Solving matrix models using holomorphy

TL;DR

This paper explores how holomorphy controls quantum corrections in supersymmetric gauge theories with moduli spaces, and how these corrections relate to solving associated matrix models via generalized resolvents and algebraic curves.

Contribution

It introduces a method to solve matrix models linked to supersymmetric gauge theories by leveraging holomorphic quantum deformations and their algebraic relations.

Findings

Quantum corrections are governed by holomorphy.

Generalized resolvents satisfy non-trivial relations.

Matrix models can be solved using algebraic curves.

Abstract

We investigate the relationship between supersymmetric gauge theories with moduli spaces and matrix models. Particular attention is given to situations where the moduli space gets quantum corrected. These corrections are controlled by holomorphy. It is argued that these quantum deformations give rise to non-trivial relations for generalized resolvents that must hold in the associated matrix model. These relations allow to solve a sector of the associated matrix model in a similar way to a one-matrix model, by studying a curve that encodes the generalized resolvents. At the level of loop equations for the matrix model, the situations with a moduli space can sometimes be considered as a degeneration of an infinite set of linear equations, and the quantum moduli space encodes the consistency conditions for these equations to have a solution.