Gluing Z₂-Harmonic Spinors on 3-Manifolds

Abstract

This thesis develops the analytic foundations of a gluing result for Z₂-harmonic spinors on 3-manifolds. Z₂-harmonic spinors are a singular version of classical harmonic spinors and naturally arise as limits of sequences of solutions to generalized SeibergWitten equations. The gluing problem studied here addresses the reverse question in the case of the two-spinor Seiberg-Witten equations on a 3-manifold. This thesis is divided into two parts.

Part I: This part constructs model solutions for the two-spinor Seiberg-Witten equations in a neighborhood of the singular set of a Z₂-harmonic spinor, and analyzes the linearized Seiberg-Witten equations at these. The model solutions are shown to converge to a given Z₂-harmonic spinor in a suitable sense, and the linearization is shown to be invertible with near-uniform control on suitable function spaces. The proofs rely on a detailed analysis of degenerating families of elliptic operators.

Part II: This part studies the deformation theory of Z₂-harmonic spinors. For a fixed smooth singular set, the deformation theory of 𝑍₂-harmonic spinors is obstructed by infinite-dimensional cokernel of the semi-Fredholm Z₂-Dirac operator. It is shown that the component of the first variation of the Z₂-Dirac operator with respect to deformations of the singular set in this obstruction space is an elliptic pseudo-differential operator of order 1/2. Consequently, the resulting deformation theory is Fredholm up to a loss of regularity phenomenon which may be addressed by Nash-Moser theory.