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 $({\mathcal{Z}}_0,A_0,{\Phi }_0)$ 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 $Y$.

104 pages, 3 figures. Minor revisions to original version. To appear in Asian Journal of Mathematics