χ-Crossed ribbon categories (Bangxin Wang)

Orateur : Bangxin Wang

Lieu : salle des Séminaires M3

Résumé :

We introduce the notion of χ-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, ω), where M is a 3-manifold, T ⊂ M is an embedded ribbon graph, and ω ∈ H^1(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 a homotopy QFT whose algebraic input consists of G-crossed ribbon categories, expected to recover the BCGP theory. This is 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, the pair (G,H) must form a crossed module. We present two constructions of such categories: one arising from representations of suitable Hopf algebras, and another from twisted-local modules in a given ribbon category. An explicit example is provided by variations of quantum sl2. This part is joint work with A. Gainutdinov and I. Runkel.

If time permits, I will also discuss how our notion can be realized as E2-algebras in 2Fun(G//H), and its connection to 2-group (crossed) extensions of (braided )tensor categories. This is ongoing work with H. Xu