Non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ and their geometric applications

The $\mathbb{Z}_{2}$-harmonic $1$-forms arises in various compactification problems in gauge theory, deformation problem in special holonomy and calibrated geometry, including those involving flat $PSL(2,\mathbb{C})$ connections, Hitchin equation, Fueter sections, branched deformations of special Lagragians and Donaldson's branched maximal sections into indefinite spaces.

In this talk, I will describe a recent construction of a family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^3$ and describe their relation to Lawlor's neck, a family of special Lagrangian in $\mathbb{C}^{n}$. We will also discuss a gluing construction, in which these examples are glued to a regular zero of a harmonic $1$-form on a compact manifold. This gluing construction verified a folklore conjecture on the existence of $\mathbb{Z}_2$ harmonic 1-forms on the manifolds with $b^{1}>0$.

This talk is based on my work on the construction of non-degenerate $\mathbb{Z}_{2}$-harmonic 1-forms on Euclidean space and on the gluing construction of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-form on compact manifolds.