Vertex operator algebras associated to modified regular representations of affine Lie algebras

Abstract

Let $G$ be a simple complex Lie group with Lie algebra $\mf g$ and let $\af$ be the affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of $\mathbb{N}$-graded vertex operator algebras associated to $\mf g$. They are $\af \oplus \af$-modules of dual levels $k, \bar k \notin \Q$ in the sense that $k+\bar{k}=-2h^{\vee }$ where $h^{\vee }$ is the dual Coxeter number of $\mf g$. Its conformal weight 0 component is the algebra of regular functions on $G$. This family of vertex operator algebras were previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that the vertex envelope of the vertex algebroid associated to $G$ and level $k$ is isomorphic to the vertex operator algebra we constructed above when $k$ is irrational. The case of integral central charges is also discussed.

29 pages

Cited by in corpus (2)

• Algebras of twisted chiral differential operators and affine localization of $\mathfrak{g}$-modules
• On the semi-regular module and vertex operator algebras