Thursday, August 31, 2006

Ahrenshoop update

As said earlier, I would like to talk about a few talks here at the Ahrenshoop conference. Let's see how far I get before the afternoon sessions start.

The first talk I would like to mention is Matthias Gaberdiel's about closed string moduli influencing open string moduli. As an example consider strings on a circle. Generically, you can have D0 and D1 branes. D0 sit at a point on the circle and correspond to Dirichlet boundary conditions of open strings while D1 branes wrap the circle and correspond to Neumann conditions. However, if the circle has exactly the self dual radius (fixed under T-duality), the generic U(1) symmetry is enhanced to SU(2) (at level 1 to be specific), thus there is a full SU(2) worth of D-branes. A similar thing happens if the radius is rational in string units R=M/N R_sd say. Then, there besides the genereic branes there are SU(2)/Z_M x Z_N branes.

Thus the spectrum of branes depends critically on R. But R itself is a closed string modulus! You can change it by exciting closed string fields and the obvious question is what happens to the additional branes if you tune the radius away from the special values. Matthias and friends worked out the details and found that because of a bulk boundary 2 point function, in the presence of the special D-brane the operator changing the radius is no longer marginal. Thus changing the radius kicks of an RG flow which they can in fact integrate and show that the special brane decays into either a D0 or D1 brane depending on whether the radius is increased or reduced. They can fill all this prose with calculations which are quite neat and do more general cases. So, go and read their paper!

The next talk I would like to report on was by Niklas Beisert about the spin chain/integrability business. I must admit, in the past I was not following these developments closely and was quite confused. People wrote papers and gave talks reporting that they had done more and more loops for larger and larger subgroups and compared that to many different stringy calculations. But I was lost and had no real idea about where the real progess was happening.

Now Niklas seems to have cleared up a lot of the supergroup theory and the dust has settled considerably. He presented the situation as follows: Both the gauge theory and the stringy side of dilatations operators seem to be integrable in the sense that the S-matrix factorises into products of two particle S matrices. As both sides have N=4 Susy the superalgebra SU(2,2|4) is a symmetry and it seems to restrict this 2 particle S-matrix considerably: The dispersion relation with the square root and the sin is completely fixed by the symmetry and the S-matrix is determined up to a scalar function (diagonal in flavour space). Thus, everything except this function is kinematics and the function contains all the dynamics.

The gauge and the string side of things are different expansions of this function (one from the weak and one from the strong coupling side). On the gauge side, the function to all perturbative orders that have been worked out vanishes while on the other side, the function vanishes at low orders but is non-zero from coupling^3 on. This explains that up to two loops the matching worked (it just tested the kinematics) and why there are discrepancies from 4 loops (where the function starts to matter). I should add that this is not leathal to AdS/CFT since you should not expect a functions expanded around two different regimes to look the same.

Chairwoman just clapped hands, have to go.

Short update:
Internet connectivity is a bit tricky here since all connections go through two ISDN lines and some people use it to do skype effectively stopping connectivity for everybody else. But let me just add to Niklas talk that the reason for the strong conclusions he can draw from the group theory can be traced back to the unusual fact that for that supergroup it happens that the tensor product of two fundamental representations is itself irreducible thus there is no branching. I should also have mentioned that Niklas and friends have a guess for the exact form of that dressing function involving Gamma functions and Betti numbers.

Second update: The paper by Gaberdiel and friends is out.


Anonymous said...

I think the solution involves Bernoulli numbers rather than Betti number, just as Schnabl's exact string field theory solution did.

Robert said...

Betti Schmetti. Of course, I meant Bernoulli numbers. There is no manifold involved so where should Betti numbers come from?