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Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds

Gregory J. Parker

TL;DR

This work analyzes deformations of $Z_2$-harmonic spinors on a closed 3-manifold by introducing a universal Dirac operator that couples deformations of the singular set with spinor data. The authors develop edge-operator techniques, polyhomogeneous asymptotics, and a detailed obstruction theory to capture infinite-dimensional cokernels, showing the linearization projects to a codimension-1 wall in the parameter space. Because of a loss of regularity, they establish a Nash–Moser framework to obtain a nonlinear deformation theory, proving existence of a local Kuranishi-type chart with virtual codimension 1 and proving wall-crossing phenomena for these spinors. The results connect to gauge theory and Fueter sections, providing analytic tools toward compactness and gluing problems in related settings and offering structural insight into wall-crossing behavior of singular spinor configurations.

Abstract

A $\mathbb Z_2$-harmonic spinor on a 3-manifold $Y$ is a solution of the Dirac equation on a bundle that is twisted around a submanifold $\mathcal Z$ of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of $\mathbb Z_2$-harmonic spinors over the space of parameters $(g,B)$ consisting of a metric and perturbation to the spin connection. The main result states that near a $\mathbb Z_2$-harmonic spinor with $\mathcal Z$ smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set $\mathcal Z$, necessitating the use of the Nash-Moser Implicit Function Theorem.

Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds

TL;DR

This work analyzes deformations of -harmonic spinors on a closed 3-manifold by introducing a universal Dirac operator that couples deformations of the singular set with spinor data. The authors develop edge-operator techniques, polyhomogeneous asymptotics, and a detailed obstruction theory to capture infinite-dimensional cokernels, showing the linearization projects to a codimension-1 wall in the parameter space. Because of a loss of regularity, they establish a Nash–Moser framework to obtain a nonlinear deformation theory, proving existence of a local Kuranishi-type chart with virtual codimension 1 and proving wall-crossing phenomena for these spinors. The results connect to gauge theory and Fueter sections, providing analytic tools toward compactness and gluing problems in related settings and offering structural insight into wall-crossing behavior of singular spinor configurations.

Abstract

A -harmonic spinor on a 3-manifold is a solution of the Dirac equation on a bundle that is twisted around a submanifold of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of -harmonic spinors over the space of parameters consisting of a metric and perturbation to the spin connection. The main result states that near a -harmonic spinor with smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set , necessitating the use of the Nash-Moser Implicit Function Theorem.
Paper Structure (45 sections, 61 theorems, 374 equations)

This paper contains 45 sections, 61 theorems, 374 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Relations to Gauge Theory
  4. Gauge Theory in Low-Dimensions.
  5. Fueter Sections.
  6. Outline
  7. Semi-Fredholm Properties
  8. Edge Sobolev Spaces
  9. Mapping Properties
  10. The Adjoint Operator.
  11. The Second Order Operator.
  12. Higher Regularity
  13. Local Expressions
  14. The Model Operator
  15. Local Expressions
  16. ...and 30 more sections

Key Result

Theorem 1.4

Let $\text{d}_{(\mathcal{Z}_0, \Phi_0)}\slashed {\mathbb D}$ denote the linearization of the universal Dirac operator at a regular $\mathbb{Z}_2$-harmonic spinor $(\mathcal{Z}_0,\Phi_0)$. Then the cokernel component of the partial derivative with respect to the singular set is an elliptic pseudo-differential operator, and its Fredholm extension has index $-1$.

Theorems & Definitions (136)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 126 more