Non-integrability of strings in background and its 5D holographic duals

G. Alencar; M. O. Tahim

doi:10.1140/epjc/s10052-023-11225-3

2024 Impact factor 4.8

Particles and Fields

Open Access

Eur. Phys. J. C (2023) 83: 189
https://doi.org/10.1140/epjc/s10052-023-11225-3

Regular Article - Theoretical Physics

Non-integrability of strings in $A d S_{6} \times S^{2} \times Σ$ $AdS_{6}\times S^{2}\times \Sigma$ background and its 5D holographic duals

G. Alencar¹^,3^a and M. O. Tahim²^,3

¹ Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici, 60455-760, Fortaleza, Ceará, Brazil
² Faculdade de Educação, Ciências e Letras do Sertão Central, Universidade Estadual do Ceará, Campus Itaperi, Caixa Postal 703, 60714-903, Fortaleza, Ceará, Brazil
³ International Institute of Physics, Federal University of Rio Grande do Norte, Campus Universitário, Lagoa Nova, 59078-970, Natal, RN, Brazil

^a geova@fisica.ufc.br

Received: 9 May 2022
Accepted: 13 January 2023
Published online: 2 March 2023

Abstract

In this manuscript we study Liouvillian non-integrability of strings in $A d S_{6} \times S^{2} \times Σ$ $AdS_{6}\times S^{2}\times \Sigma$ 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 $η$ $\eta$ and $σ$ $\sigma$ 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 $_{N}$ $_{N}$ , $+_{MN}$ $+_{MN}$ theories and the ${\tilde{T}}_{N, P}$ ${\tilde{T}}_{N,P}$ and $+_{P, N}$ $+_{P,N}$ quivers. We show that all of them are not integrable. Finally, we consider the general case at the boundary $σ = σ_{0}$ $\sigma =\sigma _0$ for large $σ_{0}$ $\sigma _0$ 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.

© The Author(s) 2023

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.