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.
This post imported from StackExchange MathOverflow at 2014-10-25 10:42 (UTC), posted by SE-user Mitchell Porter