Zhu's algebra and the C2-algebra of a classically free vertex operator algebra

Abstract

We prove that, for a classically free vertex algebra, the C2-algebra is naturally isomorphic to the associated graded algebra of Zhu's algebra. As an application, we show that non-trivial holomorphic VOAs of CFT type are not classically free. This includes the affine vertex operator algebra of E8 at level one, as well as the Moonshine VOA.

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