Probing analytical and numerical integrability: The curious case of $(Ad{S}_5\times S^5{)}_{\eta }$

Abstract

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 $(Ad{S}_5\times S^5{)}_{\eta }$. We start by revisiting conclusions from earlier studies on string motion in $(\mathbb{R}\times S^3{)}_{\eta }$ and $(Ad{S}_3{)}_{\eta }$ and then move on to more complex problems of $(\mathbb{R}\times S^5{)}_{\eta }$ and $(Ad{S}_5{)}_{\eta }$. Discussing both analytically and numerically, we deduce that while $(Ad{S}_5{)}_{\eta }$ 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.

31 pages, 3 figures, references updated, analysis for Spiky strings in section (4.1) have been revised, version to appear in JHEP

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