Let's say we have a principal bundle $(P,B,\pi;G)$ and associated bundle $E=P \times_{(G,\rho)}V$and $Ad(P)=P\times_{(G,Ad)} \mathfrak{g}$ the adjoint bundle. The Yang-Mills-Higgs action (without potential) is

\begin{equation} \int_M(- \frac{1}{2}\langle F^A, F^A \rangle_{\operatorname{Ad}(P)} +\langle d_A \phi, d_A \phi \rangle_E - m^2 \langle \phi, \phi \rangle_E )d\nu_g \end{equation} with $\phi \in \Gamma(E)$. If we vary the equation, i.e. add $A+t \omega$ and $\phi+t \alpha$ and then take $d/dt$ the equations of motion are

\begin{equation} \delta_A F^A=j \quad \delta_A d_A \phi=0 \end{equation} with the codifferential $\delta_A$ and the implicit defined current

\begin{equation} \langle j, \alpha \rangle = -\langle d_A \phi,\rho_*(\alpha) \phi \rangle \end{equation} In Differential Geometry And Mathematical Physics 2 by Gerd Rudolph and Matthias Schmidt it says that if we take the associated bundle, i.e. $E$ to be the adjoint bundle, the first equation of motion becomes

\begin{equation} \delta_A F^A=[d_A \phi,\phi] \end{equation}

I have two equations:

- How can I derive $j=[d_A \phi,\phi]$ form the general equation for the current. And how do I express it in local coordinates to get something similar to the currents in physics.
- Is there some Binachi-identity for the curvature form with the codifferential, i.e. $\delta_A \delta_A F^A=0$. This somehow should be the case, since the current should be conserved, i.e. $\delta_A j=0$

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