Contents

cohomology

# Contents

## Idea

The Ochanine elliptic genus $\Omega^{SO}_\bullet \to M_\bullet = \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]$ lifts to a map of ring spectra

$M SO \longrightarrow Ell$

(Landweber-Ravenel-Stong 93). Here $Ell[\tfrac{1}{6}]$ is equivalently tmf0(2) (Behrens 05) and as such this lift is analogous to the string orientation of tmf $M String \to tmf$.

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).

## Definition

After inversion of the prime number 2, the oriented cobordism ring is a polynomial ring over $\mathbb{Z}[\tfrac{1}{2}]$ on generators in degrees $4k$

$\Omega^{SO}_\bullet[\tfrac{1}{2}] \simeq \mathbb{Z}[\tfrac{1}{2}] [x_4, x_8, x_{12}, \cdots ]$

where $x_4$ is the class of the complex projective space $\mathbb{C}P^2$ and $x_8$ that of $\mathbb{H}P^2$ and where all elliptic genera vanish on all the other generators (Landweber-Ravenel-Stong 93, prop. 3.2).

From this one gets that the quotient by the ideal generated by these higher elements is

$\Omega^{SO}_\bullet[\tfrac{1}{2}]/(x_{4(k \geq 3)}) \simeq MF_0(2)_\bullet \coloneqq \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]$

where the right hand side here is naturally identified as the ring of those modular forms for the congruence subgroup $\Gamma_0(2)$ which have half-integral coefficients in their $q$-expansion at the nodal curve (Landweber-Ravenel-Stong 93, theorem 1.5).

Now by a general construction due to (Baas 73) this induces a generalized homology theory

$\Omega^{SO}_\bullet[\tfrac{1}{2}](-)$

represented by some spectrum $Ell$,whose coefficient ring is as above

$Ell_\bullet \simeq MF_0(2)_\bullet \,.$

By construction, this comes with a quotient map

$M SO[\tfrac{1}{2}] \longrightarrow Ell$

which is a map of ring spectra by (Mironov 78). This maplifts the universal elliptic genus (in that it reproduces it on homotopy groups) (Landweber-Ravenel-Stong 93, section 4.6, 4.7)

## Properties

### Induced relation between cobordism and homology

The SO orientation of elliptic cohomology makes it expressible in terms of the cobordism cohomology theory, see at cobordism theory determining homology theory (Landweber-Ravenel-Stong 93, theorem 1.2).

### Relation to the Atiyah-Bott-Shapiro Spin orientation of KO

There are maps of spectra

$Ell [\epsilon^{-1}] \longrightarrow KO[\tfrac{1}{2}]$

and

$Ell [(\delta^2- \epsilon)^{-1}] \longrightarrow KO[\tfrac{1}{2}]$

such that postcomposition of the above SO-orientation with reproduces the signature genus and the Atiyah-Bott-Shapiro orientation of KO, respective, hence the A-hat genus (Landweber-Ravenel-Stong 93, prop. 4.9).

Notice that here the second localization correponds again to including the nodal curve:

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

coveringby of level-n structures (modular curve)
$\ast = Spec(\mathbb{Z})$$\to$$Spec(\mathbb{Z}[ [q] ])$$\to$$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack$\mathcal{M}_{1dTori}$$\hookrightarrow$$\mathcal{M}_{Tate}$$\hookrightarrow$$\mathcal{M}_{\overline{ell}}$ (M_ell)$\hookrightarrow$$\mathcal{M}_{cub}$$\to$$\mathcal{M}_{fg}$ (M_fg)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$KU$KU[ [q] ]$elliptic spectrumcomplex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf(KO $\hookrightarrow$ KU) = KR-theoryTate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)tmf$\mathbb{S}$

### Homotopy type of the spectrum $Ell$

After suitable localization the spectrum Ell is a wedge sum of suspensions of the Morava E-theory $E(2)$ (Baker 97).

Specifically after K(2)-localization and inversion of 6 it coincides with TMF0(2)

$L_{K(2)} TMF_0(2) \simeq L_{K(2)}(E(2) \vee \Sigma^8 E(2)) \,.$
$d$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 is due to

based on general constructions of multiplicative homology theories from cobordism theories due to

• Nils Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973), 279–302.

• O. K. Mironov, Multiplications in cobordism theories with singularities, and Steenrod-tom Dieck operations_, Izv. Akad. Nauk SSSR, Ser. Mat. 42 (1978), 789–806; English transl. in Math. USSR Izvestiya 13 (1979), 89–106.

Analysis of $Ell$ is in

• Andrew Baker, The homotopy type of the spectrum representing elliptic cohomology, Proceedings of the American Mathematical Society 107.2 (1989): 537-548. (pdf)

• Andrew Baker, On the Adams $E_2$-term for elliptic cohomology, 1997 (pdf)

The interpretation of $Ell$ in terms of TMF0(2) is discussed in

More is in

Last revised on February 26, 2016 at 11:18:16. See the history of this page for a list of all contributions to it.