Quantum Gravity Seminar - Spring 2005
Gauge Theory and Topology
John Baez and Derek Wise
In the 2004-2005 academic year, our seminar is about
gauge theory and topology. In the Fall
we showed how to construct topological quantum field theories
in which the gauge group is a finite group.
These so-called Dijkgraaf-Witten models are a warmup for
more interesting theories having a full-fledged Lie group as
gauge group: in particular, BF theory and Chern-Simons 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:
(Mar. 29, 31) - The frame bundle of a manifold. The set of
frames as a G-torsor where G is the Lie group
GL(n). More generally, principal G-bundles as locally
trivial bundles of G-torsors.
(Apr. 5, 7) - Connections on principal G-bundles. Parallel
transport. Connections as Lie-algebra valued 1-forms on the
total space of the principal bundle.
(Apr. 12, 14) - Trivializations and connections: using a trivialization
of a principal bundle to think of connections as Lie-algebra
valued 1-forms on the base space. Associated bundles.
(Apr. 19, 21) - Parallel transport and covariant derivatives.
Calculating covariant derivatives with the help of a trivialization.
(Apr. 26, 28) - Covariant derivatives and curvature.
Exterior covariant derivatives. Ad(P)-valued differential forms.
(May 5, 10) - Gauge transformations: how they act on principal
bundles, associated bundles, sections, connections and their curvatures.
(May 10, 12) - Towards EF theory (usually called
BF theory). Discrete versus smooth connections.
(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
(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 Dijkgraaf-Witten model to 2d
EF theory. The moduli stack of flat bundles.
(May 31) -
The 2d Dijkgraaf-Witten model and EF theory: measures and
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.
© 2005 John Baez and Derek Wise