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) \equiv (9 Tr R^3-3 Tr R)/32. The maximal value of a_{trial} is then, by a result of Anselmi et. al., the central charge \it{a} of the SCFT. Our a_{trial} maximization principle almost immediately ensures that the central charge \it{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 \it{a} and \it{c}, are always algebraic numbers.