Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra

TL;DR

This paper constructs a categorified version of the $sl(2,C)$-Knizhnik-Zamolodchikov connection using a novel infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra, leading to new insights in 2-connection theory.

Contribution

It introduces an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra and shows how it yields flat 2-connections categorifying the KZ connection.

Findings

Constructed a flat 2-connection in configuration space

Established the existence of an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra

Demonstrated that categorical representations produce categorified KZ connections

Abstract

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.