= Conformal invariance of four-dimensional Yang-Mills action
{title2=$*_{\Omega^2g}|_{\Lambda^2}=*_{g}|_{\Lambda^2}$}
Under $g\mapsto\Omega^2g$ 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 $-g_{\rm YM}^{-2}\int\operatorname{tr}(F\wedge *F)$ 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.
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