aainstitutetext: School of Physics, Institute for Research in Fundamental
Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran
bbinstitutetext: Institut für Theoretische Physik und Astrophysik, Julius-
Maximilians-Universität Würzburg,
Am Hubland, 97074 Würzburg, Germany
ccinstitutetext: Würzburg-Dresden Cluster of Excellence ct.qmat
ddinstitutetext: Bethe Center for Theoretical Physics and Physikalisches
Institut der Universitaet Bonn, Nussallee 12, 53115 Bonn, Germany

# Complexity via Replica Trick

Mohsen Alishahiha<EMAIL_ADDRESS>b,c Souvik Banerjee
<EMAIL_ADDRESS>d Joshua Kames-King<EMAIL_ADDRESS>
###### Abstract

We consider the complexity of a single-sided AdS black hole as modelled by an
end-of-the-world brane. In addition we present multi-boundary partition
functions and matter correlation functions for such a setting. We compute the
complexity using a modified replica trick corresponding to the “quenched
geodesic length” in JT gravity. The late time behaviour of complexity shows a
saturation to a constant value of order $e^{S_{0}}$ following a period of
linear growth. Furthermore, we show that our approach leads to an improved
result for the variance of complexity, namely it being time-independent at
late times. We conclude by commenting on the introduction of dynamical end-of-
the-world branes.

##  1 Introduction

In the context of the AdS/CFT correspondence Maldacena:1997re ; Witten:1998qj
; Gubser:1998bc  it is believed that the interior of a black hole may be
systematically studied via the notion of quantum computational complexity.
This field of study quantifies the difficulty of constructing a specific
“target state” by use of a simple set of “universal gates”. More specifically
in a holographic setting it is conjectured that for a chaotic CFT the growth
of complexity has a simple geometric description in terms of the growth of the
black hole interior.

One of the arguments for this conjecture is that for a fast-scrambling system
with finite entropy $S$, complexity is expected to grow for exponentially
large times in the entropy, long after thermal equilibrium has been reached
Susskind:2014rva ; Susskind:2018pmk . Remarkably, the same growth holds for
the black hole interior. Therefore a concrete instantiation of this
conjectured duality is the “complexity=volume” (CV) conjecture, which proposes
that the complexity equals the volume of a maximal slice in the black hole
interior Susskind:2014rva ; Stanford:2014jda . There is also another competing
proposal known as the “complexity=action” (CA) conjecture, in which the on-
shell action on a Wheeler-de Witt patch is determined Brown:2015bva ;
Brown:2015lvg .

We note however, that for chaotic Hamiltonians (as can, for example, be seen
in simple circuit models) after the aforementioned period of growth, at times
$t\sim\left(\mathcal{O}(e^{S})\right)$ we expect saturation to a plateau of
size $C\sim\left(\mathcal{O}(e^{S})\right)$ Susskind:2015toa ; Brown:2016wib ;
Balasubramanian:2019wgd ; Susskind:2020wwe ; Balasubramanian:2021mxo ;
Haferkamp:2021uxo . While semi-classical contributions both in form of the CV
and CA conjectures indeed furnish the period of growth, the saturation to the
plateau, until recently, has been illusive.

To understand complexity better one may study this concept in Jackiw-
Teitelboim (JT) gravity; a theory of two-dimensional dilaton gravity,
including arbitrary genus, hyperbolic Riemann surfaces and therefore also
exponentially small corrections to semi-classical gravity calculations
Teitelboim:1983ux ; Jackiw:1984je ; Maldacena:2016upp ; Engelsoy:2016xyb ;
Saad:2018bqo ; Saad:2019lba . Actually extending the gravitational sector by
including such geometries with an arbitrary number of asymptotic boundaries
and arbitrary genus corrects the partition function to be equivalent to a
specific double-scaled Hermitian matrix integral. This implies that JT gravity
follows RMT universality at late times and therefore exhibits spectral
statistics with a dip-ramp-plateau structure Saad:2018bqo ; Saad:2019lba ;
Saad:2019pqd ; Altland:2020ccq .111For work on the relationship between chaos
universality and Euclidean wormholes in higher dimensions see Belin:2020hea ;
Cotler:2020ugk . By use of the same theory it has also been shown that the
inclusion of higher topologies gives a unitary Page curve Almheiri:2019qdq ;
Penington:2019kki .

Recently, holographic complexity was calculated in JT gravity using the CV
conjecture in Iliesiu:2021ari  where it was shown that including higher genus
geometries (as mentioned above) gives the correct late-time behaviour for
complexity. More precisely, in this paper the authors compute complexity in
terms of a non-perturbative geodesic length in JT gravity as follows

$\langle\ell\rangle=\lim_{\Delta\rightarrow
0}\left\langle\sum_{\gamma}\ell_{\gamma}e^{-\Delta\ell_{\gamma}}\right\rangle_{\text{JT}}\,,$
(1.1)

where $\gamma$ refers to non self-intersecting geodesics, $\Delta$ is a
regulator and $\langle\rangle_{\text{JT}}$ a correlator in JT gravity defined
over arbitrary genus. It is then argued that in practice (1.1) is calculated
by

$\langle\ell(t)\rangle=-\lim_{\Delta\rightarrow
0}\frac{\partial\langle\chi(t)\chi(0)\rangle_{\text{non-
int.}}}{\partial\Delta}\,,$ (1.2)

where $\langle\chi(t)\chi(0)\rangle_{\text{non-int.}}$ is obtained in the
Euclidean JT theory and then analytically continued. Here $\Delta$ is the
scaling dimension of the operator $\chi$. Eq. (1.2) then of course involves
(on surfaces with $g\geq 1$) an infinite number of geodesics which can be
taken care of by the moduli space volume of hyperbolic surfaces
Mirzakhani:2006fta . It was, then, demonstrated that the above definition
results in the following expression for complexity

$\langle\ell(t)\rangle=-\frac{2e^{-S_{0}}}{Z(\beta)}\int_{0}^{\infty}\frac{\langle\rho(s_{1})\rho(s_{2})\rangle}{\bar{s}\sinh(2\pi\bar{s})\omega\sinh(\pi\omega)}\exp\left(-\beta\left(\frac{\bar{s}^{2}}{2}+\frac{\omega^{2}}{8}\right)-i\bar{s}\omega
t\right)\,.$ (1.3)

with the definitions of $\omega=s_{1}-s_{2}\,,\bar{s}=\frac{s_{1}+s_{2}}{2}$
and $s_{1,2}=\sqrt{E_{1,2}}\;$. The quantity (1.3) was called “spectral
complexity” in Iliesiu:2021ari , which can be calculated for any quantum
theory by use of its spectral correlation
$\langle\rho(s_{1})\rho(s_{2})\rangle$.

Due to the usual arguments regarding quantum chaos PhysRevA.43.2046 ;
PhysRevE.50.888 , one would suspect that for chaotic systems, (1.3) would
reduce to RMT predictions at late times. For the case of JT gravity, the
spectral two-point function can be shown to take on the standard RMT sine-
kernel structure Saad:2019lba ; Okuyama:2020ncd ; Altland:2020ccq  by use of
doubly non-perturbative effects. This in turn leads to the aforementioned,
expected behaviour for the quantity $\langle\ell(t)\rangle$: early linear
growth followed by a late-time plateau saturation.

In the present work, we are interested in studying two aspects of complexity
for JT gravity. First, we would like to use an approach which removes the
worrisome behaviour of the variance obtained in Iliesiu:2021ari , as we will
explain in greater detail below. Secondly, we would like to study the
introduction of an end-of-the-world (EOW) brane. Recently, these objects have
played a crucial role in understanding quantum aspects of black holes in a
two-dimensional setting as they can be used to model black hole microstates in
JT gravity Penington:2019kki . Since a black hole with an EOW brane behind the
horizon may be understood as a $\mathcal{Z}_{2}$ quotient of the two-sided
scenario, it corresponds to a pure state Kourkoulou:2017zaj ; Maldacena:2001kr
. However, according to the eigenstate thermalisation hypothesis (ETH)
PhysRevA.43.2046 ; PhysRevE.50.888 , a pure state is in many ways
indistinguishable from a thermal state.

It is also worth mentioning that EOW branes may also be used in a dynamical
manner, which means they appear as loops and are summed over in the path
integral. In this approach they may provide an ingredient in defining a UV
completion of JT gravity and solve the factorisation problem Gao:2021uro ;
Harlow:2018tqv . 222For other approaches to possible non-perturbative
completions of JT gravity see Johnson:2019eik ; Johnson:2020exp ;
Johnson:2022wsr .

Motivated by this, we consider the computation of multi-boundary partition
functions and matter correlation functions in the presence of an EOW brane.
While we adopt the techniques developed in Saad:2019lba  and Yang:2018gdb
respectively, the modified result we obtain due to the presence of the EOW
brane is expected to represent the aforementioned quantities in a single-sided
black hole geometry.

Indeed the main concern of the present paper is the computation of the late
time behaviour of complexity. We define this as the geodesic length connecting
the EOW brane and the asymptotic boundary.333In the Lorentzian picture this
replaces the bridge-to-nowhere of Susskind:2014jwa . More concretely, this is
calculated in JT gravity as a quenched expectation value. The qualitative
behaviour remains the same as in the case of a two-sided black hole, namely,
the complexity grows linearly at late times up to a time $t\sim e^{S_{0}}$ and
subsequently saturates to a constant value. The value of this constant which
is of ${\cal O}(e^{S_{0}})$ depends crucially on the tension of the EOW brane.

Although we adopt the non-perturbative definition444This is non-perturbative
by virtue of an analytic continuation of the Euclidean path integral. of
complexity from Iliesiu:2021ari , we refrain from rewriting it in terms of the
correlators as in (1.2). The reason is, although the quantity structurally
looks similar to the aforementioned correlators, the limits on $\Delta$
appearing in the definition are counter-intuitive and do not agree with the
standard geodesic approximation to the two-point function.

Therefore we rather use a modified version of the replica trick in order to
compute the quenched expectation value of the length of the
geodesic.555Following Iliesiu:2021ari , we only consider non self-intersecting
geodesics. This avoids the aforementioned ambiguity. Moreover using the
definition of variance engendered by the modified replica approach, we observe
time-independent results at late times both for the two-sided and the one-
sided geometries. This is in contrast with the result for the variance
presented in Iliesiu:2021ari  where the complexity is defined in terms of a
two-point function (1.2).

Our paper is organised as follows. We will start by introducing the theory of
interest in section 2. By use of the quantisation procedure in the presence of
a boundary brane Gao:2021uro , we construct various wavefunctions needed in
building up different partition functions and of course the path integral,
which describes the volume of the black hole interior for our setting. In this
section we also consider matrix elements in the geodesic length basis on the
Hilbert space produced by the EOW brane. More specifically, we calculate the
off-diagonal elements showing that while we are describing a pure state, they
still obey the ETH. In sections 3 and 4 we construct the multi- boundary
partition function and the quantum gravity matter correlation functions
respectively. We put the pieces together in section 5, where we compute the
complexity using the definition mentioned above. Then we also consider the
variance of this quantity. We conclude in section 6 with a couple of
interesting questions and comments on work in progress.

##  2 Lorentzian JT gravity with EOW Branes and Wavefunctions

In this section we use the canonical quantisation procedure first introduced
in Harlow:2018tqv , to construct different wavefunction expressions for JT
gravity in the presence of an EOW brane. After reviewing the quantisation
procedure in presence of a boundary brane Gao:2021uro , we generalise the
construction to compute wavefunctions for different configurations of the EOW
brane on the disk and then for the trumpet. These quantities are the essential
building blocks in the calculation of correlation functions as well as
complexity in our setup.

###  2.1 The classical solution

JT gravity is a two-dimensional theory of gravity with the Lorentzian action
Jackiw:1984je ; Teitelboim:1983ux

$\displaystyle
S_{\text{JT}}=\frac{S_{0}}{2\pi}\left(\int\sqrt{-g}R+2\int\sqrt{|h|}K\right)+\int\sqrt{-g}\phi\left(R+2\right)+2\int\sqrt{|h|}\phi\left(K-1\right)\,,$
(2.1)

where the first term is the topological Gauss-Bonnet term and $S_{0}$ is the
ground state entropy. In addition, we add the action of an EOW brane, which is
of the form:

$S_{\text{Brane}}=\mu\int_{\text{Brane}}ds\,,$ (2.2)

with $\mu$ being the brane tension. In two spacetime dimensions, the eq (2.2)
boils down to the action of a particle with mass $\mu$. The overall action is
given by

$S=S_{\text{JT}}+S_{\text{Brane}}\,.$ (2.3)

The corresponding equations of motion are

$R+2=0\,,\quad\nabla_{\mu}\nabla_{\nu}\phi-
g_{\mu\nu}\nabla^{2}\phi+g_{\mu\nu}\phi=0\,.$ (2.4)

At the asymptotic AdS boundary, the boundary conditions are set by fixing the
induced metric and the dilaton value Maldacena:2016upp ; Jensen:2016pah ;
Engelsoy:2016xyb

$ds^{2}|_{\partial
M}=-\frac{dt^{2}}{\epsilon^{2}}\,,\;\;\;\;\;\;\;\;\;\;\;\;\phi|_{\partial
M}=\frac{\phi_{b}}{\epsilon}\,,$ (2.5)

where $\epsilon$ is a holographic renormalisation parameter and we are
interested in the limit $\epsilon\rightarrow 0$. Additionally, at the EOW
brane the following conditions are set Penington:2019kki

$K=0\,,\;\;\;\;\;\;\;\;\;\;\partial_{n}\phi=\mu\,.$ (2.6)

Here $\partial_{n}$ denotes the derivative normal to the EOW brane. The latter
condition is essential in ensuring dynamical gravity on the EOW brane.

###  2.2 Quantisation in presence of a brane

Let us denote the normalised geodesic distance between the AdS boundary and
the EOW brane by $L$. The Hilbert space may be constructed in terms of
$L_{2}$-normalisable functions of $L$. 666This is referred to as the “L-basis”
in Harlow:2018tqv . The choice of this basis avoids the subtlety of defining a
“time operator” whose dual Hamiltonian is bounded from below. Furthermore,
this choice also allows for a full phase space $\mathbb{R}^{2}$ without any
restrictions on the phase space coordinates. As the system may be thought of
as a particle in a Morse potential, the Hamiltonian amounts to Gao:2021uro

$H=\frac{2}{\phi_{b}}\left(\frac{P^{2}}{4}+\mu e^{-L}+e^{-2L}\right)\,,$ (2.7)

such that the Schrödinger equation is given by Harlow:2018tqv ; Gao:2021uro

$\left(-\partial^{2}_{L}+4\mu
e^{-L}+4e^{-2L}\right)\psi_{\mu,E}(L)=2E\;\psi_{\mu,E}(L)\,.$ (2.8)

In going from (2.7) to (2.8) we have set $\phi_{b}=1$ and replaced
$P\rightarrow-i\partial_{L}$. In solving (2.8), we are generally assuming
$\mu>0$. Setting $k^{2}=2E$ and $z=4e^{-L}$ the corresponding normalised
wavefunction Gao:2021uro  is 777Due to the fact that $W_{a,b}=W_{a,-b}$ we are
restricted to $k\geq 0$.

$\psi_{k,\mu}(z)=\sqrt{f_{\mu}(k)}\;\frac{W_{-\mu,ik}(z)}{\sqrt{z}},\;\;\;\;\;\;\;\;{\rm
with}\;\;f_{\mu}(k)=\gamma_{\mu}(k)r(k)\,,$ (2.9)

where we have defined

$\gamma_{\mu}(k)=\left|\Gamma\left(\frac{1}{2}+\mu+ik\right)\right|^{2},\;\;\;\;\;\;\;\;\;\;r(k)=\frac{k\sinh(2\pi
k)}{\pi^{2}}\,.$ (2.10)

The normalisation of $\psi_{k,\mu}(z)$ requires the use of the orthogonality
relation for Whittaker functions of the second kind of imaginary order
Szmytkowski:2009

$\int_{0}^{\infty}\frac{dz}{z^{2}}\;W_{-\mu,ik}(z)\;W_{-\mu,ik^{\prime}}(z)=\frac{1}{f_{\mu}(k)}\delta(k-k^{\prime})\,.$
(2.11)

The quantum mechanical propagator is Gao:2021uro

$\displaystyle G_{\beta}(z_{1},z_{2})=\langle L_{2}|e^{-\beta
H}|L_{1}\rangle=\int dk\,e^{-\frac{\beta
k^{2}}{2}}\,f_{\mu}(k)\,\frac{W_{-\mu,ik}(z_{1})}{\sqrt{z_{1}}}\frac{W_{-\mu,ik}(z_{2})}{\sqrt{z_{2}}}\,.$
(2.12)

Let us now come to a more geometric description in terms of the Euclidean path
integral of JT gravity. In the Euclidean picture, the time coordinate $\tau$
is periodic with $\tau\sim\tau+\beta$. The Euclidean action is given by

$S=-\frac{S_{0}}{2\pi}\left(\int\sqrt{g}R+2\int\sqrt{|h|}K\right)-\int\sqrt{g}\phi\left(R+2\right)-2\int\sqrt{|h|}\phi\left(K-1\right)\,,$
(2.13)

where we set the following boundary conditions for an asymptotic AdS boundary

$ds^{2}|_{\partial
M}=\frac{d\tau^{2}}{\epsilon^{2}}\,,\;\;\;\;\;\;\;\;\;\;\;\phi|_{\partial
M}=\frac{\phi_{b}}{\epsilon}\,.$ (2.14)

Again, the first term of (2.13) is purely topological and accounts for the
Euler characteristic of the Riemann surface $\chi=2-2g-n$, where $g$ is the
genus and $n$ the number of boundaries. The integration over the dilaton
localises the path integral to surfaces of constant negative curvature with an
asymptotic boundary length determined by the boundary conditions (2.14). The
extrinsic curvature term gives a Schwarzian action to the asymptotic boundary
fluctuations on the hyperbolic space Maldacena:2016upp ; Engelsoy:2016xyb .

The complete path integral includes an integral over the moduli of such
surfaces and the boundary fluctuations. Briefly stated, the higher genus
surfaces for one asymptotic boundary may be viewed as consisting of two parts,
namely, one asymptotic boundary of fixed length and a geodesic boundary of
length $b$ and a remaining genus $g$ Riemann surface with geodesic boundary of
the same length $b$. The genus expansion of JT then takes on the form
Saad:2019lba :

$\langle
Z(\beta)\rangle=e^{S_{0}}\hat{Z}_{D}(\beta)+\sum_{g=1}e^{(1-2g)S_{0}}\int_{0}^{\infty}bdbV_{g,1}(b)\hat{Z}_{T}(\beta,b)\,,$
(2.15)

where $V_{g,1}$ is the Weil-Petersson volume of genus $g$ and one geodesic
boundary Mirzakhani:2006fta ; Eynard:2007fi  and the integration over $b$
glues the two parts of the surface together. Here $\hat{Z}_{D}(\beta)$ refers
to the disk topology partition function and $\hat{Z}_{T}(\beta,b)$ to the
“trumpet” partition function Stanford:2017thb ; Saad:2019lba

$\hat{Z}_{D}(\beta)=\frac{e^{\frac{2\pi^{2}}{\beta}}}{\sqrt{2\pi}\beta^{3/2}}\,,\quad\quad\hat{Z}_{T}(\beta,b)=\frac{e^{\frac{-b^{2}}{2\beta}}}{\sqrt{2\pi}\beta^{1/2}}\,.$
(2.16)

This construction can be generalised to $n$ asymptotic boundaries with the
connected contribution being of the form Saad:2019lba :

$\langle
Z(\beta_{1})...Z(\beta_{n})\rangle_{\text{C}}=\sum_{g=0}e^{(2-2g-n)S_{0}}\hat{Z}_{g,n}(\beta_{1},...,\beta_{n})\,,$
(2.17)

with the definition

$\hat{Z}_{g,n}(\beta_{1},...,\beta_{n})=\int_{0}^{\infty}b_{1}db_{1}...b_{n}db_{n}V_{g,n}(b_{1},...,b_{n})\hat{Z}_{T}(\beta_{1},b_{1})...\hat{Z}_{T}(\beta_{n},b_{n})\,.$
(2.18)

Moreover, the hats, $\hat{}$ denote quantities without manifest topological
weighting. Incorporating the latter, one defines
$Z_{D}(\beta)=e^{S_{0}}\hat{Z}_{D}(\beta)$. In our construction, we
additionally consider the addition of an EOW brane via the action (2.2) and
the boundary conditions (2.6). This modifies the partition function as we
explain in the next sections.

At various points we will compute the expectation value of geodesic length in
the Euclidean JT path integral. In contrast to the disk, on hyperbolic
surfaces of genus $g\geq 1$ there are an infinite number of geodesics. Let us
consider the case of non self-intersecting geodesics as in Iliesiu:2021ari .
The moduli space of hyperbolic, bordered Riemann surfaces
$\mathcal{M}_{g,n}(b_{1},...,b_{n})$ comes with a symplectic form, the Weil-
Petersson form $\Omega=\sum_{i=1}^{3g+n-3}db\wedge d\tau$, which in principle
allows the calculation of the corresponding moduli space volume if restricted
to a fundamental domain. Similarly, as first argued for in the $g=1$ case in
Saad:2019pqd , and elaborated upon in Blommaert:2020seb ; Iliesiu:2021ari ,
the integral of the geodesics over moduli space may be calculated by modding
via the mapping class group, which we denote $\text{MCG}_{g,n}$. This leads to
the expression Saad:2019pqd ; Iliesiu:2021ari ; Mirzakhani:2006fta ;
Iliesiu:2021ari

$\int_{\frac{\mathcal{M}_{g,1}}{\text{MCG}_{g,1}}}\Omega\sum_{\gamma}e^{-\Delta\ell_{\gamma}}=e^{-\Delta\ell}\int_{\frac{\mathcal{M}_{g-1,2}}{\text{MCG}_{g-1,2}}}\Omega+\sum_{h\geq
0}e^{-\Delta\ell}\int_{\frac{\mathcal{M}_{h,1}}{\text{MCG}_{h,1}}}\Omega\int_{\frac{\mathcal{M}_{g-h,1}}{\text{MCG}_{g-h,1}}}\Omega\,.$
(2.19)

This formula may be visualised as cutting along the geodesic and considering
the resulting geometries.

###  2.3 The disk wavefunctions

Let us start by quickly revisiting some results we need from the two-sided AdS
system. A natural procedure to prepare the states in the Hilbert space of the
two-sided system is via the Hartle- Hawking construction Harlow:2018tqv ,
which is depicted in fig.1(a).

(a)

(b)

(c)

Figure 1: Three possible disk configurations corresponding to different
wavefunctions. Figure (a) is the wavefunction of the Hartle-Hawking state of
the two-sided AdS system in JT gravity. Figure (b) and figure (c) are two
options in the presence of an EOW brane. While in figure (b) we see a geodesic
connecting the EOW brane to the asymptotic boundary, for figure (c) the
geodesic connects to two different points on the asymptotic boundary. An
orange curve corresponds to the former geodesic and a violet curve to the
latter. Green denotes an AdS boundary and blue an EOW brane, respectively.

We denote the fixed geodesic length between two parts of the AdS boundary by
$\ell$. Then the Hartle-Hawking wavefunction $\Phi_{D,\beta}(\ell)$
corresponds to the integral over all Euclidean geometries with disk topology
and asymptotic AdS boundary of renormalised length $\beta$. Explicitly it
amounts to

$\Phi_{D}(\beta,\ell)=2e^{S_{0}/2}\int_{0}^{\infty}dke^{-\frac{\beta
k^{2}}{2}}\,r(k)\;K_{2ik}(y)\,,$ (2.20)

where $y=4e^{-\frac{\ell}{2}}$. In this formalism the disk partition function
is given as

$\displaystyle
Z_{D}(\beta)=\int_{0}^{\infty}\frac{dy}{y}\;\Phi_{D}(\beta/2,\ell)\;\Phi_{D}(\beta/2,\ell)=\frac{e^{S_{0}}}{2}\int_{0}^{\infty}dk\,e^{-\frac{\beta
k^{2}}{2}}\,r(k)$ (2.21) (2.22)
$\displaystyle\;\;\;\;\;\;\;\;\;=e^{S_{0}}\int_{0}^{\infty}dE\;e^{-\beta
E}\hat{\rho}_{D}(E),$ (2.23)

where $\hat{\rho}_{D}(E)$ is the disk density of states, which is given as
Cotler:2016fpe ; Bagrets:2017pwq ; Stanford:2017thb ; Mertens:2017mtv ;
Kitaev:2018wpr ; Yang:2018gdb

$\hat{\rho}_{D}(E)=\frac{\sinh(2\pi\sqrt{2E})}{2\pi^{2}}\,.$ (2.24)

From (2.21) we see that the wavefunction is normalised in such a way to give
the correct expression for (2.21) and (2.24).

Before moving on to more complicated hyperbolic surfaces, let us now introduce
the EOW brane already in this setting and construct the disk wavefunction in
its presence. We can interpret the resulting wavefunction as the Hartle-
Hawking wavefunction in the $L$ basis for the case of a one-sided black hole.
This wavefunction is associated to a region enclosed by an asymptotically AdS
boundary of renormalised length $\beta$, an EOW brane and a geodesic of length
$L$ connecting them. 888In contrast to the geodesic length connecting two
points on the AdS boundary which we denoted by $\ell$. This configuration is
depicted in fig.1(b).

We will denote the corresponding wavefunction by $\Psi_{D}(\beta,L)$ 999We
denote wavefunctions associated to the two-sided black hole via $\Phi$ and
those in the presence of EOW branes by $\Psi$. which should satisfy

$\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{D}(\beta/2,L)\Psi_{D}(\beta/2,L)=\int_{0}^{\infty}\frac{dz_{1}}{z_{1}}\frac{dz_{2}}{z_{2}}\Psi_{D}(x,L_{1})G_{\beta-2x}(z_{1},z_{2})\Psi_{D}(x,L_{2})\,,$
(2.25)

where the variable of integration is $z=4e^{-L}$. It is straightforward to see
that (2.25) is fulfilled for the following expression

$\Psi_{D}(\beta,L)=\frac{e^{S_{0}/2}}{\sqrt{2}}\int_{0}^{\infty}dk\;e^{-\frac{\beta
k^{2}}{2}}\,\gamma_{\mu}(k)r(k)\;\frac{W_{-\mu,ik}(z)}{\sqrt{z}}\,.$ (2.26)

The disk partition function in the presence of an EOW brane therefore amounts
to

$\displaystyle Z_{D,\mu}(\beta)$
$\displaystyle=\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{D}(\beta/2,L)\Psi_{D}(\beta/2,L)=\frac{e^{S_{0}}}{2}\int_{0}^{\infty}dk\;e^{-\frac{\beta
k^{2}}{2}}\,\gamma_{\mu}(k)r(k)$
$\displaystyle=e^{S_{0}}\int_{0}^{\infty}dE\;e^{-\beta
E}\gamma_{\mu}(E)\hat{\rho}_{D}(E),$ (2.27)

Comparing (2.3) to (2.21) we see that the effect of the EOW brane is
encompassed by an additional $\Gamma$-function expression defined in (2.10).
The above expressions also allow us to calculate the wavefunction
${\Psi}_{{D}}(\zeta_{1},\zeta_{2},\ell)$ for region $I$ depicted in fig. 1(c):
a region enclosed by an EOW brane and a geodesic connecting points on the
asymptotic AdS boundary. This wavefunction can be derived from the
identification,

$Z_{D,\mu}(\beta)=\int_{0}^{\infty}\frac{dy}{y}\;{\Psi}_{{D}}(\zeta_{1},\zeta_{2},\ell)\Phi_{D}(\beta-\zeta_{1}-\zeta_{2},\ell),$
(2.28)

by which, using (2.20), one arrives at

${\Psi}_{{D}}(\zeta_{1},\zeta_{2},\ell)=2e^{S_{0}/2}\int_{0}^{\infty}dk\;e^{-\frac{k^{2}}{2}(\zeta_{1}+\zeta_{2})}\;\gamma_{\mu}(k)r(k)\;K_{2ik}(y)\,.$
(2.29)

###  2.4 The trumpet wavefunctions

The most important ingredients of our study are the wavefunctions on the
trumpets whose asymptotic boundaries are either pinched off by the disk
regions considered in fig.1 or replaced in some parts by the EOW brane.

While more complicated hyperbolic surfaces require the use of Riemann surfaces
with geodesic boundaries, the simplest configuration on the trumpet is
depicted in fig. 2(a).

(a)

(b)

(c)

(d)

Figure 2: Four different possible trumpet geometries corresponding to four
distinct wavefunctions. The closed geodesic boundary is depicted in red. In
figure (a) we see the generalisation of the disk configuration, figure 1(a) to
the trumpet. Figure (b) corresponds to the wavefunction on a trumpet geometry
with both an EOW brane and an asymptotically AdS boundary. Figure (c) shows
the wavefunction of a geodesic connecting two points on an asymptotically AdS
boundary which contains an EOW brane. Lastly, in figure (d) we see a geodesic
connecting EOW brane and AdS boundary on a trumpet geometry.

The corresponding wavefunction $\Phi_{T}(\beta,b,\ell)$ can be realised as the
trumpet wavefunction pinched off by the disk wavefunction shown in fig. 1(a).
This is obtained through the identity

$\Phi_{T}(\beta,b)=\frac{1}{\pi}\int_{0}^{\infty}dk\,\cos(kb)\,e^{-\frac{\beta
k^{2}}{2}}=\int_{0}^{\infty}\frac{dy}{y}\;\Phi_{T}(\beta-x,b,\ell)\;\Phi_{D}(x,\ell)\,,$
(2.30)

which results in

$\Phi_{T}(\beta,b,\ell)=\frac{4e^{-S_{0}/2}}{\pi}\int_{0}^{\infty}dk\;e^{-\frac{\beta
k^{2}}{2}}\,\cos(kb)\;K_{2ik}(y)\,.$ (2.31)

Let us now come to the geometry depicted in fig.2(b). This can be computed by
gluing the above geometry with a region enclosed by a geodesic and EOW brane
as shown in fig.1(c). This yields the wavefunction $\Psi_{T}(\beta,b)$
associated with this diagram

$\Psi_{T}(\beta,b)=\int_{0}^{\infty}\frac{dy}{y}\;\Phi_{T}(\beta,b,\ell)\;{\Psi}_{{D}}(0,0,\ell)=\frac{1}{\pi}\int_{0}^{\infty}dk\;\cos(kb)\;\gamma_{\mu}(k)\,e^{-\frac{\beta
k^{2}}{2}}.$ (2.32)

The wavefunction (2.32) is in a perfect agreement with the corresponding
wavefunction presented in Gao:2021uro .

This in turn allows for the calculation of
$\Psi_{T}(\zeta_{1},\zeta_{2},b,\ell)$, the wavefunction associated with
fig.2(c) and obtained through the equation

$\Psi_{T}(\beta,b)=\int_{0}^{\infty}\frac{dy}{y}\;\Psi_{T}(\zeta_{1},\zeta_{2},b,\ell)\Phi_{D}(\beta-\zeta_{1}-\zeta_{2},\ell)\,.$
(2.33)

Using (2.20) and (2.32), this yields

$\Psi_{T}(\zeta_{1},\zeta_{2},b,\ell)=\frac{4e^{-S_{0}/2}}{\pi}\int_{0}^{\infty}dk\;\cos(kb)\,\gamma_{\mu}(k)\,e^{-\frac{k^{2}}{2}(\zeta_{1}+\zeta_{2})}\,K_{2ik}(y)\,.$
(2.34)

Finally, the wavefunction corresponding to the geometry shown in the panel (d)
of fig. 2, namely a trumpet geometry with geodesic of length $L$ from the EOW
brane to the asymptotic boundary, can be computed by pinching-off the
wavefunction $\Psi_{D}(\beta,L)$ from the above wavefunction. Therefore the
structure

$\Psi_{T}(\beta,b)=\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{T}(\beta-x,b,L)\;\Psi_{D}(x,L)\,,$
(2.35)

by use of (2.26) results in the wavefunction

$\Psi_{T}(\beta,b,L)=\frac{\sqrt{2}e^{-S_{0}/2}}{{\pi}}\int_{0}^{\infty}dk\,\cos(kb)\gamma_{\mu}(k)\,e^{-\frac{\beta
k^{2}}{2}}\,z^{-1/2}\,W_{-\mu,ik}(z)\,.$ (2.36)

###  2.5 Pure vs. Thermal States

As already mentioned in the introduction, by considering an EOW brane we are
describing a pure state. However, to establish its interpretation as a typical
boundary state, it is essential to try and delineate differences to a thermal
state. We can check the expectation value of the energy. Indeed at disk level
this amounts to

$\langle
E\rangle=\frac{\int_{0}^{\infty}\frac{dz}{z}\,\Psi_{D}(\beta/2,L)\,H\,\Psi_{D}(\beta/2,L)}{\int_{0}^{\infty}\frac{dz}{z}\,\Psi_{D}(\beta/2,L)\,\Psi_{D}(\beta/2,L)},$
(2.37)

with $H$ being the Hamiltonian defined in (2.7). As the corresponding system
may be thought of as a particle in a Morse potential, by use of the
Schrödinger equation, one arrives at

$\langle E\rangle=-\frac{\partial}{\partial\beta}\ln Z_{\mu}(\beta)\,,$ (2.38)

which is in agreement with the expectation value of a thermal ensemble with
temperature $\frac{1}{\beta}$. This may be readily generalised to higher
genus. Therefore the wavefunctions in the presence of an EOW brane indeed
correspond to states which are indistinguishable from thermal states.

On the other hand we note that the ETH delineates between diagonal and off-
diagonal matrix elements. More explicitly, the matrix elements of observables
in the eigenstate of the Hamiltonian are given by 1999

$\mathcal{O}_{mn}=\mathcal{O}(\overline{E})\delta_{mn}+e^{-\frac{S(\overline{E})}{2}}f_{\mathcal{O}}\left(\overline{E},\omega\right)R_{mn}\,,$
(2.39)

where $\overline{E}=\frac{E_{m}+E_{n}}{2}$, $\omega=E_{m}-E_{n}$ and
$S(\overline{E})$ is the entropy. Moreover, $\mathcal{O}(\overline{E})$ is the
expectation value in the microcanonical ensemble,
$f_{\mathcal{O}}\left(\overline{E},\omega\right)$ is a smooth function and
$R_{mn}$ a random variable with zero mean and unit variance.

One observes that off-diagonal elements are suppressed by the Hilbert space
size. In order to show that the wavefunction we consider also satisfies ETH,
we need to calculate off-diagonal elements of the inner product in the length
basis $|L\rangle$ used in the quantisation of (2.7). Actually the inner
product we need for this analysis was already considered in Gao:2021uro ,
where the importance of higher topologies was stressed. First we need to
define a building block, which is shown in fig.3.

Figure 3: One important ingredient in the calculation of the leading order
correction to the inner product $\langle L_{1}|L_{2}\rangle$ via Euclidean
path integral. We see the two geodesics $L_{1}$ and $L_{2}$ in orange, the EOW
brane in blue and a closed geodesic $b$ in red. The wavefunction of this
geometry is denoted by $\Psi_{T}(b,L_{1},L_{2})$. Topologies beyond the disk
are important in recovering ETH-like behaviour.

Denoting the corresponding wavefunction by $\Psi_{T}(b,L_{1},L_{2})$, one has

$\Psi_{T}(\beta,b)=\int_{0}^{\infty}\frac{dz_{1}}{z_{1}}\;\frac{dz_{2}}{z_{2}}\Psi_{D}(\beta-x,L_{1})\Psi_{T}(b,L_{1},L_{2})\Psi_{D}(x,L_{2})\,,$
(2.40)

which in combination with the expression (2.26) may be used to find

$\Psi_{T}(b,L_{1},L_{2})=\frac{2e^{-S_{0}}}{\pi}\int_{0}^{\infty}dk\;\cos(kb)\,\gamma_{\mu}(k)\;(z_{1}z_{2})^{-1/2}W_{-\mu,ik}(z_{1})\;W_{-\mu,ik}(z_{2})\,,$
(2.41)

in agreement with the result obtained in Gao:2021uro . The wavefunction (2.41)
plays an important role in recovering ETH behaviour, as the standard canonical
quantisation condition

$\langle L_{1}|L_{2}\rangle=\delta\left(L_{1}-L_{2}\right)\,,$ (2.42)

is corrected via higher genus contributions to the expression

$\langle
L_{1}|L_{2}\rangle=\delta(L_{1}-L_{2})+\int_{0}^{\infty}b\,db\,X(b)\,\Psi_{T}(b,L_{1},L_{2}),$
(2.43)

where we have introduced the notation $X(b)$ as in Gao:2021uro . Here $X(b)$
is an integration measure which corresponds to all topologies ending on a
single closed geodesic length $b$, such that the weighting by the Euler
characteristic and the Weil-Petersson volumes are included in this quantity.
We could also consider it to include an arbitrary number of EOW brane loops as
in Gao:2021uro . By use of (2.41), (2.43) takes on the form

$\langle
L_{1}|L_{2}\rangle=\delta(L_{1}-L_{2})+\frac{2e^{-S_{0}}}{\pi}\int_{0}^{\infty}dk\;\chi(k)\,\gamma_{\mu}(k)\;\frac{W_{-\mu,ik}(z_{1})\;W_{-\mu,ik}(z_{2})}{\sqrt{z_{1}z_{2}}}\,,$
(2.44)

where

$\chi(k)=\int_{0}^{\infty}b\,db\,X(b)\,\cos(kb)\,.$ (2.45)

The leading contribution to the off-diagonal term comes from surfaces with
genus one for which $\chi(k)\sim e^{-S_{0}}$, which results in

$\langle
L_{1}|L_{2}\rangle\approx\delta(L_{1}-L_{2})+(\cdots)_{L_{1},L_{2}}\,e^{-2S_{0}}\,,$
(2.46)

in agreement with Miyaji:2021ktr . Here $(\cdots)_{L_{1},L_{2}}$ refers to the
$g=1$ contribution, where we have already pulled out the topological
weighting. We therefore see that off-diagonal terms are suppressed
exponentially just as in (2.39).

##  3 Partition Function

In this section we construct the partition function in the presence of an EOW
brane via the wavefunction formalism developed in section 2. The most natural
quantity to analyse is the two-point function or the spectral form factor.
More specifically, we require the trumpet wavefunction (2.32). We may
visualise the connected contribution to the two-point function as gluing two
trumpet geometries of the type illustrated in fig.2(a) together along their
closed geodesic boundaries, which results in the geometry shown in fig.4.

Figure 4: Two trumpet geometries glued together along their closed geodesic
boundaries. This geometry corresponds to the connected part of the spectral
form factor.

In analogy to (2.17), the overall contribution including connected and
disconnected structures gives the following expression:

$\displaystyle\langle Z(\beta_{1})Z(\beta_{2})\rangle_{\mu}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}b_{1}db_{1}\;b_{2}db_{2}\;\Psi_{T}(\beta_{1},b_{1})X(b_{1},b_{2})\Psi_{T}(\beta_{2},b_{2})$
(3.1) $\displaystyle=$
$\displaystyle\frac{e^{-S_{0}}}{\pi^{2}}\int_{0}^{\infty}dk_{1}dk_{2}\;e^{-\frac{\beta_{1}k_{1}^{2}}{2}-\frac{\beta_{2}k_{2}^{2}}{2}}\gamma_{\mu}(k_{1})\gamma_{\mu}(k_{2})\int_{0}^{\infty}b_{1}db_{1}b_{2}db_{2}X(b_{1},b_{2}){\cos(k_{1}b_{1})\cos(k_{2}b_{2})}.$

Here we have introduced the function $X(b_{1},b_{2})$ that denotes the
topologically weighted sum over the Weil-Petersson volumes associated to
surfaces with two geodesic boundaries parametrised by $b_{1}$ and $b_{2}$. It
is of the form

$X(b_{1},b_{2}):=\sum_{g=0}e^{(2-2g)S_{0}}\left({V}_{g-1,2}(b_{1},b_{2})+\sum_{a\geq
0}{V}_{g-a,1}(b_{1}){V}_{a,1}(b_{2})\right)\,.$ (3.2)

We note that the first term of (3.2) corresponds to the connected
contribution, whereas the second term corresponds to the disconnected
contribution. There are two contributions in (3) which must be put in “by
hand” as the moduli space volumes $V_{g=0,1}(b)$ and $V_{g=0,2}(b_{1},b_{2})$
in (3.2) are undefined.101010These two volumes constitute input values for the
topological recursion Eynard:2004mh ; Eynard:2007kz . For the disconnected
contributions involving $V_{g=0,1}(b)$, the correct result is given by (2.3)
ands the two boundary $g=0$ connected contribution is defined as

$\displaystyle Z(\beta_{1},\beta_{2})_{g=0,n=2,\mu}$
$\displaystyle=\int_{0}^{\infty}b_{1}db_{1}b_{2}db_{2}\Psi_{T}(\beta_{1},b_{1})\Psi_{T}(\beta_{2},b_{2})\,.$
(3.3)

Comparing (3) to the two-sided expression of Saad:2019lba , one observes that
the distinction to (3) lies in the factor
$\gamma_{\mu}(k_{1})\gamma_{\mu}(k_{2})$. Analytically continuing (3) to the
spectral form factor and rewriting in terms of energy variables one arrives at

$\langle Z(\beta+it)Z(\beta-
it)\rangle_{\mu}=\int_{0}^{\infty}dE_{1}dE_{2}e^{-\beta(E_{1}+E_{2})-it(E_{1}-E_{2})}\;\gamma_{\mu}(E_{1})\gamma_{\mu}(E_{2})\;\langle\rho(E_{1})\rho(E_{2})\rangle$
(3.4)

where

$\langle\rho(E_{1})\rho(E_{2})\rangle=\int
b_{1}db_{1}\,b_{2}db_{2}\,X(b_{1},b_{2})\;\frac{\cos(b_{1}\sqrt{2E_{1}})\cos(b_{2}\sqrt{2E_{2}})}{2\pi^{2}\sqrt{E_{1}E_{2}}},$
(3.5)

which is the density of states corresponding to two boundary case of (2.17).
At late times, the integral (3.4) is dominated by small energy ranges, and it
can be shown that for $|E_{1}-E_{2}|\ll 1$, non-perturbative contributions
give the following expression for (3.5)Saad:2019lba 111111See also
Altland:2020ccq  based on the elegant approach of Wegner ; Efetov:1983xg .

$\langle\rho(E_{1})\rho(E_{2})\rangle\approx
e^{2S_{0}}\hat{\rho}_{D}(E_{1})\hat{\rho}_{D}(E_{2})+e^{S_{0}}\hat{\rho}_{D}(E_{2})\delta(E_{1}-E_{2})-\frac{\sin^{2}\left(\pi
e^{S_{0}}\hat{\rho}_{D}(E_{2})(E_{1}-E_{2})\right)}{\pi^{2}(E_{1}-E_{2})^{2}}\,,$
(3.6)

where ${\hat{\rho}}_{D}(E)$ refers to the genus zero contribution to the
density of states (2.24). The last term in (3.6) is the so-called sine-kernel.
The non-perturbative nature of this contribution can be spotted by noting the
factor of $e^{S_{0}}$ inside the “$\sin$”. As should be expected, plugging
(3.6) into (3.4), gives a ramp-plateau structure for the connected and
decaying behaviour for the disconnected contribution.

##  4 Correlation Functions

Following the procedure of Yang:2018gdb  we will now determine full quantum
gravity expressions for the matter correlation functions in the presence of an
EOW brane. The idea of Yang:2018gdb  is to construct a certain Kernel which
can be used to dress quantum field theory correlation functions on AdS2 to
produce gravity correlators. For the two-sided case, the Kernel essentially
amounts to the Hartle-Hawking wavefunction (2.20). More concretely, let us
denote the coordinates by ${\bf x}=(\xi,x)$, where $\xi$ is the holographic
coordinate and $x$ the boundary coordinate. The regularised geodesic distance
between two points is given by

$e^{\frac{\ell}{2}}=\frac{|x_{1}-x_{2}|}{\sqrt{\xi_{1}\xi_{2}}}.$ (4.1)

In terms of this expression the Kernel is

$K(u_{12},{\bf x}_{1},{\bf
x}_{2})=2e^{S_{0}/2}\frac{4\sqrt{\xi_{1}\xi_{2}}}{|x_{1}-x_{2}|}\int_{0}^{\infty}dke^{-\frac{u_{12}k^{2}}{2}}\,r(k)\;K_{2ik}\left(\frac{4\sqrt{\xi_{1}\xi_{2}}}{|x_{1}-x_{2}|}\right)\,.$
(4.2)

The quantum gravity correlators constructed in Yang:2018gdb  then amount to

$\displaystyle\langle\mathcal{O}_{1}(u_{1})\cdots\mathcal{O}_{n}(u_{n})\rangle_{D}=\int_{x_{1}>\cdots>x_{n}}\frac{\prod_{i}d\xi_{i}dx_{i}}{{\rm
Vol}\left(\text{SL}(2,R)\right)}\\!$ $\displaystyle\\!K(u_{12},{\bf
x}_{1},{\bf x}_{2})\cdots K(u_{1n},{\bf x}_{n},{\bf x}_{1})$
$\displaystyle\\!\times\,\prod_{i}\xi_{i}^{\Delta_{i}-2}\langle\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{n}(x_{n})\rangle_{\rm
CFT}\,,$

where $\Delta_{i}$ is the scaling dimension of the operator $\mathcal{O}_{i}$.
${{\rm Vol}\left(\text{SL}(2,R)\right)}$ reminds us that one needs to fix the
$\text{SL}(2,R)$ gauge symmetry. In our case, while the general logic leading
to the structure of (4) is preserved, now two different Kernels have to be
used. In addition to (4.2), a Kernel must be introduced due to the presence of
the EOW brane. A quick look at fig.5 suggests that this Kernel corresponds to
the wavefunction (2.29), which results in the expression

Figure 5: This figure corresponds to the two-point function of two operators
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ on the Euclidean disk in the presence
of an EOW brane. The corresponding wavefunction is given in (2.29).

$M(\zeta_{1},\zeta_{2},{\bf x}_{1},{\bf
x}_{2})=2e^{S_{0}/2}\frac{4\sqrt{\xi_{1}\xi_{2}}}{|x_{1}-x_{2}|}\int_{0}^{\infty}\\!dk\;e^{-\frac{k^{2}}{2}(\zeta_{1}+\zeta_{2})}\;\gamma_{\mu}(k)r(k)\;K_{2ik}\left(\frac{4\sqrt{\xi_{1}\xi_{2}}}{|x_{1}-x_{2}|}\right)\,.$
(4.4)

Using this kernel and (4.2) the quantum gravity correlators in the presence of
an EOW brane is

$\displaystyle\langle\mathcal{O}_{1}(u_{1})\cdots\mathcal{O}_{n}(u_{n})\rangle_{D,\mu}$
$\displaystyle=\int_{x_{1}>\cdots>x_{n}}\frac{\prod_{i}d\xi_{i}dx_{i}}{{\rm
Vol}\left(\text{SL}(2,R)\right)}K(u_{12},{\bf x}_{1},{\bf x}_{2})\cdots
K(u_{n-1\,n},{\bf x}_{n-1},{\bf x}_{n})$ $\displaystyle\times
M(\zeta_{1},\zeta_{n},{\bf x}_{n},{\bf
x}_{1})\prod_{i}\xi_{i}^{\Delta_{i}-2}\langle\mathcal{O}_{1}(x_{1})\cdots\mathcal{O}_{n}(x_{n})\rangle_{\rm
CFT}.$ (4.5)

The above expressions are for disk topology as indicated by the index $D$. Let
us briefly describe how to generalise to arbitrary topology by use of the two-
point function as a concrete example. For the disk the two-point function is
shown in fig.5. The variables of fig.5 are related to those of formula (4) via
$u=\zeta_{1}+\zeta_{2}$. Keeping in mind that the CFT two-point function is
given by:

$\langle\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})\rangle=e^{-\Delta\ell}\,,$
(4.6)

we arrive at the quantum gravity two-point function at disk level (according
to (4))

$\displaystyle\langle\mathcal{O}_{1}(\zeta_{1}+\zeta_{2})\mathcal{O}_{2}(0)\rangle_{D,\mu}\\!$
$\displaystyle\\!=\\!$
$\displaystyle\\!\int_{0}^{\infty}\frac{dy}{y}\;\Phi_{D}(\beta-\zeta_{1}-\zeta_{2},\ell)\Psi_{D}(\zeta_{1},\zeta_{2},\ell)\;\left(\frac{y}{4}\right)^{2\Delta}$
$\displaystyle\\!=\\!$
$\displaystyle\\!e^{S_{0}}\int_{0}^{\infty}dk_{1}\,dk_{2}\;e^{-\frac{k_{1}^{2}}{2}(\beta-u)-\frac{k_{2}^{2}}{2}u}\;r(k_{1})r(k_{2})\,\gamma_{\mu}(k_{2})\;{\cal
N}(\Delta,k_{1},k_{2})\,,$

where

${\cal
N}(\Delta,k_{1},k_{2})=4\int_{0}^{\infty}\frac{dy}{y}\;K_{2ik_{1}}\left(y\right)K_{2ik_{2}}\left(y\right)\left(\frac{y}{4}\right)^{2\Delta}=\frac{|\Gamma(\Delta+i(k_{1}+k_{2}))\Gamma(\Delta+i(k_{1}-k_{2}))|^{2}}{2^{2\Delta+1}\Gamma(2\Delta)}\,.$
(4.8)

Comparing (4) to the expression for the two-sided AdS black hole Yang:2018gdb
we again see the new factor $\gamma_{\mu}\left(k\right)$ due to the presence
of the EOW brane.

In order to generalise (4) to higher genus, the wavefunctions (2.31) and
(2.34) are needed. By making use of these wavefunctions and formula (2.19) the
two-point function of arbitrary genus is

$\displaystyle\langle\mathcal{O}_{1}(\zeta_{1}+\zeta_{2})\mathcal{O}_{2}(0)\rangle_{\mu}$
$\displaystyle=$ $\displaystyle\int b_{1}db_{1}\,b_{2}db_{2}\,X(b_{1},b_{2})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\times\int_{0}^{\infty}\frac{dy}{y}\;\Phi_{T}(\beta-\zeta_{1}-\zeta_{2},b_{1},\ell)\Psi_{T}(\zeta_{1},\zeta_{2},b_{2},\ell)\;\left(\frac{y}{4}\right)^{2\Delta}$
$\displaystyle=$
$\displaystyle\frac{16e^{-S_{0}}}{\pi^{2}}\int_{0}^{\infty}dk_{1}dk_{2}\;\gamma_{\mu}(k_{2})e^{{-\frac{k_{1}^{2}}{2}(\beta-u)}{-\frac{k_{2}^{2}}{2}u}}{\cal
N}(\Delta,k_{1},k_{2})$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\times\;\int
b_{1}db_{1}\,b_{2}db_{2}\,X(b_{1},b_{2})\;{\cos(k_{2}b_{2})\cos(k_{1}b_{1})}.$

We note, however, that the disk contribution is a particular case and it is
understood that the genus zero contribution is defined to be (4). Altogether
one gets

$\langle\mathcal{O}_{1}(u)\mathcal{O}_{2}(0)\rangle_{\mu}=16e^{-S_{0}}\int_{0}^{\infty}dE_{1}dE_{2}e^{-E_{1}(\beta-u)-E_{2}u}\;\gamma_{\mu}(E_{2})\;\langle\rho(E_{1})\rho(E_{2})\rangle\;{\cal
N}(\Delta,E_{1},E_{2})\,,$ (4.10)

where we are using (3.5). The late-time behaviour of the two-point function
amounts to considering the analytic continuation $u=\beta+it$, which gives

$\langle\mathcal{O}_{1}(t)\mathcal{O}_{2}(0)\rangle_{\mu}=16e^{-S_{0}}\int_{0}^{\infty}dE_{1}dE_{2}e^{-\frac{\beta}{2}(E_{1}+E_{2})+it(E_{1}-E_{2})}\;\gamma_{\mu}(E_{2})\;\langle\rho(E_{1})\rho(E_{2})\rangle\;{\cal
N}(\Delta,E_{1},E_{2})\,.$ (4.11)

Comparing this expression to (3.4) shows that the late-time behaviour is
essentially the same as that of spectral form factor. Indeed as far as the
ramp and the plateau are concerned the extra ${\cal N}(\Delta,E_{1},E_{2})$
plays no essential role.

##  5 The late time behaviour of complexity

In this section we would like to study the late time behaviour of complexity
in our setup. It is conjectured that the holographic quantum complexity is
given by the volume of the Einstein-Rosen bridge Stanford:2014jda . In our
language in two dimensions it translates into the length of a geodesic
connecting two boundaries. This definition was used to compute the late time
behaviour of complexity of a two-sided black hole in Iliesiu:2021ari . In that
work it was shown that the complexity exhibits linear growth at late times
before it eventually saturates to a finite value. As detailed in the
introduction, the most essential step in this construction was the use of the
non-perturbative expression (3.6) to furnish the saturation at late times.

In this section we adopt the same logic to work out the late time behaviour of
complexity for a single-sided black hole. Crucially however, we do not relate
the geodesic length to a matter two-point function but use the quenched
expectation value. For the calculation of the complexity itself this leads to
the same expressions for the two-sided case but a decisively different result
for the variance. For the one-sided case, we need to compute the quenched
expectation value of a geodesic suspended between the AdS boundary and the EOW
brane. Note that, in our notation, classically the geodesic distance between
boundary and EOW brane is denoted by $L=-\ln z/4$. The complexity is therefore
proportional to the expectation value of the geodesic ${\cal C}\sim\langle
L\rangle_{\rm QG}$ in quantum gravity. It is also worth noting that in the
present case one could also compute the expectation value of a geodesic length
connecting two points on the boundary, $\langle\ell\rangle_{QG}$. In what
follows we will study the time dependence of these quantities using the
wavefunction formalism we developed in the previous sections.

###  5.1 The geodesic $\ell$

To proceed, let us start with the geodesic $\ell$ which is used in the two-
sided case and compute its “quantum expectation” value. At the disk level one
has

$\langle\ell(u)\rangle=-\frac{1}{Z_{D,\mu}(\beta)}\int_{0}^{\infty}\frac{dy}{y}\;\Psi_{D}(\zeta_{1},\zeta_{2},\ell)\;\Phi_{D}(\beta-u,\ell)\;(2\ln\frac{y}{4}),\;\;\;\;\;\;\;\;{\rm
with}\;u=\zeta_{1}+\zeta_{2}\,.$ (5.1)

To evaluate this quantity, we will use a trick which is inspired by the
replica trick used e.g. in computing the quenched free energy. We write the
logarithm in terms of the following limit 121212In the context of JT gravity,
see Engelhardt:2020qpv ; Johnson:2020mwi ; Johnson:2021rsh ;
Alishahiha:2020jko .

$\ln A=\lim_{N\rightarrow 0}\frac{A^{N}-1}{N}=\lim_{N\rightarrow
0}\frac{d}{dN}\;A^{N}\,.$ (5.2)

We normalise by multiplying with a factor of ${Z_{D,\mu}^{-1}(\beta)}$, where
$Z_{D,\mu}(\beta)$ is the disk partition function, given in (2.3). Using this
definition one may define complexity as

$\langle\ell(u)\rangle=-\lim_{N\rightarrow 0}\frac{\langle
y^{2N}\rangle_{u}-1}{N},$ (5.3)

where

$\langle
y^{2N}\rangle_{u}=\frac{1}{Z_{D,\mu}(\beta)}\int_{0}^{\infty}\frac{dy}{y}\;\Psi_{D}(\zeta_{1},\zeta_{2},\ell)\;\Phi_{D}(\beta-u,\ell)\;\left(\frac{y}{4}\right)^{2N}$
(5.4)

and it is understood that an analytic continuation must still be performed.
Expressions such as (5.4) may then be calculated via (2.19). It is very
interesting that in this context, the complexity, similar to entanglement
entropy, can also be computed via a replica trick. To be clear, while the
expression (5.4) is calculated in the Euclidean path integral, we have not
explicitly shown the existence of replicated geometries. Perhaps one should
take the validity of (5.4) as an indication on the existence of some kind of
broader approach involving replica geometries. It is also worth noting that
the above expression found by use of a replica trick is identical to the
expression of the matter two-point function (4) with the identification of
$\Delta=N$. However, although they are the same expression, conceptually they
play different roles as (5.4) is used in (5.3). This is where our approach
deviates significantly from Iliesiu:2021ari .

Indeed, it is not clear if one could interpret (5.4) as a matter two-point
function since the corresponding matter two-point function is obtained from an
opposite limit, namely, in the limit of large scaling dimension. On the
contrary, in our case, we need the limit, $N\rightarrow 0$ by which we lose
the semiclassical interpretation of the two-point function. Nonetheless, as
long as the computations are concerned, both yield the same result.

In particular from (4) by use of (2.19) one gets

$\langle
y^{2N}\rangle_{u}=\frac{e^{S_{0}}}{Z_{D,\mu}(\beta)}\int_{0}^{\infty}dk_{1}\,dk_{2}\;e^{-\frac{k_{1}^{2}}{2}(\beta-u)-\frac{k_{2}^{2}}{2}u}\;r(k_{1})r(k_{2})\,\gamma_{\mu}(k_{2})\;{\cal
N}(N,k_{1},k_{2})\,.$ (5.5)

Of course this expression in itself does not yet furnish late time linear
growth as (5.5) is not the end of the story and needs to be plugged into the
replica formula (5.3) and analytically continued to find complexity.
Performing the analytic continuation $u=\frac{\beta}{2}+it$ and using energy
variables we arrive at

$\langle
y^{2N}\rangle_{t}=\frac{e^{S_{0}}}{Z_{D,\mu}(\beta)}\int_{0}^{\infty}dE_{1}\,dE_{2}\;e^{-\frac{\beta}{2}(E_{1}+E_{2})+i(E_{1}-E_{2})t}\;{\hat{\rho}}_{D}(E_{1}){\hat{\rho}}_{D}(E_{2})\,\gamma_{\mu}(E_{2})\;{\cal
N}(N,E_{1},E_{2})\,.$ (5.6)

Now we have to simply plug this equation into the replica formula (5.3).
Moreover since we are interested in the behaviour at late times, the main
contribution should come from the coincident limit, $E_{1}\rightarrow E_{2}$.
In this limit, using the change of variables,

$E=\frac{E_{1}+E_{2}}{2},\;\;\;\;\;\;\;\omega=E_{1}-E_{2},$ (5.7)

one gets

$\langle\ell(t)\rangle\sim{\rm const.}-\frac{e^{S_{0}}}{2\sqrt{2}\pi
Z_{D,\mu}(\beta)}\int_{0}^{\infty}dEe^{-\beta
E}\sqrt{E}{\hat{\rho}}_{D}(E)\gamma_{\mu}(E)\int_{-\infty}^{\infty}d\omega\;\frac{e^{i\omega
t}}{\omega^{2}}\,,$ (5.8)

which results in the linear growth $\langle\ell(t)\rangle\sim t$. Of course,
one still needs to perform the integral over $E$, though we will not do it
here. Here our aim was only to show that the linear growth at the disk level
could be thought of as the consequence of our replica trick. Performing the
calculation of the quenched length on a two-boundary topology and using (3.6)
would lead to the results already obtained in Iliesiu:2021ari  and we will
therefore not do this explicitly.

###  5.2 The geodesic $L$

It is straightforward to compute the late time behaviour of the quantum
expectation value of the length of the geodesic connecting a point on the
boundary to one on the EOW brane

$\langle
L(u)\rangle=-\frac{1}{Z_{D,\mu}(\beta)}\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{D}(\beta-u,L)\,\Psi_{D}(u,L)\,\ln\frac{z}{4}=-\lim_{N\rightarrow
0}\frac{\langle z^{N}\rangle_{u}-1}{N},$ (5.9)

where

$\displaystyle\langle
z^{N}\rangle_{u}=\frac{1}{Z_{D,\mu}(\beta)}\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{D}(\beta-u,L)\,\Psi_{D}(u,L)\,\left(\frac{z}{4}\right)^{N}$
(5.10)
$\displaystyle\;\;\;\;\;\;\;\;\;\;=\frac{e^{S_{0}}}{{2}Z_{D,\mu}(\beta)}\int_{0}^{\infty}dk_{1}\,dk_{2}\;e^{-\frac{k_{1}^{2}}{2}(\beta-u)-\frac{k_{2}^{2}}{2}u}\,\gamma_{\mu}(k_{1})\gamma_{\mu}(k_{2})r(k_{1})r(k_{2})\;{\cal
M}(N,k_{1},k_{2})\,.$

Here we have introduced

${\cal
M}(N,k_{1},k_{2})=\int_{0}^{\infty}\frac{dz}{z^{2}}\;W_{-\mu,ik_{1}}(z)\;W_{-\mu,ik_{2}}(z)\left(\frac{z}{4}\right)^{N}\,.$
(5.11)

At this point, one could perform a computation similar to what was done in the
case of $\langle\ell(u)\rangle$ in the previous section to find the late time
behavior of $\langle L(u)\rangle$. In general, we would expect to get the same
linear growth as before, although in this case we will have to deal with the
Whittaker functions. However, we will postpone this computation for a little
while and will first study the higher genus corrections to the late time
behaviour of complexity. The reason for changing the order of computation is
as follows. The computation of complexity as the quantum expectation value of
the geodesic length at the disk level yields a late time linear growth which
keeps growing forever. However, on general grounds it is expected that
complexity saturates at late times. Therefore the disk level computation
should not constitute the entire story. It is natural to expect that the
inclusion of higher topologies and connected geometries plays an important
role. Thus, in order to see the saturation phase, one needs to compute the
quantum expectation of geodesic length taking into account surfaces of higher
genus Iliesiu:2021ari . By making use of the trumpet wavefunctions we have
found in section 2.4, one has

$\langle L(u)\rangle=-\frac{1}{Z_{\mu}(\beta)}\int
b_{1}db_{1}\;b_{2}db_{2}\;X(b_{1},b_{2})\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{T}(\beta-u,b_{1},L)\;\Psi_{T}(u,b_{2},L)\;\ln\frac{z}{4}\,,$
(5.12)

where we have used the notation (3.2) again. In this case we compute the
following quantity to be used in the replica formula

$\langle z^{N}\rangle_{u}=\frac{1}{Z_{\mu}(\beta)}\int
b_{1}db_{1}\;b_{2}db_{2}\;X(b_{1},b_{2})\int_{0}^{\infty}\frac{dz}{z}\;\Psi_{T}(\beta-u,b_{1},L)\;\Psi_{T}(u,b_{2},L)\;\left(\frac{z}{4}\right)^{N}\,.$
(5.13)

Using equation (2.36) and expression (5.11) one finds

$\displaystyle\langle z^{N}\rangle_{u}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\frac{2e^{-S_{0}}}{{\pi^{2}}Z_{\mu}(\beta)}\int_{0}^{\infty}dk_{1}\,dk_{2}\;e^{-\frac{k_{1}^{2}}{2}(\beta-u)-\frac{k_{2}^{2}}{2}u}\,\gamma_{\mu}(k_{1})\;\gamma_{\mu}(k_{2})$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\int
b_{1}db_{1}\;b_{2}db_{2}\;X(b_{1},b_{2}){\cos(k_{1}b_{1})\cos(k_{2}b_{2})}{\cal
M}(N,k_{1},k_{2})\,,$

which in the energy variable may be reexpressed as

$\langle
z^{N}\rangle_{u}=\frac{2e^{-S_{0}}}{Z_{\mu}(\beta)}\int_{0}^{\infty}dE_{1}\,dE_{2}\;e^{-E_{1}(\beta-u)-E_{2}u}\,\gamma_{\mu}(E_{1})\;\gamma_{\mu}(E_{2})\;\langle\rho(E_{1})\rho(E_{2})\rangle{\cal
M}(N,E_{1},E_{2})\,,$ (5.15)

$\langle\rho(E_{1})\rho(E_{2})\rangle$ being the spectral correlation. The
main part of the above equation is ${\cal M}(N,E_{1},E_{2})$ which is an
integral involving Whittaker functions. This can be evaluated using the
integral identity Gradshteyn2014

$\displaystyle\int_{0}^{\infty}x^{\rho-1}$ $\displaystyle
W_{k,m}(x)W_{\lambda,n}(x)=\frac{\Gamma(2n)\,\Gamma(-m-n+\rho+1)\,\Gamma(m-n+\rho+1)}{\Gamma\left(n-\lambda+\frac{1}{2}\right)\Gamma\left(-k-n+\rho+\frac{3}{2}\right)}$
${}_{3}F_{2}\left(-n-\lambda+\frac{1}{2},-m-n+\rho+1,m-n+\rho+1;1-2n,-k-n+\rho+\frac{3}{2};1\right)$
$\displaystyle+\frac{\Gamma(-2n)\,\Gamma(-m+n+\rho+1)\,\Gamma(m+n+\rho+1)}{\Gamma\left(-n-\lambda+\frac{1}{2}\right)\Gamma\left(-k+n+\rho+\frac{3}{2}\right)}$
${}_{3}F_{2}\left(n-\lambda+\frac{1}{2},-m+n+\rho+1,m+n+\rho+1;2n+1,-k+n+\rho+\frac{3}{2};1\right).$
(5.16)

In order to evaluate the late time behaviour of complexity, the scheme is as
follows. First we need to make the analytic continuation
$u=\frac{\beta}{2}+it$ as before. Then plugging the resulting expression in
the replica formula and taking the $N\rightarrow 0$, limit one can find the
quantum expectation value of the geodesic length or equivalently, the
complexity. Now since we are only interested in late time behaviour, the main
contribution comes from the coincident limit, namely, $E_{1}\rightarrow
E_{2}$. It is convenient to use $E$ and $\omega$ variables as defined in
(5.7). Using (5.2) for our case, we get a nice expansion of the function
${\cal M}(N,E_{1},E_{2})$ in the limit $\omega\rightarrow 0$

$\lim_{N\rightarrow 0}\frac{d}{dN}{\cal
M}(N,E_{1},E_{2})=\frac{\sqrt{2E}}{2\pi\gamma_{\mu}(E){\hat{\rho}}_{D}(E)}\frac{1}{\omega^{2}}+{\rm
local\,terms}\,.$ (5.17)

However, in order to obtain the late time behaviour of complexity, we still
need to perform the integrations over $E$ and $\omega$. Using this and the
replica trick detailed above, one arrives at131313Since we are interested in
the time dependence of complexity, in this expression we have dropped a local
term leading to a time independent term in the complexity and added all terms
into the constant term. The corresponding term is divergent and has the form
of $\delta(\omega)/\omega$.

$\displaystyle\langle L(t)\rangle={\rm const.}-\frac{e^{S_{0}}}{\pi
Z_{\mu}(\beta)}\int_{0}^{\infty}dE\,e^{-\beta
E}\,\sqrt{2E}\gamma_{\mu}(E){\hat{\rho}}_{D}(E)$ (5.18) $\displaystyle\hskip
199.16928pt\times\int_{-\infty}^{\infty}d\omega\,\frac{e^{i\omega
t}}{\omega^{2}}\,\left(1-\frac{\sin^{2}\left(\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\omega\right)}{(\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\omega)^{2}}\right)\,.$

It is worth stressing here that in order to derive the expression given in
(5.18), one needs to take into account the non-perturbative effects explicitly
through the sine-kernel appearing in the spectral correlation given in (3.6)
Saad:2019lba .

It is now clear that the $\omega$-integral may be performed exactly. In
particular the expression in brackets on the right hand side of (5.18)
corresponds to the disk contribution that results in linear growth. As was
observed in Iliesiu:2021ari , the disk linear growth is cancelled by the non-
perturbative term as long as $2\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\ll t$. It is
easy to check that in this regime the integral vanishes identically.

On the other hand for $2\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\gg t$, expanding the
“sin”-contribution in terms of exponentials and deforming the pole one finds
Iliesiu:2021ari

$\int_{-\infty}^{\infty}d\omega\,\frac{e^{i\omega
t}}{\omega^{2}}\,\left(1-\frac{\sin^{2}\left(\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\omega\right)}{(\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\omega)^{2}}\right)=\frac{2\pi^{2}{\hat{\rho}}_{D}(E)e^{S_{0}}}{3}\,\left(1-\frac{t}{2\pi{\hat{\rho}}_{D}(E)e^{S_{0}}}\right)^{3}\,.$
(5.19)

Therefore overall

$\langle L(t)\rangle={\rm const.}-\frac{2\pi
e^{2S_{0}}}{3Z_{\mu}(\beta)}\int_{E_{0}}^{\infty}dE\,e^{-\beta
E}\,\sqrt{2E}\gamma_{\mu}(E){\hat{\rho}}^{2}_{D}(E)\,\left(1-\frac{t}{2\pi{\hat{\rho}}_{D}(E)e^{S_{0}}}\right)^{3}\,.$
(5.20)

Here $E_{0}$ is implicitly obtained via the equation
$\pi{\hat{\rho}}_{D}(E_{0})e^{S_{0}}=t$. Finally we have to perform the
integral over $E$. To proceed, it is instructive to consider particular values
of $\mu$ for which the above expression is simplified further. In what follows
we will consider the case of $\mu=\frac{1}{2}$ as an illustrative example. In
this case using the fact that

$\gamma_{\frac{1}{2}}(E)=\frac{\pi\sqrt{2E}}{\sinh(\pi\sqrt{{2E}})}\,,$ (5.21)

one gets

$\langle L(t)\rangle={\rm
const.}-\frac{4\pi^{2}e^{2S_{0}}}{3Z_{\frac{1}{2}}(\beta)}\int_{E_{0}}^{\infty}dE\,e^{-\beta
E}\,\frac{E\,{\hat{\rho}}_{D}^{2}(E)}{\sinh(\pi\sqrt{{2E}})}\left(1-\frac{t}{2\pi{\hat{\rho}}_{D}(E)e^{S_{0}}}\right)^{3}\,,$
(5.22)

where

$Z_{\frac{1}{2}}(\beta)=e^{S_{0}}\int_{0}^{\infty}dE\,e^{-\beta
E}\gamma_{\frac{1}{2}}(E){\hat{\rho}}_{D}(E)=\frac{e^{\pi^{2}/2\beta}e^{S_{0}}}{\sqrt{2\pi}\beta^{3/2}}\left(1+\frac{\pi^{2}}{\beta}\right)\,.$
(5.23)

For times $t\ll e^{S_{0}}$ one may expand the r.h.s of (5.22) and evaluate the
integral which at leading order takes the form

$\langle L(t)\rangle\approx{\rm const.}-C_{0}e^{S_{0}}+C_{1}\,t\,,$ (5.24)

where

$\displaystyle C_{0}$ $\displaystyle=$
$\displaystyle\frac{\pi^{2}+3\beta+9e^{\frac{4\pi^{2}}{\beta}}\left(\beta+3\pi^{2}\right)}{6\beta\left(\beta+\pi^{2}\right)}\,,$
(5.25) $\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}e^{-\frac{\pi^{2}}{2\beta}}\sqrt{\beta}\left(2\beta+\pi^{2}\right)+\pi^{3/2}\left(3\beta+\pi^{2}\right)\text{erf}\left(\frac{\pi}{\sqrt{2}\sqrt{\beta}}\right)}{\sqrt{\pi}\beta\left(\beta+\pi^{2}\right)}\,.$
(5.27)

For large $t$ ( $t\sim e^{S}$) the lower limit of the integral becomes large
as well; $E_{0}\rightarrow\infty$. Taking into account that the integrand
itself has a factor of $e^{-\beta E}$ results in the fact that the integral
decays and therefore the quantum expectation value of the geodesic length
becomes constant. This can be interpreted as the saturation of complexity. For
large $t$, one can estimate the rate by which the integral decays. For large
$t$ the lower limit of integral reads $E_{0}=\frac{1}{8\pi^{2}}\ln^{2}(2\pi
e^{-S_{0}}t)$. In this limit, approximating the “sinh” by an exponential
function one arrives at

$\langle L(t)\rangle\approx{\rm
const.}-\frac{2\beta^{3/2}e^{-\frac{\pi^{2}}{2\beta}}}{3\pi^{2}\left(\beta+\pi^{2}\right)}\,e^{S_{0}}\,e^{-\frac{\beta}{8\pi^{2}}\ln^{2}(2\pi
e^{-S_{0}}t)}\,\left(e^{-S_{0}}t\right)^{3/2}\,\ln^{2}(2\pi e^{-S_{0}}t)\,.$
(5.28)

To summarise, our computation shows that the complexity grows linearly at late
times up to $t\sim e^{S_{0}}$ and then saturates to a constant value of order
$e^{S_{0}}$. Although we have demonstrated this behaviour explicitly only for
a particular value of $\mu$, the qualitative late time behaviour of complexity
is the same for any value of $\mu$.

###  5.3 The variance of complexity

Although the results of section 5 and the results of Iliesiu:2021ari  exhibit
late time behaviour in line with general expectations for complexity, this can
be probed further by calculating the variance $\sigma$. Based on the procedure
of computing the complexity in terms of the boundary-to-boundary two-point
function, the variance of complexity has been evaluated in Iliesiu:2021ari
where it was observed that the fluctuations exhibit linear growth at late
times that is in tension with general expectations. In particular, this
becomes especially problematic as the “noise” grows to the same size as the
“signal” at $t\sim e^{2S_{0}}$.

Here we would like to use our approach based on the replica trick to compute
the variance. To proceed let us focus on the two-sided case first to draw a
direct comparison. Its generalisation to the one-sided case is then evident.

The variance has the structure

$\sigma_{\ell}^{2}=\langle\ell^{2}(u)\rangle-\langle\ell(u)\rangle^{2}=\langle\ell^{2}(u)\rangle_{\text{C}}\,,$
(5.29)

where we denote the connected contribution by C. Now in line with the rest of
this section, it is clear that the quantity we have to determine is

$\langle\ell^{2}(u)\rangle_{\text{C}}=\frac{4}{Z(\beta)}\int_{0}^{\infty}db_{1}b_{1}db_{2}b_{2}X(b_{1},b_{2})\int_{0}^{\infty}\frac{dy}{y}\Phi_{T}(\beta-u,b_{1},\ell)\Phi_{T}(u,b_{2},\ell)\left(\ln\frac{y}{4}\right)^{2}\,.$
(5.30)

In order to calculate this we have to apply a replica type formula. We utilise
the simple relation

$\ln^{2}A=\lim_{N\rightarrow 0}\frac{d^{2}}{dN^{2}}\;A^{N}\,.$ (5.31)

By which the equation (5.30) may be recast into the following form

$\displaystyle\langle\ell^{2}(u)\rangle_{\text{C}}=\frac{1}{Z(\beta)}\lim_{N\rightarrow
0}\frac{d^{2}}{dN^{2}}\int_{0}^{\infty}b_{1}db_{2}\;b_{2}db_{2}X(b_{1},b_{2})$
(5.32) $\displaystyle\hskip
170.71652pt\times\int_{0}^{\infty}\frac{dy}{y}\Phi_{T}(\beta-u,b_{1},\ell)\Phi_{T}(u,b_{2},\ell)\left(\frac{y}{4}\right)^{2N}\,.$

This of course has a structure similar to the calculations of sections 5.1 and
5.2 and it is therefore clear that by making use of the trumpet wavefunction
(2.31) one arrives at

$\langle\ell^{2}(u)\rangle_{\text{C}}=\frac{4e^{-S_{0}}}{Z(\beta)}\int_{0}^{\infty}dE_{1}\;dE_{2}e^{-E_{1}(\beta-u)-E_{2}u}\langle\rho(E_{1})\rho(E_{2})\rangle\left(\lim_{N\rightarrow
0}\frac{d^{2}}{dN^{2}}{\cal N}(N,E_{1},E_{2})\right)\,,$ (5.33)

which we analytically continue to

$\langle\ell^{2}(t)\rangle_{\text{C}}=\frac{4e^{-S_{0}}}{Z(\beta)}\int_{0}^{\infty}dE\int_{-\infty}^{\infty}\;d\omega
e^{-\beta E+i\omega
t}\langle\rho(E+\frac{\omega}{2})\rho(E-\frac{\omega}{2})\rangle\left(\lim_{N\rightarrow
0}\frac{d^{2}}{dN^{2}}{\cal N}(N,E,\omega)\right)\,,$ (5.34)

where we are using the coordinates (5.7). At late times, taking the limit
$\omega\rightarrow 0$, we have

$\lim_{N\rightarrow 0}\frac{d^{2}}{dN^{2}}{\cal
N}(N,E,\omega)=\frac{\sqrt{E}}{8\pi{\hat{\rho}}_{D}(E)}\;\left(\psi(2i\sqrt{2E})+\psi(-2i\sqrt{2E})-\ln
4\right)\;\frac{1}{\omega^{2}}+{\cal O}(\omega^{0})\,,$ (5.35)

where we have introduced the Polygamma function $\psi(x)$. This may then be
used together with (3.6) to arrive at the final result

$\displaystyle\langle\ell^{2}(t)\rangle_{\text{C}}=\frac{e^{S_{0}}}{2\pi
Z_{D}(\beta)}\int_{0}^{\infty}dEe^{-\beta
E}(\psi(2i\sqrt{2E})+\psi(-2i\sqrt{2E})-\ln 4){\hat{\rho}}_{D}(E)\sqrt{E}$
(5.36) (5.37)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\times\int_{-\infty}^{\infty}d\omega\frac{e^{i\omega
t}}{\omega^{2}}\left(1-\frac{\sin^{2}(\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\omega)}{(\pi{\hat{\rho}}_{D}(E)e^{S_{0}}\omega)^{2}}\right)\,.$
(5.38)

We can see that the $\omega$ integration is of the same form as the one which
appears in the calculation of the complexity itself.141414Note that in this
expression we have not considered a contact term that is proportional to a
delta function. As we mentioned in the calculation of complexity, this term
being of the form of $\delta(\omega)/\omega$ leads to a time independent term
which does not contribute to complexity growth. In the present case this term
gives a divergent term which could be removed by subtracting $\ell(0)$.
Although it is important to consider this term in the computation of variance,
since our aim was to show how the replica trick results in a reasonable
variance, we have just considered $\ell^{2}(t)$ and dropped the corresponding
term by hand. Indeed the only difference is the additional Polygamma
structure. This is a pleasing result. The expression (5.36) circumvents the
problematic late time growth of noise observed in Iliesiu:2021ari . The result
saturates to a constant value and we therefore recover time-independent
fluctuations before the recurrence time. We also observe that (5.36) implies a
signal-to-noise ratio of order $\sim e^{-\frac{S_{0}}{2}}$ at $t\sim
e^{S_{0}}$.

For the one-sided black hole the procedure is the same and indeed we recover a
similar expression with a rather more complicated $E$-dependent function that
comes from the fact that

$\lim_{N\rightarrow 0}\frac{d^{2}}{dN^{2}}{\cal
M}(N,E,\omega)=\frac{1}{{\hat{\rho}}_{D}(E)}\frac{F(E,\mu)}{\omega^{2}}+{\cal
O}(\omega^{0})\,,$ (5.39)

where we introduced $F(E,\mu)$, which is a complicated function of $E$ and
$\mu$ containing hypergeometric and polygamma functions and their derivatives.

##  6 Conclusion and Outlook

In this work, we have considered a fixed EOW brane which plays the role of a
cutoff by removing a part of boundary. This setup provides a holographic model
for a one-sided black hole. We have computed the multi-boundary partition
functions and the matter correlation functions in this model. However, the
most important result in this work is the computation of complexity.

To compute complexity we have employed a modified version of the well-known
replica trick used to study the quenched free energy. This avoids the
ambiguity of defining complexity in terms of boundary-to-boundary correlation
functions as advocated for in Iliesiu:2021ari . The tension between the limit
of scaling dimensions and the geodesic approximation is therefore not present
in this work. We have retrieved the expected non-perturbative plateau regime
in the late time growth of complexity, which follows an early period of
perturbative linear growth in time. Although the result is qualitatively
similar to that of a two-sided black hole, except for the coefficients being
sensitive to the tension of the EOW brane now, the replica trick employed in
our work yields a more satisfactory result for the variance. The emergence of
only time-independent fluctuations in the variance compared to the late-time
linear growth of Iliesiu:2021ari  would seem an advancement in the calculation
of the black hole volume in JT gravity. Of course in our approach the
geometric picture is less obvious.

We will now conclude with a couple of interesting and related questions which
are in progress.

#### Dynamical EOW branes

So far we have considered a fixed EOW brane without any associated dynamics.
However, it is interesting to consider a dynamical EOW brane. This requires
considering a certain EOW brane that contributes to the path integral. In
other words, one could imagine a general hypersurface with some of geodesics
capped by EOW branes.

To start with we can start with a toy model where the geodesic of a trumpet
geometry is capped off by an Fateev-Zamolodchikov-Zamolodchikov-Teschner
(FZZT) anti-brane Fateev:2000ik ; Teschner:2000md  as shown in fig. 6.
Following the prescription of Okuyama:2021eju , what we need to do is insert a
factor of $-\frac{1}{b}e^{-\xi b}$ in the path integral on a trumpet with
parameter $b$.

In order to see the effect of this brane on the behaviour of complexity as a
function of time, following the procedure we adopted for the EOW brane, one
first needs to construct the corresponding wavefunction in presence of the
FZZT anti-brane. In what follows, for simplicity, we shall consider two-sided
black holes. Starting from $\Phi_{T}(\beta,b,\ell)$ given in (2.31), one can
compute the wavefunction associated with fig. 6 as

$\Phi_{T}(\beta,\ell)=-\int_{0}^{\infty}db\,e^{-\xi
b}\,\Phi_{T}(\beta,b,\ell)=-\frac{4e^{-S_{0}/2}}{\pi}\int_{0}^{\infty}dk\;e^{-\frac{\beta
k^{2}}{2}}\,\frac{\xi}{\xi^{2}+k^{2}}K_{2ik}(y).$ (6.1)

Figure 6: Trumpet capped by the FZZT brane shown by a brown circle and
parametrized by $\xi$.

With this result in hand, we need to employ our modified replica method
defined through (5.3) which yields, at late time,

$\displaystyle\langle\ell(t)\rangle={\rm
Const.}+\frac{1}{\pi^{3}{\tilde{Z}}(\beta)}\int_{0}^{\infty}\frac{dE}{2E}\,e^{-\beta
E}\,$
$\displaystyle\bigg{[}\pi\xi\,\kappa\frac{2\sqrt{2E}{\hat{\rho}}_{D}(E)}{\xi^{2}+2E}-\frac{\kappa^{2}\xi^{2}e^{-S_{0}}}{(\xi^{2}+2E)^{2}}\bigg{]}$
(6.2)
$\displaystyle\times\frac{\sqrt{2E}}{{\hat{\rho}}_{D}(E)}\,\int_{-\infty}^{\infty}d\omega\frac{e^{i\omega
t}}{\omega^{2}},$

where $\kappa$ is the number of FZZT anti-branes. Since we are only interested
in the late time behaviour, we have used the $E$ and $\omega$ variables (5.7)
in the coincident limit, $E_{1}\rightarrow E_{2}$.

The $\omega$-integral of (6.2) can be readily performed and yields

$\displaystyle\langle\ell(t)\rangle={\rm
Const.}-\frac{t}{\pi^{2}{\tilde{Z}}(\beta)}\int_{0}^{\infty}\frac{dE}{2E}\,e^{-\beta
E}\,\bigg{[}\pi\xi\,\kappa\frac{2\sqrt{2E}{\hat{\rho}}_{D}(E)}{\xi^{2}+2E}-\frac{\kappa^{2}\xi^{2}e^{-S_{0}}}{(\xi^{2}+2E)^{2}}\bigg{]}\frac{\sqrt{E}}{{\hat{\rho}}_{D}(E)}.$
(6.3)

From (6.3) it is clear that whether the above contribution results in a
decreasing or increasing behaviour of complexity at late times depends on the
$E$ integral. Note that the disk contribution is proportional to $e^{S_{0}}$
whereas the above contribution is given in terms of the number of branes
$\kappa$, therefore one might naively expect an interesting competition
between $\kappa$ and $e^{S_{0}}$ that is similar to that of entanglement
entropy. We hope to report the final conclusion, both for the two-sided and
one-sided black hole geometries, soon inprogress . We expect this computation
to shed light on the physical interpretation of the replica procedure we
employed to compute complexity.

#### UV cutoff

In this paper we discussed EOW branes playing the role of cutoffs. In the
Lorentzian version of the theory, the cutoff EOW brane lies behind the event
horizon of the black hole. In holographic theories, there is an interesting
correspondence between a UV cutoff near the boundary of AdS spacetime and a
conformal field theory deformed by a particular irrelevant operator quadratic
in the stress-energy tensor McGough:2016lol ; Taylor:2018xcy ; Hartman:2018tkw
, namely, the $T{\bar{T}}$ deformation Zamolodchikov:2004ce ; Smirnov:2016lqw
; Cavaglia:2016oda . The wavefunction technique we used for the EOW brane will
also be useful in computing complexity for a $T{\bar{T}}$-deformed CFT.

The partition function of $T{\bar{T}}$ deformed JT gravity may be written as
Iliesiu:2020zld

$Z_{D,\lambda}(\beta)=\int_{-\infty}^{\infty}dE\;e^{-\beta
f(E)}{\hat{\rho}}_{D}(E),$ (6.4)

where $f(E)=\frac{1-\sqrt{1-8\lambda E}}{4\lambda}$, $\lambda$ is the
deformation parameter and $E$, the energy of the undeformed theory. Clearly
for $\lambda\rightarrow 0$ one finds the standard partition function.

Our aim is to compute the complexity for this deformed version of JT gravity.
As mentioned above, we will use the wavefunction formalism. To do so, one
needs to write down the corresponding disk wave function for the deformed
theory. Using the formalism developed in Iliesiu:2020zld  for $\lambda<0$ one
can easily find the deformed wavefunction as

$\displaystyle\Phi_{D,\lambda}(\beta,\ell)=4e^{S_{0}/2}\int_{0}^{\infty}dEe^{-\beta
f(E)}\,{\hat{\rho}}_{D}(E)\,K_{2i\sqrt{2E}}(y)\,.$ (6.5)

which exactly reproduces the partition function (6.4).

Once we have the wavefunction (6.5), we can once again use the modified
replica method (5.3) to compute complexity. In the late time limit, using the
coincident variables (5.7), we obtain

$\langle\ell(t)\rangle\sim{\rm const.}-\frac{2e^{S_{0}}}{\sqrt{2}\pi
Z_{\lambda}(\beta)}\int_{0}^{\infty}dEe^{-\beta
f(E)}\,\sqrt{E}\,{\hat{\rho}}_{D}(E)\,\int_{-\infty}^{\infty}d\omega\;\frac{e^{\frac{it\omega}{\sqrt{1-8\lambda
E}}}}{\omega^{2}}\,.$ (6.6)

The integral over $\omega$ can be performed exactly and we arrive at the
following expression at late time showing linear growth of complexity, as
expected from the disk level computation.

$\langle\ell(t)\rangle\sim{\rm
const.}+\frac{2e^{S_{0}}t}{\sqrt{2}Z_{\lambda}(\beta)}\int_{0}^{\infty}dEe^{-\beta
f(E)}\,\frac{\sqrt{E}\,{\hat{\rho}}_{D}(E)}{\sqrt{1-8\lambda E}}\,.$ (6.7)

While obtaining the plateau regime of complexity in this setup can be done
straightforwardly by adding higher genus contributions as before, it will be
interesting to study the saturation of complexity in this deformed JT setup in
presence of an EOW brane. We postpone this for future work.

###### Acknowledgements.

We would like to thank Moritz Dorband, Johanna Erdmenger and Misha Usatyuk for
helpful discussions.

## References

  * (1) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (2) E. Witten, _Anti-de Sitter space and holography_ , _Adv. Theor. Math. Phys._ 2 (1998) 253 [hep-th/9802150].
  * (3) S.S. Gubser, I.R. Klebanov and A.M. Polyakov, _Gauge theory correlators from noncritical string theory_ , _Phys. Lett. B_ 428 (1998) 105 [hep-th/9802109].
  * (4) L. Susskind, _Computational Complexity and Black Hole Horizons_ , _Fortsch. Phys._ 64 (2016) 24 [1403.5695].
  * (5) L. Susskind, _Three Lectures on Complexity and Black Holes_ , SpringerBriefs in Physics, Springer, 10, 2018, DOI [1810.11563].
  * (6) D. Stanford and L. Susskind, _Complexity and Shock Wave Geometries_ , _Phys. Rev. D_ 90 (2014) 126007 [1406.2678].
  * (7) A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Holographic Complexity Equals Bulk Action?_ , _Phys. Rev. Lett._ 116 (2016) 191301 [1509.07876].
  * (8) A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, _Complexity, action, and black holes_ , _Phys. Rev. D_ 93 (2016) 086006 [1512.04993].
  * (9) L. Susskind, _The Typical-State Paradox: Diagnosing Horizons with Complexity_ , _Fortsch. Phys._ 64 (2016) 84 [1507.02287].
  * (10) A.R. Brown, L. Susskind and Y. Zhao, _Quantum Complexity and Negative Curvature_ , _Phys. Rev. D_ 95 (2017) 045010 [1608.02612].
  * (11) V. Balasubramanian, M. Decross, A. Kar and O. Parrikar, _Quantum Complexity of Time Evolution with Chaotic Hamiltonians_ , _JHEP_ 01 (2020) 134 [1905.05765].
  * (12) L. Susskind, _Black Holes at Exp-time_ , 2006.01280.
  * (13) V. Balasubramanian, M. DeCross, A. Kar, Y.C. Li and O. Parrikar, _Complexity growth in integrable and chaotic models_ , _JHEP_ 07 (2021) 011 [2101.02209].
  * (14) J. Haferkamp, P. Faist, N.B.T. Kothakonda, J. Eisert and N.Y. Halpern, _Linear growth of quantum circuit complexity_ , 2106.05305.
  * (15) C. Teitelboim, _Gravitation and Hamiltonian Structure in Two Space-Time Dimensions_ , _Phys. Lett. B_ 126 (1983) 41.
  * (16) R. Jackiw, _Lower Dimensional Gravity_ , _Nucl. Phys. B_ 252 (1985) 343.
  * (17) J. Maldacena, D. Stanford and Z. Yang, _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ , _PTEP_ 2016 (2016) 12C104 [1606.01857].
  * (18) J. Engelsöy, T.G. Mertens and H. Verlinde, _An investigation of AdS 2 backreaction and holography_, _JHEP_ 07 (2016) 139 [1606.03438].
  * (19) P. Saad, S.H. Shenker and D. Stanford, _A semiclassical ramp in SYK and in gravity_ , 1806.06840.
  * (20) P. Saad, S.H. Shenker and D. Stanford, _JT gravity as a matrix integral_ , 1903.11115.
  * (21) P. Saad, _Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity_ , 1910.10311.
  * (22) A. Altland and J. Sonner, _Late time physics of holographic quantum chaos_ , _SciPost Phys._ 11 (2021) 034 [2008.02271].
  * (23) A. Belin and J. de Boer, _Random statistics of OPE coefficients and Euclidean wormholes_ , _Class. Quant. Grav._ 38 (2021) 164001 [2006.05499].
  * (24) J. Cotler and K. Jensen, _AdS 3 gravity and random CFT_, _JHEP_ 04 (2021) 033 [2006.08648].
  * (25) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, _Replica Wormholes and the Entropy of Hawking Radiation_ , _JHEP_ 05 (2020) 013 [1911.12333].
  * (26) G. Penington, S.H. Shenker, D. Stanford and Z. Yang, _Replica wormholes and the black hole interior_ , 1911.11977.
  * (27) L.V. Iliesiu, M. Mezei and G. Sárosi, _The volume of the black hole interior at late times_ , 2107.06286.
  * (28) M. Mirzakhani, _Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces_ , _Invent. Math._ 167 (2006) 179.
  * (29) J.M. Deutsch, _Quantum statistical mechanics in a closed system_ , _Phys. Rev. A_ 43 (1991) 2046.
  * (30) M. Srednicki, _Chaos and quantum thermalization_ , _Phys. Rev. E_ 50 (1994) 888.
  * (31) K. Okuyama and K. Sakai, _Multi-boundary correlators in JT gravity_ , _JHEP_ 08 (2020) 126 [2004.07555].
  * (32) I. Kourkoulou and J. Maldacena, _Pure states in the SYK model and nearly- $AdS_{2}$ gravity_, 1707.02325.
  * (33) J.M. Maldacena, _Eternal black holes in anti-de Sitter_ , _JHEP_ 04 (2003) 021 [hep-th/0106112].
  * (34) P. Gao, D.L. Jafferis and D.K. Kolchmeyer, _An effective matrix model for dynamical end of the world branes in Jackiw-Teitelboim gravity_ , _JHEP_ 01 (2022) 038 [2104.01184].
  * (35) D. Harlow and D. Jafferis, _The Factorization Problem in Jackiw-Teitelboim Gravity_ , _JHEP_ 02 (2020) 177 [1804.01081].
  * (36) C.V. Johnson, _Nonperturbative Jackiw-Teitelboim gravity_ , _Phys. Rev. D_ 101 (2020) 106023 [1912.03637].
  * (37) C.V. Johnson, _Explorations of nonperturbative Jackiw-Teitelboim gravity and supergravity_ , _Phys. Rev. D_ 103 (2021) 046013 [2006.10959].
  * (38) C.V. Johnson, _The Microstate Physics of JT Gravity and Supergravity_ , 2201.11942.
  * (39) Z. Yang, _The Quantum Gravity Dynamics of Near Extremal Black Holes_ , _JHEP_ 05 (2019) 205 [1809.08647].
  * (40) L. Susskind and Y. Zhao, _Switchbacks and the Bridge to Nowhere_ , 1408.2823.
  * (41) K. Jensen, _Chaos in AdS 2 Holography_, _Phys. Rev. Lett._ 117 (2016) 111601 [1605.06098].
  * (42) R. Szmytkowski and S. Bielski, _An orthogonality relation for the whittaker functions of the second kind of imaginary order_ , 0910.1492.
  * (43) B. Eynard and N. Orantin, _Weil-Petersson volume of moduli spaces, Mirzakhani’s recursion and matrix models_ , 0705.3600.
  * (44) D. Stanford and E. Witten, _Fermionic Localization of the Schwarzian Theory_ , _JHEP_ 10 (2017) 008 [1703.04612].
  * (45) A. Blommaert, _Dissecting the ensemble in JT gravity_ , 2006.13971.
  * (46) J.S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S.H. Shenker et al., _Black Holes and Random Matrices_ , _JHEP_ 05 (2017) 118 [1611.04650].
  * (47) D. Bagrets, A. Altland and A. Kamenev, _Power-law out of time order correlation functions in the SYK model_ , _Nucl. Phys. B_ 921 (2017) 727 [1702.08902].
  * (48) T.G. Mertens, G.J. Turiaci and H.L. Verlinde, _Solving the Schwarzian via the Conformal Bootstrap_ , _JHEP_ 08 (2017) 136 [1705.08408].
  * (49) A. Kitaev and S.J. Suh, _Statistical mechanics of a two-dimensional black hole_ , _JHEP_ 05 (2019) 198 [1808.07032].
  * (50) M. Srednicki, _The approach to thermal equilibrium in quantized chaotic systems_ , _Journal of Physics A: Mathematical and General_ 32 (1999) 1163–1175.
  * (51) M. Miyaji, T. Takayanagi and T. Ugajin, _Spectrum of End of the World Branes in Holographic BCFTs_ , _JHEP_ 06 (2021) 023 [2103.06893].
  * (52) B. Eynard, _Topological expansion for the 1-Hermitian matrix model correlation functions_ , _JHEP_ 11 (2004) 031 [hep-th/0407261].
  * (53) B. Eynard and N. Orantin, _Invariants of algebraic curves and topological expansion_ , _Commun. Num. Theor. Phys._ 1 (2007) 347 [math-ph/0702045].
  * (54) F. Wegner, _The mobility edge problem: Continuous symmetry and a conjecture_ , _Zeitschrift fur Physik B Condensed Matter_ 35 (1979) 207.
  * (55) K.B. Efetov, _Supersymmetry and theory of disordered metals_ , _Adv. Phys._ 32 (1983) 53.
  * (56) N. Engelhardt, S. Fischetti and A. Maloney, _Free energy from replica wormholes_ , _Phys. Rev. D_ 103 (2021) 046021 [2007.07444].
  * (57) C.V. Johnson, _Low Energy Thermodynamics of JT Gravity and Supergravity_ , 2008.13120.
  * (58) C.V. Johnson, _On the Quenched Free Energy of JT Gravity and Supergravity_ , 2104.02733.
  * (59) M. Alishahiha, A. Faraji Astaneh, G. Jafari, A. Naseh and B. Taghavi, _Free energy for deformed Jackiw-Teitelboim gravity_ , _Phys. Rev. D_ 103 (2021) 046005 [2010.02016].
  * (60) I.S. Gradshteyn, I.M. Ryzhik, D. Zwillinger and V. Moll, _Table of integrals, series, and products; 8th ed._ , Academic Press, Amsterdam (Sep, 2014), 0123849330.
  * (61) V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, _Boundary Liouville field theory. 1. Boundary state and boundary two point function_ , hep-th/0001012.
  * (62) J. Teschner, _Remarks on Liouville theory with boundary_ , _PoS_ tmr2000 (2000) 041 [hep-th/0009138].
  * (63) K. Okuyama and K. Sakai, _FZZT branes in JT gravity and topological gravity_ , _JHEP_ 09 (2021) 191 [2108.03876].
  * (64) M. Alishahiha, S. Banerjee and J. Kames-King, _To appear soon_ , .
  * (65) L. McGough, M. Mezei and H. Verlinde, _Moving the CFT into the bulk with $T\overline{T}$_, _JHEP_ 04 (2018) 010 [1611.03470].
  * (66) M. Taylor, _TT deformations in general dimensions_ , 1805.10287.
  * (67) T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, _Holography at finite cutoff with a $T^{2}$ deformation_, _JHEP_ 03 (2019) 004 [1807.11401].
  * (68) A.B. Zamolodchikov, _Expectation value of composite field T anti-T in two-dimensional quantum field theory_ , hep-th/0401146.
  * (69) F.A. Smirnov and A.B. Zamolodchikov, _On space of integrable quantum field theories_ , _Nucl. Phys. B_ 915 (2017) 363 [1608.05499].
  * (70) A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, _$T\bar{T}$ -deformed 2D Quantum Field Theories_, _JHEP_ 10 (2016) 112 [1608.05534].
  * (71) L.V. Iliesiu, J. Kruthoff, G.J. Turiaci and H. Verlinde, _JT gravity at finite cutoff_ , _SciPost Phys._ 9 (2020) 023 [2004.07242].