Supersymmetric localisation of $\mathcal{N}=(2,2)$ theories on a spindle

TL;DR

This work extends supersymmetric localisation to two-dimensional ${ m N}=(2,2)$ theories on the spindle orbifold $ ext{WCP}^1_{[n_1,n_2]}$, deriving the exact partition function for a simple abelian setup with a vector multiplet, a charged chiral multiplet, and a Fayet–Iliopoulos term. Building on the spindle solution of five-dimensional minimal gauged supergravity, the authors place a real 2d ${ m N}=(2,2)$ theory on the spindle via the anti-twist and perform localisation to the real BPS locus, with the action on this locus reduced to FI and topological contributions. The one-loop determinants are computed exactly using both the unpaired-eigenvalue method and the fixed-point index, yielding Gamma-function ratios that match in both approaches and with the S^2 limit, after accounting for scale dependence through a renormalised FI parameter. The resulting partition function exhibits a natural factorisation into north/south contributions and provides a concrete setting to study two-dimensional orbifold dualities, including potential mirror symmetry, and invites extensions to non-abelian theories and holographic applications. This framework thus offers a precise, tractable handle on 2d theories on orbifold backgrounds and their dualities.

Abstract

We consider two-dimensional $\mathcal{N}=(2,2)$ supersymmetric field theories living on a weighted projective space $\mathbb{WCP}_{[n_1,n_2]}^1$, often referred to as a spindle. Starting from the spindle solution of five-dimensional minimal gauged supergravity, we construct a theory on a spindle which preserves supersymmetry via the anti-twist mechanism and admits two Killing spinors of opposite $R$-charge. We apply the technique of supersymmetric localisation to compute the exact partition function for a theory consisting of an abelian vector multiplet and a chiral multiplet, finding that the path integral localises to a real moduli space of vector multiplet fluctuations. We compute the one-loop determinants via the equivariant index, using both the method of unpaired eigenvalues and the fixed point theorem, finding agreement between the two approaches. We conclude with the explicit partition function for an example of a charged chiral multiplet in the presence of a Fayet-Iliopoulos term and comment on its dependence on the overall length scale of the geometry. This work paves the way towards uncovering two-dimensional dualities, such as mirror symmetry, for field theories defined on orbifold backgrounds.