Aspects of Yang-Mills Theory

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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.


Problem sheets

Recommended reading

The course is based on the papers:

You might also find the following books useful. For those interested in further directions in the subject, here are some good places to look.