The Exact Superconformal R-Symmetry Extremizes Z

TL;DR

This work proposes and tests Z-extremization: the exact IR superconformal R-symmetry in 3d ${ m cal N}=2$ theories is determined by extremizing the three-sphere partition function Z, computed exactly via localization. By coupling to curved space with an R-symmetry and exploiting holomorphy in real mass parameters, Z becomes a function whose magnitude |Z|^2 is extremized to fix the IR R-charges, assuming no mixing with accidental symmetries. The method is validated through several examples, including SQED and Chern-Simons-matter theories, reproducing known IR dimensions and matching dualities like 3d mirror symmetry. The results provide a practical, exact route to determine operator dimensions in IR SCFTs and connect to holographic and RG-flow expectations for Z as a measure of degrees of freedom.

Abstract

The three sphere partition function, Z, of three dimensional theories with four supercharges and an R-symmetry is computed using localization, resulting in a matrix integral over the Cartan of the gauge group. There is a family of couplings to the curved background, parameterized by a choice of R-charge, such that supersymmetry is preserved; Z is a function of those parameters. The magnitude of the result is shown to be extremized for the superconformal R-charge of the infrared conformal field theory, in the absence of mixing of the R-symmetry with accidental symmetries. This exactly determines the IR superconformal R-charge.