Holographic duals of large-c torus conformal blocks

We study CFT2 conformal blocks on a torus and their holographic realization. The classical conformal blocks arising in the regime where conformal dimensions grow linearly with the large central charge are shown to be holographically dual to the geodesic networks stretched in the thermal AdS bulk space. We discuss the n-point conformal blocks and their duals, the 2-point case is elaborated in full detail. We develop various techniques to calculate both quantum and classical conformal block functions. In particular, we show that exponentiated global torus blocks reproduce classical torus blocks in the specific perturbative regimes of the conformal parameter space.


Introduction
The study of the AdS 3 /CFT 2 correspondence in the large central charge approximation has lead to the simple formula relating the classical conformal block f class and the length of geodesic network L dual in the dual spacetime f class ( ,˜ |z) ∼ = L dual ( ,˜ |w) , (1.1) where coordinates z = z(w) are given by the conformal map from the AdS boundary to the plane CFT, ,˜ are conformal dimensions, the equality symbol ∼ = means that both sides are equal modulo logarithmic terms defined by the conformal map [1][2][3][4][5][6][7][8] (for the further development see [9][10][11][12][13][14][15][16][17][18][19][20]). The large-c conformal blocks can be defined by a number of heavy or light operators with dimensions growing linearly with c. The corresponding bulk geometry can be either conical singularity/BTZ [2] or thermal AdS [15] in the case of the boundary spherical or toric CFT, respectively. In this paper we continue to study large-c CFT 2 on a torus from the holographic perspective. Basically, we consider the 2-point torus blocks generalizing to the n-point case wherever possible. We calculate the corresponding classical blocks within various approximations and show that they are holographically realized as geodesic networks on the thermal AdS in keeping with the block/length correspondence formula (1.1).
The outline of the paper is as follows. In Section 2 we first shortly discuss n-point correlation functions on a torus and then focus on the n = 2 case. In order to calculate the block functions we develop two methods, (i) a straightforward evaluation of the constituent matrix elements, (ii) a combinatorial (AGT) representation, see Appendices A, B, and C, respectively. 1 In Section 3-5 we discuss various limiting torus blocks paying particular attention to the so called classical and global blocks. 2 In Section 6 we discuss the classical global torus blocks that arise as exponentiation of the global blocks in the regime of large conformal dimensions. We also show that classical global blocks describe the linearized part of the standard classical blocks within particular perturbation theory when some conformal dimensions are larger than the others. This yields the method to calculate approximate classical blocks that bypass the full quantum Virasoro block analysis.
The holographic interpretation of classical torus blocks is discussed in Section 7. Here we propose the general scheme and formulate the system of differential and algebraic equations that describes the dual network. However, similar to the n-point sphere case, exact solutions to the equation system are not known yet. Instead, in Section 8 we propose a particular perturbation theory and find approximate solutions in the 2-point case.
We close with a brief conclusion in Section 9. All technicalities and subsidiary discussions are collected in Appendices A -E.
An n-point torus correlation function of arbitrary primary operators φ i (z i ,z i ) with conformal dimensions (∆ i ,∆ i ) is given by φ 1 (z 1 ,z 1 ) · · · φ n (z n ,z n ) = (qq) − c 24 Tr q L 0qL0 φ 1 (z 1 ,z 1 ) · · · φ n (z n ,z n ) , (2.1) where q = e 2πiτ cf t , τ cf t ∈ C is the torus modular parameter, L 0 andL 0 are the Virasoro generators, c is the central charge. Since a torus can be realized as a cylinder with the edges rotated and glued together, the torus correlation functions can be defined using the plane CFT notation (we always assume that (2.1) is supplemented by the conformal map from the plane to the cylinder) [22]. Assuming that the space of states is generated by primary operators with dimensions denoted as (∆ 1 ,∆ 1 ) and evaluating the trace we find that the torus correlation functions is given by a power series in the modular parameter, where the matrix element of n primary operators F (∆ i ,∆ i ,∆ 1 ,∆ 1 , M, N,M ,N |z,z) = ∆ 1 ,∆ 1 , M,M |φ 1 (z 1 ,z 1 ) · · · φ n (z n ,z n )|N,N ,∆ 1 ,∆ 1 , (2.3) is given in the standard basis |M,M ,∆ 1 ,∆ 1 =L j 1 −n 1 · · ·L j l −n l L i 1 −m 1 · · · L i k −m k |∆ 1 ,∆ 1 . Here, descendant vectors in the Verma module are generated from the primary state |∆ 1 ,∆ 1 , indices M,M label basis monomials, |M | = i 1 m 1 + . . . + i k m k and |M | = j 1 n 1 + . . . + j l n l . The matrix B M |N is the inverse of the Gram matrix.
In what follows we focus on the n = 2 point case and expand the matrix elements (2.3) into conformal blocks. Different exchanged channels can be obtained by plugging resolutions of identity into the matrix element (2.3) and/or using the OPE. Note that conformal dimensions (∆ 1 ,∆ 1 ) associated to the q-expansion (2.2) partially describe possible exchanged channels and, in fact, define an (un)closed loop part of the corresponding diagrams.
In the 2-point case, the torus correlation functions can be expanded in two channels that we call s-channel and t-channel, see Fig. 1. They directly follow from (2.1): the trace over the space of states can be understood as a sum of 4-point correlation functions on a sphere over two outermost descendant operators in points ∞ (left) and 0 (right). Then, using the OPE and recalling that on a sphere there are three different 4-point exchanged channels we can see that identifying two external legs we get just two topologically non-equivalent configurations, Fig. 1. s-channel. Two-point torus conformal block in the s-channel (left diagram on Fig. 1) is defined by inserting the resolution of identity between two primary operators in the matrix element (2.3), where the inverse Gram matrix B S|T enters the formula because the standard basis is nondiagonal. Then (2.3) splits into products of two 3-point functions of primary operator with two descendant operators. Recalling that ∆ m |φ k (z k )|∆ l = C∆ m∆k∆k z∆ m−∆k −∆ l k , where C∆ m∆k∆k are the structure constants, we decompose the 2-point correlation function as follows where the s-channel (holomorphic) conformal block is 3 (2.6) where∆ 1,2 are exchanged conformal dimensions. Given that all inner products in (2.6) have been explicitly calculated we arrive at the double power series in the modular parameter q and the ratio x = z 2 /z 1 with the expansion coefficients being rational functions of conformal parameters, (2.7) Using explicit formulas from Appendix A we write down the block function in the lowest orders, 4 (2.8) Setting ∆ 2 = 0, ∆ 1 ≡ ∆ and equating∆ 1 =∆ 2 ≡∆ we reproduce the 1-point torus block [25] with external dimension ∆ and exchanged dimension∆. From the form of coefficients in (2.8) it follows that vacuum (∆ 1 = 0 or∆ 2 = 0) blocks in this channel are absent.
t-channel. Alternatively, we can use the OPE of two primary operators in the matrix element (2.3). In this case (2.3) reduces to summing over 3-point functions of three descendant operators with the OPE coefficients. Namely, we fuse two primaries where the resulting operator is given by Plugging the OPE (2.9) into the the correlation function (2.2) we find In principle, the matrix approach can be used to find coefficients in any order. Instead, in Appendix C we develop the combinatorial representation of the n-point conformal torus blocks in the s-channel. Using Mathematica we apply this representation to compute the block coefficients up to high enough order in q and x.
where the t-channel (holomorphic) conformal block is (2.12) where∆ 1,2 are exchanged conformal dimensions. Given that 3-point functions of three descendants in (2.12) have been explicitly calculated we arrive at the double power series in the modular parameter q and the ratio w = (z 1 − z 2 )/z 2 with the expansion coefficients being rational functions of conformal parameters, Note that the t-channel block is the power series in w-variable, while the s-channel block is the Laurent series in x-variable, cf. (2.7). Using explicit formulas from Appendix B we write down the block function in the lowest orders, (2.14) The 1-point block with dimensions ∆,∆ is reproduced by setting ∆ 1 = 0, ∆ 2 =∆ 2 ≡ ∆, and∆ 1 ≡∆. Let us note that there exist vacuum t-channel conformal blocks which arise when the second exchanged operator (an intermediate straight line on the right diagram on Fig. 1) is the unity operator. Thus, setting∆ 2 = 0 supplemented by the fusion condition ∆ 1 = ∆ 2 ≡ ∆ we find the vacuum block function which depends on the external dimension ∆, and the loop exchanged dimension∆.

Classical two-point torus blocks
The parameter space of the conformal block functions V ∆,∆ c (q|z) includes external and intermediate dimensions, modular parameters, and the central charge. In this section we shall discuss semiclassical blocks that correspond to different asymptotics in the parameter space when the conformal dimensions scale differently with the central charge. Near the point c = ∞ we distinguish between heavy and light dimensions depending on how a given dimension scales with the central charge: ∆ light ≈ or ∆ heavy ≈ c , where are classical dimensions. Assuming that a conformal block depends on a number of light and heavy operators and expanding around c = ∞ we find that where the expansion coefficients are power series in the modular parameter q with coefficients being rational functions of classical dimensions ,˜ . The principal part of (3.1) vanishes in the large-c limit, while the form of the regular part defines a particular semiclassical block. It is known that there are different types of semiclassical blocks, including the light blocks, various heavy-light blocks, and the classical block (see e.g. discussion in [7,21,26,27]). 5 Using the expansion (3.1) we see that e.g. the light block is given by v ,˜ 0 (q, z), while other coefficients vanish, v ,˜ n (q, z) = 0, n > 0. The classical block has non-vanishing coefficients v ,˜ n (q, z) = 0, ∀n ∈ N 0 . In particular, the regular part of the classical block is claimed to be an exponential functional linear in c.
In what follows we focus on the classical torus block which therefore is given by where function f 1,2 ,˜ 1,2 is the corresponding classical conformal block conveniently parameterized by the classical conformal dimensions s. Using (2.8) we find the s-channel block function where a few lowest level coefficients are given by t. Using (2.14) we find the t-channel block function where a few lowest level coefficients are given by 4 Perturbative classical s-channel torus blocks The analysis of classical blocks simplifies within various approximations where some of dimensions are much larger than the others, see, e.g. [2,6,8,13,15]. In the torus case, already from the first coefficients (3.5) (for simplicity, we choose pairwise equal dimensions) f 0 = 2 1 2˜ 1 we immediately conclude that it blows up when˜ 1 1 and is smooth in the opposite regime In what follows we study the torus s-channel blocks assuming that the exchanged channels with equal dimensions˜ ≡˜ 1 =˜ 2 are much heavier than the external operators 1 and 2 , i.e. i /˜ 1. This step provides the first level perturbation expansion because the external dimensions remain unrelated to each other. On the second level, we find two possible perturbation expansions. In the first version called the superlight expansion [8,10,26] we assume that the second external dimension is much less than the first one, 1 / 2 1. In the second version that we call the double leg expansion the external dimensions are equated.
Superlight expansion. In this case the dimensions are restricted as The first constraint is the fusion rule guaranteeing that 1 = 0 and/or 2 = 0 are consistent. The corresponding conformal block is parameterized by three parameters˜ , δ, ν so that we arrive at the triple deformation theory: we expand around˜ = ∞ and then around δ = 0 and ν = 0. Keeping terms linear in˜ we arrive at the quartic series expansion where the second term is the perturbative 1-point block with coefficients [15,27] f (1) because setting ν = 0 we automatically reproduce the 1-point case, and the third term gives the ν-correction, where a few first coefficients read More higher order terms can be found in (D.1).
Double leg expansion. The conformal dimensions are pairwise equal and satisfy the con-straints˜ The corresponding block depends on two parameters˜ and δ and, therefore, we consider the double perturbation theory: we expand around˜ = ∞ and then around δ = 0. Keeping terms linear in˜ we arrive at the triple series expansion where a few first coefficients read The higher order expression can be found in (D.2).

Global torus blocks
Global blocks of CFT 2 are associated to sl(2, R) ⊂ V ir. Equivalently, the global blocks can be obtained by considering particularly contracted Virasoro algebra at c → ∞ while keeping both external and exchanged conformal dimensions independent of the central charge, [27]. Note that just sending the central charge to infinity while keeping the dimensions fixed yields the so called light blocks which are generically different from the global blocks by the truncated Virasoro character prefactor (see [27] and [21] for details, and our discussion below (3.1)). In what follows we calculate the global blocks from scratch using the definitions (2.6) and (2.12) and restricting the Virasoro generators to sl(2) ones. 6 s-channel. Associating the 2-point s-channel block (2.6) to the sl(2) algebra we find where x = z 1 /z 2 and the coefficients τ m,n = τ m,n (∆ a , ∆ b , ∆ c ) defining the sl(2) 3-point function of a primary operator ∆ b and descendant operators ∆ a,c on the levels n, m are given by [26] . Setting ∆ 2 = 0, ∆ 1 ≡ ∆ and equating∆ 1 =∆ 2 ≡∆ we will reproduce the 1-point torus block. Indeed, in this case τ n,m (∆, 0,∆) = δ n,m m!(2∆) m and therefore the n-th global block coefficient (5.1) is given by τn,n(∆,∆,∆) n!(2∆)n . It defines the 1-point global block with external/exchanged dimensions ∆ and∆ and can be expressed as the hypergeometric function coefficients [25].
In the n-point case we can consider the generalized s-channel block defined by a diagram consisting of a loop with n external legs. Let x i = z i /z i−1 , where x i = 2, ..., n and q = x 1 x 2 · · · x n . With the identification ∆ 1 = ∆ n+1 ,∆ 1 =∆ n+1 and s 1 = s n+1 the n-point global block in the generalized s-channel is given by where τ -coefficients are given by (5.2).
t-channel. To find the 2-point global block in the t-channel (2.12) we use the OPE for quasi-primary fields, see e.g. [28][29][30], where the OPE coefficients are now packed into the confluent hypergeometric function, cf. (2.9)-(2.10). Substituting the above OPE into the block function (2.12) restricted to the sl(2) subalgebra we find the global t-channel block Setting ∆ 1 = 0, ∆ 2 =∆ 2 ≡ ∆, and∆ 1 ≡∆ we will reproduce the 1-point torus block because in this case σ m (0, ∆, ∆) = δ m,0 and, therefore, the n-th global block coefficient (5.5) is given by τn,n(∆,∆,∆) n!(2∆)n which is exactly the 1-point block coefficient. Let us note that the 1-point torus block factorizes from the 2-point expression. Moreover, (5.5) is a product of two hypergeometric functions in accordance with the analysis in [19]. The higher-point generalizations can be obtained in the standard fashion by successively applying the OPE (5.4) in the torus correlation function (2.2) associated to the sl(2) algebra.

Exponentiating global blocks
Let us consider the regime of large dimensions, when all conformal dimensions are rescaled using a large parameter κ in a coherent manner, where σ i andσ j can be referred to as classical global dimensions. We expect that in this regime the global blocks are exponentiated where g σ i ,σ j (x) is a classical global block. Indeed, n-point global blocks satisfy the Casimir channel equations which are second order partial differential equations with coefficients being rational functions of the conformal dimensions ∆ i , ∆ j [19,26,27,31]. At the same time from the general theory of differential equations it follows that once the equation coefficients depend on some large parameter in a specific way then the leading asymptotics in the solution space are given by exponentials. 7 Matching the central charge c of the Virasoro block and the scale κ of the global blocks we see that to some extent the classical global blocks (6.2) are similar to the standard classical blocks, cf. (3.2). 8 Most probably, the standard classical block and classical global blocks are not related for general values of conformal dimensions. However, there is a lot of evidence that they can be related to each other in particular perturbative regimes when some conformal dimensions ∆ bgr are much larger than the others ∆ prt , (Note that this assumption is equally translated both to the standard classical and global classical dimensions.) For example, in the sphere case with two background external operators the perturbative classical block coincides with respective perturbative classical global block [7,26]. In the 1-point torus case, the relevant perturbation theory assumes that the exchanged channel (loop) dimension is much large than the external dimension. In this case we also observe that the perturbative classical block can be reproduced from the global block [27].
Superlight expansion. Using the superlight expansion assumption (4.1) rewritten in terms of classical global dimensions σ i andσ j we find out that modulo the logarithmic term the respective perturbative classical block coincides with the perturbative classical global block. Indeed, the corresponding classical global block is given by where the first term defines the 1-point block with coefficients (4.3), and the second term is the first order ν-correction, where a few first coefficients read 3 (x, q) = 7 See e.g. the monograph [32]. The 1-point case was considered in [27], the n-point case was discussed in [19]. 8 Contrary to the case of global blocks, the Virasoro blocks do not satisfy any differential equations (except for special values of conformal dimensions that results in the BPZ equation). In particular, this is why the exponentiation of the Virasoro blocks has not been rigorously proven yet (for related considerations see [33,34]).
We observe that in a given order this expression coincides with the perturbative conformal block calculated using the combinatorial representation (D.1). Also, as a consistency check, setting x = 1 we reproduce the 1-point perturbative torus block with the external conformal dimension ( 1 + 2 ). Technically, at this point the second term in (6.4) coincides with the first term up to a prefactor fixed in the linear order in ν by that ( 1 + 2 ) n ≈ (1 + nν)δ n . Indeed, we notice that the polynomial in the square bracket of f 3 has a root x = 1, and, therefore, we find that f (2) 2k+1 (x = 1) = const and f (2) 2k (x = 1) = 2kf (1) 2k + const. Double leg expansion. Similarly, using the double leg expansion assumption (4.5) rewritten in terms of the classical global dimensions σ i andσ j we find out that the corresponding block is given by where a few first coefficients are found to be (6.9) In a given order, the above expression reproduces the perturbative conformal block calculated using the combinatorial representation (D.2). Noticing that g 3 = 2f (2) 3 and setting x = 1 we are left with the 1-point perturbative torus block with the external conformal dimension 2 because g (1) 2k+1 (x = 1) = const and g (1) 2k (x = 1) = 2 2k f (1) 2k + const. Indeed, going to the 1-point case produces factors ( 2 ˜ ) n ≈ 2 n δ n . t-channel case. The relevant perturbation theory in the t-cannel leads to imposing the following constrains, where the first constraint is the fusion rule guaranteeing that˜ 2 = 0 approximation is consistent. The corresponding perturbative classical global block is found to be where a first few coefficients can be represented as We notice that the first coefficient h 1 does not depend on q, while higher coefficients h n at n ≥ 2 do not depend on w and coincide with the 1-point coefficients (4.3).
7 Holographic duals of the classical s-channel torus blocks In this section we advocate that in the s-channel the perturbative classical block function f class considered in Section 4 can be represented as 9 where S therm is the holomorphic part of the 3d gravity action evaluated on the thermal AdS space, L dual (y, β) is the length of the dual necklace graph attached to the boundary points y i , i = 1, ..., n, see Fig. 2. The gravitational action expressed in terms of the rescaled central charge c/6 is S thermal = iπτ /2 [35][36][37]. In terms of the modular parameter the action is S thermal = 1/4 log q that reproduces the classical conformal block (3.4) at zeroth dimensions. The block/length correspondence (7.1) is supplemented with the identification of the modular parameter and coordinates of the primary operators as Let us consider the thermal AdS space that is a solid torus with time running along the non-contractible cycle, 3) 9 The case of 1-point classical torus blocks was considered in [15]. The holographic correspondence for higher point global blocks was discussed recently in [19].
where t ∼ t + β, ϕ ∼ ϕ + 2π, r ≥ 0, the AdS radius is set l = 1. The time period β defines the modular parameter τ ads = iβ/2π and the temperature β ∼ T −1 . The equivalent form of the metric can be obtained by rescaling the time coordinate t → −iτ ads t so that the metric coefficients explicitly depend on the modular parameter while t ∼ t + 2π. However, the metric (7.3) is more convenient in practice because the modular parameter shows up only when integrating along the non-contractible cycle. Otherwise, the local dynamics is the same as in the AdS space with non-periodic time.
Let us notice that in order to describe the correspondence between holomorphic conformal blocks and geodesic networks on the two-dimensional slice (7.1) we assume that the modular parameter q and coordinates of the primary operators are real (7.2). In this way we obtain that the holomorphic conformal block, being by definition a complex function, is equated to the real geodesic length on the thermal AdS with pure imaginary modulus τ ads . 10 Since q = e 2πiτ cf t we find that up to the modular transformations the parameter τ cf t can take one of two values τ cf t = 0 + ia or τ cf t = 1 2 + ia, where ∀a ∈ R. Therefore, In the first case, the torus where our CFT lives is indeed the conformal boundary of the bulk space. This is realized by the map (7.2). In the second case, the modular parameters are different but in the low-temperature approximation τ ads → ∞, we again arrive at the standard duality with τ cf t ≈ τ ads [15]. The two corresponding bulk solutions are discussed below in Section 8.1.1.

Worldline formulation
In what follows we shortly review the worldline formulation on the thermal AdS background with the metric (7.3), see [6,8,15] for more details. A geodesic segment can be described by the action where local coordinates are x m = (t, φ, r) and the metric coefficients g mn (x) are read off from (7.3). The reparametrization invariance allows us to impose the normalization condition |g mn (x)ẋ mẋn | = 1 so that the on-shell action is now given by S = λ 2 − λ 1 . Using the Killing vectors of (7.3) we can restrict the dynamics to the constant angle ϕ = 0 surface (the annulus on Fig. 2). The corresponding conserved momentum is p φ = 0, while the other conserved momentum p t is the motion constant that defines the shape of geodesics. In this case, the normalization condition is given bẏ The geodesics network stretched between boundary points on different two-dimensional slices computes the full classical conformal block which is the sum of holomorphic and antiholomorphic conformal blocks. See [6] for an extended discussion in the sphere case.
• The length of a loop segment is given by where r(t) being the radial deviation, and t 1,2 are initial/final time positions.
• The length of an external leg with one endpoint attached to the conformal boundary is where ρ is the vertex radial coordinate, and the cutoff parameter Λ → ∞ is introduced to regularize the conformal boundary position.
The radial deviation is governed by the following evolution equation that can be explicitly integrated. Using (7.9) we can express the momentum s in terms of the radius and its time derivative and substitute then into (7.7). The resulting action is (7.5) on the φ = const slice, where the metric is given by (7.3). The corresponding equations of motion are second-order ODE having a general solution that depends on two integration constants. The formulation described above is partially integrated form of these equations. Indeed, one of integration constants is given by the time momentum that drastically simplifies the analysts of geodesic lines, while the residual dynamical equation (7.9) is first-order ODE with a general solution depending on the other integration constant.
Integrating the evolution equation (7.9) we get 10) where r(t 0 ) = r 0 are initial conditions. Solving this equation we can express the radial coordinate as a function of parameters r 0 and s, i.e. r = r(t|r 0 , s).

Dual geodesic networks
Let us first describe the kinematics of the dual networks in the n-point case. The graph on The geodesic length can be associated to the mechanical system of massive test particles propagating in the bulk space. Hence, the total length of the geodesic network is given by where classical conformal dimensions ∼ masses, and the length of each geodesic segment (7.7), (7.8) reads where R m (t) is a radius of m-th loop segment, s m ands m are the leg and loop momenta, ρ m is the radial vertex coordinate, the integration limits γ m 1,2 parameterize endpoints of each loop segment: To find the total length of the geodesic network we have to know the radial functions R m (t) explicitly as well as the vertex position ρ m , the momenta s m ,s m , and the times γ m . supplemented with the condition (7.14). Whence, we have 2m equations for 2m variables that allows us to solve the evolution equations in terms of the vertex radial positions.
Time intervals. In order to find initial/final time positions (7.13)-(7.14) explicitly we use the general formula for time intervals stretched between given two radial positions (7.10). Let us consider the m-th leg stretched from the vertex point (ρ m , γ m ) to the boundary attachment point (∞, y m ). In this case, using formula (7.10) we find the time interval relation which can be explicitly solved as γ m = γ m (ρ m , s m , y m ).
Momentum conservation conditions. Both the vertex radial positions and momenta are constrained by the conservation conditions following from the least action principle for the total action (7.11). There are three possible configurations of external legs according to values of their momenta: radial, convex, or concave legs. In what follows we consider the case of one concave and n − 1 convex legs. Other possible options can be shown to be equivalent to this configuration or be inconsistent. The momentum conservation condition in each vertex reads˜ mpm +˜ m+1pm+1 + m p m = 0, where m = 1, ..., n and p m ,p m are two-component conserved momenta on the annulus (for the general discussion of trivalent vertices on the hyperbolic spaces see [38]). The corresponding time and radial components are given by where indices run m = 1, ..., n with the identification (n + 1) → 1. Remarkably, these constraints can be explicitly solved as where we introduced Following the general analysis of trivalent vertices on hyperbolic spaces [38] we can show that there are real vertex solutions ρ 2 m ≥ 0 provided that the conformal dimensions satisfy the triangle inequalities m +˜ m+1 ≥ m ,˜ m + m ≥˜ m+1 ,˜ m+1 + m ≥˜ m . (7.22) Final assembling. Now, we gather all ingredients discussed above. In general, there are 2n equations (7.16), 2n equations (7.18) and (7.19), and n equations (7.17) for 5n variables ρ m , s m ,s m , γ m , and integration constants c m , m = 1, ..., n. Thus, we unambiguously fix all these variables in terms of the modular parameter β, the boundary attachment positions y m , and classical dimensions m ,˜ m , m = 1, ..., n. For convenience, using the rotational invariance of the boundary circle we can set y 1 = 0. Finally, we will obtain the total length as L dual = L dual (β, y m , m ,˜ m ).

Dual networks for the 2-point block
We shall apply the general scheme described above to the 2-point case. The respective geodesic network is shown on Fig. 3. It consists of four different segments: two loop segments and two legs stretched from the boundary attachment points to trivalent vertex points. The total length is given by where the segments are described by (7.12)-(7.14). The radial coordinates of each loop segment R m (t) in (7.12) are described by the evolution equation (7.15) subjected to the vertex boundary conditions (7.16), which in this case take the form The general solutions R 1 = R 1 (t | c 1 ,s 1 ) and R 2 = R 2 (t | c 2 ,s 2 ) are parameterized by two integrations constants c 1 and c 2 .
On top of that there are four momentum conservation conditions (7.18) and (7.19). Thus, we have to solve 8 equations to fix 8 variables ρ 1,2 , s 1,2 ,s 1,2 , and c 1,2 . Exact solutions to this algebraic systems are not known that is a common problem arising in the bulk analysis, for a review see, e.g. [38].˜ In what follows we use different approximations analogous to those developed for the classical conformal blocks in Section 4. In general, we assume that the loop segments are more massive than the external legs,˜ m / k 1 .

(8.3)
To simplify our consideration we assume that˜ k =˜ m ≡˜ that gives rise to the natural heavy parameter. Expanding around˜ = ∞ we arrive at the perturbative geodesic network length function which is linear in˜ and depends on δ i = i /˜ 1. In the 2-point case, there are two small parameters that define further possible perturbation expansions: (i) δ 2 δ 1 , (ii) δ 1 = δ 2 . In this section we explicitly consider these two opposite regimes referring to them respectively as superlight and double leg expansions.

The seed solution: 1-point block
The seed solution arising at ν = 0 is given by the tadpole geodesic graph corresponding to the 1-point perturbative torus block [15]. However, it is not exact in δ because the remaining geodesic equations are still too complicated to be solved exactly.
In the 1-point case the conservation conditions are reduced tõ Following [15] we expands 1 ≡ s, R 1 (t) ≡ R(t), and ρ 1 ≡ ρ as We note that the time coordinate of the vertex defined by (7.17) does not depend on δ. Indeed, s 1 = 0, i.e. the leg is stretched along the radial direction, and, therefore, from (7.17) it follows that γ 2 = y 2 = 0. Now, we see that the boundary conditions (7.16) in the 1-point case are given by R(0) = ρ and R(0) = R(β). Using the first condition we find that the time interval equation (7.10) can be reduced to (8.7) Expanding (8.5), (8.7) in δ using (8.6) we find in the first non-trivial order that there are two possible sets of solutions which can be used to calculate the corresponding first order correction R 1 (t).
The solution set (8.9) and the length of the corresponding tadpole graph has been analyzed in [15]. One can show that in this case the loop goes through the point t = β/2 and r = 0 in all orders in δ. It implies that the deformation caused by the leg pulling the loop is rapidly damped near the vertex point. In what follows we use the solution set (8.8). In this case the radial deviations of the loop segments are not vanishing near t = β/2 meaning that the leg produces perturbations along the whole loop. 11 Most importantly, these two types of solutions correspond to boundary CFTs with two different modular parameters, see our discussion below (7.4).
To summarize, the seed solution is given by We note that the zeroth order s 2 , ρ 2 , and γ 2 associated to the second vertex turn out to be non-vanishing power series in δ, see below. 12 They can also be treated as the seed quantities not seen at ν 0 order.

First order corrections
Following Appendix E we find the first non-trivial corrections Momenta s 1 and s 2 can be calculated using the conservation condition (E.2). Substituting the above expansions into the total action (8.1) and evaluating the integrals (7.12) we find that where the leading contribution is the known length of the 1-point graph, terms O(νδ) can be shown to be absent, while the first non-trivial correction is O(νδ 2 ). This decomposition is consistent with what we would expect from the boundary side, cf. (D.1).
Using the map (7.2) which in our case takes the form β = − log q and y = − log x we find that the total length (8.14) goes to Adding the thermal AdS term S thermal = 1 4 log q and expanding in the modular parameter q we find that up to unimportant additive constant S thermal + L dual (β, y) = −f˜ ,δ,ν (q, x) , where the perturbative block is given by (D.1), or, in a more refined form by (6.4). In this way we reproduce the identification formula (7.1).

Double leg expansion
We will now move on to consider the second perturbation theory. In this case we assume that conformal dimensions are constrained by (4.5). Most of the steps are similar to that of the previous section so here we only write down the resulting expressions. A more detailed analysis is relegated to Appendix E.
The radial functions of two loop segments and the corresponding momenta are and We note that in the first order the external momenta coincide. Moreover, both s α = 0, i.e. contrary to the 1-point case the legs are curved, cf. Fig. 3. It follows that the loop segments have different momenta and, therefore, the whole loop is inhomogeneously stretched despite that the loop dimensions are equal. Substituting the above expansions into (7.11) and (7.12) we obtain that in the first nontrivial order the total action is given by, 19) and, finally, using the map (7.2) we arrive at Expanding in the modular parameter q we find that modulo an additive constant S thermal + L dual (β, y) = −f δ,˜ (q, x), where the perturbative block is given by (D.2), or, in a more refined form by (6.7). This proves the identification formulas (7.1) and (7.2).

Concluding remarks
We considered the semiclassical holographic duality in the case of Virasoro CFTs on the torus. We studied classical torus blocks from various perspectives, in particular, we showed that they are dual to geodesic networks stretched in the thermal AdS bulk space, Fig. 2. In the n-point case we formulated the system of differential and algebraic equations that govern the dual network. Using various approximation schemes we explicitly solved the system in the 2-point case.
Our analysis in this paper was mainly focused on the perturbative classical torus blocks in the s-channel. We explicitly showed that the perturbative classical blocks within one or another perturbation theory are equal to the classical global blocks in the limit of large dimensions. Nevertheless, we also analyzed different forms of the t-channel torus blocks: quantum, global and classical. In particular, we calculated the perturbative classical global block in the case of equal external dimensions and heavy loop channel. In this respect our results prove one of the main conjectures of [19] that global blocks are related to the perturbative classical blocks.
In this paper the geodesic networks are described using the first order formulation which results from partially integrating the standard second-order geodesic equations. In this case the integration constants arising when going to the first order formulation are conserved momenta. In the sphere CFT case, it turns out that momenta are holographically related to the accessory parameters of the monodromy method and this observation is instrumental in proving the holographic duality between classical blocks and geodesic lengths in the npoint case [6,10,38]. In this respect, the monodromy method on the torus [39][40][41][42] which is essentially based on Virasoro symmetry provides an interesting possibility to go beyond the sl(2) Casimir equation analysis of [19].
The study of the large-c torus CFT and semiclassical duality can be extended in many ways. For example, conformal blocks considered in this paper were defined in two channels only. The next natural step would be to generalize to other channels and identify the corresponding bulk backgrounds that is intimately related to perturbation schemes we use. Also, it would be interesting to analyze the torus correlation functions with heavy insertions in the Liouville theory along the lines of Refs. [43,44].

A The s-channel torus block
For illustrative purposes, using the general formula (2.6) we explicitly find the block coefficients in the linear order in the modular parameter, .
A-coefficients. We easily find the first coefficient Now, we calculate the second coefficient defined as A 2 x 2 = κ TrLB −1 2 R, where the Gram matrix on the second level is given by (A.4), and matrices L and R are given by where κ 1,2 are defined below (A.5). Using (A.9) we can finally assemble the coefficient A 2 as follows (A.10) B-coefficients. Analogously, we find where new matrices L and R are given by 12) The matrix elements are given by Finally, we assemble the coefficient B 2 as follows (A.14) We note that the above calculations can be effectively extended to any order using a recursive technique elaborated in [45].
where κ −1 = ∆ 1 |φ∆ 2 (z 2 )|∆ 1 . Let w = (z 1 − z 2 )/z 2 . Then, the expansion coefficients (B.4) up to the second order in w can be represented as The matrix elements are found to be (B.6) For the sake of simplicity, we represent the coefficients in terms of the β-coefficients as follows and

C Combinatorial representation of the s-channel block
Here we explicitly elaborate the combinatorial AGT [47] representation of the torus conformal multi-point blocks. First, we set convenient Liouville-like parametrization. Instead of central charge c and conformal dimensions ∆ k we will use parameters b and p k according to Let F (q), where q = (q 1 , ..., q N ), be the conformal block of N primary fields φ ∆ k (z k ), k = 1, ..., N , on the torus with the modular parameter q, associated with the (left) diagram on Fig. 2 and with the intermediate channel parameters between two external lines p k and p k+1 beingp k . The modular parameter q and the holomorphic coordinates z k are expressed in terms of According to AGT the Virasoro conformal blocks obey the following factorization property where F H (q) is the N -point conformal block of Heisenberg primariesφ k (z k ) associated with the same diagram. The Heisenberg algebra is defined by [a n , a m ] = n 2 δ n+m,0 , n, m ∈ Z , (C.4) and p k is the conformal parameter of the corresponding Virasoro primary field φ ∆ k (z k ). Using commutation relations (C.4), (C.5) and diagonal form of the Heisenberg Gram matrix the Heisenberg conformal block F H (q) coefficients can be computed order by order. 13 For example, on the two lower levels we find and ∼ q i q i+1 : where M 11 stands for the inverse (first-level block) Gram matrix and the normalization φp i |φ p j |φp k = 1 is taken into account.
In the case of 2-point s-channel torus Heisenberg block F H 2pt (q) which contributes to (C.3), we find up to forth degree in q i the following expression ∞ n=0 1 − q n q 1 α 1,2 1 − q n q 2 α 2,1 1 − q n+1 α 1,1 +α 2,2 , (C.8) where w z = q 1 and q = q 1 q 2 , and α i, The second factor in (C.3) has the following combinatorial representation where N 0 stands for non-negative integers, and . (C.10) Here, the sum goes over N pairs of Young tableaux λ j = (λ j ) 14 with the total number of cells | λ j | ≡ |λ (1) j | + |λ (2) j | = k j and we identifyp 0 =p N , λ 0 = λ N . The explicit form of Z reads For a cell s = (m, n) such that m and n label a respective row and a column, the arm-length function a λ (s) = (λ) m − n and the leg-length function l λ (s) = (λ) T n − m, where (λ) m is the length of m-th row of the Young tableau λ, and (λ) T n the height of the n-th column, where (λ) T stands for the transposed Young tableau.
Using the final combinatorial expression for the 2-point s-channel torus block we find the block coefficients reproducing those in Appendix A.

D Perturbative classical s-channel torus blocks
Using the combinatorial representation of the 2-point torus block elaborated in Appendix C we find the perturbative classical s-channel block: where the ellipses denote higher orders in q, x and˜ , δ, ν.
Double leg expansion.

E Perturbation theory in the bulk
Superlight expansion. Let F = {s α ,s α , γ α , ρ α , R α , L α loop , L α leg | α = 1, 2} be a double series in the deformation parameters ν and δ, Using the general formulas (7.20) we solve the conservation conditions as and In what follows we discuss several peculiarities of geodesic equations not seen previously in the 1-point torus case and in the n-point sphere case. The main observation is that the angle (time) positions must be known explicitly because they are the integration limits in the loop segment integrals.
(I) First of all, we note that defining the radial vertex positions as ρ α = R α (γ α ) and then decomposing in ν we find in the lowest orders relations like where the dot denotes a time derivative. Such decompositions do not arise in the 1-point case because the external leg is radial, and, therefore, the angle position is fixed and coincides with the boundary attachment position. When defining the superlight expansions (E.1) we have to assume that ν-corrections start with δ 1 . It follows that setting δ = 0 we are guaranteed that all ν-dependent terms vanish, i.e. once the first leg is switched off than the second leg is vanished identically. Hence, Using these formulas we find that in the first non-trivial orders In other words, the radial vertex positions are defined by radial functions at the seed angle values, γ α[00] ≡ y α = {0, y}. This property holds only in the first non-trivial order, while in higher orders we will have to use the general expansion formulas like (E.4). Note that relations (E.6) considerably simplify the analysis of the evolution equation and the vertex boundary conditions. To find how the angles flow with the deformation parameters we use the interval equation (7.17). Assuming that the seed solution is given by (8.5)  where the ellipses denote higher order terms. The above relations say that the angle (time) positions of the boundary attachment points and vertex points are generally different: in the zeroth order they coincide, but switching the interaction on they are starting to fall apart.
(II) The parameter flow of the angle positions (E.7) makes integrating along the loop segments (7.12) technically involved because the integration limits are now double power series in δ and ν. Indeed, for a given integral (7.12) there is the standard expansion formula , where each term is to be further expanded in powers of δ. We can show that in the first nontrivial orders the right-hand said of (E.8) reduces to the first and second terms, i.e. terms ∂γ α do not contribute. Analogously, expanding the remainder in δ we find that corrections in γ α do not contribute as well. In other words, in the first non-trivial order in ν and δ we can assume that the integration limits are of the zeroth order, γ α ≈ γ α[0,0] , while the integrands are expanded in the standard fashion. In higher orders this is not generally true.