Topological entanglement entropy in Euclidean AdS3 via surgery
JHE
AdS3
ZhuXi Luo 0 1 3
HaoYu Sun 0 1 2
0 266 LeConte Hall, MC 7300 , Berkeley, CA 94720 , U.S.A
1 201 James Fletcher Bldg. , 115 South 1400 East, Salt Lake City, UT 841120830 , U.S.A
2 Department of Physics, University of California , USA
3 Department of Physics and Astronomy, University of Utah , USA
We calculate the topological entanglement entropy (TEE) in Euclidean asymptotic AdS3 spacetime using surgery. The treatment is intrinsically threedimensional. In the BTZ black hole background, several di erent bipartitions are applied. For the bipartition along the horizon between two singlesided black holes, TEE is exactly the BekensteinHawking entropy, which supports the ER=EPR conjecture in the Euclidean case. For other bipartitions, we derive an EntanglingThermal relation for each singlesided black hole, which is of topological origin. After summing over genusone classical geometries, we compute TEE in the hightemperature regime. In the case where k = 1, we nd that TEE is the same as that for the Moonshine double state, given by the maximallyentangled superposition of 194 types of anyons" in the 3d bulk, labeled by the irreducible representations of the Monster group. We propose this as the bulk analogue of the thermo eld double state in the Euclidean spacetime. Comparing the TEEs between thermal AdS3 and BTZ solutions, we discuss the implication of TEE on the HawkingPage transition in 3d.
AdSCFT Correspondence; Black Holes; Anyons; Topological Field Theories

HJEP12(07)6
1 Introduction 2
Review of relevant components
Thermal AdS3
BTZ black hole
BTZ geometry
\Surgery" and replica trick
3
Conformal boundary and H =
Solid tori classi ed as Mc;d
Bipartition into two disks
Two disjoint thermal AdS3
3
4
5
6
2.1
2.2
2.3
3.1
3.2
4.1
4.2
4.3
5.1
5.2
Summation over geometries
TEE for the full partition function
di as quantum dimensions
Discussion and outlook
A Bipartition for the full partition function
B TEE from the whole J (q) function
TEE between two onesided black holes and mutual information
The entanglingthermal relation
1
Introduction
Topological entanglement entropy (TEE), rst introduced in condensed matter physics [
1,
2
], has been widely used to characterize topological phases. It is the constant
subleading term (relative to the arealaw term) in the entanglement entropy, only dependent on
universal data of the corresponding topological phase.
At low energy, a large class of topological phases can be e ectively described using
ChernSimons gauge theory with a compact, simple, simplyconnected gauge group. When
this is the case, TEE can be found using surgery [3] and replica trick [4] by computing the
partition function on certain 3manifolds. For compact gauge groups, TEE is expressed [3]
in terms of modular S matrices of WessZuminoWitten rational conformal eld theory
(RCFT) on a 2d compact Riemann surface, following the CS/WZW correspondence rst
described in geometric quantization by [5].
{ 1 {
ratio between the interval length on the boundary circle that is contained in subregion A
and the circumference of the full circle. After applying the replica trick, the glued manifold
is a genusn handlebody. Using oneloop partition function on this handlebody [10{15],
we derive an explicit expression for TEE, which vanishes in the lowtemperature limit.
Then we consider two disjoint thermal AdS3's and calculate the TEE between them, which
turns out to be the thermal entropy of one thermal AdS3. However, this does not mean
any nontrivial entanglement between the two solid tori, and we support this argument by
calculating the mutual information between them, which gives zero.
We also compute TEEs in an eternal BTZ background. In the Euclidean picture
there is only one asymptotic region for the eternal BTZ black hole [17], which corresponds
to the gluing of the two asymptotic regions of the two singlesided black holes in the
Lorentzian picture. We show that TEE between the two singlesided black holes is equal
to the BekensteinHawking entropy of one singlesided black hole. The mutual information
between them does not vanish and again equals to the BekensteinHawking entropy, which
guarantees the explanation of the result as supporting the ER=EPR conjecture in 3d bulk
to be true [18{20].
stating
Focusing on one singlesided black hole, we then derive an EntanglingThermal relation,
lim
relation is similar to but di erent from the thermal entropy relation [
24
] derived from the
RyuTakayanagi formula [25], in that our result is topological and does not depend on
geometrical details.
The full modularinvariant genus one partition function of threedimensional pure
gravity is a summation of classical geometries or gravitational instantons, which includes both
thermal AdS3 and the BTZ black hole. At high temperatures, the full partition function is
dominated by the SL(2; Z) family of black hole solutions, whereas the lowtemperature
solution is dominated by the thermal AdS3. We compute TEE for the full partition function
with a bipartition between the two singlesided black holes in the high temperature regime
and again observe ER=EPR explicitly. When ChernSimons level kR = kL = l=16G = 1,
after de ning the quantum dimension data on the boundary Monster CFT with orbifolding,
we see from the TEE calculation that the black hole geometries correspond to a
topological phase in the bulk which contains a maximallyentangled superposition of 194 types
of \anyons", labeled by the irreducible representations of the Monster group. This state,
dubbed as Moonshine double state, has the similar property as the thermo eld double
state on the asymptotic boundary in that TEE between the anyon pairs is equal to the
BekensteinHawking entropy. The rest of the paper is organized as follows. In section 2 we
give a minimal introduction to the knowledge that facilitates the TEE calculation,
including replica trick and Schottky uniformization. In section 3 we show the calculation of TEE
in thermal AdS3, which amounts to the computation of the partition function on a genus
nhandlebody. We also compute the TEE between two disjoint thermal AdS3 and show their
mutual information vanishes. Section 4 illustrates the TEE calculation for BTZ black holes
for several di erent bipartitions. We discuss the relations with ER=EPR and show that
mutual information between the two singlesided black holes is equal to the
BekensteinHawking entropy. We further propose an EntanglingThermal relation for singlesided black
holes. Then in section 5 we demonstrate the TEE of the full modularinvariant partition
function after summing over geometries and present the quantum dimension interpretation.
The system is mapped to a superposition of 194 types of \anyons". Comments on the
implication of TEE on the HawkingPage transition and the outlook can be found in section 6.
2
Review of relevant components
In this section we will introduce basic concepts that are essential to understanding the rest
of the paper.
2.1
\Surgery" and replica trick
Surgery was originally invented by Milnor [28] to study and classify manifolds of dimension
of techniques used to produce a new nitedimensional manifold from an existing one in a
controlled way. Speci cally, it refers to cutting out parts of a manifold and replacing it by
a part of another manifold, matching up along the cut.
As a warmup, we review the usage of surgery in the entanglement calculation of
2d CFT for a single interval at
nite temperature T = 1=
[4]. The interval A lies on
an in nitely long line whose thermal density matrix is denoted as . The reduced density
matrix of subregion A is then de ned as A = trA , where the trace trA over the complement
of A only glues together points that are not in A, while an open cut is left along A.
Entanglement entropy between A and its complement A is then SA =
tr A ln A. The
matrix logarithm is generally hard to compute, so alternatively one applies the replica
trick to obtain an equivalent expression, with proper normalization (so that the resultant
quantity is 1 when being analytically continued to n = 1):
ds2 =
dy2 + dzdz
y2
;
{ 4 {
Now the problem reduces to the computation of tr( nA). Using surgery, one can interpret
it as the path integral on the glued 2manifold [29]. An example for n = 3 is shown in
gure 2, where the left panel sketches 3A, and the right panel is tr( 3A). In this case with
a nite temperature, SA is not necessarily equal to SA.
This operation can be extended to 3manifolds in a straightforward way, as shown in
ref. [3]. The authors calculated examples where the constant time slices are closed surfaces
and restricted to ground states, so that the
cycle is in nitely long.
The constant time slices that we are interested in for Euclidean AdS3 are all open
surfaces with asymptotic conformal boundaries, and the quantum states do not necessarily
belong to the ground state Hilbert subspace. Details will be presented in sections 3 and 4.
2.2
Conformal boundary and H3=
We now introduces the hyperbolic threespace H3 that describes the Euclidean AdS3. It is
the 3d analogue of hyperbolic plane, with the standard Poincarelike metric
(2.1)
(2.2)
where y > 0 and z is a complex coordinate.
Any 3manifold M having a genus n Riemann surface
n as its conformal boundary
that permits a complete metric of constant negative curvature can be constructed using
( ) = S2
1
Schottky uniformization. The idea is to represent the 3manifold M as the quotient of H
by a Kleinian group
[30], which is a discrete subgroup of SL(2; C) as well as a discrete
group of conformal automorphisms of n.
The conformal boundary of H3 is a sphere at in nity, S2 , on which
acts discretely,
except for a limit set of accumulation points of
denoted by
( ). The complement
1
( ) is called the domain of discontinuity. Then the 3manifold M has
boundary ( )= , a wellde ned quotient.
In particular, when M is a handlebody,
reduces to a Schottky group, which is freely
nitely generated by the loxodromic elements 1; : : : ; n 2 SL(2; C), that acts on S12 as a
fractional linear transformation. Among these generators, there are 3n
3 independent complex parameters, which are coordinates on the Schottky space, a covering space of the complex moduli of the Riemann surface. Each
2
is completely characterized by its xed points and its multiplier q . An
eigenvalue q is de ned through the unique conjugation of
jq j < 1. More explicitly, denoting ;
as the xed points of , one has
under SL(2; C): z 7! q z with
(z)
(z)
= q
z
z
Within the Schottky group , there are primitive conjugacy classes h 1; : : : ; ni of ,
with \primitive" meaning that
is not a positive power of any other element in .
2.3
Solid tori classi ed as Mc;d
The physical spacetimes we are concerned about in this paper are all solid tori, i.e. the
n = 1 case in the previous subsection. They have toroidal conformal boundaries, so the
Schottky group actions is relatively simple.
After these topological constructions, we can further classify them into the Mc;d family
according to their geometries. This family rst appeared in the discussion of classical
gravitational instantons which dominate the path integral in ref. [31], and is further explained
in refs. [14] and [32].
boundaries T 2 =
be isomorphic to Z
In this case, ( ) composes of the north and south poles of S2 . Since solid tori have
1
( )= , 1( ( )) must be a subgroup of 1(T 2), so 1( ( )) can only
Z, Z, or the trivial group. When
1( ( )) = Z
Z, ( ) has to
be a Riemann surface of genus 1, which cannot be isomorphic to an open subset of S2 .
When 1( ( )) is trivial, ( ) is a simplyconnected universal cover of T 2, so that
1
has
to be Z
Z. It is easily seen from (2.2) that if
= Z
Z, then although H3=(Z
Z) has
a toroidal boundary at y = 0, there is a cusp at y ! 1, whose subPlackian length scale
invalidates semiclassical treatments.
The only possibility is thus 1( ( )) = Z, where
can be either Z or Z
Zn. The
latter yields M to be a Znorbifold, indicating the existence of massive particles, which are
not allowed in pure gravity. To avoid undesirable geometries such as cusps and orbifolds
in the contributions to path integral [10, 14], we restrict our Schottky group to be
= Z,
{ 5 {
generated by the matrix
W =
0
q
where jqj < 1.
The boundary torus is thus obtained by quotiening the complex zplane without the
origin by Z. Rede ne z = e2 i!, so ! is de ned up to ! ! ! + 1, and W acts by ! ! ! +
ln q=2 i. Hence, the complex modulus of the torus is
ln q=2 i, de ned up to a PSL(2; Z)
Mobius transformation
(a + b)=(c + d), where integers a; b; c; d satisfy ad
bc = 1.
When constructing a solid torus from its boundary torus,
is de ned only up to
+ Z by a choice of solid lling, completely determined by the pair (c; d) of relatively
prime integers. This is because the ip of signs (a; b; c; d) ! ( a; b; c; d) does not
a ect q, and once (c; d) are given, (a; b) can be uniquely determined by ad
bc = 1 up
to a shift (a; b) ! (a; b) + t(c; d); t 2 Z which leaves q una ected. We call these solid
tori Mc;d's, and any Mc;d can be obtained from M0;1 via a modular transformation on
. Physically, M0;1 is the Euclidean thermal AdS3 and M1;0 is the traditional Euclidean
BTZ black hole obtained from Wick rotating the original metric in [8]. Excluding M0;1,
Mc;d's are collectively called the SL(2; Z) family of Euclidean black holes, to be discussed
in section 5.
3
The Euclidean thermal AdS3 has the topology of a solid torus M0;1, whose noncontractible
loop is parametrized by the Euclidean time. The constant time slice is thus a disk D2 with
a boundary S1, perpendicular to the noncontractible loop.
3.1
Bipartition into two disks
We bipartite the disk into upper and lower subregions A and B, both having the topology
of a disk. The solid torus is then turned into a sliced bagel as in
gure 3. Boundary of
each subregion contains an interval lying on the S1. In the following we will denote the
ratio between the length of one interval and the circumference of the boundary S1 to be a,
satisfying 0
a
1. Except for the symmetric case where a = 1=2 and the two subregions
are equivalent, generally SA 6= SB.
As introduced in section 2, one then glues each of n copies of subregion B separately
while gluing the n copies of subregion A together. The resultant 3manifold is an
nhandlebody, which is a lled genusn Riemann surface, shown in gure 3. (In the special
case of n = 1, the handlebody reduces to a solid torus.)
With a proper normalization, the entanglement entropy corresponding to subregion A
is then
takes the form
CFT [12],1
Contribution to the path integral around a classical saddle point for an nhandlebody
where k i+1Si(n) is the iloop free energy of boundary graviton excitations. At tree
level (i = 0), Ztree(nhandlebody) can be derived assuming the dual CFT is an extremal
Z(n) = exp kS0(n) + X k i+1Si(n) ;
"
#
i
Ztree(n) =
Y
1
Y
with the product running over primitive conjugacy classes of , q being the multiplier of
introduced in section 2, and k = l=16G.
In general the two products are hard to evaluate. However, in the lowtemperature
regime when thermal AdS3 dominates, the leading contribution to the in nite product over
m comes from m = 1. Furthermore, the product over
is dominated by a singleletter
contribution [15, 16],
q j
j
1
q1j2n. Combining these, we obtain
with q1 a function of n and a, having the form
Q
prim.
j
1
Ztree(n)
Y
prim.
j
1
q1j24k = j1
q1j48nk;
q1 =
sin2( a)
n2 sin2( a=n)
e 2 :
At oneloop (i = 1) level, the general expression for Zloop(nhandlebody) can be derived
from either the boundary extremal CFT [12, 13] or the bulk heat kernel method [10]. They
both depend on the Schottky parametrization of the boundary genus nRiemann surface.
The result is
Zloop(n) =
1
Y
Y
1This partition function is motivated by the Liouville action of a single free boson on a handlebody, and
is conjectured in [12] as a weight 12k modular form to avoid singularities of special functions.
{ 7 {
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
in the lowtemperature regime q1
into (3.1), we obtain
ST AdS (a)
The terms containing k come from treelevel, while others are oneloop contributions. The
entire expression approaches to zero very fast in the lowtemperature regime
! 1 for
any k. The dependence of the above result on a distinguishes itself from the original
de nition [
1, 2
] of TEE, which is a universal constant.
We note that a enters as the boundary condition on the constant time slice, and has nothing to do with the leading
arealaw term in usual expressions of entanglement entropies.
When subregion A is \nothing", i.e. a ! 0, a cot( a) ! 1, thus the TEE between
subregions A and B vanishes. When A is instead \everything", i.e. a ! 1, a cot( a) !
1, balanced by the smaller e 2
1 at low temperatures. We observe that apart from
the a ! 0 case, the TEE for thermal AdS3 is always negative. Another important case is
when a = 1=2 so that the two subregions are symmetric. In this case we have
ST AdS
a =
1
2
Now we take two noninteracting thermal AdS3's as the whole system, represented by two
disjoint solid tori M0;1. There are two noninteracting, nonentangled, identical CFTs living
on their asymptotic boundaries. One would naively expect the TEE between these two
solid tori to be zero, which is not really the case. To calculate the entanglement entropy
between these two solid tori, one can simply use
In the low temperatures, we can approximate q = e2 i = e 2
as a small number and
thus at leading order Z0;1( )
q 2k(1
After straightforward calculations we obtain
We have used the shorthand notation Z0;1( ) = Z0;1( ; ) to take into account both
holomorphic and antiholomorphic sectors. The partition function Z0;1(n ) comes from gluing
n copies of solid torus A, which is a new solid torus with modular parameter n .
Meanwhile, Z0;1( )n comes from gluing individually the n copies of solid torus B. We
can simply multiply the contributions from A and B together because they are disjoint.
Then we can plug these into the expression for the solid torus partition function, i.e. the
1handlebody result from (3.3) and (3.5),
Z0;1( ) = jqj2k Y
j
1
qmj 2:
1
m=2
ST AdS
This contains only the loop contribution, i.e. the semiclassical result is zero. For
comparison, we also calculate the canonical ensemble thermal entropy of a single thermal AdS3 at
temperature
1: STthAerdmSal = ln Z(1handlebody)
has the lowtemperature form
Z(1handlebody) 1 @Z(1handlebody) : It
STthAerdmSal
mal entropy of a single thermal AdS3 is the same as the TEE between two independent
thermal AdS3's.
This does not imply that there are nontrivial topological entanglement between the
two copies of thermal AdS3, but simply reveals the insu ciency of using entanglement
entropy as an entanglement measure at
nite temperatures. For example, consider two
general subsystems A and B with thermal density matrices A and B and combine them
into a separable system,
= A
B:
These two subregions are thus obviously nonentangled. But if one attempts to calculate
the entanglement entropy between A and B by tracing over B, one can still get an arbitrary
result depending on the details of A. If we choose
state, then the entanglement entropy will be zero. If instead we choose
A = j ih j where j i is some pure
the proper normalized identity matrix, then the entanglement entropy will be ln(dim(HA)).
So depending on the choice of A, one can obtain any value of the entanglement entropy
between these minimum and maximum values. This shortcoming is due to the fact that
now the entanglement entropy calculation involves undesired classical correlations in mixed
1
A = dim(HA)
1 as
To address this issue, we look at the topological mutual information between the two
states.
solid tori,
eternal BTZ black hole.
4
BTZ black hole
black hole.
4.1
BTZ geometry
I(A; B) = S(A) + S(B)
S(A [ B);
(3.14)
so that the thermal correlations can be canceled. Following similar replica trick
calculations, one easily obtain S(A [ B) = 2S(A) = 2S(B), thus the mutual information vanishes
and there exists no nontrivial topological entanglement between the two disjoint thermal
AdS3's. We will observe in the next section that this statement no longer holds true for an
We will explore in this section the topological entanglement in the bulk of Euclidean BTZ
It has been speculated for a long time that the 3d gravity is rather trivial because there is
no gravitational wave besides local uctuations. However in 1992, authors of [8] proposed
{ 9 {
a new type of AdSSchwarzschild black hole with Lorentzian metric
where the lapse and shift functions have the form NL2 =
4Gr2JL : G is the threedimensional Newton constant, l the curvature radius of AdS3, and
M , JL are the mass and angular momentum of the black hole, respectively. The outer and
8GML + rl22 + 16Gr22JL2 ; NL =
inner horizons are de ned by
r
2 = 4GMLl
2
1
s
1
J 2 !
L
ML2l2
:
Let tL = it and JL = iJ , and we do the Wick rotation to get
ds2 = N 2dt2 + N 2dr2 + r2(d + N dt)2;
with N 2 =
2
8GM + rl2
16Gr22J2 ; N (r) =
4rG2J . The horizons are now given by
r
2 = 4GM l2
1
r
1 +
J 2 !
M 2l2
:
The Euclidean BTZ black hole is locally isometric to the hyperbolic threespace H3 and is
3
globally described by H =
with
= Z. The topology is a solid torus, and one can make
it explicit by doing the following coordinate transformations [34]
x =
y =
z =
s r2
s r2
where
variable
They bring the metric (4.3) to the upper halfspace H
3 with z > 0. Further changing to
the spherical coordinates (x; y; z) = (R cos cos ; R sin cos ; R sin ), we nally arrive at
ds2 =
l
2
sin2
dR2
R2 + cos2 d 2 + d 2 :
To ensure that the above coordinate transformation is nonsingular (contains no conical
singularities) at the z axis r = r+, we must require periodicity in the arguments of the
trigonometric functions. That is, we must identify
1
2 l
( ; t)
1
2 l
( + ; t + );
= r+2jr rj2 ;
= r+2r+rl2 : We recombine the real pair ( ; ) into a single complex
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
x
y
clidean BTZ black hole is a solid torus. Horizon is the blue dashed line threading the central cord
of the solid torus. The Euclidean time runs in the meridian direction.
which is the complex modular parameter of the boundary torus. In terms of metric (4.6),
this corresponds to the global identi cations
(R; ; )
Re2 r+=l; +
2 jr j
l
;
:
(4.9)
A fundamental region for (4.6) is the lling of the slice between inner and outer
hemispheres centered at the origin having radii R = 1 and R = e2 r+=l respectively, with an
opening 2 jr j=l or 2
(if r
= 0) in azimuthal angle, as shown by
gure 4, and two
hemispheres are identi ed along the radial lines with a twist of angle 2 jr j=l or 2 (if
r
= 0). Hence, the segment on zaxis between two hemispheres corresponds to the outer
horizon, and is mapped to the central cord of solid torus at
=
=2 (the boundary torus
is at
= 0).
For convenience, in the rest of the paper, unless stated otherwise, we only focus on
nonrotating Euclidean BTZ black hole, so that
is pure imaginary and r
= 0.
4.2
TEE between two onesided black holes and mutual information
Following refs. [18{20], an eternal Lorentzian AdS black hole has two asymptotic regions
and can be viewed as two black holes connected through a nontransversable wormhole. It
is also suggested from the dual CFT perspective that the entanglement entropy between the
CFTs living on the two asymptotic boundaries is equal to the thermal entropy of one CFT.
Motivated by this, we are interested in calculating the TEE between the two singlesided
black holes in the bulk.
However, for the Euclidean BTZ black hole (4.3) and (4.6), the metrics only cover
the spacetime outside the horizon of one singlesided black hole. Everything inside the
horizon is hidden, so is another singlesided black hole. In order to make the computation
of TEE between two singlesided black holes possible, we take an alternative view of the
solid torus M1;0, as in
gure 5. In the left panel, we sketch the constant time slice of the
right singlesided black hole, called R. It is the constant
slice in metric (4.6) with an
annulus topology, whose inner boundary is identi ed with the horizon. In the right panel,
we glue the two constant time slices for black holes L and R along the horizon. Then there
The inner boundary in blue denotes the horizon. Time evolution of this slice corresponds to rotating
angle
around the inner blue boundary. Right: gluing the constant time slices of singlesided black
holes R (light grey) and L (dark grey) along the horizon (blue line) in the middle.
parts A~ and B~ in spacetime are respectively formed by rotating both spatial subregions A and B
by
. Right: the graphical representation of A, with a wedge missing in spacetime subregion A.
comes the most important step: we fold the annulus of black hole L along the horizon, so
that it coincides with the annulus of black hole R. To obtain the full spacetime geometry,
one rotates the constant time slice of L about the horizon counterclockwise by , while
rotating the constant time slice of R about the horizon clockwise by . Namely, the two
annuli meet twice: once at angle 0, the other at . The resultant manifold is a solid torus,
same as M1;0 introduced before. Hence one can view this solid torus either as one
singlesided black hole R with modular parameter
= i , or as two singlesided black holes L
and R, each contributing 0 = i =2.
It might concern some readers that the CFTs living on the asymptotic boundaries
of L and R in the Lorentzian picture are now glued together.
We note that this is a
feature of the Euclidean picture: due to the di erent direction of evolutions, we have
CFTL(t) =CFTR( t). At t = 0, these obviously coincide. Then at t =
=2, they give
CFTL(t =
=2) =CFTR(t =
=2). Using the fact that in the Euclidean picture we have
=2 =
= 2 +
=
=2, we arrive at CFTL(t =
=2) =CFTR(t =
=2), thus they
coincide again and the two CFTs are glued together. This is consistent with the fact that
in the Euclidean signature, there should only be one asymptotic region, as shown in [17].
Now we can calculate the TEE between the constant time slices of L and R, which we
denote as A and B. Importantly, since in general the result can be time dependent, we
specify the cut to be done at t = 0. As shown in the left panel of gure 6, each subregion
contributes 0 to the modular parameter of the solid torus. We sketch one copy of A in
the right panel.
~
~
B2
~
B1
~
A2
cutaway wedge runs along the longitude (noncontractible loop) of the solid torus, with its vertex
on the horizon. Right: graphical representation of tr nA. The disk is perpendicular to the horizon.
To nd S(A), we need to calculate the partition function of the 3manifold that
corresponds to tr nA. We rst enlarge the missing wedge in the right panel of gure 6 and shrink
the size of A~, B~. To add the second copy of A, one should glue A~1 to B~2, with B~2 glued
with A~2, as shown in
gure 7. Note that this di ers from the usual way of doing replica
tricks, where A~1 is always glued to A~2. This is again a result of the opposite directions of
time evolutions for L and R: the B spatial slice at t =
=2 should always be identi ed with
the A spatial slice at t =
=2. One can then follow this procedure and glue ncopies of A.
The resultant 3manifold is a solid torus with modular parameter 2n 0, since each copy
of A~ contributes 0 and the same goes for B~. Replica trick then gives
(4.10)
(4.11)
1, the
(4.12)
Partition function Z1;0( ) can be obtained from that of the thermal AdS3 by a modular
transformation
!
where the rst term comes from tree level and is identi ed with the BekensteinHawking
entropy. The above expression matches with the thermal entropy of one singlesided black
hole at oneloop,
SBthTerZmal(A) = ln Z1;0( )
= SBT Z (A):
(4.13)
Remarkably, this equation holds true regardless of Z1;0( )'s speci c form.
It might be confusing at rst that the BekensteinHawking entropy, usually viewed
as an arealaw term, appears in the calculation of topological entanglement entropy. To
~
~
B
~
C
surrounding the lower half circle corresponds to C~.
Right: one copy of A. The picture shows the disk perpendicular to the horizon. The thin layer
make it explicit that the results above are TEEs instead of the full entanglement
entropy, alternatively we can use Z1;0( ) derived from supersymmetric localization method
in ChernSimons theory on 3manifolds with boundaries [22]. Following the replica trick,
we nd exactly the same expression.2 Since ChernSimons theory is a topological quantum
eld theory, the resulting entanglement entropy is a TEE. The horizon area r+ should be
understood as a topological quantum number of the theory. In the calculation of TEE
between two disjoint thermal AdS3's, as stated in section 3, we have seen that a nonzero
TEE is not enough to guarantee true nontrivial entanglement between two subregions
because of the possible contribution from classical correlations. So we resort to the mutual
information I(A; B) between two singlesided black holes. We then need to nd S(A [ B).
Since in the Euclidean picture we are no longer at a pure state, it is not necessary that
S(A [ B) vanishes, although A [ B consists the entire system.
We start with bipartiting the system into A [ B and C at t = 0, as shown in gure 8.
C is a very small region whose area will nally be taken to zero.
The glued manifold is a solid torus with modular parameter 2n 0, exactly the same
form as that in gure 6. The contributions from C vanish because C is still contractible in
the glued manifold and we can safely take their area to be zero. Plugging (4.11) into the
replica trick formula (4.10), we again obtain
SBT Z (A [ B) = SBthTerZmal(A):
So indeed the TEE of A [ B does not vanish. Combining these, we nd that the mutual
information is the same as the BekensteinHawking entropy for a singlesided black hole:
I(A; B) = SBT Z (A) + SBT Z (B)
SBT Z (A [ B) = SBthTerZmal(A):
Note that, had we naively taken the full partition function of the eternal BTZ black
hole to be Z1;0( )2, namely, the two singlesided black holes are independent and
nonentangled so that their partition functions can be multiplied together, then SBT Z (A [ B)
would have been twice SBthTerZmal(A) and the mutual information would have vanished. So
the nonzeroness of mutual information indicates nontrivial entanglement between L and R.
2The supersymmetric localization method involves boundary fermions. We need to remove the
contribution from the boundary fermions to match with the partition function (4.11).
(4.14)
(4.15)
R lead to Z1;0(n ) after gluing. The gray area corresponds to subregion A, and the width of the
annulus B will be taken to zero.
There is still another surgery that can yield SBthTerZmal(A): (1) restrict to the right
singlesided black hole R as the full spacetime, which is a solid torus with modular parameter ,
obtained from rotating the constant time slice of it by 2 ; (2) thicken the horizon S1 to a
narrow annulus inside the spatial slice of the solid torus R; (3) calculate the TEE between
the thin solid torus generated by thickened horizon, denoted by B^, and the rest, denoted
by A^; (4) and
nally take the limit that thickness of solid torus B^ goes to zero.
The bipartition of the constant slice in this case is sketched in gure 9. In this
bipartition, the obtained TEE is between the exterior and the interior of horizon, rather than that
between two singlesided black holes. The glued manifold is again represented by Z1;0(n )
and the replica trick yields the BekensteinHawking entropy.
We have thus come to a conclusion that the followings are equal:
(a) TEE between the two singlesided black holes,
(b) TEE between the exterior and the interior of the horizon for a singlesided black hole,
(c) thermal entropy of one singlesided black hole,
(d) mutual information between the two singlesided black holes.
The equivalence of (a) and (c) supports the ER=EPR conjecture [18{20] in the Euclidean
AdS3 case. The equivalence between (b) and (c) shows explicitly from the bulk perspective
that one should view the thermal entropy of a black hole as entanglement entropy (see for
example ref. [21]). In general for a rotating BTZ black hole, although there is an inner
horizon at r = r , the zaxis still represents the outer horizon at r = r+ in the spherical
3
coordinates (4.5) for the upper H . Hence, the replica trick described earlier still applies
to a rotating BTZ black hole with modular parameter
=
+ i , where
is the angular
potential, the conjugate variable to angular momentum. Geometrically, we just need to
put r = jr j \inside" the inner edge of the constant time slice, so that it is not observable.3
4.3
The entanglingthermal relation
In ref. [
24
], the authors showed a relation (4.16) for a singlesided BTZ black hole between
the entanglement entropy of CFT on the conformal boundary and the BekensteinHawking
3A similar situation will be described in appendix A.
ends of the grey region. Middle: the front view of tr A for the \ring"
con guration. Right: the side view of tr 4A inside the \ring" of the rst tr A.
entropy:
l!0
lim(SA(L
l)
SA(l)) = Sthermal;
(4.16)
where SA(L
l) is the entanglement entropy of a subregion A on the boundary 1+1d CFT
with an interval length (L
l), and Sthermal is the thermal entropy in the bulk. In this
section, we propose another similar but di erent EntanglingThermal relation.
We rst consider the bipartition of the constant time slice as in gure 10 for a
singlesided black hole. We put the separation between two subregions away from the horizon,
so that region B generates the white contractible region in the left panel. The right panel
is equivalent to the left one, and will be convenient for visualization of the gluing. We
will call the glued manifold as the \ring", because after time evolution, region B = A (the
complement) will glue to itself and form a ring around the solid torus, as shown in the
middle panel of gure 11, where the small white part corresponds to the unglued part in
the left panel. Hence, a single copy is the middle panel: away from the ring, the open
wedge running around the longitude is the same as that in the left panel of gure 7.
Naively it seems that one is unable to glue n copies of the above geometry, since the
ring blocks a portion of the wedge's opening. However, there do exist a unique embedding
from n copies to R3 up to homotopy equivalence, as shown in the right panel of gure 11:
one rst stretches the grey region in the left panel to the blue area in right panel, and glue
a second light grey copy so that its t = 0 edge are glued to the t =
edge of the blue copy;
now one repeats this process for green and yellow regions and so on, still preserving the
replica symmetry. Notice that rings from gray, green and yellow copies (color online) are
not in this piece of paper, but on parallel planes above or below. Then one puts rings from
each copy side by side on the boundary torus, which requires each ring to be in nitesimally
thin since n is arbitrarily large. The resultant manifold is again a solid torus of modular
parameter n . So the replica trick calculation follows the previous equation (4.10) and gives
lim
Area(A)!0
S(A) = SBthTerZmal:
(4.17)
For completeness, we note that gure 11 has another limiting case, where the width
of the ring covers almost the entire longitudinal direction of the solid torus, and its depth
occupies a considerable portion of the radial direction, as shown in gure 12. Now in order
to put rings side by side upon gluing n copies, we need to stretch the noncontractible
direction for n times to accommodate them, so that the resultant manifold is approximately
a solid torus with modular parameter =n. Now plug Z1;0( =n) into (4.10):
lim
lim
Area(A)!0
S(A) = 2
n=1
4
+ 1 e 4 = ;
which vanishes at high temperature. Note that here is no kdependence, meaning we can
observe the oneloop e ect directly.
Now we consider the complementary bipartition to gure 11, as shown in gure 13,
where the grey region is generated by B in
gure 10. The gluing here is simple: since
the unglued cut in the grey region A~ is parallel to the longitude, n copies should be
arranged around a virtual axis tangent to the annulus. The resultant manifold is a vertical
One can calculate the corresponding TEE following a parallel procedure in the
calculation of thermal AdS3 in section 3. The partition function of the glued manifold is
Z(n) =
Y
1
Y
j
q j
m 24k
1
Y
Y
where the rst and second factors come from tree level and oneloop, respectively. The
products are over primitive conjugacy classes of
. In the hightemperature regime,
2
(4.19)
(4.20)
this expression can be simpli ed by the singleletter word approximation
j
1
q10j2n, so that
Q
prim.
j
1
Here q10 can be obtained from q1 in (3.5) using a modular transformation,
Z(n; q10)
j
1
j
1
q10j48nk
q102j2n :
q10(n; a) =
sinh2( a= )
n2 sinh2( a=n )
e 2 = :
d
dn
Z(n; q10(n))
Z(1; q10(1))n
n=1
:
The replica trick then gives
This is explicitly written as S(A) = 96k
a
q j
(4.21)
(4.22)
(4.23)
a
2 e 2 = + 8(12k
1)
2 e 4 = + O(e 6 = ):
(4.24)
We now take the limit a ! 0 because this corresponds to the limit where the grey region
in gure 13 goes to zero, so that:
lim
which vanishes at high temperature. The in nitesimally negative value is a quirk due to
approximation on q 's.
Combining equations (4.17) and (4.25), one obtains the EntanglingThermal relation:
lim
Area(A)!0
[S(A)
We give this relation a di erent name from the twodimensional thermal entropy relation
in the dual CFT calculation (4.16) because this is not merely a generalization of it in one
higher dimension. The thermal entropy relation (4.16) relates the entanglement entropy
on the dual CFT with the thermal entropy of black hole in the bulk, while the
entanglingthermal relation connects the topological entanglement entropy and thermal entropy both
in the bulk gravitational theory. Additionally, the explanation for thermal entropy relation
relies on the geometrical detail (minimal surfaces) in the bulk [
24
], while the
entanglingthermal relation is of topological origin. In the rst bipartition in
gure 11, subregion A
sees the noncontractible loop and the nontrivial ux threading through the hole inside the
annulus. In the second bipartition in gure 13, subregion A does not completely surround
the noncontractible circle, i.e. the horizon. The di erence between them thus characterizes
the noncontractible loop.
Finally we remark that there are several cases in which gluing procedures are not
available. The nogluing criterion is that, as long as the boundary of a subregion is contractible
and not anchored on the boundary S1, the spatial slice is not nglueable. Also, a single
copy in which glued region B completely surrounds region except for the inner edge is not
nglueable.
5
Summation over geometries
The partition functions of thermal AdS3, Z0;1( ), and BTZ black hole, Z1;0( ), are not
modularinvariant by themselves. To obtain the full modularinvariant partition function,
one needs to sum over the pair of parameters (c; d) for Zc;d. This can alternatively be
written as the summation over modular transformations of Z0;1 as follows:
Z( ) =
Zc;d( ) =
X
1 SL(2;Z)
X
1 SL(2;Z)
Z0;1
a + b
c + d
:
1 r
and Schottky parametrization are invariant under
make the full partition function invariant under both T :
1, and the summation over coset is to
!
+ 1 and S :
!
1= .
Note that in the previous sections we have used Zc;d( ) = Zc;d( ; ) as the shorthand
for the product of holomorphic and antiholomorphic pieces, whereas in this section we
return to the notation that Zc;d( ) describes the holomorphic part of the partition function
only. The antiholomorphic part can easily be found as Z( ) and Z( ; ) = Z( )Z( ).
Modularinvariant partition function of the form (5.1) is unique for the most negative
cosmological constant (k = 1) [11, 35] and was investigated in more general situations
(k > 1) in [14]. An important theorem due to [35] is that the moduli space of Riemann
surfaces of genus one is itself a Riemann surface of genus zero, parametrized by the
jfunction. Consequently, any modularinvariant function can be written as a function of it.
The J function is de ned as
J ( )
1728g2( )3
g2( )3
27g3( )2
744
(5.1)
(5.2)
where q = e2 i as usual, and g2( )
morphic Eisenstein series of weight 2k; k
60G4( ) and g3( )
140G6( ), where G2k are
holo2, de ned as G2k
P(m;n)6=(0;0)(m + n ) 2k:
Since the pole in the full partition function Z(q) at q = 0 is of order k (due to the
holomorphic treelevel contribution of thermal AdS3, q k), it must be a polynomial in J
of degree k,
k
j=0
Z(q) =
X aiJ i =
X c(k; n)qn:
n
For k = 1 we simply have Z(q) = J (q). The coe cients of J (q) in front of qn was known
to be intimately related to the dimensions of irreducible representations of the monster
group M, the largest sporadic group. It has 246 320 5
9 7
1053 group elements and 194 conjugacy classes. Dimensions of
the irreducible representations of the monster group can be found in the rst column of its
character table [36]: 1, 196883, 21296876, 842609326, 18538750076, 19360062527 : : : .
After John McKay's observation 196884 = 1 + 196883, Thompson further noticed [37]:
(5.3)
This phenomenon is dubbed \monstrous moonshine" by Conway and Norton [38], later
proved by Borcherds [39].
Ref. [11] conjectures that for cosmological constant k
l=16G 2 Z, quantum 3d
Euclidean pure gravity including BTZ black holes can be completely described by a
rational CFT (RCFT) called extremal selfdual CFT (ECFT) with central charge (cL; cR) =
(24k; 24k), which is factorized into a holomorphic and an antiholomorphic pieces. An
ECFT is a CFT whose lowest dimension of primary eld is k + 1, and it has a sparsest
possible spectrum consistent with modular invariance, presenting a
nite mass gap. The
only known example is the k = 1 one with a monster symmetry, constructed by
FrenkelLepowskyMeurman (FLM) [40] to have partition function as J (q), but its uniqueness has
not been proved. The existence of ECFTs with k > 1 is conjectured to be true [11] and is
still an active open question [41, 42].
In this section we will mainly focus on the k = 1 case.
5.1
TEE for the full partition function
The modularinvariant partition function is still de ned on a solid torus. We will again
consider the bipartition that separate the two singlesided black holes, similar to the story
in section 4.2. It is justi ed in appendix A that one can still cut SL(2; Z) family of BTZ
black holes along their outer horizons, which lie in the core of the solid torus. So one just
needs to plug the partition function J (q) into the replica trick formula.
At low temperatures, q = e 2
is small, so that the full partition function will be
dominated by the q 1 term with almost trivial thermal entropy and TEE, trivial in the
sense that there are no treelevel contributions. At high temperatures, richer physics is
allowed. Below we calculate the TEE of the full partition function in this regime.
Generally, the coe cient in front of qn in the partition function Z(q) for any k can be
written as
193
X
i=0
c(k; n) =
mi( k; n)di;
where each di is the dimension of the corresponding irreducible representations Mi of M,
and mi( k; n) is the multiplicity of the irreducible representation Mi in the decomposition
similar to (5.4) so c(k; n) is guaranteed to be a nonnegative integer. At large n, mi( k; n)
has the following asymptotic form [43],
mi( k; n)
p
dijkj1=4
2jMjjnj3=4
e
4 pjknj:
Now we restrict to the k = 1 case and let n to be a variable. After taking care of the
antiholomorphic part, the replica trick (4.10) gives the following TEE
Sfull(A) = Sftuhlelrmal = 2 ln J (q)
:
Note that this is again the same as the expression for calculation of thermal entropy in the
canonical ensemble. (Using
= l=r+ = 1=p
M = 1=pn, n is viewed as a function of
so the second term in (5.7) is nonzero.) The computation of SA[B for the entire SL(2; Z)
family of black holes is also similar to that of M1;0 calculated in section 4.2. The result
is again equal to the thermal entropy, based on the fact that the SL(2; Z) family of black
holes are all solid tori with horizons living in the core. This implies that the system is again
in a mixed state due to Euclideanization, as expected in [44, 45]. The mutual information
I(A; B) is also the thermal entropy, parallel to the discussion in section 4.
In the hightemperature expansion, we only take the qn term Jn(q) from the summation
in J (q) to calculate TEE because this desired term has a coe cient exponentially larger
than those at lower temperatures:4
Jn(q) =
X
Mathematically the two copies of di in di2 are both the dimension of irreducible module
Mi of the monster group, which will be explained in detail later in section 5.2.
But
physically they have di erent origins: one is the contribution from a single Mi as shown in
equation (5.5), while the other is probability amplitude for Mi to appear in the summation
as in equation (5.6). Namely, there is a correspondence between the partition function J (q)
and a pure state in the bulk, which is a superposition of all di erent Mi's:
In analogy to topological phases, the state is a maximallyentangled state of 194 types of
\anyons" labelled by the irreducible representations of the Monster group M. The di that
4We will take into account all terms of J(q) in appendix B.
j i =
193
X
i=0
di
pjMj
ji; i i:
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
i
Wilson line corresponding to the quasiparticleantiquasiparticle pair i, i intersects with horizon
both on the constant time slice and in the 3d bulk.
appears explicitly in (5.9) corresponds to that in (5.6), whereas ji; i i means a
quasiparticleantiquasiparticle pair labeled by Mi and contributes another di, which correspond to the
one in (5.5). In ref. [27], the authors proposed from abstract category theory, that the
ER=EPR realization in the context of TQFT should be exactly of the form (5.9). We will
show later that this speci c maximallyentangled superposition is the bulk TQFT version
of the thermo eld double state on the dual CFTs.
Applying to equation (5.8) the identity for nite groups: Pi di2 = jMj, we arrive at
Jn(q) = p2n3=4
e
4 pn
qn = p
1
2
3=2e2 = :
8
Sfull(A) =
+ 3 ln
ln 2
3:
Plugging it into (5.7) and taking into account the antiholomorphic part, we again recover
the BekensteinHawking entropy:
The rst three terms agree with Witten's asymptotic formula for BekensteinHawking
entropy [11], and provides an additional term
3. Remarkably, the \anyons" become
invisible in TEE after the summation over i. This is exactly due to the appearance of the
maximallyentangled superposition in equation (5.9). Had we taken another state where
only one single Mj appears with probability amplitude 1 and all the others appear with
amplitude 0, the corresponding contribution would have been proportional to ln dj =pjMj
instead of 0. The latter matches with the entanglement entropy calculations in refs. [46{48]
for an excited state labeled by j in a rational CFT.5
In our case, the creation of the quasiparticleantiquasiparticle pair i and i can be
represented by a Wilson line, as shown in
gure 14. The Wilson line intersects the
noncontractible loop of the solid torus, i.e. the horizon, which is the reason why it can be
detected by a cut along the horizon.
To make full understanding of the \anyon" picture, we rewrite state (5.9) as
j i =
1
pJ (q) i=0
193
X e 2 Ei ji; i i;
5This disappearance of \anyons" in the TEE for a maximallyentangled superposition is also expected
in the context of topological phases, see equation (40) of ref. [3], where one takes j jj there to be dj=D.
(5.10)
(5.11)
(5.12)
where the energy level corresponding to the \anyon" pair i; i is described by the quantum
dimension of Mi:
Denoting ji; i i
matrix
Ei =
1
ln
d
2
jMj
i Jn(q) :
A =
X e
i
Ei jiihij;
jiiji i, one can trace over all the ji i's and obtain the reduced density
which is just the thermal density matrix for \anyons", and di erent types of anyons i
form an ensemble. Using the expression for energy levels (5.13), the entanglement entropy
between the \anyon" pair can be easily calculated as
S (A) = Sthermal(A) = Sfull(A);
where we have added the antiholomorphic contribution. Thus the state (5.12) has the
similar property as the thermo eld double state does in that the entanglement entropy
between the quasiparticleantiquasiparticle pair is equal to the thermal entropy of one
quasiparticle. We call this state in the 3d bulk as the Moonshine double state, in which the
pair of \anyons" are separated by the horizon, just like the two singlesided black holes L
and R are separated by it.
Unfortunately it has a shortcoming: as a pure state, the Moonshine double state above
cannot reproduce the result of nonzero S(A [ B) (4.14). To account for this, one could
modify the nal total quantum state as
= j ~ ih ~ j
th;
where the modi ed moonshine double state now reads j ~ i = p4J(q)
1
Pi1=930 e 2 E~i ji; i i with
Ei =
1 ln h jdMi2j Jn(q)1=2i. These energy levels lead to the partition function Z(q) =
J (q)1=2. When one bipartites the system into two singlesided black holes A and B, one can
see from straightforward computation that j ~ i will contribute half of BekensteinHawking
entropy. The newly introduced th is purely thermal and exhibits no nonlocal correlations
between A and B, so that its von Neumann entropy is extensive and scales with volume.
When one bipartites the system into the two singlesided black holes A and B, it will give
half of the BekensteinHawking entropy. Combining the contribution from j ~ i, we recover
S ~ (A) = Sthermal(A), the BekensteinHawking entropy. When considering S(A [ B), the
modi ed moonshine double state contributes nothing as a pure state, while the result for
th is simply Sthermal(A), matching with the calculations in (4.14).
Another caveat is that since ln J is approximately the BekensteinHawking entropy,
the leading term in Ei scales with
2
n. So in order to have a genuine quantum
theory, our theory has to have a UV cuto scale at a certain n. Furthermore, apart from the
asymptotic expression (5.6) which gives rise to the treelevel BekensteinHawking entropy,
there is the remainder formula [49] for coe cients of qn in the whole partition function
(5.13)
(5.14)
(5.15)
(5.16)
j(1; p)j + 62p2e 2 pnnp=2;
the remainder formula reads
c(k;n) = p2(kn)3=4
ke4 pkn "
p 1
1+ X ( 1)m(1;m)
(kn)p=2 +
p2n3=4
e4 pn S(k;n)
where p(x) is the integer partition of x 2 Z+, and
m=1 (8
p 1
1+ X ( 1)m(1;m)
(kn)p=2 +
p2n3=4
e4 pn S(k;n) 5
;
ar(k)
p(r + k)
p(r + k
1);
(5.18)
0 <
p2n3=4
e4 pn S(k; n)
To check this claim, one could restrict to the k = 1 monstrous case and plug this expression
into (5.7). Alternatively one may
x n and view the c(k; n) as the number of possible
microstates at
xed energy, i.e. in the microcanonical ensemble. One then performs a
unilateral forward Laplace transform to return to canonical ensemble and then plug it
into (5.7). Computations in both methods are in general complicated, and we do not
pursue it here.
We provide another perspective towards the loop contribution in appendix B by
plugging in the whole J function instead of only one large n term. We observe that the loop
correction is negative, consistent with both the thermal AdS3 case in section 3 and the
BTZ case in section 4.
5.2
di as quantum dimensions
In this section we provide more mathematical details and show that di equals the quantum
dimension of the irreducible module Mi of jMj. An ECFT at k = 1 is a special vertex
operator algebra (VOA) V \ whose automorphism group is the Monster group M. This VOA,
also known as the moonshine module [40], is an in nitedimensional graded representation
of M with an explicit grading:
where every Vn\ is an Mmodule, called a homogeneous subspace. It can be further
decomposed into
V \ =
1
M
n= 1
V \;
n
193
M
i=0
V \
n '
M
i
mi( 1;n);
with Mi labeling the irreducible Mmodules, and mi( 1; n) is the multiplicity of Mi. This
is the same multiplicity that appears in (5.5). (For ECFTs with general k, we have a
tower of moonshine modules [43] V ( k) = L1
n= k Vn
( k), where Vn
( k)'s are all irreducible
(5.17)
!3
(5.19)
(5.20)
sectors:
where M
and M
the Mmodules Mi in Vn
( k), so that Vn
( k)
' Li1=930 Mi
mi( k;n):)
Mmodules. For each summand, one can similarly de ne mi( k; n) as the multiplicity of
Since we restrict to the holomorphic part of Z( ; ) in this section, the entire dual
CFT contains the ECFT above as a holomorphic piece. Furthermore, it is diagonal, i.e.
its Hilbert space is a graded sum of tensor products of holomorphic and antiholomorphic
H =
M
2C
M
M ;
as V M
Mmodules, for the 194 V Msubmodules ViM in V \ with V M = V1M, where Mi
denotes an irreducible module for M with character di. This V M is a subVOA of V \ of
CFT type [54], and is called the monster orbifold, because it is obtained from orbifolding
V \ by its automorphism group M [83], in the same sense as orbifolding the Leech lattice
VOA by Z=2Z in the FLM construction.
The standard de nition of the quantum dimension of a VOAmodule N with respect
to a general VOA V is [52, 53]
qdimV N = lim
q!1 chqV
chqN
:
The quantum dimensions of submodules of orbifold VOA V G obtained from orbifolding
V by a subgroup G
Aut(V ) only recently found their applications in quantum Galois
theory [52, 53]. In our case, the quantum dimensions of all ViM's with respect to V M were
are indecomposable representations of right and left Virasoro algebras.
HJEP12(07)6
Since Virasoro action is built into the VOA axioms [50], these are also modules of the right
and left monstrous VOAs, so V \ admits induced representations from representations of the
Virasoro algebra [51]. Obviously there are in nite number of Virasoro primaries, and V \
is not an RCFT in this sense. However, V \ is a typical example of a holomorphic/selfdual
VOA, i.e. there is only one single irreducible V \module which is itself. Knowing that there
is only one VOAprimary, one can reorganize Virasoro elds in M
and M
representations of V \, by introducing the graded dimension of the V \module N , de ned as
into irreducible
chqN
trN qL0 =
X dim Nnqn;
1
n=0
where L0 is the usual Virasoro generator and Nn's are homogeneous subspaces of N
labelled by eigenvalues of L0. (Note that we have omitted the overall prefactor q c=24 often
appeared in literature.) The above procedure is similar to regourpong in nite number of
Virasoro primaries in WZW models into nite number of KacMoody primaries.
To explain the di appearing in (5.8), it is natural to consider quantum dimensions
associated to V M consisted of xed points of the action by M on V \. By theorem 6.1
in [52, 53], we have the following decomposition of V \
V \
194
M ViM
i=1
Mi
(5.21)
(5.22)
(5.23)
(5.24)
rst calculated to be qdimV M ViM = di in [43], using the asymptotic formula for multiplicities
of Mmodules Mi in Fourier coe cients of jinvariant, bypassing the knowledge of V M's
rationality, which is still only conjectured to be true.
The remaining question is to de ne in parallel a quantum dimension for the Mmodules
in the above pair ViM; Mi . The de nition (5.24) does not directly apply to an Mmodule,
but one can extend the de nition using the ngraded dimension of Mmodules Mi's. We
de ne chqMi as6
chqMi
X j
i( ):
n= 1 Vn\ ( )qn is the monstrous McKayThompson series for each
as
well as the unique Hauptmodul for a genus0 subgroup
of SL(2; R) for each
belongs to an index set with order 171, deduced from the 194 conjugacy classes of M. The
di erence 194
171 = 23 can be understood from the onetoone correspondence between
conjugacy classes and irreducible representations of M: most of the 194 irreducible
representations have distinct dimensions, except for 23 coincidences. 's are only sensitive to the
dimensions of the corresponding irreducible representations. i( ) is complex conjugation
of the character of the irreducible representation Mi of the 171 \conjugacy classes"
At large n, summation in chqMi is dominated by the rst Hauptmodul for the identity
.
7
element of M, which is exactly the Klein's invariant j(q), so that
q!1
lim chqMi
In other words, one can view chqMi as a function chqMi(g) on group M, and when de ning
the quantum dimension in (5.25), we take the value when its argument is the identity
With this, we can de ne the quantum dimension of Mmodules Mi in (5.20) relative
qdimV \ Mi
limq!1 cchhqqMV\i = lim
n!1
dim(Mi)n
dim Vn
\ :
Here chqV \ = J (q) by applying (5.24) to V \, which is a V \module of itself. Combining
the discussions above, the quantum dimension is just
qdimV \ Mi = di:
The di's that appeared explicitly in (5.8) of the TEE calculation are quantum
dimensions of Mi, while those in (5.6) are quantum dimensions of ViM. They coincide numerically.
As we mentioned before, the rationality of V M is widely conjectured to be true,8 and by a
theorem of Huang [55], the module category of any rational, C2co nite VOA is modular,
i.e. it is a modular tensor category with a nondegenerate Smatrix. If one believes in
6We are deeply grateful to Richard E. Borcherds for suggesting this alternative formula. It is similar to
the generating function of multiplicity mi( 1; n) in section 8.6 of [43], but without normalization by 1=jMj.
7In literature this is often denoted by tr( jMi) or tr(Mi( )) or chMi ( ) as well.
8Unfortunately, the conjecture has only been proved only when the subgroup of the automorphism group
is solvable [84, 85], which is not our case.
(5.25)
(5.26)
(5.27)
(5.28)
the rationality conjecture, then qdimV M ViM's have a wellde ned interpretation in terms of
modular Smatrices of the orbifold CFT V M:
di = Si0=S00:
(5.29)
Note that these 194 \anyons" are the pure charge exitations in the corresponding
topological ordered system described by the modular tensor category associated with the orbifold
VOA V M.
6
Discussion and outlook
In the hightemperature regime, the full modularinvariant partition function (5.1) is
dominated by the black hole solution Z1;0( ), while in the lowtemperature regime, it is
dominated by Z0;1( ), the thermal AdS3 solution [14, 32]. It is widely believed that there exists
a HawkingPage [56, 57] transition at the critical temperature
1, or r+
l. However,
there is no consensus on whether this transition really exists [14, 58, 59], or if it exists,
whether it is a rstorder or a continuous phase transition [60{65], or something else that
is more subtle. In this section we o er a clue from the TEE perspective.
We compare the a = 1 (de ned in
gure 1) case in (3.7) of thermal AdS3 and the
gure 9 case of a singlesided black hole, for their subregion A's both cover the whole
space. One then observes that even at the tree level, TEE of BTZ and thermal AdS3 have
di erent signs. A natural guess would thus be that, if the transition exists, it should be
topological and happen at where the TEE changes sign.
Our de nition of topological entanglement entropy is the constant subleading term in
the expression for entanglement entropy, which is in general di erent from the tripartite
information as used in [
1
]. For topological phases in condensed matter physics, these two
formulations di er by a factor of two and are both negative. For gravitational theories in
the bulk, our topological entanglement entropies can be either positive (as in BTZ black
hole case) or negative (as in the thermal AdS3 case). To calculate the tripartite
information, one can use the surgery method presented in this paper and nd its timedependence,
which at late times is negative of the BekensteinHawking entropy [76]. This matches with
the results in CFTs with gravitational duals, it is expected that the tripartite information
should be negative [77] and that for thermo eld double state, it equals the negative of the
BekensteinHawking entropy [78].
Quantum dimensions also appears in the calculation of leftright entanglement in
RCFT [86]. One might perform similar computations in the orbifold VOA V M appeared
in section 5.2, by using the Ishibashi boundary CFT states that were constructed in [87]
for open bosonic strings ending on Dbranes.
Given the \anyonic" interpretation in section 5, one natural question to ask is that,
to what extent 3d pure quantum gravity can be described as a theory of topological order.
Naively one would expect the corresponding topological order to be the 3d DijkgraafWitten
theory of the monster group M, which gives rise to the same modular tensor category as
the one given by orbifold CFT V M as explained in section 5.2. On the other hand, it is
also natural to expect the corresponding topological order to be the one which is e ectively
described by the double SL(2; C) ChernSimons theory. It would be highly nontrivial to
nd a mechanism that reconciles these two theories.
Another remark is that we have speci ed the bipartitions to be done at t = 0 in
section 4, while in general the result can be timedependent. In the latter case one can still
use the surgery method proposed in this paper to nd the TEE or Renyi entropies, which
can serve as an indicator of scrambling [79].
A
nal mathematically motivated direction is the following. Vaughn Jones considered
how one von Neumann algebra can be embedded in another and developed subfactor
theory [80]. In general, the Jones program is about how to embed one in nite object into
another, reminiscent of eld extensions in abstract algebra, and quantum dimension is
dened exactly in this spirit. It would be interesting to see how subfactor theory in general
can help connect topological phases and pure quantum gravity [81].
Acknowledgments
We are deeply grateful to Richard E. Borcherds for teaching us quantum dimensions of
Mmodules over V \. We appreciate Song He and Mudassir Moosa's suggestions on the
manuscript, and thank Ori J. Ganor and YongShi Wu for remarks on HawkingPage
transition. We thank Norihiro Iizuka and Seiji Terashima for explaining their work, Andreas
W. W. Ludwig and Zhenghan Wang for extremely helpful comments on the moonshine
module. We thank Diptarka Das, Shouvik Datta and Sridip Pal for explaining their work
and pointing out ref. [87] to us. ZhuXi thanks Herman Verlinde for comments on the sign
of BTZ TEE, and ZhengCheng Gu, Muxin Han, Jiandong Zhang for helpful discussions.
We also appreciate the workshop \Mathematics of Topological Phases of Matter" at SCGP,
where part of the work was completed.
A
Bipartition for the full partition function
In this appendix we justify that inputting jinvariant into the replica trick formula is a
legal operation. We need to make sure that the horizon in the SL(2; Z) family of Euclidean
BTZ black holes is still at the central cord of their solid tori, so that we can cut along
it. Although jfunction contains contribution from thermal AdS3 which contains no black
holes, we will see later that this con guration contributes nothing at a high enough nite
temperature. For convenience we set l = 1.
To see how Euclidean BTZ Schwarzschild coordinates transform under the SL(2; Z)
action on , we need an intermediate FRW metric for the unexcited (before being quotiented
by ) AdS3 with cylindrical topology, similar to the one mainly used in [14]:
ds2 = cosh2
d 2 + d 2
=
sinh2 (du
du)2 + cosh2 (du + du)2 + d 2
= sinh2
d 2 + cosh2
dt02 + d 2;
(A.1)
where 2u
i
t and 2u
indicates the radial direction.
i
t parametrize the domain of discontinuity
, and
To obtain a Euclidean BTZ from this, we demand 2u
1= =
+ i the modular parameter for BTZ black hole, and
the modular parameter of thermal
AdS3. The identi cation in the BTZ spatial direction is automatic due to the periodicity
in the H
3 metric; Im 0 represents the time identi cation because it is the length of the
time cycle, and Re 0 o ers a spatial twist upon that identi cation, inducing an angular
momentum by \tilting" the meridian.9 De ne the Schwarzschild radial coordinate r:
sinh2
=
r
2
(Im(1= 0))2
j 0j2
;
we obtain the Euclidean BTZ black hole in Schwarzschild coordinates for r
Im(1= 0):
HJEP12(07)6
ds2 = N 2dt2 + N (r) 2dr2 + r2[d + N (r)dt]2;
where
N 2(r)
(Re(1= 0))(Im(1= 0))=r2.
[r2
(Im(1= 0))2][r2 + (Re(1= 0))2]=r2,
and
N (r)
=
Now the outer horizon is at r+
= Im(1= 0).
When an
SL(2; Z) transformation is applied 0 !
00 = 1=(c 0 + d) = =(d
c), r becomes
r002 !
(c Re 0 + d)2 sinh2
+ (c Im 0)2 cosh2
jc 0 + dj4
:
It is enough to just think of 1=(c 0 + d) because there are only three independent
parameters in (a; b; c; d) due to the constraint ad
bc = 1. One has the freedom to choose
a = 0, which
xes
bc = 1, consequently (a 0 + b)=(c 0 + d) =
1=(c2 0 + cd). Rede ne
c2 = c and
cd = d, then we arrive at 1=(c 0 + d). The minus sign in both c and d is not
a problem, because (c; d) is equivalent to ( c; d).
Since sinh2
= r
2 2
1, we have Im 00 =
c =(c2 2 + d2), Re 00 = d=(c2 2 + d2),
implying a rotating black hole. Now we need to see if the new r00 is still at the horizon
in the Schwarzschild coordinates associated to 0, and it su ces to check that r+00 = Im 00.
This is indeed true. Hence no matter what (c; d) we change into, as long as
and 00 are
SL(2; Z)equivalent, r00 = r00
+
coordinate system for the upper half H3, so our cut is still valid.
Im
will be mapped to a segment on zaxis of spherical
B
TEE from the whole J (q) function
Now we plug the entire J function as the canonical partition function into (4.10). We start
from the de nition of jinvariant j( ) = J ( )
744
E3( )= ( ), where
4
=
24( ) is
the normalized modular discriminant. To nd the derivative of J ( ), we make use of the
Jacobi theta function #(f )
f 0
m
12 E2( )f [73], where Ej ( ) is Eisenstein series of weight
j and m is the weight of an arbitrary modular form f . Substituting j( ) for f , we obtain
(A.2)
(A.3)
(A.4)
d
j( ) = #(j( )) + E2( )j( ):
(B.1)
9Situation is almost identical in the thermal AdS3 (A.1), where Im speci es the time identi cation,
upon which Re indicates a spatial twist.
d
j( ) =
2 i E6( )
E4( )
j( ):
Sfull( ) = ln J ( ) + 2
j( ) E6( )
J ( ) E4( )
:
Plugging into the replica trick equation (4.10) we obtain for the holomorphic part
HJEP12(07)6
Einstein series Gs( )
positive integer N [74]:
To calculate the ration E6=E4, we use the asymptotic formula for the holomorphic
2 (s)Es( ), assuming 0 < j arg j <
and Re(s) >
N + 1 for any
(B.2)
(B.3)
(B.4)
J (i ),
(B.5)
We have made use of the fact that the weight of j( ) is three times the weight of E4( )
by de nition. One easily observes from the right hand side of above equation that the
weight of j( ) becomes 12 + 2 = 14 after di erentiation. Since the vector space of SL(2; Z)
plugging in the rst several terms of the j( ) function and we nally arrive at10
modular forms of weight 14 is spanned by E2( )E6( ) and has complex dimension 1, we
4
must have dd j( ) / EE46(( )) j( ), up to a constant prefactor. This factor can be found by
[2] M. Levin and X.G. Wen, Detecting topological order in a ground state wave function, Phys.
10It is also a consequence of applying Ramanujan's identities on E2, E4 and E6 [16].
1)
s)(1 + e is) (s) + 2 sin(s )
(1 + cos(s )) (s)
X
k=1; k odd
2 sin(s )
s
k
(s + k) ( k) k + O(j jN );
j j
1:
For both s = 4; 6, the second term vanishes at high temperatures j j ! 0, and sin(s ) in
the summation over k vanishes as well. Switching to the real variable
=
i , we have
G4(i )
2 4 (4) and G6(i )
where we have taken into account the antiholomorphic part.
Now we see that if we consider the entire SL(2; Z) family of black holes as well as
thermal AdS3 (the later contributes little at small ), the oneloop contribution to TEE is
negative, agreeing with our previous calculations.
Open Access.
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