Historically, elliptic cohomology arose from the study of
elliptic genera. It was known by Atiyah and Hirzebruch that if 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 -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 even periodic if for i odd and there is an invertible element . 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 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 over . The usual construction of such elliptic cohomology theories uses the
Landweber exact functor theorem. If the formal group law of is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by
Franke has identified the condition needed to fulfill Landweber exactness:
needs to be flat over
There is no irreducible component of , where the fiber is
supersingular for every
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
over 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 forms[1]pg 20
as the homotopy limit of this presheaf over the previous site.
Landweber, Peter S. (1988), "Elliptic genera: An introductory overview", in Landweber, P. S. (ed.), Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, vol. 1326, Berlin: Springer, pp. 1–10,
ISBN3-540-19490-8.
Landweber, Peter S. (1988), "Elliptic cohomology and modular forms", in Landweber, P. S. (ed.), Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics, vol. 1326, Berlin: Springer, pp. 55–68,
ISBN3-540-19490-8.
Landweber, P. S.; Ravenel, D. & Stong, R. (1995), "Periodic cohomology theories defined by elliptic curves", in Cenkl, M. & Miller, H. (eds.), The Čech Centennial 1993, Contemp. Math., vol. 181, Boston: Amer. Math. Soc., pp. 317–338,
ISBN0-8218-0296-8.
Lurie, Jacob (2009), "A Survey of Elliptic Cohomology", in Baas, Nils; Friedlander, Eric M.; Jahren, Björn; et al. (eds.), Algebraic Topology: The Abel Symposium 2007, Berlin: Springer, pp. 219–277,
doi:
10.1007/978-3-642-01200-6,
hdl:2158/373831,
ISBN978-3-642-01199-3.