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Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds

Gregory J. Parker

Abstract

Given a $\mathbb Z_2$-harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg-Witten monopoles converging to it after renormalization. The proof is a gluing construction beginning with model solutions on a neighborhood of the $\mathbb Z_2$-harmonic spinor's singular set. The gluing is complicated by the presence of an infinite-dimensional obstruction bundle for the singular limiting linearized operator. This difficulty is overcome by introducing a generalization of Donaldson's alternating method in which a deformation of the $\mathbb Z_2$-harmonic spinor's singular set is chosen at each stage of the alternating iteration to cancel the obstruction components.

Gluing $\mathbb Z_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds

Abstract

Given a -harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg-Witten monopoles converging to it after renormalization. The proof is a gluing construction beginning with model solutions on a neighborhood of the -harmonic spinor's singular set. The gluing is complicated by the presence of an infinite-dimensional obstruction bundle for the singular limiting linearized operator. This difficulty is overcome by introducing a generalization of Donaldson's alternating method in which a deformation of the -harmonic spinor's singular set is chosen at each stage of the alternating iteration to cancel the obstruction components.
Paper Structure (58 sections, 59 theorems, 268 equations, 1 figure)

This paper contains 58 sections, 59 theorems, 268 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Seiberg--Witten Equations
  3. $\mathbb{Z}_2$-Harmonic spinors
  4. The Gluing Problem
  5. Main Results
  6. Wall-Crossing Formulas
  7. Outline
  8. Gluing by the Alternating Method
  9. The Structure of Gluing Problems
  10. The Alternating Method
  11. Gluing as Non-linear Excision
  12. The Semi-Fredholm Alternating Method
  13. Key Technical Difficulties
  14. Loss of Regularity
  15. Non-local Deformation Operator
  16. ...and 43 more sections

Key Result

Theorem 1.2

(PartII) Let $(\mathcal{Z}_0, A_0,\Phi_0)$ be a regular $\mathbb{Z}_2$-harmonic spinor, and let $\Pi_0$ denote the projection onto the cokernel of (prelimsingulardirac). Then the cokernel component of the linearization of the universal Dirac operator with respect to deformations of $\mathcal{Z}_0$ is an elliptic pseudo-differential operator of order $\tfrac{1}{2}$ and its Fredholm extension has in

Figures (1)

  • Figure 1: An illustration of the alternating iteration procedure in Steps (1)--(4) above. (Top) The cut-off functions $\chi^\pm$, (red) the error terms with alternating support and decreasing norm, (blue/green) the decay of solutions across the neck region.

Theorems & Definitions (135)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 125 more