Index theorem for $\mathbb{Z}/2$-harmonic spinors

Abstract

Let M denote a compact 3-manifold. The author proved in [8] that there exists a Kuranishi structure for the moduli space of pairs consisting of a Riemannian metric on M and a non-zero Z/2-harmonic spinor subject to certain natural regularity assumptions. This paper proves that the virtual dimension of Z/2-harmonic spinors for a generic metric is equal to zero. The paper also computes the virtual dimension of certain Z/2-harmonic spinors on 4-manifolds using an index theorem developed by Jochen Bruning and Robert Seeley and, independently, Fangyun Yang.

Deformations of multivalued harmonic functions

“…Questions of a similar character to the problem we consider here arise in recent work on non-compactness phenomena for various equations in gauge theory over 3-manifolds and 4-manifolds. This development began with the work of Taubes [9] and has been studied further by a number of other authors, for example [5], [8]. It seems possible that the methods in this paper could have useful applications to these other equations.…”