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

We apply the method of Higgs branch localization to the $\mathcal{N}=1$ supersymmetric partition function on $S^3\times S^1\mathrm{.}$ 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.