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The twisted index of 3d $\mathcal{N}=2$ gauge theories on $S^1 \times \Sigma$has an algebro-geometric interpretation as the Witten index of an effectivesupersymmetric quantum mechanics. In this paper, we consider the contributionsto the supersymmetric quantum mechanics from topological saddle points insupersymmetric localisation of abelian gauge theories. Topological saddles areconfigurations where the matter fields vanish and the gauge symmetry isunbroken, which exist for non-vanishing effective Chern-Simons levels. Wecompute the contributions to the twisted index from both topological andvortex-like saddles points and show that their combination recovers theJeffrey-Kirwan residue prescription for the twisted index and itswall-crossing.

The Twisted Index and Topological Saddles