The Exact Superconformal R-Symmetry Extremizes Z

TL;DR

This paper demonstrates that the superconformal R-charge in three-dimensional theories with four supercharges can be exactly determined by extremizing the three-sphere partition function Z, computed via localization, which depends on R-charge parameters.

Contribution

It establishes that extremizing the partition function Z over R-charge parameters precisely yields the superconformal R-charge without mixing with accidental symmetries.

Findings

Z is computed as a matrix integral via localization.

The magnitude of Z is extremized at the superconformal R-charge.

This extremization uniquely determines the IR superconformal R-charge.

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.