The (anti-)holomorphic sector in $mathbb{C}/Lambda$-equivariant cohomology, and the Witten class

Atiyahs classical work on circular symmetry and stationary phase shows how the $hat{A}$-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 $mathbb{C}/Lambda$-equivariant cohomology of the conformal double loop space $mathrm{Maps}(mathbb{C}/Lambda,X)$ 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.