# More From Davis

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.

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### 3 Responses to More From Davis

1. Peter says:

SU(n,n) means special unitary transformations that preserve a metric of signature (n,n) (i.e. 2n-dimensional complex space with metric with n + signs and n – signs). For supergroups people write them with a bar, like SU(n|n) for a group with n bosonic and n fermionic generators.

Part of the twistor story is that SU(2,2) is the group of conformal transformations in 4d.

There are a lot of different ways of thinking about twistors, but they are pretty natural objects if you are thinking about the geometry of spinors in 4d in a way that makes explicit the conformal symmetry.

I’m also mystified about why Witten remains such a believer in string theory. I think one reason he keeps working on it is that there continue to be interesting relations between string theory and gauge theory, so even if it doesn’t work as a unification idea, string theory can still lead to interesting new ways of thinking about gauge theories (e.g. AdS/CFT).

His recent work has certainly lead to interesting results about perturbative gauge theory amplitudes, although the relation to string theory seems somewhat tenuous to me even though it is clearly a big part of his motivation.

2. erinj says:

I saw Witten’s paper and thought that this was another attempt to concretely link string theory with gauge theory, or to indicate that gauge theory is contained within string theory in a convoluted manner and therefor subsumes it – thus demonstrating the correctness or superiority of string theory. This has been attempted before, hasn’t it, with things like AdS/CFT, dualities, etc. Surely if this strategy was correct, they would have found much simpler and more obvious connections between string theory and gauge theory than all this stuff about bulks and boundaries, SUSY, AdS, SUGRA and twistors.

It seems to me as if the string theorists are grasping at straws with this approach – that is, try and dig up gauge theory quantities out of string theory in peculiar situations. They just keep re-hashing the same old arguments and try and squeeze gauge theory out of strings all the time. I’m a fan of Witten’s marvellous non-string theory work (e.g. SUSY QM & Morse theory, TQFT & knot theory, bosonization, Seiberg-Witten/Donaldson theory, QFT & elliptic genera, etc.), and I thought his Fields Medal was a crowning achievement, but why he keeps on spending so much cerebral energy on string theory mystifies me.

Also, it would seem preferable, and surely much simpler, to build representations on higher cohomology spaces using the Kostant Dirac operator and spinors than with twistors, which appear to be far less intuitive objects and are harder to work with.

Incidentally, Peter, I’ve always had some confusion over the notation SU(n,n) – isn’t the (n,n) notation something to do with the signs of the metric components? And isn’t SU(n,n) a `supergroup’, meaning that it’s Lie algebra is modified to include anti-commutators (as with the super-Poincare group)? I ask as I’ve never been able to find a proper definition for this notation or a reference which defines it clearly.

3. Pyracantha says:

I am looking forward with a kind of eschatological eagerness to the time when I will be able to recognize and understand what you are talking about in this entry, and what the speakers at that conference are talking about. However, by the time I get that far in my physics and math self-education project, all of that will probably be very obsolete.

Pyracantha at “Electron Blue”

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