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You can see it happening in these talks. For now it's just d=4 N=4 super-Yang-Mills and d=4 N=8 supergravity, but there is every reason to think that the relationships being discovered there will be extended (in more complex forms) to other gauge theories such as the standard model.

AdS/CFT revealed that a gauge theory could be a string theory. The twistor string revealed that a gauge theory could be simpler as a twistor theory. Arkani-Hamed et al rewrote planar N=4 theory (only planar Feynman diagrams) in terms of a new object, a Grassmannian in twistor space, with an extremely simple definition. Zvi Bern in his talk says you should be able to get non-planar N=4 (the rest of N=4 SYM) by transforming planar N=4, and then from there it's just a trivial step to N=8 supergravity. And behind it all is the theory of motives, and other aspects of the mathematical universe created by Grothendieck and others. In his talk, Marcus Spradlin exhibits how a 17-page formula for a "remainder function" (part of a scattering amplitude) was reduced to a single line using the motivic theory of polylogarithms, and says that probably the whole theory (N=4 SYM again, in this case) is a Hopf algebra. The latter perspective was already developed by Connes and Kreimer when they studied renormalization theory, so I take all of this to mean that there is an algebra of diagrams (Feynman diagrams and other diagrams) in which amplitudes are derived from a combinatorial motivic cohomology. Yet somehow these quantities are also equivalent to volumes of polytopes in AdS space - this talk by Maldacena might be a useful complement to the KITP talks:

http://pirsa.org/10040040/

So, in short, quantum field theory, the workhorse of particle physics for at least sixty years, turns out to have a hidden side. There was already a big conceptual revolution in QFT in the 1970s, one that still isn't in the textbooks - the revolution associated with Wilson's renormalization group, which made conformal field theories secretly important to the structure of QFT. This revolution looks to be at least as big. Of course, it has been brewing for years, at least since AdS/CFT (which took off in 1998), and the revival of twistor theory dates from the end of 2003. But now it looks to be reaching a new culmination, perhaps the complete solution of d=4 N=4 super-Yang-Mills, in terms of a motivic twistor theory. It's amazing to watch it happening.