More on Wilson toroidal networks and torus blocks

We consider the Wilson line networks of the Chern-Simons $3d$ gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus $2d$ CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of $sl(2,\mathbb{R})$ algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of $sl(2,\mathbb{R})$ representations: (1) $3mj$ Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental $sl(2,\mathbb{R})$ representation.

The Wilson line networks under consideration are typical Penrose's spin networks [63,64]. Formally, such a network is a graph in AdS space with a number of boundary endpoints, edges associated with sl(2, R) representations and vertices given by 3-valent intertwiners. For a fixed background gravitational connection the Wilson line network is a gauge covariant functional of associated representations. To gain the conformal block interpretation one calculates the matrix element of the network operator between specific boundary states which are highest(lowest)-weight vectors in the respective sl(2, R) representations. 2 In this paper we revisit the holographic relation between the Wilson line networks and conformal blocks focusing on the case of finite-dimensional sl(2, R) representations. Our primary interest are toroidal Wilson networks in the thermal AdS 3 space and corresponding torus blocks. We formulate and calculate one-point and two-point Wilson network functionals which are dual to one-point and two-point torus conformal blocks for degenerate quasi-primary operators. The paper is organised as follows: -in Section 2 we review what is known about Wilson networks and how they compute conformal blocks. Here, we briefly recall some necessary background about Chern-Simons description of 3d gravity with the cosmological term. Then, on the basis of the findings of Refs. [21,56,61], we attempt to rethink the whole approach focusing on key elements that would allow one to study higher-point conformal blocks of (quasi-)primary and secondary operators as well as extension to toroidal Wilson networks which are dual to torus conformal blocks.
-in Section 3 we define toroidal Wilson network operators with one and two boundary attachments. They are the basis for explicit calculations of one-point blocks and two-point blocks in two OPE channels in the following sections.
-in Section 4 we consider torus conformal blocks for degenerate quasi-primary operators which are dual to the Wilson networks carrying finite-dimensional representations of the gauge algebra.
-Section 5 contains explicit calculation of the one-point toroidal Wilson network operator in two different representations, using 3j Wigner symbols and symmetric tensor product representation. In particular, in the later representation we find the character decomposition of one-point torus block for degenerate operators.
-Section 6 about two-point torus blocks is less detailed and contains two-point Wilson network operators in the symmetric tensor product representation. Explicit demonstration that these operators calculate 2-point blocks is given by one simple example for each OPE channel contained in Appendix D.
-concluding remarks and future perspectives are shortly discussed in Section 7. Technical details are collected in Appendices A-D.

Wilson networks vs conformal blocks
In this section we mainly review Wilson line approach to conformal blocks proposed and studied in different contexts in [21,27,56,61,62]. Here, we rephrase the whole construction in very general terms underlying key elements that finally allow direct passing from concrete calculations of sphere blocks in the above references to calculation of torus blocks. We will discuss only the case of (non-unitary) finite-dimensional sl(2, R) representations (see Appendix A). The Wilson networks carrying (unitary) infinite-dimensional representations and the corresponding global sphere blocks are considered in [21,62].

Brief review of 3d Chern-Simons gravity theory
The Chern-Simons formulation of 3d gravity with the cosmological term is obtained by combining the dreibein and spin-connection into the o(2, 2)-connection A [65,66] (for extensive review see e.g. [17,67]). Decomposing the gauge algebra as o(2, 2) ≈ sl(2, R) ⊕ sl(2, R) one introduces associated (anti)-chiral connections A,Ā in each simple factor sl(2, R) with basis elements J 0,±1 , see the Appendix A for more details. Then, the 3d gravity action is given by the o(2, 2) Chern-Simons action where k is related to the 3-dimensional Newton constant G 3 through k = l/(4G 3 ) and l is the AdS 3 radius, and Tr stands for the Killing invariant form. Equivalently, the action can be factorized as , where each chiral component is the sl(2, R) Chern-Simons action. The convenient choice of local coordinates is given by x µ = (ρ, z,z) with radial ρ 0 and (anti)holomorphic z,z ∈ C.
The equations of motion that follow from the CS action (2.1) are generally solved by flat o(2, 2)-connections A. After imposing appropriate boundary conditions the solutions yielding flat boundary metric can be written as the gauge transformed (chiral) connection A = U −1 ΩU +U −1 dU with the gauge group element U (x) = exp ρJ 0 [68] and the holomorphic gravitational connection given by where T (z) is the holomorphic boundary stress tensor, the central charge c is defined through the Brown-Henneaux relation c = 3l/(2G 3 ) [69]. The same anti-holomorphic connection Ω =Ω(z) arises in the anti-chiral sl(2, R) sector. Considering a path L connecting two points x 1 , x 2 ∈ M 3 we can associate to L the following chiral Wilson line operators , where the gauge group elements are U R = exp J R with generators J R in the representation R. As we deal with the flat connections, the Wilson line operators depend only on the path L endpoints and on the topology of the base manifold M 3 . (Anti-)chiral Wilson operators (2.3) are instrumental when discussing (anti-)holomorphic conformal blocks in the boundary conformal theory.

General construction
The Euclidean AdS 3 space metric can be obtained from (2.2) by taking a constant boundary stress tensor. In what follows we discuss spaces with both periodic and non-periodic time directions. In the non-periodic case, the stress tensor can be chosen as T (z) = 0 so that the chiral gravitational connection (2.2) takes the form Ω = J 1 dz , (2.4) and the corresponding AdS 3 metric is given in the Poincare coordinates. In the periodic case (thermal AdS 3 ), the stress tensor is T (z) = −c/48π so that the chiral connection is given by along with the standard identifications w ∼ w + 2π and w ∼ w + 2πτ , where iτ ∈ R. The boundary (rectangular) torus is defined by the modular parameter τ while the conformal symmetry algebra in the large-c limit is contracted to the finite-dimensional sl(2, R)⊕sl(2, R).
In the chiral sector, the Wilson line (2.3) for the connections (2.4) or (2.5) is the holonomy of the chiral gauge field along the path L with endpoints x 1 and x 2 : In order to realize conformal blocks through the Wilson networks we need the following ingredients.
1) The Wilson line W a [z, x] in a spin-j a representation D a of sl(2, R) algebra, connecting the external operator O ∆a (z,z) on the boundary with some point x in the bulk. The conformal dimension of the boundary operator is ∆ a = −j a .
2) The Wilson line W a [x, y] connecting two bulk points x and y. In the thermal AdS 3 , the thermal cycle yields the Wilson loop W α [x, x + 2πτ ].
3) The trivalent vertex in the bulk point x b connects three Wilson line operators associated with three representations D b , D c , and D a by means of the 3-valent intertwiner operator which satisfies the defining sl(2, R) invariance property where U α labelled by α = a, b, c are linear operators acting in the respective representation spaces. In other words, the intertwiner spans the one-dimensional space of sl(2, R) invariants Inv(D * a ⊗ D b ⊗ D c ), where * denotes a contragredient representation.
4) The Wilson line attached to the boundary acts on a particular states |a ∈ D a .
In general, n-point global conformal blocks F(∆, ∆|q, z) on Riemann surface of genus g with modular parameters q = (q 1 , ..., q g ), with external and intermediate conformal dimensions ∆ and ∆, can be calculated as the following matrix element (2.10) Here, the Wilson network operator Φ[W a , I b;c,d ] is built of Wilson line operators W a associated to a particular (bulk-to-bulk or bulk-to-boundary) segments joined together by 3-valent intertwiners I b;c,d to form a network with the boundary endpoints z = (z 1 , ..., z n ). The double brackets mean that one calculates a particular matrix element of the operator Φ between specific vectors of associated sl(2, R) representations in such a way that the resulting quantity is sl(2, R) gauge algebra singlet. Using general arguments one may show that the matrix element (2.10): (a) does not depend on positions of bulk vertex points due to the gauge covariance of the Wilson operators, (b) transforms under sl(2) conformal boundary transformations as n-point correlation function.

Vertex functions
In 10) which depend on positions of boundary operators. Thus, it is their properties that completely define how the resulting CFT correlation function (block) transforms with respect to the global conformal symmetry algebra. One can show that depending on the choice of particular W a and |a ∈ D a , the conformal invariance of the correlation function of quasi-primary operators is guaranteed by the following basic property [56,61,62]

3-point vertex function.
Following the general definition of the Wilson network operator (2.10) we use the intertwiner (2.8) and introduce a trivalent vertex function (see Fig. 1) as the following matrix element where z = (z 1 , z 2 , z 3 ) stands for positions of boundary conformal operators, |a , |b , |c are arbitrary boundary vectors. Using the invariance condition (2.9) in the form  Two comments are in order. First, in order to have a non-trivial intertwiner with the property (2.9) the weights of three representations must satisfy the triangle inequality. Indeed, tensoring two irreps as in (2.8) we find out the Clebsch-Gordon series If a representation D c of a given spin j c arises in the Clebsch-Gordon series then the intertwiner is just a projector, otherwise it is zero. Equivalently, j a + j b − j c 0. Second, one may rewrite In this form the trivalent vertex function is represented as a product of four matrix elements which can be drastically simplified when |a , |b , and |c are chosen in some canonical way like lowest-weight or highest-weight vectors. The last three factors are matrix elements of the Wilson operators, or, equivalently, coordinates of tilded vectors in the standard basis. The first factor is the matrix element of the intertwiner which in fact is the Wigner 3j symbol. 3 Indeed, let us denote the matrix element of the intertwiner and 3j symbol as Here, each of magnetic numbers m, n, k runs its domain. Then, the two tensors are related as where (a)ml is the Levi-Civita tensor in the D a representation. Obviously, both tensors are sl(2, R) invariant, while the 3j symbol spans Inv(D a ⊗ D b ⊗ D c ). The Levi-Civita tensor in D a is given by The last equality introduces the 1jm Wigner symbol which is considered as an invariant metric relating the standard and contragredient standard bases. In particular, this object allows introducing 2-point vertex function as where I a;a is 2-valent intertwiner belonging to Inv(D * a ⊗ D a ) which definition directly follows from (2.8), (2.9) at D c = 1. Thus, (2.23) Coming back to the 3-point vertex functions one may explicitly check that choosing the boundary vectors as highest-weight elements of the respective spin-j η representations along with the Wilson line operator in Euclidean AdS 3 space defined by the connection (2.4), one reproduces the 3-point function of quasi-primary operators on the plane [21,61]:  Fig. 1 Using the transition property we can represent W e [y, x] = W e [y, 0]W e [0, x] and then: (1) for the left factor we repeat arguments around (2.14) to neglect dependence on y, (2) for the right factor we use the intertwiner transformation property to neglect dependence on x. The result is that positions x, y fall out of (2.27). Effectively, it means that we set x = y = 0 so that the intermediate Wilson line operator trivializes W e [x, x] = 1. All in all, we find that the vertex function can be cast into the form Similarly to the previous consideration of the trivalent function one may reshuffle the Wilson operators using the intertwiner transformation property and by inserting the resolutions of identities represent the final expression as contractions of six matrix elements. Choosing |a , |b , |c , and |d to be highest-weight vectors in their representations one directly finds 4-point conformal block on the sphere [21,61]. For our further purposes, the 3-point function (2.22) or (2.15) along with the 4-point function (2.27) will prove convenient to build conformal blocks on the torus (see Section 3). Building the operator Φ (2.10) for the Wilson networks with more endpoints and edges is expected to give higher point conformal blocks on the sphere though this has not been checked explicitly (except for 5-point sphere block in the comb channel [21]). In the next section we discuss Φ in terms of n-valent intertwiners.

Further developments
Here we extend the general discussion in the previous section by considering some novel features of the Wilson network vertex functions.
Descendants. Let us demonstrate that choosing the boundary vectors as descendants of highest-weight vectors we reproduce 3-point function of any three (holomorphic) secondary operators O where L −1 is one of three conformal generators on the plane, L n = z n+1 ∂ + (n + 1)∆z n , n = 0, ±1. Taking descendants as and using that: (1) the gravitational connection is given by (2.4) so that [W a [x, y], J 1 ] = 0, (2) the property J 1 ∼ L −1 (2.26), we find that the respective (holomorphic) 3-point correlation function is given by where superscript (i) in the last line refers to z i coordinates. Similarly, 4-point functions of secondary conformal operators can be obtained by choosing the boundary states to be descendants vectors in the respective representations. Indeed, 4point correlation function of quasi-primary conformal operators decomposes as Fixing the order of a, b, c, d one can introduce one more 4-valent intertwiner with shuffled edges I ac,bd|e = I d;b,e I e;a,c , (2. 35) or, in components, The two intertwiners provide two bases in Inv(D * d ⊗ D a ⊗ D b ⊗ D c ). In the standard basis, one intertwiner is expressed in terms of the other by the following relation where, by definition, the expansion coefficients are given by the 6j Wigner symbol. In terms of the conformal block decomposition of the 4-point correlation function we say about exchanges in two OPE channels, while the change of basis (2.37) is the crossing relation. We see that the crossing matrix is given by the 6j Wigner symbol which in its turn can be expressed from (2.37) as a contraction of two distinct 4-valent intertwiners or, equivalently, four 3-valent intertwiners. 4 Intertwiners of arbitrary higher valence can be introduced in the same manner to build n-point conformal blocks. Fixing the order of representations, an n-valent intertwiner can be defined by contracting n − 2 copies of the 3-valent intertwiner by means of n − 3 intermediate representations with representations ordered in different ways, I a 1 ,a 2 ,...,an|e 1 ,...,e n−3 = I a 1 ;a 2 ,e 1 I e 1 ;a 3 ,a 4 ...I e n−3 ;a n−1 ,an . (2.38) Each of possible contractions defines a basis in Inv(D * j 1 ⊗ · · · ⊗ D jn ) which can be changed by an appropriate Wigner 3(n − 2)j symbol. E.g. in the five-point case the crossing matrices are given by Wigner 9j symbol, etc.
The corresponding n-point blocks of conformal (quasi-primary/secondary) operators in where the intertwiner is built by a particular ordering the representations that corresponds to given exchange channels in CFT 2 with dimensions ∆ l = −j e l , l = 1, ..., n − 3. In this way, we explicitly obtain the Wilson network operator on the sphere (2.10).

Toroidal Wilson networks
As discussed in Section 2.2, due to the gauge covariance, the Wilson networks do not depend on positions of vertex points in the bulk. It follows that bulk-to-bulk Wilson lines are effectively shrunk to points so that all diagrams with exchange channels expanded in trees and loops are given by contact diagrams only. However, on non-trivial topologies like (rigid) torus we discuss in this paper, there are non-contractible cycles. Then the associated Wilson networks will contain non-contractible loops given by non-trivial holonomies.
The general idea is that we can build torus blocks from the Wilson networks described in the sphere topology case simply by gluing together any two extra edges modulo 2πτ (see Fig. 2), and then identifying the corresponding representations. Taking a trace in this representation one retains the overall sl(2, R) gauge covariance. More concretely, one takes (n + 2)-point sphere function (2.39) with n + 2 boundary states in D j l , l = 1, ..., n + 2 with any two of them belonging to the same representation, say D j 1 ≈ D j k for some k. Then, taking a trace over D j 1 naturally singles out a part of the original (n + 2)−valent intertwiner involving two Wilson line operators and a number of constituent 3-valent intertwiners (k, for the above choice). By means of the intertwiner invariance property, the two Wilson operators can be pulled through the intertwiners to form a single Wilson loop-like operator, schematically, Tr j 1 W j 1 [2πτ, 0] I j 1 ;a,b . . . I c;d,j 1 . A true Wilson loop is obtained only when one starts from 2-point sphere function and in this case we get the sl(2, R) character (see below), while for higher-point functions an operator under the trace necessarily contains at least one intertwiner. Let us demonstrate how this works for the trivalent function (2.15) on the torus T 2 with local coordinates (w,w) giving rise to a toroidal one-point Wilson network. The Wilson operators here are built using the respective background gravitational connection (2.5). We identify any two endpoints of the trivalent graph on Fig. 2, which means that points w 2 = 0 and w 1 = 2πτ lie on the thermal cycle. Identifying D a ∼ = D b , then choosing |a = |b = |j a , m and summing up over all basis states in D a (recall that the standard basis is orthonormal) we find from (2.15) that where by • V we denote the resulting toroidal vertex function with some |c ∈ D c . If D c is a trivial representation (i.e. j c = 0), then the Wilson line operator W c = 1 c and the intertwiner I a;a,0 = 1 a so that (3.1) goes to the Wilson loop operator, which is known to be a character of the representation D a . For non-trivial representations D c we can choose |c to be a lowest-weight vector in D c and obtain the expression conjectured in [56]. In Section 5 we explicitly check that the expression (3.1) reproduces the 1-point torus block (4.7).
Let us now turn to the two-point toroidal Wilson networks and consider the rightmost graph on Fig. 2. Here, the representations labelled by a, b, c, d are associated with endpoints ordered as w 1 , w 2 , w 3 , w 4 . In terms of the vertex function (2.28) we identify representations D d ∼ = D c and respective endpoints w 4 = 0 and w 3 = 2πτ . Now, choosing |d = |c = |j c , m and summing up over all m to produce a trace over D c , from (2.28) we directly obtain The other possible toroidal two-point Wilson network corresponds to the middle graph on Fig. 2. We fix endpoints as w 2 = 0 and w 4 = 2πτ . Identifying representations D d ∼ = D b and then summing up over states |d = |b = |j b , m we find

Global torus blocks
In this section we shortly review one-point and two-point torus global blocks and find their explicit form when quasi-primary operators are degenerate, which through the operator-state correspondence are described by finite-dimensional representations of the global conformal algebra. 5 In what follows we work in the planar coordinates (z,z) ∈ C related to the cylindrical coordinates (w,w) ∈ T 2 on the torus by the conformal mapping w = i log z.

One-point blocks
The global one-point block in the torus CFT 2 is defined as the holomorphic contribution to the one-point correlation function of a given (quasi-)primary operator, where the trace Tr is taken over the space of states, and the right-hand side defines the OPE expansion into (anti)holomorphic torus blocks, the expansion coefficients are the 3point structure constants with two dimensions identified that corresponds to creating a loop (see Fig. 3). The parameter q = exp 2πiτ , where τ ∈ C is the torus modulus, (holomorphic) conformal dimensions ∆, ∆ parameterize the external (quasi-)primary operator and the OPE channel, respectively. The convenient representation of the torus block is given by using the hypergeometric function [78] The one-point block function is z-independent due to the global u(1) translational invariance of torus CFT 2 . Note that at ∆ = 0 the one-point function becomes the sl(2, R) character showing that for generic dimension ∆ there is one state on each level of the corresponding Verma module. The block function (4.2) can be shown to have poles at ∆ = −n/2, where n ∈ N 0 which is most manifest when the block is represented as the Legendre function (see Appendix C). In general, conformal dimensions ∆, ∆ are assumed to be arbitrary. The corresponding operators are related to (non)-unitary infinite-dimensional sl(2, C) representations. For integer negative dimensions 6 ∆ = −j 1 and ∆ = −j p , these representations contain singular submodules on level 2j with the conformal dimension ∆ = −∆+1 so that one may consider quotient representations D j which are finite-dimensional non-unitary spin-j representations of dimension 2j + 1.
Note that the degenerate dimension ∆ of the loop channel (4.4) defines poles of the block function. It follows that in order to find torus blocks for D j one may proceed in two equivalent ways. The first one is to use the BPZ procedure [1] to impose the singular vector decoupling condition on correlation functions. E.g., for the zero-point blocks which are characters (4.3) the BPZ condition is solved by subtracting the singular submodule character F ∆ ,0 from the original character F ∆,0 to obtain the well-known expression for finite-dimensional sl(2, C) character Similarly, one may formulate and solve the respective BPZ condition for one-point blocks.
The alternative way is to define the torus block as (anti)holomorphic constituents of the correlation functions built by evaluating the trace already in the finite-dimensional quotient modules. In this case, the holomorphic one-point block is simply where the quasi-primary operator O j 1 corresponds to D j 1 , and the trace is taken over D jp , cf. (4.1). It becomes the order-2j p polynomial in the modular parameter, where the coefficients are given by

Two-point blocks
The global two-point torus correlation functions can be expanded in two OPE channels that below are referred to as s-channel and t-channel, see Fig. 4.
s-channel. The two-point s-channel global block is given by [57]: where the coefficients τ m,n = τ m,n (∆ i , ∆ j , ∆ k ) defining the sl(2) 3-point function of a primary operator ∆ j and descendant operators ∆ i,k on the levels n, m are [11] τ n,m (∆ i,j,k ) = The conformal dimensions of the degenerate (external and internal) quasi-primary operators read These values of the intermediate channel dimensions define poles of the block function. It follows that in order to use (4.10) for the degenerate operators the summation is to be restricted to the region n < −2 ∆ 1 + 1 and m < −2 ∆ 2 + 1 which is an implementation of the BPZ decoupling condition. t-channel. The two-point global block in the t-channel can be calculated either by solving the sl(2, C) Casimir equation [56] or by summing over 3-point functions of three descendant operators [57]: 13) where the τ -function is given by (4.11) and σ m (∆ 1 , ∆ 2 , The t-channel block for the degenerate quasi-primary operators is obtained by applying arguments given around (4.12).

One-point toroidal Wilson networks
In this section we explicitly calculate the one-point toroidal vertex function (3.1) in the thermal AdS 3 (2.5) which is equal to the one-point block function (4.7)-(4.8).

Diagonal gauge
Let us choose the boundary state as the lowest-weight vector in the respective spin-j a representation D a (see Appendix A) |a = |lw a ∈ D a : The Wilson loop traversing the thermal circle can now be given in terms of the modular parameter as where J 0 is taken in the representation D a . Due to the intertwiner transformation property, using the Wilson operators in the diagonal gauge does not change toroidal vertex functions of Section 3 except for that the boundary lowest-weight vectors (5.1) are to be transformed accordingly, |lw a → | lw a = U −1 a |lw a ∈ D a , (5.4) with the gauge group (constant) element U a given above and evaluated in the representation D a . In the diagonal gauge, the conformal invariance of the toroidal vertex functions is guaranteed by the property (2.12) which is now given by the identity matrix C m n = δ n m and the holomorphic conformal algebra generators L n = − exp (inw)(i∂ w − n∆) in the cylindrical coordinates on T 2 .
In what follows we will need the transformed boundary state in the fundamental (twodimensional) representation D 1 2 which is read off from (5.6), (5.8) as (5.9)

Wigner 3j symbol representation
Let us consider the one-point toroidal Wilson network in the diagonal gauge. To this end, using the translation invariance we set w = 0 so that W c = 1 c , then insert the resolutions of identities (2.17) that allows representing the vertex function (3.1) in the form n ≡ j a , m| q J 0 |j a , n = δ m n q n , m, n = −j a , −j a + 1, ..., j a , (5.11) where the last equality is obtained by using the standard basis (A.2). The last factor is the l-th coordinate of the transformed boundary vector in the standard basis, It is given by (5.6), (5.8). Finally, the second factor is the matrix element of the intertwiner which is related to the Wigner 3j symbol by (2.20). Gathering all matrix elements together we obtain The last two properties allow reducing three sums to one. Also, it follows that only the middle (the magnetic number =0) component of the boundary transformed vector contributes by giving an overall factor. Now, we adjust representations D a ≈ D jp and D c ≈ D j 1 to the loop and the external leg. In this case, the sum in the right-hand side of (5.13) can be cast into the form 14) which is obtained by changing n = m + j p . On the other hand, the expansion coefficients of the one-point block (4.7) are given by the hypergeometric function (4.8). Hence, in order to verify the representation (5.13) we need to check the identity where κ is an n-independent factor. This relation holds for To see this we use the explicit representation for the Wigner 3j symbol [80] 17) which gives for the right-hand side of (5.15) (5.18) The right-hand side here can be transformed by making use of the (Euler-type) transformation for the generalized hypergeometric function (see e.g. [81]), 19) so that we obtain the equality (5.15) that proves the representation (5.13) for the one-point torus block (4.7).

Symmetric tensor product representation
Yet another possible realization of the intertwiners is given when finite-dimensional sl(2, R) representations D j are realized as components of the symmetric tensor products of fundamental (spinor) representation D 1 2 (for notation and conventions, see Appendix B). This multispinor technique was used in [61] to calculate three-point and four-point sphere blocks. In what follows we explicitly calculate the one-point torus block for degenerate operators using the toroidal vertex function (3.1) realized via multispinors. In particular, this realization of the Wilson network formulation brings to light an interesting decomposition of the torus block function in terms of sl(2, R) characters (see Section 5.4).
We start with the toroidal one-point vertex function given in the form (5.10) or (5.13), which is a product of three matrix elements which are now to be calculated using the multispinor approach.
(I) The first matrix element in (5.10) is the Wigner D-matrix, where λ a = 2j a and D β α is the Wigner D-matrix of the Wilson line wrapping the thermal cycle in the fundamental representation, where |e α = | 1 2 , (−) 1+α 1 2 denote the standard basis elements.
(II) The third matrix element in (5.10) is coordinates of the transformed boundary state |c = | lw c defined by (5.5). Then, using the product formulas for spinors and representing | lw c as a product of elements (5.9) we find where a spinor V γ is now coordinates of the transformed boundary vector (5.9) in the fundamental representation, V γ = e γ | |e 1 + |e 2 = δ γ,1 + δ γ,2 . so that the vertex function (5.10) is given by In the rest of this section we show that the vertex function calculates the one-point torus block with degenerate operators as In order to calculate (5.27) we parameterize 2j p = p and j 1 = k along with the fusion condition k p (4.9). This expression can be simplified by introducing new spinor and scalar functions, which are calculated using the definitions (5.21) and (5.23). Then, (5.27) can be cast into the form where coefficients are given by the triangle sequence where C l k are binomial coefficients k l . We omit a combinatorial consideration that leads us to this formula. As a consistency check, one can show that the coefficients satisfy the natural condition k l=0 a l = 1 meaning that we enumerated all possible permutations of originally symmetrized indices by re-organizing them in terms of the matrix elements (5.31). 8 Now, using explicit form of the matrix elements (5.31) we find (up to an overall normalization) where factors C n are given by (5.29). Expressing C n in terms of the modular parameter q the multiple summation can be reduced to just four sums. To this end, we split the multiple sum in two parts, which take into account two terms in the last factor C p−k−m 1 −...−ms , Here, (−) n+r C r n q r J 1 (n, r, q) , (−) n+r C r n q r J 2 (n, r, q) , (5.40) We find where the first binomial coefficient takes into account different ways to choose a set of (n − r) elements m j not arising in the summand and the second coefficient counts the restricted compositions (see e.g. [82]) for the set of the rest r elements m j . We note that for J 2 (n, r, q) the evaluation differs only by interchanging the two sets and simultaneously replacing q by 1/q: This gives 43) that can be directly manipulated and after a somewhat tedious but straightforward resummation yields the conformal block function (4.7)-(4.8).
Let us note the representation (5.37) has the triangle degree of complexity in the sense that it is simple when k = 0 (j 1 = 0, the 0-point function, i.e. the character (4.5)) or k = p (j 1 = 2j p , maximal admissible value of the external dimension (4.9)) while the most complicated function arises at k = p/2, when all multiple sums contribute.

Character decomposition
The one-point block in the form (5.37) can be represented as a combination of zero-point blocks, i.e. characters, of various dimensions. To this end, we note that the character (4.5) can be expressed in terms of variables C n as where, in its turn, C 0 can be interpreted as the inverse character of the weight ∆ = 1/2 representation, see (4.3). Then, rewriting (5.37) in terms of the characters we arrive at the following representation of the one-point block (up to a prefactor, recall that p = 2j p and k = j 1 ) This form suggests that one can alternatively evaluate this expression by using the Clebsch-Gordon rule for characters. As an example, let the external dimension take the minimal admissible (bosonic) value, j 1 = k = 1. In this case, from (5.47) we obtain the relation (recall that p is even) Now we recall the Clebsch-Gordon series (2.16) in terms of the characters Substituting this expression into (5.48) we find which after substituting the explicit form of characters in terms of q gives back the one-point torus block function.
To perform this calculation for general values of external dimension j 1 = k one needs to know the Clebsch-Gordon series for tensoring any number n of irreps of weights j = {j i = m i /2}. It essentially reduces to knowing the Clebsch-Gordon numbers N j (j) which are multiplicities of occurrence of a spin-j in the range [j min , j max ], where min(max) weights are simply defined as 2j max = n i=1 m i and 2j min = n i=1 (−) n+i−1 m i . 9 However, a closed formula for the general Clebsch-Gordon numbers is an unsolved mathematical problem (for recent developments see e.g. [83]). This consideration leads to the following representation F jp,j 1 (q) = 1 which realizes the one-point block as the linear combination of characters. Here, unknown constant coefficients d m and the summation range D(j p , j 1 ) depend on the Clebsch-Gordon numbers N j (j) for strings of characters and factorial coefficients arising when re-summing multiple sums in the original formula (5.47). An example is given by (5.50).

Two-point toroidal Wilson networks
In what follows we represent two-point toroidal vertex functions of Section 3 in terms of the symmetric tensor products along the lines of Section 5.3.

s-channel toroidal network
Let us consider the toroidal vertex function (3.4) and insert resolutions of identities to express all ingredients as matrix elements in the standard basis (see Appendix A) c,e |j e , k ⊗ |j c , l j e , k|I e;a,b |j b , m ⊗ |j a , r j c , l| c j a , r| a , where the last two matrix elements are given by coordinates of the tilded boundary vectors, where the transformed boundary vectors are defined by expressions (5.6), (5.8) in the respective representations. Now, we identify representations in two internal edges as D b ≈ D jp 1 and D e ≈ D jp 2 , and two external edges as D c ≈ D j 1 and D a ≈ D j 2 . Following the discussion in Section 5.3 we explicitly calculate each of the matrix elements entering (6.1)-(6.2). The Wigner D-matrix (5.20) in the present case reads as where λ p 1 = 2j p 1 , while the projections of the boundary states are calculated as where V (1,2) γ are coordinates of the tilded transformed boundary vectors (6.2) in the basis of the fundamental representation, cf. (5.23). Now, we gather the above matrix elements together contracting them by means of the two intertwiners (B.4)-(B.5). Using the first intertwiner we obtain . The second intertwiner gives where κ = . Finally, the overall contraction yields the two-point torus block in the s-channel as F The cylindrical coordinates on the torus can be related to the planar coordinates by w = i log z so that the block function (4.10) given in the planar coordinates is obtained from (6.8) by the standard conformal transformation for correlation functions. Explicit calculation of this matrix representation along the lines of the one-point block analysis in Section 5.3 will be considered elsewhere. Here, we just give one simple example with spins j p 1 = j p 2 = j 1 = j 2 = 1 demonstrating that the resulting function is indeed (4.10), see Appendix D.

t-channel toroidal network
Let us consider the toroidal vertex function (3.3) and insert resolutions of identities to express all ingredients as matrix elements in the standard basis where the last two matrix elements are given by j a , r| a = j a , r|W a [0, w 1 ]| lw a , (6.10) The representations are identified as D c ≈ D jp 1 and D e ≈ D j p2 for intermediate edges, D a ≈ D j 1 and D b ≈ D j 2 for external edges. Using the matrix elements (6.3) and (6.4), (6.5) and then evaluating the second intertwiner in (6.9) we obtain 2 . Finally, the overall contraction yields the two-point torus block in the t-channel (4.13) in the cylindrical coordinates as F ∆ 1,2 , ∆ 1,2 t (q, w 1,2 ) = (W 2 ) γ 1 ···γ λp 1 γ 1 ···γ λp 1 . (6.13) Just as for s-channel blocks, we leave aside the straightforward check of this matrix representation and explicitly calculate just the simplest example given by spins j p 1 = j p 2 = j 1 = j 2 = 1 to demonstrate that the resulting function is indeed (4.13), see Appendix D.

Concluding remarks
In this work we discussed toroidal Wilson networks in the thermal AdS 3 and how they compute sl(2, R) conformal blocks in torus CFT 2 . We extensively discussed the general formulation of the Wilson line networks which are actually SU (1, 1) spin networks, paying particular attention to key features that allow interpreting these networks as invariant parts of the conformal correlation functions, i.e. conformal blocks, on different topologies. We explicitly formulated toroidal Wilson line networks in the thermal AdS 3 and built corresponding vertex functions which calculate one-point and two-point torus conformal blocks with degenerate quasi-primary operators. In particular, in the one-point case we described two equivalent representations: the first is in terms of symmetric tensor products (multispinors), while the second involves 3j Wigner symbols. It turned out that the calculation based on the 3j symbols is obviously shorter than the one based on multispinors. However, this is because the multispinor approach makes all combinatorial calculations manifest while using the 3j symbols we package this combinatorics into the known relations from the mathematical handbook.
Our general direction for further research is to use the spin network approach, which is a quite developed area (for review see e.g. [64]), in order to generalize Wilson line network representation of the sl(2, R) conformal blocks to the full Virasoro algebra V ir 2 conformal blocks. In this respect, recent papers [22][23][24][25]62] dealing with 1/c corrections to the sphere CFT 2 global blocks are interesting and promising. It would be tempting, for instance, to formulate the Wilson line network representation of quantum conformal blocks in 1/c perturbation theory for CFT 2 on general Riemann surfaces Σ g . Obviously, this problem is quite non-trivial already in the leading approximation since even global blocks on Σ g are unknown. It might be that the Wilson line approach will prove efficient to calculate such block functions because the underlying spin networks are essentially the same as in the sphere topology case except for loops corresponding to non-trivial holonomies of the bulk space. It would be an alternative to the direct operator approach to calculating conformal blocks in CFT on Riemann surfaces. multispinor with λ = 2j, spinor indices α = 1, 2, and |e α 1 · · · e α λ are basis elements. The highest-weight element is by construction |h = |e 1 · · · e 1 .
A tensor product of representations D j 1 and D j 2 can be conveniently expressed in terms of multiplying two Young diagrams of respective lengths λ 1 = 2j 1 and λ 2 = 2j 2 as The tensor product of two representations of weights h 1 and h 2 can be conveniently expressed in terms of multiplying two Young diagrams of respectively lengths 1 = 2h 1 and 2 = 2h 2 as where on the right-hand side there are 2 +1 di↵erent two-row Young diagrams (here, without loss of generality we supposed that 1 > 2 ). However, The fundamental matrix element of the Wilson line wrapping the thermal cycle (6.7) Using the form of the special state (6.6), we write the fundamental matrix element for the Wilson line connecting to the external operator on the boundary where we used z 1 = x b .

1-point block calculation
We now define the tensors representing the matrix elements of the Wilson lines carrying representations R p and R 1 : By definition, the symmetrization of n indices is weighted with 1/n!. In order to get the singlet we have to perform the following steps: 1) Using symmetrization S 1 and contraction C 1 with the Levi-Civita symbols we build a tensor M 12 of p lower and p upper symmetric indices out of M 1 ⌦ M 2 .
2) Contraction C 2 of the upper and lower indices in M 12 gives the one-point block.
From   where, without loss of generality, we supposed that λ 1 λ 2 . The last equality on Fig. 5 follows from the fact that 2d Levi-Civita symbol αβ = equals a scalar by Hodge duality that lead to cutting off the two-row part of any sl(2, R) Young diagram.
There are two technical tools that we use in the sequel. First, a totally symmetrized product of λ spinors realizes a rank-λ multispinor, or, in terms of ket vectors |e α 1 · · · e α λ = |e α 1 ⊗ · · · ⊗ |e α λ sym . (B.2) Here, |e α = | 1 2 , 1 2 , | 1 2 , − 1 2 are the standard basis elements of the spinor sl(2, R) representation. The basis elements of D j parameterized by λ = 2j read |j, m ∼ |e α 1 · · · e α λ . (B.3) Second, the intertwiner I j 3 ;j 1 ,j 2 or the projector on the rank λ 3 multispinor on the righthand side of Fig. 5 is defined by contracting indices by the Levi-Civita symbols (cf. (2.20)) and subsequent symmetrization, The projector formula (B.4) directly follows from the decomposition on Fig. 5 by requiring that the right-hand side contains a representation with the weight λ 3 . The two-row part of a given Young diagram gives a product of k Levi-Civita symbols. The convention λ 1 λ 2 guarantees the triangle inequalities λ 1 −λ 2 λ 3 λ 1 +λ 2 which define the summation domain in (2.16), and so that k 0.

C One-point block via Legendre functions
The one-point torus block expressed in terms of the hypergeometric function (4.2) can be equally represented in terms of the Legendre functions (see e.g. [84]) as Here, P µ ν [z] are the Legendre functions with arbitrary parameters µ, ν ∈ C. For integer parameters µ, ν ∈ Z we get the associated Legendre polynomials which at µ = 0 become the standard Legendre polynomials.
Using the hyperbolic parameterization with x = 1 2 log q = iπτ , where τ ∈ C is the modulus, we can introduce yet another representation