On Inherited Duality in N=1 d=4 Supersymmetric Gauge Theories

TL;DR

This paper reviews a field theory-based argument demonstrating SL(2,Z) invariance in certain 4D N=1 supersymmetric gauge theories with two adjoints, extending known duality properties without relying on AdS/CFT correspondence.

Contribution

It presents a novel, purely field-theoretic method to establish SL(2,Z) invariance in N=1 gauge theories with two adjoints, generalizing previous approaches.

Findings

Shows the N=1 theory shares the same auxiliary torus as N=4 gauge theory.

Confirms the SL(2,Z) invariance through the complexified flavor rotation technique.

Demonstrates the method's consistency with earlier SU(2) case results.

Abstract

Four-dimensional N=1 supersymmetric gauge theories with two adjoints and a quartic superpotential are believed, from AdS/CFT duality, to have SL(2,Z) invariance. In this note we review an old, unpublished argument for this property, based solely on field theory. The technique involves a complexified flavor rotation which deforms an N=2 supersymmetric gauge theory with matter to an N=1 theory, leaving all holomorphic invariants unchanged. We apply this to the N=1 gauge theory with two massless adjoints and show that it has the same auxiliary torus as that of N=4 gauge theory, from which SL(2,Z) invariance follows. In an appendix, we check that our arguments are consistent with earlier work on the SU(2) case. Our technique is general and applies to many other N=1 theories.