The exact superconformal R-symmetry maximizes a

Abstract

An exact and general solution is presented for a previously open problem. We show that the superconformal R-symmetry of any 4d SCFT is exactly and uniquely determined by a maximization principle: it is the R-symmetry, among all possibilities, which (locally) maximizes the combination of 't Hooft anomalies a trial( R)≡(9Tr R 3−3Tr R)/32. The maximal value of a trial is then, by a result of Anselmi et al. the central charge a of the SCFT. Our a trial maximization principle almost immediately ensures that the central charge a decreases upon any RG flow, since relevant deformations force a trial to be maximized over a subset of the previously possible R-symmetries. Using a trial maximization, we find the exact superconformal R-symmetry (and thus the exact anomalous dimensions of all chiral operators) in a variety of previously mysterious 4d N=1 SCFTs. As a check, we verify that our exact results reproduce the perturbative anomalous dimensions in all perturbatively accessible RG fixed points. Our result implies that N=1 SCFTs are algebraic: the exact scaling dimensions of all chiral primary operators, and the central charges a and c, are always algebraic numbers.