Notes by Jurčo on generalized Bundle Gerbes
Posted by Urs Schreiber
Branislav Jurčo asks me to draw attention here to the following notes which he provides on his website:
a) Associated 2-vector bundle construction using Baez-Crans 2-vector spaces
b) 2-crossed module bundle 2-gerbes
c) Differential geometry of 2-crossed module bundle 2-gerbes
These are generalizations building on his work on nonabelian bundle gerbes as presented in
P. Aschieri, L. Cantini B. Jurčo
Nonabelian Bundle Gerbes, their Differential
Geometry and Gauge Theory
hep-th/0312154
and
Branislav Jurčo
Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry
math/0510078.
The first one discusses the bundle gerbe analog of how a vector bundle may be associated to a principal bundle. The second and third generalize bundle (1-)gerbes to bundle 2-gerbes. This are then gadgets that are to 3-bundles with some structure 3-group like transitions functions are to ordinary principal bundles.
(A crossed module is the realization of a strict 2-group in the world of complexes of groups. A “crossed module bundle gerbe” is to an abelian bundle gerbe like a 2-bundle for a general strict 2-group is to a 2-bundle for the 2-group “shifted ”. A 2-crossed module is a realization of a suitably well behaved 3-group in the world of complexes of groups.)
See also the entry
Jurčo on Gerbes and Stringy Applications
Unfortunately I am quite busy at the moment. But if I find the time, I might try to write a summary of Brano’s notes. Unless sombody else beats me to it…