= Eleven-dimensional three-form action variation
{title2=$d\star F+3F\wedge F=0$}
For a <differential three-form> $A$ with $F=dA$ on an oriented eleven-dimensional <manifold>, varying $\int(\tfrac12F\wedge\star F+F\wedge F\wedge A)$ with fixed metric gives $\delta S=\int\delta A\wedge(d\star F+3F\wedge F)$ up to a boundary term. Indeed $d(\delta A\wedge F\wedge A)=d\delta A\wedge F\wedge A-\delta A\wedge F\wedge F$, so the two differentiated $F$ factors and the differentiated $A$ factor contribute equally. The shift $A\mapsto A+d\Lambda$ changes the density by $d(F\wedge F\wedge\Lambda)$, preserving the field equation.
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