Math 235: Topics in Manifold Theory

The Equations of Gauge Theory

Winter 2015


Outline:

Mathematical gauge theory is the study of several elliptic partial differential equations that arose in physics, and that are invariant under the action of a group of bundle automorphisms (called the gauge group). The prototype are the anti-self-dual Yang-Mills equations. In the early 1980's, Donaldson used the Yang-Mills equations to prove deep results about the topology of 4-dimensional manifolds. For example, one consequence of his work was the existence of exotic smooth structures on R^4.

The first part of the course will consist in a study of the Yang-Mills equations. We will sketch the proof of Uhlenbeck's compactness theorem, Donaldson's diagonalizability theorem, and give the ADHM description of instantons on S^4. We will also describe various dimensional reductions of the Yang-Mills equations, and the construction of instanton Floer homology.

In the second part of the course we will discuss other gauge-invariant equations, such as the Seiberg-Witten, Vafa-Witten, Kapustin-Witten and Haydys-Witten equations. The goal will be to put in context Witten's proposal for a gauge theoretic interpretation of Khovanov's knot homology.

Prerequisites:

Knowledge of differential topology, algebraic topology and differential geometry, at the level of the Math 225 sequence. Math 226A recommended. Also recommended is some familiarity with Hodge theory and characteristic classesbut these two topics will be briefly reviewed in class.

Textbook:

S. K. Donaldson and P. B. Kronheimer, "The Geometry of Four-Manifolds," Oxford University Press, New York, 1990.

We will also use material from some additional sources, such as: