The one-loop QCD scale anomaly on ADE orbifolds as an index

Abstract

We derive the one-loop scale anomaly of $\SU(N)$ Yang--Mills on the orbifold $\R^4/Γ$, with $Γ\subset \SU(2)$ a finite subgroup, from an equivariant index theorem on twistor space. The anomaly separates into the QCD $β$-function and a contribution localized at the orbifold fixed point. This localized term inherits the gravitational Weyl anomaly of the smooth ALE resolution, which concentrates under the singular degeneration, along with an additional unprotected heat-kernel coefficient.