Generalized $ \widetilde{W} $ algebras

TL;DR

This paper explores a new family of infinite-dimensional $ ilde{W} $ algebras linked to elements of a commutative subalgebra of $ W_{1+ ablafty} $, providing proofs, examples, and applications to matrix models.

Contribution

It offers a comprehensive account and proofs of the generalized $ ilde{W} $ algebras and demonstrates their use in deriving Ward identities for WLZZ matrix models.

Findings

Derived Ward identities for specific matrix models

Expanded $ W $-operators in terms of infinite variables

Provided illustrative examples of the algebraic structures

Abstract

Recently, a new generalized family of infinite-dimensional $ \widetilde{W} $ algebras, each associated with a particular element of a commutative subalgebra of the $ W_{1+\infty} $ algebra, was described. This paper provides a comprehensive account of the aforementioned association, accompanied by the requisite proofs and illustrative examples. This approach allows a derivation of Ward identities for selected WLZZ matrix models and the expansion of corresponding $ W $-operators in terms of an infinite set of variables $ p_k $.