On rationality for $C_2$-cofinite vertex operator algebras

Abstract

Let V be an N-graded, simple, self-contragredient, C2-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 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 C is rigid, that 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 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 C2-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 C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U ⊗ V ֒→ 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 C2-cofinite. ROBERT MCRAE 6. Applications 61 6.1. Rationality for C 2 -cofinite W -algebras 61 6.2. Rationality for C 2 -cofinite cosets 66 Appendix A. Differential equations 69 Appendix B. The assumption in Theorem 6.4 for odd nilpotent orbits 74 B.1. Classical types 74 B.2. Exceptional types 76 References 82

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

“…Since all the coefficient rational functions a n (z; λ, α 0 − λ, λ) in this theorem are analytic at z = 0, the differential equation has a regular singular point at z = 0. We will apply the theorem to Y λ 1 = Ω 0 (Y F λ ) and Y λ 2 = E λ to prove Theorem 5.2; we will also use the following result from the theory of ordinary differential equations (see for example [McR2, Appendix A] for a proof):…”

On the tensor structure of modules for compact orbifold vertex operator algebras

“…It follows from [Hu3] (see also the discussion in [CM,Section 3.1] and [McR3,Lemma 2.10]) that the category Rep(V ) of all grading-restricted generalized V -modules is a finite abelian category and a braided tensor category. In particular, if V is N-graded and C 2 -cofinite, then Rep(V ) is a finite braided ribbon category if and only if it is rigid.…”

“…Suppose a simple U -module M ∈ Ob(U ) has a projective cover p M : P M ։ M in U . It is not difficult to show that P M is singly-generated by any m ∈ P M \ Ker p M (see for example [McR3,Lemma 5.13]). That is, Ker p M is the unique maximal proper submodule of P M .…”

“…If V is an N-graded C 2 -cofinite vertex operator algebra, then the category Rep(V ) of grading-restricted generalized V -modules is a finite abelian category and a braided tensor category [Hu3], and it is easy to see from the spanning set of [Mi1,Lemma 2.4] that Rep(V ) = C 1 V . If V is also simple and self-contragredient, and Rep(V ) is a rigid tensor category, then Rep(V ) is a (generally non-semisimple) modular tensor category [McR3,Main Theorem 1]. Thus Theorem 4.13 specializes to C 2 -cofinite vertex operator algebras as follows, where we use induction on n to generalize from the tensor product of two vertex operator algebras to the tensor product of finitely many:…”

On the nilpotent orbits arising from admissible affine vertex algebras

“…† The rationality of exceptional 𝑊-algebras has been recently proved in full generality by McRae[32].…”