The last two talks at the Davis conference were quite interesting. Alexandre Givental gave one entitled “Twisted Loop Groups and Gromov-Witten Theory”, which went by way too fast. He has an interpretation of the generating functional for Gromov-Witten invariants that uses a loop group. Some infinite dimensional symplectic geometry involving this group gives a conjectural explanation of the properties of these invariants. The talk covered a lot of material so, while intriguing, was quite hard to follow.
Witten gave the last talk, and his philosophy was the opposite of Givental’s. He covered only a little material, at a level that was easy to follow. The nominal topic of his talk was his recent work on the relation of strings in twistor space to gauge theory scattering amplitudes, but he didn’t really get to this. He began by saying that there were at least a couple different possible talks he could give about the background to his recent work. One was the one he gave a couple weeks ago at a conference at NYU which covered gauge theory scattering amplitudes. Luckily for me since I had heard that one, he decided to give a different one at Davis, mostly covering some of the ideas about twistors used in his work. He discussed the twistor theory construction of an SU(2,2) representation using the massless single particle solutions of a fixed helicity. This is related to a non-compact version of the Borel-Weil-Bott theorem, constructing a representation on a higher cohomology space. Presumably one could also do this using the Kostant Dirac operator techniques I mentioned recently.