\beta-deformed matrix model and Nekrasov partition function

TL;DR

This paper explores the connection between eta-deformed Penner matrix models and Nekrasov partition functions for certain 4D N=2 SU(2) gauge theories, explicitly calculating corrections and matching instanton contributions.

Contribution

It explicitly constructs the first half-genus correction to the free energy for general eta and demonstrates its agreement with Nekrasov partition functions, including higher instanton effects.

Findings

First half-genus correction matches Nekrasov partition function

Explicit construction of corrections for general eta

Provides a complementary approach using loop equations

Abstract

We study Penner type matrix models in relation with the Nekrasov partition function of four dimensional \mathcal{N}=2, SU(2) supersymmetric gauge theories with N_F=2,3 and 4. By evaluating the resolvent using the loop equation for general \beta, we explicitly construct the first half-genus correction to the free energy and demonstrate the result coincides with the corresponding Nekrasov partition function with general \Omega-background, including higher instanton contributions after modifying the relation of the Coulomb branch parameter with the filling fraction. Our approach complements the proof using the Selberg integrals directly which is useful to find the contribution in the series of instanton numbers for a given deformation parameter.