Generalized W(infinity) Higher-Spin Algebras and Symbolic Calculus on Flag Manifolds

TL;DR

This paper introduces a new class of infinite-dimensional Lie algebras W_(p,q) that generalize W_, exploring their structure, representation theory, and geometric aspects on flag manifolds, with potential applications in gauge theories and sigma models.

Contribution

It develops the theory of W_(p,q) algebras, including their representations, geometric structures, and connections to symbol calculus on flag manifolds, extending the understanding of higher-spin symmetries.

Findings

Defined W_(p,q) as a tensor operator algebra of U(p,q)

Calculated higher-spin representations and coherent states

Established geometric structures on flag manifolds and links to symbol calculus

Abstract

We study a new class of infinite-dimensional Lie algebras W_\infty(p,q) generalizing the standard W_\infty algebra, viewed as a tensor operator algebra of SU(1,1) in a group-theoretic framework. Here we interpret W_\infty(p,q) either as an infinite continuation of the pseudo-unitary symmetry U(p,q), or as a "higher-U(p,q)-spin extension" of the diffeomorphism algebra diff(p,q) of the N=p+q torus U(1)^N. We highlight this higher-spin structure of W_\infty(p,q) by developing the representation theory of U(p,q) (discrete series), calculating higher-spin representations, coherent states and deriving K\"ahler structures on flag manifolds. They are essential ingredients to define operator symbols and to infer a geometric pathway between these generalized W_\infty symmetries and algebras of symbols of U(p,q)-tensor operators. Classical limits (Poisson brackets on flag manifolds) and quantum (Moyal) deformations are also discussed. As potential applications, we comment on the formulation of diffeomorphism-invariant gauge field theories, like gauge theories of higher-extended objects, and non-linear sigma models on flag manifolds.