Concentrating Local Solutions of the Two-Spinor Seiberg-Witten Equations

Given a compact 3-manifold $Y$ and a $\mathbb Z_2$-harmonic spinor $(\mathcal Z_0, A_0,\Phi_0)$ with singular set $\mathcal Z_0$, I will explain a construction of 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. I will explain how these model solutions are used in a gluing construction showing a $\mathbb Z_2$-harmonic spinor satisfying mild assumptions necessarily arises as the limit of a sequence of two-spinor Seiberg-Witten solutions on Y.

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