Supersymmetry and Localization on Three-Dimensional Orbifolds

Abstract

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 \emph{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.

v3: published version, 67 pages