Friday, November 10, 2006 ... //

Juan Maldacena and integrability

Talks by Juan Maldacena (IAS) and discussions with him are always special. He gave a very nice Duality Seminar about the planar limit of the "N=4" gauge theory. As you should know, Juan is a key person behind many layers of these fascinating ideas, for example (in chronological order)

The focus was on the last topic. He has also explained how the transcendentality calculus makes predictions about the matching of the weak 't Hooft coupling and the strong 't Hooft couplings and what can be extrapolated and what cannot be extrapolated.

At large "N", the planar diagrams of the "SU(N)" gauge theory dominate and the corresponding string theory should be described by a nice weakly coupled string theory on a geometry whose radius is fixed if you keep the 't Hooft coupling unchanged. Almost no one knows how to define this string theory as a conformal field theory because of the technical difficulties with including the Ramond-Ramond field strength into the "mainstream" NS-R formalism. Nathan Berkovits probably knows how to define the theory in his pure spinor formalism but he needs more people who would follow him and derive many more consequences of his approach that could be compared with the work on the gauge theory side.

As we have indicated, you can equivalently describe this theory either in terms of a limit of gauge theory or, using holography, by the hypothetical worldsheet description. It has such a high supersymmetry that many people expect - and have evidence - that this theory is integrable.

What does the adjective "integrable" exactly mean? It is, informally speaking, equivalent to "solvable". How much can you solve? In principle, one should be able to find the spectrum of operators and their dimensions, together with their correlators. But the actual quantities that are being solved by the specialists are different: you look at the worldsheet of a string that looks like a spin chain and study "scattering theory" in this setup.

Unlike the BMN case, in this integrability setup, you want to consider traces or spin chains that are infinitely long. The cyclicity of the trace is lost due to the infinite length and you can therefore have BMN-like operators with a single impurity that carries a nonzero momentum along the worldsheet. Such single-impurity states are in fact supersymmetric. The BPS bound is staturated because of the existence of an additional term on the right-hand side of the SUSY algebra: the worldsheet momentum. Recall that this quantity must vanish for the finite traces, because of cyclicity, so it has never appeared in the BMN formulae. However, for infinite worldsheets, this additional momentum or "central charge" is nonzero and must be added.

The excitations propagating on this spin chain - or the string - are called magnons. One can study their scattering - again, this is a scattering of impurities propagating on infinitely long operators (traces) in the Yang-Mills theory. The low-momentum expansion of these magnons gives the two-dimensional Lorentz-symmetric worldsheet theory. However, at finite momentum, this Lorentz symmetry is broken - something that you might find strange but it is not as strange if you realize that the Yang-Mills theory gives you a gauge-fixed version of the hypothetical worldsheet conformal field theory.

The momentum of the impurities on the string effectively becomes periodic, after some center-of-mass contributions are subtracted. This is morally related to the discrete positions of the letters in the traces.

The theory is arguably integrable which can also be understood as the existence of infinitely many additional conserved charges. The existence of these charges is enough to make the scattering essentially trivial. For "two to two" scattering, only the processes that preserve the momenta of each particle can have a nonzero amplitude, and this amplitude is a pure phase (multiplied by a particular 256 x 256 matrix encoding the polarizations) that is now probably known, at least up to some number of loops.

There was some discussion whether it was nice to believe that this theory is integrable and whether we should be happy if it is. Also, we understood again why the radius of convergence of the expansion in terms of the 't Hooft coupling of the planar limit of gauge theories is finite, unlike the generic field theories whose convergence radius is zero. Dyson's argument about the unstable vacuum for negative couplings becomes harmless in the planar limit because the tunneling rate controlling Dyson's instability goes like "exp(-C/g^2)=exp(-CN/lambda)" which brutally disappears in the large N planar limit for a fixed lambda. "C" was just a numerical constant.

There have been many other insights and discussions. But because this topic has been covered on this blog, let me stop and recommend you to click at some of the links at the beginning of this article or other links.