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topological modular form

Context

Stable Homotopy theory

Elliptic cohomology

String theory

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

Write ω\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 kk (as discussed there).

Then there is the descent spectral sequence

H s(M¯,ω t)tmf 2ts H^s(\overline{M}, \omega^{\otimes t}) \Rightarrow tmf_{2t-s}

and since the ordinary modular forms embed on the left as

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

this induces an edge morphism

tmf 2MF 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

Revised on November 17, 2014 22:45:27 by maming? (134.76.62.130)