The uses of the refined matrix model recursion

TL;DR

This paper explores the refined recursion relations in beta-ensemble matrix models, providing explicit corrections and applications to supersymmetric gauge theories and topological string models, revealing limitations of certain deformations.

Contribution

It introduces explicit first-order beta-deformed corrections in matrix models and applies these to gauge theories and topological strings, highlighting new insights and limitations.

Findings

Explicit beta-deformed corrections for one-cut and two-cut cases

Applications to superpotentials and surface operators in supersymmetric theories

Beta deformation does not accurately describe Omega-deformed topological strings

Abstract

We study matrix models in the beta ensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first beta-deformed corrections in the one-cut and the two-cut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in N=1 gauge theories, and the calculation of vevs of surface operators in superconformal N=2 theories and their Liouville duals. Finally, we study the beta deformation of the Chern-Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Omega-deformed topological string on the resolved conifold, and therefore that the beta-deformation might provide a different generalization of topological string theory in toric Calabi-Yau backgrounds.