Double Yang-Baxter deformation of spinning strings
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Springer
Received: March 17, Revised: May 2, Accepted: June 5, Published: June 18,
2020 2020 2020 2020
Rafael Hern´ andez and Roberto Ruiz Departamento de F´ısica Te´ orica and Instituto de F´ısica de Part´ıculas y del Cosmos, IPARCOS, Universidad Complutense de Madrid 28040 Madrid, Spain
E-mail: [email protected], [email protected] Abstract: We study the reduction of classical strings rotating in the deformed threesphere truncation of the double Yang-Baxter deformation of the AdS3 ×S 3 ×T 4 background to an integrable mechanical model. The use of the generalized spinning-string ansatz leads to an integrable deformation of the Neumann-Rosochatius system. Integrability of this system follows from the fact that the usual constraints for the Uhlenbeck constants apply to any deformation that respects the isometric coordinates of the three-sphere. We construct solutions to the system in terms of the underlying ellipsoidal coordinate. The solutions depend on the domain of the deformation parameters and the reality conditions of the roots of a fourth order polynomial. We obtain constant-radii, giant-magnon and trigonometric solutions when the roots degenerate, and analyze the possible solutions in the undeformed limit. In the case where the deformation parameters are purely imaginary and the polynomial involves two complex-conjugated roots, we find a new class of solutions. The new class is connected with twofold giant-magnon solutions in the degenerate limit of infinite period. Keywords: AdS-CFT Correspondence, Gauge-gravity correspondence ArXiv ePrint: 2003.05724
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)115
JHEP06(2020)115
Double Yang-Baxter deformation of spinning strings
Contents 1 Introduction
1
2 Deformation of the Neumann-Rosochatius system 2.1 Integrals of motion
2 5 6 8 15
4 Conclusions
21
1
Introduction
The representation of classical strings rotating in the AdS5 × S5 background in terms of effective integrable mechanical systems is a renowned trait of the integrable structure underlying the AdS5 /CFT4 correspondence [1, 2]. In reference [1], it was shown that the usage of a periodic ansatz for which the coordinates of the Cartan subalgebra of the AdS5 × S5 background are proportional to the world-sheet time reduces the action of the corresponding spinning string to the Neumann integrable system. The latter is a mechanical model that consists of a collection of N + 1 simple harmonic oscillators restricted to lie in a N dimensional sphere. The existence of the Uhlenbeck constants, a set of N independent first integrals in involution, proves the integrability of the system by virtue of the Liouville theorem. The analysis was broadened in [2] by allowing non-trivial winding numbers along the compact coordinates of the Cartan subalgebra in the ansatz. The associated mechanical model is accordingly extended to the Neumann-Rosochatius integrable system, which involves an additional centrifugal potential for each oscillator. The integrability of
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