Dynamics of Abelian Vortices Without Common Zeros in the Adiabatic Limit

TL;DR

This paper investigates the behavior of Abelian vortices on a Riemann surface as a parameter tends to infinity, establishing a precise correspondence with holomorphic maps and confirming a conjecture about their metric properties.

Contribution

It provides an explicit description of the vortex moduli space, analyzes the convergence of vortex solutions, and proves the isometric correspondence conjectured by Baptista.

Findings

Vortex solutions without common zeros converge to holomorphic maps as the parameter grows.

The $s$-dependent correspondence becomes an isometry in the limit $s o fty$.

Explicit moduli space descriptions facilitate understanding of vortex dynamics.

Abstract

On a smooth line bundle $L$ over a compact K\"ahler Riemann surface $\Sigma$, we study the family of vortex equations with a parameter $s$. For each $s \in [1,\infty]$, we invoke techniques in \cite{Br} by turning the $s$-vortex equation into an $s$-dependent elliptic partial differential equation, studied in \cite{kw}, providing an explicit moduli space description of the space of gauge classes of solutions. We are particularly interested in the bijective correspondence between the open subset of vortices without common zeros and the space of holomorphic maps. For each $s$, the correspondence is uniquely determined by a smooth function $u_s$ on $\Sigma$, and we confirm its convergent behaviors as $s \to \infty$. Our results prove a conjecture posed by Baptista in \cite{Ba}, stating that the $s$-dependent correspondence is an isometry between the open subsets when $s=\infty$, with $L^2$ metrics appropriately defined.