Classical torus conformal block, $ \mathcal{N} $ = 2∗ twisted superpotential and the accessory parameter of Lamé equation

Abstract

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.

Toward the pole

“…On the complex plane ν ∈ C with branch cuts, extending from the points n + 2iq n /ζ n , n ∈ N to infinity and passing through the points ν = n, n − 2iq n /ζ n ( Fig. 1), one can choose branches of all the multivalued functions entering the expression (15) in such a way that all the square roots will be positive at real ν, and all the logarithms will be real. On this branch the expansion of the double series (15) in q reproduces the expansion of the conformal block (14).…”

“…Namely, the problem of finding the semiclassical 1-point torus conformal block is equivalent to the spectral problem of the Lame equation [14]. From the point of view of conformal field theory, the Lame equation arises as the null-vector decoupling equation for the 2-point correlator on the torus [15]. In a proper limit the Lame equation turns into the Mathieu equation, whose solution defines the semiclassical irregular conformal block [16,17].…”

Holographic interpretation of 1-point toroidal block in the semiclassical limit

“…(Coefficients f 1 and f 2 can also be found in [21].) The 1-point linearised classical block on a torus can be defined as in the spherical case [6,9,22,23].…”

“…In this paper we are interested in toroidal conformal blocks and their dual realization. For the previous studies of the toroidal conformal blocks in the framework of CFT see [14][15][16][17][18][19][20][21].…”

Comments on Rényi entropy in AdS3/CFT2

“…One may be able to prove (6.3) from CFT by studying torus conformal blocks. [48] and references therein may be useful. Relatedly, we expect that, in analogy to the case of two intervals on the plane [27], the torus uniformization equations (6.6) can be understood as null decoupling equations obeyed by Liouville three-point functions involving two heavy operators and one light, degenerate operator with a null state at level two.…”