Spinning strings inAdS5×S5:New integrable system relations

A general class of rotating closed string solutions in ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ is shown to be described by a Neumann-Rosochatius one-dimensional integrable system. The latter represents an oscillator on a sphere or a hyperboloid with an additional “centrifugal” potential. We expect that the reduction of the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ sigma model to the Neumann-Rosochatius system should have further generalizations and should be useful for uncovering new relations between integrable structures on two sides of the AdS/conformal field theory (CFT) duality. We find, in particular, new circular rotating string solutions with two ${\mathrm{AdS}}_5$ and three ${\mathrm{S}}^5$ spins. As in other recently discussed examples, the leading large-spin correction to the classical energy turns out to be proportional to the square of the string tension or the ’t Hooft coupling $\lambda ,$ suggesting that it can be matched onto the one-loop anomalous dimensions of the corresponding “long” operators on the super-Yang-Mills side of the AdS/CFT duality.