= Self-dual primitive of an exact three-form
{title2=$\sigma=\delta\psi+*\delta\psi,\quad d\sigma=\beta$}
On a compact oriented <Riemannian manifold> of dimension four without boundary, every exact three-form has a <self-dual two-form> primitive. Write $\beta=d\omega$ and use the <Hodge decomposition theorem> to split $\omega=h+d\xi+\delta\psi$. Then $\beta=d\delta\psi$. For $a=\delta\psi$, the <codifferential> identity $\delta=-*d*$ and the fact $*^2=1$ on two-forms give $*a=-d*\psi$, so $d*a=0$. Thus $\sigma=a+*a$ satisfies $*\sigma=\sigma$ and $d\sigma=\beta$. It is twice the self-dual projection of $a$, rather than the projection itself.
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