Semiclassical torus blocks in the t-channel

We explicitly demonstrate the relation between the 2-point t-channel torus block in the large-c regime and the geodesic length of a specific geodesic diagram stretched in the thermal AdS3 spacetime


Introduction
Nowadays the AdS 3 /CFT 2 correspondence is an active field of study. There are many results, particularly in the case of the semi-classical limits where the central charge tends to infinity or, equivalently, the gravitational coupling is small [1]. Classical (large-c) conformal blocks have been studied in the Riemann sphere, where they were identified with lengths of geodesic networks embedded in the asymptotic AdS spacetime with the angle deficit or BTZ black hole . Recently, in works [30][31][32][33][34] the holographic dual interpretation of the 1-point and 2points torus conformal blocks was considered.
It is known that the general solution to the classical euclidean AdS 3 gravity is topologically associated with a solid torus (see e.g. [35]) so that the corresponding boundary CFT 2 lives on a torus [36]. In this paper, we take up again the discussion in [30,33] about the duality of the linearized classical torus conformal block. For the previous studies of the torus conformal blocks in the framework of CFT see [37][38][39][40][41][42][43][44].
The duality proposed in [30,32,33] describes large-c torus conformal block as the geodesic length of a specific geodesic diagram embedded in the thermal AdS 3 spacetime. This duality translates to the following relatioń where the LHS is the large-c torus conformal block, while the first term of RHS is the holomorphic part of the 3d gravity action evaluated on the thermal AdS spacetime, and the second term of RHS represents the sum of the geodesic lengths (S i ) over all parts of the diagram (we denote each part of the diagram by index i), each geodesic length (S i ) in the sum is multiplied by its respective mass-parameter (ǫ i ) (these parameters are associated with the conformal dimensions of the block). Fig.1 represents the diagram, which is associated with the 2-point t-channel torus conformal block.
Thermal AdS 1 2 3 4 Figure 1. The black interior and exterior circles represent the boundary of the thermal AdS. Numbers represent each geodesic trajectory. The first trajectory is a closed trajectory, the second one is a radial trajectory, the third and fourth one are external tajectories attached to the boundary.
In [30,33] the large-c duality was demonstrated by explicit computation in the first order in external conformal dimensions of both sides in (1.1) for the 1-point torus conformal block and 2-point s-channel torus conformal block cases respectively. In [32] the duality was demonstrated for n-point global blocks with large dimensions by establishing that the block and length functions satisfy the same equation both on the boundary and in the bulk. In this paper, having in mind these general arguments, we explicitly calculate and compare the 2-point block function in the t-channel and the length of the corresponding geodesic network in the bulk thereby demonstrating the relation (1.1) explicitly (for simplicity, we take equal conformal dimensions). In particular, we will see that the validity of (1.1) for the 2-point t-channel torus global block case induces the validity of (1.1) for the 1-point torus conformal block case. This is because the linearized classical 1-point torus conformal block is a part of linearized classical 2-point t-channel torus global block. In [28] the non-perturbative computation of the RHS of (1.1) has been developed for the 1-point torus conformal block case, and we will see that our results agree with those of [28].
The order of this paper is the following. In section 2 we review the 2-point torus correlation function and the corresponding 2-point torus conformal blocks, focusing specifically on the tchannel case. In section 3 we review the computation of the torus global block from the 2-point torus conformal block. In section 4 we compute the global block with large dimensions which we refer to as large-∆ global block. Our computation is based on the following assumptions: (a) all conformal dimensions except for the loop intermediate dimension are equal to each other (b) the loop intermediate dimension is much larger than other dimensions. Thus, expanding in large loop dimension, we can calculate a linearized large-∆ global block. In section 5 we establish the holographic duality for the 2-point linearized large-∆ global block. We review the prescription of how to calculate the geodesic length in the thermal AdS 3 spacetime and characterize the geodesic diagram which length calculates this block. In section 6, we describe in detail the geodesic diagram and find its total geodesic length. Then, the resulting function is compared with the 2-point linearized large-∆ global block in t-channel. We close with a brief conclusion in section 7. All technical details are in appendices A, B, C.

Torus conformal block
Let us consider conformal primary fields Φ ∆ 1 ,∆ 2 , Φ ∆ 2 ,∆ 2 with conformal weights (∆ 1 ,∆ 1 ), (∆ 2 ,∆ 2 ). The 2-point correlation function on a torus takes the form where p H " 2πpL 0`L0 q´π c 6 is the Hamiltonian of the system, p P " 2πpL 0´L0 q is the momentum operator, q " e 2πiτ , τ is the torus modular parameter. The trace in (2.1) can be calculated in the standard basis of Verma module: where | r ∆,r ∆ 1 y is a primary state, pj k , n k , i k , m k q P N, @k P N and M " ř k i k m k ,M " ř k j k n k . The ket | r ∆ 1 ,r ∆ 1 , M,M y is the M-th level descendant state in the Verma module generated from the primary state | r ∆ 1 ,r ∆ 1 y. Taking the trace (2.1) over the states (2.2), we obtain xΦ ∆ 1 ,∆ 1 pz 1 ,z 1 qΦ ∆ 2 ,∆ 2 pz 2 ,z 2 qy " pqqq´c 24 where G M |N is the inverse of the Gram matrix, and the function F is defined as: We can expand matrix elements (2.4) in two different channels, in the so-called s-channel and t-channel. We will focus on the t-channel expansion which is defined by replacing Φ ∆ 1 ,∆ 1 pz 1 ,z 1 qΦ ∆ 2 ,∆ 2 pz 2 ,z 2 q with its OPE, holomorphic part of which is defined as: Plugging the OPE (2.5) into (2.4) and then placing (2.4) in (2.3), we obtain where r ∆ 1 , r ∆ 2 are intermediate conformal dimensions and the t-channel (holomorphic) conformal block is (2.8)

Torus global blocks
The global t-channel torus block is given by (2.8), where we consider only the generators L´1, L 1 , L 0 (and their antiholomorphic counterparts), which are the generators of SLp2, Cq group. In [33] the general formula for the global t-channel torus block is given . The 2-point t-channel global block is paq pnq is the Pochhammer symbol for a number a P C, and [45] τ n,m px 1 , x 2 , x 3 q " minrn,ms ÿ paq pnq " C a n n! and C n k is the combinatorial coefficient. Hereafter we omit the factor q´c {24 . Using the hypergeometric functions we can express (3.1) in a concise form [32],

4)
We must note that the first line of (3.4) is exactly the 1-point global global block given in [31,39].

Large-∆ global blocks
We begin this section by making some observations regarding equation (3.4), and some previous results of the conformal blocks. In works [30,33] was shown perturbatively that the linearized classical version of the 1-point torus conformal block (linearized classical 1-point torus conformal block) was associated with geodesic length of a certain diagram embedded in the euclidean thermal AdS 3 spacetime. On the other hand, in work [31] the relation that exists between the linearized classical (in our terms linearized large-∆) version of the 1-point torus global block with the linearized classical 1-point torus conformal block was conjectured, that relation is given by equation (5.9) of work [31], that equation defines a direct relation between the linearized classical (linearized large-∆) 1-point torus global block and the linearized classical 1-point torus conformal block. In this sense, the holographic correspondence of the linearized classical 1-point torus conformal block of works [30,33] is at the same time a correspondence for the linearized classical (linearized large-∆) 1-point torus global block.
The main objective of this work is to find a holographic correspondence of the global block (3.4), more precisely, to find a correspondence for a certain version of global block (3.4).
Since block (3.4) is factorized in terms of 1-point torus global block, and based on the above mentioned with respect to the linearized classical (linearized large-∆) 1-point torus global block, we consider that it is necessary to look for this correspondence in the large-∆ version of the global block (3.4) (more precisely in the linearized large-∆ version). In this sense, we will find the linearized large-∆ version of block (3.4).
By the definition of the 2-point large-∆ global block we have [31,33] where g la 2pt pǫ i , r ǫ i , q, wq is the large-∆ global block, and where ǫ i , r ǫ i are rescaled conformal dimensions (In [31] they are called classical global dimensions ), in our case i " 1, 2, and k as we defined it in (4.1) is a large dimensionless parameter.
In what follows we consider one of the most simple cases when the conformal dimensions are constrained as We want to analyze the behavior of g la 2pt under these conditions and then obtain the linearized large-∆ global block g lin 2pt , which is the r ǫ 1 -linear part of g la 2pt , Calculating g la 2pt using the definitions (3.4) and (4.1) and taking its r ǫ 1 -linear part, we obtain that g lin 2pt pr ǫ 1 , δ, q, , wq " g lin 1pt pr ǫ 1 , δ, qq`δr ǫ 1 plog where g lin 1pt pr ǫ 1 , δ, qq is the 1-point linearized large-∆ global block given in [30,31],(see appendix A). In [31] g lin 1pt pr ǫ 1 , δ, qq was obtained by taking the large-k asymptotic expansion of the 1point global block, and it was conjectured that g lin 1pt pr ǫ 1 , δ, qq is equal to the linearized classical 1-point conformal block of [30]. Coming back to complex variables z 1 and z 2 in (4.5) through (3.2), and representing z 1 " re iy 1 , z 2 " re iy 2 we obtain the final expression g lin 2pt pr ǫ 1 , δ, q, y 1 , y 2 q " g lin

Dual interpretation
In analogy to works [30,32,33], the holographic dual interpretation of the linearized large-∆ global block is that the 2-point linearized large-∆ global block (4.6) is associated with the geodesic length of a diagram (trajectories) embedded in the AdS 3 spacetime and shown in Fig.2. We call this diagram the t-channel diagram for the 2-point conformal block. It consists of four parts: a loop part, a radial part, and two external parts. There is another diagram associated with the 2-point conformal block, it is the s-channel diagram treated in [33]. All these diagrams are embedded in the Euclidean thermal AdS 3 spacetime where t " t`β, ϕ " ϕ`2π, r ě 0. We set l " 1. The time period β is associated with the modular parameter τ ads " iβ{2π and the temperature β " T´1.
In the geodesic approximation, the gravity functional integral is calculated near the saddlepoint given by a particular solution. At low temperatures, or in other words for Impτ ads q " 1, the thermal AdS predominates in the functional integral [46], and therefore, in this case, the classical action is determined mainly by the gravitational part. It has been calculated in [35,46,47] to be S thermal " iπτ ads 2 . Furthermore, there is a matter part of the action, which we associate with geodesic lengths of worldlines of massive particles (the masses are the rescaled conformal dimensions r ǫ 1,2 , ǫ 1,2 q. As a result, the total on-shell classical action is given by the gravitational and matter parts as Here, the first term is the gravitational part of the action in the thermal AdS, the second, third and fourth terms correspond to the material part of the action, they are the geodesic lengths of the loop trajectory, radial line and external trajectories respectively, see Fig. 2.

Worldline approach
The worldline approach has been effective for the calculation of lengths of various geodesic networks [6,9,14]. Each geodesic segment is given by the following action: S " where 1 and 2 are initial/final positions, local coordinates are x m " pt, φ, rq, the metric coefficients g mn pxq we can obtain from (5.1), and the velocity 9 x is defined with respect to the evolution parameter λ. The action is reparametrization invariant, therefore we can set the following normalization condition: |g mn 9 x m 9 x n | " 1, so that from (5.3) we obtain the on-shell value of the action S " λ 2´λ1 . Since the metric coefficients do not depend on time and angular variables the respective momenta are constant 9 p t " 0, 9 p ϕ " 0. It follows that the motion can be constrained on a surface with constant angle ϕ " 0 and constant momentum p ϕ " 0. From the normalization condition (5.4) we have g tt 9 t 2`g rr 9 r 2 " 1 and taking into account g tt g rr "´τ 2 , we have: the overall sign˘tells us if r decreases (sign´) or increases (sing`) along the proper time λ.
The length of the loop trajectory can be calculated using the definition of the time momentum p t " g tt 9 t, from which we have where s is the loop momentum, and rptq is the radial deviation. From equation (5.5) and (5.6) we can calculate the derivative of r with respect to time t, dr dt " dr dλ dλ dt , so we have the following equation of evolution: Here, the signs˘are from (5.5), the selected sign says us if the function rptq increases or decreases. We can find also the length of trajectory contained between two points r 1 and r 2 through the equation

Equilibrium condition on vertex points
In each vertex point (an intersection of three geodesic lines) there is a sum of three terms of the type (5.3). Minimizing the combination of these terms we find out that the time momenta satisfy the following weighted equilibrium condition (see [33] for more details): where p 1,2 are the ingoing/outgoing momenta of geodesic segments with the same mass parameter and p 0 is an external momentum (geodesic segments with a different mass parameter in relation to the previous two), and m " pt, rq.

Geodesic length and t-channel global block
In the t-channel case, the whole diagram of trajectories is composed of four parts, each part is interpreted as the path of a classical particle that moves in the thermal Euclidean AdS 3 spacetime with a mass equal to rescaled conformal dimension (ǫ or r ǫ). The first part is the loop trajectory beginning from point (t 0 , ρ 1 ) and going around t (in the clockwise direction), finally ending at the same point (t 0 , ρ 1 ). In Fig. 2 we denote the radius of this trajectory, its momentum and its mass-parameter by R, s and r ǫ 1 respectively. The second part is the radial trajectory that is connected to the loop trajectory at the point pt 0 , ρ 1 q and from this point goes radially to the point (t 0 , ρ 2 ), its momentum and mass-parameter are denoted by s 0 and r ǫ 2 . The third and fourth parts are two external trajectories, both of them start at the point (t 0 , ρ 2 ), the fourth one ends at point (y 2 , R 2 " 8) and the third one ends at the point (y 1 , R 1 " 8) (y 2 ă y 1 ). In the general case their masses can be different, but we will just analyze the case when these two parameters are the same. We denote this mass by ǫ, in the Fig. 2, and their radius and momenta by R 1 , R 2 , s 1 , s 2 , respectively. Quantities Rptq, R 1 ptq, R 2 ptq, t 0 , s, s 0 , s 1 , s 2 , ρ 1 , ρ 2 are functions of variables r ǫ 1 , r ǫ 2 , ǫ, y 1 , y 2 .

System of equations
Thermal AdS In this section we basically follow [33]. Fig. 2

and the respective system of equations describe the t-channel geodesic diagram for the 2-point global block.
The equation of motion for the radius is given by Integrating (6.1) for the loop trajectory and taking into account the initial condition (6.3) we find e 2pt´t 0 q " " pi`ρ 1 qpRptq´iqpρ 1´s a ρ 2 1´s 2`1`i s 2´i qps a Rptq 2´s2`1`R ptq´is 2`i q pi´ρ 1 qpRptq`iqpρ 1`s a ρ 2 1´s 2`1´i s 2`i qps a Rptq 2´s2`1´R ptq´is 2`i q , The momenta equilibrium equations at the vertex point pt 0 , ρ 1 q are For the external trajectories we fix the initial point of R 1 ptq and R 2 ptq as t " t 0 , while R 1 py 1 q " 8, R 2 py 2 q " 8 along with R 1 pt 0 q " R 2 pt 0 q " ρ 2 are boundary conditions. Integrating (6.1) for radial functions R 1 ptq, R 2 ptq and imposing the above mentioned conditions, we arrive at The momentum equilibrium equations at the point pt 0 , ρ 2 q are where we used (4.3).
In order to calculate geodesic length we need explicit expressions for Rptq, s, ρ 1 , s 0 , t 0 , s 1 , s 2 , ρ 2 that solve the defining system (6.2)-(6.7). In the next sections we consider these equations in more detail and give solutions.

Solving the defining system
Firstly, we are going to compute the lengths of the loop and radial trajectories. Since these lengths do not depend on t 0 , we can set t 0 " 0 for simplicity so that (6.3) becomes Rpβq " Rpt 0 " 0q " ρ 1 . (6.8) From (6.4) we have s 0 " 0 and We have three equations (6.2), (6.8), (6.9) to find three unknown functions Rptq, ρ 1 , s. We will refer to this system of equations as 1-point system of equations because it describes the geodesic diagram of 1-point conformal block [30]. There are two independent solution to this system of equations, both yielding the same geodesic length, see Appendix B for more details.
Here we present only one solution, Rptq "´δ psinhpt´βq´sinh ptqq a p1´cosh pβqqp4´δ 2`δ2 cosh p2t´βq´4 cosh pβqq . (6.10) Finding the geodesic length of this part of the diagram, we obtain L " r ǫ 1 pL loop`δ L radial q, (6.11) r 2`1 dr " arcsinhrρ 2 s´arcsinhrρ 1 s, (6.13) and, finally, L " log q 4`r ǫ 1 r2 arccoth 1`q a´1`2 q´q 2´q δ 2`δ log δp1`qq`2 a 1´2q`q 2`q δ 2 p´1`qq ? 4´δ 2 sr ǫ 1 δ arcsinhrρ 2 s . (6.14) For further convenience, we separate the 1-point part of this expression as (6.15) In the limit ρ 2 Ñ 8 in (6.14) the first line of this equation correspond to the geodesic length of the 1-point diagram, this result coincides with those given in [28,30]. Now, we are going to compute the geodesic lengths of the external trajectories. We know that s 0 " 0, so that using this relation in (6.7) we obtain s 1 "´s 2 . Thus, taking the ratio (6.5){(6.6) and using ρ 2 ps 1 q found from (6.7), finally, we obtain an equation for s 1 . Solving this equation we find s 1 and then ρ 2 as The lengths of external trajectories are the same due to the reflection symmetry of the geodesic diagram (for equal external dimensions only). It is given by (up to an infinite constant)

Relation between geodesic length and global block
Using (6.16), (6.17), (6.14) and (6.15) we calculate the total geodesic length of the entire diagram which is given by We see the following relation between (6.19) and (4.6) S thermal`Ltotal "´g lin 2pt pr ǫ 1 , δ, q, y 1 , y 2 q| y i Ñ´iy i`p r ǫ 1 δq constant, (6.20) where the last term does not depend on coordinates y i and q. The equation (6.20) is our main result.

Conclusions
In this work, the torus block/length duality proposed in [30,32,33] has been confirmed in the 2-point case by finding explicit expressions for the torus block with large dimensions in t-channel and the corresponding length of dual geodesic diagram. This duality was demonstrated non-perturbatively in conformal dimensions. More precisely, it was possible to verify the duality relation (1.1) for the linearized large-∆ 2-point torus global block in the t-channel, the demonstration of this case also implied non-perturbative check of (1.1) for 1-point block. It follows that these blocks have a dual interpretation in terms of certain geodesic diagrams in the thermal AdS 3 spacetime. More specifically, the linearized large-∆ 2-point torus global block is calculated by the geodesic length of the diagram shown in Fig. 2 by means of the equation (6.20).
To conclude, let us shortly mention that explicit calculations in the s-channel case entails a greater degree of complexity. This is because the respective algebraic equation system of section 6.1 is of degrees greater than four.
Acknowledgments. The author thanks the following people very deeply: K.B. Alkalaev for having proposed the problem discussed here and for his constant ideas that helped to improve this work; M.M. Pavlov for useful discussions; his family for their unconditional support.

B.2 Geodesic lengths
In this section, we compute the geodesic lengths of the loop-trajectory and radial trajectory of the diagram of Fig.2. The part containing ρ 2 in (6.13) will not be considered. We calculate the geodesic length of the loop-trajectory and radial trajectory with the equations (6.12) and ( Let us express S 1pt´diagram in terms of q " e 2iπτ (τ is the CFT modular parameter of the torus ) provided that β "´log q ads and q ads " e 2πiτ ads . For this solution we set τ ads " τ , 1 which means that q " q ads and β "´log q. Then, we obtain For the Second solution, we have (B.14) For this solution we set τ " τ ads`1 {2, this means that β "´log q ads "´log p´qq. Making this change of variables for τ ads we obtain the following equation for S 1pt´diagram