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

TL;DR

This review discusses the connection between spinning string solutions in AdS_5 x S^5 and integrable spin chains in N=4 super Yang-Mills theory, highlighting matching results and discrepancies at higher loops.

Contribution

It provides a detailed comparison of semiclassical string energies with gauge theory scaling dimensions using integrable spin chain models and Bethe ansatz techniques.

Findings

Matching of string energies with one-loop gauge theory predictions.

Identification of discrepancies at three-loop order.

Proposal of an all-loop asymptotic Bethe ansatz for gauge theory.

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.