Mathematical Research Letters

Volume 25 (2018)

Number 5

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

Pages: 1645 – 1671



Ryosuke Takahashi (Institute of Mathematical Scences, Chinese University of Hong Kong, Shatin, N.T., Hong Kong)


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.

Received 31 May 2017

Accepted 1 May 2018

Published 1 February 2019