Conformal Transformations and Strings for an Accelerating Quark-Antiquark Pair in AdS3

From a simple moving open string solution dual to a moving heavy quark with constant velocity in the Poincare AdS_3 spacetime, we construct an accerlerating open string solution dual to a heavy quark-antiquark pair accelerated in opposite directions by performing the three mappings such as the SL(2,R)_L x SL(2,R)_R isometry transformation, the special conformal transformation and the conformal SO(2,2) transformation. Using the string sigma model action we construct two open string solutions staying in two different regions whose dividing line is associated with the event horizon appeared on the string worldsheet and obtain the accelerating open string solution by gluing two such solutions.


Introduction
of the accelerating open string dual to a uniformly accelerating quark-antiquark pair, there have been several studies where the quark-antiquark pair is concretely regarded as a color singlet EPR pair in the N = 4 SYM theory and its entanglement is encoded in a nontraversable wormhole on the worldsheet of the flux tube connecting the pair [21,22,23,24]. The entanglement entropy of the quark-antiquark pair has been investigated [21,25,26] and the relation between the entanglement entropy and the string surface describing gluon scattering in position space has been studied [27].
We will use the Nambu-Goto action in the static gauge for the open string in the Poincare AdS 3 spacetime and make an ansatz for the string profile expressed by three parameters to reconstruct the two string solutions associated with the one-cusp Wilson loop [28] and the accelerating string solution dual to an accelerating quark-antiquark pair [8].
We will consider the SL(2, R) L ×SL(2, R) R isometry group of the AdS 3 spacetime [29,30] and make this isometry transformation for a simple moving string solution dual to a moving quark with constant velocity to construct the accelerating string solution dual to a uniformly accelerating quark-antiquark pair. This isomery transformation will be applied further to the accelerating string solution. For the moving string solution with constant velocity and the accelerating string solution we will perform the special conformal transformation in the Poincare coordinates and the conformal SO (2,2) transformation in the embedding coordinates to see what kinds of string solutions appear.
Based on the string sigma model action we will make a special ansatz for the string profile in the factorized form and construct the accelerating string solution on which the event horizon appears.

The accelerating string in the Nambu-Goto action
Based on the Nambu-Goto action we consider an open string in AdS 3 with the Poincare metric where we have set the AdS radius R to unity. We use the static gauge to express the string action in the Lorentzian worldsheet coordinates we make an ansatz to have For B = 0 there is one solution specified by A = C = 1 which yields the string configuration expressed in terms of B ≡ b 2 as This string solution was found in ref. [8] where the infinitely massive quark and antiquark are located on the hyperbolic trajectories x = ± √ t 2 + b 2 at the AdS boundary z = 0 such that the plus/minus sign of (7) represents the right and left half of the accelerating string. The quark and antiquark first approach to each other in decelerating and stop to return back in accelerating away from each other with proper acceleration 1/b. For B = 0 there are two solutions which are provided by A = 0, 1 with C = 0 and C = 0, 1 with A = 1. The former gives a simple solution while the latter is complementary to (7) and leads to a string solution whose C is fixed as C = A/2 = 1/2. The latter string configuration is described by which is the one-cusp Wilson loop solution of [28], where the open string surface ends on two semi infinite lightlike lines. This solution yields pure imaginary Lagrangian so that the amplitude shows the exponential suppression. The planar four-gluon scattering amplitude was computed by using the four-cusp Wilson loop solution in the T-dual AdS spacetime which was obtained from the one-cusp Wilson loop solution by perfoming the conformal SO(2,4) transformation [31].
In [32] the following two-cusp Wilson loop solution was constructed by applying the conformal SO (2,4) transformation to the one-cusp Wilson loop solution whose surface ends on four lines t = x±1, t = −x±1 which meet at two cusps (t, x) = (0, ±1) for the plus sign and two cusps (t, x) = (±1, 0) for the minus sign. The appropriate square of (11) yields an equation for x in the fourth order whose solution is given by As the string solution in the Poincare coodinates can be rescaled as x µ = (t, x) → x µ /b, z → z/b, the expression (12) becomes which shows a suggestive expression that contains two polynomials t 2 + b 2 − z 2 and t 2 − z 2 /2 of (7) and (9). The expression (13) as a convolution of two square roots is confirmed indeed to solve the string equation of motion (4) through D = −b 2 σ 2 /4x 2 (τ 2 − σ 2 /2).

Conformal transformations
We use w ± ≡ x ± t to rewrite the Poincare metric (1) as ds 2 = (dz 2 + dw + dw − )/z 2 . The SL(2, R) L × SL(2, R) R isometry group for this AdS 3 metric was investigated in ref. [29] (see also [33]). The SL(2, R) L transformation is given by with real α, β, γ, δ obeying αδ − βγ = 1, while the SL(2, R) R transformation is In view of (14) and (15) α and δ have no dimension whereas β and 1/γ have the same dimension as w ± . Both transformations map the AdS 3 boundary to itself and act on the boundary as the usual conformal transformations of (1+1)-dimensional Minkowski spacetime. Let us consider a string configuration which extends straight from the AdS boundary at z = 0 to the Poincare horizon at z = ∞ in the z direction and moves with constant velocity v in the x direction which simply satisfies the string equation (4) as is seen in (8), and is dual to an isolated infinitely-massive quark moving with constant velocity v. We perform the SL(2, R) L transformation with γ = 0 for the straight moving string solution to obtain an accelerating string configuration associated with proper acceleration γ (1 + v)/(1 − v) of a quark-antiquark pair in the following form where there are some constant shifts in x ′ and t ′ compared with (7). Under a particular SL(2, R) L transformation with γ = 0 the string becomes to move with the different velocity Now we apply the SL(2, R) L transformation to the accelerating string solution (7) which is figured out by the expanding semicircle x 2 + z 2 = t 2 + b 2 for 0 ≤ z in the (x, z) plane. The first relation in (14) gives x + t expressed in terms of x ′ + t ′ , which is substituted into the third relation in (14) to obtain z expressed in terms of x ′ + t ′ and z ′ . We combine with the second relation in (14) to derive a third-order equation for x ′ which has two solutions The former (19) corresponds to (8) with A = 1 so that it does not satisfy the string equation, while the latter (20) shows the expanding string where the acceleration of a quark-antiquark pair changes from 1/b to δ/b. In a particular γ = 0 case the transformed string configuration becomes a second-order equation for x ′ which is also given by (20) with γ = 0. In a particular δ = 0 case the accelerating string solution is transformed to a moving string solution with constant velocity and the expression (19) which does not obey the string equation. When the SL(2, R) R transformation (15) is applied to the moving string with constant velocity (16) and the expanding string (7), the generally mapped string configurations are described by (17) with t ′ , v replaced by −t ′ , −v and (19), (20) with t ′ replaced by −t ′ respectively.
Here we consider the special conformal transformation of the Poincare AdS 3 spacetime coordinates z, x µ = (t, x) which was studied for the circular Wilson loop [34]. The two cases with a µ = (−a, a) and a µ = (a, a) coincide with the particular SL(2, R) L and SL(2, R) R transformations with α = 1, β = 0, γ = 2a, δ = 1 respectively. Let us make a special conformal transformation with a µ = (0, 1/l) for the straight string moving with constant velocity v (16) to have where P is expressed in terms of t ′ , z ′ through These expressions lead to the expanding string with acceleration 2/l On the other hand a special conformal transformation with a µ = (1/l, 0) generates the expanding string with acceleration 2v/l √ 1 − v 2 in the same way For the expanding string (7) with acceleration 1/b we perform a special conformal transformation with a µ = (0, 1/l) to obtain The second relation in (27) together with (28) leads to which becomes through the solution P of (28) to be for b/l = ±1. Thus the magnitude of acceleration changes from 1/b to |b 2 /l 2 − 1|/b. In particular l = ±b cases the expanding string turns back to a static straight string located at x ′ = ±b/2 stretching from the AdS boundary to the Poincare horizon, which is dual to a static isolated quark. The other special conformal transformation with a µ = (1/l, 0) is applied to the expanding string (7) as with P = 1 − 2t/l − b 2 /l 2 . The first relation in (31) reads which is substituted into the third relation in (31) to generate Combining (32), (33) with the second relation in (31) we have again the expanding string configuration with acceleration Here we restore the AdS radius R to express the following relations between the Poincare coordinates in AdS 3 and the embedding coordinates X M (M = −1, 0, 1, 2) on which the conformal SO(2,2) transformation is acting linearly For the moving string with constant velocity v, which is described by X 1 = vX 0 , we perform one conformal SO(2,2) transformation which interchanges X −1 and X 0 . The transformed configuration is specified by X 1 ′ = −vX −1 ′ that is expressed in terms of the Poincare coordinates as which represents the expanding string with acceleration v/R √ 1 − v 2 . The other conformal SO(2,2) transformation defined as the interchange between X 1 and produces X 2 ′ = vX 0 ′ which becomes Thus the expanding string solution with acceleration 1/R √ 1 − v 2 is constructed. Now performing the conformal SO(2,2) transformation (36) for the expanding string (7) which is expressed as we obtain a curve The mapped expression is a polynomial of x ′ in the fourth order which is compared with the second-order plynomial of x in (40). It, however, is expressed in terms of y ≡ x ′ 2 + z ′ 2 as so that we have two solutions The former (43) corresponds to (7) with b = 0 so that it does not solve the string equation, while the latter (44) is the expanding string solution with acceleration (b 2 + R 2 )/2bR 2 . The other conformal SO(2,2) transformation (38) applied to the expanding string solution (40) produces a curve which is similarly expressed in terms of y ≡ t ′ 2 − z ′ 2 as For R = b two solutions are obtained by The former (47) also does not obey the string equation, while the latter (48) yields the expanding string solution with acceleratioin |b 2 − R 2 |/2bR 2 as shown by In a particular R = b case the equation (46) leads to whose solutions are given by x ′ = 0 and x ′ = R ± √ t ′ 2 − z ′ 2 . The former shows the static string solution, while the latter does not satisfy the string equation.
The off-diagonal Virasoro constraint gives In this section we use the dot and prime as the derivatives with respect to τ and σ respectively.
Here we choose the following ansatz in the factorized form The off-diagonal Virasoro constraint (52) reads which is solved by with an integration constant N. First we consider the plus case to parameterize x τ and t τ in terms of a positive parameter p as with −∞ < τ < ∞. The integration constant N is absorbed into f (σ) so that N can be set to unity. The diagonal Virasoro constraint gives Substituting the ansatz (53) with (56) into the equations of motion for t and x ∂ τ ṫ we have an identical equation The equation of motion for z We sum (61) and (59) multiplied by f to derive a differential equation Here we consider a simple solution Owing to 0 ≤ z ≤ b the solution of (64) is expressed as where we take the range of σ as 0 ≤ σ < ∞. In the region 0 ≤ z ≤ b we have a string profile expressed by (65) and Hereafter the plus and minus solutions are called by the I and II solutions respectively. These I and II solutions are confirmed to satisfy the string equations (58) and (60). Eliminating the dependences of the worldsheet coordinates we reproduce the accelerating string solution (7) for the region 0 ≤ z ≤ b. Alternatively we replace σ by z through (65) to have If we choose p = 1/b, then in the AdS boundary z = 0, that is, σ = 0 the solutions (67) become x = ±b cosh τ /b and t = ±b sinh τ /b, which yield x = ± √ t 2 + b 2 and represent the accelerating quark and antiquark trajectories for plus and minus signs respectively with proper time τ and proper acceleration 1/b. The expressions of (65) and (67) with p = 1/b agree with ones in ref. [8] which are described as a static solution in the generalized Rindler spacetime that is derived from the AdS spacetime by a coordinate transformation. The string in the generalized Rindler spacetime was analyzed in [13,17] where the thermodynamics associated with the worldsheet horizon which has the Unruh temperature is further studied.
For fixed σ, that is, fixed z in the I solution the limit τ = −∞ leads to t = −∞, x = ∞ and τ = 0 gives t = 0, x = b/ cosh(σ/b) = √ b 2 − z 2 that is the position of the string bit at depth z and at time t = 0. The limit τ = ∞ leads to t = ∞, x = ∞, and there is a restriction |t| < x for each τ through t/x = tanh τ /b. On the other hand in the II solution, For fixed t the I string extends from the quark location (x, z) = ( √ t 2 + b 2 , 0) specified by σ = 0 to (x, z) = (|t|, b) in an arc, while the II string extends from the antiquark location (x, z) = (− √ t 2 + b 2 , 0) to (x, z) = (−|t|, b) similarly. Here let us consider the minus case for (55) with N = 1 and represent x τ and t τ as The diagonal Virasoro constraint yields In this case the equations of motion for t and x lead to the same equation as (59), however, the equation of motion for z gives Combining together we derive which has two simple solutions z 2 − f 2 = ±b 2 , where the upper sign case has a restriction b ≤ z. The equation (69) with f 2 = z 2 ∓ b 2 can be expressed as This equation of z for the lower sign has no real solution, while for the upper sign it has the following solution which yields with 0 ≤ σ < ∞, −∞ < τ < ∞. To the plus and minus solutions we call the III and IV solutions. These solutions are expressed as which also reproduce the accelerating string solution (7) through the elimination of τ for the region b ≤ z ≤ √ t 2 + b 2 . In the IV solution of (74) and (75) with p = 1/b for fixed τ , t is negative and changes from t = −∞ at σ = 0 to t = 0 at σ = ∞, where the roles of τ and σ are exchanged in comparison with the I and II solutions. Owing to x/t = tanh τ /b, x varies such that x = t = −|t| at τ = ∞, x = 0 at τ = 0 and x = −t = |t| at τ = −∞. At fixed t < 0 the IV string extends from (x, z) = (−|t|, b) to (x, z) = (|t|, b) in an arc and shrinks to zero at t = 0 On the other hand in the III solution for fixed τ , t is positive and changes from t = 0 at σ = ∞ to t = ∞ at σ = 0. Owing to x/t = tanh τ /b, x varies such that x = −t at τ = −∞, x = 0 at τ = 0 and x = t at τ = ∞. At fixed t, z is described by z = b(1 + t 2 /(b cosh τ /b) 2 ) 1/2 so that z becomes z = b at τ = ±∞ that implies σ = ∞ for t to be fixed, and z = √ t 2 + b 2 at τ = 0. Thus at t = 0 the III string starts as a point at (x, z) = (0, b) and at fixed t > 0 extends from ( ). Now we calculate the induced metric on the string surface, (65) and (66) in the region 0 ≤ z ≤ b to obtain a conformally flat expression which has a horizon at σ = ∞ on the worldsheet that yields z = b as a dividing line in the bulk spacetime. The induced metric on the string surface, (73) and (74) in the region b ≤ z is described by a different expression which has also a horizon at σ = ∞ on the worldsheet that corresponds to z = b.
If we make a coordinate transformation from σ to z using (65) and (73) to rewrite (76) and (77) in terms of τ and z respectively, we obtain a single expression where there is a horizon at z = b. In the interior region b ≤ z the roles of τ and z are exchanged such that τ becomes a spacelike coordinate and z becomes a timelike coordinate, which corresponds to the exchange of the roles of τ and σ between (76) and (77). Combining the string solutions I, II, III and IV derived from the string sigma model action, we have the following picture. In the early time specified by t < 0, the right string I and the left string II staying in the exterior region 0 ≤ z ≤ b are decelerated and connected at z = b by the middle string IV staying in the interior region b ≤ z. At t = 0 the interior string IV shrinks to zero and the two exterior strings I and II stop and directly touch at z = b. In the late time t > 0, the exterior strings I and II return back and are accelerated in opposite directions, where the two exterior strings are connected by the interior string III.

Conclusion
For the open string in the Poincare AdS 3 spacetime we have used the Nambu-Goto action in the static gauge to make an ansatz in a square root expression characterized by three parameters for the string profile. We have observed that if three parameters are appropriately chosen, there appear two open string solutions in a complementary pair, the string solution associated with the one-cusp Wilson loop [28] and the expanding string solution associated with a uniformly accelerating quark-antiquark pair [8].
We have constructed the expanding string solution by applying the SL(2, R) L ×SL(2, R) R isometry transfomations, the special conformal transformations and the conformal SO (2,2) transformations to a simple moving string solution dual to a moving quark with constant velocity. We have demonstrated that under the three kinds of transformations the expanding string solution is usually mapped to the same expanding string with the different acceleration. It has been observed that some particular transformations make the expanding string solution change back to the moving string solution with constant velocity or the static string solution.
Based on the string sigma model action we have made an ansatz for the open string profile in the factorized form and constructed two kinds of string solutions, the exteror strings and the interior strings that stay in the two different bulk spacetime regions. We have observed that on each string worldsheet there appear the horizon which is associated with the dividing line which separates the two different bulk spacetime regions. We have demonstrated that the expanding string solution is constructed by connecting two separated exterior strings with one interior string.