Maxwell–Chern–Simons one-form equation ($d\ast F+kF=0$)

For a differential one-form $A$ on an oriented three-dimensional spacetime, varying $\frac{1}{2}\int (F\wedge \ast F+kA\wedge F)$, with $F=dA$, gives $d\ast F+kF=0$. With signature $(-,+,+)$, ${\ast }^2=-1$ on one- and two-forms; writing $B=\ast F$ gives $dB-k\ast B=0$ and the identity $d\ast B=0$. An Abelian gauge transformation changes the action by $\frac{k}{2}{\int }_{\partial M}\Lambda F$. Thus the bulk field equation is invariant, while invariance of the integrated action depends on boundary conditions.