The (anti-)holomorphic sector in [formula omitted]-equivariant cohomology, and the Witten class

Abstract

Atiyah's classical work on circular symmetry and stationary phase shows how the Aˆ-genus is obtained by formally applying the equivariant cohomology localization formula to the loop space of a simply connected spin manifold. The same technique, applied to a suitable “antiholomorphic sector” in the C/Λ-equivariant cohomology of the conformal double loop space Maps(C/Λ,X) of a rationally string manifold X produces the Witten genus of X. This can be seen as an equivariant localization counterpart to Berwick-Evans supersymmetric localization derivation of the Witten genus.