Ian Swanson from Caltech has been visiting us. He has told us a lot of interesting stuff, and was also giving a special seminar yesterday (Friday) at 3:00 pm.
He used PowerPoint with a lot of sophisticated equations, tables, and some animations. His topic was String integrability and the AdS/CFT correspondence.
Let's describe Ian's talk and the current status of the post-BMN business. Because I am writing this for the second time, let me be shorter:
- the industry of the pp-wave limit has transmuted into the spin chains and integrable systems
- Ian Swanson et al. (which includes John Schwarz, but also Jonathan Heckman who is now at Harvard) studied the physics of strings beyond the pp-wave - which means that AdS5 x S5 is described as the pp-wave deformed by some 1/R^2 corrections
- the gauge theory side of the calculation more or less reduces to the spin chains and other tools of integrable systems
- the string theoretical side is described by the Green-Schwarz string that, at least classically, becomes the supercoset SU(2,2|4) / ( SO(5) x SO(4,1) ), if I remember well - and as long as you forget about stringy loops, the ghosts (like Berkovits' pure spinors) are not necessary
- the integrable structure on both sides agree, even though it is not manifest
- most of the calculations are done at the leading order of the 1/N expansion (or the J^2/N expansion), and the stringy loop corrections are more or less open questions
- the results are still expanded in lambda'=lambda/J^2, and in 1/J
- the gauge theory (spin chains) agrees with the string (supercoset) at the one-loop and the two-loop level
- there is a discrepancy at the three-loop level, and Ian says that it is believed that the problem is a subtlety neglected by the spin chain models
- many states are combined into various multiplets, and there is indirect evidence of the existence of an infinite number of conserved charges that make up the integrable structure
Symplectic field theory
Before Ian's talk, we also saw another talk in the Science Center C (Shing-Tung Yau invited us), namely a talk by the mathematician Helmut Hofer from NYU. It was about symplectic field theory, and already the meaning of the term is not quite clear.
- instead of a manifold, Hofer was discussing a concept of a polyfold
- it seems that the dimension of the polyfold may jump from point to point
- this technology should be powerful for infinite-dimensional systems, and it should give a unified framework to study Gromov-Witten theory, Morse theory, and many other theories like that
- nevertheless one may draw finite-dimensional examples. Hofer's example is a heart connected with Jupiter by three tubes, while Jupiter is surrounded with red Saturn's rings :-)
- this mathematical machinery should explain how Riemann surfaces bubble off - all such phenomena have a local flavor in his formalism
- in the second part of his lecture, Hofer discussed applications, and Sergei Gukov told us a couple of words about this second part, too
- the first 30 minutes were spent with deriving the equations analogous to cubic string field theory, namely QF+F*F=0
- then he discussed how to construct Riemann surfaces from pieces - namely from Riemann surfaces with holes p and antiholes q - where he formally defines the commutator of [p,q] to be hbar, whatever it exactly means - this commutator is probably the source of the word "symplectic"
- it's not clear what's the relation between the different things he was discussing