Simple Toroidal Vertex Algebras and Their Irreducible Modules

TL;DR

This paper constructs and classifies simple toroidal vertex algebras associated with toroidal Lie algebras, establishing a correspondence between modules and algebra homomorphisms, and determining conditions for integrability.

Contribution

It introduces a new construction of toroidal vertex algebras linked to toroidal Lie algebras and classifies their irreducible modules, extending the understanding of their structure.

Findings

Constructed an (r+1)-toroidal vertex algebra $V(T,0)$.

Classified simple quotient toroidal vertex algebras parametrized by graded ring homomorphisms.

Determined conditions under which modules are integrable and classified 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 $L_{r}(\hat{\frak{g}})=\hat{\frak{g}}\otimes L_r$, 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 $\psi:S_c\rightarrow L_r$ such that Im$\psi$ is a $\mathbb{Z}^r$-graded simple $S_c$-module. Denote by $L(\psi,0}$ the simple $(r+1)$-toroidal vertex algebra of $V(S_c,0)$ associated to $\psi$. We determine for which $\psi$, $L(\psi,0)$ is an integrable $L_{r}(\hat{\frak{g}})$-module and we then classify irreducible $L(\psi,0)$-modules for such a $\psi$. For our need, we also obtain various general results.