nLab SO orientation of elliptic cohomology

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:

covering					by 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)
of	1d tori		Tate curves		elliptic curves		cubic curves		1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$	KU		$KU[ [q] ]$		elliptic spectrum				complex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf	(KO $\hookrightarrow$ KU) = KR-theory		Tate 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)

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

Related concepts

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

The construction is due to

• Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)

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

• Mark Behrens, A modular description of the K(2)-local sphere at the prime 3 (arXiv:math/0507184)

More is in

• Dylan Wilson, Orientations and Topological Modular Forms with Level Structure (arXiv:1507.05116)