})}{\Gamma(1+a_{1}+a)\Gamma(1+a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}+\frac{1}{2}\partial_{a_{0}})F}$
$\displaystyle+\frac{\Gamma(1+2a_{1})}{\Gamma(a_{1}+a)\Gamma(a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}-\frac{1}{2}\partial_{a_{0}})F}z\big{[}-2\psi(1)-1+\frac{1}{2}\psi(1+a_{1}+a)+\frac{1}{2}\psi(1+a_{1}-a)$
$\displaystyle+\frac{1}{2}\psi(a_{1}+a)+\frac{1}{2}\psi(a_{1}-a)-\frac{1}{2}\partial_{a_{0}}^{2}F+\log
z-\frac{t+t(1-t)\partial_{t}\partial_{a_{0}}F}{2(-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u)}\big{]}+\ldots$
$\displaystyle=$
$\displaystyle\frac{\Gamma(1+2a_{1})}{\Gamma(1+a_{1}+a)\Gamma(1+a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}+\frac{1}{2}\partial_{a_{0}})F}w^{(0)}_{-}$
$\displaystyle+\frac{\Gamma(1+2a_{1})}{\Gamma(a_{1}+a)\Gamma(a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}-\frac{1}{2}\partial_{a_{0}})F}\big{[}-2\psi(1)-1+\frac{1}{2}\psi(1+a_{1}+a)+\frac{1}{2}\psi(1+a_{1}-a)$
$\displaystyle+\frac{1}{2}\psi(a_{1}+a)+\frac{1}{2}\psi(a_{1}-a)-\frac{1}{2}\partial_{a_{0}}^{2}F-\frac{t+t(1-t)\partial_{t}\partial_{a_{0}}F}{2(-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u)}\big{]}w^{(0)}_{+}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(1+2a_{1})}{\Gamma(a_{1}+a)\Gamma(a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}-\frac{1}{2}\partial_{a_{0}})F}\big{[}\frac{t}{-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u}w^{(0)}_{-}$
$\displaystyle+\big{(}-2\psi(1)-1+\frac{1}{2}\psi(1+a_{1}+a)+\frac{1}{2}\psi(1+a_{1}-a)+\frac{1}{2}\psi(a_{1}+a)+\frac{1}{2}\psi(a_{1}-a)$
$\displaystyle-\frac{1}{2}\partial_{a_{0}}^{2}F-\frac{t+t(1-t)\partial_{t}\partial_{a_{0}}F}{2(-\frac{t}{2}+t(a_{0}^{2}+a_{1}^{2}+a_{t}^{2}-a_{\infty}^{2})+(1-t)u)}\big{)}w^{(0)}_{+}\big{]}$
(92)

In general, the coefficient $c_{N}$ in the series solution
$z^{\frac{1}{2}-a_{0}}\sum_{k=0}^{\infty}c_{k}z^{k}$ and the connection
coefficient for $w^{(0)}_{+}$ on the right hand side of (84) simultaneously
take $a_{0}=\frac{N}{2}$ as a pole, so the limit $a_{0}\to\frac{N}{2}$ always
becomes a differentiation with respect to $a_{0}$. For example, for $a_{0}=1$
we have

$\displaystyle
w^{(1)}_{+}=\frac{\Gamma(1+2a_{1})}{2\Gamma(-\frac{1}{2}+a_{1}+a)\Gamma(-\frac{1}{2}+a_{1}-a)}e^{(\frac{1}{2}\partial_{a_{1}}-\frac{1}{2}\partial_{a_{0}})F}\big{[}-\frac{1}{(2-2a_{0})c_{2}|_{a_{0}=1}}w^{(0)}_{\widetilde{-}}$
$\displaystyle+(2\psi(1)+\frac{5}{2}-\frac{1}{2}\psi(-\frac{1}{2}+a_{1}+a)-\frac{1}{2}\psi(-\frac{1}{2}+a_{1}-a)-\frac{1}{2}\psi(\frac{3}{2}+a_{1}+a)-\frac{1}{2}\psi(\frac{3}{2}+a_{1}-a)$
$\displaystyle+\frac{1}{2}\partial_{a_{0}}^{2}F+\frac{\partial_{a_{0}}((2-2a_{0})c_{2})}{2(2-2a_{0})c_{2}}|_{a_{0}=1})w^{(0)}_{\widetilde{+}}\big{]}$
(93)

We use Mathematica to compute the connection relation in the degenerate case
for higher values of $N$.

##  Appendix B The Nekrasov-Shatashvili function

The Nekrasov-Shatashvili function is defined as

$\displaystyle F({a_{t}\atop a_{\infty}}a{a_{1}\atop
a_{0}};\frac{1}{t})=\lim_{\epsilon\to 0}\epsilon\log\big{(}$
$\displaystyle(1-\frac{1}{t})^{-2\epsilon^{-1}(\frac{1}{2}+a_{t})(\frac{1}{2}+a_{1})}\sum_{\vec{Y}}(\frac{1}{t})^{|\vec{Y}|}z_{v}(\vec{a},\vec{Y})$
$\displaystyle\prod_{\theta=\pm}z_{h}(\vec{a},\vec{Y},a_{1}+\theta
a_{0})z_{h}(\vec{a},\vec{Y},a_{t}+\theta a_{\infty})\big{)}$ (94)

where $\vec{a}=(a,-a)$ and $\vec{Y}=(Y_{1},Y_{2})$ is a vector with two Young
tableau as the components. $|\vec{Y}|=|Y_{1}|+|Y_{2}|$ is the sum of the sizes
of the two tableau. $z_{h}$ and $z_{v}$ are defined as

$\displaystyle z_{h}(\vec{a},\vec{Y},b)=$
$\displaystyle\prod_{I=1,2}\prod_{(i,j)\in
Y_{I}}(a_{I}+b+(i-\frac{1}{2})+\epsilon(j-\frac{1}{2}))$ $\displaystyle
z_{v}(\vec{a},\vec{Y})=$ $\displaystyle\prod_{I,J=1,2}\prod_{(i,j)\in
Y_{I}}\prod_{(i^{{}^{\prime}},j^{{}^{\prime}})\in
Y_{J}}\frac{1}{a_{I}-a_{J}-L_{Y_{J}}((i,j))+\epsilon(A_{Y_{I}}((i,j))+1)}$
$\displaystyle\frac{1}{a_{I}-a_{J}+(L_{Y_{I}}((i^{{}^{\prime}},j^{{}^{\prime}}))+1)-\epsilon
A_{Y_{J}}((i^{{}^{\prime}},j^{{}^{\prime}}))}$ (95)

where $L$ stands for the leg length and $A$ stands for the arm length of the
site in the tableau.

Our definition of the Nekrasov- Shatashvili function is essentially the
definition in Dodelson:2022yvn , with $a_{t}$ and $a_{1}$ are swapped and $t$
replaced by $\frac{1}{t}$ in the argument. That’s because we consider the
connection relation between local solutions around $z=0$ and $z=1$, while in
Dodelson:2022yvn  the pair $z=0$ and $z=t$ was considered.

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