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

We consider two-dimensional \( \mathcal{N} \) = (2, 2) supersymmetric field theories living on a weighted projective space \( {\mathbbm{WCP}}_{\left[{n}_1,{n}_2\right]}^1 \), 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.