The local renormalization of super-Yang-Mills theories

We show how to consistently renormalize $mathcal{N} = 1$ and $mathcal{N} = 2$ super-Yang-Mills theories in flat space with a local (i.e. space-time-dependent) renormalization scale in a holomorphic scheme. The action gets enhanced by a term proportional to derivatives of the holomorphic coupling. In the $mathcal{N} = 2$ case, this new action is exact at all orders in perturbation theory.