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Existence of abelian BPS vortices on surfaces with Neumann boundary conditions

Rene Garcia-Lara

TL;DR

This work addresses the existence and structure of abelian BPS vortices on compact Riemannian surfaces with boundary under Neumann boundary conditions. Using a Taubes equation with Neumann boundary and a regularisation via Green functions, it proves a Bradlow-type condition $N+\frac{M}{2}<\frac{A}{4\pi}$ guaranteeing a unique vortex solution for prescribed interior and boundary zeros, up to gauge. It shows energy and flux quantisation $E=(2N+M)\pi$, $\Phi=(2N+M)\pi$, and that boundary zeros contribute half-vortex amounts. Numerical computations on the disk illustrate both interior and boundary vortices and reveal that the $L^2$ metric on moduli space can depend on boundary data, signaling a departure from the purely local localization observed in closed surfaces or Dirichlet problems.

Abstract

Existence of abelian BPS vortices on a manifold with boundary satisfying Neumann boundary conditions is proved. Numeric solutions are constructed on the Euclidean disk, and the L^2-metric of the moduli space of one vortex located at the interior of a rotationally symmetry disk is studied. The results presented extend previous work of Manton and Zhao on quotients of surfaces that admit a reflection.

Existence of abelian BPS vortices on surfaces with Neumann boundary conditions

TL;DR

This work addresses the existence and structure of abelian BPS vortices on compact Riemannian surfaces with boundary under Neumann boundary conditions. Using a Taubes equation with Neumann boundary and a regularisation via Green functions, it proves a Bradlow-type condition guaranteeing a unique vortex solution for prescribed interior and boundary zeros, up to gauge. It shows energy and flux quantisation , , and that boundary zeros contribute half-vortex amounts. Numerical computations on the disk illustrate both interior and boundary vortices and reveal that the metric on moduli space can depend on boundary data, signaling a departure from the purely local localization observed in closed surfaces or Dirichlet problems.

Abstract

Existence of abelian BPS vortices on a manifold with boundary satisfying Neumann boundary conditions is proved. Numeric solutions are constructed on the Euclidean disk, and the L^2-metric of the moduli space of one vortex located at the interior of a rotationally symmetry disk is studied. The results presented extend previous work of Manton and Zhao on quotients of surfaces that admit a reflection.

Paper Structure

This paper contains 8 sections, 11 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. The Taubes equation with Neumann boundary conditions
  4. The regular elliptic problem
  5. Numerical results
  6. Comparing the geometry of the moduli space of one vortex
  7. Conclusion and outlook
  8. Deforming Green functions on a surface with boundary

Key Result

Theorem 1

Let $S$ be the interior of a compact Riemannian surface with boundary $\partial S$ and area $A$. Given two finite sets of points with multiplicities, $\mathcal{N}$ or $\mathcal{M}$ possibly empty but $\mathcal{N} \cup \mathcal{M} \neq \emptyset$, with total multiplicities $N$ and $M$ respectively, then the following statements are equivalent:

Figures (2)

  • Figure 1: Energy, magnetic flux and Higgs field modulus of a single vortex located at the center of a disk with Neumann boundary conditions.
  • Figure 2: Energy, magnetic flux and Higgs field modulus of a half-vortex located at the boundary of a radius 3 disk with Neumann boundary conditions.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm:existence-vortices']}
  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Proposition 2
  • ...and 10 more