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

TL;DR

This paper explores the structure of $W_{1+ abla}$ and $ ilde{W}$ algebras, revealing their connections to integrable models, matrix models, and Ward identities, and providing new definitions and realizations of these algebras.

Contribution

It introduces a generalized definition of $ ilde{W}$ algebras associated with $W_{1+ abla}$ algebra elements, linking them to integrable systems and matrix models.

Findings

Generalized $ ilde{W}$ algebras are associated with all integer slope rays in $W_{1+ abla}$.

The simplest commutative subalgebra relates to rational Calogero Hamiltonians.

Positive integer rays correspond to Ward identities in WLZZ matrix models.

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 $\widetilde W$ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the $\widetilde W$ algebras studied earlier. We suggest a definition of the generalized $\widetilde W$ 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 $\widetilde W$ 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.