Representations of toroidal extended affine Lie algebras

We show that the representation theory for the toroidal extended affine Lie algebra is controlled by a VOA which is a tensor product of four VOAs: a sub-VOA of a hyperbolic lattice VOA, affine and VOAs and a Virasoro VOA. A tensor product of irreducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also show that for N=12, becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0.