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.

## Construction via Cubical structure

The construction proceeds via the relation between orientations in complex orientable cohomology theory and cubical structures on line bundles, see there for more.

## Properties

### Relation to twists of $tmf$

(…) relation to the twists of tmf-cohomology theory (…)

$\array{ && B String \\ & \swarrow && \searrow \\ \ast && && B Spin \\ & \searrow && \swarrow \\ && B^3 U(1) \\ && \downarrow \\ && B GL_1(tmf) }$
$d$partition function in $d$-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin \to KO$
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

## References

The construction/identification 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 (pdf)

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

In fact the construction there is a refinement of the orientation of just $tmf$ to one of all $E_\infty$-rings $A$ carrying a derived elliptic curve $E \to Spec(A)$.

Discussion in relation to the twists of tmf-cohomology is in

with some related chat in Quantization via Linear homotopy types.

Revised on October 28, 2014 17:47:40 by Urs Schreiber (141.0.8.158)