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

References in corpus (4)

• Higher-Dimensional Algebra V: 2-Groups
• Higher Gauge Theory: 2-Connections on 2-Bundles
• Higher Yang-Mills Theory
• Categorical representations of categorical groups

Cited by in corpus (9)

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• Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy
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• 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