nLab topological modular form

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.

Related concepts

• modular form

• Jacobi form

References

See also the references at tmf.

An introductory exposition is in

• Aaron Mazel-Gee, You could’ve invented $tmf$, April 2013 (pdf slides)

A collection of resources is in

• Nora Ganter, Topological modular forms literature list

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

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

and for more on the sigma-orientation see

• Matthew Ando, The sigma orientation for analytic circle-equivariant elliptic cohomology, Geom. Topol. 7 (2003) 91-153 (arXiv:math/0201092)

Equivariant topological modular forms are discussed in

• David Gepner, Lennart Meier, On equivariant topological modular forms, (arXiv:2004.10254)