\def\Z{{\bf Z}}
\def\U{{\rm U}}

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 $\exp(-i\hbar^{-1}q\int_\gamma A)$ 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: $\gamma\mapsto\exp(i\int_\gamma A)$.
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:
$D'(\gamma)=g(\gamma(t))D(\gamma)g^{-1}(\gamma(0))$.
Here $\gamma\colon[0,t]\to M$ is a path and $g\colon M\to U(1)$ is a gauge transformation.
We have $D'(\gamma)=\exp(i\int_\gamma(A+df))$, where $g=\exp(if)$.

Notice that holonomy is gauge invariant
(a)~because $\int_\gamma df=0$ by Stokes;
(b)~if $\gamma(t)=\gamma(0)$, then $D'(\gamma)=D(\gamma)$ 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 $B\to M$ we have $D(\partial B)=\exp(i\int_B F)$.

We have exact sequences $0\to H^1(\U(1))\to R\to\Omega^2_\Z\to 0$.
$0\to\Omega^1_\Z\to\Omega^1\to R\to H^2(\Z)\to 0$.
We have maps $H^1(\U(1))\to H^2(\Z)$ and $d\colon\Omega^1\to\Omega^2_\Z$.

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

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