Matrix Models as Integrable Systems

TL;DR

This paper reviews the connection between matrix models and integrable systems, discussing determinantal formulas, conformal field models, and group-theoretical interpretations of tau-functions to extend integrable hierarchy frameworks.

Contribution

It provides a comprehensive overview of matrix models' relation to integrable hierarchies, including new insights into tau-functions and their group-theoretical interpretations.

Findings

Relation of matrix models to integrable hierarchies clarified

Determinantal formulas and conformal models discussed

Group-theoretical interpretation of tau-functions developed

Abstract

The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Determinantal formulas, relation to conformal field models and the theory of Generalized Kontsevich model are discussed in some detail. Attention is also paid to the group-theoretical interpretation of $\tau$-functions which allows to go beyond the restricted set of the (multicomponent) KP and Toda integrable hierarchies.