group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The Ochanine elliptic genus $\Omega^{SO}_\bullet \to M_\bullet = \mathbb{Z}[\tfrac{1}{2}][\delta, \epsilon]$ lifts to a map of ring spectra
(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).
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 $4k$
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
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
represented by some spectrum $Ell$,whose coefficient ring is as above
By construction, this comes with a quotient map
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)
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).
There are maps of spectra
and
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 | $\mathbb{S}$ |
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)
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
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.