Spin chains and classical strings in rotating Rindler-AdS space

In this paper, we study the spin chain and string excitation in the rotating Rindler-$AdS_3$ proposed in [12]. We obtain a one-parameter deformed $SL(2)$ spin chain at the fast spin limit. Two-spin GKP-like solutions are studied at short and long string limits. General ansatz for the giant magnons and the spiky strings are analyzed in detail for various $\beta$. At last, we explore its counterpart in analytic continuation and pp-wave limit.


I. INTRODUCTION AND SUMMARY
AdS/CFT correspondence [1][2][3] opens a new window for us to understand the fundamental physics of the nature. This conjecture had been tested from many aspects.
One of them is the duality between the integrable spin chains arising from the single trace operators in N = 4 super Yang-Mills (SYM) theory [4] and the rotating strings in AdS 5 × S 5 [5][6][7]. This duality can be checked by the agreement between the spin chain effective action and the rotating string sigma model action [6,7], and from the view point of integrability such that the Bethe equation for the spin chain and that for the classical string sigma model in AdS 5 × S 5 are equivalent [8]. The spiky string solution dual to the giant magnon excitation of the spin chain are also constructed [9,10].
One may wonder the possibility of generalizing the AdS/CFT correspondence to the non-AdS/non-CFT cases. A way to investigate this problem is to consider deformations on both sides and see how the correspondence works. One example in the context of the super Yang-Mills spin chain is the β-deformed N = 4 SYM and its gravity dual [11]. In contrast to [11] where a smooth deformation from S 5 in the original AdS 5 ×S 5 background is accounted for, in this paper we explore the spin chain dual solution in a gravitational background which switches to another conjugacy class of AdS 3 once the deformation is turned on.
One-parameter stationary vacua 1 in AdS 3 belonging to the loxodromic conjugacy class of the Lorentz group (i.e. rota-boosts) were constructed in [12] by twisting the Lorentz group in the embedding flat space. In particular, a combination of two boosts M 01 −βM 23 (where x 0 and x 3 are time-like coordinates) gives rise to the rotating Rindler AdS space, which is the universal cover for the BTZ black hole but with less symmetry. There is an event horizon and an ergosphere due to this rota-boost. As the deformation (or rotation) parameter β vanishes, it reduces to the Rindler spacetime, with the acceleration identified as the inverse of the AdS radius, a ∼ L −1 . On the other hand, a combination of a temporal and a spatial rotations M 03 − βM 12 gives rise to the rotating global AdS, where some region around the center is hidden from a co-rotating observer. As β vanishes, it reduces to the AdS in global coordinates. Although the boundary theory dual to the rotating global AdS still satisfies the Virasoro algebra, the conformal symmetry in static vacuum is broken for uneven deformation in the left and the right sectors. While the computation power seems out of control on the field theory side due to lost symmetry, one is hoping that computation from gravity side is still tractable either analytically (at certain limits) or numerically. As β vanishes, the rotating Rindler AdS 3 falls in the conjugacy class of hyperbolic transformations (i.e. pure boosts), while the global rotating AdS 3 switches to the class of elliptic transformations (i.e. pure rotations). Note that the rota-boosts cannot reduce to the pure boosts or the pure rotations by Lorentz transformations.
The goal of this paper is to study the rotating global AdS vacuum by probing a classical string and observe the effect of deformation to the string excitation and dispersion. We solve the classical string solutions dual to the spin chain in the β-vacua of global rotating AdS 3 × S 1 embedded in AdS 5 × S 5 . At certain limits, we can also obtain the analytic expressions for spin chain model and dispersion relation. Their implications on the dual field theory side, however, remains to be investigated.
The structure of this paper is outlined as follows. In Section II, we derive a deformed SL(2) spin chain Hamiltonian for a fast spinning string, and some simple excitation is examined for nonzero deformation. In Section III, we first study the dispersion relation for GKP string at different limits, and then the general solutions for giant magnons and spiky strings are analyzed in detail. In Section IV, we study the analytically continued version of the geometry and its sin(h)-Gordon model. In Section V, the dispersion relation for a spiky string is studied in the pp-wave limit.

II. A SPIN CHAIN FROM ADS β-VACUA
It was shown in [6] that in the fast spinning limit, one was able to obtain the Heisenberg spin chain by using the sigma model approach, which agrees with the one-loop calculation of anomalous dimensions in N = 4 super Yang-Mills. Although this quantity is no longer protected by the symmetry in the deformed theory, one can still study the effect of deformation to the spin chain Hamiltonian and dynamics from gravity side. In order to reach sensible spinning limit, we will include additional circle in the background metric.
This circular dimension is easily obtained from dimensional reduction, say from type IIB theory in ten dimensions to AdS 3 × S 3 × T 4 . After deforming to the rotating global AdS 3 [12] , the global metrics reads: Note that in this coordinate, cosh 2 ρ < 1 1−β 2 is inaccessible due to the deformation, since in this region the rotation generator ∂ ∂θ becomes time-like. Now we would like to show that at the fast spinning limit, a spin chain Hamiltonian can be obtained from the string worldsheet. Without loss of generality, we apply the following embedded ansatz: where (τ, σ) are worldsheet coordinates. A change of coordinates brings the Polyakov action into 2 where we use X ′ to denote the derivative of X with respect to σ andẊ with respect to τ . Then we take fast spinning limit [6], by sending κ → ∞ andẊ µ → 0, such that κẊ µ remains finite. After taking the limit, the above action simplifies as Taking account of the Virasoro constraint: G µνẊ µ X ν ′ = 0, that is one reaches the classical action of spin chain where we defineλ ≡ λ 8π 2 and L ≡ κ √ λ 2π , and the spin chain Hamiltonian density reads 2 The general form of the equations of motion and the Virasoro constraints arising from the Polyakov action is presented in the Appendix.
This will reduce to SL(2) XXX spin chain at the limit β → 0 as expected [7].
To illustrate the effect of deformation, we will examine a rotated string solution against the deformed spin chain action (7). First, the equations of motion are derived: The simplest solution is obtained for θ = ωt and ρ = ρ(σ). Then the equation of motion for ρ can be integrated to give for some constant a. This solution describes that a folded string stretching between ρ = ±ρ max = cosh −1 a b rotates in uniform speed at the center of AdS 3 . The total energy E and spin S, defined as follows, can be written in terms of complete elliptic integral of the first kind K(x) and second kind E(x): where x = b−a 2b . We plot the energy and spin against ω for several β's in the Figure (1) and find out that deformation increases the energy but slows down the spin.
Several comments are in order: first, the horizon censoring the AdS center seems boosted away in the fast spin limit such that the folded string is able to pass through ρ = 0. Secondly, the equation (10) implies that the apparent string tension is enhanced by a factor (1 −β 2 ) −1/2 . We recall in the earlier studies of turning on the NSNS field B for spinning string in S 3 [13,14], the apparent tension is reduced by a factor (1 − B 2 ) 1/2 . This might be some kind of electric-magnetic duality or strong-weak duality between the metric component G tθ and its analytic continued counterpart G ′ ϕ 1 ϕ 2 , which acts as a nontrivial B ϕ 1 ϕ 2 on the string worldsheet 3 .
While a U(1) is identified to the global time generator, one can still turn on maximal two spins charged under the remaining U(1) 2 . In the following sections, we will first study the spinning closed string solution to understand its leading Regge trajectory behavior at short and long string limits. Then we will construct general ansatz for giant magnons and spiky strings and obtain their dispersion relations.

GKP solution
First we consider a two-spin classical solution in β-deformed AdS 3 × S 1 background 4 .
We apply the following ansatz to the metric (1): where κ, ω and χ are integers. In the conformal gauge, the Polyakov action leads to From the Virasoro constraint, we get the following equation We will present the solution later. For now, by integrating (14), we obtain the following here ρ max and η are defined as We calculate three conserved quantities of this system, the energy E and two angular momenta S and J which are associated to the coordinates θ and φ respectively Using (14), the energy E and the spin S can be evaluated in terms of the elliptic integral In the following we consider the short string limit ρ max ≪ 1 and the long string limit ρ max ≫ 1, and evaluate a relation between the energy E and the spins S and J for each case.

A. Short string limit
First we consider the short string limit ρ max ≪ 1. From (16), this corresponds to the limit η ≫ 1. From (15) and (16), by taking the limit, we get the relations Also, by a suitable combination of (20) and (21), we obtain a simple expansion relation It is natural to identify E − βS and S − βE as the twisted energy and spin in the rotating global AdS background, because they correspond to generators ∂ t − β∂ θ and ∂ θ − β∂ t respectively. By substituting this, we obtain the relation of the twisted energy and the twisted spin for any value of χ as In order to describe this relation in terms of the spin J, let us first consider the limit of χ ≪ 1. At this limit, (24) leads to and (25) is simplified to Then, using (19), we obtain the relation Next let us see another limit χ ≫ 1. At this limit, (24) leads to and (25) becomes or, using (19), this corresponds to the relation In the case of β = 0, this result reduces to (3.24) in [17] as expected.

B. Long string limit
Next we consider the case of the long string limit ρ max ≫ 1, i.e. η ≪ 1. From (15) and (16), by taking the limit, we get the relations Also, from the expansion of the spin S in (21), we obtain We find that the spin is large S ≫ 1 for any value of χ. Since it is not easy to spot a dispersion relation in this complicated expression, we consider several limits of χ in the following. First let us consider the case for χ ≪ ln 1 η . For the limit (34) leads to and using (19) we obtain the following relation In the case of the opposite limit χ ≫ ln 1 η , (34) leads to From this equation we find that S ≫ χ. Comparing an expansion of the energy (20) with (37), and using (19), we obtain the relation which also leads to (3.32) of [17] when β = 0.

A. The solution
To describe the 2-spin giant magnon/spiky string solutions, we take the ansatz following [10]: where the worldsheet coordinates are changed from (τ, σ) to (τ, y), with y = σ − vτ and 0 < v < 1. In the following, we will set Ω = 1 in our analysis for convenience, and 0 < ω < 1. Then the equations of motion (A.2) and (A.3) are rewritten into the differential equations for h 1 (y), h 2 (y), 5 {v where now ′ stands for d/dy. These equations further reduce to where c 1 , c 2 are two integration constants arising from integrating (40), (41). As β = 0, With the string profile ansatz (39), the Virasoro constraints are given by Eliminating (ρ ′ ) 2 by equating the LHS of these two equations and substituting in the h ′ 1 , h ′ 2 expressions from (42) and (43), one obtains the following relation for the two integration constants. 5 We set √ λ 2π = 1 in the following numerical analysis of the giant magnon/spiky sting solutions.
In order to proceed to obtain the explicit string solutions, we need to assign specific values to c 1 , c 2 . In this paper we set 6 This choice yields a constraint by requiring forward propagation of the strings, Note that the constraint (50) is regarded as a natural β-deformation from the original v 2 < 1. Moreover, (47) and (49) reduce to the corresponding results in [10] at β = 0.
With the given c 1 , Substituting these expressions into the Virasoro constraints, one obtains the differential equation for ρ(y): It is straightforward to check that ρ ′2 is indeed an integral of ρ ′′ according to (44) and (54).
The next step is to solve ρ(y) profile. (54) can be rewritten into 6 As being demonstrated later in this paper, the choice of c 1 , c 2 in (49) gives rise to consistent β = 0 reduction. One may choose other c 1 , c 2 , for example But when β is taken to zero, such a choice does not yield the same condition to distinguish the hanging string and the spiky string profiles as the β-free case in [10], despite that β = 0 in (49) and (48) reduce to the c 1 , c 2 choice of [10] . (1), with the corresponding β values, where the strings end. ρ 0 increases with β.
(2) ρ + ≤ ρ 0 . The string extends all the way from ρ 0 to the asymptotic infinity, and it is a spiky solution depicted in Fig. 3.
One finds that the range of y for the spiky string becomes finite due to the β- coordinates still have a remaining Weyl symmetry in the Polyakov action, one can rescale σ according to β to get rid of this issue. As for the bulging stings in Fig. 2(b), the length of the string segment increases with β.
The parameter regions of (β, ω, v) ∈ (0, 1) for all three types of the string solutions are shown in Fig. 4. The boundary separating the spiky and the hanging profiles are obtained comparing ρ + and ρ 0 , and as β is infinitely small, it cannot be obtained as a smooth deformation from the profile distinguishing condition at β = 0 derived from (55). The transition from vanishing to non-vanishing β is discontinuous in this aspect. At β = 0, the two roots in for f (ρ) becomes cosh 2 ρ 1 = 1 and cosh 2 ρ 2 = v 2 1−ω 2 , while the origin is at ρ 0 = 0 = ρ 1 . Here ρ 2 can be greater or smaller than ρ 1 . If ρ 2 > 1, i.e. v 2 1−ω 2 > 1, it is a hanging string; otherwise the solution is spiky (for v 2 1−ω 2 ≤ 1), as predicted by [10]. This can be seen in Fig. 4

B. Dispersion relation
The rotating strings in the β-vacua in AdS 3 × S 1 carry the following energy E and spins S, J which associate with θ, φ respectively: They satisfy a relation where K is a β-dependent correction term and given by the expression K vanishes identically as β = 0, and (60) reduces to E − J = S ω , consistent with [10]. (57)∼(59) diverge as they are integrated to ρ → ∞, and require regularization. However, due to lack of analytic results out of these integrals, we are unable to obtain analytic regularized expressions for E, S and J. We refer to the numerical computation to reveal their dependence on ω, such that the dispersion relation (60) is satisfied. We take a cutoff at Λ = 50 while integrating these quantities over ρ, and denote them by E Λ , S Λ , J Λ , and K Λ . The behaviors of E, S, J against ω remains the same whether they are regularized