Spinning strings in the η -deformed Neumann-Rosochatius system

Abstract

The sigma model of closed strings spinning in the η deformation of AdS 5 × S 5 leads to an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical system. In this article we construct general solutions to this system that can be written in terms of elliptic functions. The solutions correspond to closed strings with nonconstant radii rotating with two different angular momenta in an η-deformed three-sphere. We analyze the reduction of the elliptic solutions for some limiting values of the deformation parameter. For the case of solutions with constant radii we find the dependence of the classical energy of the string on the angular momenta as an expansion in the 't Hooft coupling.

Neumann-Rosochatius system for strings in ABJ model

“…Reduction of the string-sigma model on AdS 5 × S 5 , AdS 4 × CP 3 , etc. to the NR system has been quite useful in unravelling new relationships between the integrable structures of the two sides of the AdS/CFT duality [18,[23][24][25][26][27]. Such an approach has previously been used to study the finite size effect to the giant magnon (GM) solutions in AdS 4 × S 7 background [28].…”

Double Yang-Baxter deformation of spinning strings

“…First and foremost, the undeformed Neumann-Rosochatius system [2] is recovered when we set κ ± = 0. We can also retrieve a truncation of the η-deformation of the Neumann-Rosochatius system [18][19][20], which describes classical strings spinning in the ηdeformation of the AdS 5 × S 5 background, for κ − = 0 and κ + = κ. Such a limit corresponds to the case in which the two factors of the global symmetry group of the semisymmetric space are deformed in the same way [4,5].…”

“…Such a limit corresponds to the case in which the two factors of the global symmetry group of the semisymmetric space are deformed in the same way [4,5]. This implies that all considerations regarding the limits κ = i and κ = ∞ studied in [20] can be recovered starting from (2.9) for special choices of the deformation parameters. In particular, the limit κ = i deserves a special mention, since in this case the spinning string rotates in a truncation of a tendimensional pp-wave background [8] and it is possible to find explicit solutions to the Neumann-Rosochatius system in terms of hyperbolic functions [20].…”

“…Therefore, we may address the problem of proving the integrability of a deformation of the N -dimensional Neumann-Rosochatius system, provided that it preserves the directions of the Cartan subalgebra of the (2N − 1)-dimensional sphere, attaining a deformation of the Uhlenbeck constants. This task may be fulfilled, for instance, trying to deform the Uhlenbeck constants directly [20,22] or drawing on the associated Lax connection in order to generate them [18,19]. Nevertheless, the problem we study here is simplified by the low dimensionality of the model, which allows us to construct straightforwardly these first integrals in a systematic and symmetric fashion.…”

Finite size effect from classical strings in deformed AdS3× S3

“…It is constructed by Yang-Baxter deformation of super currents which has Cartan subgroup [U (1)] 6 [43][44][45][46][47][48][49]. To explore the integrability beyond the original AdS 5 × S 5 various rigidly rotating string, spinning string and oscillating string have been studied in this deformed background [50][51][52][53]. The infinite size giant magnon and single spiky string solution in κ-deformed background have been studied in [54][55][56].…”