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

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 $\mathbb{Z}/2$-harmonic spinor subject to certain natural regularity assumptions. This paper proves that the virtual dimension of $\mathbb{Z}/2$-harmonic spinors for a generic metric is equal to zero. The paper also computes the virtual dimension of certain $\mathbb{Z}/2$-harmonic spinors on $4$-manifolds using an index theorem developed by Jochen Bruning and Robert Seeley and, independently, Fangyun Yang.

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Received 31 May 2017

Accepted 1 May 2018

Published 1 February 2019