resources for classical gauge theory

Question

As a prospective grad student, I would like to get an entry level introduction to classical (i.e. non-quantum) gauge theory. Please direct me to resources suitable for a novice.

Answer 1

Before saying what I think are good introductions to mathematical gauge theory, I should say what I think gauge theory is. What gauge theory means to me is the application of certain PDEs, relevant in physics, to the topology and geometry of manifolds. I note here that one can be ignorant of the actual physics (as I am). Here is one example.

Given a Riemannian 4-manifold $(M,g)$, one can write down a certain PDE called the "ASD equation", whose input is a connection on a $G$-bundle over $M$. (Usually $G=SU(2), U(2), SO(3).$) This equation has a rather large group of symmetries: that is, there is an (infinite-dimensional) Lie group $\mathcal G$ with an action on the space of connections $\mathcal A$ such that, if $A$ is a solution to the ASD equation, so is $g(A)$; so we usually consider it as an equation on the quotient space $\mathcal A/\mathcal G$. (The appearance of the word gauge in all this is that $\mathcal G$ is called the group of gauge transformations, and we're interested in solutions modulo gauge equivalence.) If you're lucky, the space of solutions in this quotient is 0-dimensional, and you can take a (signed) count of the points. Essentially, this is an invariant of the underlying smooth manifold $M$ - it did not depend on the metric $g$! This allows us to tell apart manifolds that are homeomorphic but not diffeomorphic; for instance, there is a countable infinity of smooth manifolds $D_q$, pairwise non-diffeomorphic, but all homeomorphic to $\Bbb{CP}^2 \# 9\overline{\Bbb{CP}^2}.$ Invariants from gauge theory or its cousins are thusfar the only known way to tell apart smooth 4-manifolds that are homeomorphic.

In addition to the ASD equation, also useful are the Seiberg-Witten equations; equations that might turn out to be useful (but are still in the early stages) include the Vafa-Witten equations and the Haydys-Witten equations. There are associated theories in 3 dimensions called Floer theories that are active areas of research; in 2013 Manolescu disproved the Triangulation conjecture with a Floer-type theory with just slightly more symmetry than the better-known Seiberg-Witten Floer homology. Donaldson has a program to apply these ideas to the study of Riemannian manifolds with $SU(3), \text{Spin}(7)$, and $G_2$ holonomy; this is also still early (no actual invariants have been defined yet, I don't think) but active.

If one wanted to learn some of mathematical gauge theory, I think the best place to start is with the Seiberg-Witten equations, which have abelian gauge group (which simplifies quite a lot of the technical difficulty). My favorite introduction is Salamon's extensive notes on Seiberg-Witten theory; it's self-contained, well-written, and contains a lot of material that's harder to find in other introductions. The canonical (well, only) reference for the ASD equations in 4 dimensions is Donaldson and Kronheimer's book "The Geometry of 4-Manifolds". I would say this is significantly harder than the Salamon notes.

As a last point, note again that this is highly influenced by physics, but everything I've said here is pretty disjoint from having to know it - I know no physics except for the mechanics course I took in undergrad 6 years ago, and this hasn't been an impairment.