Electrodynamics on a principal bundle II « Gauge Theory and All Her Friends

Electrodynamics on a principal bundle II

By Eitan

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.

This entry was posted on August 4, 2008 at 2:48 PM and is filed under Gauge theory, Geometry. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

2 Responses to “Electrodynamics on a principal bundle II”

mbrandonw Says:
August 5, 2008 at 8:58 AM | Reply
You need the action of G to be transitive too. For example, Z acts freely (but not transitively) on R^2 by translating vertically. Let pi : R^2 –> R be the quotient map of this action. Then pi is a fiber bundle with fiber [0,1), but clearly is not a principal Z-bundle.
Eitan Says:
August 5, 2008 at 9:48 AM | Reply
Yeah, guesso. I was hoping transitivity comes out of it somehow. But there’s no reason it should.

Gauge Theory and All Her Friends

Electrodynamics on a principal bundle II

2 Responses to “Electrodynamics on a principal bundle II”

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