Integrable Hamiltonian for Classical Strings on AdS5 × S5

We find the Hamiltonian for physical excitations of the classical bosonicstring propagating in the AdS_5 x S^5 space-time. The Hamiltonian is obtainedin a so-called uniform gauge which is related to the static gauge by a 2dduality transformation. The Hamiltonian is of the Nambu type and depends on twoparameters: a single S^5 angular momentum J and the string tension \lambda. Inthe general case both parameters can be finite. The space of string statesconsists of short and long strings. In the sector of short strings the large Jexpansion with \lambda'=\lambda/J^2 fixed recovers the plane-wave Hamiltonianand higher-order corrections recently studied in the literature. In the strongcoupling limit \lambda\to \infty, J fixed, the energy of short strings scalesas \sqrt[4]{\lambda} while the energy of long strings scales as \sqrt{\lambda}.We further show that the gauge-fixed Hamiltonian is integrable by constructingthe corresponding Lax representation. We discuss some general properties of themonodromy matrix, and verify that the asymptotic behavior of the quasi-momentumperfectly agrees with the one obtained earlier for some specific cases.Comment: 30 pages, LaTex; v2: a few comments added, misprints corrected, references adde