Chern-Simons current ($K^{\mu }$)

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.