Integrability and Matrix Models

TL;DR

This paper reviews matrix models and their connection to integrable hierarchies, covering various models, their properties, and future research directions in the field.

Contribution

It provides a comprehensive overview of matrix models, their integrability aspects, and detailed analysis of different types including Kontsevich models.

Findings

Analysis of discrete 1-matrix and 2-matrix models

Discussion of Ward identities and W-constraints

Exploration of continuum limits and model interrelations

Abstract

The theory of matrix models is reviewed from the point of view of its relation to integrable hierarchies. Discrete 1-matrix, 2-matrix, ``conformal'' (multicomponent) and Kontsevich models are considered in some detail, together with the Ward identites (``W-constraints''), determinantal formulas and continuum limits, taking one kind of models into another. Subtle points and directions of the future research are also discussed.