Noncommutative N=1 super Yang-Mills, the Seiberg-Witten map and UV divergences

TL;DR

This paper investigates the one-loop UV divergences of noncommutative N=1 super Yang-Mills theory under the Seiberg-Witten map, revealing renormalisability in the large N limit and for SU(N) theories, but not for finite N.

Contribution

It provides the first detailed one-loop analysis of UV divergences in noncommutative super Yang-Mills theories mapped to ordinary gauge theories, highlighting conditions for renormalisability.

Findings

Large N limit yields a renormalisable theory.

Finite N gauge sector is non-renormalisable.

SU(N) noncommutative theories are renormalisable.

Abstract

Classically, the dual under the Seiberg-Witten map of noncommutative U(N), {\cal N}=1 super Yang-Mills theory is a field theory with ordinary gauge symmetry whose fields carry, however, a \theta-deformed nonlinear realisation of the {\cal N}=1 supersymmetry algebra in four dimensions. For the latter theory we work out at one-loop and first order in the noncommutative parameter matrix \theta^{\mu\nu} the UV divergent part of its effective action in the background-field gauge, and, for N>=2, we show that for finite values of N the gauge sector fails to be renormalisable; however, in the large N limit the full theory is renormalisable, in keeping with the expectations raised by the quantum behaviour of the theory's noncommutative classical dual. We also obtain --for N>=3, the case with N=2 being trivial-- the UV divergent part of the effective action of the SU(N) noncommutative theory in the enveloping-algebra formalism that is obtained from the previous ordinary U(N) theory by removing the U(1) degrees of freedom. This noncommutative SU(N) theory is also renormalisable.