A Survey of Elliptic Cohomology

This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in Sect. I with an overview of the classical theory of elliptic cohomology. In Sect. 2 we review the theory of E(infinity)-ring spectra and introduce the language of derived algebraic geometry. We apply this theory in Sect. 3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In Sect. 4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In Sect. 5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2-equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who arc not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [1].