  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.

High Energy Physics - Theory;Algebraic Topology