Chaos Rules out Integrability of Strings in AdS_5 x T^{1,1}

TL;DR

This work demonstrates that classical string dynamics on $AdS_5 \times T^{1,1}$ is not integrable by analyzing a simple winding configuration that reduces to two coupled gravitational pendula. Chaos is diagnosed through Poincaré sections and a positive largest Lyapunov exponent, with the KAM theorem illustrating progressive destruction of invariant tori as energy grows. The findings imply integrability is a special feature of $AdS_5 \times S^5$, with notable implications for the dual ${\cal N}=1$ Klebanov-Witten theory and the spectrum of large-charge operators. The authors also suggest generalizations to other Sasaki–Einstein manifolds, where a nontrivial $U(1)$ fibration underpins the chaotic dynamics.

Abstract

We show that certain classical string configurations in AdS_5 x T^{1,1} are chaotic. This answers the question of integrability of string on such backgrounds in the negative. We consider a string localized in the center of AdS_5 that winds around two circles of T^{1,1}. The corresponding dynamical system is equivalent to two coupled gravitational pendula and allows a very intuitive understanding. We find conclusive evidence of chaotic behavior by systematically analyzing the workings of the KAM theorem. We also show that the largest Lyapunov exponent is positive.

Figures (4)

• Figure 1: Plot of the potential $V(\theta_1,\theta_2)$ with $\alpha_1=1$ and $\alpha_2=2$.
• Figure 2: Plot of $\sin(\theta_1(t))$ with $\alpha_1=1$ and $\alpha_2=2$ and a resting initial condition with $\theta_1(0)=0.2,\theta_2(0)=0.1$. The time evolution shows chaotic motion.
• Figure 3: Poincaré Sections.
• Figure 4: Motion of the string and the corresponding Lyapunov indices for a chaotic motion. We have chosen the initial condition $\theta_1(0)=0.1,\theta_2=0.1,\dot \theta_1=0,\dot \theta_2=0.8$. We see a convergent Lyapunov index with $\lambda \approx 0.245$.