Topological expansion of the chain of matrices

TL;DR

This paper solves the loop equations for the chain of matrices model to all orders, revealing that the topological expansion of the free energy is given by symplectic invariants of the spectral curve, and explores various limits and properties.

Contribution

It extends the topological expansion framework to the chain of matrices model, including external fields, and explicitly derives the double scaling limit and modular properties.

Findings

Topological expansion expressed via symplectic invariants

Explicit double scaling limit derived

Discussion of large N asymptotics and infinite chain limit

Abstract

We solve the loop equations to all orders in $1/N^2$, for the Chain of Matrices matrix model (with possibly an external field coupled to the last matrix of the chain). We show that the topological expansion of the free energy, is, like for the 1 and 2-matrix model, given by the symplectic invariants of the associated spectral curve. As a consequence, we find the double scaling limit explicitly, and we discuss modular properties, large $N$ asymptotics. We also briefly discuss the limit of an infinite chain of matrices (matrix quantum mechanics).