The Kempf-Ness theorem (see e.g. arXiv:0912.1132) - that the algebraic quotient of geometric invariant theory is also a symplectic quotient - suggests (to me) that certain physical constructions used to compute equivariant cohomologies might be useful to Mulmuley et al's geometric complexity theory. Edward Witten is the wellspring of these ideas and his latest paper (arXiv:1009.6032) continues to develop them. My main concern is that they might not carry across to the objects of interest to complexity theory (e.g. the "class varieties" of arXiv:cs/0612134). But the power of the quantum techniques, and the diversity of approaches possible within GCT, leads me to keep looking...

Cross-posted to TCS StackExchange.

Edit: Witten maps a nonrelativistic quantum theory (on a 2n-dimensional phase space) onto a (1+1)-dimensional field theory (actually, an "A-model" topological string theory whose target space is the complexification of the previous theory's phase space). The objective is just to make path integrals of the first theory tractable. But we end up working in the loop space of the complexified phase space, and this looks like a promising domain in which to prove properties of interest to GCT. (Here I draw inspiration from Ben-Zvi & Nadler, arXiv:1004.5120.) The challenge is to see if any of the conjectures in GCT (e.g. about quantum groups) can be posed in a form amenable to such a mapping.

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    $\begingroup$ Witten's paper came out last Thursday, so this is almost certainly an open question. $\endgroup$ – userN Oct 3 '10 at 3:08
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    $\begingroup$ It might be useful if you provide a bit more background for readers like me that find the question intriguing, but find the preprints cited daunting. $\endgroup$ – j.c. Oct 3 '10 at 16:59
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    $\begingroup$ You can always ask questions of the form, "What happens if you rub together well-known researchers A and B?" What if Deligne talked to Perelman? Do Gromov's ideas apply to the Tate conjecture? It is true that Mulmuley seems a bit pretentious in this sort of way, but Witten (incredibly) is not really pretentious at all. No, Witten is not THE wellspring of THESE ideas, although he is A wellspring of many ideas. So, I don't see a real question here. $\endgroup$ – Greg Kuperberg Oct 7 '10 at 7:37
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    $\begingroup$ Very few mathematicians currently really understand Mulmuley's program. Very few people understand Witten's brand new paper. The intersection of these sets, which is the number of people who can answer your question, is extremely likely to be 0. Maybe there's a connection, but it is likely to be years before anything really comes out of it. $\endgroup$ – Peter Shor Mar 19 '11 at 15:32
  • $\begingroup$ "algebraic quotient of geometric invariant theory is also a symplectic quotient" may be he is looking at enlarging the possible tools to use in GCT! $\endgroup$ – user16007 Sep 7 '11 at 11:53

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