Large-$c$ conformal $(n \leq 6)$-point blocks with superlight weights and holographic Steiner trees

In this note we study CFT$_2$ Virasoro conformal blocks with heavy operators in the large-$c$ limit in the context of AdS$_3$/CFT$_2$ correspondence. We compute the lengths of the holographic Steiner trees dual to the $5$-point and $6$-point conformal blocks using the superlight approximation when one or more dimensions are much less than the others. These results are generalized for $N$-point holographic Steiner trees dual to $(N+1)$-point conformal blocks with superlight weights.


Introduction
The study of the AdS/CFT correspondence [1,2] provides many new ideas and fruitful observations related to computations in QFT. In the case of AdS 3 /CFT 2 correspondence it is essential to consider the large-c limit which corresponds to the weak gravitational coupling in the bulk according to Brown-Henneaux formula [3]. One of the most elaborated issues is the correspondence between Virasoro conformal blocks with heavy operators in the large-c limit and probe particles propagating in the AdS 3 background with conical defects originally obtained for lower-point blocks [4][5][6][7][8][9][10]. 1 The large-c n-point conformal blocks were studied in [17,[19][20][21]. However, exact expressions for large-c conformal blocks are still unknown.
In this work, we continue to study large-c conformal blocks as holographic Steiner trees on the Poincare disk [22]. We consider holographic Steiner trees with N = 4 and N = 5 endpoints in the superlight approximation where one of more weights are much less than the others. Their lengths are calculated by making use of the hyperbolic trigonometry relations. On the boundary, such Steiner trees are dual to the large-c conformal blocks with superlight operators [8]. Also, we find the lengths of (2M + 1)-point holographic Steiner trees in the superlight approximation corresponding to the (2M + 2) large-c conformal blocks with superlight operators.
The paper is organized as follows. In Section 2 we study the Steiner tree problem on the Poincare disk and calculate the lengths of the holographic Steiner trees with N = 4 and N = 5 endpoints in the superlight approximation. Section 3 applies the monodromy method to calculate the large-c conformal blocks in the heavy-light approximation extended further by the superlight approximation. Here, we show the holographic correspondence relation between large-c conformal blocks and the lengths of the Steiner trees obtained in Section 2.2. Concluding Section 4 summarizes our results.

Holographic Steiner trees on the Poincare disk
In the context of AdS 3 /CFT 2 correspondence the Poincare disk with an angle deficit arises as a constant-time slice of AdS 3 space [6,8]. In this section we focus on the Steiner tree problem on the Poincare disk 2 for the special class of trees called holographic [22]. We use hyperbolic trigonometry to calculate particular holographic Steiner trees with N = 3, 4, 5 endpoints and then generalize these results to N -point Steiner trees.
In what follows we do all calculations on the Poincare disk and then recover parameter α. The length of a geodesic segment between two points z 1 and z 2 is given by The regularized length (see e.g. appendix A in [22] for details) of the geodesic connecting two boundary endpoints z i = exp[iw i ] and z j = exp[iw j ] takes the form where the regulator ε → 0+. The regularized length of the geodesic connecting the bulk point z = r exp[iϕ] and the boundary point z i = exp[iw i ] is given by We denote by L D the finite part of the regularized length on the Poincare disk which is obtained by discarding the ε-dependent terms in (2.2) and (2.3).
Steiner trees. Given N points (outer vertices) belonging to D or ∂D we consider a connected tree G N with N outer edges attached to outer vertices and N − 3 inner edges. The outer and inner edges are connected to each other at N − 2 trivalent inner vertices. The weighted length of G N reads The Euclidean Steiner tree problem in context of QFT is considered in [23].
where i and˜ j are weights of outer and inner edges, respectively. The Steiner problem is to find positions of inner vertices for given tree and weights such that the weighted length (2.4) is minimal. 3 In this case the inner vertices are called Fermat-Torricelli (FT) points and G N is called the Steiner tree (see Fig. 1). Also, for further purposes one can consider a hyperbolic N -gon with corners at the outer vertices of the Steiner tree (outer polygon).  One can show that the angles between edges with weights a , b , c intersecting at FT point are given by supplemented by the triangle inequalities The relations (2.5) and (2.6) follow from the requirement that the Steiner tree has a minimal length and fix the positions of the FT points.
In what follows we focus on two types of Steiner trees [22]: 1) N boundary endpoints, 2) N − 1 boundary endpoints and one endpoint in the center of D. We will refer to them as ideal and non-ideal holographic Steiner trees, respectively.
Superlight approximation. Suppose now that one of the three weights in (2.5) is much less than the other two, which are assumed to be equal, We see that two edges of the vertex merge into a single geodesic segment while the third edge stretches in a perpendicular direction. Then, the case of three arbitrary weights can be regarded as a perturbation of this configuration in the small parameter c .
Hyperbolic trigonometry. The lengths of the edges of the Steiner tree in (2.4) are determined by the coordinates of the FT points. For N = 3 Steiner trees the coordinates of the FT point can be calculated explicitly but for case N ≥ 4 the analysis becomes much more complicated. However, the lengths of the edges of Steiner trees can be found using hyperbolic trigonometry. For example, N = 3 Steiner trees cut the outer triangle into three triangles and the edges of the trees can be considered as the sides of the hyperbolic triangles. Here we provide the hyperbolic trigonometry relations that will be useful in calculating the edge lengths of holographic Steiner trees.
Given a hyperbolic triangle with sides A, B, C and interior angles α, β, γ opposite to A, B, C the first and second cosine theorem, and the sine theorem read as (2.8) When one of the vertices is on the boundary (β = 0), the first cosine law can be cast into the form where A and C denote the regularized lengths of sides connected to the vertex. For two vertices on the boundary (β = γ = 0) the regularized lengths A, B, C are related as (2.10)

Examples of Steiner trees
In this section, the lengths of N = 3 ideal and non-ideal Steiner trees and N = 4 ideal Steiner tree are found for arbitrary weights. On the other hand, N = 4, 5 non-ideal trees are considered in the superlight approximation. We generalize this approach to higher N and consider a particular example of the N -point non-ideal Steiner tree.
N=3 trees. Let us consider N = 3 ideal Steiner tree with three boundary endpoints w i and outer edges of lengths X i , i = 1, 2, 3 (see (a) Fig. 2). Since the Steiner tree splits the outer triangle into three triangles with two vertices on the boundary we apply (2.10) to each of them and find where a ij and γ ij are given by (2.2) and (2.5). Solving this system of linear equations we find the weighted length defined by (2.4) as A similar analysis in the case of non-ideal N = 3 tree is a bit more complicated (see (b) Fig. 2). Let Y and Z be the lengths of outer edges of weights 1,2 and X be the length of the radial line of weight 3 . The outer triangle has a vertex in the center of D and two boundary vertices w 1,2 . In this case, the outer triangle is cut by the Steiner tree into two triangles with one boundary vertex and one triangle with two boundary vertices. Again, using (2.9) and (2.10) we find (2.14) -5 - The weighted length of the N = 3 non-ideal tree is found to be 4 andC is given bỹ . (2.17) Ideal N = 4 tree. Here we consider an N = 4 ideal tree with two FT points (see (a) Fig.  3). This Steiner tree has four outer edges with weights i , i = 1, ..., 4 and one inner edge with weight˜ connecting two FT points. 5 The minimum length condition here is encoded by six angles α k (three at each of the FT points) given by (2.5). Let A, B, C, D denote the regularized lengths of outer edges and R be the length of the inner edge. Consider an auxiliary triangle whose vertices are two boundary endpoints w 1 and w 2 and the FT point (see (b) Fig. 3). Here, K 1 and K 2 are the regularized lengths of the sides attached to the boundary points w 1 and w 2 and λ and λ are the angles between R and 4 Originally, this length was obtained in the context of the wordline approach [6]. For the analysis in the context of Steiner trees see [22]. 5 The particular case of the tree with weights 1 = 2 and 3 = 4 was studied in [27].

N = 4 and N = 5 non-ideal trees in the superlight approximation
The lengths of N ≥ 4 non-ideal Steiner trees with arbitrary weights are unknown. The N = 4 non-ideal tree can be considered as a perturbation of the N = 3 non-ideal tree with respect to one of outer weights [8]. In this section we calculate N = 4 and N = 5 non-ideal trees in the superlight approximation by perturbing N = 3 ideal tree and disconnected N = 4 trees.
Non-ideal N = 4 tree from disconnected N = 4 tree. Let us consider a N = 4 nonideal tree as a perturbation of a disconnected N = 4 tree (see (a) and (b) Fig.4). The resulting N = 4 non-ideal tree has one inner edge with the weight˜ 1 1,3 and two pairs of outer edges: the first one with weights 1 = 2 is a geodesic connecting w 1 and w 2 according to (2.7), and the second one is a radial line with weight 3 =˜ 2 . However, the radial length is a weight-dependent constant so that it can be omitted. The N = 4 non-ideal tree without the radial line can be obtained by cutting an auxiliary N = 4 ideal tree as shown on (c) of Fig.4. Such an auxiliary tree has four outer edges with weights 1 and the outer vertices of the tree are located at points (w 1 , w 2 , w s 2 , w s 1 ), where w s 2 = w 1 + 2w 3 − w 2 and w s 1 = 2w 3 + w 1 are identified by reflecting endpoints w 1 and w 2 relative to the radius connecting the center of D and the endpoint w 3 . Using (2.24) we find that the length of the N = 4 non-ideal tree takes the form Non-ideal N = 4 tree from ideal N = 3 tree. Another example of a N = 4 non-ideal tree is obtained by adding an outer edge with superlight weight˜ 2 to the N = 3 ideal tree (see (b) Fig.5). According to (2.18) the outer edge (denoted by K) is the perpendicular to the third edge of the N = 3 ideal tree. Let us consider an auxiliary triangle with two boundary vertices and a third vertex in the center of D (see (c) Fig.5). An auxiliary Steiner tree of the triangle consists of edges X 1 , X 2 of the N = 3 ideal tree and the edge A stretched to the center of D. To simplify the calculations here we assume 1 = 2 . Using the trigonometric relations (2.8) and (2.9) we find (2.26) where α is the angle between edges X 3 and A. Solving equations (2.26) in the variable K we obtain Then, the length of the non-ideal N = 4 tree takes the form

is given by (2.12).
Non-ideal N = 5 tree from N = 4 disconnected tree. Here, we consider a N = 5 non-ideal tree with two superlight weights˜ 1 ,˜ 3 , see Fig.6. The unperturbed N = 4 disconnected tree is given by two geodesics with weights 1 , 3 connecting pairs w 1 , w 2 and w 3 , w 4 , respectively.   Since the lengths of the radial and bridge lines for the tree are given by (2.29) and (2.30), then the weighted length of the N = 2M + 1 non-ideal tree takes the form Note that this analysis can be generalized to other cases of non-ideal Steiner trees in the superlight approximation. For example, one can consider a disconnected M = 3N tree consisting of N ideal Steiner trees with three boundary endpoints as an unperturbed tree. However, the example is more complicated from a computational point of view.

Large-c conformal blocks
Here, we discuss the n-point large-c conformal blocks with heavy operators in the heavy-light approximation. To this end, we use the monodromy method [4,5,9,19,21,28] to demonstrate the holographic correspondence relation

Large-c conformal blocks and monodromy method
Consider primary operators O i (z i ,z i ), i = 1, ..., n at fixed points (z,z) = {(z 1 ,z 1 ), ..., (z n ,z n )}. Let F n (z|∆ i ,∆ p , c) be the corresponding holomorphic conformal block which depends on conformal dimensions ∆ i and exchange dimensions∆ p , p = 1, ..., n − 3 and the central charge c [29]. Assuming that in the limit c → ∞ dimensions ∆ and∆ are proportional to the central charge one can check perturbatively up to a sufficiently high order that the conformal block takes the exponential form [30] where f (z| i ,˜ p ) is a large-c block, i ,˜ p are classical dimensions which are finite in the large-c limit. In what follows, we work within the heavy-light approximation [5-7, 21, 22] when two external operators with n = n−1 = h are assumed to be heavier than the other external and exchange operators 7 (see Fig.8) h i ,˜ p , i = 1, ..., n − 2 , p = 1, ..., n − 3 . (

3.2)
Monodromy method and heavy-light approximation. This method is discussed in details in [5,6] for 4-point conformal blocks and generalized to n-point blocks in [4,8,19,21]. Below we summarize the main steps. Let Ψ(y|z) be an auxiliary n+1-point conformal block with one degenerate operator V (1,2) inserted in the point (y,ȳ) and n primary operators O i . In the large-c limit the auxiliary block is factored into a product of the form

3)
6 Conformal blocks beyond these limits limit are considered in [31,32]. For recent study of the block exponentiation see [33]. 7 The case of three or more heavy operators is considered in [27,34].
where ψ(y|z) is a semiclassical contribution of the operator V (1,2) . On the other hand, the auxiliary block satisfies the BPZ equation which is reduced to the Fuchsian-type equation with n singular regular points where gradients c j are accessory parameters. In the first order in the heavy-light approximation it leads to the monodromy equations [21] I (n|k) where I (n|k) These are n − 3 quadratic equations that can be explicitly solved for lower-point conformal blocks.
Holographic correspondence relation. The duality between large-c conformal blocks and Steiner trees on the Poincare disk is given by the holographic correspondence relation where L (n−1) D (αw k | k ,˜ p ) is the weighted length of the Steiner tree corresponding to the npoint block with weights k ,˜ p that are equal to the classical dimensions of the block. Note that the length L (n−1) D (αw k | k ,˜ p ) depends on the rescaled coordinates αw k due to the fact that α is the angle deficit of the Poincare disk (see Section 2.1).

Examples of conformal blocks
In this section we calculate 5-point and 6-point large-c conformal blocks dual to the lengths of Steiner trees computed in Section 2.2. By virtue of (3.7) the relations (2.6) define the fusion rules for such blocks. We use the following variables and set w 1 = 0 in the lengths of Steiner trees due to the condition P 1 = 1.

5-point non-identity blocks with superlight operators.
Here we suppose that one of the exchange operator dimensions˜ 1 or˜ 2 is superlight:˜ 1,2 1,2,3 . 8 The first example corresponds to˜ 1 1,2,3 . The weighted length of the dual Steiner tree is given by (2.25) and from the fusion rules (2.6) we get 1 = 2 and˜ 2 = 3 . Then, using the holographic correspondence relation (3.7) we find (3.9) The accessory parameters corresponding to the conformal block (3.9) must satisfy monodromy equations (3.5) A few comments are in order. Since the block (3.9) is obtained in approximation˜ 1 1,3 we require the equation (3.10) to be fulfilled in the first order in˜ 1 . The first equation does not contain terms linear in˜ 1 but after explicit substitution of the accessory parameters corresponding to the conformal block (3.9) we can see that it is satisfied exactly. The second equation is valid up to the first order in˜ 1 . Next we consider the case˜ 2 1,2,3 . The block is dual to the Steiner tree of the length (2.28) so that we assume 1 = 2 . The fusion rules (2.6) require˜ 1 = 3 . According to (3.7) the confromal block has the form 11) and, substituting the accessory parameters associated with the conformal block into the monodromy equations (3.5), we find As in the previous case the monodromy equations are satisfied in the first order in˜ 2 .
6-point identity block with light operators. Let us consider the 6-point identity block with˜ 3 = 0 and denote˜ =˜ 1 . According to the fusion rules (2.6) it follows that˜ 2 = 4 . The length of the corresponding Steiner tree is given by (2.23) and according to (3.7) the conformal block takes the form and one can explicitly show that the corresponding accessory parameters satisfy these equations without using superlight approximation.
6-point non-identity block with superlight operators.
(3.19) It can be explicitly shown that this block satisfies the monodromy equations (3.5) up to the first order in the dimensions of superlight operators˜ j .

Conclusion
In this paper we explicitly computed the weighted lengths of the N = 4, 5, 6 Steiner trees on the Poincare disk and demonstrated that they calculate the dual CFT 2 large-c conformal blocks. On the boundary side, the superlight approximation corresponds to superlight operators. Our results along with previously known are shown in the table below. HL and SL denote the heavy-light and the superlight approximations, respectively.