The vortex equation on affine manifolds

Abstract

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, \phi), consisting of a flat vector bundle E over M and a flat nonzero section \phi of E, admits a solution to the vortex equation if and only if it is polystable. To prove this, we adapt the dimensional reduction techniques for holomorphic pairs on K\ahler manifolds to the situation of flat pairs on affine manifolds.