G. Moore

First Meeting: Wednesday September 5, 2001, 9:50am, Serin 372

The past 20 years have been an exciting time for the interaction
of mathematics and physics.

Remarkable new mathematical discoveries have been made
using the methods of quantum field

theory. Conversely, powerful new mathematical techniques
have been successfully applied to

gain nontrivial insights in sophisticated theories like
supersymmetric gauge theories and string theory.

The purpose of this course is to provide some of the mathematical
background which one needs

in order to learn about these modern developments. The
course surveys some aspects of topology and differential geometry
of manifolds, with an emphasis on relations to modern mathematical physics.
The course will also cover some aspects of supersymmetry.

This course is primarily intended for physics graduate students specializing in theoretical physics.

Some knowledge of manifolds, differential forms, and cohomology
will be assumed, but we

will try to keep the course material self-contained.

An optimistic goal is to end with the proofs of the index theorems for elliptic operators using supersymmetric quantum mechanics. An approximate sequence of topics will be as follows:

1. Riemannian geometry. Orthonormal frames.

2. Clifford algebras and spinors. Supersymmetry
algebras.

3. Hodge *, Hodge theory, harmonic forms

4. Maxwell theory and generalized Maxwell theories. Electric-magnetic duality of gauge theories of forms. Self-dual equations of motion.

5. Connections and curvature on fiber bundles.

6. Supersymmetric quantum mechanics and the index.

7. Proofs of the index theorems using supersymmetric
quantum mechanics.