Re: surface holonomy from connections on gerbes?

From: Urs Schreiber (Urs.Schreiber_at_uni-essen.de)
Date: 09/24/04 

Date: Fri, 24 Sep 2004 10:46:03 -0400

"Michael Murray" <mmurray@maths.adelaide.edu.au> schrieb im Newsbeitrag
news:mmurray-C11612.19293523092004@duster.adelaide.on.net...

> Associated to this is a holonomy over a surface which we
> define in 3.2.

Many thanks for the reply. So I see that my first impression was quite wrong
and there is indeed a detailed prescription how to obtain that surface
holonomy.

I see that the formulas which you are referring to are also summarized in
eqs. (27)-(29) of Aschieri&Jurco, hep-th/0409200, which is what I am
currently studying in order to understand what is known about the
non-abelian case.

Aschieri and Jurco have a very nice method (whether invented by thems or
recalled by them I am not quite sure) to obtain non-abelian gerbes. They
note that "twisting" an non-abelian principal bundle (=0-gerbe) produces
data that defines an abelian 1-gerbe. Conversely, the non-abelian 0-gerbe
can be regarded as a "gerbe module" of the abelian 1-gerbe.

By increasing the dimension by one this suggests/implies that twisted
non-abelian 1-gerbes are similarly gerbe modules of abelian 2-gerbes, and
the authors spell out on pp. 12-13 what this means explicitly in terms of
hypercohomology. In particular, an ordinary (non-twisted) non-abelian
1-gerbe is defined by setting the left-hand-side of (56-59) to 0 (or 1,
respectively).

As far as I understand that's a very nice result, which improves on previous
attempts to make gerbes non-abelian as proposed by J. Kalkkinen in
hep-th/9910048.

But Aschieri&Jurco don't seem to mention how to insert the data

 (f_ijk, phi_ij, a_ij, A_i, B_i, d_ij, H_i)

defining a (twisted) non-abelian 1-gerbe into any formula for non-abelian
1-gerbe surface holonomy.

Is any such non-abelian generalization of the formulas that you mentioned
known?

Using loop space or 2-group reasoning I'd know how to cook up from
non-abelian 1-form and 2-form data such a surface holonomy which is
rep-inavriant - but what I don't see yet is how to ensure that the result is
also independent of the chosen cover. This cover-independence is what fixes
the abelian 1-gerbe surface holonomy to the surface integral over the "error
2-form" in the terminology of Chatterjee, which equivalently gives rise to
the formulas that you mentioned, as far as I understand. If it were not for
cover independence we could for instance just drop one term in the error
2-form, I believe, or change the relative coefficients.

When one uses 2-group or loop space technology to cook up rep-inavriant
non-abelian surface holonomy one encounters the constraint "B+F=0", relating
the 2-form B to the field strength of the 1-form A. (Curiously, B+F is, up
to relative factors, precisely the structure of the error 2-form.) It seems
to be interesting to see if anything like this appears in non-abelian
gerbes.

The first interesting observation in this respect is that, at least
according to the construction used by Aschieri and Jurco, the data for a
non-abelian 1-gerbe

 (f_ijk, phi_ij, a_ij, A_i, B_i, d_ij, H_i)

contains *two* 2-forms, B_i and d_ij, as well as two 1-forms, A_i and a_ij.
On single overlaps, where the "transition bundles" of the gerbe reside, they
get the hypercohomology condition (their (58))

 phi_ij(B_j) - B_i - d_ij + da_ij + a_ij /\ a_ij + a_ij /\ T_A_i = 0

where phi_ij acts in the automorphism group of G and T_A_i is some 1-form.

What I am wondering about if maybe this is related to the loop space/2-group
condition B+F = 0, because, tweaking one's eyes, both look very similar. For
instance one could tentatively identify B with phi_ij(B_j) - B_i - d_ij +
da_ij .

In fact, despite the apparent complexity of the above formula, it is more
easily related to the loop space/2-group expression, because of the presence
of the new 2-form d_ij, which lives on double overlaps, just like the 1-form
a_ij.

All this seems to make it rather natural to speculate that non-abelian
1-gerbe surface holonomy on single overlaps U_ij (using the principal
bundles over them) is equal to the loop space/2-group holonomy of the given
surface with respect to the 2-form phi_ij(B_j) - B_i - d_ij + da_ij and the
1-form a_ij.

I think it could make sense because the "purpose" of the 1-form contribution
in your eqaution (3.11) in hep-th/0204199 for the 1-gerbe surface holonomy
is sort of to relate 2-form contributions at various points, just like the
purpose of the 0-form in the ordinary connection (e.g. (27) in
Aschieri&Jurco) is to relate 1-forms in different patches and it is
precisely such a "transporting purpose" of the 1-form which appears in
loop-space/2-group holonomy.

Does this make sense to anyone? :-)

Sorry, now I have mostly talked about Aschieri&Jurco's paper. I'll have a
closer look at "Holonomy on D-branes" and see if maybe that helps me see
clearer.