Re: surface holonomy from connections on gerbes?

From: Urs Schreiber (Urs.Schreiber_at_uni-essen.de)
Date: 09/22/04 

Date: Wed, 22 Sep 2004 16:01:38 -0400

I wrote:

>Given some non-abelian
> gerbe with connection over a manifold M and given a simply connected open
> subset U of M. Is there a way to turn the crank such that the gerbe
> connection spits out the "surface holonomy" of U in some sense? If so,
how?

>>From looking at

M. Nackaay & R. Picken:
Holonomy and parallel transport for Abelian gerbes,
math.DG/0007053

as well as

A. Carey, S. Johnson, M. Murray:
Holonomy on D-branes,
hep-th/0204199

it seems that the answer is rather simple:

According to p. 11 of math.DG/0007053 a 0-connection on a gerbe is a 1-form
on each triple overlap and a 1-connection is a 2-form on each double overlap
of elements of an open cover such that the well-known cocycle conditions
hold and apparently given these two forms one can forget that they come from
a gerbe and just try to construct a notion of surface holonomy from them.

At least that's what is done in section 6 of math.DG/0007053, where nothing
but the well understood relation between abelian connections on loop space
and abelian 2-groups is described and surface holonomy is obtained by
integrating a 2-form over a surface. There seems to be no further input from
gerbe theory about *how* to do that integration, notably when it comes to
the non-abelian case (or is there?).

As far as I can see from having read these references (which is possibly not
far enough) it seems that what gerbes do for us concerning surface holonomy
is to tell us how a collection of 1-forms and 2-forms associated with an
open cover of the manifold have to be related on intersections of open sets,
while for actually integrating up the 1-form and the 2-form to obtain a
group element for a given surface we have to use the well known local loop
space and 2-group methods, which in particular tell us that this only works
when either dt(B)+F_A = 0 or a slightly less restrictive condition holds.

If that assessment is correct it might explain why one never sees any such
condition discussed in the gerbe literature, because a "connection on a
gerbe" is apparently not usually defined as something that can be
"integrated" to yield a surface holonomy.