Math 235: Topics in Manifold Theory
The Equations of Gauge Theory
Time and Place: MWF 1-1:50 pm in MS 6201
Instructor: Ciprian Manolescu
Office: MS 6921
Office hours: Fri 11am-12pm and by appointment
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.
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:
- H. Blaine Lawson, Jr., The Theory of Gauge Fields in Finite Dimensions,
AMS, Providence, 1985
- B. Booss and D. D. Bleecker, Topology and Analysis: The Atiyah-Singer
Index Formula and Gauge-Theoretic Physics, Springer, New York, 1989
- M. F. Atiyah and R. Bott, The Yang-Mills Equations over Riemann
Surfaces, Phil. Trans. R. Soc. Lond. A 308, 523-615 (1982)
- M. F. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic
Monopoles, Princeton University Press, Princeton, 1988
- N. Hitchin, The Self-Duality Equations on a Riemann Surface, Proc.
London Math. Soc. S3-55:1, 59-126 (1987)
- S. K. Donaldson, Floer Homology Groups in Yang-Mills Theory, Cambridge
University Press, Cambridge, 2002
- J. D. Moore, Lectures on the Seiberg-Witten Invariants, Springer,
- J. W. Morgan, The Seiberg-Witten Equations and Applications to the
Topology of Smooth Four-Manifolds, Princeton University Press, Princeton, 1996
- E. Witten, Fivebranes and Knots, Quantum Topology 3:1, 1-137 (2012)