Topological entanglement entropy in Euclidean AdS3 via surgery

Journal of High Energy Physics, Dec 2017

We calculate the topological entanglement entropy (TEE) in Euclidean asymptotic AdS3 spacetime using surgery. The treatment is intrinsically three-dimensional. In the BTZ black hole background, several different bipartitions are applied. For the bipartition along the horizon between two single-sided black holes, TEE is exactly the Bekenstein-Hawking entropy, which supports the ER=EPR conjecture in the Euclidean case. For other bipartitions, we derive an Entangling-Thermal relation for each single-sided black hole, which is of topological origin. After summing over genus-one classical geometries, we compute TEE in the high-temperature regime. In the case where k = 1, we find that TEE is the same as that for the Moonshine double state, given by the maximally-entangled 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 thermofield double state in the Euclidean spacetime. Comparing the TEEs between thermal AdS3 and BTZ solutions, we discuss the implication of TEE on the Hawking-Page transition in 3d.

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Topological entanglement entropy in Euclidean AdS3 via surgery

JHE AdS3 Zhu-Xi Luo 0 1 3 Hao-Yu 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 84112-0830 , 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 three-dimensional. In the BTZ black hole background, several di erent bipartitions are applied. For the bipartition along the horizon between two single-sided black holes, TEE is exactly the BekensteinHawking entropy, which supports the ER=EPR conjecture in the Euclidean case. For other bipartitions, we derive an Entangling-Thermal relation for each single-sided black hole, which is of topological origin. After summing over genus-one classical geometries, we compute TEE in the high-temperature regime. In the case where k = 1, we nd that TEE is the same as that for the Moonshine double state, given by the maximally-entangled 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 Hawking-Page transition in 3d. AdS-CFT 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 one-sided black holes and mutual information The entangling-thermal 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 area-law 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 Chern-Simons gauge theory with a compact, simple, simply-connected 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 3-manifolds. For compact gauge groups, TEE is expressed [3] in terms of modular S matrices of Wess-Zumino-Witten 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 genus-n handlebody. Using one-loop partition function on this handlebody [10{15], we derive an explicit expression for TEE, which vanishes in the low-temperature 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 single-sided black holes in the Lorentzian picture. We show that TEE between the two single-sided black holes is equal to the Bekenstein-Hawking entropy of one single-sided black hole. The mutual information between them does not vanish and again equals to the Bekenstein-Hawking 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 single-sided black hole, we then derive an Entangling-Thermal relation, lim relation is similar to but di erent from the thermal entropy relation [ 24 ] derived from the Ryu-Takayanagi formula [25], in that our result is topological and does not depend on geometrical details. The full modular-invariant genus one partition function of three-dimensional 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 low-temperature solution is dominated by the thermal AdS3. We compute TEE for the full partition function with a bipartition between the two single-sided black holes in the high temperature regime and again observe ER=EPR explicitly. When Chern-Simons 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 maximally-entangled 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 Bekenstein-Hawking 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 single-sided black holes is equal to the BekensteinHawking entropy. We further propose an Entangling-Thermal relation for single-sided black holes. Then in section 5 we demonstrate the TEE of the full modular-invariant 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 Hawking-Page 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 nite-dimensional 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 warm-up, 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 2-manifold [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 3-manifolds 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 three-space H3 that describes the Euclidean AdS3. It is the 3d analogue of hyperbolic plane, with the standard Poincare-like metric (2.1) (2.2) where y > 0 and z is a complex coordinate. Any 3-manifold 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 3-manifold 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 3-manifold M has boundary ( )= , a well-de 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 simply-connected 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 sub-Plackian length scale invalidates semi-classical treatments. The only possibility is thus 1( ( )) = Z, where can be either Z or Z Zn. The latter yields M to be a Zn-orbifold, 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 z-plane 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 non-contractible loop is parametrized by the Euclidean time. The constant time slice is thus a disk D2 with a boundary S1, perpendicular to the non-contractible 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 3-manifold is an nhandlebody, which is a lled genus-n 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 n-handlebody where k i+1Si(n) is the i-loop free energy of boundary graviton excitations. At tree level (i = 0), Ztree(n-handlebody) 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 low-temperature 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 single-letter 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 one-loop (i = 1) level, the general expression for Zloop(n-handlebody) 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 n-Riemann 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 low-temperature regime q1 into (3.1), we obtain ST AdS (a) The terms containing k come from tree-level, while others are one-loop contributions. The entire expression approaches to zero very fast in the low-temperature 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 area-law 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 non-interacting thermal AdS3's as the whole system, represented by two disjoint solid tori M0;1. There are two non-interacting, non-entangled, 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 anti-holomorphic 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 1-handlebody 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 semi-classical result is zero. For comparison, we also calculate the canonical ensemble thermal entropy of a single thermal AdS3 at temperature 1: STthAerdmSal = ln Z(1-handlebody) has the low-temperature form Z(1-handlebody) 1 @Z(1-handlebody) : 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 non-entangled. 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 AdS-Schwarzschild black hole with Lorentzian metric where the lapse and shift functions have the form NL2 = 4Gr2JL : G is the three-dimensional 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 three-space 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 half-space 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 non-singular (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 z-axis 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 non-rotating Euclidean BTZ black hole, so that is pure imaginary and r = 0. 4.2 TEE between two one-sided 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 non-transversable 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 single-sided 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 single-sided black hole. Everything inside the horizon is hidden, so is another single-sided black hole. In order to make the computation of TEE between two single-sided 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 single-sided 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 single-sided 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 single-sided 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 (non-contractible 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 3-manifold 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 n-copies of A. The resultant 3-manifold 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 Bekenstein-Hawking entropy. The above expression matches with the thermal entropy of one single-sided black hole at one-loop, 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 Bekenstein-Hawking entropy, usually viewed as an area-law 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 Chern-Simons theory on 3-manifolds with boundaries [22]. Following the replica trick, we nd exactly the same expression.2 Since Chern-Simons 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 single-sided 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 Bekenstein-Hawking entropy for a single-sided 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 single-sided 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 single-sided black holes. The glued manifold is again represented by Z1;0(n ) and the replica trick yields the Bekenstein-Hawking entropy. We have thus come to a conclusion that the followings are equal: (a) TEE between the two single-sided black holes, (b) TEE between the exterior and the interior of the horizon for a single-sided black hole, (c) thermal entropy of one single-sided black hole, (d) mutual information between the two single-sided 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 z-axis 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 entangling-thermal relation In ref. [ 24 ], the authors showed a relation (4.16) for a single-sided BTZ black hole between the entanglement entropy of CFT on the conformal boundary and the Bekenstein-Hawking 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 Entangling-Thermal 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 non-contractible 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 k-dependence, meaning we can observe the one-loop 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 one-loop, respectively. The products are over primitive conjugacy classes of . In the high-temperature regime, 2 (4.19) (4.20) this expression can be simpli ed by the single-letter 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 Entangling-Thermal relation: lim Area(A)!0 [S(A) We give this relation a di erent name from the two-dimensional 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 non-contractible 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 non-contractible circle, i.e. the horizon. The di erence between them thus characterizes the non-contractible loop. Finally we remark that there are several cases in which gluing procedures are not available. The no-gluing 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 n-glueable. Also, a single copy in which glued region B completely surrounds region except for the inner edge is not n-glueable. 5 Summation over geometries The partition functions of thermal AdS3, Z0;1( ), and BTZ black hole, Z1;0( ), are not modular-invariant by themselves. To obtain the full modular-invariant 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 anti-holomorphic pieces, whereas in this section we return to the notation that Zc;d( ) describes the holomorphic part of the partition function only. The anti-holomorphic part can easily be found as Z( ) and Z( ; ) = Z( )Z( ). Modular-invariant 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 modular-invariant 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 tree-level 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 self-dual CFT (ECFT) with central charge (cL; cR) = (24k; 24k), which is factorized into a holomorphic and an anti-holomorphic 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 FrenkelLepowsky-Meurman (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 modular-invariant partition function is still de ned on a solid torus. We will again consider the bipartition that separate the two single-sided 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 tree-level 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 non-negative 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 anti-holomorphic 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 high-temperature 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 maximally-entangled 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 quasiparticle-antiquasiparticle 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 maximally-entangled 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 anti-holomorphic part, we again recover the Bekenstein-Hawking entropy: The rst three terms agree with Witten's asymptotic formula for Bekenstein-Hawking 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 maximally-entangled 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 quasiparticle-antiquasiparticle 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 maximally-entangled 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 anti-holomorphic contribution. Thus the state (5.12) has the similar property as the thermo eld double state does in that the entanglement entropy between the quasiparticle-antiquasiparticle 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 single-sided 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 single-sided black holes A and B, one can see from straightforward computation that j ~ i will contribute half of Bekenstein-Hawking entropy. The newly introduced th is purely thermal and exhibits no non-local 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 single-sided black holes A and B, it will give half of the Bekenstein-Hawking entropy. Combining the contribution from j ~ i, we recover S ~ (A) = Sthermal(A), the Bekenstein-Hawking 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 Bekenstein-Hawking 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 tree-level Bekenstein-Hawking 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 micro-canonical 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 nite-dimensional graded representation of M with an explicit grading: where every Vn\ is an M-module, 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 M-modules, 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 M-modules Mi in Vn ( k), so that Vn ( k) ' Li1=930 Mi mi( k;n):) M-modules. 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 anti-holomorphic H = M 2C M M ; as V M M-modules, for the 194 V M-submodules ViM in V \ with V M = V1M, where Mi denotes an irreducible module for M with character di. This V M is a sub-VOA 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 VOA-module 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/self-dual VOA, i.e. there is only one single irreducible V \-module which is itself. Knowing that there is only one VOA-primary, 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 Kac-Moody 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 M-modules Mi in Fourier coe cients of j-invariant, 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 M-modules in the above pair ViM; Mi . The de nition (5.24) does not directly apply to an M-module, but one can extend the de nition using the n-graded dimension of M-modules Mi's. We de ne chqMi as6 chqMi X j i( ): n= 1 Vn\ ( )qn is the monstrous McKay-Thompson series for each as well as the unique Hauptmodul for a genus-0 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 one-to-one 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 M-modules 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, C2-co nite VOA is modular, i.e. it is a modular tensor category with a non-degenerate S-matrix. 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 well-de ned interpretation in terms of modular S-matrices 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 high-temperature regime, the full modular-invariant partition function (5.1) is dominated by the black hole solution Z1;0( ), while in the low-temperature regime, it is dominated by Z0;1( ), the thermal AdS3 solution [14, 32]. It is widely believed that there exists a Hawking-Page [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 rst-order 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 single-sided 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 time-dependence, which at late times is negative of the Bekenstein-Hawking 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 Bekenstein-Hawking entropy [78]. Quantum dimensions also appears in the calculation of left-right 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 D-branes. 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 Dijkgraaf-Witten 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) Chern-Simons theory. It would be highly non-trivial 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 time-dependent. 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 M-modules over V \. We appreciate Song He and Mudassir Moosa's suggestions on the manuscript, and thank Ori J. Ganor and Yong-Shi Wu for remarks on Hawking-Page 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. Zhu-Xi thanks Herman Verlinde for comments on the sign of BTZ TEE, and Zheng-Cheng Gu, Muxin Han, Jian-dong 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 j-invariant 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 j-function 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 z-axis 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 j-invariant 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 anti-holomorphic part. 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Zhu-Xi Luo, Hao-Yu Sun. Topological entanglement entropy in Euclidean AdS3 via surgery, Journal of High Energy Physics, 2017, 116, DOI: 10.1007/JHEP12(2017)116