A general class of rotating closed string solutions in AdS5 × S 5 is shown to be described by a Neumann-Rosochatius one-dimensional integrable system. The latter represents an oscillator on a sphere or a hyperboloid with an additional “centrifugal” potential. We expect that the reduction of the AdS5 × S 5 sigma model to the Neumann-Rosochatius system should have further generalizations and should be useful for uncovering new relations between integrable structures on two sides of the AdS/CFT duality. We find, in particular, new circular rotating string solutions with two AdS5 and three S 5 spins. As in other recently discussed examples, the leading large-spin correction to the classical energy turns out to be proportional to the square of the string tension or the ’t Hooft coupling λ, suggesting that it can be matched onto the one-loop anomalous dimensions of the corresponding “long” operators on the SYM side of the AdS/CFT duality.

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Spinning strings in AdS 5 × S 5 : New integrable system relations

A procedure is developed for constructing deformations of integrable σ-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter σ-model introduced a few years ago by C. Klimyc´ok. In the case of the symmetric space σ-model on F/G we obtain a new one-parameter family of integrable σ-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical q-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset σ-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the SU(2)/U(1) coset σ-model which interpolates all the way to the SU(1,1)/U(1) coset σ-model.

/pdf/on-classical-q-deformations-of-integrable-s-models-2q36glt0qb.pdf

On classical q-deformations of integrable σ-models

Hamiltonian structures and Lax equations

New integrable canonical structures in two-dimensional models

We flnd the hamiltonian for physical excitations of the classical bosonic string propagating in the AdS5£S 5 space-time. The hamiltonian is obtained in a so-called uniform gauge which is related to the static gauge by a 2d duality transformation. The hamiltonian is of the Nambu type and depends on two parameters: a single S 5 angular momentum J and the string tension ‚. In the general case both parameters can be flnite. The space of string states consists of short and long strings. In the sector of short strings the large J expansion with ‚ 0 = ‚=J 2 flxed recovers the plane-wave hamiltonian and higher-order corrections recently studied in the literature. In the strong coupling limit ‚ ! 1, J flxed, the energy of short strings scales as 4 p ‚ while the energy of long strings scales as p ‚. We further show that the gauge-flxed hamiltonian is integrable by constructing the corresponding Lax representation. We discuss some general properties of the monodromy matrix, and verify that the asymptotic behavior of the quasi-momentum perfectly agrees with the one obtained earlier for some speciflc cases.

Integrable Hamiltonian for Classical Strings on AdS5 x S5

Quadratic algebras and integrable systems

Kac-Moody algebra and extended Yang-Baxter relations in the O(N) non-linear σ-model

Hamiltonian structures for integrable classical theories from graded Kac-Moody algebras

We investigate the algebras of the non-local charges and their generating functionals (the monodromy matrices) in classical and quantum non-linear σ models. In the case of the classical chiral σ models it turns out that there exists no definition of the Poisson bracket of two monodromy matrices satisfying antisymmetry and the Jacobi identity. Thus, the classical non-local charges do not generate a Lie algebra. In the case of the quantum O(N) non-linear σ model, we explicitly determine the conserved quantum monodromy operator from a factorization principle together withP,T, and O(N) invariance. We give closed expressions for its matrix elements between asymptotic states in terms of the known two-particleS-matrix. The quantumR-matrix of the model is found. The quantum non-local charges obey a quadratic Lie algebra governed by a Yang-Baxter equation.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103940928

J. M. Maillet

Papers

Quadratic algebras and integrable systems

Kac-Moody algebra and extended Yang-Baxter relations in the O(N) non-linear σ-model

Hamiltonian structures for integrable classical theories from graded Kac-Moody algebras

Classical and quantum algebras of non-local charges in σ models

On classical and quantum integrable field theories associated to Kac-Moody current algebras☆