关于模-2椭圆属的评论

IF 2.2 1区物理与天体物理 Q1 PHYSICS, MATHEMATICAL

Communications in Mathematical Physics Pub Date : 2024-12-18 DOI:10.1007/s00220-024-05202-4

Yuji Tachikawa, Mayuko Yamashita, Kazuya Yonekura

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引用次数: 0

摘要

对于物理学家来说对于超对称量子力学来说，有些情况下可以定义模-2维滕指数，即使更普通的（\mathbb {Z}\）值维滕指数消失了。同样，对于二维超对称量子场论，即使更普通的椭圆属消失了，也有可以定义模-2椭圆属的情况。我们在 $\mathcal {N}{=}(0,1)$ 超对称的背景下研究了这种模-2椭圆属，并证明在一些假设条件下，它们的特征是积分模形式的模-2还原。对于数学家我们研究了标准同态 $$\begin{aligned} 的图像。\pi _n\textrm{TMF}\rightarrow \pi _n\textrm{KO}((q))\simeq \mathbb {Z}/2((q))\end{aligned}$$对于（n=8k+1\）或（8k+2\），通过将它们与积分模形式的 mod-2 还原联系起来。

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Remarks on Mod-2 Elliptic Genus

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.

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来源期刊

Communications in Mathematical Physics 物理-物理：数学物理

CiteScore

4.70

自引率

8.30%

发文量

226

审稿时长

3-6 weeks

期刊介绍： The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.