Direct Integration and Non-Perturbative Effects in Matrix Models

TL;DR

This paper develops a method using direct integration to solve multi-cut matrix models with polynomial potentials, explicitly solving the cubic case and exploring non-perturbative effects and large genus asymptotics.

Contribution

It introduces a direct integration approach for multi-cut matrix models and explicitly solves the cubic case, connecting spectral curves to Seiberg--Witten theory and non-perturbative effects.

Findings

Explicit expressions for non-holomorphic modular objects in cubic matrix models.

Holomorphic anomaly equations solved up to genus four.

Identification of non-perturbative sectors influencing large genus asymptotics.

Abstract

We show how direct integration can be used to solve the closed amplitudes of multi-cut matrix models with polynomial potentials. In the case of the cubic matrix model, we give explicit expressions for the ring of non-holomorphic modular objects that are needed to express all closed matrix model amplitudes. This allows us to integrate the holomorphic anomaly equation up to holomorphic modular terms that we fix by the gap condition up to genus four. There is an one-dimensional submanifold of the moduli space in which the spectral curve becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic modular ring of the group $\Gamma(2)$. On that submanifold, the gap conditions completely fix the holomorphic ambiguity and the model can be solved explicitly to very high genus. We use these results to make precision tests of the connection between the large order behavior of the 1/N expansion and non-perturbative effects due to instantons. Finally, we argue that a full understanding of the large genus asymptotics in the multi-cut case requires a new class of non-perturbative sectors in the matrix model.