nLab
tmf

Idea

The generalized (Eilenberg-Steenrod) cohomology theory called $tmf$ is the one represented by the spectrum that is obtained as the homotopy limit of the spectra of all elliptic cohomology theories.

The abbreviation ”$tmf$” stands for the ring of topological modular forms as this is the cohomology ring that $tmf$ assigns, essentially, to the point.

One of the greatest recent achievements in algebraic topology is the construction of the $tmf$ spectrum as the global section of a certain (infinity,1)-sheaf of commutative ring spectra over the moduli stack of elliptic curves.

From this sheaf, one can recover the Adams-type spectral sequence associated to $tmf$. According to SEC, this sheaf is actually the structure sheaf of the moduli stack classifying “oriented elliptic curves” over commutative ring spectra, or, to be in the correct variance, over derived affine schemes.

Witten genus

The $\mathrm{tmf}$-spectrum is the codomain of the Witten genus

$${\mathrm{Bord}}_{(\mathrm{\infty },\mathrm{\infty })}^{\mathrm{String}}\to \mathrm{tmf}.$$Bord_{(\infty,\infty)}^{String} \to tmf \,.

References

• Jacob Lurie, A Survey of Elliptic Cohomology

• Mike Hopkins and others: Talbot workshop on TMF (web)

On this side one can find diagrams showing the Adams spectral sequence for tmf.