Yang-Mills theory coupled to an arbitrary scalar representation ($\mathcal{L}=-B(\mathcal{F},\mathcal{F})/(4g^2)+h(D\Phi ,D\Phi )-V$)

For a compact simple gauge group, use $B=-\kappa$ on its algebra and an averaged positive invariant Hermitian form $h$ on any supplied finite-dimensional complex representation. With anti-Hermitian connection and signature $(+---)$, $\mathcal{L}=-B({\mathcal{F}}_{\mu \nu },{\mathcal{F}}^{\mu \nu })/(4g^2)+h(D_{\mu }\Phi ,D^{\mu }\Phi )-V(\Phi )$ is gauge-invariant whenever $V$ is invariant. A real scalar representation instead has a symmetric form and kinetic prefactor $1/2$.

Take $D_{\mu }X={\partial }_{\mu }X+[A_{\mu },X]$, signature $(+---)$ and the trace Lagrangian density $\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })/4-\mathrm{Tr}(D_{\mu }\Phi D^{\mu }\Phi )/2+m^2\mathrm{Tr}({\Phi }^2)/2$. The variation of the gauge curvature is $D_{\mu }\delta A_{\nu }-D_{\nu }\delta A_{\mu }$. Cyclicity of the trace and covariant integration by parts give gauge coefficient $-D_{\mu }{F}^{\mu \nu }+[D^{\nu }\Phi ,\Phi ]$ and scalar coefficient $D_{\mu }{D}^{\mu }\Phi +m^2\Phi$. Their vanishing gives the displayed Euler-Lagrange field equations. In static temporal gauge at zero mass they reduce to $D_j{F}_{ji}=[\Phi ,D_i\Phi ]$ and $D_i{D}_i\Phi =0$.

// Destination: integrable-systems.bigb

A real potential $V$ for a scalar field is invariant when $V(R(U)\Phi )=V(\Phi )$ for every gauge transformation. In a unitary representation, $m^{2}h(\Phi ,\Phi )+\lambda h(\Phi ,\Phi {)}^2$ is always an example. Particular representations may admit additional invariant interactions.