Pulsating strings with mixed three-form flux

Journal of High Energy Physics, Apr 2018

Abstract Circular strings pulsating in AdS3 × S3 × T 4 with mixed R-R and NS-NS three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in AdS3 with pure NS-NS three-form flux, where the action reduces to the SL(2, ℝ) WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow by Maldacena and Ooguri. We use our elliptic solutions in AdS3 to extend the dispersion relation beyond the limit of pure NS-NS flux.

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Pulsating strings with mixed three-form flux

JHE Pulsating strings with mixed three-form flux Rafael Hern´andez 0 1 Juan Miguel Nieto 0 1 Roberto Ruiz 0 1 0 Madrid , 28040 Spain 1 Departamento de F ́ısica Teo ́rica I, Universidad Complutense de Madrid Circular strings pulsating in AdS3 × S3 three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in AdS3 with pure NS-NS three-form flux, where the action reduces to the SL(2, R) WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow by Maldacena and Ooguri. We use our elliptic solutions in AdS3 to extend the dispersion relation beyond the limit of pure NS-NS flux. The integrable structure that underlies the AdS3/CFT2 correspondence has provided a deep understanding of numerous aspects of string theory in backgrounds with an AdS3 factor and two-dimensional conformal field theories with maximal supersymmetry [1]-[22]. Integrability has also been shown to remain a symmetry of general string backgrounds that support a mixture of R-R and NS-NS three-form fluxes [23]. This discovery has led to an insightful view of many features of the AdS3/CFT2 correspondence in the presence of mixed fluxes [24-26]-[42]. One of these developments was the demonstration that the sigma-model of type IIB closed strings spinning in AdS3 ×S3 with mixed fluxes corresponds to an integrable deformation of the Neumann-Rosochatius mechanical system [43, 44]. The identification of the Lagrangian that describes spinning strings with mixed fluxes with a deformation of the Neumann-Rosochatius system allows the use of a systematic approach to the construction of general classes of solutions. In this letter we will extend the problem to the case of an ansatz corresponding to closed strings pulsating in AdS3 × S3. In particular, we will show that pulsating strings with mixed fluxes can also be treated using the integrable AdS-CFT Correspondence; Gauge-gravity correspondence - × T 4 with mixed R–R and NS–NS deformation of the Neumann-Rosochatius system obtained from the spinning string ansatz. We will make use of the flux-deformation of the Uhlenbeck constants of the model to integrate the equations of motion in terms of Jacobi elliptic functions. We will also study the problem in the limit of pure NS–NS three-form flux, where the AdS3 piece of the system reduces to the SL(2, R) WZW model. On the basis of our class of elliptic solutions we will derive a general form of the dispersion relation valid beyond the WZW point. In what follows we will first present the Neumann-Rosochatius system that arises from the motion of closed strings pulsating in AdS3 × S3 × T 4 with non-vanishing NS–NS flux. We will consider no dynamics along the torus, and thus the background metric will be ds2 = − cosh2 ρ dt2 + dρ2 + sinh2 ρ dφ2 + dθ2 + sin2 θdφ12 + cos2 θdφ22 , HJEP04(218)7 together with btφ = q sinh2 ρ , bφ1φ2 = −q cos2 θ , for the NS–NS B-field, where 0 ≤ q ≤ 1. The limit q = 0 corresponds to the case of pure R–R flux, while setting q = 1 we are left with pure NS–NS flux. In the case of pure R–R flux the sigma-model for closed strings rotating in AdS3 × S3 reduces to the NeumannRosochatius system [45, 46]. The presence of the NS–NS flux term leads to an integrable deformation of the Neumann-Rosochatius model [43, 44]. In this letter we will extend the analysis in that reference for the spinning string ansatz to the case of pulsating strings. In order to introduce the pulsating ansatz it will be convenient to use the embedding coordinates of AdS3 and S3, which are related to the global angles by Y1 + iY2 = sinh ρ eiφ , X1 + iX2 = sin θ eiφ1 , Y3 + iY0 = cosh ρ eit , X3 + iX4 = cos θ eiφ2 . (1) (2) (3) (4) (5) (6) (7) + 1 gab z˙az˙b + zazaβ˙b2 − z1 k1 − qk1z12β˙0 − 2 2 2 Λ 2 ˜ gabzazb + 1 # where the dot stands for derivatives with respect to τ , the Lagrange multipliers Λ and Λ˜ are needed to impose that the solutions lie, respectively, on S3 and AdS3, and we have taken g = diag(−1, 1), with a = 0, 1. We must note that the flux term in (7) appears with the – 2 – In these coordinates the ansatz for a pulsating string is Y1 + iY2 = z1(τ ) eiβ1(τ)+ik1σ , X1 + iX2 = r1(τ ) eiα1(τ)+im1σ , Y3 + iY0 = z0(τ ) eiβ0(τ) , X3 + iX4 = r2(τ ) eiα2(τ)+im2σ , where we have excluded the winding along the time coordinate because the time direction has to be single-valued. When we enter this ansatz in the world-sheet action in the conformal gauge we find L = √ 2π λ " X2 1 i=1 2 r˙i2 + ri α˙ i − ri2mi2 + qr22 (m2α˙ 1 − m1α˙ 2) − 2 2 2 (r12 + r22 − 1) Λ opposite sign than that in the Lagrangian coming from the spinning string ansatz [43, 44]. The equations of motion for the radial coordinates following from (7) are given by for the ri coordinates, while for the za coordinates they are The cyclic nature of the spherical and hyperbolic angular coordinates in the Lagrangian implies that its equations of motion can be easily integrated once, z¨0 = z0β˙02 − Λ˜ z0 , z¨1 = z1β˙12 − z1k12 − Λ˜ z1 − 2qz1k1β˙0 . α˙ 1 = v1 − qr22m2 2 , r 1 β˙0 = − u0 + qk1z12 , z 2 0 α˙ 2 = v2 + qr22m1 2 , r 2 β˙1 = u1 z 2 1 , (8) (9) ( 10 ) (11) (12) (13) (14) (15) (16) (17) where vi and ua are some integration constants. We can use equations (12) and (13) to write the energy, the Lorentzian spin and the two angular momenta of the string as E = −√λu0 , S = √λu1 , J1 = √λv1 , J2 = √λv2 . The equations of motion coming from (7) must be supplied with the Virasoro constraints, which are responsible for the coupling between the AdS3 and the S3 pieces of the system. They take the form X r˙i2 + α˙ i2 + mi2 ri2 + Xgii(z˙i2 + zi2β˙i2) + k12z12 = 0 , 2 i=1 2 i=1 z12k1β˙1 + Xri2miα˙ i = 0 . 2 i=1 – 3 – We will now move to the construction of general solutions to the above system. In order to proceed, we will follow [45, 46] and introduce ellipsoidal coordinates ζ and μ for the sphere and the Anti-de Sitter factors defined, respectively, as the roots of the equations r 2 1 ζ − m21 + r 2 2 ζ − m22 = 0 , z 2 1 μ − k12 − μ z 2 The ranges of the ellipsoidal coordinates are m21 ≤ ζ ≤ m22 and k12 ≤ μ. When we enter directly ζ and μ into the equations of motion for ri and za we are left with a second order differential equation for each ellipsoidal coordinate. However, we can also use the Uhlenbeck constants of the system to reduce the problem to a pair of independent first order differential equations [45, 46]. The Uhlenbeck constants for the pulsating NeumannRosochatius system in the presence of the flux deformation can be obtained immediately from the ones for the spinning Neumann-Rosochatius system in [43, 44] just by replacing q by −q, because of the change in the sign of flux terms. Thus the Uhlenbeck constants corresponding to motion of the pulsating string either on the sphere or on the Anti-de Sitter factor are, respectively, given by1 where the third order polynomials P3(ζ) and Q3(ζ) are given by P3(ζ) = −(1 − q2)(m22 − ζ)(ζ − m12)2 + I¯1(m22 − m12)(m22 − ζ)(ζ − m12) +(v1 − qm2)2(m22 − ζ)2 + v22(ζ − m12)2 ≡ (1 − q2)Y(ζ − ζi) , Q3(μ) = (1 − q2)(μ − k12)2μ − F¯1k12(μ − k12)μ + (u0 − qk1)2(μ − k12)2 + u1μ 2 2 3 i=1 ≡ (1 − q2)Y(μ − μi) . 3 i=1 (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) The solution to equations (20) can be written in terms of the Jacobian elliptic functions (see [43, 44] for details in the case of the spinning string ansatz). We find ζ(τ ) = ζ3 + (ζ2 − ζ3) sn2 p(1 − q2) (ζ3 − ζ1) τ + τ0, κ , μ(τ ) = μ2 + ′ (μ3 − μ2)(μ2 − μ1) sd2 μ3 − μ1 p(1 − q2) (μ3 − μ1) τ + τ0′ , ν , where τ0 and τ0 are integration constants that can be set to zero by performing a rotation, and the elliptic moduli are We therefore conclude that2 κ = ζ3 − ζ2 ζ3 − ζ1 , ν = μ3 − μ2 . μ3 − μ1 r12(τ ) = 1The integrability of the Neumann-Rosochatius system follows from the existence of a set of integrals of motion in involution, the Uhlenbeck constants [47]. In the case of strings spinning in S integrals I1 and I2, constrained to satisfy I1 + I2 = 1. In the presence of flux they are deformed to I¯1 and I¯2, with the condition I¯1 + I¯2 = 1 [43, 44]. A similar set of constants, F0 and F1, with the constraint 3 there are two F1 − F0 = −1, and their corresponding deformation by the flux term, arises when the string spins in AdS3. 2Pulsating string solutions in AdS3 × S 3 with mixed R–R and NS–NS fluxes have been considered before in [37]. To contact with the notation in that reference we just need to identify the roots R+ and R− in there with our choice of roots through R+ = μ3/k12 − 1 and R− = μ1/k12 − 1. – 4 – We must stress that we need to order the roots of the ellipsoidal coordinate ζ for the sphere in such a way that ζ1 < ζ3. On the contrary, the ellipsoidal coordinate μ for the Anti-de Sitter factor is unbounded and symmetrical under permutation of the roots μ1 and μ3 and thus there is no need to choose the roots μ1 and μ3 according to any particular ordering. Moreover, we should choose μ2 ≥ k 12 because z 12 ≥ 0. Furthermore, we will restrict the elliptic moduli (25) to their fundamental domains 0 ≤ κ, τ ≤ 1, which implies that ζ1 < ζ2 < ζ3, and either μ1 < μ2 < μ3 or μ3 < μ2 < μ1. We will now focus on the analysis of string solutions restricted to pulsate in AdS3 × S1 in the limit of pure NS–NS three-form flux. We will therefore set r1 = α1 = 0, and r2 = 1 and α2 = ω. We must first note that the cubic term in Q3(μ) is dressed with a factor 1−q2. Thus in the case of pure NS–NS flux the degree of the polynomial reduces to two, and the solution can be written in terms of trigonometric functions. In order to understand the reduction of the problem from the point of view of the roots of the polynomial, we will first present them for general values of q. The Virasoro constraints reduce now to (u0 − k1)2 − k12F¯1 sin2 q(u0 − k12)2 − k12F¯1 τ . We can therefore write z12(τ ) = sinh2 ρ(τ ) = sinh2 ρ0 sin(ατ ), so that cosh ρ0 = is given by μ0 = k12. The remaining roots are given by (u0 − qk1)2 ∓ q (u0 − qk1)2 − (1 − q2)f1 2 + 4(1 − q2)k12(u0 − qk1) If we choose (u0 − k12)2 > k12F¯1, the roots become If we exclude the case with F¯1 = 0, which corresponds to the trivial limit where the solution collapses to a point, it is immediate to check that the roots satisfy μ− < μ0 < μ+. It is also clear that the limit of pure NS–NS flux depends on the sign of the term (u0 − k1) − k12F¯1. 2 2 μ− → −∞ , can reach. On the contrary, in the case where k12F¯1 > (u0 − k12)2 the roots become , Now the parameter ρ0 does not have the interpretation of the maximum size of the solution. It is also worth to consider the threshold case, with k12F¯1 = (u0 − k12)2, where the ellipsoidal coordinate μ displays a parabolic behaviour. In this case both μ− and μ+ diverge and z12(τ ) = k12F¯1τ 2 . and the hyperbolic radius is unbounded. This is the long string regime of [48], and the solution reduces now to k12F¯1 k12F¯1 − (u0 − k1)2 sinh2 qk12F¯1 − (u0 − k1)2 τ . In this case we will introduce ρ0 using z12(τ ) = sinh2 ρ(τ ) = cosh2 ρ0 sinh(ατ ). Accordingly, sinh ρ0 = . In the same way as the short and long string regimes are, respectively, constructed by means of spectral flow on time-like and space-like geodesics, the parabolic behaviour can be understood as the result of performing spectral flow on light-like geodesics. In order to fix the sign of α in the previous expressions we need to find the dependence of the time coordinate t on τ . We can obtain that dependence by direct integration of the equation of motion (13) for β˙0. In the short string regime, we conclude that3 tan t(τ ) = tan(−k1τ ) + cosh ρ0 tan − sign(u0 − k1)p(u0 − k1)2 − F¯1k12τ 1 − cosh ρ0 tan(−k1τ ) tan − sign(u0 − k1)p(u0 − k1)2 − F¯1k12τ In the long string regime we can proceed identically, or continue analytically expression (39), to find that tan t(τ ) = tan(−k1τ ) + sinh ρ0 tanh − sign(u0 − k1)pk12F¯1 − (u0 − k1)2τ 1 − sinh ρ0 tan(−k1τ ) tanh − sign(u0 − k1)pk12F¯1 − (u0 − k1)2τ This is indeed the result in [48] provided we read α = −sign(u0 − k1)p|(u0 − k1)2 − F¯1k12| and identify the winding number k1 with −w in that reference. Besides, the threshold case corresponds simply to t(τ ) = −k1τ . We will now use our solutions to extend the dispersion relation to general values of the flux parameter, beyond the WZW limit of the action. At these point we should emphasize 3We must note that when we move away from the limit of pure NS–NS flux the division of the solutions in three different regimes breaks down. However, we can still find tan t(τ) in terms of complete elliptic integrals. – 6 – . . (35) (36) (37) (38) (39) (40) (41) that, despite we have chosen F¯1 and ω as constants of motion, any other set of first integrals is valid as long as they remain independent in phase space. In particular, we can replace F¯1 by n¯ in the argument of the elliptic sine, sn(n¯τ, ν) = sn p(1 − q2)(μ3 − μ1) τ, ν . Thus which in the limit of pure NS–NS flux becomes qn¯4 − (1 − q2)(k12 − ω2) corresponds to the short string regime, while the lower one is the long string regime, in accordance with [48]. A natural question arising from this letter is the derivation of (43) by extending the analysis of the conformal field theory beyond the limit of the WZW model. It would also be interesting to investigate in more detail the relation between the WZW model and the reduction of the Neumann-Rosochatius system with pure NS–NS flux. Acknowledgments grant GR3/14-A 910770. 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Rafael Hernández, Juan Miguel Nieto, Roberto Ruiz. Pulsating strings with mixed three-form flux, Journal of High Energy Physics, 2018, 78, DOI: 10.1007/JHEP04(2018)078