Topics in Gauge Theory

Goals: This course covers mathematical gauge theory. In content and organization, it will adapt the perspective of physicists. The aim is to integrate the physics viewpoint and intuition into the mathematical theory. The terminology of physics permeates the subject, but if often not understood by mathematicians. In particular, we will discuss quantum gauge theories.     

Prerequisites: Familiarity with manifolds and vector bundles. It will be very useful, but not strictly necessary, to have some knowledge of PDEs and Riemannian geometry.

Handouts:   Course Outline

Homework:     HW Set 1     HW Set 2     HW Set 3    HW Set 4    HW Set 5

Preliminary list of topics:

1. Classical fields and Maxwell's equations as a gauge theory
2. Spinors and the Dirac equation
3. The Seiberg-Witten equations
4. Second quantization
5. Yang-Mills and general gauge theories
6. Examples

Helpful reference books:

• Mirror Symmetry, by K. Hori, S. Katz, A. Klemm, R. Pandharipande, R.~Thomas, C. Vafa, R. Vakil and E. Zaslow.
• The geometry of four-manifolds, by S. Donaldson and P. Kronheimer.
• The Seiberg-Witten equations and applications to the topology of four-manifolds, by J. Morgan
• An introduction to the Seiberg-Witten equations on symplectic manifolds, by M. Hutchings and C. Taubes

Other reference articles:

• Advanced Quantum Mechanics, by Freeman Dyson.
• Notes of Geometric Topology by Selman Akbulut.
• Notes on Principal bundles and connections, from "Foundations of Differential Geometry, Vol. 1 by Kobayashi and Nomizu.
• Harmonic Spinors, by N. Hitchin, Advances in Math 1974.