categorified gauge theory
Urs Schreiber
> Some of us have spent time here talking about connections on nonabelian
> gerbes and their relation to physics. Lately Urs Schreiber and I have made > a bunch of progress on these issues. So, some of you may be interested in a > talk I'm giving in Paris next Wednesday, at a conference in honor of Larry
> Breen's 60th birthday.
The paper describing these issues has now appeared:
John Baez & Urs Schreiber,
Higher Gauge Theory: 2-Connections on 2-Bundles
hep-th/0412325 .
Some discussion of it can be found at the String Coffee Table at
http://golem.ph.utexas.edu/string/archives/000488.html
as well as on sci.physics.strings in the thread "Towards nonabelian
membrane action".
The crux of the paper is to show that 2-bundles with 2-connecions
define nonabelian gerbes with connection and curving PLUS a notion of
surface holonomy.
This is interesting for certain aspects of string theory, but also for
ordinary gauge theory.
Some gauge theories in four dimensions are well known to exhibit a
certain weak/strong duality with respect to their coupling constant.
This duality can be understood geometrically by thinking of the
four-dimensional spacetime as a six-dimensional spacetime which is
compactified on a torus. Certain symmetry operations on that torus
appear as weak/strong dualities in the field theory in four
dimensions.
A beautiful elementary description of this fact can be found in this
talk by Witten:
http://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/
as was pointed out by Peter Woit on his blog at
http://www.math.columbia.edu/~woit/blog/archives/000122.html .
Incidentally, these six-dimensional theories are also known to be
understandable as the worldvolume theory of 5-branes that appear in
string theory. On these 5-branes 2-branes may end, and the boundary of
these 2-branes in general couples to a nonabelian 2-form, in much the
same way as a point (the boundary of a string) couples to a nonabelian
1-form.
Or is expected to. In order to write down an action which expresses
this coupling one will need to have an idea of "nonabelian surface
holonomy" in a globally defined way. This has not been known so far.
But in 2-bundles with 2-connections it does exist. There are
indications that such a nonabelian surface holonomy also plays a role
in the, so far missing, Lagrangian description of these 6-dimensional
gauge theories that arise as a certain decompactification limit of
ordinary gauge theory.
Hence 2-bundles with 2-connections should be relevant, indirectly, for
the understanding of strongly coupled four-dimensional gauge theory.
But there are several things to be worked out before this is more than
a hope.
One thing is the point that Thomas Larsson keeps emphasizig: There is
that constraint of vanishing 'fake curvature' which appears
unexpectedly restrictive.
whop...@csd.uwm.edu
One can very transparently characterize most of the key structures in
principal bundles, and gauge theory, in terms of a quotient operation.
Let P be a principal bundle, G the group acting on it, with base space
Q = P/G. Then, the action P x G -> P extends naturally to one on the
tangent spaces, TP x G, P x TG -> TP, such that:
(p(t)g(t))' = p'(t)g(t) + p(t)g'(t).
The signature of the operations are:
T_p(P) x {g}, {p} x T_g(G) -> T_{pg}(G).
Corresponding to this is the QUOTIENT, defined over each orbit by the
properties:
p\q defined if pG = qG
p\pg = g; p p\q = q.
Likewise, one naturally wants an ability to take quotients under the
differential operator with
d(p\q)/dt = p'\q + p\q'.
This is EXACTLY the conditions that define a connection. The
connection, here is just p\v, for v in T_p(P).
Similarly, one can define an OUTER quotient p/q: T_q(P) -> T_p(P) that
acts as an equivaiant map between separate fibres. This defines the
gauge groupoid.
I'll let you explore the issues further without further comment.