Spinning strings in AdS_5 x S^5: new integrable system relations

TL;DR

This work shows that a broad class of rotating strings in AdS_5 × S^5 reduces to the Neumann-Rosochatius integrable system, enabling explicit construction of new circular multi-spin solutions with two AdS_5 and three S^5 spins. The authors derive the NR Lagrangian, its integrals of motion, a Lax representation, and analyze both constant-radius and non-constant solutions, including pulsating and ellipsoidal-coordinate cases. In the large-spin regime, the leading energy corrections scale with the 't Hooft coupling, supporting potential matching to one-loop SYM anomalous dimensions via Bethe Ansatz techniques. They further show that stable hybrid S/J circular configurations exist, and discuss how these results illuminate the integrable structures on both sides of the AdS/CFT correspondence, while outlining directions for future gauge/string correspondence checks.

Abstract

A general class of rotating closed string solutions in AdS_5 x 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 AdS_5 x 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 the two sides of the AdS/CFT duality. We find, in particular, new circular rotating string solutions with two AdS_5 and three 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 λ, suggesting that it can be matched onto the one-loop anomalous dimensions of the corresponding ``long'' operators on the SYM side of the AdS/CFT duality.