the abstract (especially point 2) seems to suggest that this paper is if anything a reason to believe that these methods will not work.

I don't know if you mean, won't work to separate P from NP - Mulmuley's goal - or won't work on the subproblems of FAI - which is what I was proposing.

as far as I am aware from prior work I've seen, there's no substantial connection between where string theory interacts with algebraic geometry and where complexity theory interacts with algebraic geometry. I'd be very interested in seeing anything that showed a connection.

Quantization via brane -> symplectic quotients -> partial stability.