TY - JOUR

T1 - Categorifying the Knizhnik-Zamolodchikov connection

AU - Martins, João Nuno Gonçalves Faria

PY - 2012/1/1

Y1 - 2012/1/1

N2 - 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.

AB - 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.

KW - Differential crossed module

KW - Infinitesimal braiding

KW - Categorical representation

KW - Braided surface

KW - Higher gauge theory

KW - 4-term relation

KW - Two-dimensional holonomy

KW - Chord diagrams

KW - Knizhnik-Zamolodchikov equations

U2 - 10.1016/j.difgeo.2012.03.004

DO - 10.1016/j.difgeo.2012.03.004

M3 - Article

SN - 0926-2245

VL - 30

SP - 238

EP - 261

JO - Differential Geometry And Its Applications

JF - Differential Geometry And Its Applications

IS - 3

ER -