Advances in Theoretical and Mathematical Physics

Volume 21 (2017)

Number 1

Categorifying the $\mathfrak{sl}(2, \mathbb{C})$ Knizhnik-Zamolodchikov connection via an infinitesimal $2$-Yang-Baxter operator in the string Lie-$2$-algebra

Pages: 147 – 229

DOI: http://dx.doi.org/10.4310/ATMP.2017.v21.n1.a3

Authors

Lucio Simone Cirio (Mathematisches Institut, Georg-August Universität Göttingen, Germany)

João Faria Martins (Department of Pure Mathematics, School of Mathematics, University of Leeds, United Kingdom)

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 $\mathfrak{sl}(2, \mathbb{C})$-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of $\mathfrak{sl}(2, \mathbb{C})$. 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 $\mathfrak{sl}(2, \mathbb{C})$-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.

Keywords

higher gauge theory, two-dimensional holonomy, categorification, crossed module, braid group, braided surface, configuration spaces, Knizhnik-Zamolodchikov equations, Zamolodchikov tetrahedron equation, infinitesimal braid group relations, infinitesimal relations for braid cobordisms, categorical representation

2010 Mathematics Subject Classification

Primary 16T25, 20F36. Secondary 17B37, 18D05, 53C29, 57M25, 57Q45.

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