Quantum Gravity Seminar  Spring 2005
Gauge Theory and Topology
John Baez and Derek Wise
In the 20042005 academic year, our seminar is about
gauge theory and topology. In the Fall
and Winter
we showed how to construct topological quantum field theories
in which the gauge group is a finite group.
These socalled DijkgraafWitten models are a warmup for
more interesting theories having a fullfledged Lie group as
gauge group: in particular, BF theory and ChernSimons theory.
In the Spring, we hope to tackle these more interesting theories.
But, we're starting out with some basic concepts from
geometry that all mathematical physicists need to know: connections on
principal bundles, curvature, and so on.
For the big picture, read this:
As usual, Derek Wise is
writing notes for the seminar, based on lectures by John Baez:

Week 1
(Mar. 29, 31)  The frame bundle of a manifold. The set of
frames as a Gtorsor where G is the Lie group
GL(n). More generally, principal Gbundles as locally
trivial bundles of Gtorsors.

Week 2
(Apr. 5, 7)  Connections on principal Gbundles. Parallel
transport. Connections as Liealgebra valued 1forms on the
total space of the principal bundle.

Week 3
(Apr. 12, 14)  Trivializations and connections: using a trivialization
of a principal bundle to think of connections as Liealgebra
valued 1forms on the base space. Associated bundles.

Week 4
(Apr. 19, 21)  Parallel transport and covariant derivatives.
Calculating covariant derivatives with the help of a trivialization.

Week 5
(Apr. 26, 28)  Covariant derivatives and curvature.
Exterior covariant derivatives. Ad(P)valued differential forms.

Week 6
(May 5, 10)  Gauge transformations: how they act on principal
bundles, associated bundles, sections, connections and their curvatures.

Week 7
(May 10, 12)  Towards EF theory (usually called
BF theory). Discrete versus smooth connections.

Week 8
(May 17, 19)  The map from smooth to discrete connections;
the map from smooth to discrete gauge transformations.
The moduli space of flat connections versus the moduli space of
flat bundles.

Week 9
(May 24, 26) 
The moduli space of flat bundles: examples when the gauge group is
U(1), SU(n), and SO(3). From the 2d DijkgraafWitten model to 2d
EF theory. The moduli stack of flat bundles.

Week 10
(May 31) 
The 2d DijkgraafWitten model and EF theory: measures and
Hilbert spaces.
If you discover any errors in the course notes
please email me, and we'll try to correct them.
We'll keep a list of errors that
haven't been fixed yet.
You can also download
the LaTeX, encapsulated
PostScript and xfig files if for some bizarre reason you want them.
However, I reserve all rights to this work.
baez@math.removethis.ucr.andthis.edu
© 2005 John Baez and Derek Wise