Remarks on Mod-2 Elliptic Genus

Abstract

For physicists: For supersymmetric quantum mechanics, there are cases when a mod-2 Witten index can be defined, even when a more ordinary \(\mathbb {Z}\)-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 \(\mathcal {N}{=}(0,1)\) 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

$$\begin{aligned} \pi _n\textrm{TMF}\rightarrow \pi _n\textrm{KO}((q))\simeq \mathbb {Z}/2((q)) \end{aligned}$$

for \(n=8k+1\) or \(8k+2\), by relating them to the mod-2 reductions of integral modular forms.