2
$\begingroup$

In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable mathematical statements that I can see from my eyes to relate them -- or to learn the relation between TMF and below:

$\Omega_3^{fr}=\Omega_3^{string}=Z_{24}$

http://www.map.mpim-bonn.mpg.de/String_bordism

http://www.map.mpim-bonn.mpg.de/Framed_bordism

Do you know such relations or formulas?

Note that Wikipedia provides a background story, BUT THIS IS NOT what I asked for -- because there is no direct mathematical derivation in this paragraph:

"Some interest in tmf comes from string theory and conformal field theory. Graeme Segal first proposed in the 1980s to provide a geometric construction of elliptic cohomology (the precursor to tmf) as some kind of moduli space of conformal field theories, and these ideas have been continued and expanded by Stephan Stolz and Peter Teichner. Their program is to try to construct TMF as a moduli space of supersymmetric Euclidean field theories.

In work more directly motivated by string theory, Edward Witten introduced the Witten genus, a homomorphism from the string bordism ring to the ring of modular forms, using equivariant index theory on a formal neighborhood of the trivial locus in the loop space of a manifold. This associates to any spin manifold with vanishing half first Pontryagin class a modular form. By work of Hopkins, Matthew Ando, Charles Rezk and Neil Strickland, the Witten genus can be lifted to topology. That is, there is a map from the string bordism spectrum to tmf (a so-called orientation) such that the Witten genus is recovered as the composition of the induced map on the homotopy groups of these spectra and a map of the homotopy groups of tmf to modular forms. This allowed to prove certain divisibility statements about the Witten genus. The orientation of tmf is in analogy with the Atiyah–Bott–Shapiro map from the spin bordism spectrum to classical K-theory, which is a lift of the Dirac equation to topology."

Improve this question
$\endgroup$
1
  • $\begingroup$ There are maps of ring spectra $\mathbb{S} \to \mathrm{MString} \to \mathrm{TMF}$, inducing a map $\pi_3\mathbb{S} \to \pi_3\mathrm{MString} \to \pi_3\mathrm{TMF}$, which happen to be isomorphisms. (Indeed, the first map is an isomorphism by the connectivity of $\mathrm{BString}$; in general, $\pi_*\mathrm{MString} \to \pi_*\mathrm{tmf}$ is only a surjection and not an isomorphism; here, $\mathrm{tmf}$ is a connective variant of $\mathrm{TMF}$. See arxiv.org/abs/1911.10534) $\endgroup$
    – Lennart Meier
    Commented Nov 30, 2023 at 8:56

0

Reset to default

You must log in to answer this question.