Holographic variables for CFT$_2$ conformal blocks with heavy operators

We consider large-$c$ $n$-point Virasoro blocks with $n-k$ background heavy operators and $k$ perturbative heavy operators. Conformal dimensions of heavy operators scale linearly with large $c$, while splitting into background/perturbative operators assumes an additional perturbative expansion. Such conformal blocks can be calculated within the monodromy method that basically reduces to solving auxiliary Fuchsian second-order equation and finding monodromy of solutions. We show that there exist particular variables that we call holographic, use of which drastically simplifies the whole analysis. In consequence, we formulate the uniformization property of the large-$c$ blocks which states that in the holographic variables their form depends only on the number of perturbative heavy operators. On the other hand, the holographic variables encode the metric in the bulk space so that the conformal blocks with the same number of perturbative operators are calculated by the same geodesic trees but on different geometries created by the background operators.


Introduction
Virasoro conformal block functions F(x|∆,∆, c) [1] are not known in closed form for general values of conformal dimensions ∆,∆ and the central charge c. On the other hand, the AdS/CFT correspondence motivates the study of the regime when the central charge tends to infinity, c → ∞. If external and intermediate conformal dimensions ∆,∆ are heavy, i.e. they scale linearly with large c, then the original conformal block takes a simpler, exponential form [2]. However, such large-c conformal blocks with heavy operators are still quite complicated functions. In the bulk, the Brown-Henneaux relation [3] says that the large-c blocks can be reproduced from the three-dimensional quantum gravity path integral evaluated in the semiclassical approximation.
Further simplification can be achieved by considering the so-called heavy-light expansion [4], when a number of original heavy primary operators forms a background for other heavy primary operators, i.e. ∆ p /∆ b 1, where ∆ b and ∆ p are dimensions of the background and perturbative operators, respectively. The resulting perturbative conformal blocks are much simpler as compared to the original large-c blocks. From the holographic perspective, the perturbative blocks are calculated by lengths of geodesic trees stretched in the bulk space created by the background heavy operators [4][5][6][7][8][9][10][11].
In this paper, we continue the study of L k H n−k perturbative conformal blocks by revealing previously hidden structure that underlies the heavy-light expansion. By that we mean that the perturbative blocks allow for very special parameterization that we call holographic variables. Since n coordinates of n-point perturbative blocks are naturally split into two parts, one can transform coordinates of the perturbative operators by means of a particular mapping function, while keeping coordinates of the background operators intact. The mapping function can be explicitly defined by using solutions to the auxiliary Fuchsian equation. It is parameterized by coordinates of the background operators. Such a transformation allows to reorganize the original coordinate dependence of perturbative blocks so that now they depend on the holographic variables only. In fact, the holographic variables realize the general observation of [7] that, owing to fact that the stress tensor is not primary, the dependence on the background operators can be absorbed by performing a particular conformal transformation.
A remarkable consequence of using the holographic variables is the uniformization of perturbative conformal blocks already noticed in [7,12] in the case of vacuum 4-point LLHH blocks. For n-point blocks, it may be formulated as follows: the perturbative blocks of L k H n−k and L k H m−k types being represented in terms of the holographic variables have the same form at m = n. From the holographic perspective, the uniformization is quite natural. Indeed, the background operators define the bulk space while the perturbative operators are realized via dual geodesic trees. The shape of geodesic trees is defined by perturbative operators only and not by the background operators.
The outline of this paper is as follows. In Section 2 we discuss the monodromy method and formulate the heavy-light expansion which finally defines n-point L k H n−k perturbative blocks. In Section 3 we introduce the holographic variables and formulate the uniformization property of the perturbative blocks. Section 4 contains examples of LLHH and LLHHH blocks which demonstrate the use of the holographic variables. In Section 5 the holographic variables are explicitly related to building the dual three-dimensional geometry created by the background operators. Here, using the holographic variables we identify a dual geodesic tree which length calculates the perturbative LLHHH block. Section 6 summarizes our findings.

Classical conformal blocks and heavy-light expansion
We consider holomorphic Virasoro n-point conformal block F(x|∆,∆, c) in a given OPE channel [1]. Here, x = {x 1 , .., x n } denotes coordinates of primary operators with holomorphic conformal dimensions ∆, intermediate holomorphic conformal dimensions are denoted by∆, and c is the central charge. Let all external and intermediate conformal dimensions be heavy, i.e. grow linearly with the central charge, ∆ = O(c) and∆ = O(c). In the large-c regime the conformal block behaves exponentially [2,18] where f (x| ,˜ ) is the classical conformal block which depends on the central charge only through the classical dimensions ,˜ . A convenient way to calculate large-c conformal blocks is the monodromy method. 1 To this end, one considers an auxiliary (n + 1)-point conformal block with an additional degenerate operator of light conformal dimension O(c 0 ). Due to the fusion rules the auxiliary block in the large-c regime factorizes as where f (x| ,˜ ) is n-point classical block (2.1) and ψ(y|x) stands for the large-c contribution of the degenerate operator. Imposing the BPZ condition one obtains the Fuchsian type equation [1]  ensuring that the algebraic part of (2.3) has no singularity at y → ∞. Knowing all the accessory parameters one can integrate the gradient equations to obtain the classical block.
Heavy-light expansion. Finding classical blocks can be drastically simplified by employing the so-called heavy-light expansion [4]. Suppose now that n − k heavy operators with classical dimensions j are much heavier than other k heavy operators, i j , i = 1, .., k , j = k + 1, .., n . (2.5) Then, the positions of all operators can be split into two subsets: perturbative sector and background sector x = {z , z} ≡ {z 1 , .., z k , z k+1 , .., z n }. Now, we implement the heavy-light expansion 6) where m = 1, ..., n. By construction, the zeroth-order accessory parameters of the perturbative operators are zero, c Zeroth-order solutions. In the zeroth-order, the Fuchsian equation (2.3) takes the form and its solutions are given by two independent branches Here, c j , j = k+1, ..., n, are independent parameters that can be found by solving constraints (2.7) and the gradient equations Since the background conformal block is assumed to be known, then c ± (y|z, ). 2 Usually, the linear constraints are solved in the very beginning to isolate n − 3 independent accessory parameters. However, equally one can keep n accessory parameters independent and solve the three constraints (2.4) at later stages. In that case, solutions to the Fuchsian equation in the zeroth order are explicitly parameterized by the background accessory parameters (2.10).
Two comments are in order. First, the solution (2.10) near singular points z j behaves as that follows from that the leading asymptotics are defined by the most singular terms in (2.9). The exponents are restricted as Second, the zeroth-order solutions are hard to find for any number of heavy background insertions except for two and three background operators in which case the Fuchsian equation can be solved explicitly (see the footnote 2). More than three background operators require the knowledge of higher-point classical conformal blocks f (0) (z| ,˜ ), which can be calculated only as power series in coordinates z.
First-order solutions. In the first-order the equation (2.3) is reduced to (2.14) The solution is given in terms of the zeroth-order solutions (2.10) as where the Wronskian is The Wronskian is independent of y that is the general property of Fuchsian equations.
Monodromy analysis. The monodromy method consists of comparing the monodromy of solutions to the Fuchsian equation against that of the original correlation function. This yields a system of algebraic equations on the accessory parameters. In principle, the system can be solved and then the problem of finding the classical block can be reduced to solving the gradient equations (2.3).
To this end, let us consider contours Γ p encircling points {z 1 , ..., z k , z k+1 , ..., z p+1 }, where p = 1, ..., n − 3. The monodromy matrices along Γ p are defined as and, within the heavy-light expansion, the monodromy matrices can be decomposed as ab (Γ p |z, z) is defined by the first-order solution (2.15). Due to the form of (2.15) the first-order correction factorizes as (2.20) The above integrals are straightforward to calculate since the stress tensor T (1) (2.14) has a simple pole structure, and with the exponents satisfying (2.13).
On the other hand, traversing the light degenerate operator V (2,1) (y) in the original (n+1)point correlation function along contours Γ p we find the respective monodromy matrices Equating the eigenvalues of these matrices with those of (2.19) yields a system of n − 3 algebraic equations on perturbative accessory parameters. Recalling that there are three additional constraints (2.8) we conclude that in total there are n equations on n accessory parameters.

Holographic variables
Let us consider the holographic function and its derivative defined as where ψ (0) ± (y|z) are solutions to the zeroth-order Fuchsian equation (2.10) and the prime denotes a derivative with respect to y. The second relation follows from the first one by virtue of (2.16). Note that function w(y|z) is determined up to the Möbius transformation since in (3.1) we can equally take linear combinations of solutions. Recalling (2.12) we find that the functions (3.1) behave near the singular points z j as where the exponents are restricted by (2.13). Now, we consider a partial conformal map such that coordinates of the perturbative operators are replaced by values of the holographic function w(y|z), i.e.
We leave the coordinates of the background operators intact, otherwise w(z j |z) = 0, j = k + 1, ..., n due to the singular behaviour (3.2). Evaluating functions (3.1) at y = z i we denote The values w i can be called holographic coordniates because of the special role they play in the dual bulk geometry (see Section 5.2). Equivalently, the holographic function (3.1) defines the map of k-dimensional complex spaces C k → C k , which is parameterized by z. This map is invertible. Indeed, the Jacobi matrix is diagonal J ij = w i δ ij , where w i = w (z i |z) are derivatives (3.1) evaluated at y = z i . Since w (y|z) can have zeros/poles only at points y = z j (3.2), then the Jacobi matrix is non-degenerate.
Monodromy integrals. Using the holographic variables the monodromy integrals along contours Γ p (2.20) can be represented as and explicitly calculated by means of the residue theorem, where instead of original first-order accessory parameters we introduced j , j = k + 1, ..., n . (3.7) A few comments are in order. Firstly, it is crucial that the upper limit value min{p+1, k} leads to that all integrals over contours Γ p , p > k are equal to I (k) ±± . This is why the integrals are independent of the accessory parameters Y j (3.7). Secondly, the monodromy integrals explicitly depend on the holographic variables only, while dependence on z i and z j is implicit. Thirdly, the integrals are simple linear functions of new parameters X i and remarkably mimic the linear constraints (2.4).
where the block on the left-hand side is given in the original z-coordinates, while the block on the right-hand side is given in the new w-coordinates. Since the accessory parameters are the gradients of the conformal block (2.3) then j , j = k + 1, ..., n , (3.12) which is exactly the definition (3.7). Indeed, the prefactor in c (1) i is the Jacobian while the second term in the brackets is the derivative of the i log w i .
The gradient equations in the sector of the perturbative operators now read as Thus, the conformal block function depends on n independent variables x = {z, z} only through k < n holographic coordinates w i , i = 1, ..., k (3.4).
Two and more background operators. In order to move further we recall that up to now all coordinates of the primary operators were kept arbitrary. Now, let the last three coordiniates be fixed, x f ix = (x n−2 ,x n−1 ,x n ). Since we always have two or more heavy background operators, then x f ix = (ẑ n−2 ,ẑ n−1 ,ẑ n ) or x f ix = (ẑ n−2 ,ẑ n−1 ,ẑ n ). In a given OPE channel coordinates x m with m ≤ n − 3 must be separated from x f ix through a particular OPE ordering.
Supplementing the monodromy equations (3.9), (3.10) with the linear constraints (2.8) we obtain the equation system of k + 3 independent conditions for n accessory parameters. It follows that the first-order accessory parameters of the background operators remain unfixed by the monodromy equations. In this respect, let us consider two different situations: • Two background operators, i.e. k = n−2 perturbative operators. In this case, the equations (3.10) are absent and we have n−3 equations (3.9) for n−2 variables X i , i = 1, ..., n−2. Adding the three constraints (2.8) along with two accessory parameters of the background operators Y n−1 , Y n we obtain in total n equations for n accessory parameters. The three constraints (2.8) can be solved for three accessory parameters X n−2 , Y n−1 , Y n (3.7) of the operators located at x f ix = (ẑ n−2 ,ẑ n−1 ,ẑ n ) = (z n−2 , 1, ∞). We notice that then the three constraints depend on coordinates z i , i = 1, ..., n − 2 only. On the other hand, since the holographic map is invertible (see our comments below (3.4)) we can introduce inverse functions z i = z i (w|z) such that w i • z i = 1. Then, the three constraints can be rewritten in terms of the holographic variables, hence the accessory parameters still depend on the holographic variables only.
• For three or more background operators, i.e. k ≤ n − 3 perturbative operators. In this case, there are exactly k equations (3.9), (3.10) for k variables X i , i = 1, ..., k. The three constraints (2.8) can be solved for three accessory parameters Y m (3.7) with m = n − 2, n − 1, n of the operators located at x f ix = (ẑ n−2 ,ẑ n−1 ,ẑ n ) = (0, 1, ∞). Other parameters X i , i = 1, ..., k and Y j , j = k + 1, ..., n − 3 are independent. Then, recalling that holographic variables (3.1) are functions of coordinates of the background insertions z j and using (3.7), we can evaluate the first derivatives of the perturbative block function to find the first-order accessory parameters of the background operators, (3.14) Uniformization property. To summarise this section, we formulate the uniformization property of L k H n−k perturbative conformal blocks: in holographic variables, the form of npoint block function f (1) (w| ,˜ ) is defined by k perturbative operators only. In particular, using the holographic parameterization, instead of n equations on the accessory parameters we essentially have k < n equations. So, for instance, the calculation of the (known) 4point LLHH block and n-point LLH n−2 block is essentially the same and gives the same expression in the holographic variables, see Sections 4.1 and 4.3. The only difference is that the holographic function (3.1) is different for different numbers of the background operators so that the perturbative block functions in the z-parameterization will be different as well.

4-point LLHH conformal block
Here, x = (z 1 , z 2 , 1, ∞). In this case, the block is determined by two holographic variables w 1 = w(z 1 ), w 2 = w(z 2 ) (3.4), where w(y) is given by (4.3), and two accessory parameters X 1 , X 2 . The monodromy integrals (3.6) read (4.5) The only monodromy equation (3.9) in this case reads Solving the constraints (2.8) yields the relation, which can be rewritten in the form I Thus we have two equations (4.6) and (4.7) for X 1 and X 2 which are solved as , where a sign of the radical term is fixed by the asymptotic behaviour of the resulting conformal block. Integrating the gradient equations (3.13) we find the perturbative 4-point LLHH block function (4.9) In particular, the vacuum block (by definition,˜ 1 = 0, whence, 1 = 2 ) reads Going back to the z-parameterization by performing the conformal transformation (3.11) we reproduce the 4-point block functions found in [4,6]. E.g., the vacuum block (4.10) will be given by with the holographic function (4.3) (the same expression was obtained in [12] by a different method).

4-point LHHH conformal block
Here, x = (z 1 , z 2 , z 3 , z 4 ). The holographic variable is given by w 1 = w 1 (z 1 ) (3.4), where w(y) is given by (4.4), and one accessory parameter X 1 . The only monodromy equation (3.10) and its solution read Also, the background external and intermediate dimensions are restricted by (3.8): The respective block is found by integrating (3.13), Going back to the z-parameterization by using (4.4) the above function reproduces the 4-point LHHH block found in [17].

(4.16)
Integrating the gradient equations (3.13) we find the perturbative 5-point LLHHH block function (4.17) In particular, the vacuum block (by definition,˜ 1 = 0, whence, 1 = 2 ) reads Note that the above monodromy equations are exactly the same as those in the LLHH case (4.6) and (4.7), hence, the accessory parameters (4.8) and (4.16) are also the same. In this way, we demonstrate the uniformization property formulated in the end of Section 3: in the holographic variables the LLHH block (4.9) and LLHHH block (4.17) have the same form. On the other hand, substituting functions (4.3) and (4.4) we will obtain, of course, different block functions in z-coordinates.

Conformal blocks as geodesic trees
The holographic function introduced earlier to describe the perturbative blocks also occurs when describing the dual bulk geometry. It allows to identify the dual space as threedimensional AdS 3 [n − k] space with n − k conical singularities created by the background heavy operators. We explicitly show that the 5-point LLHHH perturbative block is calculated by the length of particular geodesic tree in AdS 3 [3]. The geodesic tree is the same as for the 4-point LLHH perturbative block but in AdS 3 [2].

Dual geometry
Let us consider the three-dimensional metric in the Bañados form [22] where u ≥ 0, z,z ∈ C, and H,H are (anti)holomorphic functions on C , and the AdS radius is set to one. In the context of the AdS 3 /CFT 2 correspondence, the function H(z) is related to the stress tensor of background operators in CF T 2 by where the central charge c = 3R/2G N [3,23]. Under the boundary conformal transformations z → w(z) the stress tensor changes as The Bañados metric (5.1) can be cast into the Poincare form with v ≥ 0, q,q ∈ C, by changing the coordinates as follows [24] q(z,z, u) = w(z) − 2u 2 w (z) 2w (z) 4w (z)w (z) + u 2 w (z)w (z) , where the function w(z) solves the equation (see [5,7,17,25] for more details) It is remarkable that the solution w(z) can be constructed by means of two independent solutions to the auxiliary Fuchsian equation ψ (z) + H(z)ψ(z) = 0 as w(z) = Aψ 1 (z) + Bψ 2 (z) Identifying the stress tensor of the background operators with the metric-defining function H(z) according to (5.2) we immediately conclude that ψ 1,2 (z) can be considered as solutions (2.10) to the auxiliary Fuchsian equation of the monodromy method in the zeroth-order (2.9). It follows that the mapping function (5.7) is exactly the holographic function (3.1): its values at points of the boundary primary operators define the holographic variables (3.4).
Finally, the length of a geodesic stretched between two points (q 1 ,q 1 , v 1 ) and (q 2 ,q 2 , v 2 ) is given by

Geodesic trees
In this section we consider the geometry created by three background operators AdS 3 [3] and geodesic trees dual to LHHH and LLHHH perturbative blocks. Since the zeroth-order stress tensor (5.2) has three singular points, then in the Bañados coordinates (z,z, u) these operators create three singular lines: (0, 0, u), (1, 1, u) and (∞, ∞, u) stretched along u ≥ 0.
In the Poincare coordinates, the geometry is completely determined by the properties of the function w(z) given by By construction, this is the same function as (4.4). Near the singular points (0, 1, ∞) it can be represented where ∼ means that the coefficients in the Laurent series near these points are omitted. The function (5.9) is known as the Schwarz triangle function which maps the complex plane (z,z) onto a curvilinear Schwarz triangle on the plane (w,w) with vertices at points w(0), w(1), w(∞) [26], . (5.11) The asymptotic behaviour (5.10) suggests that near the singular points corresponding to the background operators the Schwarz triangle describes angle excesses/deficits: angle deficits β and α at 0 and ∞, an angle excesses −α at 1.
Moreover, the geodesic tree is required to lie on the surface qq + v 2 = 1. 3 . Then, using (5.8) we compose lengths of three geodesic segments as Representing q = t exp[iφ] and minimizing (5.16) with respect to (t, φ) we find that the FermatTorricelli point is given by , cos φ = a 3 t 2 + 2t + a 3 t 2 + 2a 3 t + 1 , Substituting these expressions into (5.16) we obtain the holomorphic part of the legnth function which reproduces the 5-point LLHHH perturbative block (4.17).

Summary
We showed that using the holographic variables allows to formulate the uniformization property of n-point L k H n−k perturbative blocks which claims that their form essentially depends on the number of the perturbative operators k and not on the background operators. In other words, the perturbative conformal block function can be reorganized so that all coordinates are packed into k functions of original coordinates. In this new parameterization, the n-point block function has the same form for any given number k.
The holographic variables are indeed holographic as they reappear in the bulk analysis as the boundary coordinates of the perturbative operators in the three-dimensional space AdS 3 [n − k] with n − k conical singularities produced by the background operators. From this perspective, the uniformization property is more obvious because it is quite natural that k perturbative operators produce the same geodesic tree no matter how many background operators created the bulk space. In fact, the background operators with dimensions ∆ < c/24 (cf. (2.13)) produce conical singularities so that AdS 3 [n − k] is locally AdS 3 . By casting the original Bañados metric to the Poincare form, all dependence on positions of the background operators is now hidden inside the mapping function and its domain of definition. It turns out that the same function defines the holographic coordinates in the boundary CFT 2 because of the same Fuchsian equation that underlies both bulk and boundary calculations. Here, the Schwarz triangle function (5.9) which is the mapping function in AdS 3 [3] and the holographic function (4.4) in CFT 2 clearly illustrates all details.
We have explicitly demonstrated this machinery for LLHH and LLHHH perturbative blocks. Going beyond more than three background operators faces the problem of lacking explicit expressions for higher-point conformal blocks 4 that define the background part H n−k of the original n-point large-c block function. Nonetheless, the uniformization property claims that the perturbative L k part will be the same.