Conformal invariance of four-dimensional Yang-Mills action (${\ast }_{{\Omega }^{2}g}{|}_{{\Lambda }^2}={\ast }_g{|}_{{\Lambda }^2}$)

Under $g\mapsto {\Omega }^{2}g$ in dimension $d$, the Hodge star on $p$-forms scales by ${\Omega }^{d-2p}$. For gauge gauge curvature, $d=4,p=2$, so the star and the action $-g_{\mathrm{YM}}^{-2}\int \mathrm{tr}(F\wedge \ast F)$ are unchanged. Dilation therefore changes the instanton size modulus without changing the classical action. Quantum running of the coupling can break this classical scale invariance.