Introduction to Yang-Mills theories

Yang-Mills theories are a class of classical field theory generalizing Maxwell’s equations. When quantized, Yang-Mills theories form the basis for all successful modern quantum field theories, including the standard model of particle physics, and grand unified theories (GUTs) that attempt to go beyond the standard model.

This post contains working notes (32 pages, pdf only) that I wrote in an attempt to come to a satisfactory personal understanding of Yang-Mills theory. They are part of a larger project of understanding the standard models of particle physics and of cosmology – some related earlier notes are here.

Caveat: The current notes take a geometric approach to Yang-Mills theory, and include quite a bit of background on differential geometry. After completing a first draft, I realized that if I was to write either a pedagogical introduction or a review of Yang-Mills theory, this geometric approach is not the approach I’d prefer to take. Rather, I’d start with a bare statement of the Yang-Mills equations, considered as a generalization of Maxwell’s equations, and then work through a series of examples, only gradually mixing in the geometric approach. This would have the advantage of bringing readers up to speed much more quickly, without needing to absorb reams of differential geometry upfront.

Because of this, I haven’t polished these notes – they remain primarily my personal working notes, and there are various inaccuracies and shortcomings in the notes. I’m content to ignore these – why spend time polishing when you know a better approach is possible – but would appreciate advisement if you spot any serious misconceptions.

Despite these caveats, I believe the notes may be useful to some readers. In particular, if you’d like to understand the approach to Yang-Mills theory from differential geometry, these notes may serve as a useful first step, to be supplemented by additional reading such as the book by Baez and Muniain (“Knots, Gauge Fields and Gravity”, World Scientific 1994) on which the notes are primarily based.


Update: If you’re reading the notes in detail, then you might want to take a look at the comments, esepcially those by David Speyer and Aaron Bergman, who provide some important corrections and extensions.