Faces of matrix models

TL;DR

This paper explores the multifaceted nature of matrix models in mathematical physics, highlighting their various representations and their foundational role as special functions connecting different theoretical frameworks.

Contribution

It provides a comprehensive overview of the diverse descriptions and roles of matrix models, emphasizing their central importance in modern mathematical physics.

Findings

Matrix models can be described as integrals, equations, tau-functions, and geometric prepotentials.

They act as fundamental special functions linking various mathematical physics concepts.

Matrix models serve as a paradigm for unifying different theoretical approaches.

Abstract

Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as tau-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of W-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic "special functions" to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.