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

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.