In their seminal paper uncovering the integrable spin chain structure, Minahan and Zarembo [92] actually considered the full scalar sector of the gauge theory at one-loop order. This gives rise to an integrable SO(6) magnetic quantum spin chain of which the discussed SU(2) Heisenberg model arises in a subsector. This work was generalized in [11] to all local operators of the planar one-loop theory, leading to an integrable super-spin chain with SU(2,2|4) symmetry discussed in [24]. The excitations of this super-spin chain consist of scalars, field strengths, fermions, and an arbitrary number of covariant derivatives of these three, leading to an infinite number of spin degrees of freedom on a single lattice site. The thermodynamic limit of this super-spin chain was later on constructed in larger supersymmetric subsectors in [105] and in [19] for the full system leading to spectral curves, which reproduce the results of the classical string at one-loop order.

The conjectured form of the asymptotic higher-loop Bethe ansatz for the PSU(2) subsector was generalized to the full theory recently in [25] in form of a long-range SU(2,2|4) Bethe ansatz. In this paper, the corresponding generalization for the quantum string Bethe equations, generalizing [6] relevant for the SU(2) sector, was also provided. A novel feature of leaving the minimal SU(2) sector at higher loop orders is that the spin chain begins to fluctuate in length [12]: The Hamiltonian (or dilatation operator) preserves the classical scaling dimensions but not the length of the chain, e.g. two fermions have the same classical scaling dimension as three scalars and can mix if they carry identical charges.

An alternative route for comparing string energies to gauge theory scaling dimensions lies in the coherent-state effective action approach pioneered by Kruczenski [79]. Here one establishes an effective action for the string whose center of mass moves along a big circle of the with large angular momentum in the “weak coupling” limit . This action is then shown to agree with the long-wave length approximation of the gauge theory spin chain at one-loop. In this approach there is no need to compare explicit solutions any longer, however, considering higher-loop effects and fermions becomes more challenging in this language. For details see [80, 71, 111, 81, 70, 112, 29, 28] and also Tseytlin’s review [116].

In view of the reviewed insights, an obvious next question to address is what can be said about the non-planar sector of the gauge theory dual to string interactions. The gauge theory dilatation operator is indeed known for the first two-loop orders in the SU(2) sector exactly, that is including all non-planar contributions. However, extracting physical data from it, such as amplitudes for the decay of single trace operators to double trace ones is nontrivial. This has been performed with great success for the case of two or three magnon excitations in the BMN limit being dual to the interacting plane-wave superstring (reviewed in [94]). Performing the same computation for a macroscopic number of magnons, thus describing the quantum decay of the discussed spinning strings, is complicated enormously by the complexity of the Bethe wave-function (62) for large . The analysis on the string side, however, can be performed by considering a semiclassical decay process [96]. For a related discussion on the non-planar gauge theory aspects see [30].

An interesting toy model for Super Yang-Mills is its dimensional reduction on a three sphere to a quantum mechanical system [76], which turns out to be the plane-wave matrix theory of [33, 49, 77, 50] related to M-theory on the plane-wave background. The Hamiltonian of this matrix quantum mechanics reduces to an integrable spin chain in the large limit, which - remarkably - is identical to the full system up to the three-loop level in the overlapping SU(2|3) sector [78] (via a perturbative redefinition of the coupling constant). However, a recent four-loop study displays a breakdown of BMN scaling at this level of perturbation theory while integrability is stable [58]. What this finding implies for the full model remains to be seen.

A recent line of research concerns the study of deformations of Super Yang-Mills, which maintain the quantum conformal structure known as the Leigh-Strassler or deformations [82]. The one-loop dilatation operator was constructed in subsectors of the theory in [99, 32]. Moreover, the explicit construction of the supergravity background dual to the deformed gauge theory was achieved by Lunin and Maldacena [84]. Again, the classical bosonic string theory in this background is integrable and exhibits a Lax pair [65] yielding a string Bethe equation, which agrees with the thermodynamic limit of the one-loop Bethe equation for the gauge theory dilatation operator [59]. So, the complete discussion of this review lifts to the deformed case. As a matter of fact, even larger (three-parameter) families of generically non-supersymmetric deformations of Super Yang-Mills have been considered with known supergravity duals [59]. The corresponding “twisted” gauge theory spin chain and Bethe ansatz was constructed in [23].

Open integrable spin chains have also appeared in the AdS/CFT setting where the boundaries of the spin chain correspond to fields in the fundamental representation; see [110, 47, 48, 51] for such constructions. In [34], open integrable spin chains emerged within subdeterminant operators in Super Yang-Mills dual to so-called “giant gravitons”.

First investigations on the role of integrability for the three-point functions in the gauge theory were performed in [93, 101, 3].

Finally, let us mention that integrable structures are known to also appear in QCD, such as in high-energy scattering processes and other instances [83, 56, 40, 26], see [27] for a recent review.

In conclusion, the emergence of integrable spin chains in the AdS/CFT correspondence has led to great insights into dynamical aspects of the duality and might hold the key to a complete determination of the spectrum of both theories. Recent developments point towards integrability being a generic property of conformal gauge theories in the planar limit not necessarily connected to supersymmetry. Finally, a great challenge for the future is to understand the integrable spin chain nature of the quantum string in and related backgrounds.

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