Kavli Affiliate: Yuji Tachikawa
| First 5 Authors: Yuji Tachikawa, Mayuko Yamashita, , ,
| Summary:
We construct a morphism of spectra from $mathrm{KO}((q))/mathrm{TMF}$ to
$Sigma^{-20}I_{mathbb{Z}} mathrm{TMF}$, which we show to be an equivalence
and to implement the Anderson self-duality of $mathrm{TMF}$. This morphism is
then used to define another morphism from $mathrm{TMF}$ to
$Sigma^{-20}I_{mathbb{Z}}(mathrm{MSpin}/mathrm{MString})$, 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 $pi_bulletmathrm{TMF}$. Our analysis leads to conjectures
concerning certain self-dual vertex operator superalgebras and some specific
torsion classes in $pi_bulletmathrm{TMF}$.
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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