Nate Watson. Yang-Mills theories. Last time: Electromagnetic field comes from the curvature of a connection D on a U(1)-bundle. Idea: Bohm-Aharonov effect comes from exp(−(i/h̄)qS(γ). Electromagnetic field assigns phase to a path in spacetime like a connection on a trivial bundle. The interference depends on holonomy along the difference of paths. If B is a disk in M we have exp(i ∫_{B} F) = D(∂B). Action ∫_{M} F ∧∗F gives Maxwell’s equation d∗F = 0.

dF = 0 is automatic.

Recall the character diagram from the last talk. Holonomies are maps LM → U(1) that are holonomies of some connection on a U(1)-bundle. Theorem: f: LM → U(1) is a holonomy if there exists a 2-form F such that for any disk B we have

exp(i ∫_{B} F) = f(∂B).

Charge quantization. Given an electromagnetic field F with some magnetic charge at 0 we expect flux ∫_{S_{2}} B·dη = ∫_{S_{2}} F, which is impossible for contractible manifold because F is closed. Thus the theory excludes magnetic charges for contractible manifolds. But for non-contractible manifolds we can have magnetic charges. Now ∫_{S_{2}} F can be nonzero and sees nontriviality of the bundle and is a multiple of 2π. We set h̄/q = 1 so that m = N · 2π · h̄/q hence magnetic charge is quantized in terms of any electric charge and vice versa.

Now we want to replace U(1) by a non-abelian Lie group G. Take an associated bundle (need representation of G on a generic fiber). We have a notion of a G-connection on the associated bundle E → M. The curvature of this connection is an End(E)-valued 2-form with values in g (the Lie algebra of G). Curvature is not gauge invariant. We need a number, but F ∧ ∗F is an End(E)-valued n-form. We should apply the trace. Yang-Mills action is ∫_{M} tr(F ∧ ∗F). This action is gauge invariant!