Canonical normalization of a gauge kinetic term ($\mathcal{A}=\sqrt{\mathcal{N}}A,g_c=g/\sqrt{\mathcal{N}}$)

For $\mathcal{L}=-\mathcal{N}{F}_{\mu \nu }^a{F}^{a\mu \nu }/4$ with $\mathcal{N}>0$, rescale ${\mathcal{A}}^a=\sqrt{\mathcal{N}}{A}^a$ and $g_c=g/\sqrt{\mathcal{N}}$. The gauge field strength then has canonical component normalization. Masses inferred from matter couplings must use $g_c$, not an unnormalized coefficient. A matrix-trace convention $\mathrm{Tr}(t^a{t}^b)=\mathcal{N}{\delta }^{ab}$ fixes this factor.