The vortex equation on affine manifolds

TL;DR

This paper establishes a correspondence between solutions to the vortex equation and polystability for flat vector bundles with sections on special affine manifolds, adapting techniques from Kähler geometry.

Contribution

It introduces a new existence criterion for vortex equations on affine manifolds, extending stability concepts and dimensional reduction methods beyond Kähler settings.

Findings

Solutions to the vortex equation exist if and only if the pair is polystable.

The approach adapts dimensional reduction techniques to affine manifolds.

Provides a stability criterion for flat vector bundles with sections.

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.