Fine Structure of Jackiw-Teitelboim Quantum Gravity

We investigate structural aspects of JT gravity through its BF description. In particular, we provide evidence that JT gravity should be thought of as (a coset of) the noncompact subsemigroup SL$^+$(2,R) BF theory. We highlight physical implications, including the famous sinh Plancherel measure. Exploiting this perspective, we investigate JT gravity on more generic manifolds with emphasis on the edge degrees of freedom on entangling surfaces and factorization. It is found that the one-sided JT gravity degrees of freedom are described not just by a Schwarzian on the asymptotic boundary, but also include frozen SL$^+$(2,R) degrees of freedom on the horizon, identifiable as JT gravity black hole states. Configurations with two asymptotic boundaries are linked to 2d Liouville CFT on the torus surface.


Introduction
When considering models of two-dimensional gravity, the Jackiw-Teitelboim (JT) theory plays a privileged role [1,2]: It consists of a 2d metric g µν , whose only physical degree of freedom is the Ricci scalar R, and a dilaton field Φ. It is the spherical dimensional reduction from pure 3d gravity with cosmological constant Λ and as such, it is the closest one can get in two dimensions to a dynamical pure quantum gravity theory. 1 This model also appears as the universal low-energy gravitational sector in SYK-type models [3,4,5,6,7,8,9,10,11,12,13,14,15,16].
Being the spherical sector of 3d gravity, the JT model (1.1) does not have any bulk propagating degrees of freedom, but it does have black hole solutions. 2 When considering JT gravity (1.1) on a manifold with a boundary, one finds that the dynamics is governed by Schwarzian quantum mechanics [18,19,20]: 3 the Schwarzian derivative of f , the boundary time reparametrization. Schwarzian amplitudes can be explicitly computed and indeed exhibit virtual intermediate virtual black hole states [21], see also [22,23,24,25,26,27]. We set C = 1/2 from here on out.
Ever since the early work by Jackiw in the model [28], the JT action (1.1) has been known to be identical to the action of a SL(2, R) BF theory, which in turn is the dimensional reduction of 3d SL(2, R) CS theory. 4 In [29] we claimed that JT quantum gravity is in fact a SL + (2, R) BF theory, and not a SL(2, R) BF theory, with SL + (2, R) the subsemigroup of SL(2, R) obtained be restricting SL(2, R) matrices to matrices with all elements positive. In the first part of this work (section 3), we substantiate this claim. BF theory for compact groups is understood rather well [30,31]. JT gravity is different from this in a number of ways: the relevant group is noncompact, it is in fact not a group but a subsemigroup, and finally gravitational boundary conditions constrain the group theoretic degrees on the boundary resulting in a coset construction. We will deal with each of these issues one by one throughout sections 2 and 3, gradually working our way up to JT gravity. This completes the precise BF formulation of JT gravity initiated in [32,29].
The remainder of this work is devoted to the study of JT gravity on more generic manifolds. The main focus is on JT gravity on a strip (Lorentzian) or equivalently an annulus (Euclidean), as this configuration is relevant for black hole physics. This is discussed in section 5. More general Euclidean topologies are discussed in Appendix B. In particular in section 5 we explain how cutting manifolds assigns edge dynamics or JT edge modes to entangling surfaces, in the spirit of [33]. The boundary surface can be made transparent, or equivalently the manifolds can be glued together by taking the trace in the extended Hilbert space associated with the edge degrees of freedom (see e.g. [34,35,36,37,38,39,40,33,41] and references therein). As a byproduct we establish that the spectrum of JT gravity contains one-sided black hole states; unlike the Schwarzian theory which is only holographically dual to the disk and at its own is insufficient to describe the Hilbert space of one-sided JT black holes. 5 Including edge modes then allows the theory to factorize across horizons, and in this sense solves the factorization problem posed in [42,43], in a similar fashion as it does for Maxwell theory. Within a BF formulation of JT gravity, factorization of the Hilbert space follows from basic group-theoretic properties. We highlight the factorization of BF at the very beginning of this work in section 2, 6 and come back to this for JT gravity in section 5.
As a warm-up for the JT edge mode story of section 5 we consider compact group BF in section 4. Furthermore, we repeat the edge mode story for CS in Appendix C and compare the BF formulas of section 4 with known formulas of 2d CFT. At the end of section 5 we elaborate on the identification of JT annulus amplitudes as a specific limit of Liouville torus amplitudes.
A natural class of operator insertions in JT and BF are boundary-anchored Wilson lines. Generic correlation functions with Wilson lines inserted, possibly crossed, can be written down using a diagrammatic construction. 7 Though the emphasis in this work is not on such correlation functions, at several instances we will write down some amplitudes, with the goal of showing that the BF perspective on JT allows us to understand dynamics of JT quantum gravity on generic manifolds.

Holography for Quantum Mechanics on Groups and Cosets
We start this section with a review on how quantum mechanics on the group manifold G appears when studying 2d BF theory on a disk [32,29], with compact gauge group G. Later we generalize the boundary conditions to incorporate coset models for a subgroup H ⊂ G. Finally we discuss how to generalize to noncompact groups.

Review: Compact Groups
Consider BF theory on a disk with boundary labeled by t: the boundary term can be dealt with by imposing: Path integrating over χ forces A = g −1 dg, with g periodic g(t + β) = g(t) and we are left with the untwisted particle on a group action: studied e.g. in [45,46]. This theory will henceforth be refered to as quantum mechanics on the group manifold. More generally we can include a puncture in irrep λ in the disk. Path integrating our χ now imposes a non-trivial holonomy on A: A = g −1 dg + λ. The result is the action: S[g, λ] = − dt Tr g −1 ∂ t g + λ 2 , (2.5) with partition function [47]: in terms of the weight λ ≡ λ · H, with H the Cartan generators. The Peter-Weyl theorem implies the Hilbert space of both BF on an interval and that of quantum mechanics on the group manifold consists of all matrix elements of all irreducible representations R of G: with normalized coordinate space wavefunctions: One way of formulating this conclusion, is that a quantum particle on the group manifold can be written in terms of an emergent 2d spacetime. In this sense, this is a form of holography on the worldline (see also [48]), albeit one without propagating bulk degrees of freedom, in perfect analogy with the situation for 2d WZW CFTs.

Factorization of the Thermofield Double
In [29] we introduced several useful families of time-slicings of the BF disk. Next to the defect channel slicing (Figure 1 left), in this paper we introduce two more slicings that turn out to be very useful. These are an angular slicing of the disk, and a circular slicing ( Figure 1 middle and right). The angular slicing is analogous to Schwarzschild time slicing in Euclidean signature. As we will be mostly interested in the Lorentzian continuation in this time coordinate, we will adhere to this slicing throughout most of this work. The disk partition function can be computed in either of these channels: (2.9) The thermofield double (TFD) state is a semi-disk amplitude and can accordingly be calculated using either of these slicings. The defect channel slicing is most reminiscent of the definition of the TFD state as preparing the vacuum: The disk calculation results using (2.8) in: Consider now the wavefunction g 1 · g 2 |R, a, a in combination with the defining property of representation matrices R aa (g 1 · g 2 ) = R ab (g 1 )R ba (g 2 ). We find the factorization of the wavefunction: Using this, we can equivalently write the thermofield double state as: Using (2.13) we now find: This corresponds to a state defined on the t = 0 slice with a predefined bifurcation in two pieces H L ⊗ H R . This formula is very suggestive and shows the purification of a thermal ensemble of states |R, a, b associated with the submanifold obtained by cutting a two-sided geometry on the horizon. We will make this picture explicit in section 4, where we identify the states |R, b as the edge states associated with the horizon.

Cosets G/H
The boundary condition (2.3) can be generalized into 8 for some subset of generators labeled b. This leads to a restricted particle on a group action: We will focus on the case where the generators τ b span a subalgebra h ⊂ g. The resulting theory then describes a particle on the right coset G/H. The extreme case of H = G sets all boundary values of χ = 0 and removes all boundary dynamics: as a result the theory G/G only contains topological data such as knots contained in the BF bulk.
The Peter-Weyl theorem for groups G is readily extended to right cosets G/H. Functions on the coset G/H are restricted by right H-invariance: ψ(g) = ψ(g · H). In terms of the matrix element basis functions (2.8), this leads to the constrained basis: R a0 (g) = R, a| g |R, 0 = R, a| g · H |R, 0 (2.18) with right-states constrained by invariance under H denoted by a label 0: H |R, 0 = |R, 0 . For homogeneous spaces (to which we restrict from now on), there is only one such basis vector |R, 0 within each irrep R. Thus the Hilbert space is spanned by the orthonormal basis of so-called spherical functions: We can now directly write down the propagator on the coset manifold from g = 1 to g = U : As highlighted in Appendix A.1, the angular slicing ( Figure 2 left) in BF theory is mani- 8 One can generalize this further by including sign changes as A a | bdy = ±χ a | bdy . These sign changes correspond to changing the signature of the bilinear form on the algebra g at the boundary; this boils down to switching between different real forms of the complex algebra. The magnitude of the proportionality factor can be absorbed by a field redefinition. festly equal to the boundary particle-on-a-coset evaluation. The second way of writing the amplitude in (2.20) on the other hand is interpreted as closed channel propagation between initial and final states ( Figure 2 middle and right). The matrix element is both left-and right-H-invariant and is called a zonal spherical function.
As shown in Appendix A.1, regions in the bulk diagrams enclosed by Wilson lines are weighed by dim R reminiscent of inserting a complete set of wavefunctions of the parent G theory. The deep interior does not know about the modding by H and is insensitive to the choice of boundary conditions (2.16). Indeed: interior points come with free labels a, whereas boundary labels are constrained to be 0. Accordingly, the 6j-symbols that appear at the bulk crossing of Wilson lines are those of the parent group G. For JT gravity there is a similar scenario [29]: the gravitational constraints are genuine boundary conditions and do not affect the theory in the deep bulk, as we discuss in section 3.4.
As an illuminating example of a quantum particle on a coset manifold, take the two sphere S 2 ∼ = SU (2)/U (1). In this case, the full matrix element is the Wigner D-function, the spherical functions are the standard spherical harmonics, and the zonal spherical function is the Legendre function. We provided details and some more discussion in Appendix A.2.1.
We end by remarking that cosets are conjectured to be quite dense in the space of all manifolds, and the fact that we can directly generalize our conclusion from section 2 to this case, is hence a vast expansion of the number of available models of this kind.

Noncompact Groups
Consider next quantum mechanics on a noncompact group manifold. The Peter-Weyl theorem (or equivalently the Plancherel decomposition) states how square integrable functions on the group manifold can be decomposed into representation matrix elements: The difference with compact groups is that now continuous irrep labels k will appear, as well as infinite-dimensional representations. The irrep matrix elements are orthogonal with respect to the Plancherel measure: We read off the normalized eigenfunctions: The propagator on the group manifold is written down using these ingredients as: In BF language, this is the amplitude for a disk-shaped region, so each such region is weighted with the Plancherel measure ρ(k). For several irreps, including the unitary irreps of relevance in the Peter-Weyl decomposition, the representation space is infinite-dimensional. Its dimension is found as the character evaluated at the identity element. We will prove further on that this is also equal to the Plancherel measure: 9 26) but for this we must first discuss non-compact cosets.
The propagator on coset manifolds G/H with both G and H noncompact is well-understood and described in detail in [49]. It is the generalization of (2.20): where ρ G (k) is the Plancherel measure on G and where we used R k 00 (1) = 1. Let's consider some instructive examples of this formula.
• G = SL(2, R) and H = U (1). The resulting space is the Euclidean hyperbolic plane H + 2 . The propagator on Euclidean AdS 2 is again well-known: and we recover the SL(2, R) Plancherel measure ρ(s) ∼ s tanh(πs). 10 • G = G × G and H = G diag . This is the coset realization of the group G itself. 11 For a direct product of groups Hence for the diagonal coset which is just the group, the partition function can be rewritten as: Comparing this equation with (2.25), completes the proof of (2.26).
As a further example, in Appendix A.2.2 we consider quantum mechanics on SL(2, C).

The Subsemigroup Structure of JT Gravity
In this section, we build up towards describing JT gravity as a SL + (2, R) BF theory. In section 3.3 we show that one can consistently describe quantum mechanics on the subsemigroup SL + (2, R). In section 3.4, we work out the coset perspective on the JT disk amplitudes. In order to appreciate the difference between SL(2, R) and SL + (2, R), we present a short recap of the relevant representation theory in Appendices D and E.

Density of States, Plancherel Measure and Shockwaves
Let us first present an argument in favor of the SL + (2, R) structure. For SL + (2, R) the Plancherel measure is sinh 2π √ E (E.15) whereas for SL(2, R) the Plancherel measure is tanh π √ E (D. 30). The former has a Cardy rise at large energies, consistent with the semiclassical Bekenstein-Hawking entropy formula, the latter doesn't. So an SL(2, R) BF theory will not result in a correct calculation of the black hole entropy [43], as there are simply not enough microstates. 12 Let us briefly touch on a second physical application for which it is pivotal that we describe JT gravity as a BF theory with Plancherel measure sinh 2π √ E, attributing this weight to each disk-shaped region. Recently, the semi-classical limit of the exact JT correlation functions was investigated in [51]. Analyzing generic diagrams with crossing bilocal lines, the eikonal shockwave expressions were reproduced [52,53], where the corresponding shockwave diagram in real time is topologically the same as the crossing lines disk diagram. When performing such a calculation, it is crucial that each region in the Euclidean bulk carries a measure factor sinh 2π √ E, as these factors ultimately determine the saddle point that represents the mass of the original black hole on which the shockwaves propagate. This is even more crucial for regions that are completely sealed off from the holographic boundary ( Figure 3), as no coset conditions are imposed at all for such region and the theory is sensitive to the full BF theory. √ E measure to agree with a semi-classical shockwave computation of the same topology (Right).

Limits of 3d Gravity and Quantum Groups
Next, we elaborate on a much deeper structural reason for the group-theoretic SL + (2, R) structure of JT gravity. Jackiw-Teitelboim gravity is unambiguously defined as a suitable dimensional reduction of 12 We will elaborate on the microstates further on. 3d gravity. The dynamics of 3d gravity are governed in essence by the Virasoro modular bootstrap, which in turn is governed by the representation theory of the quantum group SL + q (2, R). This was discussed in detail by Ponsot and Teschner in [54,55]. By taking suitable limits of their formulas we end up uniquely with the representation theory of SL + (2, R).
In discussing the harmonic analysis on the quantum group SL + q (2, R) in the context of the Virasoro modular bootstrap, Ponsot and Teschner write down the following Plancherel decomposition: 13 where P P are the self-dual representations of U q (sl(2, R)), dµ(P ) is the Plancherel measure on SL + q (2, R) and P > 0. Explicitly, the measure reads: where we recognize the Virasoro modular S matrix element S P 0 . In the classical limit b → 0, with P = bk, this becomes the Sklyanin measure: which is just the Plancherel measure on SL + (2, R). The objects appearing on the r.h.s. in (3.1) are viewed more naturally as representations of the modular double U q (sl(2, R)) ⊗ Uq(sl(2, R)). The classical limit q → 1 of these representations does not yield a double copy of the classical group SL(2, R), instead the representations are self-dual, and form a basis of functions on SL + (2, R) [56]. Hence the classical limit of (3.1) is just the Plancherel decomposition of SL + (2, R): Note that no discrete representation are present. The Plancherel decomposition (3.5) is to be read as the statement that the matrix elements K ++ s 1 s 2 (g) (E.13) of SL + (2, R) are complete in SL + (2, R) in the sense that: for uniquely determined expansion coefficients c k,s 1 s 2 , with the associated orthonormality and completeness relation: As a consistency check on the limiting procedure from (3.1) to (3.5), recall from Appendix E the SL + (2, R) gravitational wavefunction: which is the mixed parabolic matrix element of the Cartan element φ. In the mathematics literature, this is the so-called Whittaker function (or coefficient) [57,58,59,60]. The JT result (3.9) matches with the classical limit b → 0 of the Whittaker function of U q (sl(2, R)) ⊗ Uq(sl(2, R)) [61].
When considering out-of-time ordered correlation functions in JT gravity, 6j-symbols of SL + (2, R) pop up [29]. Alternatively, these 6j symbols are obtained as the classical limit b → 0 of the braiding matrices of Virasoro conformal blocks. The fusion matrices of Virasoro are given as 6j-symbols of the quantum group SL + q (2, R). 14 As a consistency check, the orthogonality relation of the quantum 6j symbols [62]: is taken in the b → 0 limit to (3.4): dp p sinh 2πp Because JT gravity is a TQFT, gravitational Wilson lines can be uncrossed in the bulk at no cost. This can be proven directly in the path integral before initiating an explicit calculation [29]. The above formula which includes the 6j-symbols that appear at bulk crossings of Wilson lines in JT, expresses precisely this operation, given that we work with a BF theory whose Plancherel decomposition is precisely (3.5). So on top of identifying the 6j-symbols as those of SL + (2, R), (3.11) also proves that the Plancherel decomposition of the BF theory associated to JT gravity is precisely (3.5).
A related point is that in [21,51] Schwarzian OTO correlators were obtained by applying the braiding R-matrix in 2d Virasoro CFT for each line crossing. The double-scaling Schwarzian limit then demonstrated that each such procedure generates an additional momentum integral, with the k i sinh(2πk i ) measure accompanying it. This includes regions that end up being completely enclosed in the interior of the bulk.

Quantum Mechanics on SL + (2, R)
Motivated by the previous subsections, we will now prove that the particle on the subsemigroup SL + (2, R) or equivalently SL + (2, R) BF on a disk is a mathematically consistent model. The contents of this section build on some SL + (2, R) representation theory summarized in Appendix E. The consistency hinges on the fact that the SL + (2, R) manifold is a submanifold of the SL(2, R) manifold.

Partition Function
The particle on SL + (2, R) is defined by the path integral: on the thermal manifold g(t + β) = g(t) and constrained to the SL + (2, R) patch γ − , γ + > 0 (E.10). Within a Hamiltonian context, we obtain the propagator (or twisted partition function) on the SL + (2, R) manifold: Here, α and β label the hyperbolic basis of SL + (2, R). Because we are considering propagation on the SL + (2, R) submanifold, obviously g and U λ are restricted to be positive. The matrix elements of SL + (2, R) are a subset of the hyperbolic basis matrix elements of SL(2, R): with composition property K(g 1 · g 2 ) = K(g 1 ) · K(g 2 ) and inverse K(g −1 ) = K(g) −1 . Using the explicit expressions for the matrix elements [63,64], one readily finds Similarly, the matrix representation can be shown to be unitary: 15 For g positive, the property (3.15) can be used to show that group composition of SL(2, R) implies (3.17) and hence: 16 Using (3.15), one furthermore proves that the following property holds: for any g ∈ SL(2, R). Putting the pieces together we get Hence the propagator on the SL + (2, R) manifold becomes: Notice that we recover the fact that the SL + (2, R) manifold is homogeneous, simply because the SL(2, R) manifold is.
Let's now give an explicit expression of the characters, exploiting its embedding within SL(2, R). The SL(2, R) character χ k (U λ ) = Tr K ++ (U λ ) + Tr K −− (U λ ) Using formulas (9) and (10) on p358 of [63] one finds K −− (λ) = K ++ (λ), and hence χ k (U λ ) = 2χ + k (U λ ). 17 The net factor 2 is irrelevant and the appropriate finite characters for SL(2, R) are 18 (3.25) 16 Notice here that it is crucial that SL + (2, R) is not just a semigroup, but a subsemigroup of SL(2, R). In particular the embedding of SL(2, R) allows us to give meaning to g −1 for g positive. Elements of SL + (2, R) do have an inverse, but it lies outside of SL + (2, R). 17 In fact we can use [63] to prove a more generic property. The general character of SL(2, R) can be rewritten as: with e = diag(−1, 1). The action of e on wavefunctions f µ (x) defined on the positive axis is: Explicitly for the relevant wavefunctions we obtain −x|s = e πs x|s and s|−x = e −πs s|x where we used −1 = e iπ since we cannot go through the branch cut. Writing the character as χ + µ (e · g · e) = ds s| e · g · e |s , inserting a completeness relation in the x-basis and using the above properties one finds that Using this in (3.22) and again dropping an irrelevant factor 2 we obtain for all positive g. 18 See Appendix G.
Equation (3.21) can then be written more explicitly as: The vacuum character on the other hand is the Plancherel measure by (2.26): So the partition function of a particle on SL + (2, R) is:

Correlation Functions
We can now use the methodology of [29] to calculate a generic SL + (2, R) disk correlation function, decomposing the full amplitude into propagators and 3j-symbols. 19 This decomposition can also be appreciated by starting solely with the boundary theory and realizing that this immediately gives a particular bulk slicing of the amplitude, the coset slicing. We provide details on this argument in Appendix A.1.
By (3.5), a complete set of states of SL + (2, R) BF theory is given by the semigroup element states |g with g ∈ SL + (2, R) resulting in the resolution of the identity: Amplitudes of SL + (2, R) BF including several Wilson line insertions can be constructed as usual by cutting the manifold into disk-shaped regions, inserting completeness relations (3.29) on the edges of the regions, calculating the amplitude for each disk-like region with fixed g i on the boundaries, and then gluing the disk back together including the external Wilson lines by performing integrals over g i of the type: where we used the crucial property (3.18). On the right hand side one recognizes the vertex functions of interest as the SL + (2, R) (hyperbolic) 3j symbols.
There is still the question of mathematical consistency of this calculation to be answered. For SL + (2, R), within each disk-like region, the calculation only works as explained around (3.13) if the disk can be written as Hamiltonian propagation from positive group elements to other positive group elements. 20 Positivity of a group element along a certain line requires the choice of an orientation on this line. As illustrated for example in Figure 4, this is accomplished by choosing a set of oriented Cauchy surfaces within the disk.

Figure 4:
We evolve a set of oriented Cauchy slices (black) through the disk. In this way, an orientation is associated to each of the boundaries of the smaller disks (blue) that allows for an SL + (2, R) BF calculation in each of these disks. The black dot represents the horizon.

Constrained Asymptotic States
The Schwarzian theory dual to JT gravity on a disk, can be viewed as quantum mechanics on a particular coset of SL + (2, R), inherited from the coset constraints to obtain 2d Liouville CFT from WZW CFT [65,66,67]. It is instructive to see that we can obtain the JT disk amplitudes from this coset construction using the results of section 2.3.
Explicitly, gravitational disk wavefunctions are associated with the parabolic state |i + defined in Appendix E to satisfy J + |i + = i |i + [29,67]. In terms of functions f on SL + (2, R), the condition is a basis for the gravitational coset. Indeed, the functions R s s (g) = s | g |s are complete in L 2 (SL + (2, R)). Of these, only those linear combinations of the form fulfill the gravitational constraints (3.31). The Hilbert space can be written in the form: or in the dual group basis as the states |φ, γ − . The Schwarzian states |k, i, i respectively |φ used in [42,29,43,21] live on the defect slices of Figure 1 left.

Edge States of BF
Next, we will explain the precise nature of the edge degrees of freedom in Jackiw-Teitelboim gravity that appear at entangling surfaces (or black hole horizons). To obtain these edge dynamics we follow the logic of [33].
As earlier, we start by focusing on compact BF theory; the generalization to JT gravity becomes straightforward with the previous section in mind.

Edge Dynamics from the Path Integral
The correct way to split the BF Lorentzian path integral of a surface Σ in two pieces L and R proceeds by introducing a functional delta constraint as in [33] ( Figure 5): Integrating out χ L and χ R , forces the connections to be flat in the bulk of L and R and the path integral over A L (and A R ) is reduced to a path integral over independent boundary group element configurations on all boundaries of Σ as well as on the gluing boundaries. 21 Figure 5: Gluing two BF sectors along one boundary in terms of two particle-on-a-group models g L and g R .
The path integral over A in general also includes an integral over holonomies A = U λ along the gluing boundary. For example, if L is a disk, the holonomy is fixed, but if L is an annulus, the holonomy is an additional degree of freedom to be integrated over.

Explicitly, localization on flat connections results in
In the path integral (4.1), the functional delta becomes: and two twisted particle on a group actions pop up associated with the gluing surface (one for L and one for R). The action on the left boundary, is minus the right one, as the orientation of the boundary surface in L respectively R is opposite. 22 As a result, the two actions cancel when we enforce the functional delta constraints and set λ R = λ L ≡ λ: 23 The dots represent the other degrees of freedom in L and R that are irrelevant for this argument. This procedure consistently glues the submanifolds together.
Notice that the argument of the functional delta in (4.2) is just the current density on 22 This descends from the parity transformation on the Chern-Simons action taking k → −k to flip the orientation. 23 The final equality uses that Z S 2 = 1. There are several ways to argue for this. Performing the doublescaling large k limit on the Chern-Simons partition function on S 2 × S 1 is trivial since [68] Alternatively, the volume of the moduli space of flat gauge connections on S 2 is trivial: since there is only 1 gauge orbit on S 2 .
the boundary, so it can be read as δ(J L − J R ). The theory associated with the submanifold R only is obtained from (4.5) as in [33] by dropping all reference to L: (4.6) As shown by the second equality, this formula can be interpreted as the path integral on the right manifold sourced by a boundary current J R , including an additional path integral over the boundary charges J R to account for the edge degrees of freedom, in the spirit of [33]. In canonical language, this means there is an extended Hilbert space that accounts for edge states on the dividing surface. The gluing condition δ(J L − J R ) acts as a Gubta-Bleuler constraint that extracts the physical subsector from the extended Hilbert space. 24 The path-integral over J L = J R glues the manifolds together.

Two-Boundary Models
As an application of the above, we will show how to split a spatial interval in two pieces.
Consider first the BF model on a Lorentzian strip I. The Euclidean configuration associated with this setup in I × S 1 with two circular boundaries that break topological invariance. This manifests itself as the dependence of the path integral on a choice of metric / einbein on the boundary curves, through its circumferences β L respectively β R ( Figure 6). Flatness of F = 0 implies A = dgg −1 + λ where λ is an unspecified holonomy: the time circle is not contractable hence λ is a physical degree of freedom of the theory to be integrated over. We obtain the path integral for this configuration as: (4.7) The result of the Euclidean path integral is then:  where one recognizes the twisted particle on a group partition functions (2.6). Writing this out using orthogonality of the finite characters 25 this becomes ( Figure 7): Two interesting limits are the thermal cylinder where we take β L = β R and the disk obtained by β L = 0. They are shown in Figure 6. For the thermal case, (4.10) implies that the spectrum of the theory consists of the states |R, a, b ≡ |R, a ⊗ |R, b and the Hamiltonian is H L + H R . Due to the Peter-Weyl theorem, the Hilbert space of 2d BF on an interval is indeed given by these states, to be interpreted as open strings with one endpoint on each boundary (Figure 6 left). The latter case β L = 0 comes into play when constructing the thermofield double from the Rindler Hilbert space or equivalently when computing vacuum entanglement entropy of an interval with an adjacent interval. As shown by the modular flow in Figure 8, the particle on a group on the inner boundary is frozen and does not contribute to the modular Hamiltonian: K = βH R . We recover the disk amplitude: which includes a sum over edge modes, and is comparable to (4.6). The edge degrees of freedom associated with the horizon or inner boundary are identified as the states |R, a . The precise microstate |R, a contributes zero energy and does not affect any of the bulk observations a right-observer would perform, which translates to the fact that the correlation functions in a pure microstate |R, a ⊗ H R are independent of a. c R a b Figure 8: Splitting an interval in two pieces using the modular Hamiltonian.
Formula (4.11) is a consistency check: including the correct edge degrees of freedom to a one-sided theory ensures that the trace in the Rindler Hilbert space equals the thermal disk path integral. Graphically, summing over edge degrees of freedom a stuffs the hole in the annulus (Figures 6 and 7 right). This proves the claims made around (2.15). From the above we can directly purify the density matrix to re-obtain the thermofield double state: The conclusion here is that whereas (2.11) and (4.12) describe the same state, only the latter makes manifest the factorization of the theory, as it can be directly read as a purification of the Rindler thermal density matrix, which crucially includes an edge sector on the horizon.

Two-Boundary Models and Black Hole States of JT gravity
In this section we generalize the BF discussion of the previous section to JT gravity. We consider two different two-boundary models. There is a distinction to be made between a holographic boundary, where gravitational constraints are to be imposed, and entanglement boundaries where no such constraints are imposed.
First we discuss a configuration with two holographic boundaries. Second, we consider one holographic boundary and one entangling boundary, which describes a one-sided black hole configuration. Finally, in sections 5.3 and 5.4 we return to the configuration of two holographic boundaries, and identify the correlation functions with the direct Schwarzian double-scaling limit of Liouville torus amplitudes.

Wormhole States
Consider first Jackiw-Teitelboim gravity between two holographic (Schwarzian) boundaries, L and R, on which the gravitational boundary conditions are to be enforced [69,70,71,29]: in terms of a dynamical function T (τ ) and the generators (D.2). These boundary conditions act by constraining the boundary theory from a particle on SL + (2, R) to the Schwarzian theory ( Figure 9) [29], in terms of the time reparametrizations f L and f R of the left-respectively right holographic boundary, defined as: The Hilbert space of this gravitational coset system is of the form |k, i, i , as we will demonstrate. The thermal path integral for this configuration is the analogue of (4.7) and includes an integral over conjugacy class elements (or orbits) λ: with the twisted Schwarzian action where f (τ + β) = f (τ ) + β, and the twisted Schwarzian partition function [32,21]: Explicitly to derive this one simply takes the Schwarzian double-scaling limit of a Virasoro character χ λ (τ ) [32,21]. 26 The Virasoro modular S-matrices are given by S k λ = cos 2πλk, (5.6) Using S-matrix unitarity We deduce that only the constrained states |k, i, i make up the Hilbert space of this theory. We will call these the wormhole states of JT gravity. The states in this Hilbert space are labeled in the same way as in the defect channel slicing of Figure 1 left and as in the Hilbert space of a single Schwarzian theory [21]; as in each of these scenarios we are considering a Cauchy surface connecting two constrained boundaries ( Figure 9).

Black Hole States and Factorization
The question arises how Jackiw-Teitelboim gravity behaves away from the asymptotic boundary. Does it behave as an unconstrained SL + (2, R) BF theory or does it still feel the constraints? In particular, when cutting a manifold in the sense of (4.5), do we get Schwarzian actions on the gluing boundaries or particle on SL + (2, R) actions? For cosets in section 2.3, we found that interior regions are insensitive to the constraints and behave as if they are part of the parent G theory. We provided arguments in section 3.4 that the gravitational theory should be viewed as a specific example of a coset model. This suggests the edge dynamics of JT on a gluing surface is that of a particle on SL + (2, R).
Following the logic around Figure 8, the edge theory is frozen on the horizon. Using the twisted SL + (2, R) partition function Z + (β, λ) from (3.21), we can write: The finite characters of SL + (2, R) (3.23) are χ + k (U λ ) = cos 2πkλ. Notice that these are identical to the classical b → 0 limit of the Virasoro S-matrix (5.6) appearing in Z(β, λ). 27 This means we can use SL + (2, R) character orthogonality to rewrite (5.10) as From this one finds the spectrum of states as |k, s, i ≡ |k, s ⊗ |k, i , with s a hyperbolic SL + (2, R) label as introduced in Appendix E. The result (5.12) is the JT disk amplitude (5.5), proving that we have included precisely the correct edge states by postulating a particle on SL + (2, R) lives on the entangling surface. 28 In the context of Section 2.2, this is just the statement that an SL + (2, R) representation matrix factorizes using its defining property as So the Hartle-Hawking calculation already illustrates the edge states should be the states |k, s , and this is confirmed by (5.12).
From (5.12) we can directly write down the purification of the thermal ensemble: The Von Neumann entropy of the thermal state was calculated in [43] and gives the Bekenstein Hawking entropy in the limit where the bulk is classical. In writing (5.15) we have 27 What we have proven here is a non-compact generalization of a well-known result. Consider the modular S-matrices associated to two compact groups G and G/H for G and H compact. It is an elementary result that these are identical S G = S G/H . In particular this carries through in the classical double-scaling k → ∞ limit to: pinpointed the gravitational states responsible for this entropy, so the conclusion is that the states |k, s ⊗ |k, i are the black hole microstates of JT gravity.
Using (5.14) we can rewrite the TFD state of JT (5.15) in terms of wormhole states as: This is the form that appeared in the literature [42,43], where factorization is not manifest.
Projecting it onto a g-eigenstate, one writes: (5.17) The group variable ϕ = −d can be geometrically interpreted as a bulk length parameter between both sides, as shown in [26]. This is a direct geometric interpretation of the abstract group variable.

Two-Boundary Correlation Functions
Let us return to the situation with two asymptotic boundaries discussed in section 5.1. In this setup, we encounter a new type of Wilson line operators with endpoints on different boundaries. 29 In the dual boundary theory one is led to studying correlators of the type: for one or more bilocal operators connecting both boundaries O λ,LR (τ 1 , τ 2 ). In the BF formulation of JT gravity this is easy. But let us first give a more precise holographic expression for the bulk crossing Wilson line O λ,LR (τ 1 , τ 2 ). After integrating out χ, we find: where {f L,R , τ } = T L,R (τ ), for possibly different time reparametrizations f L and f R at the endpoints. The proof can be found in Appendix F. 30 Performing a final reparameterization 29 Such operators are SL(2, R) covariant under SL(2, R) L or SL(2, R) R separately, but invariant under only the diagonal combination. W.r.t. each boundary, these operators are of the form of those discussed in Appendix D of [21], which were analyzed in terms of KZ equations. 30 In earlier work [29], we demonstrated this for a Wilson line with both endpoints on the same boundary (with hence f L = f R ), where the Wilson line could be deformed to lie entirely within the boundary. This proof no longer holds for bulk-crossing Wilson lines, or for Wilson lines encircling punctures such as those discussed in Appendix A of [29].
to the variables used in the action (5.4) f L,R → tanh π β LR λf L,R , we find: Let us emphasize that the two asymptotic boundary model discussed here is very different from the TFD. The model, unlike the TFD, has two independent clocks f L and f R running on each of its boundaries, reflected in the separate temperatures β L and β R . 31 Symmetries of the model on both boundaries. The independence of both boundary times shows that the amplitude for a single such bulk crossing Wilson line will be time-independent: the time t L of an incoming pulse in the L system learns the L observer nothing about the time t R at which the pulse left the R system.
As an application of the BF perspective on JT gravity let us write down two single Wilson line correlation functions in this model.
Taking a particle on SL + (2, R) on the inner boundary, we find the correlator for a single bilocal straddling the annulus: The special case of β L = 0 can be interpreted as a Wilson line stretching from the holographic boundary to the black hole horizon, the fact that the resulting amplitude vanishes is a manifestation of the fact that bulk operators do not couple to horizon degrees of freedom 31 The annulus amplitude contains two separate boundary theories at finite temperature simultaneously, whereas the TFD configuration is only thermal upon tracing out half of the theory. The two sides of the TFD state are mirror images of one another and hence it takes as many degrees of freedom to describe the dynamical clock for a TFD configuration than for a single-sided configuration. [40,33]. Taking the inner boundary to be the Schwarzian instead, we find: 32 the numerator indeed reduces to (5.9) in the → 0 limit. Continuing to real-time is trivial and in terms of the time-ordered and anti-time-ordered two-point correlators G ± (t L , t R ) ≡ O ,± LR (t L , t R ) , one readily has: The dual spacetime is connected since the correlator (5.24) is non-zero, but no communication can occur between both boundaries.
Similarly, a Wilson line correlator with both endpoints on the same boundary is: (5.26) Taking the infinite redshift β L → 0 limit, the result is the same as the Schwarzian disk computation, demonstrating exterior observables are insensitive to the precise microphysics in the edge sector. This conclusion is readily generalized to arbitrary correlation functions, and is qualitatively the same conclusion as that obtained in dynamical theories such as Maxwell in arbitrary dimensions [40]. Taking instead a Schwarzian to live on the inner boundary, one finds These kinds of computations can be readily generalized to multi-boundary Euclidean JT configurations, we provide an example in Appendix B.

A Liouville Perspective
We demonstrate here that as an alternative to the BF calculations, JT correlators of the type (5.18) can alternatively be obtained by taking the Schwarzian double-scaling limit of Liouville CFT on a torus surface. Insertions of Liouville primary vertex operators then corresponds to the Schwarzian wormhole-crossing bilocals (5.20). This is a direct generalization 32 We used the known expression for the Schwarzian 3j-symbol: of the argument used in [21,32] where Schwarzian disk correlators were obtained by taking the Schwarzian double-scaling limit on Liouville on the cylinder between ZZ-branes.
The Liouville torus partition function is well-known [72]: and is identical to that of a 2d free boson due to the KPZ scaling law [73]. It famously contains only the continuous Virasoro primaries at with the vacuum h = 0 being left out, a well-known argument against a gravity dual of Liouville CFT. It is modular invariant since +∞ 0 dP S P P 1 S P 2 P = +∞ 0 dP cos(4πP P 1 ) cos(4πP P 2 ) = 1 2 δ(P 1 − P 2 ). (5.30) We will reproduce this partition function (5.28) from the Liouville path integral perspective, by deconstructing it into Virasoro coadjoint orbits. Consider the phase space Liouville path integral on the torus surface: with the Liouville Hamiltonian: We perform the following field redefinition (f = ∂ σ f ): 33 in terms of fields f L and f R , which are quasiperiodic in the sense: with λ labeling orbits or conjugacy class elements. 34 The path integral over φ and π φ is replaced by a path integral over f L and f R as well as an integral over λ, since λ labels physically inequivalent configurations: with the unit measure on the space of conjugacy class elements (see Appendix G). The Jacobian in this transformation follows from the Pfaffian of the symplectic form. It was computed in this setup explicitly in [32] and will not be written explicitly here. The Hamiltonian (5.32) is transformed into: 35 Rescaling the fields as f L → λf L and f R → λf R , one finds that the Liouville path integral (5.28) decomposes into a diagonal sum (integral) over coadjoint orbit actions: 36 with [80,81] In the double-scaling Schwarzian limit of interest, one takes the central charge c ∼ 1/b → +∞ along with the circumference in the t-direction to go to zero, keeping the product fixed (for more details see [21,32]). This eliminates the π φφ term in the action (the term in square brackets in (5.40)), and leaves only the Hamiltonian (5.37). Setting σ → τ , this 34 One can appreciate the appearance of this extra parameter λ by noting that (5.33) and (5.34) describe periodic Liouville fields φ and π φ for any value of λ. This parameter should hence be included in the phase space description of the theory. This is analogous to what happens in compact WZW theories [79,32,78]. 35 There is a renormalization effect here that should be found by treating the Liouville determinant more carefully. We have effectively set c = 6/b 2 , which is the classical result. Tracking this effect more carefully will not bother us here, as we are interested in the Schwarzian double scaling limit that includes c → +∞. 36 There is a common U (1) redundancy in the field redefinition (5.33) and (5.34), f L,R → f L,R + α, so the integration space is reduces precisely to (5.3). 37 It furthermore follows that the field redefinition (5.33) maps Liouville vertex operators e 2 φ to Wilson lines stretched between the two asymptotic boundaries (5.20).
We conclude that the Schwarzian limit of Liouville torus correlation functions compute correlation functions of the type (5.18). The two Schwarzian sectors interact indirectly through modular invariance of the torus, and directly by bilocal operator insertions ( Figure  10). An immediate check is on the partition function itself. The Liouville torus partition function (5.28) reduces to the JT gravity partition function (5.9) in the Schwarzian limit. 38 For the correlation function (5.24), one can use the q → 0 Schwarzian double-scaling limit of the torus conformal block expansion of the one-point function (V (z,z) = e 2 φ ) [82,83,84]: As in [21], setting P s = bk s , the block H Ps, (q) reduces in this limit to the primary propagation and the DOZZ coefficient C(−P s , , P s ) then precisely yields (5.24). The independence 37 In [32], this system was studied between ZZ-branes. The latter are dealt with with the doubling trick, combining the f L -and f R -degrees of freedom into a single periodic field F , directly reproducing the Virasoro vacuum character. Changing branes amounts to changing the character to any Liouville primary of interest. 38 Note that the absence of the sinh-measure here is in direct unison with the flat measure on Liouville theory itself. of the correlator on the bilocal times τ 1 and τ 2 originates in this language from the independence on the location of the Liouville primary vertex operator V . Generalizations to multiple such insertions is then straightforward using Liouville techniques by inserting complete sets of Liouville states and reducing all conformal blocks to primary propagation as in [21]. Within our choice of variables, the torus conformal blocks are graphically: Notice that calculating bilocals with endpoints on the same asymptotic boundary seems to be impossible within the Liouville language. In that respect, the BF formulation of JT gravity developed above and in [32,29] is more versatile.

Discussion
We summarize the main lessons learned about the BF structure of JT gravity: • JT quantum gravity is precisely equal to an SL + (2, R) BF theory with coset boundary constraints. The ubiquitous sinh 2π √ E density of states in the theory is simply the Plancherel measure of SL + (2, R). For almost all purposes, neither the fact that SL + (2, R) is noncompact, nor the fact that it is only a subsemigroup affect any of the diagrammatic rules for constructing BF amplitudes. The gravitational boundary conditions can be viewed as a coset construction in the BF language.
• The only real subtlety arising from the noncompactness of the group, is that BF calculations on manifolds with handles or more than two boundaries may diverge. One can deal with this as explained in Appendix B by isolating annuli with Schwarzian boundaries from the remaining amplitude, using known results for both 39 and then gluing the pieces back together. In all, we are now in principle able to calculate generic JT gravity correlation functions on arbitrary manifolds.
In the second part of this work we investigated edge dynamics and entanglement in JT gravity. Let us summarize the results.
• By cutting the JT path integral on a given manifold we learned that an SL + (2, R) quantum mechanics lives on all entangling boundaries, whereas the asymptotic boundaries are described by Schwarzian quantum mechanics.
• From the perspective of a Rindler observer, the SL + (2, R) quantum mechanics on the horizon is frozen due to infinite redshift. Its degrees of freedom represent the JT black hole microstates and account for the Bekenstein Hawking entropy [43]. Alternatively these new degrees of freedom simply arise in the factorization of a BF state on an interval into smaller intervals. 40 The extended Hilbert space associated with the resulting subregion includes the edge states [40,33,36,37] or black hole microstates.
Finally, we discussed JT gravity on a manifold with two Schwarzian boundaries, where the full path integral of the system can be written in terms of Schwarzian quantum mechanics on both boundaries. The resulting theory is identical to the double-scaling Schwarzian limit of the full Liouville path integral. This identification is strengthened by the fact that amplitudes of wormhole-crossing Wilson lines match with the double-scaling limit of Virasoro torus conformal block expansions. Besides providing an alternative perspective on JT amplitudes, this provides the torus conformal block literature [82,83,84] with an interesting limit, and connects it to the SYK literature. This may come as somewhat of a surprise. Though Virasoro coadjoint orbit models are the building blocks of 3d quantum gravity, the role of full-fledged Liouville theory in 3d quantum gravity is less clear [86,87,88,89,90]. However, in the double-scaling limit, we do find a 2d quantum gravity application of full Liouville.
We end with some speculation about entanglement and black hole entropy in 3d pure gravity. We saw in Appendix C that the modular partition function for CS theory in a Rindler wedge was just calculating the solid torus amplitude χ 0 (S · β). Accordingly, to compute the modular partition function for 3d gravity (which consists roughly of two copies of SL(2, R) CS of opposite chirality), we would naively write: where one recognizes the Virasoro vacuum character. The resulting density of states is ρ(λ, λ ) = S λ 0 S λ 0 , which is the expression written down in [91] and which matches the semiclassical BTZ black hole entropy. A state counting interpretation in terms of one-sided states along the lines of (C.13) is far less obvious. For compact cosets G/H the conclusion would be that a frozen G WZW model lives on the horizon and accounts for the edge states. For 3d gravity, there is an immediate obstruction: the SL(2, R) WZW model is nonunitary. The precise resolution in this case certainly deserves further study.

A BF Amplitudes
We review the Feynman rules for correlation functions of boundary-anchored Wilson lines in BF [32,29].
• Draw a disk with the Wilson line insertions.
• Each disk-shaped region is assigned an irrep R i , and contributes a weight dim R i . A label m i denoting eigenvalues of a maximal set of commuting generators is assigned to each boundary segment. One sums over these labels R i and m i to obtain the amplitude.
• Each boundary segment carries a Hamiltonian propagation factor proportional to the length L i of the relevant segment i (depending on the chosen einbein). Each intersection of an endpoint of a Wilson line with the boundary is weighted with a 3j-symbol.
• A Wilson line crossing in the bulk comes with a 6j-symbol of the group.

A.1 Coset Slicing
We demonstrate next that the slicing of coset models can be identified with angular slicing in the BF model directly.
In [29] we computed a generic correlation function directly within the particle-on-a-group model by inserting complete sets of states in between all operator insertions. E.g. for three bilocals, one inserts complete sets of 1 = dg i |g i g i |, i = 1 . . . 6 in between all legs of operators, followed by complete sets of |R, a, b to diagonalize the Hamiltonian propagation factors e −t ij H . The computation can then be manifestly identified with a computation in BF in angular slicing ( Figure 11 left and middle) [29]. This identification immediately extends to Introducing complete sets in coordinate space x like this, we can use precisely the same construction as above to get the generic correlator.
In [29], we explained that these pie-shaped bulk diagrams may be freely deformed into diagrams with enclosed regions (see e.g. Figure 11 right). In particular, it can be shown that enclosed interior regions obtained in this manner are to be weighted with dim R coming from the G parent theory; the interior of the disk does not know about the modding by H.

A.2 Examples
To illuminate the more abstract discussion of section 2 we work out two examples.
The partition function is (2.20): which indeed matches the spectrum of the rigid rotor quantum mechanical system. The matrix elements R ab (g) of SU (2)  θ| j, 0, 0 = j, 0| g(θ, φ) |j, 0 = P j (cos θ). (A.10) Using these, we can e.g. write down the correlator with a single boundary-anchored Wilson line: Slicing this amplitude using Cauchy surfaces with both endpoints on the outer boundary, requires using the R 00 (θ) zonal spherical functions. Using the angular slicing where only one endpoint touches the boundary, requires using spherical functions R i0 (θ, φ) instead. Formula (A.11) is obtained using the well-known identities: As explained above, regions that are in the deep interior and closed off from the boundary, see the full SU (2) BF model with matrix elements the Wigner D-functions.
We can give a complementary perspective on this by looking at the Casimir differential equation. The left-and right regular representation (realization) of the su(2) algebra in Euler angles (A.6), found by imposingD a L g = τ a g andD a R g = gτ a is given by the sets of differential operators: The su(2) Casimir equation is then directly found as solved by the Wigner D-functions D j m,m (θ, φ, ψ). Setting J 3 R =D 3 R = 0, one finds 16) in terms of the spherical harmonics Y j m (θ, φ). Additionally setting J 3 L =D 3 L = 0, one finds ∂ 2 θ + cot θ∂ θ P j (cos θ) = j(j + 1)P j (cos θ), (A. 17) solved in terms of the Legendre functions P j (cos θ). This process of imposing the coset conditions J 3 R = 0 and J 3 L = 0 is the direct analogue of the gravitational / Liouville constraints discussed in Appendix F of [29]. The left-and right-regular representation operators act on the bra, respectively the ket of the matrix element R ab (g) ≡ R, a| g |R, b .

A.2.2 Quantum Mechanics on SL(2, C)
As a second instructive example we consider particle on SL(2, C). From (2.31) we obtain the partition function: To obtain a basis of the representation, one conventionally diagonalizes the generator J 3 = m, or after Fourier transforming to a continuous 2-sphere of labels (x,x): 41 Within this basis, inserting a single boundary-anchored Wilson line O s x,x (τ 1 , τ 2 ) of SL(2, C) gives the correlator: are the well-known 3j-symbols of SL(2, C) [95], identifiable as conformal three-point functions as recently discussed in [96].

B Other Euclidean Topologies
As an application of the BF perspective on JT gravity, we show how to calculate amplitudes of generic JT gravity Euclidean manifolds. We start with a discussion on one-boundary manifolds with handles and later on generalize to multi-boundary models.
Consider a disk with multiple handles attached as for example in Figure 12 left. The goal here is to explain how to calculate such contributions, within the BF framework of section 3.3 and [29].
If we naively apply the BF cutting and gluing rules to such surfaces by evolving SL + (2, R) states through the manifold, we obtain for a disk with h handles: (B.1) 41 See e.g. [92,93,94]. This integral diverges in the k → 0 region, an artifact of the non-compactness of the group. This divergence knows its origin in the infinite volume of the moduli space of flat connections. This contribution can be isolated by separately evolving through an annulus that includes the Schwarzian boundary, and through the remaining handlebody. The two are then glued together by introducing the resolution of the identity on a circular Cauchy slice: with λ conjugacy class elements. The path integral hence decomposes as: in terms of the twisted Schwarzian path integral: which is the divergent volume of the moduli space of flat connections. 42 A more careful treatment [85] however results in a finite answer for this topological gravity amplitude V h (λ) as the Weil-Petersson volume.
The conclusion is that we can obtain a sensible JT handlebody result by cutting off annuli at all the asymptotic boundaries, calculating the corresponding amplitude using the by now familiar BF machinery, and using the Weil-Petersson volume for the remaining purely topological piece.
In [97], the authors noted that it might be challenging to find JT amplitudes on disks with handles, as there is no classical solution. As the above calculation demonstrates, in BF language there seems to be no issue in calculating such contributions. 43 The generalization to multiple gravitational boundaries is straightforward. One glues several twisted Schwarzians together, using the Weil-Petersson volume whenever handles appear.
As an example, consider three Schwarzian boundaries with no handles attached ( Figure  12 right). The result again diverges due to the IR k → 0: One again cuts off the annuli at the three boundaries and glues these to a three-holed sphere. The latter again can be written in terms of a Weil-Petersson volume V 0 (λ 1 , λ 2 , λ 3 ) in terms of the conjugacy classes of the three gluing cycles.
It is in principle possible to calculate generic correlation functions on these multi-boundary manifolds. As an example consider the Wilson line stretching between boundary 1 and boundary 2 in figure 12 right. We find: Notice that the geometry of the manifold minus the Wilson line M \ W is topologically an annulus, which comes with a flat measure and hence we do not require the topological gravity machinery to calculate this amplitude. And indeed, the above is finite for > 0.

C Edge States in Chern-Simons theories
BF theory is defined as the dimensional reduction of 3d CS. The goal in this Appendix is to repeat the discussion of section 4 for CS. By comparing famous CS formulas with some of the BF formulas obtained in section 4 we provide with an alternative proof of the latter.

C.1 Edge Dynamics From the Path Integral
We first review how Chern-Simons on a manifold with boundary, leads to a Wess-Zumino-Witten 2d CFT on the boundary [68,99], in parallel to the BF argument of Section 2.1.
Focusing on a manifold with cylindrical topology, we write The background-dependence is only in the orientation of the chosen coordinate axes which we choose trφ = 1. We parameterize the spatial disk D as in Figure 13. Variation results in the boundary condition A φ | bdy ∼ A t | bdy . Rescaling the coordinates is a symmetry of the problem hence we can bring the proportionality factor to ±1. Changing sign corresponds to changing orientation and with our ordering of the coordinates, only the +-sign leads to a positive Hamiltonian: CS on a manifold with boundary is only consistent with the boundary conditions: Path integration over the Lagrange multiplier A t results in with g a G-valued field, in general twisted in the φ-direction: g(φ + 2π) = U λ g(φ), with U λ determined by a possible Wilson line insertion in irrep λ in the t-direction. Bulk values of g are redundant and only its boundary profile is a physical degree of freedom. Moreover there is a global G redundancy in (C.4), (C.5) under g → V g with V constant. 44 The path integral over A in (C.7) is reduced to a path integral over boundary configurations g. 45 Making the substitution g(φ) → Λ(φ)g(φ) with Λ(φ + 2π) −1 Λ(φ) = U λ untwists g(φ). Using partial integration combined with the boundary conditions A φ | bdy ± A t | bdy , the CS action (C.2) becomes a right-moving chiral WZW model, or a (right-moving) affine coadjoint orbit action: Let's now take two such Chern-Simons theories on spatial disks, and glue them into a single S 2 along the equator (figure 13 right) [38,39]. The correct way to split the Chern-Simons Lorentzian path integral is by the introduction of a functional delta constraint: The CS path integral on S 2 hence decomposes as: the final equality being true because the Hilbert space of CS on S 2 is just the vacuum [68]. Upon taking the t-dimensional reduction, the chiral WZW model (C.6) reduces precisely to the twisted particle on group (2.5) and (C.8) goes to (4.5), upon renaming φ → t. Notice again that the two actions will cancel upon gluing. The left action is minus the right one, or k → −k. 46 The argument in the functional delta in (C.7) becomes the WZW current density upon path integrating out A t : δ(J L − J R ). The theory associated with the submanifold R only is obtained from (C.8) by dropping all reference to L and is just the chiral WZW model: which can also be interpreted as path integrating over all boundary sources J R with a suitable action. In terms of Hilbert spaces this means there is a extended Hilbert space construction that accounts for edge states on the dividing surface, with again the gluing condition δ(J L − J R ) acting as a Gubta-Bleuler constraint, projecting onto the physical subspace. Explicitly, the Gubta-Bleuler constraint selects just one state in the extended Hilbert space associated with the entangling surface, to be written as an Ishibashi state [38,39]: of the left-and right sectors of the 2d WZW CFT on the entangling surface. In the BF limit, this becomes the factorization property (2.13) or (5.14), as only primaries survive.

C.2 Two-Boundary Models
As an application of the above consider the annulus path integral: Since the φ-direction is non-contractible in the annulus A, the holonomy of the connection cannot be specified and hence the path integral includes a sum over conjugacy class elements. 47 The Euclidean configuration associated with this setup in A × S 1 whose boundaries are two tori. The boundaries break topological invariance, as made explicit by the dependence of the theory on a choice of modular parameter β i on both tori ( Figure 14 left): This is equivalent to the statement that the spectrum of the theory consists of the states |λ, m ⊗ |λ, n and the Hamiltonian is β 1 H 1 + β 2 H 2 . This should be compared to (4.10).
In the special (thermal) case that β 1 = β 2 , (C.12) is just the partition function of a nonchiral WZW model: the fields g L , g R and λ can be recombined into a single non-chiral field g as described for example in [29] and applied to Liouville in section 5.4. Notice in particular that this expression is modular invariant.
The case that β 1 = 0 is closely related to the construction of the thermofield double state from the single-sided Hilbert space, and corresponds geometrically to (the exterior of) a solid torus (Figure 14 right). The amplitude (C.12) can then be interpreted as TFD|TFD by analyzing the relation between Rindler (or modular or one-sided) time τ and Kruskal (or 47 The holonomy on the inner boundary equals the holonomy on the outer boundary because g L and g R are derived from a single field g with an r-independent holonomy. Similarly, the global redundancy in the parameterization is the diagonal g L,R → V g L,R . (C.14) In the BF limit, the CFT Hamiltonian goes to the Casimir L 0 (λ) → C λ as shown by the Sugawara construction and we recover the factorized (4.12) and non-factorized (2.11) thermofield double states respectively.
One can match the norm of the thermofield double with the torus partition function: with the S-transform reflecting that the Hamiltonian now generates evolution along the A-cycle of the torus. Indeed, the β 1 → 0 limit yields the exterior of the original torus as the partition function. The latter is then related to a usual torus precisely by a modular S-transform [68].

D Some Representation Theory of SL(2, R)
We review some of the representation theory of SL(2, R) that is used in the main text. We will be mainly concerned with the parabolic basis which paves the way for the representation theory of SL + (2, R) in Appendix E. The emphasis here is on the continuous series irreps for which we derive the matrix elements and the Plancherel measure in a down-to-earth manner. This section is largely based on [64].
Group elements of SL(2, R) can be represented as the set of matrices: The (self-adjoint) generators of the group are the traceless matrices J a : satisfying the sl(2, R) algebra: The Casimir is: C = J 2 0 + 1 2 {J + , J − }. A set of basis functions of SL(2, R) is obtained by diagonalizing the Casimir and one of the generators J a . Suppose we label the eigenvalues of the Casimir as C = j(j + 1). For each fixed j, a spin j representation is defined as a basis for the corresponding eigenspace of J a . Labeling the eigenvalues of the diagonalized generator of choice as J a = ν, we end up with the orthonormal states |jν . To make this more explicit we must specify a realization of the algebra or the group. We will be discussing functions on the real line x ∈ R with the usual inner product. A spin j representation is obtained by defining the action of the group element g on the basis functions f j ν (x) as: Infinitesimally, using g = 1 + i a J a , we observe that this is the Borel-Weil realization of the sl(2, R) algebra: This also confirms that (D.4) is a spin j representation: using (D.5) the Casimir is immediately calculated to be C = j(j + 1). Representation matrices are as always the Fourier components of transformed states: The above can be viewed as introducing a complete set of states |x with One immediately verifies that these satisfy the composition property: indeed demonstrating the defining property of a representation: R(g 1 ) · R(g 2 ) = R(g 1 · g 2 ).
From the definition of the adjoint action g † : we obtain: such that the adjoint action is obtained by acting with the inverse group element g −1 .

D.1 Mixed Parabolic Basis
In the harmonic analysis on SL(2, R) two sets of unitary irreducible representations of SL(2, R) appear: the discrete ones with j = and 2 ∈ N, and the continuous ones j = − 1 2 + ik with k ∈ R. The goal of this section is to find explicit formulas for the associated matrix elements and Plancherel measure. With one eye on SL + (2, R) we choose to focus on only the continuous irreps here, and we choose to construct the matrix elements in the mixed parabolic basis. 48 Suppose one chooses to diagonalize J − or equivalently the subgroup h − (t) = exp itJ − with t ∈ R. A basis of the spin j representation is then the plane wave basis f k ν (x) = e iνx : with J − = ν ∈ R. We will denote the associated state by |ν − , suppressing the j index, such that x|ν − = e iνx . (D.12) Alternatively one may choose to diagonalize J + or equivalently the subgroup h + (t) = exp itJ + . A basis of the irrep is now formed by f k ν (x) = |x| 2ik−1 e i ν x with J + = ν. Denoting the associated states by |ν + we obtain: One can transform J − eigenstates into J + eigenstates by applying the group element as s transforms h − into h + : s · h − · s −1 = h + . And indeed, from the property x| s |ν − = x|−ν + we find: This property will prove pivotal in the construction that follows.
Mixed parabolic matrix elements are defined as ν − | g |λ + , (D. 16) and form a basis of wavefunctions for the continuous spectrum of quantum mechanics on SL(2, R). Indeed, ordinary matrix elements for example in the basis |ν − are orthogonal: with dg the Haar measure and ρ(k) the Plancherel measure. Using the property (D.15) and the invariance of the Haar measure under g → g · s proves that mixes parabolic matrix elements are orthogonal in precisely the same way: The same argument can be used to show that the 3j-symbols in the J + basis are the same as those calculated in the J − basis, modulo some sign changes.

D.2 Matrix Elements
Let us then continue to compute the matrix elements explicitly. As a first instructive example consider the g = 1 matrix elements or the overlap ν − |λ + . From the definition (D.6) we find: To evaluate these integrals we use the integral representation of the modified Bessel function of the second kind: This values of ν and λ can be taken to the imaginary axis, analytically continuing from the positive real axis as in Figure 15. Taking ν → e iπ/2 ν and λ → e −iπ/2 λ results in: Similarly by taking ν → e −iπ/2 ν and λ → e iπ/2 λ to rotate in the other direction, we find: Combining both we obtain: 49 More generically we are interested in computing ν − | g |λ + . For this, we choose to parameterize the SL(2, R) group element by its Gauss decomposition: This covers the Poincaré patch of SL(2, R) with metric: Since |ν ± diagonalizes J ± we find: where the second equality follows from a change of integration variables x → xe −φ in (D.6). Inserting (D.23) now directly results in the relevant matrix elements:

D.3 Plancherel measure
Finally, we would like to read off the Plancherel measure using (D.18) and the orthogonality relation of the Bessel functions. To do this, we must make a detour on the coordinatization of the SL(2, R) manifold as the Gauss parameterization (D.24) does not cover the entire SL(2, R) manifold but only the Poincaré patch. Any integral over the full group manifold (such as (D.18)) is a sum of four terms. The whole SL(2, R) group is covered by four patches of the form [100]: These patches give 2 by 2 the same result as an overall sign of ω gives the same matrix elements. 50 This means the group integral splits in two a priori distinct pieces: one over 50 As we are in fact studying PSL(2, R).
g(φ, γ − , γ + ) and one over g(φ, γ − , γ + ) · s. We demonstrate in the next subsection that an elementary substitution is sufficient to show that the second term equals the first one, both for the grand orthogonality as for the 3j-integral. We end up with: which is indeed known to be the Plancherel measure on SL(2, R). Of course, this result can be obtained in any basis of interest. Most discussions utilize the hyperbolic basis to deduce this (see e.g. [101] for a recent account).

D.4 Covering of the SL(2, R) manifold by Gauss Patches
To go to the ω = s patch, we set e φ → −e φ /γ + , γ + → −1/γ + and γ − → γ − + e 2φ /γ + in the matrix element (D.28) [102], and we obtain: from which we read off the contribution to orthonormality of the ω = s patch to be: where in the first equality we did the γ − -integral, in the second one we first shifted φ → φ + ln |γ + |, used γ + → 1/γ + and did the γ + -integral, while in the third equality we did the final φ-integral. This is just the same answer as the Gauss ω = 1 patch, and the only effect of considering all four patches is a quadrupling of the result, leading to the Plancherel measure (D.30).

E Some Representation Theory of SL + (2, R)
We present the representation theory of SL + (2, R). It is very closely related to that of SL + (2, R) itself, and large parts of it can be found in the available literature [63,64,67].
The semigroup SL + (2, R) is defined as the set of positive SL + (2, R) matrices with the usual matrix operations: In spite of the lack of an inverse, hence the name semi group, it is possible to set up a meaningful representation theory in the sense that It has an action on L 2 (R + ) in the same way as (D.4), but restricted to x > 0: Due to the positivity of all matrix entries, this operation is internal in R + and is well-defined. The sl(2, R) algebra is still relevant.
A matrix element in a representation j is defined as the overlap:

E.2 Unitarity of the Matrix Elements
We can use the explicit expressions (E.12) and (E.13) to prove that the continuous representation K ++ (g) is unitary. We compute: Using successively

E.3 Gravitational Matrix Elements
Gravitational matrix elements are associated with the parabolic states |i ± defined as before to satisfy J ± |i ± = ±i |i ± . In the coset slicing we are interested in obtaining s| g(φ, γ − ) |i + .
In the Schwarzian slicing we are interested in obtaining i − | g(φ) |i + . In the coordinate basis we obtain a damped exponential: Notice that neither the damped e −νx nor the oscillating exponentials e iµx are orthogonal on R + . This is because J + as defined in (D.5) is not self-adjoint on R + , so its eigenfunctions are not necessarily orthogonal and the vectors |µ + do not form a basis, in sharp contrast with the situation in SL(2, R) above. J 0 on the other hand is self-adjoint on R + , and leads to the hyperbolic basis (E.5) we constructed above.
These states can be decomposed in the hyperbolic basis using the Cahen-Mellin integral: This transition is the same as that linking Minkowski eigenmodes to Rindler modes. 52 The matrix element of the middle Cartan element e 2iφJ 0 is called the Whittaker function (or coefficient). The elementary basis functions f ν and f λ are called the Whittaker vectors in the mathematics literature [57,58,59,60]. The matrix element between these states is easily found as F Schwarzian Bilocals from SL + (2, R) BF We are interested in evaluating the Wilson line in the lowest weight representation of a discrete irrep of SL + (2, R). This is easy in the Borel-Weil realization where we know the lowest weight state | , 0 to be of the form [70]: x| , 0 = 1 x 2 , , 0|x = δ(x), (F.9) and the generators (F.8) exponentiate to one-parameter subgroups of SL(2, R), acting as: corresponding to translation, scaling and special conformal transformations respectively. Hence the Wilson line can be written as: O (τ 1 , τ 2 ) = , 0| g(t f )g −1 (t i ) | , 0 = dx δ(x) g(t f )g −1 (t i ) · 1 x 2 , (F.13) with the differential operator g −1 = e γ − ∂x e 2φ(−x∂x+ ) e γ + (−x 2 ∂x−2 x) , (F.14) with parameters (F.7). Explicitly, the wavefunction transforms under the action of g(z f )g −1 (z i ) as: with f 1 and f 2 possibly different functions associated with the respective holographic boundaries on which t i and t f are located. Setting x = 0, and specifying to (5.19) we obtain: (F.16)

G Measure on the Space of Conjugacy Class Elements
We distill some formulas relevant for this work from [30,31] regarding the precise choice of integration measure on the space of conjugacy class elements (or orbits).
When one usually talks about finite characters, one uses the integration measure on the space of conjugacy class elements, inferred from the Haar measure on the group manifold. 54 The resulting characters, orthogonal with respect to these measures, are: SU (2) : χ n (θ) = sin nθ sin θ , SL(2, R) : χ µ (λ) = cos µλ sinh λ (G.1) And indeed, with the Haar measure inferred from the group manifold, orthogonality can be checked to hold. The point made in [30] is however, that this integration measure is a choice, and depending on the situation a different normalization might be required.