The gauge potential and field strength

We can then define the parallel transporter for matter fields to be a linear map \({\parallel_{C}\colon V_{p}\rightarrow V_{q}}\), where \({C}\) is a curve in \({M}\) from \({p}\) to \({q}\). Choosing a gauge, the parallel transporter can be viewed as a (gauge-dependent) map \({\parallel^{\beta}{}_{\alpha}\colon\left\{ C\right\} \rightarrow GL(n,\mathbb{C})}\). This determines the (gauge-dependent) matter field connection 1-form \({\Gamma^{\beta}{}_{\alpha}\left(v\right)\colon T_{x}M\rightarrow gl(n,\mathbb{C})}\), which can also be written when acting on a \({\mathbb{C}^{n}}\)-valued 0-form as \({\check{\Gamma}\left(v\right)\vec{\Phi}}\). The values of the parallel transporter are again viewed as a rep of the gauge group \({G}\), so that the values of the connection are a rep of the Lie algebra \({\mathfrak{g}}\), and if \({G}\) is compact we can choose a unitary gauge so that \({\mathfrak{g}}\) is represented by anti-hermitian matrices. We then define the gauge potential (AKA gauge field, vector potential) \({\check{A}}\) by

\(\displaystyle \check{\Gamma}\equiv-iq\check{A}, \)

where \({q}\) is called the coupling constant (AKA charge, interaction constant, gauge coupling parameter). Note that \({A^{\beta}{}_{\alpha}}\) are then hermitian matrices in a unitary gauge. The covariant derivative is then

\(\displaystyle \nabla_{v}\Phi=\mathrm{d}\vec{\Phi}\left(v\right)-iq\check{A}\left(v\right)\vec{\Phi}, \)

which can be generalized to \({\mathbb{C}^{n}}\)-valued \({k}\)-forms in terms of the exterior covariant derivative as

\(\displaystyle D\vec{\Phi}=\mathrm{d}\vec{\Phi}-iq\check{A}\wedge\vec{\Phi}. \)

For a matter field (0-form), this is often written after being applied to \({e_{\mu}}\) as

\(\displaystyle D_{\mu}\vec{\Phi}=\partial_{\mu}\vec{\Phi}-iq\check{A}{}_{\mu}\vec{\Phi}, \)

where \({\mu}\) is then a spacetime index and \({\check{A}_{\mu}\equiv\check{A}(e_{\mu})}\) are \({gl(n,\mathbb{C})}\)-valued components.

This connection defines a curvature \({\check{R}\equiv\mathrm{d}\check{\Gamma}+\check{\Gamma}\wedge\check{\Gamma}}\), which lets us define the field strength (AKA gauge field) \({\check{F}}\) by

\(\displaystyle \begin{aligned}\check{R} & \equiv-iq\check{F}\\ \Rightarrow\check{F} & =d\check{A}-iq\check{A}\land\check{A}. \end{aligned} \)

An Illustrated Handbook