Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 46 / 4 / a / Solution

Choose Lie-algebra generators with $[T_b,T_c]=f^a{}_{bc}{T}_a$ and an invariant orthonormal color metric. Since the coupling is placed outside the gauge kinetic term, it is absorbed into $A_{\mu }$ in the connection convention. The gauge field strength and adjoint covariant derivative areThe structure constants are antisymmetric in $b,c$ and obey the Jacobi identity. These definitions give $[D_{\mu },D_{\nu }]v=[F_{\mu \nu },v]$ on an adjoint-valued field. The field $c$ is a Grassmann-odd Faddeev-Popov ghost field; the Lie bracket in $D_{\mu }c$ contracts its color components in the same way as for any adjoint field.