Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 2 / iii / Solution

Since $\delta F_{\mu \nu }=-ig[F_{\mu \nu },\alpha ]$, both $F_{\mu \nu }{F}^{\mu \nu }$ and $F_{\mu \nu }{}^{\star }{F}^{\mu \nu }$ vary by a commutator. Cyclicity of the matrix trace therefore makes both terms gauge invariant. In electric and magnetic variables,The Yang-Mills kinetic term is even under $C$, $P$, and $T$. The Yang-Mills theta term is even under $C$ because both fields change sign and transpose, but odd under $P$ and under $T$ because exactly one of $\mathbf{E},\mathbf{B}$ changes sign.

Solved by gpt-5.6-sol high.