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

Kavli Affiliate: Yuji Tachikawa

| First 5 Authors: Yuji Tachikawa, Mayuko Yamashita, , ,

| Summary:

Inspired by consideration in heterotic string theory, 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 isomorphism and to implement the Anderson
self-duality of $mathrm{TMF}$. We will use this to factor a related morphism
from $mathrm{TMF}$ to
$Sigma^{-20}I_{mathbb{Z}}(mathrm{MSpin}/mathrm{MString})$. This latter
morphism induces a differential geometric pairing, which captures not only the
invariant of Bunke and Naumann but also a finer invariant which detects subtle
Anderson dual pairs of elements of $pi_bullet mathrm{TMF}$. Our analysis
leads to a conjecture concerning certain self-dual vertex operator
superalgebras and some specific torsion classes in $pi_bullet mathrm{TMF}$,
which we will detail in an appendix.
This paper is written in the rigorous mathematical style, except in a section
which summarizes and translates the content into a language more amenable to
string theorists. 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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