The Spindle Index from Localization

We present a new supersymmetric index for three-dimensional $\mathcal{N}=2$ gauge theories defined on $\mathrm{\Sigma }\times S^1$, where $\mathrm{\Sigma }$ is a spindle, with twist or anti-twist for the $R$-symmetry background gauge field. We start examining general supersymmetric backgrounds of Euclidean new minimal supergravity admitting two Killing spinors of opposite $R$-charges. We then focus on $\mathrm{\Sigma }\times S^1$ and demostrate how to realise twist and anti-twist. We compute the supersymmetric partition functions on such backgrounds via localization and show that these are captured by a general formula, depending on the type of twist, which unifies and generalises the superconformal and topologically twisted indices.