Simple Toroidal Vertex Algebras and Their Irreducible Modules

Abstract

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 $L_r(\hat{\mathfrak{g}})=\hat{\mathfrak{g}}\otimes L_r$, where $\hat{\mathfrak{g}}$ is an untwisted affine Lie algebra and $L_r=$\mathbb{C}[t_{1}^{\pm 1},\ldots,t_{r}^{\pm 1}]$.Wefirstconstructan$(r+1)$-toroidalvertexalgebra$V(T,0)$andshowthatthecategoryofrestricted$L_{r}(\hat{\frak{g}})$-modulesiscanonicallyisomorphictothatof$V(T,0)$-modules.Let$c$denotethestandardcentralelementof$\hat{\frak{g}}$andset$S_c=U(L_r(\mathbb{C}c))$.Wefurthermorestudyadistinguishedsubalgebraof$V(T,0)$,denotedby$V(S_c,0)$.Weshowthat(graded)simplequotienttoroidalvertexalgebrasof$V(S_c,0)$areparametrizedbya$\mathbb{Z}^r$-gradedringhomomorphism$ψ:S_c\rightarrow L_r$suchthatIm$ψ$isa$\mathbb{Z}^r$-gradedsimple$S_c$-module.Denoteby$L(ψ,0}$thesimple$(r+1)$-toroidalvertexalgebraof$V(S_c,0)$associatedto$ψ$.Wedetermineforwhich$ψ$,$L(ψ,0)$isanintegrable$L_{r}(\hat{\frak{g}})$-moduleandwethenclassifyirreducible$L(ψ,0)$-modulesforsucha$ψ$. For our need, we also obtain various general results.

50 pages