Remarks on Mod-2 Elliptic Genus

For physicists: For supersymmetric quantum mechanics, there are cases when a mod-2 Witten index can be defined, even when a more ordinary \(\mathbb {Z}\) $\mathbb{Z}$ -valued Witten index vanishes. Similarly, for 2d supersymmetric quantum field theories, there are cases when a mod-2 elliptic genus can be defined, even when a more ordinary elliptic genus vanishes. We study such mod-2 elliptic genera in the context of \(\mathcal {N}{=}(0,1)\) $\mathcal{N}=(0,1)$ supersymmetry, and show that they are characterized by mod-2 reductions of integral modular forms, under some assumptions. For mathematicians: We study the image of the standard homomorphism \(\begin{aligned} \pi _n\textrm{TMF}\rightarrow \pi _n\textrm{KO}((q))\simeq \mathbb {Z}/2((q)) \end{aligned}\) $$\begin{array}{c}\hfill {\pi }_n\text{TMF}\to {\pi }_n\text{KO}((q))\simeq \mathbb{Z}/2((q))\end{array}$$ for \(n=8k+1\) $n=8k+1$ or \(8k+2\) $8k+2$ , by relating them to the mod-2 reductions of integral modular forms.