Gluing ${\mathbb{Z}}_2$-Harmonic Spinors and Seiberg-Witten Monopoles on 3-Manifolds

Given a ${\mathbb{Z}}_2$-harmonic spinor satisfying some genericity assumptions, this article constructs a 1-parameter family of two-spinor Seiberg-Witten monopoles converging to it after renormalization. The proof is a gluing construction beginning with model solutions on a neighborhood of the ${\mathbb{Z}}_2$-harmonic spinor’s singular set. The gluing is complicated by the presence of an infinite-dimensional obstruction bundle for the singular limiting linearized operator. This difficulty is overcome by introducing a generalization of Donaldson’s alternating method in which a deformation of the ${\mathbb{Z}}_2$-harmonic spinor’s singular set is chosen at each stage of the alternating iteration to cancel the obstruction components.