Multi-Matrix Models: Integrability Properties and Topological Content

TL;DR

This paper explores multi-matrix models, their integrability, and topological properties, providing exact solutions, classifying models including discretized string theories, and establishing their topological field theory structure.

Contribution

It introduces a framework for analyzing multi-matrix models with integrability and topological features, including exact solutions and classification of models.

Findings

Exact solutions for Gaussian-type models

Classification of discretized $c=1$ string theory models

Establishment of topological field theory structure in multi-matrix models

Abstract

We analyze multi--matrix chain models. They can be considered as multi--component Toda lattice hierarchies subject to suitable coupling conditions. The extension of such models to include extra discrete states requires a weak form of integrability. The discrete states of the $q$--matrix model are organized in representations of $sl_q$. We solve exactly the Gaussian--type models, of which we compute several all-genus correlators. Among the latter models one can classify also the discretized $c=1$ string theory, which we revisit using Toda lattice hierarchy methods. Finally we analyze the topological field theory content of the $2q$--matrix models: we define primary fields (which are $\infty^q$), metrics and structure constants and prove that they satisfy the axioms of topological field theories. We outline a possible method to extract interesting topological field theories with a finite number of primaries.