Abstract For physicists: For supersymmetric quantum mechanics, there are cases when a mod-2 Witten index can be defined, even when a more ordinary $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -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 $$\mathcal {N}{=}(0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\begin{aligned} \pi _n\textrm{TMF}\rightarrow \pi _n\textrm{KO}((q))\simeq \mathbb {Z}/2((q)) \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mtext>TMF</mml:mtext> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>π</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mtext>KO</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≃</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> for $$n=8k+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> or $$8k+2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>8</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , by relating them to the mod-2 reductions of integral modular forms.

Yuji Tachikawa

Mayuko Yamashita

Kazuya Yonekura

Communications in Mathematical Physics

COMMUNICATIONS IN MATHEMATICAL PHYSICS 的使命是为受现代物理学的远见和挑战激发的作品提供一个高论坛，同时满足最高的数学标准。

The mission of "Communications in Mathematical Physics" is to provide a high-level forum for works inspired by the foresight and challenges of modern physics, while adhering to the highest mathematical standards.

《Communications in Mathematical Physics》的使命是为受到现代物理学的前瞻性思维和挑战所启发的研究提供一个高水平的交流平台，同时确保满足最高的数学标准。

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物理与天体物理	1
物理与天体物理, 物理数学物理	1

Remarks on Mod-2 Elliptic Genus

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