Generalized W∞ Higher-Spin Algebras and Symbolic Calculus on Flag Manifolds

Abstract

We study a new class of infinite-dimensional Lie algebras W1(N+,N−) generalizing the standard W1 algebra, viewed as a tensor operator algebra of SU(1, 1) in a grouptheoretic framework. Here we interpret W1(N+,N−) either as an infinite continuation of the pseudo-unitary symmetry U(N+,N−), or as a “higher-U(N+,N−)-spin extension” of the diffeomorphism algebra diff(N+,N−) of the N = N++N− torus U(1)N. We highlight this higher-spin structure of W1(N+,N−) by developing the representation theory of U(N+,N−) (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 W1 symmetries and algebras of symbols of U(N+,N−)-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.