AdS/CFT spin chain integrability

Until tomorrow, I am at DESY for a two day north German (including Copenhagen) string meeting. Urs is here as well, don't know if he's blogging as well. The first two talks by Matthias Staudacher and Sakura Schäfer-Nameki on the latest on the relation between anomalous dimensions in the gauge theory and string energies in AdS. I do not intend to give you a full scale coverage but instead would like to mention one recent result that Matthias reported on.

In brief, the story about the spin chain integrability is the following: The chiral primary operators of the N=4 gauge theory are single trace operators that contain the the six scalar fields and their covariant derivatives as well as fermions and field strengths. One can compute two point functions of these operators and they are basically given by one number for each pair of operators. The aim is then to diagonalize the matrix of these numbers and the eigenvalues are the anomalous dimensions to be compared with string energies in AdS x S^5. Over recent years, there were a lot of papers reporting on "integrability" of this problem. So what is meant by this?

The matrix of two point functions has some block structure that allows one to study in restricted subsectors (like containing only opertors with two of the complex scalar fields and a fixed number of insertions of these fields). There the matrices are finite. But everybody knows that any finite symmetric matrix can be diagonalized. So what's the hype?

On the other hand, the experts talk about "three loop integrability" etc, but in truely integrable models, integrability is not a perturbative statement, it usually refers to the full, non-perturbative S-matrix.

Of course, all this is a confusion of language. The point of the integrability is that there are algebraic formulas for the eigenvalues (Bethe Ansätze) which do not explicitly contain the length of the operators but work simultaneously for a large class of subsectors (for example for all lengths of operators or at least asymptotically). And "three loop" refers to the fact that the matrix of two point functions was calculated using three loop Feynman diagrams (or equivalent methods). This matrix is the interpreted as the Hamiltonian of an auxillary problem, typically a spin chain and that Hamiltonian is integrable.

So far, all this was just about the integrability of this auxilliary Hamiltonian and was not refereing to the physical space-time S-matrix of the original N=4 YM theory.

But now the news is, that there is now a connection, although still quite a weak one: In a paper on MHV amplitudes (an upshot of the twistor story), Bern etal conjecture a form for all loop gluon scattering amplitudes. This contains three (not really known) functions. But one of these functions (the one called f) also appears in the Bethe Ansatz business as Matthias reports. Thus there is hope that the spin chain results yield information on space-time scattering beyond two point functions!