Categorifying the Knizhnik-Zamolodchikov Connection

Abstract

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.

30 pages, 2 figures; v3: final version to be published in Differential Geometry and its Applications

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Cited by in corpus (9)

• Poincare 2-group and quantum gravity
• Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy
• Infinitesimal 2-braidings and differential crossed modules
• Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra
• Infinitesimal 2-braidings from 2-shifted Poisson structures
• Holonomies of gauge fields in twistor space 5: amplitudes of gluons and massive scalars
• Holonomies of gauge fields in twistor space 6: incorporation of massive fermions
• A Kontsevich integral of order 1
• Categorical generalization of spinfoam models