Elliptic solutions in the Neumann–Rosochatius system with mixed flux

Abstract

Closed strings spinning in AdS 3 × S 3 × T 4 with mixed Ramond-Ramond and Neveu-Schwarz-NeveuSchwarz three-form fluxes are described by a deformation of the one-dimensional Neumann-Rosochatius integrable system. In this article we find general solutions to this system that can be expressed in terms of elliptic functions. We consider closed strings rotating either in S 3 with two different angular momenta or in AdS 3 with one spin. To find the solutions, we will need to extend the Uhlenbeck integrals of motion of the NeumannRosochatius system to include the contribution from the flux. In the limit of pure Neveu-Schwarz-NeveuSchwarz flux, where the problem can be described by a supersymmetric Wess-Zumino-Witten model, we find exact expressions for the classical energy in terms of the spin and the angular momenta of the spinning string.

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].…”

On N-spike strings in conformal gauge with NS-NS fluxes

“…Subsequently finite-gap solutions were proposed in [49,50] and a large class of folded [51], rotating [52,53,54], pulsating [55,56] and GKP-like multi-spike [57,58] strings have been studied in the literature. Also, well known Neumann-Rosochatius dynamical model of strings has been generalised to AdS 3 × S 3 with NS-NS fluxes together with relevant classical solutions [59,60,61] and one-loop quantization [62] being elucidated in a large number of papers. Moreover, probe D1 strings exploring the mixed flux background have been discussed in [63].…”

The AdS3× S3× S3× S1worldsheetSmatrix

“…The Mathematica package grassmann.m by M. Headrick and J. Michelson was very useful when performing the calculations presented in the first part of this paper. 2 For recent work on integrable AdS 3 string solutions involving NS-NS flux see [45][46][47][48][49]. 3 The appearance of such central extensions when level-matching is relaxed is similar to what happens in AdS 5 × S 5 [27,28] and AdS 3 × S 3 × T 4 [29][30][31].…”