aa r X i v : . [ h e p - t h ] J un Prepared for submission to JHEP
CCTP-2015-10CCQCN-2015-80
Turbulent strings in AdS/CFT
Takaaki Ishii and Keiju Murata Crete Center for Theoretical Physics, Department of Physics, University of Crete,PO Box 2208, 71003 Heraklion, Greece Keio University, 4-1-1 Hiyoshi, Yokohama 223-8521, Japan
E-mail: [email protected] , [email protected] Abstract:
We study nonlinear dynamics of the flux tube between an externalquark-antiquark pair in N = 4 super Yang-Mills theory using the AdS/CFT duality.In the gravity side, the flux tube is realized by a fundamental string whose endpointsare attached to the AdS boundary. We perturb the endpoints in various ways andnumerically compute the time evolution of the nonlinearly oscillating string. As aresult, cusps can form on the string, accompanied by weak turbulence and powerlaw behavior in the energy spectrum. When cusps traveling on the string reach theboundary, we observe the divergence of the force between the quark and antiquark.Minimal amplitude of the perturbation below which cusps do not form is also inves-tigated. No cusp formation is found when the string moves in all four AdS spacedirections, and in this case an inverse energy cascade follows a direct cascade. Keywords:
AdS/CFT ontents Z -symmetric quench 20 – 1 – Introduction
The gauge/gravity duality [1–3] is successfully applied to investigating strongly cou-pled gauge theories. Through this duality, it is hoped that one can access theirnontrivial aspects that are hard to be handled because of the strong coupling. Inparticular, among advantages in using gravity duals, it is worth notifying that dy-namics in time-dependent systems can be powerfully computed from time evolutionin classical gravity. Applications range over far-from-equilibrium dynamics governedby nonlinear equations, and using numerical techniques for solving them attractsmuch attention. For instance, physics of strongly coupled plasma of quarks and glu-ons at RHIC and LHC brings motivations to numerically study dual gravitationaldynamics; a series of seminal works is in Refs. [4–8].Far-from-equilibrium processes in the D3/D7 brane system dual to N = 2 super-symmetric QCD have been recently studied in Refs. [9–13], where partial differentialequations for time evolution were solved numerically. As a phenomena characteristicin non-linear dynamics, it has been found that long time evolution of the D7-branegenerates a singularity on the brane, and this is understood from the viewpoint ofweak turbulence on the D7-brane: The energy in the spectrum is transferred fromlarge to small scales [10–12]. Small scale fluctuations there correspond to excitedheavy mesons in the dual gauge theory. The turbulent behavior of the D7-brane canbe interpreted as production of many heavy mesons in the dual gauge theory, andthe singularity formation is interpreted as deconfinement of such mesons.To gain a deep insight into this kind of nonlinear dynamics in the gauge/gravityduality, in this paper, we will consider a string in AdS dual to the flux tube betweena quark-antiquark pair in N = 4 super Yang-Mills theory, and fully solve its nonlin-ear time evolution with a help of numerical techniques. This setup corresponds tofocusing on a Yang-Mills flux tube compared with the collective mesons describedin the D3/D7 system. In addition, working in this setup is simpler than using theD3/D7 system and will provide a clear understanding of the turbulent phenomenaand instabilities in probe branes in the gauge/gravity duality. To initiate the timeevolution, we will perturb the endpoints of the string for an instant. Our setup isschematically illustrated in Fig. 1. The endpoints are forced to move momentarilyand then brought back to the original locations. We will loosely use the terminology“quench” for expressing this process. This action introduces waves propagating onthe string, and we will be interested in their long time behavior where nonlinear-ity in the time evolution of the string plays an important role. When we performnumerical computations, we will use the method developed in [9], which turned outto be efficient for solving the time evolution in probe brane systems. We are alsomotivated by the weakly turbulent instability found by Bizo´n and Rostworowski [14].Our setup may give a simple playground to study that kind of phenomenon.The string hanging from the AdS boundary is one of the most typical probes– 2 – d S bounda r y Figure 1 . A schematic picture of our setup. Perturbing the endpoints induces fluctuationson the string. in the gravity dual. This gives the gravity dual description of a Wilson loop corre-sponding to the potential between a quark and an antiquark [15, 16]. Although thepotential in a conformal theory is different from that in real QCD, linear confine-ment can be realized in nonconformal generalization [17]. In finite temperatures, astring extending to the AdS black hole corresponds to a deconfined quark [18, 19],and this has been utilized for studying the behavior of moving quarks in Yang-Millsplasma [20–26]. Moving quark-antiquark pairs were also considered [27, 28]. In far-from-equilibrium systems, holographic Wilson loops have also been used as probes forthermalization [29, 30]. Nevertheless, veiled by these applications to QGP, nonlinear(and non-dissipative) dynamics of the probe hanging string in AdS has not been shedlight on so much. Some analytic solutions of non-linear waves on an extremal surfacein AdS have been studied in Ref. [31]. Notice that that configuration corresponds toa straight string in the Poincare coordinates.When it comes to nonlinear dynamics of a string, formation of cusps would beprimarily thought of. In fact, it is well known in flat space in the context of closedcosmic strings that cusp formation is ubiquitous [32]. Cusp formation of fundamentalstrings ending on D-branes has been also found in Ref. [33]. We will turn our attentionto whether there is such formation of cusps also in AdS. The organization of the rest of this paper is as follows. We start from reviewingthe static solution and linear perturbations in Section 2 and 3, where we introducea parametrization convenient for our use. In Section 4, we explain the setup forour time dependent computations. We introduce four patterns of quenches that weconsider, derive the evolution equations and the boundary conditions, prepare initial A development of a cusp in a decelerating trailing string was discussed in Ref. [34]. – 3 –ata, and explain measures for evaluating the time evolution. Sections 5, 6, and 7 arereserved for numerical results: In Section 5, we discuss two of the four quenches wherethe oscillations of the boundary flux tube are restricted to compression waves andtherefore we call them longitudinal. We evaluate cusp formation, turbulent behaviorin the energy spectrum, and the forces acting on the quark endpoints. In Section 6,we show results of a quench where the flux tube oscillates in one of its transversedirections. Finally, in Section 7, we examine the last quench where the motion of theflux tube is in all three spatial directions of the boundary 3+1 dimensions. Section 8is devoted to summary and discussion. In appendices, we explain details for thenumerical computations and derive some formulae used in the main text.
We briefly review the holographic calculation of the static quark-antiquark potentialin the near horizon limit of extremal D3-branes [15, 16]. The background metric isAdS × S , ds = ℓ z (cid:0) − dt + dz + d x (cid:1) + ℓ d Ω , (2.1)where ℓ = 4 πg s N c α ′ and x ≡ ( x , x , x ). We consider a rectangular Wilson loopwith the quark-antiquark separation L along the x -direction where the quark andantiquark are located at x = ± L/
2. The dynamics of the string is described byNambu-Goto action, S = − πα ′ Z dτ dσ √− γ , (2.2)where γ = det( γ ab ) and γ ab is the induced metric on the string. It is convenientto take a static gauge where the worldsheet coordinates ( τ, σ ) coincide with targetspace coordinates as ( τ, σ ) = ( t, z ). The static solution is then described by a singlefunction x = X ( z ). The Nambu-Goto action becomes S = − √ λ π Z dtdz z p X ′ , (2.3)where λ = 4 πg s N c is the ’t Hooft coupling.Solving the equation of motion of X ( z ) gives the bulk string configuration. Let z = z specify the bulk bottom of the string reached at x = 0, where the regularboundary condition ∂ x z = 0 is imposed. The embedding solution is given by X ( z ) = ± z Z z/z dw w √ − w = ± z (cid:2) Γ + F ( z/z ; i ) − E ( z/z ; i ) (cid:3) , (2.4) We will use small letters for coordinates and capital letters for functions specifying the stringposition. – 4 – .50.40.30.20.10-0.1-0.2-0.3-0.4-0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Figure 2 . The string profile of the static solution. where Γ ≡ √ π / / Γ(1 / ≃ . F and E are the incomplete ellipticintegrals of the first and second kinds defined as F ( x ; k ) = Z x dt p (1 − t )(1 − k t ) , E ( x ; k ) = Z x dt r − k t − t . (2.5)Setting z = 0 in (2.4), we obtain L/ z Γ . In Fig. 2, we show the profile of thestatic string in the ( z, x )-plane. We will use this solution as the initial configurationfor our time evolution.It is also known that the dependence of the potential energy on L is Coulomb.The energy is evaluated from the on-shell action, which in general diverges at theboundary, but this divergence can be regulated by comparing with the divergingenergy of two strings straightly extending to the Poincare horizon. The regularizedenergy is then given by E reg = − π √ λ Γ(1 / L . (2.6)In this sense, the quark-antiquark potential in the AdS background does not cor-respond to the confining potential of real QCD. This, however, is considered as ahandy playground for testing nonlinear evolution in the gauge/gravity duality.
Linearized fluctuations and stability of holographic quark-antiquark potentials havebeen studied in Refs. [35–38]. Here, we solve the linearized fluctuations of the staticsolution in coordinates convenient for our use. In (2.4), we parametrized the locationof the string in terms of the z -coordinate. In this coordinate, however, the staticsolution X ( z ) becomes multi-valued, and this may not be suitable for considering– 5 – (a) Profiles of special functions (b) Polar-like coordinates Figure 3 . (a) The profiles of the functions f ( φ ) and g ( φ ). (b) r = const. and φ = const.surfaces are shown in the ( z, x )-plane. linear perturbations. Instead, we introduce new polar-like coordinates ( r, φ ) in whichthe static embedding is expressed by single-valued functions, z = rf ( φ ) , x = rg ( φ ) , (3.1)where we define the functions f and g as f ( φ ) ≡ sn( φ ; i ) , (3.2) g ( φ ) ≡ − Z φβ / dφ ′ f ( φ ′ ) = ( φ − E (sn( φ ; i ); i ) + Γ ( φ ≤ β / φ + E (sn( φ ; i ); i ) − Γ − β ( φ > β / , (3.3)where sn( x ; k ) is a Jacobi elliptic function defined as the inverse function of F ( x ; k )given in Eq. (2.5): F (sn( x ; k ); k ) = x . For k = i = √−
1, the Jacobi elliptic functionhas roots at x = β n ( n ∈ Z ), where β = π/ (2Γ ) ≃ . f and g , f ′ ( φ ) + g ′ ( φ ) = 1 . (3.4)With these functions f and g , the static embedding (2.4) is simply given by r = z .The profiles of the functions f ( φ ) and g ( φ ) are shown in Fig. 3(a). We also depictthe r = const. and φ = const. surfaces in the ( z, x )-plane in Fig. 3(b).Using t and φ as the worldsheet coordinates, we can describe the dynamics ofthe string in terms of three functions, r = R ( t, φ ) , x = X ( t, φ ) , x = X ( t, φ ) . (3.5)We consider perturbations around the static solution, R = z , X = X = 0, as R ( t, φ ) = z { χ ( t, φ ) } , X ( t, φ ) = z χ ( t, φ ) , X ( t, φ ) = z χ ( t, φ ) , (3.6)– 6 – -1-0.8-0.6-0.4-0.200.20.40.60.811.2 0 0.5 1 1.5 2 2.5 (a) Longitudinal modes -1-0.8-0.6-0.4-0.200.20.40.60.811.2 0 0.5 1 1.5 2 2.5 (b) Transverse modes Figure 4 . Normal mode frequencies of the longitudinal and transverse modes. Theeigenfunctions for n = 1 , , , where χ i ( i = 1 , ,
3) are dimensionless perturbation variables. We will refer to χ as the longitudinal mode and χ i ( i = 2 ,
3) as the transverse modes. Then, in thesecond order in χ , the Nambu-Goto action becomes S = √ λ πz Z dtdφ " h ( φ ) ( z ˙ χ − χ ′ ) + X i =2 , f ( φ ) ( z ˙ χ i − χ ′ i ) , (3.7)where we define ˙ ≡ ∂ t and ′ ≡ ∂ φ , and introduce h ( φ ) ≡ [( g/f ) ′ f ] . To derivethe above expression, we used the relation of f and g (3.4) and omitted the totalderivative terms. The equations of motion for ( χ , χ , χ ) are( ∂ t + H ) χ = 0 , H ≡ − z h ∂ φ h∂ φ , ( ∂ t + H ′ ) χ i = 0 , H ′ ≡ − f z ∂ φ f ∂ φ ( i = 2 , . (3.8)Operators H and H ′ are Hermitian under the inner products( α, β ) ≡ Z β dφ h ( φ ) α ( φ ) β ( φ ) , ( α, β ) ′ ≡ Z β dφ f ( φ ) α ( φ ) β ( φ ) , (3.9)respectively. We denote the eigenvalues and eigenfunctions of H and H ′ as { ω n , e n ( φ ) } and { ω ′ n , e ′ n ( φ ) } , respectively. These are labeled by integers n ≥ ω n and ω ′ n . The eigenfunctions are orthonormalized as ( e n , e m ) = δ nm and( e ′ n , e ′ m ) ′ = δ nm .It is easy to check the linear stability of the static embedding against the longi-tudinal perturbation as ω n = ( e n , H e n ) = 1 z Z β dφ h ( ∂ φ e n ) ≥ . (3.10)– 7 –n the same way, the stability against the transverse perturbations can be alsochecked, ω ′ n ≥ ω n , ω ′ n ∝ n for n → ∞ .In fact, from the WKB analysis, we can obtain z ω n ≃ ( n + 1) for n → ∞ [35, 36].Our numerical results are consistent with the WKB approximation. The main focus of this paper is to study nonlinear dynamics of the string, where wemake use of numerical techniques for solving the time evolution. In this section, thesetup for this is prepared.
We consider AdS × S (2.1) as the background spacetime, and take the static solu-tion (2.4) as the initial configuration. We then consider “quench” on the endpoints ofthe string: We move their positions momentarily and put them back to the originalpositions. A schematic picture of this setup is depicted in Fig. 1.Let us denote the two endpoints of the string as x q ( t ) and x ¯ q ( t ), correspondingto the locations of the quark and antiquark, respectively. In this paper, we considerthe following four kinds of quenches on x q ( t ) and x ¯ q ( t ): (i) Longitudinal one-sided quench : x q ( t ) = (cid:18) L ǫLα ( t ) , , (cid:19) , x ¯ q ( t ) = (cid:18) − L , , (cid:19) , (4.1)where α ( t ) is a compactly supported C ∞ function defined by α ( t ) = ( exp (cid:2) (cid:0) ∆ tt − ∆ t − ∆ tt + 4 (cid:1)(cid:3) (0 < t < ∆ t )0 (else) . (4.2)The profile of this function is shown in Fig. 5. The flux tube vibrates in itslongitudinal direction, and motions are not induced in the transverse directionsby this quench. Thus, in this case, the motion of the string is restricted in(2+1)-dimensions spanned by ( t, z, x ). (ii) Longitudinal Z -symmetric quench : x q ( t ) = (cid:18) L ǫLα ( t ) , , (cid:19) , x ¯ q ( t ) = (cid:18) − L − ǫLα ( t ) , , (cid:19) . (4.3)We simultaneously quench both endpoints in the opposite directions along theflux tube. The string motion induced by this quench is invariant under x →− x and restricted in the same (2+1)-dimensions as (i).– 8 – Figure 5 . A compactly supported C ∞ function for quench. (iii) Transverse linear quench : x q ( t ) = (cid:18) L , ǫLα ( t ) , (cid:19) , x ¯ q ( t ) = (cid:18) − L , , (cid:19) . (4.4)We shake one of the endpoints along x -direction. String fluctuations in thisdirection are induced by the quench but not in x -direction. Thus, the stringoscillates in (3+1)-dimensions spanned by ( t, z, x , x ). (iv) Transverse circular quench : x q ( t ) = (cid:18) L , ǫLα ( t ) , ± ǫL p α ( t )(1 − α ( t )) (cid:19) , x ¯ q ( t ) = (cid:18) − L , , (cid:19) , (4.5)where we choose the upper and lower signs for t ≤ ∆ t/ t > ∆ t/
2, respec-tively. The orbit of x q is a circle given by ( x − ǫL/ + x = ( ǫL/ . Stringfluctuations along both x - and x -directions are induced by this quench, andhence the string moves in all (4+1)-dimensions spanned by ( t, z, x , x , x ).In Fig. 6, we show schematic pictures of the quenches (i)-(iv). These patterns arechosen to represent typical string motions, particularly with different dimensionality.String dynamics is specified by two parameters ǫ and ∆ t once we choose a quenchtype.In this paper, we will take modest values for the amplitude of the quench ( ǫ ∼ .
01) since we are interested in nonlinear evolution starting from small deviationfrom the linear theory and focus on weak turbulence and cusp formation driven bythe nonlinearity. For a large value of ǫ , we expect some effect similar to the overeagereffect found in Ref. [9]: The string will be able to plunge into the Poincare horizon z = ∞ because of the strong perturbation. Such strong quenches will be studied indetail elsewhere [39]. – 9 – ongitudinal Z - symmetricquenchLongitudinal one-sided quench Transverselinear quench Transversecircular quench Figure 6 . Schematic pictures of the quenches we will consider.
To calculate the time evolution on the string worldsheet, we find it convenient to usedouble null coordinates. With worldsheet coordinates ( u, v ), the string position isparametrized as t = T ( u, v ) , z = Z ( u, v ) , x = X ( u, v ) . (4.6)Using these expressions with Eq. (2.1), we obtain the induced metric as γ uu = ℓ Z ( − T ,u + Z ,u + X ,u ) , γ vv = ℓ Z ( − T ,v + Z ,v + X ,v ) ,γ uv = ℓ Z ( − T ,u T ,v + Z ,u Z ,v + X ,u · X ,v ) . (4.7)The reparametrization freedom of the worldsheet coordinates allows us to imposethe double null condition on the induced metric as C ≡ γ uu = 0 , C ≡ γ vv = 0 . (4.8)Notice that these conditions do not fix the coordinates completely: There are residualcoordinate freedoms, u = u (¯ u ) , v = v (¯ v ) . (4.9)These will be fixed by boundary conditions and initial data.In the double null coordinates, the Nambu-Goto action (2.2) becomes S = − πα ′ Z dudv p γ uv − γ uu γ vv = 12 πα ′ Z dudvγ uv = ℓ πα ′ Z dudv Z ( − T ,u T ,v + Z ,u Z ,v + X ,u · X ,v ) , (4.10)– 10 –here at the second equality we eliminate the square root in the action using thedouble null conditions (4.8). (Note that γ uv is negative.) From the action, we obtainthe evolution equations of the string as T ,uv = 1 Z ( T ,u Z ,v + T ,v Z ,u ) ,Z ,uv = 1 Z ( T ,u T ,v + Z ,u Z ,v − X ,u · X ,v ) , X ,uv = 1 Z ( X ,u Z ,v + X ,v Z ,u ) . (4.11)Using these, we find that the constraints (4.8) are preserved under the time evolution: ∂ v C = ∂ u C = 0 . (4.12)Hence if we impose C = C = 0 on the initial surface and the boundaries, theconstraints (4.8) are automatically satisfied in the whole computational domain.Our numerical method for solving the evolution equations is briefly summarizedin Appendix A and its numerical error is estimated in Appendix B. For more detailsof the numerical method, see also Appendix A in [9]. On the worldsheet, there are two time-like boundaries that correspond to the twoendpoints of the string attaching on the AdS boundary. We need boundary conditionsthere. Using the residual coordinate freedoms (4.9), we can fix the locations of theboundaries to u = v and u = v + β . The boundary conditions for the spatial partsof the target space coordinates are given by Z | u = v = 0 , X | u = v = x q , Z | u = v + β = 0 , X | u = v + β = x ¯ q . (4.13)We also need boundary conditions for T . To derive them, we solve Eq. (4.11) and(4.8) near the boundaries. Defining τ = u + v and σ = u − v ( τ ∈ ( −∞ , ∞ ), σ ∈ [0 , β ]), we obtain the asymptotic solutions around σ = 0 as T = t ( τ ) + (cid:20)
12 ¨ t − γ v · a ˙ t (cid:21) σ + v · x σ + · · · ,Z = ˙ t γ σ + (cid:20) ... t γ − γ v · a ¨ t ˙ t − γ γ a + v · j ) ˙ t (cid:21) σ + · · · , X = x q ( τ ) + (cid:20) v ¨ t − (cid:18) γ − (cid:19) a ˙ t (cid:21) σ + x ( τ ) σ + · · · . (4.14)where ˙ = d/dτ , v = ˙ x q / ˙ t , a = ˙ v / ˙ t , j = ˙ a / ˙ t and γ = 1 / √ − v . The asymptoticsolutions near σ = β can be given by replacing σ → β − σ and x q → x ¯ q in the above– 11 –xpressions. From the second equation in Eq. (4.14), we have ∂ σ Z | σ =0 = q ˙ t − ˙ x q .Using this, we obtain T ,τ | σ =0 = q ( Z ,σ | σ =0 ) + ˙ x q , T ,τ | σ = β = q ( Z ,σ | σ = β ) + ˙ x q . (4.15)These equations determine the time evolution of T at the boundaries σ = 0 , β .Their numerical implementation is explained in Appendix A.For consistency, the speed of the quark endpoints during the quench should beslower than light, | v | <
1. Otherwise, the Lorentz factor γ becomes imaginary.Solving this condition, we obtain constraints for the quenches (i)-(iii), ǫL ∆ t < (19 − √ e √ − p √ − ≃ . , (4.16)and for the transverse circular quench (iv), ǫL ∆ t < √ ≃ . . (4.17)The parameter values examined in this paper satisfy these conditions. Before the quenches are applied, we assume that the string is static, namely, we usethe static solution (2.4) as the initial configuration. For numerical computations,we need initial data written in the ( u, v )-coordinates. Substituting Eq. (2.4) intoEqs. (4.8) and (4.11), we can express the static solution in terms of ( u, v ) as T ( u, v ) = z [ φ ( u ) + φ ( v )] , Z ( u, v ) = z f ( φ ( u ) − φ ( v )) ,X ( u, v ) = z g ( φ ( u ) − φ ( v )) , X ( u, v ) = X ( u, v ) = 0 , (4.18)where φ and φ are arbitrary functions associated with the residual coordinatefreedom (4.9), and the functions f and g are defined in Eqs. (3.2) and (3.3).Locating the initial surface at v = 0 on the worldsheet, we parametrize the initialconfiguration as T ( u,
0) = z u , Z ( u,
0) = z f ( u ) ,X ( u,
0) = z g ( u ) , X ( u,
0) = X ( u,
0) = 0 , (4.19)where we set the free functions φ ( u ) = u and φ (0) = 0 so that the boundaries areat u = 0 , β . If the string endpoints are not perturbed, our numerical calculationsdescribe the static evolution of the exact solution (4.18). In the non-linear dynamics of the string, we expect to observe formation of cusps.For detailed analyses, we will in particular use the following quantities for evaluation.– 12 – .5.1 Cusp formation
Let us consider the profile of the string on a surface with t =constant. There isa cusp if the string does not change its target space position when the parameteron the string is varied. Using the u -coordinate as a parameter on the string in the t =constant surface, we obtain the conditions for the cusp formation as ∂ u X I | t =const =0 where X I ≡ ( Z, X i ) and i = 1 , ,
3. These conditions are rewritten as J I ≡ T ,u X I,v − T ,v X I,u = 0 . (4.20)In our numerical computations, we monitor the roots of J I , which typically appear ascurves on the ( u, v )-plane. If these curves overlap at a point, we find cusp formation,and if the overlap continues in the time evolution, it is implied that the cusps continueto exist.As an obvious corollary, if (4.20) is satisfied, there is a necessary condition that X I ,u X J ,v − X I ,v X J ,u = 0 (4.21)is also satisfied. This condition is conveniently utilized for a consistency check of thecusp formation detected by (4.20).
We will also study the energy spectrum of the non-linear fluctuations of the string be-cause, from the spectrum, it is expected to find weak turbulence on the string. Oncea dynamical solution ( T ( u, v ) , Z ( u, v ) , X ( u, v )) is calculated, we can convert it to thepolar-like coordinates introduced in Eq. (3.1): r = R ( u, v ) and φ = Φ( u, v ). Elim-inating the worldsheet coordinates ( u, v ) from ( T ( u, v ), R ( u, v ), Φ( u, v ), X ( u, v ), X ( u, v )), we can express the dynamical solution using target space coordinates ( t, φ )as r = R ( t, φ ) , x = X ( t, φ ) , x = X ( t, φ ) . (4.22)As in Eq. (3.6), we define the non-linear version of the “perturbation” variablesˆ χ , ˆ χ , ˆ χ as ˆ χ ( t, φ ) = R ( t, φ ) z − , ˆ χ i ( t, φ ) = X i ( t, φ ) z ( i = 2 , . (4.23)We then decompose ˆ χ , ˆ χ and ˆ χ with the eigenfunctions of the linear theory e n ( φ )and e ′ n ( φ ), which were introduced below Eq. (3.8), asˆ χ = ∞ X n =1 c n ( t ) e n ( φ ) , ˆ χ i = ∞ X n =1 c in ( t ) e ′ n ( φ ) ( i = 2 , . (4.24) Solving Z ( u, v ) /X ( u, v ) = f ( φ ) /g ( φ ) for φ , we have φ = Φ( u, v ). Then, we can obtain R ( u, v )from R ( u, v ) = Z ( u, v ) /f (Φ( u, v )). – 13 –sing the mode coefficients c n and c in , we define the energy contribution from the n -th mode ε n ( t ) and the total energy in terms of the linear theory ε ( t ) as ε n ( t ) = √ λz π " ˙ c n + ω n c n + X i =2 , ( ˙ c in + ω ′ n c in ) , ε ( t ) = ∞ X n =1 ε n . (4.25)The quantify ε n is conserved in linear theory. Therefore, if we find time dependencein ε n , it is a fully non-linear effect. Note that the total energy ε defined in the lineartheory is also time dependent in the non-linear theory although the time dependenceis suppressed by the amplitude of the quench: ˙ ε/ε = O ( ǫ ). Since we consider onlysmall ǫ in this paper ( ǫ ∼ . n = 50 forevaluating ε in Eq. (4.25). Its cutoff dependence is also not essential for our followingarguments on the energy spectrum on each t -slice. From the dynamical solution of the string, we can read off the time dependence ofthe forces acting on the quark and antiquark located at x = x q and x ¯ q . The forceacting on the quark can be evaluated from the on-shell Nambu-Goto action as h F i ( t ) i = √ λ π γ − ( δ ij + γ v i v j ) ∂ z X j | z =0 , (4.26)where γ = 1 / p − ( d x q /dt ) . The same formula can be applied to the force acting onthe antiquark h ¯ F ( t ) i by replacing x q → x ¯ q . The derivation of this expression is sum-marized in Appendix D. For our numerical analysis, it is convenient to rewrite thisexpression in terms of the ( τ, σ )-coordinates. Using the asymptotic expansions (4.14),we obtain h F ( t ) i = 3 √ λ π γ x ˙ t . (4.27)We need to extract the third order coefficient x from our numerical data. For thispurpose, it is convenient to define Y ≡ X − x q − ( T − t ) v − γ (cid:20) γ ( v · a ) v − (cid:18) γ − (cid:19) a (cid:21) Z . (4.28)Using Eq. (4.14), we find that this function behaves as Y ≃ [ x − ( v · x ) v ] σ nearthe boundary. To evaluate x , we fit numerical data for Y by a function c σ in σ ∈ [0 , . From this section to Section 7, we discuss results of numerical computations for ourquenches (i)-(iv). In this section, we firstly treat the longitudinal one-sided quench(i) and then discuss the longitudinal Z -symmetric quench (ii).– 14 – .1 Cusp formation We start from the longitudinal one-sided quench (4.1). In Fig. 7, we show snapshotsof the time evolution of the string for ∆ t/L = 2 and ǫ = 0 .
03. The left panel isjust after the quench, t/L = 2 , . , .
4. We observe that perturbations are inducedon the string by the quench. Late time behavior is shown in the right panel for t/L = 7 , . , .
4, where it is seen that cusps are formed on the string. The time forthe cusp formation is evaluated as t/L ∼ z, x )-plane looks singular on top of the cusps, the fields T ( u, v ), Z ( u, v ) and X ( u, v ) are indeed smooth on the ( u, v )-plane. Therefore, even afterthe formation of the cusps, the time evolution can continue without a breakdown ofnumerical computations. Physically, however, finite- N c effects can be important atcusp singularities, and the time evolution after the cusp formation may be consideredas unphysical without such corrections. We will discuss possible finite- N c effects atcusps in section 8. For the present, the dynamics of cusps are discussed withouttaking into account these corrections. We find that the cusps appear as a pair. (Thecusps at t/L = 7 . t/L = 2 is fixed and ǫ is varied. In the left panel, the times for the cusp formationfor each ǫ are plotted, and the corresponding locations in the φ -coordinate are shownin the right panel. The time for the cusp formation becomes longer as the quenchamplitude becomes smaller. It is also seen that the cusp formation times are mildlydiscretized, as well as the locations of the formation away from the boundary. Thisdiscretization indicates that the cusps might be difficult to form near the boundary.The times and locations of the cusp formation tend to degenerate in the veryearly time before the first reflection of the initial perturbation wave at X = − L/ t/L ∼ .
5. Features in this region seem to be influenced by the largeness ofthe quench and would be different from late time dynamics. Quenches with largeamplitudes will be reported elsewhere [39].The cusp formation time becomes longer as the amplitude gets smaller. A naturalquestion then is if the cusps can be formed for any small amplitude ǫ . As ǫ decreases,however, the variations in the fields are tinier and tinier, and eventually it becomesnumerically difficult to use the cusp condition (4.20) for finding the cusp formation.To obtain a reasonable inference, we fit results and extrapolate to smaller ǫ . A resultis given in Fig. 9, where we fit data points at ǫ ≥ .
01 and t cusp /L ≥
10 by a The points marked with the purple boxes were not derived from directly computing Eq. (4.20)because of difficulty in small ǫ . We are, however, able to find a suspect of cusp formation by lookingat the plot of the necessary condition (4.21). We do not use these points for the fit, but they look – 15 – (a) Just after the quench (b) After the cusp formation Figure 7 . Snapshots of the string time evolution for the longitudinal one-sided quenchwith ∆ t/L = 2, ǫ = 0 . (a) Cusp formation time (b) Corresponding formation points Figure 8 . Cusp formation for various ǫ when ∆ t/L = 2. The formation points in (b)correspond to those at the same time in (a). In the plots, the points are computed at every0.001 variation of ǫ for t cusp /L < . t cusp /L > polynomial a ǫ + b ǫ / + c . From the extrapolation, we find that there is a criticalvalue ǫ crit below which the cusp formation time would be infinity. In Fig. 9, we obtain ǫ crit ≃ . × − .We repeat this procedure to estimate the critical ǫ for different ∆ t/L and seehow it changes. For each ∆ t/L , we fit data by a ǫ + b ǫ / + c and extrapolate tothe limit of infinite cusp formation time to read off the value of critical ǫ . Resultsare shown in Fig. 10. We find that the critical value scales as (∆ t/L ) . A fit of ourresults is ǫ crit = 9 . × − (∆ t/L ) . Note that this scaling may be altered if ∆ t/L becomes very long and the wavelength of the induced wave is comparable with the consistent with the fit. This fitting function is chosen by our intuition, and other choices for extrapolation could beutilized. For instance, the data can be also fit with a ǫ + b ǫ + c , and a qualitatively consistentresult can be obtained. – 16 – igure 9 . Cusp formation in small ǫ for ∆ t/L = 2. The dashed line is a fit of the redpoints. The points marked with the boxes are not used for the fit but are consistent withthe fit curve. The fit curve reaches ( t cusp /L ) − = 0 at ǫ ≃ . × − . Figure 10 . The dynamical phase diagram for cusp formation. The red points areestimated values of critical ǫ at different ∆ t/L . The dashed curve is a fit given by ǫ = 9 . × − (∆ t/L ) . Cusps are formed above this curve, but not below. length of the hanging string. The critical amplitude in such a case, however, will bealso big. For this reason, we do not focus on larger ∆ t/L in this paper.The cusp formation found here is similar to the weakly turbulent instability inthe global AdS [14]: Small fluctuations in that AdS propagate between the boundaryand the center and, eventually, the perturbations collapse into a black hole afterseveral bounces. A difference between our cusp formation and that AdS instabilityis their critical amplitude of initial perturbations. The AdS instability occurs forarbitrary small initial perturbations while our cusps are formed only for ǫ > ǫ crit > ω n and ω ′ n , are commensurable onlyfor n → ∞ . Because of the non-commensurable spectrum of the string, we needfinite perturbations for the cusp formation.– 17 – .2 Energy spectrum in the non-linear theory Given the formation of cusps, we look into the time dependence in the energy spec-trum following the procedure described in Section 4.5.2. For samples, we focus on thefollowing four cases: (a) ∆ t/L = 2, ǫ = 0 . t/L = 2, ǫ = 0 .
01, (c) ∆ t/L = 2, ǫ = 0 .
03, and (d) ∆ t/L = 4, ǫ = 0 .
07. With the parameters (a), cusps do not form onthe string, while cusps are created for (b), (c) and (d). In Fig. 11, we show the timedependence of the energy spectra for these parameters. The dashed curves are theenergy spectra computed in the linear theory; see Appendix C for the calculations.Although the spectra are defined only for integer n , for visibility of the plot theseresults are generalized to continuous n .High frequencies in the energy spectra are suppressed when the cusps are notformed. In Fig. 11(a), the spectrum can be well approximated by the linear theoryjust after the quench ( t/L = 2). Although it slightly deviates from the linear theoryas the time increases, we do not find any remarkable change in the late time.In contrast, the energy spectra show power law behaviors in the cases of cusp for-mation. In Fig. 11(b), although the spectrum can be well approximated by the lineartheory just after the quench, we see the growth of the spectrum in high frequencies astime passes, apparently because of the nonlinearity in the time evolution equations.In Fig. 11(c) and 11(d), the spectra deviate from those in the linear theory even at t = ∆ t , and this indicates that the nonlinearity evolves even in the quenching time,0 < t < ∆ t . For these three cases, we find a direct energy cascade: The energyis transferred to higher n -modes during the time evolution. Eventually, power lawspectra are observed, and these behaviors persist until the time of cusp formation.(See magenta points.) In particular, as seen in 11(b), the time for reaching thepower law behavior can be rather earlier than that for the cusp formation, and oncerealized, the behavior lasts until the cusps are formed. Hence, the cusp formationon the string can be regarded as a variation of weak turbulence [14]. We fit the power law spectra by ε n ∝ n − a . Just before the cusp formation, weobtain a = 1 . ± . a = 1 . ± .
128 and a = 1 . ± .
017 for (b), (c) and(d), respectively. The exponents are distributed around a ∼ . Following the procedure in Section 4.5.3, we compute the forces acting on the quarkand antiquark as functions of time. Since the string motion is now restricted in the After the cusp formation, the function R ( t, φ ) becomes multi-valued and the energy spectrumis ill-defined. Therefore, we only show spectra before the cusp formation. The string is smooth before the cusp formation, and the energy spectrum ε n must fall off fasterthan any power law function as n → ∞ . Hence, it is indicated that the power law spectrum willbe no longer maintained at n ≫
50, while many numerical efforts are necessary for computing thespectrum in such a region. The AdS weak turbulence was found to be characterized by a Kolmogorov-Zakharov scaling [40]. – 18 – inear theory e- e- e- (a) ∆ t/L = 2, ǫ = 0 .
005 (no cusp) linear theory e- e- e- (b) ∆ t/L = 2, ǫ = 0 .
01 (cusp) linear theory e- e- e- (c) ∆ t/L = 2, ǫ = 0 .
03 (cusp) linear theory e- e- e- (d) ∆ t/L = 4, ǫ = 0 .
07 (cusp)
Figure 11 . Time dependence of the energy spectrum. For the parameters (a), cuspsdo not form on the string. For the parameters (b), (c) and (d), cusps are created on thestring. The dashed curves are the energy spectra computed in the linear theory. Redpoints are spectra just after the quench. Magenta points correspond to the time for cuspformation. For (b), (c) and (d), the energy is transferred to higher modes as time increases,and eventually power law spectra are reached. In (b), light blue points indicate the timewhen the spectrum realizes the power law, which lasts until the cusp formation. We fit themagenta points by ε ∝ n − a , and the results are plotted with dotted lines. ( z, x )-plane, only the x -components of the forces are non-zero. In Fig. 12, we plotthe forces h F ( t ) i and h ¯ F ( t ) i for ǫ = 0 .
005 and ∆ t/L = 2. For these parameters,cusps do not form on the string. Figure 12(a) is for the early stage in the timeevolution (0 ≤ t/L ≤ ≤ t/L ≤ h F ( t ) i < h ¯ F ( t ) i > (a) 0 ≤ t/L ≤ - (b) 0 ≤ t/L ≤ Figure 12 . Time dependence of the forces acting on the quark and the antiquark for thelongitudinal one-sided quench with ǫ = 5 . × − and ∆ t/L = 2. In this case, cusps donot form on the string. For the parameters where the cusps form on the string, the time evolution of theforces is different from the previous example. In Fig. 13, we show h F ( t ) i and h ¯ F ( t ) i for ǫ = 1 . × − and ∆ t/L = 2, where figures (a) and (b) are for 0 ≤ t/L ≤ ≤ t/L ≤
30, respectively. In figure (b), we take the absolute values ofthe forces, and plot in the log scale in the vertical axis. We find that the pulse-likeoscillations are getting sharp and amplified as the time increases. The forces canchange the sign because of their large oscillations, and this implies that the forcebetween the quark and antiquark can be repulsive temporarily. Eventually, when acusp arrives at the boundary after the cusp formation, the force diverges.A mathematical explanation why the forces diverge after the cusp formationis given as follows. Near the AdS boundary, the time dependent solution is wellapproximated by Eq. (4.14). The conditions for the cusp are then given by ∂ σ Z = 0and ∂ σ X = 0 as discussed in Section 4.5.1. The latter condition is automaticallysatisfied near the boundary because of ∂ σ X ∼ σ , while the former gives ˙ t = 0.In Eq. (4.27), the denominator has ˙ t , and it appears that the numerator does notcancel the zero in the denominator. Hence, it is natural that the force diverges whenthe cusp arrives at the boundary. Z -symmetric quench When the two endpoints of the string are simultaneously quenched in the oppositedirections with the same amplitude, there is a new contribution from the one-sidedquench that the propagating waves collide at the Z -symmetric point X = 0, andcusps are expected to form on the collision. This case is equivalent to impose theNeumann boundary condition at X = 0. Such a condition is typically imposed inprobe D-brane embeddings. We would like to emphasize that understanding thedifference between this case and the case without the Neumann condition would be– 20 – (a) 0 ≤ t/L ≤ e +0061e+0081e+010 0 5 10 15 20 25 30 (b) 0 ≤ t/L ≤ Figure 13 . Time dependence of the forces acting on the quark and the antiquark for thelongitudinal one-sided quench with ǫ = 1 . × − and ∆ t/L = 2. For these parameters,cusps form on the string. Pulse-like oscillations in the forces are getting amplified as timeincreases and diverge eventually. important for distinguishing mechanisms for cusp formation whether the cusps areformed spontaneously as discussed in previous sections or formed with a help of thecollisions or the Neumann condition.We investigate the cusp formation by computing the condition (4.20). Resultsare shown in Fig. 14 for varying ǫ with ∆ t/L = 2 fixed. The left panel shows thetimes for (4.20) to be satisfied for the first time, and the corresponding φ -coordinatesare plotted in the right panel. We find that there are two kinds of cusp formationand wave collisions at the Z -symmetric point: One is cusp formation by the wavecollision, and the first cusp formation in this case is marked with red points in theplots, since in this case these cusps disappear once the colliding waves pass. Theother is that cusps are formed on the propagating waves in the same way as inthe one-sided quench, and the cusp formation for this case is marked with greentriangles. The times for the cusp formation are clearly discretized, and the locationsare concentrated to φ = β /
2. The small change in the formation time is becausethe propagating speed of the wave slightly varies for different ǫ .Closely looking at the results in evaluating (4.20), we find that even numbers ofcusps are created for the first case (red points). In the bulk coordinates, the cuspformation is seen at very close to φ = β / φ = β /
2, we find that a pair of cusps whose orientations are oppositeare created, and then these cusps pair-annihilate shortly when the waves pass by.Hence, the cusps created by the collision do not propagate away from that point, andtherefore these instantaneous cusps are not observed in the force at the boundary. Inthis case, other formation of cusps in the same way as that in the one-sided quenchalso happens afterward. – 21 – a) Cusp formation times (b) Corresponding formation points
Figure 14 . Cusp formation times for the Z -symmetric quench with ∆ t = 2. Cusp creationby wave collisions around φ = β / In the cases of the green triangles in Fig. 14, the cusps are formed slightly beforethe collision point, and these cusps subsequently collide at the Z -symmetric point.In fact, it is seen in Fig. 14(a) that these points go ahead of the cusp formation bythe collisions. These cusps then continue traveling on the string, inferring that thewaves are already magnified enough for forming cusps.In the Z -symmetric case, it is convenient to evaluate the time evolution of theworldsheet Ricci scalar at φ = β / R = 2( γ uv,u γ uv,v − γ uv γ uv,uv ) γ uv . (5.1)For the static configuration, this monotonically changes from R = − /ℓ at φ = 0to R = − /ℓ at φ = β /
2, and because of this coordinate dependence it mightbe desirable to compare the Ricci scalar at a fixed φ . The Ricci scalar divergeson top of a cusp, and the cusp formation condition (4.20) is consistent with thedivergence of the Ricci scalar since γ uv in the denominator becomes zero when (4.20)is satisfied. Practically, as we use discretized computations, the Ricci scalar does notexactly become infinity, while at least it becomes huge. Results of the Ricci scalarrepresenting the cusp formation at second, third, and forth collisions are shown inFig. 15. In these results, the absolute value of the Ricci scalar suddenly becomeshuge at the cusp formation, and come back to of order one when the waves pass. Inour setup, the numerical evolution does not breakdown after the Ricci scalar divergesin contrast to the D3/D7 case [10–12], although finite- N c corrections may have to betaken into account. – 22 – igure 15 . Ricci scalar evaluated at φ = β / t/L = 2. - . - (cid:0)4 (cid:1) (cid:2) (cid:3) (cid:4)(cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:13) (cid:14)(cid:15) (cid:16) (cid:17) (a) Just after the quench - (cid:18) (cid:19) (cid:20)(cid:21) (cid:22) (cid:23) (cid:24) (cid:25)(cid:26) (cid:27) (cid:28) (cid:29) (cid:30) (cid:31) ! " $ % (b) After the cusp formation Figure 16 . Snapshots of the string in the transverse linear quench. We set the parametersas ∆ t/L = 2, ǫ = 0 . In this section, we study the string dynamics induced by the transverse linearquench (4.4). With such a quench, the string moves in the (3+1)-dimensions spannedby ( t, z, x , x ). Snapshots of string configurations under a quench with parameters∆ t/L = 2 and ǫ = 0 .
03 are shown in Fig. 16. The left panel is just after the quench, t/L = 2 , . , .
4, where we do not find cusps. However, in the right panel for latetime t/L = 15 , . , .
4, we find cusps on the string. This demonstrates that cuspscan form in the transverse linear quench. For quenches with smaller amplitudes( ǫ . . We expect to see nonlinear origin for the cusp formation in the energy spectrum alsoin the case of the transverse linear quench. In Fig. 17, we show the time dependenceof the energy spectrum when the parameters are ∆ t/L = 2 and ǫ = 0 .
03. The– 23 – inear theory & ’ ( ) * +
11 1 2 5 10 20 50
Figure 17 . Time dependence of the energy spectrum in the transverse linear quench with∆ t/L = 2 and ǫ = 0 .
03. Magenta points correspond to the time for the cusp formation.We fit them by ε ∝ n − a and show the result by a dotted line. time for the cusp formation is t/L = 14 .
45 for these parameters. Red points arejust after the quench, and magenta points correspond to the time at cusp formation.We find the direct energy cascade as in Section 5.2, and eventually the spectrumobeys a power law until the time of cusp formation. Thus, also in the transverselinear quench, we find the turbulent behavior toward the cusp formation. Fittingthe spectrum by ε ∝ n − a , we obtain a = − . ± . We turn to the time-dependence of the forces acting on the quark and the antiquarkin the transverse linear quench. Since the motion of the string is in the ( z, x , x )-space, the x - and x -components of the forces can be non-zero. In Fig. 18(a), weshow h F i and h ¯ F i as functions of time for ∆ t/L = 2 and ǫ = 0 .
01, with which cuspsdo not appear on the string. Similar to the longitudinal quench, pulse-like oscillationsare repeated at intervals. Although there are sharp peaks, they are always O (1) inunits of λ − / L . We also find that h F ( t ) i < h ¯ F ( t ) i >
0. Thus, the forcebetween quarks is always attractive.In Fig. 18(b), we show the absolute values of the forces for ∆ t/L = 2 and ǫ = 0 .
03, where cusps are formed as seen in Section 6.1. The pulses are getting sharpand amplified as time increases, and eventually after the cusp formation, the forcesdiverge when the cusps arrive at the boundaries. We also monitored ˙ t , which isin the denominator in Eq. (4.27), at the boundaries, and found that it is consistentwith zero at t/L ≃ . . x = x q and x ¯ q , respectively.– 24 – , / (a) ∆ t/L = 2, ǫ = 0 .
01 (no cusp) (b) ∆ t/L = 2, ǫ = 0 .
03 (cusp)
Figure 18 . Forces acting on the quark and the antiquark in the transverse linear quench.We fix the time scale of the quench as ∆ t/L = 2. Left and right panels are for ǫ = 0 . ǫ = 0 .
03, respectively. In the right panel, we take the absolute values of the forces,and the vertical axis is log scale.
Finally, we study the string dynamics induced by the transverse circular quench (4.5).The string moves in all (4+1)-dimensions spanned by ( t, z, x , x , x ). For the trans-verse circular quench, we did not find any cusp formation at least for modest pa-rameters: around at ǫ ∼ .
01 and ∆ t/L ∼
1. Nevertheless, we found an interestingbehavior in the energy spectrum. In Fig. 19, we show the time dependence of theenergy spectrum for parameters ∆ t/L = 2 and ǫ = 0 .
02. In the early time evolutionuntil t/L .
14, there is a direct energy cascade: the energy is transferred from largeto small scales, and eventually the spectrum obeys a power law at t/L ∼
14. Fittingthe numerical data at t/L = 14, we obtain ε n ∝ n − . ± . . For t/L &
14, however,we find that this turns into an inverse energy cascade: The energy is transferred tothe large scale. Thus, in contrast to the previous low dimensional cases, the powerlaw once realized at an intermediate time t/L = 14 is not maintained in the latetime.In Fig. 20, we show snapshots of string configurations around the “turning point”of the energy cascade: t/L = 14 , . , .
4. Since the string motion is in the (4 +1)-dimensions, we project the string profile into ( x , x , z )- and ( x , x , z )-spaces.Although cusp-like points can be seen in the right figure, these are not real cusps:We find that although roots of J Z , J X and J X become close at t/L ≃ J X is notzero at the point. (See Eq. (4.20).) However, the perturbation variable ˆ χ defined inEq. (4.23) becomes cuspy, namely, its energy is transferred to the small scale. Hencethe direct energy cascade appears in t/L .
14. After t/L ≃
14, the cuspy shape getsloose because of the dispersive spectrum in the linear perturbation.In Fig. 21, we show the time dependence of the forces acting on the quark and– 25 – inear theory - e- - (a) Direct energy cascade - (b) Inverse energy cascade Figure 19 . Time dependence of the energy spectrum for the transverse circular quenchwith ∆ t/L = 2 and ǫ = 0 . - - (a) ( x , x , z )-space - - (b) ( x , x , z )-space Figure 20 . Snapshots of the string in the transverse circular quench with ∆ t/L = 2 and ǫ = 0 .
02. In the left and right figures, we project the profiles of the string into ( x , x , z )-and ( x , x , z )-spaces, respectively. the antiquark for ǫ = 0 .
02 and ∆ t/L = 2. We do not find the divergence of theforces. However, the forces are magnified until t/L ≃
14 and can be repulsive atsome time intervals ( h F i > h ¯ F i <
0) even though there is no cusp formation.The magnitude of the forces in the late time is not as big as that period, reflectingthe looseness in the cuspy shape.
We studied nonlinear dynamics of the flux tube between an external quark-antiquarkpair in N = 4 SYM theory using the AdS/CFT duality. We numerically computedthe time evolution of the string in AdS dual to the flux tube when we perturbed thepositions of the string endpoints to induce string motions. We considered four kindsof quenches that were chosen to represent typical string motions: (i) longitudinal one-sided quench, (ii) longitudinal Z -symmetric quench, (iii) transverse linear quench,– 26 – -4-202 0 10 20 30 -404812 0 10 20 30 Figure 21 . Time dependence of the forces acting on the quark and the antiquark for thetransverse circular quench with ǫ = 0 .
02 and ∆ t/L = 2. and (iv) transverse circular quench. (See Eqs. (4.1-4.5) and Fig. 6.) For (i)-(iii), wefound cusp formation on the string. In the time evolution of the energy spectrum,we observed the weak turbulence, that is, the energy was transferred to the smallscale, and the energy spectrum eventually obeyed a power law until the time of thecusp formation. The cusp formation occurred only when the amplitude of the quenchwas larger than a critical value, ǫ > ǫ crit , and the dependence of its magnitude onthe quench duration ∆ t was given by a simple form ǫ crit ∝ (∆ t/L ) in small ∆ t/L .When the cusps arrived at the AdS boundary, we observed the divergence of theforce between the quark pair. For (iv), we found no cusp formation. Nevertheless, weobserved a direct energy cascade and the power law spectrum for a while. However, inlate time the direct cascade turned into an inverse energy cascade, where the energywas transferred to the large scale. There was no divergence of the force between thequark pair.How can we understand the weak turbulence of the string in view of gaugetheory? Eigen normal modes e n of the fundamental string studied in section 3 canbe regarded as the excited states | n i of the flux tube in the gauge theory side. Hence,the fluctuating string solution, such as R ( t, φ ) = z + P n c n ( t ) e n ( φ ), corresponds to | ψ i = | i + ∞ X n =1 c n ( t ) | n i (8.1)in the boundary theory, where | i is the ground state. The weak turbulence impliesthat | c n ( t ) | with n ≫ N = 2 supersymmetric QCD [10–12], where a direct energy cascade was found inthe fluctuations on the D7-brane and regarded as production of many heavy mesons– 27 –n the SQCD. In that paper, this phenomenon was referred as “turbulent mesoncondensation”. Although the endpoints of the flux tube we considered are regarded asnondynamical and infinitely heavy quarks, the string turbulence found in this paperwould be regarded as the microscopic picture of the turbulent meson condensation.We found cusp formation when the motion of the string is restricted in (2 +1)- and (3 + 1)-dimensions. The divergence of the forces acting on the quarks isaccompanied by the cusp singularities, where we expect that finite- N c effects willbecome important. These will contain quantum effects of the string, and such effectsmay resolve the cusp singularities and the divergence of the forces. Nevertheless, thecusp formation in the classical sense can give us observable effects: Finite- N c effectswill also appear as gravitational backreactions. If these are taken into account, astrong gravitational wave will be emitted at the onset of the cusp formation. Forcosmic strings in flat spacetime, gravitational wave bursts from cusps have beenstudied in Ref. [42], and it has been found that their spectra obey a power law in thehigh-frequency regime. It would be nice to compute the gravitational waves from thestring in AdS and find the power law spectrum. The description in the dual fieldtheory may be the power law spectrum in gluon jets from the flux tube.When the motion of the string is in (4 + 1)-dimensions, we did not find cuspformation. Hence, in AdS spacetime, the cusp formation on the string is not ageneral phenomenon but accidental one. This implies that the dual phenomenonto the cusp formation is not ubiquitous in the (3 + 1)-dimensional boundary fieldtheory. However, if we consider the many-body system of quark-antiquark pairs, itis possible that the time evolution of some flux tubes happens to be restricted inlower dimensional spaces approximately, and such flux tubes would be able to emitthe gluon jets with the approximately power law spectrum, which is characteristicto the cusp formation. Besides, gravitational wave bursts in the presence of extradimensions were discussed in [43, 44]. Even though real cusps are not formed, theremay be some gluon emission from cuspy shapes.There are some future directions in our work. In this paper, we only consideredmodest values for the amplitude of the quench, ǫ ∼ .
01. For a large value of ǫ ,we expect that the string can even plunge into the Poincare horizon because of thestrong perturbation. This will demonstrate a non-equilibrium process of breakingof the flux tube. It is also straightforward to take into account finite temperatureeffects in this process. It will be also interesting to consider the string motionsin confined geometries [17]. In the theories dual to these backgrounds, the quark-antiquark potential is linear, and the presence of such potential may affect conditionsfor cusp formation. Studying nonlinear string dynamics in such geometries may give In the asymptotically flat spacetime, gravitational self-interaction of cosmic strings has beenperturbatively studied in Ref. [41]. This work suggests that cusps survive the backreaction. We thank Claude Warnick for pointing out this argument. – 28 –ew insights into understanding the QCD flux tubes and non-equilibrium processesin realistic QCD.Closed strings rotating in AdS and having cusps were constructed in Ref. [45].Although our dynamical cusp formation on an open string is different from the ex-istence of cusps in those steady solutions, it may be interesting to obtain useful in-formation from such configurations. In [10], the universal exponent in the power lawwas deduced from a stationary solution called critical embedding in the D3/D7-branesystem in the presence of a constant electric field, and results in time dependent com-putations obeyed that universal value. In our setup, we do not have a correspondingstatic cuspy configuration, but our power law exponents, distributed around 1.4, maybe naturally understood from cuspy stationary strings in AdS.Ultimately, it will be important to understand the mechanism relevant for thethe turbulent behavior. For the integrable Wilson loops such as those in AdS × S ,the turbulent behavior may be studied with the techniques of integrability.In non-linear systems, there may be underlying chaos. In [46], closed stringsmoving in Schwarzschild-AdS background were studied from the viewpoint of chaos.It may be interesting if chaos is seen also in the motion of open strings and theturbulent behavior is understood, particularly in non-integrable situations. Acknowledgments
The authors thank Constantin Bachas, Koji Hashimoto, Shunichiro Kinoshita, ShinNakamura, Vasilis Niarchos, So Matsuura, Giuseppe Policastro, Harvey Reall, Christo-pher Rosen and Claude Warnick for valuable discussions and comments. The workof T.I. was supported in part by European Union’s Seventh Framework Programmeunder grant agreements (FP7-REGPOT-2012-2013-1) no 316165, the EU program“Thales” MIS 375734 and was also cofinanced by the European Union (European So-cial Fund, ESF) and Greek national funds through the Operational Program “Educa-tion and Lifelong Learning” of the National Strategic Reference Framework (NSRF)under “Funding of proposals that have received a positive evaluation in the 3rd and4th Call of ERC Grant Schemes”.
A Numerical methods
In this appendix, we explain our numerical method for solving the equations ofmotion of the string (4.11). Basic ideas are explained in Appendix A in [9].We found that the original form of the equations of motion (4.11) is numeri-cally unstable. To stabilize the numerical evolution, it is effective to use the con-straints (4.8). From them, we have T ,u = q Z ,u + X ,u , T ,v = q Z ,v + X ,v , (A.1)– 29 – igure 22 . Discretization of the world volume. where we choose the positive signs for the square roots since we take ∂ u and ∂ v asfuture directed vectors. Eliminating T ,u and T ,v from Eq. (4.11), we obtain T ,uv = 1 Z [( Z ,u + X ,u ) / Z ,v + ( Z ,v + X ,v ) / Z ,u ] ,Z ,uv = 1 Z [( Z ,u + X ,u ) / ( Z ,v + X ,v ) / + Z ,u Z ,v − X ,u · X ,v ] , X ,uv = 1 Z ( X ,u Z ,v + X ,v Z ,u ) . (A.2)The evolution equations in these expressions are found numerically stable.To numerically solve (A.2), we discretize the world volume ( u, v )-coordinateswith the grid spacing h as shown in Fig. 22. Let us denote the fields ( T, Z, X ) by Ψ.At a point C apart from the boundary, the fields and their derivatives are discretizedwith second-order accuracy asΨ ,uv | C = Ψ N − Ψ E − Ψ W + Ψ S h , Ψ ,u | C = Ψ N − Ψ E + Ψ W − Ψ S h , Ψ ,v | C = Ψ N + Ψ E − Ψ W − Ψ S h , Ψ | C = Ψ E + Ψ W . (A.3)Discretization error is O ( h ). Substituting these into the evolution equations (A.2),we obtain nonlinear equations to determine Ψ N by using known data of Ψ E , Ψ W ,and Ψ S . We use the Newton-Raphson method for solving the coupled nonlinearequations.The equations for the boundary time evolution (4.15) become coupled nonlinearequations of T N and X N . ( Z N = 0 is trivially imposed.) The form at σ = 0 is T N = T S + q Z W + ( X N − X S ) , X N = x q ( T N ) , (A.4)where we used a relation for Z , Z E = − Z W , derived from the boundary condition Z ,uv = 0. At the other boundary σ = β , Z W and x q in (A.4) are replaced with Z E and x ¯ q . These equations are also solved by using the Newton-Raphson method.– 30 – e - - (a) Longitudinal - - (b) Transverse linear - - (c) Transverse circular Figure 23 . Constraint violation for several resolutions, N = 200 , , , t/L = 2 and ǫ = 0 .
01. (b) Transverse linear quench with∆ t/L = 2 and ǫ = 0 .
03. (c) Transverse circular quench with ∆ t/L = 2 and ǫ = 0 .
02. For(a) and (b), the right ends of the figures correspond to the times for the cusp formation.
B Error analysis
In this section, we estimate errors in our numerical calculations. We define˜ C = 1 L ( − T ,u + Z ,u + X ,u ) , ˜ C = 1 L ( − T ,v + Z ,v + X ,v ) . (B.1)These constraints should be zero for exact solutions. Hence, these can be nice indi-cators of our numerical errors. For visibility of the constraint violation, we introduce C max ( v ) = max fixed v ( | ˜ C | , | ˜ C | ) , (B.2)where we take the maximum value when we vary u on a fixed v surface. We alsochoose the bigger of the two constraints, | ˜ C | and | ˜ C | . Introducing an integer N such that the mesh size is given by h = β /N , we plot C max ( v ) for several valuesof N in Fig. 23. We see that the constraint violation is small ( C max ∼ − evenfor N = 200) and behaves as C max ∝ /N . This is consistent with the fact thatour numerical method has the second order accuracy. In this paper, we mainly set N = 800. Then, the constraint violation is O (10 − ). C Energy spectrum in the linear theory
In this appendix, we derive the energy spectrum induced in the linear theory ofsection 3 when a quench is added on the boundary. The equations of motion forthe perturbation variables ( χ , χ , χ ) are given in Eq. (3.8). Here, we focus on thelongitudinal mode χ for simplicity and denote it as χ = χ . Application to trans-verse modes is straightforward. For the quench, we consider the following boundaryconditions for χ : χ ( t, φ = 0) = χ b ( t ) , χ ( t, φ = β ) = 0 , (C.1)– 31 –here χ b ( t ) is the quench function assumed to have a compact support at 0 < t < ∆ t .We also assume that the solution is trivial before the quench, χ ( t ≤ , φ ) = 0.We firstly consider a time independent solution to (3.8), H S ( φ ) = 0 . (C.2)A solution is given by S ( φ ) = − A Z φβ dφ ′ h ( φ ′ ) , (C.3)where A ≡ R β dφ ′ /h ( φ ′ ) is a constant for normalization. Near the boundaries thefunction S behaves as S = 1 − A Γ φ + · · · ( φ ∼ , S = 13 A Γ ( β − φ ) + · · · ( φ ∼ β ) . (C.4)Let us introduce the quench χ b ( t ). Using the function S , we define ˜ χ as χ ( t, φ ) = ˜ χ ( t, φ ) + χ b ( t ) S ( φ ) . (C.5)The new variable ˜ χ satisfies trivial boundary conditions, ˜ χ ( t, φ = 0) = ˜ χ ( t, φ = β ) =0. The equation of motion for ˜ χ becomes( ∂ t + H ) ˜ χ = − ¨ χ b ( t ) S ( φ ) , (C.6)where we used Eq. (C.2).To solve the equation, we consider a Green’s equation:( ∂ t + H ) G ( t, t ′ ; φ, φ ′ ) = δ ( t − t ′ ) δ ( φ − φ ′ ) , (C.7)where G is the Green’s function. By using G , a special solution to Eq. (C.6) can bewritten in the form˜ χ = − Z ∞−∞ dt ′ Z β dφ ′ G ( t, t ′ ; φ, φ ′ ) ¨ χ b ( t ′ ) S ( φ ′ ) . (C.8)Operating R t ′ + ǫt ′ − ǫ dt on both sides of the Green’s equation and taking the limit of ǫ →
0, we obtain the junction condition as ∂ t G | t = t ′ +0 t = t ′ − = δ ( φ − φ ′ ) . (C.9)For t < t ′ , we assume the Green’s function is trivial: G = 0. For t > t ′ , the Green’sfunction can be written as G = X n [ a n sin ω n ( t − t ′ ) + b n cos ω n ( t − t ′ )] e n ( φ ) , (C.10)– 32 –rom the continuity of G at t = t ′ we have b n = 0, and then from the junctioncondition (C.9), we find that a n satisfies X n a n ω n e n ( φ ) = δ ( φ − φ ′ ) . (C.11)Operating ( e n , ∗ ) to the above equation, we obtain a n = 1 ω n γ ( φ ′ ) e n ( φ ′ ) . (C.12)Thus, the Green’s function can be written as G ( t, t ′ ; φ, φ ′ ) = ( t < t ′ ) , P n ω − n sin ω n ( t − t ′ ) γ ( φ ′ ) e n ( φ ′ ) e n ( φ ) ( t > t ′ ) . (C.13)Since the Green’s function is zero at t < t ′ , the special solution obtained fromEq. (C.8) is also zero before the quench, t <
0, and this is nothing but the solutionwe are looking for. After the quench t > T , the solution becomes χ = − X n ω − n S n e n ( φ ) Z T dt ′ ¨ χ ( t ′ ) sin ω n ( t − t ′ )= X n ω n S n e n ( φ ) Z T dt ′ χ b ( t ′ ) sin ω n ( t − t ′ ) , (C.14)where S n ≡ ( S, e n ). Note that ˜ χ = χ after the quench. At the second equality, weintegrated by parts twice.It is then straightforward to compute the energy spectrum. The mode coefficient c n = ( χ, e n ) is computed by using (C.14) as c n ( t ) = ω n S n Z T dt ′ χ b ( t ′ ) sin ω n ( t − t ′ ) , (C.15)and then from Eq. (4.25) the energy spectrum for the longitudinal quench in thelinear theory becomes ε n = √ λz π (cid:2) ˙ c n + ω n c n (cid:3) = √ λz ω n S n π | ˆ χ ( ω n ) | . (C.16)where we define ˆ χ ( ω ) = R dtχ b ( t ) e − iωt . The energy spectrum does not depend on t as we expect. Taking into account the transverse modes, we obtain the energyspectrum as ε n = √ λz π " ω n S n | ˆ χ ( ω n ) | + X i =2 , ω ′ n S ′ n | ˆ χ i ( ω n ) | , (C.17)– 33 –hereˆ χ i ( ω ) = Z dtχ ib ( t, φ = 0) e − iωt , S ′ = g ( φ ) + Γ , S ′ n = ( S ′ , e n ) ′ , (C.18)and χ ib are the quench functions for χ i . The total energy is ε = P ∞ n =1 ε n .The spectrum (C.17) is defined only for integer n . However, in Figs. 11, 17 and19, we generalize ε n to a continuous number by interpolating ω n , S n , ω ′ n , and S ′ n andregarding them as function of continuous number n for visibility. D Forces acting on the quark and the antiquark
In this appendix, we derive the formula for the forces acting on the quark endpoints(4.27). We denote the on-shell Nambu-Goto action as S [ x q , x ¯ q ], where x q and x ¯ q are the locations of the string endpoints regarded as the quark and the antiquark,respectively, at the AdS boundary: X ( t, z →
0) = x q , x ¯ q . The on-shell action relatesto the partition function of the boundary theory as Z CFT [ x q , x ¯ q ] = e iS [ x q , x ¯ q ] . (D.1)In the field theory, the partition function is written as Z CFT [ x q , x ¯ q ] = Z D φ exp ( iS SYM [ φ ] + iS q [ φ ( x q ) , x q ] + iS ¯ q [ φ ( x ¯ q ) , x ¯ q ]) , (D.2)where φ represents the set of the fields in N = 4 super Yang-Mills theory and S SYM is its action. S q and S ¯ q denote the actions for the quark and the antiquark,respectively. We regard x q ( t ) and x ¯ q ( t ) as external fields. The quark action isschematically written as S q [ φ ( x q ) , x q ] = Z dt h − m √ − v + L int [ φ ( x q ) , x q ] i , (D.3)where m is the quark mass and L int corresponds to the interaction term with SYMfields, and the velocity of the quark is introduced as v ≡ d x q /dt . We define the forceacting on the quark as F = δδ x q Z dtL int [ φ ( x q ) , x q ] . (D.4)The variation of the on-shell Nambu-Goto action is then related to the field theoryterms as δSδ x q = − i δδ x q ln Z CFT = − m ( γ v ) · + h F i , (D.5)where h F i = Z − R D φ F e iS SYM + iS q + iS ¯ q , γ = 1 / √ − v , and the dot in the upperright of the parentheses denotes · ≡ d/dt . In the first equality, we used the AdS/CFTduality (D.1). – 34 –ow, we evaluate δS/δ x q in the gravity side. For this purpose, it is convenient touse target space coordinates ( t, z ) as the world sheet coordinates. Then, the positionof the string is specified by x = X ( t, z ), and the Nambu-Goto action becomes S = − √ λ π Z dtdz z (cid:2) (1 − ˙ X )(1 + X ′ ) + ( ˙ X · X ′ ) (cid:3) / . (D.6)The equations of motion are given by " − (1 + X ′ ) ˙ X + ( ˙ X · X ′ ) X ′ z √ ξ · + " (1 − ˙ X ) X ′ + ( ˙ X · X ′ ) ˙ X z √ ξ ′ = 0 , (D.7)where · ≡ ∂/∂t , ′ ≡ ∂/∂z , and ξ ≡ (1 − ˙ X )(1 + X ′ ) + ( ˙ X · X ′ ) . Solving thesenear the AdS boundary, we obtain X = x q ( t ) − γ a z + x z + O ( z ) , (D.8)where a = d x q /dt . Let us consider the variation of the on-shell action with respectto one of the endpoints of the string, x q , and the other endpoint is fixed, δ x ¯ q = 0.By defining Lagrangian as S = R dτ dσ L , the variation of the action becomes δS [ x q , x ¯ q ] = Z dtdz (cid:20) δ ( ∂ a X ) · ∂ L ∂ ( ∂ a X ) + δ X · ∂ L ∂ X (cid:21) = Z dtdz ∂ a (cid:18) δ X · ∂ L ∂ ( ∂ a X ) (cid:19) = − Z dt δ X · ∂ L ∂ X ′ (cid:12)(cid:12)(cid:12)(cid:12) x = x q ,z = ǫ , (D.9)where we take the cutoff at z = ǫ . At the second equality, we used Euler-Lagrangeequation, ∂ L /∂ X = ∂ a [ ∂ L /∂ ( ∂ a X )]. There is no contribution from the other bound-ary X = x ¯ q and z = ǫ since δ X = 0 there. Substituting Eq. (D.8) into Eq. (D.9),we obtain δS [ x q , x ¯ q ] = Z dt δ x q · − √ λ πǫ ( γ v ) · + 3 √ λ πγ ( x + γ ( v · x ) v ) ! . (D.10)Hence, the upshot for δS/δ x q is δSδ x q = − √ λ πǫ ( γ v ) · + 3 √ λ πγ ( x + γ ( v · x ) v ) . (D.11)Comparing above expression with Eq. (D.5), we can see that the first term in (D.11)corresponds to a diverging quark mass m ∼ /ǫ . This is a natural consequence sincewe are considering an infinitely extended string. Setting m = √ λ/ (2 πǫ ), we obtainthe force acting on the quark from Eqs. (D.5) and (D.11) as h F i ( t ) i = √ λ πγ ( δ ij + γ v i v j ) ∂ z X j | z =0 , (D.12)– 35 –here we replaced x with ∂ z X | z =0 /
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