Math 235: Topics in Manifold Theory
The Equations of Gauge Theory
Winter 2015

Time and Place: MWF 11:50 pm in MS 6201

Instructor: Ciprian Manolescu

Email: cm_at_math.ucla.edu

Office: MS 6921

Office hours: Fri 11am12pm and by appointment
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
antiselfdual YangMills equations. In the early 1980's, Donaldson used the
YangMills equations to prove deep results about the topology of 4dimensional
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 YangMills
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
YangMills equations, and the construction of instanton Floer homology.
In the second part of the course we will discuss other gaugeinvariant
equations, such as the SeibergWitten, VafaWitten, KapustinWitten and
HaydysWitten 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 FourManifolds,"
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 AtiyahSinger
Index Formula and GaugeTheoretic Physics, Springer, New York, 1989
 M. F. Atiyah and R. Bott, The YangMills Equations over Riemann
Surfaces, Phil. Trans. R. Soc. Lond. A 308, 523615 (1982)
 M. F. Atiyah and N. Hitchin, The Geometry and Dynamics of Magnetic
Monopoles, Princeton University Press, Princeton, 1988
 N. Hitchin, The SelfDuality Equations on a Riemann Surface, Proc.
London Math. Soc. S355:1, 59126 (1987)
 S. K. Donaldson, Floer Homology Groups in YangMills Theory, Cambridge
University Press, Cambridge, 2002
 J. D. Moore, Lectures on the SeibergWitten Invariants, Springer,
Berlin, 2001
 J. W. Morgan, The SeibergWitten Equations and Applications to the
Topology of Smooth FourManifolds, Princeton University Press, Princeton, 1996
 E. Witten, Fivebranes and Knots, Quantum Topology 3:1, 1137 (2012)