Dilatation operator and Cayley graphs

The most relevant results include the increasingly detailed correspondence between states in this theory with string states on AdS5×S, the technical improvements in the evaluation of the anomalous dimension of operators which led to the discovery of quantum integrability, some unexpected relations with high energy sectors of quantum chromodynamics. We cannot possibly quote the pertinent vast literature and we refer the reader to the papers [1] [2] [3] for introduction to the subject and to the original literature. Each of these outstanding results were obtained in the large-N ’t Hooft limit, in several sectors of the theory, at several orders in loop expansion, then suggesting the possibility of a complete understanding of the theory. It seems important both for a more complete understanding of the N = 4 Super Yang Mills SU(N) theory at large-N and for further tests of the AdS/CFT Maldacena conjecture to evaluate the eigenvalues of the dilatation operator for all states product of a small number of fields and for sequences made of an arbitrary number of fields. To this goal, an essential progress was obtained by expressing the dilatation operator in a way that translates the evaluation of anomalous conformal dimension of states into a diagonalization problem in finite dimensional spaces, then avoiding the previous cumbersome evaluation by Feynman graphs.