More on torus wormholes in 3d gravity

We study further the duality between semiclassical AdS3 and formal CFT2 ensembles. First, we study torus wormholes (Maldacena-Maoz wormholes with two torus boundaries) with one insertion or two insertions on each boundary and find that they give non-decaying contribution to the product of two torus one-point or two-point functions at late-time. Second, we study the ℤ2 quotients of a torus wormhole such that the outcome has one boundary. We identify quotients that give non-decaying contributions to the torus two-point function at late-time. We comment on reflection (R) or time-reversal (T) symmetry vs. the combination RT that is a symmetry of any relativistic field theory. RT symmetry itself implies that to the extent that a relativistic quantum field theory exhibits random matrix statistics it should be of the GOE type for bosonic states and of the GSE type for fermionic states. We discuss related implications of these symmetries for wormholes.


Introduction
It was conjectured that there is an approximate duality between semiclassical 3d gravity and an ensemble of formal CFT 2 s with large central charge c and a sparse low-energy spectrum [1,2,3] 1 .This is analogous to the duality between Jackiw-Teitelboim (JT) gravity [6,7,8] and random matrix theory [9,10].These dualities are both different from earlier examples of AdS/CFT and their boundary ensembles satisfy the Eigenstate Thermalization Hypothesis (ETH) [11,12].
This proposed duality provides a playground to study various quantities.One thing is a BTZ black hole [13] in AdS 3 with light operator insertions.Here light and heavy are respectively distinguished by whether the operator dimension (h, h) is below or above the black hole threshold c  24 where c = 3 2G is the central charge.In particular, we study averaged one-point and two-point correlators on a torus on the CFT 2 ensemble side.Their variances involve a torus wormhole on the 3d gravity side.More precisely, the averaged product of two torus correlators are dual to a Maldacena-Maoz wormhole [14] with insertions.A Maldacena-Maoz wormhole has two asymptotic boundaries with identical topology and constant moduli.The boundaries are Riemann surfaces, with the simplest options Riemann spheres or tori.In order for a Riemann sphere to be stable, there need to be at least three insertions.For torus, at least one insertion.If we take the operator dimension to zero for a torus wormhole, we can connect back to [15,16] which calculates the torus wormhole partition function in 3d pure gravity [17].
The BTZ black hole can be reduced to JT gravity by Kaluza-Klein mechanism [18,19].In this paper we study 3d gravity using methods inspired by 2d gravity.In particular, there is an analogy between a solid torus in 3d and a disk in 2d as shown in figure 1(a).This can be extended to the analogy between a torus wormhole in 3d and a cylinder (or double-trumpet) in 2d as shown in figure 1(b).We can take the analogy further by taking a Z 2 quotient of a torus wormhole in 3d and compare that to a Z 2 quotient of a cylinder in 2d.In 2d, identifying antipodal points on a cylinder gives a disk with a crosscap inserted as shown in figure 2(a).Similarly in 3d, we identify the torus wormhole as shown in figure 2(b) and get a solid torus with a thinner solid torus carved out in the middle.Every point on the blue torus is identified with another point on that blue torus.We find that there are two ways of doing the Z 2 quotient; one orientable and one nonorientable.Each configuration gives a non-decaying contribution to torus two-point function reminiscent to that a disk with a crosscap gives a non-decaying contribution to the thermal two-point function in 1d [20].We can understand these two contributions using Random Matrix Theory (RMT) from the boundary side perspective.In particular, the orientable geometry comes from RT symmetry contained in the proposed formal CFT 2 ensemble.In 2d, CPT symmetry is equivalent to reflec-tion+time reversal (RT).While the non-orientable geometry arises from adding time reversal (T) symmetry to the formal CFT 2 ensemble.Noticing this RT symmetry has bigger consequences, one is that it tells us that the torus wormhole partition function calculated in [15,16] should be multiplied by a factor of two.The other is that we notice that actually any relativistic quantum field theory with random matrix statistics has energy eigenvalue distribution a GOE ensemble for bosonic states and a GSE ensemble for fermionic states.
We show the 3d calculations in the main text and the parallel 2d calculations in Appendix A. In section 2, we consider one insertion on each boundary torus.First in section 2.1, we find that if we take the mass of the insertions to zero, we get a result similar to the partition function found in [15,16] but with a factor off. Second in section 2.2, we find that if we analytically continue the two boundary tori, the average product of one-point functions do not decay over time.In section 3, we consider two insertions on each boundary torus.First in section 3.1, we find that if we analytically continue the location of one of the insertions on each boundary, the average product of two-point functions do not decay over time.Second in section 3.2 we discuss the Z 2 quotients of a torus wormhole and their contributions to torus two-point function both from a bulk perspective and from a boundary perspective, and examine the effect of RT and T symmetry.Finally, in section 4 we look again at the partition function found by [15,16] and also examine a generic relativistic quantum field theory with random matrix statistics.

Review: CFT 2 ensemble
We review some key properties of unitary compact CFT 2 following the presentation of [3] and state the conjectures of [1] that we will also assume in this paper.A CFT 2 is defined by its central charge c and a list of primary operators with known scaling dimensions (h, h) and operator product expansion (OPE) coefficients c ijk .We reparametrize in terms of "Liouville parameters" Also denote the spin as s = h − h.Viewed as an ensemble, CFT 2 's have some universal features depending only on the central charge, we list two such features below.Both can be formulated in two ways which are equivalent to each other because of modular invariance.

(a)
The identity operator has (b) Cardy's formula for the density of primary states 2. (a) For any operator O i , the OPE coefficient of two O i 's and the identity is (b) Averaged value of the OPE coefficients over all primary operators is given by when at least one of the three operator is heavy.Here C 0 is given by where From now on, following the conventions of [1], we assume that our CFTs are holographic, i.e. with a large central charge and a sparse spectrum of low-lying operators.We also extend 1(b) to hold above the black hole threshold and 2(b) to hold for all primaries.In addition assume and where the remaining four terms are cyclic permutations of the first two terms.In section 3.2.2,we discuss a way to understand this ensemble from the prospective of Random Matrix Theory.
We introduce a graphical way of representing the OPE coefficients.Consider a CFT 2 on a Riemann sphere.Let O 1 , O 2 , and O 3 be three insertions.Instead of thinking of these as point insertions we can expand the point out to form cycles and the Riemann sphere with three insertions now look like a pair of pants.This just gives the OPE coefficient c 123 .
(1.10) [1] showed that the averaged product of two three-point-functions of two CFT 2 's on Riemann spheres matches the semiclassical action of the sphere wormhole shown in figure 3 on the 3d gravity side.
2 Torus one-point function wormhole In this section, we focus on studying a torus wormhole with one insertion on each boundary.In section 2.1 we take the mass of the insertions to zero and compare with partition function in [15,16], in section 2.2 we show that the averaged product of two torus one-point functions does not decay over time.We consider the averaged product of two one-point functions.
Note that pictorially, for each torus one-point function we can think of it as a torus with a hole.Then this can be decomposed as a sum of the product of OPE coefficient c 1pp and conformal block |F g=1 1 (h p ; τ )| 2 over primary operators p.The sum over descendents of each primary operator is encoded in the conformal block.
The proposed ensemble of formal CFT 2 (1.11) predicts that the averaged product of two torus one-point functions is given by where ) are torus Liouville one-point functions.For a derivation of this result see Appendix B.1.
It was shown in [1] that in the large c limit (to one-loop order), (2.7) agrees with the semiclassical limit of the 3d gravity calculation of a torus wormhole with a defect insertion.It is interesting to contemplate that the RHS of (2.6) could actually be the exact answer for this wormhole in 3d gravity.Our understanding is that this has indeed been established as part of the work of [21].This motivates us to take the formula seriously enough to consider a non-semiclassical limit where ∆ → 0.

∆ → 0 limit
In this section, we take the mass of the insertions to zero ∆ → 0. There are a couple of interesting features that we want to point out about this limit.First, the expression is simple and given by where the character is given by Here η(τ ) is the Dedekind eta function.Second, naively we would guess that in the ∆ → 0 limit, the averaged product of two one-point functions ⟨O⟩ ⟨O⟩ should reproduce the partition function predicted by Cotler-Jensen [15,16].This turns out to not to be the case, and the discrepancy is simple, just a factor inside our integral over k and k.The same kind of discrepancy also appears in 2d because even though we take ∆ → 0 there is a still a sum over windings of the particle around the cylinder which contributes a divergent overall factor (see Appendix A.2 for a detailed 2d analysis).
Let us begin with our formula of the averaged product of two one-point functions (2.10) where C 0 is given by (1.6).Now we take the weight of O 1 to zero, i.e. ∆ = 2h 1 → 0. In this limit O 1 becomes the identity operator.Then the above equation becomes for derivation see B.3.
From (2.5) we know that But we also know that and the averaged product of two torus one-point function in the limit ∆ → 0 is given by On the other hand, the partition function given in Cotler&Jensen [15,16] is where so the lim ∆→0 ⟨O⟩ ⟨O⟩ differ by a factor k k inside the integral compared to the partition function Z before the sum over PSL(2, Z).However, not like in 2d, in 3d we do not know a precise way to calculate this factor yet.
We can also compare the expressions after integration which are

Lorentzian torus limit
In this section, we examine the late-time behavior of the product of two torus one-point functions.In 2d, the simplest quantity to probe the late-time behavior is the spectral form factor . This is two thermal partition functions, one with β analytically continued to β + it and the other to β − it.We can do a similar analytical continuation in 3d.
Writing the torus one-point function in operator form, we get where τ = iβ + s and τ = −iβ + s are complex conjugates since the torus T 2 is a Euclidean torus.Also by definition H = h + h − c/12 and P = h − h.
The above is one boundary of the Maldacena-Maoz wormhole.The other side T 2 (τ ′ , τ ′ ) is another torus that look like the reflection of T 2 (τ, τ ).Thus τ ′ = iβ − s and τ ′ = −iβ − s.Then the torus one-point function is For simplicity, we take s = 0 so the product of the two torus one-point function becomes Now we can do analytical continuation similar to ⟨Z(β + it)Z(β − it)⟩ [22], and make the two tori into Lorentzian tori by taking But this is just saying now so we are calculating the quantity But now we can see that the Liouville one-point functions on the right-hand-side are on Euclidean tori instead of Lorentzian tori.Another way to present the answer is On the LHS we start with an average over the formal ensemble of CFTs of a quantity similar to the spectral form factor, but with an operator inserted in each factor.The RHS is the answer for this ensemble average, or equivalently, it is the wormhole contribution in 3d gravity -either way it boils down to a computation in Liouville theory involving an operator insertion with the same dimension.The key point is that on the RHS, the large time parameter t multiplies the momentum operator P , not the Hamiltonian H. Now, P is quantized so the RHS is periodic in time with a short period, and in particular does not decay for large t.

Torus two-point function wormholes
In this section we focus on studying a torus wormhole with two insertions on each boundary.In section 3.1, we show that the averaged product of two torus two-point functions does not decay over time and in section 3.2 we study Z 2 quotients of this torus wormhole.These quotients give non-decaying contributions to the torus two-point function.We calculate these contributions from the bulk in section 3.2.1 and from the boundary in section 3.2.2where we justify and extend the proposed formal CFT 2 ensemble (1.9) using RMT.
We consider the averaged product of two torus two-point functions Each two-point function is given by Thus the averaged product of two torus two-point functions is given by where

Large time-separation limit
In this section, we examine the late-time behavior of the product of two torus two-point functions.
Note that here the notion of late-time is different from section 2.2.In section 2.2, at late time the two tori on the boundary becomes large in time direction.Here two operators inserted on each torus becomes far away from each other.We take the twist of the Euclidean torus s = 0 but we do not analytically continue β.Instead, we just analytically continue the location of the insertion O 1 on both boundaries plugging these into (3.7)we get Observe that one torus has one insertion at 0 and another insertion at i β 2 −⌊t⌋.The other torus has one insertion at 0 and another insertion at i β 2 +⌊t⌋.Here ⌊•⌋ means the fractional part of t.Thus, as t becomes large, the arguments don't change much, so does not decay at late time.

Single-boundary quotients of a torus wormhole
So far we have discussed two-boundary configurations with large Lorentzian separation that do not decay with time.In this section we calculate Z 2 quotients of a torus wormhole from both the bulk side (section 3.2.1)and the boundary side (section 3.2.2).These configurations each has one boundary which is a torus, so they contribute to the torus two-point function.And then using the result from 3.1, we observe that these contributions do not decay at late time.

Semi-classical gravity calculation
Let us classify ways of doing Z 2 quotient of a torus wormhole that gives a smooth geometry with one torus boundary.In other words, we classify Z 2 symmetries of the torus wormhole such that the two boundaries are mapped to each other.Our trick of doing the classification is to focus on the zero-curvature slice in the middle of the torus wormhole.This zero-curvature slice is a torus that is mapped to itself under the Z 2 symmetry.We represent this torus as a square with sides identified as shown in figure 4, i There are two inequivalent Z 2 symmetries and doing a quotient results in either a torus T2 or a Klein bottle K 2 as shown in table 1.Without loss of generality, the fundamental region is the part to the left of the blue dashed line.
Table 1: Ways of doing a Z 2 quotient of a torus Going back to 3d, the above analysis tells us that there are two ways of doing Z 2 quotient of a torus wormhole.Let us use the coordinate (x, y, z) ∈ T 2 × [−1, 1] for the torus wormhole.Thus these two Z 2 quotient corresponds to (1) (x, y, z) ∼ (x + 1/2, y, −z) and (2) (x, y, z) ∼ (x + 1/2, −y, −z) respectively. 2We should note that both these configurations contain a Mobius band (x, 0, z) ∼ (x + 1/2, 0, −z).Now let us examine whether our two kinds of Z 2 quotients are orientable or not.To do that we go back to the torus wormhole which is a 3d smooth manifold embedded in R 3 .We can think of the identifications as transition maps between two different elements in the atlas of the torus wormhole.The Z 2 quotient is orientable iff the atlas is oriented.The atlas is oriented (unoriented) if the Jacobi determinant is positive (negative).
(2) (x, y, z) ∼ (x + 1/2, −y, −z).This has Jacobi det = 1, which means it's an orientable 3d geometry.Intuitively, it has a twist in the z direction and another twist in the y-direction, they cancel each other.
Notice that there is an analogy between Z 2 quotients of a 3d torus wormhole and the Z 2 quotient of a 2d cylinder as shown in figure 5. Z 2 quotient of a 2d cylinder is the same as a disk with a crosscap inserted as shown in figure 5(a).Z 2 quotients of a 3d torus wormhole are also equivalent to carving out a solid torus and then identify points on this torus as shown in figure 5(b).We should note that if we insert operators on the boundaries of torus wormhole and then take the Z 2 quotient, the insertions need to be compatible with the corresponding Z 2 symmetry.Before analyzing the insertions in 3d torus wormhole, let us recall a similar situation in 2d where we insert four operators on the cylinder that respect the antipodal map so that we can do the antipodal identification for the cylinder later.The configuration of insertions compatible with the Z 2 symmetry is shown in figure 6.The two V 's are antipodal of each other and the two W 's are also antipodal of each other.We analytically continue the Euclidean distance between V and W to β 2 + it and β 2 − it.In particular, we can check that the configuration shown in figure 6 makes sense by observing that the left boundary and right boundary of the cylinder gives the same two-point correlator.

LHS = tr(e
Analogously in 3d we again insert four operators so that they satisfy the Z 2 symmetries, i.e.There is one more subtlety to the configuration.For the operators to satisfy the Z 2 symmetries, we should actually insert T V T −1 and T W T −1 for (1) and (RT )V (RT ) −1 and (RT )W (RT ) −1 for (2) where T denotes time-reversal symmetry and R denotes reflection.
Thus the torus wormhole contribution to the averaged product of two torus two-point functions we are calculating is Now we take V = O 1 and discuss the two cases separately (1) (3.18) is only nonzero when Using (3.7), we get This is given by two quotient wormhole contributions glued together and semiclassically we can write e −S wormhole = e −2S quotient (3.22) Thus quotient (1) contributes to the torus two-point function by and this is non-decaying over time.We should note that this equation is only valid in the leading (zero loop) classical approximation, because we are just using the relationship between the classical actions (3.22).Also since we can use saddle-point approximation (for a justification see Appendix B.4) to get Using (3.7), we get Quotient (2) contributes to the torus two-point function by and this is non-decaying over time.Again this is only valid in the leading (zero loop) classical approximation.And again using saddle-point approximation But RT acting on O 1 is equivalent to a 180 • rotation so (3.27) and (3.35) are results of a classical approximation.However, it's interesting to contemplate that whether they are actually exact equations.

CFT calculation
Before calculating the Z 2 quotients of a torus wormhole from the boundary side, let us examine the boundary ensemble in [1] more closely.Here all operators that are Hermitian.We should note that We now show that an OPE coefficient c 123 is real if s 1 + s 2 + s 3 is even and purely imaginary if s 1 + s 2 + s 3 is odd.If the operator O is a symmetric traceless tensor of spin s = h − h, we can write it in component form as Let Rot denote rotation by 180 • , define |1⟩ = O 1 (x 1 , J 1 ) |0⟩ and |3⟩ = O 3 (x 3 , J 3 ) |0⟩, then threepoint function is given by where in the second line the prefactor comes from rotating the spin while the braket comes from rotating operators keeping spin fixed.Thus The ensemble of OPE coefficients can be justified by considering a simple model of Haar random matrices.We introduce a change of basis by unitary matrix u and write where u is a unitary matrix and they form a Gaussian unitary ensemble (GUE) [23].We can justify writing the change of basis u by thinking that there are two sets of basis: local basis and energy eigenbasis.In local basis, the operator O α is simple but in the energy eigenbasis O α need to be transformed by some rather complicated change of basis, so we can take those complicated matrix u to be random.We can then model averaging over ensemble of OPE coefficients as integral over random unitary matrices.Recall we have the leading order contribution to Weingarten's formula [24] du u Now let us review how to derive this formula.The general idea is we observe that u is a projector onto invariants.Thus we can first find the invariant states then we can just write the integral formula immediately.More specifically, denote our Hilbert space by H, then our random matrices as maps u, u † : H → H and u * , u T : (3.46)Note in particular that |13⟩ is the identity operator which is basis-independent, and so welldefined.The same hold for |24⟩ , |14⟩ , |23⟩ so For two boundary dimensions, we should modify (3.43) by adding reflection + time-reversal, i.e.RT symmetry. 4We want to know how the ensemble of OPE coefficients would change if we add RT symmetry (i.e.RT commuting with u or u † RT u = RT ) to GUE.Recall that Lorentzian time-reversal T is antilinear and antiunitary.Since we know that complex conjugate K is antilinear and antiunitary 5 , it is natural to model T with a factor of K in it.With RT symmetry, there are more invariant states Therefore, time-reversal symmetry adds a term to the Weingarten's formula Thus the averaged product of two OPE coefficients with one light and two heavy operators is where in the last line we used two facts: First, RT acting on O α by conjugation is equivalent to rotation by 180 • so Second, we can write (RT K) −1 il and (RT K) jk in braket notation as Thus we have reproduced (3.36).
Now we want to know how the ensemble of OPE coefficients would change if we add timereversal symmetry without reflection R (i.e. the time-reversal operator T commuting with u i.e. u † T u = T ) to GUE.With T symmetry, we have invariant state Therefore, time-reversal symmetry adds a term to the Weingarten's formula This would add an additional term to the averaged product of OPE coefficients There are two kinds of anomalies T 2 = ±1.For T 2 = 1, we can just take T = K and the condition T uT −1 = u reduces u † u = 1 to u T u = 1, which is equivalent to saying u is orthogonal, and the ensemble becomes GOE.Thus the added term to ensemble simplifies to For T 2 = −1, we can take T = Kω where ω = 0 1 −1 0 .The condition T uT −1 = u reduces u † u = 1 to u T ωu = ω, which is equivalent to saying u is symplectic, and the ensemble becomes GSE.Thus the added term to ensemble simplifies to Now we can use results from last subsection to calculate torus two-point functions.Note that on our 2d boundary, we always have RT symmetry so the ensemble is given by (3.55), which we now use to contract indices of the OPE coefficients = dh p dh q ρ 0 (h p )ρ 0 (h q )C 0 (h 1 , h p , h q )F g=1 11 (h p , h q ; τ, v) If we also add time-reversal symmetry, the ensemble becomes (3.65).This would add another term to the two-point function

Comments on RT symmetry
In this section, we first show that a generic relativistic quantum field theory with random matrix statistics should be of the GOE type for bosonic states and GSE for fermionic states, then we point out that the partition function of a torus wormhole calculated in [15,16] needs another multiplicative factor of 2.
Section 3.2.2tells us that the CFT 2 ensemble proposed by [1] is inherently GOE for bosonic states and GSE for fermionic states since it contains RT symmetry, and RT symmetry is an anti-linear, anti-unitary symmetry that squares to (−1) F [25].We should note that in 2d, RT symmetry always exists (this can be understood as coming from the CPT theorem6 ).Inspired by the above observation, we claim that to the extent that a relativistic quantum field theory exhibits random matrix statistics it should be of the GOE type for bosonic states and of the GSE type for fermionic states.To start with, a relativistic quantum field theory exhibits random matrix statistics is of GUE, if the system has no additional symmetry, adding RT symmetry would make it into GOE or GSE.One caveat is that if the Hamiltonian has an additional symmetry which block diagonalize it, the RT symmetry could potentially exchange different blocks instead of acting on each individual block.We now show that the above situation does not happen.We can block diagonalize the Hamiltonian into different momentum blocks In order to show the energy eigenvalue distribution is GOE or GSE, we need to make sure that each individual subblock of H commutes with RT.This is equivalent to showing that momentum commutes with RT.Let T µν be the stress-energy tensor then the momentum is given by From here, we can show that RT p(0) = p(0) ( Therefore, momentum commutes with CPT and we conclude that any chaotic CFT 2 has energy eigenvalue distribution a GOE for bosonic states and a GSE for fermionic states. RT symmetry has implications for torus wormhole partition function studied in [15,16].In Euclidean AdS 3 bulk, having RT symmetry means that in addition to the usual torus wormhole that can be obtained by gluing together two torus trumpets as shown in figure 8(a), there should always exists a configuration that first act on the right torus trumpet with RT, i.e. rotate the right torus trumpet by 180 • , and then glue it to the left torus trumpet as shown in figure 8(b).This implies that the partition function of a torus wormhole calculated in [15,16] needs another multiplicative factor of 2. Intuitively, this can be understood by focusing on how the two cycles of a torus α and β get mapped.Recall that a torus is a quotient of the complex plane by a 2d lattice as shown in figure 9.
Figure 9: T 2 is a quotient of the complex plane by a 2d lattice The lattice has α and β as basis vectors, and we think of them as complex numbers.We want to determine the number of ways this torus can be mapped to itself ignoring simply zooming in or out.(α, β) gets mapped to (pα, qβ) where p and q are relatively prime integers with the same sign.We know that any such pair (p, q) can be mapped from (1, 1) by a unique element of SL(2, Z) a b c d Ignoring the sign of (p, q).The complex structure of the torus is given by However, we should note that this representation of MCG ignores the case where (α, β) gets mapped to (−α, −β).This is the case where the parallelogram formed by (α, β) get rotated by 180 • .This 180 • rotation is not included in PSL(2, Z) because it is identified with the identity element.However, with RT symmetry, the directions of α and β are important as shown in figure 10.Thus we should add in this element back, i.e. we should modify (2.16) to This gives an additional multiplicative factor of two in the result.In large spin, this result would again reduce to a double-trumpet in JT but with an additional factor of two, which is consistent with JT with time-reversal symmetry added7 [10].Also note that this is more consistent with (2.15), if we take s 1 even and large.
In 2d, correlators are studied in the context of Jackiw-Teitelboim (JT) gravity [6,7,8] which is dual to an ensemble of quantum mechanical systems on the boundary [9,10] and can be described by Random Matrix Theory (RMT).Saad [26] 8 computed bosonic two-point correlation functions using the techniques developed by Yang [28] 9 on the bulk side and compared with RMT predictions for operators satisfying Eigenstate Thermalization Hypothesis (ETH) [12,11] on the boundary side.

A.1 Introduction
The 2d gravity theory we will study consists of the Einstein-Hilbert action + JT gravity action + action from matter.JT gravity on a 2d manifold M has Euclidean action Classically, the equation of motion fixes the bulk geometry to be AdS 2 with R = −2 and the action reduces to a Schwarzian action on the boundary [33].In 2d, the Einstein-Hilbert action is purely topological and can be written as where χ = 2 − 2g − n is the Euler character for manifold M with g the genus and n the number of boundaries, and S 0 is the zero-temperature bulk entropy which is a constant.The Einstein-Hilbert action then contributes an overall factor e χS 0 to the partition function.In all of our figures the orange disks represent infinite hyperbolic space (or its quotient) and yellow geometries inside represent the physical Euclidean spacetimes, with wiggly regularized boundaries described by the Schwarzian theory [33].
The two main shapes of Euclidean AdS we consider in this review are a hyperbolic disk which has one asymptotically boundary with renormalized length β, and a hyperbolic trumpet which has one asymptotic boundary with renormalized length β and one geodesic boundary with length b (see figure 11).That is because a disk is the simplest hyperbolic geometry with one asymptotic boundary and a trumpet can be thought of as a building block of more complicated geometries via attaching a Riemann surface with one geodesic boundary to the geodesic boundary of the trumpet.
JT path integrals without operator insertions can be computed directly by doing the path integral over the wiggly boundary of the disk and the trumpet explicitly.Disk [34,35,36,37,38,39,40,28,41] and trumpet partition functions [37,26] are given respectively by where ρ 0 (E) denotes the density of state.

A.2 Product of two one-point functions
In this section, we focus on the product of two one-point functions, which on the gravity side corresponds to a cylinder with one operator insertion on each side as shown in figure 13.
Take the limit ∆ → 0 we get We should note that here we only care about the saddle point (i.e.only the integrand of the above integral).Thus we get In [20], crosscap contribution to the 2-point function was calculated directly using propagator We start from the torus partition function Z(τ, τ ) = (q q) −c/24 Tr(q L 0 q L0 ) (B.1) where q = e 2πiτ .Based on this the one-point function can be written as where we decompose a sum over all operators into a sum over primaries and a sum over all descendents of a primary.Thus, we get an expression for the conformal block since complex conjugation gives a minus sign while τ → −τ gives another minus sign, these two minus signs cancel.
Thus the averaged product of two torus one-point functions is given by

Figure 1 :
Figure 1: (a) Analogy between a 3d solid torus and a 2d disk (b) Analogy between a 3d torus wormhole and a 2d cylinder

Figure 2 :
Figure 2: (a) Z 2 quotient of a cylinder in 2d (b) Z 2 quotients of a torus wormhole in 3d

Figure 3 :
Figure 3: A Maldacena-Maoz wormhole with two boundaries both Riemann spheres with three light insertions Figure 4: A torus represented as a square with sides identified.

Figure 5 :
Figure 5: (a) Z 2 quotient of a cylinder in 2d (b) Z 2 quotients of a torus wormhole in 3d

Figure 6 :
Figure 6: In 2d, we get a cylinder from the hyperbolic disk by identifying the two brown geodesics.Then we insert two pairs of operators V and W on the boundary.

)
Note that both (1) and (2) Z 2 quotients are compatible with these operator insertions because the y-coordinates of all four insertions are zero and the only difference between (1) and (2) is from the y-coordinates.Again the distance between V and W are β 2 + it and β 2 − it.If the tori are given by T 2 (τ, τ ) = T 2 (iβ, −iβ) and T 2 (τ ′ , τ ′ ) = T 2 (iβ, −iβ) respectively.The insertions should be V at 0 and W at (v, v) = i( β 2 + it), −i( β 2 + it) for the left torus, and should be W at 0 and V at (v ′ , v′ ) = i( β 2 − it), −i( β 2 − it) for the right torus.

Figure 7 :
Figure 7: We insert operators the same way for a 3d torus wormhole as for a 2d cylinder.

Figure 8 :
Figure 8: (a) gluing two torus trumpets to get a torus wormhole (b) act on the right torus trumpet with RT and then glue to the left trumpet

Figure 10 :
Figure 10: A 180 • rotation flips the directions of both α and β cycles of a torus.