## On a $\mathbb{Z}_3$-valued discrete topological term in 10d heterotic string theories

Yuji Tachikawa, Hao Y. Zhang

SciPost Phys. 17, 077 (2024) · published 11 September 2024

- doi: 10.21468/SciPostPhys.17.3.077
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### Abstract

We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a $\mathbb{Z}_3$-valued discrete topological term even after we turn off the $E_8×E_8$ and $Spin(32)/\mathbb{Z}_2$ gauge fields. This will be demonstrated by considering the inflow of normal bundle anomaly to the respective NS5-branes from the bulk. This result will be used to show further that the $Spin(16)×Spin(16)$ non-tachyonic non-supersymmetric heterotic string has the same non-zero $\mathbb{Z}_3$-valued discrete topological term. We will also explain the relation of our findings to the theory of topological modular forms. The paper is written as a string theory paper, except for an appendix translating the content in mathematical terms. We will explain there that our finding identifies a representative of the $\mathbb{Z}/3$-torsion element of $\pi_{-32}\mathrm{TMF}$ as a particular self-dual vertex operator superalgebra of $c=16$ and how we utilize string duality to arrive at this statement.

### Authors / Affiliation: mappings to Contributors and Organizations

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^{1}Yuji Tachikawa, -
^{1}Hao Zhang