# Reflected Entropy in AdS3/WCFT

Bin Chen1,2,3, Yuefeng Liu1 and Boyang Yu2

###### Abstract

Reflected entropy is a newly proposed notion in quantum information. It has
important implications in holography. In this work, we study the reflected
entropy in the framework of the AdS3/WCFT correspondence. We determine the
scaling dimensions and charges of various twist operators in non-Abelian
orbifold WCFT by generalizing the uniformization map and taking into account
of the charge conservation. This allows us to compute the reflected entropy,
logarithmic negativity and odd entropy for two disjoint intervals in
holographic WCFT. We find that the reflected entropy can be related
holographically to the pre-entanglement wedge cross-section, which is given by
the minimal distance between the benches in two swing surfaces.

1Department of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871,
P.R. China

2Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

3 Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

###### Contents

  1. 1 Introduction
  2. 2 Reflected Entropy and Holographic 2D CFT
    1. 2.1 Reflected entropy: finite dimensional case
    2. 2.2 Reflected entropy in conformal field theory
    3. 2.3 Reflected entropy: holographic 2D CFT
    4. 2.4 Entanglement wedge cross section
  3. 3 Reflected entropy in 2D WCFT
    1. 3.1 Warped conformal field theory
    2. 3.2 Twist operators in orbifold WCFT
      1. 3.2.1 Twist operator in Abelian orbifold WCFT
      2. 3.2.2 Twist operators in non-Abelian orbifold WCFT
    3. 3.3 Reflected entropy in 2D holographic WCFT
  4. 4 Pre-Entanglement Wedge and AdS3/WCFT
    1. 4.1 Approximate modular flow and the swing surface
    2. 4.2 Pre-entanglement wedge in AdS3/WCFT
    3. 4.3 Pre-entanglement wedge cross section and two interval entanglement phase transition
  5. 5 Logarithmic negativity and odd entropy
    1. 5.1 Logarithmic negativity
    2. 5.2 Odd entropy
  6. 6 Conclusion and Discussion

##  1 Introduction

The quantum entanglement may play a fundamental role in the emergence of
spacetime. In the past fifteen years, there have been a lot of works on the
holographic entanglement entropy and its implications, since the seminal works
in [1, 2](see [3] for a nice review). It is widely believed that the
entanglements in holographic states encode the information of dual (quantum)
gravity. Various hard-core problems in AdS/CFT have been addressed from
entanglement point of view.

The entanglement entropy is defined as the von Neumann entropy of the reduced
density matrix, and it smooths out the details of the reduced density matrix.
As a result, the entanglement entropy does not capture the full entanglement
structure in a quantum state. In particular, for the mixed state, it is not a
proper entanglement measure: it captures classical (thermal) correlations as
well as purely quantum ones.

One can purify a mixed state by combining it with an auxiliary system such
that the combined system is a pure state. There are infinite number of ways to
purify a mixed state. One may define the entanglement of purification (EoP) to
be the minimal entanglement in all purification. Even though it is difficult
to find EoP in the quantum systems, it was proposed in [4, 5] that EoP could
be holographically dual to the entanglement wedge cross section.

The entanglement wedge is defined to be the bulk causal domain of a co-
dimension one surface interpolating between the boundary subregion and its
holographic entanglement surface. It plays an important role in studying the
bulk reconstruction and the subregion/state duality[6]. The entanglement wedge
cross-section (EWCS) is the minimal area surface in the bulk that divides the
entanglement wedge of two disconnected subregions. Remarkably, besides the
EoP, several quantum information measures have been proposed to be related to
EWCS, including the logarithmic entanglement negativity[7, 8], odd entropy[9]
and reflected entropy[10].

The reflected entropy $S_{R}(A:B)$ is defined for a bipartite quantum system
$A\cup B$ and a mixed state denoted by a density matrix $\rho_{AB}$ on the
Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Analogous
to the relationship between the thermofield double state and thermal density
matrix, one can define the canonical purification of this density matrix in a
doubled Hilbert space
$\mathcal{H}\otimes\mathcal{H}^{\prime}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}}$.
Then the reflected entropy is defined as the entanglement entropy or the von
Neumann entropy of the density matrix $\rho_{AA^{\prime}}$:

$S_{R}(A:B)=S(AA^{\prime})_{\sqrt{\rho_{AB}}}=S(BB^{\prime})_{\sqrt{\rho_{AB}}}=S_{vN}(\rho_{AA^{\prime}})$
(1.1)

where
$\rho_{AA^{\prime}}=\mbox{Tr}_{BB^{\prime}}|\sqrt{\rho_{AB}}\rangle\langle\sqrt{\rho_{AB}}|$
is the reduced density matrix after tracing over
$\mathcal{H}_{B}\otimes\mathcal{H}_{B^{\prime}}$. For the recent studies on
the reflected entropy in quantum information and quantum field theories, see
[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. For holographic CFT, it was
proposed [10] that the reflected entropy is related to the entanglement wedge
cross-section (EWCS)

$S_{R}(A:B)=2\frac{\mbox{Area}(EWCS)}{4G_{N}}+\mbox{(subleading terms)},$
(1.2)

where $G_{N}$ is the Newton’s constant and the subleading terms include the
quantum corrections in the small $G_{N}$ expansion. Another remarkable point
is that the reflected entropy can be useful in understanding the requirement
of tripartite entanglement in holographic states[22, 16].

In this work, we would like to study the reflected entropy in two-dimensional
(2D) holographic warped conformal field theory (WCFT) and its bulk dual within
the framework of AdS3/WCFT. The 2D warped CFT is a field theory whose symmetry
is generated by Virasoro-Kac-Moody algebra[23, 24]. The holographic WCFT could
be dual to semiclassical AdS3 gravity with the Compère-Song-Strominger (CSS)
boundary condition[25] or the 3D topological massive gravity in warped AdS
spacetime. The resulting AdS3/WCFT and WAdS3/WCFT correspondences provide
nontrivial windows to study the holographic duality beyond the AdS/CFT
correspondence. We focus on the AdS3/WCFT case in this work. The AdS3/WCFT
correspondence has been studied from various points of view in [26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

In the study of holographic entanglement entropy in AdS/WCFT, there appears
some novel features. For a single interval, one may use the generalized
Rindler method to find the swing surface, whose area gives the entanglement
entropy[40, 27]. Quite recently, it was proposed in [41, 38] to use the
modular Hamiltonian to find the swing surface. In contrast to the usual
AdS/CFT correspondence, the definition of the entanglement wedge in AdS3/WCFT
is subtle, since the homology surface interpolating between boundary interval
and the bench in the swing is not always well-defined. Nevertheless, one can
still define pre-entanglement wedge and moreover the pre-entanglement wedge
cross section (pre-EWCS) of two disjoint intervals.

As the first step, we need to compute the reflected entropy in a WCFT. The
computation turns out to be challenging. The main difficulty is how to
determine the dimensions and charges of the twist operators in the non-Abelian
orbifold. We need to generalize the uniformization map proposed in the Abelian
orbifold[34] to the case at hand. Moreover we choose the monodromy conditions
on the twist fields in a consistent way such that the charge conservation is
kept. As a result, not only can we compute the reflected entropy, but also
other two information quantities, the logarithmic negativity and odd entropy.
For the holographic WCFT we will show that the conjectured relation (1.2)
between the reflect entropy and EWCS is still true even in the AdS3/WCFT
correspondence.

The remaining parts of the paper are organized as follows. In section 2, we
review the properties of the reflected entropy, its computations in 2D CFT and
its implication in the AdS3/CFT2 correspondence. In section 3, in order to
compute the reflected entropy in a WCFT, we show how to determine the
dimensions and the charges of various twist operators in a non-Abelian
orbifold, by using a generalized uniformization map and charge conservation.
In particular we calculate the reflected entropy of two disjoint intervals in
holographic WCFT. In section 4, we discuss the pre-entanglement wedge cross
section in AdS3 gravity with CSS boundary condition, and its relation with the
reflected entropy in the dual field theory. In section 5, with the general
discussions on the twist operators in section 3, we compute two other
information quantities, logarithmic negativity and odd entropy, in the
holographic WCFT. We end with conclusion and some discussions in section 6.

##  2 Reflected Entropy and Holographic 2D CFT

In this section we would like to give a concise review on the reflected
entropy $S_{R}(A:B)$ of two disjoint intervals $A,B$ and its holographic
description in the standard AdS3/CFT2 correspondence. The review is mainly
based on [10, 17].

###  2.1 Reflected entropy: finite dimensional case

It is easier to start with the definition of reflected entropy in finite
dimensional Hilbert space. Suppose that we have a bipartite quantum system
$A\cup B$ and a mixed state denoted by a density matrix $\rho_{AB}$ on the
Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Analogous
to the relationship between the thermofield double state and thermal density
matrix, we can define the canonical purification of this density matrix in a
doubled Hilbert space
$\mathcal{H}\otimes\mathcal{H}^{\prime}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}}$.
Actually there is a natural mapping between the space of linear operators
acting on $\mathcal{H}$ and the spaces of states on this doubled Hilbert space
$\mathcal{H}\otimes\mathcal{H}^{\prime}$ with the inner product

$\langle\rho|\sigma\rangle_{ABA^{\prime}B^{\prime}}=\mbox{Tr}_{AB}(\rho^{\dagger}\sigma).$
(2.1)

Thus, the operator $\sqrt{\rho_{AB}}$ is mapped to a state
$|\sqrt{\rho_{AB}}\rangle_{ABA^{\prime}B^{\prime}}$ which is the canonical
purification of $\rho_{AB}$. It is not hard to show that the original density
matrix $\rho_{AB}$ can be recovered by tracing out the subregions
$A^{\prime}\cup B^{\prime}$

$\mbox{Tr}_{A^{\prime}B^{\prime}}|\sqrt{\rho_{AB}}\rangle\langle\sqrt{\rho_{AB}}|=\rho_{AB}.$
(2.2)

So the above construction does represent a genuine purification. The reflected
entropy $S_{R}(A:B)$ is defined as the entanglement entropy or the von Neumann
entropy of the density matrix on the subregion $A\cup A^{\prime}$ in the pure
state $|\sqrt{\rho_{AB}}\rangle$

$S_{R}(A:B)\equiv
S(AA^{\prime})_{\sqrt{\rho_{AB}}}=S(BB^{\prime})_{\sqrt{\rho_{AB}}}=S_{vN}(\rho_{AA^{\prime}}),$
(2.3)

where
$\rho_{AA^{\prime}}=\mbox{Tr}_{BB^{\prime}}|\sqrt{\rho_{AB}}\rangle\langle\sqrt{\rho_{AB}}|$
denotes the reduced density matrix after tracing over
$\mathcal{H}_{B}\otimes\mathcal{H}_{B^{\prime}}$. One remarkable properties of
reflected entropy is that it is bounded by the mutual information $I(A:B)$ and
original entanglement entropies $S(A)$, $S(B)$,

$I(A:B)\leq S_{R}(A:B)\leq 2\mbox{min}(S(A),S(B)).$ (2.4)

For more properties of the reflected entropy, please refer to [10].

###  2.2 Reflected entropy in conformal field theory

Although the above definition is adapted to the finite dimensional Hilbert
space, we can directly generalize the definition of $S_{R}(A:B)$ to the
continuous field theories with an infinite dimensional Hilbert space, where
the situation is very similar to the case of defining the entanglement entropy
in field theory. By using the replica trick, we may first define the Rényi
reflected entropy $S_{R}^{(n)}(A:B)$, and then taking the limit that the
replica number $n$ goes to $1$ to read the reflected entropy, under the
assumption that the analytical continuation in $n$ is feasible. In general,
the Rényi reflected entropy is given by the partition functions of original
field theory on a replicated manifolds, which may have nontrivial topology and
geometry. In 2D CFT, due to the infinite dimensional conformal symmetry and
the fact that the twist operators inducing the identification of the fields in
doing the replication are local, the partition function can be transformed
into the multi-point correlation functions of the twist operators in an
orbifold CFT.

However, unlike the case of entanglement entropy, here we need to do double
replication by making $n\times m$ copies of the original theories in order to
both represent the purified state
$|\sqrt{\rho_{AB}}\rangle_{ABA^{\prime}B^{\prime}}$ and evaluate its Renyi
reflected entropy in a path integral language. The first replication replace
the canonically purified state by its $m$ copies

$|\psi_{m}\rangle=\frac{1}{\sqrt{\mbox{Tr}\rho_{AB}^{m}}}|\rho_{AB}^{m/2}\rangle,$
(2.5)

which is normalized. This state can be described by a path-integral when $m\in
2\mathbf{Z}^{+}$. The second replication is necessary to compute the
$(m,n)$-Rényi reflected entropy for a positive integer $n$. We would like to
compute

$S^{(m,n)}_{R}(A:B)=\frac{1}{1-n}\log{\mbox{Tr}\big{(}\rho_{AA^{\prime}}^{(m)}\big{)}^{n}}$
(2.6)

where

$\rho_{AA^{\prime}}^{(m)}=\mbox{Tr}_{BB^{\prime}}|\psi_{m}\rangle\langle\psi_{m}|=\frac{1}{\mbox{Tr}\rho_{AB}^{m}}\mbox{Tr}_{BB^{\prime}}|\rho_{AB}^{m/2}\rangle\langle\rho_{AB}^{m/2}|.$
(2.7)

Introducing $Z_{n,m}$ to represent the un-normalized partition functions

$Z_{n,m}=\mbox{Tr}_{AA^{\prime}}\big{(}\mbox{Tr}_{BB^{\prime}}|\rho_{AB}^{m/2}\rangle\langle\rho_{AB}^{m/2}|\big{)}^{n},$
(2.8)

which satisfy $Z_{1,m}=\mbox{Tr}\rho_{AB}^{m}$, then we find

$S^{(m,n)}_{R}(A:B)=\frac{1}{1-n}\log{\frac{Z_{n,m}}{(Z_{1,m})^{n}}}.$ (2.9)

Note that the partition function $Z_{n,m}$ can be evaluated by a path integral
on a replicated manifold $\Sigma_{n,m}$ for $n\in\mathbf{Z}^{+}$ and $m\in
2\mathbf{Z}^{+}$. Note that the number of replica sheets in the replicated
manifold of $|\rho_{AB}^{m/2}\rangle$ is $m/2$ which should be a positive
integer. Then by using the analytic continuation in $m$ and $n$, we can
evaluate the reflected entropy by taking the limit:

$S_{R}(A:B)=\lim_{n,m\to 1}S^{(m,n)}(A:B).$ (2.10)

Another way to obtain the partition function (2.8) is to compute the
correlation function of the (generalized) twist operators, which induce the
identifications between the fields at different replicas. For disjoint
intervals $A$ and $B$ in general dimensions, we can rewrite and reinterpret
the partition functions $Z_{n,m}$ in (2.8) as follows [10]

$Z_{n,m}=\langle\Sigma_{g_{B}}(B)\Sigma_{g_{A}}(A)\rangle_{\mbox{CFT}^{\otimes
nm}}$ (2.11)

where $\Sigma$’s are twist operators located at $\partial A$ or $\partial B$
in the product theory $\mbox{CFT}^{\otimes nm}$. The meaning of the twist
operators needs some clarifications. Usually in higher dimensions ($D\geq 3$),
the twist operators are truly non-local and hard to work with, while in two
dimensions the twist operators can be seen as the local fields located at each
endpoints. Moreover, if the original 2D field theory is conformal invariant,
the twist operators in it are actually primary fields, and hence their
correlation functions can be discussed using various analytical CFT
techniques. However, there is an additional complication when actually
calculating the reflected entropy, which is related to the fact that we choose
not to gauge the product theory $\mbox{CFT}^{\otimes nm}$. We would clarify
this point more later. Now we are taking double replication and thus have
$n\times m$ copies of original CFT, each living on a flat Euclidean spacetime.
There is a symmetric permutation group $S_{nm}$ related to these replicas. We
can label each replica with
$(\alpha,\beta)\in(\mathbf{Z}_{n}^{+},\mathbf{Z}_{m}^{+})$, and represent the
actions of the special group elements of $S_{nm}$ which are needed in the
definition of $\Sigma$ operators by

$\tau_{m}^{(\nu)}(\alpha,\beta)=(\alpha,\beta+\delta_{\alpha,\nu}),\hskip
12.91663pt\tau_{n}^{(\mu)}(\alpha,\beta)=(\alpha+\delta_{\beta,\mu},\beta).$
(2.12)

The first action $\tau_{m}^{(\nu)}(\alpha,\beta)$ defines the cyclic
permutation on the fixed $n$-replica along $m$ direction, while the second one
$\tau_{n}^{(\mu)}(\alpha,\beta)$ defines the cyclic permutation on the fixed
$m$-replica along $n$ direction. The full $m$-cyclic permutation is simply the
product of all $\tau_{m}^{(\nu)}(\alpha,\beta)$, i.e.,
$g_{m}=\prod_{\nu=0}^{n-1}\tau_{m}^{(\nu)}$, and similarly for $g_{n}$. From
the Euclidean path integral in computing the partition functions $Z_{nm}$, we
can read the group elements which identify the fields in different replicas in
an appropriate way:

$\displaystyle\partial A:$ $\displaystyle
g_{A}=(\tau_{n}^{(0)})^{-1}\tau_{n}^{(m/2)}g_{m},$ (2.13)
$\displaystyle\partial B:$ $\displaystyle g_{B}=g_{m}.$ (2.14)

These two group elements help us to define co-dimension-two twist operators
$\Sigma(A)$ at $\partial A$ and $\Sigma(B)$ at $\partial B$ in (2.11). Note
that the two group elements $g_{A}$ and $g_{B}$ defined in (2.13,2.14) are
conjugate to each other. For more complete discussions and properties about
the group elements defined here, see [10].

Let us specialize to 2D CFT and still choose two disjoint intervals
$A=(x_{1},x_{2})$ and $B=(x_{3},x_{4})$ on the same time-slice with
$x_{1}<x_{2}<x_{3}<x_{4}$. Here the two twist operators $\Sigma(A)$,
$\Sigma(B)$ become four (quasi) local twist operators $\sigma_{g_{A}}$,
$\sigma_{g_{A}^{-1}}$, $\sigma_{g_{B}}$ and $\sigma_{g_{B}^{-1}}$ located at
each endpoints separately as mentioned above. Then the partition functions
(2.8) is captured by the following four-point function

$\displaystyle Z_{n,m}$
$\displaystyle=\langle\sigma_{{g}_{A}}(x_{1})\sigma_{{g}^{-1}_{A}}(x_{2})\sigma_{{g}_{B}}(x_{3})\sigma_{{g}^{-1}_{B}}(x_{4})\rangle_{CFT^{\otimes
mn}},$ (2.15) $\displaystyle Z_{1,m}$
$\displaystyle=\langle\sigma_{{g}_{m}}(x_{1})\sigma_{{g}^{-1}_{m}}(x_{2})\sigma_{{g}_{m}}(x_{3})\sigma_{{g}^{-1}_{m}}(x_{4})\rangle_{CFT^{\otimes
m}}.$ (2.16)

The twist operators $\sigma_{g_{A}}$, $\sigma_{g_{A}^{-1}}$, $\sigma_{g_{B}}$
and $\sigma_{g_{B}^{-1}}$ are defined by the specific boundary conditions on
the replica sheets, and the twist operators $\sigma_{g_{m}}$ and
$\sigma_{g_{m}^{-1}}$ represent the usual cyclic and anti-cyclic permutations
among $m$ replica sheets, which can be read from the $n\to 1$ limit of
$\sigma_{g_{A}}$, $\sigma_{g_{B}}$ operators:

$\sigma_{g_{m}}=\lim_{n\to 1}\sigma_{g_{A}}=\lim_{n\to 1}\sigma_{g_{B}}.$
(2.17)

In Figure 1, we show explicitly the monodromy conditions or field
identifications of various twist operators in the case $m=4,n=2$. It can be
interpreted as a path integral representation of the partition function
$Z_{2,4}=\mbox{Tr}_{{\cal H}_{A}{\cal H}_{A^{\prime}}}\left(\mbox{Tr}_{{\cal
H}_{A}{\cal
H}_{A^{\prime}}}|\rho_{AB}^{2}\rangle\langle\rho_{AB}^{2}|\right)^{2}$ with
replicated $2\times 4$ sheets.

Figure 1: (Half of) different monodromy conditions or field identifications
related to the four twist operators $\sigma_{g_{A}}$, $\sigma_{g_{A}^{-1}}$,
$\sigma_{g_{B}}$ and $\sigma_{g_{B}^{-1}}$ in the case $m=4,n=2$. The twist
operator $\sigma_{g_{A}}$ can be equivalently represented in another useful
notation $\sigma_{(1278)(3456)}$ for this particular picture, where $(1278)$
denotes the cyclic permutation of these four numbers. Here to keep the picture
tidy and clean, we only show half of the identifications $(1278)$.

There are a few remarkable features in the calculation of reflected entropy
$S_{R}(A:B)$ of two disjoint intervals, compared with the ones of entanglement
entropy or mutual information:

  * •

In calculating the mutual information, the twist operators can be taken as the
primary operators in an Abelian orbifold CFT
$\mbox{CFT}^{\otimes}/\mathbf{Z}^{n}$, while loosely speaking for the
reflected entropy the twisted operators are in a non-Abelian orbifold CFT. The
twist operators induce the field identifications in different replicas. In the
Abelian case, the field identifications are simply

$\phi_{j}(w+(z-w)e^{i\theta})\sigma_{n}(z,\bar{z})\to\phi_{j+1}(z)\sigma_{n}(z,\bar{z}),\quad\theta\in(0,2\pi),\quad
j\in[1,n]\in\mathbf{Z}^{+},$ (2.18)

while in the non-Abelian case, the field identifications may take more
involved forms

$\phi_{j}(w+(z-w)e^{i\theta})\sigma_{p}(z,\bar{z})\to\phi_{p(j)}(z)\sigma_{p}(z,\bar{z}),\quad\theta\in(0,2\pi),\quad
j\in[1,n]\in\mathbf{Z}^{+}$ (2.19)

where $p\in S_{n}$ is an arbitrary group element in permutation group $S_{n}$
and $p(j)$ is the image of number $j$ under the group element $p$. A crucial
point is that the permutation group element $p$ can have complicated multi-
cycle structures. For example in Figure 1 the operator $\sigma_{g_{B}}$ can be
rewritten in the form $\sigma_{(1234)(5678)}$ with $(1234)$ representing
cyclic permutation of $(1,2,3,4)$. In any case, the holomorphic weight of the
twist operator is

$h_{p}=\sum_{j=1}^{m}h_{p_{j}}=\frac{c}{24}\sum_{j=1}^{m}(p_{j}-\frac{1}{p_{j}})$
(2.20)

which is simply the summation of the holomorphic weights of the “twist”
operators in separated cycles. This is closely related to the locality of
stress tensor $T_{\mu\nu}$. Another important features is that the OPE of
these arbitrary twist operators must respect the group symmetry of $S_{n}$.

  * •

The twist operators $\sigma_{g_{A}}$, $\sigma_{g_{A}^{-1}}$, $\sigma_{g_{B}}$
and $\sigma_{g_{B}^{-1}}$ are defined in a product theory $\mbox{CFT}^{\otimes
mn}$, without gauging the permutation group $S_{nm}$. Due to the specific
boundary conditions of the replica manifolds in the path integral
representation, or more precisely, due to the multi-cycle structure of
$g_{A}$, $g_{B}$ group elements in defining the relevant twist operators,
gauging the permutation group $S_{nm}$ would not only generate local
operators, but also introduce some complicated non-local objects connecting
the boundaries of the twist operators. Thus we choose to directly work in the
tensor product theory $\mbox{CFT}^{\otimes nm}$, and at the same time be
careful with various subtleties originating from this choice.

  * •

One subtlety related to the product theory $\mbox{CFT}^{\otimes nm}$ is that
unlike the true non-Abelian orbifold, where different twist operators are
labelled by different group conjugacy classes, the operator spectrum of the
this product theory contain much more “new” twist operators in the “new”
twisted sectors, that would be identified as the same one after gauging the
group $S_{n}$. Thus we should be careful about the evaluation of the OPE
coefficients of the twist operators, due to the great richness of spectrum.
Note that usually in the evaluation of entanglement entropy $S_{A}$ or mutual
information $I(A:B)$ in holographic 2D CFT, the OPE coefficient in the
expansion

$\sigma_{n}\tilde{\sigma_{n}}\sim\mathrm{1}+...$ (2.21)

is of order $1$, and thus this OPE coefficient do not play an important role
in the final answer of $S_{A}$ or $I(A:B)$. While in the reflected entropy
case, as explicitly shown below, the OPE coefficients indeed contribute
significantly to the final answer. Actually the twist operators in the product
theory are not the truly local ones. More precisely, they always carry the
disconnected multi-cycle structures in their definition such that the four-
point correlation functions of twist operators are not single valued.
Nevertheless, there exist operator product expansions (OPE) of these twist
operators as well. The price we pay is that the fusion rule for the OPE now is
as follows

$\sigma_{g_{A}^{-1}}\sigma_{g_{B}}\to\sigma_{g_{B}g_{A}^{-1}}+...$ (2.22)

where the lowest-weight operator appearing in the expansion is not the
identity operator, although $g_{A}$ and $g_{B}$ are conjugate to each other.
The ellipsis here denote the primary operators with the same monodromy
conditions as the twist operator $\sigma_{g_{B}g_{A}^{-1}}$ but with higher
weights.

Now coming back to the evaluation about the reflected entropy $S_{R}(A:B)$, we
can get directly from (2.20) the weights of various twist operators

$h_{\sigma_{g_{A}}}=h_{\sigma_{g_{B}}}=nh_{m},\quad
h_{\sigma_{g_{B}}\sigma_{g_{A}}^{-1}}=2h_{n}.$ (2.23)

This comes from the fact that as we go through the replicated manifold with
$mn$ sheets as shown in Figure 1, we would find that $g_{A}$ and $g_{B}$ both
contain $n$ $m$-cycles, and $g_{B}g_{A}^{-1}$ contain 2 $n$-cycles. Next let
us turn to the OPE coefficients in (2.22). We have

$\sigma_{g_{A}^{-1}}\sigma_{g_{B}}=C_{n,m}\sigma_{g_{B}g_{A}^{-1}}+...\quad,\quad\quad\quad\mbox{with}\hskip
8.61108ptC_{n,m}=(2m)^{-4h_{n}}$ (2.24)

which can be proved by using the method presented in [42, 10].

###  2.3 Reflected entropy: holographic 2D CFT

As is well known, in any 2D CFT with the Virasoro symmetries, a four-point
correlation function of primary operators can be expanded by the Virasoro
conformal blocks

$\langle{\cal O}_{1}(0){\cal O}_{2}(x){\cal O}_{3}(1){\cal
O}(\infty)\rangle=\sum_{{\cal O}_{p}}c_{12p}c_{34p}{\cal
F}(c,h_{p},h_{i},z)\bar{{\cal F}}(c,\bar{h}_{p},\bar{h}_{i},\bar{z})$ (2.25)

where $z=z_{12}z_{34}/z_{13}z_{24}$ is the standard cross ratio and we have
expanded in $s$-channel, i.e. $z\to 0$ limit. Here $h_{i},\bar{h}_{i}$ are the
holomorphic and anti-holomorphic weights of the primary operator ${\cal
O}_{i}$, the summation is over all the primary operators ${\cal O}_{p}$
appearing in the OPE, and $c_{ijp}$ denote the corresponding OPE coefficients.
The Virasoro blocks ${\cal F}$ capture all the contributions from the Verma
module of ${\cal O}_{p}$. In general, the Virasoro blocks do not have simple
analytical expressions, and a way to manipulate them is to compute their forms
in a series expansion by using the recursive relations first developed in
[43]. However, if we focus on the holographic CFT with a large central charge
in the semi-classical limit, the Virasoro conformal block is expected to be
exponentiated [43, 44]

${\cal
F}(c,h_{p},h_{i},z)\simeq\exp\left\\{-\frac{c}{6}f\left(\frac{h_{p}}{c},\frac{h_{i}}{c},z\right)\right\\},$
(2.26)

where the function $f$ can be determined by the solution of certain well-
defined monodromy problem.

The holographic CFT is expected to own a sparse light spectrum, which would
lead to the vacuum block dominance when calculating the four-point function of
twist operators.In the following we would like to show the calculation of the
reflected entropy $S_{R}(A:B)$ for two disjoint intervals sitting on the same
time slice in both the $t$-channel and $s$-channel, and in both the vacuum
state and thermal state on the Lorentzian plane for later convenience.

##### Vacuum state on the plane

Let us first consider the conformal block expansion of four-point correlation
in the $t$-channel, i.e., $z\to 1$, which corresponds to the situation that
the two disjoint intervals are quite close to each other compared with their
own lengths. As shown in (2.25),

$\displaystyle\langle\sigma_{{g}_{A}}(x_{1})\sigma_{{g}^{-1}_{A}}(x_{2})\sigma_{{g}_{B}}(x_{3})\sigma_{{g}^{-1}_{B}}(x_{4})\rangle_{\mbox{CFT}^{\otimes
nm}(\mathbf{C})}$
$\displaystyle=\frac{1}{x_{41}^{4h_{g_{A}}}x_{32}^{4h_{g_{A}}}}\sum_{{\cal
O}_{p}}C_{ABp}^{2}\mathcal{F}(mnc,h_{i},h_{p},1-z)\overline{\mathcal{F}}(mn\bar{c},\bar{h}_{i},\bar{h}_{p},1-\bar{z})$
$\displaystyle\approx\frac{1}{x_{41}^{4nh_{m}}x_{32}^{4nh_{m}}}C_{mn}^{2}e^{-2(2h_{n})\ln{\frac{1+\sqrt{z}}{4(1-\sqrt{z})}}}.$
(2.27)

where $z$ is still the standard conformal invariant cross ratio, which is a
real quantity in the symmetric setup. Now we are considering the product
theory so the central charge are $mnc$ and $mn\bar{c}$. In the last step, we
assume the conformal block dominance from the single Virasoro block of the
primary twist operator $\sigma_{g_{B}g_{A}^{-1}}$ in the fusion rule of OPE
(2.22). In addition, we have used the fact that in the semi-classical limit
defined by the conditions

$mnc\to\infty,\qquad\text{while}\quad\frac{h_{i}}{mnc},\frac{h_{p}}{mnc}\quad\text{fixed,
}$ (2.28)

the Virasoro conformal block $\mathcal{F}$ would be exponentiated into the
following form

$\log{\mathcal{F}(mnc,h_{i},h_{p},1-z)}\sim-\frac{mnc}{6}f(\frac{h_{i}}{mnc},\frac{h_{p}}{mnc},1-z)\sim-
h_{p}\ln{\frac{1+\sqrt{z}}{4(1-\sqrt{z})}}.$ (2.29)

From the above expression we can find the reflected entropy for the vacuum
state in 2D holographic CFTs at the leading order in the large $c$ limit

$\displaystyle S_{R;vac}^{CFT}(A:B)$
$\displaystyle=\frac{c}{3}\ln{\frac{1+\sqrt{z}}{1-\sqrt{z}}}.$ (2.30)

For convenience, we set the two disjoint intervals $A,B$ to be symmetric about
the origin and sitting on a fixed time slice $t=0$

$A=(-b,-a),\quad B=(a,b),\quad z=(\frac{b-a}{b+a})^{2},$ (2.31)

then we have the final result:

$S_{R;vac}^{CFT}(A:B)=\frac{c}{3}\ln\frac{b}{a}.$ (2.32)

##### Thermal state on the plane

To represent the thermal state of 2D CFT at $t=0$ timeslice on a Lorentzian
plane, we should construct an Euclidean path integral representation where the
imaginary time direction should be compactified on an infinite long cylinder
with circumference $\beta=T^{-1}$. Using the map $z=e^{\frac{2\pi w}{\beta}}$,
which takes the complex plane $(z,\bar{z})$ to the Euclidean cylinder
$(w,\bar{w})$, we can get the resulting four point functions of twist
operators on the thermal Euclidean cylinder

$\displaystyle\langle\sigma_{{g}_{A}}(w_{1})\sigma_{{g}^{-1}_{A}}(w_{2})\sigma_{{g}_{B}}(w_{3})\sigma_{{g}^{-1}_{B}}(w_{4})\rangle_{\mbox{\small
CFT}^{\otimes nm}}^{cylinder}$ (2.33) $\displaystyle=$
$\displaystyle|z^{\prime}(w_{1})z^{\prime}(w_{2})z^{\prime}w_{3})z^{\prime}(w_{4})|^{2h_{g_{A}}}\langle\sigma_{{g}_{A}}(z_{1})\sigma_{{g}^{-1}_{A}}(z_{2})\sigma_{{g}_{B}}(z_{3})\sigma_{{g}^{-1}_{B}}(z_{4})\rangle_{\mbox{\small
CFT}^{\otimes nm}(\mathbf{C})}.$

Suppose we consider the symmetric configuration (2.31) on the Euclidean
cylinder at $\tau=0$ slice

$A=(-b,-a),\quad B=(a,b),$ (2.34)

which under the conformal mapping would lead to a configuration on the plane
with a cross ratio

$z=\frac{z_{12}z_{34}}{z_{13}z_{24}}=\frac{\sinh{\frac{\pi
z_{12}}{\beta}}\sinh{\frac{\pi z_{34}}{\beta}}}{\sinh{\frac{\pi
z_{13}}{\beta}}\sinh{\frac{\pi
z_{24}}{\beta}}}=\left(\frac{\sinh{\frac{\pi(b-a)}{\beta}}}{\sinh{\frac{\pi(b+a)}{\beta}}}\right)^{2}.$
(2.35)

Then the reflected entropy on the thermal state $S_{R;thermal}^{CFT}(A:B)$ at
the temperature $T=\beta^{-1}$ with the configuration (2.34) would be

$\displaystyle
S_{R;thermal}^{CFT}(A:B)=\frac{c}{3}\ln{\frac{1+\sqrt{z}}{1-\sqrt{z}}}=\frac{c}{3}\ln(\coth\frac{\pi
a}{\beta}\tanh\frac{\pi b}{\beta}).$ (2.36)

##### Phase transition

On the other hand, the $s$-channel conformal block expansion of the four point
correlator (2.25) comes from the OPE

$\sigma_{g_{A}}\sigma_{g_{A}^{-1}}\sim\textbf{1}+....$ (2.37)

When taking the limit $x\to 0$ and the dominant Virasoro block is just the one
related to the identity operator with dimension $h=\bar{h}=0$. Then the
reflected entropies at the leading order of $c$ are simply

$S_{R;vac}^{CFT}(A:B)=S_{R;thermal}^{CFT}(A:B)=0.$ (2.38)

As a result, there is a first-order phase transition when the cross ration $x$
goes from $0$ to $1$, which corresponds to the change of different dominant
blocks in the conformal block expansion. This is similar to the situations
happened in the mutual information $I(A:B)$ of two disconnected intervals.

###  2.4 Entanglement wedge cross section

It was proposed in [10] that for a holographic CFT, the reflected entropy
$S_{R}(A:B)$ is captured holographically by the area of a simple geometric
object called the entanglement wedge cross-section (EWCS)

$S_{R}(A:B)=2\frac{Area(EWCS)}{4G}+S_{R}(bulk)+...,$ (2.39)

in which the first term is the semi-classic contribution in gravity and the
other terms are from quantum corrections. In this subsection, we give a brief
review on entanglement wedge and entanglement wedge cross-section.

The notion of entanglement wedge, which plays an essential role in the bulk
reconstruction of AdS/CFT, was introduced in the literature when discussing
the subregion/state duality[6, 45, 46], which aims to find the bulk region
that is dual to the boundary reduced density matrix $\rho_{AB}$. In a
generically time-dependent setting, the holographic entanglement entropy of a
subregion $AB$ is given by the area of so-called HRT surface[47]
$\gamma_{AB}$. The entanglement wedge is the bulk domain of dependence of a
co-dimension one surface, which interpolates between $AB$ and the extremal
surface $\gamma_{AB}$ anchored at $\partial(AB)$. Another closely related
notion in the bulk reconstruction is the so-called bulk causal wedge, which is
defined to be the intersection of the causal past and future of boundary
domain of dependence $D[AB]$. In the case of pure AdS spacetime, these two
wedges coincide with each other, while in more general situations, the
entanglement wedge contains the causal wedge.

Figure 2: Entanglement wedges related to single boundary interval and two
boundary intervals (with large cross ratio) in Poincare coordinates of AdS3.
The blue plane denote the asymptotic boundary, the red line is the Cauchy
surface on the boundary. The closed region bounded by yellow lightsheets and
blue boundary is the entanglement wedge, whose intersection with the boundary
is the domain of dependence of boundary subregion.

The entanglement wedge cross-section (EWCS) is the minimal cross-section area
of the entanglement wedge of two disconnected intervals. It is related to a
bulk co-dimension 2 extremal surface whose endpoints lie on two separated null
surfaces composing the boundary of entanglement wedge. Actually, this extremal
surface can only exist on the surface $\Sigma_{AB}$ whose boundary is the
union of the boundary intervals and its corresponding HRT surfaces
$\partial\Sigma_{AB}=AB\cup\gamma_{AB}$, and its endpoints must lie on two
separated HRT surfaces. More generally, as a simple fact of differential
geometry, there is no extremal spacelike geodesics interpolating between two
non-intersecting null surfaces, unless they have spacelike boundaries.

Figure 3: Two dynamical constructions of EWCS. The left figure (a) shows the
cutting and gluing procedure. The field theory living on $AA^{*}$, $BB^{*}$
independently is the same as the original one. The Green lines denote
$\gamma_{AB}$ which is the RT surface of $AB$. The right figure (b) shows that
viewing the union of blue part of $\gamma_{AB}$ and $A$ as the new subregion
(the union of green part of $\gamma_{AB}$ and $B$ is its complement), and EWCS
is just the minimal RT surface.

Although the kinematical definition of EWCS is clear, which is good enough to
give the leading semi-classical contribution in (2.39), there actually exist
two dynamical constructions in the literature [4, 5, 10], which could lead to
different first-order quantum corrections in (2.39). These two constructions
both get intuition from the sub-region duality. The first one [4, 5] take the
RT surface $\gamma_{AB}$ as part of the new boundary and remove the geometry
outside the entanglement wedge of $AB$ so that the field theory defined on
$AB\cup\gamma_{AB}$ is in a pure state. Separating $\gamma_{AB}$ into two
parts and combining $A$ or $B$ to get two regions, one may try to minimize the
entanglement between two regions by using the RT formula and then find that
the minimal entanglement would give us the EWCS, as shown in Figure 3 (b). The
second construction [10] glues the entanglement wedge to its CPT conjugate
using the Engelhardt-Wall procedure [48, 49] and take the entangling surface
between the two boundary fields in this new two-sided black hole as twice the
EWCS, as shown in Figure 3 (a).

##  3 Reflected entropy in 2D WCFT

In the following sections we would like to consider the reflected entropy in
the framework of AdS3/WCFT[25]. As it has been shown in [27, 41], the
holographic entanglement entropy in this correspondence presents some novel
features, it would be interesting to investigate the holographic description
of the reflected entropy. As the first step, we study the computation of the
reflected entropy in 2D warped conformal field theories in this section. To
compute the reflected entropy, we have to study carefully the properties of
the relevant twist operators in the orbifold theory of WCFT.

###  3.1 Warped conformal field theory

2D WCFT is a non-Lorentzian invariant field theory, which possess an infinite
dimensional symmetry named the warped conformal symmetry. The symmetry is
generated by a Virasoro algebra plus a $U(1)$ Kac-Moody algebra, with the
global part being $SL(2,R)\times U(1)$. This warped conformal symmetry is
constraining and powerful, and allows us to obtain the Cardy-like formula
[24], one-loop partition functions [50] and to do the modular bootstrap[30].

Let us first consider a 2D WCFT on the reference plane with coordinates
$(x,y)$. In position space a general warped conformal transformation can be
written as

$x^{\prime}=f(x),\quad y^{\prime}=y+g(x)$ (3.1)

where $f(x)$ and $g(x)$ are two arbitrary functions. Let $T(x)$ and $P(x)$ be
the Noether currents associated with the translations along $x$ and $y$
directions respectively. Under the warped conformal transformations (3.1),
their transformation properties are given by

$\begin{split}T^{\prime}(x^{\prime})&=\Big{(}\frac{\partial x}{\partial
x^{\prime}}\Big{)}^{2}\Big{(}T(x)-\frac{c}{12}\\{f,x\\}\Big{)}+\frac{\partial
x}{\partial x^{\prime}}\frac{\partial y}{\partial
x^{\prime}}P(x)-\frac{k}{4}\Big{(}\frac{\partial y}{\partial
x^{\prime}}\Big{)}^{2},\\\ P^{\prime}(x^{\prime})&=\frac{\partial x}{\partial
x^{\prime}}\Big{(}P(x)+\frac{k}{2}\frac{\partial y^{\prime}}{\partial
x}\Big{)},\end{split}$ (3.2)

where $\\{f,x\\}$ denotes the standard Schwarzian derivative, $c$ is the
central charge and $k$ is the $U(1)$ Kac level. We can construct infinitely
many conserved charges out of these two currents

$\displaystyle L_{n}=-\frac{i}{2\pi}\int dxx^{n+1}T(x),\quad
P_{n}=-\frac{1}{2\pi}\int dxx^{n}P(x).$ (3.3)

The commutation relations for these charges form a warped conformal algebra
consisting of a Virasoro algebra and a Kac-Moody algebra,

$\displaystyle=(n-m)L_{n+m}+\frac{c}{12}n(n^{2}-1)\delta_{n,-m},$ (3.4)
$\displaystyle[L_{n},P_{m}]$ $\displaystyle=-mP_{n+m},$
$\displaystyle[P_{n},P_{m}]$ $\displaystyle=k\frac{n}{2}\delta_{n,-m}.$

When describing 2D WCFT, it is important to account for spectral flow
transformations that induce a redefinition of charges $L_{n}$ and $P_{n}$
while leaving the Virasoro-Kac-Moody algebra unchanged. On the reference plane
$(x,y)$ of WCFT the vacuum expectation values of conserved currents are zero
$\langle T(x)\rangle=\langle P(x)\rangle=0$, which means that this vacuum
state is invariant under the global $SL(2,R)\times U(1)$ transformation.
Moreover the vacuum state on the reference plane satisfies

$L_{n}|0\rangle=P_{n+1}|0\rangle=0,\quad n\geq-1.$ (3.5)

We can also define WCFT on a canonical plane with coordinates
$(x^{\prime},y^{\prime})$ by introducing a spectral flow parameter $\mu$
together with a specific warped conformal transformation

$x^{\prime}=x,\quad y^{\prime}=y+\mu\log{(x)}.$ (3.6)

Then the spectral flowed charges $L^{\prime}_{n}$ and $P^{\prime}_{n}$ satisfy
following relations,

$L^{\prime}_{n}=L_{n}-i\mu P_{n}-\frac{\mu^{2}k}{4}\delta_{n},\quad
P^{\prime}_{n}=P_{n}-\frac{i\mu k}{2}\delta_{n}$ (3.7)

which lead to the non-vanishing vacuum expectation values of zero modes
$L^{\prime}_{0}$ and $P^{\prime}_{0}$ on this canonical plane

$\langle L^{\prime}_{0}\rangle=-\frac{\mu^{2}k}{4},\quad\langle
P^{\prime}_{0}\rangle=-\frac{i\mu k}{2}.$ (3.8)

Note that if the spectral flow parameter $\mu$ is real, then the charge
$\langle P^{\prime}_{0}\rangle$ is imaginary, which is a special feature of
WCFT with holographic dual.

To match with the holographic calculations in asymptotic AdS3 spacetime, we
need to define WCFT having specific vacuum charges on the canonical cylinder
with the coordinates $(\hat{x}^{\prime},\hat{y}^{\prime})$. This can be
accomplished by using an exponential map in the direction $x^{\prime}$, while
keeping the direction $y^{\prime}$ unchanged

$x^{\prime}=e^{i\hat{x}^{\prime}},\quad\hat{y}^{\prime}=y^{\prime}.$ (3.9)

The vacuum charges on this canonical cylinder can be obtained by adding a term
$-\frac{c}{24}$ to the Virasoro zero mode on the canonical plane, like what
happens in 2D CFT. To be complete, we can also define the reference cylinder
$(\hat{x},\hat{y})$ by the same exponential map related to the reference
plane. These two cylinders are related by a spectral flow transformation

$\hat{x}=\hat{x}^{\prime},\quad\hat{y}=\hat{y}^{\prime}-i\mu\hat{x}^{\prime}$
(3.10)

where the parameter $\mu$ is the same one as in (3.6). Because of boundary
conditions, we start from the canonical cylinder with the identification
$(\hat{x},\hat{y}^{\prime})\sim(\hat{x}+2\pi,\hat{y}^{\prime})$, which leads
to a corresponding identification on the reference plane

$(x,y)\sim(xe^{2\pi i},y-2\pi i\mu).$ (3.11)

Note that (3.11) is consistent with the modular flow transformation (3.6).

Different setups of the WCFT play their own specific roles in exploring
various properties of WCFT and its holographic dual. We summarize here for
later convenience,

  * •

Reference plane:

$(x,y)\sim(xe^{2\pi i},y-2\pi i\mu),\quad\langle
L^{vac}_{0}\rangle=0,\quad\langle P^{vac}_{0}\rangle=0.$ (3.12)

  * •

Canonical plane:

$(x^{\prime},y^{\prime})\sim(x^{\prime}e^{2\pi i},y^{\prime}),\quad\langle
L^{vac}_{0}\rangle=-\frac{\mu^{2}k}{4},\quad\langle
P^{vac}_{0}\rangle=-\frac{i\mu k}{2}.$ (3.13)

  * •

Reference cylinder:

$(\hat{x},\hat{y})\sim(\hat{x}+2\pi,\hat{y}-2\pi i\mu),\quad\langle
L^{vac}_{0}\rangle=-\frac{c}{24},\quad\langle P^{vac}_{0}\rangle=0.$ (3.14)

  * •

Canonical cylinder:

$(\hat{x}^{\prime},\hat{y}^{\prime})\sim(\hat{x}^{\prime}+2\pi,\hat{y}^{\prime}),\quad\langle
L^{vac}_{0}\rangle=-\frac{c}{24}-\frac{\mu^{2}k}{4},\quad\langle
P^{vac}_{0}\rangle=-\frac{i\mu k}{2}.$ (3.15)

  * •

The coordinate transformations among these setups are

$\displaystyle x=x^{\prime},$ $\displaystyle
y=y^{\prime}-\mu\log(x^{\prime}),$ $\displaystyle\hat{x}=\hat{x}^{\prime},$
$\displaystyle\hat{y}=\hat{y}^{\prime}-i\mu\hat{x}^{\prime},$ $\displaystyle
x=e^{i\hat{x}},$ $\displaystyle y=\hat{y}.$ (3.16)

Note that as the underlying geometry of WCFT is not the usual Riemann
surfaces, neither the plane nor the cylinder in which a WCFT is defined is the
complex plane or Euclidean cylinder in the usual sense.

Another remarkable point is that there exist spectral-flow invariant Virasoro
generators

$L^{inv}_{n}=L_{n}-\frac{1}{k}\big{(}\sum_{m\leq-1}P_{m}P_{n-m}+\sum_{m\geq
0}P_{n-m}P_{m}\big{)},$ (3.17)

which commute with all the Kac-Moody generators $P_{n}$, and form a Virasoro
algebra with central charge $c-1$. A primary state is defined to be in the
highest weight representation, and it is characterized by the weight $h$ and
the charge $q$, which are eigenvalues of $L_{0}$ and $P_{0}$ respectively.
Then the primary state $|h,q\rangle$ under $L_{n}$ and $P_{n}$ is also a
primary state under $L_{n}^{inv}$ and $P_{n}$, but with the conformal weight
being shifted by a $q$-dependent quantity

$h^{inv}=h-\frac{q^{2}}{k}.$ (3.18)

Since the spectral flow invariant Virasoro $L_{n}^{inv}$ commute with $P_{n}$,
it is more convenient to label the state basis, i.e. the primary states and
their warped conformal descendants, by $(h^{inv},q)$ when considering the
conformal block expansion.

Due to the vanishing of the vacuum charges on the reference plane, we can
completely determine the two- and three-point functions on the vacuum state of
WCFT by using the global $SL(2,R)\times U(1)$ symmetries. Moreover, the four-
point function is determined up to a function of the cross-ratio
$\mathrm{x}=\frac{x_{12}x_{34}}{x_{13}x_{24}}$ with $x_{ij}\equiv
x_{i}-x_{j}$,

$\langle\phi_{1}\phi_{2}\phi_{3}\phi_{4}\rangle=\big{(}e^{i\sum_{j}q_{j}y_{j}}\delta({\sum_{i}q_{i}})\big{)}\big{(}\frac{x_{24}}{x_{14}}\big{)}^{h_{12}}\Big{(}\frac{x_{14}}{x_{13}}\Big{)}^{h_{34}}\frac{\mathcal{G}(\mathrm{x})}{x_{12}^{h_{1}+h_{2}}x_{34}^{h_{3}+h_{4}}},$
(3.19)

where $\phi_{i}\equiv\phi_{i}(x_{i},y_{i})$ is the primary operator with
conformal weight $h_{i}$ and $U(1)$ charge $q_{i}$ respectively, and
$\mathcal{G}(\mathrm{x})$ is an undetermined function of the cross ratio
$\mathrm{x}$. The function $\mathcal{G}(\mathrm{x})$ can be decomposed into
the warped conformal blocks $A^{h,q}(\mathrm{x})$

$\mathcal{G}(\mathrm{x})=\sum_{\mathcal{O}}C_{12\mathcal{O}}C^{\mathcal{O}}_{34}A^{h,q}(\mathrm{x}),$
(3.20)

where the sum runs over the primary states with weight $h$ and charge
$q=q_{3}+q_{4}=-q_{1}-q_{2}$, and $C_{12\mathcal{O}},C^{\mathcal{O}}_{34}$ are
OPE coefficients. Actually the full warped conformal block is just the
Virasoro-Kac-Moody block whose expression has been found in chiral CFTs with
an internal $U(1)$ symmetry in [51]. More precisely, the warped conformal
block can be factorized as

$\displaystyle
A^{h,q}(\mathrm{x})=\mathcal{V}_{p}(k,q_{i},\mathrm{x})\mathcal{F}(c-1,h_{i}^{inv},h^{inv},\mathrm{x})=\mathrm{x}^{\frac{2q_{1}^{2}}{k}}(1-\mathrm{x})^{\frac{2q_{1}q_{2}}{k}}\mathcal{F}(c-1,h_{i}^{inv},h^{inv},\mathrm{x})$
(3.21)

where $\mathcal{V}_{p}(k,q_{i},\mathrm{x})$ is the Kac-Moody block,
$\mathcal{F}(c-1,h_{i}^{inv},h^{inv},\mathrm{x})$ is the standard Virasoro
conformal block with central charge $c-1$.

Under the warped conformal transformation (3.1), the $n$-point functions of
primary fields transform as

$\langle\prod_{i=1}^{n}\phi^{\prime}_{i}(x^{\prime}_{i},y^{\prime}_{i})\rangle=\prod_{i=1}^{n}\Big{(}\frac{\partial
x^{\prime}_{i}}{\partial
x_{i}}\Big{)}^{-h_{i}}\langle\prod_{i=1}^{n}\phi_{i}(x_{i},y_{i})\rangle.$
(3.22)

Note that it only depends on the diffeomorphism of $x$.

###  3.2 Twist operators in orbifold WCFT

To compute the reflected entropy, it is essential to understand the properties
of twist operators in non-Abelian orbifold WCFT. Let us first consider the
Abelian orbifold, which appears in the study of entanglement entropy.

####  3.2.1 Twist operator in Abelian orbifold WCFT

In computing the single-interval entanglement entropy, one way is to compute
the two-point function of twist operators. In this case, the resulting
orbifold is an Abelian orbifold. The conformal weight $h_{n}$ and the $U(1)$
charge $q_{n}$ of the twist field $\sigma_{n}$ for WCFT were determined in
[40, 28] by using a generalization of Rindler method developed in [52]. The
results are as follows

$h_{n}=n(\frac{c}{24}+\frac{L_{0}^{vac}}{n^{2}}+\frac{iP_{0}^{vac}\alpha}{2n\pi}-\frac{\alpha^{2}k}{16\pi^{2}}),\quad
q_{n}=n(\frac{P_{0}^{vac}}{n}+i\frac{k\alpha}{4\pi}),$ (3.23)

where $L_{0}^{vac}$ and $P_{0}^{vac}$ denote the vacuum expectation values of
$L_{0}$ and $P_{0}$ on the cylinder respectively. The variable $\alpha$ was
introduced in [27] as a free parameter when using the generalized Rindler
method, and its value was later determined by matching the single interval
boundary entanglement entropy with the holographic ones in BTZ black holes. In
the $n\to 1$ limit, the Abelian orbifold WCFT reduces to the original theory,
which require that $h_{1}=q_{1}=0$. This fact implies the values of the vacuum
charges

$L_{0}^{vac}=-\frac{c}{24}-\frac{\alpha^{2}k}{16\pi^{2}},\quad
P_{0}^{vac}=-\frac{i\alpha k}{4\pi},$ (3.24)

which is precisely the spectral flowed charges defined on the canonical
cylinder (3.15) if we make an identification between the parameter $\alpha$
and the spectral flow parameter $\mu$ (3.6) of the WCFT

$\mu=\frac{\alpha}{2\pi}.$ (3.25)

Substituting (3.24) and (3.25) into (3.23), we find

$h_{n}=\frac{n-1}{n}[(n+1)\frac{c}{24}-(n-1)\frac{\mu^{2}k}{24}],\quad
q_{n}=(n-1)\frac{i\mu k}{2},$ (3.26)

which are proportional to $n-1$ as expected. For later convenience, we mention
two things here:

  * •

In the conformal block expansion of the 4-point function (3.21), we use
$h^{inv}$ as the convenient parameter because of its invariance under the
spectral flow. In the case at hand, there is

$h_{n}^{inv}=h_{n}^{2}-\frac{q_{n}^{2}}{k}=\frac{c}{24}\frac{n^{2}-1}{n},$
(3.27)

which is exactly the holomorphic weight of the twist operator in 2D orbifold
CFT. This relation will help us explicitly see the entropy relations between
CFT and WCFT.

  * •

To reproduce the values of $L_{0}^{vac}$ and $P_{0}^{vac}$ found in
holographic entanglement entropy calculation of WCFT (where $k$ is assumed to
be negative), the parameter $\mu$ must take the following value

$\mu=\sqrt{\frac{-ck}{6}}\quad\to\quad L_{0}^{vac}=0,\quad
P_{0}^{vac}=\frac{i}{2}\sqrt{\frac{-c}{6k}}.$ (3.28)

By using (3.23), we can get the entanglement entropy of single interval $\cal
A$ with endpoints $\partial\mathcal{A}=\\{(x_{1},y_{1}),(x_{2},y_{2})\\}$ on
the reference plane

$S_{\mathcal{A}}=-iP_{0}^{vac}y_{21}-2(i\mu
P_{0}^{vac}+2L_{0}^{vac})\log{\left(\frac{x_{21}}{\epsilon}\right)},$ (3.29)

where the parameter $\epsilon$ is a UV regulator in the $x$ direction. In
contrast, the entanglement entropy of the same interval $\cal A$ on the
canonical plane is

$S_{\mathcal{A}}=-iP_{0}^{vac}\left(y^{\prime}_{21}-\mu\log{\frac{x_{2}}{x_{1}}}\right)-2\left(i\mu
P_{0}^{vac}+2L_{0}^{vac}\right)\log{\left(\frac{x_{21}}{\epsilon}\right)}.$
(3.30)

To compute the entanglement entropy of single interval at a finite
temperature, we consider WCFT on the thermal cylinder with the general
coordinate identification
$(w^{+},w^{-})\sim(w^{+}+i\beta,w^{-}-i\bar{\beta})$, which can be obtained
via a special warped conformal transformation from the reference plane

$x=e^{\frac{2\pi}{\beta}w^{+}},\quad
y=w^{-}+\frac{\bar{\beta}-2\pi\mu}{\beta}w^{+}.$ (3.31)

Using the transformation property of correlation functions (3.22), we obtain

$S_{\mathcal{A}}^{\beta,\bar{\beta}}=-iP_{0}^{vac}\left(\bar{l}+\frac{\bar{\beta}-2\pi\mu}{\beta}l\right)-2(i\mu
P_{0}^{vac}+2L_{0}^{vac})\log{\big{(}\frac{\beta}{\pi\epsilon}\sinh{\frac{\pi
l}{\beta}}\big{)}},$ (3.32)

where $l=|w^{+}_{21}|$ and $\bar{l}=|w^{-}_{21}|$. Note that when taking the
zero-temperature limit of (3.32), it does not simply reduce to the vacuum one
(3.30) due to the presence of the spectral flow parameter $\mu$.

####  3.2.2 Twist operators in non-Abelian orbifold WCFT

As reviewed in section 2, to compute the reflected entropy, one may consider
the correlation function of twist operators in a non-Abelian orbifold.
Different from the entanglement entropy case, the twist operators here are
more subtle. In this subsection, our goal is to give a detailed analysis on
the field identifications induced by different twist operators in the orbifold
WCFT, and find a way from these specific monodromy conditions to calculate the
quantum numbers of the twist operators that should be consistent with the
charge conservation. We would give several non-trivial examples to show the
effectiveness of our method.

In 2D CFT, the holomorphic and anti-holomorphic weight of local primary twist
and anti-twist operators $\sigma_{p}$ and $\tilde{\sigma}_{p}$ locating at the
endpoints of the subregions can be determined by evaluating the expectation
values of the conserved current $T(X)$ on the replicated Riemannian manifold
$\Sigma_{r}$

$\langle T(z^{j},\bar{z}^{j})\rangle_{\Sigma_{r}}=\frac{\langle
T(z,\bar{z})\sigma_{r}(z_{1},\bar{z}_{1})\tilde{\sigma}_{r}(z_{2},\bar{z}_{2})\rangle_{\mbox{CFT}(\cal
C)}}{\langle\sigma_{r}(z_{1},\bar{z}_{1})\tilde{\sigma}_{r}(z_{2},\bar{z}_{2})\rangle_{\mbox{CFT}(\cal
C)}}$ (3.33)

where $T(z^{j})$ represents the stress tensor on the $j$-th replica sheet. The
left-hand side can be evaluated by the transformation rule of the stress
tensor $T(z)$ under a uniformization map from replicated manifold $\Sigma_{r}$
to the complex plane $\mathcal{C}$. The other side can be evaluated by using
the Ward identity and the two-point correlator of the twist operators. A 2D
CFT is defined on Riemann surfaces with a complex structure. Thus the anti-
holomorphic coordinate $\bar{z}$ must transforms in accord with the
holomorphic one $z$. More explicitly when $z\to ze^{2\pi i}$, there must be
$\bar{z}\to\bar{z}e^{-2\pi i}$. Due to this relation, the uniformization map
about anti-holomorphic coordinate is similar to the ones of holomorphic
coordinate, and so is the anti-holomorphic weight.

In 2D WCFT, the local primary twist operator is a charged field under the
$U(1)$ Kac-Moody current. Thus, the Ward identities related to the holomorphic
energy momentum tensor $T(X)$ and the $U(1)$ current $P(X)$ would help us to
find the conformal dimensions and charges of various twist fields. Analogues
to the case of 2D CFT, we determine the dimensions and charges of the twist
operators in WCFT by computing the expectation value of the stress tensor
$T(\bar{x}^{i})$ and the current $P(\bar{x}^{i})$ on the replicated manifold
$\Sigma_{r}$

$\displaystyle\langle T(\bar{x}^{i})\rangle_{\Sigma_{r}}=\frac{\langle
T(\bar{x})\sigma_{g_{r}}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{g_{r}}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}}{\langle\sigma_{g_{r}}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{g_{r}}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}},$
$\displaystyle\langle P(\bar{x}^{i})\rangle_{\Sigma_{r}}=\frac{\langle
P(\bar{x})\sigma_{g_{r}}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{g_{r}}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}}{\langle\sigma_{g_{r}}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{g_{r}}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}}$
(3.34)

where $r$ denotes the replica number of the twist operators $\sigma_{g_{r}}$
and $g_{r}$ is the group element in the definition of twist operator
$\sigma_{g_{r}}$, which in the Abelian orbifold case equals to $(12...r)$ for
the twist operator (the anti-twist operator is related to the group element
$(r(r-1)...1)$). However, unlike 2D CFT, the background geometry of 2D WCFT is
a type of Newton-Cartan geometry [53, 40] with a scaling structure
$J^{a}_{b}$. The two axes $x$ and $y$ are independent with each other. The two
subgroups of global group $SL(2,R)\times U(1)$ of a 2D WCFT act on $x$ and $y$
respectively. For the coordinate $x$, it plays a similar role in 2D WCFT as
the holomorphic coordinate $z$ in complex plane of 2D CFT. While for the other
axis $y$, it behaves in a special manner under the spectral flow. Actually, it
needs special attention to find the transformation of the $y$ coordinate in
defining various twist operators. Moreover, there is another subtlety in
computing the dimensions and charges of twist operators in WCFT by using the
uniformization map method. Due to the distinct property of Newton-Cartan
geometry on which WCFT are defined, there is a whole family of planes ${\cal
C}_{p}$ parametrized by the spectral flowed parameter $p$ with different
vacuum expectation values of conserved charges. The coordinates
$(\bar{x},\bar{y})$ on ${\cal C}_{p}$ are related to the coordinates $(x,y)$
on the reference plane by

$\bar{x}=x,\quad\bar{y}=y-p\,x.$ (3.35)

More concretely, there are three kinds of geometries appearing in our
calculation, see Figure 4: 1). The original physical plane ${\cal C}_{p}$ with
coordinates $(\bar{x},\bar{y})$, whose properties are determined by parameter
$p$; 2). The replicated geometry $\Sigma_{r}$ with coordinates
$(\bar{x}^{i},\bar{y}^{i}),\,i=1...r$, which is composed of $r$ original
physical planes ${\cal C}_{p}$; 3). The image plane ${\cal C}_{img}$ with
coordinates $(\tilde{x},\tilde{y})$, which could be chosen as a canonical
plane, a reference plane or other more general spectral flowed ones under the
uniformization map (3.42).

In evaluating the left-hand side of (3.33), the replicated manifold
$\Sigma_{r}$ and the image plane $\mathcal{C}_{img}$ are relevant. While in
order to evaluate the right-hand side of (3.33), we need to use the physical
plane ${\cal C}_{p}$ where the original WCFT lives, because we have

$\Sigma^{r}/\mathbf{Z}^{r}=\text{Original physical plane}\;\;{\cal C}_{p}$
(3.36)

Thus the vacuum charges related to the operators $T$ and $P$ are distinct on
the evaluation of the two sides of (3.33), which would be shown explicitly in
the following calculations.

Let us evaluate the right-hand side of (3.34). The two-point function on the
physical plane ${\cal C}_{p}$ has the following form

$\langle\phi_{1}\phi_{2}\rangle_{\mathcal{C}_{p}}=\big{(}e^{i\sum_{j}q_{j}(\bar{y}_{j}+p\bar{x}_{j})}\delta({\sum_{j}q_{j}})\big{)}\frac{\delta(h_{\phi_{1}}-h_{\phi_{2}})}{(\bar{x}_{12})^{2h_{\phi_{1}}}},$
(3.37)

and the vacuum expectation values of charges are given by

$\langle T(\bar{x})\rangle_{\mathcal{C}_{p}}=-\frac{kp^{2}}{4},\quad\langle
P(\bar{x})\rangle_{\mathcal{C}_{p}}=-\frac{kp}{2}.$ (3.38)

Note that the above result can be computed by (3.35), (3.2) and (3.7). The
Ward identities for the energy momentum tensor $T(\bar{x})$ and the $U(1)$
current $P(\bar{x})$ on the physical plane $\mathcal{C}_{p}$ are [43]

$\displaystyle\langle
T(\bar{x})\sigma_{r}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{r}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}$
$\displaystyle\hskip
21.52771pt=\sum_{i=1}^{2}\left(\frac{\Delta_{r}/r}{(\bar{x}-\bar{x}_{i})^{2}}+\frac{1/r}{\bar{x}-\bar{x}_{i}}\frac{\partial}{\partial\bar{x}_{i}}+\langle
T(\bar{x})\rangle_{\mathcal{C}_{p}}\right)\langle\sigma_{r}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{r}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}},$
$\displaystyle\langle
P(\bar{x})\sigma_{r}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{r}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}$
$\displaystyle\hskip
21.52771pt=\sum_{i=1}^{2}\left(\frac{1/r}{\bar{x}-\bar{x}_{i}}\frac{\partial}{\partial\bar{y}_{i}}+\langle
P(\bar{x})\rangle_{\mathcal{C}_{p}}\right)\langle\sigma_{r}(\bar{x}_{1},\bar{y}_{1})\tilde{\sigma}_{r}(\bar{x}_{2},\bar{y}_{2})\rangle_{\mathcal{C}_{p}}.$
(3.39)

We place the twist operator $\sigma_{r}$ and the anti-twist operator
$\tilde{\sigma}_{r}$ at points $(0,0)$ and $(l_{0},0)$ separately.
Substituting the two-point function (3.37) and the Ward identities (3.2.2)
into (3.34), we find

$\displaystyle\langle P(\bar{x}^{i})\rangle_{\Sigma_{r}}$
$\displaystyle=\frac{l_{0}}{\bar{x}(\bar{x}-l_{0})}\frac{-iq_{r}}{r}-\frac{kp}{2},$
(3.40) $\displaystyle\langle T(\bar{x}^{i})\rangle_{\Sigma_{r}}$
$\displaystyle=\frac{l_{0}^{2}}{\bar{x}^{2}(\bar{x}-l_{0})^{2}}\frac{h_{r}}{r}+p\frac{l_{0}}{\bar{x}(\bar{x}-l_{0})}\frac{-iq_{r}}{r}-\frac{kp^{2}}{4}$
(3.41)

where $q_{r}$, $h_{r}$ are undetermined quantum numbers of the twist operator
$\sigma_{r}$.

Figure 4: Three physical sheets in coordinates $(\bar{x}^{j},\bar{y}^{j})$
with $j=1,2,3$ constitute the replicated manifold $\Sigma_{m=3}$. When the
parameter $p$ in (3.42) is zero, each physical sheet is a reference plane with
identification $(\bar{x}^{j},\bar{y}^{j})\sim(\bar{x}^{j}e^{2\pi
i},\bar{y}^{j}-2\pi i\mu)$. The dotted blue lines show the shifts in $y$
direction on each replica sheet, and the dotted black lines help us understand
their relations. Under a uniformization map (3.42), the image plane with
coordinates $(\tilde{x},\tilde{y})$ can be chosen as a reference plane, i.e.,
$\tilde{y}=y$ where $y$ is defined in (3.12), or a canonical plane, i.e.,
$\tilde{y}=y^{\prime}$ where $y^{\prime}$ is defined in (3.13). Here we choose
the image plane as a reference plane for simplicity.

Now we turn to the computation of left-hand side of (3.34). The uniformization
map has the form

$\tilde{x}=\left(\frac{\bar{x}}{\bar{x}-l_{0}}\right)^{\frac{1}{r}},\quad\tilde{y}-\mathrm{t}\mu\log{\tilde{x}}+\kappa\mu\log{\tilde{x}}=\bar{y}+p\bar{x}$
(3.42)

where $\mu$ (the same parameter in (3.11)) and $p$ are two parameters both
related to the spectral flow in 2D WCFT, and they parametrize the image plane
${\cal C}_{img}$ and physical plane ${\cal C}_{p}$ respectively. The parameter
$\mathrm{t}$ is equal to the total shift in $\bar{y}$ direction (as we follow
the closed contour around the twist operator $\sigma_{g_{r}}$ counter-
clockwise on the replicated geometry $\Sigma_{r}$) divided by $-2\pi i\mu$.
The parameter $\kappa$ denotes the freedom we own to choose the image plane
${\cal C}_{img}$. In the following context we would like to fix the image
plane ${\cal C}_{img}$ to be the reference plane with coordinates $(x,y)$.
More precisely, we require that when the total number of replica sheets $l$
equal to $1$, we have

$\bar{x}=x,\quad\tilde{y}=\bar{y}=y,\quad\mathrm{when}~{}l=1.$ (3.43)

This requirement, combined with (3.35), gives the following constraint on the
parameters at $l=1$

$\quad\left(\kappa-\mathrm{t}\right)\lvert_{l=1}=0$ (3.44)

Using the transformation rule (3.2), we can get the left-hand side of (3.34)

$\displaystyle\langle P(\bar{x}^{i})\rangle_{\Sigma_{r}}$
$\displaystyle=\langle\frac{\partial\tilde{x}}{\partial\bar{x}}\left(P(\tilde{x})+\frac{k}{2}\frac{\partial\bar{y}}{\partial\tilde{x}}\right)\rangle_{{\cal
C}_{img}}$
$\displaystyle=\frac{l_{0}}{\bar{x}(\bar{x}-l_{0})}\frac{k}{2r}\left(\mathrm{t}-\kappa\right)\mu-\frac{kp}{2},$
(3.45) $\displaystyle\langle T(\bar{x}^{i})\rangle_{\Sigma_{r}}$
$\displaystyle=\langle\left(\frac{\partial\tilde{x}}{\partial\bar{x}}\right)^{2}\left(T(\tilde{x})-\frac{c}{12}\\{f,\tilde{x}\\}\right)+\frac{\partial\tilde{x}}{\partial\bar{x}}\frac{\partial\tilde{y}}{\partial\bar{x}}P(\tilde{x})-\frac{k}{4}\left(\frac{\partial\tilde{y}}{\partial\bar{x}}\right)^{2}\rangle_{{\cal
C}_{img}}$ $\displaystyle=$
$\displaystyle\frac{l_{0}^{2}}{\bar{x}^{2}(\bar{x}-l_{0})^{2}}\frac{1}{r}\left(\frac{c}{24}\frac{r^{2}-1}{r}-\frac{k}{4r}\left(\mathrm{t}-\kappa\right)^{2}\mu^{2}\right)$
$\displaystyle+p\frac{k}{2r}\frac{l_{0}}{\bar{x}(\bar{x}-l_{0})}\left(\mathrm{t}-\kappa\right)\mu-\frac{kp^{2}}{4},$
(3.46)

where in the final step of (3.45) and (3.46) we have used the fact that the
image plane denoted by ${\cal C}_{img}$ is chosen to be the reference plane,
which have vanishing vacuum charges, $\langle P(\tilde{x})\rangle_{img}=0$ and
$\langle T(\tilde{x})\rangle_{img}=0$. By comparing (3.40), (3.41) with (3.45)
(3.46), we obtain the conformal dimension and charge of the twist operator
$\sigma_{g_{r}}$

$\displaystyle h_{g_{r}}$
$\displaystyle=\frac{c}{24}\frac{r^{2}-1}{r^{2}}-\frac{k}{4r}\left(\mathrm{t}-\kappa\right)^{2}\mu^{2}=\frac{c}{24}\frac{r^{2}-1}{r}+\frac{q_{g_{r}}^{2}}{rk},$
(3.47) $\displaystyle q_{g_{r}}$
$\displaystyle=i\frac{k}{2}\left(\mathrm{t}-\kappa\right)\mu.$ (3.48)

In the following we would use the spectral flow invariant conformal dimension
$h^{inv}_{g_{r}}$ for convenience

$h_{g_{r}}^{inv}=h_{g_{r}}-\frac{q_{g_{r}}^{2}}{rk}=\frac{c}{24}\frac{r^{2}-1}{r}$
(3.49)

where $rk$ appears in the denominator because the level related to the
orbifold WCFT defined on $\Sigma_{r}$ is $r\times k$.

Before we consider the general non-Abelian case, let us first study a simple
case: the twist operator $\sigma_{(12...r)}$ in the Abelian orbifold
$\text{WCFT}^{\otimes r}/\mathbf{Z}^{r}$. In this case, as we follow the
contour we have

$\bar{x}^{j}=x^{j}\to\bar{x}^{j}e^{2\pi i}=x\,e^{2\pi
i},\quad\bar{y}=y\to\bar{y}-2\pi i\mu=y-2\pi i\mu,\quad i=1...r$ (3.50)

so the total shift of $y$ direction is $-2\pi i\mu r$, then $\mathrm{t}=r=l$.
The condition in (3.44) tell us $\kappa=1$. We can see easily that the
uniformization map (3.42) reduce to the one appearing in [34]

$\tilde{x}=\left(\frac{\bar{x}}{\bar{x}-l_{0}}\right)^{\frac{1}{r}},\quad
y-r\,\mu\log{\tilde{x}}+\mu\log{\tilde{x}}=\bar{y}+p\bar{x},$ (3.51)

and reproduce the dimension $h_{r}^{inv}$ and charge $q_{r}$ of the twist
operator $\sigma_{(12...r)}$

$h_{r}^{inv}=\frac{c}{24}\frac{r^{2}-1}{r},\quad q_{r}=i\frac{k}{2}(r-1)\mu,$
(3.52)

which are the same as the ones in (3.26) obtained from the Rindler method.

For a generic non-Abelian orbifold, the monodromy conditions is tricky, as the
field identifications induced by the twist operators are involved. To
determine the monodromy condition of all twist operators in an orbifold WCFT,
the guideline is to keep charge conservation. More explicitly, here is the
guideline for the monodromy condition.

Guideline: The monodromy condition of twist operators in $\bar{x}$ direction
is the same as the holomorphic coordinate $z$ of 2D CFT; For the $\bar{y}$
coordinate, when the shifts along $\bar{y}$ axis at one branching point
$(\bar{x}_{0},\bar{y}_{0})$ have been fixed by several independent twist
operators $\sigma_{g_{i}}(\bar{x}_{0},\bar{y}_{0})$ as we go around
$(\bar{x}_{0},\bar{y}_{0})$, then all the other twist operators which may
appear as the anti-twist operators or in the OPE of the above twist operators
can be determined completely.

In the following, we present several non-trivial examples to show in detail
how this guideline works. These examples appear in the computations of
entanglement entropy, reflected entropy, entanglement negativity and odd
entropy.

##### Example 1:

Anti-twist operator $\tilde{\sigma}_{(12...l)}$

This example appears in the calculation of single interval entanglement
entropy $S({\cal A})$. The relations of the sheets in the replicated geometry
of Abelian orbifold WCFT have been fixed by the single operator
$\sigma_{(12...l)}$ located at the left end-point of the interval ${\cal A}$,
which is just (3.50). Thus for the anti-twist operator
$\tilde{\sigma}_{(12...l)}$ located at the right end-point of ${\cal A}$
should behave as follows:

$\bar{x}^{j}\to\bar{x}^{j}e^{2\pi i},\quad\bar{y}^{j}\to\bar{y}^{j}+2\pi
i\mu,\quad j=1,2,...,r.$ (3.53)

Here we have identified the anti-twist operator $\tilde{\sigma}_{(12...l)}$
with $\sigma_{(l(l-1)...1)}$ inserted at $(l_{0},0)$ on each sheet. Therefore,
the route is going around the branch point $(l_{0},0)$ counterclockwise from
the sheet $i$ to the sheet $i-1$. As a result, the phase acquired by $\bar{x}$
is the same as (3.50) but the shift along $\bar{y}$ is opposite to (3.50). The
total shift in $y$ direction is $2\pi i\mu r$. Then the parameters in (3.42)
have been determined to be

$r=l,\quad\mathrm{t}=-l,\quad\kappa=-1$ (3.54)

Then from (3.48) and (3.49) we get the dimension and charge of anti-twist
operator

$q=-i\frac{k}{2}(l-1)\mu,\quad h^{inv}=\frac{c}{24}\frac{l^{2}-1}{l}$ (3.55)

which is consistent with charge conservation of correlation function
$\langle\sigma_{(12...l)}\tilde{\sigma}_{(12...l)}\rangle$.

More generally, the anti-twist operator locating at the other end-point always
has the same dimension but opposite charge with respect to a twist operator.
This simply comes from the monodromy condition.

##### Example 2:

OPE of twist operators $\sigma_{(12...l)}\sigma_{(12...l)}$ with even $l$

This example would appear in the calculation of entanglement negativity ${\cal
E}(\rho_{AB})$. When $l$ is an even number, the OPE of two same $l$-fold twist
operators would generate two $l/2$-fold twist operators as the leading order
contribution. For example when $l=4$, we have
$\sigma_{(1234)}\sigma_{(1234)}\sim\sigma_{(13)(24)}+...$, see Figure 5 (a).
We view the twist operator $\sigma_{(13)(24)}$ as two independent twist
operators $\sigma_{(13)}$ and $\sigma_{(24)}$ located at the same point to
compute the quantum numbers of $\sigma_{(13)(24)}$ for clarity. The replicated
geometry have been fixed by the single twist operator $\sigma_{(12...l)}$ for
general $l$, then the twist operators $\sigma_{(13...l-1)}$ and
$\sigma_{(24...l)}$ appearing in the OPE have the following monodromy
conditions

$\bar{x}^{j}\to\bar{x}^{j}e^{2\pi i},\quad\bar{y}^{j}\to\bar{y}^{j}-4\pi
i\mu,\quad j=1,3,...,l-1\;\;\text{or}\;\;\;2,4,...,l.$ (3.56)

The total shift in $y$ direction is $-4\pi i\mu r$. This implies the
parameters in (3.42) for both twist operator $\sigma_{(13...l-1)}$ and
$\sigma_{(24...l)}$ have the following relations

$r=l/2,\quad\mathrm{t}=l,\quad\kappa=1,$ (3.57)

so from (3.48) and (3.49) we get their dimensions and charges

$h^{inv}=\frac{c}{24}\frac{(l/2)^{2}-1}{l/2},\quad q=i\frac{k}{2}(l-1)\mu.$
(3.58)

Then the total charge of twist operator $\sigma_{(13...l-1)(24...l)}$ are
twice the above result, which is consistent with the charge conservation of
the three-point correlation functions
$\langle\sigma_{(12...l)}(\bar{x}_{1},\bar{y}_{1})$
$\sigma_{(12...l)}(\bar{x}_{2},\bar{y}_{2})$
$\tilde{\sigma}_{(13...l)(24...l)}((\bar{x}_{3},\bar{y}_{3}))\rangle$.

Figure 5: Contour configurations representing the OPE of two twist operators
$\sigma_{l}\sigma_{l}$. The red lines show the shifts in $y$ direction, which
is fixed by $\sigma_{l}$ operator. (a): A typical example for $l$ even ($l=4$)
showing how the sheets decouples into two independent sets, each of $l/2$
replica sheets, and how the boundary condition in $y$ direction (3.56) are
obeyed; (b): A typical example for $l$ odd ($l=5$) showing the net effect is
not just a reshuffling of the sheet numeration due to the non-trivial boundary
condition in $y$ direction (3.59).

##### Example 3:

OPE of twist operators $\sigma_{(12...l)}\sigma_{(12...l)}$ with odd $l$

This example would appear in the calculation of the odd entropy ${\cal
S}_{o}(AB)$. When $l$ is an odd number, the OPE of two $l$-fold twist
operators would generate another $l$-fold twist operator. For example when
$l=5$, we have $\sigma_{(12345)}\sigma_{(12345)}\sim\sigma_{(13524)}+...$, see
Figure 5 (b). Again the relations of $y$ direction of all the $l$ replica
sheets have been fixed by the single twist operator $\sigma_{12...l}$, so for
the twist operator $\sigma_{(13...(l)24...(l-1))}$ the monodromy conditions
are

$\bar{x}^{j}\to\bar{x}^{j}e^{2\pi i},\quad\bar{y}^{j}\to\bar{y}^{j}-4\pi
i\mu,\quad j=1,2,...,l.$ (3.59)

The total shift in $y$ coordinate now is $-4\pi i\mu l$. Then the parameters
are

$r=l,\quad\mathrm{t}=2l,\quad\kappa=2$ (3.60)

which tell us the dimension and charge for the twist operator
$\sigma_{(13...(l)24...(l-1))}$ is

$h^{inv}=\frac{c}{24}\frac{l^{2}-1}{l},\quad q=i\frac{k}{2}(2l-2)\mu.\quad$
(3.61)

Note that due to the non-trivial monodromy conditions of the twist operator
$\sigma_{(13...(l)24...(l-1))}$ in the $y$ direction, the net effect is not
just a reshuffling of the sheet numeration. This is the key difference between
the twist operators in orbifold CFT and orbifold WCFT. The result is
consistent with the charge conservation of three-point correlation functions
$\langle\sigma_{(12...l)}(\bar{x}_{1},\bar{y}_{1})$
$\sigma_{(12...l)}(\bar{x}_{2},\bar{y}_{2})$
$\tilde{\sigma}_{(13...(l)24...(l-1))}(\bar{x}_{3},\bar{y}_{3})\rangle$.

##### Example 4:

OPE of twist operators $\sigma_{g_{A}}\sigma_{g_{B}^{-1}}$ in the reflected
entropy

Let us consider a much more complicated example, the OPE of twist operator
$\sigma_{g_{A}}\sigma_{g_{B}^{-1}}$ in (2.22) in the non-abelian orbifold WCFT
containing $mn$ replica sheets, which shows the power of our formulation. The
independent twist operators determining the relations of all the $mn$ replicas
can be chosen as $\\{\sigma_{g_{A}},\sigma_{g_{B}}\\}$ or
$\\{\sigma_{g_{A}^{-1}},\sigma_{g_{B}}\\}$. See Figure 1 for more concrete
showing, where the single twist operator $\sigma_{g_{B}}=$
$\sigma_{(1234)(5678)}$ determine the relations of all the replicas in the
same $n$-column and the single twist operator $\sigma_{g_{A}}=$
$\sigma_{(1278)(5634)}$ allow us to determine the relations between replica
sheets in different $n$-columns. In addition, we have

$\sigma_{(1278)(5634)}\tilde{\sigma}_{(1234)(5678)}=\sigma_{(26)(48)}+...,\quad
g_{B}^{-1}g_{A}=(\tau_{k}^{0})^{-1}\tau_{k}^{m/2}$ (3.62)

where $(\tau_{k}^{0})^{-1}$ and $\tau_{k}^{m/2}$ are two $n$-cyclic
permutation group elements appearing in (2.13). As in Example 2, to compute
the dimension of twist operator $\sigma_{(26)(48)}$, we separate it into two
twist operators $\sigma_{(26)}$ and $\sigma_{(48)}$ located at the same point.
Closely following the contours in Figure 1, we would get a conclusion that as
we go around counter-clockwise by a $2\pi$ angle from $j$-th replica sheet to
$k$-th replica sheet with

$(j,k)=(1,5)\;\text{or}\;(2,6)\;\text{or}\;(3,7)\;\text{or}\;(4,8)$ (3.63)

we have $\bar{y}^{j}\to\bar{y}^{k}$ with no shift, which means the total shift
in $y$ direction vanishes. Then we have the parameters for both twist
operators $\sigma_{(\tau_{k}^{0})^{-1}}$ and $\sigma_{\tau_{k}^{m/2}}$ for
general $m$ and $n$

$r=n,\quad\mathrm{t}=0,\quad\kappa=0,\quad l=m\times n.$ (3.64)

These give us the dimension and charge of $\sigma_{g_{B}^{-1}g_{A}}$

$h^{inv}=2\frac{c}{24}\frac{n^{2}-1}{n},\quad q=0$ (3.65)

This case is a true non-abelian orbifold calculation, and the $l=1$ condition
is equivalent to $m=n=1$.

Figure 6: A (brown) closed contour without touching the (blue) branching cuts
representing the boundary interval ${\cal A}$ in the abelian orbifold
$\Sigma_{3}$ for computing entanglement entropy. The total shift in $y$
direction is zero if we go around the contour counter-clockwise following the
arrows.

There is a nice interpretation of the guideline on the above monodromy
conditions. It implies that any closed contour not crossing any branch cut in
the replicated geometry $\Sigma_{l}$ has a total zero shift in $y$ direction.
Such configuration is relevant to the computation of the correlation function
of the twist operators $\sigma_{g_{i}}$, whose related group elements satisfy

$\prod_{i}g_{i}=\mathbf{1}.$ (3.66)

In order that the correlation function is non-vanishing, the charges must be
conserved. The charge conservation then requires that the total amount of the
shifts in $\bar{y}$ direction must be vanishing. See Figure 6 for an explicit
example.

Technically, as shown in the above examples, the amount of shifts in
$\tilde{y}$, the U(1) direction of the image place, is not always $-2\pi
i\mu$, as what we have for the twist operator $\sigma_{(12...l)}$ when
computing the entanglement entropy. Instead, the amount of shifts varies case
by case. This can be seen immediately from the uniformization map (3.42). Due
to the definition of $\mathrm{t}$, we can see that $\tilde{y}$ is shifted by
$2\pi\kappa i\mu$, i.e. $\kappa$ times the periodicity of the reference plane,
with $\kappa$ taking different values in specific examples. This suggests that
when $\bar{y}$ is shifted by $-2\pi i\mathrm{t}\mu$ following the contour
suggested by the twist operator, the image of this contour encircles the
$\tilde{y}$ direction by $\kappa$ times.

###  3.3 Reflected entropy in 2D holographic WCFT

In this subsection, let us compute the reflected entropy $S_{R}(A:B)$ in
holographic 2D WCFT. The results can be summarized as follows:

  * •

The reflected entropy $S_{R}(A:B)$ of two disjoint intervals $A$, $B$ in WCFT
is independent of the $U(1)$ direction, similar to the case of mutual
information $I(A:B)$. This means that it is a spectral-flow invariant quantity
with the same value on any spectral flowed plane for the same entangling
configuration.

  * •

For the same entangling configuration of two disjoint intervals $A$, $B$ on
both the vacuum state and thermal state, we have the following relation
between WCFT and CFT,

$S_{R}^{WCFT}(A:B)=\frac{1}{2}S_{R}^{CFT}(A:B).$ (3.67)

We now show the computational process to reach the above statements. From the
study in the above subsection, we can read easily the conformal dimensions
$h^{inv}$ and $U(1)$ charges $q$ of various twist operators in the relevant
orbifold WCFT. Since we are dealing with WCFT on a replicated geometry of $nm$
sheets, both $c$ and $k$ need to be multiplied by the number of copies $nm$.
Therefore, the relation between $h$ and $h^{inv}$ for twist operators is
$h^{inv}=h-\frac{q^{2}}{nmk}$, where $k$ is the level of the original WCFT
defined on a single sheet. In the end, we get the following results.

  * •

$\sigma_{g_{A}}$, $\sigma_{g_{B}}$ :

$\displaystyle
h_{\sigma_{g_{A}}}^{inv}=h_{\sigma_{g_{B}}}^{inv}=nh_{m}^{inv}=\frac{nc}{24}(m-\frac{1}{m}),$
(3.68) $\displaystyle
q_{\sigma_{g_{A}}}=q_{\sigma_{g_{B}}}=nq_{m}=n(m-1)\frac{i\mu k}{2}.$

  * •

$\sigma_{g_{A}^{-1}}$, $\sigma_{g_{B}^{-1}}$ :

$\displaystyle
h_{\sigma_{g^{-1}_{A}}}^{inv}=h_{\sigma_{g^{-1}_{B}}}^{inv}=nh_{m}^{inv}=\frac{nc}{24}(m-\frac{1}{m}),$
(3.69) $\displaystyle
q_{\sigma_{g^{-1}_{A}}}=q_{\sigma_{g^{-1}_{B}}}=-nq_{m}=-n(m-1)\frac{i\mu
k}{2}$

  * •

$\sigma_{g_{A}g_{B}^{-1}}$,

$h_{g_{A}g_{B}^{-1}}^{inv}=2h_{n},\quad q_{g_{A}g_{B}^{-1}}=0.$ (3.70)

For the important OPE coefficients, we have

$\sigma_{g_{A}^{-1}}\sigma_{g_{B}}=C_{n,m}\sigma_{g_{B}g_{A}^{-1}}+...\quad,\quad
e^{\pi\mu q_{n}(n-1)}C_{n,m}=(2m)^{-2h_{n}}.$ (3.71)

This relation can be obtained by using the same method as in the CFT case. We
choose to compute this particular three-point function on the reference plane
${\cal C}_{ref}$, which corresponds to choosing $p=0$ in (3.42) for the
original physical plane to do the replication. With this choice, the
coordinate $\bar{x}$ for ${\cal C}_{p}$ becomes $x$ for the reference plane
hereafter. The image plane can be chosen to be the reference plane without
loosing generality. Then we find

$\displaystyle\langle\sigma_{g_{A}^{-1}}(x_{1},y_{1})\sigma_{g_{B}}(x_{2},y_{2})\sigma_{g_{A}g_{B}^{-1}}(x_{3},y_{3})\rangle_{\mbox{\small
WCFT}^{\otimes nm}(ref)}$ $\displaystyle=$ $\displaystyle
e^{S_{L}(\phi)}\underbrace{\Big{\lvert}\frac{\partial s^{+}}{\partial
x}\Big{\lvert}^{2h_{n}}_{s=s_{1}^{+}}\Big{\lvert}\frac{\partial
s^{+}}{\partial
x}\Big{\lvert}^{2h_{n}}_{s=s_{2}^{+}}\langle\sigma_{(\tau_{n}^{0})^{-1}}(s_{1}^{+},s_{1}^{-})\sigma_{\tau_{n}^{m/2}}(s_{2}^{+},s_{2}^{-})\rangle_{\mbox{\small
WCFT}^{\otimes n}(ref)}}_{\left({\cal D}\right)}$ $\displaystyle=$
$\displaystyle\left.\left(\langle\sigma_{g_{A}^{-1}}(x_{1},y_{1})\sigma_{g_{B}}(x_{2},y_{2})\rangle_{\mbox{\small
WCFT}^{\otimes m}(ref)}\right)^{n}\right|_{A=B}\times\left({\cal D}\right)$
$\displaystyle=$ $\displaystyle e^{-\pi\mu
q_{n}(n-1)}e^{inq_{m}(y_{1}-y_{2})}(2m)^{-2h_{n}}|x_{32}|^{-2h_{n}}|x_{31}|^{-2h_{n}}|x_{12}|^{-2nh_{m}+2h_{n}}.$
(3.72)

The first equality follows from a warped conformal transformation

$s^{+}=\frac{(x-x_{1})^{1/m}}{(x-x_{2})^{1/m}},\hskip
12.91663pts^{-}-(n-1)\mu\log{s^{+}}=y,$ (3.73)

which maps the replicated manifold composed of $nm$ reference planes to a
replicated manifold composed of $n$ reference planes. In addition, the twist
operators $\sigma_{(\tau_{n}^{0})^{-1}}(s_{1}^{+},s_{1}^{-})$ and
$\sigma_{\tau_{n}^{m/2}}(s_{2}^{+},s_{2}^{-})$ have the same conformal
dimensions $h_{n}$ but opposite charges $q_{n}$ and $-q_{n}$, with the
following coordinates

$\displaystyle s^{+}_{1}=\frac{(x_{3}-x_{1})^{1/m}}{(x_{3}-x_{2})^{1/m}},$
$\displaystyle s^{-}_{1}-(n-1)\mu\log{s^{+}_{1}}=y_{3},$ (3.74) $\displaystyle
s^{+}_{2}=e^{i\pi}\frac{(x_{3}-x_{1})^{1/m}}{(x_{3}-x_{2})^{1/m}},$
$\displaystyle s^{-}_{2}-(n-1)\mu\log{s^{+}_{2}}=y_{3}.$ (3.75)

From (3.3), we can easily read off the OPE coefficient $C_{n,m}$ as well as
the conformal dimension and the charge of $\sigma_{g_{A}g_{B}^{-1}}$ as
claimed in (3.70) and (3.71). Compared to the CFT case, there is an additional
factor $e^{-\pi\mu q_{n}(n-1)}$ coming from the correlation functions of
$\sigma_{\tau}$’s with the coordinates (3.74) and (3.75) and the fact that
twist operator $\sigma_{(\tau_{n}^{0})^{-1}}$ and $\sigma_{(\tau_{n}^{m/2})}$
have charge $-q_{n}$ and $q_{n}$ respectively. More precisely there is

$\langle\sigma_{\tau}\sigma_{\tau}\rangle\sim
e^{iq_{n}(s_{1}^{-}-s_{2}^{-})}=e^{iq_{n}\mu(n-1)\log({s^{+}_{1}}/{s^{+}_{2}})}=e^{-\pi\mu
q_{n}(n-1)}$

However this factor would not affect the reflected entropy, since its exponent
scales like $(n-1)^{2}$ that would vanish in the $n\to 1$ limit. Apart from
this extra factor, the OPE coefficient is actually the square root of the one
in the CFT case. This is due to the fact that we have only the uniformization
map on $x$, half of the ones in CFT.

Now we are ready for the computation of the reflected entropy in the
holographic WCFT. Let us first consider the reflected entropy in the vacuum
state. We need to compute the four-point function of the twist operators on
the plane

$\langle\sigma_{{g}_{A}}(x_{1},y_{1})\sigma_{{g}^{-1}_{A}}(x_{2},y_{2})\sigma_{{g}_{B}}(x_{3},y_{3})\sigma_{{g}^{-1}_{B}}(x_{4},y_{4})\rangle_{\mbox{\small
WCFT}^{\otimes nm}({\cal C}_{p})}$ (3.76)

As in the CFT case, we care about the $t$-channel expansion, which is
equivalent to take the limit of cross ratio $\mathrm{x}\to 1$. One essential
assumption is the dominance of a single block in the large $c$ limit of
holographic WCFT. This allows us to find

$\displaystyle\langle\sigma_{{g}_{A}}(x_{1},y_{1})\sigma_{{g}^{-1}_{A}}(x_{2},y_{2})\sigma_{{g}_{B}}(x_{3},y_{3})\sigma_{{g}^{-1}_{B}}(x_{4},y_{4})\rangle_{\mbox{\small
WCFT}^{\otimes nm}(\cal C_{\mu})}$ $\displaystyle=$
$\displaystyle\frac{e^{iq_{\sigma_{g_{A}}}(y_{12}+y_{34}-p(x_{12}-x_{34}))}}{x_{41}^{2h_{\sigma_{g_{A}}}}x_{32}^{2h_{\sigma_{g_{A}}}}}\sum_{p}C_{p12}C^{p}_{34}(1-\mathrm{x})^{\frac{2q_{\sigma_{g_{A}}}^{2}}{mnk}}\mathrm{x}^{-\frac{2q_{\sigma_{g_{A}}}^{2}}{mnk}}\mathcal{F}(mnc-1,h_{i}^{inv},h_{p}^{inv},1-\mathrm{x})$
$\displaystyle\approx\underbrace{\frac{e^{inq_{m}(y_{12}+y_{34}-p(x_{12}-x_{34}))}}{x_{41}^{4nh_{m}}x_{32}^{4nh_{m}}}(1-\mathrm{x})^{\frac{2(nq_{m})^{2}}{mnk}}\mathrm{x}^{-\frac{2(nq_{m})^{2}}{mnk}}}_{\big{(}\cal
D\big{)}}(C_{n,m})^{2}e^{-(2h_{n})\ln{\frac{1+\sqrt{\mathrm{x}}}{4(1-\sqrt{\mathrm{x}})}}}.$
(3.77)

Note that in the $\cal D$ part of expression (3.77), the exponent of each term
is proportional to $n$. It is exactly this fact that makes the $\cal D$ part
cancel out between the numerator and the denominator in evaluating the
reflected entropy. Apart from the $\cal D$, the form of the rest part of
(3.77) is similar to the square root of the contribution part of CFT in
(2.27), which leads us to the final result

$S^{WCFT}_{R;vac}(A:B)=\frac{1}{2}S^{CFT}_{R;vac}(A:B).$ (3.78)

Next, we consider the refleted entropy in thermal state. Now the four-point
function is defined on the thermal cylinder
$(w^{+},w^{-})\sim(w^{+}+i\beta,w^{-}-i\bar{\beta})$. From the transformation
property (3.22) of primary operators, we have the relation:

$\displaystyle\langle\sigma_{{g}_{A}}(w_{1}^{+},w_{1}^{-})\sigma_{{g}^{-1}_{A}}(w_{2}^{+},w_{2}^{-})\sigma_{{g}_{B}}(w_{3}^{+},w_{3}^{-})\sigma_{{g}^{-1}_{B}}(w_{4}^{+},w_{4}^{-})\rangle_{\mbox{\small
WCFT}^{\otimes nm}}^{cylinder}$ $\displaystyle=$
$\displaystyle|x^{\prime}(w_{1}^{+})x^{\prime}(w_{2}^{+})x^{\prime}(w_{3}^{+})x^{\prime}(w_{4}^{+})|^{h_{g_{A}}}\langle\sigma_{{g}_{A}}(x_{1},y_{1})\sigma_{{g}^{-1}_{A}}(x_{2},y_{2})\sigma_{{g}_{B}}(x_{3},y_{3})\sigma_{{g}^{-1}_{B}}(x_{4},y_{4})\rangle_{\mbox{\small
WCFT}^{\otimes nm}}^{ref;plane}.$ (3.79)

A straightforward computation leads to

$S^{WCFT}_{R;thermal}(A:B)=\frac{1}{2}S^{CFT}_{R;thermal}(A:B).$ (3.80)

As in the holographic CFT, and the $s$-channel expansion at the leading order
of large $c$ gives

$S^{WCFT}_{R;vac}(A:B)=S^{WCFT}_{R;thermal}(A:B)=0.$ (3.81)

Consequently, we may expect a first-order phase transition when the cross
ratio increases from $0$ to $1$.

##  4 Pre-Entanglement Wedge and AdS3/WCFT

In this section we would identify the bulk geometric dual of boundary
reflected entropy in the AdS3/WCFT correspondence, by using the swing surface
proposal described in [38, 41]. A crucial step is to find which bulk region in
AdS3/WCFT is the natural analogue of the entanglement wedge in AdS/CFT. In
what follows, we briefly review the holographic correspondence between the
AdS3 gravity with the Compère-Song-Strominger (CSS) boundary conditions and 2D
WCFT.

In three dimensions, a set of consistent asymptotically boundary conditions is
essential to define a gravity theory. The transformations keeping these
boundary conditions form an asymptotic symmetry group(ASG), which restrict the
boundary field theory dual. In AdS3 gravity, under the well-known Brown-
Henneaux boundary conditions[54], the ASG is generated by two copies of the
Virasoro algebra, suggesting a CFT dual. In contrast, under the CSS boundary
conditions, the asymptotic symmetry group (ASG) is generated by a left-moving
Virasoro and a $U(1)$ Kac-Moody algebra [25], leading to the so-called
AdS3/WCFT correspondence. The similar Virasoro-Kac-Moody algebra generates the
ASG of 3D topological massive AdS3 gravity under appropriate boundary
conditions as well[55, 56], leading to the (W)AdS3/WCFT correspondence. Here
we focus on the AdS3/WCFT correspondence.

Denoting the lightcone coordinates of global AdS3 on a cylinder by

$\left(u=\phi+t,\;v=\phi-t,\;r\right),\quad\phi\sim\phi+2\pi,$ (4.1)

then the CSS boundary conditions are as follows

$\displaystyle g_{rr}=\frac{1}{r^{2}}+{\cal O}(\frac{1}{r^{4}})$
$\displaystyle,\quad g_{ru}={\cal O}(\frac{1}{r^{3}}),\quad g_{rv}={\cal
O}(\frac{1}{r^{3}}),\quad g_{uv}=\frac{r^{2}}{2}+{\cal O}(1),$ $\displaystyle
g_{uu}=r^{2}J^{\prime}(u)+{\cal O}(1),\quad g_{vv}=T_{v}^{2}+{\cal
O}(\frac{1}{r}),$ (4.2)

where $J(u)$ is an arbitrary function and

$c=\frac{3}{2G},$ (4.3)

with $G$ being the 3D Newton’s constant. Here we have set the AdS scale to be
$1$, as before. The parameter $c$ appears as the central charge in the dual
WCFT. In the Fefferman-Graham gauge, (4.2) is sufficient to determine the
phase space of solutions in pure Einstein AdS3 gravity in terms of two
functions $L(u)$, $J(u)$ and a constant $T_{v}$,

$\displaystyle ds^{2}=\frac{dr^{2}}{r^{2}}$
$\displaystyle+r^{2}du\left[dv+J^{\prime}(u)du\right]+L(u)du^{2}+T_{v}^{2}\left[dv+J^{\prime}(u)du\right]^{2}$
$\displaystyle+\frac{1}{r^{2}}T_{v}^{2}L(u)du\left[dv+J^{\prime}(u)du\right].$
(4.4)

We will mainly work in the zero mode backgrounds with $L(u)=T_{u}^{2}$ and
$J(u)$ being constants. For these backgrounds, at the $r\to\infty$ conformal
boundary, the coordinates $(u,v)$ are related to the coordinates
$(\hat{x}^{\prime},\hat{y}^{\prime})$ of the canonical cylinder by a state-
dependent map [24]

$u=\hat{x}^{\prime},\quad v=\hat{x}^{\prime}+\frac{\hat{y}^{\prime}}{2\mu
T_{v}},$ (4.5)

where $\mu$ is a parameter characterizing the WCFT given by (3.28). In these
cases, (4.4) describes a subset of the space of solutions of three dimensional
gravity that are also compatible with Brown-Henneaux boundary conditions, and
the energy and angular momentum associated with such backgrounds are
respectively given by $M=(T_{u}^{2}+T_{v}^{2})/4G$ and
$J=(T_{u}^{2}-T_{v}^{2})/4G$. These zero mode solutions include the the global
$AdS_{3}$ vacuum with $M=-1/8G$ and $J=0$, the conical defect geometries with
$-1/8G<M<0$, $-1/8G<J<1/8G$, and the BTZ black holes with $M\geq|J|\geq 0$.
When $L(u)$ or $J^{\prime}(u)$ are not constants, the corresponding
backgrounds are interpreted as the solutions dressed with additional boundary
gravitons.

###  4.1 Approximate modular flow and the swing surface

In [41], it was proposed that the holographic entanglement entropy in the
AdS3/WCFT and Flat3/BMSFT correspondences could be described by a swing
surface. In this subsection, let us review this proposal briefly.

For the case where is an exact boundary modular flow generator $\zeta$, the
assumptions that the Killing vectors of bulk vacuum state corresponds to the
symmetry generators of WCFT vacuum state make us have the ability to extend
the boundary modular flow generator into the bulk spacetime with no obstacle.
However, for more general field states the modular Hamiltonian is generally
nonlocal, and the modular flow generator can no longer be expressed as the
linear combinations of boundary symmetry generators $h_{i}$ or the bulk
Killing vector fields $H_{i}$ which may not exist. In [41] the authors
generalized the approach in [57] by constructing the modular flow near the
swing surface in AdS3/WCFT holography. The key steps of their construction can
be summarized as follows,

  * •

Find approximate boundary modular flow generator
$\zeta^{(p)}=\sum_{i}a_{i}^{(p)}h_{i}=2\pi\partial_{\tau_{(p)}}$ around the
endpoints of the subregion $p\in\partial{\cal A}$. For 2D WCFT, this can be
obtained simply from the ones for the vacuum state by sending the other
endpoint to infinity.

  * •

Extend the $d$-dimensional vector field $\zeta^{(p)}$ into a
$(d+1)$-dimensional vector field $\xi^{(p)}_{\infty}$ near the asymptotic
boundary,

$\xi^{(p)}_{\infty}=\sum_{i}a_{i}^{(p)}H_{i},\quad\text{with}\;\;H_{i}|_{\partial{\cal
M}}=h_{i},\;\;\xi^{(p)}_{\infty}|_{\partial{\cal M}}=\zeta^{(p)},$ (4.6)

where $H_{i}$ are the approximate Killing vectors near $p\in{\cal A}$.

  * •

Extend the $(d+1)$-dimensional vector field $\xi^{(p)}_{\infty}$ defined near
the asymptotic boundary into the bulk following the null geodesics
$\gamma_{(p)}$, which defines a null vector field $\xi^{(p)}$ along
$\gamma_{(p)}$ satisfying

$\xi^{(p)\mu}\xi^{(p)}_{\mu}=0,\quad\xi^{(p)}|_{\partial{\cal
M}}=\zeta^{(p)},\quad\xi^{(p)\mu}\nabla_{\mu}\xi^{(p)\nu}=\pm
2\pi\xi^{(p)\nu}.$ (4.7)

  * •

Extend the vector field $\xi^{(p)}$ defined on the null geodesics
$\gamma_{(p)}$ into the vector field on the whole $(d+1)$-dimensional
spacetime by requiring that

$\xi|_{\gamma_{(p)}}=\xi^{(p)},\quad\xi|_{\gamma_{\xi}}=0,\quad\nabla_{\mu}\xi^{\nu}|_{\gamma_{\xi}}=2\pi
n^{\mu\nu},\quad\left({\cal L}_{\xi}g\right)_{\mu\nu}\sim{\cal
O}(r^{-\alpha_{\mu\nu}})\;as\;r\to\infty$ (4.8)

where $\gamma_{\xi}$ is the set of fixed points of $\xi$ and $\alpha_{\mu\nu}$
are constant. These conditions make sure that $\xi$ is a correctly constructed
asymptotic Killing vector at conformal boundary $r\to\infty$ and behaves like
the boost generator in the local Rindler frame near $\gamma_{\xi}$.

With the above constructions we can identify the swing surface which gives the
holographic entanglement entropy. The swing surface is the union of the ropes
and the bench

$\gamma_{{\cal
A}}=\gamma\cup\gamma_{b\partial},\quad\text{where}\quad\gamma_{b\partial}\equiv\cup_{p\in\partial{\cal
A}}\gamma_{(p)},$ (4.9)

where $\gamma_{b\partial}$ are called the ropes of the swing surface, which is
a collection of null geodesics tangent to the approximate bulk modular flow
around the point $p$ at the conformal boundary, and $\gamma$ is called the
bench of the swing surface, which are subset of the fixed points of the
constructed approximated bulk modular flow bounded by the null ropes.
Actually, there exist another equivalent prescription for this swing surface
which is very similar to the construction of HRT surface in the standard
AdS/CFT case: the swing surface [41] of single boundary interval ${\cal A}$ is
the extremal surface, which is the one with minimal area, that bounded by
$\gamma_{b\partial}$. Then the holographic entanglement entropy is finally
given by

$S_{\cal A}=\min\underset{\gamma_{\cal A}\sim{\cal
A}}{\mathrm{ext}}\frac{\mbox{Area}(\gamma_{\cal A})}{4G}.$ (4.10)

###  4.2 Pre-entanglement wedge in AdS3/WCFT

In this subsection we try to find the analogue of entanglement wedge and
causal wedge in AdS3/WCFT. During this process we would also like to emphasize
some geometric properties of the swing surface and discuss the phenomena
related to single interval entanglement phase transition.

Now it is not possible to simplify the calculation by choosing constant time
slice respecting time reversal symmetry like the case in AdS/CFT although the
bulk spacetime is static. We choose to work in the following gauge in 3D
Lorentzian spacetime directly. The line element takes the form

$ds^{2}=\frac{d\rho^{2}}{4(\rho^{2}-4T_{u}^{2}T_{v}^{2})}+\rho\,dudv+T_{u}^{2}du^{2}+T_{v}^{2}dv^{2},$
(4.11)

which is related to the Fefferman-Graham gauge by a redefinition of the radial
coordinate $\rho=r^{2}+T_{u}^{2}T_{v}^{2}/r^{2}$, and to the Boyer-Lindquist-
like coordinates $(t,r,\phi)$ by the following coordinate transformations

$\rho=r^{2}-4M,\quad u=\phi+t,\quad v=\phi-t.$ (4.12)

The boundary interval ${\cal A}$ is parametrized by the bulk coordinates
(4.11) as

$\partial{\cal A}=\\{(u_{-},v_{-}),(u_{+},v_{+})\\},\quad
u_{+}-u_{-}=l_{u},\quad v_{+}-v_{-}=l_{v},$ (4.13)

which is related to the boundary parametrization by the bulk-boundary map
(4.5).

The parametrization of the swing surfaces can be described as follows. The
determination of null ropes turns out to be the most crucial step in the
entire construction because it enables the boundary information to come in.
With the help of two conservation laws related to two commuting Killing
vectors $\partial_{u}$, $\partial_{v}$ of the zero-mode background (4.11) and
the null geodesic condition, the two ropes emitted from the boundary endpoints
should satisfy

$T_{u}^{2}u^{\prime}+\frac{\rho v^{\prime}}{2}=p_{u},\quad
T_{v}^{2}v^{\prime}+\frac{\rho u^{\prime}}{2}=-p_{v},\quad\frac{\rho^{\prime
2}}{4(\rho^{2}-4T_{u}^{2}T_{v}^{2})}+T_{u}^{2}u^{\prime 2}+T_{v}^{2}v^{\prime
2}+\rho u^{\prime}v^{\prime}=0,$ (4.14)

where the null ropes $\left(\rho(\lambda),u(\lambda),v(\lambda)\right)$ are
parametrized by the affine parameter $\lambda$, and $p_{\mu}$, $p_{\nu}$
denote the momenta along the $u$, $v$ directions. Rewriting the above
equations as

$\rho^{\prime}=4\sqrt{\rho
p_{u}p_{v}+T_{u}^{2}p_{v}^{2}+T_{v}^{2}p_{u}^{2}},\quad
u^{\prime}=-\frac{4T_{v}^{2}p_{u}+2\rho
p_{v}}{\rho^{2}-4T_{u}^{2}T_{v}^{2}},\quad
v^{\prime}=\frac{4T_{u}^{2}p_{v}+2\rho p_{u}}{\rho^{2}-4T_{u}^{2}T_{v}^{2}},$
(4.15)

we can see that if we require the ropes to reach the asymptotic boundary
$\rho\to\infty$ at large $\lambda>0$, then the momenta $p_{u}$ and $p_{v}$
should satisfy $p_{u}p_{v}\geq 0$. For the case of $p_{u}p_{v}>0$, the
solutions to (4.15) are given by

$\displaystyle\rho=\frac{p_{u}^{2}p_{v}^{2}(2\lambda+\lambda_{0})^{2}-p_{u}^{2}T_{v}^{2}-p_{v}^{2}T_{u}^{2}}{p_{u}p_{v}},$
$\displaystyle
u=-\frac{1}{4T_{u}}\log{\\{\frac{p_{v}^{2}[T_{u}+p_{u}(2\lambda+\lambda_{0})^{2}]^{2}-p_{u}^{2}T_{v}^{2}}{p_{v}^{2}[T_{u}-p_{u}(2\lambda+\lambda_{0})^{2}]^{2}-p_{u}^{2}T_{v}^{2}}\\}}+u_{0},$
(4.16) $\displaystyle
v=\frac{1}{4T_{v}}\log{\\{\frac{p_{u}^{2}[T_{v}+p_{v}(2\lambda+\lambda_{0})^{2}]^{2}-p_{v}^{2}T_{u}^{2}}{p_{u}^{2}[T_{v}-p_{v}(2\lambda+\lambda_{0})^{2}]^{2}-p_{v}^{2}T_{u}^{2}}\\}}+v_{0},$

where $\lambda_{0}$, $u_{0}$ and $v_{0}$ are integration constants. There are
two remarkable points: 1) $p_{u}=0$ or $p_{v}=0$ is the singular limit of the
solutions (4.16), which means that we should solve them directly from (4.15);
2) For $p_{u}\neq 0$ and $p_{v}\neq 0$, the above solutions are symmetric
about $u$ and $v$ coordinates (up to an irrelevant minus sign), and this
symmetry is not broken at the cutoff surface $\rho=\rho_{\infty}$. The second
one would force the projection of tangent vectors of the null geodesics on the
asymptotic boundary to the symmetric one, which is the dilatation generator.
However in a WCFT, the modular flow is not symmetric about $u$ and $v$
directions, thus the possible solutions of the swing surface can only exist in
the singular limit of (4.16), i.e., $p_{u}=0$ or $p_{v}=0$ (but not both). It
has been shown in [41] that when choosing $T_{v}^{2}>0$, the null ropes are
the ones with $p_{v}=0$. If we denote the future directed one as $\gamma_{+}$,
then it has momentum $p_{u}<0$. Similarly, the past directed one has
$p_{u}>0$.

Setting $p_{v}=0$ in (4.15), we get

$\rho^{\prime}=4\sqrt{T_{v}^{2}p_{u}^{2}},\quad
u^{\prime}=-\frac{4T_{v}^{2}p_{u}}{\rho^{2}-4T_{u}^{2}T_{v}^{2}},\quad
v^{\prime}=\frac{2\rho p_{u}}{\rho^{2}-4T_{u}^{2}T_{v}^{2}},$ (4.17)

which have the solutions

$\gamma_{\pm}:\quad\rho=4|p_{u}|T_{v}\lambda+\rho_{0},\;\;u=\mp\frac{1}{4T_{u}}\log(\frac{\rho+2T_{u}T_{v}}{\rho-2T_{u}T_{v}})+{u}_{\pm},\;\;v=\mp\frac{1}{4T_{u}}\log(\frac{\rho^{2}-4T_{u}^{2}T_{v}^{2}}{\rho_{\infty}^{2}})+{v}_{\pm}.$
(4.18)

In the above solution, $\rho_{0}$ is the integration constant, $u_{\pm}$,
$v_{\pm}$ are the coordinates of boundary interval endpoints (4.13) that are
determined by the bulk-boundary matching condition, and $\rho_{\infty}$
denotes the radial location of the cutoff surface. The null ropes have the
following properties which can be seen easily from (4.17) and (4.18):

  * •

For fixed zero-mode background (4.11) with constant $T_{u}$ and $T_{v}$, the
rate of change of null ropes $\frac{d\rho}{du}$ and $\frac{d\rho}{dv}$ related
to any single boundary interval are always the same, which can be seen clearly
from (4.17).

  * •

As we go toward the boundary, i.e., increase the value of $\rho$, the radial
projection of the slope $|du/dv|$ of null ropes on the cut-off surface
$\rho=\rho_{\infty}$ becomes smaller

$\frac{du}{dv}=-\frac{2T_{v}^{2}}{\rho}<0.$ (4.19)

In addition, at each endpoint of the boundary interval, there are two null
geodesics with $p_{v}=0$ and opposite values of $p_{u}$. However, only one of
them can be the candidate of the ropes in the construction of swing surface,
due to the matching condition (4.6) and (4.7). More precisely, the future
outgoing lightray has $p_{u}<0$ and past outgoing lightray has $p_{u}>0$,
which can be checked by

$dt=du-
dv=-\frac{2p_{u}(2T_{v}^{2}+\rho)}{\rho^{2}-4T_{u}^{2}T_{v}^{2}}d\lambda\propto-
p_{u}.$ (4.20)

  * •

The condition $p_{v}=0$ of the null ropes ensures that they are perpendicular
to both the boundary of the domain of dependence of the boundary interval and
the bench of the swing surface, which all purely extend along the $v$
direction.

Figure 7: A cartoon picture showing the basic elements of a swing surface in
the global coordinates of 3D Lorentzian BTZ black hole spacetime: the boundary
interval in blue is on the (cyan) cutoff surface with $\rho=\rho_{\infty}$,
two red lines are the boundary of its domain of dependence, the null ropes are
in yellow and the bench is in black. The yellow surface is the true spacelike
conformal boundary with $\rho=\infty$, the purple surface denotes the black
hole horizon located at constant radius $\rho=2T_{u}T_{v}$. The coordinate $t$
in the Boyer-Lindquist-like coordinates $(t,r,\phi)$ is explicitly shown in
the picture, and the $\phi$ direction is anticlockwise when viewing from the
above.

Finally, the parametrization equations for the bench, which is an extremal
(not maximal or minimal) spacelike geodesic lying between the two null ropes,
are given by

$\gamma:u=\frac{u_{+}+u_{-}}{2},\quad v\in\bigg{[}\frac{v_{+}+v_{-}-\Delta
v}{2},\frac{v_{+}+v_{-}+\Delta
v}{2}\bigg{]},\quad\rho=2T_{u}T_{v}\coth{(T_{u}l_{u})},$ (4.21)

where the value of $\Delta v$ is

$\Delta v\equiv
l_{v}+\frac{1}{T_{v}}\log\bigg{[}\frac{\sinh{T_{u}l_{u}}}{2T_{u}T_{v}\rho_{\infty}^{-1}}\bigg{]}.$
(4.22)

Thus the single interval holographic entanglement entropy on the BTZ black
hole background (4.10) is given by

$S_{A}=\frac{T_{v}l_{v}}{4G}+\frac{1}{4G}\log{[\frac{\sinh{(T_{u}l_{u})}}{2T_{u}T_{v}\rho_{\infty}^{-1}}]},$
(4.23)

which matches the entanglement entropy formula (3.32) on the thermal cylinder
by the holographic map (4.5), (3.28) and the relations from [27]

$\beta=\frac{\pi}{T_{u}},\quad\bar{\beta}=2\pi\mu\left(\frac{T_{v}}{T_{u}}+1\right),\quad\mu=\sqrt{\frac{-1}{4Gk}},\quad
c=\frac{3}{2G},\quad\rho_{\infty}=\frac{2T_{v}}{\epsilon}.$ (4.24)

Compared with the case in AdS/CFT, there are several unusual features about
the holographic entanglement entropy in AdS3/WCFT:

  * •

Causality and unitarity. For a Lorentzian invariant 2D local CFT, causality is
a key property which can put constraints on the behavior of entanglement
entropy. For the subregion ${\cal A}$ on a boundary Cauchy surface
$\Sigma\in{\cal B}$, whose future and past domain of dependence
$D^{+}[\Sigma]$, $D^{-}[\Sigma]$ together make up the whole spacetime ${\cal
B}$, there is a reduced density matrix $\rho_{{\cal A}}$ whose entanglement
spectrum depends only on the domain of dependence $D[{\cal A}]=D^{+}[{\cal
A}]\cup D^{-}[{\cal A}]$ of the subregion ${\cal A}$. The whole spacetime can
be decomposed into four regions

${\cal B}=D^{+}[\Sigma]\cup D^{-}[\Sigma]=D[{\cal A}]\cup D[{\cal A}^{c}]\cup
J^{+}[\partial{\cal A}]\cup J^{-}[\partial{\cal A}]$ (4.25)

where $J^{+}(p)$ and $J^{-}(p)$ denote the boundary causal future and causal
past of the point $p$. The Hilbert space ${\cal H}$ can be decomposed into
${\cal H}_{{\cal A}}\otimes{\cal H}_{{\cal A}^{c}}$. If we unitarily evolve
the reduced density matrix $\rho_{{\cal A}}$ by the transformations that are
solely defined on ${\cal H}_{{\cal A}}$, then the eigenvalues of $\rho_{{\cal
A}}$ are not affected and so is the entanglement entropy.

For a 2D WCFT, the background geometry is not Riemannian geometry and there is
no notion about causal curves. Through the Rindler method we may identify the
domain of dependence of an interval as an infinite long strip along $v$
direction, and instead of (4.25) we now have

${\cal B}=D[{\cal A}]\cup D[{\cal A}^{c}]$ (4.26)

in 2D WCFT. In addition, in 2D WCFT the entanglement spectrum does not only
depend on the domain of dependence of subregion, and different subregions
having the same domain of dependence would have different entanglement
entropies. Note that in the case of 2D WCFT, the modular flow is not equal to
zero at the endpoints of ${\cal A}$, which means that we can construct an
evolution operator from the modular Hamiltonian related to the subregion
${\cal A}$ to change the length of interval in $v$ direction and thus to
change the entanglement entropy.

  * •

Holographic UR and IR properties: A remarkable feature of the AdS/CFT
holography is that it geometrize the energy scale of boundary field theory by
bulk radial direction. From holographic entanglement entropy (HEE) point of
view, the UV divergence in the area law of EE is related geometrically to the
local near-boundary behavior of RT/HRT surface ending normally on the
asymptotic AdS boundary, while the IR volume law contribution to EE at a
finite temperature corresponds holographically to the straddle behavior of HRT
surface lying close to the horizon of black hole which is deep in the
interior. For both UV and IR properties of EE in 2D WCFT, we only have similar
behaviors in the $u$ direction, while in the $v$ direction the relation is
always linear due to the non-local property of WCFT. Thus we focus only on the
$u$ part of HEE in the following discussions. The swing surface touches the
boundary through null ropes which have vanishing contributions to the HEE,
thus the divergent UV behavior of HEE in AdS3/WCFT shows up geometrically by
pushing the whole bench toward to asymptotic boundary, which is very different
from the way in the AdS/CFT case. While the IR property of HEE in AdS3/WCFT
case is behaved geometrically in the similar way as the AdS/CFT case by mainly
extending the bench along the black hole horizon.

  * •

Applicability: The holographic formula (4.23) can be applied only to the zero
mode background of asymptotically AdS3 spacetime with parameter range
$T^{2}_{v}\geq 0$, i.e., the BTZ black holes with non-zero charges. When
$T_{v}$ has an imaginary value, it would cause troubles in several aspects.
First, in (4.5) ,(4.24) and (4.23) the meaning for imaginary $T_{v}$ is
unclear. Second, from (4.17) we can see there would be no solutions111When
$T_{u}^{2}=T_{v}^{2}=0$, there is still no sensible solution of the null ropes
because this requires $u^{\prime}=0,$ $\rho^{\prime}=0,$
$v^{\prime}=\frac{2p_{u}}{\rho}$ whose solutions can not reach the asymptotic
boundary. of null ropes. This means that we cannot use this formula to the
global AdS3 spacetime and the BTZ black hole with $M=J=0$.

Figure 8: The figure shows schematically the infinite long pre-entanglement
wedge ${\cal W}_{\xi}$ of (blue) single boundary interval bounded by two
(purple) null sheets $N_{\pm}$ and boundary domain of dependence (bounded by
two red lines) in planar BTZ spacetime. For clarity we do not show explicitly
the horizon here. Note that the whole ${\cal W}_{\xi}$ lies completely outside
the black hole horizon.

Let us explore the entanglement wedge ${\cal W}_{{\cal E}}[{\cal A}]$ of
single boundary interval in AdS3/WCFT. In AdS3/WCFT, the bulk spacetime has
well defined causal structure but the boundary field theory does not. This
would lead to a tension between the cutoff surface at $\rho=\rho_{\infty}$ and
the asymptotic boundary at $\rho=\infty$, which we can see from the point of
view of entanglement entropy (EE). As can be seen easily from Figure 7 which
is consistent with bulk to boundary state-dependent map (4.5), the boundary
interval can be a timelike line on the cutoff surface that has well defined
induced metric. In addition, it is difficult to specify a homology surface
${\cal R}_{{\cal A}}$ interpolating between the boundary interval ${\cal A}$
and the bench $\gamma$, i.e., $\partial{\cal R}_{{\cal A}}={\cal
A}\cup\gamma$, such that ${\cal R}_{{\cal A}}$ is a subregion of the bulk
Cauchy surface of the whole spacetime. So the definition of entanglement wedge
described in Section 2.4 can not be applied here directly.

However, there is another equivalent definition of ${\cal W}_{{\cal E}}[{\cal
A}]$ in AdS/CFT: it is the region containing the set of bulk points that are
spacelike-separated from HRT surface $\gamma$ and connected to boundary domain
$D[{\cal A}]$. This definition has the advantage of freeing us from having to
specify an homology surface ${\cal R}_{{\cal A}}$, and it depends only on
$\gamma$ and $D[{\cal A}]$. We would apply this definition to AdS3/WCFT. In
this case, it requires starting from the bulk swing surface to grow the null
normal congruence with non-positive expansion. The collection of such null
normal geodesics span $N_{\pm}$, which are the future and past directed part
of the light-sheets. There are potentially two candidates from which we can
grow null surface, one is the whole modular flow invariant bifurcating horizon
$\gamma_{\xi}$ and the other one is the finite bench $\gamma$ that is a
subregion of $\gamma_{\xi}$. The reason to choose the infinite long
$\gamma_{\xi}$ is that we hope the entanglement wedge cover the whole region
where bulk modular flow sitting in. Therefore we prefer the first one, and
refer to the region bounded by the resulting null sheets $N_{\pm}$ and the
boundary of domain $\partial D[{\cal A}]$ as the pre-entanglement wedge ${\cal
W}_{\xi}$, see Figure 8. It satisfies

${\cal W}_{\xi}\cap{\cal B}=D[{\cal A}]$ (4.27)

for single boundary interval.

Figure 9: The relative location of the pre-entanglement wedge ${\cal W}_{{\cal
E}}[{\cal A}]$ (bounded by two purple surfaces) and the causal wedge ${\cal
W}_{{\cal C}}[{\cal A}]$ (bounded by two yellow surfaces) for (blue) single
boundary interval in planar BTZ spacetime. Two red lines on the (gray) black
hole horizon are the intersection of the horizon and the yellow surfaces.
Obviously the causal wedge ${\cal W}_{{\cal C}}[{\cal A}]$ contains the pre-
entanglement wedge ${\cal W}_{{\cal E}}[{\cal A}]$ in AdS3/WCFT.

Let us turn to the causal wedge in AdS/WCFT. The definition of causal wedge in
AdS/CFT is

${\cal W}_{{\cal C}}[{\cal A}]=\tilde{J}^{+}[D[{\cal
A}]]\cap\tilde{J}^{-}[D[{\cal
A}]]=\tilde{J}^{+}[P_{1}]\cap\tilde{J}^{-}[P_{2}],$ (4.28)

where $\tilde{J}^{\pm}[p]$ denote the bulk causal future/past of a point $p$,
and $P_{1,2}$ are the timelike future and past tips of $D[{\cal A}]$. In WCFT
the boundary domain of dependence $D[{\cal A}]$ of interval ${\cal A}=AB$ is
an infinite long strip, which can arise from a limiting procedure defined by

$\lim_{P_{u_{+}},P_{u_{-}}}\tilde{J}^{+}[P_{u_{+}}]\cap\tilde{J}^{-}[P_{u_{-}}]\cap{\cal
B}=D[{\cal A}]$ (4.29)

where $\lim_{P_{u_{+}},P_{u_{-}}}$ denote the process of moving the two points
$P_{u_{+}}$ and $P_{u_{-}}$, which live on the upper boundary and lower
boundary of $D[{\cal A}]$ respectively, along the $v$ direction to the future
and past null infinity respectively. Then the bulk causal wedge is

${\cal W}_{{\cal C}}[{\cal
A}]=\lim_{P_{u_{+}},P_{u_{-}}}\tilde{J}^{+}[P_{u_{+}}]\cap\tilde{J}^{-}[P_{u_{-}}].$
(4.30)

Roughly speaking, under this limiting process, the causal wedge of ${\cal A}$
would cover as much points as it can. As shown in Figure 9, the causal wedge
contains the whole pre-entanglement wedge in AdS/WCFT, in contrast with the
situation in AdS/CFT.

###  4.3 Pre-entanglement wedge cross section and two interval entanglement
phase transition

In this subsection, we would like to consider the pre-entanglement wedge
related to two disjoint intervals in the AdS3/WCFT holography. Generally,
there are two different ways of pairing the endpoints which are both
consistent with the homology constraint in computing the holographic
entanglement entropy of two disconnected intervals. Among them we should
choose the configuration with minimal swing surface area and the corresponding
entanglement wedge would follow. Like the case in AdS/CFT, an entanglement
phase transition would happen in AdS3/WCFT case as we change the separation of
two intervals while keeping their sizes fixed. For convenience, we choose the
symmetric configuration of two boundary intervals222The result also holds for
general non-symmetric configurations., as shown in Figure 10:

$\partial{\cal A}_{1}=\\{A=(-b,-c),B=(-a,-d)\\},\quad\partial{\cal
A}_{2}=\\{C=(a,d),D=(b,c)\\},$ (4.31)

with $b>a>0,c>d>0$. If the swing surface configuration pairs $A$ to $B$ and
$C$ to $D$, then the area is given by

$S_{{\cal A}_{1}{\cal
A}_{2}}^{(1)}=\frac{T_{v}(c-d)}{2G}+\frac{1}{2G}\log\Big{[}\frac{\sinh(T_{u}(b-a))}{2T_{u}T_{v}\rho_{\infty}^{-1}}\Big{]}.$
(4.32)

If the swing surface configuration pairs $A$ to $D$ and $B$ to $C$, then the
area is given by

$S_{{\cal A}_{1}{\cal
A}_{2}}^{(2)}=\frac{T_{v}(c+d)}{2G}+\frac{1}{2G}\log\Big{[}\frac{\sqrt{\sinh(2T_{u}b)\sinh(2T_{u}a)}}{2T_{u}T_{v}\rho_{\infty}^{-1}}\Big{]}.$
(4.33)

The difference of the area between these two configurations is given by

$\displaystyle\delta S_{{\cal A}_{1}{\cal A}_{2}}$ $\displaystyle\equiv
S_{{\cal A}_{1}{\cal A}_{2}}^{(2)}-S_{{\cal A}_{1}{\cal
A}_{2}}^{(1)}=\frac{T_{v}d}{G}+\frac{1}{2G}\log\Big{[}\frac{\sqrt{\sinh(2T_{u}b)\sinh(2T_{u}a)}}{\sinh(T_{u}(b-a))}\Big{]}$
$\displaystyle=\frac{T_{v}d}{G}+\frac{1}{4G}\log\frac{1-w}{w}.$ (4.34)

where $w$ denotes the cross ratio on the thermal cylinder. This suggests that
for fixed $d$, if $w\ll 1$ we would have $\delta S_{{\cal A}_{1}{\cal
A}_{2}}>0$, and we need to choose the first paring pattern. In this case, the
pre-entanglement wedge ${\cal W}_{\xi}[{\cal A}]$ determined by the swing
surface is disconnected and there is no non-trivial pre-entanglement wedge
cross section. For the other limit where $1-w\ll 1$ with $d$ fixed, we have
$\delta S_{AB}<0$ and we should choose the second paring pattern. In this
case, the corresponding pre-entanglement wedge is connected, which supports a
nontrivial cross section. In addition, there always exists a non-unique
spacelike homology surface ${\cal R}_{{\cal A}}$ interpolating between the two
infinite “benches”, which gives the identification of connected pre-
entanglement wedge in this situation as the bulk domain of dependence of
${\cal R}_{{\cal A}}$

${\cal W}_{\xi}[{\cal A}]=D[{\cal R}_{{\cal A}}].$ (4.35)

Figure 10: The figure shows the non-trivial connected pre-entanglement wedge
${\cal W}_{\xi}[{\cal A}]$ and its cross section $E_{W}$ (green) related to
two disconnected (blue) boundary intervals $AB$ and $CD$. It is drawn in the
planar BTZ black hole spacetime. For clarity, we choose the two intervals
sitting on the same line and omit the black hole horizon. $E_{W}$ has a $U(1)$
translational symmetry along the $v$ direction, which is parallel to the
infinite bifurcation horizon $\gamma_{\xi(BC)}$ and $\gamma_{\xi(AD)}$. ${\cal
W}_{\xi}[{\cal A}]$ would never intersect the asymptotic boundary, which is a
new phenomena in AdS3/WCFT.

In order to compute the cross section of the connected entanglement wedge
${\cal W}_{\xi}[{\cal A}]$, we choose the symmetric boundary intervals
satisfying (4.31) with the cross ratio $1-w\ll 1$. The line with extremal
length can only exist between two spacelike benches, $\gamma_{AD}$ and
$\gamma_{BC}$. To begin with, we write down the general expression of geodesic
distance $L(p_{1},p_{2})$ in BTZ black hole spacetime (4.11) between two
spacelike separated points $p_{1}=(\rho_{1},u_{1},v_{1})$ and
$p_{2}=(\rho_{2},u_{2},v_{2})$ for later convenience

$L(p_{1},p_{2})=\cosh^{-1}\left(D(p_{1},p_{2})\right),$

with

$\displaystyle D(p_{1},p_{2})=D_{+}(p_{1},p_{2})+D_{-}(p_{1},p_{2}),$ (4.36)
$\displaystyle D_{\pm}(p_{1},p_{2})=\frac{e^{\pm
T_{v}v_{12}}}{8T_{u}T_{v}}\bigg{[}e^{\pm
T_{u}u_{12}\sqrt{(\rho_{1}+2T_{u}T_{v})(\rho_{2}+2T_{u}T_{v})}}-e^{\mp
T_{u}u_{12}\sqrt{(\rho_{1}-2T_{u}T_{v})(\rho_{2}-2T_{u}T_{v})}}\bigg{]}.$
(4.37)

The two independent functions $D_{+}(p_{1},p_{2})$ and $D_{-}(p_{1},p_{2})$,
which are invariant under $SL(2,R)_{L}$ $\times U(1)_{R}$ transformations,
together constitute a quantity $D(p_{1},p_{2})$ that is invariant under the
transformations of the group $SL(2,R)_{L}$ $\times SL(2,R)_{R}$, the isometry
group of AdS3. The final form of the $L(p_{1},p_{2})$ can be determined from
the normalization condition[41].

The parametrizations of two benches are respectively

$\displaystyle\gamma_{a}:u_{a}=0,\quad v_{a}\in\bigg{[}\frac{-\Delta
v_{a}}{2},\frac{\Delta
v_{a}}{2}\bigg{]},\quad\rho_{a}=2T_{u}T_{v}\coth(2T_{u}a),$ (4.38)
$\displaystyle\gamma_{b}:u_{b}=0,\quad v_{b}\in\bigg{[}\frac{-\Delta
v_{b}}{2},\frac{\Delta
v_{b}}{2}\bigg{]},\quad\rho_{b}=2T_{u}T_{v}\coth(2T_{u}b),$ (4.39)

where

$\displaystyle\Delta v_{a}$ $\displaystyle=$ $\displaystyle
2d+\frac{1}{T_{v}}\log\left(\frac{\sinh(2T_{u}a)}{2T_{u}T_{v}\rho_{\infty}^{-1}}\right),$
$\displaystyle\Delta v_{b}$ $\displaystyle=$ $\displaystyle
2c+\frac{1}{T_{v}}\log\left(\frac{\sinh(2T_{u}b)}{2T_{u}T_{v}\rho_{\infty}^{-1}}\right).$

We take the point $p_{1}$ on $\gamma_{a}$ and the point $p_{2}$ on
$\gamma_{b}$, and find the geodesic distance between them

$\displaystyle D(p_{1},p_{2})$
$\displaystyle=D_{+}(p_{1},p_{2})+D_{-}(p_{1},p_{2})$
$\displaystyle=D_{+}(u_{12}=0,v_{12},\rho_{12}=const.)+D_{-}(u_{12}=0,v_{12},\rho_{12}=const.)$
$\displaystyle=\frac{A_{0}}{2}\cosh(T_{v}v_{12}).$ (4.40)

where

$A_{0}=\sqrt{(\coth(2T_{u}a)+1)(\coth(2T_{u}b)+1)}-\sqrt{(\coth(2T_{u}a)-1)(\coth(2T_{u}b)-1)}.$
(4.41)

It is easy to see that there is only one minimal value of $D(p_{1},p_{2})$
which corresponds to $v_{12}=0$

$D_{min}(p_{1},p_{2})=\frac{A_{0}}{2}=\left|\frac{\sinh(T_{u}(a+b))}{\sqrt{\sinh(2T_{u}a)\sinh(2T_{u}b)}}\right|.$
(4.42)

Because the distance $L(p_{1},p_{2})$ is a monotonically increasing function
of $D(p_{1},p_{2})$, then the minimal value of $D(p_{1},p_{2})$ directly gives
the minimal value of $L(p_{1},p_{2})$. Using the following identities

$\cosh^{-1}(x)=\log(x+\sqrt{x^{2}+1}),\quad\sinh(b+a)^{2}-\sinh(b-a)^{2}=\sinh{2a}\sinh{2b},$
(4.43)

we find the minimal length between $\gamma_{a}$ and $\gamma_{b}$

$\displaystyle L_{min}(p_{1},p_{2})$
$\displaystyle=\cosh^{-1}(D_{min}(p_{1},p_{2}))=\log\big{(}D_{min}+\sqrt{D_{min}^{2}-1}\big{)}$
$\displaystyle=\frac{1}{2}\log\left(\left|\coth{T_{u}a}\tanh{T_{u}b}\right|\right).$
(4.44)

In conclusion, we get the extremal pre-entanglement wedge cross section

$E_{W}({\cal
A})=\frac{L_{min}(p_{1},p_{2})}{4G}=\frac{1}{8G}\log\left(\left|\coth(T_{u}a)\tanh(T_{u}b)\right|\right).$
(4.45)

Comparing it with the reflected entropy of WCFT (3.80), we find that under the
holographic dictionary $c=\frac{3}{2G}$ in (4.24), they satisfy

$E_{W}({\cal A})=\frac{1}{2}S^{\text{WCFT}}_{R;thermal},$ (4.46)

which is precisely the conjectured holographic relation (2.39) in AdS/CFT
case. Moreover, if we take $T_{u}\to 0$ in (4.24) which corresponds to
$\beta\to\infty$, we would have

$D_{min}\rightarrow\frac{1}{2}\left(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}\right)\geq
1,\qquad L_{min}\to\frac{1}{2}\log(\frac{b}{a}).$ (4.47)

Then the pre-EWCS in $T_{u}=0$ zero mode background is half of the zero
temperature reflected entropy $S^{\text{WCFT}}_{R;vac}$ on the vacuum state of
WCFT.

In AdS/CFT, entanglement wedge ${\cal W}_{{\cal E}}[{\cal A}]$ satisfies the
entanglement wedge nesting property, which mainly states that if the two
boundary subregions have the relation ${\cal A}^{\prime}\subset{\cal A}$ then
the entanglement wedge of ${\cal A}^{\prime}$ should lie within the
entanglement wedge of ${\cal A}$. It is consistent with subregion duality and
implies that the sub-algebra of operators localized in ${\cal W}_{{\cal
E}}[{\cal A}^{\prime}]$ must be a subset of the sub-algebra of operators
localized in ${\cal W}_{{\cal E}}[{\cal A}]$. In AdS3/WCFT, we still have this
nice property for pre-entanglement wedge as can be seen clearly from Figure
11. The domains of dependence for the sub-regions $AB$, $BC$ and $CD$ are
disconnected, and their corresponding pre-entanglement wedges are also
disconnected. Furthermore, the boundary domains of dependence for the
subregions $AB$, $BC$ and $CD$ are all contained in the domain of dependence
of $AD$, and the same happens for their corresponding pre-entanglement wedges.

Figure 11: This cartoon picture shows the pre-entanglement wedge nesting
property in AdS3/WCFT holography. For two (blue) disjoint boundary intervals
$AB$ and $CD$, the region bounded by the yellow null surfaces and (cyan)
cutoff surface corresponds to the disconnected ${\cal W}_{{\cal E}}[{\cal
A}]$. The pre-entanglement wedge related to single interval $BC$ are bounded
by red null surfaces, and the one related to interval $AD$ are bounded by
purple null surfaces.

##  5 Logarithmic negativity and odd entropy

Additional information quantities measuring different entanglement structures
of mixed state $\rho_{AB}$ in many body systems have also been shown to be
holographically related to the entanglement wedge cross section in AdS3/CFT2.
They include the logarithmic negativity ${\cal E}(\rho_{AB})$ in [58] which is
shown to be a suitable [59] and tractable measure of entanglement for mixed
states, and the odd entropy which was recently introduced in [9] as a new
information quantity. It is an interesting question to ask how these two
entanglement measures are related to the entanglement wedge cross section in
the AdS3/WCFT correspondence.

###  5.1 Logarithmic negativity

The (logarithmic) negativity was proposed as a suitable measure of quantum
entanglement for mixed states[7]. It is derived from the positive partial
transpose criterion for the separability of mixed states[7, 59, 60, 61]. Given
a bipartite density matrix $\rho_{AB}$, its partial transpose
$\rho_{AB}^{T_{B}}$ is defined in terms of its matrix elements as

$\langle i_{A},j_{B}|\rho_{AB}^{T_{B}}|k_{A},l_{B}\rangle=\langle
i_{A},l_{B}|\rho_{AB}|k_{A},j_{B}\rangle,$ (5.1)

where $i_{A},j_{B},k_{A}$ and $l_{B}$ are the bases for subsystems $A$ and
$B$. The logarithmic negativity is then defined as

$\mathcal{E}(\rho_{AB})\equiv\log\Big{|}\rho_{AB}^{T_{B}}\Big{|}_{1},$ (5.2)

where $|\mathcal{O}|_{1}=\mathrm{Tr}\sqrt{\mathcal{O}\mathcal{O}^{\dagger}}$.
When the subregion is spherically symmetric and the state is pure, the
negativity $\mathcal{E}$ is simply the Rényi entropy with Rényi index 1/2. In
this special case, people has shown that it satisfies[62, 63]

$\mathcal{E}=\mathcal{X}_{d}S,$ (5.3)

where $S$ is the entanglement entropy and $\mathcal{X}_{d}$ is a constant
which depends on the dimension of the CFT. For 2D CFTs, its value is
$\mathcal{X}_{2}=3/2$.

It was conjectured in [8] and proved in [58] that the logarithmic negativity
${\cal E}(\rho_{AB})$ for two disjoint intervals in the vacuum of holographic
CFT is dual to a back-reacted entanglement wedge cross section. The key point
in the proof is the relation between negativity and $n=\frac{1}{2}$ Renyi
reflected entropy in holographic theories,

$\mathcal{E}=\frac{S_{R}^{1/2}}{2}.$ (5.4)

Generally the back-reacted entanglement wedge cross section is hard to
evaluate, however when considering spherically symmetric subregion
configurations in the vacuum state in two dimensions we have

$\mathcal{E}=\frac{3}{2}E_{W}.$ (5.5)

This relation is consistent with (5.3) with $d=2$ once we note that the
entanglement wedge cross section $E_{W}$ reduces to the entanglement entropy
for a single interval.

The logarithmic negativity of two disjoint intervals can be computed by a
correlation function of twist operators

$\mathcal{E}=\lim_{n_{e}\to
1}\log{\langle\sigma_{n_{e}}(z_{1})\tilde{\sigma}_{n_{e}}(z_{2})\tilde{\sigma}_{n_{e}}(z_{3})\sigma_{n_{e}}(z_{4})\rangle},$
(5.6)

where $n_{e}$ represent the fact that we analytically continue the even
numbers of the replica sheets, and $z_{i}$ are the endpoints of these
intervals. In the $t$-channel OPE expansion of four-point correlators, there
appear the twist operators $\sigma_{n}^{(2)}$ and $\tilde{\sigma}_{n}^{(2)}$
which relate the $n$-th replica sheet to the $(n\pm 2)$-th ones and have
holomorphic dimensions that depend on the parity of $n$ [64],

$h_{\sigma_{n}^{(2)}}=h_{\tilde{\sigma}_{n}^{(2)}}=\left\\{\begin{array}[]{rl}\frac{c}{24}(n-\frac{1}{n})&\text{if
$n$ is odd},\\\ \frac{c}{12}(\frac{n}{2}-\frac{2}{n})&\text{if $n$ is
even}.\end{array}\right.$ (5.7)

When we consider holographic CFTs, the Virasoro block related to
$\sigma_{n}^{(2)}$ operator would be the dominant one for some specific region
$x_{c}<x<1$ in the $t$-channel block expansion of the four-point correlator.
We can check the above relation (5.4) (5.5) directly in symmetric interval in
vacuum state

$\mathcal{E}=\frac{S_{R}^{n=1/2}}{2}=\frac{3}{2}E_{W}=\frac{3}{4}S_{R}^{n=1}=\frac{c}{2}\ln{\frac{1+\sqrt{z}}{1-\sqrt{z}}}.$
(5.8)

In the case of WCFT, the logarithmic negativity is determined by (5.6) as
well. The twist operators appearing in the four-point correlators have the
following dimensions and $U(1)$ charges.

  * •

External twist operator $\sigma_{n_{e}}$:

$h_{\sigma_{n_{e}}}^{inv}=\frac{c}{24}\left(n_{e}-\frac{1}{n_{e}}\right),\quad
q_{\sigma_{n_{e}}}=\left(n_{e}-1\right)\frac{i\mu k}{2}.$ (5.9)

  * •

External twist operator $\tilde{\sigma}_{n_{e}}$:

$h_{\tilde{\sigma}_{n_{e}}}^{inv}=\frac{c}{24}\left(n_{e}-\frac{1}{n_{e}}\right),\quad
q_{\tilde{\sigma}_{n_{e}}}=-\left(n_{e}-1\right)\frac{i\mu k}{2}.$ (5.10)

  * •

Internal dominant twist operator $\sigma_{n_{e}}^{(2)}$:

$h_{\sigma_{n_{e}}^{(2)}}^{inv}=\frac{c}{12}\left(\frac{n_{e}}{2}-\frac{2}{n_{e}}\right),\quad
q_{\sigma_{n_{e}}^{(2)}}=2(n_{e}-1)\frac{i\mu k}{2}.$ (5.11)

Working in the holographic WCFTs on the vacuum state of the plane with general
spectral flow parameter $\mu$, we can evaluate (5.6) assuming the WCFT block
relating to the operator $\sigma_{n_{e}}^{(2)}$ is the dominant one as in the
case of reflected entropy by using (3.19) and (3.21). We choose the symmetric
configuration

$A=\\{(-x_{2},-y_{2}),(-x_{1},-y_{1})\\},\quad
B=\\{(x_{1},y_{1}),(x_{2},y_{2})\\},$ (5.12)

and find

$\displaystyle\mathcal{E}_{WCFT}$ $\displaystyle\sim\lim_{n_{e}\to
1}\left(\underbrace{i\sum_{j=1}^{4}q_{j}(y_{j}-\mu
x_{j})+\frac{2q_{\sigma_{n_{e}}}^{2}}{k}\log{\mathrm{x}}+\frac{2q_{\sigma_{n_{e}}}q_{\tilde{\sigma}_{n_{e}}}}{k}\log{(1-\mathrm{x})}}_{D_{1}=0}+\underbrace{\log{\left(\frac{\mathcal{F}(nc-1,h_{i}^{inv},h_{\sigma_{n_{e}}^{(2)}}^{inv},\mathrm{x})}{x_{21}^{2h_{\sigma_{n_{e}}}}x_{43}^{2h_{\tilde{\sigma}_{n_{e}}}}}\right)}}_{D_{2}}\right)$
$\displaystyle=\lim_{n_{e}\to
1}\left(\underbrace{-\frac{q_{\sigma_{n_{e}}}^{2}}{nk}\log{\left(x_{21}x_{43}\right)}}_{D_{2,1}}+\underbrace{\log{\left(\frac{\mathcal{F}(nc-1,h_{i}^{inv},h_{\sigma_{n_{e}}^{(2)}}^{inv},\mathrm{x})}{x_{21}^{2h_{1}^{inv}}x_{43}^{2h_{3}^{inv}}}\right)}}_{D_{2,2}}\right)$
$\displaystyle=\frac{1}{2}\mathcal{E}_{\text{CFT}}$ (5.13)

where $q_{1,4}=q_{\sigma_{n_{e}}}$, $q_{2,3}=q_{\tilde{\sigma}_{n_{e}}}$,
$h_{1,4}^{inv}=h_{\sigma_{n_{e}}}^{inv}$ and
$h_{2,3}^{inv}=h_{\tilde{\sigma}_{n_{e}}}^{inv}$. In the first equality, we
use (3.19) (3.21) and the block dominance; in the second equality, $D_{1}$
part vanish due to von Neumann limit $n_{e}\to 1$ and we use (3.18); in the
third equality, $D_{2,1}$ part vanish due to von Neumann limit and $D_{2,2}$
part is the same expression as the holomorphic part of negativity in
holographic CFT. Thus we have proved that in AdS3/WCFT2, (5.4) (5.5) still
holds

$\mathcal{E}_{WCFT}=\frac{S_{R;WCFT}^{1/2}}{2}=\frac{3}{2}E_{W}^{WCFT}=\frac{c}{4}\ln{\frac{1+\sqrt{\mathrm{x}}}{1-\sqrt{\mathrm{x}}}}.$
(5.14)

It seems that the logarithmic negativity in holographic WCFTs is still dual to
a backreacted entanglement wedge cross section.

###  5.2 Odd entropy

Odd entanglement entropy was proposed to be another measure of mixed state[9].
The definition of Rényi version of odd entropy is

$S_{o}^{n_{o}}(\rho_{AB})=\frac{1}{1-n_{o}}\left(\mbox{Tr}(\rho_{AB}^{T_{B}})^{n_{o}}-1\right),$
(5.15)

where $T_{B}$ is the partial transposition with respect to the subsystem $B$,
and $n_{o}$ represents the analytic continuation of an odd integer from which
the name “odd entropy” comes from. We can use the replica trick as in the
logarithmic negativity, and evaluate it in holographic CFTs of two disjoint
intervals in vacuum state,

$\displaystyle S_{o}$ $\displaystyle=\lim_{n_{o}\to
1}\frac{1}{1-n_{o}}\left[\langle\sigma_{n_{o}}(z_{1})\tilde{\sigma}_{n_{o}}(z_{2})\tilde{\sigma}_{n_{o}}(z_{3})\sigma_{n_{o}}(z_{4})\rangle-1\right]$
$\displaystyle\sim-\partial_{n_{o}}\left[b_{\sigma_{n}^{(2)}}(z_{14}z_{23})^{-4h}\mathcal{F}(nc,h_{i},h_{\sigma_{n}^{(2)}},z)\bar{\mathcal{F}}(n\bar{c},\bar{h}_{i},\bar{h}_{\sigma_{n}^{(2)}},\bar{z})\right]_{n_{o}=1}$
$\displaystyle=\frac{c}{3}\log{\frac{z_{14}}{\epsilon}}+\frac{c}{3}\log{\frac{z_{23}}{\epsilon}}+\frac{c}{6}\ln{\frac{1+\sqrt{z}}{1-\sqrt{z}}}$
$\displaystyle=S_{\rho_{A}\otimes\rho_{B}}+E_{W}(\rho_{AB})$ (5.16)

where we used the odd integer part of (5.7) and $S_{\rho_{A}\otimes\rho_{B}}$
represents the entanglement entropy of the product state
$\rho_{A}\otimes\rho_{B}$. The effect of OPE coefficients
$b_{\sigma_{n}^{(2)}}$ can be evaluated in the pure state limit [9].

For WCFT, the external twist operators in computing the odd entropy are the
same as the ones in computing negativity, and their dimensions and charges are
given by (5.9) (5.10) with $n_{e}$ being replaced by $n_{o}$. But the internal
dominant twist operators $\sigma_{n_{o}}^{(2)}$ have different dimensions and
$U(1)$ charges,

$h_{\sigma_{n_{o}}^{(2)}}^{inv}=h_{\sigma_{n_{o}}}^{inv}=\frac{c}{24}\left(n_{o}-\frac{1}{n_{o}}\right),\quad
q_{\sigma_{n_{o}}^{(2)}}=2(n_{o}-1)\frac{i\mu k}{2}$ (5.17)

which is consistent with (5.7). Assuming the block dominance in the
holographic WCFT, we get the odd entropy on the reference plane333The odd
entropy of WCFT is not spectral flow invariant and depends on the plane on
which it is evaluated.

$\displaystyle S_{o}^{WCFT}$ $\displaystyle=\lim_{n_{o}\to
1}{\frac{1}{1-n_{o}}}\left[\langle\sigma_{n_{o}}(x_{1},y_{1})\tilde{\sigma}_{n_{o}}(x_{2},y_{2})\tilde{\sigma}_{n_{o}}(x_{3},y_{3})\sigma_{n_{o}}(x_{4},y_{4})\rangle_{{\mbox{\small
WCFT}^{\otimes n_{o}}(ref)}}-1\right]$
$\displaystyle\sim-\partial_{n_{o}}\left[\underbrace{e^{iq_{\sigma_{n_{o}}(y_{1}+y_{4}-y_{2}-y_{3})}}\left(x_{14}x_{23}\right)^{-2h_{\sigma_{n_{o}}}}}_{D_{1}}\underbrace{b_{\sigma_{n_{o}}^{(2)}}^{2}\mathcal{F}(nc-1,h_{i}^{inv},h_{\sigma_{n_{o}}^{(2)}}^{inv},\mathrm{x})}_{D_{2}}\underbrace{\left(\frac{\mathrm{x}}{1-\mathrm{x}}\right)^{2q_{\sigma_{n_{o}}}^{2}}}_{D_{3}}\right].$

Here the $D_{3}$ part has vanishing contribution and $D_{2}$ leads to
$E_{W}(\rho_{AB})$, but the $D_{1}$ part does not give the expected
$S_{\rho_{A}\otimes\rho_{B}}$ which should come from a term like