Power operations in elliptic cohomology and representations of loop groups

Abstract

Abstract. Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.

Conformal field theory and elliptic cohomology

“…In discussing the motivation for Proposition 17, we used the cohomology theory K[[q]][q −1 ] and Adams operations. However, it would be nice to phrase these ideas is in terms of the power operations in elliptic cohomology considered by Ando [1]. We also refer the reader to [18] for more background and further considerations.…”

“…This might suggest K [[q]][q −1 ] as the first candidate for elliptic cohomology. This approach was indeed pursued by Ando [1], and leads to some valuable conclusions; in particular, K [[q]][q −1 ] is the "homotopical counterpart" of the Tate curve. However, we want to go further: the coefficient ring of the Tate curve does not consist of modular forms of any kind, so this approach does not explain the modularity of the Thompson series.…”

The elliptic Weyl character formula

“…In a sense, the paper at hand is a following to Ando's [And00] and [And03], taking apart the ideas presented in those papers and rearranging them into a new picture. The most important new feature is that our derivation of the Weyl-Kac formula as a push-forward is entirely a compactmanifold argument.…”

“…Let G be a simple and simply connected compact Lie group with weight lattice Λ and Weyl group W . In [Loo77], Looijenga considers two graded rings (see also [And00]).…”

Comparison of power operations in Morava $E$-theories

“…where W (F p n ) is the ring of Witt vectors with coefficients in the finite field F p n with p n elements. Power operations in Morava E-theory were first studied by Ando in [1,2], where he used power operations in complex cobordism to produce power operations in Morava E-theory. In [6] Goerss-Hopkins showed that Morava E-theory supports an E ∞ -ring spectrum structure unique up to homotopy, which implies the existence of an H ∞ -ring spectrum structure on Morava E-theory.…”