Gauge theories in anti-selfdual variables

Some years ago the Nicolai map, viewed as a change of variables from the gauge connection in a fixed gauge to the anti-selfdual part of the curvature, has been extended by the first named author to pure YM from its original definition in N=1 SUSY YM. We study here the perturbative 1PI effective action in the anti-selfdual variables of any gauge theory, in particular pure YM, QCD and N=1 SUSY YM. We prove that the one-loop 1PI effective action of a gauge theory mapped to the anti-selfdual variables in any gauge is identical to the one of the original theory. This is due to the conspiracy between the Jacobian of the change to the anti-selfdual variables and an extra functional determinant that arises from the non-linearity of the coupling of the anti-selfdual curvature to an external source in the Legendre transform that defines the 1PI effective action. Hence we establish the one-loop perturbative equivalence of the mapped and original theories on the basis of the identity of the one-loop 1PI effective actions. Besides, we argue that the identity of the perturbative 1PI effective actions extends order by order in perturbation theory.