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Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

William L. Blair, Minh Lam Nguyen

TL;DR

This work analyzes the Seiberg-Witten vortex equations on the plane arising from dimensional reduction, establishing that the moduli space contains both polynomial-growth and exponentially decaying solutions. It develops a variational Kazdan–Warner framework to construct exponentially decaying solutions and uses systems of Vekua equations to control zero sets, obtaining a finite-zero result under the decay condition. A key achievement is a surjective map from the polynomial-growth and exponential-decay moduli to the symmetric products $\bigcup_n Sym^n(\mathbf{C})$, linking vortex data to complex-geometric invariants. Overall, the paper reveals richer asymptotics for Seiberg-Witten vortices on $\mathbf{R}^2$ and integrates gauge theory with generalized analytic methods, offering a pathway to non-flat connections with exponential decay and suggesting avenues for extending to Higgs-coupled variants.

Abstract

Clifford Taubes showed that the moduli space of the variational equation of the Yang-Mills-Higgs functional on the plane is non-empty, and its elements correspond to "vortices". Inspired by this result, in this paper, we show that the moduli space of the Hitchin-type dimensional reduction of the Seiberg-Witten equations on the plane contains both exponentially decayed solutions and polynomial growth solutions. Furthermore, we show that there is correspondence from the moduli space of exponentially decayed and polynomial growth solutions to the symmetric products of complex numbers. The correspondence restricted to the latter is a surjective map.

Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

TL;DR

This work analyzes the Seiberg-Witten vortex equations on the plane arising from dimensional reduction, establishing that the moduli space contains both polynomial-growth and exponentially decaying solutions. It develops a variational Kazdan–Warner framework to construct exponentially decaying solutions and uses systems of Vekua equations to control zero sets, obtaining a finite-zero result under the decay condition. A key achievement is a surjective map from the polynomial-growth and exponential-decay moduli to the symmetric products , linking vortex data to complex-geometric invariants. Overall, the paper reveals richer asymptotics for Seiberg-Witten vortices on and integrates gauge theory with generalized analytic methods, offering a pathway to non-flat connections with exponential decay and suggesting avenues for extending to Higgs-coupled variants.

Abstract

Clifford Taubes showed that the moduli space of the variational equation of the Yang-Mills-Higgs functional on the plane is non-empty, and its elements correspond to "vortices". Inspired by this result, in this paper, we show that the moduli space of the Hitchin-type dimensional reduction of the Seiberg-Witten equations on the plane contains both exponentially decayed solutions and polynomial growth solutions. Furthermore, we show that there is correspondence from the moduli space of exponentially decayed and polynomial growth solutions to the symmetric products of complex numbers. The correspondence restricted to the latter is a surjective map.
Paper Structure (16 sections, 38 theorems, 126 equations)

This paper contains 16 sections, 38 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Organization
  4. Preliminaries
  5. Classical Vekua Equation
  6. System of Vekua Equations
  7. Vortex Equations
  8. Gauge Theory Set-Up
  9. Seiberg-Witten Gauge Theory
  10. Dimensional Reduction
  11. Existence Of Solutions With Exponential Decay
  12. A Kazdan-Warner Equation
  13. Some Estimates for $\mathscr{A}$
  14. Infinimum of $\mathscr{A}$
  15. Zeroes of Exponential Decay Solutions
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\{z_1,\cdots, z_k\}$ be a finite collection of points on the plane. For any $(c_1,c_2) \in \mathbf{C}^* \times \mathbf{C}^*$ and $\theta \in \mathbf{R}$, $(A_0, A_1, \psi_1, \psi_2)$ given by is always a solution of eq:maineq, where $\mathbf{C}^*$ denotes $\mathbf{C}\setminus \{0\}$.

Theorems & Definitions (73)

  • Theorem 1.1: cf. Proposition \ref{['Prop3.9']}, Corollary \ref{['Cor3.10']}
  • Theorem 1.2: cf. Theorem \ref{['Th3.11']}
  • Definition 1.3
  • Theorem 1.4: cf. Theorem \ref{['Existenceofexponentialdecaysolution']}
  • Theorem 1.5: cf. Theorem \ref{['Existenceofsolution']}
  • proof : Acknowledgement
  • Theorem 2.1: $\bar{\partial}$-Poincare Lemma MR0507725; Cauchy-Pompeiu Theorem, Theorem 20 BegBook
  • Theorem 2.2: Theorem 1.16 Vek
  • Theorem 2.3: Theorem 1.19 Vek
  • Theorem 2.4: "The Basic Lemma" Vek
  • ...and 63 more