The non-integrability of strings in massive type IIA and their holographic duals

JHEP06(2018)078 Published for SISSA by Springer Received:April 20, 2018 Accepted:June 10, 2018 Published:June 15, 2018 The non-integrability of strings in massive type IIA and their holographic duals Carlos N´u˜nez,aJos´e Manuel Pen´ın,b,c Dibakar Roychowdhuryaand Jeroen van Gorsela aDepartment of Physics, Swansea University, Swansea SA2 8PP, U.K. bDepartamento de F´ısica de Part´ıculas Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain cInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), E-15782 Santiago de Compostela, Spain E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: In this work we study various aspects of six-dimensional N= (1,0) SCFTs. We consider the construction of their string duals in Massive IIA and discuss some observables in given examples. We study the dynamics of string solitons wrapping and rotating on the Massive IIA background and show that the associated Hamiltonian system is both non-integrable and chaotic, implying the non-integrability of the dual CFT. Our procedure is analytic, using well developed mathematical techniques, and numerical, by the explicit calculation of power spectra, Lyapunov coefficients and Poincar´e sections. Keywords: AdS-CFT Correspondence, Brane Dynamics in Gauge Theories ArXiv ePrint: 1802.04269 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP06(2018)078 JHEP06(2018)078 Contents 1 Introduction 1 2 Six-dimensional SCFTs and their holographic description 3 2.1 Cremonesi-Tomasiello formulation of the holographic duals to N= (1,0) SCFTs 4 2.1.1 Examples of SCFTs and their holographic type IIA backgrounds 5 2.2 Page charges and central charge 9 2.3 A formal elaboration 11 3 Dynamics of strings on AdS7×M3backgrounds 12 3.1 Liouvillian integrability 14 3.2 Analytical study of the (non) integrability of the SCFTs 15 3.3 Integrability for the quivers of figures 1and 5 16 4 Numerical analysis 17 4.1 Numerical evolution and power spectra 18 4.2 The Lyapunov spectrum 20 4.3 Poincar´e sections and non-integrability 23 5 Conclusions and future work 26 A The function α(z) for the quivers in figures 5and 6 27 B On Liouvillian integrability and Kovacic’s algorithm 29 B.1 One example 31 C Integrability of the configuration in type IIA and M-theory 32 D Computation of the Lyapunov spectrum 33 E A relation with non-Abelian T-duality? 35 1 Introduction Quantum field theories in six dimensions were objects of curiosity before the 1990’s. Reasoning based on perturbation theory around a Gaussian fixed point (which implied the existence of a continuum Lagrangian) suggested that no such theories existed by themselves, needing some UV completion. But, observations based on the possibility of encountering a strongly coupled fixed point and having at the same time anomaly cancellations, gave credibility to the existence of these theories, making them objects of interest [1]. These ideas – 1 – JHEP06(2018)078 were supported by the construction of Hanany-Witten brane set-ups [2] for six-dimensional field theories. Indeed, the papers [3,4], found the first of those realisations. These ideas were subsequently developed and in the past few years, the different versions of the (2,0) six-dimensional SCFT were used as an effective way to organise and understand various features of lower dimensional CFTs — see for example [5]. The Maldacena conjecture [6] gave another important piece of evidence to ascertain the existence of these theories. For six-dimensional CFTs with minimal N= (1,0) SUSY, the same holographic ideas were used by the authors of [7]–[17]. These authors constructed backgrounds in Type II supergravity that realise the SO(2,6) (with an AdS7factor) and the necessary SU(2) R-symmetry. These backgrounds are dual to the SCFTs realised at low energies, on the six-dimensional intersection among NS5-D6-D8-branes. The six-dimensional N= (1,0) SUSY CFTs provides us with a well defined holographic pair (quiver CFT-background) on which different ideas and methods developed in the last twenty years can be tested. The extension of these ideas to the case in which orientifold planes are present have been carefully discussed in [18]. One of these developments is the possibility of finding integrability of the field theory. Classical integrability, first formalised by Liouville, is a frame to solve Hamiltonian problems via quadratures. Indeed, if the system has the same number of conserved quantities as coordinates in phase space, one could move to action-angle variables (Ik,θk) and solve θk=Ikt. This was further developed later under the name of the classical inverse scattering method. Of course, integrability does not equal solvability. Integrability refers to the property of systems to exhibit regular orbits in phase space (in contrast to chaotic ones). When a system displays classical integrability, there are various methods to find the exact solution to the system. These properties extend, with some limitations to the quantum case. In contrast, in systems that do not display integrability one cannot in general, write a closed analytic solution. Some non-integrable systems, present the feature of a strong sensitivity to the initial conditions. This notion is formalised by the (largest) Lyapunov exponent, a quantity that characterises the rate of separation of two trajectories that start being arbitrarily close together in phase space. Another quantity characterising chaotic systems (or the transition from integrable to chaotic by a growing non-integrable perturbation) are the Poincar´e sections [19]. For integrable systems, the phase space is foliated by the so-called KAM tori [19], but these start to disappear as the non-integrable perturbation grows in influence. Showing the integrability of a system is usually quite a difficult task. In some situations, it may be easier to prove that it is non-integrable. There are different methods developed to this end. We will use those in the papers [20,21]. The idea is to find a string soliton (in the holographic language, this soliton captures the dynamics of a long, spinning, heavy operator in the dual CFT) and show that the dynamics of such an object is non-integrable in the sense defined by Liouville. Various techniques developed by mathematicians studying this problem are explained and used in this paper. This can be complemented numerically, by studying the chaos indicators (Poincar´e sections, Lyapunov exponent) of the Hamiltonian system in question. – 2 – JHEP06(2018)078 In the rest of this work, we deal with six-dimensional N= (1,0) SUSY CFTs. We discuss the constructions of the dual backgrounds and study some of their properties and observables. After that, we focus on their (non-) integrability properties. The plan of the paper is the following: in section 2, we will discuss generalities of the CFTs of interest in this work. We shall briefly review the formalism developed in the papers [7]- [10]. We give new examples of pairs (CFT-backgrounds) and present some developments that might be useful in future studies of these systems. In section 3, we discuss the (non-) integrability of these CFTs. By focusing on the generic examples developed in section 2, we analytically show their non-integrable behaviour. In section 4, we provide a careful numerical analysis of the material in section 3, showing that indeed, those systems are chaotic. We summarise and give an outlook for future developments in section 5. Various interesting appendices complement our technical presentation. 2 Six-dimensional SCFTs and their holographic description In dimension higher than four, when flowing up in energies, a Yang-Mills theory becomes strongly coupled and non-renormalisable. Hence, the field theory needs a UV-completion. It was suggested in [1], that such a completion is given in terms of a conformal field theory at strong coupling. The existence of these field theories was supported by the computation of anomalies in [22], that showed that the inclusion of vector, hyper and tensor multiplets gives place to anomaly-free field theories. Such expectations were realised with the brane construction of six-dimensional field theories [3,4]. These constructions are based on intersections of NS5-D6-D8-branes. All the branes extend along R1,5. The NS5-branes are located at fixed x6ipositions. The D6branes extend on the x6intervals (between pairs of NS5-branes) and are point-like objects in the [x7, x8, x9] directions. Finally, the D8-branes extend on [x7, x8, x9] being localised in x6. The system preserves eight supercharges, which is associated with the chiral N= (1,0) super-algebra. The isometries of the brane set-up are SO(1,5)×SO(3). The SO(3) ∼SU(2) is the R-symmetry algebra of the N= (1,0) conformal algebra. The anomaly cancellation implies that for every gauge group the number of flavour fields doubles the number of gauge fields Nf= 2Nc. In the brane set-up Ncis the number of D6-branes in a given interval [x6,i, x6,i+1] between two NS five-branes. On the other hand, the number Nf=ND6,i+1 +ND6,i−1+ND8counts the D6-branes in the two adjacent intervals and the number of D8-branes in the interval (these are hypermultiplets in the six-dimensional low energy field theory realised on the D6-branes). Unlike all other Hanany-Witten brane set-ups realising field theories in lower dimensions, the tensor multiplets living on the NS5-branes play an important role in the cancellation of anomalies and provide self-dual two-forms, which give place (in the case in which the NS5-branes become coincident) to tensionless strings. In fact, the positions of the NS5-branes are not fixed like in lower dimensional set-ups, but they are represented by a real scalar field Φi, which gets a VEV. This scalar field couples to the gauge field strength on the D6-branes, leading to a term in the Lagrangian L∼(Φi+1 −Φi)F2 mn +. . .. When the five-branes become coincident, the effective gauge coupling 1 g2 6=hΦi+1 −Φii – 3 – JHEP06(2018)078 diverges. In this limit, the field theory flows to the conjectured CFT [1]. Some pieces of evidence support this proposal: the number of supercharges preserved by the brane set-up, the isometries — associated with the unique N= (1,0) superconformal algebra, the massless string solitons corresponding with D2-branes that extend in [x0, x1, x6] and end on the five-branes. A detailed explanation of the story summarised above can be found in [23]. Important evidence for the existence of the N= (1,0) SCFTs, comes from holography. Indeed, the paper [7] started a very fertile line of research, searching for supersymmetric AdS7solutions in Type II supergravities. The metric was proposed to be of the form, ds2=f1(z)ds2 AdS7+f2(z)dz2+f3(z)(dχ2+ sin2χdξ2), with Neveu-Schwarz and Ramond-Ramond fields preserving the isometries SO(2,6)×SO(3) and eight supercharges. In the particular case of Massive type IIA supergravity a system of BPS equations was written and a family of solutions found. The paper [8], pointed out the type of field theories these Massive IIA solutions are holographically dual to. Various efforts to solve the BPS equations and interpret the solutions followed [9]. These lead to the formulation by Cremonesi and Tomasiello [10], where a precision test was put forward, calculating the a-central charge, both in the CFT and in holography. Other interesting developments deal with flows away from, and compactifications of the six-dimensional SCFTs, see for example [11–17]. In the following, we summarise the Massive IIA backgrounds as written by Cremonesi and Tomasiello. This is the language in which we shall present the different findings of this paper. 2.1 Cremonesi-Tomasiello formulation of the holographic duals to N= (1,0) SCFTs After various manipulations, Cremonesi and Tomasiello [10] wrote the Massive IIA backgrounds dual to the six-dimensional conformal field theories as, ds2=f1(z)ds2 AdS7+f2(z)dz2+f3(z)dΩ2(χ, ξ), B2=f4(z)VolS2, F2=f5(z)VolS2, eφ=f6(z).(2.1) We have defined dΩ2(χ, ξ) = dχ2+ sin2χ dξ2and VolS2= sin χ dχ ∧dξ. The functions fi(z) are written in terms of another function α(z) and its derivatives, f1(z) = 8√2πr−α α00 , f2(z) = √2πr−α00 α, f3(z) = √2πr−α00 αα2 α02−2αα00 , f4(z) = π−z+αα0 α02−2αα00 , f5(z) = α00 162π2+πF0αα0 α02−2αα00 , f6(z) = 25 4π5 234(−α/α00)3 4 pα02−2αα00 .(2.2) The different geometries specified by α(z) are supersymmetric solutions of the Massive IIA equations of motion (with mass parameter F0), if α(z) solves the differential equation α000 =−162π3F0.(2.3) – 4 – JHEP06(2018)078 SU(N) SU(2N) SU(3N) SU(4N) Figure 1. The quiver encoding the dynamics of our first example CFT. Given a six-dimensional N= (1,0) super-conformal field theory encoded in a quiver diagram, Cremonesi and Tomasiello [10] gave a recipe to find the precise solution to eq. (2.3), such that when replaced in eq. (2.1) gives the holographic dual to the SCFT. We discuss now some interesting examples. 2.1.1 Examples of SCFTs and their holographic type IIA backgrounds Given a quiver diagram encoding the dynamics of a six-dimensional N= (1,0) SCFT, we revise here the prescription of [10] to construct the function α(z). It is clear that solutions to the eq. (2.3) are of the form α(z) = c0+c1z+c2z2−27π3F0z3. In general the solutions we search for, are continuous, differentiable and piecewise defined in the interval (i, i + 1) with ian integer. The first region is (0,1), here the function α(z) must satisfy that α(z= 0) = 0. In the last region, the interval (P, P + 1), the other boundary condition is α(z=P+ 1) = 0. In general, this allows for a solution of the form, α(z) = −81π2             a1z+a2 2z2+a3 6z30≤z≤1 b0+b1(z−1) + b2 2(z−1)2+b3 6(z−1)31≤z≤2 c0+c1(z−2) + c2 2(z−2)2+c3 6(z−2)32≤z≤3 . . . i ≤z≤i+ 1 p0+p1(z−P) + p2 2(z−P)2+p3 6(z−P)3P≤z≤P+ 1 where the constants (a2, a3), (b2, b3),. . . , (p2, p3) are determined by inspecting the function R(z) that describes the ranks of the gauge groups. In fact, this implies that a2= 0 and a3is the slope corresponding to the first node in the quiver. The other coefficients are determined by imposing continuity of the functions α(z), α0(z), and that α(z=P+1) = 0. The procedure to be followed is better understood by inspecting some examples. Let us first consider the quiver depicted in figure 1, with three gauge groups SU(N)× SU(2N)×SU(3N) ending with a flavour group SU(4N). Notice that each node satisfies Nf= 2Nc. The function R(z) describing the ranks of this first quiver is R1(z) = N(z0≤z≤3 12 −3z3≤z≤4, indicating the presence of gauge groups SU(N) at z= 1, SU(2N) at z= 2 and SU(3N) at z= 3. In this sense, the z-direction of the supergravity background encodes the field – 5 – JHEP06(2018)078 theory information. The slope of the first three nodes is s=N, which translates to a3=b3=c3=N. Similarly p3=−3N. The change in slope ∆s=−4Nindicates the presence of the SU(4N) flavour group. On the other hand, in the first interval 0 ≤z≤1, there is no gauge group, hence a2= 0, while the gauge group in the second interval is SU(N), indicating that b2= 1. Similarly c2= 2 and p2= 3, reflecting the presence of the SU(2N) and SU(3N) gauge groups. With this, we can write, α(z) = −81π2N         a1z+1 6z30≤z≤1 b0+b1(z−1) + 1 2(z−1)2+1 6(z−1)31≤z≤2 c0+c1(z−2) + 2 2(z−2)2+1 6(z−2)32≤z≤3 p0+p1(z−3) + 3 2(z−3)2−1 2(z−3)33≤z≤4. where the remaining constants are determined by imposing continuity of α, α0and that α(z= 4) = 0. This gives, a1=−5 2, b1=−2, c1=−1 2, p1= 2; b0=−7 3, c0=−11 3, p0=−3. (2.4) The function α(z) describing the background in eq. (2.1), dual to the quiver CFT in figure 1 reads α1(z) = −81π2N         −5 2z+1 6z30≤z≤1 −7 3−2(z−1) + 1 2(z−1)2+1 6(z−1)31≤z≤2 −11 3−1 2(z−2) + 2 2(z−2)2+1 6(z−2)32≤z≤3 −3 + 2(z−3) + 3 2(z−3)2−1 2(z−3)33≤z≤4. (2.5) We have worked with a quiver with three colour nodes and one flavour node. Strictly speaking, the supergravity description is valid if the number of colour nodes is taken to be large [10]. Our example in eq. (2.5) illustrates the procedure. In order to have a better holographic description of the CFT, we should work with a quiver with SU(N)×SU(2N)× SU(3N)×SU(4N)×. . . ×SU(PN) closed by an SU(PN +N)-flavour group (and taking Pto be large). In that case, we write the function, −α1(z) 81π2N=     a1z+1 6z30≤z≤1 (ka1+k3 6)+(a1+k2 2)(z−k)+k 2(z−k)2+1 6(z−k)3k≤z≤(k+1), (Pa1+P3 6)+(a1+P2 2)(z−P)+ P 2(z−P)2−P 6(z−P)3P≤z≤P+1, where k= 1, . . . ,P −1,−6a1=P2+ 2P. (2.6) The case of a quiver with increasing ranks, not closed by the flavour group, is described holographically by the function α1(z) = −81π2N(a1z+z3 6), being a1a free parameter. It is instructive to plot the function −α1(z) 81π2and its derivatives for the background defined by eq. (2.5), see figure 2. We also plot the fields defining the background and the Ricci scalar, see figures 3and 4. None of these functions are divergent for the α(z) in eq. (2.5). We shall use this background in the coming sections to study the dynamics of a string configuration that rotates and winds on it. – 6 – JHEP06(2018)078 Figure 2. The function ˆα(z)≡α(z) 81π2Nand its derivatives, that describe the CFT associated with the quiver in figure 1. Figure 3. From top-left to bottom-right, the functions f1(z), . . . , f6(z), that describe the CFT associated with the quiver in figure 1. Figure 4. The Ricci scalar associated to the quiver in figure 1. – 7 – JHEP06(2018)078 SU(N) SU(N) SU(N) SU(N) SU(N) Figure 5. The quiver encoding the dynamics of our second example CFT. As a second example, we can work out the function α(z) for the quiver in figure 5. This quiver starts with an SU(N)-flavour node followed by three nodes SU(N)-colour, and it is closed by a final SU(N)-flavour node. The function describing the ranks is, R2(z) = N     z0≤z≤1 1 1 ≤z≤3 4−z3≤z≤4, and the function that determines the holographic description α2(z) is, α2(z) = −81π2N         −3 2z+1 6z30≤z≤1 −4 3−(z−1) + 1 2(z−1)21≤z≤2 −11 6+1 2(z−2)22≤z≤3 −4 3+ (z−3) + 1 2(z−3)2−1 6(z−3)33≤z≤4. (2.7) The holographic description of this CFT is trustable when the number of nodes is large. We take the above quiver to be long enough for the illustrative purposes we aim at. Finally, we shall consider an endless quiver. The quiver starts with an SU(N)-flavour group and is continued by an infinite tail of SU(N)-colour groups. As a consequence, the z-coordinate is unbounded. There is one integration constant that remains undetermined. The function describing the ranks is R3(z) = N(z0≤z≤1 1 1 ≤z≤ ∞, The function α3(z) reads, α3(z) = −81π2N                            a1z+1 6z30≤z≤1 (a1+1 6)+(a1+1 2)(z−1)+ 1 2(z−1)21≤z≤2 (2a1+7 6)+(a1+3 2)(z−2)+ 1 2(z−2)22≤z≤3 (3a1+19 6)+(a1+5 2)(z−3)+ 1 2(z−3)23≤z≤4 (4a1+37 6)+(a1+7 2)(z−4)+ 1 2(z−4)24≤z≤5 ... (Pa1+3P2−3P+1 6)+(a1+2P−1 2)(z−P)+ 1 2(z−P)2P≤z≤P+1 .... (2.8) Ending the quiver with a flavour SU(P+ 1) node, reflects in a cap-off of the geometry at z=P+ 1. This is achieved by adding a term −1 6(z−P)3to the last line and choosing – 8 – JHEP06(2018)078 More explicitly, using the expressions in eq. (2.2) the coefficients Aand Bare A=ν2−Eν 4π√2 1 √−αα00 (−3α02α00 + 6αα002−2αα0α000) (−α02+ 2αα00)zsol , B=E 8π3α0 α+(α02+ 2αα00) (α02−2αα00) α000 α00 zsol .(3.13) The Liouvillian integrability of the string soliton depends on the function α(z) defining the background. Below, we shall study this integrability. 3.2 Analytical study of the (non) integrability of the SCFTs In this section, we apply the Kovacic algorithm [43] to the eqs. (3.12)–(3.13). For a summary of Kovacic’s procedure see appendix B. We particularise in the cases studied in section 2, more concretely for the quivers in figures 1and 5. The whole problem boils down to study the presence — (or not) of Liouvillian solutions to eqs. (3.12), given the different functions α(z). Consider first the case in which the function α(z) is α(z) = −81π2k1 2z2−2R2 0 81π2k2.(3.14) This function corresponds to a background in (massless) Type IIA — since α000 = 0. In this background there are kD6-branes. Once lifted to eleven dimensions the metric is AdS7×S4/Zk. Notice that the coordinate range is |z| ≤ 2R0 9πk . The background is singular at the ends of the space. We find that the NVE equation (3.12) reads in this case, ¨x(τ)−243E2k2π2τ 16π24R2 0−81k2π2(Eτ 4π)2˙x(τ) + ν ν+27kEπ 4πq4R2 0−81k2π2(Eτ 4π)2 x(τ) = 0. (3.15) This equation is hard to solve exactly. We observe that for large values of the parameter R0(or for very short times), the eq. (3.15) reduces to an oscillator equation. This indicates that in such a regime of parameters, the string soliton is Liouville integrable and possibly the full CFT is integrable in that limit too. This may be reminiscent of the ‘islands of integrability’ discussed in [37]. In fact they appear in the regime in which Eτ →0. Nevertheless, for finite R0(or Eτ ∼R0) we failed to find a Liouvillian solution. Let us perform a more refined analysis — the details of the logic behind the analysis are in appendix B. The first step is to write the NVE as a second order differential equation with rational coefficients. We choose 64R2 0= 81k2E2= 1 to ease the algebra (not loosing generality). The NVE equation reads, ¨x−3τ 1−τ2˙x+1 + 3 √1−τ2x= 0. – 15 – JHEP06(2018)078 We change variables to τ=√1−v2. The NVE differential equation in this new variable reads, x00(v) + C(v)x0(v) + D(v)x(v) = 0,C=1 dv dτ B(v) + d dv dv dτ ,D=A(v) (dv dτ )2.(3.16) Where, in this particular case we have v=p1−τ2,dv dτ =−√1−v2 v,d dv dv dτ =1 v2√1−v2, C=3v2−4 v−v3,D=v2+ 3v 1−v2.(3.17) Following the analysis detailed in appendix B, we construct a function 4V(v)=4D−C2− 2C0, 4V=−4 + 3 4(v−1)2−17 4(v−1) −24 v2+3 4(v+ 1)2−31 4(v+ 1).(3.18) The pole structure of this function is analysed according to the criteria in appendix B. The existence of poles of order-one and the fact that the function Vis of order-one at infinity, implies that none of the three possible cases detailed in appendix Bcan be satisfied. Therefore, the equation has a non-Liouvillian solutions. The string soliton is non-integrable, and also is non-integrable the associated CFT. For a study of the integrability of a membrane equations in the eleven dimensional lift of this solution see appendix C. 3.3 Integrability for the quivers of figures 1and 5 Consider now the quiver CFT with holographic dual defined by the function α1(z) in eq. (2.5). Finding an exact solution to eqs. (3.12)–(3.13) is challenging. We may attempt to rewrite eq. (3.12) by redefining x(τ) = e−1 2RBdτ f(τ), leading to an equation in the Schr¨odinger form, f00(τ) + V(τ)f(τ)=0, V (τ) = A− 1 4B2−1 2B0|zsol .(3.19) To solve exactly this last equation is a daunting task. Nevertheless, we can simplify matters if we study the problem very close to z∼τ∼0, that is for short times. Indeed, choosing E= 4πand ν= 1 to avoid cluttered expressions we find for a series expansion in τ, A ∼ γ1−γ2τ2,B ∼ γ4 τ−γ3τ. (3.20) The explicit expression of the coefficients γ1, γ2, γ3, γ4is not important for this analysis. Using the leading terms in eq. (3.20), the differential eqs. (3.12), (3.19) admit Liouvillian solutions. Nevertheless, when the subleading terms are included, both equations present solutions that contain Hermite polynomials, and hypergeometric functions 1F1. This implies the non-Liouvillian character of the solution, indicating non-integrability of the string soliton of eq. (3.3). A more refined study is presented in appendix B. We change variables to have an NVE with rational coefficients. The necessary conditions for this NVE to admit Liouvillian – 16 – JHEP06(2018)078 solutions are not satisfied — the details are given below eq. (B.3). This translates into the non-integrability of the N= (1,0) SCFT described by the quiver in figure 1. The same can be concluded about any background whose defining function α(z) starts as α(z)∼ a1z+a3z3close to z= 0. The non-Liouvillian integrability can also be studied numerically. In what follows, we provide a detailed numerical analysis of different observables that suggest that the system of equations (3.5), (3.8) is non-Liouvillian and chaotic. 4 Numerical analysis In this section, we carry out some explicit numerical computations that provide a solid backup to our findings of analytic non-integrability associated with N= (1,0) SCFTs in six dimensions. We study the dynamics of classical strings on the backgrounds in eqs. (2.1)–(2.3). We demonstrate that the phase space dynamics of classical strings on these backgrounds is chaotic (and hence non-integrable). Let us mention that non-integrability does not necessarily imply chaos. However, as far as the Gauge/Gravity duality is concerned, all the examples encountered, that have been found to be non-integrable were also chaotic in general. The evolution of a dynamical system is given by a set of deterministic differential equations that allows us to calculate the state of a system at a time t, knowing an earlier state of the system at some initial time t0. A dynamical system is said to be chaotic when it is exponentially sensitive to its initial conditions, making it practically impossible to accurately predict the long term dynamical behaviour. Indeed, when we have two adjacent initial conditions x1(t0) and x2(t0) = x1(t0) + , we say the system exhibits chaotic dynamics when |x1(t)−x2(t)| ∼ eλt, provided that the trajectory of our system in phase-space remains bounded. This boundedness of the trajectories is to rule out the trivial case where the trajectories move off to infinity and only diverge exponentially because they are moving apart [19]. In our case we are studying the motion of classical strings that sit at the centre of AdS7spacetime, while moving and rotating in an internal space of the form R×S2. This is described by the system in eqs. (3.5)–(3.6) or their analog (3.8). The coordinates z, χ are bounded and the respective momenta along the pzand pχ-directions are bounded also due to the conserved Hamiltonian in eq. (3.7). The trajectory of this string embedding in the phase space will therefore be bounded if the z-coordinate itself is bounded. In this case, the Lyapunov exponent (that measures the exponential divergence of initial conditions) indeed provides a good observable to determine whether the dynamics of this classical string embedding is chaotic or not. The numerical analysis that follows is quite dense. The plan is the following: first, in section 4.1, we examine the motion of classical strings over the background solutions (2.5) and (2.7), by numerically evaluating the equations of motion. We calculate the corresponding power spectra and discuss how this is indicative of chaotic dynamics. In section 4.2, we explore the Lyapunov spectrum [44], demonstrating that the dynamical behaviour is indeed chaotic. We end the analysis in section 4.3, discussing the Poincar´e sections of these solutions and their implications on the chaotic dynamics associated to classical string con- – 17 – JHEP06(2018)078 50 100 150 200 250 t -1 1 2 3 4 (a) z(t) and pz(t) in blue and yellow respectively. 50 100 150 200 250 t -1.0 -0.5 0.5 1.0 cos(χ) (b) χ(t). Figure 7. Numerical evolution for a string on the background solution (2.7) (quiver 3) with initial conditions χ(0) = 0.001, pχ(0) = 0, z(0) = 2 and pz(0) = 1, corresponding to an energy E≈3.83. figurations considered in this paper. Appendix Dcomplements the numerical analysis with some rigorous definitions. 4.1 Numerical evolution and power spectra The equations of motion for the classical strings (3.5)–(3.6) are considerably simplified with the choice ¨χ= ˙χ=χ= 0 (or χ=π), i.e. when the string stays fixed at the north or the south pole of the 2-sphere. With this choice, the remaining equation for the motion along zis, 2f2(z)¨z+ ˙z2f0 2(z) + f0 1(z) f1(z)2E2= 0,(4.1) which is eq. (3.9). Let us study how the classical string dynamics becomes increasingly disorganised as we allow the strings to move further away from the poles of the two-sphere. First, consider a classical string on the background solutions that were already discussed in eqs. (2.5) and (2.7). Below, we refer to them as quiver 1 and quiver 3 respectively. In figure 7a, we see that when the string stays very close to the poles of the two-sphere, it moves along the z-direction until it hits the end of the z-domain. Then, it turns around and moves back along the z-direction. On the two-sphere, the string starts out located near the north pole (at, χ=0.01). However, when the string turns around along the z-direction it moves almost instantaneously from the north-pole to the south-pole (see, figure 7b). Now, we move the initial position of the string away from the poles (that are located at χ= 0 or, χ=π) to the middle of the two-sphere, located at χ(0) = π/2. We keep all the other initial conditions the same. In figure 8b, we consider an initial χ(0) = 0.1, corresponding to E≈6.75. We see that the square shaped trajectory that the string traces out in the (z, χ)-plane gets deformed. The dynamics of the system becomes more complicated as the additional χ-dependence weighs in eqs. (3.5). Similar conclusions could be drawn for the background in eq. (2.5) — quiver 1. Like in the previous example, the trajectories start looking unstructured as we increase χ(0). – 18 – JHEP06(2018)078 1 2 3 4 z -1.0 -0.5 0.5 1.0 cos(χ) (a) χ(0) = 0.01, E≈3.83, tmax = 400. 1234 z -1.0 -0.5 0.5 1.0 cos(χ) (b) χ(0) = 0.10, E≈6.75, tmax = 400. 1 2 3 4 z -1.0 -0.5 0.5 1.0 cos(χ) (c) χ(0) = 0.25, E≈14.31, tmax = 400. 1 2 3 4 z -1.0 -0.5 0.5 1.0 cos(χ) (d) χ(0) = 0.9, E≈43.82, tmax = 100. Figure 8. Different trajectories in the (z, χ)-plane for a string-embedding of the form in eq. (3.3) on the background in (2.7), we run the evolution up to t=tmax and only change the initial condition χ(0). We now discuss the power spectra [19]. By taking the Fourier transform of the numerical evolutions in figure 8, we can distinguish whether z(t) and χ(t) are periodic, quasiperiodic or chaotic. When a signal is perfectly periodic with a frequency ω, its Fourier spectrum will show a vertical line at the characteristic frequency of the system. Notice that, for very low values of χ(0), the string moves almost periodically along the z-direction with a jigsaw motion (see figure 7a) and along the χ-direction with a squarewave profile (see figure 7b). From figure 8a, we can see that this motion is not exactly periodic, as the path of the string in the (z, χ)-plane does not exactly close on itself. We can see a confirmation of this in the corresponding Fourier spectrum — see figure 9a. In fact, we clearly see a fundamental frequency of value 0.02 and the corresponding oscillations along the z-axis that have a period of roughly 55t. In addition to this, we see the higher harmonics of the jigsaw and box shaped waveforms. The finite width of these peaks however suggests that this is not a periodic signal but that there is instead some noise present. Such a noisy power spectrum is a typical characteristic of a deterministic chaotic system. As we increase the value of χ(0), we first see that the jigsaw and box-shaped waveforms become distorted — see figure 8b. This is reflected by the corresponding power spectrum, loosing their higher harmonics as shown in figure 9b. Increasing χ(0) even further, we see that a broad band of noise around a frequency 0.35 starts to overpower the spectrum — see figures 9b–9c. At even higher values of χ(0), we even loose the initial peak at frequency 0.02. The spectrum becomes primarily dominated by noise. – 19 – JHEP06(2018)078 0.0 0.2 0.4 0.6 0.8 1.0 10-6 10-5 10-4 0.001 0.010 0.100 Frequency Spectral Level (a) χ(0) = 0.01, E≈3.83. 0.0 0.2 0.4 0.6 0.8 1.0 10-7 10-6 10-5 10-4 0.001 0.010 0.100 Frequency Spectral Level (b) χ(0) = 0.10, E≈6.75. 0.0 0.2 0.4 0.6 0.8 1.0 10-6 10-5 10-4 0.001 0.010 0.100 Frequency Spectral Level (c) χ(0) = 0.25, E≈14.31. 0.0 0.2 0.4 0.6 0.8 1.0 10-5 10-4 0.001 0.010 0.100 Frequency Spectral Level (d) χ(0) = 0.90, E≈43.82. Figure 9. Power spectra for the trajectories in figures 8a–8d. Here the spectra for both z(t) and χ(t) are shown in yellow and blue respectively. To calculate these spectra we ran a numerical evolution up to t= 5000 (roughly 100 oscillations along the z-direction), with a resolution of 10 data-points per time-unit. These plots and its analysis have been done for the background in eq. (2.7) — quiver 3. For the background defined by eq. (2.5), we see exactly the same qualitative behaviour, for roughly the same values of χ(0). 4.2 The Lyapunov spectrum The Lyapunov spectrum is generally introduced as a measure of the dynamical information loss in a chaotic system. This loss of information is what sources the dynamical Kolmogorov-Sinai (KS) entropy production within a chaotic system. Typically, for dynamical systems with a non-zero Lyapunov, the time evolution associated with two nearby trajectories in the phase space turns out to be highly sensitive to a tiny change in the initial conditions that is eventually amplified exponentially at sufficiently late times. In other words, the existence of a non-zero Lyapunov exponent, for a point X= (q, p) in the phase space with initial condition X0= (q(t= 0), p(t= 0)) is, λ= lim τ→∞ lim ∆X0→0 1 τlog ∆X(X0, τ) ∆X(X0,0) (4.2) – 20 – JHEP06(2018)078 0 100 200 300 400 500 -1.0 -0.5 0.0 0.5 1.0 Steps LCEs (a) LCE massless: t= 0, z= 0.05, χ= 0.05, pt= 0.5, pz= 4.60155, pχ= 0.01. Integration: τ= 0.1, K= 500, T= 0.009. 0 20 40 60 80 100 120 140 -1.0 -0.5 0.0 0.5 1.0 Steps LCEs (b) LCE quiver 1: t= 0, z= 0.15, χ= 0.15, pt= 15, pz= 0.845631, pχ= 0.01. Integration: τ= 0.1, K= 150, T= 0.01. 0 50 100 150 200 -1.0 -0.5 0.0 0.5 1.0 Steps LCEs (c) LCE quiver 2 t= 0, z= 0.09, χ= 0.09, pt= 15, pz= 0.491105, pχ= 0.22. Integration: τ= 0.5, K= 200, T= 0.02. 0 50 100 150 200 -1.0 -0.5 0.0 0.5 1.0 Steps LCEs (d) LCE quiver 3 t= 0, z= 0.09, χ= 0.09, pt= 9, pz= 0.821405, pχ= 0.22. Integration: τ= 0.5, K= 200, T= 0.02. Figure 10. LCEs for different quivers. is intimately related to the degree of randomness associated with the dynamical phase space of a Hamiltonian system. It is typically introduced as a quantitative measure of the rate of separation between two infinitesimally close trajectories in the phase space. The function ∆X(X0, τ) measures the separation between two infinitesimally close trajectories (at very late times) as a function of this initial location. Typically, for chaotic systems, one ends up with ∆X(X0, τ)∼∆X(X0,0)eλτ .(4.3) Below, we provide a detailed analysis of the computation of the Lyapunov exponents [44] corresponding to various background solutions, characterised by a function α(z). We will also be interested in the massless background solution described in eq. (3.14) and in the solution corresponding to the quiver that never ends described by eq. (2.8) — without the ‘closure’. We denote this last one as quiver 2 in the plots below. We set P= 10 to be the length of the quiver for the purposes of the numerical analysis. – 21 – JHEP06(2018)078 0 50 100 150 200 250 300 -1.0 -0.5 0.0 0.5 1.0 Steps LCEs Figure 11. LCEs for the massless solution with large R0. The computation of the Lyapunov exponents is solely based on the prescription of [44].2 The initial conditions are fixed to satisfy the Hamiltonian constraint, H= 0. This in turn implies that for a 2Ndimensional phase space, 2N X i=1 λi= 0 (4.4) where we denote by λithe i-th Lyapunov Characteristic Exponent (LCE). It is the measure of the exponential growth associated to the ith direction in the phase space. For some systems, the sum of all positive Lyapunov exponents measures the KS entropy production during the dynamic evolution in the phase space. We substitute some appropriate initial conditions into the dynamical equations in order to generate a solution. Choosing these initial conditions for the phase space variables to satisfy the vanishing of the Hamiltonian, we find the corresponding Lyapunov spectrum for each of the quiver configurations, which clearly have non-zero LCEs — see figures 10a–10d. Notice that, in our analysis, we are eventually computing four LCEs (λi) that characterise a four-dimensional dynamical phase space. The addition of all them gives a vanishing number as it corresponds to a Hamiltonian system. At this stage, it is worthwhile to mention an interesting limit associated with the massless solution mentioned above. This is the limit in which we set the parameter R0→ ∞, which as can be seen from eq. (3.15) yields the NVE, ¨x(τ) + ν2x(τ) = 0 (4.5) which is therefore trivially integrable. To perform the corresponding (numerical) computation on the LCEs, we choose the parameters k= 1, R0= 500, ν = 1 (4.6) taking as initial conditions for the phase space variables: t= 0, z = 0.05, χ = 0.05, pt= 100, pz= 0.159689, pχ= 0.01 (4.7) 2The details of the numerical techniques and the precise definition of Kand Tare provided in the appendix D. – 22 – JHEP06(2018)078 that satisfy the vanishing of the Hamiltonian. From the plot in figure 11, it is easy to notice that the largest LCE is substantially reduced for large enough times. 4.3 Poincar´e sections and non-integrability An N-dimensional integrable system possesses Nindependent integrals of motion that are in involution, namely, the Poisson bracket of any two of these conserved quantities vanishes. As a consequence of this, the corresponding phase space trajectories are confined to the surface of an N-dimensional KAM torus [19]. When we change our variables to action-angle variables (qi, pi)→(φi, Ji), such that our Hamiltonian only depends on Ji, the corresponding trajectories on this KAM torus are completely specified in terms of N frequencies (ωi) that specify the velocities along the different angles on this torus. When there is no set of integers nisuch that ωini= 0, these trajectories are said to be quasiperiodic, they do not close on themselves but fill the surface of a KAM torus. As a consequence of this, we can see whether a system is integrable or not, by taking cross-sections of its phase-space trajectories. When we plot for example (φ1, J1) every time φ2= 0 we will see a large number of foliated circular KAM curves associated with the 2-dimensional cross-sections of these foliated KAM tori. Such a cross-section is known as a Poincar´e section [19]. The KAM theorem tells us how these KAM curves will change when we perturb an integrable Hamiltonian with a small deformation H0, where 1. The resonant tori — for which these trajectories close on themselves, will be destroyed by this perturbation. However, a large number of these non-resonant KAM tori will survive. As we continue to increase the strength of this perturbation, more and more of these tori are destroyed until the motion becomes seemingly random and we loose all of the KAM curves in our Poincar´e section. In order to generate Poincar´e sections for the background solutions in eqs. (2.5) and (2.7) we first choose a set of different initial conditions, all having the same energy E. We do this by setting, z(0) = 2, pχ(0) = 0, and varying pz(0) ∈[0,10] and χ(0) in such a way that the Virasoro constraint in eq. (3.6) is always satisfied for a given value of the energy. We then run the numerical evolution for these initial points, and plot the points (z, pz) every time χ(t) = 0 — see figure 12. If the string motion were integrable, the corresponding trajectories would have been constrained to a 2-dimensional torus in this (z, pz, χ, pχ) phase-space. The Poincar´e crosssections of the phase-space would then show the different resonant tori as embedded circles. The absence of such embedded circular KAM curves — in figures 12,13 and 14 — indicates that we are dealing with a non-integrable system, in agreement with the results we found in the earlier sections. From our earlier study of the numerical evolution in section 4.1, we know that for low energies the string oscillates between the different poles of the two-sphere when turning around along the z-axis. As we explained, the point χ(t) = 0 corresponds to the string being on the north pole of the two-sphere. We noticed that for low enough energies the string stays localised at this pole while moving along the z-axis. This is clearly seen from the horizontal lines in figure 12a. Also, notice that for very low momenta the string does not reach the other side of the z-domain and stays localised around one of the endpoints. – 23 – JHEP06(2018)078 1 2 3 4 z 0.5 1.0 pz (a) E= 3. 1 2 3 4 z -0.5 0.5 1.0 1.5 pz (b) E= 5. 1 2 3 4 z -1 1 2 3 pz (c) E= 10. 1 2 3 4 z -1 1 2 3 4 pz (d) E= 15. Figure 12. Poincar´e sections for the (z, pz)-plane at χ(t) = 0, for the quiver in eq. (2.7) at different energies. As we increase the energy (and consequently choose a higher value for χto satisfy the Virasoro constraint) the string is no longer fixed at the pole but starts to oscillate quasi-periodically around the poles as it transverses the z-direction. This is the nature of the circles that we see appearing along the horizontal lines in figures 12b and 12c. Finally, as we increase the energy even further we see that the Poincar´e section looses all of its structure, since the string seems to move randomly along the 2-sphere as the string moves along the z-direction. Similar Poincar´e sections for the background in eq. (2.5) — quiver 1, can be seen in figure 14. Though this second quiver solution is not left-right symmetric along the z-direction the behaviour of its numerical evolution is in both cases very similar. Finally, a different Poincar´e cross-section for the quiver in eq. (2.7) is shown in figure 14. Here we choose our initial conditions in a similar manner but we now plot the points (χ, pχ) every time z= 2. We see again that for low energies the string stays located at the poles where cos χ= 1 or cos χ=−1. As we go to higher energies the string is located at random points on the two-sphere every time we cross z= 2. – 24 – JHEP06(2018)078 B.1 One example Let us work out an example to see the criteria at work. We study the NVE in eqs. (3.12)– (3.13). To simplify matters, we will just study the NVE in the interval 0 ≤z≤1, so that the function α(z) = −81π2N(a1z+z3 6). In this case the coefficients are A= 1 −√3(z4+ 20a1z2−60a2 1) √−6a1−z2(z4+ 12a1z2−12a2 1), B=2 z+3z (6a1+z2)−46a1z+z3 (z4+ 12a1z2−12a2 1).(B.3) To avoid cluttering the expressions, we have chosen the coefficients E= 4π(such that z=τ) and ν= 1. The coefficients of this NVE are not rational functions. To amend this, we change from zto a new variable v, z=p−6a1−v2(B.4) denoting x0=dx dv , the NVE reads x00(v) + Cx0(v) + Dx(v) = 0,C=B(v) + d dv dv dz  dv dz ,D=A(v) (dv dz )2.(B.5) According to what was explained around eq. (B.2), we now need to analyse the principal part of the potential 4V= 2dC dv +C2−4D. For the particular case of the Aand B, and the change of variables in eqs. (B.3)–(B.4) we find, C=v6−12a1v4−240a2 1v2−576a3 1 v(v2+ 6a1)(v4−48a2 1), D=−1 + 6a1+ 5√3v v2+ 6a1−4√3v(v2−4a1) v4−48a2 1 , 4V= 4 + γ0 v2+γ1 (v2+ 6a1)2+γ2+γ3v (v2+ 6a1)+γ4v2 (v4−48a2 1)2+ +γ5+γ6v+γ7v2+γ8v3 (v4−48a2 1).(B.6) The coefficients γiare numerical constants, not very relevant for our analysis below. Notice that the potential has a pole of order one at v=±i√6a1. The order of the potential (leading power of the degrees of the denominator minus numerator) is one. Hence, V(x) does not fall in any of the three allowed cases. The solution to the equation should then be non-Liouvillian. Notice that this analysis covers the cases of our quivers in figures 1and 5. Indeed, both these quivers start with the function α(z) = −81π2N(a1z+z3 6). Hence, both backgrounds and both quiver CFTs are non-integrable. – 31 – JHEP06(2018)078 C Integrability of the configuration in type IIA and M-theory In this short appendix (the details of which will be fully worked out elsewhere), we will indicate the steps that lead to the a more detailed study of our string configuration for the case in which the function α(z) = (R2−µ2z2), characterising a background in Type IIA (with mass parameter m= 0) that lifts to and AdS7×S4/Zkin eleven dimensional supergravity. This is the case we studied around eq. (3.14). We shall lift the background to eleven dimensions, and then study the dynamics of a membrane that mimics our type IIA string. The lifted background is, ds2 11 =f−2/3 6f1ds2 AdS7+f2dz2+f3dΩ2(χ, ξ)+f4/3 6(dy −f5cos χ dξ)2, 6C3=f4sin χ dχ ∧dξ ∧dy. (C.1) We define γij =Gµν∂iXµ∂jXν, where i, j =τ, σ, ρ are the world-volume coordinates of the membrane, the functions fiare all functions of zand are defined in eq. (2.2). We use the action and constraints for a membrane (see for example [48]), S=Zdτdσdρ γττ +L2det(γαβ)+2LijkCµνδ∂iXµ∂jXν∂kXδ. γτα = 0, γττ +L2det γαβ = 0.(α, β =σ, ρ) (C.2) We propose a membrane configuration that is the natural lift of the string soliton we proposed in the main part of the paper, t=t(τ), z =z(τ), χ =χ(τ), ξ =kσ, y =λρ. (C.3) We find an effective Lagrangian and constraint that can be written as, L=f−2s/3 6hf1˙ t2−f2˙z2−f3˙χ2+L2k2λ2f3f4s/3 6sin2χ+Lkλf4f2s/3 6˙χsin χi, 0 = −f1˙ t2+f2˙z2+f3˙χ2+L2k2λ2f3f4s/3 6sin2χ. (C.4) One can see that choosing s= 0 (and identifying Lλk =ν), we recover the expressions for strings written below eq. (3.3). On the other hand, for s= 1, we have the Lagrangian and constraint for the membrane. We now study the equations of motion derived from eq. (C.4). We find, ˙ t=E f1 f2s/3 6,(C.5) 2f3¨χ=−2L2k2λ2f3cos χsin χf4s/3 6−2 ˙z˙χf0 3+ 2Lkλ sin χ˙zf0 4f2s/3 6+4s 3 f3f0 6 f6 ˙z˙χ. ¨z+E2f4s/3 6 2f1f2f0 1 f1−2s 3 f0 6 f6+ ˙z2f0 2 2f2−s 3 f0 6 f6+ ˙χ2f3 2f2−f0 3 f3 +2s 3 f0 6 f6+ +Lkλ sin χ˙χf0 4 f2 f2s/3 6+L2k2λ2f3f4s/3 6sin2χ 2f2f0 3 f3 +2s 3 f0 6 f6= 0. The reader can check that for s= 0 the equation of motion of the string are recovered. – 32 – JHEP06(2018)078 We now apply the same algorithmic procedure as in the main body of the paper. The configuration with χ(τ) = ˙χ(τ) = ¨χ(τ) = 0 solves the χ−equation of motion and leaves the z−equation as, ¨z+E2f4s/3 6 2f1f2f0 1 f1−2s 3 f0 6 f6+ ˙z2f0 2 2f2−s 3 f0 6 f6= 0.(C.6) Calculating explicitly for the function α(z) = µ(1 −z2) (after choosing constants appropriately), using the explicit expression for fi(z), we find that z−equation is solved by zs(τ) = cosh τ. (C.7) Fluctuating the χ−equation as χ(τ) = 0 + f(τ), we find the NVE, ¨ f+B˙ f+Af= 0,(C.8) B= ˙z(τ)f0 3 f3−2s 3 f0 6 f6|zs= 2 coth τ, A=L2k2λ2f4s/3 6−Lλkf2s/3 6 f0 4 f3 ˙z(τ)|z=zs=n1sinh τ+n2sinh2τ. With n1, n2two numbers. In what follows we take s= 1 to discuss the case of the membrane only. It is convenient to change the variable v=e−τ, to have an NVE that reads, f00 +3v2+ 1 v(v2−1)f0+n1 2v3(1 −v2) + n2 4v4(1 −v2)2f= 0.(C.9) We denoted f0=df dv . We can construct the effective potential of the associated Schr¨odinger problem, as indicated in eq. (B.2), V(v) = 2B0+B2−4A 4=3 4v2−n1 2v3+n1 2v−n2 4v4−n2 4+n2 2v2.(C.10) We observe that the first of the necessary conditions discussed in appendix Bis satisfied. The Kovacic algorith should produce a Liouvillian solution for the membrane, making the membrane configuration in eq. (C.3) Liouville-integrable. We see that the problem with the string is that it ‘misses’ the effects of the dilaton, represented above by the various powers of f2/3 6. It is the presence of the dilaton (that the ‘classical limit’ of the Polyakov action misses), what changes the equation to introduce integrability. Note that the dilaton goes very large at the ends of the interval z=±1 (in these units), hence it cannot be neglected. D Computation of the Lyapunov spectrum In the following we discuss the algorithm used to compute the Lyapunov characteristic exponents (LCEs) for a generic system, in particular for a system of canonical equations such as the ones we have previously studied. We will make use of the prescription described in [44]. Let us consider a generic n-dimensional smooth dynamical system, which can generically be written as: ˙q=V(q) (D.1) – 33 – JHEP06(2018)078 where q(τ) is the n-dimensional state vector q=~ X(τ),~ P(τ)at time τ, ˙q=dq dτ and Vis a vector field on an open set Uof the phase space manifold, which generates a flow f: ˙ fτ(q) = V(fτ(q)) for all q∈U, τ ∈R (D.2) where fτ(q) = f(q, τ). Consider the evolution under the flow of two nearby points in the phase space, q0and q0+δ0, being δ0a small perturbation of the initial point q0. After a time τ, the perturbation δτwill become: δτ≡fτ(q0+δ0)−fτ(q0)≈Dq0fτ(q0)·δ0(D.3) The average exponential rate of divergence (or convergence) of two trajectories is defined by: λ(q0, δ0) = lim τ→∞ 1 τlog ||δτ|| ||δ0|| = lim τ→∞ 1 τlog ||Dq0fτ(q0)·δ0|| (D.4) being ||δ|| the length of the vector δ. If λ(x, u)>0, we have exponential divergence of nearby orbits. Under weak conditions on the nature of the dynamical system, the limit D.4 exists, it is finite and it is equal to the largest LCE λ1, see [49] for reference. The LCEs of order p, 1 ≤p≤n, are introduced to describe the mean rate of growth of a p-dimensional volume in the tangent space. Considering a parallelepiped U0in the tangent space whose edges are the pvectors δ1, . . . , δp, the LCEs of order pare defined by: λp(q0, U0) = lim τ→∞ 1 τlog[Volp(Dq0fτ(U0))] (D.5) being Volpthe p-dimensional volume defined in the tangent space. It can be seen [49] that there exist plinearly independent vectors u1, . . . , upsatisfying: λp(q0, U0) = λ1+. . . +λp(D.6) The tangent vector δτdefined in D.3 evolves in time satisfying: ˙ Φτ(q0) = DqV(fτ(q0)) ·Φτ(q0),Φ0(q0) = I(D.7) where Φτ(q0) = Dq0fτ(q0). To calculate the trajectory, we have to integrate the system: (˙q ˙ Φ)=(V(q) DqV(q)·Φ),(q(τ0) Φ(τ0))=(q0 I)(D.8) To compute the spectrum of LCEs, we will use the algorithm discussed in [50], based on the calculation of the order-pLCEs defined in equation (D.6) and on a repeated application of a Gram-Schmidt orthonormalization procedure (which avoids technical difficulties that arise in the implementation of the recipe described in [49]) that we briefly summarize here. Recall that if we compute an orthonormal set of vectors {ˆ δi}out of the original set of vectors {δi}, by using the Gram-Schmidt orthogonalisation procedure, the volume of the parallelepiped spanned by δ1, . . . , δpis Vol{δ1, . . . , δp}=||ˆ δ1||. . . ||ˆ δp|| (D.9) – 34 – JHEP06(2018)078 Then the algorithm starts by choosing an initial condition q0and a n×nmatrix ∆0= [δ0 1, . . . , δ0 n]. Using the Gram-Schmidt procedure, we calculate the corresponding matrix of orthonormal vectors ˆ ∆0= [ˆ δ0 1,...,ˆ δ0 n] and integrate the equation (D.8) from {q0,∆0}for a short interval T, to obtain q1=fT(q0) and ∆1≡[δ1 1, . . . , δ1 n] = Dq0fT(∆0)=ΦT(q0)·[δ0 1, . . . , δ0 n] (D.10) The algorithm proceeds by repeating this integration-orthonormalization procedure K times. During the k-th step, the p-dimensional volume Volpdefined in D.5 increases by a factor of ||wk 1||. . . ||wk p||, where {wk 1, . . . , wk p}is the set of orthogonal vectors calculated from Ukusing the Gram-Schmidt technique. Then: λp(q0,∆0) = lim k→∞ 1 kT k X i=1 log(||ˆ δi 1||. . . ||ˆ δi p||) (D.11) From which we can derive λp= lim k→∞ 1 kT k X i=1 log ||ˆ δi p|| (D.12) To obtain the Lyapunov spectrum, we continue calculating the quantities: 1 KT K X i=1 log ||ˆ δi 1|| ≈ λ1,..., 1 KT K X i=1 log ||ˆ δn 1|| ≈ λn(D.13) for a suitable value of T, until they show convergence. E A relation with non-Abelian T-duality? Let us briefly discuss a possible (and quite weak at the moment of this writing!) relation between the metrics in the Cremonesi-Tomasiello backgrounds (see section 2.1) and nonAbelian T-duality. Since the work of Sfetsos and Thompson [51], the non-Abelian version of the usual Tduality has regained interest and played a role as a solution generating technique. Various papers taking a perspective inspired by holography, have made clear that when applied to symmetric enough backgrounds (characteristically AdSp+1 ×M9−pbackgrounds) the generated solutions correspond to CFTs in pdimensions, realised on a Dp−NS5−Dp+2 system — for a sample of results, see the papers [52–56] for early attempts and [57–60] for more recent and precise connections between non-Abelian T-duality and brane set-ups. It is natural to ask if the Massive IIA AdS7backgrounds studied in this paper can be thought of as the non-Abelian T-dual of some background, conjecturally in Type IIB, with dilaton and F3flux. Let us give some comments in the direction of realising the previous idea. We consider a solution in Massive IIA that is the simplest possible. Consider, for example, the case in which the function α(z) in Massive IIA is α(z) = Asin ωz. – 35 – JHEP06(2018)078 This is a solution to the equation of motion if α000 =−162π2F0=−Aω3cos ωz, which implies that the mass-parameter is actually position dependent. A possible way to understand this, suggests a position dependent smearing of D8-branes. Since α00 =−ω2α, we have that the background and the Ricci scalar read, ds2= 8π√2 ωAdS7+√2πωdz2+√2π ωsin2ωz 1 + sin2ωz dΩ2,(E.1) e−2φ=e−2φ0(1 + sin2ωz), B2=π−z+sin ωz cos ωz ω(1 + sin2ωz)dΩ2, F0=−Aω3cos ωz 162π2 F2=−Aω2 81π2sin3ωz 1 + sin2ωz dΩ2, R =ωsin4ωz 4√2π12 + 100 cot2ωz + 75 cot4ωz 1 + sin2ωz  Notice that the background is non-singular. We expand this Massive IIA solution close to z→0 and we find, ds2∼8π√2 ωAdS7+√2πω dz2+z2dΩ2,(E.2) e−2φ∼e−2φ0(1 + ω2z2), B2∼ −5πω2 3z3dΩ2, F2∼ −Aω5 81π2z3dΩ2, F0∼ − Aω3 162π2. We want to think about this background as obtained by non-Abelian T-duality applied on some seed-solution. We can consider a space-time of the form AdS7×S3in Type IIB. Notice that this is not a solution to the equations of motion. Hence we need to consider a more complicated background, depending on the coordinates of S3. Importantly, notice that this background needs not be SUSY, it may be the case that the duality creates the supersymmetry. This putative background should have warp factors that can be decomposed in harmonics of S3. Consider the s-wave and perform non-Abelian T-duality on it. We insist, this s-wave should not be a solution of the equations of motion. We shall obtain, ds2=L2 AdSAdS7+L2dr2+r2 r2+ 1dΩ2,(E.3) e−2φ=e−2φ0(1 + r2), B2=µ0 r3 1 + r2dΩ2, F2=ν0 r3 1 + r2z3dΩ2, F0=f0. Where µ0, ν0, LAdS, L, f0, φ0are some constants. Consider an expansion of this background near r→0. We find that after appropriately choosing the constants, it has the form in eq. (E.2). Notice that the same result would be obtained by starting with the background described around eq. (2.6) and expanding for z→0. This is obvious as close to z= 0 the function α=a1z+z3 6and α=Asin ωz coincide for some choice of A, ω, a1. This is reminiscent of the observations in the paper [57]. In fact, there the non-Abelian T-dual of AdS5×S5(the Sfetsos-Thompson solution [51]) was considered. A solution in type IIA with a linear charge density λ=Nz is found and a completion of the geometry is proposed. Analogously, for the background defined around eq. (2.6), we have linear charge density, and for small values of the z-coordinate, the charge density (that is the rank function R(z) = −α00 ) is also linear for the background in eq. (E.1). – 36 – JHEP06(2018)078 In other words, the backgrounds in eq. 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