On a Z₃-valued discrete topological term in 10d heterotic string theories

We show that the low-energy effective actions of two ten-dimensional supersymmetric heterotic strings are different by a Z₃-valued discrete topological term even after we turn off the E₈× E₈ and Spin(32)/Z₂ 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)× Spin(16) non-tachyonic non-supersymmetric heterotic string has the same non-zero Z₃-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 Z/3-torsion element of π₋₃₂TMF as a particular self-dual vertex operator superalgebra of $c=16$ and how we utilize string duality to arrive at this statement.