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SO orientation of elliptic cohomology

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Elliptic cohomology

Cohomology

cohomology

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Special notions

Variants

Extra structure

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Theorems

Contents

Idea

The Ochanine elliptic genus Ω SOM =[12][δ,ϵ]\Omega^{SO}_\bullet \to M_\bullet = \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon] lifts to a map of ring spectra

MSOEll M SO \longrightarrow Ell

(Landweber-Ravenel-Stong 93). Here Ell[16]Ell[\tfrac{1}{6}] is equivalently tmf0(2) (Behrens 05) and as such this lift is analogous to the string orientation of tmf MStringtmfM 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 [12]\mathbb{Z}[\tfrac{1}{2}] on generators in degrees 4k4k

Ω SO[12][12][x 4,x 8,x 12,] \Omega^{SO}_\bullet[\tfrac{1}{2}] \simeq \mathbb{Z}[\tfrac{1}{2}] [x_4, x_8, x_{12}, \cdots ]

where x 4x_4 is the class of the complex projective space P 2\mathbb{C}P^2 and x 8x_8 that of P 2\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

Ω SO[12]/(x 4(k3))MF 0(2) [12][δ,ϵ] \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 Γ 0(2)\Gamma_0(2) which have half-integral coefficients in their qq-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

Ω SO[12]() \Omega^{SO}_\bullet[\tfrac{1}{2}](-)

represented by some spectrum EllEll,whose coefficient ring is as above

Ell MF 0(2) . Ell_\bullet \simeq MF_0(2)_\bullet \,.

By construction, this comes with a quotient map

MSO[12]Ell 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[ϵ 1]KO[12] Ell [\epsilon^{-1}] \longrightarrow KO[\tfrac{1}{2}]

and

Ell[(δ 2ϵ) 1]KO[12] 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)
*=Spec()\ast = Spec(\mathbb{Z})\toSpec([[q]])Spec(\mathbb{Z}[ [q] ])\to ell¯[n]\mathcal{M}_{\overline{ell}}[n]
structure group of covering /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} /2\downarrow^{\mathbb{Z}/2\mathbb{Z}} SL 2(/n)\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})} (modular group)
moduli stack 1dTori\mathcal{M}_{1dTori}\hookrightarrow Tate\mathcal{M}_{Tate}\hookrightarrow ell¯\mathcal{M}_{\overline{ell}} (M_ell)\hookrightarrow cub\mathcal{M}_{cub}\to fg\mathcal{M}_{fg} (M_fg)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value 𝒪 Σ top\mathcal{O}^{top}_{\Sigma} of structure sheaf over curve Σ\SigmaKUKU[[q]]KU[ [q] ]elliptic spectrumcomplex oriented cohomology theory
spectrum Γ(,𝒪 top)\Gamma(-, \mathcal{O}^{top}) of global sections of structure sheaf(KO \hookrightarrow KU) = KR-theoryTate K-theory (KO[[q]]KU[[q]]KO[ [q] ] \hookrightarrow KU[ [q] ])(Tmf \to Tmf(n)) (modular equivariant elliptic cohomology)tmf𝕊\mathbb{S}

Homotopy type of the spectrum EllEll

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

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

L K(2)TMF 0(2)L K(2)(E(2)Σ 8E(2)). L_{K(2)} TMF_0(2) \simeq L_{K(2)}(E(2) \vee \Sigma^8 E(2)) \,.

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

ddpartition function in dd-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 MSpinKOM 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 MSpin cKUM 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 MSpin cKUM 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 EllEll 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 2E_2-term for elliptic cohomology, 1997 (pdf)

The interpretation of EllEll 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.