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

TL;DR

The paper proves that the 4d superconformal index on $S^3\times S^1$ reduces to the 3d $S^3$ partition function in the small-$S^1$ limit, with the index matrix model smoothly mapping to the Kapustin–Willett–Yaakov 3d matrix model. It shows that hypermultiplet and vector multiplet contributions map cleanly under $q\to1$, and that the 4d index is a $q$-deformation of the 3d partition function, highlighting a deep link between 4d and 3d dualities. Through explicit SU(2) examples and discussions of S-duality and mirror symmetry, the work reveals how 4d dualities translate into 3d fixed-point data and dualities, offering a route to probe non-Lagrangian theories via dimensional reduction. The results bolster the view of the $S^3$ partition function as a robust probe of IR physics and dualities across dimensions.

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.