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Probing analytical and numerical integrability: the curious case of \(\left(\mathrm{AdS}_5 \times S^5 \right)_{\eta}\). (English) Zbl 1404.83110

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

MSC:

83E30 String and superstring theories in gravitational theory
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems
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