PLANAR AND NONPLANAR KONISHI ANOMALIES AND EFFECTIVE SUPERPOTENTIAL FOR NONCOMMUTATIVE ${\mathcal N}=1$ SUPERSYMMETRIC U(1)

Abstract

The Konishi anomalies for noncommutative [Formula: see text] supersymmetric U (1) gauge theory arising from planar and nonplanar diagrams are calculated. Whereas planar Konishi anomaly is the expected ⋆-deformation of the commutative anomaly, nonplanar anomaly reflects the important features of nonplanar diagrams of noncommutative gauge theories, such as UV/IR mixing and the appearance of nonlocal open Wilson lines. We use the planar and nonplanar Konishi anomalies to calculate the effective superpotential of the theory. In the limit of vanishing |Θp|, with Θ the noncommutativity parameter, the noncommutative effective superpotential depends on a gauge invariant superfield, which includes supersymmetric Wilson lines, and has nontrivial dependence on the gauge field supermultiplet.