Matrix model partition function by a single constraint

TL;DR

This paper shows that a single W-constraint can uniquely determine the partition functions of various matrix models, simplifying previous multi-requirement approaches and revealing deep connections to integrability.

Contribution

It introduces a single W-constraint framework that replaces multiple previous conditions for defining matrix model partition functions, applicable to a broad class of models.

Findings

Single W-constraint suffices to specify partition functions.

Equivalent to W-representation and related to superintegrability.

Applicable to models with higher-degree potentials and larger W-algebras.

Abstract

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of $\hat w$-operators. In this letter, we demonstrate that even more is true: a {\it single} $w$-constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of {\it two} requirements: either a string equation imposed on the KP/Toda $\tau$-function or a pair of Virasoro generators. This mysterious {\it single}-entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to {\it super}\,integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with the potential of higher degree, when the constraint algebra is a larger $W$-algebra, and neither W-representation nor superintegrability are understood well enough.