Expanding $F_{\mu\nu}$, using antisymmetry of $\epsilon^{\mu\nu\rho\sigma}$, and cyclically rearranging the <matrix trace> gives
$$
\operatorname{Tr}(F_{\mu\nu}{}^\star F^{\mu\nu})
=\partial_\mu K^\mu,
$$
where the <Chern-Simons current> is
$$
K^\mu=2\epsilon^{\mu\nu\rho\sigma}
\operatorname{Tr}\left(
A_\nu\partial_\rho A_\sigma-\frac{2ig}{3}A_\nu A_\rho A_\sigma
\right).
$$
The theta contribution to the action is therefore a boundary term. For variations that vanish at the spacetime boundary, its variation vanishes, so it contributes nothing to the classical <Euler-Lagrange field equations>. Its integral can still distinguish topologically nontrivial gauge configurations in the quantum theory.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-305/2/iv/solution