Current Correlators and AdS/CFT Geometry

We consider current-current correlators in 4d $\mathbb{N}=1$ SCFTs, and also 3d $\N =2$SCFTs,inconnectionwithAdS\mathrm{/}CFTgeometry\mathrm{.}Thesuperconformal$U(1)_R$ symmetry of the SCFT has the distinguishing property that, among all possibilities, it minimizes the coefficient, ${\tau }_{RR}$ of its two-point function. We show that the geometric Z-minimization condition of Martelli, Sparks, and Yau precisely implements ${\tau }_{RR}$ minimization. This gives a physical proof that Z-minimization in geometry indeed correctly determines the superconformal R-charges of the field theory dual. We further discuss and compare current two point functions in field theory and AdS/CFT and the geometry of Sasaki-Einstein manifolds. Our analysis gives new quantitative checks of the AdS/CFT correspondence.