On manifolds with rational string structure the Witten genus takes values in modular forms. On manifolds with actual string structure this refines further to a ring of “topological modular forms”. This ring is at the same time the ring of homotopy groups of an E-∞ ring spectrum, called tmf.
Write $\overline{\mathcal{M}}$ for the Deligne-Mumford compactification of the moduli stack of elliptic curves regarded as a derived scheme, such that tmf is defined as the global sections of the derived structure sheaf
Write $\omega$ for the standard line bundle on $\overline{\mathcal{M}}$ such that the sections of $\omega^{\otimes k}$ are the ordinary modular forms of weight $k$ (as discussed there).
Then there is the descent spectral sequence
and since the ordinary modular forms embed on the left as
this induces an edge morphism
from topological modular forms to ordinary modular forms.
The kernel and cokernel of this map are 2-torsion and 3-torsion and hence “away from 6” this map is an isomorphism.
See also the references at tmf.
An introductory exposition is in
A collection of resources is in
The original identification of topological modular forms as the coefficient ring of the tmf E-∞ ring and the refinement of the Witten genus to a morphism of E-∞ rings, hence to the string orientation of tmf is due to
Michael Hopkins, Topological modular forms, the Witten Genus, and the theorem of the cube, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf)
Michael Hopkins, Algebraic topology and modular forms, Proceedings of the ICM, Beijing 2002, vol. 1, 283–309 (arXiv:math/0212397)
Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)
see also remark 1.4 of
and for more on the sigma-orientation see
Last revised on April 6, 2019 at 10:22:09. See the history of this page for a list of all contributions to it.