Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V . We employ tensor categories to study untwisted (also called local) Amodules in C, using results of Huang-Kirillov-Lepowsky that show that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang.Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V -modules to A-modules is a vertex tensor functor. Two applications are given:First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z-graded components, respectively.Second, we analyze parafermionic cosets C = Com(VL, V ) where L is a positive definite even lattice and V is regular. If the vertex tensor category of either V -modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples. * T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460).

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

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We prove a general mirror duality theorem for a subalgebra U of a simple vertex operator algebra A and its coset V = ComA(U ), under the assumption that A is a semisimple U ⊗ V -module. More specifically, we assume that A ∼ = i∈I Ui ⊗ Vi as a U ⊗ V -module, where the U -modules Ui are simple and distinct and are objects of a semisimple braided ribbon category of U -modules, and the V -modules Vi are semisimple and contained in a (not necessarily rigid) braided tensor category of V -modules. We also assume that U and V form a dual pair in A, so that U is the coset ComA(V ). Under these conditions, we show that there is a braid-reversing tensor equivalence τ : UA → VA, where UA is the semisimple category of U -modules with simple objects Ui, i ∈ I, and VA is the category of V -modules whose objects are finite direct sums of the Vi. In particular, the V -modules Vi are simple and distinct, and VA is a rigid tensor category.

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Let O25 be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge 25 whose composition factors are the irreducible quotients of reducible Verma modules. We show that O25 is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl2-modules. We also show that this sl2type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge 1. As an application, we obtain a new conformal vertex algebra of central charge 26 as an extension of the tensor product of Virasoro vertex operator algebras at central charges 1 and 25, analogous to the modified regular representations of the Virasoro algebra constructed previously from Virasoro vertex operator algebras at generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.