The n-Category Café

I have only just arrived, by night train over the Alps from Venezia . Am a bit tired. But now Chris Schommer-Pries is talking about a nice result that he told me about a few weeks back in Lisbon.

I don’t think there is anything in print, but the abstract on his webpage is slightly more detailed than the talk abstract:

The Structure of Fusion Categories via 3D TQFTs

Fusion categories arise in several areas of mathematics and physics - conformal field theory, operator algebras, representation theory of quantum groups, and others. They have a rich and fascinating structure. In this recent work, joint with Christopher Douglas and Noah Snyder, which ties this structure to the structure of 3D TQFTs.

In particular we show that fusion categories are fully-dualizable objects in a certain natural 3-category and identify the induced $O(3)$-action on the `space’ of fusion categories, as predicted by the cobordism hypothesis. In light of Hopkins’ and Lurie’s work on the cobordism hypothesis, this provides a fully local 3D TQFT for arbitrary fusion categories. Moreover understanding various homotopy fixed point spaces uncovers how many familiar structures from the theory of fusion categories are given a natural explanation from the point of view of 3D TQFTs.

• Dualizable Tensor Categories I: The Structure of Fusion Categories via 3D TQFTs. In preparation.
• Dualizable Tensor Categories II: Partial TQFTs and finite tensor categories. In preparation.
• Dualizable Tensor Categories III: Fully Dualizable Tensor Categories.

So the idea is: there are plenty of component-based constructions in the literature that produce 3-dimensional TQFTs from the data of a fusion category. So far no particularly deeper reason for why all these constructions exist for fusion categories has been known. The theorem now says: simply because fusion categories are, in their natural ambient 3-category, fully dualizable objects: precisely the kind of objects that the general abstract cobordism hypothesis says arise as the value of extended QFTs on the point and indeed fully characterize these extended QFTs.

More is true: there are slight variants of the cobordism hypothesis, depending on which extra topological structure the cobordisms are equipped with, for instance: framing, or orientation. There are also variants of notions of fusion categories equipped with extra property and structure: pivotal categories and spherical categories (though there is a conjecture saying that these are all the same). Chris et al.’s theorem says that under the above correspondence, these extra structures match on both sides.

In a little while, more details are here.

Now Alan Carey is speaking, about (cocycle models for) twisted differential K-theory (see the reference linked to there) and twisted differential string structures.

The first of these two $n$Lab entries needs expansion…

Later today there was also a regular seminar talk at the math institute:

Giorgio Trentinaglia spoke about his work on Tannaka duality for Lie groupoids.

He obtains a Tannaka duality for proper Lie groupoids over a fixed manifold of objects by generalizing from representations on the stack of smooth vector bundles to representation on a notion of vector bundles whose fiber dimensions may jump, which he calls “smooth Euclidean fields”.

For the analogous situation not in the differential geometric but in the algebraic context there is a general Tannaka duality for algebraic stacks, which of course works in terms of representations on quasicoherent sheaves. Makes me wonder if that construction and theorem cannot be moved to the differential geometric context. But I haven’t really thought about it.