$W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities

Abstract

It was demonstrated recently that the $W_{1+\infty }$ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized $\tilde{W}$ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the $\tilde{W}$ algebras studied earlier. We suggest a definition of the generalized $\tilde{W}$ algebra as differential operators in variables $p_k$ basing on the matrix realization of the $W_{1+\infty }$ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the $W_{1+\infty }$ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with $\tilde{W}$ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric $\tau$-functions corresponding to the completed cycles.

arXiv comment11 pages, 1 figure

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Cited by in corpus (5)

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• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Generalized $\tilde{W}$ algebras
• Phases in WLZZ Matrix Models