Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry

Details Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry

2005-10-04

arxiv.org/abs/math/0510078v2

Int.J.Geom.Methq.Mod.Phys.08 2011:1079-1095,2011

Mathematics

Differential Geometry

Scientific paper

10.1142/S0219887811005555

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| are shown to be one-to-one with stable equivalence classes of what we call crossed module gerbes bundle gerbes. We can also associate to a crossed module a 2-category C'. Then there are two equivalent ways how to view classifying spaces of NC-bundles and hence of |NC|-bundles and crossed module bundle gerbes. We can either apply the W-construction to NC or take the nerve of the 2-category C'. We discuss the string group and string structures from this point of view. Also a simplicial principal bundle can be equipped with a simplicial connection and a B-field. It is shown how in the case of a simplicial principal NC-bundle these simplicial objects give the bundle gerbe connection and the bundle gerbe B-field.

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