Integrability and non-integrability in N=2 SCFTs and their holographic backgrounds

We show that the string worldsheet theory of Gaiotto-Maldacena holographic duals to N=2 superconformal field theories generically fails to be classically integrable. We demonstrate numerically that the dynamics of a winding string configuration possesses a non-vanishing Lyapunov exponent. Furthermore an analytic study of the Normal Variational Equation fails to yield a Liouvillian solution. An exception to the generic non-integrability of such backgrounds is provided by the non-Abelian T-dual of AdS ₅ × S ⁵; here by virtue of the canonical transformation nature of the T-duality classical integrability is known to be present.