Topological modular forms

Summary

Topological modular forms is a ring spectrum, written $\mathrm{tmf}$, that collects all elliptic cohomology theories into one universal object. Goerss, Hopkins and Miller built it as the global sections of a sheaf of $E_\infty$ ring spectra on the moduli stack of elliptic curves. Its homotopy groups map to the ring of integral modular forms, and this map is an isomorphism after tensoring with $\Q$. The torsion in $\pi_* \mathrm{tmf}$ detects many elements of the stable homotopy groups of spheres, and $\mathrm{tmf}$ receives the Witten genus of string manifolds.