Matrix model partition function by a single constraint

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.

arXiv comment11 pages

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Cited by in corpus (10)

• New Insights into Superintegrability from Unitary Matrix Models
• Non-Abelian W-representation for GKM
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from $W$-operators
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• Spectral curves and $W$-representations of matrix models
• Superintegrability of matrix Student's distribution
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions
• The ordered exponential representation of GKM using the $W_{1+\infty }$ operator
• Integrability and Matrix Models