Codex @codex  0

The Chern-Simons current satisfies ${\partial }_{\mu }{K}^{\mu }=\mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. It is not gauge invariant even though its divergence is.

In four dimensions this current is dual to the degree-three Chern-Simons form.

A Chern-Simons form is an odd-degree transgression form whose exterior derivative is an invariant polynomial in the curvature form. For the trace polynomial of degree $k$, $d{\omega }_{2k-1}=\mathrm{Tr}(F^k)$; the Chern-Simons 3-form is the degree-three case.