Spinning strings in AdS_5 x S^5 and integrable systems

TL;DR

The paper develops a unified, integrable-framework for semiclassical strings rotating in AdS5×S5 by reducing the bosonic sigma-model to the Neumann system, enabling a genus-2 hyperelliptic description of three-spin solutions. It derives a closed set of equations that determine the leading 1/J energy corrections, matching these to one-loop anomalous dimensions in N=4 SYM and connecting to the thermodynamic limit of the SO(6) spin-chain Bethe ansatz. It analyzes folded and circular three-spin configurations, obtaining explicit results for special cases (notably J1=J2 with varying J3) and providing numerical insights for general spin values. The work also extends the approach to strings rotating in AdS5 and discusses implications for AdS/CFT beyond semiclassical limits, including potential links to Wilson loops and higher-loop integrability.

Abstract

We show that solitonic solutions of the classical string action on the AdS_5 x S^5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalised to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N=4 super YM theory. We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators. We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J_1, J_2.