On Zhu's Associative Algebra as a Tool in the Representation Theory of Vertex Operator Algebras

TL;DR

This paper explores the use of Zhu's associative algebra to classify and construct representations of vertex operator algebras, demonstrating explicit calculations for lattice theories and revealing new non-unitary representations.

Contribution

It introduces a novel approach using Zhu's algebra for classification and construction of VOAs representations, including explicit examples and new non-unitary representations.

Findings

Explicit calculation of Zhu's algebra for lattice theories

Identification of new non-unitary representations of Z_3-invariant subtheories

Demonstration of Zhu's algebra as a tool for classification and construction

Abstract

We describe an approach to classify (meromorphic) representations of a given vertex operator algebra by calculating Zhu's algebra explicitly. We demonstrate this for FKS lattice theories and subtheories corresponding to the Z_2 reflection twist and the Z_3 twist. Our work is mainly offering a novel uniqueness tool, but, as shown in the Z_3 case, it can also be used to extract enough information to construct new representations. We prove the existence and some properties of a new non-unitary representation of the Z_3-invariant subtheory of the (two dimensional) Heisenberg algebra.