 Yang-Mills theory coupled to an arbitrary scalar representation

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 $1/2$.