  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
$\pi_{-32}\mathrm{TMF}$ as a particular self-dual vertex operator superalgebra
of $c=16$ and how we utilize string duality to arrive at this statement.
