††institutetext: Institut für Theoretische Physik und Astrophysik
and Würzburg-Dresden Cluster of Excellence ct.qmat,
Julius-Maximilians-Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

# Monodromy methods for torus conformal blocks and entanglement entropy at
large central charge

Marius Gerbershagen<EMAIL_ADDRESS>
###### Abstract

We compute the entanglement entropy in a two dimensional conformal field
theory at finite size and finite temperature in the large central charge limit
via the replica trick. We first generalize the known monodromy method for the
calculation of conformal blocks on the plane to the torus. Then, we derive a
monodromy method for the zero-point conformal blocks of the replica partition
function. We explain the differences between the two monodromy methods before
applying them to the calculation of the entanglement entropy. We find that the
contribution of the vacuum exchange dominates the entanglement entropy for a
large class of CFTs, leading to universal results in agreement with
holographic predictions from the RT formula. Moreover, we determine in which
regime the replica partition function agrees with a correlation function of
local twist operators on the torus.

##  1 Introduction

Entanglement entropy is a measure for the amount of entanglement between two
parts of a quantum system. It is defined as the von Neumann entropy of the
reduced density matrix $\rho_{A}$ for a subsystem $A$. In general, the
entanglement entropy depends on details of the theory and state in question
such as the spectrum and operator content. However, certain universal features
are common to all quantum field theories. For example, the leading order
divergence in the UV cutoff usually scales with the area of the boundary of
the subregion $A$ Bombelli:1986rw ; Srednicki:1993im . Conformal field
theories in two dimensions admit more general universal features. In
particular, the entanglement entropy of a single interval $A$ at zero
temperature is given by Calabrese:2004eu

$S_{A}=\frac{c}{3}\log(l/\epsilon_{\text{UV}}),$ (1)

depending only on the central charge, irrespective of any other details such
as the OPE coefficients or the spectrum of the theory. For subsystems $A$
consisting of multiple intervals, the entanglement entropy is no longer
universal for any CFT. However, as shown in Hartman:2013mia , in the
semiclassical large central charge limit and at zero temperature, the
entanglement entropy again becomes universal for a large class of conformal
field theories. These CFTs are characterized by a sparse spectrum of light
operators and at most exponentially growing OPE coefficients. By using
conformal transformations, the universal results of Calabrese:2004eu ;
Hartman:2013mia  translate to the case of either finite temperature or finite
size. This publication is dedicated to the study of universal features of the
entanglement entropy in a system with both finite size and finite temperature.

The computational approach most commonly used to determine entanglement
entropies is the replica trick. It is based on the calculation of the Rényi
entropies

$S_{A}^{(n)}=\frac{1}{1-n}\log{\mathrm{Tr}}\rho_{A}^{n},$ (2)

via a partition function $Z_{n}$ on a higher genus Riemann surface
${\mathcal{R}}_{n}$ obtained by gluing $n$ copies of the complex plane
cyclically together along the entangling interval $A$. This partition function
is then mapped to a correlation function of twist operators, i.e. local
operators with scaling dimension $h=\bar{h}=c/24(n-1/n)$ inserted at the
endpoints of the entangling interval. For a subsystem $A$ consisting of $N$
disjoint intervals, the correlation function contains $2N$ twist operator
insertions. Finally, the entanglement entropy is obtained by analytically
continuing $n$ to the real numbers and taking the limit $n\to 1$. The
universality of the entanglement entropy for a single interval follows
immediately from the universality of the two-point function in any conformal
field theory Calabrese:2004eu . For multiple intervals, the Rényi entropy is
mapped to a higher-point correlation function of twist operators which
decomposes into a sum over conformal blocks. The universality in the
semiclassical limit observed in Hartman:2013mia  is explained by the fact that
only a single conformal block (the vacuum block) contributes to the
entanglement entropy. More precisely, in the semiclassical limit of large
central charge, the contribution of other conformal blocks is exponentially
suppressed in the central charge, assuming the aforementioned restrictions on
the theory, i.e. a sparse spectrum of light operators and at most
exponentially growing OPE coefficients.

In the case of a system with both finite size and finite temperature, the
replica trick instructs us to calculate the partition function on a higher
genus surface constructed by gluing $n$ copies of the torus along the
entangling interval $A$. In the small interval limit, this partition function
can be obtained analogously to the zero temperature case as a correlation
function on the torus of two (local) twist operators inserted at the endpoints
of the entangling interval (see e.g. Azeyanagi:2007bj ; Calabrese:2009qy ;
Cardy:2014jwa ; Chen:2016lbu ). For large intervals, on the other hand, the
replica partition function does not agree with a correlation function of local
twist operators, as can be seen by the following argument. It is well known
that for a pure state $\rho$, the entanglement entropy for $A$ is equal to the
entanglement entropy of the complement $A^{c}$, $S_{A}=S_{A^{c}}$. However,
for mixed states such as the thermal states described by the CFT on the torus,
this property no longer holds. Since the correlation function of local twist
operators contains no information about whether we compute the entanglement
entropy for $A$ or for $A^{c}$ (the location of the branch cuts between the
twist operators is not fixed by the twist correlator), it cannot give the
correct answer on the torus111In the case of a free CFT, a discrepancy between
the correlator of local twist operators and the higher genus partition
function has been observed in Lokhande:2015zma ; Mukhi:2017rex  based on
previous work Headrick:2012fk ; Datta:2013hba ; Chen:2015cna . In particular,
it was found in Lokhande:2015zma  that the twist operator result is not
modular covariant and violates Bose-Fermi equivalence.. This issue can be
resolved for instance by defining non-local twist operators on the torus222We
would like to thank the referee for pointing this out to us. as in
Chen:2014hta , however we will not do this and instead phrase the calculation
directly in terms of the replica partition function. Therefore, in the
following the term “twist operator” will always refer to a local operator of
scaling dimension $h=\bar{h}=c/24(n-1/n)$.

A convenient way to calculate the entanglement entropy on the plane works by
expanding the twist correlator in conformal blocks and calculating these using
the well-known monodromy method first described in Zamolodchikov1987 . This
monodromy method is derived as follows. By inserting a degenerate field into
the correlation function, one obtains a differential equation for an auxiliary
function $\Psi(z)$. This differential equations contains derivatives of the
sought after conformal block as accessory parameters. The accessory parameters
are fixed by demanding a certain monodromy of the solution $\Psi(z)$ around
cycles which encircle a number of operator insertion points. Which insertion
points are encircled depends on the channel in which the correlation function
is expanded. We review this method in detail in sec. 2.1 before generalizing
it to the case of finite temperature and finite size.

A different perspective on this method was offered in Faulkner:2013yia , where
it was related to a uniformization problem on the replica surface
${\mathcal{R}}_{n}$. In general, a compact Riemann surface $\Sigma$ can be
obtained as the quotient of the complex plane by a subgroup of
$PSL(2,{\mathbb{C}})$ Zograf_1988 . Thus, there exists a single valued map
$w\to z$ from the complex plane to ${\mathcal{R}}_{n}$. This uniformization
map is given as the quotient $w=\Psi_{1}(z)/\Psi_{2}(z)$ of two independent
solutions of a differential equation. It turns out that this differential
equation is equal to the one from the monodromy method for the conformal
blocks of the twist correlator on the plane Faulkner:2013yia . Using the
equivalence between the CFT partition function and the gravitational action in
the dual AdS space, this yields a proof of the RT formula at zero temperature
Faulkner:2013yia . We use similar arguments from the uniformization problem on
the replica surface of the torus to determine a monodromy method for the zero-
point conformal blocks of the replica partition function at finite
temperature. This new monodromy method is closely connected to the monodromy
method for the conformal blocks of the correlator of local twist operators on
the torus, with the crucial difference being that the new method allows for
choosing a larger set of cycles around which to impose the monodromy. These
new cycles are necessary to reproduce the $S_{A}\neq S_{A^{c}}$ property for
thermal states.

In the context of the AdS/CFT correspondence, universal features of the
entanglement entropy for holographic CFTs at large central charge are
predicted by the Ryu-Takayanagi formula (RT formula for short) Ryu:2006bv .
The RT formula states that the entanglement entropy of a subregion $A$ in the
boundary field theory corresponds to the area of a minimal surface
$\gamma_{A}$ in the bulk, anchored at $\partial A$ on the boundary of the AdS
space. Apart from reproducing the universal results obtained in
Calabrese:2004eu ; Hartman:2013mia , the RT formula also predicts interesting
universal features of the entanglement entropy in the case of both finite
temperature and finite size. In particular, there are two phase transitions
Azeyanagi:2007bj . First, there is a Hawking-Page transition in the bulk from
thermal AdS3 to the BTZ black hole phase as the temperature increases. This
induces a corresponding phase transition in the entanglement entropy. Second,
the entanglement entropy in the BTZ phase also shows a phase transition as the
size of the entangling interval increases. We explain how these features
appear from the CFT side.

Related work on the entanglement entropy in conformal field theories at finite
size and finite temperature includes Chen:2014hta ; Chen:2014ehg ;
Chen:2014unl ; Barrella:2013wja ; Chen:2015kua . Chen:2014hta ; Chen:2014ehg ;
Chen:2014unl  is concerned with the entanglement entropy in various limits of
high and low temperature or size of the entangling interval $A$, in which case
universal results for arbitrary values of the central charge can be obtained.
In Barrella:2013wja ; Chen:2015kua , the holographic entanglement entropy for
a single entangling interval on the boundary of a thermal AdS3 space and the
BTZ black hole was calculated using a monodromy method on the gravity side.
The monodromy method derived from the CFT side in this publication will turn
out to be equivalent to the monodromy method on the gravity side used in
Barrella:2013wja ; Chen:2015kua . Related work on torus conformal blocks
includes KashaniPoor:2012wb  which derived the monodromy method for the
special case of one-point Virasoro conformal blocks on the torus and
Alkalaev:2016fok ; Alkalaev:2017bzx  which performed explicit calculations of
one- and two-point Virasoro conformal blocks in various limits including the
semiclassical one which we study in this publication.

Our paper is organized as follows. In sec. 2, we derive the monodromy methods
used in this publication. After a review of the standard monodromy method for
conformal blocks on the plane in sec. 2.1, we generalize to torus conformal
blocks in sec. 2.2. Sec. 2.3 explains how to obtain a monodromy method for
zero-point conformal blocks of the partition function on the replica surface
and the differences between it and the monodromy method for conformal blocks
on the torus. Following this, we apply the newly derived monodromy methods to
the calculation of the entanglement entropy in sec. 3. Assuming that the
vacuum exchange dominates the partition function on the higher genus Riemann
surface, we find universal results in agreement with the RT formula. For the
partition function on the replica surface, we find in particular agreement
with the phase transition in the entanglement entropy at large interval size
and high temperature. This feature cannot be reproduced from the correlator of
local twist operators. We check the assumption on the dominance of the vacuum
exchange numerically in sec. 3.4. Finally, we conclude with a brief discussion
and outlook in sec. 4.

##  2 Monodromy methods

This section contains an overview over the monodromy methods used in this
publication. We start with a review of the standard monodromy method for
conformal blocks on the plane, then turn to the case of conformal blocks on
the torus and finally explain how to derive a monodromy method for zero-point
blocks of the partition function on the replica surface relevant to the
computation of entanglement entropy on the torus.

###  2.1 Conformal blocks on the plane

In this section, we review the monodromy method for the calculation of four-
point semiclassical conformal blocks on the plane first derived in
Zamolodchikov1986  (see also Harlow:2011ny  for a more detailed explanation).
The starting point of the derivation is the correlation function of four
primary fields ${\mathcal{O}}_{i}$

$\langle{\mathcal{O}}_{1}(z_{1},\bar{z}_{1}){\mathcal{O}}_{2}(z_{2},\bar{z}_{2}){\mathcal{O}}_{3}(z_{3},\bar{z}_{3}){\mathcal{O}}_{4}(z_{4},\bar{z}_{4})\rangle.$
(3)

Let us parametrize the central charge as $c=1+6(b+1/b)^{2}$ and take the
semiclassical limit $c\to\infty,b\to 0$ in which the conformal weights $h_{i}$
of the operators ${\mathcal{O}}_{i}$ as well as the internal conformal weight
$h_{p}$ scale proportional to the central charge. In the correlation function
(3), we insert a degenerate operator $\Psi(z,\bar{z})$ with conformal weight
$h_{\Psi}=-1/2-3b^{2}/4\sim{\mathcal{O}}(c^{0})$ obeying

$\left(\hat{L}_{-2}+\frac{1}{b^{2}}\hat{L}_{-1}^{2}\right)\Psi(z,\bar{z})=0$
(4)

From the conformal Ward identities, this yields the following differential
equation known as the decoupling equation,

$\left[\frac{1}{b^{2}}\partial_{z}^{2}+\sum_{i}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{\partial_{z_{i}}}{z-z_{i}}\right)\right]\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle=0.$
(5)

To get to the $s$-channel conformal block, we then insert the operator product
expansion

${\mathcal{O}}_{1}(z_{1},\bar{z}_{1}){\mathcal{O}}_{2}(z_{2},\bar{z}_{2})=\sum_{p}C^{p}_{21}\sum_{k,\bar{k}}(z_{2}-z_{1})^{h_{p}-h_{1}-h_{2}+|k|}(\bar{z}_{2}-\bar{z}_{1})^{\bar{h}_{p}-\bar{h}_{1}-\bar{h}_{2}+|\bar{k}|}\beta^{pk}_{21}\beta^{p\bar{k}}_{21}{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}(z_{1},\bar{z}_{1})$
(6)

into the correlation function which yields terms containing
$\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$.
At large central charge, these terms can be approximated by

$\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle\approx\Psi_{p}\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle,$
(7)

where $\Psi_{p}$ is defined by

$\Psi_{p}=\frac{\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle}{\langle{\mathcal{O}}_{p}{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle}.$
(8)

This can be shown by employing the form of
$\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$
in terms of a string of differential operators
${\mathcal{L}}_{k_{i}},\bar{\mathcal{L}}_{k_{i}}$ acting on
$\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$, where

${\mathcal{L}}_{-k_{i}}^{\Psi}=-\sum_{j=3,4,\Psi}\left(\frac{(1-k_{i})h_{j}}{(z_{j}-z_{1})^{k_{i}}}+\frac{1}{(z_{j}-z_{1})^{k_{i}-1}}\partial_{z_{j}}\right).$
(9)

Now $\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$
scales as $e^{-c/6S_{\text{cl.}}}$ in the semiclassical limit where
$S_{\text{cl.}}\sim{\mathcal{O}}(c^{0})$ while
$\Psi_{p}\sim{\mathcal{O}}(c^{0})$ and $h_{\Psi}\sim{\mathcal{O}}(c^{0})$.
Hence, we can neglect the derivatives acting on $\Psi_{p}$ and on the
$h_{\Psi}$ term to obtain (7) in the leading order in $c$. Now, use a
conformal transformation to send $z_{1}\to 0$, $z_{3}\to 1$, $z_{4}\to\infty$
and $z_{2}$ to the cross ratio $x$. This implies

$\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle\approx\sum_{p}\Psi_{p}(z,x,\bar{x})C^{p}_{21}C^{p}_{43}{\mathcal{F}}^{p}_{12,34}(x)\bar{\mathcal{F}}^{p}_{12,34}(\bar{x}),$
(10)

where ${\mathcal{F}}^{p}_{12,34}(x)$ is the desired conformal block which in
the semiclassical limit scales as ${\mathcal{F}}^{p}_{12,34}(x)\sim
e^{-\frac{c}{6}f_{\text{cl.}}(x)}$ as was conjectured in Zamolodchikov1986
and recently shown in Besken:2019jyw . The semiclassical conformal block
$f_{\text{cl.}}$ depends only on the cross ratio $x$ and on
$b^{2}h_{i},b^{2}h_{p}$. The decoupling equation (5) then yields at leading
order in $c$

$\left[\partial_{z}^{2}+\sum_{i}\left(\frac{b^{2}h_{i}}{(z-z_{i})^{2}}-\frac{\partial_{z_{i}}f_{\text{cl.}}(x)}{z-z_{i}}\right)\right]\Psi_{p}=0.$
(11)

There is one separate decoupling equation for each term in the sum over $p$
since generically, each term has a different monodromy and thus must vanish
separately. All terms involving derivatives of $\Psi_{p}$ vanish to leading
order due to $\Psi_{p}\sim{\mathcal{O}}(c^{0})$. From the expression for the
cross ratio $x=\frac{(z_{1}-z_{2})(z_{4}-z_{3})}{(z_{4}-z_{2})(z_{1}-z_{3})}$,
we obtain linear relations among the $\partial_{z_{i}}f_{\text{cl.}}$

$\sum_{i}\partial_{z_{i}}f_{\text{cl.}}=\sum_{i}(z_{i}\partial_{z_{i}}f_{\text{cl.}}-b^{2}h_{i})=\sum_{i}(z_{i}^{2}\partial_{z_{i}}f_{\text{cl.}}-2z_{i}b^{2}h_{i})=0.$
(12)

These follow from $\partial_{z_{i}}f_{\text{cl.}}=(\frac{\partial x}{\partial
z_{i}})\partial_{x}f_{\text{cl.}}$ and $\sum_{i}\frac{\partial x}{\partial
z_{i}}=\sum_{i}\frac{\partial x}{\partial z_{i}}z_{i}=\sum_{i}\frac{\partial
x}{\partial z_{i}}z_{i}^{2}=0$ as can easily be shown from the definition of
$x$ and the conformal transformation properties of correlation functions of
primary operators. This yields the final form of the decoupling equation,

$\left[\partial_{z}^{2}+\frac{b^{2}h_{1}}{z^{2}}+\frac{b^{2}h_{2}}{(z-x)^{2}}+\frac{b^{2}h_{3}}{(z-1)^{2}}-\frac{b^{2}(h_{1}+h_{2}+h_{3}-h_{4})}{z(z-1)}+\frac{x(1-x)\partial_{x}f_{\text{cl.}}}{z(z-x)(z-1)}\right]\Psi_{p}=0.$
(13)

To obtain $f_{\text{cl.}}$ from this equation, we use the fact that the
solutions $\Psi_{p}$ must have a certain monodromy when $z$ is taken in a loop
around $0,x$. This monodromy can be derived from the decoupling equation of
$\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$,

$\biggl{[}\frac{1}{b^{2}}\partial_{z}^{2}+\left(\frac{h_{p}}{(z-z_{1})^{2}}+\frac{1}{z-z_{1}}\partial_{z_{1}}\right)+\sum_{i=3,4}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}\partial_{z_{i}}\right)\biggr{]}\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle=0.$
(14)

As $z\to z_{1}$, the leading coefficient of the OPE between $\Psi$ and
${\mathcal{O}}_{p}$ is given by
$(z-z_{1})^{\kappa}{\mathcal{O}}_{p}^{\prime}(z_{1})$ where $\kappa$ can be
determined by inserting this coefficient into (14),

$\displaystyle\biggl{[}$
$\displaystyle\frac{1}{b^{2}}\kappa(\kappa-1)(z-z_{1})^{\kappa-2}+\sum_{i=3,4}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}\partial_{z_{i}}\right)(z-z_{1})^{\kappa}$
(15)
$\displaystyle+h_{p}(z-z_{1})^{\kappa-2}-\kappa(z-z_{1})^{\kappa-2}\biggr{]}\langle{\mathcal{O}}_{p}^{\prime}{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle=0.$

The leading contribution in $z\to z_{1}$ is given by

$\biggl{[}\frac{1}{b^{2}}\kappa(\kappa-1)+h_{p}-\kappa\biggl{]}(z-z_{1})^{\kappa-2}=0.$
(16)

Thus as $b^{2}\to 0$ and $z\to z_{1}$,

$\kappa=\frac{1}{2}\left(1\pm\sqrt{1-4h_{p}b^{2}}\right)\text{\leavevmode\nobreak\
\leavevmode\nobreak\ and\leavevmode\nobreak\ \leavevmode\nobreak\
}\Psi_{p}\sim(z-z_{1})^{\frac{1}{2}(1\pm\sqrt{1-4h_{p}b^{2}})}.$ (17)

Therefore, the monodromy matrix around $0,x$ is given by

$M_{0,x}=\left(\begin{array}[]{cc}e^{i\pi(1+\sqrt{1-4h_{p}b^{2}})}&0\\\
0&e^{i\pi(1-\sqrt{1-4h_{p}b^{2}})}\\\ \end{array}\right).$ (18)

The trace of the monodromy matrix, which is independent of the basis in which
the two solutions of (13) are decomposed, is given by

${\mathrm{Tr}}M_{0,x}=-2\cos\left(\pi\sqrt{1-4h_{p}b^{2}}\right).$ (19)

Thus, the torus conformal block can be extracted from (13) by choosing
$\partial_{x}f_{\text{cl.}}$ such that the monodromy of the solution
$\Psi_{p}$ around a loop enclosing $z_{1}$ and $z_{2}$ is given by (19) 333The
loop needs to enclose both $z_{1}$ and $z_{2}$ in order for the OPE between
${\mathcal{O}}_{1}(z_{1})$ and ${\mathcal{O}}_{2}(z_{2})$ to converge..
Finally, the conformal block is obtained by integrating the chosen
$\partial_{x}f_{\text{cl.}}$.

The four point conformal block in other channels is obtained from the same
decoupling equation by imposing different monodromy conditions. For example,
for the $t$-channel block we impose the monodromy condition around the
insertion points of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{3}$,
${\mathrm{Tr}}M_{1,x}=-2\cos\left(\pi\sqrt{1-4h_{p}b^{2}}\right)$. Higher
point conformal blocks on the plane are computed from similar monodromy
methods derived analogously to the four-point case. For $n$ point blocks, the
decoupling contains $n-3$ independent derivatives fixed by $n-3$ monodromy
conditions around the operator insertion points which are contracted in the
OPE.

###  2.2 Conformal blocks on the torus

We now continue with the derivation of a monodromy method for conformal blocks
on the torus. The derivation of this monodromy method proceeds in a very
similar way to the one on the plane. We illustrate the derivation using the
two-point function on the torus

$\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\rangle_{\tau}={\mathrm{Tr}}[e^{2\pi
i\tau(L_{0}-c/24)}e^{-2\pi
i\bar{\tau}(\bar{L}_{0}-c/24)}{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})],$
(20)

however conformal blocks for other correlation functions on the torus are
obtained in a similar fashion, as we briefly discuss as the end of this
section. The modular parameter of the torus is denoted by $\tau$, related to
the inverse temperature by $\beta=-2\pi i\tau$. We also introduce the
parameter

$Q=e^{-\beta}=e^{2\pi i\tau},$ (21)

written with an uppercase $Q$ instead of the standard lowercase $q$ to
distinguish it from the conformal dimension $h_{q}$ of the internal index of
the conformal block which we are about to derive.

As on the plane, we insert the degenerate operator $\Psi(z,\bar{z})$ into
(20). To derive the corresponding decoupling equation, we use the conformal
Ward identity on the torus Eguchi:1986sb  444Note that Eguchi:1986sb  uses a
convention where correlation functions on the torus are normalized by the
inverse of the partition function $Z(\tau)$ and thus the expression for the
conformal Ward identity there contains an additional term $(2\pi
i\partial_{\tau}Z(\tau))\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\rangle_{\tau}$.,

$\displaystyle\langle
T(z)\prod_{i}{\mathcal{O}}_{i}(z_{i})\rangle_{\tau}=\left[\sum_{i}\bigl{(}h_{i}\left(\wp(z-z_{i})+2\eta_{1}\right)+\left(\zeta(z-z_{i})+2\eta_{1}z_{i}\right)\partial_{z_{i}}\bigr{)}+2\pi
i\partial_{\tau}\right]\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\rangle_{\tau}.$
(22)

Here, $z\sim z+1\sim z+\tau$ are the coordinates on the torus with modular
parameter $\tau$ and $\wp(z),\zeta(z)$ denote Weierstraß elliptic functions
with associated $\eta_{1}$ parameter (see App. A for more details on the
Weierstraß functions). Using the definition of the Virasoro generators,

$\hat{L}_{-n}\Psi(z)=\int\frac{dw}{2\pi i}\frac{1}{(w-z)^{n-1}}T(w)\Psi(z).$
(23)

and the conformal Ward identity (22), we see that

$\displaystyle\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\hat{L}_{-2}\Psi(z)\rangle_{\tau}=\biggl{[}$
$\displaystyle\sum_{i}\left(h_{i}(\wp(z-z_{i})+2\eta_{1})+(\zeta(z-z_{i})+2\eta_{1}z_{i})\partial_{z_{i}}\right)$
(24) $\displaystyle+2\eta_{1}z\partial_{z}+2h_{\Psi}\eta_{1}+2\pi
i\partial_{\tau}\biggr{]}\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}$

and

$\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\hat{L}_{-1}\Psi(z)\rangle_{\tau}=\partial_{z}\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}.$
(25)

Thus $\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}$ obeys the
decoupling equation

$\displaystyle\biggl{[}$
$\displaystyle\frac{1}{b^{2}}\partial_{z}^{2}+\sum_{i}\left(h_{i}(\wp(z-z_{i})+2\eta_{1})+(\zeta(z-z_{i})+2\eta_{1}z_{i})\partial_{z_{i}}\right)$
(26) $\displaystyle+2\eta_{1}z\partial_{z}+2h_{\Psi}\eta_{1}+2\pi
i\partial_{\tau}\biggr{]}\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}=0.$

To relate
$\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z)\rangle_{\tau}$
to a conformal block, we decompose the trace over states into contributions
from a primary ${\mathcal{O}}_{q}$ and its descendants and insert the
appropriate OPE contractions.

12$p$$q$12$p$$q$ Figure 1: Conformal blocks for the two point function on the
torus. Left: OPE channel, right: projection channel.

For the two-point function, there are two possible channels (see fig. 1). The
projection block is obtained by OPE contracting ${\mathcal{O}}_{2}$ and
${\mathcal{O}}_{q}$. On the other hand, for the OPE block we contract
${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{1}$,

$\displaystyle\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z)\rangle_{\tau}=\sum_{q}\sum_{l}Q^{h_{q}-c/24+|l|}\langle{\mathcal{O}}_{q}^{\\{l\\}}(z_{0}){\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}$
(27)
$\displaystyle=\sum_{p,q}C^{p}_{21}\sum_{k,l}(z_{2}-z_{1})^{h_{p}-h_{1}-h_{2}+|k|}Q^{h_{q}-c/24+|l|}\beta^{pk}_{21}\langle{\mathcal{O}}^{\\{l\\}}_{q}(z_{0}){\mathcal{O}}^{\\{k\\}}_{p}(z_{1})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)},$

where (c.c.) denotes schematically the antiholomorphic parts of the expression
and $z_{0}\to-i\infty$ while $z_{\infty}\to+i\infty$. We define

$\Psi_{pq}=\frac{\langle{\mathcal{O}}_{q}(z_{0})\Psi(z){\mathcal{O}}_{p}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}{\langle{\mathcal{O}}_{q}(z_{0}){\mathcal{O}}_{p}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}.$
(28)

As on the plane, in the large $c$ limit

$\langle{\mathcal{O}}^{\\{l\\}}_{q}(z_{0}){\mathcal{O}}^{\\{k\\}}_{p}(z_{1})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}\approx\Psi_{pq}\langle{\mathcal{O}}^{\\{l\\}}_{q}(z_{0}){\mathcal{O}}^{\\{k\\}}_{p}(z_{1}){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}$
(29)

This yields

$\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi\rangle_{\tau}\approx\sum_{p,q}C^{p}_{21}C^{q}_{pq}\Psi_{pq}{\mathcal{F}}_{21,pq}\bar{\mathcal{F}}_{21,pq},$
(30)

where ${\mathcal{F}}_{21,pq}$ is the conformal block which we want to compute.
Assuming that exponentiation of the conformal blocks in the semiclassical
limit holds, ${\mathcal{F}}_{21,pq}\sim e^{-c/6\,f_{\text{cl.}}}$, and using
that $\partial_{z_{1}}f_{\text{cl.}}=-\partial_{z_{2}}f_{\text{cl.}}$, we
obtain

$\biggl{[}\partial_{z}^{2}+\sum_{i=1,2}\left(b^{2}h_{i}(\wp(z-z_{i})+2\eta_{1})+\partial_{z_{2}}f_{\text{cl.}}(-1)^{i+1}(\zeta(z-z_{i})+2\eta_{1}z_{i})\right)-2\pi
i\partial_{\tau}f_{\text{cl.}}\biggr{]}\Psi_{pq}=0.$ (31)

From the definition of $\Psi_{pq}$ we derive the monodromy conditions in the
same way as on the plane. For the OPE block, these are

${\mathrm{Tr}}M_{z_{1},z_{2}}=-2\cos(\pi\sqrt{1-4h_{p}b^{2}}),\leavevmode\nobreak\
{\mathrm{Tr}}M_{z_{0}}=-2\cos(\pi\sqrt{1-4h_{q}b^{2}}).$ (32)

The subscripts of the monodromy matrices show around which cycles the
monodromy is taken. The derivation for the projection block works analogously.
Here we contract ${\mathcal{O}}_{q}(z_{0}){\mathcal{O}}_{2}(z_{2})$:

$\displaystyle\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z)\rangle_{\tau}$
(33)
$\displaystyle=\sum_{p,q}C^{p}_{2q}\sum_{k,l}(z_{2}-z_{0})^{h_{p}-h_{q}-h_{2}+|k|}Q^{h_{q}-c/24+|l|}\beta^{pk}_{2q}\langle{\mathcal{O}}^{\\{k\\}}_{p}(z_{0}){\mathcal{O}}_{1}(z_{1})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}.$

Using $\Psi_{pq}$ defined by

$\Psi_{pq}=\frac{\langle{\mathcal{O}}_{p}(z_{0})\Psi(z){\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}{\langle{\mathcal{O}}_{p}(z_{0}){\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}$
(34)

and related to the two point correlator by

$\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi\rangle_{\tau}\approx\sum_{p,q}C^{p}_{2q}C^{q}_{1p}\Psi_{pq}{\mathcal{F}}_{2q,1p}\bar{\mathcal{F}}_{2q,1p}$
(35)

we find the same decoupling equation (31). However, the monodromy conditions
differ. They are given by

${\mathrm{Tr}}M_{z_{0},z_{2}}=-2\cos(\pi\sqrt{1-4h_{p}b^{2}}),\leavevmode\nobreak\
{\mathrm{Tr}}M_{z_{\infty}}=-2\cos(\pi\sqrt{1-4h_{q}b^{2}}).$ (36)

To solve the decoupling equation, it is useful to perform a coordinate
transformation $u=e^{-2\pi iz}$. Using the transformation of primary operators
under conformal transformations as well as the series representations of the
Weierstrass elliptic functions from app. A, the decoupling equation becomes

$\displaystyle\biggl{[}$
$\displaystyle\partial_{u}^{2}+y(h_{2}-(1+y)\partial_{y}f_{\text{cl.}})\sum_{m=-\infty}^{\infty}\frac{Q^{m}}{u(u-Q^{m})(u-Q^{m}(1+y))}+\frac{1/4-Q\partial_{Q}f_{\text{cl.}}}{u^{2}}$
(37)
$\displaystyle+h_{1}\sum_{m=-\infty}^{\infty}\frac{Q^{m}}{u(u-Q^{m})^{2}}+h_{2}\sum_{m=-\infty}^{\infty}\frac{Q^{m}(1+y)}{u(u-Q^{m}(1+y))^{2}}\biggr{]}\Psi_{pq}=0,$

where we have chosen w.l.o.g. $z_{1}=0$ and $e^{-2\pi iz_{2}}=1+y$. In these
coordinates, the monodromy conditions become

$\displaystyle{\mathrm{Tr}}M_{1,1+y}$
$\displaystyle=-2\cos(\pi\sqrt{1-4h_{p}b^{2}})$
$\displaystyle,\leavevmode\nobreak\ {\mathrm{Tr}}M_{0}=$
$\displaystyle-2\cos(\pi\sqrt{1-4h_{q}b^{2}})$ (OPE block) (38)
$\displaystyle{\mathrm{Tr}}M_{0,1+y}$
$\displaystyle=-2\cos(\pi\sqrt{1-4h_{p}b^{2}})$
$\displaystyle,\leavevmode\nobreak\ {\mathrm{Tr}}M_{\infty}=$
$\displaystyle-2\cos(\pi\sqrt{1-4h_{q}b^{2}})$ (projection block)

This representation of the decoupling equation is immediately applicable for
the calculation of the OPE block, which is defined through a series expansion
in $y$ and $Q$. Using this series expansion as well as a WKB approximation for
large $h_{p},h_{q}$, (37) can be solved order by order. For example, to first
order in $y$ and $Q$ we get

$f_{\text{cl.}}^{\text{OPE}}=-b^{2}(h_{p}-h_{1}-h_{2})\log
y-(b^{2}h_{q}-1/4)\log
Q+\frac{1}{2}b^{2}(h_{p}+h_{2}-h_{1})y-b^{2}\frac{h_{p}^{2}}{2h_{q}}Q+...$
(39)

The projection block, on the other hand, can be expanded in a series in
$q_{1}=Q/(1+y)$ and $q_{2}=1+y$. The decoupling equation can then be solved in
the same way as for the OPE block order by order in $q_{1}$ and $q_{2}$. For
example, to first order in $q_{1}$ and $q_{2}$ we obtain

$\displaystyle f_{\text{cl.}}^{\text{projection}}=$
$\displaystyle-(b^{2}(h_{p}-h_{2})-1/4)\log q_{2}-(b^{2}h_{q}-1/4)\log q_{1}$
(40)
$\displaystyle-b^{2}\frac{(h_{1}-h_{p}+h_{q})(h_{2}-h_{p}+h_{q})}{2h_{q}}q_{1}-b^{2}\frac{(h_{1}+h_{p}-h_{q})(h_{2}+h_{p}-h_{q})}{2h_{p}}q_{2}+...$

We have checked that the results for both the OPE and the projection block are
in agreement with the recursion formulas derived in Cho:2017oxl  (see app. B
for detailed expressions) as well as explicit calculations up to third order.

It is clear that the above derivation can be easily generalized to other
conformal blocks on the torus. The simplest case is the zero-point block on
the torus, i.e. the Virasoro character. Performing a similar derivation as
above or equivalently taking the limit $h_{1,2,p}\to 0$ in (37), we arrive at
the following decoupling equation

$\left[\partial_{u}^{2}+\frac{1/4-Q\partial_{Q}f_{\text{cl.}}}{u^{2}}\right]\Psi_{q}=0,$
(41)

together with the monodromy condition
${\mathrm{Tr}}M_{0}=-2\cos(\pi\sqrt{1-4h_{q}b^{2}})$. In this case, we can
give the full solution. The decoupling equation is solved by
$\Psi_{q}=u^{1/2\pm\sqrt{Q\partial_{Q}f_{\text{cl.}}}}$, from which we obtain
$f_{\text{cl.}}=(1/4-b^{2}h_{q})\log Q$ which correctly reproduces the leading
order contribution in $c$ of the Virasoro character
$\chi_{q}=\frac{1}{\eta(\tau)}Q^{h_{q}-(c-1)/24}\sim e^{-c/6f_{\text{cl.}}}$.
For a general $n$-point conformal block, the decoupling equation is given by

$\biggl{[}\partial_{z}^{2}+\sum_{i=1}^{n}\left(b^{2}h_{i}(\wp(z-z_{i})+2\eta_{1})+\partial_{z_{i}}f_{\text{cl.}}(\zeta(z-z_{i})+2\eta_{1}z_{i})\right)-2\pi
i\partial_{\tau}f_{\text{cl.}}\biggr{]}\Psi=0,$ (42)

and there are $n$ monodromy conditions around non-trivial cycles determined by
the OPE contractions. By conformal transformations, the insertion point of one
of the operators can be fixed, for example to $z_{1}\to 0$. Then, there are
$n$ independent accessory parameters $\partial_{\tau}f_{\text{cl.}}$ and
$\partial_{z_{i}}f_{\text{cl.}}$ for $i\geq 2$ fixed by these monodromy
conditions.

###  2.3 Partition function on the replica surface

We now turn to the computation of the partition function $Z_{n}$ on the
replica surface ${\mathcal{R}}_{n}$. In general, the partition function on any
higher genus Riemann surface can be expanded in zero-point conformal blocks,
which can again be calculated via a monodromy method. This monodromy method
can be derived by inserting the degenerate operator directly on the higher
genus Riemann surface – in contrast to the last section, where we inserted it
in a correlation function on the torus – and inserting projection operators in
the appropriate places. The difficulty of this approach is of course that
deriving the decoupling equation for an arbitrary Riemann surface is quite
hard. However, we will see that things simplify for the special higher genus
surface that we are interested in, that is the replica surface
${\mathcal{R}}_{n}$. Assuming that the dominant contribution to the partition
function depends only on the temperature and size of the entangling interval
and not on any other moduli of ${\mathcal{R}}_{n}$, we find the same
decoupling equation as for the twist operator correlator on the torus. The
difference to the last section lies in the monodromy conditions. The zero-
point block on ${\mathcal{R}}_{n}$ admits more general monodromy conditions
(corresponding to different channels) than the conformal block on the torus.
One of these more general monodromy conditions will give the dominant
contribution to the entanglement entropy for large intervals.

Before deriving the decoupling equation on ${\mathcal{R}}_{n}$, we collect
some facts about the topology and moduli of ${\mathcal{R}}_{n}$. For
simplicity, we specialize again to the single interval case. The replica
surface is given by $n$ copies of a torus with modular parameter $\tau$,
joined at a branch cut along the entangling interval $A$. We use coordinates
$z,\bar{z}$ to parametrize ${\mathcal{R}}_{n}$ with identifications $z\sim
z+1$ and $z\sim z+\tau$. In these coordinates, ${\mathcal{R}}_{n}$ is
described by a branched cover of the torus with branch points located at
$z=z_{1,2}+k+l\tau$ for $k,l\in{\mathbb{Z}}$. Near the branch points, the
covering map is given by $y^{n}\propto(z-z_{1}-k-l\tau)$ and $y^{n}\propto
1/(z-z_{2}-k-l\tau)$. The genus of ${\mathcal{R}}_{n}$ is then obtained by the
Riemann-Hurwitz theorem. The ramification index at each branch point is equal
to $n$, yielding $g=n$. Since the Euler characteristic is $\chi<0$, there are
no conformal Killing vectors. This implies by the Riemann-Roch theorem that
there exist $3(n-1)$ holomorphic quadratic differentials $\omega_{zz}^{(i)}$
parametrizing deformations of the complex structure of the Riemann surface.
The $\omega_{zz}^{(i)}$ are meromorphic doubly periodic functions that are
regular everywhere on the covering surface, i.e.
$\omega_{yy}^{(i)}dy^{2}=\omega_{zz}^{(i)}\bigl{(}\frac{dz}{dy}\bigr{)}^{2}dz^{2}$
is non-singular for all $y$. Simple examples include
$\omega_{zz}^{(1)}=\text{const.}$ which is trivially regular and doubly
periodic as well as
$\omega_{zz}^{(2)}=\zeta(z-z_{1})-\zeta(z-z_{2})+2\eta_{1}(z_{1}-z_{2})$.
$\omega_{zz}^{(2)}$ is regular since near $z=z_{1}+k+l\tau$ we have
$\omega_{yy}^{(2)}\propto y^{n-2}$ which is regular at $y=0$ for $n\geq 2$.
Near $z=z_{2}+k+l\tau$, regularity can be shown in an analogous way. In fact,
these two examples are the only ones relevant for the following arguments
since they are the only ones that respect the ${\mathbb{Z}}_{n}$ replica
symmetry permuting the different copies of the torus with each other555This
can be seen as follows. The replica symmetry acts as $y\to ye^{2\pi i/n}$.
Therefore, only the $\omega_{zz}\sim(z-z_{i})^{\alpha_{i}}$ with
$\alpha_{i}\in{\mathbb{Z}}$, $i=1,2$ are invariant under this symmetry. The
case $\alpha_{i}<-1$ is singular at $z=z_{i}$. $\alpha_{i}>0$ is singular at
some other point since any non-constant elliptic function has at least two
poles inside the fundamental parallelogram, which lead to singularities in
$\omega_{yy}$. This leaves only $\alpha_{i}=0,-1$ which are the two examples
described above..

The derivation of the decoupling equation on the replica surface then proceeds
in a similar fashion as in the previous section. Assuming exponentiation of
the zero-point block in the semiclassical limit, the conformal Ward identities
for a general Riemann surface Eguchi:1986sb  imply a decoupling equation of
the form

$\left[\partial_{z}^{2}+\langle
T_{zz}\rangle+\sum_{i=1}^{n}\omega_{zz}^{(i)}\partial_{w_{i}}f_{\text{cl.}}\right]\Psi(z)=0,$
(43)

where $w_{i}$ are the modular parameters associated to $\omega_{zz}^{(i)}$.
$\langle T_{zz}\rangle$ is the expectation value of the energy momentum
tensor. It can be derived along the lines of Faulkner:2013yia : $\langle
T_{zz}\rangle$ transforms with a Schwarzian derivative,

$\langle T_{yy}\rangle=\left(\frac{\partial z}{\partial y}\right)^{2}\langle
T_{zz}\rangle+\frac{nc}{12}\\{z,y\\},$ (44)

and $\langle T_{yy}\rangle$ must be regular. The Schwarzian derivative term
comes with a $nc/12$ prefactor since the stress-energy tensor on the replica
surface is given as the sum of the stress-energy tensors of the $n$ tori.
Therefore, the Schwarzian for the transformation of the stress-energy tensor
on the replica surface is given by the sum of $n$ identical Schwarzian terms
with prefactor $c/12$. Together with the requirement that $\langle
T_{zz}\rangle$ be doubly periodic, regularity of $\langle T_{yy}\rangle$
implies

$\langle
T_{zz}\rangle=\frac{c}{24}\left(n-\frac{1}{n}\right)\sum_{i}(\wp(z-z_{i})+2\eta_{1}).$
(45)

The $1/(z-z_{i})^{2}$ poles in $\wp(z-z_{i})$ give a $1/y^{2}$ contribution to
$\langle T_{yy}\rangle$ that cancels with the Schwarzian derivative term666For
ease of comparison with the previous section we have also added a constant
term $\frac{c}{24}\left(n-\frac{1}{n}\right)4\eta_{1}$ to $\langle
T_{zz}\rangle$ which is not strictly necessary for regularity and could be
absorbed into the prefactor of $\omega_{zz}^{(1)}$.. Letting the sum over $i$
in (43) run only over $i=1,2$, we recover the decoupling equation (31) for the
twist correlator. Restricting the sum to this range means that we assume
$\partial_{w_{i}}f_{\text{cl.}}=0$ for $i>2$, i.e. we assume that the result
for the partition function on the replica surface does not depend on other
moduli of the replica surface than the size of the torus $\tau$ and the length
of the entangling interval $z_{2}-z_{1}$.

$\langle(P_{p}{\mathcal{O}}_{1}{\mathcal{O}}_{2}P_{p}){\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle\sim$$4$$3$$p$$1$$2$$\sim$$P_{p}$${\mathcal{O}}_{1}$${\mathcal{O}}_{2}$${\mathcal{O}}_{3}$${\mathcal{O}}_{4}$$\langle{\mathcal{O}}_{1}(P_{p}{\mathcal{O}}_{2}{\mathcal{O}}_{3}P_{p}){\mathcal{O}}_{4}\rangle\sim$$4$$1$$p$$3$$2$$\sim$${\mathcal{O}}_{1}$$P_{p}$${\mathcal{O}}_{2}$${\mathcal{O}}_{3}$${\mathcal{O}}_{4}$
Figure 2: Inserting projection operators $P_{p}$ onto the Verma module of a
primary ${\mathcal{O}}_{p}$ into a correlator yields the conformal block with
internal weight $h_{p}$.

However, as mentioned in the beginning of this section, the admitted monodromy
conditions for the decoupling equation (43) are more general than those of
(31) for the twist correlator. To see this, recall that conformal blocks of
any correlation function can be obtained in two equivalent ways. Either we can
perform OPE contractions between two or more operators and then keep only
terms of particular primaries and their descendants in the OPE or equivalently
we can insert projection operators onto the Verma modules of these primaries
in the correlation function at appropriate places. The projectors of the
latter approach can be thought of as non-local operators acting in a closed
line around the operators whose OPE contractions are performed in the former
approach (see fig. 2). For the zero-point block on an arbitrary higher genus
Riemann surface, there are in general $3(n-1)$ projectors to be inserted
corresponding to $3(n-1)$ monodromy conditions. However, as mentioned above we
assume that the partition function on the higher genus Riemann surface depends
only on two of the moduli and thus we consider only two of the $3(n-1)$
monodromy conditions.

Which monodromy conditions are appropriate for the calculation of the
entanglement entropy? For the conformal block, the monodromy conditions must
be taken around the spatial circle and around $z_{1},z_{2}$ 777It is also
possible to calculate the modular transformed block for the twist correlator,
which we expect to be the dominant contribution at large temperature and small
intervals. In this case, the monodromy conditions are taken around
$z_{1},z_{2}$ and around the time circle.. On the other hand, for the zero-
point block on the replica surface the prescription described in this section
still leaves open the question of where to put the monodromy conditions – i.e.
which channel to choose – in order to obtain the dominant contribution to the
partition function from the vacuum block. Taking the limits of high and low
temperature, it is clear that for small intervals one of the monodromy
conditions must be taken around the spatial circle for low temperatures and
the time circle for high temperatures, while the other monodromy condition
must be imposed around the entangling interval $A$ between $z_{1}$ and
$z_{2}$. For large intervals, the correct monodromy condition is obtained by
reformulating the problem along the lines of Cardy:2014jwa ; Chen:2015kua . We
separate the branch cut on the torus along $A$ yielding the replica surface
into a branch cut along the full spatial circle and a branch cut in the
opposite direction along $A^{c}$ (see fig. 3). We then impose trivial
monodromy around $A^{c}$ to fix the dependence on the size of the entangling
interval. For small temperatures, the monodromy condition around the spatial
circle remains unchanged. However, for high temperatures the monodromy
condition around the time circle is now transformed into a monodromy condition
around a time circle of size $n\tau$, since the branch cut along the full
spatial circle connects all $n$ replica copies together to effectively create
a torus with modular parameter $n\tau$.

$A$$A^{c}$ Figure 3: Branch cut structure for large intervals. We can
decompose the branch cut along $A$ (denoted in red on the left) into a branch
cut along the full spatial circle (denoted in red on the right) and a branch
cut in the opposite direction along $A^{c}$ (denoted in blue on the right).

Note again that it is perfectly valid to use any of the above monodromy
conditions for all values of the temperature and entangling interval size.
However, outside of the regimes of validity of the monodromy conditions
described above, we don’t expect the vacuum block to give the dominant
contribution in the semiclassical limit and thus the partition function in
this case would be obtained by summing up all of the conformal blocks for
different values of the dimensions of the exchanged operators. The cross-over
point between the regimes must be determined by an analysis of these
contributions from the exchange of non-identity operators.

Let us also note that explicit calculations for the free fermion case support
the arguments presented in this section with regards to the differences
between twist correlators and partition functions on ${\mathcal{R}}_{n}$ and
with regards to the monodromy condition for large intervals. Namely, in
Lokhande:2015zma  it was observed that the twist correlator on the torus does
not give the correct answer for the entanglement entropy and in particular
violates Bose-Fermi equivalence and modular covariance. This was traced back
in Mukhi:2017rex  to the way in which different spin structures of the replica
surface ${\mathcal{R}}_{n}$ combine to give the total answer for the partition
function $Z_{n}$. The replica surface, being composed of $n$ copies of a
torus, has $2n$ nontrivial cycles around which the fermions have either
periodic or antiperiodic boundary conditions. To calculate the total partition
function $Z_{n}$ on the replica surface, it is necessary to sum over all
possible spin structures each of which corresponds to a particular choice of
boundary conditions around the nontrivial cycles of the replica surface. For
small intervals, the replica surface essentially factorizes into $n$
unconnected torus copies. In this limit, it was shown in Mukhi:2017rex  that
$Z_{n}$ is given by an “uncorrelated” sum where the summation over spin
structures is performed for each of the $n$ tori separately. For large
intervals, $Z_{n}$ is given by a “correlated” sum where only one sum over spin
structures is performed, i.e. we take equal boundary conditions around the
time resp. space circles of each of the $n$ replica copies Mukhi:2017rex . In
this case, the partition function $Z_{n}$ of the replica surface is
essentially given by the partition function on a torus with modular parameter
$n\tau$.

##  3 Entanglement entropy at large central charge

In this section, we present the calculation of the entanglement entropy on the
torus at large central charge. As explained in the previous section, the
entanglement entropy can be obtained from the partition function $Z_{n}$ on
the replica surface ${\mathcal{R}}_{n}$ which decomposes into zero-point
conformal blocks. The claim we want to investigate is that at large central
charge $c\to\infty$ and for $n\to 1$ the dominant contribution to $Z_{n}$
comes from the vacuum block with $h_{p}=h_{q}=0$. The derivation of this
statement proceeds as follows. First, it is necessary to show that the
semiclassical limit is well-defined not only for $h_{p,q}={\mathcal{O}}(c)$
but also for $h_{p,q}={\mathcal{O}}(c^{0})$. This means that for
$h_{p,q}=\gamma c$ the limits $\lim_{c\to\infty}$ and $\lim_{\gamma\to 0}$ of
the conformal block commute. A discussion of this point starting from the
recursion relation for torus conformal blocks is relegated to app. B. In the
next step we solve the decoupling equation (31) perturbatively in
$\varepsilon=n-1$ up to first order. Imposing the monodromy conditions derived
in sec. 2.2 then yields the conformal block from which we finally extract the
entanglement entropy.

We first consider the limits of high and low temperature as well as small and
large entangling interval size in sec. 3.1. Each combination of these limits
comes with different monodromy conditions as explained in the previous
section. We find agreement with the known universal results in the limits
where the torus degenerates into a cylinder. We then examine the conditions on
the CFT spectrum and OPE coefficients under which these results extend to
intermediate temperature and interval size regimes in sec. 3.2, leading to the
conclusion that for holographic CFTs the results from sec. 3.1 are valid for
all temperatures and interval sizes. Moreover, we consider the case of
multiple intervals on the torus. Finally, we numerically check the assumption
on the dominance of the vacuum block in sec. 3.4.

###  3.1 Limiting cases

####  3.1.1 Low temperature and small intervals

In the low temperature limit $\beta\to\infty$ the torus degenerates into a
cylinder with periodic space direction. For the cylinder, the entanglement
entropy of a single interval can be obtained directly by mapping this cylinder
to the plane and using the known formula for the entanglement entropy on the
plane Calabrese:2004eu ,

$S_{A}=\frac{c}{3}\log\left(\sin(\pi(z_{2}-z_{1}))\right)+\text{const.}$ (46)

To obtain the same result from the monodromy method, we series expand
$\Psi_{pq}$ and $f_{\text{cl.}}$ in $n-1$:
$\Psi_{pq}=\sum_{k}\Psi_{pq}^{(n)}(n-1)^{k}$ and
$f_{\text{cl.}}=\sum_{k}f_{n}(n-1)^{k}$. The decoupling equation (31) at
zeroth order in $n-1$ becomes

$[\partial_{z}^{2}-2\pi i\partial_{\tau}f_{0}]\Psi_{pq}^{(0)}(z)=0.$ (47)

This is solved by

$\Psi^{(0)}_{pq}(z)=\exp(\pm\sqrt{2\pi i\partial_{\tau}f_{0}}z).$ (48)

By a coordinate transformation $u=\exp(-2\pi iz)$, we obtain
$\tilde{\Psi}_{pq}^{(0)}(u)=u^{h_{\Psi}}\Psi_{pq}^{(0)}(z=i\log(u)/(2\pi))$.
Imposing trivial monodromy of $\tilde{\Psi}_{pq}^{(0)}(u)$ around $u=0$ is
equivalent to antiperiodic monodromy conditions for $\Psi_{pq}^{(0)}(z)$
around the spatial circle of the torus,
$\Psi_{pq}^{(0)}(z+1)=-\Psi_{pq}^{(0)}(z)$. As expected, this implies that
$f_{0}$ is equal to the leading order in $c$ of the vacuum character on the
torus,

$f_{0}=\pi i\tau/2=-\beta/4\hskip 28.45274pt\Leftrightarrow
e^{-c/6f_{0}}=e^{c/24\beta}=\left.\chi_{h=0}(\beta)\right|_{c\to\infty}.$ (49)

At first order in $n-1$, the decoupling equation is given by

$[\partial_{z}^{2}-2\pi
i\partial_{\tau}f_{0}]\Psi_{pq}^{(1)}(z)+m(z)\Psi_{pq}^{(0)}(z)=0,$ (50)

yielding

$\Psi_{pq}^{(1)}(z)=\frac{e^{-i\pi z}}{2\pi i}\int^{z}dx\,m(x)e^{i\pi
x}\Psi_{pq}^{(0)}(x)-\frac{e^{i\pi z}}{2\pi i}\int^{z}dx\,m(x)e^{-i\pi
x}\Psi_{pq}^{(0)}(x),$ (51)

where $m(z)$ is given by

$m(z)=\sum_{i}\left(\frac{1}{2}(\wp(z-z_{i})+2\eta_{1})+(-1)^{i+1}(\zeta(z-z_{i})+2\eta_{1}z_{i})\partial_{z_{2}}f_{1}\right)-2\pi
i\partial_{\tau}f_{1}.$ (52)

To compute the conformal block we impose trivial monodromy around
$z_{1},z_{2}$ which is equivalent to the vanishing of

$\oint_{z_{1},z_{2}}dx\,m(x)e^{\pm i\pi x}\Psi_{pq}^{(0)}(x).$ (53)

This gives the $z_{1},z_{2}$ dependence of $f_{1}$,

$f_{1}=\log(\sin(\pi(z_{2}-z_{1})))+C_{1}(\tau).$ (54)

From trivial monodromy around $u=0$ we find the $\tau$ dependence to be
$\partial_{\tau}f_{1}=0$, which implies that $C_{1}(\tau)=\text{const.}$ is
independent of $\tau$. The antiholomorphic conformal block $\bar{f}_{1}$ gives
the same result as the holomorphic one. Then the entanglement entropy is given
by $S_{A}=\frac{c}{6}(f_{1}+\bar{f}_{1})$, in agreement with (46). In this
limit, the OPE vacuum block of the twist correlator gives the same results,
since it is given by the same monodromy method as the zero-point vacuum block
of the replica partition function computed in this section.

####  3.1.2 Low temperature and large intervals

In this limit, we demand trivial monodromy around the spatial circle (i.e.
around $u=e^{-2\pi iz}=0$) as well as trivial monodromy around $A^{c}$ (i.e.
around $z_{1},z_{2}-1$). This gives the same vacuum block and thus the same
entanglement entropy (46) as in the small interval case at low temperature.
Also in this case, the twist correlator gives the correct result.

####  3.1.3 High temperature and small intervals

In the high temperature limit $\beta\to 0$ the torus again degenerates into a
cylinder, now with periodic time direction. As in the low temperature case,
the entanglement entropy of a single interval can be obtained by mapping this
cylinder to the plane Calabrese:2004eu ,

$S_{A}=\frac{c}{3}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(z_{2}-z_{1})\right)\right)+\text{const.}$
(55)

In the monodromy method, we impose trivial monodromy around the time circle
and around $z_{1},z_{2}$. As in the low temperature limit, we solve the
decoupling equation (31) in a series expansion around $n-1$. An analogous
calculation as above yields

$f_{\text{cl.}}=-\frac{\pi
i}{2\tau}+\epsilon\log\left(\tau\sinh\left(\frac{\pi
i}{\tau}(z_{2}-z_{1})\right)\right)+\text{const.}=f_{0}+\epsilon f_{1}$ (56)

At zeroth order in $\epsilon$ we recover the leading order in $c$ of the
vacuum character
$\chi_{h=0}(\beta)=e^{\frac{c}{24}\frac{4\pi^{2}}{\beta}}=e^{-\frac{c}{6}(-\frac{\pi
i}{2\tau})}$. The entanglement entropy given by the first order contribution,
$S_{A}=\frac{c}{6}(f_{1}+\bar{f}_{1})$, agrees with the result from the
cylinder (55).

From the twist correlator point of view, we obtain the high temperature limit
by a modular transformation $\tau\to-1/\tau$ from the low temperature result.
The two point function on the torus transforms covariantly,

$\langle{\mathcal{O}}_{1}(-z_{1}/\tau,-\bar{z}_{1}/\bar{\tau}){\mathcal{O}}_{2}(-z_{2}/\tau,-\bar{z}_{2}/\bar{\tau})\rangle_{-1/\tau}=(-\tau)^{h_{1}+h_{2}}(-\bar{\tau})^{\bar{h}_{1}+\bar{h}_{2}}\langle{\mathcal{O}}_{1}(z_{1},\bar{z}_{1}){\mathcal{O}}_{2}(z_{2},\bar{z}_{2})\rangle_{\tau}.$
(57)

If the two point function of twist operators is dominated by a single
conformal block, the conformal block for the modular transformed $\tau$ is
obtained from the modular transformation properties of the correlation
function as

$f_{\text{cl.}}^{-1/\tau}(-z_{1}/\tau,-z_{2}/\tau)=-b^{2}(h_{1}+h_{2})\log(-\tau)+f_{\text{cl.}}^{\tau}(z_{1},z_{2}).$
(58)

Therefore, we can immediately read off the high temperature behavior of the
twist correlator from

$\left.f_{\text{cl.}}^{\tau}(z_{1},z_{2})\right|_{\tau\to\infty}=\left[f_{\text{cl.}}^{-1/\tau}(-z_{1}/\tau,-z_{2}/\tau)+b^{2}(h_{1}+h_{2})\log(-\tau)\right]_{\tau\to\infty}$
(59)

For $h_{1}=h_{2}=1/2\epsilon$,
$\left.f_{\text{cl.}}^{\tau}(z_{1},z_{2})\right|_{\tau\to 0}=\frac{\pi
i\tau}{2}+\epsilon\log(\sin(\pi(z_{2}-z_{1})))+\text{const.}$ we again obtain
(56). Thus the twist correlator agrees with the replica partition in this
limit.

Let us note that is also possible to determine the high temperature expansion
of twist correlator by applying the modular transformation to the monodromy
conditions. At low temperatures, we demand trivial monodromy around the
spatial circle of the torus at time. Since the modular transformation
$\tau\to-1/\tau$ exchanges the time and space directions of the torus, we can
obtain the high temperature behavior of the twist correlator by demanding
trivial monodromy around the time circle of the torus, showing directly the
equivalence between the twist correlator result and the replica partition
function in the limit of high temperature and small entangling intervals.

####  3.1.4 High temperature and large intervals

As explained in sec. 2.3, the correct monodromy condition of the zero-point
block on the replica surface for large intervals imposes trivial monodromy
around $A^{c}$ and $z\to z+n\tau$. To zeroth order in $n-1$ the decoupling
equation (31) is solved by $\Psi_{pq}^{(0)}(z)=\exp(\pm\sqrt{2\pi
i\partial_{\tau}f_{0}})$. Imposing
$\Psi_{pq}^{(0)}(z+n\tau)=-\Psi_{pq}^{(0)}(z)$ yields

$f_{0}=-\frac{\pi i}{2n^{2}\tau}.$ (60)

To first order, the solution reads

$\Psi_{pq}^{(1)}(z)=n\tau\frac{e^{-\frac{i\pi z}{n\tau}}}{2\pi
i}\int^{z}dx\,m(x)e^{\frac{i\pi
x}{n\tau}}\Psi_{pq}^{(0)}(x)-n\tau\frac{e^{\frac{i\pi z}{n\tau}}}{2\pi
i}\int^{z}dx\,m(x)e^{-\frac{i\pi x}{n\tau}}\Psi_{pq}^{(0)}(x).$ (61)

Imposing trivial monodromy around $A^{c}$ and expanding in $n-1$ we obtain

$f_{\text{cl.}}=-\frac{\pi
i}{2\tau}+(n-1)\left(\frac{i\pi}{\tau}+\log\left(\tau\sinh\left(\frac{i\pi}{\tau}(1-(z_{2}-z_{1}))\right)\right)\right)+\text{const.}$
(62)

giving

$S_{A}=\frac{c}{3}\left(\frac{i\pi}{\tau}+\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(1-(z_{2}-z_{1}))\right)\right)\right)+\text{const.}$
(63)

As expected from general arguments Azeyanagi:2007bj ; Cardy:2014jwa , the
difference between the entanglement entropy for $A$ and for the complement
$A^{c}$ in the limit of large interval size is given by the thermal entropy
$S(\beta)=\beta^{2}(-\beta^{-1}\partial_{\beta}\log
Z(\beta))=\frac{c}{3}\frac{2\pi^{2}}{\beta}=\frac{c}{3}\frac{i\pi}{\tau}$,
using the partition function $Z(\beta\to
0)=\exp(\frac{c}{12}\frac{4\pi^{2}}{\beta})$ from the Cardy formula. The twist
correlator, on the other hand, cannot reproduce this feature as is easy to see
by applying the modular transformation argument from the last section which
gives (56) in disagreement with (62).

###  3.2 Holographic CFTs

We now argue that the results in the limits considered in the previous section
are valid for all temperatures and interval sizes in holographic CFTs. This
statement holds if the vacuum block computed in the previous section gives the
dominant contribution to the partition function on the replica manifold
${\mathcal{R}}_{n}$ in the large central charge limit.

Let us first consider the case $n=1$, i.e. the zeroth order in the $n-1$
expansion. In this order, the monodromy method calculates the vacuum character
to leading order in $c$. If this vacuum character dominates the partition
function, $Z(\beta)$ takes on the universal form

$Z(\beta)=\left\\{\begin{array}[]{ll}\exp\bigl{(}\frac{c}{12}\beta\bigr{)}&,\leavevmode\nobreak\
\beta>2\pi\\\
\exp\bigl{(}\frac{c}{12}\frac{4\pi^{2}}{\beta}\bigr{)}&,\leavevmode\nobreak\
\beta<2\pi\end{array}\right.$ (64)

for all temperatures. For consistency of the computation method, dominance of
the conformal block to first order in $n-1$ also requires dominance of the
zeroth order in $n-1$. It has been argued in Hartman:2014oaa  that a partition
function of the form (64) is characteristic of a holographic CFT. Thus it is a
necessary condition that the CFT in question be holographic in order for the
results of sec. 3.1 to hold at arbitrary temperatures.

More explicit conditions on the CFT in question can be given following an
argument for dominance of the vacuum block in the zero temperature case which
goes as follows Hartman:2013mia . The arguments below hold for CFTs with OPE
coefficients $C^{p}_{21}$ that grow at most exponentially with $c$ 888This is
equivalent to the requirement that correlation functions obey the cluster
decomposition principle and are smooth in a neighborhood of the point where
multiple operator insertion points coincide Hartman:2013mia . and a density of
states for light operators that does not grow with $c$. Here light operators
mean operators with conformal weight $h,\bar{h}<h_{\text{gap}}$ where
$h_{\text{gap}}$ is of the order of the central charge. In the large central
charge limit, the conformal block expansion in the $s$-channel of the four-
point twist correlator on the plane takes on the form

$\langle\sigma_{n}(0)\sigma_{n}(x)\sigma_{n}(1)\sigma_{n}(\infty)\rangle=\sum_{p}C^{p}_{21}\exp\left(-\frac{c}{6}\bigl{(}f_{\text{cl.}}(h_{\sigma_{n}}/c,h_{p}/c,x)+\bar{f}_{\text{cl.}}(\bar{h}_{\sigma_{n}}/c,\bar{h}_{p}/c,\bar{x})\bigr{)}\right).$
(65)

On account of the cluster decomposition principle, the vacuum exchange is the
leading contribution in the $s$-channel around $x=0$ and in the $t$-channel
around $x=1$. This implies that the contribution of the heavy operators with
$h,\bar{h}>h_{\text{gap}}$ in the $s$-channel in some finite region around
$x=0$ is suppressed exponentially and similarly for the $t$-channel around
$x=1$. The sparse spectrum of the light operators allows ignoring the
multiplicity factors for $h,\bar{h}<h_{\text{gap}}$. Consider first the
scenario that the OPE coefficients grow subexponentially with $c$. In this
case, we can also ignore the coefficient $C^{p}_{21}$ and the sum over the
light operators in (65) is dominated exponentially in $e^{-c}$ by its largest
term. If the semiclassical conformal block $f_{\text{cl.}}$ is a monotonically
increasing function with $h_{p}/c$, this implies that the vacuum block with
the lowest possible $h_{p}/c=0$ dominates. In the case that the OPE
coefficients grow exponentially with $c$, the sum over the light operators
gives

$\int_{0}^{h_{\text{gap}}}dh_{p}d\bar{h}_{p}\exp\left(\frac{c}{6}\bigl{(}g(h_{p}/c,\bar{h}_{p}/c)-f_{\text{cl.}}(h_{\sigma_{n}}/c,h_{p}/c,x)-\bar{f}_{\text{cl.}}(\bar{h}_{\sigma_{n}}/c,\bar{h}_{p}/c,\bar{x})\bigr{)}\right),$
(66)

where $g(h_{p}/c,\bar{h}_{p}/c)$ contains the contribution of the OPE
coefficients and multiplicities. This integral is dominated either by the
endpoints of the integration or by a saddle point. Near coincident points, the
leading universal term in any correlation function is given by the vacuum
exchange. Therefore, in a finite region around $x=0$ and $x=1$, the integral
in (66) is dominated by the endpoint at $h_{p}=0$. However, now – unlike for
subexponentially growing OPE coefficients – it is not possible to exclude
saddle points dominating the integral (66) in some finite subset of
$x\in[0,1]$. Examples of such saddle points have been found in Belin:2017nze
for genus two partition functions computing the third Rényi entropy on the
plane.

In summary, the vacuum block dominates correlation functions in a finite
region around coincident operator insertion points if the conformal block
$f_{\text{cl.}}$ increases monotonically with the internal conformal weights
$h_{p,q}$. In sec. 3.4, we present numerical evidence that this is indeed the
case. Therefore, the results of sec. 3.1 are valid for all temperatures and
interval sizes assuming subexponentially growing OPE coefficients and a sparse
spectrum of light operators. For exponentially growing OPE coefficients, the
results are still valid in a finite region around the respective limiting
points. The points at which the different limits exchange dominance are
obtained as follows. From the requirement of consistency of our results at
$n=1$ with the partition function (64) we see that the low and high
temperature regimes exchange dominance at the Hawking-Page phase transition
point $\beta=2\pi$. The transition between small and large interval behavior
in the high temperature regime is estimated by equating the results for the
conformal block in the small and large interval regime. The dominant
contribution comes from the smaller conformal block.

Assuming the aforementioned restrictions on the CFT spectrum and OPE
coefficients, the results match perfectly with the predictions from the RT
formula Ryu:2006bv ; Azeyanagi:2007bj . There is also a direct way to
implement the monodromy computation for the calculation of the entanglement
entropy in the dual gravity theory, as was shown in Faulkner:2013yia  at zero
temperature. This was generalized to the finite temperature case in
Barrella:2013wja ; Chen:2015kua . We obtain the same monodromy method as
Barrella:2013wja ; Chen:2015kua  from the CFT side. This clearly shows that
our results are valid for holographic CFTs.

Furthermore, the high temperature and large interval size result (62) agrees
with a CFT calculation of the vacuum sector contribution to the entanglement
entropy done in Chen:2015kua  using complementary techniques to obtain the
vacuum block. Specifically, the conformal block is obtained in Chen:2015kua
by an explicit construction of the Virasoro generators and descendant states
in the twisted sector of a ${\mathbb{Z}}_{n}$ orbifold. This allows for a
series expansion of the Rényi entropy. Moreover, in the limit $n\to 1$, the
authors of Chen:2015kua  find that the leading order of the semiclassical
vacuum block is given by an expression in terms of the four-point function of
twist-operators on the plane, which gives the same conformal block (62) that
we obtain using the monodromy method.

Let us also note that the universal form of the partition function (64) in
holographic theories can be derived from the same vacuum block dominance
argument as the universal form of the entanglement entropy. The partition
function can be expanded in zero-point blocks on the torus either in a low
temperature expansion (zero-point blocks are Virasoro characters) or in a high
temperature expansion (zero-point blocks are modular transformed Virasoro
characters). From the known form of the Virasoro characters, the leading order
contribution in the central charge of the characters is given by $\chi\sim
Q^{h_{q}-c/24}$ where $Q=e^{2\pi i\tau}$ resp. $Q=e^{-2\pi i/\tau}$ in the low
and high temperature expansions. Since $(-6/c)\log\chi$ is an increasing
function of $h_{q}$, the same arguments as for the entanglement entropy given
above apply to the partition function which is dominated by the vacuum
character with $h_{q}=0$.

###  3.3 Multiple intervals

The generalization to an entangling interval consisting of the union of
multiple intervals, $A=[z_{1},z_{2}]\cup[z_{3},z_{4}]...[z_{2N-1},z_{2N}]$, is
straightforward. The decoupling equation is given by

$\left[\partial_{z}^{2}+\sum_{i=1}^{2N}\left(\frac{1}{4}(n-1/n)(\wp(z-z_{i})+2\eta_{1})-\partial_{z_{i}}f_{\text{cl.}}(\zeta(z-z_{i})+2\eta_{1}z_{i})\right)-2\pi
i\partial_{\tau}f_{\text{cl.}}\right]\Psi(z)=0.$ (67)

We impose trivial monodromy around $N$ pairs $(i,j)$ of interval endpoints
$z_{i},z_{j}$ to fix the $\partial_{z_{i}}f_{\text{cl.}}$. The temperature
dependence is fixed by demanding trivial monodromy around either the spatial
circle, a time circle of size $\tau$ or a time circle of size $n\tau$
depending on the temperature and total entangling interval size
$|A|=\sum_{i}|z_{2i}-z_{2i-1}|$. This yields

  * •

low temperature: trivial monodromy for $z\to z+1$

$S_{A}=\frac{c}{3}\sum_{(i,j)}\log\left(\sin(\pi(z_{i}-z_{j}))\right)+\text{const.}$
(68)

  * •

high temperature and small total interval size: trivial monodromy for $z\to
z+\tau$

$S_{A}=\frac{c}{3}\sum_{(i,j)}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(z_{i}-z_{j})\right)\right)+\text{const.}$
(69)

  * •

high temperature and large total interval size: trivial monodromy for $z\to
z+n\tau$

$S_{A}=\frac{c}{3}\left(\frac{i\pi}{\tau}+\sum_{(i,j)}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(z_{i}-z_{j})\right)\right)\right)+\text{const.}$
(70)

Which monodromy condition and which combination of pairs $(i,j)$ to take, i.e.
in which channel to expand the conformal block, depends on the interval size.
We expect the dominant contribution to come from the channel with the smallest
$f_{\text{cl.}}$, in which case we find agreement with the RT formula.
However, we caution that this argument depends on the vacuum block dominating
the partition function, which we have checked only for a single interval.

One particular interesting special case of the above calculation is the time
dependence of the entanglement entropy between two intervals on opposite
boundaries of a two-sided BTZ black hole. The two-sided BTZ black hole is dual
to the thermofield double state. On the CFT side, the entanglement entropy is
obtained by positioning the two intervals at a distance $\tau/2$ in the time
coordinate on the torus. To obtain the time evolution of the entanglement
entropy, it is necessary to perform an analytic continuation of the endpoints
Hartman:2013qma ,

$\displaystyle z_{1}=\bar{z}_{1}=0$ $\displaystyle z_{2}=\bar{z}_{2}=L$ (71)
$\displaystyle z_{3}=2t+L+\tau/2,\bar{z}_{3}=-2t+L+\bar{\tau}/2$
$\displaystyle z_{4}=2t+\tau/2,\bar{z}_{4}=-2t+\bar{\tau}/2,$

where $L$ is the size of the entangling interval taken to be equal on both
boundaries and $t$ is the time coordinate at which both parts of the
entangling interval are placed on the asymptotic boundaries of the
wormhole999Note that this time coordinate has nothing to do with the euclidean
time coordinate on the torus on which we calculate the finite temperature
correlator in the euclidean CFT. (see Hartman:2013qma  for more details on the
setup). For small interval size $L$, there are two conformal blocks to
consider. At early times, i.e. for small $t$, the dominant contribution comes
from imposing trivial monodromy around $z_{1},z_{4}$ and $z_{2},z_{3}$, while
at late times the vacuum block with trivial monodromy around $z_{1},z_{2}$ and
$z_{3},z_{4}$ dominates. Taking into account that due to the analytic
continuation $f_{\text{cl.}}\neq\bar{f}_{\text{cl.}}$, the corresponding
entanglement entropy is given by

$S_{A}=\frac{2c}{3}\log\left(\frac{\tau}{i\pi}\cosh\left(\frac{2\pi
i}{\tau}t\right)\right)+\text{const.}$ (72)

at early times and by

$S_{A}=\frac{2c}{3}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}L\right)\right)+\text{const.}$
(73)

at late times. This reproduces the phase transition in the dual RT surfaces
from geodesics that connect the two boundaries through the interior of the
two-sided black hole at early times to disconnected geodesics on opposite
boundaries that do not enter the black hole interior at late times
Hartman:2013qma .

###  3.4 Vacuum block dominance

In this section, we provide numerical evidence that the vacuum block
exponentially dominates the twist correlator in the large $c$ limit. For
simplicity, we restrict to the single interval case. Assuming the same
conditions on the spectrum and OPE coefficients of the CFT detailed in the
last section, we need to show that the conformal block monotonically increases
with the weight of the internal operators $h_{p,q}$.

For the zero temperature case, this was done numerically in Hartman:2013mia
for arbitrary $n$, giving evidence that the Rényi entropies are given by the
vacuum conformal block contribution only. However, the calculation is much
simpler if we restrict to $n$ close to one which implies $h_{p}/c\gg
h_{i}/c\to 0$ for $i=1,2,3,4$. In this limit, the conformal block can be
obtained in closed form from the monodromy calculation by a WKB expansion in
$1/(h_{p}/c)$ Zamolodchikov1987 ,

$\frac{c}{6}f_{\text{cl.}}(0,h_{p}/c,x)=h_{p}\left(\pi\frac{K(1-x)}{K(x)}-\log
16\right),$ (74)

where $K(x)$ is the complete elliptic integral of the first kind. Thus,
$f_{\text{cl.}}$ is an increasing function of $h_{p}/c$ if $\pi
K(1-x)/K(x)-\log 16>0$ which can easily be checked to be fulfilled for
$x<1/2$. In fact, $f_{\text{cl.}}(0,h_{p}/c,x)-f_{\text{cl.}}(0,h_{p}/c,1-x)$
reaches its crossover point exactly at $x=1/2$, confirming that at this point
dominance is exchanged from the $s$ to the $t$-channel block.

Applying the same arguments as on the plane to the case of the torus, it is
clear that the vacuum block dominates if the semiclassical block is an
increasing function of $h_{p}$ and $h_{q}$. Without loss of generality, we
take $z_{1}=0$ in the following. Restricting again to $n\approx 1$, we need to
find the semiclassical block in the limit $h_{p,q}/c\gg h_{1,2}/c\to 0$.
Unlike on the plane, however, $f_{\text{cl.}}$ can not easily be obtained in a
closed form expression from the monodromy calculation in this limit101010The
reason for this is that the solution of the decoupling equation on the torus
takes on the schematic form of an integral over
$\sqrt{A\partial_{z_{2}}f_{\text{cl.}}+B\partial_{\tau}f_{\text{cl.}}}$ for
some functions $A$ and $B$, from which it is not easily possible to extract
$\partial_{z_{2}}f_{\text{cl.}}$ and $\partial_{\tau}f_{\text{cl.}}$. The
solution of the decoupling equation on the plane, on the other hand, is given
by an integral over $\sqrt{A\partial_{x}f_{\text{cl.}}}$ from which
$\partial_{x}f_{\text{cl.}}$ can be factored out immediately.. Thus, we apply
a series expansion in $y=e^{-2\pi iz_{2}}-1$ and $Q=e^{2\pi i\tau}$ on top of
the WKB approximation in $1/(h_{p,q}/c)$. While this yields a very precise
numerical approximation to the true value of the conformal block if enough
terms are included in the series, the expansion has a restricted domain of
validity in the $y,Q$ plane. In particular, the series expansion in the cross-
ratio $x$ of the four-point block on the plane converges for $|x|<1$
Zamolodchikov1987 , therefore we expect the series expansion of the two-point
block on the torus to have a convergence radius of $|y|=1$ (the torus block
reduces to the block on the plane in the limit $Q\to 0$). The numerics confirm
this expectation. For $|y|>1$, we observe large fluctuations in the value of
the conformal block as we include more terms in the series expansion. The
numerics for the convergence radius in $Q$ is less clear, but also in this
variable we observe large fluctuations close to $Q=e^{-2\pi}$. Thus, we can
check the vacuum block dominance only in a restricted region around the origin
in $y$, corresponding to small intervals. The restricted convergence radius in
$Q$ is not limiting since above the Hawking-Page transition temperature given
by $Q=e^{-2\pi}$, we expect the block in the high temperature expansion to
dominate. The high temperature expansion of the conformal block is given by a
series expansion in $e^{2\pi iz_{2}/\tau}-1$ and $e^{-2\pi i/\tau}$. In the
$h_{p,q}/c\gg h_{1,2}/c$ limit, the series coefficients are equal to those of
the low temperature expansion. Moreover, in the same limit at high
temperatures and for $n\to 1$ the conformal block in the large interval limit
(where the monodromy condition is taken around a time circle of size $n\tau$)
is equivalent to the conformal block in the small interval limit with the
replacement $z_{2}\to 1-z_{2}$.

We show some plots of $\frac{\partial f_{\text{cl.}}}{\partial h_{p}}$ and
$\frac{\partial f_{\text{cl.}}}{\partial h_{q}}$ in fig. 4 for small
temperatures and values of $y$ and $Q$ inside the convergence radius.

Figure 4: Derivative of the semiclassical conformal block $f_{\text{cl.}}$
w.r.t. $h_{p}$ (left) and $h_{q}$ (right) for $z_{2}=0.1$, $\beta=4\pi$ and
different values of $h_{p}$ and $h_{q}$ in the range $[0,c/24]$. Note that the
plotted value is greater than zero in all cases, showing that the
semiclassical conformal block increases with $h_{p}$ and $h_{q}$. The series
expansion used in this figure was truncated at order 10 in both $y$ and $Q$.

The plots for the conformal block in the high temperature limit show no
significant differences from the ones in the low temperature limit. We find in
all cases that inside the convergence radius of the series expansion
$\frac{\partial f_{\text{cl.}}}{\partial h_{p,q}}>0$, i.e. the assumption of
vacuum block dominance is fulfilled

While it is not possible to find an analytic continuation for the conformal
block from a truncated series expansion, we can use a Padé approximant to get
a heuristic approximation of the series outside its convergence radius. The
Padé approximation works by replacing the truncated series expansion by a
rational function whose Taylor expansion agrees with the series expansion up
to the order in which the truncation was performed BakerGraves-Morris . In
many cases, this approximation has a better radius of convergence than the
original series expansion due to poles in the function limiting the radius of
convergence of its Taylor expansion being taken into account in the Padé
approximation. We plot the Padé approximant of $f_{\text{cl.}}$ in the special
case $h_{p}=h_{q}$ for different orders of the denominator polynomial in fig.
5 depending on $z_{2}$ (the size of the entangling interval).

Figure 5: Derivative of the Padé approximation of the semiclassical conformal
block w.r.t. the internal conformal weight plotted against $z_{2}$. The upper
plot shows the conformal block at $\beta=4\pi$ in the low temperature
expansion and the lower plot the block at $\beta=\pi$ in the high temperature
expansion. For plotting convenience we set $h_{p}=h_{q}$ which gives a
derivative of $f_{\text{cl.}}$ that is constant for all $h_{p}$. As we
increase the order of the denominator polynomial in the Padé approximation,
the fluctuations in $f_{\text{cl.}}$ outside of the convergence radius of the
series expansion decrease. Moreover, multiple Padé approximants converge to
the same value outside of the convergence radius. The convergence radius
$|y|=1$ is reached at $z_{2}=1/6$ in the upper plot and
$z_{2}=\frac{\log(2)}{4\pi}=0.0552$ in the lower plot. The series expansion
approximated by the Padé approximants in this figure was truncated at order 10
in both $y$ and $Q$.

We find that different Padé approximants for $f_{\text{cl.}}$ converge to the
same value and yield $\frac{\partial f_{\text{cl.}}}{\partial h_{p,q}}>0$ in a
finite region outside of the convergence radius $|y|=1$ of the series
expansion of $f_{\text{cl.}}$. While this is not a formal proof, it does
indicate that vacuum block dominance holds also outside of the convergence
radius.

##  4 Discussion

Let us briefly summarize the main points investigated in this publication. In
the first part in sec. 2, we took a detailed look at monodromy methods for the
computation of arbitrary conformal blocks on the torus as well as zero-point
conformal blocks on the special higher genus Riemann surface relevant to the
calculation of the entanglement entropy via the replica trick. We then applied
the derived monodromy methods to the calculation of the entanglement entropy
in sec. 3. We found that for holographic CFTs, the vacuum conformal block
dominates the partition function on the replica surface and therefore the
entanglement entropy takes on a universal form in agreement with the RT
formula.

The main advancement compared to previous work on entanglement entropy for
finite size and finite temperature systems at large central charge is as
follows.

First of all, we clarified the role of the correlation function of twist
operators on the torus compared to the partition function $Z_{n}$ on the
replica surface ${\mathcal{R}}_{n}$. The twist correlator only gives the
correct results for small intervals or low temperature, while the replica
partition function $Z_{n}$ must give the entanglement entropy in all cases by
construction. The reason why the twist correlator agrees with $Z_{n}$ for
small intervals or low temperature is evident from the monodromy method. The
conformal blocks of the twist correlator obey the same monodromy method as the
zero-point conformal blocks for $Z_{n}$ apart from the set of allowed
monodromy conditions, which for the twist correlator blocks is a subset of
those of the replica partition function.

Secondly, our derivation of the monodromy method is based solely on CFT
techniques and thus provides a non-trivial check of both the RT formula as
well as previous calculations of Barrella:2013wja ; Chen:2015kua  using the
monodromy method on the gravity side. Moreover, the monodromy method presented
here is derived from first principles, further strengthening the heuristic
arguments that were used to justify the monodromy method in Barrella:2013wja ;
Chen:2015kua . The derivation from the CFT side furthermore allows for a
determination of the restrictions on the operator content and OPE coefficients
that must be obeyed by the CFT in question in order for the vacuum block to
give the dominant contribution and the RT results for the entanglement entropy
to be valid. Perhaps unsurprisingly, these restrictions are equivalent to
those imposed in Hartman:2013mia  for the entanglement entropy at zero
temperature. That is, the entanglement entropy for any CFT with large central
charge, sparse spectrum of light operators and at most exponentially growing
OPE coefficients agrees with the RT formula not only at zero temperature as
was already known from the results of Hartman:2013mia , but also at finite
temperature and finite size. In addition, the same arguments that imply the
universal form of the entanglement entropy in these CFTs also imply a
universal form of the partition function at all temperatures, another feature
of holographic CFT investigated also in Hartman:2014oaa .

We close with a short outlook on possible future directions. One interesting
direction is an investigation in full generality of the issue under which
conditions the twist correlator yields the same results as the replica
partition function $Z_{n}$. Previous work in the free fermion case
Lokhande:2015zma ; Mukhi:2017rex  indicates that for short intervals the twist
correlator is equivalent to the replica partition function, a result which we
can corroborate for large $c$ CFTs. Does this hold in general? Other
directions can be found in further applications of the monodromy method. Since
conformal blocks are the basic building blocks of any correlation function,
the derivation of the monodromy methods from first principles lays the
groundwork for a number of applications. In particular, the monodromy methods
found in this paper may prove useful in deriving recursion relations in the
dimensions of the exchanged operators for torus conformal blocks with better
convergence properties than the known recursion relations in the central
charge found in Cho:2017oxl . Moreover, the monodromy methods also pave the
way to the study of other information theoretic quantities closely connected
to the entanglement entropy such as entanglement negativity Calabrese:2012nk ;
Calabrese:2012ew ; Calabrese:2014yza , symmetry resolved entanglement
Goldstein:2017bua ; Bonsignori:2019naz ; Murciano:2020lqq ; Capizzi:2020 ;
Zhao:2020qmn  or entwinement Balasubramanian:2014sra ; Lin:2016fqk ;
Balasubramanian:2016xho ; Balasubramanian:2018ajb ; Erdmenger:2019lzr  for
finite temperature states, providing opportunities for new insights into the
physics of the dual AdS black holes. We hope to be able to report on further
results in this direction in the future.

##### Acknowledgements

I would like to thank Johanna Erdmenger and Christian Northe for useful
discussions. I acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy through Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id
390858490).

##  Appendix A Conventions for elliptic functions

This appendix contains an overview over conventions and useful identities for
the Weierstraß elliptic functions used in the rest of the publication (see
e.g. NIST:DLMF  for more details about these functions). All elliptic
functions are defined for a lattice $\Lambda$ generated by the identifications
$z\sim z+1$ and $z\sim z+\tau$. The Weierstraß elliptic functions
$\wp(z),\zeta(z)$ and $\sigma(z)$ are defined by

$\displaystyle\wp(z)$
$\displaystyle=-\zeta^{\prime}(z)=\frac{1}{z^{2}}+\sum_{(m,n)\neq(0,0)}\left(\frac{1}{(z+n+m\tau)^{2}}-\frac{1}{(n+m\tau)^{2}}\right),$
(75) $\displaystyle\zeta(z)$
$\displaystyle=\frac{\sigma^{\prime}(z)}{\sigma(z)}=\frac{1}{z}+\sum_{(m,n)\neq(0,0)}\left(\frac{1}{(z+n+m\tau)}-\frac{1}{(n+m\tau)}+\frac{z}{(n+m\tau)^{2}}\right),$
$\displaystyle\sigma(z)$
$\displaystyle=z\prod_{(m,n)\neq(0,0)}\left(1-\frac{z}{n+m\tau}\right)\exp\left(\frac{z}{n+m\tau}+\frac{1}{2}\frac{z^{2}}{(n+m\tau)^{2}}\right).$

$\wp(z)$ is a true elliptic (i.e. doubly periodic) function while $\zeta(z)$
and $\sigma(z)$ are quasiperiodic:

$\displaystyle\wp(z+1)=\wp(z+\tau)=\wp(z)$ (76)
$\displaystyle\zeta(z+1)=\zeta(z)+2\eta_{1},$
$\displaystyle\zeta(z+\tau)=\zeta(z)+2\eta_{3}$
$\displaystyle\sigma(z+1)=-\exp(2\eta_{1}(z+1/2))\sigma(z),$
$\displaystyle\sigma(z+\tau)=-\exp(2\eta_{3}(z+\tau/2))\sigma(z),$

where $\eta_{1}=\zeta(1/2)$ and $\eta_{3}=\zeta(\tau/2)=\tau\eta_{1}-\pi i$.
Another useful definition of $\wp(z)$ and $\zeta(z)$ is given by

$\displaystyle\wp(z)+2\eta_{1}=(2\pi
i)^{2}\sum_{m=-\infty}^{\infty}\frac{Q^{m}u}{(u-Q^{m})^{2}}=\left(\frac{2\pi
i}{\tau}\right)^{2}\sum_{m=-\infty}^{\infty}\frac{\tilde{Q}^{m}\tilde{u}}{(\tilde{u}-\tilde{Q}^{m})^{2}}$
(77)
$\displaystyle\zeta(z)-2\eta_{1}z=i\pi\sum_{m=-\infty}^{\infty}\frac{Q^{m}+u}{Q^{m}-u}=-\frac{i\pi}{\tau}\sum_{m=-\infty}^{\infty}\frac{\tilde{Q}^{m}+\tilde{u}}{\tilde{Q}^{m}-\tilde{u}},$

where $u=e^{-2\pi iz},Q=e^{2\pi i\tau}$ and $\tilde{u}=e^{2\pi iz/\tau}$,
$\tilde{Q}=e^{-2\pi i/\tau}$.

##  Appendix B Recursion relations for torus conformal blocks

For completeness, this appendix shows the recursion formulas for the two-point
conformal blocks on the torus following as a special case from the general
method derived in Cho:2017oxl . For details of the derivation, see Cho:2017oxl
and also Hadasz:2009db  for the one-point torus block. For the OPE block, the
conformal block is given by the following recursion scheme

$\displaystyle{\mathcal{F}}^{\text{OPE}}_{21,pq}(h_{p},h_{q},c)$
$\displaystyle=U^{\text{OPE}}(h_{p},h_{q},c)$ (78) $\displaystyle-\sum_{r\geq
2,s\geq 1}$ $\displaystyle\frac{\partial c_{rs}(h_{q})}{\partial
h_{q}}Q^{rs}\frac{A^{c_{rs}(h_{q})}_{rs}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\
h_{q}+rs\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\
h_{q}\end{array}\biggr{]}}}{c-c_{rs}(h_{q})}{\mathcal{F}}^{\text{OPE}}_{21,pq}(h_{q},h_{q}+rs,c_{rs}(h_{q}))$
$\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial
c_{rs}(h_{p})}{\partial
h_{p}}y^{rs}\frac{A^{c_{rs}(h_{p})}_{rs}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{q}\\\
h_{q}\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{2}\\\
h_{1}\end{array}\biggr{]}}}{c-c_{rs}(h_{p})}{\mathcal{F}}^{\text{OPE}}_{21,pq}(h_{p}+rs,h_{q},c_{rs}(h_{p}))$

where the fusion polynomials are given by

${P^{rs}_{c}\biggl{[}\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\biggr{]}}=\prod_{m=1-r,m\in
1-r+2{\mathbb{N}}}^{r-1}\prod_{n=1-s,n\in
1-s+2{\mathbb{N}}}^{s-1}\frac{\lambda_{1}+\lambda_{2}+mb+nb^{-1}}{2}\frac{\lambda_{1}-\lambda_{2}+mb+nb^{-1}}{2}$
(79)

with $\lambda_{i}=\sqrt{(b+b^{-1})^{2}-4h_{i}}$ while the prefactor is

$A^{c}_{rs}=\frac{1}{2}\prod_{\stackrel{{\scriptstyle(m,n)=(1-r,1-s)}}{{(m,n)\neq(0,0),(r,s)}}}^{(r,s)}(mb+nb^{-1})^{-1}.$
(80)

$c_{rs}$ denotes the value of the central charge where degenerate
representations of the Virasoro algebra appear,

$\displaystyle c_{rs}(h)=1+6(b_{rs}(h)+b_{rs}^{-1}(h))^{2},$ (81)
$\displaystyle
b_{rs}(h)^{2}=\frac{rs-1+2h+\sqrt{(r-s)^{2}+4(rs-1)h+4h^{2}}}{1-r^{2}}.$

The $c$-regular part $U$ is given by

$U^{\text{OPE}}(h_{p},h_{q},c)=\left[\prod_{n=2}^{\infty}\frac{1}{1-Q^{n}}\right]\sum_{i,j\geq
0}Q^{i}y^{j}\frac{s_{ij}(h_{q},h_{p},h_{q})(1-h_{p}-j)_{j}(h_{p}+h_{1}-h_{2})_{j}}{i!(2h_{q})_{i}j!(2h_{p})_{j}},$
(82)

where we define the rising and falling Pochhammer symbols by

$(a)_{n}=\prod_{k=0}^{n-1}(a+k)\hskip
56.9055pt(a)^{(n)}=\prod_{k=0}^{n-1}(a-k)$ (83)

and

$\displaystyle s_{ij}(h_{1},h_{2},h_{3})=\langle
h_{1}|(L_{-1}^{i})^{\dagger}{\mathcal{O}}_{h_{2}}(1)L_{-1}^{j}|h_{3}\rangle$
(84) $\displaystyle=\left\\{\begin{aligned}
&\sum_{p=0}^{i}\biggl{(}\begin{array}[]{c}i\\\
p\end{array}\biggr{)}(j)^{(p)}(2h_{3}+j-1)^{(p)}(h_{1}+h_{2}-h_{3}-(j-p))_{i-p}(h_{3}+h_{2}-h_{1})_{j-p}\leavevmode\nobreak\
,\leavevmode\nobreak\ j\geq i\\\
&\sum_{p=0}^{j}\biggl{(}\begin{array}[]{c}j\\\
p\end{array}\biggr{)}(i)^{(p)}(2h_{1}+i-1)^{(p)}(h_{3}+h_{2}-h_{1}-(i-p))_{j-p}(h_{1}+h_{2}-h_{3})_{i-p}\leavevmode\nobreak\
,\leavevmode\nobreak\ i\geq j\end{aligned}\right.$

The recursion formula for the projection block is given by 111111The authors
of Cho:2017oxl  also claim to have found a different recursion formula in the
conformal dimensions of the exchanged primaries $h_{p,q}$ for a class of
conformal blocks termed the “necklace blocks” which include the projection
block considered here as a special case. We cannot confirm this claim since
the formula presented in Cho:2017oxl  agrees neither with the recursion
relation in the central charge derived in Cho:2017oxl  nor with an explicit
calculation in the first few orders of the series expansion in $q_{1},q_{2}$
for the projection block.

$\displaystyle{\mathcal{F}}^{\text{projection}}_{2p,1q}(h_{p},h_{q},c)=$
$\displaystyle U^{\text{projection}}(h_{p},h_{q},c)$ (85)
$\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial
c_{rs}(h_{q})}{\partial
h_{q}}q_{1}^{rs}\frac{A^{c_{rs}(h_{q})}_{rs}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\
h_{1}\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\
h_{2}\end{array}\biggr{]}}}{c-c_{rs}(h_{q})}{\mathcal{F}}^{\text{projection}}_{2p,1q}(h_{p},h_{q}+rs,c_{rs}(h_{q}))$
$\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial
c_{rs}(h_{p})}{\partial
h_{p}}q_{2}^{rs}\frac{A^{c_{rs}(h_{p})}_{rs}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{q}\\\
h_{1}\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{q}\\\
h_{2}\end{array}\biggr{]}}}{c-c_{rs}(h_{p})}{\mathcal{F}}^{\text{projection}}_{2p,1q}(h_{p}+rs,h_{q},c_{rs}(h_{p})),$

where the $c$-regular part is given by

$U^{\text{projection}}(h_{p},h_{q},c)=\left[\prod_{n=2}^{\infty}\frac{1}{1-Q^{n}}\right]\sum_{i,j\geq
0}q_{1}^{i}q_{2}^{j}\frac{s_{ij}(h_{q},h_{2},h_{q})s_{ji}(h_{p},h_{1},h_{q})}{i!(2h_{q})_{i}j!(2h_{p})_{j}}.$
(86)

By explicit calculation, it is easy to check in the first few orders of the
series expansion that for $h_{1}=h_{2}$ and $h_{p,q}=\gamma c$, the limits
$\lim_{\gamma\to 0}$ and $\lim_{c\to\infty}$ of the OPE block commute. We have
checked this up to fourth order in $y,Q$. A more convenient proof is possible
with a recursion relation in the conformal weights of the exchanged operators
$h_{p,q}$, as derived for the conformal block on the plane in
Zamolodchikov1987 . In fact, the singular parts proportional to $\sim
1/(h_{p,q}-h_{rs})$ of such a recursion relation are proportional to the
singular parts $\sim 1/(c-c_{rs}(h_{p,q}))$ of the above recursion relations
in $c$ Cho:2017oxl . Using these known singular parts, one can show that the
limits $\gamma\to 0$ and $c\to\infty$ commute for the singular parts of this
recursion relation to all orders, assuming the above conditions on
$h_{1,2,p,q}$. Together, these calculations provide some evidence that the
semiclassical limit is well-defined for the vacuum block on the torus derived
in sec. 3. Since the vacuum block on the torus in most limits is equivalent to
a zero-point block on the replica surface ${\mathcal{R}}_{n}$, this also
provides evidence that the semiclassical limit for the zero-point vacuum block
on ${\mathcal{R}}_{n}$ is well-defined.

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