Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras

We construct a morphism of spectra from KO((q))/TMF to Σ^-20I_ℤTMF, which we show to be an equivalence and to implement the Anderson self-duality of TMF. This morphism is then used to define another morphism from TMF to Σ^-20I_ℤ(MSpin/MString), which induces a differential geometric pairing and captures not only the invariant of Bunke and Naumann but also a finer invariant which detects subtle Anderson dual pairs of elements of π_∙TMF. Our analysis leads to conjectures concerning certain self-dual vertex operator superalgebras and some specific torsion classes in π_∙TMF. This paper is written as an article in mathematics, but much of the discussions in it was originally motivated by a study in heterotic string theory. As such, we have a separate appendix for physicists, in which the contents of the paper are summarized and translated into a language more amenable to them. In physics terms, our result allows us to compute the discrete part of the Green-Schwarz coupling of the B-field in a couple of subtle hitherto-unexplored cases.