On marginal deformations and non-integrability

We study the interplay between a particular marginal deformation of 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.Poincare 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.