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“Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations). By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis.
I was asked recently to explain what a gauge theory was, and so I will try to do so in this post. For simplicity, I will focus exclusively on classical gauge theories; quantum gauge theories are the quantization of classical gauge theories and have their own set of conceptual difficulties (coming from quantum field theory) that I will not discuss here. While gauge theories originated from physics, I will not discuss the physical significance of these theories much here, instead focusing just on their mathematical aspects. My discussion will be informal, as I want to try to convey the geometric intuition rather than the rigorous formalism (which can, of course, be found in any graduate text on differential geometry).
Next week (starting on Wednesday, to be more precise), I will begin my class on Perelman’s proof of the Poincaré conjecture. As I only have ten weeks in which to give this proof, I will have to move rapidly through some of the more basic aspects of Riemannian geometry which will be needed throughout the course. In particular, in this preliminary lecture, I will quickly review the basic notions of infinitesimal (or microlocal) Riemannian geometry, and in particular defining the Riemann, Ricci, and scalar curvatures of a Riemannian manifold. (The more “global” aspects of Riemannian geometry, for instance concerning the relationship between distance, curvature, injectivity radius, and volume, will be discussed later in this course.) This is a review only, in particular omitting any leisurely discussion of examples or motivation for Riemannian geometry; it is impossible to compress this subject into a single lecture, and I will have to refer you to a textbook on the subject for a more complete treatment (I myself am using the text “Riemannian geometry” by my colleague here at UCLA, Peter Petersen).