Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model

TL;DR

This paper develops a beta-deformed matrix model approach to compute q-expansion coefficients of conformal blocks and Nekrasov functions, providing exact solutions and new formulas, including for the second Nekrasov coefficient.

Contribution

It introduces a novel matrix model framework to derive q-expansion coefficients and provides exact solutions for the finite N loop equations and Nekrasov functions.

Findings

Derived the second Nekrasov coefficient for SU(2) with N_f=4.

Solved the finite N loop equations at q=0 and in the planar limit.

Computed the planar free energy to the lowest non-trivial order.

Abstract

We observe that, at beta-deformed matrix models for the four-point conformal block, the point q=0 is the point where the three-Penner type model becomes a pair of decoupled two-Penner type models and where, in the planar limit, (an array of) two-cut eigenvalue distribution(s) coalesce into (that of) one-cut one(s). We treat the Dotsenko-Fateev multiple integral, with their paths under the recent discussion, as perturbed double-Selberg matrix model (at q=0, it becomes a pair of Selberg integrals) to construct two kinds of generating functions for the q-expansion coefficients and compute some. A formula associated with the Jack polynomial is noted. The second Nekrasov coefficient for SU(2) with N_f =4 is derived. A pair of Young diagrams appears naturally. The finite N loop equation at q=0 as well as its planar limit is solved exactly, providing a useful tool to evaluate the coefficients as those of the resolvents. The planar free energy in the q-expansion is computed to the lowest non-trivial order. A free field representation of the Nekrasov function is given.