The Spindle Index from Localization

TL;DR

This paper introduces a new supersymmetric index called the Spindle Index for 3D ${ m N}=2$ gauge theories on spindle backgrounds, unifying superconformal and topologically twisted indices through localization techniques.

Contribution

It develops a novel supersymmetric index for theories on spindle geometries, extending the understanding of partition functions in supersymmetric gauge theories.

Findings

Derived a general formula for the Spindle Index depending on twist type.

Unified superconformal and topologically twisted indices within a single framework.

Demonstrated the computation of partition functions on spindle backgrounds using localization.

Abstract

We present a new supersymmetric index for three-dimensional ${\cal N}=2$ gauge theories defined on $\Sigma \times S^1$, where $\Sigma$ 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 $\Sigma \times S^1$ 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.