cohomology

spin geometry

string 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

$\sigma \;\colon\; M String \longrightarrow tmf$

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

$w \;\colon\; \Omega^{String,rat}_\bullet \longrightarrow MF_\bullet$

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

$w \simeq \pi_\bullet(\sigma) \,.$

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 9, 2014 02:19:47 by Urs Schreiber (77.251.114.72)