Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations on 3-Manifolds

TL;DR

This paper constructs localized solutions to the two-spinor Seiberg-Witten equations on 3-manifolds that concentrate near singular sets and converge to given harmonic spinors, advancing understanding of solution limits and gluing techniques.

Contribution

It introduces a method to generate local solutions concentrating near singular sets, linking them to harmonic spinors and enabling new gluing constructions.

Findings

Solutions concentrate near the singular set as epsilon approaches zero.

Renormalized solutions converge locally to the original harmonic spinor.

The approach facilitates constructing solutions with prescribed singular behavior.

Abstract

Given a compact 3-manifold $Y$ and a $\mathbb Z_2$-harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ with singular set $\mathcal Z_0$, this article constructs a family of local solutions to the two-spinor Seiberg-Witten equations parameterized by $\epsilon \in (0,\epsilon_0)$ on tubular neighborhoods of $\mathcal Z_0$. These solutions concentrate in the sense that the $L^2$-norm of the curvature near $\mathcal Z_0$ diverges as $\epsilon\to 0$, and after renormalization they converge locally to the original $\mathbb Z_2$-harmonic spinor. In a sequel to this article, these model solutions are used in a gluing construction showing that any $\mathbb Z_2$-harmonic spinor satisfying some mild assumptions arises as the limit of a family of two-spinor Seiberg-Witten solutions on $Y$.