2011/06/30 by Lucio S. Cirio, João Faria Martins · 1 citation
Physics and Astronomy · Mathematics · #hep-th #math.CT #math.QA #msc:16T25 #msc:20F36 #msc:18D05 #msc:17B37 #msc:53C29 #msc:57Q45
paper · pdf · doi:10.1016/j.difgeo.2012.03.004
published as Differential Geometry and its Applications 30 (2012), 238-261 · 30 pages, 2 figures; v3: final version to be published in Differential Geometry and its Applications
arxiv created 2012/05/16 · arxiv updated 2017/05/23
In the context of higher gauge theory, we construct a flat and fake flat 2-connection, in the configuration space of n particles in the complex plane, categorifying the Knizhnik-Zamolodchikov connection. To this end, we define the differential crossed module of horizontal 2-chord diagrams, categorifying the Lie algebra of horizontal chord diagrams in a set of n parallel copies of the interval. This therefore yields a categorification of the 4-term relation. We carefully discuss the representation theory of differential crossed modules in chain-complexes of vector spaces, which makes it possible to formulate the notion of an infinitesimal 2-R matrix in a differential crossed module.