Superintegrability of Kontsevich matrix model

TL;DR

This paper demonstrates superintegrability in the Kontsevich matrix model and related models, showing explicit calculability of averages of characters, extending known results to new models with external fields.

Contribution

It introduces superintegrability in the cubic Kontsevich model and a complex model with external fields, expanding the class of models exhibiting this property.

Findings

Superintegrability observed in the Kontsevich model and related models.

Explicit formulas for character averages in new models.

Extension of superintegrability to models with external fields.

Abstract

Many eigenvalue matrix models possess a peculiar basis of observables which have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb potentials are distinguished among other central potentials, and we call it superintegrability. Aa a peculiarity of matrix models, the relevant basis is formed by the Schur polynomials (characters) and their generalizations, and superintegrability looks like a property $<character>\,\sim character$. This is already known to happen in the most important cases of Hermitian, unitary, and complex matrix models. Here we add two more examples of principal importance, where the model depends on external fields: a special version of the complex model and the cubic Kontsevich model. In the former case, straightforward is a generalization to the complex tensor model. In the latter case, the relevant characters are the celebrated $Q$ Schur functions appearing in the description of spin Hurwitz numbers and other related contexts.