1202.2280 (David Viennot)

Non-abelian higher gauge theory and categorical bundle    [PDF]

David Viennot

A gauge theory is associated with a principal bundle endowed with a
connection permitting to define horizontal lifts of paths. The horizontal lifts
of surfaces cannot be defined into a principal bundle structure. An higher
gauge theory is an attempt to generalize the bundle structure in order to
describe horizontal lifts of surfaces. A such attempt is particularly difficult
for the non-abelian case. Some structures have been proposed to realize this
goal (twisted bundle, gerbes with connection, bundle gerbe, 2-bundle). Each of
them uses a category in place of the total space manifold of the usual
principal bundle structure. Some of them replace also the structure group by a
category (more precisely a Lie crossed module viewed as a category). But the
base space remains still a simple manifold (possibly viewed as a trivial
category with only identity arrows). We propose a new principal categorical
bundle structure, with a Lie crossed module as structure groupoid, but with a
base space belonging to a bigger class of categories (which includes
non-trivial categories), that we call affine 2-spaces. We study the geometric
structure of the categorical bundles built on these categories (which is a more
complicated structure than the 2-bundles) and the connective structures on
these bundles. Finally we treat an example interesting for quantum dynamics
which is associated with the Bloch wave operator theory.

View original: http://arxiv.org/abs/1202.2280