Super Yang-Mills on Branched Covers and Weighted Projective Spaces

In this work we conjecture the Coulomb branch partition function, including flux and instanton contributions, for the $\mathcal{N}=2$ vector multiplet on weighted projective space ${\mathbb{C}\mathbb{P}}_{\mathit{N}}^2$ for equivariant Donaldson-Witten and ``Pestun-like’’ theories. We claim that this partition function agrees with the one obtained from dimensional reduction of the 5d $\mathcal{N}=1$ vector multiplet on a certain branched cover of $S^5$. More precisely, the branch locus and indices have to be such that they match the singular locus and deficit angles in ${\mathbb{C}\mathbb{P}}_{\mathit{N}}^2$. Our conjecture is substantiated by checking that partition functions on spindles are similarly obtained from dimensional reduction of the 3d $\mathcal{N}=2$ vector multiplet on branched covers of $S^3$. This work paves the way for obtaining partition functions on more generic symplectic toric orbifolds.