Elliptic cohomology

In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
History and motivation.
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. This led to certain other problems concerning formula_1-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In some sense it can be seen as an approximation to the K-theory of the free loop space.
Definitions and constructions.
Call a cohomology theory formula_3 even periodic if formula_4 for i odd and there is an invertible element formula_5. These theories possess a complex orientation, which gives a formal group law. A particularly rich source for formal group laws are elliptic curves. A cohomology theory formula_6 with
is called "elliptic" if it is even periodic and its formal group law is isomorphic to a formal group law of an elliptic curve formula_8 over formula_9. The usual construction of such elliptic cohomology theories uses the Landweber exact functor theorem. If the formal group law of formula_8 is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by
Franke has identified the condition needed to fulfill Landweber exactness:
These conditions can be checked in many cases related to elliptic genera. Moreover, the conditions are fulfilled in the universal case in the sense that the map from the moduli stack of elliptic curves to the moduli stack of formal groups
is flat. This gives then a presheaf of cohomology theoriesformula_19over the site of affine schemes flat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking global sections has led to the construction of the topological modular formspg 20formula_20as the homotopy limit of this presheaf over the previous site.