2012/07/31 by Lucio S. Cirio, João Faria Martins · 1 citation
Physics and Astronomy · Mathematics · #hep-th #math.GT #math.QA #msc:16T25 #msc:20F36 #msc:18D05 #msc:17B37 #msc:53C29 #msc:57M25 #msc:57Q45
published as Adv. Theor. Math. Phys. Volume 21, Number 1, 147 - 229, 2017 · The main result was considerably sharpened. Title, abstract and introduction updated. 50 pages
arxiv created 2012/12/07 · arxiv updated 2017/05/23
We construct a flat (and fake-flat) 2-connection in the configuration space of n indistinguishable particles in the complex plane, which categorifies the sl(2,C)-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of sl(2,C). This will be done by considering the adjoint categorical representation of the string Lie 2-algebra and the notion of an infinitesimal 2-Yang-Baxter operator in a differential crossed module. Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra, proving that any (strict) categorical representation of the string Lie-2-algebra, in a chain-complex of vector spaces, yields a flat and (fake flat) 2-connection in the configuration space, categorifying the sl(2,C)-Knizhnik-Zamolodchikov connection. We will give very detailed explanation of all concepts involved, in particular discussing the relevant theory of 2-connections and their two dimensional holonomy, in the specific case of 2-groups derived from chain complexes of vector spaces.