Many-point classical conformal blocks and geodesic networks on the hyperbolic plane

We study the semiclassical holographic correspondence between 2d CFT n-point conformal blocks and massive particle configurations in the asymptotically AdS3 space. On the boundary we use the heavy-light approximation in which case two of primary operators are the background for the other (n-2) operators considered as fluctuations. In the bulk the particle dynamics can be reduced to the hyperbolic time slice. Although lacking exact solutions we nevertheless show that for any n the classical n-point conformal block is equal to the length of the dual geodesic network connecting n-3 cubic vertices of worldline segments. To this end, both the bulk and boundary systems are reformulated as potential vector fields. Gradients of the conformal block and geodesic length are given respectively by accessory parameters of the monodromy problem and particle momenta of the on-shell worldline action represented as a function of insertion points. We show that the accessory parameters and particle momenta are constrained by two different algebraic equation systems which nevertheless have the same roots thereby guaranteeing the correspondence.


Introduction
The AdS/CFT correspondence gives an effective prescription how to calculate CFT correlators from AdS action, at least in the saddle-point approximation [1][2][3]. It is interesting that the correspondence can be understood at a more structural level. CFT correlation functions can be decomposed into the theory-independent conformal blocks completely fixed by conformal symmetry. It is natural to question what are the bulk counterparts of conformal blocks. Recently, such dual objects were described in the case of 2d CFT conformal blocks considered in the limit of infinite conformal parameters ∆, c → ∞ what corresponds to the semiclassical approximation on the gravity side. It was shown that on the Riemann sphere the limiting conformal blocks called classical are equally described as lengths of particular geodesic networks stretched in the asymptotically AdS 3 space [4][5][6][7][8][9][10][11][12][13][14][15][16]. The essential ingredient here is the heavy-light approximation, where two of primary operators form the background for the other operators [5]. Depending on their conformal dimensions the background operators produce the angle deficit or BTZ black hole in the bulk so that perturbative operators correspond to massive test particles.
The basic idea behind this kind of semiclassical AdS/CFT correspondence is quite simple. The semiclassical regime assumes that both the central charge and conformal dimensions tend to infinity such that ratios ∆/c are kept fixed. Then, the conformal block F (z|∆, c) is exponentiated F (z|∆, c) ∼ exp 1 c f (z| ∆ c ) to yield the classical conformal block f (z| ∆ c ) [17]. On the gravity side, the semiclassical path integral is dominated by classical paths describing geodesic motions of massive particles corresponding to primary operator insertions on the boundary [18][19][20].
The relation between classical conformal blocks and classical mechanics goes far beyond the AdS/CFT correspondence. Indeed, conformal blocks are known to be solutions of the Virasoro singular vector decoupling condition [21]. On the other hand, the conformal blocks with arbitrary conformal dimensions in the semiclassical limit were shown to satisfy the Painlevé VI equation which is just the decoupling condition represented in the Hamilton-Jacobi form [22]. It follows that constraining conformal dimensions within the heavy-light approximation the Painleve VI equation can be perturbatively reduced to the equation of motion for massive particles propagating in the asymptotically AdS 3 space.
The study of the singular vector decoupling condition brings to light many rich algebraic structures (see, e.g., [22,23] and references therein). For example, the monodromy problem for the semiclassical decoupling condition expressed by the second order Fuchsian equation and the classical conformal blocks are deeply related [17,21]. The Fuchsian equation ψ (z) + T (z)ψ(z) = 0 has the monodromy group fixed by the form of the stress-energy tensor T (z), where conformal dimensions ∆ are residues at the second order poles, while the accessory parameters c i are residues at the simple poles. On the other hand, the accessory parameters are gradients of the classical conformal block c i = ∂ ∂z i f (z| ∆ c ). It follows that fixing the monodromy allows finding the classical conformal block.
Classical conformal blocks in the heavy-light approximation can be calculated within various approaches, e.g., using the monodromy method [4,5,9,14,15], the Zamolodchikov recursion or the FKW global block method [9,10,24], or the AGT technique [12]. One way or another, the resulting block reproduces the length of dual geodesic network. However, it seems that only the monodromy method conceptually explains why the correspondence holds, see, e.g., the discussion in [9]. For example, the accessory parameter of the monodromy method can be interpreted as the momentum of a particle in the bulk whose worldline is attached to the conformal boundary. 1 This is quite natural because the accessory parameter is the gradient of the classical conformal block, while external particle's momentum is the gradient of the on-shell worldline action. Exact relation based on the analysis of the accessory parameter/particle's momentum equations was given in [14].
Our analysis in this paper focuses on proving the AdS/CFT correspondence between classical conformal blocks and dual geodesic networks in the n-point case in the heavy-light approximation. We show that many-point blocks and dual lengths coincide up to logarithmic terms related to the conformal map from the complex plane to cylinder. It is important to note that we do not find these functions explicitly. Instead, using the monodromy method on the boundary and the worldline approach in the bulk we prove that the two descriptions of n-point configurations are equivalent.
Intrinsically, the proof of the equivalence is reduced to considering a potential vector field , where x i are n − 2 coordinates and U (x) is a potential. Potential vector field systems underlie both the bulk and boundary analysis. On the boundary, using the monodromy method we identify the vector field components with the accessory parameters and the potential with the conformal block, while x i are insertion coordinates of n − 2 perturbative primary operators. In the bulk, the network stretched in the angle deficit space has n − 2 boundary attachment points x i , the vector field components are particular components of canonical momenta of massive test particles, while the potential is the on-shell worldline action identified with the geodesic length.
Both the accessory parameters and momenta are subjected to complicated algebraic equations arising respectively as the monodromy conditions on the boundary and the least action principle in the bulk. Their exact solutions are known only in the 4-point case because the corresponding equation systems are reduced to just a quadratic equation [5,9]. In manypoint case only approximate solutions are available [4,12,14,15]. In this paper we prove that the two algebraic equation systems have the same roots. Therefore, the correspondence is guaranteed.
The paper is organized as follows. In Section 2 we review the monodromy method for conformal blocks and fix our notation and conventions. In Section 3 the monodromy problem is analyzed using the heavy-light perturbation theory. In Section 4 we discuss the bulk space with the angle deficit and the dual geodesic network, see also Appendix A.1. Here, we consider the Routh reduction of the bulk dynamics to the hyperbolic time slice, we study a vertex of three lines on the hyperbolic plane, and analyze the minimal length condition and the so-called angular balance condition. The bulk/boundary correspondence in the n-point case is shown in Section 5. Proofs of lemmas and propositions formulated in Sections 4.2 and 5 are collected in Appendices A.2 -A.5. In the concluding Section 6 we discuss our main results and some future directions.

Classical blocks and the monodromy problem
There are two basic ideas to compute n-point classical conformal blocks using the monodromy method. First, conformal blocks in a given channel are eigenfunctions of the monodromy operator associated with a particular contour on the punctured region. Second, in the c → ∞ limit a particular degenerate operator of the quantum (n+1)-point conformal block effectively decouples yielding the classical n-point conformal block. Comparing the monodromy matrices along particular contours computed before and after taking the semiclassical limit defines all gradients of the n-point classical conformal block in terms of coordinates of punctures. In this way, the problem is reduced to solving first order differential equations (for review, see, e.g., [4,8,22]). There are 2(n − 3) blocks of this type arising as solutions of the decoupling condition. In the limit c → ∞ the degenerate operator decouples and therefore the only surviving block is given by that one shown on Fig. 2. We fix z 1 = 0, z n−1 = 1, z n = ∞.
We consider the (n + 1)-point correlation function V (1,2) (y)V 1 (z 1 ) · · · V n (z n ) on the Riemann sphere with one second level degenerate operator of dimension ∆ (1,2) in point y and n general primaries of dimensions ∆ i in points z i , i = 1, ... , n. The correlation function satisfies the second order differential equation originating from the Virasoro singular vector decoupling condition [21]. The same is true for conformal blocks because the decoupling condition follows from Virasoro algebra only. The space of solutions is two-dimensional, hence the monodromy operators are 2 × 2 matrices. On the other hand, any (n + 1)-point conformal block in a given channel has n singular points and therefore there are n independent monodromies.
Remarkably, the OPE ties monodromy of solutions around particular contours to dimensions of the exchanged operators in a particularly simple way. For the degenerate primary inserted as on Fig. 1 we find that the conformal block is dominated by (z m −y)∆ m+1 −∆ (1,2) −∆m . By the OPE argument, moving y around z m is equivalent to moving around insertion points of those operators which have been fused into the exchanged operator. Thus, computing the monodromy of the above power-law function we easily find the monodromy along the contour γ k (2.1).
Indeed, using the Liouville parameterization 2 we find that ∆ (1,2) = −1/2 − 3b 2 /4, while conformal dimensions of exchanged operators are related by the fusion rule as∆ m+1 −∆ m = −b 2 /4 ± ibP m [21]. Then, the monodromy matrix associated with γ k is given by The classical conformal blocks arise in the limit when the central charge and conformal dimensions simultaneously tend to infinity. Both external and exchanged dimensions ∆ m and ∆ n grow linearly with the charge c in such a way that ratios m = 6∆ m /c and˜ n = 6∆ n /c called classical dimensions remain fixed in c → ∞. Then, the quantum conformal block is represented as an exponential of the classical conformal block [17]. Operators with fixed classical dimensions are heavy, while those with vanishing classical dimensions are light. In our case of the (n + 1)-point conformal block all operators are supposed to be heavy while the degenerate operator is light, lim b→0 ∆ (1,2) = 1/2. Thus, in the semiclassical limit it decouples from the other operators, while adjacent exchanged dimensions get equal lim b→0 (∆ m −∆ m+1 ) = 0, see Fig. 1. The limiting (n + 1)-point conformal block factorizes as where we denoted z = {z 1 , ... , z n }, function ψ(y|z) describes the semiclassical contribution of the degenerate operator, while the exponential factor f (z| i ,˜ j ) is the n-point classical conformal block which depends on external and exchanged classical conformal dimensions i and˜ j [17] (see also [4,8,17,22,26]). We note that (n + 1)-point conformal blocks can be considered in different channels arising from different ways to insert the degenerate operator between other operators, see Fig. 1. Such an ordering singles out a particular channel of the limiting n-point block (2.3), see Fig. 2.
Function ψ(y|z) satisfies the Fuchsian equation arising from the decoupling condition The accessory parameters c i are gradients of the classical n-point conformal block and satisfy the linear constraints It follows that expanding the stress-energy tensor around y = ∞ we find no terms 1/y l , l = 1, 2, 3 and therefore near the infinity T (y) ∼ 1/y 4 . Choosing c 2 , ... , c n−2 as independent parameters and fixing, accordingly, z 1 = 0, z n−1 = 1, z n = ∞, the above constraints are solved as [14,15] Then, the stress-energy tensor T (y|z) takes the form The corresponding Fuchsian equation still has n regular singular points around which we compute the monodromy. Continuing solutions ψ(y|z) along the contours γ k (2.1) on the punctured y-plane and comparing the resulting monodromy matrices with (2.2) we can find the accessory parameters as functions of classical conformal dimensions and insertion points.

Heavy-light perturbation theory
Though the general solution to the Fuchsian equation with n singular points is unknown we can try to use perturbation theory. The idea is to consider k heavy operators as the background for the rest n − k heavy operators. It follows that the corresponding dimensions are constrained as pert / back 1, where back are dimensions of the background operators, pert are dimensions of the perturbative operators. Obviously, the simplest cases are given by k = 2 or k = 3 background operators. The case of k = 1 background operator is trivial because one-point functions on the sphere are vanishing. In the k = 2 case, the background conformal dimensions should be equal to each other for the corresponding two-point function to be non-vanishing. In order to apply perturbation theory in the k ≥ 4 case we have to know exact solutions of the Fuchsian equation with k singularities. Following [5,9,14,15] we consider the case of k = 2 background operators (bold black lines on Fig. 2). Let n−1 = n ≡ h be the background heavy dimension, while i , i = 1, ... , n − 2 be perturbative heavy dimensions. It is assumed that i / h 1. Then, the Fuchsian equation (2.4) with the stress-energy tensor (2.9) can be solved perturbatively. We expand all functions as where expansion parameters are perturbative heavy dimensions. The accessory parameter expansion starts with terms linear in the conformal dimensions so that c For the sake of simplicity, from now on we denote c (3.4) in the way consistent with expansion (3.3) and relation (2.5). The zeroth approximation corresponds to the classical conformal block of 2-point functions of the background operators. Hence, f (0) (z) = 0 and the first non-trivial correction is given by f (1) (z). By analogy with the first order accessory parameters we denote f (1) (z) := f (z) and therefore the relation (2.5) remains unchanged.

Using (3.1) -(3.3) we find that the perturbatively expanded Fuchsian equation yields a chain of inhomogeneous linear equations
.., with a unique differential part given by the operator D 0 = d 2 /dy 2 + T (0) (y). The lowest order equations are given by where the stress-energy components are read off from (2.9) with the convention z 1 = 0. There are two zeroth order branches ψ Using the zeroth order solution we find in the first order that ± have branch points inherited from those of ψ (0) ± (y) and T (y, z).

Computing monodromies
Now, we continue the perturbative solution ψ = ψ (0) +ψ (1) +... along particular contours (2.1). Encircling the branch points of the solution we define the monodromy as M(γ) : ψ → M(γ)ψ, where the monodromy matrix is also expanded as M = M 0 + M 1 + . . . . In components, Expanding both the solution and the monodromy matrix as above we find that the zeroth order matrix M 0 defines the monodromy of ψ (0) (y) with the branch point y = 1. However, the contours γ k (2.1) do not enclose this point, and, therefore, M 0 = I. It follows that the perturbative solution represented as a linear combination of the zeroth order solutions with the integral coefficients (3.8) fits well the monodromy computation. Indeed, the whole computation is reduced to evaluating the integrals along the contours γ k , Substituting the stress-energy tensor correction T (1) (y, z) given by (3.6) and using the residue theorem we have For a given number of insertion points n integrals I ++ and I −− over the maximal contour γ n−3 are always zero, I (3.14)

Eigenvalue condition
Matrices M(γ k ) (3.14) on the one hand and matrices M(γ k ) (2.2) at infinite central charge and small perturbative classical dimensions on the other hand describe the same monodromy associated to continuation of the degenerate operator along the contours γ k . Equating the corresponding eigenvalues we arrive at the system of equations on conformal dimensions of exchanged operators, insertion points, and accessory parameters. We consider the monodromy (2.2) semiclassically. The limiting matrix reads Within the perturbation theory, the eigenvalues of (3.15) up to linear order in dimensions of the exchanged operators are given by λ ±k = 1 ± 2πi˜ k . To diagonalize matrices (3.14) we solve the characteristic equation det (M(γ k ) − λ k I) = 0. Using (3.13) we find that the eigenvalues are defined by the quadratic equation ( Equating two sets of eigenvalues we arrive at the following system In what follows, equations (3.16) supplemented by equations (2.7) are referred to as the accessory equations. Recalling the form of the contour integrals (3.12)-(3.13) we find out that (3.16) is the system of quadratic equations with coefficients depending on punctures z k and conformal dimensions α, i ,˜ j . Solving them we can unambiguously express the accessory parameters as functions of conformal dimensions and coordinates of the punctures. We discuss the accessory parameters equations in Section 5.2 and their dual interpretation in Section 5. Here we stress that classical conformal dimensions of external and exchanged operators in (3.16) are arbitrary. However, we can impose various constraints to facilitate solving the accessory parameters equations using approximation techniques [5,9,14,15,24].

Geodesic networks on the hyperbolic plane
Perturbative conformal blocks can be represented in terms of massive point particles propagating in the three-dimensional space described by the metric [5] The space contains a conical singularity measured by the angle deficit 2π(1 − α), where α ∈ (0, 1]. The metric (4.1) describes the constant negative curvature space with the topology R × C 2 . Two-dimensional slices φ = const and t = const are also negative curvature spaces identified with the punctured hyperbolic plane, where the puncture corresponds to the conical singularity projected on two dimensions. The conformal boundary reached at ρ → π/2 is the Euclidean cylinder with local coordinates t and φ. The presence of conical singularity breaks the global AdS 3 isometry (α = 1) down to Abelian isometry R ⊕ o(2) generated by two Killing vectors ∂ t and ∂ φ . On the conformal boundary, the Abelian isometry is enhanced to full Virasoro algebra. A massive particle on the angle deficit space with the interval (4.1) is described by the worldline action S = dλ g ttṫ 2 + g φφφ 2 + g ρρρ 2 , where is a classical conformal dimension identified with a mass, the metric coefficients are read off from (4.1), λ is the evolution parameter and the dot denotes differentiation with respect to λ, see Appendix A.1 for more details. The Abelian isometry guarantees that coordinates φ and t are cyclic, i.e. δS/δφ ≡ 0 and δS/δt ≡ 0. It follows that the original mechanics can be reduced to a simpler system described by the Routhian function, which means that we have to perform a partial Legendre transformation with respect toφ andṫ. Choosing the partial constraintṫ = 0 we arrive at the Routhian action, which describes a massive particle moving on the punctured hyperbolic disk, The residual isometry is given by sl(2, R) at α = 1 and o(2) at α = 1.
We consider a set of massive point particles propagating around the conical singularity. They interact to each other forming cubic vertices of worldlines. Of course, there are other possible types of interaction including quartic and higher vertices. However, the block/length correspondence singles out only cubic vertices. There are 2n − 5 particles corresponding to the total number of external/exchanged lines of the dual n-point conformal block diagram shown on Fig. 2. External n − 2 worldlines are attached to the conformal boundary at fixed points w = (w 1 , ... , w n−2 ), where w = φ + it. Exchanged n − 3 worldlines are stretched between vertices except for the radial line which ends at the center of the disk. The resulting geodesic network on the hyperbolic disk is shown on Fig. 3 [12]. It can be obtained by copy-pasting the conformal block diagram on Fig. 2 into the disk so that two background operator lines shrink to the point identified with the center of the disk, while the insertion points of conformal primaries go to the boundary attachment points. 3 Geodesics on the hyperbolic time slice (4.2) are most easily described using the Poincare disk model. In this case, these are segments of circles perpendicular to the boundary, including circles of infinite radius (the radial line on Fig. 3). Geodesic lengths are explicitly known as functions of endpoints, see, e.g., formula (A.4).

Cubic vertex and triangle inequalities
Any vertex of the geodesic network on Fig. 3 connects three external and/or exchanged lines, where the vertex point is locally given by three coinciding inner endpoints while outer endpoints are free, see Fig. 4. The vertex action for three distinct lines has the form where each term is the worldline action on the hyperbolic disk The equilibrium condition is conveniently parameterized by classical conformal dimensions A and angular parameters The parameter s A is an integration constant characterizing the form of a particular segment (see Appendix A.1). Since P (A) φ = ±αs A , where the overall sign depends on the direction of the flow, we find that the radial and angular projections of (4.4) are given by where we expressed radial velocities through the vertex position according to (A.3).
Using the linear relation (4.7) we solve the radical equation (4.6) as follows Remarkably, there are no other roots. Indeed, the radical equation (4.6) can be solved by isolating one of radicals on one side and then squaring both sides. The resulting equation is linear in η. For example, the representation (4.8) is obtained by isolating the first radical in (4.6). Other equivalent forms of the vertex position can be obtained by isolating the second and third radicals, Note that equations (4.6) and (4.7) are linear combinations of the same type terms, but with different signs. This is why σ JK in (4.10) has a different sign. The radicals in (4.6) impose restrictions on the radial vertex position The region η < 0 is unphysical corresponding to imaginary values of the radial position. Examining the region η ≥ 0 we find out that the classical conformal dimensions necessarily satisfy the triangle inequalities.

Proposition 4.1
The reality condition η ≥ 0 is satisfied iff The proof is given in Appendix A.2. 4 From the triangle inequalities (4.12) it follows that the sigmas (4.8)-(4.10) can be parameterized as cosines σ AB = cos γ AB , where γ AB is the 4 The triangle inequalities (4.12) are also satisfied by the vertex of two background heavy operators of dimension h ≡ n−1 = n and the perturbative heavy exchanged operator of dimension˜ n−3. In this case, 2 h ˜ n−3 and˜ n−3 ≥ 0.
angle between A and B sides of the triangle (4.12) in the space of conformal dimensions. Introducing s 2 AB = s 2 A + s 2 B ± 2σ AB s A s B according to (A.9) we represent the vertex position (4.8)-(4.10) as (4.13) In this form, it is similar to the law of sines in planar trigonometry. Nevertheless, unlike the conformal dimensions, the angular parameters s I , s J , s K do not form a triangle because, generally, s AB = s C . It would be interesting to understand the role of the ratio (sin γ)/s as an invariant of the geometry of cubic vertices on the hyperbolic space. Here we just note that the triangle inequalities (4.12) are analogous to triangle inequalities satisfied by conformal dimensions of primary operators in the semiclassical limit of the DOZZ three-point correlation function [28]. Note that the triangle inequalities in the Liouville theory are supplemented by the Seiberg bound [29] and the Gauss-Bonnet constraint [28]. All together they guarantee that the Liouville field solution is real.

Dual geodesic network
Gluing together n − 3 vertices η 1 , ... , η n−3 with endpoints attached to n − 2 boundary points w = (w 1 , ... , w n−2 ) and the center of the disk we obtain the network shown on where Lagrangian L A describes A-th geodesic external/exchanged segment with endpoints •A and •A defined by the form of the network. Any vertex η i , i = 1, ... , n − 3 joins lines with labels I = i + 1, J =ĩ, K = i − 1, where, for convenience, we equated0 = 1. Given that the action functional S is stationary we find the equilibrium conditions (4.4) at each vertex point and out-flowing momenta in all attachment points on the boundary and at the center of the disk, where the last equality is assumed to be weak, i.e. the action S is evaluated on-shell. All the attachment points have limiting radial positions, ρ = π/2 for the boundary points and ρ = 0 for the center of the disk. It follows that radial components of (4.17) trivialize because the corresponding variation terms do not contribute to the variation δS. The on-shell action of the network depends on angles of the boundary attachment points and the center of the disk, S = S(w, φ o ). Using (4.5) we can represent non-vanishing components of (4.17) as angular gradients of the on-shell action It follows that the corresponding worldline is radial. Recalling that the center contains the conical singularity we conclude that the radial fall of the outer exchanged particle is quite natural. Angular components of the equilibrium conditions (4.16) can be explicitly written as We can then solve (4.21) to obtaiñ and therefore all exchanged momenta can be expressed in terms of the external ones. Taking k = n − 3 we find out that the total outflowing angular parameter is zero. Indeed, in this case the right-hans side of (4.22) is the sum of outflowing parameters at the boundary attachment points, while the left-hand side is the outflowing parameter at the center of the disk (4.20). Thus, all external parameters are linearly related as Choosing again I = k − 2, J = k − 1, K = k we find that radial components of the equilibrium conditions (4.16) can be explicitly written as Vertex positions η k−1 are directly read off from the general formula (4.8), namely Equivalent representations of η k−1 can be obtained using (4.9) and (4.10).

Angular balance condition
We study angular positions of the endpoints which define the limits of integration in the worldline action (4.15). Let ψ i be angles of vertices η i , i = 1, ... , n−3 and w i , i = 1, ... , n−2 be angles of the boundary attachment points, see Fig. 5. Generally, exchanged lines are stretched between two neighboring vertices while external lines connect vertices with boundary points. From Fig. 5 we find that the angular separation of the i-th external segment is while the angular separation of the i-th exchanged segment is Both the rightmost and leftmost parts of the network on Fig. 3 are different from the general pattern on Fig. 5. Therefore, we identify w 1 = ψ 0 and hence ∆φ 0 = ψ 1 − ψ 0 is the angular separation of the first external line. Also, ∆φ n−3 = 0 because the outer exchanged line is radial. From Fig. 5 we find that angular positions satisfy the balance equation Angular separation of the geodesic segment with two endpoints having radial and angular positions (φ , η ) and (φ , η ) and characterized by the angular parameter s can be represented as [12] iα cf. (A.5). Boundary attachment points have η = 0, radial vertex positions η k are given by (4.25). From the technical perspective, the logarithmic representation (4.29) is well suitable for the bulk/boundary correspondence analysis of Section 5 because the conformal map from the complex plane to the boundary cylinder is also logarithmic (5.1).
Substituting (4.29) into the angular balance equation (4.28) we obtain the system of radical equations where we introduced notation (4.32) The left-hand side of (4.30) is unimodular so that the corresponding real and imaginary parts are not independent. Thus, the complex equations (4.30) are equivalent to original real equations (4.28). 5 The angular balance equation (4.30) is too complicated because of radicals depending on vertex radial positions, which in turn depend non-trivially on angular paprameters (4.25). The roots of the system are yet unknown except for the simplest n = 4 case [5,9], where the solution can be found exactly, and the n = 5 case [12,14], where only perturbative solutions are available. 6 In the next section we show that the dual network equations (4.21), (4.25), (4.30) have roots coinciding with those of the accessory parameter equations (2.7), (3.16).

Bulk/boundary correspondence
The correspondence is most clear in the case of 2-point correlation function of heavy background operators. Two operators inserted in z = 1 and z = ∞ produce an asymptotically AdS 3 geometry (4.1) in the bulk with cylindrical conformal boundary. Angle deficit α in the metric (4.1) is related to the background operator conformal dimension by (3.7). This can be explicitly seen using the three-dimensional metric of asymptotically AdS 3 spacetime in the Banados form [30], where the stress-energy tensor is taken to be the background value T (0) (z) (3.6) (see [6,10] for more details). Note that the CFT we started with is originally defined on the sphere. However, puncturing the sphere twice we obtain the once punctured complex plane, which can be conformally mapped on the cylinder. Opposite ends of the resulting infinitely long cylinder correspond to the background operator insertions. 7 Let (z,z) be coordinates on the punctured complex plane and (w,w) be coordinates on the boundary cylinder. Then, the conformal map is given by In our case, the boundary attachment points w i are located on a circle obtained by slicing the boundary cylinder, see Fig. 3. It follows that insertion points z i belong to the unit circle In particular, z 1 = 0 fixed by projective sl(2, C) isometry of the plane goes to w 1 = 0 fixed by o(2) isometry transformation of the boundary circle. We argue that the correspondence can be shown without knowing explicit expressions for conformal blocks/geodesic lengths. We have seen that both classical the conformal block and the geodesic length are defined through auxiliary parameters subjected to algebraic equation systems. It is natural to question whether two equation systems are equivalent provided that the conformal block and geodesic length are related by the conformal map. We might hope that they are literally the same. However, it turns out that the equivalence is weaker. It is claimed only that being generally different the two systems have the same roots. This section addresses such an equivalence in the n-point case.
The correspondence between a CFT with two background operators in the infinite central charge limit and dual geodesic networks on the angle deficit background claims that the perturbative classical n-point block (2.3) and the on-shell worldline action of the dual network (4.15) be related to each other as Here, the last term results from conformal transformations (5.1) of n − 2 perturbative heavy operators. Indeed, the corresponding correlation function gains the factors (dw/dz) − i in the insertion points, while the quantum conformal block is exponentiated to give the classical conformal block (2.3).
To prove the proposition we recall that both accessory parameters of perturbative heavy operators and angular parameters of the corresponding external worldlines are uniformly defined as the gradients, To complete the proof of the correspondence we show that the algebraic equation systems imposed on the accessory and angular parameters have the same roots. To this end, we find the same structures on both sides and partially solve the bulk system to show that the resulting equations coincide with the boundary equations.

Weak equivalence
On the formal level, the problem is as follows. We consider a potential vector field on a manifold with n − 2 local coordinates x i and potential U (x). Let the vector components A i be subjected to algebraic equations given implicitly by where B k are possible auxiliary variables, k = 1, ... , N − (n − 2). The coefficients in (5.7) may explicitly depend on coordinates x and some additional parameters. We assume that the algebraic system (5.7) is non-degenerate and therefore the auxiliary variables can be completely expressed in terms of the potential vector field components, B = B(A). We consider two potential vector field systems defined by two different sets In our case, the boundary system has no B-type variables which are characteristic of the bulk system. This is quite natural from the AdS/CFT perspective in the sense that not all bulk degrees of freedom are fundamental. Integrating out the local degrees of freedom identified here with B-type variables we are left with A-type variables which are fundamental boundary variables.
Given the conformal map (5.1) we introduce the following notation Insertions points satisfy the unit circle constraint (5.2) so thatā p = 1/a p . Moreover, in order to simplify the accessory equations we recover the exchanged angular momenta expressed via the external ones (4.22). As a result, we rewrite the accessory equations in terms of the external and exchanged momenta.
The proof is given in Appendix A.3.
In the bulk there are two types of redundant variables compared to those on the boundary: exchanged angular parameterss k and vertex radial positions η k . These are B-type variables of Section 5.1. The A-type variables in this case are accessory parameters and external momenta. It follows that a straightforward way to compare two descriptions is to express η k in terms of s i ands j =s j (s i ) by (5.18), (5.19) using (4.25) and then substitute them into the angular balance equations (5.20). However, the resulting equations depending only on s i turn out to be very complicated. In what follows we partially solve the momentum equations that drastically simplifies the analysis of the correspondence.
To this end we notice that functions D ± k are building blocks of the angular balance equation (5.20). We study their properties given that the equilibrium equations (5.18) and (5.19) are satisfied and find out that they are proportional to the contour integrals I The proof is given in Appendix A.4.
Lemma 5.5 Given the equilibrium conditions (5.18) and (5.19), the real part of D + k can be chosen in the form where σ k is given by (4.25).
The proof is given in Appendix A.5.
The following proposition states that the momentum equations can be reformulated in terms of new variables D ± k . where D − k and D + k+1 , k = 1, ... , n − 2 satisfy conditions The proof directly follows from Lemmas 5.4 and 5.5. Here, the vertex radial positions have been eliminated by means of introducing new independent variables D ± k satisfying relations (5.25) and (5.26). In this form the momentum equations are similar to the accessory equations of Proposition 5.3.

Comparing two systems
Putting Propositions 5.3 and 5.6 together we find out that the two equation systems can be weakly equivalent if the contour integrals I (k−1) −+ and functions D + k are related asĀ k+1 I On the other hand, we might conclude that this cannot be the case because given the linear relation (5.4) the contour integrals are quadratic functions of accessory parameters, while D ± k are composite functions with radicals and rational functions of momenta, cf. (4.31), (4.32). However, from Proposition 5.6 it follows that D ± k are now independent variables subjected to linear equations (5.25) and (5.26) with coefficients at most quadratic in A k and A k . Below we explicitly solve these linear relations.
First, in Section 4.2 we showed that the relation (5.23) can be solved in terms of external parameters to give (5.15), see (4.21)-(4.23).
Second, recalling that the lowest indices can be identified as0 = 1 implying˜ 0 ≡ 1 we find that equations (5.25) are explicitly written as Now, recalling (5.17) we find the relation The weak equivalence assumes that the bulk system has more roots compared to the boundary system. Our choice of roots is hidden in Lemma 5.5. In Appendix A.5 we show that there is the other possible value of ReD + k . However, the corresponding root (A.25) does not give rise to known dual counterparts. In other words, it seems that there are admissible bulk configurations which cannot be realized through the classical conformal block.

Conclusion
In this work we showed that n-point classical conformal blocks in the heavy-light approximation are equal (modulo the conformal map) to the lengths of dual geodesic networks for any n. To this end we reformulated both bulk/boundary systems as the potential vector field equations, where vector components are subjected to the algebraic equations. Given the conformal map from the complex plane to cylinder we demonstrated that both algebraic systems share the same roots. This guarantees the correspondence even though explicit block and dual length functions are not presently known (except for various approximations). Moreover, using the notion of the weak equivalence we showed that the roots of the boundary system is the subset in the roots of the bulk system. The role of extra roots in the bulk will be studied elsewhere.
A possible future direction is to apply our technique to a semiclassical CFT on higher genius Riemann surfaces starting with the torus. The classical toroidal blocks were considered, e.g., in [32,33], while their holographic interpretation in the heavy-light semiclassical approximation was proposed in [34]. However, the perturbative monodromy approach for toroidal and higher genius CFTs and its holographic interpretation similar to that in the spherical case have not yet been elaborated.
Also, the semiclassical correspondence considered in this paper can be extended by including 1/c corrections. The 4-point case was studied in [35][36][37][38][39]. It would be interesting to understand how our results for n-point blocks connect with going beyond the leading 1/c order. a segment stretched between endpoints λ and λ . Choosing the constant time slice t = 0 we find that the normalization condition is cast into the form where p φ is the conserved angular momentum associated with the cyclic coordinate φ. A tricky point here is that we can avoid solving the geodesic equations of motion explicitly because the normalization condition (A.2) is sufficient to express a proper parameter λ as a function of radius and angular momentum. From (A.2) we find the radial velocitẏ Recallingρ = dρ/dλ we find that equation (A.3) can be directly integrated to obtain the on-shell value of S on the hyperbolic disk where η = cot 2 ρ and η = cot 2 ρ are initial/final radial positions. Here we used parameter s = |p φ |/α which is an integration constant describing the shape of a geodesic segment, cf. (4.5).
Using p φ = g φφφ and recalling that the angular momenta are integration constants we find that the angle increment of the geodesic segment is given by Taking the integral we arrive at the logarithm representation (4.29).

A.2 Proof of Proposition 4.1
To prove the proposition we introduce the classical fusion polynomial [12] Π IJK = ( I + J + K )(− I + J + K )( I − J + K )( I + J − K ) . where the sigmas are given by (4.8)-(4.10). We introduce s IJ = s 2 I + s 2 J − 2σ IJ s I s J , s IK = s 2 I + s 2 K − 2σ IK s I s K , (A.9) s JK = s 2 J + s 2 K + 2σ JK s J s K .
The vertex positions (4.8), (4.9), (4.10) are then given by (A.10) Assume that I + J < K . Squaring this inequality and using (A.8) we find that σ IJ < −1, and Π IJK < 0. On the other hand, the angular momenta are non-negative s I,J,K ≥ 0, so that if σ IJ < −1 then s IJ ≥ 0. Using for η the first representation in (A.10) we conclude that η < 0. It follows that I + J ≥ K is necessary to have real vertex positions.
The second triangle inequality is proved along the same lines. Suppose that I + K < J . Using for η the second representation in (A.10) we conclude that η ≥ 0 provided I + K ≥ J . Now, suppose that J + K < I . In this case, it is convenient to represent η by the third formula in (A.10), where σ JK in s JK has an opposite sign compared to s IJ and s IK , cf. (A.9). Substituting the other two established triangle inequalities I + J ≥ K and I + K ≥ J into Π JKI we find that Π JKI ≥ 0. On the other hand, squaring J + K ≥ I yields σ JK > 1, hence s JK ≥ 0.
We conclude that having the triangle inequalities (4.12) is a necessary condition for η ≥ 0.