= Abelian Chern-Simons five-form field equation
{c}
{title2=$d{*F}=3cF\wedge F$}
For $F=dA$ on a five-dimensional oriented spacetime, the action $S=-\frac12\int F\wedge *F+c\int A\wedge F\wedge F$ has bulk variation $\int\delta A\wedge(-d*F+3cF\wedge F)$. Two integrations by parts make the three variations of the <Chern-Simons 5-form> contribute equally. Under $A\mapsto A+d\lambda$, the extra action is $c\int d(\lambda F\wedge F)$, a boundary term by the <Generalized Stokes theorem>. Its bulk field equation is gauge invariant. The Chern-Simons term is metric-independent and has no bulk contribution to the <stress-energy tensor> when the differential-form field is held fixed in metric variation.
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