On rationality for $C_2$-cofinite vertex operator algebras

Abstract

Let $V$ be an $\mathbb{N}$-graded, simple, self-contragredient, $C_2$-cofinite vertex operator algebra. We show that if the $S$-transformation of the character of $V$ is a linear combination of characters of $V$-modules, then the category $\mathcal{C}$ of grading-restricted generalized $V$-modules is a rigid tensor category. We further show, without any assumption on the character of $V$ but assuming that $\mathcal{C}$ is rigid, that $\mathcal{C}$ is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of $V$ is semisimple, then $\mathcal{C}$ is semisimple and thus $V$ is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to $V$. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that $C_2$-cofinite affine $W$-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such $W$-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of $C_2$-cofiniteness for the coset. That is, given a vertex operator algebra inclusion $U\otimes V\hookrightarrow A$ with $A$, $U$ strongly rational and $U$, $V$ a pair of mutual commutant subalgebras in $A$, we show that $V$ is also strongly rational provided it is $C_2$-cofinite.

78 pages, final version to appear in Cambridge Journal of Mathematics, some major changes from previous version: Appendix A from previous version and proofs of some results already in the literature have been removed for brevity; Section 3.1 from previous version has been moved and combined with material from Section 5.2 to form new Section 5.3; new Section 1.1 on previous work has been added

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Cited by in corpus (1)

• Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras