Non-Integrability of Strings in $AdS_{6}\times S^{2}\times\Sigma$ Background and its 5D Holographic Duals

TL;DR

This paper investigates the non-integrability of string dynamics in a specific AdS background, demonstrating through analytical and numerical methods that various related models are non-integrable, with implications for their holographic duals.

Contribution

The study introduces a systematic approach to assess integrability of strings in complex backgrounds, applying it to multiple recent models and establishing their non-integrability.

Findings

All considered models are non-integrable.

Long quivers generally exhibit non-integrability.

Numerical chaos analysis supports non-integrability conclusions.

Abstract

In this manuscript we study Liouvillian non-integrability of strings in $AdS_{6}\times S^{2}\times\Sigma$ background. We consider soliton strings and look for simple solutions in order to reduce the equations to only one linear second order differential equation called Normal Variation Equation (NVE). We study truncations in $\eta$ and $\sigma$ variables showing their applicability or not to catch (non) integrability of models. With this technique we are able to study many recent cases considered in the literature: the abelian and non-abelian T-duals, the $(p,q)$-5-brane system, the T$_{N}$, $+_{MN}$ theories and the $\tilde{T}_{N,P}$ and $+_{P,N}$ quivers. We show that all of them are not integrable. Finally, we consider the general case at the boundary $\sigma=\sigma_0$ for large $\sigma_0$ and show that we can get general conclusions about integrability. For example, beyond the above quivers, we show generically that long quivers are not integrable. In order to establish the results, we numerically study the string dynamical system seeking by chaotic behaviour. Such a characteristic gives one more piece of evidence for non-integrability of the background studied.