Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 4 / i

Choose Hermitian Lie-algebra generators satisfying $[T^a,T^b]=i{f}^{abc}{T}^c$, with real totally antisymmetric Lie algebra structure constants in an orthonormal compact-group basis. The Adjoint representation generators are $(T_{\mathrm{ad}}^b{)}_{ac}=-i{f}^{bac}$. For a canonically normalized Yang-Mills action, the coupling is in the gauge covariant derivative, rather than an overall $1/g^2$ kinetic prefactor. The conventions $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }^b{T}_{\mathrm{ad}}^b$ giveEquivalently, for the Lie-algebra-valued spinor, $D_{\mu }\psi ={\partial }_{\mu }\psi -ig[A_{\mu },\psi ]$. This is the adjoint covariant derivative with the Hermitian-generator and explicit-coupling convention specified. Changing the sign convention for $D$ changes the accompanying gauge-transformation convention consistently.