Z/2 harmonic 1-forms, R-trees, and the Morgan-Shalen compactification

This paper studies the relationship between an analytic compactification of the moduli space of flat ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan-Shalen compactification of the ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ character variety of the fundamental group of $M$. We exhibit an explicit correspondence between $\mathbb{Z}\mathrm{/}2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to $\mathbb{R}$-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}\mathrm{/}2$ harmonic 1-forms exist on all Haken manifolds with respect to all Riemannian metrics. We also show that there exist manifolds that support singular $\mathbb{Z}\mathrm{/}2$ harmonic 1-forms but have compact ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ character varieties, which resolves a folklore conjecture.