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

TL;DR

This paper investigates the integrability of string motion in the deformed background $(AdS_5)_{\eta}$, revealing that while the $AdS_5$ part remains regular, the deformed five-sphere can exhibit chaotic trajectories, challenging assumptions about integrability.

Contribution

The study compares analytical and numerical methods to analyze string integrability in deformed backgrounds, highlighting the unexpected chaotic behavior in the deformed five-sphere.

Findings

$AdS_5$ strings do not show irregular trajectories.

String motion in the deformed five-sphere can be chaotic.

Analytical and numerical procedures reveal differences in integrability.

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 $(AdS_5\times S^5)_{\eta}$. 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.