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Solutions to the Seiberg-Witten equations in all dimensions

Partha Ghosh

TL;DR

This work introduces higher-dimensional generalizations of the Seiberg–Witten equations on oriented manifolds with spin$^\mathbb{C}$ structures and constructs explicit solutions in dimensions $6$, $8$, and $5$, including on Kähler manifolds and product geometries. By exploiting two-spinor formulations, holomorphic line bundles, and vortex equations, the authors relate higher-dimensional solutions to vortex moduli spaces and identify when nontrivial solutions exist via cohomological and Chern-class constraints. A key outcome is that, unlike the compact moduli spaces in low dimensions, higher-dimensional spaces can be noncompact due to harmonic forms, with concrete examples in dimension $6$. Perturbations by harmonic terms yield additional solution families and connect to perturbed variants that admit Calabi–Yau or Fano structures, while Taubes-type limits suggest a link to pseudo-holomorphic curves in favorable settings. Overall, the paper provides a structured bridge between higher-dimensional SW theory, vortex geometry, and the topology of Kahler and complex manifolds, highlighting both constructive methods and fundamental obstructions.

Abstract

This article explores solutions to a generalised form of the Seiberg--Witten equations in higher dimensions, first introduced by Fine and the author. Starting with an oriented $n$ dimensional Riemannian manifold with a spin$^\mathbb{C}$-structure, we described an elliptic system of equations that recovers the traditional Seiberg-Witten equations in dimensions $3$ and $4$. The paper focuses on constructing explicit solutions of these equations in dimensions $5, 6$ and $8$, where harmonic perturbation terms are sometimes required to ensure solutions. In dimensions $6$ and $8$ we construct solutions on Kähler manifolds and relate these solutions to vortices. In dimension $5$, we construct solutions on the product of a closed Riemann surface and $\mathbb{R}^3$. The solutions are invariant in the $\mathbb{R}^3$ directions and can be related to vortices on the Riemann surface. A key issue in higher dimensions is the potential noncompactness of the space of solutions, in contrast to the compact moduli spaces in lower dimensions. In our solutions, this noncompactness is linked to the presence of certain odd-dimensional harmonic forms, with an explicit example provided in dimension $6$.

Solutions to the Seiberg-Witten equations in all dimensions

TL;DR

This work introduces higher-dimensional generalizations of the Seiberg–Witten equations on oriented manifolds with spin structures and constructs explicit solutions in dimensions , , and , including on Kähler manifolds and product geometries. By exploiting two-spinor formulations, holomorphic line bundles, and vortex equations, the authors relate higher-dimensional solutions to vortex moduli spaces and identify when nontrivial solutions exist via cohomological and Chern-class constraints. A key outcome is that, unlike the compact moduli spaces in low dimensions, higher-dimensional spaces can be noncompact due to harmonic forms, with concrete examples in dimension . Perturbations by harmonic terms yield additional solution families and connect to perturbed variants that admit Calabi–Yau or Fano structures, while Taubes-type limits suggest a link to pseudo-holomorphic curves in favorable settings. Overall, the paper provides a structured bridge between higher-dimensional SW theory, vortex geometry, and the topology of Kahler and complex manifolds, highlighting both constructive methods and fundamental obstructions.

Abstract

This article explores solutions to a generalised form of the Seiberg--Witten equations in higher dimensions, first introduced by Fine and the author. Starting with an oriented dimensional Riemannian manifold with a spin-structure, we described an elliptic system of equations that recovers the traditional Seiberg-Witten equations in dimensions and . The paper focuses on constructing explicit solutions of these equations in dimensions and , where harmonic perturbation terms are sometimes required to ensure solutions. In dimensions and we construct solutions on Kähler manifolds and relate these solutions to vortices. In dimension , we construct solutions on the product of a closed Riemann surface and . The solutions are invariant in the directions and can be related to vortices on the Riemann surface. A key issue in higher dimensions is the potential noncompactness of the space of solutions, in contrast to the compact moduli spaces in lower dimensions. In our solutions, this noncompactness is linked to the presence of certain odd-dimensional harmonic forms, with an explicit example provided in dimension .

Paper Structure

This paper contains 18 sections, 16 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Overview of the main results
  4. Solutions to the $6$-dimensional Seiberg--Witten equations
  5. Spin$^\mathbb{C}$-bundle, Dirac operator and Clifford multiplication
  6. Solution on closed Kähler $3$-folds
  7. Relationship with vortices
  8. Failure of compactness
  9. Taubes' limit
  10. Solution on $\Sigma\times\mathbb{C}^2$
  11. Solutions to the $8$-dimensional Seiberg--Witten equations
  12. Spin$^\mathbb{C}$-bundle, Dirac operator and Clifford multiplication
  13. Solution on closed Kähler $4$-folds
  14. Relationship with vortices
  15. Solutions to the $5$-dimensional Seiberg--Witten equations
  16. ...and 3 more sections

Key Result

Theorem 1.1

For a fixed pair of holomorphic structures on $\mathcal{L}_0$ and $\mathcal{L}_1$, there exists a non-trivial solution to the Seiberg--Witten equations Dirac 6d +,Curvature 6d +,Dirac 6d - and Curvature 6d - under the following conditions:

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1
  • Lemma 2.1
  • Remark 2
  • Lemma 2.2
  • ...and 9 more