= Maxwell–Chern–Simons one-form equation
{c}
{title2=$d{*F}+kF=0$}
For a <differential one-form> $A$ on an oriented three-dimensional spacetime, varying $\frac12\int(F\wedge*F+kA\wedge F)$, with $F=dA$, gives $d*F+kF=0$. With signature $(-,+,+)$, ${*}^2=-1$ on one- and two-forms; writing $B=*F$ gives $dB-k*B=0$ and the identity $d*B=0$. An Abelian <gauge transformation> changes the action by $\frac k2\int_{\partial M}\Lambda F$. Thus the bulk field equation is invariant, while invariance of the integrated action depends on boundary conditions.
Back to article page
