Various semiclassical limits of torus conformal blocks

Journal of High Energy Physics, Apr 2017

We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation.

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Various semiclassical limits of torus conformal blocks

Received: January Various semiclassical limits of torus conformal blocks Konstantin Alkalaev 0 1 3 4 6 7 Roman Geiko 0 1 3 5 7 Vladimir Rappoport 0 1 2 3 6 7 Open Access 0 1 3 7 c The Authors. 0 1 3 7 0 Usacheva str. 6, Moscow, 119048 Russia 1 Institutskiy per. 7, Dolgoprudnyi, Moscow region, 141700 Russia 2 Department of Quantum Physics, Institute for Information Transmission Problems 3 Leninsky ave. 53 , Moscow, 119991 Russia 4 Department of General and Applied Physics, Moscow Institute of Physics and Technology 5 Mathematics Department, National Research University Higher School of Economics 6 I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute 7 Bolshoy Karetny per. 19, Moscow, 127994 Russia We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation. AdS-CFT Correspondence; Conformal Field Theory 1 Introduction 2 3 4 Global torus block Large conformal dimensions Light torus block Virasoro algebra contractions Torus c-recurrence Heavy-light torus block Linearized classical torus block A Torus block coefficients B Proof of proposition 3.1 B.1 Proof of lemma B.5 C Recursive representation Introduction The large central charge CFTs have been receiving much attention recently in the context charge and GN is the gravitational constant suggests that the large-c regime corresponds to the semiclassical approximation in the gravity path integral calculations [1]. This brings to light many issues regarding studying the leading saddle point gravity description of the large-c CFT and finding the subleading 1/c corrections (see, e.g., [2–11] and references In this paper we study one-point conformal blocks in various large-c regimes. We distinguish between different large-c torus blocks: the global, light, heavy-light, and classical blocks. Three of them are known in the spherical case. We add another type of torus block to this list: the light block. We show that all of them are related by a particular chain of We shortly remind the definition of one-point torus conformal block (see, e.g., [12– V(Δ, Δ˜, c|q) = X qnVn(Δ, Δ˜ , c) = 1 + 1 + where the expansion coefficients are defined as denotes the sum of the Virasoro generator indices. The matrix BM|N is the inverse of the of the Ward identity. We consider the large-c expansions of the block function (1.1). All associated conformal blocks are referred to as semiclassical. There are different semiclassical blocks because (1.1) charge. We distinguish between the following four types. ∞. Depending on the particular 1/c → 0 contraction of the Virasoro algebra we distinguish between the global and light blocks. The light block is the leading asymptotic in the c-recursive representation of the torus heavy external operator and light exchanged operator does not exist for the one-point torus block, cf. the first coefficient in (1.1). It follows that the concept of vacuum block approximation used to calculate conformal blocks on the sphere [2, 4, 5, 17–19] (see [20] for review) is not directly applicable to one-point torus blocks because the exchanged operators wrap around non-vanishing cycle and, therefore, cannot be set function (1.1) is the exponential of the classical conformal block. The linearized that the linearized classical block admits the holographic interpretation in the thermal AdS3 space [21]. In the present paper we find the concise integral representation of the linearized classical block. The paper is organized as follows. Each of the following sections from 2 to 5 considers a particular type of semiclassical torus blocks listed above and establishes their mutual relations. In section 6 we discuss our conclusions and further perspectives. Appendix A collects first four block coefficients, appendix B shows the relation between the global and describes the c-recursive representation of the one-point torus block. The Virasoro symmetry algebra Vir on any Riemann surface contains a finite-dimensional to that of the sl(2, C) subalgebra. The resulting global block is a simpler function which depends on the conformal dimensions and not on the central charge. Note that the general conformal block of higher dimensional CFT being restricted to two dimensions yields the In the torus case, the general formula (1.2) simplifies to yield the global torus block where L0,±1 are sl(2, C) basis elements. The coefficients can be found explicitly1 The global block coefficients are related to the hypergeometric function [15] q−1 = 1+ 1+ limiting case. In this case we arrive at the zero-point block coinciding with the sl(2, C) 1 − q = 1 + q + q2 + q3 + . . . , which indicates that there is just one state on each level of the corresponding sl(2, C) Large conformal dimensions The expression (2.3) suggests that the global block function satisfies the Gauss hypergeometric equation. To this end, using the Pfaff transformation the global torus block (2.3) can be cast into the form2 F (Δ, Δ˜ |q) = (1 − q)−Δ 2F1(2Δ˜ − Δ, 1 − Δ, 2Δ˜ | q) . 1A detailed derivation of sl(2, C) matrix elements can be found in [22]. 2Recall that the 4-point global block on the sphere is also given by the hypergeometric function. It is tempting to speculate that the recently established correspondence between the one-point torus block and the four-point spherical block [13–15] is exact in the sl(2, C) case once the conformal dimensions satisfy appropriate linear relations, while q and z are related in the standard way. Then, we find the second order differential equation q − 1 − q F − q(1 − q)2 + q(1 − q) F = 0 , by (2.5). We note that global blocks on the sphere satisfy the analogous second order differential equations following from imposing the sl(2, C) Casimir eigenvalue equations in the exchanged channels [22, 23]. The torus case is more intricate because the global On the other hand, the equation (2.6) should be realized as the torus Casimir channel Proposition 2.1. The large-k asymptotic expansion of the global block function is expoqn qn−1 + γn,n−2qn−2 + . . . + γn,0 , (2.9) (1 − q)2n−1 where γn,i = (−1)n−i+1 2nn+−i1 , i = 0, 1, . . . , n − 1 are binomial coefficients, and κn = 4n(2n−1)n . 2 The proof directly follows from the theory of differential equations with a large parameter. by substituting (2.10) into the original differential equation (2.6). The resulting recurrence where the prime denotes the q-derivative. All other coefficients in (2.10) are expressed in " ∞ − x x(1 − x)2 We first decompose the right-hand side of (2.12) with respect to the small parameter Light torus block block. For example, the limiting 4-point function on the sphere is known to be the hypergeometric function [24]. In the 5-point case this is the Horn hypergeometric series of two variables [22], the n-point case was studied in [25]. The light blocks are also known in the WN case [26, 27]. The light and global blocks on the sphere are identical. regime the quantum torus block (1.1) is represented as V(Δ, Δ˜ , c|q) = L(Δ, Δ˜ |q) + O(c−1) , where the leading term is the one-point light torus block. It turns out that global and light blocks are different on a torus. The light block is a formal power series = 1+ 1+ cf. (2.3). There is the following Proposition 3.1. The global and light blocks are related as The proof is given in appendix B. The component form of (3.3) reads 1 − q Ln(Δ, Δ˜ ) = X p1(n − k)Fk(Δ, Δ˜ ) , In brief, to prove proposition 3.1 we analyze the primary operator matrix and the Gram matrix elements which are polynomial functions of the central charge.3 Note that the n-th level block coefficients can be represented as a trace of the [n × n] matrix product, see (1.2). Thus, the block coefficients are rational functions of c. In appendix B we show elements are exactly the global block coefficients weighted with numbers of the sl(2, C) basis Virasoro algebra contractions We consider the Ino¨nu-Wigner contraction [29] for the Virasoro algebra, where the defortypes of contractions underlying the global and light torus blocks. One of contractions gives rise to the Heisenberg algebra that explains the weight prefactor in (3.3). We recall that the Virasoro algebra commutation relations are given by [Lm, Ln] = (m − n)Lm+n + m, n ∈ Z , while primary operators transform as L0,±1 → l0,±1 = L0,±1 , |m| ≥ 2 , In the case (A), the contracted Virasoro algebra splits into sl(2) algebra and the infinitedimensional Abelian algebra A, [lm, ln] = (m − n)lm+n , [lm, an] = (m − n)am+n , |m + n| ≥ 2 ; [am, an] = 0 , [lm, an] = 0 , |m + n| ≤ 1 . are lowest weight sl(2)-modules. 3See the analogous discussion of the global blocks on the sphere using the projector technique [17]. Also, In the case (B), the contracted Virasoro algebra splits into sl(2) algebra and the infinitedimensional Heisenberg algebra H, [lm, ln] = (m − n)lm+n , [am, an] = − 1) [lm, an] = (m − n)am+n , |m + n| ≥ 2 ; [lm, an] = 0 , |m + n| ≤ 1 . branches of H are lowest weight sl(2)-modules. Comparing (3.9) and (3.10) we see that the only difference is the central term in (3.10), which refers to that the Heisenberg algebra is the centrally extended Abelian algebra. We argue that the contracted algebras VirA and VirB underlie the global and light torus blocks, respectively. Let the basis monomials be represented as |M | = s + |M¯ | , conjugation rules a−m = (am)† are assumed. The global torus block corresponds to the case (A). We compute the block coeffimodule, while the primary operator satisfies (3.8). The Abelian factor A is trivially realized so that the resulting expression is just the global block (2.1). We note that truncating the Virasoro algebra to the sl(2, C) subalgebra of section 2 is equivalent to considering the contracted VirA with trivially realized Abelian factor. The light torus block corresponds to the case (B). To compute the VirB block we δMN αM hΔ˜ |l1sl−s1|Δi and hΔ˜ , M |φΔ|N, Δ˜i = δMN βM hΔ˜ |l1sφΔl−1|1Δ˜ i, where αM = βN are ˜ s some non-zero constants, the bra and ket vectors are given in the basis (3.11). Using that equality above. The second equality follows from the first one and the commutator (3.8). It follows that the number s of lm generators in the matrix elements on the n-th level varies from 0 to n. We see that, contrary to the case (A), the Heisenberg factor H is non-trivially realized and, therefore, the global block associated to the sl(2) factor and the block of the full the n-th level the infinite-dimensional Heisenberg algebra H can be effectively reduced to the finite-dimensional Heisenberg algebra Hp(n)+1 with p(n) + 1 basis elements, where k(n) is a number of partitions of n. Then, using the diagonal matrix relations we find that the corresponding conformal block is given by the sum of the global block coefficients Fi, in the basis monomials on the n-th level, cf. (3.4). Indeed, the basis (3.11) has a natural identified with the light block expressed in terms of the global block (3.3). Torus c-recurrence The light block can be considered as the leading O(c0) term of the so-called c-recursive representation of the quantum conformal block, where sub-leading O(1/c) terms are simple with the light block (3.1). The recursive relations are given by Heavy-light torus block The heavy-light limit of conformal blocks assumes that some dimensions scale linearly with the central charge, while the others remain fixed [3, 17]. Thus, in the large-c regime we distinguish between heavy and light operators. contrary to the spherical blocks the external operator is light and the exchanged operator is heavy. In the opposite regime the torus block does not exit because the block function diverges. It follows that there is just one heavy-light torus block defined as high enough orders are given by H(Δ, ǫ˜|q) = 1 + q + 2q2 + 3q3 + 5q4 + . . . = where ϕ(q) is the Euler function, cf. (3.3). Here, the last equality is the conjecture based on the middle terms following from explicit calculations of the first few levels. Below we argue that the heavy-light block is indeed the inverse Euler function. In this case, the heavy-light block is just the zero-point block, or, equivalently, the Virasoro character. Note that the Proposition 4.1. The heavy-light block is the limiting case of the light block totic of the global block (2.4). The relation (4.3) is analogous to that between global and heavy-light blocks on the sphere originally found in the 4-point case [3] and further developed in the higher-point case [22]. Recall that two heavy external operators in spherical blocks can be generated by Then the heavy-light block is just the global block with the light operator insertions evaluated in the new coordinates supplemented with the corresponding Jacobian factors. From the holographic perspective, the conformal map generates heavy insertions on the boundary that corresponds to the conical singularity/BTZ in the bulk. In the torus case, the external operator is lighter than the exchanged operator and therefore it cannot produce a singularity like in the spherical case. The corresponding bulk geometry is the thermal AdS space which is the standard AdS with periodic time [21]. Thus, there is no additional Jacobian factors and the light torus block which replaces the global block in the spherical case is related to the heavy-light block as in (4.3). Similarly to the case of the global and light blocks, the heavy-light block (4.2) is related to the particular contracted Virasoro algebra. Indeed, we rescale the Virasoro generators as L0 → l0 = L0/c , Lm → lm = Lm/√2c , m 6= 0 , and find that in the limit c → ∞ the contracted Virasoro algebra commutation rela − 1) m, n ∈ Z , while the primary operator transformation law is given by m ∈ Z . The contracted Virasoro algebra is the infinite-dimensional Heisenberg algebra H. The commutation relations (4.5) can be cast into the standard form by linearly transforming the basis elements lm. Note that the original sl(2) subalgebra is contracted4 into the threeH-singlet. It follows that the resulting block coefficients (1.2) are simply the traces of the [p(n) × p(n)] unit matrices, Hn = Tr 1 p(n) ≡ p(n) , where p(n) is the partition of the level number n, cf. (4.2) and (B.19). The particular form of the corresponding Gram matrix makes no difference. Linearized classical torus block As opposed to the limiting block functions considered in the previous sections, here we study the large-c asymptotic expansion of the torus block, where all conformal dimensions 4See also ref. [30]. external and exchanged operators are heavy. Decomposing the block function (1.1) around c = ∞ we arrive at the following Laurent series, V(Δ, Δ˜, c|q) = X vn(ǫ, ǫ˜|q) n∈N ǫ = where finite parameters the rescaling (5.2). are classical conformal dimensions, and vn(ǫ, ǫ˜|q) are formal power series in the modular parameter q with expansion coefficients being rational functions in ǫ and ǫ˜. Note that the expansion (5.1) is essentially different from the c-recursive representation (3.12) because of At large c the principle part of (5.1) tends to zero. Less obvious is the fact that the regular part exponentiates [31]. It follows that the one-point torus block is asymptotically c → ∞ . Here, the function f (ǫ, ǫ˜|q) is the classical conformal block [32], other lower level coefficients can be found in [21, 32]. Note that coefficients fn(ǫ, ǫ˜) are The torus one-point linearized classical block is defined by introducing the lightness qn qn−1 + γn,n−2qn−2 + . . . + γn,0 , (1 − q)2n−1 Keeping terms at most linear in ǫ˜ in the original classical block we obtain the linearized linearized classical torus block f lin(δ, ǫ˜|q) ≡ (ǫ˜ − 1/4) log q + ǫ˜X fn(1)(q)δ2n . conjecture the general formula [21] 3 = 480, κ4 = 3584, κ5 = 23040, etc.5 Remarkably, the linearized classical torus block is related to the global block at large dimensions. Namely, we put forward the following earized classical block coefficients fn(1)(q) (5.6) are equal to each other, Conjecture 5.1. Coefficients g0n(q) of the exponentited global block (2.9) and the linwe find that the linearized classical block is exactly equal to the exponential factor of the global block at large dimensions, fn(1)(q) = g0n(q) . g0(ǫ, ǫ˜|q) = f lin(ǫ, ǫ˜|q) , cf. proposition 2.1. This relation re-expressed in the logarithmic form is analogous to that between the global and linearized blocks on the sphere originally proposed in the 4-point case [3] and further developed in the higher point case [22]. Using conjecture 5.1 we can explicitly find coefficients κn in (5.7) as κn = 4n(2n−1)n , 2 cf. first 5 coefficients listed below (5.7). Using the integral representation of the global block (2.12) and the conjecture 5.1 we suggest the closed formula for the linearized classical f lin(ǫ, ǫ˜|q) = − x (1 − x)2x From the AdS/CFT perspective, the linearized block is naturally realized as the tadpole geodesic graph in the thermal AdS space [21]. It would be interesting to reproduce this integral formula in the bulk terms thereby proving the correspondence in all orders of the The one-point torus block depends on three conformal parameters whose particular asymptotic behavior defines different type limiting block functions. In this paper we explicitly described four torus blocks calculated at different scalings of conformal dimensions with respect to the central charge which is either large or infinite. We showed that such semiclassical blocks are related to each other. In particular, we found the integral representation of the linearized classical torus block that is important from the AdS/CFT perspective. We showed that semiclassical torus blocks corresponding to the infinite central charge are associated to differently contracted Virasoro algebra. These are the global, light, and heavy-light blocks. 5Typically, the linearized classical blocks both on the sphere and torus are infinite series in the dimensionless lightness parameter [5, 21, 22]. An exception to this is the 4-point linearized classical block on the sphere which is a linear function in conformal dimensions [4, 17]. On the other hand, the (linearized) classical block one defines as large central charge asymptotic expansion of the original Virasoro block can be related to 1/c deformations of one of the contracted Virasoro algebras considered in this paper. Indeed, the representation representations. In other words, the dimension infinitely grows when the contraction pasee (5.2). We expect that the semiclassical blocks can be entirely reformulated in terms of the representation theory of the contracted Virasoro algebras and their deformations. The same contraction procedure can be equally applied in the case of the sphere as well as in CFTs on surfaces of any genus. We note also that the Virasoro algebra contractions and associated blocks studied in this paper has much in common with the so-called irregular conformal blocks and their semiclassical limits, see, e.g., [33–36]. In that case the irregular blocks are associated to the particularly truncated Virasoro algebra and demonstrate the exponential behavior in the limit c → ∞. Hopefully, further study of the symmetry arguments underlying various semiclassical blocks will improve our understanding of the AdS/CFT duality in the large-c regime. In particular, that will help to clarify how the contracted Virasoro algebras are realized in terms of the bulk geometry and the associated geodesic networks. We are grateful to I. Tipunin for useful discussions. The work of K.A. was supported by the RFBR grant No. 14-01-00489. The work of V.R. was performed with the financial support of the Russian Science Foundation (Grant No. 14-50-00150). Torus block coefficients A few first expansion coefficients of the torus block (1.2) are given by Proof of proposition 3.1 We analyze the c-dependence of the first two block coefficients (1.2). The explicitly calculate the second level coefficient, V2 = L2−1|Δ˜ i. The Gram matrix B and its inverse B−1 are given by B = The matrix elements read hΔ˜ , 0, 2|φΔ|1, 1, Δ˜i = 2(Δ−1)Δ(Δ+1)+6Δ˜ Note that the last matrix element is proportional to the second coefficient of the global block (2.3). Noting that any block coefficient (1.2) can be represented as a trace of the matrix product we find that the second level coefficient can be represented as follows V2 = O(c−1) O(c−1) ! O(c−1) O(c0) Now, we analyze the general behavior of the Gram matrix and its inverse Kac determinant) and the leading part of the product of diagonal elements are the same. Lemma B.1. Let BM|N be elements of the Gram matrix B. Then, QM BM|M = 1 , where det B is the Kac determinant. The proof is given in appendix B.1. level case (B.4). We note that the Gram matrix elements and the primary operator matrix have the same behavior with respect to the central charge c. By this we mean that the commutation relations of primary operators with any Virasoro generators do not depend on c so that generators, cf. (3.5) and (3.6). We conclude that estimating powers of the central charge in the block coefficients we can trade the primary operator matrix elements for the Gram matrix elements (cf. (B.2) and (B.3)). We consider the inverse Gram matrix elements, BM|N = det B = (−)M+N BM|N AM|N , Vn = X where we separated diagonal from off-diagonal contributions. We similarly represent the deing (B.5) we conclude that off-diagonal contribution in the determinant expansion is of less Now, we are in a position to show that the off-diagonal contribution in (B.7) is of the off-diagonal contribution to the block coefficient is suppressed at least as 1/c. On the same power of c as QM BM|M . Finally, we can explicitly compute the diagonal contribution in (B.7). Indeed, from and, therefore, we can easily find its inverse which is also a diagonal matrix. In particular, n-th level light block coefficient given by Ln(Δ, Δ˜ ) = X p1(n − k)Fk(Δ, Δ˜ ) , the global block (2.1) and using (B.8) and (B.18) we find that 1 − q and thereby show the relation between the light and global torus blocks (3.3). Proof of lemma B.5 Kac determinant. In what follows we use the Liouville parametrization of the central det B = Y ((2r)ss!)m(r,s)(Δ˜ − Δr,s)p(n−rs) , p(n − r(s + 1)). det B = Y i2 − 1 p(n−ij) rs≤n Y ((2r)ss!)m(r,s) . Diagonal elements. We consider diagonal elements of the Gram matrix in the b → 0 limit. In a given diagonal element we take one Lm to the right, = hΔ˜ |Li11 . . . Limm−1[Lm, L−m]Li−mm−1 . . . Li−m1|Δ˜ i + hΔ˜ |Li11 . . . Limm−1L−m[Lm, Li−mm−1 . . . Li−m1]| Δ˜i . 1)/(2b2), cf. the central extension term in (3.5). Commuting all Lm to the right we find using partitions of n. It follows that the product of diagonal elements can be conveniently represented in terms of restricted partitions as Y BM|M = Y p1(n−k) n Y (s!)pr(n−rs) , where pd(n) is the number of partitions of n which does not contain d as a part and #(i, n) is the number of all parts i in all partitions of n. The first product in (B.13) accounts for partitions of n which contain exactly k units. The second product accounts for generators the restricted partitions discussed above in terms of the standard partitions p(n). Auxiliary relations. The following relations are useful in practice, #(i, n) = X kpi(n − ki) , p(n) = X pd(n − kd) , m(r, s) = pr(n − rs) , X p(n − ki) = X kpi(n − ki) . To show (B.14) we note that the total number of parts in all partitions of n can be calculated as follows: a number of partitions containing exactly one part d is equal to etc. Indeed, fixing a part d and and its number in a given partition k we split the number integer such that n − kd ≥ 0. 1 − q Qk∞=1(1 − qk) = pd(0) + pd(1)q + pd(2)q2 + pd(3)q3 + . . . , see, e.g., [39], along with the inverse Euler function, The function (B.19) can be represented as a product of two series Qk∞=1(1 − qk) = 1 − qd 1 − qd Qk∞=1(1 − qk) = pd(0)q0 +. . .+ X pd(n − kd) qn +. . . . Comparing the expansion coefficients in (B.19) and (B.20) we arrive at the relation (B.15). The relation (B.16) follows from (B.15), m(r, s) := p(n−rs)−p(n−r(s+1)) = pr(n−rs) . Finally, to show (B.17) we write down (B.15) at different arguments, p(n − d) = p(n − 2d) = pd(n − d − kd) = X pd(n − kd) , pd(n − 2d − kd) = X pd(n − kd) , · · · · · · · · · · · · · · · · · · · · · · · · p(n − rd) = pd(n − kd) , Equivalent representations. We prove that (B.11) and (B.13) are equal. On the detB=Y =Y =Y i2−1 p(n−ij)rs≤n i2−1 p(n−ij) n i2−1 p(n−ij) n Y ((2r)ss!)m(r,s) Y2sp1(n−s) Y2p(n−s) Y (s!)pr(n−rs). p(n−k) ij≤n i3−i p(n−ij)rs≤n (2r)spr(n−rs) Y (s!)pr(n−rs) (2r)p(n−rs) Y (s!)pr(n−rs) = Y = Y = Y Pin=k p1(n−i) n p(n−k) n p(n−k) ij≤n (s!)pr(n−rs) (s!)pr(n−rs) i3 − i p(n−ij) rs≤n Y (s!)pr(n−rs) . The last lines in (B.23) and (B.24) coincide, and, therefore, the relation (B.5) holds true. Recursive representation The recursive representations split conformal blocks into the limiting function and the singular part with respect to a given conformal parameter which is either the central In what follows we carefully derive the c-recursive representation. c(b) = 1 + 6(b + b−1)2 , (b + b−1)2 − 4 be degenerate with dimensions (b + b−1)2 (rb + sb−1)2 Δ˜ − Δrs(b) Vn−rs(λ, Δrs + rs, b) , by the inverse Euler function, cf. (B.19), and On the other hand, Y BM|M = Y p1(n−k) n i3 − i #(i,n) rs≤n Y (s!)pr(n−rs) Ars(b) = k=1,k=1 (mod 2) l=1,l=1 (mod 2) Y (pb + qb−1)−1 , paring the residua, where the leading term is the light block (3.2). The Jacobian prefactor follows from com 1 I ∂br,s The minus sign of the Jacobian in (C.8) is due to interchanging terms of the pole part Finally, using the standard parametrization b2 = cr,s = 1 + 6(br,s + br−,s1)2 , c − 13 + √25 − 26c + c2 ∂cr,s Using the recursive formula (C.3) we can elaborate the other recursive representation, Consequently, the leading asymptotic in this case is the c → ∞ limiting block which is the light block (3.2). To elaborate the c-recursive representation we first re-parameterize the conformal block poles on the b-plane (br,s + br−,s1)2 (rbr,s + sbr−,s1) , ∂br,s we find that poles arise when the central charge b takes particular “degenerate” values, 1 − r2 2Δ˜ + rs − 1 + q(r − s)2 + 4(rs − 1)Δ˜ + 4Δ˜ 2 . 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Various semiclassical limits of torus conformal blocks, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP04(2017)070