# Aspects of Yang-Mills Theory

This was a graduate course I taught at ETH in 2011. Sadly the videos are no longer available from ETH. All teaching materials available from this site are released under a CC-BY-SA 4.0 Licence. That means you're free to use them as long as you give appropriate attribution and release derivatives under an isomorphic licence.

## Summary of course

Maxwell's equations form the basis of modern physics and have inspired much of modern geometry. That is where this course will begin, illustrating how one arrives naturally at the concept of a gauge theory starting with Maxwell's equations and how one can solve the resulting field equations (though we will cheat and work over the compact Riemannian manifolds and hence only with magnetostatics). This Riemannian version of Maxwell theory is variously called 'abelian gauge theory' or the 'Hodge theory of harmonic forms' and is a beautiful application of linear elliptic PDE. For instance we can deduce that each de Rham cohomology class contains a unique harmonic form.

Next we will progress to non-abelian gauge theory, (the eponymous Yang-Mills theory) which underlies the chromodynamic and electroweak theories in physics. Not being so ambitious we concentrate on the case of Yang-Mills over a compact Riemann surface (a real 2-dimensional manifold). Through the Narasimhan-Seshadri theorem we will see that the solutions to the field equations are intimately related to objects of algebraic geometry: stable holomorphic vector bundles. This allows us to probe the topology of the space of stable holomorphic vector bundles e.g. to compute its Betti numbers, a computation which before Atiyah and Bott did it this way required the Weil conjectures in characteristic p.

## Lectures

• Abelian gauge theory
1. Introduction (notes)
2. Magnetostatics (notes)
3. U(1)-bundles (notes)
4. Harmonic forms (notes)
5. The Hodge theorem (notes)
• Non abelian gauge theory
1. Principal bundles (notes)
2. Yang-Mills functional (notes)
3. The Kempf-Ness theorem (notes)
4. The moment map for Yang-Mills (notes)
5. Holomorphic bundles I (Existence) (notes)
6. Holomorphic bundles II (Stability) (notes)
7. Holomorphic bundles III (Harder-Narasimhan) (notes)
12. Gauge-fixing I (notes)
13. Gauge-fixing II (notes)
• Topology of the moduli space
1. Equivariant cohomology I (notes)
2. Equivariant cohomology II (notes)
3. Equivariant cohomology III (notes)
4. The Harder-Narasimhan stratification (notes)
5. The Atiyah-Bott formula (notes)
6. Equivariant perfection and Morse strata (notes)

## Problem sheets

The course is based on the papers:

• M. F. Atiyah and R. Bott “The Yang-Mills equations over Riemann sur- faces” Phil. Trans. R. Soc. Lond. A. 308 (1983), 523–615
• K. Uhlenbeck “Connections with L p -bounds on curvature” Commun. Math. Phys. 83 (1982), 31–42
• S. K. Donaldson “A new proof of a theorem of Narasimhan and Seshadri” J. Diff. Geom. Volume 18, Number 2 (1983), 269–277.

You might also find the following books useful.
• S. K. Donaldson and P. B. Kronheimer “The geometry of 4-manifolds” OUP (1990)
• S. K. Donaldson “Riemann surfaces” OUP (2011)
• D. Gilbarg and N. S. Trudinger “Elliptic partial differential equations of second order” Springer Comprehensive Studies in Mathematics 224, 2nd Edition (1998)
• M. F. Atiyah “Geometry of Yang-Mills fields” Scuola Normale Superiore Pisa (1979)
• G. Naber “Topology, geometry and gauge fields, Foundations” Springer Texts in Applied Mathematics 25 (1997)
• M. Nakahara “Geometry, topology and physics” IOP Graduate Student Series in Physics (1990)
• Some nice lecture notes of Figueroa-O'Farrill on principal bundles which lay things out in perhaps a more ordered way than I did.

For those interested in further directions in the subject, here are some good places to look.
• Daskalopoulos's paper which develops the Yang-Mills flow on Riemann surfaces and completes the Morse-theoretic picture sketched by Atiyah-Bott. The main result is that the Harder-Narasimhan strata deformation retract along the Yang-Mills flow onto the subset of Yang-Mills connections of a fixed Harder-Narasimhan type.
• See here for an eloquent overview of the subject, some of its open problems and its influence on subsequent mathematics by Simon Donaldson.
• Notes from a course taught by Tim Perutz at DPMMS in 2006 from which I learned Donaldson theory. They are a light introduction to the four-dimensional theory.
• For a more serious study of 4-dimensional gauge theory, the canonical reference is Donaldson and Kronheimer's (1990) "The Geometry of Four-Manifolds" Oxford University Press. I cannot overemphasise the brilliance and clarity of this book. It is my favourite maths book.
• For further directions in 2-d Yang-Mills theory, Hitchin's paper on Higgs bundles is an excellent starting place. Alas I didn't have time in the course to talk about Higgs bundles, but the theory is of central importance in an exciting circle of ideas known as the geometric Langlands program. The higher-dimensional version of Higgs bundle theory (developed notably by Carlos Simpson) leads to interesting restrictions on the fundamental groups of Kähler manifolds and a novel proof of Yau's uniformisation theorem for surfaces of general type on the Bogomolov-Miyaoka-Yau line. See Simpson's paper for more details.