Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation

In this work the correspondence between the semiclassical limit of the DOZZ quantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of the N=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose new formulae for the accessory parameter of the Lame equation. This quantity is in particular crucial for solving the problem of uniformization of the one-punctured torus. The computation of the accessory parameters for torus and sphere is an open longstanding problem which can however be solved if one succeeds to derive an expression for the so-called classical Liouville action. The method of calculation of the latter has been proposed some time ago by Zamolodchikov brothers. Studying the semiclassical limit of the four-point function of the quantum Liouville theory on the sphere they have derived the classical action for the Riemann sphere with four punctures. In the present work Zamolodchikovs idea is exploited in the case of the Liouville field theory on the torus. It is found that the Lame accessory parameter is determined by the classical Liouville action on the one-punctured torus or more concretely by the torus classical block evaluated on the saddle point intermediate classical weight. Secondly, as an implication of the aforementioned correspondence it is obtained that the torus accessory parameter is related to the sum of all rescaled column lengths of the so-called"critical"Young diagrams extremizing the instanton"free energy"for the N=2* gauge theory. Finally, it is pointed out that thanks to the known relation the sum over the"critical"column lengths can be expressed in terms of a contour integral in which the integrand is built out of certain special functions.


Introduction
The name "Lamé equation" denotes in fact a class of related ordinary second-order differential equations in the complex or real domain which contain (explicitly or implicitly) certain elliptic functions [1]. One of the most suitable forms of the Lamé equation in practical applications is the so-called Jacobian form: (1.1) The potential is parameterized by the elliptic modular parameter 1 m of the Jacobi sn-function sn(u|m) and the constant κ. In order to classify the solutions it is convenient to write κ = ℓ(ℓ + 1). In particular, in the real domain if ℓ is a nonnegative integer the energy spectrum consists of bands. There are 2ℓ + 1 eigenfunctions called Lamé polynomials 2 associated with the boundaries of the energy gaps. For these solutions the values of E are the solutions of a certain algebraic equation [1,2]. 3 Another important representation of the Lamé equation contains the Weierstrass ℘function: The substitution η = ℘(z) converts eq. (1.2) to the third version of the Lamé equation, commonly encountered in the literature, the so-called algebraic form: Eq. (1.4) "lives" on the Riemann sphere C ∪ {∞}. Its (regular) singular points are located at η = e 1 , e 2 , e 3 , ∞. The algebraic form of the Lamé equation is the most appropriate form for further generalizations, cf. [2]. Historically, the Lamé equation has first been obtained (by Lamé) by applying the method of separation of variables to the Laplace equation in ellipsoidal coordinates [1]. More recently it has been noticed that the Lamé equation arises in various physical contexts. First, the Lamé potential can be considered as a good candidate for a realistic model of a one-dimensional crystal [13]. Other areas, where the Lamé equation is applicable, are superconductivity [14], certain version of the Ginzburg-Landau theory [15], and the cosmological models [16][17][18][19][20]. 5 In mathematical physics applications the Lamé equation occurs in the so-called Lie-algebraic approaches to the Schrödinger equation [22].
In pure mathematics the Lamé equation arises in the uniformization theory of tori [23]. More concretely, let T denotes the Teichmüller space for the one-punctured torus. It is well known that T has at least two distinct models. The first one is the upper half plane U = {τ ∈ C | Im τ > 0}. The second model for T is a subset of R 3 , namely F = (x, y, z) ∈ R 3 | x, y, z are positive and x 2 + y 2 + z 2 = xyz . The question arises what is a relationship between U and F. It turns out that the mapping ϕ : U → F is determined by the monodromy of the linearly independent solutions (Ψ 1 , Ψ 2 ) of the equation: [ ℘(z|L) + C(L) ] Ψ = 0, (1.5) where z ∈ C − L and L is a period lattice. However, an explicit construction of ϕ is difficult and still an open problem since the accessory parameter C in the above equation is an undetermined constant, cf. [23]. The Weierstrassian-form Lamé equation appears also in a two-dimensional conformal field theory (2d CFT) as a classical limit of the null vector decoupling equation satisfied by a torus two-point correlation function with a degenerate field. 6 Moreover, one can observe that the accessory parameter B in this equation is expressed in terms of the socalled classical Liouville action on the one-punctured torus. The latter quantity can be computed by means of 2d CFT technics and recently discovered dualities, in particular applying the correspondence between the classical limit of the quantum DOZZ Liouville theory on the torus and the Nekrasov-Shatasvili limit of the N = 2 * U(2) super-Yang-Mills theory.
The aim of the present work is to find an analytical expression of the Liouville classical action on the one-punctured torus employing aforementioned technology and apply it to compute the Lamé accessory parameter B (⇔ A and/or C). The main motivation for this line of research is the above mentioned monodromy problem for the Lamé equation. Its solution is crucial not only for finding the correspondence between models of the Teichmüller space for punctured torus but also for constructing a solution of the Liouville equation on such surface, cf. [29][30][31][32].
The organization of the paper is as follows. In section 2 we briefly review interrelationships between Liouville theory and the problem of computation of the accessory parameters 5 For instance, in the theory of preheating the Lamé equation has been recognized to determine quantum fluctuations of the inflaton field [17][18][19][20]. Let us note, that in several listed above applications the Lamé equation plays the same role. Indeed, regardless of the specific physical contexts one can observe that the Lamé equation serves as the stability equation or the equation of small fluctuations around classical configurations associated with a certain class of basic potentials, cf. [21]. 6 Surprisingly, it seems to be possible to obtain the Lamé system (Hamiltonian and eigenfunctions) from a certain matrix model. Indeed, as has been observed in [24], in a suitable limit the so-called loop equation of certain generalized matrix model yields the equation closely related to the KZB equation [25][26][27] (see also [28]). The latter is known to reduces to Lamé equation in some cases.
of the Fuchsian uniformization of the punctured Riemann sphere. The principal purpose of this section is to recall the concept of the classical Liouville action and the classical conformal block [33]. For a long time the motivations to study classical blocks were mainly confined to applications in pure mathematics, in particular to the celebrated uniformization problem, which roughly speaking is related to the construction of conformal mappings between Riemann surfaces (RS) admitting a simply connected universal covering and the three existing simply connected RS, the sphere, the complex plane and the upper half plane. The uniformization problem is well illustrated by the example of the uniformization of the Riemann sphere with n punctures. Its uniformization may be associated to a Fuchsian equation whose form is known up to some constants that are called accessory parameters. Their computation is an open longstanding problem, which can however be solved if we succeed to derive an analytical expression of the classical block obtained by performing the classical limit of the n-point correlation function of the quantum Liouville field theory. The importance of the classical blocks is not only limited to the uniformization theorem, but gives also information about the solution of the Liouville equation on surfaces with punctures. Recently, an interesting mathematical application of classical blocks emerged in the context of Painlevé VI equation [34]. Due to the recent discoveries the classical blocks are also relevant for physics, since they are related to integrable models and to the instantonic sector of certain N = 2 supersymmetric gauge field theories [35][36][37][38]. Moreover, lately classical conformal blocks have been of use to studies of holographic principle and AdS/CFT correspondence [39].
In section 3 we exploit the idea of brothers Zamolodchikov (see [33]) in order to propose the form of the Liouville classical action S torus L on the one-punctured torus. Our conjecture is that (i) S torus L decomposes into a sum of the three-point Liouville action on the sphere and the torus classical block; (ii) factorization holds on the saddle point intermediate classical conformal weight. Next, we consider the classical limit of the null vector decoupling equation satisfied by the torus two-point function with a degenerate field and find an expression for the Lamé accessory parameter. As has been already mentioned the latter is determined by the (one-punctured) torus classical action.
In section 4 we employ the AGT correspondence and express the toroidal classical block and then the eigenvalue/accessory parameter of the Lamé equation in terms of the so-called effective twisted superpotential of the N = 2 * U(2) supersymmetric gauge theory. In order to compute the latter quantity, i.e. the twisted superpotential, we use a straight-forward generalization of the calculation performed by Poghossian in [40]. We check that on the "classical level" the eigenvalue computed by means of the WKB method exactly coincides with that obtained from the classical torus block. Finally, using a relation between the instantonic sector of the twisted superpotential of the N = 2 * U(2) SYM theory and the classical toroidal block we find that the Lamé accessory parameter is related to the sum of column lengths of the so-called "critical" Young diagrams. It is shown that such sum can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions.
In section 5 we present our conclusions. The problems that are still open and the possible extensions of the present work are discussed.
2 Liouville theory and accessory parameters 2.1 Quantum and classical conformal blocks Let C g,n denotes the Riemann surface with genus g and n punctures. The basic objects of any two-dimensional conformal field theory living on C g [41,42] are the n-point correlation functions of primary vertex operators defined on C g,n . Given a marking 7 σ of the Riemann surface C g,n any correlation function can be factorized according to the pattern given by a pants decomposition of C g,n and written as a sum (or an integral for theories with a continuous spectrum) which includes the terms consisting of holomorphic and antiholomorphic conformal blocks times the three-point functions of the model for each pair of pants. The Virasoro conformal block F . . , β n ) depends on the cross ratios of the vertex operators locations denoted symbolically by Z and on the 3g − 3 + n intermediate conformal weights Moreover, it depends on the n external conformal weights ∆ βa = β a (Q − β a ) and on the central charge c which can be parameterized as follows Conformal blocks are fully determined by the underlying conformal symmetry. These functions possess an interesting, although not yet completely understood analytic structure. In general, they can be expressed only as a formal power series and no closed formula is known for its coefficients. Let us write down two canonical examples which illustrate this fact.
Let q = e 2πiτ be the elliptic variable on the torus with modular parameter τ then the conformal block on C 1,1 is given by the following q-series: Let x be the modular parameter of the four-punctured sphere then the s-channel 7 A marking of the Riemann surface Cg,n (for definition see [43]) is a pants decomposition of Cg,n together with the corresponding trivalent graph.
conformal block on C 0,4 is defined as the following x-expansion: In the above equations G c,∆ IJ is the inverse of the Gram matrix G c,∆ IJ = ν ∆,I , ν ∆,J of the standard symmetric bilinear form in the Verma module The operator V ∆ in the matrix elements is the normalized primary chiral vertex operator acting between the Verma modules In order to calculate the matrix elements in (2.2) and (2.4) it is enough to know the covariance properties of the primary chiral vertex operator with respect to the Virasoro algebra: As the dimension of V n ∆ grows rapidly with n, the calculations of conformal blocks coefficients by inverting the Gram matrices become very laborious for higher orders. A more efficient method based on recurrence relations for the coefficients can be used [44][45][46][47][48].
Among the issues concerning conformal blocks which are still not fully understood there is the problem of their semiclassical limit. This is the limit in which all parameters of the conformal blocks tend to infinity in such a way that their ratios are fixed. It is commonly believed that such limit exists and the conformal blocks behave in this limit exponentially with respect to Z. This last property can be heuristically justified in the case of conformal blocks on C 0,4 and C 1,1 .
Indeed, the existence of the semiclassical limit of the Liouville four-point correlation function with the projection on one intermediate conformal family implies a semiclassical limit of the quantum conformal block with heavy weights ∆ = b −2 δ, ∆ i = b −2 δ i , with δ, δ i = O(1) in the following form: The function f δ δ 3 δ 2 δ 4 δ 1 ( x) is called the classical conformal block [33] or with some abuse of terms, the "classical action" [44,49]. The existence of the semiclassical limit (2.5) has been postulated first in [44,49] where it has been pointed out that the classical block is related to a certain monodromy problem of a null vector decoupling equation in a similar way in which the classical Liouville action is related to the Fuchsian uniformization. This relation has been further used to derive the ∆ → ∞ limit of the four-point conformal block and its expansion in powers of the so-called elliptic variable.
Analogously, the existence of the semiclassical limit of the projected Liouville torus one-point function implies that the semiclassical limit of the torus one-point block with heavy weights ∆ = b −2 δ,∆ = b −2δ with δ,δ = O(1) has the form: The function fδ δ (q) we shall call the classical torus (or toroidal) conformal block. It should be stressed once again that the exponential behavior (2.5) and/or (2.6) is a nontrivial statement concerning the quantum conformal blocks. Although there is no proof of this property, it seems to be well confirmed together with its consequences by sample numerical calculations [50] and recent discoveries.
The classical conformal blocks are in general again available only as power series with coefficients calculated from the semiclassical asymptotics and the power expansions of the quantum blocks. The question arises how to sum up these series. Surprisingly, one can find closed formulae for at least the four-point spherical [38] and the one-point toroidal classical blocks employing the AGT correspondence.
Indeed, a considerable progress in the theory of conformal blocks and their applications has been achieved recently. This is mainly due to the discovery of the Liouville/N = 2 gauge theories correspondence by Alday, Gaiotto and Tachikawa in 2009 [51]. The AGT conjecture states that the LFT correlators on the Riemann surface C g,n with genus g and n punctures can be identified with the partition functions of a class T g,n of four-dimensional N = 2 supersymmetric SU(2) quiver gauge theories. A significant part of the AGT conjecture is an exact correspondence between the Virasoro blocks on C g,n and the instanton sectors of the Nekrasov partition functions of the gauge theories T g,n . Very soon after its discovery, the AGT hypothesis was extended to the SU(N)-gauge theories/conformal Toda correspondence [52][53][54]. 8 Let us recall that originally the Nekrasov partition functions have been introduced to calculate the low energy effective N = 2 SUSY gauge theories prepotentials [64,65]. The Seiberg-Witten prepotentials [66,67] determine the low energy effective dynamics of the four-dimensional N = 2 super-Yang-Mills theories and can be recovered from the Nekrasov partition functions in the appropriate limit, i.e. when the so-called Ω-background parameters: ǫ 1 , ǫ 2 appearing in the Nekrasov functions tend to zero.
The Nekrasov functions lead also to an interesting application when one of the two Ω-background parameters is non-zero. Such situation has been considered by Nekrasov and Shatashvili in [10]. They observed that in the limit when one of the Ω-background parameters, say ǫ 2 , is zero and the second one, i.e. ǫ 1 , is kept finite (Nekrasov-Shatashvili limit) then one can extract from the Nekrasov functions in this limit the so-called effective twisted superpotentials. These quantities determine the low energy effective dynamics of the two-dimensional (Ω-deformed) supersymmetric gauge theories. Twisted superpotentials play also a prominent role in another context, namely in the so-called Bethe/gauge correspondence [10][11][12] which maps supersymmetric vacua of the N = 2 two-dimensional theories to Bethe states of quantum integrable systems (QIS's). A result of that duality is that twisted superpotentials are identified with Yang's functionals [68] which describe a spectrum of QIS's.
Looking at the AGT relations it is not difficult to realize that the Nekrasov-Shatashvili limit of the Nekrasov instanton partition functions corresponds to the classical limit of conformal blocks. Let us note that by combining the AGT duality and the Bethe/gauge correspondence it is possible to link classical blocks to Yang's functionals, cf. [35][36][37].
Using correspondence identifying classical blocks and twisted superpotentials one can find a closed formulae for at least spherical and toroidal classical blocks. As has been already mentioned, the relevant technical problem of this strategy consists in the summation of the series defining the classical block. This problem can be tackled on the gauge theory side by means of the saddle point method [40,64,65,69].

Classical Liouville action
Let C 0,n be the n-punctured Riemann sphere with complex coordinates chosen in such a way that z = ∞. Consider the Liouville equation with one of the following asymptotic behaviors of the Liouville field φ(z,z) near the punctures: 1. case of elliptic singularities: It is known that it exists a unique solution of eq. (2.7) if one of the conditions (2.8) [70][71][72] or (2.9) [73] is satisfied. One can define the Liouville action S L [φ] on C 0,n . Because of the singular nature of the Liouville field at the punctures such action has to be properly regularized: The prescription given in eqs. (2.10) and (2.11) is valid for parabolic singularities (corresponding to ξ j = 0) as well.
It is well known mathematical fact that the critical value S cl L [φ] of the Liouville action functional S L [φ] on C 0,n (the classical Liouville action) is the generating function for the accessory parameters c j of the Fuchsian uniformization of the punctured Riemann sphere, i.e.: Primarily, this formula has been derived within the so-called geometric path integral approach to the quantum Liouville theory by analyzing the quasi-classical limit of the conformal Ward identity [74]. Then, for parabolic singularities formula (2.12) has been proved by Takhtajan and Zograf. The details can be found in [75]. In ref. [76] the extension of [75] to compact Riemann surfaces has been presented. For general elliptic singularities eq. (2.12) has been proved in [77] and non rigorously derived in [78]. It is also possible to construct the Liouville action functional satisfying (2.12) for the so-called hyperbolic singularities (holes) on the Riemann sphere, see [79].
On the other hand one can observe that in the case of the four-punctured sphere with singularities located at z 4 = ∞, z 3 = 1, z 2 = x, z 1 = 0 the Fuchsian differential equation with an accessory parameter c 2 (x) given by the derivative w.r.t. x of the four-point classical action can be obtained from the classical limit b → 0 of certain null vector decoupling equation. Concretely, from the equation satisfied by the five-point function and four heavy primary operators Analogously, the Weierstrassian-form Lamé equation with the accessory parameter determined by the torus classical one-point action can be recovered from the null vector decoupling equation satisfied by the torus two-point correlation function with one degenerate field (see subsection 3.2).

Zamolodchikov's conjecture
Hence, one can compute the accessory parameters once the classical action is known. The latter can be derived by performing the classical limit of the DOZZ quantum LFT correlation functions. In particular, Zamolodchikov brothers [33] studying the classical limit of the four-point function of the quantum Liouville theory on the sphere argued that the classical Liouville action with four elliptic/parabolic singularities located at z 4 = ∞, z 3 = 1, z 2 = x, z 1 = 0 can be expressed as follows: Indeed, the four-point function of the DOZZ theory can be defined as an integral of schannel conformal blocks and DOZZ couplings over the continuous spectrum of the theory. In the semiclassical limit b → 0 the integrand can be expressed in terms of three-point classical Liouville actions and the classical block, and the integral itself is dominated by the saddle point ∆ s = 1 b 2 δ s (x). One thus gets the factorization (2.13). Concluding, in order to construct the four-point classical action via Zamolodchikov's prescription one needs the following data: (a) the classical three-point Liouville action S cl L (δ 3 , δ 2 , δ 1 ) for the classical weights δ 1 , δ 2 , δ 3 at the locations 0, 1, ∞; (b) the four-point classical conformal block on the sphere f δ Let us stress that we will exploit the above idea in the present work in order to compute the classical action on the one-punctured torus.
The semiclassical limit should be independent of the choice of the channel in the representation of the DOZZ four-point function. Therefore, one gets the consistency conditions known as the classical bootstrap equations [50]: The saddle weights δ t (x), δ u (x) in the t-and u-channel are simply related to the s-channel saddle point classical weight: There is a nice geometric interpretation of the saddle point conformal weight δ i (x). Let us recall that the classical solution describes a unique hyperbolic geometry with singularities at the locations of conformal weights. For elliptic, parabolic and hyperbolic weights one gets conical singularities, punctures and holes with geodesic boundaries respectively [79][80][81]. In the latter case the classical conformal weight δ is related to the length ℓ of the corresponding hole by where the scale of the classical configuration is set by the condition R = −̺/2 imposed on the constant scalar curvature R.
In the case of four singularities at the standard locations 0, x, 1, ∞ there are three closed geodesics Γ s , Γ t , Γ u separating the singular points into pairs (x, 0|1, ∞), (x, 1|0, ∞) and (x, ∞|0, 1) respectively. Since the spectrum of DOZZ theory is hyperbolic the singularities corresponding to the saddle point weights δ i (x) are geodesic holes. One may expect that these weights are related to the lengths ℓ i of the closed geodesics Γ i in corresponding channels [50]: 3 Lamé accessory parameter from Liouville theory

Semiclassical Liouville one-point function on the torus
Let τ be the torus modular parameter and q = e 2πiτ ,q = e −2πiτ . The Liouville one-point function on the torus (expressed in terms of 2d CFT quantities defined on the complex plane) reads as follows The trace in (3.1) is taken over the basis of the Liouville Hilbert space [82]: The operator V β in (3.1) is the primary Liouville vertex operator with the conformal weight ∆ β . It has been assumed that V β is a heavy field (∆ β b→0 const.). Moreover, the operator V β corresponds to the so-called elliptic or parabolic (ξ = 0) singularity (cf. condition (3.4)). The integrand in (3.2) is built out of the DOZZ structure constant [33,83]: and the torus one-point conformal block F ∆ β c,∆ (q) defined in (2.1)-(2.2). Now, we want to find the limit b → 0 of the one-point function (3.1)- (3.2) in the case when all the conformal weights ∆ α P , ∆ β are heavy [33], i.e.: As has been already assumed the external weight ∆ β is heavy, The corresponding classical conformal weightδ is defined as follows In the case of the intermediate weight ∆ α P , if we rescale the integration variable P = p b , then Let us recall that the elliptic/parabolic (ξ = 0) classical weights 0 <δ ≤ 1 4 are related to the parabolic/elliptic singularities, i.e. conical singularities with an opening angle 2πξ. The hyperbolic weights δ > 1 4 correspond to the hyperbolic singularities -holes with geodesic boundary (as has been already mentioned the classical hyperbolic weight is related to the length of the corresponding hole).
Let us turn to the problem of finding the b → 0 limit of the one-point function (3.19)- (3.20). First let us determine the asymptotical behavior of the integrand in (3.2) when b → 0.
In [81] it has been found that for the hyperbolic spectrum: the DOZZ three-point function in the limit b → 0 behaves as follows At this point, a few comments are in order. The expression in the square brackets should correspond to the known expression for the classical Liouville action S L [ϕ] on the sphere with three hyperbolic singularities (holes). Such classical action has been constructed in [81] (see also [79]). The construction of S L [ϕ] relies on a solution of a certain monodromy problem for the Fuchsian differential equation: with hyperbolic singularities (δ k 's are hyperbolic). In this way one can find the form of the n-point classical action S In the case when n = 3 the Fuchsian accessory parameters c k are known and the classical action can be determined from eq. (3.14). For the standard locations of singularities z 1 = 0, z 2 = 1, z 3 = ∞ this yields [81]: where the constant on the r.h.s. is independent of z j , λ j and πµb 2 . Comparing (3.13) and (3.15) we see that the classical limit of the DOZZ structure constant differs from the classical three-point action by an additional imaginary term. As has been observed in [81] this inconsistency occurs due to the fact that the classical Liouville action is by construction symmetric with respect to the reflection α → Q − α, (λ → −λ) whereas the DOZZ three-point function is not. Under this reflection the DOZZ three-point function changes according to the formula [33]: is the so-called reflection amplitude [33]. The discrepancy between (3.13) and (3.15) can be overcome if we consider the symmetric three-point functionC (α 1 , α 2 , α 3 ) [81]: instead of C (α 1 , α 2 , α 3 ). Indeed, taking into account the classical limit of the reflection amplitude for λ ∈ R: one can easily verify that the symmetric three-point function in the limit b → 0 behaves as follows [81] C where α j , j = 1, 2, 3 are given by (3.12). Hence, in order to obtain consistent semiclassical one-point function from the quantum one it is more convenient to take as a starting point the quantum one-point function with the symmetric DOZZ structure constant: where c, ∆ β , ∆ α P are given by (3.5)-(3.7). The operator: in (3.19) is the primary Liouville vertex operator V β rescaled by the square root of the reflection amplitude (3.16). Also the primary vertex operators V α P 's which generate intermediate highest weight states are assumed to be rescaled by the square root of the reflection amplitude, in accordance with (3.21). Now, for heavy insertions (3.9)-(3.11) using (3.18) one gets It is reasonable to assume that the torus one-point block has the b → 0 asymptotic similar to the DOZZ three-point function, i.e. given by the eq. (2.6).
Finally, in the limit b → 0 the integral (3.20) with heavy insertions is determined by the saddle point approximation, i.e., by the critical value of the function: where δ = 1 4 + p 2 andδ = 1 4 1 − ξ 2 . Indeed, when b → 0 then the symmetric one-point function behaves as follows where S torus L (ξ; q) = S(ξ, p * ; q) and the saddle point momentum p * = p * (ξ, q,q) is determined by the equation: (3.23) One thus gets the factorization As a final remark in this paragraph let us note that the modular invariance of the torus Liouville one-point function [84,85] implies the classical modular bootstrap equation:

Accessory parameter from torus one-point classical action
Consider the null fields [41] χ ± (z) = L −2 (z) − 3 2(2∆ ± + 1) which correspond to the null vectors appearing on the second level of the Verma module. The operators V ± in (3.25) are the primary degenerate fields with the following conformal weights: The correlation functions with null fields must vanish. In particular, for the two-point function on a torus 9 with the null field χ + ≡ χ α + =− b 2 one has Using the torus Ward identities [42] one can convert the condition (3.27) to the second order differential equation: where Z(τ ) is a partition function and ζ(z|τ ) = ∂ z log θ 1 (z|τ ) + 2η 1 z, (1 − q n ), Let us introduce For w = 0 from (3.28) one gets Recall, that the primary vertex operators in the two-point function above are defined on the cylinder V σ ≡ V cyl σ . These operators are related to the operators living on the plane (used in the previous paragraph) in the standard way V cyl σ (z,z) = e z∆σ ez ∆σ V C σ (e z , ez). One can think of (3.29) as the equation obeyed by the Liouville two-point function on the torus where the "alpha" α + or equivalently the conformal weight ∆ α + of the operator located at z "has been continued" to the degenerate value. 10 If so, let us multiply both sides of the eq. (3.29) by [S(iᾱ P − iQ/2) S(iβ − iQ/2) S(iα P − iQ/2)] 1 2 =: R. Next, let us assume that α P and β are heavy. On the other hand for b → 0 the operator V + remains light (∆ + = O(1)) and its presence in the correlation function has no influence on the 10 The idea which makes use of the degenerate representation of the Virasoro algebra in the Liouville field theory is not new. In particular, such trick has been used by Teschner in order to re-derive the DOZZ formula [86] (for reviews, see [82,87]). classical dynamics. Also log Z(τ ) is of order O(1). Hence, one can expect that for b → 0 11 After substituting (3.30) into the eq. (3.29) and taking the limit b → 0 one gets From eqs. (3.23) and (3.24) we have More in general, i.e. for the elliptic singularities −κ =δ ∈ (0, 1 4 ) one gets Let us note that η 1 depends on the modular parameter τ according to the formula η 1 = 4π 2 24 E 2 (τ ) where E 2 (τ ) is the second Eisenstein series. Hence, finally one can express B (and C, C ell ) in terms of functions depending on τ : light vertices on the sphere, see for instance [88]. It is reasonable to expect that the same holds on the cylinder, cf. [89].

Accessory parameter from twisted superpotential
In order to compute the torus classical block fδ δ (q) entering the expressions for B and/or C, C ell one can exploit the "chiral" AGT relation on the torus and the correspondence between the classical limit of the conformal blocks and the Nekrasov-Shatashvili limit of the Nekrasov instanton partition functions. The "chiral" AGT relation on the torus identifies the torus quantum block with the Nekrasov instanton partition function [64,65] of the N = 2 * , SU(2) gauge theory (which equals to [Z inst as it is written in the second line of the equation below): In eq. (4.1) η(q) = q Above µ is the mass of the adjoint hypermultiplet, a is the vacuum expectation value of the complex scalar of the gauge multiplet and ǫ 1 , ǫ 2 are Ω-background parameters. The relation (4.1) is understood as an equality between the coefficients of the expansions of both sides in powers of q. For the torus conformal block such expansion has been introduced in eqs. (2.1)-(2.2). For the definition of the instanton partition function appearing in (4.1), see subsection 4.3. The identity (4.1) has been proved by Fateev and Litvinov [46]. They have shown that the coefficients of the expansions of both sides of (4.1) obey the same recurrence relation. Let us note that the relation (4.1) holds for the heavy conformal weights. Indeed, from (4.2)-(4.4) we havẽ Hence, one can consider the limit b → 0 of the AGT relation (4.1). The limit b → 0 corresponds to ǫ 2 → 0 (ǫ 1 = const.), in accordance with (4.2). As has been observed by Nekrasov and Shatashvili [10] if ǫ 2 → 0 while ǫ 1 is kept finite the Nekrasov instanton partition function has the following asymptotical behavior: W inst ( · , ǫ 1 ) is the instanton contribution to the so-called effective twisted superpotential of the corresponding two-dimensional gauge theory restricted to the two-dimensional Ωbackground. Twisted superpotentials play also a prominent role in already mentioned Bethe/gauge correspondence [10][11][12]. Therefore, taking into account the semiclassical asymptotic (2.6) of the torus quantum block and (4.7) one can get from (4.1) the "classical version" of the torus AGT relation: Using (4.8) and ∂ τ log η(τ ) = iπ 12 E 2 (τ ) one can rewrite the expression (3.34) for the parameter B to the following form: Let us stress that the twisted superpotential W N =2 * ,U(2) inst is a computable quantity. The derivation of W N =2 * ,U(2) inst directly from the Nekrasov instanton partition function shall be presented in the third subsection. In the next subsection we confront the calculation method of the eigenvalue, which employs the idea of the classical block, with another procedure based on the WKB analysis, cf. [46,[89][90][91][92].

WKB analysis
Taking into account (4.10) one can rewrite the eq. (1.2) to the following Schrödinger-like form: into the eq. (4.11) one finds Above equation can be solved iteratively by expansions in ǫ 1 : In particular, for lower orders this yields Note, that in our case V 0 (z) = µ 2 ℘(z), V 1 (z) = µ℘(z) and V k (z) = 0 for all k > 1.
The quasiclassical approximation E ≡ E| zero order in ǫ 1 to the energy eigenvalue is determined by the A-cycle integral [46,90,91]: (4.13) Physically, the equation above is nothing but the Bohr-Sommerfeld quantization condition [90]. It is convenient to introduce E = E 4π 2 a 2 , ν = µ 2 4π 2 a 2 (4.14) and rewrite the eq. (4.13) to the form Let us recall that we are working on the torus with periods 1 and τ parameterized by the complex coordinate z ≡ z + 1 ≡ z + τ . The A-cycle here is just the interval [0, 1]. Eq. (4.15) allows to compute E as an expansion in ν with coefficients depending on q = e 2πiτ : Indeed, after an expansion of the square root the eq. (4.15) becomes then, the higher terms on the r.h.s. of the eq. (4.17) must vanish. Therefore, up to ν 4 one finds [89]: where E 2n , n = 1, 2, 3 are the Eisenstein series. In above calculations appear integrals of the form A ℘(z) n dz =: K n .
For their explicit computation see appendix B. Finally, using (4.14), (4.16), (4.18)-(4.21) and the q-expansions of the Eisenstein series (A.4)-(A.6) one gets On the other hand from (3.34) and (4.12) we have The torus classical block fδ δ (q) appearing above has the following expansion fδ ,n δ q n (4.24) with coefficients fδ ,n δ determined by the semiclassical asymptotic (2.6) of the quantum block: For instance, .

Nekrasov-Shatashvili limit
A goal of this subsection is to compute an instanton contribution to the so-called effective twisted superpotential of the N = 2 * U(2) gauge theory. We closely follow here the method of the calculation developed by Poghossian in ref. [40].
Consider the instanton part of the Nekrasov partition function of the N = 2 supersymmetric U(2) gauge theory with an adjoint hypermultiplet (the N = 2 * theory) [10,64]: where . We will assume that µ, a u , ǫ 1 , ǫ 2 ∈ R. The poles which contribute to the integral (4.26) are at Recall, that these poles are in correspondence with pairs Y = (Y 1 , Y 2 ) of Young diagrams with total number of boxes |Y| = |Y 1 | + |Y 2 | = k. The index i parameterizes the columns whereas j runs over the rows of the diagram Y u . The parameters ǫ 1 , ǫ 2 describe a size of 12 Here δ * = 1 4 + p 2 * and p 2 * = − a 2 ǫ 2 1 . a box (i, j) ∈ Y u in horizontal, vertical direction respectively. The instanton sum over k in (4.26) can be rewritten as a sum over a pairs of Young diagrams as follows: The contributions Z Y to the instanton sum correspond to those obtained by performing (in some specific order) the contour integrals in (4.26).
Now we want to calculate the Nekrasov-Shatashvili limit [10] ǫ 2 → 0 (ǫ 1 is kept finite) of the instanton partition function (4.26). Based on the arguments developed by Nekrasov and Okounkov in ref. [65] it is reasonable to expect that for vanishingly small values of ǫ 2 the dominant contribution to the instanton partition function (4.26) will occur when k ∼ 1 ǫ 2 . Unfortunately, we have found no proof of that mechanism in the general case. Let us note only that this statement becomes evident in the trivial case in which Z k = 1 for all k = 1, 2, . . .. Indeed, for ǫ 2 → 0 and x = q ǫ 2 ∈ R >0 we have then from eq. (4.26): This means that the whole sum is dominated by a single term with k ∼ x → ∞. Hence, in order to compute the limit ǫ 2 → 0 of Z N =2 * ,U(2) inst let us find first the leading behavior of log q k Ω k for large k (i.e. small values of ǫ 2 and finite ǫ 1 ). After simple calculations, using log( In eq. (4.28) it is implicitly understood that the poles φ I are obtained from eq. (4.27) in the limit ǫ 2 → 0. Note that in the limit ǫ 2 → 0 the poles form a continuous distribution (cf. [69]): x ui = a u + (i − 1)ǫ 1 + ω ui .
In terms of Young diagrams the situation can be explained heuristically as follows. When ǫ 2 is very small then the number of boxes k ui in the vertical direction (the number of rows) is very large, however this number multiplied by ǫ 2 , i.e.: ǫ 2 k ui = ω ui is expected to be finite. In other words we obtain a continuous distribution of rows in the limit under consideration. Then, in order to evaluate (4.28) at the values (4.29) one can assume that the summations "over instantons" in (4.28) become continuous in the row index: The limits of integration x 0 ui and x ui are the bottom and the top ends of the i-th column in Y u respectively. Applying eq. (4.30) to eq. (4.28) one gets where F (x) = x(log |x| − 1). Finally, the Nekrasov instanton partition function in the limit ǫ 2 → 0 can be represented as follows: where the "integral" is over the infinite set of variables {x ui : u = 1, 2; i = 1, . . . , ∞}. As a consequence, the Nekrasov-Shatashvili limit of Z wherex ui denotes the "critical configuration" extremizing the "free energy" (4.31).

Saddle point equation
The extremality condition for the "action" H N =2 * ,U(2) inst given by (4.31) reads as follows: where u, v = 1, 2 and i, j = 1, . . . , ∞. This implies that either the following eq.: or its analog in which −q is replaced by +q are holding. 13 Eq. (4.34) can be regularized assuming that there is an integer L such that the length of the column ω ui is equal to zero for i > L. Analyzing eq. (4.34) in such a case, i.e. when j = 1, . . . , L, one can observe that the column lengths extremizing the "free energy" are of the order ω ui ∼ O(q i ). 14 For example, at order q L one can writê Here the symbols ω uin denote the contributions to the coefficients ω ui at the n-th order in q. Now it is possible to solve equation (4.34) starting from L = 1 and deriving recursively the ω uin 's step by step up to desired order. For instance, 13 In eq. (4.34) and below q ≡ |q|. 14 Note that the saddle point equation (or in fact a system of equations) must be solved subject to the condition that the solution is consistent with the q-expansion of the twisted superpotential obtained from the instanton partition function. This implies that the order of q in the expansions of rescaled column lengths must correlate with the index i. Indeed, first, we have On the other hand where ωui's are the column lengths of the critical diagrams (see the calculation at the beginning of the next subsection). Hence, in order to get from (4.36) the q-expansion consistent with (4.35) the expansions of ωui's must start from q i .
In order to investigate the solution of the saddle point equation in the case when L → ∞ it is helpful to convert it to other equivalent form. Indeed, as has been observed in [69] (see also [40]) the eq. (4.34) can be rewritten in terms of certain "Y-system": and ψ(z) = ∂ z log Γ(z). The product in (4.39) is convergent for arbitrary z ∈ C provided that the column lengths tend to zero for large enough i, which is equivalent to the assumption that x ui → x 0 ui . If ω ui = 0 for all i, i.e. all column lengths are zero, then Y (z) becomes The functions Y (z), Y 0 (z) have zeros located at x ui and x 0 ui respectively.

Twisted superpotential, classical block, accessory parameter
Now we are ready to compute the critical value of the "free energy" (4.33), i.e. the so-called twisted superpotential. It is convenient first to calculate the derivative of W N =2 * ,U(2) inst ( q, a, m i ; ǫ 1 ) with respect to q: Above we have used the fact that ∂H inst /∂x ui | x ui =x ui = 0. Hence, it is easy to realize that the last term in (4.41) coincides with the sum over the column lengths of the "critical" Young diagrams. More explicitly, the eq. (4.41) reads as follows Knowing the coefficients of the extremal lengths of the columns from a solution of the system of equations (4.34) one gets where δ,δ are given by (4.5)-(4.6) and fδ ,n δ , n = 1, . . . denote coefficients of the torus classical block (see (4.24)-(4.25)). Concluding, the eq. (4.44) is nothing but the expansion of both sides of the "classical" AGT relation 15 : As a final conclusion of this subsection let us write down the main result of the present work. Knowing the classical torus one-point block from (4.8) and applying eqs. (4.9) and (4.41), one arrives at the following expression of the Lamé accessory parameter: Hence, we have found that the accessory parameter B is related to the sum of column lengths of the "critical" Young diagrams. The latter can be rewritten using the contour integral representation. Indeed, let γ denotes the contour which encloses all the pointsx ui , x 0 ui , u = 1, 2, r = 1, . . . , ∞. Then, as has been noticed by Poghossian in ref. [40], the sum u,i ω ui can be expressed as follows

Concluding remarks and open problems
The main result of the present work is the expression of the Lamé accessory parameter B (or equivalently A, C, C ell ) in terms of the solution of the TBA-like eq. (4.34). This equation has been solved by a power expansion in the parameter q. It has been noticed that obtained expression for B can be rewritten in terms of a contour integral in which the integrand is built out of the functions Y and Y 0 introduced in eqs. (4.39)-(4.40). As has been observed in [40,69] also the TBA-like eq. (4.34) can be rewritten in terms of the function Y . Hence, one can relate the problem of finding the Lamé accessory parameter to the problem of searching a solution of the functional equation (4.38). Another result presented in this paper is a check that the classical limit of the torus quantum conformal block exists and yields a consistent definition of the torus classical block. Moreover, it has been verified that the torus classical block corresponds to the twisted superpotential of the N = 2 * U(2) gauge theory.
Finally, in this paper has been proposed an expression for the Liouville classical action S torus L on the one-punctured torus. It has been conjectured that S torus L can be calculated in terms of the torus classical conformal block, classical three-point Liouville action and the saddle point classical intermediate weight (the saddle point momentum). We leave as an open question (to which we return very soon) a comparison of this result with the results obtained by Menotti in [29][30][31][32].
Work is in progress in order to verify formulae (3.31)-(3.34) and (4.46). First of all let us note that these formulae pave the way for numerical studies of the function B(τ ) and the results of such investigation can be compared with that obtained by Keen, Rauch and Vasquez in [23]. It is well known that in a certain limit the Lamé equation becomes the Mathieu equation. Hence, other possible check beyond the WKB analysis performed in subsection 4.2 is to verify whether our candidate for the Lamé eigenvalue correctly reduces to the Mathieu eigenvalue, cf. [92]. 16 As a final remark concerning possible combinatorial tests of our main result let us note that it would be valuable to check whether one can recover from eqs. (3.34), (4.9) the expansions of the Lamé eigenvalue worked out by Müller-Kirsten [93] (see also [94]) and Longmann [95,96]. 17 It is a well known fact [23] that the Lamé accessory parameter is a modular form of weight 2. In order to answer the question whether proposed candidate for the Lamé accessory parameter has correct modular transformation properties one has to put a heuristic derivation of eqs. (3.34), (4.9), (4.46) on a more rigorous mathematical level. First, it has to be proved that in the classical limit the quantum conformal blocks behave exponentially. 18 Let us stress that there is still no rigorous proof of convergency of the expansions defining generic quantum conformal blocks. Secondly, it seems to be possible to derive the "classical" torus AGT relation (4.8) more rigorously exploiting the Teichmüller theory approach to Liouville theory. Indeed, relations such as (4.8) implicitly appear as a byproduct in the proof of the AGT correspondence recently proposed by Teschner and Vartanov [58]. 16 Let us stress that we have found an agreement between B(τ ) and the energy eigenvalue E computed by means of the WKB method so far only for ǫ1 = 0. It remains still to calculate the "quantum corrections" to E in the higher powers of ǫ1 and compare the result with B(τ ) obtained from the classical block. 17 Note, that since the modular properties of the Lamé accessory parameter are known [23], then the relation (4.9), if it is true, encodes some information about the modular transformation properties of the N = 2 * U(2) twisted superpotential. This observation fits in an interesting line of research which aims to answer the question how the S-duality is realized in the Ω-deformed N = 2 gauge theories, see for instance [97,98]. 18 It seems to be possible to prove the classical asymptotics of conformal blocks representing them as the Coulomb gas/Dotsenko-Fateev/β-ensemble integrals [99][100][101][102] and applying matrix models technics [24,[103][104][105][106][107][108][109][110][111][112].

Jacobi and Weierstrass elliptic functions
Let ω 1 and ω 2 be two complex numbers whose ratio is not real. A function which satisfies f (z) = f (z + 2ω 1 ) = f (z + 2ω 2 ) for all z ∈ D(f (z)) is called a doubly periodic function of z with periods 2ω 1 and 2ω 2 . A doubly periodic function that is analytic except at its poles and which has no singularities other than these poles in a finite part of the complex plane is called an elliptic function. Let Im ω 2 ω 1 ∈ R, then the points 0, 2ω 1 , 2ω 1 + 2ω 2 , 2ω 2 when taken in order are the vertices of a parallelogram, known as the fundamental parallelogram. The behaviour of an elliptic function is completely determined by its values in fundamental parallelogram.
If we wish to count the number of poles or zeros of an elliptic function in a given period parallelogram and it happens that certain poles or zeros lie on the boundaries of this parallelogram, then one can translate the period parallelogram without rotation until no pole or zero lies on its boundary. Such obtained parallelogram is called a cell.
In [1] one can find a list of general properties of elliptic functions. In particular, one can prove, that (i) the number of poles of an elliptic function in any cell is finite; (ii) the sum of the residues of an elliptic function at its poles in any cell is zero.
The number of poles of an elliptic function in any cell, counted with multiplicity, is called the order of the function. The statements (i)-(ii) written down above imply, that the order of an elliptic function is necessarily at least equal to 2. Indeed, an elliptic function of order 1 would have a single irreducible pole. If this were actually a pole its residue would not be zero. Hence, in terms of singularities, the simplest elliptic functions are those of order 2. These can be divided into two classes: (I) those which have a single irreducible double pole in each cell at which the residue is zero; (II) those which have two simple poles in each cell at which the two residues are equal in absolute value, but of opposite sign.
The Jacobi elliptic functions are examples of the second class of elliptic functions of order 2. The Jacobi function sn u is defined by means of the integral where k is a constant. By inversion of the integral we have x = sn u. From definition follows that sn 0 = 0.
Conversely, given (dy/dz) 2 = 4y 3 − g 2 y − g 3 , and if numbers ω 1 and ω 2 can be determined such that then the general solution of the differential equation is y = ℘(z + const.).
Then, we have dy dz where h is a constant. When z → 0, it is seen that h is a period, and so finally one gets the following relation between Weierstrass and Jacobi elliptic functions: ℘(z) = e 3 + (e 1 − e 3 ) ns 2 z(e 1 − e 3 )

Weierstrass ζ−function
The Weierstrass zeta function ζ(z) is defined by the equation along with the condition lim z→∞ ζ(z) − 1 z = 0. In terms of the theta function θ 1 (z|τ ) and the constant η 1 the Weierstrass zeta function can be expressed as follows

Eisenstein series
The first few Eisenstein series can be expressed as follows where q = e 2πiτ and σ k (n) is the so-called divisor function defined as the sum of the k-th powers of the divisors of n, σ k (n) = d|n d k .

B Integrals K n
The integral one can compute directly using (A.2) and (A.3). The result is K 1 = −2η 1 . In order to compute the integrals K n = A ℘(z) n dz, n > 1 one can employ the following relation between powers of ℘ and its even derivatives (see [113] and refs. therein): Then, K 2 = 1 12 g 2 , K 3 = 1 10 g 3 + 3 20 g 2 K 1 = 1 10 g 3 − 3 10 g 2 η 1 , Now, in order to get the coefficients (4.18)-(4.21) of the expansion of the energy eigenvalue one has to use the following relations: