Commutative families in $W_{\infty }$, integrable many-body systems and hypergeometric $\tau$-functions

Abstract

We explain that the set of new integrable systems generalizing the Calogero family and implied by the study of WLZZ models, which was described in arXiv:2303.05273, is only the tip of the iceberg. We provide its wide generalization and explain that it is related to commutative subalgebras (Hamiltonians) of the $W_{1+\infty }$ algebra. We construct many such subalgebras and explain how they look in various representations. We start from the even simpler $w_{\infty }$ contraction, then proceed to the one-body representation in terms of differential operators on a circle, further generalizing to matrices and in their eigenvalues, in finally to the bosonic representation in terms of time-variables. Moreover, we explain that some of the subalgebras survive the $\beta$-deformation, an intermediate step from $W_{1+\infty }$ to the affine Yangian. The very explicit formulas for the corresponding Hamiltonians in these cases are provided. Integrable many-body systems generalizing the rational Calogero model arise in the representation in terms of eigenvalues. Each element of $W_{1+\infty }$ algebra gives rise to KP/Toda $\tau$-functions. The hidden symmetry given by the families of commuting Hamiltonians is in charge of the special, (skew) hypergeometric $\tau$-functions among these.

43 pages

References in corpus (10)

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• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• Interpolating Matrix Models for WLZZ series
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Many-body integrable systems implied by WLZZ models
• New Insights into Superintegrability from Unitary Matrix Models
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Commutative subalgebras from Serre relations
• $(q,t)$-deformed (skew) Hurwitz $\tau$-functions
• Spectral curves and $W$-representations of matrix models

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• Generalized $\tilde{W}$ algebras
• Higher order constraints for the ($\beta$-deformed) Hermitian matrix models
• CMM formula as superintegrability property of unitary model
• Towards mixed phase correlators in monomial matrix models
• Weyl Mutations in Quiver Yangians