Moduli of Vortices and Grassmann Manifolds

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 Σ with target \({{\rm Mat}_{r \times n}(\mathbb{C})}\), where n ≥ r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kähler 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 Kähler class of the usual L 2 metric of gauged vortices.