Partial classification of modules for Lie-algebra of diffeomorphisms of d-dimensional torus

Abstract

We consider the Lie-algebra of the group of diffeomorphisms of a d-dimensional torus which is also known to be the algebra of derivations on a Laurent polynomial ring A in d commuting variables denoted by Der A. The universal central extension of Der A for d=1 is the so-called Virasoro algebra. The connection between Virasoro algebra and physics is well known. See, for example, the book on Conformal Field Theory by Di Francesco, Mathieu, and Senechal. In this paper we classify (A, Der A) modules which are irreducible and have finite dimensional weight spaces. Earlier Larsson constructed a large class of modules, the so-called tensor fields, based on gld modules which are also A modules. We prove that they exhaust all (A, Der A) irreducible modules.