AdS/CFT and Integrability

As I indicated, I'm not able to give a comprehensive account of the past week string school at CERN. I can't help that fluxes make me sick in my stomach, or that counting BH microstates has a similar effect on me as counting sheep. Nevertheless, I listened with somewhat unexpected pleasure to the lectures on Integrability and AdS/CFT by Nick Dorey. This is a cute topic in mathematical physics that I had known nothing about before. Nick's lectures gave me a smattering of idea of what's going on, and I'm sharing a few bits and pieces that have made their way to my long-term memory.

4D maximally supersymmetric Yang-Mills theory is dual to 10D IIB superstrings on AdS5xS5, Maldacena dixit. While many aspects of these two theories are fixed by their powerful symmetries, there is still a lot to learn about the dynamics. Some help may come from the integrable structures that have recently been discovered on both sides of the duality.

Integrability is a very non-generic feature of classical or quantum systems that there are as many conserved charges as there are degrees of freedom. In classical mechanics, this would mean that the system can be fully solved by quadratures. Quantum mechanics is more tricky, but there still exists a method called the Bethe ansatz for finding the exact solutions.

The relevance of integrability in the context of SU(N) super Yang-Mill was pointed out in the paper by Minahan and Zarembo. Integrable structures pop out in the process of computing correlation functions of certain operators in perturbation theory. For example, we can compute gauge invariant local correlators of the scalars that are present in the theory. We pick up two of the three scalars, W and Z, and compute the conformal dimension of the operator
$\langle Z^{L-M} W^M \rangle $
or similar ones with different permutations of Z and W under the trace. The classical scaling dimension of this operator is L (the length of the chain), but there are divergent loop corrections that introduce an anomalous dimension. The additional complication is that loop corrections mix operators with various M, so that we have to deal with a matrix of anomalous dimensions that has to be diagonalized. The eigenvectors correspond to operators with definite scaling dimensions.

Now, the scalars W,Z form a doublet under the SU(2) subgroup of the SO(6) R-symmetry so we can call them spin up and spin down. It looks more fashionable to represent the operators as spin chains, for example
$\langle WWZWWZ\rangle \to |up,up,down,up,up,down>$
It turns out that this analogy is more far reaching. One-loop computations simplify in the large N limit of SU(N) because the planar diagrams can only "flip one spin". One finds that the matrix of anomalous dimensions is given by
$\frac{\lambda}{8 \pi^2}\Sigma_1^L(1 - P_{l,l+1})$
where $\lambda$ is the t'Hooft coupling and P is an operator that exchanges the neighboring spins. A trained eye recognizes in the above the Hamiltonian of the Heisenberg spin chain with nearest neighbor interactions. One can see that a vector with all spins up (the ferromagnetic vacuum) is an eigenvector, but simple vectors with one spin flipped to down are not. Nevertheless, the full spectrum of this system can be found exactly and the eigenvalue problem was solved in the 1930s by Bethe with the help of the Bethe ansatz (the connection to integrability was made much later by Faddeev). The whole spectrum can be constructed out of the combinations of vectors with one spin down, the so-called magnons.

The story is continues on the string theory side duality, as shown in the paper by Hofman and Maldacena. But I stop here, since all these intricate connections make my head spinning.

The video and transperencies should be available via the school's web page. But they are not. A commenter pointed out that there are some technical problems to which string theory has no solution for the moment.