关于模2椭圆亏格的注记
Remarks on mod-2 elliptic genus
AI总结：
本文研究了模2椭圆亏格在超对称量子场论及拓扑模形式同态中的刻画，证明其在N=(0,1)超对称下由积分模形式的模2约化决定。
摘要：
致物理学家：对于超对称量子力学，在某些情况下可以定义模2 Witten指数，即使更普通的取值为Z的Witten指数消失。类似地，对于2维超对称量子场论，在某些情况下可以定义模2椭圆亏格，即使更普通的椭圆亏格消失。我们在N=(0,1)超对称的背景下研究了此类模2椭圆亏格，并证明在某些假设下，它们由积分模形式的模2约化所刻画。致数学家：我们通过将标准同态π_n TMF→π_n KO((q))≃Z/2((q))（当n=8k+1或8k+2时）与积分模形式的模2约化相关联，研究了该同态的像。
英文摘要：
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 $π_n \mathrm{TMF}\to π_n \mathrm{KO}((q))\simeq \mathbb{Z}/2((q))$ for $n=8k+1$ or $8k+2$, by relating them to the mod-2 reductions of integral modular forms.
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