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Spinning strings and integrable spin chains in the AdS/CFT correspondence. (English) Zbl 1255.81211

Summary: In this introductory review we discuss dynamical tests of the \(AdS_5 \times S^5 \text{string}/\mathcal N = 4\) 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 \(\mathcal N = 4\) 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.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
83E30 String and superstring theories in gravitational theory
83E50 Supergravity
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