Index for three dimensional superconformal field theories with general R-charge assignments

TL;DR

This paper develops a general localization-based framework to compute the ${\cal N}=2$ superconformal index for three-dimensional theories with arbitrary R-charge assignments on ${\bf S}^2\times{\bf S}^1$, including monopole sectors. It derives a master formula expressed through a letter index that encodes chiral and vector multiplet contributions, along with zero-point and CS-term data. The authors apply the formalism to mirror pairs with $N_f=1,2,3$ (QED and its mirrors), performing explicit computations and confirming index equivalence, thereby validating the approach and illustrating the utility for AdS$_4$/CFT$_3$ tests. The work opens avenues for non-Abelian and large-$N$ analyses, provides a link to IR R-charge data via index structure, and clarifies the role of monopole operators in the index.

Abstract

We derive a general formula of an index for three dimensional N=2 superconformal field theories with general R-charge assignments to chiral multiplets by using the localization method in S^2xS^1 background. As examples we compute the index for theories in a few mirror pairs, and confirm the agreement of the indices in each mirror pair.