  In my previous paper, I prove the existence of the Kuranishi structure for
the moduli space $\mathfrak{M}$ of zero loci of $\mathbb{Z}/2$-harmonic spinors
on a 3-manifold. So a nature question we can ask is to compute the virtual
dimension for this moduli space
$\mathfrak{M}_{g_0}:=\mathfrak{M}\cap\{g=g_0\}$. In this paper, I will first
prove that $v-dim(\mathfrak{M}_{g_0})=0$. Secondly, I will generalize this
formula on 4-manifolds by using a special type of index developed by Jochen
Bruning, Robert Seeley, and Fangyun Yang.

Differential Geometry