Table of contents

Remarkable progress has been achieved in the last 10 years in the quantitative understanding of gauge/string duality. For the first time it now appears that it will be possible to find the exact solution of a 4-dimensional interacting quantum field theory. This should have important implications for our understanding of other strongly coupled gauge theories such as Quantum Chromodynamics.

The most studied is the most symmetric example of a gauge-string duality—the correspondence between the maximally supersymmetric gauge theory in flat 4 dimensions and superstring theory in a curved space AdS₅×S⁵, also refered to as AdS/CFT duality. The solvability of this model, allowing us for example to compute the dimensions of gauge invariant operators for any value of the coupling, is due to its hidden integrability. At weak coupling this integrability is seen by identifying the dilatation operator which acts on single trace operators in the gauge theory with the Hamiltonian of an integrable 1-dimensional spin chain. At strong coupling, where the operators of the gauge theory are most easily identifiable as string states, integrability is manifested by the underlying integrability of the sigma model defined on the string world sheet. As a result, the exact quantum spectrum is determined by a set of thermodynamic Bethe Ansatz equations.

These remarkable developments are based on a wide range of techniques from quantum field theory, condensed matter theory and mathematical physics and were reviewed in a special issue published two years ago: Kristjansen C, Staudacher M and Tseytlin A (ed) 2009 Integrability and the AdS/CFT correspondence J. Phys. A: Math. Theor.42 250301

The present special issue is an update: it contains three long reviews by Didina Serban, Benoit Vicedo and Dmytro Volin that cover recent developments and also present a few particular directions in a detailed pedagogical manner.

It starts with a review by Serban [1], where the integrabilitity of the maximally supersymmetric AdS/CFT duality is described systematically starting from the perturbative gauge theory perspective. The spin chain Hamiltonian interpretation of the dilatation operator plays a key role and predictions of the associated asymptotic Bethe ansatz are compared in detail with perturbative string theory predictions at strong coupling. There is also a discussion of very recent developments related to a Thermodynamic Bethe Ansatz (TBA) proposal describing dimensions of operators of finite length, or energies of the corresponding quantum string states.

The second review by Vicedo [2] gives a detailed account of the mathematical formalism underlying the integrability of the classical string sigma model on symmetric curved spaces like AdS₅×S⁵. It first describes the construction of the classical string solution using the finite gap method, leading to an algebraic curve description of solutions. It then introduces a semiclassical WKB-type quantization of string solitonic solutions which determines the leading quantum corrections to the classical string energies, using the example of bosonic strings on R×S³ as a model. Further extensions and elaborations of these methods for the full AdS₅×S⁵ superstring sigma model will be important for further tests of the TBA-type ansatz for the exact quantum string spectrum.

The final review, by Volin [3], gives a detailed exposition of several topics in the theory of quantum integrable models such as spin chains and various 1+1 dimensional quantum field theories. The main focus is on functional and integral equations originating from the underlying Bethe ansatz in the thermodynamic limit, which are applied in particular to the study of the strong coupling expansion in the context of AdS/CFT.

We should stress that the methods described in this special issue can be applied to areas beyond the spectral problem in AdS/CFT duality. Scattering amplitudes in gauge theories have been studied extensively over the last few years and will be the subject of a future special issue. Recently, it has been shown that these amplitudes have a hidden integrability and one can imagine using the techniques described in this special issue to study the properties of these amplitudes. These methods may also find use in condensed matter systems, such as the Hubbard model, open string chains, or supergroup chains. They could be applied to some other field theories, as we have already started to see for certain Chern–Simons theories.

P Dorey, University of Durham, UKJ Minahan, Uppsala Universitet, SwedenA Tseytlin, Imperial College London, UKGuest Editors

[1] Serban D 2011 Integrability and the AdS/CFT correspondence J. Phys. A: Math. Theor.44 124001

[2] Vicedo B 2011 The method of finite-gap integration in classical and semi-classical string theory J. Phys. A: Math. Theor.44 124002

[3] Volin D 2011 Quantum integrability and functional equations J. Phys. A: Math. Theor.44 124003

The description of gauge theories at strong coupling is one of the long-standing problems in theoretical physics. The idea of a relation between strongly coupled gauge theories and string theory was pioneered by 't Hooft, Wilson and Polyakov. A decade ago, Maldacena made this relation explicit by conjecturing the exact equivalence of a conformally invariant theory in four dimensions, the maximally supersymmetric Yang–Mills theory, with string theory in the AdS₅ × S⁵ background. 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 8 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. In this review, I present the evolution of the concept of integrability in the framework of the AdS/CFT correspondence and the main results obtained using this approach.

In view of proving the AdS/CFT correspondence one day, a deeper understanding of string theory on certain curved backgrounds such as AdS₅ × S⁵ is required. In this review we make a step in this direction by focusing on . It was discovered in recent years that string theory on AdS₅ × S⁵ admits a Lax formulation. However, the complete statement of integrability requires not only the existence of a Lax formulation but also that the resulting integrals of motion are in pairwise involution. This idea is central to the first part of this review. Exploiting this integrability we apply algebro-geometric methods to string theory on and obtain the general finite-gap solution. The construction is based on an invariant algebraic curve previously found in the AdS₅ × S⁵ case. However, encoding the dynamics of the solution requires specification of additional marked points. By restricting the symplectic structure of the string to these algebro-geometric data we derive the action-angle variables of the system. We then perform a first-principle semiclassical quantization of string theory on as a toy model for strings on AdS₅ × S⁵. The result is exactly what one expects from the dual gauge theory perspective, namely the underlying algebraic curve discretizes in a natural way. We also derive a general formula for the fluctuation energies around the generic finite-gap solution. The ideas used can be generalized to AdS₅ × S⁵.

In this review, a general procedure to represent the integral Bethe Ansatz equations in the form of the Reimann–Hilbert problem is given. This allows us to study integrable spin chains in the thermodynamic limit in a simple way. Based on the functional equations we give the procedure that allows finding the subleading orders in the solution of various integral equations solved to the leading order by the Wiener–Hopf techniques. The integral equations are studied in the context of the AdS/CFT correspondence, where their solution allows for the verification of the integrability conjecture up to two loops of the strong coupling expansion. In the context of the two-dimensional sigma models we analyze the large-order behavior of the asymptotic perturbative expansion. The obtained experience with the functional representation of the integral equations also allowed us to solve explicitly the crossing equations that appear in the AdS/CFT spectral problem.