Integrability and Matrix Models

TL;DR

This work surveys how matrix models encode integrable-hierarchy structures through Ward identities, Virasoro and W-constraints, and determinantal representations. It connects discrete models (1- and 2-matrix, conformal/multicomponent) to continuum theories via Gross–Newman equations and Kontsevich-type formalisms, showing that partition functions are tau-functions of Toda/KP hierarchies. Key themes include the generalized Kontsevich model, Itzykson–Zuber integrals, and Miwa parametrization, which together unify eigenvalue representations with free-fermion determinants and string-equation reductions. The notes also discuss continuum limits (double-scaling, multiscaling) that bridge discrete matrix models to continuum theories like KdV/topological gravity, emphasizing the integrable structure’s role in preserving solvability across limits and model classes.

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.