Chaotic string dynamics in deformed $T^{1,1}$

Abstract

Recently, Arutyunov, Bassi and Lacroix have shown that 2D non-linear sigma model with a deformed T 1,1 background is classically integrable [arXiv:2010.05573 [hep-th]]. This background includes a Kalb-Ramond two-form with a critical value. Then the sigma model has been conjectured to be non-integrable when the twoform is off critical. We confirm this conjecure by explicitly presenting classical chaos. With a winding string ansatz, the system is reduced to a dynamical system described by a set of ordinary differential equations. Then we find classical chaos, which indicates non-integrability, by numerically computing Poincaré sections and Lyapunov spectra for some initial conditions.

Analytic (non)integrability of Arutyunov-Bassi-Lacroix model

“…When k = λ 2 , the integrability of the system is lost and it becomes chaotic. This has been confirmed very recently using numerical techniques [32].…”

“…In this paper, we complement all those previous results [31,32] by taking a third path which is based on the notion of Kovacic's algorithm [34,35]. The algorithm essentially offers a set of rules in order to verify the Liouvillian non-integrability criteria for a classical 2d sigma model over general backgrounds.…”

“…In this work, we use the methodology of the AdS/CFT duality [28,29,30] in order to study the dynamics of string solitons over Arutyunov-Bassi-Lacroix (ABL) background R × T (λ 1 ,λ 2 ,λ) k [31,32]. Along the way, we explore the classical (non)integrability of the associated phase space.…”

“…Along the way, we explore the classical (non)integrability of the associated phase space. This ABL background can be considered as a generalization of the Einsteinian T 1,1 manifold [33] and is given by [31,32]…”

“…We also observe a generalized special limit when these pendulums decouple and become integrable. Interestingly, the decoupling limit studied in [31,32] can be seen as a special case of our analysis. While achieving this later limit, we observe that the string must satisfy certain conditions that have been discussed in detail in the section 3.…”