Resources on non-Abelian gauge transformations and etc

Question

I am currently studying topological insulators and occasionally the concept of non-Abelian phases and non-Abelian gauge transformations comes up in related literature. However, my understanding of these concepts are hazy and inadequate. From what I understand they are related to gauge theory.

Can anyone recommend any introductory resources that I can read to begin to understand these topics, specifically in the context of TI? I would also appreciate it if you could outline how these concepts and any others that are pertinent, relate to TI and other subjects such as QFT.

Answer 1

To get acquainted with gauge theory, the easiest references I know are

• Aitchison, I. J. R., & Hey, A. J. G. (2004). Gauge Theories in Particle Physics (First volume: From relativistic quantum mechanics to QED ; Second volume: QCD and the electroweak theory) (Third Edition). Taylor and Francis.

• Rubakov, V. (2002). Classical theory of gauge fields. Princeton University Press.

for physicist audience. For an introduction to the topic, do not worry too much about the problem of quantizing non-Abelian gauge theory and renormalisation problems, especially if you're interested in using it in condensed matter systems. If you are nevertheless interested in these topics, I suggest

• Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Westview Press.

• Zinn-Justin, J. (2002). Quantum field theory and critical phenomena (4–th ed.). Oxford University Press.

For an esay to read introduction to the use of gauge formalism in quantum field theory, check

• Zee, A. (2003). Quantum field theory in a Nutshell (1st ed.). Princeton University Press.

If you are more mathematically inclined, perhaps you may have a look on

• DeWitt-Morette, C., & Choquet-Bruhat, Y. (1996). Analysis, Manifolds and Physics, Part I and Part II (two volumes). Elsevier Science Publishing.

• Frankel, T. (2012). The Geometry of Physics: An Introduction. (C. U. Press, Ed.) (3rd ed.).

• Nakahara, M. (2003). Geometry, topology and physics. Insitute of Physics Publishing.

where gauge theory is built on the mathematical notion of fiber bundle. The historical reference about fiber bundle in mathematics is

• Steenrod, N. (1951). The topology of fiber bundles. Princeton University Press.

and they are reviewed in

• Daniel, M., & Viallet, C. (1980). The geometrical setting of gauge theories of the Yang-Mills type. Reviews of Modern Physics, 52(1), 175–197.

• Chern, S. S., Chen, W. H., & Lam, K. S. (1999). Lectures on Differential Geometry. World Scientific Publishing Co. Pte. Ltd.

For historical survey of gauge-theory and its introduction to physics, have a look on

• O’Raifeartaigh, L. (1997). The dawning of gauge theory. Princeton University Press.

• O’Raifeartaigh, L., & Straumann, N. (2000). Gauge theory: Historical origins and some modern developments. Reviews of Modern Physics, 72(1), 1–23.

• Hooft, G. ’t. (2005). 50 Years Of Yang-Mills Theory. (G. ’t Hooft, Ed.). World Scientific Publishing Co. Pte. Ltd.

Finally, the connection to topological matter is through the concept of (an)holonomy, or geometric / Berry / Zak / Thouless / Simon phase, which allow the calculation of topological invariants (also called winding numbers, Chern numbers, Pontryajin numbers). For more details, see the pedagogical ressources from Les Houches Summer School: Topological aspects of condensed matter physics. I suggest the lectures by Moore and Bernevig to start with.

There are recent reviews about topological materials, as

• Hasan, M. Z., & Kane, C. L. (2010). Colloquium: Topological insulators. Reviews of Modern Physics, 82(4), 3045–3067.

• Qi, X., & Zhang, S. (2011). Topological insulators and superconductors. Reviews of Modern Physics, 83(4), 1057–1110. arXiv:1008.2026

• Shun-Qing Shen (2012). Topological insulators. Dirac equation in condensed matters. Springer.

• Bernevig, B. A., & Hughes, T. L. (2013). Topological Insulators and Topological Superconductors. Princeton University Press.

which are perhaps the best way to start your readings if you're interested in topological materials. The connection to gauge theory is not much clear in most of these reviews, though.

An alternative road to the topological concepts is through the book

• Volovik, G. E. (2003). Universe in a Helium Droplet. Oxford University Press. Available from the author's webpage.

Note it's not easy to read at all ...