• Corpus ID: 119161463

# On the existence of harmonic $\mathbf{Z}_2$ spinors

@article{Doan2017OnTE,
title={On the existence of harmonic \$\mathbf\{Z\}\_2\$ spinors},
author={Aleksander Doan and Thomas Walpuski},
journal={arXiv: Differential Geometry},
year={2017}
}
• Published 18 October 2017
• Mathematics
• arXiv: Differential Geometry
We prove the existence of singular harmonic ${\bf Z}_2$ spinors on $3$-manifolds with $b_1 > 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg-Witten equation with two spinors. The existence of singular harmonic ${\bf Z}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce regarding Donaldson and Segal's proposal for counting $G_2$-instantons.
1 Citations

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## References

SHOWING 1-10 OF 34 REFERENCES

### Index theorem for Z/2-harmonic spinors.

In my previous paper, I prove the existence of the Kuranishi structure for the moduli space $\mathfrak{M}$ of zero loci of $\mathbb{Z}/2$-harmonic spinors on a 3-manifold. So a nature question we can

### The moduli space of $S^1$-type zero loci for $\mathbb{Z}/2$-harmonic spinors in dimension 3

Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we will construct a moduli space consisting of the following date $\{(\Sigma, \psi)\}$ where $\Sigma$ is a $C^1$-embedding

### Rectifiability and Minkowski bounds for the zero loci of $\mathbb{Z}/2$ harmonic spinors in dimension 4

This article proves that the zero locus of a $\mathbb{Z}/2$ harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.

### Generic Metrics and Connections on Spin- and Spinc-Manifolds

Abstract:We study the dependence of the dimension h0(g,A) of the kernel of the Atyiah-Singer Dirac operator ${\cal D}_{g,A}$ on a spinc-manifold M on the metric g and the connection A. The main

### Conjectures on counting associative 3-folds in 𝐺₂-manifolds

• D. Joyce
• Mathematics
Proceedings of Symposia in Pure Mathematics
• 2018
There is a strong analogy between compact, torsion-free $G_2$-manifolds $(X,\varphi,*\varphi)$ and Calabi-Yau 3-folds $(Y,J,g,\omega)$. We can also generalize $(X,\varphi,*\varphi)$ to 'tamed almost

### The moduli space of S1-type zero loci for Z/2-harmonic spinors in dimension 3

Let M be a compact oriented 3-dimensional smooth manifold. In this paper, we will construct a moduli space consisting of the following date {(Σ, ψ)} where Σ is a C-embedding S curve in M , ψ is a

### The behavior of sequences of solutions to the Vafa-Witten equations

The Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form

### Gauge Theory in higher dimensions, II

• Mathematics
• 2009
The main aim of the paper is to develop the "Floer theory" associated to Calabi-Yau 3-folds, exending the analogy of Thomas' "holomorphic Casson invariant". The treatment in the body of the paper is

### On the behavior of sequences of solutions to U(1) Seiberg-Witten systems in dimension 4

This paper studies the behavior of sequences of solutions to Seiberg-Witten like equations for a pair consisting of a Hermitian connection on a line bundle over a 4-dimensional manifold and a section

### Generic vanishing for harmonic spinors of twisted Dirac operators

In this paper we address the problem of generic vanishing for (negative) harmonic spinors of Dirac operators coupled with variable metric connections. 0. Introduction Let M be a compact spin manifold