Supersymmetric Path Integrals, the Localization principle and the Atiyah-Singer Index Theorem

We give an overview of the (non-rigorous) path integral proof of the Atiyah-Singer theorem, based
on the work of Witten, Atiyah Alvarez-Gaumé and others. The approach postulates the validity of
the localization principle of equivariant cohomology for (purely formal) differential form integrals
over the loop space of a compact manifold, in analogy to the finite-dimensional case. In the second
part of the talk, we show how to rigorously construct an integral map on differential forms on the
loop space of a compact Riemiannian manifold which is „supersymmetric“ in a certain sense, and
for which such a localization principle holds, at least in special cases. This is joint work with
Florian Hanisch.