Probing analytical and numerical integrability: the curious case of (AdS5 × S5)η

Published for SISSA by Springer Received: July 4, Revised: October 3, Accepted: November 11, Published: November 21, 2018 2018 2018 2018 Aritra Banerjeea and Arpan Bhattacharyyab a CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P.R. China b Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan E-mail: , Abstract: Motivated by recent studies related to integrability of string motion in various backgrounds via analytical and numerical procedures, we discuss these procedures for a well known integrable string background (AdS5 × S 5 )η . We start by revisiting conclusions from earlier studies on string motion in (R × S 3 )η and (AdS3 )η and then move on to more complex problems of (R × S 5 )η and (AdS5 )η . Discussing both analytically and numerically, we deduce that while (AdS5 )η strings do not encounter any irregular trajectories, string motion in the deformed five-sphere can indeed, quite surprisingly, run into chaotic trajectories. We discuss the implications of these results both on the procedures used and the background itself. Keywords: Bosonic Strings, AdS-CFT Correspondence, Gauge-gravity correspondence, Integrable Field Theories ArXiv ePrint: 1806.10924 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2018)124 JHEP11(2018)124 Probing analytical and numerical integrability: the curious case of (AdS5 × S 5)η Contents 1 Introduction 1 2 Setup 3 5 5 7 10 4 Strings in the deformed AdS 4.1 Revisiting the case of (AdS3 )κ 4.2 Analytical strings on (AdS5 )κ 4.3 Explicit numerical hamiltonian analysis 13 13 15 17 5 Summary and conclusion 17 A Effect of κ 6= 0 in NVE via Kovacic algorithm 20 1 Introduction String motion in curved spaces, described by two-dimensional non-linear sigma models, have been studied extensively from the early days of development of the subject. This is extremely interesting due to the complicated non-linear equations of motion associated with the worldsheet fields. It comes as no surprise that these equations of motion are only ‘integrable’ for a select subclass of target space backgrounds, and hence this notion of integrability helps one to pick out the cases where a complete quantitative analysis of classical (and perhaps quantum) string motion can be performed and compared to the flat space case. One of the most widely known cases is of course that of type IIB strings in the AdS5 × S 5 space-time [1], which is dual to operators in maximally supersymmetric N = 4 Yang-Mills theory (sYM) via AdS/CFT correspondence [2]. The integrability of these strings moving in the bulk AdS5 ×S 5 , in conjunction with the integrability of the dual sYM theory, makes an exceptional example to study the AdS/CFT correspondence from the point of view of integrable systems [3]. Moreover, the finding that in the semiclassical limit, the dynamics of this correspondence indeed becomes tractable [4], has regenerated interest in the classical string solutions in AdS and related geometries. Indeed, a lot of literature has been devoted to the subject of integrability in AdS/CFT in last two decades.1 1 For recent introductions to this subject, the reader is directed to [5, 6] and references therein. –1– JHEP11(2018)124 3 Strings in deformed sphere 3.1 Revisiting a warm-up example: the case of (R × S 3 )κ 3.2 Strings in the five-sphere: analytical 3.3 The hamiltonian and numerical trajectories –2– JHEP11(2018)124 With this advent of integrability studies in the context of AdS/CFT, there have been many celebrated works relating to deformation of the symmetries on both sides of the correspondence while keeping the integrable structure intact. Most of these relied on the use of target space duality symmetries to generate new integrable backgrounds [7–10]. Recently, Klimcik’s pioneering works on novel integrable deformations of σ-models [11–13] have paved the way for their application to string σ-models and finding probable deformed versions of AdS/CFT correspondence [14]. Since then a larger family of integrable deformations of AdS × S geometries have been explored, where the deformation is given by a classical r-matrix solution to the (modified) Classical Yang-Baxter Equation (CYBE). The explicit geometry and NS-NS forms for such a ‘Yang-Baxter’ deformation of AdS5 ×S 5 first appeared in [14, 15], was analysed in detailed in [16] and various consistent truncations have been discussed in [17]. In the Yang-Baxter case, the deformation works by deforming the supercoset associated to AdS5 × S 5 itself by a continuous parameter, which is often referred to as a q-deformation, or a quantum group deformation [16]. This replaces the lie algebra of the classical charges by its q-deformed version, which is then incorporated into the superstring action for AdS5 × S 5 having a real deformation parameter η ∈ [0, 1) or equivalently another parameter called κ with κ ∈ [0, ∞). Notice that here the parameter κ 2η is related to the original deformation parameter η as κ = 1−η 2 [15]. In the rest of the paper, we would instead denote κ as being the deformation parameter in our analysis and refer the background as (AdS5 × S 5 )κ . For various avenues of exploratory works on Yang-Baxter deformations, one should have a look at [18]–[76]. As integrable string backgrounds by construction, the conserved currents associated to (AdS5 × S 5 )κ strings, in general, satisfy the Lax equations. However in the case of a random string sigma model, where the existence of Lax pair is not known, proving (non)integrability is a rather complicated task. To this note, there have been a number of works to consistently truncate the two-dimensional string equations of motion of particular circular strings into one dimensional mechanical systems and analyzing the (non)-integrability properties thereof. It has been argued that it is sufficient to show that there exists at least one truncated dynamical system of differential equations, where the corresponding string motion turns chaotic [77], i.e. small variations around the equations grow nondeterministically in time. Useful tools in these studies have mainly been the variational non-integrability techniques of Hamiltonian systems and numerical experiments in the associated phase space in general. This approach is often hailed as the equivalent of the algebraic approach of finding Lax pairs for the system and a large number of works have appeared along these lines, see for example [78]–[93]. In the following note, we seek to understand this equivalence by studying string motion in the extremely complicated but integrable background of (AdS5 × S 5 )κ . One should bear in mind, the process of Yang-Baxter deformation breaks the supersymmetries associated with (AdS5 × S 5 ) and even at the bosonic level, the isometry group of SO(2, 4) × SO(6) breaks down to U(1)3 × U(1)3 in (...truncated)