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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 thatit can be written as the product of an elliptic vortex and anti-vortexpartition function summed over a finite number of Higgs vacua.

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