Hodge star on middle-degree differential forms is conformally invariant

If a metric in dimension $n$ is rescaled by $\tilde{g}=e^{2\omega }g$, then its Hodge star on $p$-forms satisfies $\tilde{\ast }=e^{(n-2p)\omega }\ast$. The exponent vanishes for middle-degree forms with $n=2p$, so their self-dual and anti-self-dual subspaces depend only on the conformal class.