Vortices as degenerate metrics

TL;DR

This paper interprets abelian vortices as degenerate hermitian metrics satisfying a curvature condition, offering new insights and a simple superposition rule, and analyzing the L^2-metric behavior on moduli space.

Contribution

It introduces a novel geometric interpretation of vortices as degenerate metrics and explores their properties and interactions from this perspective.

Findings

Vortices correspond to degenerate hermitian metrics satisfying a curvature equation.

A simple non-linear superposition rule for vortex solutions is proposed.

Behavior of the L^2-metric on moduli space under certain restrictions is described.

Abstract

We note that the Bogomolny equation for abelian vortices is precisely the condition for invariance of the Hermitian-Einstein equation under a degenerate conformal transformation. This leads to a natural interpretation of vortices as degenerate hermitian metrics that satisfy a certain curvature equation. Using this viewpoint, we rephrase standard results about vortices and make new observations. We note the existence of a conceptually simple, non-linear rule for superposing vortex solutions, and we describe the natural behaviour of the L^2-metric on the moduli space upon restriction to a class of submanifolds.