Simple toroidal vertex algebras and their irreducible modules

In this paper, we continue the study on toroidal vertex algebras initiated in [15], to study concrete toroidal vertex algebras associated to toroidal Lie algebra Lr(gˆ)=gˆ⊗Lr, where gˆ is an untwisted affine Lie algebra and Lr=C[t1±1,…,tr±1]. We first construct an (r+1)(r+1)-toroidal vertex algebra V(T,0)V(T,0) and show that the category of restricted Lr(gˆ)-modules is canonically isomorphic to that of V(T,0)V(T,0)-modules. Let cc denote the standard central element of gˆ and set Sc=U(Lr(Cc))Sc=U(Lr(Cc)). We furthermore study a distinguished subalgebra of V(T,0)V(T,0), denoted by V(Sc,0)V(Sc,0). We show that (graded) simple quotient toroidal vertex algebras of V(Sc,0)V(Sc,0) are parametrized by a ZrZr-graded ring homomorphism ψ:Sc→Lrψ:Sc→Lr such that Imψ   is a ZrZr-graded simple ScSc-module. Denote by L(ψ,0)L(ψ,0) the simple quotient (r+1)(r+1)-toroidal vertex algebra of V(Sc,0)V(Sc,0) associated to ψ. We determine for which ψ  , L(ψ,0)L(ψ,0) is an integrable Lr(gˆ)-module and we then classify irreducible L(ψ,0)L(ψ,0)-modules for such a ψ. For our need, we also obtain various general results.