Nate Watson. Electromagnetism to Yang-Mills theory. Premotivation: Yang-Mills theory is about connecting (mediating) particles (photon, gluon, etc.). Fields

are connections on principal G-bundles, where G is the gauge group of a particle under consideration. For electromagnetism we have G = U(1). Old F is the curvature of this connection.

Bohm-Aharanov effect: We have a long solenoid with magnetic field inside and no electric field outside. However we can see an interference pattern for electrons. This comes from a term exp(−ih̄^{−1}q ∫_{γ} A) in path integral, where A is the vector potential. Since the integral around solenoid is nonzero, we have interference.

So locally we get a function that assigns phases to paths in M. This is just like a connection on a trivialized bundle: γ ↦ exp(i ∫_{γ} A). Problems: (a) A + df gives different function; (b) We cannot expect a canonical trivialization.

We have gauge symmetry for connections. We can construct a new connection as follows: D′(γ) = g(γ(t))D(γ)g^{−1}(γ(0)). Here γ: [0, t] → M is a path and g: M → U(1) is a gauge transformation. We have D′(γ) = exp(i ∫_{γ}(A + df)), where g = exp(if).

Notice that holonomy is gauge invariant (a) because ∫_{γ} df = 0 by Stokes; (b) if γ(t) = γ(0), then D′(γ) = D(γ) because U(1) is abelian.

In fact, holonomy uniquely characterizes connections up to gauge symmetry. F is a unique 2-form such that for a disc B → M we have D(∂B) = exp(i ∫_{B} F). We have exact sequences 0 → H^{1}(U(1)) → R → Ω_{Z}^{2} → 0. 0 → Ω_{Z}^{1} → Ω^{1} → R → H^{2}(Z) → 0. We have

maps H^{1}(U(1)) → H^{2}(Z) and d: Ω^{1} → Ω_{Z}^{2}.

References: Baez-Munian and Freed’s paper on differential cohomology and Maxwell’s equations.