Reducing the 4d Index to the S^3 Partition Function

TL;DR

This paper demonstrates that the 4d superconformal index reduces to the 3d S^3 partition function in the zero-circle-radius limit, establishing a direct connection between 4d and 3d gauge theories via matrix integrals.

Contribution

It shows that the 4d superconformal index simplifies to the 3d partition function, recovering known matrix integrals and revealing a q-deformation relationship.

Findings

The 4d index reduces to the 3d partition function as the circle radius approaches zero.

The matrix integral for the 4d index matches the known 3d partition function matrix integral.

The superconformal index is a q-deformation of the 3d partition function.

Abstract

The superconformal index of a 4d gauge theory is computed by a matrix integral arising from localization of the supersymmetric path integral on S^3 x S^1 to the saddle point. As the radius of the circle goes to zero, it is natural to expect that the 4d path integral becomes the partition function of dimensionally reduced gauge theory on S^3. We show that this is indeed the case and recover the matrix integral of Kapustin, Willet and Yaakov from the matrix integral that computes the superconformal index. Remarkably, the superconformal index of the "parent" 4d theory can be thought of as the q-deformation of the 3d partition function.