ct 2 00 9 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 V + Hyp of a hyperbolic lattice VOA, affine ˙ g and sl N VOAs and a Virasoro VOA. A tensor product of irre-ducible 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, V + Hyp becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0. In this paper we study representations of toroidal extended affine Lie algebras using recently developed representation theory for the full toroidal Lie algebras [B3]. Extended affine Lie algebras (EALAs) have been extensively studied during the last decade (see [N], [ABFP], [AABGP] and references therein). The main features of an extended affine Lie algebra is that it is graded by a finite root system and possesses a non-degenerate symmetric invariant bilinear form. The construction of toroidal Lie algebras parallels one for affine algebras. We start with the Lie algebra of maps from an N + 1-dimensional torus into a finite-dimensional simple Lie algebra ˙ g. This multi-loop algebra may be written as a tensor product R ⊗ ˙ g of the algebra R of Laurent polynomials in N + 1 variables with ˙ g. Next, we take the universal central extension R ⊗ ˙ g ⊕ K of this multi-loop algebra and add a Lie algebra of vector fields on the torus, possibly twisted with a 2-cocycle (see section 1 for details). If we add all vector fields, we get the full toroidal Lie algebra. However, the full toroidal Lie algebra does not possess a non-degenerate invariant form, whereas its subalgebra with the divergence zero vector fields g div = (R ⊗ ˙ g) ⊕ K ⊕ D div does. We call this last algebra the toroidal extended affine Lie algebra. 1 The representation theory of toroidal Lie algebras is best described in the framework of vertex operator algebras (VOAs). We prove in this paper that the vertex operator algebra that controls the representation theory of g div is a tensor product of an affine ˙ g VOA, a sub-VOA of a hyperbolic lattice VOA V + Hyp , affine sl N VOA and a Virasoro VOA. By taking a tensor product of irreducible modules for …