(Joint with Lucio Simone Cirio, Max Planck Institute for Mathematics)

Abstract: In the context of higher gauge theory and 2-connections on 2-bundles, we categorify the Knizhnik-Zamolodchikov connection in the configuration space of n particles in the complex plane, by means of a differential crossed module of (totally symmetric) horizontal 2-chord diagrams, categorifying the 4-term relation.

We discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, inside which we formulate the notion of an infinitesimal 2-R matrix, an infinitesimal counterpart of some of the relations satisfied by braid cobordisms.

We present several open problems.

• Talk based on the preprint : arXiv:1106.0042v2 [hep-th]