Probing analytical and numerical integrability: The curious case of (AdS5×S5)η(AdS_5\times S^5)_{\eta}

Motivated by recent studies related to integrability of string motion in various backgrounds via analytical and numerical procedures, we discuss these procedures for a well known integrable string background $(AdS_5\times S^5)_{\eta}$. We start by revisiting conclusions from earlier studies on string motion in $(\mathbb{R}\times S^3)_{\eta}$ and $(AdS_3)_{\eta}$ and then move on to more complex problems of $(\mathbb{R}\times S^5)_{\eta}$ and $(AdS_5)_{\eta}$. Discussing both analytically and numerically, we deduce that while $(AdS_5)_{\eta}$ strings do not encounter any irregular trajectories, string motion in the deformed five-sphere can indeed, quite surprisingly, run into chaotic trajectories. We discuss the implications of these results both on the procedures used and the background itself.Comment: 31 pages, 3 figures, references updated, analysis for Spiky strings in section (4.1) have been revised, version to appear in JHE

Fast spinning strings on η \eta deformed AdS5×S5 AdS_5 \times S^{5}

In this paper, considering the correspondence between spin chains and string sigma models, we explore the rotating string solutions over $\eta$ deformed $AdS_5 \times S^{5}$ in the so called fast spinning limit. In our analysis, we focus only on the bosonic part of the full superstring action and compute the relevant limits on both $(R \times S^{3})_{\eta}$ and $(R \times S^{5})_{\eta}$ models. The resulting system reveals that in the fast spinning limit, the sigma model on $\eta$ deformed $S^5$ could be $\textit{approximately}$ thought of as the continuum limit of anisotropic $SU(3)$ Heisenberg spin chain model. We compute the energy for a certain class of spinning strings in deformed $S^5$ and we show that this energy can be mapped to that of a similar spinning string in the purely imaginary $\beta$ deformed background.Comment: 30 pages; references updated; version to appear in JHE

On multi-spin classical strings with NS-NS flux

We study multi spin semiclassical strings in $AdS_3\times S^3 \times T^4$ background supported by a mixture of Neveu-Schwarz-Neveu-Schwarz (NS-NS) and Ramond-Ramond (R-R) fluxes. This `mixed flux' background has been recently proved to be classically integrable. We start with a particular rigidly spinning fundamental string in $AdS_3\times S^1$ coupled to the NS-NS flux and classify the possible profiles. We also find out how the scaling relation among the energy and angular momenta of such a string changes due to presence of these fluxes. We emphasize on pure NS-NS flux case and discuss the fate of such solutions in that limit.Comment: 22 pages, 4 figure