Moduli of vortices and Grassmann manifolds

TL;DR

This paper describes moduli spaces of gauged vortices on Riemann surfaces using Quot schemes, embeds them into Grassmannians, and relates their natural metrics to the L^2 metric under certain conditions.

Contribution

It provides a new Quot scheme-based description of vortex moduli spaces and establishes their canonical embeddings into Grassmann manifolds with compatible Kähler metrics.

Findings

Moduli spaces embed into Grassmannians with Fubini-Study metrics.

In the local case r=n, the moduli spaces are smooth.

Under a quantization condition, the Fubini-Study metric matches the L^2 metric.

Abstract

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are smooth at least in the local case r=n. For abelian local vortices we prove that, if a certain "quantization" condition is satisfied, the embedding can be chosen in such a way that the induced Fubini-Study structure realizes the Kaehler class of the usual L^2 metric of gauged vortices.