3-Dimensional TQFTs from non-semisimple modular categories

Constructing TQFTs from Categories of Sheaves

In this talk, I’ll try to synthesise some of the ideas from the earlier talks in order to describe an example of a fully extended 2d TQFT. This TQFT will be built from geometric data following the “bottom-up” approach of the cobordism hypothesis. That is, we’ll choose a category of geometric objects (quashicoherent sheaves on a stack) to assign to the point, and explore some of the consequences...

Finitely semisimple spherical categories and modular categories are self - dual

We show that every essentially small finitely semisimple k-linear additive spherical category in which k = End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as...

Two Dimensional Tqfts

On non-semisimple fusion rules and tensor categories

Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be block-dia...

On Certain Integral Tensor Categories and Integral Tqfts

We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.