Geometry of Yang-Mills theory

Question

I am looking for a preferably introductory text on Yang-Mills theory with mathematical basis and language. (Differential forms) I have a background in QCD and QED, however, I want to go more in details in geometrical interpretations of Yang-Mills theory. I am sorry if this question has asked before. Thanks in advance.

Answer 1

Hands down, the best reference book to learn about gauge theories is DeWitt's The Global Approach to Quantum Field Theory. In this book you can find the most general formulation of both classical and quantum field theories, and a thorough discussion of essentially every related subtopic. The formulation is general covariant from the outset (no $3+1$ splitting); is valid for fields of arbitrary Grassmann parity and arbitrary spin; and it is able to accommodate any gauge algebra (open, reducible, etc.). The formulation adopts the modern point of view, which is based on the functional integral as a primitive concept, thus getting rid of the old-school methods of canonical quantisation.

The two-volume set is over one thousand pages long, and it is from 2003, so it is pretty up-to-date. When it comes to gauge theories, you cannot do better than this. If you want to commit to some book, you should pick this one for sure. Good luck!

Answer 2

Some people would say that those books cited in other answers are out-of-date. I have to disagree that Yang-Mills theory as presented in the 90's is outdated. In the last 30 years particle physics has not changed drastically. Perhaps the greatest news in this period of time were the discoveries of the Higgs particle, neutrino oscillations, top quark and tau neutrino. None of these changed our knowledge of gauge theories though.

As your post suggests you are more interested in gauge field theory and its mathematics formalism than in particle physics itself, thus a book like Baez & Muniain's (BM) cannot be outdated at all. Indeed this is one of the best places for a initial contact with differential geometry and topology for physicists. It is self contained, very well written, presenting the most important concepts in mathematics that are fundamental to the understanding of gauge theories. They avoid cumbersome proofs and prefer rather to sketch some results than to present them rigorously. This lack or mathematical rigor should not be a problem though since the aim of the book is just to introduce geometry and topology.

There is a book by B. Felsager: Geometry, Particles, and Fields which covers the mathematics of gauge theories exceptionally well. As well as Baez & Muniain, it presents geometry and topology but the focus here is more in calculations. It covers more topics compared to BM and it includes the basics of path integral quantization. If you want a rather newer book with similar topics, then try the 2013 The Geometry of Physics by T. Frankel. In my opinion both books are excellent and there is not any profound difference between them due to their age difference.

My favorite book on classical gauge theories is V. Rubakov's Classical Theory of Gauge Fields. The book dates 2002 and there is nothing we know today about classical gauge theories that is not covered there. Although the book does not put much emphasizes on geometry it is unique in many other aspects. For instance it gives thorough discussions about Lie algebras and groups (including representation theory), symmetries and the role of topology in gauge theories. It is mathematically self-contained, physically rigorous and detailed. Part of the book requires previous knowledge of Quantum Mechanics though. It has the style of lecture notes and focus and explicit calculations instead of phenomenology and hand waving arguments. It also has many good exercises, which keep showing up throughout the book. The only thing that bothers me is that for some reason the author decided to use all Lorentz indices down.

If you are interested in quantum aspects of gauge theories as well as particle physics, you can try Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, 2nd ed. by C. Quigg. This is definitely up to date from particle physics point of view and covers from classical field theory to particle physics, also presenting an introduction to Grand Unified Theories. I do not think it is self-contained though. You will not learn Quantum Field Theory from this book and it has more the style of phenomenological and pragmatic book: you know the Feynmann rules and then obtain all you can from that. It is a straightforward sequence to Rubakov's book.

If you are interested in Grand Unification, one of the best places to start still is the Slansky's revision paper Group Theory for Unified Model Building. It focus mainly in lie algebras and representation theory. It might be a bit old, but those mathematical topics didn't change since 1983.

Those books, together with a good Quantum Field Theory book will give you a solid understanding of High Energy Physics and should give you condition to work in the field.

Answer 3

The hardest problem in Yang-Mills theory is the problem of reduction of the gauge symmetry (redundancy); i.e. the characterization of the orbit space of gauge potentials modulo gauge transformations.

In 3+1 dimensions, this space is awfully complicated both geometrically and topologically. The solution of this problem should shed light to the longstanding open problems in Yang-Mills theory such as the mass gap and confinement. The reduced orbit space is infinite dimensional and not even a manifold.

If we knew a solution of this problem, we could, in principle, quantize the reduced configuration space (by means of geometric quantization, which is itself a formidable problem) and obtain the quantum Yang-Mills which should reflect the above conjectured properties.

Most of the known methods (except lattice regularization) use gauge fixing (together with Feddeev-Popov ghosts and BRST) to reduce the gauge redundancy. This method reduces the gauge redundancy only approximately as it suffers from the problem of Gribov copies. It is believed that this approximation is valid only in perturbation theory. There are several non-perturbative approximations such as the Faddeev-Niemi theory, but their connection to Yang-Mills is only heuristic.

Two physicists: Gurd Rudolf and Matthias Schmidt (together with a number of collaborators) are working on several methods to tackle this hard problem by reducing the gauge redundancy without gauge fixing. They have many publications on the subject. They use mainly methods of geometry and topology. Their efforts led to certain classification results of the yang-Mills gauge orbit. They wrote a book named Differential geometry and Mathematical physics (Part 1 , Part 2).

In the book, they give a detailed account of the basics of geometry and topology relevant to the Yang-Mills theory in a rigorous mathematical presentation. The entire book can be viewed, however, as an introduction to the last two chapters of Part 2 where they give account of some of their results in the classification and quantization of the Yang-Mills theory. This subject is very hard. Their research is still ongoing. They have results for certain toy models, such as a lattice of a single plaquette. They tackle both problems of the classical characterization of the orbit space and its quantization for these models. The book covers many advanced topics and can be a useful reference for a physicist interested in Yang-Mills theory research, and quantization.

Answer 4

A detailed exposition of the mathematical background material needed for gauge theory is contained in a book by Mark Hamilton, called "Mathematical Gauge Theory; With Applications to the Standard Model of Particle Physics". The last few chapters deal with the physically relevant applications to Yang-Mills theory.

Answer 5

One of my all-time favorite books is
Baez, J. and Muniain, J. P.: Gauge Fields, Knots and Gravity. World Scientific Publishing Co Inc, 1994.
Part II of the book is what you might be looking for, it explains the mathematical (geometrical) foundations of classical Yang-Mills theory. In my opinion, the book is very easily readable and goes into just enough mathematical detail; at the same time it does not forget to mention all the physically interesting applications.

If you want the maths in its full glory with more details, the standard reference is
Nakahara, M.: Geometry, Topology and Physics. CRC Press, 2003.
This book requires a bit more fighting to get through.