Liouville theory, N=2 gauge theories and accessory parameters

TL;DR

This paper connects Liouville theory and N=2 gauge theories to compute accessory parameters of the 4-punctured sphere using saddle point methods, providing explicit formulas and linking to instanton superpotentials.

Contribution

It introduces a novel method to calculate accessory parameters via the semiclassical limit of Liouville theory and N=2 gauge theory correspondence, with explicit analytic expressions.

Findings

Derived an analytic expression for the N_f=4, U(2) instanton twisted superpotential.

Expressed the accessory parameter as the derivative of the 4-point classical block.

Linked the accessory parameter to the sum over critical Young diagram column lengths.

Abstract

The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the N = 2 (Omega-deformed) U(2) super-Yang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N_f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.