On rationality of $\mathbb{C}$-graded vertex algebras and applications to Weyl vertex algebras under conformal flow

Abstract

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $\Omega$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $\Omega$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element ${\omega }_{\mu }$ parameterized by $\mu \in \mathbb{C}$, and prove that for certain non-integer values of $\mu$, these vertex algebras, which are non-integer graded, are rational, with one dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate $\mathbb{C}$-graded Weyl vertex algebras of arbitrary ranks.

arXiv commentFinal version. Typos corrected and bibliography updated. We thank the referee for their comments and suggestions. To appear in Journal of Mathematical Physics

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Cited by in corpus (2)

• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• The level two Zhu algebra for the Heisenberg vertex operator algebra