Physicists use the "AtiyahBott formula" for path "integrals" (for instance the supersymmetric proof of the AtiyahSinger index theorem. Is there some way to make atleast some of these ideas rigorous? As in is there a rigorous localisation formula for infinite dimensional integration? (i.e. using the Weiner measure when applicable?)
1 Answer
There is the work of Johan Martens on equivariant localization of nonabelian group actions on noncompact spaces ( http://arxiv.org/abs/math/0609841 ), but I think that the general problem as you posed it is still open, seeing that there is an AIM workshop partly on this question next week: http://www.aimath.org/ARCC/workshops/localization.html
edit: Charlie Frohman is right, I didn't read the AIM workshop page carefully enough.
As for the original question, I have only encountered localization techniques in physics when calculating instanton contributions to the path integral, which gives equivariant localization on finite dimensional spaces (although badly behaved in other ways). The supersymmetric "proof" of the AtiyahSinger index theorem can be made quite rigorous (see the 1983 paper by Getzler for example), without any recourse to infinite dimensional spaces. Most of the articles I read from the physics literature skipped over the infinite dimensional case, with some reference to a nonrenormalization theorem.
To my latest knowledge (around middle of 2008) there is no general method for equivariant localization in infinite dimensional spaces. Also for most highenergy physics purposes, Wiener measures are not the right approach for doing Feynman path integrals.

$\begingroup$ I think GKM localization is most certainly a descendent of the AtiyahBott formula, but it is used in AlgebreoGeometric settings as opposed to Gauge theoretic setting. It is however, a well developed and rigorous technique. $\endgroup$ Mar 9, 2010 at 15:49