Calculating the Superconformal Index and Seiberg Duality

TL;DR

The paper develops a robust framework to compute the four-dimensional superconformal index by radial quantization on $S^3\times\mathbb{R}$ and a mass-gap argument that allows turning off interactions without changing the index. It formulates the index as a protected group character built from $\Xi=H-\tfrac{1}{2}R$ and $\Delta=H-\tfrac{3}{2}R-2J^3$, and demonstrates how to obtain it for chiral multiplets, Wess-Zumino models, and gauge theories via Plethystic exponentials and gauge integrations. The authors test the approach on RG flows and Seiberg dual pairs, obtaining agreement to several orders in the $t$-expansion and uncovering nontrivial group/number theoretic identities that equate electric and magnetic expressions. The work provides a practical, generalizable tool for validating dualities and exploring the structure of BPS spectra in 4D SCFTs, with potential implications for AdS/CFT and black hole microstate counts, and suggests avenues for a deeper geometric understanding of these indices.

Abstract

We develop techniques to calculate an index for four dimensional superconformal field theories. This superconformal index is counting BPS operators which preserve only one supercharge. To calculate the superconformal index we quantize the field theory on S^3 X R and show that the twisted theory has an appropriate mass gap. This allows for the interactions to be switched off continuously without the superconformal index being changed. We test those techniques for theories which go through a non-trivial RG flow and for Seiberg dual theories. This leads to the conjecture of some group/number theoretical identities.