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

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 $E_8\times E_8$ 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. We also find that the $Spin(16)\times Spin(16)$ 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 $\ddot{I}{€}_{-32}\mathrm{TMF}$ as a particular self-dual vertex operator superalgebra of $c=16$ and how we utilize string duality to arrive at this statement.

20 pages + two appendices, 2 figures; v2: notations changed and references added, v3: appendix B on bordism group computation added