Localisation of $\mathcal{N} = (2,2)$ theories on spindles of both twists

TL;DR

This paper constructs and analyzes 2D $ abla=(2,2)$ supersymmetric theories on spindle geometries, deriving exact partition functions for twisted and anti-twisted cases using localization techniques.

Contribution

It introduces a unified approach to study supersymmetric theories on spindles, deriving exact partition functions for both twist and anti-twist mechanisms.

Findings

Derived a general formula for partition functions on spindles.

Applied localization to theories with vector and chiral multiplets.

Compared twisted and anti-twisted cases, highlighting their differences.

Abstract

We consider two-dimensional $\mathcal{N}=(2,2)$ supersymmetric field theories living on a spindle $\mathbb{WCP}_{[n_1,n_2]}^1$. Starting from the spindle solutions of five-dimensional STU gauged supergravity, we construct theories on a spindle which preserve supersymmetry via either the twist or anti-twist mechanism and admit two Killing spinors of opposite R-charge. While the study of field theories on anti-twisted spindles has already been undertaken in some detail, the advantage of our approach allows for the derivation of analogous results in the twist case. We apply the technique of supersymmetric localisation to compute the exact partition function for a theory consisting of an abelian vector multiplet and a charged chiral multiplet in the presence of a Fayet-Iliopoulos term. We compare and contrast the results for the twisted and anti-twisted spindle and find a general formula which encompasses the partition function for both cases simultaneously.