Chaos in classical string dynamics in γ ^ deformed Ad S 5 × T 1,1
PPreprint typeset in JHEP style. - HYPER VERSION
Chaos in classical string dynamics in ˆ γ deformed AdS × T , Kamal L. Panigrahi
Department of Physics, Indian Institute of Technology Kharagpur,Kharagpur-721 302, India and
Theory Group-DESY, Hamburg, Notkestrasse 85,D-22603 Hamburg, GermanyEmail: [email protected]
Manoranjan Samal
Department of Physics, Indian Institute of Technology Kharagpur,Kharagpur-721 302, IndiaEmail: [email protected]
Abstract:
We consider a circular string in ˆ γ deformed AdS × T , which is local-ized in the center of AdS and winds around the two circles of deformed T , . Weobserve chaos in the phase space of the circular string implying non-integrability ofstring dynamics. The chaotic behaviour in phase space is controlled by energy aswell as the deforming parameter ˆ γ . We further show that the point like object ex-hibits non-chaotic behaviour. Finally we calculate the Lyapunov exponent for bothextended and point like object in support of our first result. a r X i v : . [ h e p - t h ] M a y ontents
1. Introduction 12. The ˆ γ deformed AdS × T , background 33. The string sigma-model and circular string 44. Numerical analysis 6
5. Conclusion 12
1. Introduction
The AdS/CFT correspondence is a powerful technique that provides an interplaybetween the gauge theory without gravity and a string theory (supergravity theory)with gravity[1],[2],[3]. The most studied example is the duality between type IIBstring theory on
AdS × S and N =4 supersymmetric Yang Mills (SYM) theory in D = 4. It is particularly well understood in the strong coupling limit of the fieldtheory side. In this connection integrability on both sides of the duality has playeda key role in the understanding the duality better. In particular it has helped us ingetting close to a solution of N = 4 SYM in the planar limit [4]. The fact that bothsides of the duality are integrable in the planar limit leaves us interested in lookingat the theories more closely. Over the past few years there has been enormousamount of work devoted towards the advancement of integrability and that in turnhas opened up the possibility of looking the integrability techniques in a much widercontext, e.g. looking the theories, beyond the planar limit and in the backgroundof deformationed of AdS. In this context the semiclassical strings have played veryimportant role. Semiclassical quantization is one of the most popular approach toprobe general string backgrounds with various background fluxes. Classical solutionsand trajectories of rotating strings, and D-branes, have played an important role inunderstanding the AdS/CFT correspondence which was otherwise obscure at times.Semiclassical quantization has played a vital role in the study of BMN, [5], GKP [3]and rigidly rotating strings [6] which can all be understood as classical trajectoriesof the the rotating and pulsating string. These classical trajectories have been one1f the main ingredients of the understanding of the semiclassical AdS/CFT fromthe string theory prospective. In general, the string dynamics in curved space aredescribed by the help of 2d sigma models where equations of motion in general arenon-linear. Integrability plays an major role in finding out the classical solutions ofthe non linear equations, correlation functions, scattering amplitudes and spectrum.Therefore it is important to check the integrability of string sigma model in a specificbackground.In the context of integrability on the other hand, it is a common fact that thephase space of most mechanical systems is not integrable and thus the role of chaoticclassical trajectories has been investigated in detail in the past. In general a systemis said to be integrable if the number of degrees of freedom is same as the numberof conserved charges. String sigma model in two dimensions has infinite numberof degrees of freedom and the system is integrable on arbitrary backgrounds onlywhen it has infinite number of conserved charges which happens to be the case in AdS × S [7]. The standard way to show the integrability of 2d-sigma model inarbitrary background is to construct a lax pair which generates infinite number ofconserved charges. But to show the existence of Lax pair is quite comber some. Infactthe necessary condition for a system to be integrable is when all of its subsystem areintegrable. In other words, a system is said to be non-integrable if at least one of itssubsystem is non-integrable. Therefore the general proof of the non integrability of atwo dimensional sigma model in arbitrary background can be done by first reducingit to a 1d subsystem and then showing the 1d subsystem is non-integrable. This canbe done either by doing numerical analysis of string motion in phase space or byanalytic method using normal variational equation(NVE). This numerical approachhas been particularly useful in various cosmological and black hole backgrounds.Using numerical method it has been shown that phase space of a test circular cosmicstring in Schwarzschild black hole geometry is chaotic [8],[9],[10]. It has further beenfound analytically that the Friedmann-Robertson-Walker (FRW) cosmological modelis completely integrable only for some special value of the cosmological constant [11].The evidence of chaotic behaviour has been noticed in AdS × T , background [12]and in its Penrose limit [13]. Applying the analytic technique it has been shownthe AdS × X geometries are non-integrable, where X is in a general class of five-dimensional Einstein spaces [14]. In case of non-relativistic theories it has been shownthe integrability nature depends on dynamical critical exponent [15],[16]. Takingclassical spinning string solution in various supergravity backgrounds [17] it has beenshown the phase spaces are chaotic and hence non-integrable. More recently theintegrability of curved brane backgrounds has been studied and it is found exceptfor some specific limit the extended string motion is non-integrable while the pointlike string dynamics is always integrable [18]. Apart from these there are number ofinstances where the integrability is studied by the help of either analytical methodor numerical analysis[19],[20],[21]. Motivated by the recent interest in studying the2lassical integrability of string motion in various backgrounds and its connection withchaotic motion of the test string in generic deformed background and otherwise, westudy the motion of classical circular string in ˆ γ deformed AdS × T , background.We have shown numerically the appearance of chaos for a circular string moving inthe deformed background.The rest of the paper is organised as follows. In section 2 we write down ˆ γ deformed AdS × T , background geometry and the fields. In section 3 we studya consistent string sigma model, taking a semi-classical circular string ansatz. Wewrite down the equations of motion for the test string for the given ansatz andconstruct all the conserved charges. Section 4 is devoted to the study of chaos in theclassical string dynamics by two different techniques, namely first by looking at thePoincare section and then by studying the Lyapunov exponent. Finally, in section 5we conclude with some comments.
2. The ˆ γ deformed AdS × T , background The
AdS × T , geometry is the gravity dual of N = 1 super symmetric Yang-Millstheory, which arises from the near horizon geometry of a stack of N number of D3-branes at the tip of the conifold, where the base of the conic is T , . The metric of AdS × T , is given by ds = ds AdS + ds T , ds AdS = − cosh ρdt + dρ + sinh ρd Ω ds T , = 16 (cid:88) i =1 (cid:2) dθ i + sin θ i dφ i (cid:3) + 19 [ dψ + cos θ dφ + cos θ dφ ] . (2.1)The internal manifold T , is a five dimensional Sasaki-Einstein manifold and is thecoset space ( SU (2) × SU (2)) /U (1). Applying the TsT transformation to this givesrise to the so called ˆ γ deformed AdS × T , metric and NS-NS two forms ( b mn )[22],[23]. ds = ds AdS + G (ˆ γ ) (cid:34) (cid:88) i =1 ( G − (ˆ γ ) dθ i + sin θ i dφ i )+ 19 ( dψ + cos θ dφ + cos θ dφ ) + ˆ γ sin θ θ dψ (cid:21) . (2.2) b mn = ˆ γG (ˆ γ ) (cid:20) cos θ sin θ dφ ∧ dψ − cos θ sin θ dφ ∧ dψ + (cid:18) sin θ sin θ
36 + cos θ sin θ + cos θ sin θ (cid:19) dφ ∧ dφ (cid:21) , (2.3)3here G (ˆ γ ) − ≡ γ (cid:18) sin θ sin θ
36 + cos θ sin θ + cos θ sin θ (cid:19) . The above deformed geometry has also been achieved by making a deformation ofclassical Yang-Baxter sigma model as described in [24].
3. The string sigma-model and circular string
In this section we shall start our analysis by making the following ansatz for thecircular string ρ = 0 , θ i = θ i ( τ ) , φ = α σ, φ = α σ, ψ = ψ ( τ ) . (3.1)It shows that the string is localized at the center of the AdS whereas it extends alongthe two angles( φ , φ ) of deformed T , with winding numbers α and α respectively.Here we have chosen such type of ansatz because we can truncate 2d sigma-modelto 1d dynamical Hamiltonian system and the same time we can study its dynamicsin phase space. The 2d sigma-model action in generic background is written as S = − πα (cid:48) (cid:90) dτ dσ (cid:104) √− hh αβ g mn ∂ α x m ∂ β x n − (cid:15) αβ ∂ α x m ∂ β x n b mn (cid:105) , (3.2)where m, n are the spacetime indices. Further in conformal gauge h αβ =diag(-1,1)and as usual (cid:15) τσ = − (cid:15) στ = 1. Now we can write the Lagrangian from the action as L = − ˙ t θ + ˙ θ ) − G (ˆ γ )36 ( α sin θ + α sin θ ) − G (ˆ γ )18 ( α + α ) − G (ˆ γ )9 α α cos θ cos θ + ˙ ψ G (ˆ γ ) (cid:18)
118 + ˆ γ sin θ sin θ (cid:19) + ˆ γG (ˆ γ )54 (cid:0) α cos θ sin θ − α cos θ sin θ (cid:1) ˙ ψ. (3.3)The canonical momenta are introduced as p τm = ∂ L ∂ ( ∂ τ x m ) = √− hh τα ∂ α x n g mn − (cid:15) τβ ∂ β x n b mn . (3.4)Using canonical momenta and Lagrangian density we can get the Hamiltonian den-sity, H = − E p θ + p θ ) + G (ˆ γ )36 ( α sin θ + α sin θ ) + G (ˆ γ )9 α α cos θ cos θ + G (ˆ γ )18 ( α + α ) + (cid:16) J − ˆ γG ( γ )54 ( α cos θ sin θ − α cos θ sin θ ) (cid:17) G (ˆ γ ) (cid:16) + ˆ γ sin θ sin θ (cid:17) . (3.5)4ariation of action with respect to x m gives the following equation of motion,2 ∂ α ( √− hh αβ g km ∂ β x m ) −√− hh αβ ∂ k g mn ∂ α x m ∂ β x n − ∂ α (cid:15) αβ ∂ β x m b km + (cid:15) αβ ∂ α x m ∂ β x n ∂ k b mn = 0 (3.6)Further,the variation of action with respect to metric gives the Virasoro constraints, g mn ( ∂ τ x m ∂ τ x n + ∂ σ x m ∂ σ x n ) = 0 (3.7) g mn ( ∂ τ x m ∂ σ x n ) = 0 . (3.8)The equations of motion for t and ψ leads, respectively, to˙ t = E, (3.9)˙ ψG (ˆ γ ) (cid:18)
19 + ˆ γ sin θ sin θ (cid:19) + ˆ γG (ˆ γ )54 (cid:0) α cos θ sin θ − α cos θ sin θ (cid:1) = J, (3.10)here E and J both are constants motion. Further, the equations motion of θ and θ are non-trivial and are given by,¨ θ = − G (ˆ γ ) (cid:32) α cos θ sin θ − α α cos θ sin θ − ˆ γ sin 2 θ sin θ ˙ ψ γ α sin θ sin θ + α cos θ sin 2 θ ) ˙ ψ (cid:19) − G θ F (3.11)¨ θ = − G (ˆ γ ) (cid:32) α cos θ sin θ − α α cos θ sin θ − ˆ γ sin 2 θ sin θ ˙ ψ − ˆ γ α sin θ sin θ + α cos θ sin 2 θ ) ˙ ψ (cid:19) − G θ F (3.12)where G θ = − γ (3 + cos 2 θ ) sin 2 θ (108 + 2ˆ γ cos θ sin θ + ˆ γ cos θ sin θ + 3ˆ γ sin θ sin θ ) ,G θ = − γ (3 + cos 2 θ ) sin 2 θ (108 + 2ˆ γ cos θ sin θ + ˆ γ cos θ sin θ + 3ˆ γ sin θ sin θ ) , and F = 13 ( α + α ) + 23 α α cos θ cos θ + 16 ( α sin θ + α sin θ ) − ˙ ψ (cid:18)
13 + ˆ γ sin θ sin θ (cid:19) − ˆ γ (cid:0) α cos θ sin θ − α cos θ sin θ (cid:1) ˙ ψ. E = 16 ( ˙ θ + ˙ θ ) + G (ˆ γ )18 ( α sin θ + α sin θ ) + 2 G (ˆ γ )9 α α cos θ cos θ + G (ˆ γ )9 ( α + α ) + (cid:16) J − ˆ γG ( γ )54 ( α cos θ sin θ − α cos θ sin θ ) (cid:17) G (ˆ γ ) (cid:16) + ˆ γ sin θ sin θ (cid:17) . (3.13)The Hamiltonian is fixed to zero by Virasoro constraints. Since the equations ofmotion θ and θ are complicated, it is very difficult to find out normal variationalequation(NVE) and study the integrability analytically. We will study the problemfrom a numerical analysis by showing the appearance of chaos in the next section.
4. Numerical analysis
The non integrability nature of the string dynamics can be verified numerically byshowing chaotic behavior of the string in its phase space. There are various techniquesto show the chaotic behavior of the string dynamics in a particular background. Herewe have used two methods. First we wish to study it from the point of view ofPoincar´e section and then calculating the Lyapunov Exponent.
The string trajectories in phase space are distorted torus or the famous Kolmogorov-Arnold-Moser (KAM) torus [Fig.1]. As time evolves, the trajectories wind over thetorus shows the quasi-periodic nature of the trajectories. It will be convenient to takethe projection of the trajectories over a surface for studying their dynamics. Theseprojections over a surface is called Poincar´e section or surface of section [25],[26].6igure 1In the present case, the system has four phase space coordinates ( θ , θ , p θ , p θ ).The Virasoro constraint reduces it to a three dimensional subspace. Different initialconditions to the phase space coordinates give different tori in the phase space. Tofind different set of initial conditions we take p θ = 0 , θ = π , then keeping energya constant in equation 3.13 we vary θ to get the corresponding values of p θ . Foran extended string we take the winding numbers of both θ and θ coordinates to 1.The intersection of the tori with the surface sin θ = 0 gives distorted circles.When energy is small these tori in the phase space are distinct [Fig.2(a)]. Each colourcorrespond to different set of initial conditions. As the energy of the system increasedgradually some of the tori get deformed and destroyed by making a collection ofscattered points in the phase space [fig.2(b)-2(d)]. These distorted tori are calledcantori. At some higher value of energy all the tori get distorted and phase spacebecome chaotic[Fig.2(e)]. 7 a) (b)(c) (d)(e) Figure 2: Poincar´e section for γ = 18n Fig.3 it is observed that chaotic nature of phase space not only depends onenergy but also the deformation parameter (ˆ γ ) of AdS × T , background. Keepingenergy constant E = 0 . γ is changed, the tori get distorted as earlier case andbecomes chaotic for a higher value of ˆ γ . (a) (b)(c) Figure 3
Point like string
Let us look at the fate of the point like string in the deformed background. If wemake the winding numbers tends to zero then the string is no longer extended andit becomes point like. We observe the phase space of a point like string is orderedand non-chaotic [fig.4]. This ordered behaviour of phase space remains unchangedeven with the varying energy or the deformation parameter ˆ γ .9igure 4 Chaotic nature of the trajectories can be studied more quantitatively by the so calledLyapunov exponent. Lyapunov exponent describes sensitivity of the phase space tra-jectories to the initial conditions. It measures the growth rate between two initiallynearby trajectories. Figure 5Let us consider two initially nearby orbits, one passes through the point X andother X + ∆ X . These orbits can be thought of as parametric functions of time. If10 a) Extended string (b) Point like string Figure 6: Largest Lyapunav Exponents for (a) Extended string with initial condition θ (0) = 0 . , p θ (0) = 0 , θ (0) = 0 . , p θ (0) = 0 . θ (0) = 0 . , p θ (0) = 0 , θ (0) = 0 . , p θ (0) = 0 . X ( X , τ ) is the separation between these two orbits at a later time tau , then theLyapunov exponent is defined as λ = 1 τ ln (cid:107) ∆ X ( X , τ ) (cid:107)(cid:107) ∆ X (cid:107) (4.1)It will be useful to take the largest Lyapunov exponent which can be measured whenthe interval is very large.Λ = lim τ − > ∞ τ ln (cid:107) ∆ X ( X , τ ) (cid:107)(cid:107) ∆ X (cid:107) = lim τ − > ∞ τ (cid:88) λ i τ i (4.2)The largest Lyapunov exponent generally converges to a positive value for a physicalsystem which exhibits chaotic behaviour. If Λ is zero then it indicates the system isconservative. For a non conservative system or dissipative system the Λ convergesto a negative value [27],[28]. For extended string the largest Lyapunav exponent Fig6(a) converges to a positive value close to 3.55 which verifies extended string motionin γ deformed AdS × T , is chaotic in nature. But for point like string Fig 6(b) itsvalue converges to zero which indicates non-chaotic nature of the system. We arrivedat this conclusion by studying the problem mostly numerically.11 . Conclusion In this paper we have explicitly showed through the appearance of chaos that thestring motion in the ˆ γ deformed AdS × T , geometry is not integrable. We arrived atthis conclusion by studying the problem mostly numerically. We study numericallythe motion of the system and found it to be chaotic. Non-integrability does not,necessarily, imply the chaotic motion. However, the appearance of chaos is evidenceof the breakdown of integrability. In our study the chaotic motion of the strings isfirst seen in the Poincar´e sections and also in the phase space trajectories. We havetaken the example of a particular type of circular string and showed that numericallythat the motion is chaotic. We have shown further that as soon as the strings arereplaced with point particles the integrability restores back. Hence one can concludethat while the point like solutions are integrable, the extended string equations ofmotion are not. We have further support of this result by looking at the Lyapunovexponent and proved that the string equations of motion are non integrable whilethe point like equations are integrable. There are various things that one could lookat. First whether there is any link between the marginal deformations and the chaos.Whether all marginal deformations lead to chaotic behaviour ? Further one may askwhat happens to these classical chaos at the quantum level. Of course the stringtrajectories will lead to excitations of heavy string states. Hence in the field theoryside they would correspond to operators withe very large quantum numbers. Finally,the method of nonintegrability and chaos in classical string trajectories might helpus better in understanding the gauge/gravity duality better in terms of looking athow the nonintegrability really affects the field theory operators. Acknowledgement:
KLP would like to thank DESY theory group for hospitalityunder SFB fellowship where a part of this work is done.
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