Moduli Spaces of Abelian Vortices on Kahler Manifolds

TL;DR

This paper studies the geometry of vortex moduli spaces on Kahler manifolds, providing explicit descriptions, metric properties, and connections to holomorphic map spaces, extending known results to higher dimensions and abelian gauge groups.

Contribution

It extends the description of vortex moduli spaces to abelian GLSM on higher-dimensional Kahler manifolds, including explicit metric computations and links to holomorphic map spaces.

Findings

Moduli space is a projective space for simply connected manifolds.

Moduli space is the projectivization of the Fourier-Mukai transform for abelian varieties.

Explicit formulas for volume and scalar curvature in simple cases.

Abstract

We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an abelian variety, the moduli space is the projectivization of the Fourier-Mukai transform of L. We extend this description of the moduli space to the abelian GLSM, i.e. to vortex equations with a torus gauge group acting linearly on a complex vector space. After establishing the Hitchin-Kobayashi correspondence appropriate for the general abelian GLSM, we give explicit descriptions of the vortex moduli space in the case where the manifold M is simply connected or is an abelian variety. In these examples we compute the Kaehler class of the natural L^2-metric on the moduli space. In the simplest cases we compute the volume and total scalar curvature of the muduli space. Finally, we note that for abelian GLSM the vortex moduli space is a compactification of the space of holomorphic maps from M to toric targets, just as in the usual case of M being a Riemann surface. This leads to various natural conjectures, for instance explicit formulae for the volume of the space of maps CP^m --> CP^n.