On Marginal Deformations and Non-Integrability

Abstract

We study the interplay between a particular marginal deformation of $\mathcal{N}=4$ super Yang-Mills theory, the $\beta$ deformation, and integrability in the holographic setting. Using modern methods of analytic non-integrability of Hamiltonian systems, we find that, when the $\beta$ parameter takes imaginary values, classical string trajectories on the dual background become non-integrable. We expect the same to be true for generic complex $\beta$ parameter. By exhibiting the Poincaré sections and phase space trajectories for the generic complex $\beta$ case, we provide numerical evidence of strong sensitivity to initial conditions. Our findings agree with expectations from weak coupling that the complex $\beta$ deformation is non-integrable and provide a rigorous argument beyond the trial and error approach to non-integrability.

19 pages, 9 figures

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• Chaos Rules out Integrability of Strings in AdS_5 x T^{1,1}
• Nonplanar Integrability
• Stability of string configurations dual to quarkonium states in AdS/CFT
• Near BPS Wilson Loop in beta-deformed Theories

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