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

Abstract

Given a compact 3-manifold $Y$ and a $\mathbb Z_2$-harmonic spinor $(\mathcal Z_0, A_0,Φ_0)$ with singular set $\mathcal Z_0$, this article constructs a family of local solutions to the two-spinor Seiberg-Witten equations parameterized by $ε\in (0,ε_0)$ 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 $ε\to 0$, 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 $Y$.