  Atiyah's 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.
