Orbifold genera, product formulas and power operations

Abstract

There is a formula by the string theorists Dijkgraaf, Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous p-typical statement follows as an easy corollary from the fact that the map of spectra corresponding to the genus preserves power operations. We define higher chromatic versions of the notion of orbifold genus, involving h-tuples rather than pairs of commuting elements. Using homotopy theoretic methods we are able to prove an integrality result and show that our definition is independent of the representation of the orbifold. Our setup is so simple, that it allows us to prove DMVV-type product formulas for these higher chromatic orbifold genera in the same way that the product formula for the topological Todd genus is proved. More precisely, we show that any genus induced by an H[omega]-map into one of the Morava-Lubin-Tate cohomology theories Eh has such a product formula and that the formula depends only on h and not on the genus. Since the complex H[omega]-genera into Eh have been classified in [And95], a large family of genera to which our results apply is completely understood. Loosely speaking, our result says that some Borcherds lifts have a well-known homotopy theoretic refinement, namely total symmetric powers in elliptic cohomology.