TCC course: Introduction to Gauge Theory

Schedule: Trinity 2019, Fridays 14:00 to 16:00. No lecture on 17th of May and 14th of June, replacement lectures on Thursday from 10:00 to 12:00 in week 5 and 8 (30th of May and 20th of June).

Course description: The topic we cover in these lectures is the moduli problem in gauge theory, that is, to build fundamental cycles out of gauge-theoretic moduli spaces, which typically parametrize families of vector bundles with additional structures (e.g. holomorphic or stable vector bundles etc. in algebro-geometric setting). In this course, we describe Donaldson's solutions to this problem for anti-self-dual instanton moduli spaces on closed four-manifolds, which efficiently utilize nonlinear analysis in the geometric setting. These fundamental cycles are used to produce diffeomorphic invariants of the underlying four-manifolds via Intersection Theory, i.e. by "integrating" characteristic classes of the universal bundle over the cycle, which also provoked other deformation invariants such as Gromov-Witten, Seiberg-Witten, Donaldson-Thomas ones, and theories of Floer homologies and Floer homotopy types.

In contrast with algebro-geometric approach to this problem, where one would use "virtual" techniques to overcome the tranversality problem of the moduli spaces, the analytic one allows us to use perturbation methods to get the moduli spaces transversed. Thus, the issue in the analytic setting is the compactness of the moduli spaces. Donaldson got around this problem for the anti-self-dual instanton moduli spaces by using superbly elaborate analysis by Uhlenbeck and Taubes, which is now referred to as the Uhlenbeck compactification.

The goals of these lectures are therefore to get a fair understanding of this Uhlenbeck compactificaiton; and of the relation of it with algebro-geometric counterpart, the Gieseker compactification, when the underlying four-manifolds are complex surfaces. (The latter gives many computable examples of the invariants.)

We start by introducing Yang-Mills connections and anti-self-dual instantons on a principal bundle, and proceed to proving that the moduli space of anti-self-dual instantons becomes a smooth oriented manifold of the expected dimensions for a generic choice of metrics of the underlying four-manifold. Then we describe analytic aspects of boundaries of the anti-self-dual instanton moduli spaces. We assume that the students would have certain familiarity with manifolds, vector bundles, principal bundles and the characteristic classes of them such as the Pontryagin and Chern ones.

The following is a plan of this course.

1. Connections, curvatures and gauge transformations (notes)
2. Yang-Mills connections and anti-self-dual instantons (notes)
3. Sobolev spaces, elliptic operators and linearisation (notes)
4. Atiyah-Hitchin-Singer complex, Kuranishi model, Freed-Uhlenbeck perturbation (notes)
5. Reducible connections, orientation issues of the moduli space (notes)
6. Uhlenbeck compactness, Taubes' gluing and Donaldson's theorem (notes)
7. Uhlenbeck compactification and Gieseker compactifiation (notes)
8. Donaldson invariants, Witten's conjecture and other related invariants (notes)

Prerequisite: manifolds, vector bundles, principal bundles and the characteristic classes of them such as the Pontryagin and Chern ones.

Assessment: If you'd like to get credit for this course, please hand in solutions to exercises that would come up during the lectures.

References:

• M. F. Atiyah, Geometry of Yang-Mills fields, Scuola Normale Superiore Pisa, 1979.
• S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, 1990.
• D. S. Freed and K. K. Uhlenbeck, Instantons and four-manifolds, Second edition, Mathematical Sciences Research Institute Publications, 1991.
• R. Friedman and J. W. Morgan, Smooth four-manifolds and complex surfaces, Springer-Verlag, 1994.
• R. Friedman and J. W. Morgan (eds.), Gauge theory and the topology of four-manifolds, IAS/Park City Mathematics Series, Volume 4, American Mathematical Society, 1998.
• D. D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxford Graduate Texts in Mathematics, 12, Oxford University Press, 2007.
• D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Second edition. Cambridge Mathematical Library, Cambridge University Press, 2010.
• S. Kobayashi, Differential geometry of complex vector bundles, Princeton University Press, 1987.
• M. Lübke and A. Teleman, The Kobayashi-Hitchin correspondence, World Scientific, 1995.
• C. H. Taubes, Metrics, Connections and gluing theorems. CBMS Regional Conference Series in Mathematics, 89, American Mathematical Society, 1996.
• C. H. Taubes, Differential Geometry. Bundles, connections, metrics and curvature, Oxford Graduate Texts in Mathematics, 23, Oxford University Press, 2011.
• K. Wehrheim, Uhlenbeck compactness, EMS Series of Lectures in Mathematics, European Mathematical Society, 2004.