Higgs branch localization of $\mathcal{N}=1$ theories on $S^3 \times S^1$

TL;DR

<3-5 sentence high-level summary> We apply Higgs branch localization to the four-dimensional $ N=1$ supersymmetric partition function on $S^3 imes S^1$ and show that the superconformal index can be written as a finite sum over Higgs vacua, with each term factorizing into a product of a classical piece, a modified one-loop determinant, and a pair of elliptic vortex/anti-vortex partition functions. The analysis provides a detailed map between Coulomb-branch localization and Higgs-branch factorization, including a reduction to elliptic vortex data computed from an $R^2_\epsilon imes T^2$ background in the $oldsymbol{ Omega}$-background. In explicit examples (U(1), SU(2) and associated Cartan theories) the results reproduce the expected factorized forms and reveal how vortex contributions arise from residues across crossed poles, establishing a four-dimensional uplift of holomorphic blocks. The work suggests rich modular and duality structures for the elliptic vortex sector and points to further applications to other geometries and extended supersymmetry.

Abstract

We apply the method of Higgs branch localization to the $\mathcal{N}=1$ supersymmetric partition function on $S^3\times S^1.$ As a result, we show that it can be written as the product of an elliptic vortex and anti-vortex partition function summed over a finite number of Higgs vacua.