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Integrability and the AdS/CFT correspondence. (English) Zbl 1228.81242
The author is concerned with the evolution of the concept of integrability in the framework of the AdS/CFT correspondence (relations between free IIB superstrings on the anti-de Sitter $$AdS_5 \times S^5$$ space-time background and planar $$N = 4$$ super Yang-Mills theory) and the main results obtained using this approach.
“The description of gauge theories at strong coupling is one of the long-standing problems in theoretical physics. The idea of the relation between strongly-coupled gauge theories and string theory was pioneered by G. ’t Hooft [“A planar diagram theory of strong interactions”, Nucl. Physics, B 72, No. 3, 461–473 (1974)], K. G. Wilson [“Confinement of quarks”, Phys. Rev. D 10, No. 8, 2445–2459 (1974)] and A. M. Polyakov [“From quarks to strings”, arXiv:hep-th/0812.0183]. The exact equivalence of a conformally invariant theory in four dimensions, the maximally supersymmetric Yang-Mills theory, with string theory in the AdS$$_5 \times S^5$$ background is conjectured by J. Maldacena [Adv. Theor. Math. Phys. 2, No. 2, 231–252 (1998; Zbl 0914.53047)]. Other examples of correspondence between a conformally invariant theory and string theory in an AdS background were discovered recently. The comparison of the two sides of the correspondence requires the use of non-perturbative methods. The discovery of integrable structures in gauge theory and string theory led to the conjecture that the two theories are integrable for any value of the coupling constant and that they share the same integrable structure defined non-perturbatively. The last eight years brought remarkable progress in identifying this solvable model and in explicitly solving the problem of computing the spectrum of conformal dimensions of the theory. The progress came from the identification of the dilatation operator with an integrable spin chain and from the study of the string sigma model.”
For the de Sitter space in non-critical string theory, the de Sitter solutions and instantons were discussed by A. Strominger, E. Silverstein and A. Maloney [The future of theoretical physics and cosmology. Celebrating Stephen Hawking’s contributions to physics. Papers based on the presentations at the Stephen Hawking 60th birthday workshop and symposium, Cambridge, England 2002. Cambridge: Cambridge University Press. 570–591 (2009; Zbl 1205.83063)].
For string theory background characterized by simple geometric and integrability properties, the exact solutions are presented in the thesis by D. Orlando [Fortschr. Phys. 55, No. 2, 161–282 (2007; Zbl 1152.81038)].
A new collection of reviews on integrability in the context of the AdS/CFT correspondence is presented by N. Beisert et al. [“Review of AdS/CFT integrability: an overview”, Lett. Math. Phys., published online: 27 October 2011, doi:10.1007/s11005-011-0529-2].
In the introduction the author of the paper under review gives a short survey of the history of the gauge-string correspondence. Section two contains the basic facts about the $$N = 4$$ super Yang-Mills (SYM) theory. These include: the action; symmetries; oscillator representation of $$\text{PSU}(2, 2|4)$$; the dilatation operator as the Hamiltonian of an integrable spin chain; Bethe ansatz solution for the Heisenberg model; the $$SO(6)$$ sector at one loop; the full $$\text{PSU}(2, 2|4)$$ Bethe ansatz at one loop.
The third section considers perturbative integrability in $$N = 4$$ SYM theory. It includes the definition of perturbative integrability for the dilatation operator and the comparison with the Inozemtsev model.
The fourth section is concerned with strings on AdS$$_5 \times S^5$$: the Lagrangian; integrability of the classical sigma model. The gauged linear sigma model (GLSM) description of toric stacks and a description of the physics of GLSM are given by T. Pantev and E. Sharpe [Adv. Theor. Math. Phys. 10, No. 1, 77–121 (2006; Zbl 1119.14038)].
In section five the comparison between the gauge theory and the string results is presented. This includes strings in the plane-wave background, the Berenstein-Maldacena-Nastese (BMN) result, algebraic curves and the continuum limit of the Bethe ansatz equations.
Section six contains the main results concerning the asymptotic all-loop Bethe equations together with the solution for the dressing phase. In section seven the strong coupling limit of the asymptotic Bethe ansatz equations is analyzed. Section eight is devoted to the description of finite size corrections and the thermodynamical Bethe ansatz. In the final section the recent developments and open problems are discussed. Two appendixes contain descriptions of Dynkin diagrams and Bethe ansatz equations as integral equations.

MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81R12 Groups and algebras in quantum theory and relations with integrable systems 81T20 Quantum field theory on curved space or space-time backgrounds 81T13 Yang-Mills and other gauge theories in quantum field theory 81T60 Supersymmetric field theories in quantum mechanics 81V05 Strong interaction, including quantum chromodynamics 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 47A20 Dilations, extensions, compressions of linear operators
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