Elliptic cohomology

Elliptic cohomology defines a cohomology theory in the sense of algebraic topology. This term may refer to several different, though closely related, constructions, such as topological modular forms (i.e., tmf) or more classically a periodic ring spectrum equipped with the formal group of a particular elliptic curve. tmf is the more general of these two concepts: the value of the sheaf tmf on an elliptic curve produces a periodic ring spectrum equipped with the formal group of that elliptic curve. These earlier versions of tmf were constructed by Landweber, Stong, and Ravenel. In the 1980s, this generated significant interest in mathematics and mathematical physics due to the occurrence of elliptic genera in physics. These original ideas may be found in the 1986 Princeton proceedings of a conference on elliptic curves in algebraic topology. A more modern perspective may be found in Jacob Lurie's survey.

References

*Franke, J. "On the construction of elliptic cohomology" Math. Nachr. , 158 (1992) pp. 43–65.

*Landweber, P. "Elliptic cohomology and modular forms" P.S. Landweber (ed.) , Elliptic Curves and Modular Forms in Algebraic Topology (Proc., Princeton 1986) , Lecture Notes in Mathematics , 1326 , Springer (1988) pp. 55–68

*Landweber, P., Ravenel, D. and Stong, R. "Periodic cohomology theories defined by elliptic curves" , The Čech Centennial (Boston, 1993) , Contemp. Math. , 181 , Amer. Math. Soc. (1995).

*Lurie, J. "A survey of elliptic cohomology." Available at: http://www-math.mit.edu/~lurie/papers/survey.pdf