Giant magnons

Studies of ${\cal N}=4$ super Yang Mills operators with large R-charge haveshown that, in the planar limit, the problem of computing their dimensions canbe viewed as a certain spin chain. These spin chains have fundamental``magnon'' excitations which obey a dispersion relation that is periodic in themomentum of the magnons. This result for the dispersion relation was also shownto hold at arbitrary 't Hooft coupling. Here we identify these magnons on thestring theory side and we show how to reconcile a periodic dispersion relationwith the continuum worldsheet description. The crucial idea is that themomentum is interpreted in the string theory side as a certain geometricalangle. We use these results to compute the energy of a spinning string. We alsoshow that the symmetries that determine the dispersion relation and thatconstrain the S-matrix are the same in the gauge theory and the string theory.We compute the overall S-matrix at large 't Hooft coupling using the stringdescription and we find that it agrees with an earlier conjecture. We also findan infinite number of two magnon bound states at strong coupling, while at weakcoupling this number is finite.