On inherited duality in N = 1 d = 4 supersymmetric gauge theories

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 In this note we review an old, unpublished argument for this property, based solely on field theory.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 the 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.