cohomology

# Contents

## Idea

Where tmf is obtained from the moduli stack of elliptic curves, $tmf_0(2)$ (or its versions $Tmf_0(2)$, $TMF_0(2)$) denotes the spectrum obtained from the cover by the moduli stack of elliptic curves with level structure for the congruence subgroup $\Gamma_0(2)$ (the one that preserves the NS-R spin structure on an elliptic curve, see here).

Detailed discussion is in (Behrens 05).

This plays at least roughly the role for the Ochanine elliptic genus (the partition function of the type II superstring in the NS-R sector) as tmf does for the Witten genus (the partition function of the heterotic superstring). See at spin orientation of Ochanine elliptic cohomology for more.

## Properties

### Relation to Landweber-Ravenel-Stong-Ochanine elliptic spectrum

After inversion of 6, $TMF_0(2)$ is (Behrens 05) the elliptic spectrum $Ell$ of (Landweber-Ravenel-Stong 93).

### Periodicity

The periodicity of $TMF_0(2)$ is 8.

## References

Last revised on May 13, 2014 at 09:38:35. See the history of this page for a list of all contributions to it.