Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 44 / 4 / a / Solution

Use Hermitian generators $T^a$ of the special unitary group, with $[T^a,T^b]=i{f}^{ab}{}_c{T}^c$. An adjoint field is the Lie-algebra-valued matrix $\psi ={\psi }^a{T}^a$, transforming as $\psi \mapsto U\psi U^{-1}$. The adjoint covariant derivative isIn components, $(D_{\mu }\psi {)}^a={\partial }_{\mu }{\psi }^a+g{f}^{abc}{A}_{\mu }^b{\psi }^c$. With $A_{\mu }\mapsto U{A}_{\mu }{U}^{-1}-(i/g)({\partial }_{\mu }U)U^{-1}$, direct substitution gives $D_{\mu }\psi \mapsto U(D_{\mu }\psi )U^{-1}$. This is covariance in the Adjoint representation of a Lie group. In the following BRST symmetry formulas, absorb the coupling into the connection, so $D_{\mu }={\partial }_{\mu }-i[A_{\mu },\cdot ]$.