Deformation Theory for Z_2-Harmonic Spinors
Z_2-harmonic spinors are singular generalizations of classical harmonic spinors that allow topological twisting around a submanifold of codimension 2 called the singular set. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and are expected to contribute to new gauge-theoretic invariants via wall-crossing formulas. In this talk, I will focus on a recent result showing that the universal moduli space of Z_2-harmonic spinors at sufficiently generic points is locally a codimension 1 submanifold, i.e. a "wall'' in the space of parameters. The key ingredients in the proof are differentiating the relevant Dirac operator with respect to deformations of the singular set, and an application of the Nash-Moser Implicit Function Theorem.