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 Z_3 ℤ3 -valued discrete topological term even after we turn off the E_8×E_8 E8×E8 and Spin(32)/Z_2 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. This result will be used to show further that the Spin(16)×Spin(16) Spin(16)×Spin(16) non-tachyonic non-supersymmetric heterotic string has the same non-zero Z_3 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 ℤ/3 -torsion element of π₋₃₂TMF π−32TMF as a particular self-dual vertex operator superalgebra of c=16 c=16 and how we utilize string duality to arrive at this statement.