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

TL;DR

The paper establishes a fundamental link between the 3d S^3 partition function computed by localization and the 4d superconformal index, by showing that the 3d deformation action can be obtained from the 4d action through dimensional reduction when Chern-Simons levels vanish on round S^3. It derives a general limit relation that expresses the 3d partition function as a small-radius limit of the 4d index, extended to include Weyl weights, real masses, FI terms, and a squashing parameter via SU(2)_R Wilson lines. A key step is matching the deformation actions through a Wilson line, enabling a precise mapping of FI and mass parameters between 4d and 3d, and clarifying the role of anomalies. The work also generalizes to squashed spheres, connecting to recent results on squashed S^3 partition functions, while noting the difficulty of reproducing CS terms from dimensional reduction when k_a ≠ 0.

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.