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 singlew-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 two requirements: either a string equation imposed on the KP/Toda tau -function or a pair of Virasoro generators. This mysterious 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 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 potential of higher degree, when the constraint algebra is a larger W-algebra, and neither W-representation, nor superintegrability are understood well enough.
Highlights
As old as the formal theory of matrix models is the puzzle of interplay betweenintegrability and Ward identities
We report a far more powerful statement: for a matrix model partition function that satisfies W -constraints, one can consider instead w-constraints that are linear combinations of all these W -constraints multiplied by time variables, and the lowest of these w-constraints is enough to define the partition function uniquely, neither string equation, nor integrability is needed in this approach
We reported a spectacular property of matrix models: their exact partition functions are unambiguously determined by a single equation, which appears to be a w-substitute of the string equation
Summary
As old as the formal theory of matrix models is the puzzle of interplay between (super)integrability and Ward identities. 4, we consider a series of examples, which includes Hermitian matrix model (where we reproduce the recent result of [26]), and matrix models depending on external matrix: Kontsevich model, Brézin– Gross–Witten model, and, the generalized Kontsevich model (GKM), where the constraint algebra is extended to the W -algebra. Despite this extension, even the GKM partition function is unambiguously encoded by a single equation
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