Current Correlators and AdS/CFT Geometry

TL;DR

This paper demonstrates that Z-minimization in geometry corresponds exactly to the minimization of the superconformal R-symmetry coefficient in SCFTs, providing a physical proof and new checks of AdS/CFT correspondence.

Contribution

It establishes a direct link between geometric Z-minimization and field theory R-charge determination, confirming the geometric approach's validity in AdS/CFT.

Findings

Z-minimization implements $ au_{RR}$ minimization in SCFTs.

Provides a physical proof connecting geometry and superconformal R-charges.

Offers new quantitative tests of the AdS/CFT correspondence.

Abstract

We consider current-current correlators in 4d $\N =1$ SCFTs, and also 3d $\N =2$ SCFTs, in connection with AdS/CFT geometry. The superconformal $U(1)_R$ 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.