## Electrodynamics on a Principal Bundle I

I want to switch gears and talk about some mathematical physics. Actually, I’m going to cross-post some exposition I wrote for a gauge theory seminar that we held at Stony Brook.

Maxwell’s equations in relativistically covariant form are

$\partial_{\mu}F^{\mu\nu}=J^{\nu}$
$\partial_{[\lambda}F_{\mu\nu]}=0$

Since $F_{\mu\nu}=-F_{\nu\mu}$ we can define a 2-form $F=F_{\mu\nu}dx^{\mu}dx^{\nu}$. We can also define a 1-form $J=J_\mu dx^\mu$. Then we can re-express Maxwell’s equations using exterior differentiation and the Hodge star.

$d*F=*J$
$dF=0$

The continuity equation $d*J=0$ then follows from the inhomogeneous Maxwell equation. We expect from the homogeneous Maxwell equation that $F=dA$. In fact this is only true locally. This means that for every event $m$ in our spacetime $M$ there is an open set $U$ with $m\in U\subset M$ and a 1-form $A_U$ on $U$ with $F|_U=dA_U$. This follows from Poincare’s lemma.

We cannot say the $A$ exists globally. For instance if $F=sin\phi d\phi d\theta$, the area form of the unit sphere in spherical coordinates, then $dF=cos\phi d\phi d\phi d\theta=0$ since $d\phi d\phi=0$ by antisymmetry of wedge product of 1-forms. Also, taking $\Sigma$ to be the unit sphere, we know that $\int_\Sigma F=4\pi$. However, by Stokes’ Theorem, if $F=dA$ then $\int_\Sigma F=\int_\Sigma dA=\int_{\partial \Sigma} A=0\neq 4\pi$. So, we cannot have $F=dA$ globally.

Physically we interpret this as a magnetic monopole with magnetic charge $4\pi$ and worldline, the time axis, $r=0$. Mathematically, what is happening is that the complement of the time axis has nontrivial topology. Specifically its second de Rham cohomology is nontrivial. Intuitively, there is a kind of 2-dimensional, spherical “hole” in the complement of the time axis.

In addition to being nonglobal, the potential $A$ is defined only up to addition of a closed 1-form since $d(A+d\lambda)=dA=F$. We would like to find a global mathematical object corresponding to the potential which doesn’t depend on our “choice of gauge”. This is our motivation for understanding connections on principal bundles.

We will assume $G$ is a group of matrices. A principal $G$-bundle is a smooth surjection of manifolds $\pi:P\to M$ with a free transitive right action $R$ of $G$ on $P$ such that $\pi^{-1}\pi(p)=pG$ and for any $m\in M$ there is an open set $U$ with $m\in U\subset M$ and a diffeomorphism $T_U=\pi\times t_U:\pi^{-1}(U)\to U\times G$ called a “local trivialization” such that $t_U(pg)=t_U(p)g$. Local trivializations correspond to the physical notion of “choice of gauge”.

Intuitively, $P$ is a manifold composed of copies of the group $G$ parametrized by the base space $M$. A good example is the boundary of the Mobius strip which can be thought of as a $\mathbb{Z}/2$-bundle over $S^1$.

A useful notion is that of a local section $\sigma_U:U\to P$ with $U$ an open set with $U\subset M$ such that $\pi(\sigma_U(m))=m$. It can be shown that there is a canonical 1-1 correspondence between local sections $\sigma_U$ and local trivializations $T_U$.

Define transition functions $g_{UV}:U\cap V\to G$ by $g_{UV}(m)=t_U(p)t_V(p)^{-1}$ where $\pi(p)=m$. This is well defined since $t_U(pg)t_V(pg)^{-1}=t_U(p)gg^{-1}t_V(p)^{-1}=t_U(p)t_V(p)^{-1}$. Transition functions correspond to the physical notion of “change of gauge”. We can relate any two local sections by $\sigma_V=\sigma_U g_{UV}$.

Let $\mathfrak{g}$ be the Lie algebra for $G$. A connection $\omega$ is a $\mathfrak{g}$-valued 1-form on $P$ such that if If $X\in\mathfrak{g}$ and $\tilde{X}$ is the tangent field on $P$ given by $\tilde{X}_{p}=\frac{d}{dt}pe^{tX}|_{t=0}$, then $\omega(\tilde{X})=X$. Also we require that $R(g)^{*}(\omega)=g^{-1}\omega g$.

We define local connections on $M$ by $\omega_U=\sigma_U^*\omega$. Local connections are related by $\omega_{V}=g_{UV}^{-1}\omega_{U}g_{UV}+g_{UV}^{-1}dg_{UV}$.

We define curvature $\Omega=d\omega+\frac{1}{2}[\omega,\omega]$ meaning $\Omega(X,Y)=d\omega(X,Y)+\frac{1}{2}[\omega(X),\omega(Y)]$. We can define local curvature by $\Omega_U=\sigma_U^*\Omega$. Local curvatures are then related by $\Omega_V=g_{UV}^{-1}\Omega_U g_{UV}$. The Bianchi identity says $d\Omega=[\omega,\Omega]$.

We are now in a position to define electrodynamics on a principal bundle.