# nLab spin orientation of elliptic cohomology

Contents

cohomology

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

# Contents

## Idea

The Ochanine genus lifts to a homomorphism of ring spectra $M Spin \to KO[ [q] ]$ from spin structure cobordism cohomology theory to Tate K-theory (Kreck-Stolz 93, lemma 5.8, lemma 5.4).

Moreover, localized at 2 this map factors (Kreck-Stolz 93, last line of p. 18 and cor 5.2 and page 21) through a map

$M Spin \longrightarrow el$

where $el$ is a 2-local spectrum (at least closely related to tmf0(2)) whose coefficient ring is the coefficient ring for the Ochanine elliptic genus

$el_\bullet = \mathbb{Z}[a,b,c, \epsilon]/(2a, a^3, a b, b^2-4)$

wich, after inversion of 2, surjects onto the ring $\mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]$ of modular forms for congruence subgroup $\Gamma_0(2)$

This is Kreck-Stolz 93, theorem 1 (while we follow in notation Hovey 95, page 2), which is a kind of refinement of the SO orientation of elliptic cohomology due to (Landweber-Ravenel-Stong 93).

If this map of ring spectra could be shown to be “highly structured” in that it preserves E-∞ ring structure, then it would equivalently be a universal orientation (see at relation between orientations and genera).

## Properties

### Relation between cobordism and homology

(Hovey 95), for the moment see at cobordism theory determining homology theory

$d$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

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Last revised on May 2, 2014 at 04:59:12. See the history of this page for a list of all contributions to it.