Energy-momentum tensor for the toroidal Lie algebras

Abstract

We construct vertex operator representations for the full (N+1)-toroidal Lie algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl(N) VOA and a twisted Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules for the toroidal Lie algebra g. We also construct irreducible modules for an important subalgebra gdiv of the toroidal Lie algebra that corresponds to the divergence free vector fields. This subalgebra carries a non-degenerate invariant bilinear form. The VOA that controls the representation theory of gdiv is a tensor product of an affine VOA Vaff(c) at level c, a sub-VOA of a hyperbolic lattice VOA, affine sl(N) VOA and a Virasoro VOA at level cvir with the following condition on the central charges: 2(N+1) + rank Vaff(c) + cvir = 26.