Textbook recommendations for the differential geometry of Yang-Mills fields

Question

I was wondering if anyone could recommend text books or papers that could help me really understand the math behind Yang-Mills fields? Thanks!

Answer 1

Here are some introductory textbooks on mathematical gauge theory, all of which include discussions on Yang-Mills fields:

• "Mathematical Gauge Theory" by Mark J.D. Hamilton
• "Principal Bundles" by Stephen Bruce Sontz
• "The Geometry of Physics" by Theodore Frankel
• "Topology, Geometry, and Gauge Fields: Foundations" by Gregory Naber
• "Topology, Geometry, and Gauge Fields: Interactions" by Gregory Naber

There are also more advanced textbooks on Yang-Mills theory which focus on geometric analysis and applications to four-dimensional topology, such as

• "The Geometry of Four Manifolds" by S.K. Donaldson and P.B. Kronheimer
• "Instantons and Four Manifolds" by Daniel Freed and Karen Uhlenbeck
• "The Wild World of Four Manifolds" by Alexandru Scorpan

Answer 2

The other answer already includes many good resources. Let me add the following book:

• "Differential Geometry and Mathematical Physics. Part II. Fibre Bundles, Topology and Gauge Fields" by G. Rudolph and M. Schmidt.

This book also covers basics of principal bundles and their associated vector bundles and an extensive discussion about Yang-Mills theory as well as coupling to matter.

Secondly, on google you can find lecture notes by C. Bär from the university of Potsdam called "Gauge Theory", which cover a wide range of mathematical topics including Yang-Mills theory.

Last but not least, for German speaking people, there is also a very good mathematical book on the subject, called

• "Eichfeldtheorie. Eine Einführung in die Differentialgeometrie auf Faserbündeln" by H. Baum.