Open giant magnons on LLM geometries

We compute sigma model solutions for rigidly rotating open strings suspended between giant gravitons in general LLM geometries. These solutions are confined to the LLM plane. These all have a dispersion relation for $\Delta-J$ that is consistent with saturation of a BPS bound of the centrally extended spin chain. For the special case of circularly symmetric LLM geometries, we can further evaluate the amount of angular momentum $J$ carried by these strings. This quantity diverges for string configurations that try to move between different "coloring regions" in the LLM plane. All of these quantities have a perturbative expansion in the t'Hooft coupling. For the strings suspended between AdS giants, we can compute in field theory the leading result of $J$ carried by the string via an analytic continuation of the $SU(2)$ result, with the help of the Bethe Ansatz for the $SL(2)$ sector. We thus provide additional information on how the radial direction of $AdS$ arises from (open) spin chain calculations.