Spinning strings and integrable spin chains in the AdS/CFT correspondence

TL;DR

This review surveys the dynamical tests of the AdS5×S5/CFT4 duality using semiclassical spinning strings and their gauge theory counterparts. It explains how large-spin string energies map to anomalous dimensions in the SU(2) sector via the emergent Heisenberg spin chain and the coordinate Bethe ansatz, and demonstrates agreement in the thermodynamic, one-loop regime. It then discusses higher-loop, long-range spin-chain descriptions, the Beisert–Dippel–Staudacher asymptotic Bethe ansatz, and the puzzling three-loop discrepancies attributed to wrapping effects and order-of-limit issues. The discussion situates integrability as a central organizing principle, with extensions to broader sectors, deformations, and ongoing challenges in fully quantizing the quantum string and reconciling higher-loop gauge/string data.

Abstract

In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the limit of large angular momenta on the S^5. The energies of the folded and circular spinning string solutions rotating on a S^3 within the S^5 are derived, which yield all loop predictions for the dual gauge theory scaling dimensions. These follow from the eigenvalues of the dilatation operator of N=4 super Yang-Mills in a minimal SU(2) subsector and we display its reformulation in terms of a Heisenberg s=1/2 spin chain along with the coordinate Bethe ansatz for its explicit diagonalization. In order to make contact to the spinning string energies we then study the thermodynamic limit of the one-loop gauge theory Bethe equations and demonstrate the matching with the folded and closed string result at this loop order. Finally the known gauge theory results at higher-loop orders are reviewed and the associated long-range spin chain Bethe ansatz is introduced, leading to an asymptotic all-loop conjecture for the gauge theory Bethe equations. This uncovers discrepancies at the three-loop order between gauge theory scaling dimensions and string theory energies and the implications of this are discussed. Along the way we comment on further developments and generalizations of the subject and point to the relevant literature.

Figures (5)

• Figure 1: Cartoon of the $AdS_5$ (bulk cylinder with boundary $R\times S^3$) space-time and the $S^5$ (sphere) space.
• Figure 2: The folded sting extending from $\psi=-\psi_0$ to $\psi=\psi_0$, where $\sin^2\psi_0:=q$.
• Figure 3: The one-loop energies of the folded (dark) and circular (light) string solutions plotted against the filling fraction $J_2/J$. The dashed curve is the mirrored folded string solution where one interchanges $J_1\leftrightarrow J_2$.
• Figure 4: The action of the dilatation operator on a trace operator.
• Figure 5: Bethe root distribution for the gauge dual of the folded string. For large $L$ the roots condense into two cuts in the complex plane.