nLab
string orientation of tmf

Context

String theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher linear algebra

Higher spin geometry

Contents

Idea

The string orientation of tmf is the universal orientation in generalized cohomology for tmf-cohomology (“universal elliptic cohomology), given by a homomorphism

σ:MStringtmf \sigma \;\colon\; M String \longrightarrow tmf

of E-∞ rings, from the String structure-Thom spectrum to tmf This is refinement of the Witten genus (see there for more)

w:Ω String,ratMF w \;\colon\; \Omega^{String,rat}_\bullet \longrightarrow MF_\bullet

(with values in the ring of modular forms) which it reproduces on homotopy groups

wπ (σ). w \simeq \pi_\bullet(\sigma) \,.

For this reason the string orientation of tmf is also referred to as the “topological Witten genus”.

All this is due to (Ando-Hopkins-Strickland 01, Ando-Hopkins-Rezk 10).

See the Idea-section at tmf and at Witten genus for more background.

References

The construction of the string orientation and its relation to the Witten genus is due to

following announcements of results in

  • Michael Hopkins, Algebraic topology and modular forms in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press (arXiv:math/0212397)

which in turn follows the general program outlined in

  • Michael Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨urich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043

An alternative construction using the derived algebraic geometry of the moduli stack of elliptic curves is sketched in

Revised on April 23, 2014 13:21:23 by Urs Schreiber (82.169.114.243)