Relation between the 4d superconformal index and the S^3 partition function

TL;DR

This paper explores the connection between the 4d superconformal index and the S^3 partition function, demonstrating how the 3d action emerges from the 4d action through dimensional reduction and deriving a formula relating the two in various parameters.

Contribution

It explicitly shows the dimensional reduction from 4d to 3d actions in the context of localization and derives a formula linking the superconformal index to the S^3 partition function with squashing and other parameters.

Findings

3d action obtained from 4d action via dimensional reduction

Derived a formula relating the 4d index to the 3d partition function

Established the dependence of the partition function on various parameters

Abstract

A relation between the 4d superconformal index and the S^3 partition function is studied with focus on the 4d and 3d actions used in localization. In the case of vanishing Chern-Simons levels and round S^3 we explicitly show that the 3d action is obtained from the 4d action by dimensional reduction up to terms which do not affect the exact results. By combining this fact and a recent proposal concerning a squashing of S^3 and SU(2) Wilson line, we obtain a formula which gives the partition function depending on the Weyl weight of chiral multiplets, real mass parameters, FI parameters, and a squashing parameter as a limit of the index of a parent 4d theory.