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

Klaus Lucke

hep-tharXiv:hep-th/9605020

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 Z2 reflection twist and the Z3 twist. Our work is mainly offering a novel uniqueness tool, but, as shown in the Z3 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 Z3-invariant subtheory of the (two dimensional) Heisenberg algebra.

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