The (anti-)holomorphic sector in C/Λ-equivariant cohomology, and the Witten class

Atiyah's classical work on circular symmetry and stationary phase shows how the (A) over cap 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/A-equivariant cohomology of the conformal double loop space Maps(C/A, 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. (c) 2023 Elsevier B.V. All rights reserved.