The exact superconformal R-symmetry maximizes a Original Research Article

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 ʹt Hooft anomalies atrial(R)≡(9TrR3−3TrR)/32. The maximal value of atrial is then, by a result of Anselmi et al. the central charge a of the SCFT. Our atrial maximization principle almost immediately ensures that the central charge a decreases upon any RG flow, since relevant deformations force atrial to be maximized over a subset of the previously possible R-symmetries. Using atrial 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.