A note on the bulk interpretation of the Quantum Extremal Surface formula

Defining quantum information quantities directly in bulk quantum gravity is a difficult problem due to the fluctuations of spacetime. Some progress was made recently in \cite{Mertens:2022ujr}, which provided a bulk interpretation of the Bekenstein Hawking formula for two sided BTZ black holes in terms of the entanglement entropy of gravitational edge modes. We generalize those results to give a bulk entanglement entropy interpretation of the quantum extremal surface formula in AdS3 gravity, as applied to a single interval in the boundary theory. Our computation further supports the proposal that AdS3 gravity can be viewed as a topological phase in which the bulk gravity edge modes are anyons transforming under the quantum group $\SL^{+}_{q}(2,\mathbb{R})$. These edge modes appear when we cut open the Euclidean path integral along bulk co-dimension 2 slices, and satisfies a shrinkable boundary condition which ensures that the Gibbons-Hawking calculation gives the correct state counting.


Introduction
Recent progress in the AdS/CFT correspondence has centered on the application of the quantum extremal surface (QES) formula [2][3][4].This is a prescription for computing the large N expansion of the entanglement entropy of a boundary subregion A in term of the generalized entropy of a semiclassical bulk spacetime: where γ A is the codimension 2 surface that is homologous to the region A, defined by extremizing S gen .The leading O(N 2 ) area term is a generalization of the Bekenstein Hawking entropy for black holes.Meanwhile, the subleading O(1) term S bulk computes the entanglement entropy of quantum fields across γ A in a fixed gravitational background.The QES formula is the primary tool that has been used to understand how the bulk spacetime geometry is encoded in the entanglement structure of the boundary theory.However its bulk interpretation remains mysterious because the leading area term has no straightforward state counting interpretation.The main purpose of this article is to provide a bulk canonical interpretation of the QES formula in the setting of AdS3 gravity.In particular, we will find that the area term can be viewed as counting bulk gravity edge modes.
Figure 1: A path integral on geometries with the topology of a cigar can be given a trace interpretation by replacing the tip with a shrinkable boundary condition e.This effectively replaces the cigar with an annulus, which defines a trace in the "open string " channel.
Bulk modular invariance and the shrinkable boundary condition To understand the issue at hand, recall that the generalized entropy is defined via the Euclidean gravity path integral Z(β) which sums over spacetimes with a circle S1 ∞ of length β at infinity.It is defined via the semi classical evaluation of If S 1 ∞ could be interpreted as a temperature circle, then this is just the standard formula for the thermal entropy of the partition function Z(β).However, to obtain the area term in (1.1) one must evaluate (1.2) on a saddle geometry in which S 1  ∞ contracts in the bulk, giving the topology of a cigar 1 .Ironically, the same feature of the geometry which gives the correct value of the entropy also prevents us from giving a straightforward statistical interpretation of (1.2).
The Euclidean formula [5] is quite versatile: it is independent of (and predates) the AdS/CFT correspondence, works for a variety of spinning and charged black holes in various dimensions, and reproduces the Page curve for the Hawking radiation of an evaporating black hole [6,7].Given that the Euclidean path integral is defined in the low energy effective theory, its success seems somewhat miraculous.Why does the Euclidean path integral give the correct microstate counting?
In [1,8], it was argued that the Euclidean calculation gives the right answer because it uses a bulk gravitational version of modular invariance (see also [9] for earlier renditions of this argument).Consider the gravity path integral on geometries with the cigar topology of the Hawking saddle.To obtain a bulk trace interpretation, we must remove an infinitesmal disk from the tip of the cigar, and replace it with a "shrinkable" boundary condition at the resulting boundary.This boundary condition is defined so that the path integral on the annulus after the excision equals the path integral on the cigar: see figure 1.
Read in the "open string" channel, the path integral on the annulus seems to define the trace of a reduced density matrix ρ V : this is shown in figure 2. Given such a reduced density matrix, inserting (1.3) back into (1.2) shows that the generalized entropy (1.2) is the bulk entanglement entropy between the subregion V and its complement.Our main result is an explicit realization of this formula in AdS3 gravity.Due to the shrinking cycle, evaluating the bulk path integral in the open string channel requires the introduction of bulk edge modes near the tip of the cigar: they arise from a local version of holography in which the degrees of freedom of the small disk is replaced by the gravitational edge modes at its boundary, where the metric is free to fluctuate in contrast with the asymptotic boundary.These edge modes are a manifestation of UV-IR mixing in the bulk: they represent UV degrees of freedom that are normally confined in the IR, but are exposed when we cut open the path integral along a co-dimension 2 entangling surface.As we will see, their entanglement gives the leading contribution to (1.4), which is precisely the area term in the QES formula.

Summary of previous results and outline
In this paper we will carry out the argument described above within the effective theory of AdS3 gravity defined in [1].In fact, for the special case of the generalized entropy of the two sided BTZ black hole (with A being the entire spatial slice of a single sided CFT), equation (1.4) was derived in [1].In this derivation, a fundamental role was played by a bulk edge mode symmetry that is intimately related to the modular invariance of the gravity path integral.Since this same edge mode symmetry governs the QES formula for the single interval EE of a boundary subregion, we summarize those results here.
To begin with, note that in AdS3, the bulk modular invariance described above is exactly the holographic dual of modular invariance in the boundary CFT.Here the bulk semiclassical limit corresponds to the high temperature limit when β is small.In this regime we can use the modular invariance of the CFT to evaluate the partition function in the dual channel: At small β the RHS is captured by low energy part of the spectrum, and one usually just keep the vacuum contribution Z( 4π 2 β ) ∼ exp πc 6β -this is an expression of UV-IR mixing in the boundary theory.The dual bulk calculation corresponds to evaluating the gravity action on the BTZ saddle, where the Euclidean time circle shrinks in the interior as described above.However, to understand what states are being counted, one should go back to the original channel and expand in the Boltzmann factor e −βE .In the standard CFT calculation, this is achieved by doing a Laplace transform, which gives the Cardy density of states ρ(E) = exp cE 6 .When combined with Brown-Henneaux relation [10] 2G N this famously reproduces the BH entropy as where E * is the saddle point value of the energy.In this calculation, BH entropy is naturally interpreted as counting edge modes at asymptotic infinity that transform under the asymptotic Virasoro symmetry.As explained above, to access the original channel in the bulk, we must introduce gravitational edge modes.One of the main results of [1] is that these bulk edge modes do not transform under a Virasoro symmetry.Instead, the shrinkable boundary condition implies that they transform under the quantum group SL + q (2, R) ⊗ SL + q (2, R), with q related to the cosmological constant.Moreover its co-product naturally defines a factorization of the bulk Hilbert space that leads to the reduced density matrix ρ V in (1.4).In the classical limit, equation (1.6) is then replaced by Area 4G = log dim q p * dim q p * (1.7) where p, p labels representations of SL + q (2, R) ⊗ SL + q (2, R), and dim q p, dim q p are the associated quantum dimensions which count bulk gravity edge modes.These results can be viewed as a q-deformation and dimensional uplift of those found in JT gravity in [11].
It was originally suggested in [12] and further advocated in [1] that we should interpret this quantum group symmetry via the paradigm of topological phases, which are long range entangled phases described by topological quantum field theories.In this context, quantum groups arise as the symmetry of anyons that describe the collective degrees of freedom of topological phase.These anyons are objects in a modular tensor category which defines the TQFT path integral: for Chern Simons theory with compact gauge group G, this modular tensor category is the representation category of a the loop group LG or its associated quantum group.The results of [1] for AdS3 gravity suggests that we should interpret the bulk edge modes as gravitational anyons that belong to the representation category of SL + q (2, R) ⊗ SL + q (2, R), which behaves like a "gauge group" for gravity.We will say more about this viewpoint in the conclusion.
Here is a detailed outline of the paper.In section 2 we review some salient details of [1] that will be needed for this work.In 2.1 we define the boundary partition function Z(β) and compute its thermal entropy.In 2.2, we give the bulk definition of Z(β) in terms of a modified Chern Simons path integral with PSL(2, R) ⊗ PSL(2, R) gauge group [13].In 2.3 we define the bulk 2 sided Hilbert space for the BTZ black hole geometries and in 2.4 we define SL + q (2, R) and explain how its co-product determines the factorization of the bulk Hilbert space.In section 3, we repeat these steps for the effective BCFT partition function whose thermal entropy gives the entanglement entropy of a single interval.We denote this partition function by Z ee (l), where l is a dimensionless length of an interval in the boundary theory, and plays the role of β for the black holes.We derive this partition function carefully in 3.1 by factorizing the boundary theory, and comment on the relation of our entanglement boundary conditions to FZZT branes.In 3.2 we give a bulk description of Z ee (l) using AdS/BCFT and cut this path integral along a Cauchy slice to define a bulk Hartle Hawing state.In 3.3, we give the factorization of this quantum state and show that its entanglement entropy reproduces the generalized entropy.In particular, (1.7) is now replaced by which gives the entanglement entropy of an interval of length l2 .Here the reduction to a single chiral sector relative to the two sided black hole case can be attributed to introduction of a conformal boundary conditionat the entangling surface in the boundary CFT.Finally, in the conclusion we provide an extended discussion that places our work within the paradigm of extended TQFT.

Figure 2:
The lower half of the annulus with a shrinkable boundary condition e inserted can be viewed as the path integral preparation of a factorized state in the bulk extended Hilbert space H V ⊗ H V .The subsequent Lorentzian evolution describes an entangled sum of one sided geometries 2 Review: gravitational edge modes in 3d gravity In this section we review the construction of the effective 3d gravity theory proposed in [1] and its associated gravitational edge modes.We will see how representations of SL + q (2, R) naturally arise as the subregion states needed to define the factorization of the bulk Hilbert space of the two-sided BTZ black hole.

The boundary partition function
Consider an irrational (c >> 1), holographic 1+1 d CFT with only Virasoro symmetry: the modular invariant partition function is a sum of Virasoro characters: where β ,µ is the inverse temperature and chemical potential resp.The Virasoro characters are given by: It is well known [15][16][17] that for theories with gap and a sparse spectrum, the high temperature limit of Z(τ, τ ) is dominated by the vacuum character in the dual channel Specifically, the vacuum block dominates when the inverse temperature is much smaller than the gap to the first excited state: This motivates us to define an effective theory with partition function: This partition function can be viewed as an effective theory which describes the universal dynamics of an irrational, Virasoro CFT in the high temperature limit.
To see the statistical interpretation of Z(τ, τ ), we go back to the original channel-where the temperature circle is small-by applying a modular transformation to express the vacuum character χ 0 (−1/τ ) as a sum over non-degenerate Virasoro characters χ h (τ ).It will be convenient to use the Liouville parameterization3 of the conformal dimensions, (2.6) Then we can use the Virasoro S-matrix [18] to write where As noted in [12] and further explained below, these Virasoro S-matrix elements form a measure on the space of Virasoro representations, which is identical to the Plancherel measure for SL + q (2, R) S 0 p = dim q p, q = e iπb 2 (2.9) The equality of these measures is the first hint that Rep (SL + q (2, R) )might play a role in 3d gravity.In terms of the inverse temperature β and chemical potential µ defined via τ = β 2πlADS (µ + i), we can write Z(τ, τ ) explicitly as the partition function The integral runs over primary states labelled by (p, p), with primary energy and angular momentum given by As we explain below, these can be identified with black hole states in the bulk, whereas the descendants states captured by the Dedekind eta are the boundary gravitons.
The thermal entropy of the boundary partition function (2.10) reproduces the BH entropy for Euclidean BTZ black holes in the high temperature β/l AdS << 1, and semi-classical limit c >> 1 → b >> 1 limit.For β/l AdS << 1, large p, p dominates the integral so we can approximate: This gives the expected Cardy density of states.For c >> 1, we can evaluate the entropy at the saddle point (p * , p * ), which gives Here (M * , J * ) = ( p * 2 + p * 2 , p * 2 − p * 2 ) are the saddle point values of the mass and spin, which depends on β.

The bulk gravity theory
The boundary partition function (2.10) has a dual interpretation in terms of 3d pure gravity with a negative cosmological constant.The gravity action is:

AdS
), (2.14) where the bulk and boundary parameters are related via the Brown Hennaux relation c = 3l AdS 2G N .
This duality can be derived explicitly in the first order formulation of AdS3 gravity as a Chern Simons theory with gauge group PSL(2, R) ⊗ PSL(2, R), in which the gauge fields are related to the Vielbein e and spin connection ω according to Specifically, [13] showed that the Euclidean Chern Simons path integral with gravitational boundary condition on a solid torus with a contractible Euclidean time cycle reproduces the vacuum character |χ 0 | 2 (−1/τ ).Using coordinates ϕ, t E on the boundary and ρ as the bulk radial coordinate, the gravitational boundary conditions are: 41.Asymptotic AdS3 boundary conditions that set the boundary value of the gauge field to where L(ϕ + it E ) is a classical boundary stress tensor which generates the boundary Virasoro algebra with L n = dϕ 2π e in(ϕ) L(ϕ).

2.
A winding constraint defined as follows.Writing the flat connection on the (ρ, T E ) disk as A = g −1 dg, one requires that at the asymptotic boundary , the map g(t E ) ρ=∞ : S 1 → PSL(2, R) winds once around the non contractible SO(2) subgroup of PSL(2, R).
In the standard description [13,19], the bulk gravity path integral is given a perturbative interpretation as arising from the quantization of fluctuations about a Euclidean BTZ saddle, where the Euclidean time circle contracts smoothly in the bulk: this is shown in the left of figure 3, which is a solid torus with no Wilson loops inserted.This perspective is made explicit by writing where exp( 2πi is the dominant contribution from the on shell Euclidean classical action, and the partition function for fluctuations associated with boundary gravitons.In this description, the BH entropy is captured geometrically by the horizon area of the saddle point geometry, but it does not have a state counting interpretation.This is because the bulk time slice is an annulus A, and the solid torus does not take the form A × S 1 . The bulk generalized entropy is computed by including the contributions of the one loop determinant about the BTZ saddle to the BH entropy; in this case the generalized entropy gives the exact thermal entropy of Z(τ, τ ), because the partition function is one loop exact [19]: While the perturbative description (2.17)of Z(τ, τ ) is correct, it has the undesirable feature that only a small part of the total entropy -the part due to fluctuations around the BTZ saddle-has a state counting interpretation.The heavy primaries in (2.10) which gives the dominant contribution to the entropy hidden in this bulk description.One way to incorporate these heavy primaries into a bulk Euclidean time statistical mechanical partition function is to go to the dual channel via a modular transform, as we did in (2.7).In this channel, the bulk is obtained by filling in the spatial ϕ cycle with a disk, which evolves in the Euclidean time direction to give a bulk trace (see right of figure 3).The modular transform introduces Wilson lines labelled by the primaries (p, p), which punctures the spatial disk and creates a nontrivial holonomy around ϕ, leading to the nondegenerate characters χ p (τ ).The bulk is thus a sum over solid torus path integral with Wilson loops inserted, and leads to a nonperturbative5 description of Z(τ, τ ) as a statistical sum (2.10) .For the real values of p, p which appear in (2.10), the classical phase space on the punctured disk6 consists of flat connections with holonomies in the hyperbolic conjugacy class.In the metric description, these hyperbolic holonomies correspond to Lorentzian BTZ black holes with mass an spin given by (2.11).Quantizing these solutions 7 gives rise to the black hole states which is counted by the partition function (2.10).These black holes states should not be confused with the Euclidean saddle: the former are "microstates" with zero entropy and no temperature, while the latter is geometric representation of the saddle point of (2.10) with mass and spin determined by the temperature β.Thus, the dual channel gives a perfectly consistent definition of a bulk statistical partition function.However to obtain an entanglement entropy interpretation of S gen , we need to give a trace interpretation in the channel on the left of figure 3, where we filled in the time cycle with no Wilson loops inserted.To do this, we need to remove a thin tube from the center of the of the solid torus, with a shrinkable boundary condition e (figure 4).The tube arises from the modular evolution of a stretched entangling surface: in Lorentzian signature, this would correspond to a stretched horizon as depicted in the right of figure 2. The shrinkable boundary condition ensures that the resulting path integral can be given a state counting interpretation as where ρ V is a bulk reduced density matrix on a subregion V .Note that since the dynamical spacetime geometry is encoded into the field space via (2.15), we can define the subregion V in a topological sense, without fixing the metric at the entangling surface.This is crucial for obtaining the correct gravity edge modes, which differ from the conventional gauge theory edge modes that do fix a background metric at the entangling surface.Finally, we observe that the shrinkable boundary condition (2.19) guarantees that the bulk generalized entropy, identified with the full thermal entropy of Z(τ, τ ) via (2.18), is given by the entanglement entropy of ρ V : In the next section, we review the bulk Hilbert space and factorization map which gives rise to the reduced density matrix ρ V .Before moving on the discussion of bulk factorization, we should emphasize that the gravity boundary conditions-in particular the winding constraint-alter Chern Simons gauge theory in an essential way.One is no longer using the path integral measure associated to the gauge group PSL(2, R) ⊗ PSL(2, R), which would naturally sum over integer winding numbers.In fact, the main proposal of [1] is that the bulk theory should be viewed as an extended TQFT associated with the quantum semi group SL q (2, R) + ⊗ SL q (2, R) + -we will comment on this perspective in the conclusion.To avoid confusion, we will henceforth refer to the bulk theory as the gravitational TQFT, or just the bulk TQFT for short.

The bulk Hilbert space
We start by defining the bulk Hilbert space on the time-reflection symmetric Cauchy slice of the solid torus geometry (with no Wilson loops inserted).This is an annulus (depicted in figure 5) which corresponds to the Einsten Rosen bridge in the Lorentzian continuation of the geometry.The Hilbert space is obtained by quantizing the classical phase space solutions with the same topology as the two sided BTZ black holes.These correspond to the the Kruskal extension of the single-sided Banados geometries: which are parameterized by the functions L(x + ), L(x − ).These are the the expectation values of the boundary stress tensors which determine the boundary values of the gauge fields A ϕ , Āϕ according to equation (2.16).
The Kruskal extension introduces a left (L) and right (R) asymptotic boundary, with coordinates x ± L , x ± R respectively.This gives 4 stress tensor components: The L and R copies are not quite independent, because they share the same zero mode which determines the mass and spin of the black hole.This correlation between the L and R boundaries is most easily seen in the Chern Simons representation, where the zero modes label the holonomy around the noncontractible spatial cycle according to: Heuristically, we can think of this holonomy as measuring the effect of Wilson lines that thread the wormhole in the dual channel (as in left of figure 7 ).The shared zero mode implies that the phase space does not factorize into independent copies associated to the two boundaries 8Due to the AdS3 boundary conditions (2.16), the holonomy labels (p, p) can be identified with the primary labels for the Virasoro algebra.Upon quantization of the stress tensor degrees of freedom in (2.22) , we obtain a bulk Hilbert space H bulk constructed out of Virasoro primaries: where p, p labels the black hole mass and spin according to (2.11) , while m L k , m R k and nL k , nR k are the occupation numbers of the descendant states corresponding to boundary gravitons.Explicit we can write: While the L and R edge modes corresponding to the boundary gravitons factorize, the bulk degrees of freedom (p, p) prevent a complete factorization into independent, one-sided Hilbert spaces.The Hartle Hawking state, prepared by the half solid torus path integral, can be expressed explicitly as and satisfies Z(β, µ) = Ψ(β, µ)|Ψ(β, µ) .

Factorization and subregion states
In this section we define the factorization map from the Hilbert space on the ER bridge into a bulk extended Hilbert space where H V ,H V , are the subregion Hilbert spaces on each half of the ER bridge as shown in figure.i bulk can be viewed as a bulk path integral evolution that splits an annulus into two, as shown in figure 6.It was explained in [1] that compatibility with the shrinkable boundary condition requires that each of subregions V supports Virasoro edge modes at asymptotic infinity, and quantum group edge modes at the entangling surface transforming under SL + q (2, R) ⊗ SL + q (2, R).The most direct, though formal way of defining SL + q (2, R) is via its fundamental (co) representation as a two by two "quantum" matrix: in which each matrix element a, b, c, d is an operator on L 2 (R ⊗ R) with positive spectrum.These operators have commutation relations given by ab = q 1/2 ba, ac = q 1/2 ca, bd = q 1/2 db, cd = q 1/2 dc, bc = cb, ad − da = (q 1/2 − q −1/2 )bc. (2.29) When q → 1, these matrix elements can be taken to be real numbers, and the positive spectrum condition corresponds to requiring that they are positive real numbers.This limit gives the semi group SL + (2, R) : due to positivity, not all elements have inverses.Likewise, when q = 1, the positive spectrum condition implies that the quantum matrix g does not always have an antipode, which is the analogue of an inverse for a quantum group.In this sense SL + q (2, R) is a quantum semi-group.This definition may seem a bit bizarre at first sight: in what sense do the elements g define a symmetry?The simplest answer to this question is that a symmetry G should be defined by its representation category Rep(G), and to say that SL + q (2, R) is a symmetry means there is exists a well defined category of SL + q (2, R) representations, related to the continuous series representations of SL q (2, R).In fact the (highly nontrivial) representation matrices R ab (g) a, b ∈ R for SL + q (2, R) have been worked out explicitly by [20].The indices a, b are continuous but the matrix elements satisfy the usual representation property, which says that if g = g 1 g 2 , then One of the main results of [1] is that the factorization map i bulk can be characterized algebraically as the co-product of the quantum group SL + q (2, R) ⊗ SL q (2, R) + .This co product is equivalent to the representation property (2.30), viewed as a map from functions of one quantum group element g to functions on two quantum group elements g 1 , g 2 .The matrix element R ab (g) is wavefunction of the "Wilson line" variable g that crosses the ER bridge, g 1 , g 2 are the subregion Wilson lines obtained by the factorization map.The quantum group edge modes correspond to the contract indices c.Below we give a condensed explanation of how these representation matrices and their co-product arise in the factorization of H bulk .We refer the reader to [1] for further details.
Factorization To set up the discussion, it is useful to organize our bulk Hilbert space in terms of Virasoro representations V p as follows.First, the Hilbert space decouples into a tensor product chiral and anti chiral parts, corresponding to the two PSL(2, R) components of the bulk gauge field: We will focus just on a single sector H, which takes the form with basis given in (2.25).It turns out i bulk acts only on the bulk zero mode subspace H 0 ⊂ H , i.e. the primary states |p in (2.25).The boundary gravitons labelled by m L,R k are spectactors.Moreover, no descendants are introduced at the bulk entangling surface: in this sense the shrinkable boundary condition is a gapped or topological boundary condition.This is in contrast with the usual factorization map for Chern Simons theory, which would introduce descendants at the bulk entangling surface.In the shrinking limit, these states lead to an infinite entanglement entropy associated with the UV divergence of QFT on a fixed background.Gravity regularizes this divergence.
The crucial observation that underlies our factorization map is the existence of a one to one correspondence between the Virasoro representations V p and representations Ṽp of the quantum group SL + q (2, R).This is a non compact generalization of the well known relation9 between representations of the loop group LG and corresponding quantum group, usually denoted U q (G). 10 The correspondence between V p and Ṽp is a deep fact linked to the modular bootstrap for irrational CFT's with c > 1.The modular bootstrap is a set of constraints that determines the data for the representation category Rep(Vir) of the Virasoro algebra: These include the spectrum of primaries, the fusion rules, and the OPE coefficients.Ponsot and Teschner [22] showed that there is a one to one map F ( a "functor") between representation categories such that the fusion algebra of Rep (Vir ) maps to representation ring of SL + q (2, R).In particular, this fusion rules of Rep(Vir) corresponds to the decomposition of tensor products in Rep(SL + q (2, R)), and the Virasoro Fusion matrix maps to the Racah Coefficents (6j symbols) of Rep(SL + q (2, R)).Thus, Rep(SL + q (2, R)) provides a solution to the modular bootstrap.Starting from this observation, we can define the factorization map i bulk acting on the zero mode subspace H 0 as follows 1.First we use the correspondence (2.33) to relate primaries in H 0 to representation matrix elements of SL + q (2, R).Concretely, we introduce frozen quantum group indices i L , i R to the primary states: such that the corresponding zero mode wavefunctions on the ER bridge are given by representation matrix element of SL + q (2, R) The normalization is important: it corresponds to the Plancherel measure on SL + q (2, R), which we explain below.The frozen indices originate from the PSL(2, R) Kacs-Moody zero modes of the WZW model which arises from the usual bulk-boundary correspondence for Chern Simons theory.However, the AdS3 boundary conditions (2.16) projects onto a particular linear combination of Kacs-Moody zero modes, labelled by i L,R , thus reducing the symmetry algebra to Virasoro.The projection restricts to a subspace of wavefunctions invariant (up to a phase) under the action of right (left) multiplication by elements h R/L ∈ H R/L of certain subgroups of SL + q (2, R): The projected matrix elements are called Whittaker functions, and the associated quantum state only depends on the configuration space variable g ∈ H L \SL + q (2, R)/H R in the double coset 2. We can give a more precise characterization of H 0 .This follows from the fact that the matrix elements form a complete basis on the space L 2 (SL + q (2, R)) of integrable functions on SL + q (2, R).This completeness is due to a nontrivial generalization of the Peter Weyl theorem [] that states that L 2 (SL + q (2, R)) decomposes into representations Ṽp ⊗ Ṽ * p of SL + q (2, R) (the * denotes the dual vector space): (2.38) The bulk zero mode Hilbert space is obtained from (2.38) by projecting the representation matrix elements onto invariant subspaces labelled by i L,R .This is expressed by a modified Peter Weyl theorem: where Ṽp,i L , Ṽ * p,i R denote the respective invariant subspaces.Note the similarity with the structure of (2.32): the correlation in the representation label p gives rise to the L 2 space, allowing us to interpret the zero mode subspace H 0 as wavefunctions on the quantum group: in this sense g ∈ SL + q (2, R) is a "Wilson line" variable that captures the connectedness of the ER bridge.Now some technical remarks about the measure in (2.38) and (2.39).The Plancherel measure dµ(p) = dim q (p) is defined by the norm of the matrix elements with respect to the Haar measure, and appears in the orthogonality relation: Moreover, it is identified with the character in Ṽp evaluated on the identity element: Tr Ṽp (1) = dim q p (2.42) Note that the Plancherel measure is a highly constrained part of the Rep(SL + q (2, R))data.In particular, an arbitrary rescaling of the representation matrices would spoil the representation property 3. In addition to a (non-commutative) pointwise product L 2 (SL + q (2, R)) also has a co-product obtained by pulling back the multiplication rule on SL + q (2, R) : When applied to the representation basis , this defines a factorization map that "cuts open" the Wilson line via the representation property (2.43).
4. Finally, we use the co product ∆ to define the factorization map on our bulk Hilbert space.The action on the zero modes is: Here we have defined the subregion zero mode subspaces spanned by the respective matrix elements as The group elements g, g V , g V in (2.45)belong to the respective cosets above.We can lift the factorization map onto the full Hilbert space H by tensoring Ṽ * p,i L with the descendant states, which just gives back the Virasoro representation V p .Thus we can express the spectral decomposition of the subregion Hilbert spaces in a simple way: The factorization map is then given by Summary In this section, we explain aspects of the representation theory of SL + q (2, R) and defined the factorization map as the co product on SL + q (2, R) .While the discussion above may have seemed formal, it has a very simple and intuitive pay off.Heuristically, we can view the configuration space variable g as a SL + q (2, R) Wilson line g = P exp A that crosses the ER bridge.The factorization map i bulk is then defined from splitting this Wilson line g → g V g V into subregion Wilson lines in a manner analogous to the splitting of an ordinary Wilson line according to (2.43) ( see figure 7).The novel aspect of our discussion is that this Wilson line variable is a representation of the entanglement structure of the bulk wavefunction ( analogous to the "entanglement holonomy" defined in [23] ).Moreover, the measure on the bulk wavefunctions defined by the gravity path integral requires that this Wilson line transforms under a the non -invertible quantum group symmetry SL + q (2, R).

Black hole state Entangled Subregion states
Figure A heuristic depiction of the bulk factorization map via a bulk version of ER=EPR.

The single interval QES formula from bulk edge modes
In this section we use the gravitational TQFT to give a bulk derivation of the QES formula for the vacuum state of a boundary theory on S 1 × R .At first sight, it may seem strange to apply our bulk TQFT, designed to capture the high temperature behavior of the boundary theory, to a zero temperature state.However, the computation of entanglement entropy for a subregion accesses the high temperature regime because of the Unruh effect.Observers confined to a spacetime subregion of the boundary theory must accelerate.As a result they see a thermal state, and the associated thermal entropy is dominated by the high temperature region close to the entangling surface.The entanglement entropy of the sub-region is equal to the associated thermal entropy.To show the parallel structure between the bulk derivation of the RT formula and the BH entropy, we will follow the same sequence of steps as in section We start with the properly factorized boundary calculation, and write down the effective partition function whose thermal entropy gives the entanglement entropy of an interval.We then give the bulk interpretation and define the bulk factorization map.Finally, we show how the semi-classical limit of the bulk entanglement entropy reproduces the RT formula.

The boundary calculation of entanglement entropy for one interval
We begin by reviewing the canonical calculation of entanglement entropy for an interval A in vacuum state |0 ∈ H S 1 of a CFT on a spatial circle [24] of length L. The vacuum is prepared by the disk which ends on this circle, and its norm is the sphere partition function.To properly factorize the vacuum state, we introduce a regulator surface around the entangling surface, on which we choose a shrinkable boundary condition [24] [25].The factorize state is then prepared by a path integral on a hemisphere with two semi-disks removed.This can be read as an "open string" amplitude between the intervals A and its complement Ā, with respective lengths x and L − x.This is shown in figure 8, where we have made a conformal map to a long strip of length To understand the nature of the shrinkable boundary condition, it is useful to consider the trace of the reduced density matrix ρ A on A. As shown in figure 9, Tr A ρ A is identified with an annulus, which we can view as a "closed string" amplitude between two entanglement boundary states |e .In the limit → 0, this boundary state must be defined so that annulus reproduces the sphere partition function.
Strictly speaking, any boundary state with non trivial overlap with the vacuum will satisfy the shrinkability condition in the strict → 0 limit.This is because this limit correspond to the long time propagation that projects to the vacuum, which is the path integral on a small disk that closes the hole.However, the vacuum is not a conformal invariant boundary state.In general, a conformally invariant boundary condition is given by a superposition of Ishibashi states |h satisfying with h ranging over the spectrum of the CFT on the circle.The requirements of conformal symmetry and shrinkability then implies we should choose a boundary state |e ∈ H S 1 with nonzero overlap 0|e .In a holographic CFT, the assumption of a gap and a sparse spectrum implies that projecting to |0 gives a good approximation.This is because for an arbitrary conformally invariant boundary so as q → 0, the vacuum character in the dominates.Following the approach in [24], we choose |e = |0 as the entanglement boundary state.Inserting this boundary state at the regulated entangling surface defines an effective partition function Z ee (l) given by We have thus arrived once again at the vacuum character, this time with only one chirality.Note that we have applied the BCFT analogue of the logic used in section 2, which used modular invariance and the presence of a gap to argue for the dominance of the vacuum block in the dual channel.In the BCFT setting, the dual channel is the closed string channel.
In the shrinking limit l → ∞, the long time propagation projects onto the vacuum, so we can write ( setting c = c) This is exactly the form of the sphere partition function as dictated by the conformal anomaly, showing explicitly that shrinkability is satisified.In this limit, the entanglement entropy gives the standard universal answer.Notice that in the closed string channel, replicating the cylinder actually maps the parameter l → l n , because the definition of closed string Hamiltonian L 0 + L0 requires us to rescale the geometry so that the circumference of the cylinder has length 2π.

State counting in the open string channel
To understand what states are being counted by the entanglement entropy S A we need to go back to the open string channel and define the reduced density matrix.We do so by applying a modular transformation using the Virasoro S matrix This equation defines the statistical partition function Z(l) on an interval of length l,with l playing the role of a dimensionless temperature.The sum is over Virasoro representations labelled by p with a measure given by S p 0 = dim q p.While non strictly necessary, it is useful to apply the language of Liouville theory to give a concrete interpretation of the subregion Hilbert space H A on which the partition sum (3.7) is defined.This is because Liouville theory with brane insertions gives a local field theory description for Virasoro representations.In particular, for each p, we can view χ p (q) as the Liouville partition function on a spatial interval with a ZZ brane inserted at one, and FZZT branes at the other.This corresponds to a quantization of Liouville theory on the interval with local boundary conditions [18,26].The different characters χ p in (3.7) are obtained by quantizing with different FZZT branes, also labeled by p.We can thus view the integral over p in (3.7) as an integral over boundary conditions with measure dim q p.
The end result is that Hilbert space H A on an interval with entanglement boundary conditions can be view as a direct sum of Virasoro representations where as before p labels primaries and m k the occupation number of the descendants.Equivalently, we can apply the unfolding trick 11 to the annulus and view each χ p (q) as a chiral conformal block on a torus (see figure 10)-this will be particularly useful when we consider the holographic dual.In the unfolded theory, we interpret (3.7) as a sum over states on a circle, whose Z 2 quotient reproduces the original interval.Either way we find that the factorization of the global vacuum state |0 ∈ H S 1 that is compatibility with shrinkable boundary condition is N m k = km k is the total level for the occupation numbers m k .We have denoted the factorized state as |HH , because we will see below that it corresponds to a bulk Hartle Hawking state.The reduced density matrix on A is and satisfies Z(l) = Tr A ρ A by design.Notice that l now plays the role that β/ AdS did in the black hole thermal density matrix.Since we are taking the limit l → ∞ as → 0, we can apply an analogous high temperature saddle point approximation as was done in the black hole case.In particular, in the l → ∞ limit, the p-integral for Z(l) is dominated by large p.Further taking a c >> 1 classical limit, corrsponding to b >> 1, we can approximate dim q p ∼ exp 2πbp and find the saddle point This gives Z ee (l) ∼ exp cl 6 which agrees with the closed string channel calculation (3.5).The fact that a saddle point approximation of the effective partition function Z ee (l) reproduces the RT formula was observed in [14].However, in that work, the dim q p measure was introduced in an ad hoc fashion, based on its agreement with the Cardy density of states at high energies.Our work provides a justification for the calculation in [14].

Doubling
Figure 10: Using the doubling trick replaces the long cylinder with a thin torus.The partition function Z ee (l) is the chiral vacuum conformal block on the torus.

Bulk partition function and Hilbert space
Let us now consider the bulk description of the partition function Z ee (l).If we use the unfolded representation of Z ee (l) as the chiral vacuum block on a boundary torus, we can essentially repeat our analysis of the bulk path integral for the black hole spacetimes in section 3.In particular, the relevant bulk dual is obtained by filling in the modular time circle of the torus with a smooth disk.Indeed, the same path integral derivation used in [13] shows that the solid torus path integral with a single copy of the gauge field reproduces the vacuum character χ 0 (−1/τ ) on the boundary.The bulk dual is a manifold N whose boundary is ∂N = Q ∪ M .Q represent end of the world branes in a bulk dual that is locally AdS3.However, in the original boundary cylinder description of Z ee (l), the bulk dual has a more complicated description.Denoting the cylinder manifold by M , a bulk manifold that ends on M must include end-of-the-world-branes [27].An example of such a manifold is shown in the left of figure 11.

Doubling
Here the bulk manifold is N ∪ Q, where N is the bulk spacetime and Q the end of the world branes that are caps which extend from the regulator surface ∂M into the bulk.The bulk spacetime satisfies ∂N = M ∪ Q.The end of the world branes Q are responsible for reducing the asymptotic symmetry of the bulk space time to a single copy of Virasoro, consistent with the fact that the conformal boundary conditions on M breaks the Virasoro symmetry to a single chiral copy.
A concrete description of the action for these branes has been given in the literature on AdS/BCFT [28] .In the metric formulation they modify the Einstein Hilbert action12 I EH to where h ab is the induced metric on the brane Q, T is the brane tension 13 , and K = h ab K ab is the trace of the extrinsic curvarture on Q.This is defined in terms of the unit normal n a on Q as K ab = ∇ a n b .The Gibbons Hawking term allows for a variational principle that is consistent with either Neumann or Dirichlet boundary conditions on the metric: since the ETW brane Q lives in the bulk, its metric should be free to fluctuate.Thus, Neumann BC is chosen there, corresponding to When the boundary manifold is a cylinder, the corresponding bulk gravity solution has been determined (see e.g.[29] [28]).The solution depends on the length l of the cylinder, and experiences an analogue of the Hawking-page phase transition as l increases past a critical value.Our entanglement entropy problem corresponds to the long cylinder limit, where the results of [28] imply that the bulk solution has the topology shown in the left of figure 11.Cutting this geometry at the time-symmetric slice prepares the Hartle Hawking state |HH on a bulk cauchy slice with the topology of a strip bounded by the Euclidean ETW branes (see right figure 12).The Lorentzian evolution of this state produces a part of planar two sided BTZ blackhole bounded by the Lorentzian evolution of the ETW branes [30].Heuristically, one can view the two sides of the black hole geometry as emerging from the two AdS Rindler Wedges of the original AdS3 vacuum.The introduction of the ETW branes Q is the bulk dual of the factorization map i boundary , which acts on the AdS3 vacuum as shown in the left of 12.
Figure 12: On the left, we show the bulk dual to the factorization map i boundary that acts on the AdS3 vacuum by introducing two ETW branes Q.The disk like Cauchy slice D is mapped to a strip D with boundaries given by the two intervals and the ETW branes.On the right we applied the bulk diffeomorphism that corresponds to the conformal map to the boundary cylinder.This shows how a two sided-BTZ black hole geometry arises the factorization of the AdS 3 vacuum The gravitational Chern Simons formulation of AdS3/BCFT2 was introduced in [31], where the analogue of the boundary actions I GH and I brane was determined.For RCFT's, the analogous bulk Chern Simons description of a the BCFT has also been worked out appeal to these results as evidence that one can go beyond the classical solution described above to obtain a bulk TQFT path integral on the manifold N ∪ Q which reproduces the boundary character χ 0 (−1/τ ).

Subregion states, bulk factorization map and the generalized entropy
The description of the bulk subregion Hilbert space and factorization map is most straightforward in the unfolded picture, where Z ee (l) is interpreted as a solid torus path integral.As in our discussion of BH entropy, a bulk state counting interpretation of Z ee (l) is obtained by removing an infinitesmal solid tube and introduce a shrinkable boundary condition e on the resulting boundary, which we identify as a bulk entangling surface.This gives a chiral theory on A × S 1 that is depicted in the left of figure 4.
This suggests we should define the bulk subregion Hilbert space just as in equation (2.47), using the representation matrix elements of SL + q (2, R) as the wavefunctions : then the lower half of A × S 1 ( right of figure 4) defines defines the action of a factorization map i bulk that takes the Hartle Hawking state |HH into a bulk extended Hilbert space H V ⊗ H V .We can view the total bulk factorization map as a composition of i bulk and i boundary acting on the AdS vacuum By design, this factorization map gives the bulk reduced density matrix .17)which satisfies Z ee (l) = Tr V ρ V .
In the unfolded representation, we can once again appeal to the one loop exact nature of the bulk theory to conclude that the generalized entropy is given by the bulk entanglement entropy of ρ V : What about the folded theory?Here the bulk topology is more complicated.As shown in 13, in order to separate the bulk Cauchy into V , and V , the bulk entangling surface carrying the quantum group edge modes would have to intersect the EOW branes Q.It would be useful to work out the 13: To obtain a bulk trace interpretation in the folded representation (left figure), we must excise a tube which ends on the ETQ branes denoted by Q.The lower half of this geometry would then describe the factorization of the bulk Hartle Hawking state, as shown on the right.boundary terms in the action that would accommodate such an intersection.We leave this for future work.
transform under SL + q (2, R).These edge modes are associated to bulk subregion wavefunctions given by representation matrices of SL + q (2, R), and the bulk factorization map is identified with the co-product on SL + q (2, R).This supports the perspective that AdS3 gravity can be viewed as a topological phase in which the BH entropy is identified with the entanglement entropy of gravitatonal anyons.
The fact that the same SL + q (2, R) symmetry also governs the edge modes and BH entropy of the two sided BTZ black hole suggests the existence of a bulk path integral equipped with a consistent notion of cutting and gluing along co-dimension 2 surfaces.As advocated in [1] (see also [25]), extended TQFT provides the appropriate categorical framework to describe such a path integral.Let us recall the main features of this formalism, which we will use to give a broader perspective on the computations in this paper.
Extended TQFT An extended TQFT in d spacetime dimensions is a mapping Z(•) that assigns a mathematical object to surfaces of each codimension.In particular, for surfaces of codimension zero, one and two, the assignments are:

boundary category
As we move up in co dimension, the mathematical structure becomes more refined and contains more information about the theory.In fact, given the assignments at codimension k, one can reconstruct the assignments at co dimension n < k.For the purposes of computing entanglement measures, it suffice to extend down to tier 2: the boundary category Z(M d−2 ) provides information about edge modes needed to factorize the Hilbert space and compute entanglement entropies [33].A good example of this structure in d=3 is given by Chern Simons theory with compact gauge group G, where Z(M d−2 ) = Z(S 1 ) =Rep(LG) is the representation category of the loop group LG .Figure 14 shows the assignments that follow from this choice.Note the connection between the different tiers.Once we have chosen Z(S 2 ) = H S 2 to be the sphere Hilbert space, a ball bounded by S 2 must be assigned to an element of the Hilbert space: in the usual language this is the preparation of a state by the path integral.A similar relationship appears at one level higher in codimension: Since Z(S 1 ) = Rep(LG), a spatial manifold with punctures bounded by S 1 is assigned to an element Rep(LG) ( the puncture determines which element) .The latter is identified with the Hilbert space of a disk like subregion of S 2 .We can recover Z(S 2 ) by using the "inner product" on Rep(LG) which fuses together a representation and conjugate by projecting onto the invariant subspace: this fuses the Hilbert space of two disk-like subregions on S 2 back into Z(S 2 ) [34].
Note that in right of figure 14, the annulus is mapped to the identity functor on Rep(LG), generalizing the fact that the path integral on a manifold bounded by two d − 1 spatial slices is the propagator, which is identity in a strict TQFT with zero Hamiltonian.Just as the identity linear map has a resolution of of identity 1 = R |R R| over a basis |R , the identity functor has the "resolution of of identity" given by the Hilbert space where V R are representations of LG 14 .This Hilbert space agrees with the conventional one obtained via canonical quantization [35].
To sum up, from an abstract point of view, Chern Simons theory with compact gauge group is defined by Rep LG, which determines the rules for cutting and gluing along co-dimension 2 and up.We can take an analogous perspective on the definition of the our bulk gravitational path integral.In this case, there are two types of boundary circles, one at infinity assigned to the category Rep(Vir) of Virasoro representations, and one in the bulk assigned to the category Rep SL + q (2, R) of quantum group representations.Some additional assignments are shown in figure 15.In this abstract presentation, the chiral component of the Hilbert space (2.32) on the ER bridge is simply the identity functor on Rep Vir: This is the direct analogue of (4.1)Similarly, the "half wormhole" Hilbert space is the functor F from Rep(Vir) into Rep( SL + q (2, R)) discovered by Teschner as a solution of the Liouville modular bootstrap : Thus we can view our work as providing additional evidence for that the bulk theory can be described by an extended TQFT whose boundary category is given by Rep SL + q (2, R) and Rep(Vir) Finally, note that extended TQFT formulation of the bulk provides a sharp answer to the question posed in the introduction: why does the Euclidean path integral give the correct state counting?In this formulation, the bulk SL + q (2, R) edge modes are part of the data which defines the path integral.The shrinkable boundary condition is a sewing relation that ensures that the Gibbons-Hawking calculation, defined in the "closed sector" of the TQFT (with no bulk boundaries), must be consistent with a trace calculation in the open sector of the TQFT.The area term of the QES formula is then simply the Plancherel measure which forms part of the data of Rep(SL + q (2, R)).
Future directions Much work remains to be done in establishing the validity of our proposal for the bulk extended TQFT.First we should compute entanglement entropy in more general states created with operator insertions and with more general subregions.In these cases, the entanglement entropy is no longer determined by conformal symmetry alone.However, the emergence of bulk gravity can still be attributed to the dominance of the vacuum block [36] .This suggests that our bulk TQFT remains valid, although we may have to introduce Wilson lines to account for the operator insertions.Another gap in our presentation lies in the classical description of the bulk edge modes.Conceptually, it would be illuminating if we can connected our symmetry to large diffeomorphisms associated to finite subregions in classical gravity, as described in [37].It would be useful to obtain the boundary action for these edge modes, perhaps following similar ideas in [38].which introduced a classical boundary action that led to gravity edge modes associated with U q (SL(2)).Finally, we need to work out in more detail the configurations of bulk ETW branes that were invoked in section 3 to obtain the bulk dual to the factorized boundary CFT.
Going beyond our proposal for AdS3 gravity, it is natural to ask whether a similar approach can provide a canonical interpretation of de Sitter entropy.Another natural generalization is to consider charged black holes, which corresponds to having an extended chiral algebra in the boundary theory.Given that both the structure of extended TQFT and our argument for "bulk modular invariance" carries over in arbitrary dimensions, we should also consider how these structures can be leveraged to give a better understanding of the Gibbons Hawking calculation in higher dimensions.
Finally, the ultimate question we would like to answer is how to define entanglement entropy directly in the bulk string theory.Some progress was made along these lines in the context of the A model topological string [39,40].Here, bulk modular invariance and the extended TQFT framework can be applied directly to the closed string partition function on the resolved conifold, which is a fiber bundle over a sphere.In this case, the "local holography" which replaces a small disk on the sphere with the bulk edge modes takes the form of an open-closed string duality (related but not exactly the same as the Gopakumar -Vafa duality).Perhaps not surprisingly, the bulk edge modes correspond to the D branes of the topological string, with a special value of the worldvolume holonomy needed to satisfy the shrinkable boundary condition.It would be interesting to apply these ideas to a physical string theory, and find an explicit realization of these "entanglement branes".

Figure 3 :
Figure3: On the left, we have the solid torus partition function of the bulk TQFT, with no Wilson lines inserted.This can be given a perturbative interpretation as the quantization of fluctuations around a BTZ saddle obtained by filling the Euclidean time cycle.On the right, we interpret Z(τ, τ ) in terms of the solid torus in the dual channel, where we fill in the spatial cycle with a disk punctured by a superposition of Wilson lines.The punctured disk is the spatial slice of a black hole geometry describing individual microstates of the thermal ensemble(2.10)

Figure 5 :Figure 6 :
Figure 5: On the left we show the Einsten Rosen bridge as the time refection symmetric slice of the Euclidean BTZ geometry.On the right, we show the two sided Lorentzian geometry of obtained by evolving the Hartle Hawking state produced by the path integral on the left

Figure 8 :
Figure 8: The factorization of the CFT vacuum introduces regulator surfaces near the endpoints of the interval.After a conformal map, this gives a half annulus

Figure 9 :
Figure 9: From the point of view of the effective partition function for the subregion observer, the factorization map introduces two holes in the sphere partition function, where the entanglement boundary state |e is inserted.

Figure 11 :
Figure 11: The figure shows the gravitational description of the effective partition function Z ee (l).On the right we use the doubled representation where we fill in the modular time circle.On the left, we show the bulk dual in the cylinder representation of the boundary, denoted by M in the figure.The bulk dual is a manifold N whose boundary is ∂N = Q ∪ M .Q represent end of the world branes in a bulk dual that is locally AdS3.

Figure 14 :
Figure 14: Some of the path integral assignments for Chern Simons theory with compact gauge group.

Figure 15 :
Figure 15: Path integral assignments for the gravitational extended TQFT.