††institutetext: Department of Physics, Swansea University, Swansea, SA2 8PP,
U.K.

# Replica Rényi Wormholes and Generalised Modular Entropy in JT Gravity

Timothy J. Hollowood, S. Prem Kumar and Luke C. Piper
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the problem of computing semi-classical Rényi entropies of CFT on
AdS2 backgrounds in JT gravity with nongravitating baths, for general replica
number $n$. Away from the $n\to 1$ limit, the backreaction of the CFT twist
fields on the geometry is nontrivial. For one twist field insertion and
general $n$, we show that the quantum extremal surface (QES) condition
involves extremisation of the generalised modular entropy, consistent with
Dong’s generalisation of the Ryu-Takayanagi formula for general $n$. For
multiple QES we describe replica wormhole geometries using the theory of
Fuchsian uniformisation, explicitly working out the analytically tractable
case of the $n=2$ double trumpet wormhole geometry. We determine the off-shell
dependence of the gravitational action on the QES locations and boundary map.
In a factorisation limit, corresponding to late times, we are able to relate
this action functional to area terms given by the value of the JT dilaton at
the (off-shell) QES locations, with computable corrections. Applied to the
two-sided eternal black hole, we find the $n$-dependent Page times for Rényi
enropies in the high temperature limit.

##  1 Introduction

The celebrated island formula Almheiri:2019hni  provides a semiclassical
window into the resolution of Hawking’s long-standing information puzzle for
black holes Hawking:1976ra . Lying at the heart of this mechanism and the
associated generalised entropy formula Almheiri:2019psf ; Penington:2019npb ;
Engelhardt:2014gca ; Faulkner:2013ana ; Hubeny:2007xt , is the replica trick
for dynamical gravity. The gravitational replica trick necessitates the
inclusion of replica wormhole saddle points in the sum over geometries
implemented by the gravitational path integral Almheiri:2019qdq ;
Penington:2019kki . This yields the island formula111The island formula and
its implications have been explored in a variety of different scenarios and
involving evaporating black holes  for the generalised von Neumann entropy of
Hawking radiation, a fine-grained measure222The notion of a fine-grained
entropy here is a relative one, as emphasised in Ghosh:2021axl ;
Krishnan:2020oun . of entanglement which teases out subtle quantum
correlations between early and late radiation, and reproduces the Page curve
Page:1993wv .

It is worth stressing that the generalised entropy formula is an off-shell
expression for the fine-grained entropy of Hawking radiation $R$. Obtained in
the formal limit when the number of replicas $n\to 1$, it consists of an area
term evaluated at a Quantum Extremal Surface (QES) demarcating the boundary of
a putative island region $I$, together with a QFT contribution to entanglement
entropy of $I\cup R$. The implication is that the island is in the Hilbert
space of the radiation, or more precisely, $I$ lies in the entanglement wedge
of the radiation $R$.

Our motivation is to understand if a similar generalised formula exists for
the Rényi entropies $\\{S_{n}\\}$ renyi1 , for general replica number $n$, not
reliant on the crutch of the $n\to 1$ limit. Rényi entropies are measurable
Islam:2015mom , encoding the quantum state and entanglement spectrum of the
subsystem. Our main observation is that there is a generalised Rényi entropy
formula within the Euclidean framework which, in late time or factorization
limits, has the same structure as the generalised entanglement entropy, and
where the area term involves not the Rényi entropy, but the closely related
‘modular entropy’ discovered by Dong in the setting of holography Dong:2016fnf
.

The semiclassical derivation of the island formula is most explicit within the
framework of Jackiw-Teitelboim (JT) gravity Jackiw:1984je ; Teitelboim:1983ux
which naturally arises as an effective 1+1 dimensional description of gravity,
following from reduction to $s$-waves around a higher dimensional near
extremal black hole. JT gravity has proved to be a very useful model to
explore multiple aspects of information evolution and retrieval from
evaporating black holes Almheiri:2019yqk ; Almheiri:2019qdq ;
Penington:2019kki ; Goto:2020wnk .

In particular, the saddles that contribute to computation of the matter Rényi
entropies, the so-called replica wormholes, have been constructed in the
pioneering works Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019qdq ;
Penington:2019kki , at least in certain regimes and this allows one to confirm
that the entropy of the radiation follows the Page curve. Specifically,
Almheiri:2019qdq considers a setup originally introduced in Almheiri:2019yqk ,
involving the eternal AdS2 black hole in JT gravity coupled to a CFT playing
the rôle of a radiation bath. Most attention has focussed around the
computation of von Neumann entropies for radiation subsystems in this
framework. This involves taking a limit $n\to 1$ of the replica wormholes,
which makes the problem much simpler because the gravitational back-reaction
of the CFT twist operators needed to define the Rényi entropy vanishes in this
limit. What remains is a variational problem for the positions of the twist
operators, leading to the so-called Quantum Extremal Surface (QES) formula.
Remarkably this yields an explicit expression for the microscopic, or fine-
grained entropy of the radiation in a calculation that is purely semi-
classical.

In this work, we set out to address the question whether one can write down a
variational problem for Rényi entropies that generalizes the well-known QES
formula for von Neumann entropy. To this end, we consider replica wormholes
within the above setup when the limit $n\to 1$ is not taken and the back-
reaction of twist operators must be taken into account when constructing
replica wormholes. Previous work in this direction for JT gravity appears in
Colin-Ellerin:2021jev .

The second motivation for this work arose from a potential puzzle. The
equations of motion for JT gravity (in particular, the Einstein equations)
coupled to a CFT radiation bath, show how the dilaton field $\phi$ in JT
gravity, which plays the rôle of the area in this two-dimensional effective
theory of gravity, gets sourced by the CFT stress tensor. For any number of
replicas $n$, this equation, along with CFT Ward identities leads to a set of
conditions that determine the positions of the twist fields $w=a_{j}$, i.e.
the QES,

$\begin{split}\frac{\partial_{w}\phi(a_{j},\bar{a}_{j})}{4G_{N}}+\partial_{a_{j}}S^{\text{CFT}}_{n}=0\
,\end{split}$ (1)

where $S_{n}^{\text{CFT}}$ is the Rényi entropy of the matter CFT. This
equation was inferred in Almheiri:2019qdq and then shown to arise from a
simple argument involving the dilaton equation of motion and the CFT Ward
identities in Goto:2020wnk .

Whilst tempting to argue that (1) is already the general $n$ QES formula that
we are seeking, closer inspection reveals two important distinctions:

  1. 1.

Since the dilaton plays the rôle of the area, in (1) it is expected to be the
JT gravity version of the Ryu-Takayanagi (RT) formula Ryu:2006bv in
holography.333This is exactly what one expects when one views the JT gravity
black hole as describing the s-wave sector of a near-extremal charged black
hole in $3+1$ dimensions. There is already a generalization of the RT formula
to Rényi entropy discovered by Dong Dong:2016fnf (see also Dong:2017xht ;
Dong:2018seb ). In particular, Dong argued that the RT term would naturally be
associated not to the Rényi entropy but to a close refinement called the
‘modular entropy’. The modular entropy $\widetilde{S}_{n}$ is the natural
thermodynamical definition of the entropy when the replica number $n$ is
identified with the inverse temperature and the free energy
$F_{n}=-\frac{1}{n}\log Z_{n}^{\rm CFT}$ where $Z_{n}^{\rm CFT}$ is the
replica partition function,

$\widetilde{S}_{n}=-\partial_{\frac{1}{n}}F_{n}\,=\,n^{2}\partial_{n}\Big{(}\frac{n-1}{n}S_{n}^{\rm
CFT}\Big{)}\,.$ (2)

For a single interval, wherein CFT Rényi entropies are universal, the modular
entropy has an $n$ dependence distinct from the Rényi entropy, although they
are then closely related:

$\widetilde{S}_{n}^{\text{CFT}}=\frac{2}{n+1}S_{n}^{\text{CFT}}=\frac{1}{n}{S_{n=1}^{\text{CFT}}}\,.$
(3)

  2. 2.

The well known QES formula for entanglement entropy results from extremising
the generalised entropy which includes the variation of the area term (the
dilaton) with respect to the off-shell positions of punctures
$\sim\partial_{a_{j}}\phi(a_{j})$, whereas (1) involves the derivative of the
area term evaluated at the punctures $\sim\partial_{w}\phi\big{|}_{w=a_{j}}$.

The two observations above put together suggest that the QES equation for
$n>1$ should be of the form

$\begin{split}\frac{\partial_{a_{j}}\phi(a_{j},\bar{a}_{j})}{4G_{N}}+\partial_{a_{j}}\widetilde{S}^{\text{CFT}}_{n}=0\
,\end{split}$ (4)

extremising a generalised modular entropy, $\widetilde{S}^{\rm gen}_{n}$. The
puzzle is that, taken at face value the form of (4) appears to have the wrong
$n$ dependence to match the Ward identity equation (1). For multiple intervals
the modular and Rényi entropies differ nontrivially and the puzzle is more
acute. The second item above provides the potential loophole to make the
mystery evaporate. We will argue that whilst equation (1) is certainly
correct, (4) is also true. When there are multiple QES, this will be true in a
certain factorization limit. For $n>1$ there is non-trivial gravitational
backreaction from the twist fields on the geometry and in JT gravity this
means that the dilaton evaluated at the QES will have both explicit and
implicit dependence on the QES.

We will employ uniformisation techniques to compute the replica partition
function of the gravity plus CFT system as a function of the off-shell QES
locations and the boundary map which glues the CFT radiation baths to the
gravitating AdS2 region. For a single QES in the eternal black hole setup, we
can explicitly demonstrate the validity of (4) for any $n$. Computing the on-
shell value of the generalised modular entropy for generic $n>1$ is
technically challenging, as it requires knowing the boundary map precisely.

For multiple QES, we show that there is a generalized Rényi entropy which has
the approximate form

$\begin{split}S_{n}(a_{j},\bar{a}_{j})=NS_{0}+\frac{n+1}{8G_{N}}\sum_{j}\Big{(}\phi_{j}(a_{j},\bar{a}_{j})+\Phi_{n}\Big{)}+S_{n}^{\text{CFT}}(a_{j},\bar{a}_{j})\
.\end{split}$ (5)

The piece $\Phi_{n}$ can be interpreted as the effect of back-reaction on a
particular QES from all the other QES. This term comes with a factor of $n-1$
and so does not contribute when taking the $n\to 1$ limit. Fortunately, it is
suppressed in the factorization limit that is relevant for understanding the
transition from the saddle without an island (the Hawking saddle) and the
saddle with an island (replica wormhole) that happens at late times, precisely
when the replica wormhole saddle can dominate. With this term neglected and
with knowledge that
$S_{n}^{\text{CFT}}=\frac{n+1}{2}\widetilde{S}_{n}^{\text{CFT}}$, we see that
the QES equation that follows from extremizing the above is precisely of the
form (4). In addition, since the $n$ dependence of the dilaton and the modular
entropy match, i.e. go like $1/n$, means that the position of the QES is
independent of $n$ in the high-temperature, late-time, regime. This simple
picture is valid for the two-sided external black hole which is in thermal
equilibrium with the radiation baths. It is not so simple for the case of an
evaporating black hole which will be considered in a separate work. The
difficulty is that the equation of motion for the boundary map is too
complicated to solve in general (along with the associated conformal welding
problem Almheiri:2019qdq ; Goto:2020wnk ) away from the $n\to 1$ limit. For
the set up involving the eternal black hole, there is a simplification to be
exploited in the high temperature regime wherein one can use perturbation
theory to argue that the boundary map is only a small perturbation of the
trivial map Almheiri:2019qdq .

The paper is organized as follows. In section 2, we review some relevant
aspects of JT gravity and replica wormholes. In section 3, we consider the
case with one QES which is non-trivial but also tractable because we do not
have to consider the effects of back-reaction of one QES on another. We apply
the result to analyse the eternal black hole scenario in the high temperature
limit. Section 4 reviews aspects of the theory of uniformisation relevant for
the construction of the replica wormholes, which we make very explicit for the
case of $n=2$ and with two QES. We also exploit the well-known relation of the
uniformisation problem to Liouville theory. In section 7, we turn to an
analysis of the dilaton in the replica wormhole geometry using the Green’s
function method generalized from the $n=1$ case. This gives us enough
information to compute the dominant terms in the gravitational action in the
factorisation limit, and apply the results to compute the Page times for Rényi
entropies for the two sided black hole. In section 9, we bring together our
results to draw some conclusions and outline immediate future directions.

##  2 JT gravity and replica wormholes

The basic setup used to study black hole evaporation in JT gravity involves
coupling the gravitational system to an external non-gravitating Minkowski
bath. In the semiclassical picture, massless degrees of freedom or CFT modes
propagate on the gravitational background and the Minkowski bath across a
transparent bath-gravity interface.444One can, in principle, turn on non-
vanishing reflection coefficients at the interface to capture the effects of
black hole greybody factors Kruthoff:2021vgv . The backreaction of the CFT on
gravity is treated semiclassically, so the central charge $c$ is assumed to be
large.

Although black hole evaporation and associated entanglement evolution is
intrinsically a Lorentzian phenomenon, the Euclidean continuation of the setup
described above is relevant and particularly useful when considering
entanglement dynamics of the two-sided black hole in the thermofield double
(TFD) state. Whilst the system is in local thermodynamic equilibrium, we
expect the time evolution of quantum correlations between radiation modes
across the two copies to be captured by the Euclidean setting accompanied by
appropriate analytic continuation .

So we focus attention on the Euclidean section of the AdS2 black hole (the
disc). The equation of motion for the dilaton in JT gravity constrains the
Ricci scalar $R=-2$, and therefore fixes the metric to be locally AdS2. The
Euclidean CFT lives on the gravitating disc555Since the metric is fixed to be
locally AdS2 the dynamical degrees of freedom are the dilaton and the boundary
curve of the disc. glued on to the flat non-gravitating plane. We work
initially in Euclidean signature with complex conjugate coordinates
$(w,\bar{w})$ for the locally AdS2 gravitating region and $(y,\bar{y})$ for
the flat bath:

$\displaystyle ds^{2}=e^{2\rho(w,\bar{w})}\,|dw|^{2}\,,\qquad|w|\leq 1\,,$ (6)
$\displaystyle ds^{2}_{\rm bath}=|dy|^{2}\,,\qquad{\rm Re}(y)\geq 0\,,$ (7)

with $y=\sigma+i\tau$, and $\rho(w,\bar{w})$ the conformal factor to be
determined by the equations of motion. The analytic continuation to Lorentzian
signature via the Wick rotation $\tau\to it$ involves $w\to-w^{-}$ and
$\bar{w}\to w^{+}$ and similarly $y\to-y^{-}$ and $\bar{y}\to y^{+}$ where
$y^{\pm}=t\pm\sigma$ are coordinates in the Minkowski baths. The gluing map
between the boundaries of these two regions plays an important dynamical role
below.

###  Replica geometry

When computing a CFT Rényi entropy one can either proceed via the replica
method sewing together $n$ copies of the spacetime along a set of branch cuts
Calabrese:2004eu , or by computing a correlator of twist and anti-twist
operators in the un-replicated geometry at the positions of the branch points.
In the gravitating AdS2 region in JT gravity we can also consider the problem
either in the replicated geometry $\widetilde{\cal M}_{n}$ or in the ‘base’
${\cal M}_{n}$ with the twist operators. The replicated geometry
$\widetilde{\cal M}_{n}$ can be thought of as a Riemann surface that is an
$n$-fold cover of the base with respect to the assumed ${\mathbb{Z}}_{n}$
replica symmetry, ${\cal M}_{n}=\widetilde{\cal M}_{n}/{\mathbb{Z}}_{n}$. The
base is the unit disc $|w|\leq 1$ with $N$ punctures corresponding to branch
points $\\{a_{j}\\}_{j=1,2\ldots N}$ that link the $n$ sheets of the cover.
These branch points will become the QES after solving a variational problem.
The branch cuts originating from each of the QESs end at an image point
outside the disc, in the bath.The resulting Riemann surface has genus
$g=(n-1)(N-1)$. The boundary $\partial\widetilde{\cal M}_{n}$ is a single
$S^{1}$, for $N$ odd, and $(S^{1})^{n}$, for $N$ even. An example, with $n=3$
and $N=2$ QES is shown in figure 1.

$a_{2}$$1/\bar{a}_{2}$$a_{1}$$1/\bar{a}_{1}$321 Figure 1: The shaded part is
the replica wormhole $\widetilde{\cal M}_{n}$ with $N=2$ and $n=3$ so a triple
cover of the base the twice punctured disc $|w|\leq 1$. The sheets are joined
by branch cuts between the QES as shown involving image branch points outside
the disc. The cuts are colour-coded so that one can identify their image in
the uniformized picture in figure 2.

The metric in the covering space is the AdS2 metric written in standard form

$\begin{split}ds^{2}=\frac{4|dW|^{2}}{(1-|W|^{2})^{2}}\ .\end{split}$ (8)

The uniformisation map $\pi:\ \widetilde{\cal M}_{n}\to{\cal M}_{n}$ maps the
replica geometry to the base ${\cal M}_{n}$, the unit disc $|w|\leq 1$. The
multi-valued inverse map $W(w)$ determines the metric on the base:

$\begin{split}ds^{2}=e^{2\rho}\,|dw|_{n}^{2}\ ,\qquad
e^{2\rho}=\frac{4|\partial_{w}W(w)|^{2}}{(1-|W(w)|^{2})^{2}}\ ,\end{split}$
(9)

which is single-valued on the base. Near an insertion point of a twist field
at $w=a_{j}$, the inverse map

$W(w)\sim(w-a_{j})^{\pm 1/n}+\ldots,$ (10)

determines the branch point singularity of the conformal factor $e^{2\rho}$,
the $+$ and $-$ signs in the exponent associated to twist and anti-twist
fields respectively. Taking the viewpoint of the base ${\cal M}_{n}$, the
twist operators make punctures in the geometry with conical deficits with
angles $2\pi(1-\frac{1}{n})$. These deficits (for both twist and anti-twist
fields) are manifested in the equation for the conformal factor of the metric
$ds^{2}=e^{2\rho}\,|dw|_{n}^{2}$ as source terms Almheiri:2019qdq ,

$\begin{split}-4\partial_{w}\partial_{\bar{w}}\rho+e^{2\rho}=2\pi\Big{(}1-\frac{1}{n}\Big{)}\sum_{j}\delta^{(2)}(w-a_{j})\
.\end{split}$ (11)

In the absence of source terms, the equation simply sets the Ricci scalar
$R=-2$, so the metric is clearly locally AdS2. The $|dw|_{n}^{2}$ part is the
metric on the punctured disc with angular deficits of $2\pi(n-1)/n$ at each
puncture so that the metric on the cover (8) is smooth. The conformal factor
approaches that of AdS near the boundary so that

$\begin{split}e^{-2\rho}=(1-|w|^{2})^{2}+{\mathscr{O}}((1-|w|^{2})^{4})\
.\end{split}$ (12)

123 Figure 2: The uniformisation view of the replica wormhole in figure 1. The
fundamental domain ${\cal D}$ is the shaded region, i.e. the disc with the
circles cut out. The coloured arcs of the circles are then identified and
these correspond to how the cuts are joined in figure 1 and gives rise to the
illustration of the replica wormhole in figure 3.

The replicated geometry $\widetilde{\cal M}_{n}$ is most conveniently
described by using the theory of uniformisation, in this context Fuchsian
uniformisation Hulik:2016ifr ; Hadasz:2005gk ; Hadasz:2006rb . Essentially
$\widetilde{\cal M}_{n}$ is covered by a complex coordinate $W$ defined on a
domain ${\cal D}$ that is a subset of the unit disc $|W|\leq 1$ formed by
cutting out a series of circles that straddle the boundary $|W|=1$. Details
will be given later on how the circles are defined but an illustration for the
$n=3$ and $N=2$ example is shown in figure 2. The coloured arcs on the circles
are identified so that the closed blue contour in figure 1 whose image is
shown in figure 2 is actually closed modulo the non-trivial identifications in
the covering space. Joining up the contours gives a picture of the replica
wormhole as a smooth Riemann surface as shown in figure 3.

Figure 3: The replica wormhole of figures 1 and 2 pictured as a smooth Riemann
surface showing the two branch points, the QES, and the images of the coloured
cuts.

###  The dilaton

The large $c$ matter CFT couples to the metric in the standard way and the
equations of motion for the metric, namely Einstein’s equations, are actually
the equations obeyed by the dilaton,

$\begin{split}\nabla_{\alpha}\nabla_{\beta}\phi+R(\phi-\Delta\phi)g_{\alpha\beta}=8\pi
G_{N}T_{\alpha\beta}\ ,\end{split}$ (13)

or in component form

$\begin{split}e^{2\rho}\partial_{w}\big{(}e^{-2\rho}\partial_{w}\phi)&=8\pi
G_{N}T_{ww}\ ,\\\\[5.0pt]
e^{2\rho}\partial_{\bar{w}}\big{(}e^{-2\rho}\partial_{\bar{w}}\phi)&=8\pi
G_{N}T_{\bar{w}\bar{w}}\ ,\\\\[5.0pt]
-\partial_{w}\partial_{\bar{w}}\phi+\frac{1}{2}e^{2\rho}\phi&=8\pi
G_{N}T_{w\bar{w}}\ .\end{split}$ (14)

This illustrates one of the simplifying features of JT gravity; namely, the
CFT stress tensor sources the dilaton but does not couple to it directly.
Since we work in the semiclassical approximation, on the right hand side of
Einstein’s equations are the expectation values of components of the stress
tensor. In particular, the mixed component $T_{w\bar{w}}=0$ in the
semiclassical limit.

The first of the three equations (14) leads to a condition on the location of
the QES. When the derivative hits the conformal factor, there is a simple pole
at each of the branch points $w=a_{j}$, with strength proportional to the
derivative of the dilaton evaluated at the branch point. The expectation value
of the CFT stress tensor has matching poles, as dictated by conformal Ward
identities DiFrancesco:1997nk ; Goto:2020wnk following from the OPEs of the
stress tensor with primary fields, in this case the twist fields. The residues
of the simple poles are proportional to the derivatives of the CFT Rényi
entropy with respect to $(a_{j},\bar{a}_{j})$: equating the residues gives the
QES condition quoted in the introduction (1). Importantly, putative double
poles at the locations of the branch point twist fields are cancelled by the
Weyl shifts on top of the flat space stress tensor due to the nontrivial
conformal factor of the metric on the base.

###  Boundary dynamics

The action of JT gravity reduces purely to a boundary term and all non-trivial
dynamics is determined by the boundary map

$w(\tau)=e^{i\theta(\tau)}\,,$ (15)

where $\tau$ is the (Euclidean) bath time. This boundary effective action is
related to the ADM mass of the geometry given by the Schwarzian of the
boundary map. In the context of the replica geometry, the relevant map is the
one in the covering space, namely $W(\tau)=W(w(\tau))$:666The entropy is
related to action of the gravity and CFT via, $(1-n)S_{n}=\log(Z_{n}/Z_{1})$,
where the partition function $\log
Z_{n}=-I^{(n)}_{\text{JT}}-I^{(n)}_{\text{CFT}}$.

$\begin{split}I_{\text{JT}}^{(n)}[w]&=-NS_{0}-\frac{\phi_{r}}{8\pi
G_{N}}\int_{\partial\widetilde{\cal M}_{n}}d\tau\,\\{W,\tau\\}\\\\[5.0pt]
&=-NS_{0}-\frac{n\phi_{r}}{8\pi
G_{N}}\int_{|w|=1}d\tau\,\Big{(}\\{w,\tau\\}+2{\mathcal{T}}(w)(\partial_{\tau}w)^{2}\Big{)}\
.\end{split}$ (16)

The final form of the action is defined on the boundary of the base, the
circle $|w|=1$, and this accounts for the extra factor of $n$. We have defined
the quantity

${\mathcal{T}}(w)=\frac{1}{2}\\{W,w\\}$ (17)

which is a meromorphic function of $w$, and which we also recognize as the
Liouville stress tensor

${\mathcal{T}}(w)=-(\partial_{w}\rho)^{2}+\partial_{w}^{2}\rho.$ (18)

When the gravitational sector has been reduced to the dynamics of the boundary
map $w(\tau)$ the bath CFT stress tensor then sources the boundary map. The
equation of motion for the boundary map was derived in Engelsoy:2016xyb and
also in the pioneering replica wormhole paper Almheiri:2019qdq in a different
way. Here, we summarize the original. The boundary map determines a coordinate
transformation between the near boundary AdS Kruskal-Szekeres coordinates
$(w,\bar{w})$ and the Euclidean bath coordinates
$(y=\sigma+i\tau,\bar{y}=\sigma-i\tau)$,

$\begin{split}w=w(-iy)\ ,\qquad\bar{w}=\bar{w}(i\bar{y})\ .\end{split}$ (19)

Let us write the gravitational part of the metric in terms of the bath
coordinates as

$\begin{split}ds^{2}=e^{2\tilde{\rho}}\,|dy|^{2}\ .\end{split}$ (20)

Then as one approaches boundary at $\sigma=-\epsilon$, for small $\epsilon$,
where the AdS part of the geometry is glued to the Minkowski part, the
conformal factor behaves as

$\begin{split}e^{2\tilde{\rho}}=\frac{1}{4\epsilon^{2}}-\frac{1}{6}\\{W,\tau\\}+{\mathscr{O}}(\epsilon^{2})\,,\end{split}$
(21)

where $W=W(w(\tau))$. The fact that the right-hand side of the third equation
in (14) vanishes, on account of $T_{w\bar{w}}=0$, requires the dilaton to have
the asymptotic behaviour

$\begin{split}\phi=\frac{\phi_{r}}{\epsilon}+\phi_{0}+\frac{2\epsilon\phi_{r}}{3}\\{W,\tau\\}+{\mathscr{O}}(\epsilon^{2})\,.\end{split}$
(22)

If we substitute this asymptotic form into the first two equations (14) and
perform a diffeomorphism to $(y,\bar{y})$ and then take the difference, this
gives an expression that has a non-singular limit as $\epsilon\to 0$ and is
the equation we are after, the equation of motion of the boundary map
$w(\tau)$ sourced by the CFT:

$\begin{split}\frac{\phi_{r}}{8\pi
G_{N}}\frac{d}{d\tau}\Big{[}\\{w,\tau\\}+2{\mathcal{T}}(w)\Big{(}\frac{dw}{d\tau}\Big{)}^{2}\Big{]}=i\big{(}T_{yy}(i\tau)-T_{\bar{y}\bar{y}}(-i\tau)\big{)}\
.\end{split}$ (23)

On the right, we have the expectation values of the CFT stress tensor
components in the bath coordinates $(y,\bar{y})$ evaluated at the boundary at
$y=i\tau$. In Lorentzian signature, this equation has the interpretation as an
energy conservation equation: the left-hand side is the rate of change of the
gravitational energy while the right-hand side is the inward energy flux of
the matter CFT across the boundary.

##  3 One QES example

The simplest scenario is when there is one QES in the context of the eternal
black hole Almheiri:2019qdq ; Goto:2020wnk as illustrated in figure 5. Whilst
this is not directly relevant to deriving the Page curve for the radiation it
is revealing enough to provide answers to the questions that we posed in the
introduction in the simplest of settings and so is worth analysing in detail.

$a$$b$ Figure 4: The set up in (left) Euclidean (right) Lorenztian signature
with one QES in the context of the eternal black hole. The AdS region is
shaded pink and the bath in yellow. There is a point in the right
Euclidean/Minkowski bath and one QES as shown ultimately outside the horizon
of the right black hole in the Lorentzian picture.

There is one point in the right bath with coordinates
$(e^{y},e^{\bar{y}})=(b,\bar{b})$. For simplicity, we also re-scale $y\to\beta
y/2\pi$, so that in Euclidean signature $\tau$ has period $2\pi$.

With one branch point, the unformization problem is straightforward and the
inverse uniformisation map takes the form

$\begin{split}W=\Big{(}\frac{w-a}{1-\bar{a}w}\Big{)}^{1/n}\ .\end{split}$ (24)

This is, as expected, multi-valued. The fundamental domain ${\cal D}$ is the
whole of the unit disc $|W|=1$. In this case there are no circles as in figure
2 to cut out. The boundary $|W|=1$ covers the boundary of the base $|w|=1$ $n$
times.

###  Off-shell gravity action

The action for JT gravity evaluated on the replica geometry consists of two
parts,

$\begin{split}&I_{\rm JT}^{(n)}[w;a,\bar{a}]=-NS_{0}\\\
&-\frac{n\phi_{r}}{4\beta
G_{N}}\Big{(}\int_{|w|=1}d\tau\\{w,\tau\\}+\frac{(n^{2}-1)}{2n^{2}}(1-|a|^{2})^{2}\oint_{|w|=1}dw\,\frac{\dot{w}}{(w-a)^{2}(1-\bar{a}w)^{2}}\Big{)}\,\end{split}$
(25)

where, in the above we denote

$\begin{split}\dot{w}(w)=\frac{dw}{d\tau}\Big{|}_{\tau=\tau(w)}\ ,\end{split}$
(26)

i.e. $\dot{w}$ as a function of $w$. Off-shell, the Schwarzian term is
independent of the branch point location which is moved around by Möbius
transformations on $w$. The off-shell dependence on the QES location is
captured by the second term, provided we only consider boundary maps modulo
${\rm SL}(2,{\mathbb{R}})$ variations, keeping the branch point fixed. We have
expressed the second term as a contour integral over $w$.777This expression is
re-scaled by $2\pi/\beta$ to reflect the fact that $\tau$ has been re-scaled
by $\beta/2\pi$ in order to make the thermal circle have length $2\pi$. The
purpose of writing the integral in this way is that we can relate it directly
to the dilaton.

###  The dilaton from boundary dynamics

On-shell the dilaton satisfies the wave equation (14) sourced by the CFT
stress tensor. However, we are after an expression for the dilaton in terms of
the boundary curve $w(\tau)$ rather than the CFT stress tensor. The boundary
curve responds to the energy flux and so we can trade in one for the other.
For the un-replicated problem $n=1$ where the metric is just the standard one
$ds^{2}=|dw|^{2}/(1-|w|^{2})^{2}$, this can be done by using a Green’s
function method as described in Goto:2020wnk and reviewed in appendix A.

In the context of the $n>1$ problem, we simply notice that the metric takes
this form in the covering space (8) and so the same method applies. In
particular, the stress tensor component $T_{WW}$, like $T_{ww}$ in the $n=1$
problem, is analytic in the disc, here in the covering space $|W|\leq
1$.888This follows because in flat space the CFT stress tensor has double
poles at the positions of the twist operators. However, these are cancelled by
the Weyl transformation to the physical metric in $(w,\bar{w})$ coordinates
and then a diffeomorphism to the covering space Goto:2020wnk . In addition, a
key ingredient in the derivation is the equation of motion for the boundary
curve and again we remark that in our case (23) can be written in terms of
$W(w)$ in a way that is identical to the $n=1$ equation with $w\to W$. We have
no need to repeat the steps here, but simply quote the result,

$\begin{split}\phi(W,\bar{W})=-\frac{\phi_{r}(1-|W|^{2})^{2}}{\beta}\oint_{|\tilde{W}|=1}d\tilde{W}\,\frac{\dot{W}(\tilde{W})}{(\tilde{W}-W)^{2}(1-\bar{W}\tilde{W})^{2}}\,.\end{split}$
(27)

In the above, $\dot{W}=\partial_{w}W(w)\dot{w}(w)$ and then mapped back to the
covering space using the uniformisation map $\pi:\ w\to W$. We have

$\begin{split}\partial_{w}W=\frac{(1+\bar{a}W^{n})^{2}}{n(1-|a|^{2})W^{n-1}}\,.\end{split}$
(28)

In section 7, we show how the above integral expression for the dilaton
generalizes for multiples QES, or branch points.

The integral expression (27), together with $\eqref{dWw}$, manifests the
dependence of the dilaton on the position of the QES. As we discuss below, it
can be computed by picking up the residues from the double pole at
$\tilde{W}=W$ and the pole of order $n-1$ at the branch point $\tilde{W}=0$
or, at the mirror points $\tilde{W}=1/\bar{W}$ and $\infty$.

For now, we only need the expression for the dilaton evaluated at the QES,
corresponding to the branch point. In this case the two poles collide and
leave a much simpler integral over the boundary of the base instead,

$\begin{split}\phi(a,\bar{a})\equiv\phi(W=0,\bar{W}=0)=-\frac{\phi_{r}(1-|a|^{2})^{2}}{n\beta}\oint_{|w|=1}dw\,\frac{\dot{w}}{(a-w)^{2}(1-\bar{a}w)^{2}}\
.\end{split}$ (29)

In deriving this expression, one should remember that the boundary in the
unformized space $|W|=1$ covers the boundary in the base $n$ times.

Now, we notice that the integral expression for the dilaton evaluated at the
QES (29) is proportional to the second term in the gravitational action (25),
so that

$\begin{split}I_{\text{JT}}^{(n)}[w;a,\bar{a}]=-NS_{0}-\frac{n\phi_{r}}{4\beta
G_{N}}\int_{|w|=1}d\tau\\{w,\tau\\}+\frac{(n^{2}-1)\phi(a,\bar{a})}{8G_{N}}\
.\end{split}$ (30)

It is worth stressing that as an off-shell expression for the semiclassical
gravity action on the replica geometry, the explicit dependence on the branch
point location is encoded only in the dilaton term. Formally including the
contribution from the CFT bath in the semiclassical limit, we can write full
replica partition function as

$-\log Z_{n}=I_{\rm JT}^{(n)}[w,a,\bar{a}]-\log Z_{n}^{\rm CFT}$ (31)

The CFT partition function depends on the location of the QES in the AdS2
region, and implicitly on the boundary map $w(\tau)$ in a way which is
determined by the solution to the welding problem, which we will flesh out
further below.

###  Generalised modular entropy

Varying the off-shell action with respect to the location of the branch point
yields the QES condition

$\partial_{a}\left(\frac{\phi(a,\bar{a})}{4G_{N}}+\frac{2}{n+1}S_{n}^{\rm
CFT}\right)=0\,.$ (32)

The modular entropy, defined as

$\widetilde{S}_{n}^{\rm CFT}=-n^{2}\frac{\partial}{\partial n}(n^{-1}\log
Z_{n}^{\rm CFT}),$ (33)

can be written in terms of the Rényi entropy in the case of free fermions, and
more generally for one interval problem, as

$\widetilde{S}_{n}^{\text{CFT}}=\frac{2}{n+1}S_{n}^{\rm CFT}\,.$ (34)

The QES condition above makes it clear that the appropriate generalized
modular entropy is

$\begin{split}\widetilde{S}_{n}^{\text{gen}}(a,\bar{a})=\frac{\phi(a,\bar{a})}{4G_{N}}+\widetilde{S}_{n}^{\text{CFT}}(a,\bar{a})\
,\end{split}$ (35)

in complete agreement with Dong’s holographic Rényi entropy, which generalizes
the RT formula Dong:2016fnf once we identify the dilaton as playing the rôle
of the area.

###  Consistency with equation of motion

We can reach the same conclusion by examining condition (1) following from the
JT equations of motion and the CFT Ward identity relating the simple pole
residue in the stress tensor (with twist field insertions) to the CFT Rényi
entropy.

We use the integral expression in the cover (27) to compute the derivative
$\partial_{w}\phi(w,\bar{w})$ and then evaluate it at $w=a$. The calculation
is easiest to perform in covering space coordinates, picking up the residues
at the poles of the integrand. This is straightforward provided $\dot{w}(w)$
and its analytic structure is known. In general, however, $\dot{w}(w)$ is
known only formally via a Laurent expansion,

$\dot{w}(w)=\dot{w}_{+}(w)+\dot{w}_{-}(w)$ (36)

where $\dot{w}_{+}$ and $\dot{w}_{-}$ are analytic inside and outside the unit
disc respectively,

$\dot{w}_{+}(w)=iw+\sum_{m>1}p_{m}^{+}\,w^{m}\,,\qquad\dot{w}_{-}(w)=\sum_{m>1}p_{m}^{-}\,w^{-m+1}\,.$
(37)

Viewed as functions of the covering space coordinate $W$, $\dot{w}_{+}$ has
poles outside the unit disc, whist the singularities of $\dot{w}_{-}$ lie
inside the unit disc. The integrand in (27) has double poles inside the unit
disc, at $\tilde{W}=W$, and outside the disc at $\tilde{W}=1/\bar{W}$,
assuming $|W|<1$. There is also a pole of order $n-1$ at $\tilde{W}=0$ from
$\partial_{w}W$ expressed as a function of $\tilde{W}$.

We then compute the integral (27) by splitting it in two parts. We evaluate
the piece dependent on $\dot{w}_{+}$ by picking up the residues inside the
unit disc, whilst the second integral involving $\dot{w}_{-}$ is computed by
picking up the residues from poles outside the unit disc. The dilaton and its
derivative at the branch point are both well defined and given as

$\begin{split}\phi(a,\bar{a})=-\frac{2\pi
i\phi_{r}}{n\beta}\left(\partial_{w}\dot{w}_{+}(a)-\partial_{w}\dot{w}_{-}(\bar{a}^{-1})+\frac{2\bar{a}}{1-|a|^{2}}\left(\dot{w}_{+}(a)+\dot{w}_{-}(\bar{a}^{-1})\right)\right)\end{split}$
(38)

and

$\begin{split}\partial_{w}\phi(a,\bar{a})=-\frac{n+1}{n}\cdot\frac{2\pi
i\phi_{r}}{\beta}&\Big{(}\frac{\bar{a}^{2}}{(1-|a|^{2})^{2}}(\dot{w}_{+}(a)+\dot{w}_{-}(\bar{a}^{-1})\\\\[5.0pt]
&+\frac{\bar{a}\,\partial_{w}\dot{w}_{+}(a)}{1-|a|^{2}}+\frac{1}{2}\partial_{w}^{2}\dot{w}_{+}(a)\Big{)}\,.\end{split}$
(39)

It follows immediately that

$\begin{split}\partial_{w}\phi(a,\bar{a})=\frac{n+1}{2}\partial_{a}\phi(a,\bar{a})\,.\end{split}$
(40)

Given how carefully we have to account for the singularity structure of
$\dot{w}(w)$, the match is quite nontrivial. Thus eq.(1) which is a
consequence of the CFT stress tensor Ward identity, becomes a QES equation for
the generalised modular entropy (35),

$\partial_{a}\tilde{S}^{\rm gen}(a,\bar{a})=0\,,$ (41)

valid for any $n$. This shows precisely how the puzzle posed in the
introduction is solved in detail. The dilaton $\phi(w,\bar{w})$ has implicit
dependence on the position of QES $(a,\bar{a})$ as well as explicit dependence
when evaluated at the QES. Hence, there is no contradiction between the Ward
identity equation (1) and the variational equation (4) inspired by Dong’s
Rényi generalization of the RT formula. We have thus shown that the notions of
QESs and fine grained entropies admit natural generalisations for Rényi
entropies with arbitrary $n$.

###  Welding problem and boundary map

In order to write down the equation of motion for the boundary map (23), we
need the expectation value of the stress tensor of the CFT, which for
simplicity we assume is a set of $c$ massless fermions. Therefore we need to
know what is the appropriate quantum state of the CFT.

Naïvely, one would think that the thermal state in the bath coordinates
$(y,\bar{y})$ along with the insertions of the twist operators at branch
points, would be the appropriate choice. However, with the gluing provided by
the boundary map $w(\tau)=e^{i\theta(\tau)}$ this would generally not extend
to a holomorphic map in the AdS region. So one has to choose the state to be
the vacuum state in some more general coordinates $(z,\bar{z})$ that are
related holomorphically to $(y,\bar{y})$ in the bath region,

$z=F(e^{y})\,,\qquad{\rm Re}(y)\geq 0\,,$ (42)

and holomorphically to $(w,\bar{w})$ in the the AdS region,

$z=G(\omega)\,,\qquad|w|\leq 1\,.$ (43)

Finding this coordinate frame is the conformal welding problem described in
detail in Almheiri:2019qdq . The CFT stress tensor along the boundary, in the
presence of twist and anti-twist operators at $z=G(a)$ and $z=F(b)$, can then
be expressed via the Schwarzian derivative

$\begin{split}T_{yy}=-\frac{c}{24\pi}\Big{\\{}\Big{(}\frac{F(e^{y})-F(b)}{F(e^{y})-G(a)}\Big{)}^{1/n},y\Big{\\}}\
,\end{split}$ (44)

with a similar expression for the anti-holomorphic component. This is to be
inserted on the right-hand side of the equation of motion (23).

The equation of motion depends on the conformal welding problem in a non-
trivial way and it becomes impossible to find the solution $w(\tau)$ in
general. Some form of approximation is needed in order to make progress. One
way to proceed is to work in perturbation theory in $n-1$ with a view to
taking the $n\to 1$ limit in order to compute the von Neumann entropy. This
provides a tractable limit and leads to the familiar generalized (von Neumann)
entropy and QES formula but it is not a regime open to our analysis where we
want to study the problem for integer $n$.

Fortunately there is another approximation that is open to us and which works
for any $n$; namely, the high temperature limit where the dimensionless
combination

$\begin{split}\kappa\equiv\frac{c\beta G_{N}}{6\pi\phi_{r}}\ll 1\,\end{split}$
(45)

$\kappa$ controls the strength of the backreaction due to the energy flux in
eq.(23). This is the limit analysed in Almheiri:2019qdq and corresponds to a
large Bekenstein-Hawking entropy $S_{\text{BH}}(\beta)\gg c$ and physically to
when the absorption and emission of CFT modes by the black hole is very slow,
precisely the quasi-static regime where Hawking’s original calculation is
valid.

In the small $\kappa$ regime, the boundary map is determined in perturbation
theory around the the trivial map. To linear order, with
$\delta\theta\sim{\mathscr{O}}(\kappa)$,

$\begin{split}w(\tau)=e^{i\theta(\tau)}=e^{i\tau}\big{(}1+i\delta\theta(\tau)\big{)}\
,\qquad\delta\theta(t)=\sum_{m\neq 0,\pm 1}c_{m}e^{im\tau}\ ,\end{split}$ (46)

which has the requisite periodicity in imaginary time (after the re-scaling by
$\beta/2\pi$). Crucially, we omit the modes with $m=0,\pm 1$ which are fixed
by the $\text{SL}(2,{\mathbb{R}})$ symmetry that acts as Möbius
transformations on $w$. These are precisely the transformations that move the
branch point around. Off-shell, we want to keep $w(\tau)$ independent of
variations in the location of the QES.

The solution to the welding problem can be expressed in terms of the
expansions,

$\begin{split}G(w)=w\Big{(}1-i\sum_{m>1}c_{m}w^{m}\Big{)}\ ,\qquad\quad
F(e^{y})=e^{y}\Big{(}1+i\sum_{m<-1}c_{m}e^{my}\Big{)}\,\end{split}$ (47)

valid in the domains $|w|<1$ and ${{\rm Re}(y)>0}$, respectively.

As will become apparent once we solve for the position of the QES, $a$ and
$\bar{a}$ are order $\kappa$. With this in mind, we first solve the equation
of motion (23) at order $\kappa$ for all the Fourier modes of
$\delta\theta(\tau)$ except $m=0,\pm 1$, thus keeping the QES location off-
shell for now, so

$\begin{split}\delta\theta(\tau)=-\frac{i\kappa(n^{2}-1)}{2}\sum_{m>1}\frac{m+1}{m^{2}((mn)^{2}-1)}\frac{e^{im\tau}}{b^{m}}+\text{c.c.}\end{split}$
(48)

Now we can compute the gravitational action (25) with the QES off-shell. The
term involving the Schwarzian $\\{w,\tau\\}$ vanishes to leading order in
$\kappa$ when integrated around the boundary and so we focus on the second
term, expressed as a contour integral over $w$. Following from (48), to linear
order in $\kappa$ we can determine the pieces of $\dot{w}_{\pm}(w)$ which are
analytic inside and outside the unit disc:

$\begin{split}\dot{w}_{+}(w)&=i\omega+\frac{i\kappa(n^{2}-1)}{2}\sum_{m>1}\frac{m+1}{m((mn)^{2}-1)}\frac{w^{m+1}}{b^{m}}\
,\\\\[5.0pt]
\dot{w}_{-}(w)&=-\frac{i\kappa(n^{2}-1)}{2}\sum_{m>1}\frac{m+1}{m((mn)^{2}-1)}\frac{w^{-m+1}}{\bar{b}^{m}}\
.\end{split}$ (49)

Note that the leading term $iw$ can be placed in either of $\dot{w}_{\pm}$.
Then the integral (25) can be computed by by picking up the residue at $w=a$
for the $\dot{w}_{+}$ component and $w=1/\bar{a}$ for the $\dot{w}_{-}$
component to yield

$\begin{split}\phi(a,\bar{a})&=\frac{2\pi\phi_{r}}{n\beta}\frac{1+|a|^{2}}{1-|a|^{2}}\\\\[5.0pt]
&-\frac{\pi\kappa\phi_{r}(n^{2}-1)}{n\beta(1-|a|^{2})}\sum_{m>1}\frac{(m+1)(m(1-|a|^{2})+1+|a|^{2})}{m((mn)^{2}-1)}\Big{(}\frac{a^{m}}{b^{m}}+\frac{\bar{a}^{m}}{\bar{b}^{m}}\Big{)}\
.\end{split}$ (50)

In the $n\to 1$ limit, the first term is the well-known expression for the
dilaton in the Euclidean eternal black hole.

###  QES at next-to-leading order

Since $a$ is effectively order $\kappa$, the next-to-leading terms in (50) are
those with $m=2$. Keeping these terms gives

$\begin{split}\phi(a,\bar{a})=\frac{2\pi\phi_{r}}{n\beta}\Big{(}1+2|a|^{2}+\frac{9\kappa(n^{2}-1)}{4(4n^{2}-1)}\frac{(a^{2}\bar{b}^{2}+\bar{a}^{2}b^{2})}{|b|^{4}}+\cdots\Big{)}\
.\end{split}$ (51)

To this order, the solution of the conformal welding problem involves

$\begin{split}G(w)&=w-\frac{3\kappa(n^{2}-1)}{8(4n^{2}-1)b^{2}}w^{3}+\cdots\
,\\\\[5.0pt]
F(e^{y})&=e^{y}-\frac{3\kappa(n^{2}-1)}{8(4n^{2}-1)\bar{b}^{2}}e^{-y}+\cdots\
.\end{split}$ (52)

These functions appear in the CFT modular entropy,

$\begin{split}\widetilde{S}^{\text{CFT}}_{n}(a,\bar{a})=\frac{c}{6n}\log\frac{|F(b)-G(a)|^{2}}{(1-|a|^{2})|G^{\prime}(a)F^{\prime}(b)b|}\
.\end{split}$ (53)

In order to extremize the generalized entropy we will need the effective
expansion in powers of $\kappa$

$\begin{split}\partial_{a}\widetilde{S}_{n}^{\text{CFT}}(a,\bar{a})=\frac{c}{6n}\Big{(}-\frac{1}{b}+\bar{a}-\frac{a}{b^{2}}-\frac{3\kappa(n^{2}-1)}{8(4n^{2}-1)|b|^{4}b}+\cdots\Big{)}\
.\end{split}$ (54)

Extremizing the generalized modular entropy (35) gives the position of the QES
to the next-to-leading order

$\begin{split}a=\frac{\kappa}{\bar{b}}+\frac{\kappa^{2}(1-|b|^{2})}{|b|^{2}\bar{b}}+\frac{3\kappa^{2}(n^{2}-1)}{8(4n^{2}-1)}\frac{1-6|b|^{2}}{|b|^{4}\bar{b}}+\cdots\
.\end{split}$ (55)

The modular entropy of the saddle to the order we are working is equal to

$\begin{split}\widetilde{S}^{\text{gen}}_{n}=\frac{2\pi\phi_{r}}{n\beta}\Big{(}1+\log|b|-\frac{2\kappa}{|b|^{2}}-\frac{9\kappa(n^{2}-1)}{8(4n^{2}-1)|b|^{4}}+\cdots\Big{)}\
.\end{split}$ (56)

As anticipated $a$ is of order $\kappa$ and the next-to-leading order terms
are $n$ dependent. We have computed this in Euclidean signature but we can
analytically continue to Lorentzian signature to capture the real-time entropy
dynamics. In particular, the QES is at $a^{+}=-\kappa/b^{-}$ and
$a^{-}=-\kappa/b^{+}$ and since $b^{\pm}\gtrless 0$, means that the QES will
be outside the horizon as shown in figure 5.

###  Dilaton at arbitrary point

Before we leave the single QES analysis, it is interesting also to evaluate
the dilaton at an aribtrary point in Lorentzian signature. When $n=1$, one can
confirm that the expression is simply

$\begin{split}\phi(w^{\pm})_{n=1}=\frac{2\pi\phi_{r}}{\beta}\frac{1-w^{+}w^{-}}{1+w^{+}w^{-}}\
,\end{split}$ (57)

independent of the position of the QES as expected since the back-reaction
from the QES vanishes. For general $n$, the general expression is complicated
but near the QES, $w^{\pm}\sim a^{\pm}$, we have, for $n>2$,999The $n=2$
expression is slightly more complicated.

$\begin{split}\phi(w^{\pm})=\frac{2\pi\phi_{r}}{n\beta}\Big{\\{}\frac{1-a^{+}a^{-}}{1+a^{+}a^{-}}+\frac{2[(w^{+}-a^{+})(a^{-}-w^{-})]^{\frac{1}{n}}}{(1+a^{+}a^{-})^{1+\frac{2}{n}}}+\cdots\Big{\\}}\
.\end{split}$ (58)

Of course when evaluated at the QES we return to the expression (50) continued
to Lorentzian signature. Notice that this expression is real for points space-
like separated from the QES ($w^{+}\gtrless a^{+}$ and $a^{-}\gtrless w^{-}$)
but complex for points time-like separated from the QES. This is in line with
the general analysis of Colin-Ellerin:2021jev of real-time replica wormholes
which found that the metric, or for JT gravity the dilaton, will be complex in
the causal past of the splitting surface, the QES. In fact, if we evaluate the
dilaton with the leading order position for the QES $a^{\pm}=0$, we find

$\begin{split}\phi(w^{\pm};a^{\pm}=0)=\frac{2\pi\phi_{r}}{n\beta}\frac{1-(-w^{+}w^{-})^{1/n}}{1+(-w^{+}w^{-})^{1/n}}\
,\end{split}$ (59)

which is equivalent to the expression for the dilaton in Colin-Ellerin:2021jev
.

##  4 Replica wormhole and uniformisation

In this section, we describe the geometrical set up required to construct the
replica wormhole when there is more than one QES. The mathematical setting is
the theory of Fuchsian uniformisation described in a physics context in
Hulik:2016ifr ; Hadasz:2005gk ; Hadasz:2006rb and in the slightly different
context of Schottky uniformisation and AdS/CFT for CFTs in $1+1$ dimensions
Faulkner:2013yia .

In order to describe the covering geometry $\widetilde{\cal M}_{n}$, it is
useful to define an abstract auxiliary problem in the form of a Fuchsian
equation, a second order differential equation on the full $w$ complex plane,

$\begin{split}\big{(}\partial_{w}^{2}+{\mathcal{T}}(w)\big{)}\Psi=0\
,\end{split}$ (60)

along with its complex conjugate. The relation to the replica wormhole is
expressed through the potential function ${\mathcal{T}}(w)$ that has already
appeared in the gravitational action (16) and which is identified with the
holomorphic component of the Liouville stress tensor (18).

As an analytic function this has double poles at the QES $w=a_{j}$ on the disc
and image points $w=1/\bar{a}_{j}$ outside the disc of a specific form

$\begin{split}{\mathcal{T}}(w)=\sum_{j=1}^{N}\Big{(}\frac{\varepsilon_{n}}{(w-a_{j})^{2}}+\frac{\varepsilon_{n}}{(w-1/\bar{a}_{j})^{2}}+\frac{c^{+}_{j}}{w-a_{j}}+\frac{c^{-}_{j}}{w-1/\bar{a}_{j}}\Big{)}\
,\end{split}$ (61)

where,

$\varepsilon_{n}=\frac{n^{2}-1}{4n^{2}}\,.$ (62)

The $(c^{+}_{j},c^{-}_{j})$ are the accessory parameters which depend
implicitly on the coordinates $(a_{j},\bar{a}_{j}).$101010The accessory
parameters satisfy constraints ensuring the absence of an additional
singularity at $w=\infty$. For $N>1$, the total number of undetermined
accessory parameters is $2N-3$. The dependence of the accessory parameters on
the $(a_{j},\bar{a}_{j})$ can be uncovered in an expansion around a certain
factorization limit. For the case with $N=1$, which we have studied in detail
above, the accessory parameters are completely determined by the requirement
that $w=\infty$ is a non-singular point,

$N=1:\qquad c^{\pm}=\pm\varepsilon_{n}\frac{2\bar{a}}{1-|a|^{2}}\,.$ (63)

The fact that branch points occur in pairs $(a_{j},1/\overline{a}_{j})$, one
inside and one outside the unit circle, is needed so that the conformal factor
has the required behaviour (12) near the boundary as $|w|\to 1$. On the
boundary, the stress tensor satisfies the reality conditions

$\begin{split}w^{2}{\mathcal{T}}(w)\in{\mathbb{R}}\
,\qquad{\mathcal{T}}(w)=\frac{1}{w^{4}}\,\overline{{\mathcal{T}}(1/\bar{w})}\
.\end{split}$ (64)

Let us denote a pair of two linearly independent solutions of the Fuchsian
equation (60) with constant Wronskian, the fundamental system,
$\Psi=(\psi_{1},\psi_{2})$. In order to satisfy the reality conditions (64) we
must impose the following reality conditions on the fundamental system

$\begin{split}w\,\overline{\psi_{1}(1/\bar{w})}=\psi_{2}(w)\ ,\qquad
w\,\overline{\psi_{2}(1/\bar{w})}=\psi_{1}(w)\ .\end{split}$ (65)

Given a fundamental system that does not satisfy these conditions it is always
possible to find an $\text{SL}(2,{\mathbb{C}})$ transformation $\Psi\to
V\cdot\Psi$ so that the transformed system does satisfy (65). Once (65) is
satisfied it is fixed by the subgroup $\text{SU}(1,1)$ which will play a
prominent rôle as we proceed.

This fundamental system has non-trivial monodromies
$(M_{j},M_{\bar{j}})\in\text{SU}(1,1)$ around each of the branch points
$(a_{j},1/\bar{a}_{j})$ on the complex $w$ plane. Note that $\text{SU}(1,1)$
is precisely the subgroup of Möbius transformations that preserves the unit
disc. The monodromies satisfy $M_{j}^{n}=M_{\bar{j}}^{n}=I$ to reflect the
${\mathbb{Z}}_{n}$ symmetry and the coefficients of the double poles in (61)
means that they must lie in the conjugacy class of the element

$\begin{split}\Lambda\equiv\text{diag}(-e^{\pi i/n},-e^{-i\pi/n})\
.\end{split}$ (66)

What makes the problem non-trivial is that the monodromies in general do not
commute $[M_{i},M_{j}]\neq 0$. However, the monodromies around each pairs of
points $(a_{j},1/\bar{a}_{j})$ are trivial

$\begin{split}M_{\bar{j}}M_{j}=I\ .\end{split}$ (67)

This means that the mirror points $1/\bar{a}_{j}$ have monodromies in the
conjugacy class of $\Lambda^{-1}$.

The connection to the replica wormhole is that the solution to Liouville’s
equation (11), the conformal factor of the metric, is given in terms of the
fundamental system via

$\begin{split}e^{2\rho}=\big{(}|\psi_{1}|^{2}-|\psi_{2}|^{2}\big{)}^{-2}\
.\end{split}$ (68)

This is invariant under the mondromies and so $\rho$ is well defined on the
base.

###  Fuchsian uniformisation

At this point, we can give a uniformisation perspective to the construction.
The idea is to realize the Riemann surface as the unit disc $|W|=1$ modulo the
action the Fuchsian group which is a discrete subgroup
$\Sigma\subset\text{SU}(1,1)$.111111Our discussion will not address some of
the subtleties of this construction. These transformations take the form

$\begin{split}W\longrightarrow\frac{uW+v}{\bar{v}W+\bar{u}}\
,\qquad|u|^{2}-|v|^{2}=1\ .\end{split}$ (69)

One can think of the generators of $\Sigma$ as being associated to a set of
$g$ homology cycles that identify pairs of circles $(C_{a},\tilde{C}_{a})$ in
the complex $W$ plane to create a genus $g$ surface. The unit disc with the
circles cut out describes a fundamental domain ${\cal D}$ for the cover. The
boundary $\partial{\cal D}$ consists of a subset of the boundary of the disc
$|W|=1$ along with a series of arcs, the boundaries of the circles
$(C_{a},\tilde{C}_{a})$ that lie in unit disc. See figure 2 for an example
with $n=3$ and two QES. In this case there are 3 pairs of circles.

The uniformation map is defined as $\pi:\,{\cal D}\to{\cal M}_{n}$, i.e.
$W\mapsto w$. We will often consider the inverse map $W(w)$ which is $n$-fold
valued determined by the fundamental system

$\begin{split}W(w)=\psi_{1}(w)/\psi_{2}(w)\ .\end{split}$ (70)

More generally we can exploit the $\text{SU}(1,1)$ symmetry (69) to define a
different coordinate in cover
$W=(u\psi_{1}+v\psi_{2})/(\bar{v}\psi_{1}+\bar{u}\psi_{2})$. On account of
(65), the coordinate $W$ in (70) or its $\text{SU}(1,1)$ transformation
satisfies the reality condition $\overline{W(1/\bar{w})}=1/W(w)$ and that
ensures that $W(w)$ maps the unit circle to the unit circle.

The stress tensor (18), the potential of the Fuchsian equation, is identified
with the Schwarzian derivative of the map

$\begin{split}{\mathcal{T}}(w)=\frac{1}{2}\\{W(w),w\\}\ ,\end{split}$ (71)

which is also invariant under Möbius transformations and so is valued on the
base. This is the function that appears in the gravitational action (16).

Let us now elaborate on some of the details of the Fuchsian uniformisation. By
using the freedom to perform $\text{SU}(1,1)$ transformations, we can
diagonalize the monodromy of one of the branch points, say $w=a_{j}$,
$M_{j}=\Lambda$. This has an associated choice of uniformisation coordinate
$W_{j}$ for which the image of the branch point $a_{j}$ is at $W_{j}(a_{j})=0$
and that of $1/\bar{a}_{j}$ at $W_{j}(1/\bar{a}_{j})=\infty$. The local
behaviour around the branch point is

$\begin{split}W_{j}=\lambda_{j}(w-a_{j})^{1/n}\Big{(}1+\frac{c^{+}_{j}}{2n\varepsilon_{n}}(w-a_{j})+{\mathscr{O}}(w-a_{j})^{2}\Big{)}\
.\end{split}$ (72)

Note the appearance here of the accessory parameter. There is a similar
expansion around $w=1/\bar{a}_{j}$.

With this choice of coordinate for the cover, the other branch points
$w=a_{k}$ and $1/\bar{a}_{k}$, $k\neq j$, have $n$ images related by
multiplication by $n^{\text{th}}$ roots of unity that lie on $\partial{\cal
D}$ and are identified by elements of the group $\Sigma$. These are the fixed
points of the monodromies $M_{j}^{p}M_{k}M_{j}^{-p}$, $p=0,1,\ldots,n-1$. If
we write the monodromy as

$\begin{split}M_{k}=U\Lambda U^{-1}\ ,\qquad U=\begin{pmatrix}u&v\\\
\bar{v}&\bar{u}\end{pmatrix}\in\text{SU}(1,1)\ ,\end{split}$ (73)

with $|u|^{2}-|v|^{2}=1$, then the $n$ images of $a_{k}$ are at
$W_{j}(a_{k})=e^{2\pi ip/n}v/\bar{u}$ and of $1/\bar{a}_{k}$ at
$W_{j}(1/\bar{a}_{k})=e^{2\pi ip/n}u/\bar{v}$, $p=0,1,\ldots,n-1$. The
coordinates $W_{j}$ and $W_{k}$ (i.e. one for which $W_{k}(a_{k})=0$) are
related by the $\text{SU}(1,1)$ transformation

$\begin{split}W_{j}=\frac{uW_{k}+v}{\bar{v}W_{k}+\bar{u}}\ .\end{split}$ (74)

There is a natural choice of homology cycles of the Riemann surface. There are
a set of $g$ $A$-cycles that we can take as surrounding the images of
$(A_{k},1/\bar{A}_{k})$, $k\neq j$, on the $p^{\text{th}}$ copy of the base,
$p=0,1,\ldots,n-2$. By construction the monodromy around the $A$-cycles is
trivial:

$\begin{split}\big{(}M_{j}^{p}M_{k}M_{j}^{-p}\big{)}\big{(}M_{j}^{p}M_{\bar{k}}M_{j}^{-p}\big{)}=I\
.\end{split}$ (75)

On the other hand, a suitable choice of $B$-cycles have monodromies

$\begin{split}L_{k,p}=\big{(}M_{j}^{p}M_{k}M_{j}^{-p}\big{)}M_{j}\ ,\qquad
k\neq j\ ,\qquad p\in\\{0,1,\ldots,n-2\\}\ .\end{split}$ (76)

These monodromies generate a representation of the Fuchsian group $\Sigma$ and
the Riemann surface is then defined by identifying points on boundary
$\partial{\cal D}$ with the action of the group.

As we have previously mentioned there is freedom in specifying the boundary
$\partial{\cal D}$. As in Faulkner:2013yia , which considered the related
Schottky uniformisation problem, there is choice for $\partial{\cal D}$ which
respects the $\mathbb{Z}_{n}$ symmetry of the replicas. We now describe this
choice in more detail. With (73) we have

$\begin{split}L_{k,p}=\begin{pmatrix}\gamma|u|^{2}-|v|^{2}&\gamma^{p}(\gamma^{-1}-1)uv\\\
\gamma^{-p}(\gamma-1)\bar{u}\bar{v}&\gamma^{-1}|u|^{2}-|v|^{2}\end{pmatrix}\
,\end{split}$ (77)

where $\gamma=e^{2\pi i/n}$. The pairs of circles
$\tilde{C}_{k,p}=L_{k,p}(C_{k,p})$ are then given by

$\begin{split}W_{j}=-\frac{\gamma^{-1}|u|^{2}-|v|^{2}+\mu
e^{i\phi}}{\gamma^{-p}(\gamma-1)\bar{u}\bar{v}}\
,\qquad\frac{\gamma|u|^{2}-|v|^{2}+\mu^{-1}e^{i\tilde{\phi}}}{\gamma^{-p}(\gamma-1)\bar{u}\bar{v}}\
.\end{split}$ (78)

Here, $\mu$ is parameter that fixes the scales of the circles. The choice
$\mu=1$ corresponds to the choice that has the ${\mathbb{Z}}_{n}$ symmetry. In
that case, the two circles $C_{k,p+1}$ and $\tilde{C}_{k,p}$ intersect at two
points $W_{j}=\gamma^{p}v/\bar{u}$ and $\gamma^{p}u/\bar{v}$, precisely the
images of the branch points $(a_{k},1/\bar{a}_{k})$. We remark that the
symmetric choice $\mu=1$ is a non-standard definition of the fundamental
domain ${\cal D}$. The same issue was described in the context of Schottky
uniformisation in Faulkner:2013yia . In our example in figure 2, there are 3
elements of $\Sigma$ needed which identify the coloured circles. One the
cycles is shown in blue and note how it is only closed modulo the action of
the element of $\Sigma$ on the boundary of the domain. Once the idenifications
are made we can visualize the resulting smooth surface as in figure 3.

##  5 The case $n=2$ and $N=2$

For the case with $n=2$ and $N=2$, the Riemann surface has genus one and so
the map $W(w)$ can be written explicitly in terms of elliptic functions,
following a similar analysis to the Schottky uniformisation problem studied in
Faulkner:2013yia . It is worth exploring this case in some detail. A pair of
independent solutions of the Fuchsian equation (60) take the form

$\begin{split}\psi_{\pm}(w)=\frac{1}{\sqrt{t^{\prime}(w)}}e^{\pm t(w)}\
,\end{split}$ (79)

where $t(w)$ is an incomplete elliptic integral of the first kind,

$\begin{split}t(w)=\alpha\int^{w}_{-1}\frac{d\tilde{w}}{\sqrt{\prod_{j}(\tilde{w}-w_{j})}}\
,\end{split}$ (80)

with $w_{j}=(1/\bar{a}_{1},a_{1},a_{2},1/\bar{a}_{2})$. The lower limit of the
integral is arbitrary but has been chosen for convenience to be fixed under
the transformation $w\to 1/\bar{w}$ that will be important below.121212We can
write $t(w)$ explicitly in terms of the incomplete elliptic integral
$F(\varphi,k^{2})$ as
$t(w)=\frac{i\pi}{2K(x)}\big{(}F(\varphi(w),k^{2})-F(\varphi(-1),k^{2})\big{)}$,
where $k^{2}=x$ and
$\sin^{2}\varphi(w)=\frac{w_{21}}{w_{42}}\frac{w_{4}-w}{w-w_{1}}$. The
parameter $\alpha$ is fixed by demanding that $W$ has trivial monodromy around
either of the two branch cuts along $(1/\bar{a}_{1},a_{1})$, and
$(a_{2},1/\bar{a}_{2})$. We will refer to this as the $A$-cycle of the
homology basis and the requirement is,131313In principle, the right-hand side
could be multiplied by any integer but the required solution is the minimal
one Faulkner:2013yia .

$\begin{split}\alpha\oint_{A}\frac{d\tilde{w}}{\sqrt{\prod_{j}(\tilde{w}-w_{j})}}=2\big{(}t(a_{1})-t(1/\bar{a}_{1}))=\pi
i\ ,\end{split}$ (81)

which fixes $\alpha$ as,

$\begin{split}\alpha=\frac{\pi\sqrt{w_{31}w_{42}}}{4K(x)}\,,\qquad\qquad
K(x)=\int_{0}^{1}\frac{d\theta}{\sqrt{1-x\sin^{2}\theta}}\,.\end{split}$ (82)

Here $w_{ij}=w_{i}-w_{j}$, and $K(x)$ is the complete elliptic integral, with
the conformal cross-ratio $x$ defined as,

$\begin{split}x=\frac{w_{21}w_{43}}{w_{31}w_{42}}=\frac{(|a_{1}|^{2}-1)(|a_{2}|^{2}-1)}{|a_{1}\bar{a}_{2}-1|^{2}}\
.\end{split}$ (83)

Notice that the cross-ratio is real. The monodromy around the $B$-cycle of the
homology basis that surrounds the interval $(a_{1},a_{2})$ is $\psi_{\pm}\to
q^{\pm 1}\psi_{\pm}$ where

$\begin{split}\begin{pmatrix}\psi_{+}\\\
\psi_{-}\end{pmatrix}\to\begin{pmatrix}q\,\psi_{+}\\\
\frac{1}{q}\,\psi_{-}\end{pmatrix}\,,\qquad q=e^{-\pi K(1-x)/K(x)}\
,\end{split}$ (84)

where we used the fact that $2\big{(}t(a_{2})-t(a_{1})\big{)}=\pi
K(1-x)/K(x)$. In the covering space this is the monodromy around the cycle
that leads to the identification of the circles in figure 2.

Now we turn to the reality of the fundamental system $\psi_{\pm}$. One can
easily verify that

$\begin{split}\overline{t(1/\bar{w})}=t(w)\,,\end{split}$ (85)

and so $w\,\overline{\psi_{\pm}(1/\bar{w})}=\psi_{\pm}(w)$, in contrast to
(65). However, it is straightforward to find the $\text{SL}(2,{\mathbb{C}})$
transformation that produces a fundamental system that does satisfy (65);
namely,

$\begin{split}\psi_{1}=\frac{1}{\sqrt{2}}(\psi_{+}+i\psi_{-})\
,\qquad\psi_{2}=\frac{1}{\sqrt{2}}(\psi_{+}-i\psi_{-})\ .\end{split}$ (86)

Using this fundamental system we can write down the inverse of the
uniformisation map as in (70). We can define coordinates in the cover that
vanish at the image of each of the branch points $w=a_{j}$ via

$\begin{split}W_{j}(w)=\frac{\psi_{1}(w)-\xi_{j}\psi_{2}(w)}{-\bar{\xi}_{j}\psi_{1}(w)+\psi_{2}(w)}\
,\end{split}$ (87)

where

$\begin{split}\xi_{j}=\frac{e^{2t(a_{j})}+i}{e^{2t(a_{j})}-i}\ .\end{split}$
(88)

12${W_{1}(a_{2})}$${W_{1}(1/\bar{a}_{2})}$${-W_{1}(a_{2})}$${-W_{1}(1/\bar{a}_{2})}$${\cal
D}$ Figure 5: The fundamental domain ${\cal D}$ for 2 QES and replica number
$n=2$ in the coordinate with $W_{1}(a_{1})=0$. The images of the two branch
points in the $w$-plane at $w_{3}=a_{2}$ and $w_{4}=1/\bar{a}_{2}$ are
depicted as solid black circles. The arcs of the circles cut out from the disc
are identified with their respective images. The fundamental domain is a
cylinder with 2 disjoint boundaries.

The exact expression for the accessory parameters can be computed from the
poles of the Liouville stress tensor ${\mathcal{T}}(w)$ in (61)

$\begin{split}c_{j}=-\frac{6\sum_{k}w_{jk}^{2}-\sum_{kl}w_{kl}^{2}+32\alpha^{2}}{32\prod^{\prime}_{k}w_{jk}}\
,\end{split}$ (89)

where $c_{j}=(c^{-}_{1},c^{+}_{1},c_{2}^{+},c_{2}^{-})$ and the product in the
denominator excludes the term with $k=j$. In addition, the coefficients
$\lambda_{j}$ in the expansions (72) of the covering space coordinates around
a branch point, are

$\begin{split}\lambda_{j}=-\frac{4\alpha
e^{2t(a_{j})}(e^{2\bar{t}(a_{j})}+i)}{e^{2t(a_{j})}-e^{2\bar{t}(a_{j})}}\frac{1}{\sqrt{\prod^{\prime}_{k}w_{jk}}}\
.\end{split}$ (90)

###  Map to the torus

For the genus one replica geometry one can, of course, directly construct a
map from the $w$-plane to the torus through the elliptic parametrisation in
terms of the Weierstrass $\wp$-function,

$\displaystyle{\wp}(z)=\frac{h^{\prime}(w_{4})}{4(w-w_{4})}+\frac{h^{\prime\prime}(w_{4})}{24}\,,\qquad\qquad
h(w)\equiv\prod_{j=1}^{4}(\tilde{w}-w_{j})\,.$ (91)

$A$$B$$2{\bm{\omega}}_{2}$$2{\bm{\omega}}_{1}$${{{\cal
D}}}$${\bm{\omega}_{1}}$${\bm{\omega}_{2}}$${\bm{\omega}_{3}}$ Figure 6: The
two-sheeted replica geometry with two branch cuts, can be mapped to the torus
with periods $(2{\bm{\omega}}_{1},2{\bm{\omega}}_{2})$ represented as the
fundamental parallelogram with opposite edges identified. The shaded region in
blue is the image of the fundamental domain ${\cal D}$ for $n=N=2$ on the
torus, with the $A$\- and $B$-cycles of the homology basis shown, the latter
lying entirely inside ${\cal D}$. The replica wormhole geometry is thus a
cylinder connecting two boundaries.

This transformation maps the branch points $\\{w_{i}\\}_{i=1\ldots 4}$ to
distinguished points on the parallelogram with period
$(2\bm{\omega}_{1},2\bm{\omega}_{2})$ (see figure 6):141414The two periods of
the torus are determined in terms of the branch point locations $\\{w_{i}\\}$
as $2{\bm{\omega}}_{1}=-i\pi/\alpha$ and $2{\bm{\omega}}_{2}=\pi
K(1-x)/(\alpha K(x))$, where $\alpha$ is given by (82).

$z(w_{1})=\bm{\omega}_{2}\,\qquad
z(w_{2})=\bm{\omega}_{1}+\bm{\omega}_{2}\,\qquad
z(w_{3})=\bm{\omega}_{1}\qquad z(w_{4})=0\ .$ (92)

The fundamental domain ${\cal D}$ maps to a strip along the $B$-cycle of the
torus, which is a finite cylinder with two boundaries. In the elliptic
parametrisation $t(w)$ simply yields a point on the torus and is identified,
upto a multiplicative constant, with the uniformised coordinate on the torus:

$t(w)=-\alpha\,(z-z_{-1})\,,\qquad z_{-1}\,\equiv\,z(-1)\,,$ (93)

where the constant $\alpha$ is fixed as in eq. (82). The complex structure
parameter of the torus is

$\bm{\tau}\equiv\frac{\bm{\omega}_{2}}{\bm{\omega}_{1}}=i\frac{K(1-x)}{K(x)}\,,$
(94)

which diverges in the small $x$ limit as,

$\bm{\tau}\big{|}_{x\ll 1}\approx-\frac{i}{\pi}\log x\,,$ (95)

so the $A$-cycle shrinks to zero in this regime. The metric on the fundamental
domain in uniformised coordinates is particularly simple:

$ds^{2}=\frac{dt\,d\bar{t}}{4\sin^{2}(2\,{\rm Im}\,t)}\,,$ (96)

and the two disjoint boundaries of ${\cal D}$ correspond to ${\rm Im}\,t=0$
and ${\rm Im}\,t=\pi/2$, a strip of width set by the $A$-cycle half-period
${\bm{\omega}}_{1}$. This is precisely the double trumpet wormhole geometry
(see eg. Garcia-Garcia:2020ttf ).

###  Factorization limit and late times

In the case of two QES when the cross ratio in (83) $x\ll 1$, we approach the
factorization limit. In the Euclidean set up, this happens when $a_{j}$ and
$1/\bar{a}_{j}$ are becoming close, i.e. $|a_{j}|\to 1$ and $|a_{j}-a_{k}|$ is
fixed.

In Lorentzian signature, the conformal cross-ratio as a function of the light-
cone coordinates of the QES is,

$x=\frac{(a^{+}_{1}a^{-}_{1}+1)(a^{+}_{2}a^{-}_{2}+1)}{(a^{+}_{1}a^{-}_{2}+1)(a^{-}_{1}a^{+}_{2}+1)}\,.$
(97)

The configuration with 2 QES naturally contributes to the generalised entropy
computation of the disjoint union of two semi-infinite radiation intervals
prepared in the thermofield double state Hollowood:2021wkw , and in terms of
the bath coordinates,

$a_{1}^{\pm}=\mp e^{\pm 2\pi/\beta(-t\pm y_{1})},\qquad a_{2}^{\pm}=\pm e^{\pm
2\pi/\beta(t\pm y_{2})}\,.$ (98)

factorization is naturally achieved at late times when
$x\sim{\mathscr{O}}(e^{-4\pi t/\beta})$, and when the replica wormhole saddle
is also clearly the dominant contribution to the Rényi entropy.

With the explicit integral expression for the map $W(w)$ in (87), we find that
for $w$ in the neighbourhood of, say $a_{1}$ and $1/\bar{a}_{1}$, as expected,
the map approximates the single QES situation (24) with $n=2$. In this limit,
the accessory parameters have an expansion in powers of $x$, around the $N=1$
limit (63),

$\begin{split}c^{\pm}_{j}=\pm\frac{3\bar{a}_{j}}{8(1-|a_{j}|^{2})}\big{(}1-\tfrac{1}{16}x^{2}+{\mathscr{O}}(x^{3})\big{)}\
.\end{split}$ (99)

###  The fundamental domain in the factorization limit

What is the fate of the fundamental domain ${\cal D}$ in the factorization
limit $x\ll 1$?

To answer this, it is useful to examine the two coordinates $W_{1}$ and
$W_{2}$ when $x\ll 1$. These are defined via $W_{j}(a_{j})=0$, and related by

$\begin{split}W_{1}=\frac{(1-\xi_{1}\bar{\xi}_{2})W_{2}+\xi_{2}-\bar{\xi}_{1}}{(\bar{\xi}_{2}-\xi_{1})W_{2}+1-\bar{\xi}_{1}\xi_{2}}\
.\end{split}$ (100)

This is an explicit expression for the $\text{SU}(1,1)$ transformation in
(74). It follows that the images of the branch points $w=a_{j}$ in the
covering coordinates $W_{k\neq j}$ are

$\begin{split}W_{1}(a_{2})=\frac{v}{\bar{u}}=\frac{\xi_{2}-\bar{\xi}_{1}}{1-\bar{\xi}_{1}\xi_{2}}\
,\qquad
W_{2}(a_{1})=-\frac{v}{u}=-\frac{\xi_{2}-\bar{\xi}_{1}}{1-\xi_{1}\bar{\xi}_{2}}\,.\end{split}$
(101)

The parameters have nontrivial scaling in the factorization limit $x\to 0$.
For example, with $a_{i}$ real, $a_{1}\to-1$, and $a_{2}$ fixed, we find
$|e^{t(a_{2})-t(a_{1})}|\sim x^{-1/2}$ ,

$\begin{split}|W_{1}(a_{2})|=|W_{2}(a_{1})|=1-{\mathscr{O}}(x).\end{split}$
(102)

More generally for any $n$, given a choice of coordinate $W_{i}$, with a
particular QES $a_{i}$ being at the origin in the covering space, the images
of the other QES $a_{j\neq i}$ are approaching the boundary of the disc
$|W_{i}|=1$ in the small $x$ regime. Hence, in this limit, the associated
circles of the Fuchsian uniformisation in figures 2 and 5 become very small,
151515More precisely, the radii are $|2uv|^{-1}\sim x$. We also note that the
coefficients $\lambda_{1}\sim 1/x^{3/2}$ and $\lambda_{2}\sim{1}/{\sqrt{x}}$
as $x\to 0$, taking $a_{1}\to 1/\bar{a}_{1}$ keeping $a_{2}$ fixed, and vice
versa if we swap the roles of $a_{1}$ and $a_{2}$.  and the fundamental domain
${\cal D}$ for both choices of coordinate, is essentially the whole of the
unit disc.

###  Off-shell gravity action

The basic form of the gravity action is formally easy to determine in terms of
the accessory parameters and the (undetermined) boundary map $w(\tau)$, as
this follows from the singularities of the Liouville stress tensor (61). For
general $n$ it is,

$\begin{split}&I_{\rm
JT}^{(n)}[w;a_{j},\bar{a}_{j}]=-NS_{0}-\frac{n\phi_{r}}{4\beta
G_{N}}\int_{|w|=1}d\tau\\{w,\tau\\}\\\\[5.0pt] &\qquad-\frac{i\pi
n\phi_{r}}{\beta
G_{N}}\sum_{j}\Big{\\{}\varepsilon_{n}\partial_{w}\dot{w}_{+}(a_{j})+c^{+}_{j}\dot{w}_{+}(a_{j})-\varepsilon_{n}\partial_{w}\dot{w}_{-}(1/\bar{a}_{j})-c^{-}_{j}\dot{w}_{-}(1/\bar{a}_{j})\Big{\\}}\
.\end{split}$ (103)

The main obstacle to computing this precisely is (lack of) knowledge of the
boundary map $w(\tau)$ which is determined by the energy balance equation
across the interface (23) that is generally quite complicated to solve via the
welding problem. But we can make good progress in the small $\kappa$, high
temperature limit (45) and work at leading order so that,

$w(\tau)=e^{i\tau}+{\mathscr{O}}(\kappa)\,,\qquad\dot{w}(w)=iw+{\mathscr{O}}(\kappa)\,.$
(104)

As per our conventions in (49), then $\dot{w}_{+}=iw$ and $\dot{w}_{-}=0$ so,

$I_{\rm JT}^{(n)}\big{|}_{\kappa\to 0}=-NS_{0}+\frac{n\pi\phi_{r}}{\beta
G_{N}}\left(\sum_{j=1}^{N}c_{j}^{+}a_{j}^{+}+N\varepsilon_{n}-\tfrac{1}{4}\right)\,.$
(105)

The off-shell QES dependence of the gravity action is completely determined by
the accessory parameters which depend non-trivially on the $\\{a_{j}\\}$ and
$\\{\bar{a}_{j}\\}$ as in (89) for $n=2$ with $N=2$.

###  Dilaton dependence: a first pass

Although it is easy to write down the explicit dependence of the gravity
action on the positions of the branch points, it is less obvious how the
action can be re-expressed in terms of dilaton. In particular, how does the
area contribution to generalised Rényi entropies appear when there is more
than one branch point ?

To gain some intuition, it is useful to express the gravity action in terms of
the dilaton function evaluated at the branch point (e.g. (58)), when there is
only one QES i.e. $N=1$:

${\phi}_{1}(a,\bar{a})\equiv\frac{2\pi\phi_{r}}{n\beta}\frac{1+a\bar{a}}{1-a\bar{a}}\,.$
(106)

We will treat $a$ and $\bar{a}$ as independent variables, which is the
appropriate viewpoint, particularly upon analytic continuation to Lorentzian
light-cone variables. Then, using (89) and (105), we find that the gravity
action for $n=2$, at leading order in the small $\kappa$ limit, with two
branch points is,

$\displaystyle I_{\rm
JT}^{(2)}=-2S_{0}+\frac{3}{2}\cdot\frac{1}{4G_{N}}\Big{(}\phi_{1}(a_{1},\bar{a}_{1})+\phi_{1}(a_{2},\bar{a}_{2}){\Big{)}}\big{(}1-{\cal
K}(x)\big{)}\,,$ (107) $\displaystyle{\cal
K}(x)\,\equiv\,\frac{1}{12}\left(-\frac{\pi^{2}}{K(x)^{2}}+4-2x\right)\,=\,\frac{x^{2}}{32}+\frac{x^{3}}{64}+\ldots\,.$

$x$ is the cross-ratio defined in (83), which, upon continuation to Lorentzian
signature, is replaced by (97). In the factorization limit when $x\to 0$, we
recover the expected dependence of the action as the sum of dilaton
contributions, with the correct normalisation, evaluated at each of the branch
points (30). The corrections to this expression are encoded in ${\cal K}(x)$.
Coefficients in the expansion of ${\cal K}(x)$ around $x=0$ are all positive
and the function is monotonic in the domain $0\leq x<1$.

Given that we have a closed form expression for the off-shell gravity action
for all $x$, it would be very interesting to apply the formula to compute the
generalised $n=2$ modular entropy in the time evolving evaporating black hole
setup. In section 8, we will perform a simpler analysis in the factorised
limit to compute the late time value of the generalised Rényi entropy and its
Page time as a function of the replica number $n$.

Whilst the gravity action can be determined completely as an off-shell
function of QES locations, to understand how the generalised Rényi entropy
depends on the area functional, we need the dilaton evaluated at the branch
points in the $N=2$ situation. This should be obtained by solving the dilaton
equation of motion (14) in the double trumpet background (96), written in
terms of the boundary curves, generalising (27). We leave a detailed study of
this to future work. Below we will explain how this works for general branch
point number $N$ and any $n$, and provide nontrivial confirmation of the
intuitive picture above: a simple generalised modular entropy appears
naturally in a factorization limit, corrections to which are computable and
encode the backreaction of branch points/cosmic branes on each other.

It is interesting to note that for the second Rényi entropy, cross-ratio (83)
can also be expressed in terms of the $N=1$ dilaton as,

$x=\frac{\phi_{1}(a_{1},\bar{a}_{2})+\phi_{1}(a_{2},\bar{a}_{1})}{\phi_{1}(a_{1},\bar{a}_{1})+\phi_{1}(a_{2},\bar{a}_{2})}\,.$
(108)

This yields a complicated, nonlinear dependance on the naive $(N=1)$ area
functional, encoding interactions between the branch points.

##  6 Liouville theory and the accessory parameters

For general numbers of replicas and branch points, the accessory parameters
are not known. However, there is a way to relate them to a particular
conformal block of quantum Liouville theory in an appropriate classical limit
zam2 ; Perlmutter:2015iya ; Zam1 ; Harlow:2011ny ; Hartman:2013mia ;
Litvinov:2013sxa . The block describes a $2N$-point correlation function with
insertions at the points $w_{j}$, collectively the set of all the pairs
$(a_{j},1/\bar{a}_{j})$, with operators of dimension
$h_{j}=\varepsilon_{n}c/6$. The fact that the monodromy problem has
$M_{\bar{j}}M_{j}=I$ means that the block has the identity operator and its
descendants (the Virasoro channel) on each of the internal lines as shown for
the case $N=2$ in figure 7. This completely specifies the block ${\cal
F}(w_{j};h_{j}=\varepsilon_{n}c/6,h_{a}=0)$, where the $h_{a}$ are the
dimensions of the highest weight representations exchanged on the internal
lines. In the classical limit $c\to\infty$, there is a notion of a classical
conformal block defined via

$\begin{split}f(w_{j})=\lim_{c\to\infty}\,-\frac{6}{c}\log{\cal
F}(w_{j};\varepsilon_{n}c/6,0)\ .\end{split}$ (109)

The accessory parameters are then simply given by differentiating the
classical block:

$\begin{split}c_{w_{j}}=-\partial_{w_{j}}f(w_{j})\ ,\end{split}$ (110)

with $c_{w_{j}}\equiv c^{+}_{j}$ for $w_{j}=a_{j}$ and $c_{w_{j}}\equiv
c^{-}_{j}$ for $w_{j}=1/\bar{a}_{j}$.

$1/\bar{a}_{1}$$a_{1}$$1/\bar{a}_{2}$$a_{2}$$I$ Figure 7: The fusion channel
for the Liouville correlator. The internal line with the label $I$ denotes the
descendants of the identity operator.

For example, for the case with 2 branch points $a_{j}$, $j=1,2$, the
correlator is a 4-point function and in general we can write ${\cal
F}(w_{j})=w_{21}^{-2h}w_{43}^{-2h}\tilde{\cal F}(x)$ where $x$ is the cross
ratio (83). This gives the classical block in the form161616The function
$\tilde{f}(x)$ is often defined to have a term $2\varepsilon_{n}\log x$. We
have included this in the first term.

$\begin{split}f(w_{j})=2\varepsilon_{n}\log(w_{21}w_{43})+\tilde{f}(x)\
.\end{split}$ (111)

The block can naturally be expanded in the cross ratio (83) and remarkably
$\tilde{f}(x)$ can be determined to arbitrary order in the auxiliary expansion
parameter $q$ defined in (84) via a recursion relation Zam1 (see also
Hartman:2013mia ; Perlmutter:2015iya ). Expanding the first few term of the
$q$ expansion in powers of $x$ gives

$\begin{split}\tilde{f}(x)=-\frac{\varepsilon_{n}^{2}x^{2}}{3}-\frac{\varepsilon_{n}^{2}x^{3}}{3}+\frac{\varepsilon_{n}^{2}}{270}(11\varepsilon_{n}^{2}-6\varepsilon_{n}-81)x^{4}+{\mathscr{O}}(x^{5})\
.\end{split}$ (112)

For the case $n=2$, we can compare the exact expression for the accessory
parameter in (89) with the expansion in $x$ above and find exact agreement. At
leading order in the small $x$ expansion, the accessory parameter is
determined by the the term $\sim(\log w_{21}w_{43})$ in (111),

$\begin{split}c_{j}^{\pm}=\pm\frac{2\varepsilon_{n}\bar{a}_{j}}{1-|a_{j}|^{2}}+{\mathscr{O}}(x)\
.\end{split}$ (113)

as expected for the single QES problem. The connection with Liouville theory
also provides explicit expressions for the coefficients $\lambda_{j}$ in (72)
Hadasz:2006rb .

##  7 The dilaton

In this section, we turn to the evaluation of the dilaton at branch points
when $N>1$. The dilaton at an arbitrary point in the covering space is
expressed as a contour integral over the boundary of the disc as we quoted
earlier (27) for the case $N=1$. As explained in Appendix A, when $N>1$, this
expression is not the complete story since the fundamental domain ${\cal D}$
has partial circles cut out of it reflecting additional cycles with associated
gluing conditions. However, in the factorisation limit $x\to 0$, when these
partial circles are parametrically small, we can continue to make use of the
$N=1$ expression (27) as a systematic approximation to the late time limit.

Let us evaluate the integral (27) by picking up residues as in the $N=1$ case
in section 3. Note that the derivative $\dot{W}=\partial_{w}W\dot{w}$ is to be
thought of as a function on the cover. The next step to facilitate the
evaluation of the integral, is to split $\dot{w}(w)$ as in (36) where
$\dot{w}_{\pm}(w)$ are analytic inside/outside the unit disc $|w|=1$.

Proceeding as for $N=1$, to evaluate the integral (27), we pull the contour
off to 0 for $\dot{w}_{+}$ and $\infty$ for $\dot{w}_{-}$. There are poles of
order 2 coming from the denominator and poles of order $n-1$ from the images
of the branch points $W(a_{j})$ inside the circle and $W(1/\bar{a}_{j})$
outside the circle. The new element in the discussion is that there will be
additional poles coming from the circular regions that are cut out to define
the fundamental domain ${\cal D}$ of the uniformisation, as shown in figure 2.

Firstly, consider the contribution of the double poles coming from from the
denominator of (136). For $\tilde{W}=W$ and $1/\bar{W}$, respectively, we have
a contributions

$\begin{split}\phi_{0}^{+}&=-\frac{2\pi
i\phi_{r}}{\beta}\Big{(}\frac{2\bar{W}\dot{W}_{+}(W)}{1-|W|^{2}}+\partial_{W}\dot{W}_{+}(W)\Big{)}\
,\\\\[5.0pt] \phi_{0}^{-}&=-\frac{2\pi
i\phi_{r}}{\beta}\Big{(}\frac{2\bar{W}\dot{W}_{-}(1/\bar{W})}{1-|W|^{2}}-\partial_{W}\dot{W}_{-}(1/\bar{W})\Big{)}\
.\end{split}$ (114)

Now we turn to the contribution from one of the images of the branch points,
e.g. one inside the disc $\tilde{W}=W(a_{k})$, coming from the factor
$\partial_{w}W$. It is simpler to work in the coordinate $W_{k}$ that vanishes
at $w=a_{k}$. We denote our arbitrary point as $(W_{k},\bar{W}_{k})$ in this
coordinate. The behaviour around the branch point is

$\begin{split}\dot{W}_{+}=\frac{\lambda_{k}^{n}(W_{k})^{1-n}}{n}\Big{\\{}\dot{w}_{+}(a_{k})+\big{(}\varepsilon_{n}^{-1}c^{+}_{k}\dot{w}_{+}(a_{k})+\partial_{w}\dot{w}_{+}(a_{k})\big{)}(W_{k}/\lambda_{k})^{n}+\cdots\Big{\\}}\
.\end{split}$ (115)

Picking up the residue coming from the first term in the expansion (115),
gives a contribution

$\begin{split}\phi_{k}^{+}=-\frac{2\pi
i\phi_{r}}{n\beta}(1-|W_{k}|^{2})^{2}\lambda_{k}^{n}d_{k}\dot{w}_{+}(a_{k})\
,\end{split}$ (116)

where

$\begin{split}d_{k}&=\oint_{0}\frac{dy}{2\pi
i}\,\frac{y^{1-n}}{(W_{k}-y)^{2}(1-y\bar{W}_{k})^{2}}\\\\[5.0pt]
&=\frac{W_{k}^{n}(n(|W_{k}|^{2}-1)-|W_{k}|^{2}-1)+\bar{W}_{k}^{n}(n(|W_{k}|^{2}-1)+|W_{k}|^{2}+1)}{(|W_{k}|^{3}-1)^{3}}\
.\end{split}$ (117)

There is a similar contribution from rom the branch point
$\tilde{W}=W(1/\bar{a}_{k})$. The behaviour of the coefficient $d_{k}$ is
important for our analysis. First of all, $d_{k}$ vanishes when $n=0,\pm 1$
and has the following expansion around $n=1$:

$\begin{split}d_{k}=(n-1)\frac{|W_{k}|^{2}-1-2|W_{k}|\log|W_{k}|}{W_{k}(|W_{k}|-1)^{3}}+{\mathscr{O}}(n-1)^{2}\
.\end{split}$ (118)

On the other hand, for general $n$, $d_{k}$ has the following expansion in
$1-|W_{k}|$:

$\begin{split}d_{k}=\frac{n(n^{2}-1)}{6}+{\mathscr{O}}(1-|W_{k}|)\
.\end{split}$ (119)

In general, we have an expression that consists of a sum of the contribution
from the double poles and multiple images of the QES. This is rather difficult
to handle but actually what we really want is the dilaton evaluated at one of
the QES. In this case, each double pole merges with one of the images, so for
$\tilde{W}=W$ with $\tilde{W}=W(a_{j})$, and, although separately divergent,
together they give a finite contribution

$\begin{split}\phi_{0}^{+}+\phi_{j}^{+}=-\frac{2\pi
i\phi_{r}}{n\varepsilon_{n}\beta}\Big{(}c_{j}^{+}\dot{w}_{+}(a_{j})+\varepsilon_{n}\partial_{w}\dot{w}_{+}(a_{j})\Big{)}\
,\end{split}$ (120)

which comes from the subleading term in (115). There is a similar expression
for the poles at $1/\bar{W}$ and $W(1/\bar{a}_{j})$

$\begin{split}\phi_{0}^{-}+\phi_{j}^{-}=-\frac{2\pi
i\phi_{r}}{n\varepsilon_{n}\beta}\Big{(}c_{j}^{-}\dot{w}_{-}(1/\bar{a}_{j})-\varepsilon_{n}\partial_{w}\dot{w}_{-}(1/\bar{a}_{j})\Big{)}\
.\end{split}$ (121)

Let us summarize what we have established do far. The dilaton evaluated at the
QES $(a_{j},\bar{a}_{j})$ (we use coordinates in the base here) can be
expressed in terms of the boundary map as

$\begin{split}\phi(a_{j},\bar{a}_{j})=-\frac{2\pi
i\phi_{r}}{n\varepsilon_{n}\beta}\Big{(}c_{j}^{+}\dot{w}_{+}(a_{j})+\varepsilon_{n}\partial_{w}\dot{w}_{+}(a_{j})+c_{j}^{-}\dot{w}_{-}(1/\bar{a}_{j})-\varepsilon_{n}\partial_{w}\dot{w}_{-}(1/\bar{a}_{j})\Big{)}+\Delta\phi_{j}\
.\end{split}$ (122)

The final term here schematically encodes the contribution from all the other
images of the branch points in the covering space $|W|\leq 1$ in or on the
boundaries of the uniformisation circles. Although this correction spoils the
potential simplicity of the expression for the dilaton evaluated at the branch
points, importantly, we expect it to be suppressed in the small $x$
factorization limit.

For the case of a single branch point the additional corrections
$\sim\Delta\phi_{j}$ are, of course, absent because there is only one image
$W(a)$ in the disc. In this case, the accessory parameter is exactly (113) and
the expression (122) yields (38).

###  The gravitational action

As noted previously, the gravitational action (16) consists of two pieces. The
second part contains the meromorphic function ${\mathcal{T}}(w)$ and can be
computed using Cauchy’s residue theorem after expressing $\partial_{\tau}w$ as
a function of $w$ with the decomposition
$\dot{w}(w)=\dot{w}_{+}(w)+\dot{w}_{-}(w)$, which yields (103). If we compare
this to the expression for the dilaton evaluated at the branch points (122),
we immediately have

$\begin{split}I_{\text{JT}}[w;a_{j},\bar{a}_{j}]=-NS_{0}+\frac{n^{2}-1}{8G_{N}}\Big{(}\sum_{j}\phi(a_{j},\bar{a}_{j})+\Phi_{n}\Big{)}-\frac{n\phi_{r}}{4\beta
G_{N}}\int d\tau\,\\{w,\tau\\}\,,\end{split}$ (123)

where the $\Phi_{n}$ term collects all the additional contributions
$-\sum_{j}\Delta\phi_{j}$. This is our main result since it relates the
gravitational action to the dilaton evaluated at the QES. The dependence on
the $a_{j}$ is not only in the explicit positions at the which the dilaton is
evaluated but there will also be implicit dependence when the boundary map is
put on shell.

##  8 Eternal black hole and Page times for Rényi entropies

As a concrete application we consider the case of the eternal black hole with
two QES as shown in figure 8. Once continued to Lorentzian signature, this
configuration describes the island saddle of a radiation region that spans the
left and right bath regions. We choose the end points to lie just outside the
interface between the AdS region and the baths. We also take the setup to be
symmetric in Lorentzian signature for simplicity, so with coordinates

$\begin{split}b_{1}=-b^{-}_{1}=-\bar{b}_{2}=-b^{+}_{2}=e^{-t_{0}}\
,\qquad\bar{b}_{1}=b_{1}^{+}=-b_{2}=b_{2}^{-}=e^{t_{0}}\ .\end{split}$ (124)

Again the Lorentzian time here is re-scaled in units of $\beta/2\pi$. The QES
of the island saddle will also inherit this symmetry, so
$a_{1}=-a^{-}_{1}=-\bar{a}_{2}=-a^{+}_{2}$ and
$\bar{a}_{1}=-a^{+}_{1}=-a_{2}=a_{2}^{-}$.

$a_{1}$$b_{1}$$a_{2}$$b_{2}$$I$$R$$R$$b_{1}^{\pm}$$b_{2}^{\pm}$$a_{1}^{\pm}$$a_{2}^{\pm}$
Figure 8: The set up in (left) Euclidean (right) Lorenztian signature with two
QES in the context of the eternal black hole and a radiation region $R$ that
covers the left and right bands, as shown. There is a point in the right
Minkowski bath and one QES as shown ultimately outside the horizon of the
right black hole in the Lorentzian picture and the mirror image on the left.

The next task is to solve the equation of motion (23) to find the boundary map
with the QES off shell to linear order in $\kappa$. This would be a formidable
task, dealing with the multiple poles in the Liouville stress tensor
${\mathcal{T}}(w)$ without some kind of further approximation. We already know
in the context of the $n\to 1$ limit and the von Neumann entropy that the
island saddle can only become dominant over the “no-island”, or Hawking,
saddle in the late time $t_{0}\gg 1$ limit.

So we work in the limit of small $e^{-t_{0}}\ll 1$, alongside the high
temperature limit $\kappa\ll 1$, although it is important that there is a
hierarchy of scales $e^{-t_{0}}\ll\kappa\ll 1$. This separation of scales
follows from the fact that $t_{0}$ is of the order of the Page time or
greater, when the island saddle can dominate, and $|\log\kappa|$ sets the
scrambling time (measured in units of $\beta/2\pi$).

In the late time limit, it can be seen that locations of the poles at $a_{1}$
and $1/\bar{a}_{1}$ will end up scaling like $b_{1}$ and $1/\bar{b}_{1}$, i.e.
as $e^{-t_{0}}$, whereas $a_{2}$, $1/\bar{a}_{2}$ will end up scaling like
$b_{2}$ and $1/\bar{b}_{2}$, i.e. as $e^{t_{0}}$. So the boundary map in the
vicinity of $a_{1}$ and $1/\bar{a}_{1}$ is decoupled from the poles at $a_{2}$
and $1/\bar{a}_{2}$, and vice-versa. The corrections would come in at order
$e^{-2t_{0}}$. So at lower order, from the point-of-view of the boundary map,
we can essentially use the 1 QES result in the vicinity of each pair
$(a_{j},1/\bar{a}_{j})$. There are also corrections of order $e^{-3t_{0}}$
from the expansions of the accessory parameters $c_{j}^{\pm}$ to the leading
order expressions (113). There are also corrections of the same order in
relating the gravitational action to the dilaton, the additional inter-QES
terms $\Delta\phi_{j}$ in (123) are also of the same order $e^{-2t_{0}}$.
Finally the CFT entropy will take the form of the sum of two decoupled
expressions for the left and right black holes up to terms of order
$e^{-2t_{0}}$.

We conclude that to order $e^{-2t_{0}}$, the left and right hand sides can be
treated as factorized as far as the gravitational action, boundary map and CFT
entropy is concerned. This is also true of the functions $F_{j}(w)$ and
$G_{j}(w)$ that solve the conformal welding problem which now have an
approximate form on the right, $j=1$, and the left $j=2$. These are of the
form (52) with $b\to b_{j}$ and $\bar{b}\to\bar{b}_{j}$.

The modular entropy of the no-island saddle is equal to

$\begin{split}\tilde{S}_{n}^{\text{gen}}(\emptyset)&=\frac{c}{6n}\log\frac{|F_{1}(b_{1})-F_{2}(b_{2})|^{2}}{|F^{\prime}_{1}(b_{1})F^{\prime}_{2}(b_{2})b_{1}b_{2}|}\\\\[5.0pt]
&=\frac{c}{3n}\Big{(}t_{0}-\frac{3\kappa(n^{2}-1)}{4(4n^{2}-1)}+\cdots\Big{)}\
,\end{split}$ (125)

where the corrections involve powers of $\kappa$ and terms of order
$e^{-2t_{0}}$. For the island saddle, we need only take twice the entropy of
the island saddle in the one QES scenario in (56) with $|b|=1$:

$\begin{split}\tilde{S}_{n}^{\text{gen}}(I)=\frac{4}{n+1}S_{0}+\frac{c}{3n}\Big{(}\frac{1}{\kappa}-2\kappa-\frac{9\kappa(n^{2}-1)}{4(4n^{2}-1)}+\cdots\Big{)}\
.\end{split}$ (126)

Hence, the Page time, the time when the two saddles have equal entropy, occurs
when

$\begin{split}t_{0}=\frac{6\beta nS_{0}}{\pi
c(n+1)}+\frac{\beta}{2\pi}\Big{(}\frac{1}{\kappa}-\frac{\kappa(19n^{2}-7)}{2(4n^{2}-1)}+\cdots\Big{)}\
,\end{split}$ (127)

where we have re-instated the units $\beta/2\pi$.

##  9 Discussion

We have studied the computation of generalised Rényi entropies in a CFT
radiation bath coupled to a black hole within a dynamical JT gravity
framework, using the replica trick for finite replica number $n$. Unlike the
$n\to 1$ limit wherein the back-reaction of the cosmic branes or twist fields
on the background and on each other can be neglected, we are now required to
consider these effects carefully.

Employing the theory of Fuchsian uniformisation we were able to obtain the
general form of the action functional for the off-shell locations of the QES,
in terms of accessory parameters of the associated Liouville stress tensor,
and the dynamical boundary map. The boundary map which determines the gluing
between the gravitating region and the bath is the most technically
challenging piece of the puzzle, but one which can largely be circumvented by
working in the high temperature, or small scrambling time, limit (see e.g.
Almheiri:2019qdq ). Particularly noteworthy is the second Rényi entropy for
two quantum extremal surfaces, when the underlying Riemann surface is a torus
and for which the accessory parameter and unformisation map can be written in
closed form.

Once the gravitational action functional for the QES locations is known, the
computation of generalised Rényi entropies can proceed automatically by
extremising the sum of the gravity and CFT partition functions on the replica
geometry with new branch points at the QES.

However, one of our conceptual goals was to understand how this functional can
be interpreted as a generalised Rényi entropy. The answer to this question
depends on how the dilaton in JT gravity (which plays the role of the area)
evaluated at the QES locations is related to the action functional. With one
QES, and generic $n$, the answer is clean: the dependence of the gravity
action on QES locations is, up to an $n$-dependent normalisation constant,
precisely proportional to the dilaton. The $n$-dependent proportionality
constant is important, and leads us to unambiguously identify the correct
generalised entropy for arbitrary replica number as the generalised modular
entropy exactly as might be expected from independent holographic
considerations Dong:2016fnf . For multiple QES, the relation between the
action functional and the dilaton at the branch points is more complicated.
The complications however are subleading in the late time limit, precisely
when island saddles dominate, wherein individual QESs decouple from each other
and we recover a relatively simpler extremisation problem involving the
generalised modular entropy functional for multiple QES and multiple
intervals. The correction terms collectively denoted as $\Phi_{n}$ in the
gravitational action (123) are parametrically small and governed by the size
of the Fuchsian uniformisation circles.

There are some immediate questions that we wish to address in future work:

  * •

We have focussed attention on the Euclidean problem in this paper. An
immediate pressing question is how the approach developed here can be applied
to nonequilibrium entanglement evolution within a Lorentzian setting.
Understanding the construction of replica wormholes and the computation of
time dependent Rényi entropy (arbitrary $n$) for the Hawking radiation from an
evaporating, single-sided black hole poses interesting technical and
conceptual challenges even for the single QES problem.

  * •

A key step in deducing the dilaton dependence of the gravity action on the
replica geometry, is the evaluation of the dilaton at the branch points or
QES. For the single QES problem this is achieved through a representation of
the dilation as an integral over the boundary, where the integrand depends on
the boundary map. A similar representation is possible on the covering space
in the multiple QES situation, but since the fundamental domain ${\cal D}$ is
no longer the unit disc, there are additional contributions to this integral
from within the circles associated to Fuchsian uniformisation. Determining the
additional contributions is not straightforward, but the two QES problem with
$n=2$, offers a tractable situation where these corrections can be completely
determined. In the latter case, the fundamental domain is a cylinder with two
boundaries, with the double trumpet metric (96) and the dilaton equation of
motion should be expressible as an integral over the disjoint boundaries. It
would be extremely interesting to understand the corrections to the area term
in gravity action in this case, and potentially understand their relation to
similar terms in the holographic context Dong:2016fnf .

###### Acknowledgements.

TJH and SPK would like to acknowledge support from STFC Consolidated Grant
Award ST/X000648/1. LCP is supported by a joint STFC studentship ST/V507143/1
and EPSRC DTP, Faculty of Science & Engg, Swansea U. SPK would like to thank
the Fields, Gravity & Strings group, Laboratoire de Physique de L’ École
Normale Supérieure for hospitality whilst this work was being completed. Open
Access Statement - For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising. Data access statement: no new data were generated for this
work.

##  Appendix A Dilaton from the boundary curve

We want to derive an expression for the dilaton in terms of the boundary curve
$w(\tau)$ irrespective of the CFT stress tensor subject only to some suitable
analyticity properties. The strategy is to solve the wave equation (14) in
terms of the stress tensor with a Green’s function method in the covering
space, and then to use a dispersion relation to relate the stress tensor to
its value on the boundary $\partial\widetilde{\cal M}_{n}$. Then we will use
the energy conservation equation (23) to eliminate the stress tensor to arrive
at an integral expression for the dilaton purely in terms of the boundary
curve. The steps are similar to those in appendix C in Goto:2020wnk , but we
point out additional complications arising when there are multiple branch
points, and we are working away from the $n\to 1$ limit.

###  Single QES problem: $N=1$

The single QES problem ($N$=1) offers the cleanest situation. In this case the
(inverse) uniformisation map to the covering space is simplest (24) and the
fundamental domain ${\cal D}$ is the unit disc in the $W$-plane. In order to
simplify the analysis we find it easier to map the unit disc in the covering
space $|W|\leq 1$ to the $\text{Re}\,Z\leq 0$ half of the complex $Z$ plane
using the Cayley transform ,

$\begin{split}W=\frac{Z_{0}-Z}{Z_{0}+Z}\ .\end{split}$ (128)

Here, $Z_{0}<0$ is a real arbitrary constant. The expression for the dilaton
at the point $Z=Z_{0}$ is given by the Green’s function method as

$\begin{split}\phi(Z_{0},Z_{0})=8\pi
G_{N}\int^{Z_{0}}dX\,\frac{Z_{0}^{2}-X^{2}}{Z_{0}}\,\text{Re}\,T(X)\
,\end{split}$ (129)

where $T\equiv T_{ZZ}$. The ability to add a multiple of the homogeneous
solution is hiding in the choice of the lower limit of the integral. Note that
we have chosen to evaluate the dilaton at what appears to be a distinguished
point with real $Z$ but we will be able to transform this to a generic point
using a suitable Möbius transformation later.

We can then relate the real to the imaginary part of the stress tensor using
the integral relation

$\begin{split}\text{Re}\,T(X)=\frac{1}{\pi}\int_{-\infty}^{\infty}dY\,\frac{Y}{X^{2}+Y^{2}}\,\text{Im}\,T(iY)\
.\end{split}$ (130)

The integral here is along the image of the boundary $\partial{\cal D}$ of the
disc $|W|=1$ in the $Z$ plane, i.e. the imaginary axis.

Next, we use the energy conservation equation (23) to express the energy flux
$\sim{\rm Im}\,T_{ZZ}$ at the AdS2 boundary in terms of the boundary
Schwarzian,

$-2{\rm Im}\,T_{ZZ}(iY(\tau))=\left.\frac{\phi_{r}}{8\pi
G_{N}}\frac{\partial_{\tau}\\{Z,\tau\\}}{\dot{Z}(\tau)^{2}}\right|_{Z=iY(\tau)}$
(131)

We assume that the boundary map $z(\tau)$ (in Euclidean signature) is
monotonic and therefore invertible. Hence, we can define for
$z\in\partial{\cal M}_{n}$

$\begin{split}\dot{z}(z)=\frac{dz}{d\tau}\Big{|}_{\tau=\tau(z)}\ .\end{split}$
(132)

On the boundary we can decompose $\dot{z}(z)=\dot{z}_{+}(z)+\dot{z}_{-}(z)$
and analytically continue with $\dot{z}_{+}(z)$ and $\dot{z}_{-}(z)$ analytic
in $\text{Re}\,z\gtrless 0$, respectively. This in turn allows to define the
functions in the cover $\dot{Z}(Z)=\dot{z}\partial_{z}Z$ using the
uniformisation map $z=\pi(Z)$. We then have the following identity,

$\begin{split}\frac{\partial_{\tau}\\{Z,\tau\\}}{\dot{Z}^{2}}=\partial_{Z}^{3}\dot{Z}\
.\end{split}$ (133)

Substituting in for the energy flux into the dispersion relation (130), and
(129), one can perform the $X$ integral (and just take the upper limit of the
integral fixing the homogeneous solution) and then integrate by parts three
times in $Z$ to give

$\begin{split}\phi(Z_{0},Z_{0})=-\frac{2\phi_{r}Z_{0}^{2}}{\pi}\int_{\partial{\cal
D}}dZ\,\frac{\dot{Z}}{(Z^{2}-Z_{0}^{2})^{2}}\ .\end{split}$ (134)

Finally, a Möbius transform of $(Z_{0},Z_{0})$ to an arbitrary point $({\cal
Z},\bar{\cal Z})$ yields:

$\begin{split}\phi({\cal Z},\bar{\cal Z})=-\frac{\phi_{r}({\cal Z}+\bar{\cal
Z})^{2}}{2\pi}\int_{\partial{\cal D}}dZ\,\frac{\dot{Z}}{({\cal
Z}-Z)^{2}(\bar{\cal Z}+Z)^{2}}\,,\end{split}$ (135)

on the half plane. Transforming back to the $W$ coordinate on the disc,

$\begin{split}\phi(W,\bar{W})=-\frac{\phi_{r}(1-|W|^{2})^{2}}{2\pi}\int_{\partial{\cal
D}}d\tilde{W}\,\frac{\dot{W}}{({W}-\tilde{W})^{2}(1-\bar{W}\tilde{W})^{2}}\
.\end{split}$ (136)

###  Multiple QES: $N>1$

The uniformisation problem with multiple branch points is more complex and, as
we have seen, the fundamental domain ${\cal D}$ is now a disc with circles cut
out as shown in figures 2 and 5. After identification of the arcs of the cut
out circles with their images, the boundary of the fundamental domain consists
of disjoint cycles. We can, of course, pose the dilaton Green’s function
problem in the covering space with boundary $|W|=1$, with the energy balance
equation (131), but the contributions from the regions cut out need to be
accounted for. This is a complicated and subtle point, particularly the
question whether additional singularities appear in these regions that will
contribute. We will postpone a precise investigation of these contributions to
future work. However, we note that progress can nevertheless be made, by
exploiting the limit $x\to 0$. We have shown that the partial circles
associated to the Fuchsian uniformisation become parametrically small with
radii scaling as $x$ in this limit, and therefore the corrections to the
formula (136) are correspondingly negligible at late times when the saddles
with QES dominate.

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