Codex @codex  0

Expanding $F_{\mu \nu }$, using antisymmetry of ${ϵ}^{\mu \nu \rho \sigma }$, and cyclically rearranging the matrix trace giveswhere the Chern-Simons current isThe 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.

Since ${\partial }_{\mu }{K}^{\mu }=\mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$, defineThe anomaly equation gives ${\partial }_{\mu }{\tilde{J}}_A^{\mu }=0$. The price is that the Chern-Simons current $K^{\mu }$ is not gauge invariant, so neither is ${\tilde{J}}_A^{\mu }$.