The Exact Superconformal R-symmetry Minimizes $\tau_{RR}$

Abstract

We present a new, general constraint which, in principle, determines the superconformal $U(1)_R$ symmetry of 4d $\N =1$ SCFTs, and also 3d $\N =2$ SCFTs. Among all possibilities, the superconformal $U(1)_R$ is that which minimizes the coefficient, $\tau_{RR}$, of its two-point function. Equivalently, the superconformal $U(1)_R$ is the unique one with vanishing two-point function with every non-R flavor symmetry. For 4d $\N =1$ SCFTs, $\tau_{RR}$ minimization gives an alternative to a-maximization. $\tau_{RR}$ minimization also applies in 3d, where no condition for determining the superconformal $U(1)_R$ had been previously known. Unfortunately, this constraint seems impractical to implement for interacting field theories. But it can be readily implemented in the AdS geometry for SCFTs with AdS duals.