We propose an all-order perturbative expression for the dressing phase of the AdS 5 × S 5 string S-matrix at strong coupling. Moreover, we are able to sum up large parts of this expression. This allows us to start the investigation of the analytic structure of the phase at finite coupling revealing a few surprising features. The phase obeys all known constraints including the crossing relation and it matches with the known physical data at strong coupling. In particular, we recover the bound states of giant magnons recently found by Hofman and Maldacena as poles of the scattering matrix. At weak coupling our proposal seems to differ with gauge theory. A possible solution to this disagreement is the inclusion of additional pieces in the phase not contributing to crossing, which we also study.

One-loop corrections to the energy of semiclassical rotating strings contain both analytic and non-analytic terms in the 't Hooft coupling. Analytic contributions agree with the prediction from the string Bethe ansatz based on the classical S-matrix, but in order to include non-analytic contributions quantum corrections are required. We find a general expression for the first quantum correction to the string Bethe ansatz.This expression expands in odd powers of 1/J , giving thus raise to non-analytic contributions in λ. 1 This is a generic pattern. For SL(2) circular strings we have [29] ∞ −∞

Eight easily ascertained risk factors affect elders' probability of being hospitalized repeatedly within four years. In the future, brief surveys about the presence of these factors could be used to estimate elders' risk of future hospitalization and, thereby, to identify some of those who may derive the greatest benefit from interventions designed to avert the need for hospitalization.

The ferromagnetic integrable SU(3) spin chain provides the one loop anomalous dimension of single trace operators involving the three complex scalars of N = 4 supersymmetric Yang-Mills. We construct the non-linear sigma model describing the continuum limit of the SU(3) spin chain. We find that this sigma model corresponds to a string moving with large angular momentum in the five-sphere in AdS 5 × S 5 . The energy and spectrum of fluctuations for rotating circular strings with angular momenta along three orthogonal directions of the five-sphere is reproduced as a particular case from the spin chain sigma model.where m and n are both integers or half-integers fulfilling (2.17). This ansatz solves the previous equations witḣwhere we have made use of the relationsderived from the conserved momenta (2.18), P ϕ = J 1 − J 2 and P φ = J 3 . We obtain the following expression for the energy

The planar dilatation operator of N = 4 supersymmetric Yang-Mills is the hamiltonian of an integrable spin chain whose length is allowed to fluctuate. We will identify the dynamics of length fluctuations of planar N = 4 Yang-Mills with the existence of an abelian Hopf algebra Z symmetry with non-trivial co-multiplication and antipode. The intertwiner conditions for this Hopf algebra will restrict the allowed magnon irreps to those leading to the magnon dispersion relation. We will discuss magnon kinematics and crossing symmetry on the spectrum of Z. We also consider general features of the underlying Hopf algebra with Z as central Hopf subalgebra, and discuss the giant magnon semiclassical regime.