The non-integrability of L a,b,c quiver gauge theories
TThe non-integrability of L a,b,c quiver gauge theories Konstantinos S. Rigatos ∗ School of Physics & Astronomy and STAG Research Centre, University of Southampton,Highfield, Southampton SO
17 1
BJ, UK.
We show that the
AdS × L a,b,c solution in type IIB theory is non-integrable. To do so, we considera string embedding and study its fluctuations which do not admit Liouville integrable solutions. We,also, perform a numerical analysis to study the time evolution of the string and compute the largestLyapunov exponent. This analysis indicates that the string motion is chaotic. Finally, we considerthe point-like limit of the string that corresponds to BPS mesons of the quiver theory.This work is dedicated to the memory of DavidGraeber. Academia is much poorer without him. I. Prolegomena
The gauge/string correspondence has evolved from thearchetypical duality proposal [1–3] suggesting the equiv-alence of string theory in
AdS × S and the four dimen-sional N = 4 super Yang-Mills theory to more elaborateconstructions with reduced amount of symmetry in aneffort to probe toy models for field theories that appearin nature and gain intuition for the latter. One of the de-velopments, to that end, involves the replacement of thefive-dimensional sphere of the original AdS × S back-ground geometry by a five-dimensional Sasaki-Einsteinmanifold , which we generically denote by M andtherefore we obtain a duality between type IIB stringtheory on the AdS ×M background and a quiver gaugetheory that lives on the boundary [5].If we choose to specify the five-dimensional internalmanifold to be M = T , we obtain the so-calledKlebanov-Witten model [6] which was the first one tobe studied. However, nowadays, we have at our dis-posal more general (infinite) classes of such five (and alsohigher) dimensional manifolds which are characterized byeither two or three indices and are denoted by Y p,q [7]and L a,b,c [8, 9]. These spaces possess a base topologythat is S × S and we know them explicitly in terms ofmetric descriptions.The boundary (dual) field theory descriptions havebeen obtained for both of the two different families ofSasaki-Einstein manifolds mentioned above. For the Y p,q manifolds the dual field theory description has been ob-tained [10]. The holographic dual gauge theory descrip-tion has also been obtained for the case of the L a,b,c man-ifolds [11–13]. In this work, we will be concerned withthe case of the L a,b,c models. They are more general con- ∗ [email protected] For an excellent exposition and review on Sasaki-Einstein mani-folds see [4]. structions and in fact it has been shown that the Y p,q manifolds can be obtained as special cases.In a complementary approach towards the deeper un-derstanding of gauge theories, an important role is playedby integrability as its existence uncovers an affluent struc-ture of conserved quantities. This in turn implies thesolvability of the theory for any value of the gauge cou-pling. It is related to the previous discussion, since holo-graphically we can associate the superstring worldsheetdescription to a field theory on the boundary withoutgravity, and therefore the integrability of the string sidenaturally becomes an equivalent statement for the inte-grability of the boundary gauge theory.Integrability is present in the duality between the IIBtheory in AdS × S and the N = 4 SYM in the planarlimit [14]. It is only natural to ponder upon the possi-bility of whether or not we can discover new integrablestructures in gauge theories with less symmetries. Theclassical integrability of the AdS × S string is man-ifest, since the Lagrangian equations of motion can beexpressed as a flat condition on the Lax connection [15].Similar work to the above is also available for propagatingstrings in the Lunin-Maldacena background [16]. Thisbackground is dual to the marginal Leigh-Strassler defor-mation, with a real parameter β , that preserves N = 1supersymmetry, as was shown in [17]. In the more generalcase where the β -deformation is complex, integrability isabsent [18–20].While integrable field theories possess a number of ap-pealing features, it is quite cumbersome to declare a cer-tain theory integrable. This is due to the lack of a system-atic approach in order to determine the Lax connection.Due to the aforementioned limitation, proving that a spe-cific theory is non-integrable appears to be, in principle,a more wieldy problem. The full-fledged analysis con-sists of studying the non-linear PDEs that arise from thestring σ -model. In practice, a facile approach is to studycertain wrapped string embeddings and then analyse theresulting equations of motion. Since integrability has tobe manifested universally, a single counter-example suf-fices to declare the full theory non-integrable.One approach that has been undertaken in order toderive appropriate conditions of non-integrability is theS-matrix factorization on the worldsheet [21–24]. A dif-ferent, in spirit, approach was originally developed in [25]and is based on the choice of a wrapped string embed- a r X i v : . [ h e p - t h ] N ov ding and the study of the relevant bosonic string σ -model.There is recent work [26] on the relation between thesetwo non-integrability approaches.The procedure of [25], has been used subsequently ina series of papers [20, 27–40] that studied the classical(non)-integrability of different field theories. In a nut-shell, the method consists of the following steps: writea string soliton that has D degrees of freedom and de-rive its equations of motion. Then find simple solutionsfor the ( D −
1) equations of motion. Replace in the finalequation of motion these solutions and consider fluctu-ations. Thus we have arrived at a second-order lineardifferential equation which is called the normal variationequation (NVE) and is of the form f (cid:48)(cid:48) + P f (cid:48) + Q f = 0.The existence or not of Liouville solutions is dictated bythe mathematical approach developed by Kovacic [41].If the result of the Kovacic method yields no Liouvilleintegrable solutions or no solutions for the NVE, then wecan declare the full theory as being a non-integrable.At this point we would like to stress that even if abackground is characterised as being non-integrable inall generality, this does not preclude the existence of in-tegrable subsectors in the theory. A very nice illustrativeexample of this situation is provided by the complex β -deformation. We have already mentioned that the com-plex β -deformation has been shown to be non-integrablein general, however the sub-sector that is comprised outof two holomorphic and one antiholomorphic scalar isknown to be one-loop integrable [42] as well as fast spin-ning strings in that subsector with a purely imaginary de-formation parameter [43]. Searching for integrable struc-tures within non-integrable theories is an important ques-tion. It might provide useful links and further intuitionfor the transtition from integrable to non-integrable the-ories.It is worthwhile mentioning that string solutions in the L a,b,c Sasaki-Einstein manifolds have been studied in [44]with a special emphasis on BPS configurations and differ-ent supersymmetric D-brane embeddings in the modelshave been studied in [45].The structure of this work is as follows: we begin bybriefly reviewing some basic facts regarding the L a,b,c metrics and subsequently we consider a string configu-ration positioned at the centre of the AdS space andwrapping two angles of the L a,b,c . We argue about thenon-integrability of the field theory by studying the stringdynamics. We find simple solutions of the equations ofmotion and allow the string to fluctuate around them.The study of the NVE does not yield a solution and thuswe declare the quiver gauge theory to be generally non-integrable. We also perform a numerical analysis of theequations of motion governing the string embedding andcompute the largest Lyapunov exponent. These numeri-cal studies reveal chaotic dynamics of the string motion.We finally consider the point-like limit of the strings suchthat they are related to the BPS meson states of the fieldtheory.
50 100 150 200 τ - - ( θ ( τ )) FIG. 1. The solution to the eqs. (26c) and (26d) for the L , , (red plot) and L , , (green plot). The specific values for thecoefficients α, β and the roots of ∆ x = 0 are discussed in therelevant section. The winding of the string in both cases is α = 1 , α = 2. We have also set µ = 1 in both models. Thetime evolution indicates chaotic string motion. II. The geometry
In this section we discuss the structure of L a,b,c spacesand for the reader’s convenience, we quote the necessaryrelations to obtain the Y p,q manifolds from the L a,b,c geometries. A. The L a,b,c geometry The five-dimensional L a,b,c is written as ds L a,b,c = ( dζ + σ ) + ds , (1)with the four-dimensional K¨ahler-Einstein metric ds = ρ dx x + ρ dθ ∆ θ + ∆ x ρ (cid:18) sin θα dφ + cos θβ dψ (cid:19) +∆ θ sin θ cos θρ (cid:20)(cid:18) α − xα (cid:19) dφ − (cid:18) β − xβ (cid:19) dψ (cid:21) , (2)with the relevant quantities appearing above being givenby σ = (cid:18) α − xα (cid:19) sin θdφ + (cid:18) β − xβ (cid:19) cos θdψ,ρ = ∆ θ − x, ∆ x = x ( α − x )( β − x ) − µ, ∆ θ = α cos θ + β sin θ. (3)The metrics depend on two non-trivial parameters asanyone of the α, β, µ can be set to any non-zero valueby a rescaling of the other two. The toric principal or-bits, U (1) × U (1) × U (1), are degenerate when evaluatedon the roots of ∆ x = 0 as well as at θ = 0 , π/
2. Theranges for the different coordinates are 0 ≤ θ ≤ π/ ≤ { θ, ψ } ≤ π and the x -coordinate ranges from x ≤ x ≤ x with x , the smallest roots of the equa-tion ∆ x = 0. The coordinate ζ is periodic and ranges0 ≤ ζ ≤ ˘ ζ and ˘ ζ is to be defined below. The three rootsof the ∆ x = 0 equation are related to the constants α, β and µ of the metric in the following way µ = x x x , α + β = x + x + x ,αβ = x x + x x + x x (4)where in the above x is the third root of the aforemen-tioned equation.We can find relations for x , x , α, β in terms of thequantities a, b, c, d . They have been obtained in [45],however we find it convenient and useful to repeat theanalysis here . The normalized Killing vector fields aregiven by: ∂ φ , ∂ ψ , (cid:96) i = A i ∂ φ + B i ∂ ψ + C i ∂ ζ (5)with i being valued either 1 or 2 and also, A i = αC i x i − αB i = βC i x i − βC i = ( α − x i )( β − x i )2( α + β ) x i − αβ − x i (6)Now, we are at a position to give the value ˘ ζ which isequal to ˘ ζ = 2 π k | C | b , k = gcd( a, b ) , (7)and d is defined to be: d = a + b − c (8)The constants A i , B i , C i are related to to the integers a, b, c that characterize the L a,b,c geometry through therelations aA + bA + c = 0 aB + bB + d = 0 aC + bC = (9)A consequence of eq. (9) is that the ratios of A C − A C , B C − B C , C , and C have to be rational. Morespecifically, it has been shown that cb = A C − A C C db = B C − B C C ba = − C C (10) Using eqs. (4), (9) and (10) we can derive cb = x ( x − x ) x ( x − x ) ac = ( α − x )( x − x ) α ( β − x ) cd = α ( β − x )( β − x ) β ( α − x )( α − x ) cd = α ( x − α ) β ( x − β ) (11)We will keep the parameters α, β, µ general and un-specified for most part of this work. However, in order toperform the numerical analysis of some equations we willneed to specify them. In order to do that consistently,for a particular choice of a, b, c that specifies the Sasaki-Einstein geometry, we determine d using eq. (8). We willset µ = 1 and use the first equation in eq. (4) as wellas the relations described in eq. (11) α, β, x , x , x . Wewill give a specific example below when we analyze theLagrangian equations of motion for an extended string. B. From L a,b,c to Y p,q spaces If we set a + b = 2 c , which in turn implies α = β , the L a,b,c geometry reduces to the Y p,q spaces with the useof the following relations p − q = a, p + q = b, p = c. (12)More explicitly, the transformation laws that reduce the L a,b,c metric to the Y p,q one are [12]¯ ψ = 3 ζ + ψ + φ, ¯ φ = φ − ψ, ¯ β = − ( φ + ψ ) , ¯ θ = 2 θ, ¯ y = 3 x − α α . (13)and the Y p,q metric is written explicitly as ds Y p,q = (cid:18) d ¯ ψ + ¯ σ (cid:19) + d ¯ s , (14)where in the above ¯ σ is given by:¯ σ = − (cid:0) cos ¯ θ d ¯ φ + ¯ yd ¯ β + ¯ c cos ¯ θ d ¯ φ (cid:1) , (15)with the four-dimensional metric of the Y p,q space is d ¯ s = 1 − ¯ c ¯ y (cid:0) d ¯ θ + sin ¯ θ d ¯ φ (cid:1) + 1 w (¯ y ) q (¯ y ) d ¯ y + w (¯ y ) q (¯ y )36 (cid:0) d ¯ β + ¯ c cos ¯ θ d ¯ φ (cid:1) , (16)and finally the functions w and q have the form w (¯ y ) = 2( a − ¯ y )1 − ¯ c ¯ y , q (¯ y ) = a − y + 2¯ c ¯ y a − ¯ y . (17)A comment is in order here. We saw that for specialvalues of the { a, b, c } parameters the L a,b,c models reduceto the Y p,q ones, which are known to be non-integrable[28]. However, the { p, q } parameter space is much smallerand only a subset of all the possible choices of the fullspace spanned by { a, b, c } , corresponding to the theorieswe are examining here. C. The
AdS × L a,b,c geometry The geometry that we want to consider is given by ds = L ds AdS + L ds L a,b,c . (18)For our purposes it is most convenient to describe thefive-dimensional AdS space using global coordinates, ds AdS = − cosh (cid:37) dt + d(cid:37) + sinh (cid:37) d Ω . (19)In the above, d Ω is the round metric of a unit three-sphere which is given explicitly by d Ω = dw + sin w dw + sin w sin w dw . (20)The angles are valued within the ranges 0 ≤ w ≤ π/ ≤ w , w ≤ π . III. String dynamics
The Polyakov action is given by S = − πα (cid:48) (cid:90) d σ h αβ G MN ∂ α X M ∂ β X N (21)in the conformal gauge and must be supplemented by theVirasoro constraints T τσ = T στ = G MN ˙ X M ´ X N = 0 , T ττ = 2 T σσ = G MN (cid:16) ˙ X M ˙ X N + ´ X M ´ X N (cid:17) = 0 (22)where we have used the abbreviations ˙ X ≡ ∂ τ X and´ X ≡ ∂ σ X .Since the string motion in the AdS × S backgroundis integrable and there are no NS-fluxes to deform the σ -model in our case of interest, non-integrability will bemanifested in the structure of the L a,b,c manifold. Thus,a natural choice for the string embedding is to localizethe classical string that we want to study at the centre ofthe AdS space ( (cid:37) = 0) and then wrap two directions ofthe L a,b,c space, more specifically the φ and ψ . Explicitly,we are using the ansatz: t = t ( τ ) , x = x ( τ ) , φ = α σζ = ζ ( τ ) , θ = θ ( τ ) , ψ = α σ. (23)Note that the string configuration described in eq. (23)is similar in spirit as the one used for the T , [27] as wellas the T p,q and Y p,q models [28]. A. Wrapped strings at the centre of
AdS We can now evaluate the Lagrangian density of the σ -model for our particular choice of the string embeddingdescribed by eq. (23), L = ˙ t − ˙ ζ − ρ (cid:32) ˙ x ∆ x + 4 ˙ θ ∆ θ (cid:33) + A sin θ α + A cos θα + A sin (2 θ ) α α (24)where in the above the prefactors A , , are given by A = (cid:18) α − xα (cid:19) sin θ + ∆ x sin θ + ∆ θ cos θ ( α − x ) α ρ , A = (cid:18) β − xβ (cid:19) cos θ + ∆ x cos θ + ∆ θ sin θ ( β − x ) β ρ , A = ∆ x − ( x − α )( x − β )(∆ θ − ρ )2 αβρ . (25)The equations of motion that follow from the Lagrangianread ¨ t = 0 , (26a)¨ ζ = 0 , (26b)8 (cid:18) − xα + β + ( α − β ) cos(2 θ ) (cid:19) ¨ θ =( − α + β ) sin(2 θ ) (cid:32) ˙ x ∆ x − x ˙ θ ∆ θ (cid:33) + 8∆ θ ˙ θ ˙ x − θ )( − α + β ) (cid:18) B α + B α − µαβ B α α (cid:19) , (26c)2 ∆ θ − x ∆ x ¨ x = 2 ˙ x + ( α − β ) sin(2 θ ) ˙ θ ∆ x ˙ x + 4∆ θ ˙ θ − x (cid:18) − ∆ θ − x ∆ x ( αβ − α + β ) x + 3 x ) (cid:19) ˙ x + 12(∆ θ − x ) (cid:18) C sin θα + C cos θα − µαβ C α α (cid:19) . (26d)In the above equations the B -prefactors are explicitlygiven by: B = − µα − α + β − βxα + x + 4 µ ( α − x ) α (( α − β ) cos(2 θ ) + α + β − x ) , (27a) B = α − µβ − β − αxβ + x + 4 µ ( β − x ) β (( α − β ) cos(2 θ ) + α + β − x ) , (27b) B = − α − x )( β − x )(( α − β ) cos(2 θ ) + α + β − x ) , (27c)while the C -prefactors are equal to C = − µ cos(2 θ ) α + 4 µα − β cos(2 θ ) α + β cos(4 θ ) α + 3 β α + 4 α cos(2 θ ) + α cos(4 θ )+ 3 α − β cos(4 θ ) + 2 β + 8 x α + 8 βx cos(2 θ ) α − βxα − x cos(2 θ ) − x, (28a) C = 4 α cos(2 θ ) β + α cos(4 θ ) β + 3 α β − α cos(4 θ ) + 2 α + 4 µ cos(2 θ ) β + 4 µβ − β cos(2 θ ) + β cos(4 θ ) + 3 β + 8 x β − αx cos(2 θ ) β − αxβ + 8 x cos(2 θ ) − x, (28b) C = − (2 θ ) . (28c)From the above, the equations eqs. (26a) and (26b) canbe integrated immediately˙ t = E , ˙ ζ = J , (29)with E and J being constants.The equations of motion above, eqs. (26a) to (26d),are constrained by the Virasoro conditions. We evaluateeq. (22) for our particular string conifguration eq. (23)2 T ττ = 2 T σσ = − ˙ t + ˙ ζ + ρ (cid:32) ˙ x ∆ x + 4 ˙ θ ∆ θ (cid:33) + A sin θα + A cos θα + A sin (2 θ ) α α = 0 , (30a) T τσ = T στ = (cid:18) α − xα sin θα + β − xβ cos θα (cid:19) ˙ ζ = 0 . (30b)We can express the theory under consideration in aHamiltonian formalism. The conjugate momenta aregiven by p t = 2 ˙ t, p ζ = − ζ,p x = − ρ x ˙ x, p θ = − ρ ∆ θ ˙ θ, (31) and the Hamiltonian density is equal to H = 14 ρ (cid:20) ρ ( p t − p ζ ) − x p x − ∆ θ p θ (cid:21) − A sin θ α − A cos θα − A sin (2 θ ) α α . (32)The equations of motion that follow from the Hamilto-nian are, of course, identical with the E¨uler-Lagrangeequations eqs. (26a) to (26d). Fluctuations around the simple solutions
The θ and x equations of motion are coupled eqs. (26c)and (26d). However, to prove the non-integrability ofextended string motion we can simplify this situation byfreezing one dimension and fluctuating the other arounda simple solution.To that end, it is easy to see that there exists an ob-vious and simple solution to the equation of motion for θ ( τ ) eq. (26c) which is given by θ = ˙ θ = ¨ θ = 0 . (33)We refer to it as the straight line solution. Using theabove, the equation of motion for x ( τ ), eq. (26d), simpli-fies to2 x − α ∆ x ¨ x = − β (cid:18) µβ ( α − x ) (cid:19) + ˙ x ∆ x (2 x − (4 α + β ) x + 2 α ( α + β ) x − α β + µ ) = 0 . (34)Let us denote the solution to the above equation by ¯ x ,where we have omitted the explicit time dependence fornotational convenience.We fluctuate now the x coordinate around that partic-ular solution as x = ¯ x + ε X with ε → θ coordinate frozen according to { θ = ˙ θ = ¨ θ = 0 } .We work to linear order in the small parameter ε and theresulting equation is the NVE for the x -coordinate. Itreads: ¨ X + P ˙¯ x ˙ X + 12 (cid:18) ( β − ¯ x )¯ x + µ ¯ x − α (cid:19) (cid:0) Q ¨¯ x + Q ˙¯ x + Q (cid:1) X = 0 (35)with the prefactors being given by: P = α β − µ + ¯ x ( − α ( α + β ) + (4 α + β − x ))¯ x ( α − ¯ x )( µ + ( α − ¯ x )( − β + ¯ x )¯ x ) Q = 2(( α − ¯ x )( µ + ( α − ¯ x )( − β + ¯ x )¯ x )) ( − α β + µ + 2 α ( α + β )¯ x − (4 α + β )¯ x + 2¯ x ) Q = 2 N (( α − ¯ x )( µ + ( α − ¯ x )( − β + ¯ x )¯ x )) N = (cid:20) − α β + α ( α + 2 β ) µ + ¯ x (3 α β ( α + β ) − α + β ) µ + ¯ x ( − α ( α + 3 αβ + β ) + 6 µ +¯ x (9 α + 9 αβ + β + 3¯ x ( − α − β + ¯ x )))) (cid:21) Q = − µβ ( α − ¯ x ) α . (36)We want to bring the NVE, eq. (35), in a more conve-nient form for the application of the Kovacic algorithm .With that in mind, we consider a new variable introducedvia ¯ x = z (37)and under this change, the NVE eq. (35) now becomes˙ z d X dz + (¨ z + ˙ z P ) d X dz + (cid:0) Q ¨ z + Q ˙ z + Q (cid:1) X = 0 , (38)with the P , Q , Q , Q being evaluated on ¯ x = z .We can use the worldsheet equations of motioneq. (30a) on the straight line solution and on x = ¯ x = z to solve for ˙ z . This yields˙ z =4 − µ + ( z − α )( z − β ) zα − z (cid:32) E − J + z − zβ + µz − α β (cid:33) α , (39)and use the equations of motion for x , eq. (26d), evalu-ated again on the straight line solution and on x = ¯ x = z to re-express ¨ z . We get¨ z = 2( z − α ) ( α − z ) ( z ( z − α )( z − β ) − µ ) (cid:18) + µ + β ( z − α ) β α − αβ + 3 z − z ( α + β )) β ( z − α ) (cid:0) ( z − α ) (cid:0) β ( E − J ) + z ( z − β ) α (cid:1) − α µ (cid:1) +4 (cid:32) E − J + z − βz + µ − z + α β (cid:33) ( z ( z − α )( z − β ) − µ ) (cid:19) (40) Recall that we do not know the exact form of ¯ x explicitly. We follow the analytic Kovacic algorithm, which hasbeen very thoroughly reviewed in [32], and we deducethat no combination of the parameters { a, b, c } providesa Liouville integrable solution of the NVE which suggeststhat the system is non-integrable for general values thatcharacterize the L a,b,c model. Solving the lagrangian equations of motion
While the non-integrability of a system does not implychaos necessarily, chaotic dynamics is indicative of theabsence of integrability. In this and the next section wewill perform numerical analysis of the equations of mo-tion for the extended string we have considered and allowthe system to evolve in time. This time evolution revealschaotic dynamics.The equations of motion for x ( τ ) and θ ( τ ) are coupledin general as we saw, and though we are not able to findexact analytic solutions we can solve them numerically.We choose to study the L , , manifold. We can imme-diately see that we get d = 3 using eq. (8). We have thefreedom to set any of the α, β, µ constants to any non-zerovalue and we choose to set µ = 1. We solve the systemof equations described by the first equation in eq. (4) aswell as the relations described in eq. (11) to determinethe values of α, β, x , x , x . We obtain α = 2 . , β =3 . , x = 0 . , x = 2 . , x = 3 . ζ , which is equalto ˘ ζ = 0 . θ = 0 . x = 0 .
3. For the winding of thestring along the two U (1) angles inside the L , , mani-fold we choose α = 1 and α = 2. The initial choice for x ( τ ) is such that it lies between the two smallest roots ofthe cubic equation ∆ x = 0 as required. We let the sys-tem evolve in time and we plot sin θ in a similar mannerto the AdS × T , case [27]. The result is presented inFigure 1.An interesting special case of the L a,b,c models is toconsider b = c [11, 13] with the usual condition a ≤ b .This special class of models has been dubbed generalizedconifolds. The L a,b,b case is equivalent to the L a,b,a undersome trivial reorderings as is explained in [13].We study the generalized conifold given by L , , . Fol-lowing the same steps as before for the L , , we obtainthat d = 1 and we set again µ = 1. The values that char-acterize the model for the constants α, β are 1 . , . x = 0 aregiven by x = 0 . , x = 1 . , x = 3 . ζ ranges from 0 to ˘ ζ = 1 .
501 and the remainingneeded values for the numerical solution of the equationsof motion are the same as in the L , , example. As wedid previously, we show the time evolution of the stringmotion in Figure 1.In both cases, the string motion exhibits chaos.
200 400 600 800 1000t510152025 λ FIG. 2. The Lyapunov index for the extended string motionin the L , , manifold (red) with x (0) = 0 . , θ (0) = π/ L , , model (green) with x (0) = 0 . , θ (0) = 0 .
1. For theformer, we observe a convergence with λ ≈ . λ ≈ . The Lyapunov exponent
A characteristic feature of chaos is the sensitivity ofa system to a specific choice for initial conditions. Hav-ing said that, we discuss the largest Lyapunov exponent(LLE). The sensitivity on the initial conditions can bephrased in the following way: we can consider any pointin the phase space of the theory which we call X . Thereexists at least one point which lies in an infinitesimallyclose distance to that point and that diverges from it.The said distance is denoted by ∆ X ( X , τ ) and is a func-tion of the initial position. The largest Lyapunov expo-nent is a characteristic quantity that quantifies the rateof separation of such closely laying trajectories in thetheory’s phase space. It is given by λ = lim τ →∞ lim ∆ X → (cid:18) τ log ∆ X ( X , τ )∆ X ( X , (cid:19) . (41)We compute the LLE for the systems under consid-eration. We expect that as we dynamically evolve thesystem in time and for a chaotic motion, λ will convergeto some non-zero positive value and fluctuate around thatparticular value. We have verified that such is the casefor the extended string given by eq. (23) that is movingin the L a,b,c manifolds and the result of the computationis shown in Figure 2. IV. BPS mesons and point-like strings
In [11] the authors identified the angle conjugate to theR-symmetry and argued that the BPS geodesics resultingfrom these point-like string modes are compared to theBPS mesons of the quiver theory. Below we study the(non)-integrability of point-like strings.
A. Point-like string motion
We have examined the dynamics of extended stringconfigurations so far. Now we turn our attention to thepoint-like limit of the string. This limit is obtained verystraightforwardly. The change, compared to the previouscase, is that the string now is not wrapping the two coor-dinates inside the L a,b,c ; simply put we set α = α = 0in eq. (23).The Lagrangian can be obtained readily by the previ-ous expression. It is given by: L = ˙ t − ˙ ζ − ρ (cid:32) ˙ x ∆ x + 4 ˙ θ ∆ θ (cid:33) (42)The equations of motion that follow from the La-grangian are:2(∆ θ − x )¨ x ∆ x = − ˙ x ∆ x − θ ∆ θ + (∆ θ − x )( αβ − α + β ) x + 3 x ) ˙ x ∆ x + 2 ˙ x + ( α − β ) sin(2 θ ) ˙ θ ∆ x ˙ x (43a)2(∆ θ − x )¨ θ ∆ θ = 2 ˙ x ˙ θ ∆ θ − ( α − β ) sin(2 θ ) (cid:32) ˙ θ ∆ θ + ˙ x x − θ ∆ θ + 12∆ θ ( α + β +( α − β ) cos(2 θ ) − x ) ˙ θ (cid:17) (43b)The Virasoro conditions that constrain the equationsof motion for the point-like string read2 T ττ = 2 T σσ = − ˙ t + ˙ ζ + ρ (cid:32) ˙ x ∆ x + 4 ˙ θ ∆ θ (cid:33) = 0 , (44a) T τσ = T στ = 0 . (44b)We can, of course, express the system in a Hamiltonianformalism. The canonical conjugate momenta are givenby p t = 2 ˙ t, p ζ = − ζ,p x = − ρ x ˙ x, p θ = − ρ ∆ θ ˙ θ, (45)and the associated Hamiltonian density is equal to H = 14 ρ (cid:20) ρ ( p t − p ζ ) − x p x − ∆ θ p θ (cid:21) . (46)The invariant plane of solutions on which the equationsof motion are satisfied is given by: { x = x , ˙ x = ¨ x = 0 , θ = θ , ˙ θ = ¨ θ = 0 } , (47)alongside with the simple solutions t = E τ + c , ζ = J τ + c . (48)It is quite straightforward to see that if we expand t = E τ + c + ε ˜ t as well as ζ = J τ + c + ε ˜ ζ , with ε → t and ζ respectively. Bothof them admit Liouville integrable solutions.We can also fluctuate the x -coordinate on the invariantplane as x = x + ε X with ε → α − β ) cos(2 θ ) + α + β − x x ( x − α )( x − β ) − µ ¨ X = 0 (49)which also has Liouville integrable solutions.Similarly, we can obtain the NVE for the θ -coordinate.We expand as θ = θ + εϑ in the limit ε → (cid:0) α cos ( θ ) + β sin ( θ ) − x (cid:1) α cos ( θ ) + β sin ( θ ) ¨ ϑ = 0 (50)which has Liouville integrable solutions as in the previouscases. B. Changing coordinates and the R-symmetryangle
Let us briefly describe the change of variables that wasintroduced in [11]. It is given by y = cos(2 θ ). More-over, the said change of variables makes the comparisonbetween BPS geodesics and mesons straightfroward. Inorder to be able to make a statement for the operators ofthe boundary quiver, one needs to know the angle con-jugate to the R-symmetry. This was also obtained in theaforementioned paper and it reads:Ω R = 3 ζ + φ + ψ , (51)Now one is able to re-express the geometry eqs. (2)and (3) in terms of this angle and the coordinate y . Herewe choose not to do that, however we find it useful andilluminating to have this expression explicitly in order tobe able to draw conclusions directly using our coordinatesystem - φ, ψ . C. BPS mesons from strings
It has been shown that BPS mesons correspond to theBPS geodesics [11]. These geodesics are such that x = x and y = y . This can be easily translated into thefollowing statement in our coordinates x = x and θ = θ ,where the constants are such that they respect the rangeswe have discussed. Moreover, it was argued that thenecessary minimization of the Hamiltonian is achievedfor ˙ φ = ˙ ψ = 0. This is the same string configuration thatwe examined above by taking the point-like limit of thestring. V. Epilogue
In this work we considered the motion of an extendedstring that is localized at the centre of the
AdS spaceand is wrapping two U (1) angles inside the L a,b,c space.We showed that the dynamics of that particular stringconfiguration is non-integrable, since the Kovacic algo-rithm fails to provide a solution to the fluctuation equa-tions. We also studied the coupled equations of motionthat were derived from the Lagrangian and solved themnumerically. The time evolution indicates chaotic dy-namics for the string which is another characteristic sig-nature of non-integrability. Having observed the chaoticdynamics, we computed the largest Lyapunov exponentwhich was found to converge to some positive value.Since type IIB string theory in the AdS × L a,b,c vac-uum is holographically dual to the N = 1 quiver gaugetheories and we managed to argue that the string picturein the bulk has a translation to the field theory operators,we have, essentially, argued that these particular quivergauge theories are non-integrable on general grounds.We also examined the dynamics of a string configu-ration in the point-like limit and we managed to deriveLiouville integrable solutions to the NVE. This is anothersituation where the integrability of the extended stringmotion appears to be a much more stringent statementthan the integrability of moving particles.Finally, our work here combined with the results ob-tained previously in [27, 28] suggest that the classicalstring motion in the AdS × M vacuum, with M be-ing a five-dimensional Sasaki-Einstein manifold, is non-integrable. Acknowledgements
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