Contents

Contents

Idea

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.

Properties

Relation to modular forms

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

$tmf = \mathcal{O}(\overline{\mathcal{M}}) \,.$

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

$H^s(\overline{M}, \omega^{\otimes t}) \Rightarrow tmf_{2t-s}$

and since the ordinary modular forms embed on the left as

$H^0(\overline{\mathcal{M}}, \omega^{\otimes t}) \hookrightarrow H^s(\overline{M}, \omega^{\otimes t})$

this induces an edge morphism

$tmf_{2 \bullet} \longrightarrow MF_\bullet$

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.

References

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

see also remark 1.4 of

• Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie) (pdf).

and for more on the sigma-orientation see

Equivariant topological modular forms are discussed in

Last revised on June 1, 2020 at 08:23:57. See the history of this page for a list of all contributions to it.