\def\U{{\rm U}}
\def\End{\mathop{\rm End}}
\def\tr{\mathop{\rm tr}}

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 $\exp(-(i/\hbar)qS(\gamma)$.

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 $\exp(i\int_BF)=D(\partial B)$.
Action $\int_MF\wedge*F$ gives Maxwell's equation $d*F=0$.
$dF=0$ is automatic.

Recall the character diagram from the last talk.

Holonomies are maps $LM\to\U(1)$ that are holonomies of some connection on a $\U(1)$-bundle.

Theorem: $f\colon LM\to\U(1)$ is a holonomy if there exists a 2-form~$F$
such that for any disk~$B$ we have $\exp(i\int_BF)=f(\partial B)$.

Charge quantization.
Given an electromagnetic field~$F$ with some magnetic charge at~0
we expect flux $\int_{S_2}B\cdot d\eta=\int_{S_2}F$,
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 $\int_{S_2}F$ can be nonzero and sees nontriviality of the bundle
and is a multiple of~$2\pi$.
We set $\hbar/q=1$ so that $m=N\cdot2\pi\cdot\hbar/q$
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~$E\to M$.
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 $F\wedge*F$ is an $\End(E)$-valued $n$-form.
We should apply the trace.
Yang-Mills action is $\int_M\tr(F\wedge*F)$.
This action is gauge invariant!

\bye