Supersymmetry and Localization on Three-Dimensional Orbifolds

We consider three-dimensional ${\mathcal N}=2$ supersymmetric field theories defined on general complex-valued backgrounds of Euclidean new minimal supergravity admitting two Killing spinors of opposite $R$-charges. We compute partition functions for theories defined on general circle bundles over spindles $\Sigma$, including $\Sigma\times S^1$ as well as branched and squashed lens spaces, thus obtaining novel observables characterizing three-dimensional supersymmetric gauge theories. We discuss both twisted and anti-twisted theories compactified on $\Sigma\times S^1$ and demonstrate that their partition functions are encoded by a single formula that we refer to as the spindle index, unifying and generalizing superconformal and topologically twisted indices in the limit where orbifold singularities are absent. Furthermore, we test our new index using non-perturbative dualities and obtain one-loop determinants of two-dimensional supersymmetric gauge theories compactified on the spindle.