††institutetext:  Centre for High Energy Physics,
Indian Institute of Science,
C. V. Raman Avenue, Bangalore 560012, India.

# Thermal one point functions, large $d$ and interior geometry of black holes

Justin R. David, Srijan Kumar<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study thermal one point functions of massive scalars in $AdS_{d+1}$ black
holes. These are induced by coupling the scalar to either the Weyl tensor
squared or the Gauss-Bonnet term. Grinberg and Maldacena argued that the one
point functions sourced by the Weyl tensor exponentiate in the limit of large
scalar masses and they contain information of the black hole geometry behind
the horizon. We observe that the one point functions behave identically in
this limit for either of the couplings mentioned earlier. We show that in an
appropriate large $d$ limit, the one point function for the charged black hole
in $AdS_{d+1}$ can be obtained exactly. These black holes in general contain
an inner horizon. We show that the one point function exponentiates and
contains the information of both the proper time between the outer horizon to
the inner horizon as well as the proper length from the inner horizon to the
singularity. We also show that Gauss-Bonnet coupling induced one point
functions in $AdS_{d+1}$ black holes with hyperbolic horizons behave as
anticipated by Grinberg-Maldacena. Finally, we study the one point functions
in the background of rotating BTZ black holes induced by the cubic coupling of
scalars.

##  1 Introduction

The physics of the horizon of black holes and the geometry behind the horizon
have continued to be problems of interest in quantum gravity. The AdS/CFT
correspondence Maldacena:1997re  in principle gives us a handle to study the
geometry behind the horizon using boundary correlators of the CFT at finite
temperature. Studies in this direction were initiated some time ago by
approximating 2-point functions of massive scalars in terms of space like
geodesics in the large mass limit Fidkowski:2003nf . Recently Grinberg and
Maldacena Grinberg:2020fdj  have argued in general that even one point
functions of massive scalars in $AdS_{d+1}$ black hole geometries contain some
information of the geometry behind the horizon. This has been investigated
further in Rodriguez-Gomez:2021pfh ; McInnes:2022pig ; Georgiou:2022ekc ;
Berenstein:2022nlj .

One point functions in holographic backgrounds have been studied earlier to
establish sum-rules and high frequency behaviour of transport coefficients
David:2011hy ; David:2012cd ; Myers:2016wsu . Using an example studied by
Myers:2016wsu , Grinberg and Maldacena observe the thermal one point function
in the planar $AdS_{d+1}$ Schwarzschild black hole evaluated for scalars dual
to operators of low dimensions can be analytically continued to large operator
dimensions. When this is done, the thermal one point function exponentiates
and one can read out the proper time to singularity from the event horizon.
The one point function is sourced by the coupling of the scalar to the Weyl
tensor squared term. Using the intuition from this exactly solvable example,
Grinberg:2020fdj  developed a saddle point approximation for large masses. The
saddle involved geodesics at complex radial positions and arguments justifying
the location of the saddle and the contour involved in the WKB approximation.
One of the interesting claims of their saddle point analysis is that for the
$AdS_{5}$ charged black hole which has an inner horizon, the one point
function exponentiates to the form

$\langle{\cal O}\rangle\sim\exp(-m(\ell_{\rm hor}+\ell_{\rm sing}-i\tau_{\rm
in}))\ .$ (1)

Here $\tau_{\rm in}$ is the proper time for a particle to reach the inner
horizon from the outer horizon, $\ell_{\rm hor}$ is the regularized proper
length from the boundary to the outer horizon and $\ell_{\rm sign}$ is the
proper length from the inner horizon to the singularity. The lengths are shown
on the Penrose diagram of the charged black hole in figure 2. In this
analysis, the massive scalar field is taken to be a probe and its the back
reaction is neglected. One of the aims of this paper is to provide a solvable
example similar to the $AdS_{d+1}$ Schwarzschild black hole to show indeed
that the expectation value of the thermal one point functions in the charged
black hole indeed exponentiates as in (1).

In this paper we study one point functions of massive scalars sourced by the
Gauss-Bonnet curvature as well as the Weyl tensor squared. For the solvable
case of planar Schwarzschild black hole we see that the result for the thermal
one point function sourced by the Gauss-Bonnet curvature is identical to that
of the Weyl tensor squared term. We then present examples in which the one
point function of massive scalars can be exactly obtained for small conformal
dimensions just as the example seen first in Myers:2016wsu  for the
$AdS_{d+1}$ Schwarzschild black hole. The first example is the planar charged
black hole in $AdS_{d+1}$ in a suitable large $d$ limit. Large $d$ limit has
been studied earlier by several groups, see Emparan:2020inr  for a recent
review. The large $d$ limit used in this paper also involves a simultaneous
limit on the charge of the black hole, which is distinct from that discussed
in Emparan:2020inr  111See the recent work of Giataganas:2021jbj  for another
instance in which holographic observables are exactly solvable in a large $d$
limit.. We see that the one point function can be obtained exactly and it
exponentiates for large scalar masses just as expected by arguments in
Grinberg:2020fdj . We show that the one can read out all the $3$ lengths in
(1) from the one point function. We also observe that this result is
independent of whether the one point function is sourced by Gauss-Bonnet
curvature or the Weyl tensor squared.

The second example we study are the black holes with hyperbolic horizons in
$AdS_{d+1}$. Here we show that the Gauss-Bonnet curvature sources one point
function which again behave as anticipated by Grinberg:2020fdj . We can read
out the $\tau_{s}$, the proper time from the horizon to the singularity from
the one point function. The dependence of $\tau_{s}$ on the $AdS$ radius is
different from Schwarzschild black holes with planar horizons. Therefore this
example serves another check of the general arguments of Grinberg:2020fdj .

Finally we study the one point functions in the background of the rotating BTZ
black hole. By conformal invariance, thermal one point functions of primaries
vanish in $2d$ CFT’s on a line. However if the spatial directions are
periodic, they acquire non-trivial expectation values. Holographically these
one point functions are sourced due to cubic couplings of the massive scalar
with other scalars in the theory Kraus:2016nwo . The auxiliary scalar sources
the scalar of interest due to the non-trivial windings around the circular
horizon of the BTZ. These one point functions were explored for the non-
rotating BTZ in Grinberg:2020fdj . As expected, they decay exponentially as
$\exp\big{(}-\frac{2\pi L}{\beta}\big{)}$, were $L$ is size of the spatial
identifications. However, from the coefficient of this leading term, we can
extract the proper time to singularity. We generalize this computation to the
rotating BTZ and obtain the exact one point function sourced due to the cubic
coupling of scalars. We then take a limit which retains the information of the
inner horizon, it is possible to interpret the coefficient of
$\exp\big{(}-\frac{2\pi L}{\beta}\big{)}$ geometrically. From this
coefficient, we see that we can read out both $\tau_{s}$ and $\ell_{\rm hor}$,
but there is no dependence on $\ell_{\rm sing}$. We verify that this behaviour
is also exhibited by the expectation value of composite operators
$\langle{\cal O}^{2}\rangle$ of large integer scaling dimensions in finite
temperature 2d CFT. The WKB type arguments used in Grinberg:2020fdj  assumed
radial symmetry. It will be interesting to generalize those arguments to the
case of rotating black holes in higher dimensions and as well try to obtain
the thermal one point functions exactly to see if they behave as in (1).

The organization of the paper is as follows. In section 2, we evaluate the one
point function of massive scalars in the planar Schwarzschild black hole in
$AdS_{d+1}$ sourced by the Gauss-Bonnet coupling, we observe that the result
for the one point function is identical to that of the Weyl-tensor squared
coupling. In section 3, we repeat the analysis for charged planar black holes
in $AdS_{d+1}$ in a suitable large $d$ limit. Section 4 contains the
evaluation of the one point function in hyperbolic black holes. In section 5
we study the thermal one point function sourced by cubic coupling of scalars
in the rotating BTZ black hole. Section 6 has our conclusions. Appendix A has
details regarding the evaluation of the Greens functions needed to evaluate
the thermal one point function. Appendix B discusses the details of the planar
charged black hole in $AdS_{d+1}$ in the large $d$ limit developed in the
paper. Appendix C shows that the result for the one point function sourced by
the Weyl tensor squared coupling for the charged planar black hole in the
large mass limit is the same as that obtained due to the Gauss-Bonnet
coupling.

##  2 Gauss-Bonnet induced thermal one point function

In this section we will re-visit the set-up of Grinberg:2020fdj  which results
in non-trivial thermal one point functions due to the presence of a higher
derivative coupling in the low energy effective action. Consider the minimally
coupled scalar $\varphi$ of mass $m$ in an asymptotically $AdS_{d+1}$
background. The scalar is dual to an operator ${\cal O}$ of dimensions
Witten:1998qj

$\Delta=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m^{2}R^{2}_{AdS}}\,\,,$ (2)

where $R_{AdS}$ is the radius of $AdS$. If one considers the conventional
quadratic action of $\varphi$, the symmetry $\varphi\rightarrow-\varphi$
prevents the dual operator ${\cal O}$ developing an expectation value. In
Grinberg:2020fdj , the following the low energy effective action of the scalar
was considered

$S=\frac{1}{16\pi
G_{N}}\int\sqrt{g}d^{d+1}x\left[\frac{1}{2}\left(\nabla^{\mu}\varphi\nabla_{\mu}\varphi+m^{2}\varphi^{2}\right)+\alpha\varphi
W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}\right],$ (3)

where the scalar $\varphi$ couples to the Weyl tensor squared. Such couplings
are known to be present in the low energy effective action and are dual to the
3-point function involving $2$ stress tensors and the operator ${\cal O}$.
Since it is a higher derivative coupling, it is suppressed in the strong
t’Hooft coupling limit. From dimensional analysis it is clear that

$\alpha\sim l_{s}^{2}\sim\frac{R_{AdS}^{2}}{\sqrt{\lambda}}\,,$ (4)

where $l_{s}$ is the string length and $\lambda$ is the t’ Hooft coupling and
the gauge theory. One of the motivations provided in Grinberg:2020fdj  for
this choice of the coupling is that the Weyl tensor vanishes for the pure
$AdS$, which is consistent with the one point function of the operator ${\cal
O}$ vanishing at zero temperature.

It has been observed that in the study of black hole entropy the Gauss-Bonnet
coupling results in the same corrections to black hole entropy as the Weyl
tensor squared Sen:2005iz . Indeed low energy effective actions in string
theory contain a coupling of the Gauss-Bonnet term with the dilaton. Motivated
by these observations, we will evaluate the expectation value of the one point
function by considering the action

$S=\frac{1}{16\pi
G_{N}}\int\sqrt{g}d^{d+1}x\left[\frac{1}{2}\left(\nabla^{\mu}\varphi\nabla_{\mu}\varphi+m^{2}\varphi^{2}\right)+\alpha\varphi{\cal
L}_{\rm GB}\right]\,,$ (5)

where the Gauss-Bonnet term is given by

${\cal L}_{\rm
GB}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$ (6)

We will show that the result for the one point function is identical to that
obtained by Grinberg:2020fdj . Though the Gauss-Bonnet curvature for pure AdS
is non-vanishing, the one point function evaluated using the coupling in (5)
does vanish for the pure $AdS$. Therefore the result is consistent with the
vanishing of thermal expectation values at zero temperature.

Using the rules of $AdS/CFT$, the one-point function is given by

$\langle O(t,\vec{x})\rangle=\alpha\int
dz^{\prime}dt^{\prime}d\vec{x}^{\prime}\sqrt{g}\tilde{K}(t,\vec{x};\,z^{\prime},t^{\prime},\vec{x}^{\prime}){\cal
L}_{\rm GB}(z^{\prime},t^{\prime},\vec{x}^{\prime})\,,$ (7)
222$d\vec{x}^{\prime}=(dx)^{d-1}$

where $\tilde{K}(t,\vec{x};\,z^{\prime},t^{\prime},\vec{x}^{\prime})$ is the
bulk-boundary propagator. The integral over $t^{\prime},\vec{x}^{\prime}$ is
carried over the $AdS_{d+1}$ Schwarzschild black hole. More specifically, the
range of integration for the $t^{\prime},x^{\prime}$ coordinates runs from
$-\infty$ to $\infty$ and the integral over $z^{\prime}$ runs from the horizon
to infinity. The metric of the planar Schwarzschild black hole is given by

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\bigg{(}-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+d\vec{x}^{2}\bigg{)},\
\ \ \ \ f(z)=1-\frac{z^{d}}{z_{0}^{d}}\ ,\ \ \ \ z_{0}=\frac{\beta
d}{4\pi}\,.$ (8)

Here $R$ is the radius of $AdS_{d+1}$333From this point onwards $R$ will refer
to the radius of $AdS$ and $\beta$ is the inverse temperature. The Gauss-
Bonnet curvature of the geometry in (8) is given by

${\cal L}_{\rm
GB}(z)=d(d-2)(d-1)^{2}\frac{z^{2d}}{z_{0}^{2d}R^{4}}+\frac{(d-2)(d-1)d(d+1)}{R^{4}}\,.$
(9)

It is also useful to evaluate the Weyl tensor squared

$W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}(z)=d(d-2)(d-1)^{2}\frac{z^{2d}}{z_{0}^{2d}R^{4}}\,.$
(10)

Thus the Gauss-Bonnet term and the Weyl tensor squared term differs by a
constant. We will see that this additional constant in (9) does not contribute
to the one point function.

We follow the strategy of Grinberg:2020fdj  to obtain the bulk-boundary
propagator. We first solve for the bulk-bulk Greens function in the geometry
(8) by solving the equation

$\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}G(z,z^{\prime},t,t^{\prime},x,x^{\prime})-m^{2}G(z,z^{\prime},t,t^{\prime},x,x^{\prime})=\frac{\delta(z-z^{\prime})\delta(t-t^{\prime})\delta^{d-1}(\vec{x}-\vec{x}^{\prime})}{\sqrt{-g}}\,.$
(11)

Then the bulk-boundary propagator can be obtained by Banks:1998dd ; Erbin

$K(z^{\prime},t,t^{\prime},x,x^{\prime})=\lim_{z\rightarrow
0}\frac{2\nu}{z^{\Delta}}G(z,z^{\prime},t,t^{\prime},x,x^{\prime})\,.$ (12)

Since the geometry in (8) has translation invariance in $t,\vec{x}$
directions, we can expand the Greens function in terms of the Fourier modes in
these directions 444For the Euclidean black hole $t$ is periodic, therefore
the Fourier modes in this direction are discrete, this is understood in the
paper. For convenience we will denote the sum over these modes by the integral
over $\omega$.

$G(z,z^{\prime},t,t^{\prime},x,x^{\prime})=\int\frac{d\omega
d\vec{k}}{(2\pi)^{d}}e^{i\omega(t-t^{\prime})-\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}\hat{G}(z,z^{\prime},\omega,\vec{k})\,.$
(13)

The differential equation obeyed by the Fourier coefficient is given by

$\displaystyle\partial_{z}\left(\frac{R^{d-1}}{z^{d-1}}f(z)\partial_{z}\hat{G}(z,z^{\prime},\omega,\vec{k})\right)-\frac{R^{d+1}}{z^{d+1}}m^{2}\hat{G}(z,z^{\prime},\omega,\vec{k})$
(14)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{R^{d-1}}{z^{d-1}}\left(\frac{\omega^{2}}{f(z)}-\vec{k}^{2}\right)\hat{G}(z,z^{\prime}\omega,\vec{k})=\delta(z-z^{\prime})\,.$

Therefore the bulk-boundary propagator also admits the expansion

$K(z^{\prime},t,t^{\prime},x,x^{\prime})=\int\frac{d\omega
d\vec{k}}{(2\pi)^{d}}e^{i\omega(t-t^{\prime})-\vec{k}(\vec{x}-\vec{x}^{\prime})}\hat{K}(z,z^{\prime},\omega,\vec{k})\,.$
(15)

This expansion can then be used in (7) to obtain Fourier components of the
expectation value, which can be written as

$\langle{\cal
O}\rangle_{\omega,\vec{k}}=\int\sqrt{-g}dz^{\prime}dt^{\prime}dx^{\prime}\hat{K}(z^{\prime},\omega,\vec{k})e^{i(\omega
t^{\prime}-\vec{k}\cdot\vec{x}^{\prime})}{\cal
L}_{GB}(z^{\prime},t^{\prime},\vec{x}^{\prime})\,.$ (16)

From (9) we see that the Gauss-Bonnet curvature only depends on the $z$
coordinate, This allows us to perform the $t^{\prime},x^{\prime}$ integral to
obtain

$\langle{\cal
O}\rangle_{\omega,\vec{k}}=(2\pi)^{d}\delta(\omega)\delta(\vec{k})\int
dz^{\prime}K(z^{\prime},0,0){\cal L}_{GB}(z^{\prime})\,.$ (17)

Thus the only non-zero Fourier coefficient of the expectation value is its
zero mode. This implies that the thermal one point functions $\langle{\cal
O}(t,\vec{x})\rangle$ is uniform, independent of time and position. It is
convenient to define the zero mode of the bulk-boundary Green’s function by

$\hat{K}(z^{\prime})=\hat{K}(z^{\prime},0,0)\ .$ (18)

From (12) we see that

$\hat{K}(z^{\prime})=\lim_{z\rightarrow
0}\frac{2\nu}{z^{\Delta}}\hat{G}(z,z^{\prime})\ ,\qquad{\rm
where,}\qquad\hat{G}(z,z^{\prime})=\hat{G}(z,z^{\prime},\omega,\vec{k})|_{\omega=\vec{k}=0}\
\,,$ (19)

and $\hat{G}(z,z^{\prime})$ satisfies the differential equation

$\partial_{z}\left(\frac{R^{d-1}}{z^{d-1}}f(z)\partial_{z}G(z,z^{\prime})\right)-\frac{R^{d+1}}{z^{d+1}}m^{2}G(z,z^{\prime})=\delta(z-z^{\prime})\,.$
(20)

Finally the uniform value of the thermal expectation value is given by

$\langle{\cal O}\rangle=\alpha
R^{d+1}\int_{0}^{z_{0}}\frac{dz}{z^{(d+1)}}K(z){\cal L}_{\rm GB}(z)\,.$ (21)

Though this result is obvious because of the translational invariance of the
geometry (8), we have gone through the steps in detail so that we can
generalise this discussion when we consider hyperbolic black holes in section
4.

The Green’s function $G(z,z^{\prime})$ has to be regular both at $z=0$ and at
$z=1$. The details of evaluation of this Greens function is given in the
appendix A. Let

$w=\left(\frac{z}{z_{0}}\right)^{d},\qquad\qquad\qquad h=\frac{\Delta}{d}\,,$
(22)

then

$\displaystyle G(w,w^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{\Gamma(h)^{2}}{\Gamma(2h)}\
\frac{z_{0}^{d}}{R^{d-1}d}\bigg{(}\varphi_{{\rm inf}}(w)\varphi_{{\rm
hor}}(w^{\prime})\theta(w^{\prime}-w)+\varphi_{{\rm hor}}(w)\varphi_{{\rm
inf}}(w^{\prime})\theta(w-w^{\prime})\bigg{)},$ $\displaystyle\varphi_{{\rm
inf}}(w)$ $\displaystyle=$ $\displaystyle w^{h}\
{}_{2}F_{1}(h,h,2h,w)\,,\qquad\varphi_{{\rm
hor}}(w)=w^{h}{}_{2}F_{1}(h,h,1,1-w)\,.$ (23)

Since the hypergeometric function admits a taylor series expansion around the
origin, we see that $g_{{\rm inf}}(w)$ is well behaved at infinity $w=0$,
while $g_{{\rm hor}}$ is well behaved at the horizon $w=1$. We can now use
(19) to obtain the bulk to boundary Greens function

$\displaystyle K(w)=-\frac{\Gamma(h)^{2}}{\Gamma(2h)}\
\frac{z_{0}^{d}}{R^{d-1}d}\
\bigg{(}\frac{2\nu}{z_{0}^{\Delta_{+}}}\bigg{)}w^{h}\
{}_{2}F_{1}(h,h,1,1-w)\,.$ (24)

Substituting this in the expression for the one point function in (21) we
obtain

$\displaystyle\langle O\rangle$ $\displaystyle=$
$\displaystyle-\alpha\frac{2\nu}{dz_{0}^{\Delta}R^{2}}\frac{\Gamma(h)^{2}}{\Gamma(2h)}\int_{0}^{1}dw\left[(d-2)(d-1)^{2}w^{h}\
{}_{2}F_{1}(h,h,1,1-w)\right.$ (25)
$\displaystyle\qquad\qquad\qquad\qquad\left.+(d-2)(d-1)(d+1)w^{h-2}\
{}_{2}F_{1}(h,h,1,1-w)\right],$ $\displaystyle=$
$\displaystyle\alpha\frac{(d-2)(d-1)^{2}}{R^{2}d}\frac{\Gamma(h)^{2}}{\Gamma(2h)}\bigg{(}\frac{2\nu}{z_{0}^{\Delta_{+}}}\bigg{)}\
h(h-1)\pi\csc(h\pi)\,.$

Here the second term in the integrand arising from the constant term in the
Gauss-Bonnet curvature given in (9). To obtain the last line we have used the
result

$\displaystyle\int_{0}^{1}\,{}_{2}F_{1}(h,h;1;w)(1-w)^{h+n}\,dw$
$\displaystyle=$
$\displaystyle\frac{\Gamma(-h+n+2)\Gamma(h+n+1)}{\Gamma(n+2)^{2}}$
$\displaystyle\text{ if }{\rm Re}(h-n)<2\ \text{and}\ {\rm Re}(h+n)>-1\,.$

Note that the integral vanishes for $n=-2$, therefore the constant
contribution of the Gauss-Bonnet curvature vanishes. This ensures the result
for the one point function from the Gauss-Bonnet coupling is identical to that
of the Weyl tensor squared coupling.

Having obtained the one point in (25), to extract the time to singularity, we
take the large $h$ limit with $h\rightarrow\infty-i\epsilon$, this results in

$\displaystyle\lim_{h\rightarrow\infty-i\epsilon}\langle{\cal O}\rangle$
$\displaystyle\sim$ $\displaystyle e^{-i\pi h}2^{-2h},$ $\displaystyle=$
$\displaystyle e^{-i\pi\frac{mR}{d}}4^{-\frac{mR}{d}}.$

Here we have ignored all the polynomial terms in $h$ and retained only terms
which are exponential in $h$. We use the fact that ${\rm Im}\;h<0$ to pick up
the negative phase, but we can very well pick up the positive phase when ${\rm
Im}\;h>0$.

### Comparison with geometric lengths

Maldacena and Grinberg Grinberg:2020fdj , identified the exponential terms
occurring in the expectation value of the one point function with the
following geometric lengths associated with the black hole. Consider a
radially infalling geodesic with zero energy released at $z=z_{0}$, the
horizon. The proper time, the particle takes to reach the singularity
$z=\infty$ is given by the integral

$\displaystyle t_{s}$ $\displaystyle=$
$\displaystyle\int_{z_{0}}^{\infty}\frac{Rdz}{z\sqrt{\big{(}\frac{z}{z_{0}}\big{)}^{d}-1}},$
$\displaystyle=$ $\displaystyle\frac{\pi R}{d}.$

Similarly, the regularized space like length from infinity to the horizon is
given by

$\displaystyle\hat{\ell}_{{\rm hor}}$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\int_{\epsilon}^{z_{0}}\frac{Rdz}{z\sqrt{1-\big{(}\frac{z}{z_{0}}\big{)}^{d}}},$
$\displaystyle=$ $\displaystyle\frac{R}{d}\log
4-R\log\big{(}\frac{\epsilon}{z_{0}}\big{)}+O(\epsilon).$

The regularized length is obtained by ignoring the divergence as normalised in
Grinberg:2020fdj , which results in

$\ell_{\rm hor}=\frac{R}{d}\log 4\,.$ (30)

One point worth mentioning at this stage is that both lengths are independent
of the mass of the black hole. The mass unlike these lengths can be obtained
easily by studying the behaviour of particles in the black hole geometry.
Comparing (2) and (30), we see that in the $h\rightarrow\infty$ limit we can
re-write the thermal expectation value in the following geometric form

$\lim_{m\rightarrow\infty-i\epsilon}\langle{\cal
O}\rangle\sim\exp(-im\tau_{s}-m\ell_{\rm hor})\,.$ (31)

These lengths are shown on the Penrose diagram of the $AdS$ Schwarzschild
black hole in figure 1.

Figure 1: Penrose diagram for the AdS-Schwarzschild metric: $\tau_{s}$ is the
time-like distance between the horizon and the singularity, and $\ell_{\rm
hor}$ refers to the regularised space-like distance between the boundary and
the horizon.

##  3 The charged planar black hole at large $d$

Let us consider the planar Reissner-Nördstrom black hole in $AdS_{d+1}$ whose
metric is given by Chamblin:1999tk .

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\bigg{(}-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+d\vec{x}^{2}\bigg{)},\
\ \ \ \ f(z)=1-\frac{z^{d}}{z_{0}^{d}}+q^{2}z^{2d-2}.$ (32)

In this section, we wish to evaluate the one point function of the scalar in
this background with the action (5). Maldacena and Grinberg Grinberg:2020fdj
deal with the case of charged black hole in $AdS_{5}$. Using a saddle point
analysis in terms of complex geodesics, they argue that the one point function
of the operator dual to a minimally coupled scalar is of the form

$\langle{\cal O}\rangle\sim\exp(-m(\ell_{\rm hor}+\ell_{\rm sing}-i\tau_{\rm
in}))\,,$ (33)

where $\ell_{\rm hor}$ is the regularized distance from the boundary to the
outer horizon, $\ell_{\rm sing}$ is the distance from the inner horizon to the
singularity and finally $\tau_{\rm in}$ is the time from the inner horizon to
the outer horizon. These lengths are shown in the figure 2 .

$\ell_{sing}$$\tau_{\rm in}$$\ell_{\rm hor}$Outer horizonInner
horizonBoundarySingularity Figure 2: The three lengths $\tau_{\rm
in},\ell_{\rm hor},\ell_{\rm sing}$ shown on the Penrose diagram of the
charged black hole in $AdS$.

Note that here compared to the case without the inner horizon there is an
additional dependence of $\ell_{\rm sing}$. The argument in Grinberg:2020fdj
involved a saddle with a contour that goes over to complex radial positions.
It would be more satisfying if the direct analysis similar to that performed
in for the Schwarzschild black hole in section 2 can be done for a black hole
with inner horizon. In this section we show that indeed it is possible to
arrive at an exact expression for the thermal one point function in a suitable
large $d$ limit. Just as in the Schwarzschild case in (25), the integral which
results in the one point function involves an integral from the outer horizon
to infinity.

To begin, let us examine the equation of the minimally coupled scalar (32) in
the background for the zero mode in the directions along the boundary.

$\displaystyle(1-w)w^{2}\varphi^{\prime\prime}(w)-w^{2}\phi^{\prime}(w)-h(h-1)\varphi(w)$
(34)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{q^{2}w^{3-\frac{2}{d}}z_{0}^{2d-2}}{d}\bigg{(}2(d-1)\varphi^{\prime}(w)+wd\varphi^{\prime\prime}(w)\bigg{)}=0\,.$

Here we have used the co-ordinate

$w=\left(\frac{z}{z_{0}}\right)^{d}.$ (35)

We will see that $w$ is always finite in the domain we wish to evaluate the
integral involving the one point function. To obtain a more tractable
equation, we take the following large $d$ limit

$d\rightarrow\infty,\quad{\rm with}\qquad q^{2}z_{0}^{2d-2}=Q,\qquad{\rm
held\;fixed}.$ (36)

In this limit the equation (34) 555In appendix B, we discuss how taking the
large $d$ limit as in (36), the solution in (32) satisfies the leading order
Einstein’s equation. reduces to

$\displaystyle(1-w)w^{2}\varphi^{\prime\prime}(w)-w^{2}\varphi^{\prime}(w)-h(h-1)\varphi(w)+Qw^{3}[2\varphi^{\prime}(w)+w\varphi^{\prime\prime}(w)]=0\,.$
(37)

Solutions of this differential equation can be written in terms of
hypergeometric functions. Before we go ahead, let us obtain the locations of
the horizons in the limit (36). The function $f(z)$ in (32) becomes

$\displaystyle f(w)=1-w+Qw^{2}w^{-\frac{2}{d}},$ (38)
$\displaystyle\hat{f}(w)=\lim_{d\rightarrow\infty}f(w)=1-w+Qw^{2}\,.$

Therefore the inner and the outer horizons are at

$w_{\rm in}=\frac{1+\sqrt{1-4Q}}{2Q},\qquad w_{\rm
out}=\frac{1-\sqrt{1-4Q}}{2Q}\,.$ (39)

To construct the Green’s function we need the well behaved solutions of (37)
at infinity and at the horizon. These are given by

$\displaystyle\varphi_{\rm inf}=w^{h}(w-2-\sqrt{1-4Q}\ w)^{-h}\
{}_{2}F_{1}(h,h,2h,\frac{2\sqrt{1-4Q}\ w}{2+(\sqrt{1-4Q}-1)w}),$ (40)
$\displaystyle\varphi_{\rm hor}=w^{h}(w-2-\sqrt{1-4Q}\ w)^{-h}\
{}_{2}F_{1}(h,h,1,1-\frac{2\sqrt{1-4Q}\ w}{2+(\sqrt{1-4Q}-1)w})\,.$

It is easy to observe that $\varphi_{\rm inf}$ is well behaved at the boundary
$w\rightarrow 0$, while $\varphi_{\rm hor}$ is well behaved at the outer
horizon $w_{\rm out}$. The Greens function satisfies the equation

$\partial_{w}(\hat{f}(w)\partial_{w}G(z,z^{\prime}))-\frac{h(h-1)}{w^{2}}=\frac{z_{0}^{d}}{dR^{d-1}}\delta(w-w^{\prime})\,.$
(41)

Here note that the function $f(w)$ has been replaced by its limit
$\hat{f}(w)$. The Green’s function is obtained by using the two solutions in
(40) and is given by

$\displaystyle G(w,w^{\prime})$ $\displaystyle=$
$\displaystyle-2^{2h}(1-4Q)^{h-\frac{1}{2}}\frac{(\Gamma(h))^{2}}{\Gamma(2h)}\frac{z_{0}^{d}}{R^{d-1}d}$
$\displaystyle\qquad\times\bigg{(}\varphi_{\rm inf}(w)\varphi_{\rm
hor}(w^{\prime})\theta(w^{\prime}-w)+\varphi_{\rm hor}(w)\varphi_{\rm
inf}(w^{\prime})\theta(w-w^{\prime})\bigg{)}\,.$

The bulk boundary propagator using (19) is given by

$\displaystyle K(w)$ $\displaystyle=$
$\displaystyle-2^{2h}(1-4Q)^{h-\frac{1}{2}}\frac{(\Gamma(h))^{2}}{\Gamma(2h)}\frac{z_{0}^{d}}{R^{d-1}d}(-2)^{h}\left(\frac{2\nu}{z_{0}^{\Delta}}\right)$
$\displaystyle\quad\times w^{h}(w-2-\sqrt{1-4Q}\ w)^{-h}\
{}_{2}F_{1}(h,h,1,1-\frac{2\sqrt{1-4Q}\ w}{2+(\sqrt{1-4Q}-1)w})\,.$

Finally, we need the limiting value of the Gauss-Bonnet curvature in the large
$d$ limit. Evaluating the Gauss-Bonnet curvature for the geometry in (32), we
obtain

$\displaystyle\mathcal{L}_{GB}=\frac{(d-2)(d-1)}{R^{4}}\times$ (44)
$\displaystyle\Big{(}(3d-5)(3d-4)Q^{2}w^{4-\frac{4}{d}}-2Q\big{[}d\\{d(4w-1)-10w+5\\}+6(w-1)\big{]}w^{2-\frac{2}{d}}$
$\displaystyle\qquad\qquad\qquad+d\left[(d-1)w^{2}+d+1\right]\Big{)}\,.$

Taking the limit $d\rightarrow\infty$, the leading contribution to the Gauss
Bonnet term is given by

$\displaystyle\lim_{d\rightarrow\infty}\mathcal{L}_{GB}=\frac{d^{4}}{R^{4}}\left(9Q^{2}w^{4}-8w^{3}Q+(2Q+1)w^{2}+1\right)\,.$
(45)

We now have all the necessary ingredients to evaluate the one point function
in the large $d$ limit. Substituting the leading contribution of the Gauss-
Bonnet curvature (45) and the bulk boundary propagator (3) in (21), we obtain
for the thermal one point function

$\displaystyle\begin{split}\langle{\cal O}\rangle=-&\alpha
2^{2h}\frac{d^{2}(-2)^{-h}(1-4Q)^{h-\frac{1}{2}}}{R^{2}}\frac{\Gamma(h)^{2}}{\Gamma(2h)}\bigg{(}\frac{2\nu}{z_{0}^{\Delta_{+}}}\bigg{)}\int_{0}^{w_{\rm
out}}dw\left[{9Q^{2}w^{2}}-{8wQ}+{(2Q+1)}+\frac{1}{w^{2}}\right]\\\ &\times
w^{h}(w-2-\sqrt{1-4Q}\ w)^{-h}\ {}_{2}F_{1}(h,h,1,1-\frac{2\sqrt{1-4Q}\
w}{2+(\sqrt{1-4Q}-1)w})\,.\end{split}$ (46)

One simple check of this result is to note that when $Q\rightarrow 0$, this
expression reduces to the integral obtained in the uncharged planar black hole
in (25) with $d\rightarrow\infty$. Examining (45) and (46), we see that we
need to keep the ratio $\frac{d}{R}$ to be finite in the large $d$ limit to
ensure these quantities are finite. Note that the integral is from the outer
horizon to infinity, therefore in the domain of integration in $w$ is always
finite. We can bring the range of integration from $0$ to $1$ by making the
substitution

$y=1-\frac{2\sqrt{1-4Q}\ w}{2+(\sqrt{1-4Q}-1)w}\,,$ (47)

The one point function then is given by

$\displaystyle\begin{split}\langle{\cal
O}\rangle=&\frac{4d^{2}(1-4Q)^{h/2}}{R^{2}}\frac{(\Gamma(h))^{2}}{\Gamma(2h)}\bigg{(}\frac{2\nu}{z_{0}^{\Delta}}\bigg{)}\int_{0}^{1}dy\
\,_{2}F_{1}(h,h;1;y)\times\\\ &\left[-\frac{9(1-y)^{h+2}}{4Q^{2}w_{\rm
in}^{4}(1-\chi y)^{4}}+\frac{2(1-y)^{h+1}}{Q^{2}w_{\rm in}^{3}(1-\chi
y)^{3}}-\frac{(2Q+1)(1-y)^{h}}{4Q^{2}w_{\rm in}^{2}(1-\chi
y)^{2}}-\frac{(1-y)^{h-2}}{4}\right],\end{split}$ (48)

where

$\chi=\frac{1-\sqrt{1-4Q}}{1+\sqrt{1-4Q}}\,.$ (49)

The integral of the first three terms in the square bracket in (48) can be
done using formula (7.512.9) in gradshteyn2007  666
$\displaystyle\begin{split}\int_{0}^{1}dx\ x^{\gamma-1}(1-x)^{\rho-1}(1-\chi
x)^{-\sigma}\
_{2}F_{1}(\alpha,\beta,\gamma,x)=&\frac{(1-\chi)^{-\sigma}(\Gamma(\gamma)\Gamma(\rho)\Gamma(-\alpha-\beta+\gamma+\rho))}{\Gamma(-\alpha+\gamma+\rho)\Gamma(-\beta+\gamma+\rho)}\\\
&\times\,_{3}F_{2}\left(\rho,\sigma,-\alpha-\beta+\gamma+\rho;-\alpha+\gamma+\rho,-\beta+\gamma+\rho;\frac{\chi}{\chi-1}\right).\end{split}$
. From (2), we can see that integral of the last term in the square bracket
vanishes. Using the result for the integrals we obtain

$\displaystyle\begin{split}&\langle{\cal
O}\rangle=-\frac{d^{2}(1-4Q)^{\frac{h}{2}-1}}{2R^{2}(1-\sqrt{1-4Q})}\frac{(\Gamma(h))^{2}}{\Gamma(2h)}\bigg{(}\frac{2\nu}{z_{0}^{\Delta}}\bigg{)}{\pi(h-1)h\csc(\pi
h)}\times\\\
&\bigg{\\{}\left[\left(-3h^{2}-h+10\right)Q+\sqrt{1-4Q}\left((3h^{2}-3h-2)Q+2\right)-2\right]\times\\\
&\qquad\qquad\qquad{}_{2}F_{1}\left(2-h,h+1;2;\frac{1}{2}-\frac{1}{2\sqrt{1-4Q}}\right)\\\
&\hskip
113.81102pt+4Q(h-2)\,_{2}F_{1}\left(3-h,h+1;2;\frac{1}{2}-\frac{1}{2\sqrt{1-4Q}}\right)\bigg{\\}}\,.\end{split}$
(50)

Here the ${}_{3}F_{2}$ hypergeometric function simplifies to ${}_{2}F_{1}$ for
the values of the parameters resulting from the integral.

Having obtained the leading contribution to the one point function in the
large $d$ limit we can analytically continue $h$ and take the large $h$ limit.
Note that

$h=\frac{\Delta}{d}\,.$ (51)

As we are taking the large $d$ limit, we would need to ensure that $\Delta$
grows faster than linear in $d$ so that $h$ remains large. $h$ occurs as a
parameter of the hypergeometric functions in (50), therefore we need the
asymptotic expansion for large orders. To proceed, we first write down the
asymptotic expansion obtained by Watson Watson , for hypergeometric functions
at large orders

$\displaystyle\begin{split}{}_{2}F_{1}(\alpha+\lambda,\beta-\lambda,\gamma;\frac{1}{2}-\frac{z}{2})\sim\frac{\Gamma(1-\beta+\lambda)\Gamma(\gamma)}{\pi\Gamma(\gamma-\beta+\lambda)}2^{\alpha+\beta-1}(1-e^{-\zeta})^{\frac{1}{2}-\gamma}(1+e^{-\zeta})^{\gamma-\alpha-\beta-\frac{1}{2}}\\\
\times\sum_{s=0}^{\infty}\bigg{[}c_{s}e^{(\lambda-\beta)\zeta}\Gamma(s+\frac{1}{2})\lambda^{-s-\frac{1}{2}}+c_{s}^{\prime}e^{\mp\pi
i(\frac{1}{2}-\gamma)}e^{-(\lambda+\alpha)\zeta}\Gamma(s+\frac{1}{2})\lambda^{-s-\frac{1}{2}}\bigg{]}.\end{split}$
(52)

This expansion is valid for large $|\lambda|$ with $\alpha,\beta,z$ fixed, the
constants $c_{s}$ can be found in Watson . Here

$\zeta=\operatorname{arcCosh}z$ (53)

For our purpose it is sufficient to obtain, the leading term in this
expansion. So we can retain the first term with $s=0$. All other terms are
sub-leading, they are either exponentially or polynomially suppressed in
$\lambda$. Therefore we have

$\displaystyle\begin{split}&{}_{2}F_{1}(\alpha+\lambda,\beta-\lambda,\gamma;\frac{1}{2}-\frac{z}{2})\\\
&\qquad\sim\frac{\Gamma(1-\beta+\lambda)\Gamma(\gamma)}{\pi\Gamma(\gamma-\beta+\lambda)}2^{\alpha+\beta-1}(1-e^{-\zeta})^{\frac{1}{2}-\gamma}(1+e^{-\zeta})^{\gamma-\alpha-\beta-\frac{1}{2}}e^{(\lambda-\beta)\zeta}\Gamma(\frac{1}{2})\lambda^{-\frac{1}{2}}\,,\end{split}$
(54)

with $c_{0}=1$ as given in Watson . Substituting the arguments of the
hypergeometric function that we have in (50), we obtain the following leading
contributions

$\displaystyle\lim_{h\rightarrow\infty}{}_{2}F_{1}(2-h,1+h,2,\frac{1}{2}-\frac{1}{2\sqrt{1-4Q}})$
$\displaystyle\sim$
$\displaystyle\frac{e^{h\operatorname{arcsech}(\sqrt{1-4Q})}}{h^{3/2}}\,,$
(55)
$\displaystyle\lim_{h\rightarrow\infty}{}_{2}F_{1}(3-h,1+h,2,\frac{1}{2}-\frac{1}{2\sqrt{1-4Q}})$
$\displaystyle\sim$
$\displaystyle\frac{e^{h\operatorname{arcsech}(\sqrt{1-4Q})}}{h^{3/2}}\,.$

Finally we can substitute the asymptotic form in (55) and take the large $h$
limit in the rest of the terms in the thermal one point function given in (50)
which results in

$\lim_{h\rightarrow\infty-i\epsilon}{\cal O}\rangle\sim
e^{-h\log\big{[}\frac{4}{\sqrt{1-4Q}}\big{]}+h\operatorname{arcsech}(\sqrt{1-4Q})}e^{-i\pi
h}\big{[}1+e^{-i2\pi h}+\cdots\big{]}\times(\text{\small powers of h}).$ (56)

Here we have followed the prescription in Grinberg:2020fdj  and given a small
imaginary part to $h$ so as to pick up the phase $e^{-i\pi h}$. Now replacing
$h$ by the mass in the large $h$ or mass limit we obtain

$\displaystyle\langle{\cal
O}\rangle\sim\exp\bigg{[}m\bigg{(}{{-\frac{R}{d}\log\bigg{[}\frac{4}{\sqrt{1-4Q}}\bigg{]}}+{\frac{R}{d}\operatorname{arcsech}(\sqrt{1-4Q})}}-i{\pi\frac{R}{d}}\bigg{)}\bigg{]}\,.$
(57)

### Comparison with geometric lengths

Let us now evaluate the lengths $\tau_{s},\ell_{\rm hor},\ell_{\rm sing}$. The
proper time between the outer and inner horizons is given by

$\tau_{s}=\int_{z_{\rm in}}^{z_{\rm
out}}\frac{Rdz}{z\sqrt{\frac{z^{d}}{z_{0}^{d}}-qz^{2d-2}-1}}\,.$ (58)

We re-write this integral in terms of $w$ defined in (35) and take the large
$d$ limit of (36) to obtain

$\displaystyle\tau_{s}$ $\displaystyle=$
$\displaystyle\frac{R}{d\sqrt{Q}}\int_{w_{\rm out}}^{w_{\rm
in}}\frac{dw}{w\sqrt{(w-w_{\rm out})(w_{\rm in}-w)}}\,,$ $\displaystyle=$
$\displaystyle\frac{\pi R}{d}\,.$

The proper length from the outer horizon to infinity is given by

$\displaystyle\hat{\ell}_{\rm hor}$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{R}{d\sqrt{Q}}\int_{\big{(}\frac{\epsilon}{z_{0}}\big{)}^{d}}^{w_{\rm
out}}\frac{dw}{w\sqrt{(w_{\rm out}-w)(w_{\rm in}-w)}}\,,$ $\displaystyle=$
$\displaystyle\frac{R}{d}\log\left(\frac{4}{\sqrt{1-4Q}}\right)-R\log\big{(}\frac{\epsilon}{z_{0}}\big{)}\,.$

Again the regularised length is obtained by ignoring the divergence, note that
from (30), we see that the divergence is identical to that for the planar
Schwarzschild black hole.

$\ell_{\rm hor}=\frac{R}{d}\log\left(\frac{4}{\sqrt{1-4Q}}\right)\,.$ (61)

The proper length from the inner horizon to the singularity is given by

$\displaystyle\ell_{\rm sing}$ $\displaystyle=$
$\displaystyle\frac{R}{d\sqrt{Q}}\int_{w_{\rm
in}}^{\infty}\frac{dw}{w\sqrt{(w_{\rm out}-w)(w_{\rm in}-w)}}\,,$
$\displaystyle=$
$\displaystyle\frac{R}{d}\operatorname{arcsech}(\sqrt{1-4Q})\,.$

We use these geometric lengths to rewrite the one point function obtained in
(57) in the following form

$\langle{\cal O}\rangle\sim\exp(-m(\ell_{\rm hor}+\ell_{\rm sing}-i\tau_{\rm
s}))\,.$ (63)

This is the generic structure of the one point function which was argued using
a saddle point approach in Grinberg:2020fdj . This involved a choice of
contour in the complex radial coordinate. Here we have obtained it by
explicitly evaluating the one point function which involves an integral from
the outer horizon to infinity exactly at large $d$. The one point function
contains the information of the inner horizon. In the appendix C, we show that
the same structure of the one point function is obtained for the Weyl tensor
squared coupling.

##  4 Hyperbolic black holes

In this section, we consider the hyperbolic black hole in $AdS_{d+1}$
Emparan:1998he ; Birmingham:1998nr  whose metric is given by

$\displaystyle
ds^{2}=\frac{R^{2}}{z^{2}}\bigg{[}-\bigg{(}1-\frac{z^{2}}{R^{2}}\bigg{)}dt^{2}+\frac{dz^{2}}{1-\frac{z^{2}}{R^{2}}}+R^{2}(du^{2}+\sinh^{2}u\
d\Omega_{d-2}^{2})\bigg{]},$ (64)

where $R$ is the radius of $AdS_{d+1}$ and $\Omega_{d-2}$ refers to the sphere
$S^{d-2}$. The coordinate $z$ is related to the usual radial co-ordinate by
$z=\frac{R^{2}}{r}$. This metric can be obtained from a hyperbolic slicing of
pure $AdS_{d+1}$ and can be interpreted as a black hole with a horizon at
$z=R$ with Hawking temperature

$T=\frac{1}{2\pi R}\,.$ (65)

This space is holographically dual to a conformal field theory on $R\times
AdS_{d}$ Emparan:1999gf ; Casini:2011kv . We would like to repeat the analysis
done in the previous sections for this background and check if indeed the one
point function contains information of the time from the horizon to
singularity.

We consider the effective action in (5) with Gauss-Bonnet coupling. The Gauss-
Bonnet coupling in this case is a constant which is given by

$\displaystyle\mathcal{L}_{GB}=\frac{c}{R^{4}}\,,\qquad\qquad
c=(d-2)(d-1)d(d+1)\,.$ (66)

To construct the bulk-boundary propagator, we follow the procedure discussed
in section . Since the spatial geometry of the boundary is Euclidean
$AdS_{d-1}$, we need to expand the bulk-bulk Green’s function in terms of
normalizable eigen functions on $AdS_{d-1}$. We consider the expansion

$G(z,z^{\prime},t,t^{\prime}u,u^{\prime},\Omega,\Omega^{\prime})=\int d\omega
d\lambda\sum_{l,\sigma}\hat{G}(z,z^{\prime},\lambda,l,\sigma)\phi_{\lambda
l\sigma}(u,\Omega)\phi^{*}_{\lambda
l\sigma}(u^{\prime},\Omega^{\prime})e^{-i\omega(t-t^{\prime})}\,,$ (67)

where $\phi_{\lambda l\sigma}$ are eigen functions of the Laplacian on
Euclidean $AdS_{d-1}$. They satisfy the equation

$\Box\phi_{\lambda
l\sigma}=-\Big{(}\lambda^{2}+\big{(}\frac{d-2}{2}\big{)}^{2}\Big{)}\phi_{\lambda
l\sigma}\,,$ (68)

and the quantum number $\lambda$ is a continuous variable that runs from $0$
to $\infty$. The numbers $(l,\sigma)$ refer to quantum numbers on
$\Omega_{d-2}$ and $l$ in an integer that runs from $0$ to $\infty$, $\sigma$
refer to the other quantum numbers of the spherical harmonics. These eigen
functions have been explicitly constructed in Camporesi:1994ga  and they are
given by

$\displaystyle\phi_{\lambda l\sigma}$ $\displaystyle=$ $\displaystyle
q_{\lambda,l}(u)Y_{l\sigma}(\Omega),$ (69) $\displaystyle q_{\lambda,l}(u)$
$\displaystyle=$ $\displaystyle(i\sinh
u)^{l}{}_{2}F_{1}\left(i\lambda+\frac{d-2}{2}+l,-i\lambda+\frac{d-2}{2}+l,l+\frac{d-1}{2};-\sinh^{2}\frac{u}{2}\right)\,.$

What is important for us is their orthonormality property on $AdS_{d-1}$

$\int dud\Omega_{d-2}(\sinh u)^{d-2}\;\phi_{\lambda
l\sigma}(u,\Omega)\phi_{\lambda^{\prime}l^{\prime}\sigma^{\prime}}(u,\Omega)=\delta_{l,l^{\prime}}\delta_{\sigma,\sigma^{\prime}}\delta(\lambda-\lambda^{\prime})\,,$
(70)

where the integral is on $AdS_{d-1}$ with unit radius. Substituting the
expansion of the Green’s function given in (67) and using the metric (64) we
see that the coefficients obey the differential equation

$\displaystyle\partial_{z}\left(\Big{(}\frac{R}{z}\Big{)}^{d-1}\Big{(}1-\frac{z^{2}}{R^{2}}\Big{)}\partial_{z}\hat{G}(z,z^{\prime},\omega,\lambda,l,\sigma)\right)+\Big{(}\frac{R}{z}\Big{)}^{d-1}\frac{\omega^{2}}{1-\frac{z^{2}}{R^{2}}}\hat{G}(z,z^{\prime},\omega,\lambda,l,\sigma)$
$\displaystyle-\Big{(}\frac{R}{z}\Big{)}^{d-1}\left[m^{2}\Big{(}\frac{R}{z}\Big{)}^{2}+\frac{1}{R^{2}}\Big{(}\lambda^{2}+\big{(}\frac{d-2}{2}\big{)}^{2}\Big{)}\right]\hat{G}(z,z^{\prime},\omega,\lambda,l,\sigma)=\frac{1}{R^{d-1}}\delta(z-z^{\prime}).$
(71)

Similarly the bulk-boundary Green’s function admits an eigenfunction expansion
given by

$K(z^{\prime},t,t^{\prime}u,u^{\prime},\Omega,\Omega^{\prime})=\int d\omega
d\lambda\sum_{l,\sigma}\hat{K}(z^{\prime},\omega,\lambda,l,\sigma)\phi_{\lambda
l\sigma}(u,\Omega)\phi^{*}_{\lambda
l\sigma}(u^{\prime},\Omega^{\prime})e^{-i\omega(t-t^{\prime})}.$ (72)

From (12) and the eigen function expansions we see that

$\hat{K}(z^{\prime},\omega,\lambda,l,\sigma)=\lim_{z\rightarrow
0}\frac{2\nu}{z^{\Delta}}\hat{G}(z,z^{\prime},\omega,\lambda,l,\sigma)\,.$
(73)

Then substituting the eigenmode expansion in the expression for thermal one
point function given in(7), we can obtain the expectation value of each mode

$\displaystyle\langle{\cal O}\rangle_{w,\lambda,l,\sigma}=\alpha\int
dz^{\prime}dt^{\prime}du^{\prime}d\Omega^{\prime}\sqrt{g^{\prime}}\hat{K}(z^{\prime},\omega,\lambda,l,\sigma)e^{i\omega
t^{\prime}}\phi^{*}_{\lambda l\sigma}(u^{\prime},\Omega^{\prime}){\cal
L}_{GB}(z^{\prime},t^{\prime},u^{\prime},\Omega^{\prime})\,,$

where the metric is that given in (64). From (66), we see that the Gauss-
Bonnet term is independent of time or coordinates on $AdS_{d-1}$. This implies
that we can perform the $t$ integral, furthermore since the bulk-boundary
propagator is independent of the quantum numbers $l,\sigma$ due to spherical
symmetry we can perform the integral on the sphere $S^{d-2}$. This leads to

$\displaystyle\langle{\cal
O}\rangle_{w,\lambda,l,\sigma}=\alpha\,2\pi\delta(w)\delta_{l,0}\delta_{\sigma,0}\int
dz\frac{R^{2d}}{z^{d+1}}\hat{K}(z,0,\lambda,0,0){\cal L}_{GB}(z)\int du(\sinh
u)^{d-2}q^{*}_{\lambda,0}(u)\,.$

The integral over $u$ needs to be regulated by placing a cut off in the
coordinate $u$ which is along the field theory directions. This will yield a
function which just depends on the mode $\lambda$ it does not contain the
information of the operator ${\cal O}$. Therefore, let us define the thermal
one point function of the mode $\lambda$, by scaling out the integral over
$u$.

$\displaystyle\hat{\langle{\cal O}\rangle}_{\lambda}=\alpha\int
dz\frac{R^{2d}}{z^{d+1}}\hat{K}(z,\lambda){\cal L}_{GB}(z),$ (76)
$\displaystyle\hat{K}(z,\lambda)=\hat{K}(z,0,\lambda,0,0)\,.$

Here it is understood we are in the zero mode sector $\omega=0,l=0,\sigma=0$.
It is easy to see that in case the rescaled expectation value
$\langle\hat{\cal O}\rangle_{\lambda}$ is independent of $\lambda$, then it
implies that the thermal one point function $\langle O(t,u,\Omega)\rangle$ is
uniform in the field theory directions. This is because if $\langle{\cal
O}(t,u,\Omega)$ is constant on $R\times AdS_{d-1}$ , then each mode expansion
of the expectation value is given by

$\displaystyle\langle{\cal O}\rangle_{w,\lambda,l,\sigma}$ $\displaystyle=$
$\displaystyle\int dtdud\Omega_{d-2}(\sinh u)^{d-2}\langle{\cal
O}(t,u,\Omega)\rangle e^{i\omega
t}q^{*}_{\lambda,l}(u)Y^{*}_{l\sigma}(\Omega)\,,$ $\displaystyle=$
$\displaystyle\langle{\cal O}\rangle
2\pi\delta(\omega)\delta_{l,0}\delta_{\sigma,0}R^{d-1}\int du(\sinh
u)^{d-2}q^{*}_{\lambda,0}\,.$

In the second line we have used the fact that $\langle{\cal
O}(t,u,\Omega)\rangle$ is constant on $R\times AdS_{d-1}$. Comparison of (4),
(76) and the second line of (4) shows that in case $\hat{\langle{\cal
O}\rangle}_{\lambda}$ is independent of $\lambda$, then it can be identified
with the uniform expectation value of the operator in the field theory.
Therefore we will evaluate the thermal expectation value given in (76) and
show that in the regime of our interest, that is large operator dimensions
$\Delta$, the expectation value is independent of $\lambda$ and it coincides
with the uniform expectation value on the boundary field theory on the
hyperbolic space.

To obtain the Green’s function
$\hat{G}(z,z^{\prime},0,\lambda,0,0)\equiv\hat{G}(z,z^{\prime},\lambda)$, it
is convenient to go to coordinates

$w=\frac{z^{2}}{R^{2}}\,.$ (78)

Then the equation (4) is written as

$\displaystyle
4\partial_{w}\Big{[}w^{(1-\frac{d}{2})}(1-w)\partial_{w}\hat{G}(w,w^{\prime},\lambda)\Big{]}$
(79)
$\displaystyle-w^{-(1+\frac{d}{2})}\left[m^{2}R^{2}+w\Big{(}\lambda^{2}+(\frac{d-2}{2})^{2}\Big{)}\right]\hat{G}(w,w^{\prime},\lambda)=\frac{2}{R^{d-2}}\delta(w-w^{\prime})\,.$

The Green’s function is constructed by using the solution to the homogenous
equation The solution which is regular at the boundary $z=0$ is given by

$\displaystyle\varphi_{\rm
inf}(w)=w^{\Delta/2}\,_{2}F_{1}\left(-\frac{d}{4}+\frac{\Delta}{2}-\frac{i\lambda}{2}+\frac{1}{2},-\frac{d}{4}+\frac{\Delta}{2}+\frac{i\lambda}{2}+\frac{1}{2};-\frac{d}{2}+\Delta+1;w\right),$
(80)

while the solution regular at the horizon is given by

$\displaystyle\varphi_{\rm
hor}(w)=w^{\Delta/2}\,_{2}F_{1}\left(-\frac{d}{4}+\frac{\Delta}{2}-\frac{i\lambda}{2}+\frac{1}{2},-\frac{d}{4}+\frac{\Delta}{2}+\frac{i\lambda}{2}+\frac{1}{2};1;1-w\right).$
(81)

Then the Greens function which solves (79) is obtained by the same methods
discussed in the appendix A. This results in

$\displaystyle\hat{G}(w,w^{\prime},\lambda)$ $\displaystyle=$ $\displaystyle
A\times\bigg{(}\varphi_{\rm inf}(w)\varphi_{\rm
hor}(w^{\prime})\theta(w^{\prime}-w)+\varphi_{\rm
hor}(w)\varphi_{\inf}(w^{\prime})\theta(w-w^{\prime})\bigg{)},$ (82)
$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{R^{2-d}\Gamma\left(\frac{d}{2}-\Delta\right)\Gamma\left(\frac{1}{4}(-d+2\Delta-2i\lambda+2)\right)\Gamma\left(\frac{1}{4}(-d+2\Delta+2i\lambda+2)\right)}{2\Gamma\left(\frac{1}{2}(d-2\Delta)\right)\Gamma\left(\frac{1}{2}(2\Delta-d)+1\right)}\,.$

The bulk-boundary Greens’ function can be read out using (73)

$\hat{K}(w,\lambda)=\frac{2\nu}{R^{\Delta}}A\varphi_{\rm hor}(w)\,.$ (83)

Substituting this in the expression for the thermal one point function in
(76), we obtain

$\hat{\langle{\cal
O}\rangle}_{\lambda}=\alpha\frac{2\nu}{R^{\Delta}}\frac{c}{2R^{4}}AR^{d}\int_{0}^{1}dww^{-(\frac{d}{2}+1)}\varphi_{\rm
hor}(w)\,.$ (84)

We change variables from $w$ to $1-w$ to re-write the integral as

$\displaystyle\hat{\langle{\cal O}\rangle}_{\lambda}$ $\displaystyle=$
$\displaystyle\alpha\frac{2\nu}{R^{\Delta}}\frac{c}{2R^{4}}AR^{d}\times$
$\displaystyle\quad\int_{0}^{1}dw\
(1-w)^{-\frac{\Delta}{2}-1}\,_{2}F_{1}\left(\frac{1}{4}(d-2\Delta-2i\lambda+2),\frac{1}{4}(d-2\Delta+2i\lambda+2);1;w\right),$

We can perform the integral by using the formula

$\displaystyle\int_{0}^{1}(1-w)^{a}\,_{2}F_{1}(b,c;1;w)\,dw$ $\displaystyle=$
$\displaystyle\frac{\Gamma(a+1)\Gamma(a-b-c+2)}{\Gamma(a-b+2)\Gamma(a-c+2)}\,,$
$\displaystyle\text{ if }{\rm Re}\,(a)>-1\ \text{and}\,{\rm Re}\,(a)+2>{\rm
Re}\,(b+c)\,,$

which results in

$\displaystyle\hat{\langle{\cal O}\rangle}_{\lambda}$ $\displaystyle=$
$\displaystyle-\alpha\frac{2\nu}{R^{\Delta}}\frac{c}{R^{2}}\times$
$\displaystyle\frac{\Gamma\left(-\frac{\Delta}{2}\right)\Gamma\left(\frac{\Delta-d}{2}\right)\Gamma\left(\frac{1}{4}(-d+2\Delta-2i\lambda+2)\right)\Gamma\left(\frac{1}{4}(-d+2\Delta+2i\lambda+2)\right)}{4\Gamma\left(-\frac{d}{2}+\Delta+1\right)\Gamma\left(\frac{1}{4}(-d-2i\lambda+2)\right)\Gamma\left(\frac{1}{4}(-d+2i\lambda+2)\right)}\,.$

Observe the factor $R^{\Delta+2}$ provides the right scaling dimension for the
expectation value.

Now that we have the exact expectation value of the operator, we can take the
large $\Delta\rightarrow\infty-i\epsilon$ limit with $\lambda$ fixed we obtain

$\lim_{\Delta\rightarrow\infty-i\epsilon}\hat{\langle{\cal
O}\rangle}_{\lambda}\sim\csc\bigg{(}\frac{\pi\Delta}{2}\bigg{)}\
e^{-\Delta\log(2)}\times(\text{powers of }\Delta)\,.$ (88)

From our earlier discussion, we can conclude that since this value is
independent of $\lambda$, we can conclude that in the large $\Delta$ limit the
operator acquires uniform expectation value on the field theory directions
$R\times AdS_{d-1}$. Using $\Delta\sim mR$ in the large $\Delta$ limit and
ignoring corrections due to powers of $\Delta$, we can re-write this uniform
thermal expectation value as

$\lim_{\Delta\rightarrow\infty-i\epsilon}\langle{\cal
O}\rangle\sim\exp\Big{[}m\big{(}-\frac{i\pi R}{2}-R\log 2\big{)}\Big{]}.$ (89)

### Comparison with geometric lengths

At this stage we can compare the form of the thermal expectation value in (89)
with the expression proposed in Grinberg:2020fdj  in terms of geometric
lengths. The proper time from the horizon to singularity for the hyperbolic
black hole in (64) is given by

$\tau_{s}=R\int_{R}^{\infty}\frac{dz}{z\sqrt{\frac{z^{2}}{R^{2}}-1}}=\frac{\pi
R}{2}\,.$ (90)

Recall that the radial co-ordinate $r=\frac{R^{2}}{z}$. Note that unlike the
planar Schwarzschild black hole, this length is independent of the dimension
$d$. The proper length from infinity to the horizon is given by

$\displaystyle\hat{\ell}_{\rm hor}$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}R\int_{\epsilon}^{R}\frac{dz}{z\sqrt{1-\frac{z^{2}}{R^{2}}}}\,,$
$\displaystyle=$ $\displaystyle
R\log(2)-R\log\Big{(}\frac{\epsilon}{R}\Big{)}\,.$

The regularized proper length is obtained by ignoring the divergence, so we
obtain

$\ell_{\rm hor}=R\log(2)\,.$ (92)

Using the proper time and the regularized proper length, we can write the
expectation value in (89) as

$\lim_{m\rightarrow\infty-i\epsilon}\langle{\cal
O}\rangle\sim\exp(-im\tau_{s}-m\ell_{\rm hor})\,.$ (93)

Therefore we see that the geometric form for the expectation value for
operators of large dimensions is true even in the presence of black holes with
hyperbolic horizons. One of the consequences of our calculation is that
generically, we expect operators of large dimensions on conformal field
theories on spatial hyperbolic surfaces to obtain expectation values at finite
temperature.

##  5 BTZ black holes with angular momentum

In Grinberg:2020fdj , the examples considered did not include the rotating
black holes, moreover it is not clear that the saddle point arguments to
arrive at the thermal expectation value provided in Grinberg:2020fdj  for
black holes with inner horizons generalise to that of black holes with
rotation. This is because these metrics which was used in the saddle point
arguments had spherical symmetry. In this section we use the rotating BTZ
black hole geometry to study the behaviour of thermal one point function of
scalars dual to primaries in the CFT. Though many quantities are exactly
solvable in this geometry, one apparent difficulty with this is that in the
dual field theory conformal invariance in $2d$ ensures that primary operators
do not acquire non-trivial expectation values at finite temperature if the CFT
is on a real line. In the 3 dimensional dual holographic description, this is
reflected in the fact that both the Gauss-Bonnet curvature as well Weyl tensor
curvature square vanishes in $3d$. Therefore the mechanism discussed in the
previous sections does not apply to holographic $2d$ CFTs. However it is known
that one point functions of CFTs on a torus do acquire non-trivial expectation
value. In Kraus:2016nwo , this expectation value was interpreted
holographically as arising due to the cubic coupling of the field $\varphi$
dual to the operator of interest with another bulk field $\chi$. More
explicitly one considers the following action

$\displaystyle S=\frac{1}{16\pi G_{N}}\int d^{3}x\sqrt{g}\
(\partial_{\nu}\varphi\partial^{\nu}\phi+m^{2}\varphi^{2}+\partial_{\nu}\chi\partial^{\nu}\chi+\mu^{2}\chi^{2}+g\chi^{2}\varphi)\,.$
(94)

The field $\chi^{2}$ acquires non-trivial expectation value due to the sum of
propagators that wind around the horizon of the BTZ black hole, this in turn
induces a one point function for the field $\varphi$. This is shown in figure
3.

$\varphi$$\chi$ Figure 3: One point function induced by the cubic coupling
$\chi^{2}\varphi$ in the BTZ background Kraus:2016nwo .

In Grinberg:2020fdj , the action in (94) was used to show that for the
Schwarzschild BTZ black hole, the leading contribution to the expectation
value of the operator ${\cal O}$ dual to $\varphi$ is given by

$\displaystyle\lim_{\stackrel{{\scriptstyle
m\rightarrow\infty-i\epsilon}}{{L\rightarrow\infty}}}\langle{\cal O}\rangle$
$\displaystyle\sim$ $\displaystyle({\rm
powers\;of}\;m)\exp(-im\tau_{s}-m\ell_{\rm hor})\times e^{-\frac{2\pi
L}{\beta}}$ $\displaystyle+O(e^{-\frac{4\pi L}{\beta}})\,.$

Here $L$ is the periodicity in the spatial direction, when
$L\rightarrow\infty$, the expectation value vanishes. In this section we will
show that when one considers the BTZ black hole with rotation, the thermal one
point function behaves as given in (5), where now $\tau_{s},\ell_{\rm hor}$
are evaluated in the rotating BTZ geometry. Note that we do not see the
occurrence of ${\rm\ell_{sing}}$, the proper length from the inner horizon to
the origin, unlike what was seen for the black hole with charge.

To see if this behaviour holds in other situations, we evaluate the
expectation value of the composite made out of the bilinear ${\cal O}^{2}$ in
2d CFT on a line but held at distinct left and right temperatures. The dual
description of this CFT at large central charge is the BTZ black hole with
angular momentum. Since the field is a composite, its expectation value at
finite temperature is non-vanishing and can be obtained using conformal
invariance. A closed expression can be obtained for this expectation value
when the operator $O$ has large integer dimensions. We show that this
expectation value can be written as

$\lim_{m\rightarrow\infty-i\epsilon}\langle{\cal
O}^{2}\rangle\sim\exp(-2im\tau_{s}-2m\ell_{\rm hor})\,.$ (96)

Again we do not see the appearance of the proper length $\ell_{\rm sing}$. It
will be interesting to see if this is a feature true for thermal one point
functions in rotating black holes in higher dimensions.

###  5.1 Thermal one point function from gravity

We consider the BTZ metric with angular momentum which is given by

$\displaystyle
ds^{2}=\frac{\left(-r^{2}+r_{-}^{2}+r_{+}^{2}\right)dt^{2}}{R^{2}}+\frac{r^{2}R^{2}dr^{2}}{\left(r^{2}-r_{-}^{2}\right)\left(r^{2}-r_{+}^{2}\right)}+r^{2}d\phi^{2}-\frac{2r_{-}r_{+}dtd\phi}{R}\,.$
(97)

Then by AdS/CFT, the expectation value of the operator dual to the scalar
$\varphi$ is given by

$\langle{\cal
O}(t,\phi)\rangle=g\int\sqrt{g}d^{3}x^{\prime}\langle\chi^{2}(x^{\prime})\rangle
K(x^{\prime},t,\phi)\,.$ (98)

where $K(x^{\prime},t,\phi)$ is the bulk-boundary propagator of the $\varphi$
field in the $BTZ$ background, and the integral is carried out in the BTZ
geometry outside the horizon. The expectation value
$\langle\chi^{2}(x^{\prime})\rangle$ is obtained by considering the bulk-bulk
propagator of the $\chi$ field and subtracting the coincident limit
singularity. We now provide the details for evaluation of the expectation
value $\chi^{2}$, the bulk-boundary propagator $K(x^{\prime},t,\phi)$ and then
proceed to evaluate the integral in (98).

#### Bulk-Bulk propagator for $\chi$

Bulk-Bulk propagators between 2 points in global $AdS_{3}$ are known exactly
in terms of the geodesic distance between the points. Since the $BTZ$ is a
quotient of global $AdS_{3}$ through the identification
$\phi\rightarrow\phi+2\pi$, we first write down the geodesic length between
the following two points at the same radial position and time coordinate in
BTZ but at angular separation of $\rho$.

$x_{1}:(t,r,\phi=0)\,,\qquad x_{2}:(t,r,\phi=\rho)\,.$ (99)

The easiest approach to find the geodesic length is to use the embedding of
the BTZ background into the hyperboloid defined by

$-T_{1}^{2}-T_{2}^{2}+X_{1}^{2}+X_{2}^{2}=-R^{2}.$ (100)

Then the BTZ black hole with angular momentum in (97) is obtained from the
metric

$\displaystyle ds^{2}=-dT_{1}^{2}-dT_{2}^{2}+dX_{1}^{2}+dX_{2}^{2}\,.$ (101)

with the following embedding Banados:1992gq .

$\displaystyle
T_{1}=\sqrt{\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}-r_{-}^{2}}}R\sinh\left(\frac{r_{+}t-r_{-}R\phi}{R^{2}}\right),$
(102) $\displaystyle
T_{2}=\sqrt{\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}}R\cosh\left(\frac{r_{+}R\phi-
r_{-}t}{R^{2}}\right),$ $\displaystyle
X_{1}=\sqrt{\frac{r^{2}-r_{+}^{2}}{r_{+}^{2}-r_{-}^{2}}}R\cosh\left(\frac{r_{+}t-r_{-}R\phi}{R^{2}}\right),$
$\displaystyle
X_{2}=\sqrt{\frac{r^{2}-r_{-}^{2}}{r_{+}^{2}-r_{-}^{2}}}R\sinh\left(\frac{r_{+}R\phi-
r_{-}t}{R^{2}}\right).$

From this relations, we see that the location of the 2 points in (99), in
terms of the co-ordinates
$(T_{1}(x_{i})),T_{2}(x_{i}),X_{1}(x_{i}),X_{2}(x_{i}))$ can be obtained. Then
the geodesic distance $\hat{d}$ between these points is obtained using the
relation

$-R^{2}\cosh\hat{d}=-T_{1}(x_{1})T_{2}(x_{2})-T_{2}(x_{1})T_{2}(x_{2})+X_{1}(x_{1})X_{1}(x_{2})+X_{2}(x_{1})X_{2}(x_{2})\,.$
(103)

We obtain the following simple formula for the geodesic distance between the
points (99)

$\cosh\hat{d}=\frac{\left(r^{2}-r_{-}^{2}\right)\cosh\left(\frac{\rho
r_{+}}{R}\right)+\left(r_{+}^{2}-r^{2}\right)\cosh\left(\frac{\rho
r_{-}}{R}\right)}{r_{+}^{2}-r_{-}^{2}}\,.$ (104)

The bulk-bulk propagator in $AdS_{3}$ is given by DHoker:1999mqo

$\displaystyle G_{\Delta}(u)=\frac{C_{\Delta}2^{\Delta}}{u^{\Delta}}\
_{2}F_{1}\left(\Delta,\Delta-\frac{1}{2},2\Delta-1,-\frac{2}{u}\right),$ (105)

where

$\displaystyle
u=2\sinh^{2}\frac{\hat{d}}{2}=\frac{(r_{+}^{2}-r^{2})\cosh\left(\frac{\rho
r_{-}}{R}\right)+(r^{2}-r_{-}^{2})\cosh\left(\frac{\rho
r_{+}}{R}\right)-(r_{+}^{2}-r_{-}^{2})}{r_{+}^{2}-r_{-}^{2}}\,,$ (106)
$\displaystyle\qquad
c_{\Delta}=\frac{\Gamma(\Delta)\Gamma(\Delta-\frac{1}{2})}{(4\pi)^{\frac{3}{2}}\Gamma(2\Delta-1)}\,.$

We will consider the case when the field $\chi$ has conformal dimensions
$\Delta=1$, then

$G_{1}(r,\rho)=\frac{1}{8\pi}\frac{1}{\sqrt{\frac{u}{2}(\frac{u}{2}+1)}}\,.$
(107)

Substituting for $u$ from (106) we obtain

$G_{1}(r,\rho)=\frac{1}{4\pi}\frac{(r_{+}^{2}-r_{-}^{2})}{\sqrt{\big{[}(r_{+}^{2}-r^{2})\cosh(\frac{r_{-}\rho}{R})+(r^{2}-r_{-}^{2})\cosh(\frac{r_{+}\rho}{R})\big{]}^{2}-(r_{+}^{2}-r_{-}^{2})^{2}}}\,.$
(108)

Using this bulk to bulk propagator, we obtain

$\langle\chi^{2}(r)\rangle=\sum_{n=-\infty,n\neq 0}^{\infty}G_{1}(2\pi
n)=2\sum_{n=1}^{\infty}G_{1}(2\pi n)\,.$ (109)

Here we have removed the coincident singular limit and summed over all the
windings to ensure periodicity in $\phi\rightarrow\phi+2\pi$. Note that the
expectation value of $\chi^{2}$ just depends on the radial position.

#### The bulk-boundary propagator for $\varphi$

From the fact that the expectation value of $\chi^{2}$ depends only the radial
position, it is convenient to work in the Fourier expansion of the bulk-
boundary propagator in frequency and the angle $\phi$. Let

$K(r^{\prime},t,t^{\prime},\phi,\phi^{\prime})=\sum_{n=-\infty}^{\infty}\int\frac{d\omega}{2\pi}e^{-i\omega(t-t^{\prime})+in(\phi-\phi^{\prime})}\hat{K}(r,r^{\prime},\omega,n)\,.$
(110)

Then substituting this Fourier expansion in (98) and using the fact that
$\langle\chi^{2}(r)\rangle$ just depends on the radial position, we can
perform the integral over $t^{\prime},\phi^{\prime}$. This leads to

$\langle{\cal
O}(x,t)\rangle=g\int_{r=r_{+}}^{\infty}r^{\prime}dr^{\prime}\hat{K}(r^{\prime},0,0)\langle\chi^{2}(r^{\prime})\rangle\,.$
(111)

We obtain the zero mode of the bulk-boundary propagator
$\hat{K}(r^{\prime},0,0)$ by examining the zero mode of the bulk-bulk Green’s
function, which satisfies the differential equation

$\frac{1}{r}\partial_{r}\left(\frac{(r^{2}-r_{-}^{2})(r^{2}-r_{+}^{2})}{rR^{2}}\partial_{r}\hat{G}(r,r^{\prime},0,0)\right)-m^{2}\hat{G}(r,r^{\prime},0,0)=\frac{\delta(r-r^{\prime})}{r}\,.$
(112)

We change variables to

$x=\frac{r^{2}-r_{+}^{2}}{r^{2}-r_{-}^{2}}\,.$ (113)

Then the differential equation for the Green’s function reduces to

$4(1-x)^{2}\partial_{x}\left[x\partial_{x}\hat{G}(x,x^{\prime})\right]-m^{2}R^{2}\hat{G}(x,x^{\prime})=\frac{2(1-x)^{2}R^{2}}{r_{+}^{2}-r_{-}^{2}}\delta(x-x^{\prime})\,,$
(114)

here it is understood that $\hat{G}(x,x^{\prime})$ refers to the Fourier zero
mode along $t,\phi$ directions.

We construct the Green’s function by solving the homogenous equation. Let us
define $h=\frac{\Delta}{2}$, which is related to the mass by

$h=\frac{1}{2}(1+\sqrt{1+m^{2}R^{2}})\,,\qquad\nu=\sqrt{1+m^{2}R^{2}}\,.$
(115)

The solution which is well behaved near the outer horizon $r=r_{+},x=0$ is
given by

$\varphi_{\rm hor}=(1-x)^{1-h}\,_{2}F_{1}(1-h,1-h;1;x)\,,$ (116)

and the solution which is well behaved at the boundary $r=\infty,x=1$ is given
by

$\varphi_{\rm inf}=(1-x)^{h}\,_{2}F_{1}(h,h;2h;1-x)\,.$ (117)

Then the Green’s function is given by

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
A\times\Big{[}\varphi_{\rm inf}(x)\varphi_{\rm
hor}(x^{\prime})\theta(x-x^{\prime})+\varphi_{\rm hor}(x)\varphi_{\rm
inf}(x^{\prime})\theta(x^{\prime}-x)\Big{]},$ (118) $\displaystyle A$
$\displaystyle=$
$\displaystyle-\frac{R^{2}\Gamma(h)^{2}}{2.\Gamma(2h).\left(r_{+}^{2}-r_{-}^{2}\right)}\,.$

We can now use the bulk-bulk Green’s function to obtain the bulk-boundary
Green’s function. Substituting for $z=\frac{R^{2}}{r}$ in (12) and then using
(113) we obtain

$\displaystyle K(x)$ $\displaystyle=$
$\displaystyle\lim_{x^{\prime}\rightarrow
1}\frac{2\nu}{R^{4h}}\frac{(r_{+}^{2}-r_{-}^{2})^{h}}{(1-x^{\prime})^{h}}G(x,x^{\prime}),$
$\displaystyle=$ $\displaystyle\frac{2\nu
A}{R^{4h}}(r_{+}^{2}-r_{-}^{2})^{h}(1-x)^{1-h}\,_{2}F_{1}(1-h,1-h;1;x).$

#### Thermal one point function

We are ready to substitute the necessary ingredients in (111) to obtain the
one point function Using the bulk-boundary Green’s function (5.1), the
expectation value $\langle\chi^{2}(r)\rangle$ from (108) we obtain

$\displaystyle\langle{\cal O}\rangle$ $\displaystyle=$ $\displaystyle-{\cal
K}(h)\sum_{n=1}^{\infty}\frac{1}{\sinh(\frac{2\pi
nr_{+}}{R})}\int_{0}^{1}dx\frac{(1-x)^{h-1}\
_{2}F_{1}(h,h,1;x)}{\sqrt{\big{(}1-\hat{\chi}_{n}^{(1)}x\big{)}\big{(}1-\hat{\chi}_{n}^{(2)}x\big{)}}}\,,$
(120) $\displaystyle{\cal K}(h)$ $\displaystyle=$ $\displaystyle\frac{g\nu
R^{2}}{4\pi}\frac{(r_{+}^{2}-r_{-}^{2})^{h}(\Gamma(h))^{2}}{R^{4h}\Gamma(2h)}\,.$

where we have changed the integration variables to $x$ using (113) and

$\hat{\chi}_{n}^{(1)}=\frac{\cosh^{2}(\frac{\pi
nr_{-}}{R})}{\cosh^{2}(\frac{\pi nr_{+}}{R})}\,,\ \ \ \text{and}\ \ \ \
\hat{\chi}_{n}^{(2)}=\frac{\sinh^{2}(\frac{\pi
nr_{-}}{R})}{\sinh^{2}(\frac{\pi nr_{+}}{R})}\,.$ (121)

In (120), we have used the fact that the expectation value of the operator is
uniform from (111) and dropped the dependence on $(t,\phi)$. To take various
limits it is convenient to change variables to $w=1-x$ in (120) in the
integrand, this results in

$\displaystyle\langle{\cal O}\rangle$ $\displaystyle=$ $\displaystyle-{\cal
K}(h)\sum_{n=1}^{\infty}\frac{1}{\sinh^{2}(\frac{\pi
nr_{+}}{R})-\sinh^{2}(\frac{\pi nr_{-}}{R})}\int_{0}^{1}dw\frac{w^{h-1}\
_{2}F_{1}(h,h,1;1-w)}{\sqrt{\big{(}1-\chi_{n}^{(1)}w\big{)}\big{(}1-\chi_{n}^{(2)}w\big{)}}}\,,$
(122)

where

$\displaystyle{\chi}_{n}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\hat{\chi}_{n}^{(1)}}{\hat{\chi}_{n}^{(1)}-1}=\frac{\cosh^{2}\left(\frac{\pi
nr_{-}}{R}\right)}{\sinh^{2}(\frac{\pi nr_{-}}{R})-\sinh^{2}(\frac{\pi
nr_{+}}{R})},$ (123) $\displaystyle{\chi}_{n}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\hat{\chi}_{n}^{(2)}}{\hat{\chi}_{n}^{(2)}-1}=\frac{\sinh^{2}\left(\frac{\pi
nr_{-}}{R}\right)}{\sinh^{2}(\frac{\pi nr_{-}}{R})-\sinh^{2}(\frac{\pi
nr_{+}}{R})}\,.$

We can now expand the denominator in (122) using

$\displaystyle\frac{1}{\sqrt{(1-{\chi}_{1}w)(1-{\chi}_{2}w)}}=\sum_{k=0}^{\infty}P_{k}\left(\frac{{\chi}_{1}+{\chi}_{2}}{2\sqrt{{\chi}_{1}{\chi}_{2}}}\right)\left(\sqrt{{\chi}_{1}{\chi}_{2}}\
w\right)^{k},$ (124)

where $P_{k}$ refers to the Legendre polynomial of order $k$. Substituting
this expansion in (120), and integrating term by term we obtain

$\displaystyle\langle{\cal O}\rangle$ $\displaystyle=$ $\displaystyle{\cal
K}(h)\sum_{n=1,k=0}^{\infty}\frac{1}{\sinh^{2}(\frac{\pi
nr_{+}}{R})-\sinh^{2}(\frac{\pi nr_{-}}{R})}\times$
$\displaystyle\qquad\qquad\left[P_{k}\Big{(}\frac{{\chi}_{n}^{(1)}+{\chi}_{n}^{(2)}}{2\sqrt{{\chi}_{n}^{(1)}{\chi}_{n}^{(2)}}}\Big{)}\left(\sqrt{{\chi}_{n}^{(1)}{\chi}_{n}^{(2)}}\
\right)^{k}\frac{\Gamma(-h+k+1)\Gamma(h+k)}{\Gamma(k+1)^{2}}\right].$

The above result is the closed form expression for the thermal one point
function of the operator ${\cal O}$, in the rotating BTZ background which is
induced by the cubic coupling with the field $\chi$. It will be interesting to
compare this result with that of the CFT by developing the methods in
Kraus:2016nwo . This work assumed that the black hole could be modelled as a
heavy state in the CFT, but the case when the heavy state had different left
and right conformal dimensions was not considered.

####  $r_{-}\rightarrow 0$ limit

In Grinberg:2020fdj , the one point function for the Schwarzschild BTZ was
obtained. Let us first reproduce this results from the general expression
(5.1). In the limit $r_{-}\rightarrow 0$, we see that

$\displaystyle{\chi}_{n}^{(1)}\rightarrow-\frac{1}{\sinh^{2}(\frac{\pi
nr_{+}}{R})}\ \ \ \ \text{and}\ \ \ \ {\chi}_{n}^{(2)}\rightarrow 0\,,$
$\displaystyle
P_{k}\Big{(}\frac{{\chi}_{n}^{(1)}+{\chi}_{n}^{(2)}}{2\sqrt{{\chi}_{n}^{(1)}{\chi}_{n}^{(2)}}}\Big{)}\Big{[}\sqrt{{\chi}_{n}^{(1)}{\chi}_{n}^{(2)}}\Big{]}^{k}\rightarrow\frac{\Gamma(2k+1)}{4^{k}\Gamma(k+1)^{2}}\
(\chi_{n}^{(1)})^{k},$ (126) $\displaystyle{\cal K}(h)\rightarrow\frac{g\nu
R^{2}}{4\pi}\frac{r_{+}^{2h}\Gamma(h)^{2}}{R^{4h}\Gamma(2h)}\,.$

Substituting these leading terms in (5.1), we obtain

$\displaystyle\langle{\cal O}\rangle$ $\displaystyle=$
$\displaystyle\frac{g\nu
R^{2}}{4\pi}\frac{\Gamma(h)^{2}}{z_{0}^{2h}\Gamma(2h)}\times$
$\displaystyle\qquad\sum_{n=1,k=0}^{\infty}\frac{1}{\sinh^{2}(\frac{\pi
nr_{+}}{R})}\frac{\Gamma(2k+1)}{4^{k}\Gamma(k+1)^{2}}\
(\chi_{n}^{(1)})^{k}\frac{\Gamma(-h+k+1)\Gamma(h+k)}{\Gamma(k+1)^{2}}\,,$
$\displaystyle=$ $\displaystyle\frac{g\nu
R^{2}}{4\pi}\frac{\Gamma(h)^{2}}{z_{0}^{2h}\Gamma(2h)}\sum_{n=1}^{\infty}\frac{\pi\csc(\pi
h)}{\sinh^{2}(\frac{\pi nr_{+}}{R})}\
_{3}F_{2}\left(\frac{1}{2},1-h,h;1,1;-\frac{1}{\sinh^{2}(\frac{\pi
nr_{+}}{R})}\right).$

To obtain the second line, we have used the Legendre duplication formula to
simplify the Gamma functions in the numerator and then used the definition of
the hypergeometric function $\ {}_{3}F_{2}$ to sum over $k$. We have also
replaced $\frac{R^{2}}{r_{+}}$ by $z_{0}$. The last line in (5.1), is the
expression obtained in Grinberg:2020fdj  for thermal one point function in the
Schwarzschild BTZ background.

To interpret the one point function geometrically, we first take the limit
$\frac{2\pi r_{+}}{R}\rightarrow\infty$. At this point it is instructive to
relate this limit to the one taken in Grinberg:2020fdj , in which the length
of the identification in the spatial direction of the BTZ black hole is large.
To do this let us write the metric of the BTZ Schwarzschild metric from (97)

$ds^{2}=\frac{R^{2}}{z^{2}}\Big{[}\big{(}1-\frac{z^{2}}{z_{0}^{2}}\big{)}dt^{2}+\frac{dz^{2}}{1-\frac{z^{2}}{z_{0}^{2}}}+R^{2}d\varphi^{2}\Big{]}.$
(128)

Here we have written the Euclidean BTZ, in which $t$ is identified as
$t\rightarrow t+\beta$, where

$z_{0}=\frac{R^{2}}{r_{+}}=\frac{\beta}{2\pi}\,,$ (129)

On re-defining the co-ordinates as

$z^{\prime}=\frac{z}{z_{0}}\,,\qquad t^{\prime}=\frac{t}{z_{0}}\,,\qquad
x=\frac{R}{z_{0}}\phi\,.$ (130)

the metric becomes

$ds^{2}=\frac{R^{2}}{z^{\prime 2}}\Big{[}(1-z^{\prime 2})dt^{\prime
2}+\frac{dz^{\prime 2}}{1-z^{\prime 2}}+dx^{2}\Big{]}.$ (131)

With this scaling, we have the identifications

$t^{\prime}\sim t^{\prime}+2\pi,\qquad x\sim x+2\pi\frac{2\pi R}{\beta}\,.$
(132)

To arrive at the identification in $x$ we have used the relation between
$z_{0}$ and $\beta$ in (129). This is the rescaled co-ordinates used in
Grinberg:2020fdj . The periodicity in the spatial direction can be read from
(128). It is given by to be $L=2\pi R$ and therefore $x\sim x+\frac{2\pi
L}{\beta}$. Let us now examine the ratio $\frac{r_{+}}{R}$, from (129)

$\frac{2\pi r_{+}}{R}=2\pi\frac{2\pi R}{\beta}=\frac{2\pi L}{\beta}\,,$ (133)

Therefore taking the limit

$\frac{2\pi r_{+}}{R}\rightarrow\infty$ (134)

corresponds to the same limit of large spatial periodicities of
Grinberg:2020fdj .

Examining the one point function in (5.1) in the limit (134), we see only the
$n=1$ term contributes and we can also just retain the leading term from the
hypergeometric function which results in

$\displaystyle\langle{\cal O}\rangle=\frac{g\nu
R^{2}}{2}\frac{\Gamma(h)^{2}}{z_{0}^{2h}\Gamma(2h)}\csc(\pi h)e^{-\frac{2\pi
r_{+}}{R}}.$ (135)

Note that the $z_{0}^{-2h}$ just provides the scaling dimensions for the
expectation value. In this form it is easy to see that on further taking the
limit $h\rightarrow\infty-i\epsilon$

$\displaystyle\lim_{h\rightarrow\infty-i\epsilon}\langle{\cal O}\rangle$
$\displaystyle\sim$ $\displaystyle e^{-i\pi h}2^{-2h}e^{-\frac{2\pi
r_{+}}{R}},$ $\displaystyle\sim$ $\displaystyle
e^{-i\pi\frac{mR}{2}}4^{-\frac{mR}{2}}e^{-\frac{2\pi r_{+}}{R}}.$

To obtain the last line we have used (115) to replace $h$ in terms of the mass
$m$. Comparing the coefficient of $e^{-\frac{2\pi r_{+}}{R}}$ in (2), we see
that it behaves just as seen for thermal one point functions in higher
dimensions.

####  Limit $\frac{r_{-}}{R},\frac{r_{+}}{R}\rightarrow\infty$ , with
$\frac{r_{-}}{r_{+}}$ held fixed

In this limit, we still can keep track of the dependence of the one point
function on the inner horizon Before we take this limit, it is convenient to
simplify the dependence on $r_{-},r_{+}$ in (5.1). We can write

$\frac{{\chi}_{n}^{(1)}+{\chi}_{n}^{(2)}}{2\sqrt{{\chi}_{n}^{(1)}{\chi}_{n}^{(2)}}}=\coth\frac{2\pi
nr_{-}}{R}\,,$ (137)

and

$\sqrt{{\chi}_{n}^{(1)}{\chi}_{n}^{(2)}}=\frac{\sinh\frac{2\pi
nr_{-}}{R}}{\cosh\frac{2\pi nr_{-}}{R}-\cosh\frac{2\pi nr_{+}}{R}}\,.$ (138)

We need the limit

$\lim_{\frac{r_{-}}{R}\rightarrow\infty}P_{k}(\coth\frac{2\pi
nr_{-}}{R})=1+O(\exp(-\frac{4\pi nr_{-}}{R})\,.$ (139)

We substitute (137), (138) into the expression for the one point function in
(5.1) and take the limit

$\frac{r_{-}}{R}\rightarrow\infty,\quad\frac{r_{+}}{R}\rightarrow\infty,\quad\frac{r_{-}}{r_{+}}=\delta\;\;{\rm
fixed}.$ (140)

Using (139), we see that the leading contribution arises from $n=1,k=0$ term
in the sum. All the remaining terms are exponentially suppressed compared to
this term, which is given by

$\displaystyle\langle{\cal O}\rangle$ $\displaystyle=$
$\displaystyle\frac{g\nu
R^{2}}{2}\frac{(r_{+}^{2}-r_{-}^{2})^{h}\Gamma(h)^{2}}{R^{4h}\Gamma(2h)}\csc(\pi
h)e^{-\frac{2\pi r_{+}}{R}}+O\Big{[}\exp(-\frac{2\pi
r_{+}}{R}(2-\delta))\Big{]},$ (141) $\displaystyle=$ $\displaystyle\frac{g\nu
R^{2}}{2}\frac{\Gamma(h)^{2}}{z_{0}^{2h}\Gamma(2h)}\csc(\pi
h)e^{h\log(1-\delta^{2})}e^{-\frac{2\pi r_{+}}{R}},$

with $z_{0}=\frac{R^{2}}{\sqrt{{r_{+}}^{2}-r_{-}^{2}}}$.

Finally we can take the $h\rightarrow\infty-i\epsilon$

$\displaystyle\lim_{h\rightarrow\infty-i\epsilon}\langle{\cal O}\rangle\sim
e^{-i\pi h}e^{-h(\log 4-\log(1-\delta^{2})}e^{-\frac{2\pi r_{+}}{R}},$ (142)
$\displaystyle=e^{-i\frac{\pi mR}{2}}e^{-\frac{mR}{2}(\log
4-\log(1-\delta^{2}))}e^{-\frac{2\pi r_{+}}{R}}.$

#### Comparison with geometric lengths

From the metric (97, the time to singularity is given by the integral

$\displaystyle\tau_{s}$ $\displaystyle=$ $\displaystyle
R\int_{r_{-}}^{r_{+}}\frac{rdr}{\sqrt{(r_{+}^{2}-r^{2})(r^{2}-r_{-}^{2})}}\,,$
$\displaystyle=$ $\displaystyle\frac{R\pi}{2}\,.$

The proper length from infinity to the horizon is given by

$\displaystyle\hat{\ell}_{\rm hor}$ $\displaystyle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}R\int_{r_{+}}^{\frac{R^{2}}{\epsilon}}\frac{rdr}{\sqrt{(r_{+}^{2}-r^{2})(r^{2}-r_{-}^{2})}}\,,$
$\displaystyle=$
$\displaystyle-\frac{R}{2}\log(1-\delta^{2})+R\log(2)-\log\frac{\epsilon}{z_{0}}\,,\qquad\qquad
z_{0}=\frac{R^{2}}{r_{+}}\,.$

Here we have chosen the same cut off as in (2), since $r$ and $z$ are related
by $r=\frac{R^{2}}{z}$. Therefore the regularized proper length is given by

$\hat{\ell}_{\rm hor}=-\frac{R}{2}\log(1-\delta^{2})+R\log(2)\,.$ (145)

Finally one can also evaluate the proper length from the inner horizon to the
singularity, we obtain

$\hat{\ell}_{\rm
sing}=\frac{R}{2}\log\Big{(}\frac{1-\delta}{1+\delta}\Big{)}.$ (146)

Comparing the leading contribution to the thermal one point function in (142)
with these geometric lengths, we see that it can be written as

$\langle{\cal O}\rangle\sim\exp(-im\tau_{s}-m\ell_{\rm hor})e^{-\frac{2\pi
r_{+}}{R}}.$ (147)

Note that in the limit (140) we do not see the presence of $\ell_{\rm hor}$,
the proper length from the inner horizon to the singularity.

###  5.2 One point function $\langle O^{2}\rangle$ from CFT

From the analysis of the rotating BTZ background we saw that in the large
horizon radius limit, or the large spatial length limit, the leading
contribution to the thermal one point function does not see $\ell_{\rm sing}$,
the geometric length from the inner horizon to the singularity. We know that
composite operators, or operators which do not transform as conformal
primaries gain thermal expectation value in 2d CFT on a plane. In this section
we evaluate this one point function and observe that when their conformal
dimensions are large, their expectation value can be cast into the geometric
form, but again there is no information about $\ell_{\rm sing}$.

We consider the bilinear compose ${\cal O}^{2}$ of the primary ${\cal O}$ of
dimension $2h_{\cal O}$ and let $2h_{\cal O}$ be an integer. Here the
holomorphic and the anti-holomorphic dimension of the operator is $(h_{\cal
O},h_{\cal O})$. The one point function of the operator of the composite can
be obtained by the following approach. We first look at the 2 point function
at finite temperature

$\displaystyle\langle\mathcal{O}(w,\bar{w})\mathcal{O}(0,0)\rangle$
$\displaystyle=\bigg{[}\frac{\beta_{L}}{\pi}\sinh\bigg{(}\frac{\pi
w}{\beta_{L}}\bigg{)}\bigg{]}^{-2h_{\mathcal{O}}}\bigg{[}\frac{\beta_{R}}{\pi}\sinh\bigg{(}\frac{\pi\bar{w}}{\beta_{R}}\bigg{)}\bigg{]}^{-2h_{\mathcal{O}}}.$
(148)

Here the left and right inverse temperatures of the CFT is given by
$(\beta_{L},\beta_{R})$ and since we are interested in the BTZ black hole with
angular momentum we should have $\beta_{L}\neq\beta_{R}$. The expectation
value $\langle{\cal O}^{2}\rangle$ can be obtained by examining the expansion
in $w,\bar{w}$ of (148) and extracting out the constant term. To see that this
indeed captures thermal expectation value, we know that the OPE of ${\cal O}$
with itself contains the composite ${\cal O}^{2}$.

${\cal O}(z,\bar{z}){\cal O}(0,0)\sim\frac{1}{|z|^{4h_{\cal O}}}+\cdots+C{\cal
O}^{2}(0,0)+\cdots\ .$ (149)

Here the $\cdots$ refer to other singular terms or terms which depend on
powers of $z,\bar{z}$ in the OPE which we are not interested in. The ${\cal
O}^{2}$ operator occurs in the OPE with no dependence of $z$ and $C$ is the
OPE coefficient. Now taking thermal expectation value on both sides of this
equation, we see that the constant term, independent of $z,\bar{z}$ is
proportional to the expectation value $\langle{\cal O}^{2}\rangle$. To extract
the constant term, it is sufficient to look at the holomorphic two point
function and obtain the residue in the Laurent expansion in $z$. The residue
when $2h_{\cal O}$ is an integer is given by

$a_{0}(\beta)=\frac{1}{2\pi
i}\oint\frac{dw}{w}\bigg{[}\frac{\beta}{\pi}\sinh\bigg{(}\frac{\pi
w}{\beta}\bigg{)}\bigg{]}^{-2h_{\mathcal{O}}},$ (150)

where the integral is around the origin. Then from the OPE expansion of (148),
the thermal expectation value is given by

$C\langle{\cal O}^{2}\rangle=a_{0}(\beta_{L})a_{0}(\beta_{R})\,.$ (151)

Examining the residue (150), we see that it is non-zero only when $2h_{0}$ is
an even integer or $h_{\cal O}$ is an integer. We re-write the residue as

$a_{0}(\beta)=\frac{1}{2\pi i}\Big{(}\frac{\pi}{\beta}\Big{)}^{2h_{\cal
O}}\oint dye^{-\left[\log y+2h_{\cal O}\log(\sinh y)\right]}\,.$ (152)

For large $2h_{\cal O}$ we can perform this integral by the saddle point
method. The saddle points at the leading order are at

$y_{\rm saddle}=\pm\frac{i\pi}{2}+O(\frac{1}{h_{\cal O}})\,.$ (153)

Adding the contributions at the 2 saddles along with the one loop term at each
of the saddles we obtain

$\displaystyle a_{0}(\beta)=\frac{1}{2\pi
i}\oint\frac{dw}{w}\bigg{(}\frac{\beta}{\pi}\sinh\big{[}\frac{\pi
w}{\beta}\big{]}\bigg{)}^{-2h_{\mathcal{O}}}\approx\frac{2\pi^{2h_{\mathcal{O}}-\frac{1}{2}}\cos(\pi
h_{\mathcal{O}})}{\beta^{2h_{\mathcal{O}}}\
\sqrt{2+\pi^{2}h_{\mathcal{O}}}}\,.$ (154)

We can compare the value of $a_{0}$ from the saddle point against the
numerical value, which can be obtained by explicitly evaluating the residue.
This comparison is shown in the table 1 as well as in the graph in figure 4,
we see that indeed, that the saddle point approximation is a good
approximation at large values of integer $2h_{\cal O}$.

|  |  |
---|---|---|---
$h_{\mathcal{O}}$ |  $(a_{0}\beta^{2h_{\mathcal{O}}}\pi^{\frac{1}{2}-2h_{\mathcal{O}}})$ Exact |  $(a_{0}\beta^{2h_{\mathcal{O}}}\pi^{\frac{1}{2}-2h_{\mathcal{O}}})$ Saddle point |  Error in $\%$
|  |  |
|  |  |
$5$ | $-0.280338$ | -0.279105 | 0.439624
$10$ | $0.199763$ | $0.199308$ | $0.227946$
$15$ | $-0.163527$ | $-0.163275$ | $0.153875$
$20$ | 0.141801 | $0.141637$ | $0.116138$
$25$ | $-0.126929$ | $-0.126811$ | $0.0932657$
$30$ | $0.11593$ | $0.11584$ | $0.07792$
$35$ | $-0.10737$ | $-0.107298$ | $0.0669107$
$40$ | $0.100463$ | $0.100404$ | $0.0586273$
$45$ | $-0.0947381$ | $-0.0946887$ | $0.052169$
$50$ | $0.089892$ | $0.0898497$ | $0.0469923$
$55$ | $-0.0857207$ | $-0.0856841$ | $0.0427502$
$60$ | $0.082081$ | $0.0820488$ | $0.0392106$
$65$ | $-0.0788687$ | $-0.0788402$ | $0.0362124$
|  |  |
Table 1: Numerical comparison of the exact value of the residue at integer
$h_{\cal O}$ against the saddle point given in (154). Figure 4: The blue curve
represents the plot of the function
$\log[a_{0}\beta^{2h_{\mathcal{O}}}\sec(\pi h_{\mathcal{O}})]$, where $a_{0}$
is obtained from the result of our saddle point analysis given in the equation
(154). The red points are the exact values of the
$\log[a_{0}\beta^{2h_{\mathcal{O}}}(-1)^{h_{\cal O}}]$ at the integer values
of $h_{\mathcal{O}}$ obtained by explicitly evaluating the residue (150).

From the approximation of $a_{0}(\beta)$ in (154), we see that the thermal one
point function is given by in the large $2h_{\cal O}$ limit is given by

$\displaystyle\langle{\cal O}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{C}a_{0}(\beta_{L})a_{0}(\beta_{R})\,,$ $\displaystyle=$
$\displaystyle\frac{1}{C}\bigg{(}\frac{1}{\beta_{L}\beta_{R}}\bigg{)}^{2h_{\mathcal{O}}}\
\frac{4\pi^{4h_{\mathcal{O}}-1}\cos^{2}(\pi h_{\mathcal{O}})}{\
{2+\pi^{2}h_{\mathcal{O}}}}\,.$

The $\cos^{2}(\pi h_{\mathcal{O}})$ term is just unity when $h_{{\cal O}}$ is
an integer, but we keep this factor so that it enables us to obtain the
general form of the result. Let us now recast this expression in terms of
geometric quantities in the bulk. The left and right inverse temperatures are
related to the horizon radii by Maldacena:1998bw

$\frac{1}{\beta_{L}}=\frac{r_{+}-r_{-}}{2\pi
R^{2}}\,,\qquad\frac{1}{\beta_{L}}=\frac{r_{+}+r_{-}}{2\pi R^{2}}\,.$ (156)

Using this relation in (5.2), we can write

$\lim_{h_{\rm O}\rightarrow\infty+i\epsilon}\langle{\cal
O}^{2}\rangle\sim\frac{1}{Cz_{0}^{4h_{\cal O}}}e^{-2h_{\cal O}\log 4+2h_{\cal
O}\log(1-\delta^{2})}\exp(-i2\pi h_{\cal O})\,.$ (157)

The small imaginary part selects the leading phase to be $\exp(-2\pi ih_{\cal
O})$. In gravity the $3$ OPE coefficient is given by $C=\sqrt{2}$
Belin:2017nze , therefore they does not grow exponentially in $h_{\cal O}$. We
can also identify $2h_{\cal O}=\frac{\Delta_{{\cal O}^{2}}}{2}$. With these
inputs, the thermal one point function can be written as

$\displaystyle\langle{\cal O}^{2}\rangle$ $\displaystyle\sim$
$\displaystyle\exp\Big{[}-i\pi\frac{\Delta_{{\cal
O}^{2}}}{2}-\frac{\Delta_{{\cal O}^{2}}}{2}\big{(}\log
4-\log(1-\delta^{2})\big{)}\Big{]},$ $\displaystyle\sim$
$\displaystyle\exp\big{[}-i\frac{\pi 2mR}{2}-\frac{2mR}{2}\big{(}\log
4-\log(1-\delta^{2})\big{)}\Big{]}.$

Using (5.1), (145), we can write the expectation value of the composite as

$\langle{\cal O}^{2}\rangle\sim e^{-i2m\tau_{s}-2m\ell_{\rm hor}}.$ (159)

Note that $m\rightarrow 2m$ since we are evaluating the expectation value of
the composite ${\cal O}^{2}$, and we do not see the appearance of $\ell_{\rm
sing}$.

Though our calculation in the CFT has been done only for integer conformal
dimensions, we have written the result so that it is natural to read out the
general form of the result. But, it will be interesting to see if the methods
of Iliesiu:2018fao  can be used to verify if indeed the expectation value of
the composite can be written in the form for arbitrary conformal dimensions
777 In Iliesiu:2018fao  the expectation value of the composite in which the
operator ${\cal O}$ obeys mean field theory correlators have been written for
2d CFT, but not for the correlator given in (148). We have verified that the
expectation value of the composite ${\cal O}^{2}$ using the mean field theory
correlator does not behave as in (5.2) in the large $h_{\cal O}$ limit. .

##  6 Conclusions

We have discussed two examples in which the thermal one point function of
massive scalars can be evaluated exactly. For the charged planar black hole we
introduced a large $d$ limit to achieve this. In both these examples the
result for the one point function agreed with that anticipated in
Grinberg:2020fdj  using WKB methods. In particular for the charged black hole
which has an inner horizon we observed the presence of $\ell_{hor}$, the
proper distance from the inner horizon to the singularity. The second example
involved hyperbolic black holes for which $\tau_{s}$, the time to the
singularity has a different dependence on the radius of $AdS_{d+1}$. We also
saw that these results remain the same for both the Gauss-Bonnet or the Weyl
tensor squared coupling.

We obtained the one point function of massive scalars in BTZ due to a cubic
coupling of scalars. We could interpret this result geometrically in a
suitable limit of large horizon radius, but could not find the dependence of
$\ell_{\rm sing}$ in this limit. It would be interesting to investigate the
case of rotating black hole in higher dimensions. The WKB methods of
Grinberg:2020fdj  assumed radial symmetry and it would be important to
generalize those arguments to backgrounds with axial symmetry. The recent
developments Dodelson:2022yvn ; Bhatta:2022wga  regarding the applications of
the solutions to the Heun equation to evaluate thermal 2 point functions will
be helpful in this regard.

We saw that in $2d$ we could gain some insight on thermal one point functions
by evaluating the expectation value of composite scalars at finite
temperature. For this we used integer conformal dimensions. The methods of
Iliesiu:2018fao  can be used to study this question for arbitrary conformal
dimensions. It will be important to develop such methods in field theory to
understand the features of thermal one point function. As we have seen, these
contain some information of the geometry behind the horizon and evaluating one
thermal one point functions directly in the dual field theory can give us new
insight to black hole geometry.

###### Acknowledgements.

We thank Chethan Krishnan, Suvrat Raju and Aninda Sinha for discussions during
the presentation of a preliminary version of this work. J.R.D thanks the group
at the Asia Pacific Center for Theoretical Physics, Pohang for the warm
hospitality during the completion of this work.

##  Appendix A Details for the Green’s function

The bulk-bulk Green’s function $G(z,z^{\prime})$ satisfies the equation,

$\partial_{z}\left(\frac{R^{d-1}}{z^{d-1}}f(z)\partial_{z}G(z,z^{\prime})\right)-\frac{R^{d-1}}{z^{d-1}}m^{2}G(z,z^{\prime})=\delta(z-z^{\prime})\,,$
(160)

and $G(z,z^{\prime})$ has to be well-behaved at the boundaries $z=0$ and
$z=z_{0}$; $f(z)$ was defined in the equation (8).

Now, the two linearly independent solutions of the homogeneous part of the
differential equation (160) are given by,

$\displaystyle g_{{\rm inf}}(w)=w^{h}\
{}_{2}F_{1}(h,h,2h,w)\,,\qquad\text{and,}\qquad g_{{\rm
hor}}(w)=w^{h}{}_{2}F_{1}(h,h,1,1-w)\,.$ (161) $\displaystyle\text{Where,
}\qquad w=\left(\frac{z}{z_{0}}\right)^{d},\qquad\qquad\qquad
h=\frac{\Delta}{d}\,,$ (162)

The solutions $g_{\rm inf}(w)$ and $g_{\rm hor}(w)$ are regular at $w=0$ and
at $w=1$ respectively, which indicates,

$\displaystyle\begin{split}G(w,w^{\prime})=\begin{cases}A(w^{\prime})g_{\rm
inf}(w)\ \ \text{for}\ w<w^{\prime},\\\ B(w^{\prime})g_{\rm hor}(w)\ \
\text{for}\ w>w^{\prime}.\\\ \end{cases}\end{split}$ (163)

Where $A(w^{\prime})$ and $B(w^{\prime})$ are functions of $w^{\prime}$, to be
determined.

Now, for the Green’s function to be continuous at $w=w^{\prime}$, we should
have,

$\displaystyle A(w^{\prime})g_{\rm inf}(w^{\prime})=B(w^{\prime})g_{\rm
hor}(w^{\prime})\,.$ (164)

The jump of $\partial_{w}G(w,w^{\prime})$ around $w=w^{\prime}$ is obtained by
integrating the both side of the equation (160) with respect to $z$ from
$z=z^{\prime}-\epsilon$ to $z=z^{\prime}+\epsilon$ with $\epsilon\rightarrow
0$ and then we used the equation (164) to get,

$\displaystyle\frac{B(w^{\prime})}{g_{\rm inf}(w^{\prime})}\bigg{(}g_{\rm
inf}(w^{\prime})g_{hor}^{\prime}(w^{\prime})-g_{hor}(w^{\prime})g_{\rm
inf}^{\prime}(w^{\prime})\bigg{)}=\frac{z_{0}^{d}}{R^{d-1}d(1-w)}\,.$ (165)

Note that the terms inside the parenthesis in the l.h.s is equal to the
Wronskian of the homogeneous part of the differential equation (160) written
in $w^{\prime}$ variable. And this Wronskian is calculated up to an overall
constant factor $\tilde{C}$ directly from the differential equation, thus we
have,

$\displaystyle\bigg{(}g_{\rm
inf}(w^{\prime})g_{hor}^{\prime}(w^{\prime})-g_{\rm hor}(w^{\prime})g_{\rm
inf}^{\prime}(w^{\prime})\bigg{)}=\frac{\tilde{C}}{1-w^{\prime}}\,.$ (166)

Now, the constant $\tilde{C}$ is determined by expanding the both side of the
above equation in the taylor series about $w=0$. The first term of the series
expansion of the l.h.s is equal to $\tilde{C}$. Thus one obtains,

$\displaystyle\tilde{C}=-\frac{\Gamma(2h)}{\Gamma(h)^{2}}\,.$ (167)

Hence,

$\displaystyle B(w^{\prime})=-\frac{\Gamma(h)^{2}}{\Gamma(2h)}\
\frac{z_{0}^{d}}{R^{d-1}d}\ g_{\rm inf}(w^{\prime})\ \ ,\qquad\qquad
A(w^{\prime})=-\frac{\Gamma(h)^{2}}{\Gamma(2h)}\ \frac{z_{0}^{d}}{R^{d-1}d}\
g_{\rm hor}(w^{\prime})\,.$ (168)

And finally, the bulk-bulk green’s function,

$\displaystyle G(w,w^{\prime})=-\frac{\Gamma(h)^{2}}{\Gamma(2h)}\
\frac{z_{0}^{d}}{R^{d-1}d}\bigg{(}g_{{\rm inf}}(w)g_{{\rm
hor}}(w^{\prime})\theta(w^{\prime}-w)+g_{{\rm hor}}(w)g_{{\rm
inf}}(w^{\prime})\theta(w-w^{\prime})\bigg{)}\,.$

##  Appendix B The charged planar black hole solution at large $d$

The consistency of the treatment of large $d$ limit used to evaluate the one
point function for the case of the charged planar black hole requires the
metric (32) in this limit to satisfy the Einstien-Maxwell field equation at
the leading order in $d$. In this appendix, we will verify it by
straightforward calculation.

At the limit described in (36), the metric (32) takes the form,

$\displaystyle
ds^{2}\approx\frac{R_{AdS}^{2}}{z^{2}}\bigg{(}-\hat{f}(z)dt^{2}+\frac{dz^{2}}{\hat{f}(z)}+d\vec{x}^{2}\bigg{)},\
\ \text{where}\ \ \ \ \
\hat{f}(z)=1-\frac{z^{d}}{z_{0}^{d}}+Q\frac{z^{2d}}{z_{0}^{2d}}\,.$ (169)

Note that the asymptotic behaviour of the metric at the boundary is kept
intact in this limit, thus the rules of AdS/CFT were implemented to obtain the
one-point function.

The above metric should satisfy the following Einstien-Maxwell field equation
at large $d$,

$\displaystyle\lim_{d\rightarrow\infty}\bigg{(}R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\Lambda
g_{\mu\nu}\bigg{)}=-\lim_{d\rightarrow\infty}8\pi T_{\mu\nu}\,,$ (170)

where the cosmological constant $\Lambda=-\frac{d(d-1)}{2R_{AdS}^{2}}$ for the
$AdS_{d+1}$ and $T_{\mu\nu}$ is the energy-momentum stress tensor evaluated
from the gauge potential Chamblin:1999tk ,

$\displaystyle A=\bigg{(}-\frac{qz^{d-2}}{c}+\Phi\bigg{)}dt,\qquad\Phi\
\text{is constant, and }\quad c=\sqrt{\frac{2(d-2)}{d-1}}\,.$ (171)

By calculating the l.h.s and r.h.s separately for each tensorial component of
the equation (170), after incorporating the metric (169) in it, we will show
that l.h.s$=$r.h.s for each component. For this calculation we need the
following results,

$\displaystyle
R_{00}=\frac{f(z)(-2f(z)d+z[(1+d)f^{\prime}(z)-zf^{\prime\prime}(z)])}{2z^{2}}\,,$
(172) $\displaystyle
R_{11}=\frac{2f(z)d+z[-(d+1)f^{\prime}(z)+zf^{\prime\prime}(z)]}{2z^{2}f(z)}\,,$
(173) $\displaystyle R_{ii}=\frac{df(z)-zf^{\prime}(z)}{z^{2}}\ \ \text{for}\
\ i\in[2,d]\,.$ (174)

All other components of Ricci-tensor are zero. And the Ricci-scalar is given
by,

$\displaystyle R=d(d+1)f(z)+z(-2f^{\prime}(z)d+zf^{\prime\prime}(z))\,.$ (175)

For the 00-component,

$\displaystyle\lim_{d\rightarrow\infty}\bigg{(}R_{00}-\frac{1}{2}g_{00}R-\Lambda
g_{00}\bigg{)}$
$\displaystyle=\lim_{d\rightarrow\infty}\bigg{(}-\frac{d(d-1)}{2}Qz^{2d-2}z_{0}^{-4d}\big{[}Qz^{2d}+z_{0}^{d}(-z^{d}+z_{0}^{d})\big{]}\bigg{)},$
$\displaystyle=\frac{1}{z^{2}}\bigg{[}-e^{2d\log(\frac{z}{z_{0}})}+e^{3d\log(\frac{z}{z_{0}})}-e^{4d\log(\frac{z}{z_{0}})}\bigg{]}.$
(176) $\displaystyle-\lim_{d\rightarrow\infty}8\pi T_{00}$
$\displaystyle=-\lim_{d\rightarrow\infty}\bigg{(}\frac{(d-2)(d-1)}{2}Qz^{2d-4}z_{0}^{2-4d}[Qz^{2d}+z_{0}^{d}(-z^{d}+z_{0}^{d})]\bigg{)},$
(177)
$\displaystyle=\frac{1}{z^{2}}\bigg{[}-e^{2d\log(\frac{z}{z_{0}})}+e^{3d\log(\frac{z}{z_{0}})}-e^{4d\log(\frac{z}{z_{0}})}\bigg{]}.$
(178)

For the 11-component,

$\displaystyle\lim_{d\rightarrow\infty}\bigg{(}R_{11}-\frac{1}{2}g_{11}R-\Lambda
g_{11}\bigg{)}$
$\displaystyle=\lim_{d\rightarrow\infty}\frac{d(d-1)Qz^{2d-2}}{2(Qz^{2d}+z_{0}^{2}d-z^{d}z_{0}^{d})}\,,$
(179)
$\displaystyle=\frac{1}{z^{2}}\bigg{(}\frac{1}{e^{2d\log(\frac{z_{0}}{z})}-e^{d\log(\frac{z_{0}}{z})}+Q}\bigg{)}.$
(180) $\displaystyle-\lim_{d\rightarrow\infty}8\pi T_{11}$
$\displaystyle=\lim_{d\rightarrow\infty}\bigg{(}\frac{(d-2)(d-1)Qz_{0}^{2}}{2z^{4}[Q+z^{-2d}z_{0}^{d}(-z^{d}+z_{0}^{d})]}\bigg{)}\,,$
(181)
$\displaystyle=\frac{1}{z^{2}}\bigg{(}\frac{1}{e^{2d\log(\frac{z_{0}}{z})}-e^{d\log(\frac{z_{0}}{z})}+Q}\bigg{)}.$
(182)

and, for $ii$-component,

$\displaystyle\lim_{d\rightarrow\infty}\bigg{(}R_{ii}-\frac{1}{2}g_{ii}R-\Lambda
g_{ii}\bigg{)}$
$\displaystyle=-\lim_{d\rightarrow\infty}\bigg{(}\frac{d(d+1)Qz^{2d-2}z_{0}^{-2d}}{2}\bigg{)},$
(183) $\displaystyle=-\frac{1}{z^{2}}e^{2d\log(\frac{z}{z_{0}})}.$ (184)
$\displaystyle-\lim_{d\rightarrow\infty}8\pi T_{ii}$
$\displaystyle=\lim_{d\rightarrow\infty}\bigg{(}-\frac{(d-2)(d-1)Qz^{2d-4}z_{0}^{2-2d}}{2}\bigg{)},$
(185) $\displaystyle=-\frac{1}{z^{2}}e^{2d\log(\frac{z}{z_{0}})}.$ (186)

Thus, for each tensorial component we have shown that l.h.s$=$r.h.s, i.e., the
Einstien-Maxwell field equation is satisfied by the metric given in (169) at
the leading order in $d$.

##  Appendix C Large $d$ with Weyl tensor squared coupling

We repeat the calculation for the one point function in the charged planar
black hole background at large $d$, described in section 3, but using the Weyl
tensor squared coupling in place of the GB coupling.

The rules of AdS/CFT prescribe the one point function to be,

$\langle O(t,\vec{x})\rangle=\alpha\int
dzdt^{\prime}d\vec{x}^{\prime}\sqrt{g}\tilde{K}(t,\vec{x};\,z,t^{\prime},\vec{x}^{\prime}){W^{2}}(z,t^{\prime},\vec{x}^{\prime}),$
(187)

where the $W^{2}=W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}$, and it would be
sufficient to do only the $z$ part of the above integral as was argued before.

For the metric given in (32),

$\displaystyle
W^{2}=\frac{(d-2)(d-1)^{2}w^{2-\frac{4}{d}}[(6-4d)Qw+dw^{2/d}]^{2}}{R^{4}d}\,.$
(188)

And in the limit $d\rightarrow\infty$, the leading contribution to the Weyl
tensor’s square is given by,

$\displaystyle W^{2}=\frac{w^{2}(1-4Qw)^{2}d^{4}}{R^{4}}\,.$ (189)

Now plugging the expression for the bulk-boundary green’s function $K(w)$,
given in (3), in the equation (187), and using the change of variable defined
in (47), we get,

$\displaystyle\begin{split}\langle
O\rangle=&-\frac{4d^{2}(1-4Q)^{h/2}}{R^{2}}\frac{(\Gamma(h))^{2}}{\Gamma(2h)}\bigg{(}\frac{2\nu}{z_{0}^{\Delta_{+}}}\bigg{)}\int_{0}^{1}dy(1-y)^{h}{}_{2}F_{1}(h,h,1,y)\times\\\
&\bigg{[}(2Qw_{+})^{-2}(1-\chi y)^{-2}-16Q(2Qw_{+})^{-3}(1-y)(1-\chi
y)^{-3}+64Q^{2}(2Qw_{+})^{-4}(1-y)^{2}(1-\chi y)^{-4}\bigg{]}.\end{split}$
(190)

Applying the integration formulae used in the section 3, the integrals in the
above equation are performed to get the analytic expression for the thermal
one point function,

$\displaystyle\begin{split}\langle
O\rangle=&-\frac{4d^{2}(1-4Q)^{\frac{h}{2}-1}}{R^{2}}\frac{(\Gamma(h))^{2}}{\Gamma(2h)}\frac{h(h-1)\pi\csc(\pi
h)}{12(1-\sqrt{1-4Q})}\bigg{(}\frac{2\nu}{z_{0}^{\Delta_{+}}}\bigg{)}\times\\\
&\bigg{(}\big{[}3(-1+\sqrt{1-4Q})+8Q(h-2)\\{h(-1+\sqrt{1-4Q})+\sqrt{1-4Q}\\}\big{]}\\\
&\hskip
113.81102pt\times{}_{2}F_{1}(2-h,1+h,2,\frac{1}{2}-\frac{1}{2\sqrt{1-4Q}})\\\
&\hskip 85.35826pt-8Q(h-2)\
{}_{2}F_{1}(3-h,1+h,2,\frac{1}{2}-\frac{1}{2\sqrt{1-4Q}})\bigg{)}\,.\end{split}$
(191)

An important point to note that this Weyl tensor square induced one point
function differs from the one point function enabled by the GB coupling given
in (50), by the polynomial factors in front of the hypergeometric functions
which are not so crucial in the large $h$ bahaviour of the one point function
as the more dominant contributions come from the exponential growth of the
hypergeometric functions at large $h$.

Thus, the one point function undergoes the exact similar treatment of taking
the large $h$ limit as was illustrated in section 3 to land up into the form,

$\displaystyle\langle
O\rangle\sim\exp\bigg{[}m\bigg{(}{{-\frac{R}{d}\log\bigg{[}\frac{4}{\sqrt{1-4Q}}\bigg{]}}+{\frac{R}{d}\operatorname{arcsech}(\sqrt{1-4Q})}}-i{\pi\frac{R}{d}}\bigg{)}\bigg{]}.$
(192)

So, we observe that the thermal one point function due to both the Weyl tensor
squared and GB coupling behaves identically at large $h$.

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