### 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 $U(1)$”.
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…