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The Tate curve is the formal neighbourhood of the nodal curve inside the compactified moduli stack of elliptic curves $\mathcal{M}_{\overline{ell}}$.
Over a given base ring $R$ it is an elliptic curve over the power series $R [ [ q ] ]$ which may be thought of as the quotient of the multiplicative group by $q^{\mathbb{Z}}$.
The Tate curve is naturally an oriented elliptic curve? and as such classified by a map
in derived algebraic geometry, from the spectrum of the power series over the KU E-∞ ring to the derived moduli stack of derived elliptic curves.
The generalized cohomology theory over the Tate curve by the Goerss-Hopkins-Miller theorem is Tate K-theory $KO[ [q] ]$.
This is the homotopy limit over copies $KO \wedge \{1, q, \cdots, q^n\}_+$ of KO with formal parameters adjoined.
The elliptic genus associated with the Tate curve is, according to the above, a formal power series of K-theory classes. This is the Witten genus (Hopkins 94).
Physically, Witten obtained this as the partition function of the heterotic string in perturbation theory about configurations where all the worldsheet sits in a single point of target space. This limiting case of the worldsheet is the Tate curve.
Formally this is given by the homotopy groups of the String orientation of tmf $M String \to tmf$ followed by the map to Tate K-theory above, discussed here.
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 | $\mathbb{S}$ |
Conceptual discussion is in
Details are spelled out in
Basic reviews include
See also
The relation to the Witten genus originates around
More details on this are in (Hill-Lawson 13, appendix A).
Last revised on November 13, 2020 at 13:44:24. See the history of this page for a list of all contributions to it.