  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.
