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

TL;DR

This paper proves the rationality of certain $C$-graded vertex algebras using Zhu algebra techniques and applies these results to Weyl vertex algebras with conformal flow, showing rationality for specific non-integer parameters.

Contribution

It establishes criteria for rationality of $C$-graded vertex algebras and applies these to Weyl vertex algebras under conformal flow, including non-integer graded cases.

Findings

Rationality of finitely $Omega$-generated $C$-graded vertex algebras under grading conditions.

Identification of conditions under which Weyl vertex algebras are rational for non-integer parameters.

Extension of rationality results to higher-rank $C$-graded Weyl vertex algebras.

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.