$\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds

TL;DR

This paper develops a gluing method to construct new $bZ_2$-harmonic spinors and 1-forms on connected sums and torus sums of 3-manifolds, leading to numerous examples and insights into their moduli spaces.

Contribution

It introduces a gluing technique for $bZ_2$-harmonic spinors and 1-forms on connected sums and torus sums, expanding the known examples and properties of these objects.

Findings

Existence of infinitely many $bZ_2$-harmonic spinors with distinct link singularities.

Construction of non-empty, non-compact moduli spaces for solutions to two-spinor Seiberg-Witten equations.

New examples of $bZ_2$-harmonic spinors and 1-forms on complex 3-manifolds.

Abstract

Given a pair of $\mathbb{Z}_2$-harmonic spinors (resp. 1-forms) on closed Riemannian 3-manifolds $(Y_1, g_1)$ and $(Y_2,g_2)$, we construct $\mathbb{Z}_2$-harmonic spinors (resp. 1-forms) on the connected sum $Y_1 \# Y_2$ and the torus sum $Y_1 \cup_{T^2} Y_2$ using a gluing argument. The main tool in the proof is a parameterized version of the Nash-Moser implicit function theorem established by Donaldson and the second author. We use these results to construct an abundance of new examples of $\mathbb Z_2$-harmonic spinors and 1-forms. In particular, we prove that for every closed 3-manifold $Y$, there exist infinitely many $\mathbb{Z}_2$-harmonic spinors with singular sets representing infinitely many distinct isotopy classes of embedded links, strengthening an existence theorem of Doan-Walpuski. Moreover, combining this with previous results, our construction implies that if $b_1(Y) > 0$, there exist infinitely many $\mathrm{spin}^c$ structures on $Y$ such that the moduli space of solutions to the two-spinor Seiberg-Witten equations is non-empty and non-compact.