Chern-Simons current
= Chern-Simons current
{c}
{title2=$K^\mu$}
The Chern-Simons current satisfies $\partial_\mu K^\mu=\operatorname{Tr}(F_{\mu\nu}{}^\star F^{\mu\nu})$. It is not gauge invariant even though its divergence is.
In four dimensions this current is dual to the degree-three <Chern-Simons form>.