Categorifying the sl(2, C) Knizhnik-Zamolodchikov connection via an infinitesimal 2-Yang-Baxter operator in the string Lie-2-algebra

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.