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Spiky Strings with Two Spins in AdS5

Using the reduction of the string sigma model to the 1-d Neumann integrable model we reconstruct the closed string solution with two equal spins in AdS_5 which is specified by the number of round arcs and one winding number. From the string sigma model itself as well as its reducion to the Neumann-Rosochatius system we construct a spiky string solution with two unequal spins in AdS_5 whose ratio is fixed and derive its energy-spin relation. The string configuration is characterized by the number of spikes and two equal winding numbers associated with the two rotating angular directions.


Introduction
The AdS/CFT correspondence [1] has more and more revealed the deep relations between the N = 4 super Yang-Mills (SYM) theory and the string theory in AdS 5 ×S 5 , where various types of classical string solutions play an important role. The energy spectrum of certain string states matches with the spectrum of dimensions of field theory operators in the SYM theory. There has been a mounting evidence that the spectrum of AdS/CFT is described by studying the multi-spin folded or circular rotating string solutions in AdS 5 ×S 5 in a particular large spin limit [2,3,4] and by analyzing the Bethe equation for the diagonalization of the integrable spin chain in the planar SYM theory [5,6].
Specially the energy of the one-spin folded rotating string in AdS 3 [2,7] describes the dimension of twist two gauge theory operator. The large spin limit of it is related [8] via an analytic continuation and an SO (2,4) transformation to the open string solution ending on a null cusp at the boundary which determines the planar four-point gluon amplitude at strong coupling in the SYM theory [9,10,11,12,13]. The quantum corrections to the folded string energy have been computed up to two loops in the long string limit [3,8,14,15,16] and have been tested to match with the prediction of the strong-coupling expansion of the solution [17] of the integral equation for the minimal twist anomalous dimensions as extracted from the weak-coupling all-loop asymptotic Bethe ansatz [18,6]. This matching has been extended to include the spin J in S 1 ⊂ S 5 [19,20].
The spiky string solution in AdS 3 has been constructed by using the Nambu-Goto string action as a generalization of the one-spin folded string solution to the case of many spikes [21]. This string state has been argued to correspond to a subclass of higher twist operators in the SL(2) sector of gauge theory [22]. The angular separation between each of adjacent spikes is the same. Using the string sigma-model action in the conformal gauge the Kruczenski's spiky string solution with one spin has been reproduced in [23]. The spiky string solution in AdS 3 with different angular separations has been constructed [24] by using the finite-gap formalism for the string sigma-model action [22,25], where each classical solution of string theory has an associated spectral elliptic curve which encodes the values of the higher conserved charges of the worldsheet sigma model. A different attempt to generalize the spiky string solution has been presented [26], where a construction of a general class of string solution in AdS 3 is based on a Pohlmeyer type reduction [27] with the sinh-Gordon model, whose N-soliton solutions map to the dynamical N-spike AdS string configurations.
For the spiky string in AdS 3 in the large spin limit the spikes approach the boundary of AdS 3 and their motion can be described by a string solution in an AdS 5 −pp-wave metric [28]. Further various solutions for open strings moving in the AdS 3 −pp-wave space have been produced such that they are dual to various Wilson loops in the field theory in the boundary 4-dimensional pp-wave background [29]. There has been an investigation of the spiky string solution in AdS 3 × S 1 with n spikes and spin S in AdS 3 and spin J and winding m in S 1 [30]. In a special large n limit the solution can be regarded as describing a periodic-spike string in AdS 3 − pp-wave×S 1 background. On the other hand based on the SL(2) asymptotic Bethe ansatz equations, the rational (S, J) solution has been constructed as the one-cut solution with non-trivial winding [31] and the spiky string solution with winding m, spikes n and two spins S, J has been described as the two-cut solution [32] by following the general procedure of solving integral Bethe ansatz equation at strong coupling [33]. The energy of this two-cut solution matches with that of the spiky string solution presented in [30] in a special scaling limit with large two spins and large winding where the string touches the AdS 5 boundary.
The single-spin folded string in AdS 3 has been extended so that the string solutions with two equal spins S 1 = S 2 in AdS 5 [34] and further with three spins S 1 = S 2 and J in AdS 5 ×S 1 [35] are produced by using string sigma-model actions in the conformal gauge, where the closed string configurations consisting of round arcs are characterized by the number of arcs n and the winding number m. The large spin behavior of the minimal energy specified by n = 3, m = 1 for the string solution with two equal spins S 1 = S 2 in AdS 5 has been argued [36] to match with the strong-coupling prediction from the analysis [37] of the full asymptotic Bethe ansatz equation [6,18]. In the approach of [38] that the conformal-gauge string sigma model is reduced to the 1-d Neumann integrable model, the closed string solutions in AdS 5 are parametrized by the three frequencies ω a = (ω 0 , ω 1 , ω 2 ), ω 0 = κ and the two integrals of motion b 1 , b 2 . In the special κ = ω 2 = ω 1 case an explicit solution with two unequal spins S 1 = S 2 has been constructed by making suitable change of global coordinates for the conformal-gauge string sigma-model action itself [36]. Further for the other special b 1 = b 2 case the energy-spin relation of circular string solution with two unequal spins has been derived by analyzing the 1-d Neumann integrable model.
Using the approach of [39] that the conformal-gauge string sigma model is reduced to the integrable Neumann-Rosochatius system, a solution which expresses a hanging string moving with two spins in AdS 5 has been presented [40] and shown to correspond to the double-helix Wilson loop describing a quark and an anti-quark moving on an S 3 .
We will consider the 1-d Neumann integrable model for the ω 1 = ω 2 case of the closed string in AdS 5 to reconstruct the string solution of round arc shape with two equal spins S 1 = S 2 . In order to extend the spiky string in AdS 3 to in AdS 5 we will derive a spiky string solution with two spins by analyzing the conformal-gauge string sigma model itself as well as its reduction to the integrable Neumann-Rosochatius system. The energy-spin relations for both string solutions in the large spin limit will be compared.

Strings of round arc shape with two equal spins
We consider a rigid rotating two-spin string in AdS 5 based on the reduction to the 1-d Neumann integrable model [38]. For the closed bosonic string in the conformal gauge in AdS 5 space parametrized by global coordinates we make the following ansatz In terms of the three complex embedding coordinates X a , a = 0, 1, 2 subject to a constraint with η 0 = −1, η 1 = η 2 = 1, the AdS 5 metric (1) is rewritten by where Following the approach of ref. [38,36] we use the two independent coordinates ζ i , i = 1, 2 related to r a as where In terms of the coordinates ζ i (σ) the string equations of motion read where a quintic polynomial is expressed as The b 1 , b 2 parameters are associated with the integrals of motion of the 1-d Neumann integrable model. The two-spin solution of a circular shape is specified by the following range We consider the ω 1 = ω 2 case corresponding to the rotating string with two equal spins in AdS 5 . For the equations in (7) we make the change of variables with b 21 ≡ b 2 − b 1 and then take the limit ω 2 → ω 1 to obtain the equations of motion for where the parameters h, u and t are given by The string coordinates r a in the limit ω 2 → ω 1 are expressed in terms of ξ 1 , ξ 2 as The equation (13) is integrated to be expressed in terms of the Jacobi elliptic function sn and the complete elliptic integral of the first kind K where we choose that ξ 2 (σ) decreases in the interval 0 < σ < π/2 and fix an integration constant as ξ 2 (0) = 1 so that (15) yields with (10) implies ρ 0 ≤ ρ 1 . We see that ρ(σ) starts off at its minimum ρ 0 at σ = 0, then increases to its maximum ρ 1 at σ = σ 0 in one segment. By gluing together 2n 1 such segments that are n 1 arcs to impose the periodicity condition ρ(σ + 2π) = ρ(σ), we have because the period of dn is 2K. The expression (17) is rewritten by where t and √ h are described in terms of cosh ρ 1 , cosh ρ 0 as The other equation (12) is also integrated to be Since the string configuration (15) implies we have where an integration constant is chosen such that θ(σ = σ 0 ) = 0 (i.e. θ(ρ = ρ 1 ) = 0) for convenience. Substituting the solution (16) into (23) and integrating we derive is the incomplete of elliptic integral of the third kind. From the periodicity condition θ(2π)−θ(0) = 2πn 2 for the angular coordinate an integer winding number n 2 is specified by which together with (24) yields For the ω 1 = ω 2 case we write down the energy and two spins of rotating string For the ω 1 = ω 2 case the energy and total spin are simplified to be Here from (14) we obtain b 1 , b 2 as which combine with (9) to give ω 2 0 ω 2 Gathering (18), (20) and (30) together we have Therefore the two constants of motion b 1 , b 2 can be expressed in terms of ρ 0 , ρ 1 . We use (30) and the explicit expression ξ 2 of (16) to derive where E is the complete elliptic integral of the second kind. Thus we have reproduced the same expressions for E, S, ∆θ as in [34,35,36] by means of the reduction to the 1-d Neumann integrable model. This equal two-spin string configuration consists of n 1 round arcs with equal angular separation 2∆θ. In the large spin limit u ≈ t ≈ 1 in t ≤ u which is associated with ω 0 ≤ ω 1 in (30), these expressions yield the following energy-spin relation where √ λ is the string tension and the subleading contributions are presented in [36].
To demand that the string is closed at constant global time t we substitute τ = (t − µ 0 (σ))/ω 0 into the ansatz for φ 1 , φ 2 in (41) and (34) to obtain Introducing two integer winding numbers m i for the angular coordinates φ i we have the following closeness conditions where δφ i = φ i (t, 2π) − φ i (t, 0), δµ a = µ a (2π) − µ a (0). Owing to ω 1 = ω 2 , µ 1 = µ 2 they are expressed in terms of the same winding number m 1 = m 2 = m in a single form as 2πm = 2n 1 ∆φ, Thus m is described by two parameters ρ 0 , ρ 1 for fixed n 1 and the parameter θ 0 does not appear.
From (56) and (55) the energy E and total spin S = S 1 + S 2 can be computed to be where two spins are not equal as S 1 = sin 2 θ 0 S, S 2 = cos 2 θ 0 S, but have a fixed ratio S 1 /S 2 = tan 2 θ 0 . They are further written by Thus the large spin limit is specified by t ≈ 1, that is, the long string limit ρ 1 → ∞ so that u 0 ≈ u 1 ≈ 1 and ω 0 ≈ ω 1 ≫ 1 in (69). Two suitable combinations of E and S lead to We parametrize the large spin limit as η → 0 for ω 1 /ω 0 = 1 + η (η ≥ 0) and use K(t) ≈ 1/2 ln(16/(1 − t)) − (1 − t)/8 ln(1 − t) + · · · with t ≈ 1 − 2 cosh 2ρ 0 η and S ≈ n 1 /(2πη) for (76) to have which includes the subleading contributions. In the large spin limit u 0 ≈ u 1 ≈ t ≈ 1 in t ≤ u 0 , t ≤ u 1 which are associated with the range (53), the equal angular separation ∆φ (72) for one segment turns out to be where the following formula is used for u ≈ t ≈ 1, t ≤ u. Substitution of (78) into (77) yields This expression for the spiky string solution with two spins is compared with (33) for the string solution of round arc shape with two equal spins. The coefficents of the subleading ln sin(n 2 π/n 1 ) and ln sin(mπ/n 1 ) terms are different where n 2 is the winding number in the θ direction while m is the identical winding number in both φ 1 and φ 2 directions. The energyspin relation (80) for the AdS 5 spiky string with total spin S shows the same expression as for the AdS 3 spiky string with one spin S 1 and the winding number m 1 in the φ 1 direction [24,21,23]

Conclusion
By means of reduction procedure of the string sigma model to the 1-d Neumann integrable model [38] we have demonstrated that the string solution [34,35,36] with two equal spins in AdS 5 with three angular coordinates (θ, φ 1 , φ 2 ) is reproduced to be of round arc shape in the (ρ, θ) plane, where the string stretches in both ρ and θ directions and is characterized by the arc number and the winding number in the θ direction. In this demonstration we have seen that the two constants of motion b 1 , b 2 and the three frequences ω 0 , ω 1 = ω 2 are expressed in terms of two independent parameters, the minimum and maximum AdS radial coordinates ρ 0 , ρ 1 . Based on the more general reduction procedure to the integrable Neumann-Rosochatius system [39,40] we have constructed the spiky string solution with two spins in AdS 5 by being guided from the analysis of string equations of motion for the global coordinates in the string sigma-model action itself. In spite of starting with the equal choice of two frequencies ω 1 = ω 2 associated with the two angular directions φ 1 , φ 2 we have a spiky string solution with two unequal spins whose ratio is fixed. We have observed that the two winding numbers in the φ 1 , φ 2 directions shoud be the same. We have demonstrated that the constant of motion V , the three conserved momenta C a and the three frequences ω a are also expressed in terms of two independent parameters ρ 0 , ρ 1 . The energy and total spin as well as two identical winding numbers are described by two independent parameters ρ 0 , ρ 1 and the arc number n 1 . The string configuration consisting of n 1 arcs with a fixed angular separation 2∆φ for one arc in the (ρ, φ 1 ) plane is the same as that in the (ρ, φ 2 ) plane, where the spiky string stretches in ρ as well as in both φ 1 and φ 2 directions.
We have observed that the energy-spin relation in the large spin limit for the spiky string solution with two spins and two identical winding numbers in AdS 5 shows the energy growing logarithmically with the total spin and becomes the same form as for the spiky string with one spin and one winding number in AdS 3 , while it shows a slight difference in the coefficient of a subleading term compared with the energy-spin relation for the string of round arc shape with two equal spins in AdS 5 .