Remarks on mod-2 elliptic genus

TL;DR

This paper explores mod-2 elliptic genera in supersymmetric quantum theories, showing they relate to mod-2 reductions of modular forms, and investigates their mathematical properties via homomorphisms in topology.

Contribution

It characterizes mod-2 elliptic genera in ext{(0,1)} supersymmetry as mod-2 reductions of modular forms and studies their topological image via homomorphisms from TMF to KO.

Findings

Mod-2 elliptic genera are characterized by mod-2 reductions of integral modular forms.

The image of certain homomorphisms in topology relates to these mod-2 reductions.

The study bridges physics and mathematics through modular forms and topological invariants.

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 $\pi_n \mathrm{TMF}\to \pi_n \mathrm{KO}((q))\simeq \mathbb{Z}/2((q))$ for $n=8k+1$ or $8k+2$, by relating them to the mod-2 reductions of integral modular forms.