Gauge theories in anti-selfdual variables

TL;DR

This paper proves that the one-loop 1PI effective action remains unchanged when gauge theories are reformulated in anti-selfdual variables, establishing a perturbative equivalence across different gauge theories.

Contribution

It demonstrates the perturbative equivalence of gauge theories in anti-selfdual variables and the original formulation at one-loop order, extending to all orders in perturbation theory.

Findings

One-loop 1PI effective actions are identical in original and anti-selfdual variables.

The Jacobian and functional determinants cancel each other, ensuring equivalence.

The equivalence likely extends to all orders in perturbation theory.

Abstract

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.