W1+∞ and W˜ algebras, and Ward identities

It was demonstrated recently that the W1+∞ 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 W˜ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the W˜ algebras studied earlier. We suggest a definition of the generalized W˜ algebra as differential operators in variables pk basing on the matrix realization of the W1+∞ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W1+∞ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with 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 τ-functions corresponding to the completed cycles.