QM Research Seminar: x-Crossed ribbon categories

Speaker: Bangxin Wang (​University of Hamburg)

Abstract: 
​​​​​​​​We introduce the notion of .-crossed ribbon categories, motivated by constructions in TQFT. There are two well-known non-semisimple 3d TQFTs: BCGP and DGGPR. The BCGP theory assigns invariants to triples (M, T, w), where M is a 3-manifold, T c M is an embedded ribbon graph, and w e H1(M\T;G). The DGGPR construction produces invariants for pairs (M, T). A natural question is whether one can enrich the DGGPR construction by including cohomological data. This leads to the theory of homotopy QFT whose algebraic inputs are G-crossed ribbon categories, expected to recover the BCGP theory. This is an ongoing project with A. Gainutdinov, N. Geer, B. Patureau, and I. Runkel. We generalize the notion of a G-crossed ribbon category by allowing the grading group G and the action group H to be distinct. To make sense of a crossed braiding, it turns out that the pair (G,H) should form a crossed module x. We present two constructions of such categories and prove several classification results. This part is joint work with A. Gainutdinov and I. Runkel.