 Historically, elliptic cohomology arose from the study of elliptic genera.It was known by Atiyah and Hirzebruch that if formula_1 acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes.In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant.