Super Yang-Mills on branched covers and weighted projective spaces

Abstract

Abstract In this work we conjecture the Coulomb branch partition function, including flux and instanton contributions, for the $$ \mathcal{N} $$ N = 2 vector multiplet on weighted projective space $$ {\mathbbm{CP}}_N^2 $$ CP N 2 for equivariant Donaldson-Witten and “Pestun-like” theories. More precisely, we claim that this partition function agrees with the one computed on a certain branched cover of $$ {\mathbbm{CP}}^2 $$ CP 2 upon matching conical deficit angles with corresponding branch indices. Our conjecture is substantiated by checking that similar partition functions on spindles agree with their equivalent on certain branched covers of $$ {\mathbbm{CP}}^1 $$ CP 1 . We compute the one-loop determinant on the branched cover of $$ {\mathbbm{CP}}^2 $$ CP 2 for all flux sectors via dimensional reduction from the $$ \mathcal{N} $$ N = 1 vector multiplet on a branched five-sphere along a free S1-action. This work paves the way for obtaining partition functions on more generic symplectic toric orbifolds.