On the L^2-metric of vortex moduli spaces

TL;DR

This paper derives explicit formulas for the Kaehler form of the L^2-metric on vortex moduli spaces, computes their volumes, and extends localization results to broader vortex models.

Contribution

It provides explicit expressions for the Kaehler form and volume of vortex moduli spaces, and generalizes localization techniques to non-abelian and toric vortex models.

Findings

Explicit formulas for the Kaehler form of the L^2-metric.

Computed the volume of abelian semi-local vortex moduli spaces.

Extended localization results to non-abelian vortices and gauged toric sigma-models.

Abstract

We derive general expressions for the Kaehler form of the L^2-metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kaehler class of the L^2-metric. As an application we compute the total volume of the moduli space of abelian semi-local vortices. In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. Finally we show that the localization results of Samols in the abelian Higgs model extend to more general models. These include linear non-abelian vortices and vortices in gauged toric sigma-models.