Simple Toroidal Vertex Algebras and Their Irreducible Modules

In this paper, we continue the study on toroidal vertex algebras initiated in \cite{LTW}, to study concrete toroidal vertex algebras associated to toroidal Lie algebra $L_{r}(\hat{\frak{g}})=\hat{\frak{g}}\otimes L_r$, where $\hat{\frak{g}}$ is an untwisted affine Lie algebra and $L_r=$\mathbb{C}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$. We first construct an$(r+1)$-toroidal vertex algebra$V(T,0)$and show that the category of restricted$L_{r}(\hat{\frak{g}})$-modules is canonically isomorphic to that of$V(T,0)$-modules.Let$c$denote the standard central element of$\hat{\frak{g}}$and set$S_c=U(L_r(\mathbb{C}c))$. We furthermore study a distinguished subalgebra of$V(T,0)$, denoted by$V(S_c,0)$. We show that (graded) simple quotient toroidal vertex algebras of$V(S_c,0)$are parametrized by a$\mathbb{Z}^r$-graded ring homomorphism$ψ:S_c\rightarrow L_r$such that Im$ψ$is a$\mathbb{Z}^r$-graded simple$S_c$-module. Denote by$L(ψ,0}$the simple$(r+1)$-toroidal vertex algebra of$V(S_c,0)$associated to$ψ$. We determine for which$ψ$,$L(ψ,0)$is an integrable$L_{r}(\hat{\frak{g}})$-module and we then classify irreducible$L(ψ,0)$-modules for such a$ψ$. For our need, we also obtain various general results.

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