Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras
arxiv(2023)
摘要
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.
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