$Z^N$-graded Lie algebras: Fock representations and reducibility conditions

Abstract

Manifestly consistent Fock representations of non-central (but ``core-central'') extensions of the $Z^N$-graded algebras of functions and vector fields on the $N$-dimensional torus $T^N$ are constructed by a kind of renormalization procedure. These modules are of lowest-energy type, but the energy is not a linear function of the momentum. Modulo a technical assumption, reducibility conditions are proved for the extension of $vect(T^N)$, analogous to the discrete series of Virasoro representations.