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 * (Ω-deformed) U(2) super-Yang-Mills theory is used to propose new formulae for the accessory parameter of the Lamé 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 Lamé 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 socalled "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.

Zamolodchikov's recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the classical limit of the DOZZ quantum Liouville theory are simply related to the geodesic length functions of the hyperbolic geometry on the 4-punctured Riemann sphere. Such relation provides new powerful methods for both numerical and analytical calculations of these functions. The consistency conditions for the factorization of the 4-point classical Liouville action in different channels are numerically verified. The factorization yields efficient numerical methods to calculate the 4-point classical action and, by the Polyakov conjecture, the accessory parameters of the Fuchsian uniformization of the 4-punctured sphere.

Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture and the Bethe/gauge correspondence results in a triple correspondence which identifies classical conformal blocks with twisted superpotentials and then with Yang-Yang functions. In this paper the triple correspondence is studied in the simplest, yet not completely understood case of pure SU p2q super-Yang-Mills gauge theory. A missing element of that correspondence is identified with the classical irregular block. Explicit tests provide a convincing evidence that such a function exists. In particular, it has been shown that the classical irregular block can be recovered from classical blocks on the torus and sphere in suitably defined decoupling limits of classical external conformal weights. These limits are "classical analogues" of known decoupling limits for corresponding quantum blocks. An exact correspondence between the classical irregular block and the SU p2q gauge theory twisted superpotential has been obtained as a result of another consistency check. The latter determines the spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord with the Bethe/gauge correspondence. An analogue of this statement is found entirely within 2d CFT. Namely, considering the classical limit of the null vector decoupling equation for the degenerate irregular block a celebrated Mathieu's equation is obtained with an eigenvalue determined by the classical irregular block. As it has been checked this result reproduces a well known weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new formulae for Mathieu's eigenvalue relating the latter to a solution of certain Bethe-like equation are found. 1