Seminars & Colloquia Calendar

Abstract One of the most important results in vertex operator algebras is Huang's result that if the module category of a positive-energy, self-contragredient, C_2-cofinite vertex operator algebra V is semisimple, then it is a modular tensor category. Huang also showed that if the V-module category is not necessarily semisimple, then it is at least a braided tensor category. In this talk, I will discuss my recent results on C_2-cofinite vertex operator algebras in the not-a priori-semisimple setting: First, if the modular S-transformation of the character of V is a linear combination of characters of V-modules, then the tensor category of V-modules is rigid, that is, contragredients of V-modules are duals in the tensor-categorical sense. Second, if the tensor category of V-modules is assumed to be rigid, then its braiding is non-degenerate, that is, the V-module category is a not-necessarily-semisimple analogue of a modular tensor category. Finally, if Zhu's associative algebra A(V) is semisimple, then the V-module category is in fact semisimple, so that Huang's theorem applies. Thanks to a recent Zhu algebra theorem of Arakawa and van Ekeren, this last result proves the conjecture of Kac-Wakimoto and Arakawa that all "exceptional" simple W-algebras have semisimple representation theory.