Exceptional Vertex Operator Algebras and the Virasoro Algebra

TL;DR

This paper investigates special vertex operator algebras where certain Casimir vectors are Virasoro descendants, deriving constraints and differential equations for their partition functions, with examples including Wess-Zumino-Witten models and the Moonshine Module.

Contribution

It introduces the concept of exceptional vertex operator algebras with Casimir vectors as Virasoro descendants and derives new differential equations and structural insights for these theories.

Findings

Derived explicit differential equations for partition functions.

Analyzed the structure of tensor products of exceptional Lie and Griess algebras.

Connected the theory to Wess-Zumino-Witten models and Moonshine Module.

Abstract

We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Examples include the Wess-Zumino-Witten model for Deligne's exceptional Lie algebras and the Moonshine Module. We partially verify the irreducible decomposition of the tensor product of Deligne's exceptional Lie algebras and consider the possibility of similar decompositions for tensor products of the Griess algebra. We briefly discuss some conjectured extremal vertex operator algebras arising in Witten's recent work on three dimensional black holes.