The Witten genus and topological modular forms

I will give a fairly high level introduction to topological modular forms (Tmf) and their relationship to the Witten genus. Tmf was constructed, in part, as the target of a cohomological lift of Witten genus. The Witten genus assigns a modular form to a string manifold and this assignment is invariant under string bordism. This is analogous to the A-hat genus which assigns to a spin manifold an elliptic operator which, in turn, defines a formal difference of vector spaces. Moreover the A-hat genus is invariant under spin bordism. Aityah, Bott, and Shapiro demonstrated the A-hat genus lifts to a map of cohomology theories MSpin-->KO. The analogous theory for the Witten genus is the cohomology theory tmf which has been constructed, in several ways, by Mike Hopkins, Haynes Miller, Paul Goerss, and Jacob Lurie. These lifts were constructed and shown to preserve an enormous amount of structure (they are E_\infty maps) by Ando, Hopkins, and Rezk.