Categorifying the Knizhnik-Zamolodchikov connection

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• 作者：Lucio Simone Cirio^a ; ¹ ; ^{lucio.cirio@uni.lu} ; Joã ; o Faria Martins^b ; ^{jn.martins@fct.unl.pt}
• 关键词：primary ; 16T25 ; 20F36 ; secondary ; 18D05 ; 17B37 ; 53C29 ; 57Q45
• 刊名：Differential Geometry and its Applications
• 出版年：2012
• 期刊代码：18_09262245
• 类别：mt
• 出版时间：June, 2012
• 卷：30
• 期：3
• 页码：238-261
• 文件大小：384 K

摘要

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.