Spiky strings in de Sitter space
Mitsuhiro Kato, Kanji Nishii, Toshifumi Noumi, Toshiaki Takeuchi, Siyi Zhou
PPrepared for submission to JHEP
UT-Komaba/21-1, KOBE-COSMO-21-04
Spiky strings in de Sitter space
Mitsuhiro Kato, a Kanji Nishii, b Toshifumi Noumi, b Toshiaki Takeuchi, b Siyi Zhou c a Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902, Japan b Department of Physics, Kobe University, Kobe 657-8501, Japan c The Oskar Klein Centre for Cosmoparticle Physics & Department of Physics, Stockholm Univer-sity, AlbaNova, 106 91 Stockholm, Sweden.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We study semiclassical spiky strings in de Sitter space and the correspond-ing Regge trajectories, generalizing the analysis in anti-de Sitter space. In particular wedemonstrate that each Regge trajectory has a maximum spin due to de Sitter acceleration,similarly to the folded string studied earlier. While this property is useful for the spectrumto satisfy the Higuchi bound, it makes a nontrivial question how to maintain mildness ofhigh-energy string scattering which we are familiar with in flat space and anti-de Sitterspace. Our analysis implies that in order to have infinitely many higher spin states, oneneeds to consider infinitely many Regge trajectories with an increasing folding number. a r X i v : . [ h e p - t h ] F e b ontents J J A.1 Derivation of Eq. (5.5) 25A.2 J = 0 limit 26 Understanding de Sitter space in string theory is a challenging, but definitely importantissue. In the past decades, intensive efforts have been made toward construction of deSitter space in string theory (see, e.g., Ref. [1] for a review). By evading the assumptionsof the Maldacena-Nunez no-go theorem [2], several promising scenarios such as the KKLTscenario [3] and the Large Volume Scenario [4] have been proposed. While these scenariosare based on ingredients that are individually justifiable, it is still an ongoing issue whetherone can write down an explicit and fully-controlled compactification that unifies all thenecessary components. There are also some attempts [5–7] in the swampland programwhich try to interpret the nontriviality as an obstruction to de Sitter space in string theory.Thus it is tempting to search for complementary approaches to this journey.This paper is a continuation of the previous work [8] by three of the present authors,which initiated such a complementary approach from the worldsheet theory perspective.– 1 –ore specifically, we study the semiclassical spectrum of would-be string worldsheet theoryin de Sitter space, utilizing developments on integrability in the AdS/CFT correspondence.Semiclassical spectra of worldsheet theory in various curved spacetimes have been studiedsince seminal works by de Vega and Sanchez in 80’s [9, 10] and the followups [11–13].Researches in this direction have been further boosted with the advent of the AdS/CFTcorrespondence [14], especially since the Gubser-Klebanov-Polyakov analysis [15] of foldedstrings [13]. As nicely reviewed in Ref. [16], various semiclassical solutions in AdS werethen constructed and studied by using the integrability technique [17–56]. Furthermore,progress in the past several years reaches beyond the string spectrum to include holographichigher-point correlation functions [57–63]. These developments motivate us to perform asimilar analysis in de Sitter space toward understanding of de Sitter space in string theory.In Ref. [8], as a first step in this direction, we revisited the folded string spectrum inde Sitter space [13] and studied its consistency with the Higuchi bound [64], a unitaritybound on the mass of higher-spin particles in de Sitter space. For a bosonic higher-spinparticle with the mass m and the spin s , the Higuchi bound reads m ≥ s ( s − H , where H is the Hubble scale, so that a naive extrapolation of the flat space Regge trajectoryviolates the bound at a high energy scale . Then, one might wonder if the Higuchi boundcould be an obstruction to de Sitter space in string theory. However, we demonstratedthat the Regge trajectory is modified by the curvature effects appropriately such that theHiguchi bound is satisfied. In particular, in sharp contrast to flat space and AdS, eachRegge trajectory in de Sitter space has a maximum spin because of causality and existenceof the cosmological horizon. While this property is crucial to evade a potential conflict withthe Higuchi bound, it makes a nontrivial question how to maintain mildness of high-energystring scattering which we are familiar with in flat space and AdS. It is thus intriguing toexplore this direction further, by considering different types of semiclassical string solutionsand the corresponding Regge trajectories. In this paper, we study spiky strings in de Sitterspace, generalizing the analysis in AdS [18, 27].The organization of the paper is as follows: In Sec. 2, we review basics of worldsheettheory in de Sitter space. In Sec. 3, we study folded strings with internal motion. Then,in Sec. 4 and Sec. 5, we study more general spiky string solutions. We conclude in Sec. 6with discussion of our results. Some technical details are given in Appendix. In this section we summarize basics of the worldsheet theory in de Sitter space necessaryfor our semiclassical analysis. See also Ref. [16] for a nice review on semiclassical strings inAdS. Our argument is analogous to the one there except for the fact that de Sitter spacehas an acceleration and a cosmological horizon accordingly, which turns out to bring aboutqualitative differences from the flat space and AdS cases.– 2 – igure 1 . Penrose diagram of dS : Each point represents an S subspace and the edges correspondto the north and south poles of S . For example, the planar coordinates cover a half of the wholespace (the shaded region) and the cosmological horizon for an observer sitting at the north pole is theblue line. The static coordinates cover a half of the planar coordinates Y ± Y ≥
0, that is insidethe cosmological horizon. The dotted and rigid curves are sections of constant t and r(= sin ρ ) , respectively. We study strings rotating around the center r = 0 ( ρ = 0) of the static coordinate. In this paper we study string Regge trajectories on dS × S (which may also be identifiedwith an appropriate subspace of a larger target space). The three-dimensional de Sitterspace dS with the radius R is defined by a hypersurface, − Y + Y + Y + Y = R , (2.1)embedded into a four-dimensional Minkowski space with the line element, ds = − dY + dY + dY + dY . (2.2)For our purpose, it is convenient to use the static coordinates: Y + Y R = (cid:112) − r e t , Y − Y R = (cid:112) − r e − t , Y + iY R = r e iφ , (2.3)where −∞ < t < ∞ , 0 ≤ r ≤
1, and φ has a periodicity 2 π . The corresponding metric is ds = R (cid:20) − (1 − r ) dt + d r − r + r dφ (cid:21) . (2.4)As depicted in Fig. 1, this coordinate system covers a quarter of the full de Sitter space.An observer sitting at the origin r = 0 has a cosmological horizon at r = 1, hence thiscoordinate system can be used to describe the inside of the horizon. See also Ref. [65], which pointed out a potential conflict of string theory and the Higuchi bound. – 3 –o utilize results in AdS, it is convenient to introduce a coordinate ρ defined by sin ρ = r(0 ≤ ρ ≤ π/ ds = R (cid:0) − cos ρ dt + dρ + sin ρ dφ (cid:1) . (2.5)Note that in these coordinates, the observer sitting at the origin and the cosmologicalhorizon are located at ρ = 0 and ρ = π/
2, respectively. Since global coordinates of AdSare obtained by a Wick rotation, ρ → − iρ , t → it , R → − R , (2.6)we may generalize semiclassical solutions in AdS to de Sitter space in a straightforwardmanner. Together with an internal S parameterized by the coordinate ϕ , our target spacemetric is given by ds = R (cid:0) − cos ρ dt + dρ + sin ρ dφ + dϕ (cid:1) , (2.7)where for generality we leave the periodicity of ϕ a free parameter. In other words, weabsorb the radius of the circle S into the definition of ϕ . Let us consider the Nambu-Goto string on the target space (2.7): S NG = − πα (cid:48) (cid:90) dτ dσ (cid:113) − ˙ X X (cid:48) + ( ˙ X · X (cid:48) ) , (2.8)where (2 πα (cid:48) ) − is the string tension and we defined˙ X = G AB ˙ X A ˙ X B , X (cid:48) = G AB X (cid:48) A X (cid:48) B , ˙ X · X (cid:48) = G AB ˙ X A X (cid:48) B , (2.9)with X A = ( t, ρ, φ, ϕ ) and a target space metric, G AB = R · diag (cid:0) − cos ρ, , sin ρ, (cid:1) . (2.10)Also the dot and prime stand for derivatives in the worldsheet time coordinate τ and theworldsheet spatial coordinate σ , respectively. The equation of motion for X A reads0 = ∂ τ G AB (cid:16) ˙ X B X (cid:48) − X (cid:48) B (cid:16) ˙ X · X (cid:48) (cid:17)(cid:17)(cid:114) − ˙ X X (cid:48) + (cid:16) ˙ X · X (cid:48) (cid:17) + ∂ σ G AB (cid:16) X (cid:48) B ˙ X − ˙ X B (cid:16) ˙ X · X (cid:48) (cid:17)(cid:17)(cid:114) − ˙ X X (cid:48) + (cid:16) ˙ X · X (cid:48) (cid:17) − ∂ A G BC (cid:104) ˙ X B ˙ X C X (cid:48) + X (cid:48) B X (cid:48) C ˙ X − X B X (cid:48) C (cid:16) ˙ X · X (cid:48) (cid:17)(cid:105) (cid:114) − ˙ X X (cid:48) + (cid:16) ˙ X · X (cid:48) (cid:17) . (2.11)– 4 – igid string ansatz. Classical string solutions discussed in this paper are captured bythe following ansatz for closed string configurations: t = τ , ρ = ρ ( σ ) , φ = ωτ + N σ , ϕ = ντ + ψ ( σ ) , (2.12)where σ has a periodicity 2 π and we require ρ ( σ + 2 π ) = ρ ( σ ) and ψ ( σ + 2 π ) = ψ ( σ ),assuming that the string has no winding along the circle S . Also, ω and ν are constantangular velocities, and N is an integer characterizing the “winding” number along the angle φ . Note that the case without internal space is covered simply by setting ν = ψ = 0. Asdepicted, e.g., in Fig. 2, the string at a fixed time t = τ is spreading on the two-dimensional( ρ, φ ) plane. It then rotates along φ and ϕ with angular velocities ω and ν .With the ansatz (2.12), the equations of motion (2.11) reduce to the following (gener-ally) independent three equations:0 = − ∂ σ (cid:20) ρ (cid:48) (cos ρ − ω sin ρ − ν ) √D (cid:21) + 12 sin 2 ρ [ − (1 + ω )( ρ (cid:48) + ψ (cid:48) ) + 2 N νωψ (cid:48) + N cos 2 ρ − N ν ] √D , (2.13)0 = ∂ σ (cid:20) cos ρ ( N ω sin ρ + νψ (cid:48) ) √D (cid:21) , (2.14)0 = ∂ σ (cid:20) νω sin ρ ψ (cid:48) + N (cos ρ − ν ) sin ρ √D (cid:21) , (2.15)where we introduced D = − ˙ X X (cid:48) + ( ˙ X · X (cid:48) ) R = (cos ρ − ω sin ρ − ν ) ρ (cid:48) + (cos ρ − ω sin ρ ) ψ (cid:48) + 2 N νω sin ρ ψ (cid:48) + N (cos ρ − ν ) sin ρ . (2.16)Note that reality conditions require D ≥
0, otherwise the corresponding Nambu-Gotoaction becomes imaginary. Also one may show that when both
D (cid:54) = 0 and ρ (cid:48) (cid:54) = 0 aresatisfied, Eq. (2.13) follows from Eqs. (2.14) and (2.15). Energy, spin and internal U (1) charge. To close the section, let us write down theenergy E , spin S , and internal U (1) charge J , which are of interest in the discussion of theRegge trajectory. Defining them as conjugates of R t , − φ , and − ϕ , respectively, we have E = R πα (cid:48) (cid:90) π dσ cos ρ ( ρ (cid:48) + N sin ρ + ψ (cid:48) ) √D , (2.17) S = R πα (cid:48) (cid:90) π dσ sin ρ ( ωρ (cid:48) + ωψ (cid:48) − N νψ (cid:48) ) √D , (2.18) J = R πα (cid:48) (cid:90) π dσ νρ (cid:48) + N ν sin ρ − N ω sin ρ ψ (cid:48) √D , (2.19)which satisfies the following relation: R πα (cid:48) (cid:90) π dσ √D = RE − ωS − νJ . (2.20)– 5 – Figure 2 . Illustration of ω , ν and N f : ω is the angular velocity in the φ direction. ν is the angularvelocity in the ϕ direction. N f is the folding number. We begin by generalizing our pervious work [8] on rotating folded strings to include motionalong the internal circle S . The folded string configuration is captured by the ansatz (2.12)with N = ψ = 0, under which Eqs. (2.14)-(2.15) become trivial, whereas Eq. (2.13) gives ∂ σ (cid:18) ρ (cid:48) | ρ (cid:48) | (cid:19) (cid:113) cos ρ − ω sin ρ − ν = 0 ↔ δ ( σ − σ f ) (cid:113) cos ρ − ω sin ρ − ν = 0 . (3.1)Notice that the equation of motion is localized at the folding point σ = σ f where ρ (cid:48) flipsthe sign, simply because changes in the bulk profile ρ ( σ ) ( σ (cid:54) = σ f ) are gauge degrees offreedom associated to string reparameterization. Also the folding point satisfiescos ρ − ω sin ρ − ν = 0 , (3.2)and so it propagates with the speed of light, which is essentially the same as the familiarstatement that open string end points propagate with the speed of light. Then, for given ω and ν , the radius ρ f of the folding point is determined bycot ρ f = ω + ν − ν , (3.3)which is the maximum distance dictated by causality prohibiting superluminal propagationof the string. In general, closed strings may have multiple foldings, so that the solutions areparameterized by the angular velocities ω and ν , and the folding number N f . See Fig. 2. Conserved charges.
For these folded strings, the conserved charges (2.17)-(2.19) read E = 1 √ − ν × N f R πα (cid:48) (cid:90) ρ f dρ cos ρ (cid:113) − (sin ρ/ sin ρ f ) , (3.4) S = ω √ − ν × N f R πα (cid:48) (cid:90) ρ f dρ sin ρ (cid:113) − (sin ρ/ sin ρ f ) , (3.5) J = ν √ − ν × N f R πα (cid:48) (cid:90) ρ f dρ (cid:113) − (sin ρ/ sin ρ f ) . (3.6)– 6 – .0 0.2 0.4 0.6 0.801234 S E � � = � � � = ��� � � = � � � = ��� � � = � � � = ��� Figure 3 . The left panel shows Regge trajectories for N f = 1 with different internal charge J . Theright panel shows Regge trajectories for different N f with J = 0 . R /α (cid:48) . The energy E , spin S andinternal charge J are in the units of R/α (cid:48) , R /α (cid:48) and R /α (cid:48) , respectively. We find that the Reggetrajectories always satisfy the Higuchi bound ( E ≥ S ( S − R ), which prohibits the red region. One may also rewrite them in terms of incomplete elliptic integrals, E (cid:0) ζ | k (cid:1) = (cid:90) ζ dθ (cid:112) − k sin θ , F (cid:0) ζ | k (cid:1) = (cid:90) ζ dθ (cid:112) − k sin θ , (3.7)as follows: E = 1 √ − ν × N f R πα (cid:48) (cid:2) sin ρ f E (cid:0) ρ f | csc ρ f (cid:1) + cos ρ f F (cid:0) ρ f | csc ρ f (cid:1)(cid:3) , (3.8) S = ω √ − ν × N f R πα (cid:48) sin ρ f (cid:2) −E (cid:0) ρ f | csc ρ f (cid:1) + F (cid:0) ρ f | csc ρ f (cid:1)(cid:3) , (3.9) J = ν √ − ν × N f R πα (cid:48) F (cid:0) ρ f | csc ρ f (cid:1) . (3.10)These expressions can be used to derive energy-spin relations and draw Regge trajectories. Regge trajectories.
The left panel of Fig. 3 shows Regge trajectories of one-foldedstrings ( N f = 1) with a fixed internal charge J . First, the trajectory for J = 0 accom-modates a turning point, where the string has a maximum spin. This is a consequence ofcausality and existence of the cosmological horizon, which is helpful to satisfy the Higuchibound [8]. Next, if one increases the internal charge J , the trajectory shifts upwards simplybecause the internal motion increases the energy. Also the maximum spin decreases, sothat the maximum spin of one-folded strings is the one for the J = 0 string. Then, one-folded strings scan a finite region in the energy-spin plane represented by the blue shadedregion. In particular, one needs to consider multiple folded strings ( N f = 2 , , . . . ) to haveinfinitely many higher spins (see the right panel of Fig. 3). Note that the Higuchi bound issatisfied in the entire region. In the rest of the section, we study several limits and providemore quantitative arguments. – 7 – .0 0.5 1.0 1.50.00.20.40.60.81.0 ρ f JE Figure 4 . The energy E and the internal U (1) charge J as a function of ρ f . They are plotted inthe units of N f R/α (cid:48) and N f R /α (cid:48) , respectively. J Fig. 3 implies that for a fixed folding number N f , there exists a maximum value of theinternal charge J . To see this more quantitatively, let us recall ν √ − ν ≤ (cid:114) ν + ω − ν = cot ρ f , (3.11)where the inequality is saturated for ω = 0 (for which we have ν = cos ρ f ). Then, we find J ≤ N f R α (cid:48) π (cid:90) ρ f dρ cos ρ f (cid:113) sin ρ f − sin ρ . (3.12)This simply says that the folding point has the speed of light and so for a fixed stringlength ρ f , the internal motion is maximized when the string does not rotate inside dS .As depicted in Fig. 4, the right hand side is maximized in the short string limit ρ f → J ≤ N f R α (cid:48) π (cid:90) ρ f dρ cos ρ f (cid:113) sin ρ f − sin ρ ≤ N f R α (cid:48) . (3.13)Therefore, the N f -folded string has the maximum internal charge J = N f R /α (cid:48) when ω = 0 and ν = cos ρ f →
1. Note that the energy E and the spin S in this limit are E = N f Rα (cid:48) , S = 0 , (3.14)which correspond to the upper boundary point of the shaded region in Fig. 3.– 8 – .2 Regge trajectories for fixed J Short strings.
Next, let us take a closer look at the Regge trajectory profile for a fixed J . For this, we first consider the short string limit ρ f (cid:28)
1. In this regime, we have ρ f (cid:39) (cid:114) − ν ω + ν (cid:28) , (3.15)so that the short string limit is realized for ω (cid:29) ν (cid:39)
1, or both (recall that causalityrequires 0 ≤ ν ≤ ρ f , the charges (3.8)-(3.10) are approxi-mated as E (cid:39) √ − ν × N f R πα (cid:48) (cid:90) ρ f dρ (cid:112) − ( ρ/ρ f ) = 1 √ − ν × N f Rα (cid:48) ρ f , (3.16) S (cid:39) ω √ − ν × N f R πα (cid:48) (cid:90) ρ f dρ ρ (cid:112) − ( ρ/ρ f ) = 12 (cid:114) ω ω + ν × N f R α (cid:48) ρ f , (3.17) J (cid:39) ν √ − ν × N f R πα (cid:48) (cid:90) ρ f dρ (cid:112) − ( ρ/ρ f ) = ν √ − ν × N f R α (cid:48) ρ f . (3.18)Then, in the regime ω (cid:29)
1, which implies J (cid:28) E (cid:39) J R + 2 N f α (cid:48) S . (3.19)Recall that the short string limit is also achieved when ω = O (1) and ν (cid:39)
1. In this regime,the internal charge (3.18) is not necessarily small because the prefactor ν √ − ν cancels outthe suppression by the small ρ f . Taking into account the next-to-leading order terms inEqs. (3.16), (3.18) carefully, we find a more general energy-spin relation, E (cid:39) J R + (cid:115) − (cid:18) α (cid:48) N f R J (cid:19) N f α (cid:48) S , (3.20)which is applicable for an arbitrary value of J as long as the string is short ρ f (cid:28) J increases. Long strings.
To discuss longer strings, it is convenient to rewrite Eq. (3.10) as ω = cot ρ f · (cid:104) N f R πα (cid:48) F (cid:0) ρ f | csc ρ f (cid:1)(cid:105) − J (cid:104) N f R πα (cid:48) F ( ρ f | csc ρ f ) (cid:105) + J , (3.21)where the right hand side monotonically decreases as ρ f increases (see Fig. 5). It impliesthat for a fixed J , there exists an upper bound on the angular velocity ω :0 ≤ ω ≤ N f R α (cid:48) − J J , (3.22)– 9 – .0 0.5 1.0 1.5012345 ρ f ω J = = = = Figure 5 . The angular velocity ω as a function of ρ f ( J is in the unit of N f R /α (cid:48) ). where the upper bound is saturated in the short string limit ρ f →
0. Also, for a fixed J ,the string has a maximum length when ω = 0, for which the conserved charges read E = 1sin ρ f × N f R πα (cid:48) (cid:2) sin ρ f E (cid:0) ρ f | csc ρ f (cid:1) + cos ρ f F (cid:0) ρ f | csc ρ f (cid:1)(cid:3) , (3.23) S = 0 , (3.24) J = cot ρ f × N f R πα (cid:48) F (cid:0) ρ f | csc ρ f (cid:1) . (3.25)For a given J , the maximum length is determined by solving Eq. (3.25). Then, substitutingit into Eq. (3.23) gives the energy of the longest string. See also Fig. 4. This gives theupper endpoint of each Regge trajectory with a fixed J depicted in Fig. 3. Next, we study spiky strings (see Ref. [27] for spiky strings in AdS). In this section we focuson the case without internal motion, so that our ansatz here is Eq. (2.12) with ν = ψ = 0,under which the equations of motion (2.13)-(2.15) reduce to0 = − ∂ σ (cid:34) ρ (cid:48) (cos ρ − ω sin ρ ) (cid:112) (cos ρ − ω sin ρ ) ρ (cid:48) + N cos ρ sin ρ (cid:35) + 12 sin 2 ρ [ − (1 + ω ) ρ (cid:48) + N cos 2 ρ ] (cid:112) (cos ρ − ω sin ρ ) ρ (cid:48) + N cos ρ sin ρ , (4.1)0 = ∂ σ (cid:34) cos ρ sin ρ (cid:112) (cos ρ − ω sin ρ ) ρ (cid:48) + N cos ρ sin ρ (cid:35) . (4.2)– 10 –o follow the string dynamics, it is convenient to integrate Eq. (4.2) as (cid:12)(cid:12) ρ (cid:48) (cid:12)(cid:12) = N ρ sin 2 ρ (cid:115) sin ρ − sin ρ cos ρ − ω sin ρ , (4.3)where the integration constant ρ is chosen such that ρ (cid:48) = 0 for ρ = ρ . For later use, wealso define ρ such that cot ρ = ω and 0 < ρ < π . In this language, we have | ρ (cid:48) | = N sin ρ sin 2 ρ √ ρ (cid:115) cos ρ − cos ρ cos 2 ρ − cos 2 ρ . (4.4) Three shapes.
Notice that ρ (cid:48) has to flip a sign somewhere in order for a closed stringto form a loop, otherwise the string stretches forever. Such a sign flip may appear when ρ (cid:48) = 0 or ρ (cid:48) = ∞ . Eq. (4.4) shows that ρ (cid:48) = 0 is satisfied at ρ = ρ , π − ρ . At thesepoints, the string smoothly turns back from inside to outside or vice versa. Without loss ofgenerality, we assume 0 < ρ < π in the following. On the other hand, ρ (cid:48) = ∞ is satisfiedat ρ = ρ , where the string turns back forming a spike. Based on the value of ρ relativeto ρ and π − ρ , we may classify shapes of the string into the following three classes:1. Outward spikes ( ρ < ρ < π − ρ )Recall that the inside of the square root in Eq. (4.4) has to be positive for ρ to bereal. Therefore, the reality condition implies that for this parameter set, the stringmay stretch only inside the region ρ ≤ ρ ≤ ρ . This means that the outer turningpoints are spiky, and the inner ones are smooth. We call such strings outward spikesolutions. See Fig. 7.2. Rounded spikes ( ρ < π − ρ < ρ )Similarly, for π − ρ < ρ < π , the reality condition implies that the string maystretch only inside the region ρ ≤ ρ ≤ π − ρ . In contrast to the case of outwardspikes, strings in this class have no spikes and all the turning points are smooth. Wecall such strings rounded spike solutions. See Fig. 11. Note that these strings arespecific to de Sitter space and there are no counterpart in flat space and AdS.3. Inward spikes ( ρ < ρ < π − ρ )Finally, for ρ < ρ , the string may stretch only inside the region ρ ≤ ρ ≤ ρ . Thismeans that the outer turning points are smooth, and the inner ones are spiky. Wecall such strings inward spike solutions. See Fig. 12. Periodicity conditions.
The above argument is useful enough to classify local shapesof the string. On the other hand, the full string is made of multiple segments between thespikes. In order for a closed string to form a loop, the angle ∆ φ of each segment has to bequantized appropriately. For our ansatz, an explicit form of ∆ φ is given by∆ φ = 2 N (cid:90) ρ max ρ min dρρ (cid:48) = 2 (cid:90) ρ max ρ min dρ √ ρ sin ρ sin 2 ρ (cid:114) cos 2 ρ − cos 2 ρ cos ρ − cos ρ , (4.5)– 11 – igure 6 . Contour plot of 2 π/ ∆ φ as a function of ρ and ρ : An integer on each contour representsthe value of 2 π/ ∆ φ for given ρ and ρ , which has to be n/N for n -spike strings with the windingnumber N . The two red lines, ρ = ρ and ρ = π − ρ , separate the ρ – ρ plane into three regionswhich accommodate outward spikes, rounded spikes, and inward spikes, respectively. We find inparticular that the string shape has a smooth transition from outward spikes to rounded spikes, as ρ increases from 0 to π/ n and N . Another important observation is that for inwardspikes, ρ < π at ρ = π for a finite 2 π/ ∆ φ (see the right zoom-in figure around ( ρ , ρ ) = ( π , π )). where ρ min and ρ max are the minimum and the maximum values of ρ . More explicitly,( ρ min , ρ max ) = ( ρ , ρ ) , ( ρ , π − ρ ) , ( ρ , ρ ) for outward spikes, rounded spikes, and inwardspikes, respectively. Then, the global consistency requires that∆ φ = 2 πNn , (4.6)where n is a positive integer characterizing the number of spikes. This determines the valueof ρ for given ρ , n , and N . See also Fig. 6 for a plot of 2 π/ ∆ φ as a function of ρ and ρ ,which shows a smooth transition from outward spikes to rounded spikes for fixed n and N . Energy and spin.
For later convenience, we provide the energy and the spin (2.17)-(2.18) for the present class of solutions by using Eq. (4.4) as E = ωSR + R πα (cid:48) (2 n ) (cid:90) ρ max ρ min dρ sin 2 ρ (cid:112) cos ρ − ω sin ρ (cid:112) sin ρ − sin ρ , (4.7) S = R πα (cid:48) ×
12 (2 n ) (cid:90) ρ max ρ min dρ ω sin ρ cos ρ (cid:112) sin ρ − sin ρ (cid:112) cos ρ − ω sin ρ , (4.8)where we used Eq. (2.20) to derive Eq. (4.7). In the rest of the section, we study the threetypes of string solutions in more details. – 12 – ρ - - - - - Figure 7 . Typical shapes of outward spike solutions. The left panel shows the solution with threeoutward spikes and one winding for ρ (cid:39) .
41 and ρ = π . The right panel shows the solution withtwo outward spikes and one winding for ρ (cid:39) .
54 and ρ (cid:39) .
03. The latter type of solutions arespecific to de Sitter space.
Figure 8 . Typical shapes of spiky strings with three outward spikes in flat space (the left panel)and AdS (the right panel).
We begin with outward spike solutions ( ρ < ρ < π − ρ ), whose typical shapes are givenin Fig. 7. See also the left panel of Fig. 10 for strings with more windings. To identify theshapes, first we derive a relation between ρ and ρ . If the number of spikes n and thewinding number N are specified, we may derive the relation from the periodicity condition(4.5)-(4.6) as 2 πNn = 2 (cid:90) ρ ρ dρ √ ρ sin ρ sin 2 ρ (cid:114) cos 2 ρ − cos 2 ρ cos ρ − cos ρ . (4.9)– 13 – .00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00.10.20.30.40.5 � � � ������� = �� = �� = � � � � ������� = �� = �� = � Figure 9 . Regge trajectories of outward spike solutions (the upper panel) and rounded spikesolutions (the lower panel) for the winding number N = 1. The spin S and the energy E areplotted in units of R /α (cid:48) and R/α (cid:48) , respectively, as before. For comparison, we also illustratethe Regge trajectory of one-folded strings. The dotted curves in the lower panel are the Reggetrajectories for outward spikes, which are smoothly connected with those for rounded spikes.
Now, we are left with one parameter ρ , which characterizes the size of the string. If wefurther specify ρ , we may identify the shape of the string simply by integrating dφdρ = ± √ ρ sin ρ sin 2 ρ (cid:114) cos 2 ρ − cos 2 ρ cos ρ − cos ρ . (4.10)For example, the plots in Fig. 7 are obtained in this way. It is also instructive to comparethe shapes there with those in flat space and AdS. See Fig. 8. We find that in de Sitterspace, the inner turning points shift outward compared to the flat space case due to de Sitteracceleration whereas in AdS, the inner turning points shift inward due to AdS deceleration.In particular, the n = 2 N case reflects this effect most clearly: As depicted in the rightfigure of Fig. 7, de Sitter space accommodates spiky strings which can be thought of asa fatter version of the folded strings. Both in flat space and AdS, such a spiky string isnot stable because the string tension always overcomes the centrifugal force, so that itcollapses to the folded string. In sharp contrast, de Sitter acceleration helps the spikystring to maintain the shape without collapsing into a folded string. Regge trajectories.
Using the ρ – ρ relation (4.9), we can calculate the energy E andthe spin S as a function of ρ , which defines Regge trajectories. See Fig. 9 for those ofwinding number N = 1 solutions. First, we find that each trajectory has an approximatelylinear form up to the maximum spin point and then it turns back, similarly to the foldedstring case. In particular, the spin at the turning point is smaller than that of the foldedstring. As a result, the spectrum satisfies the Higuchi bound. We also find that the tilt inthe linear region is steeper for strings with a larger number of spikes. Second, the upper– 14 – - - - S E n = = = = = = Figure 10 . The left panel shows the shape of spiky string with 8 outward spikes and 3 windingsfor ρ (cid:39) .
025 and ρ = 0 .
1. The right panel shows the Regge trajectories of different windings.The spin S and the energy E are plotted in the units of R /α (cid:48) and R/α (cid:48) , respectively. endpoint of the Regge trajectory does not touch the vertical axis S = 0 in contrast tothe folded string case. In the next subsection, we show that the trajectory is smoothlyconnected to that of rounded spike solutions, which touches the vertical axis S = 0. Third,spiky strings with a fixed winding number N scan a finite region of the energy-spin plane.Therefore, to obtain solutions with a larger spin, we need to increase the winding number N . See Fig. 10.Besides, another remark is needed on the Regge trajectory of n = 2 N solutions. Seethe red curve in the upper panel of Fig. 9 for n = 2 and N = 1. As we mentioned,the n = 2 N solutions can be thought of as a fatter version of folded strings, which aresupported by de Sitter acceleration. Then, one may expect that such solutions collapseinto folded strings when the string is small and so the support of de Sitter acceleration isnot enough. Indeed, we find that the Regge trajectory of n = 2 N outward spikes branchesfrom the turning point of the folded string trajectory. Short strings.
To provide more quantitative discussion, let us study the short stringregime of outward spike solutions: ρ , ρ (cid:28) π . (4.11)For such short strings, the ρ – ρ relation (4.9) is approximated as2 πNn (cid:39) ρ ρ (cid:90) ρ ρ dρρ (cid:112) ρ − ρ (cid:112) ρ − ρ = ρ − ρ ρ π ↔ (cid:18) − Nn (cid:19) ρ (cid:39) ρ . (4.12)This shows that ρ = 0 for n = 2 N at least under the short string approximation, whichis consistent with the fact that the n = 2 N solutions are extrapolated to folded strings asthey become smaller. Also, the energy and spin are approximated as E (cid:39) R πα (cid:48) × n (cid:90) ρ ρ dρ ρ ( ρ − ρ ) ρ (cid:112) ρ − ρ (cid:112) ρ − ρ = nR α (cid:48) ρ − ρ ρ = 2 N (cid:18) − Nn (cid:19) Rα (cid:48) ρ , (4.13) S (cid:39) R πα (cid:48) × n (cid:90) ρ ρ dρ ρ (cid:112) ρ − ρ (cid:112) ρ − ρ = nR α (cid:48) ( ρ − ρ ) = N (cid:18) − Nn (cid:19) R α (cid:48) ρ , (4.14)– 15 –rom which the energy-spin relation reads E (cid:39) α (cid:48) N (cid:18) − Nn (cid:19) S . (4.15)This correctly reproduces the linear Regge trajectory in flat space. We find that the tilt ofthe Regge trajectory is steeper for a larger number of spikes. In particular, in the limit ofinfinitely many spikes (for N fixed), the tilt approaches to α (cid:48) N . We will find in Sec. 4.3that steeper Regge trajectories are realized by inward spike solutions. Long strings.
Finally, let us take a closer look at the long string regime. First, thecondition ρ < ρ < π − ρ of outward spikes implies that ρ cannot be larger than π/ ρ approaches to π/ ρ approaches to π/ ρ depends on the number of spikes n andthe winding number N . As we mentioned earlier, we obtain the ρ – ρ relation (4.9)depicted in Fig. 6, once n and N are specified. As we increase ρ for given n and N , eachcurve on the ρ – ρ plane enters the rounded spike regime at some critical value and so thereexists a smooth transition from outward spikes to rounded spikes. For example, the criticalvalue for n = 4 and N = 1 reads ρ (cid:39) .
75, which corresponds to the upper endpoint ofthe Regge trajectory (see Fig. 9). Beyond the critical value, the Regge trajectory describesrounded spike solutions, which we study in the next subsection.
Next, we discuss rounded spike solutions ( ρ < π − ρ < ρ ). See Fig. 11 for a typicalshape of the string, which is regular everywhere. As we have just mentioned, this classof solutions are smooth continuation of outward spike solutions. Then, we may interpretthat outward spikes for ρ < π − ρ are rounded when ρ crosses the critical value definedby ρ = π − ρ (for given n and N ). Based on this interpretation, we call solutions with ρ > π − ρ rounded spike solutions.The procedure to identify the shape is parallel to the case of outward spikes. First, wespecify the number of spikes n and the winding number N , and derive a relation between ρ and ρ from the periodicity condition,2 πNn = 2 (cid:90) π/ − ρ ρ dρ √ ρ sin ρ sin 2 ρ (cid:114) cos 2 ρ − cos 2 ρ cos ρ − cos ρ . (4.16)Then, integrating Eq. (4.10), we may identify the shape of the string for each ρ . Noticethat this type of solutions do not exist for small ρ . See Fig. 6. For example, the allowedparameter range of ρ for n = 4 and N = 1 reads 0 . (cid:46) ρ < π .– 16 – π - ρ - - Figure 11 . Rounded spike solution for ρ = 0 .
72 and ρ (cid:39) .
86. We call the turning points definedby ρ = π/ − ρ rounded spikes. Regge trajectories.
Varying the value of ρ , we may draw the Regge trajectories asdepicted in the right panel of Fig. 9. There, for comparison, we also illustrate the Reggetrajectories of outward spike solutions by the dotted lines. Since rounded spikes exhibita smooth transition to outward spikes, the Regge trajectories are connected with those ofoutward spikes. We also find that each Regge trajectory touches the vertical axis S = 0,similarly to the folded string. However, as we discuss in the next paragraph, the mechanismhow the spin vanishes is different from the folded string. Circular string limit.
To see how the spin vanishes, let us consider the limit ρ → π .Recalling that ρ ≤ ρ ( σ ) ≤ π − ρ , we find that in this limit, the solution is reduced to ρ ( σ ) = ρ = π , (4.17)which is nothing but the static circular string studied in Ref. [66]. As discussed there, sucha static circular string solution exists in de Sitter space because the string tension andthe de Sitter acceleration balance and cancel each other out. Note that the equations ofmotion (4.1)-(4.2) are satisfied for an arbitrary value of ω , since rotations do not change theworldsheet profile and so they are gauge degrees of freedom. The conserved charges (2.17)-(2.18) for these circular strings read E = N R α (cid:48) , S = 0 . (4.18)In particular, the string has no spin for an arbitrary ω because the circular string has nostructures generating nonzero angular momenta. Finally, we discuss inward spike solutions ( ρ < ρ < π − ρ ), whose typical shape isillustrated in Fig. 12. The procedure to identify the shape is parallel to the case of outward– 17 – ρ - - - - - - Figure 12 . Solution with three inward spikes and one winding for ρ = π/ ρ (cid:39) . S E Foldedn = = = Figure 13 . Regge trajectories of inward spike solutions for N = 1. The spin S and the energy E are plotted in the units of R /α (cid:48) and R/α (cid:48) , respectively. For comparison, we illustrate the Reggetrajectory of the one-folded string by the dotted blue curve. and rounded spikes. First, we specify the number of spikes n and the winding number N and derive a relation between ρ and ρ from the periodicity condition,2 πNn = 2 (cid:90) ρ ρ dρ √ ρ sin ρ sin 2 ρ (cid:114) cos 2 ρ − cos 2 ρ cos ρ − cos ρ . (4.19)Then, by integrating Eq. (4.10) for a specific value of ρ , we may identify the shape. Regge trajectories.
The Regge trajectories are illustrated in Fig. 13. Similarly to theprevious cases, each Regge trajectory has the maximum energy and spin, which is helpfulfor the spectrum to satisfy the Higuchi bound. In contrast to outward spikes, the tilt inthe short string regime decreases as the number of spikes increases. However, the tilt isalways steeper than those of outward spike solutions and folded strings, as we discuss inthe next paragraph in more details. Note that the Regge trajectory does not touch thevertical axis S = 0. As far as we know, there are no solutions at least within our ansatzthat extrapolate the trajectory to S = 0, in contrast to the outward spike case.– 18 – hort strings. Then, let us take a closer look at the short string regime: ρ , ρ (cid:28) π . (4.20)For such strings, the ρ – ρ relation (4.19) is approximated as2 πNn (cid:39) ρ ρ (cid:90) ρ ρ dρρ (cid:112) ρ − ρ (cid:112) ρ − ρ = ρ − ρ ρ π ↔ (cid:18) Nn (cid:19) ρ (cid:39) ρ , (4.21)which implies that spiky strings can have an arbitrary number of inward spikes n and anarbitrary winding number N (recall that inward spike solutions in the short string regimehave a condition n > N ). Also, the energy and the spin are approximated as E (cid:39) R πα (cid:48) × n (cid:90) ρ ρ dρ ρ ( ρ − ρ ) ρ (cid:112) ρ − ρ (cid:112) ρ − ρ = nR α (cid:48) ρ − ρ ρ = 2 N (cid:18) Nn (cid:19) Rα (cid:48) ρ , (4.22) S (cid:39) R πα (cid:48) × n (cid:90) ρ ρ dρ ρ (cid:112) ρ − ρ (cid:112) ρ − ρ = nR α (cid:48) ( ρ − ρ ) = N (cid:18) Nn (cid:19) R α (cid:48) ρ , (4.23)which reproduce the linear Regge trajectories in flat space, E (cid:39) α (cid:48) N (cid:18) Nn (cid:19) S . (4.24)We find that the tilt of the Regge trajectory decreases as the number of inward spikesincrease. In particular, in the limit of infinitely many spikes, the tilt approaches to α (cid:48) N . Long strings.
Finally, let us consider the long string regime. As depicted in Fig. 6, wealways have ρ < ρ even in the limit ρ → π for a finite n . For example, for inward spikesolutions with n = 2 and N = 1, ρ is bounded as ρ (cid:46) . < π ), which is saturatedwhen ρ = π . Therefore, the string shape does not approach to a circular form as long aswe consider a finite n . This is why the upper endpoint of the Regge trajectory does nottouch the vertical axis S = 0. This is analogous to the outward spike case, but there are noanalogue of rounded spike solutions that extrapolate the Regge trajectory of inward spikesolutions to S = 0, at least within our ansatz.We conclude this section by summarizing implications of our results. First, in theshort string regime, Regge trajectories of spiky strings have a steeper tilt than that offolded strings. This means that Regge trajectories of spiky strings are subleading Reggetrajectories (whose contributions to the Regge limit amplitudes are subleading). Second,similarly to the folded string case, each Regge trajectory has the maximum spin and energy.In particular, this property is helpful for the spectra to be consistent with the Higuchibound. This also implies that a single Regge trajectory has a finite number of higher-spinstates, in contrast to flat space and AdS. Third, we found that spiky string solutions for afixed winding number N scan a finite region on the energy-spin plane. Therefore, in orderto have an infinite number of higher-spin states, we need to take into account an infinitenumber of Regge trajectories with an increasing winding number N . It would be importantto further study implications of this result for high-energy scattering in de Sitter space.– 19 – Spiky strings with internal motion
Finally, we study spiky strings with internal motion (see Ref. [52] for the correspondingsolutions in
AdS × S ). We employ the full ansatz (2.12), under which the equations ofmotion are Eqs. (2.13)-(2.15). For later convenience, we introduce a new variable r by r = sin ρ , (5.1)which will be used mainly in the rest of the section instead of ρ . To follow the stringdynamics, we first integrate the equations of motion (2.14)-(2.15) as C = ωN r + νψ (cid:48) √D (1 − r ) , (5.2) λ = ωN r + νψ (cid:48) νω ψ (cid:48) + N (1 − ν − r ) 1 − rr , (5.3)where C and λ are real integration constants. Notice here that nontrivial solutions with ν (cid:54) = 0 exist only when ψ (cid:48) is σ -dependent, otherwise r has to be a constant. Also note thatwe have four parameters ( ω, ν, C, λ ) characterizing the solutions.Then, we reformulate Eqs. (5.2)-(5.3) such that ψ (cid:48) and ρ (cid:48) are expressed in terms ofvariables without derivatives. First, Eq. (5.3) implies ψ (cid:48) = N r λ (cid:0) − r − ν (cid:1) − ω (1 − r ) ν (1 − r − λωr ) . (5.4)Second, as discussed in Appendix A.1, we can reorganize Eq. (5.2) together with Eq. (5.4)into the form, r (cid:48) = 4 r (1 − r ) ρ (cid:48) = T r (1 − r ) ( r − r A ) ( r − r B ) ( r − r C )( r − r S ) . (5.5)This shows that for generic values of ( ω, ν, C, λ ), r (cid:48) has a double pole and three zeros, inaddition to the two double zeros located at r = 0 ,
1. The location of the double pole isdetermined by ω and λ alone as r S = 11 + λω . (5.6)On the other hand, the locations of the three zeros depend on the four parameters ( ω, ν, C, λ )in a more complicated manner, which we denote by r A , r B , and r C (see Appendix A.1 fordetails). Note that r A,B,C are complex in general. Besides, the overall constant T reads T = 4 N λ (1 + ω ) C (1 + λω ) , (5.7)which is non-negative since N is a positive integer and ω , λ , and C are real numbers.Integrating Eqs. (5.4)-(5.5) gives string solutions for given ( ω, ν, C, λ ).– 20 – .1 Outward and inward spike solutions Now we are ready to study shapes and Regge trajectories of the solutions described byour ansatz (2.12). Our task is basically parallel to the one in Sec. 4, but it is morecomplicated simply because there are more parameters of the solution. In the presentpaper, for illustration, we focus on two classes of solutions that reduce to those of theprevious section in the limit J →
0, which simplifies the analysis considerably. We callthem outward spike solutions and inward spike solutions by analogy with the solutions inSec. 4. In the following, we present properties of these solutions.
Ansatz on r A , r B , r C , and r S . In Sec. 4, we demonstrated that shapes of the stringdepend on the location of zeros and poles of ρ (cid:48) . Similarly, the outward and inward spikesolutions can be classified based on the values of r A , r B , r C , and r S . First, for both classesof solutions, r A , r B , and r C are all real and positive. Without loss of generality, we assumethat r A < r B < r C . These values relative to r S are also relevant for us, based on which weperform the following classification:Outward spike solutions: r A < r B < r S < r C , (5.8)Inward spike solutions: r S < r A < r B < r C . (5.9)In Appendix A.2, we show that in the limit J →
0, these solutions indeed reduce to theircounterparts in Sec. 4.
Reality conditions.
Next let us take care of reality conditions. First, Eq. (5.5) showsthat reality of r ( σ ) requires r A ≤ r ( σ ) ≤ r B or r ( σ ) ≥ r C (recall that the overall coefficient T is positive). Also, in order for the closed string to form a loop, r (cid:48) has to flip the signsomewhere on the worldsheet, otherwise the string stretches forever. Then, for the outwardand inward spike solutions, the string has to be inside the regime r A ≤ r ( σ ) ≤ r B . Periodicity conditions.
Finally, we take into account global structures of the string.As before, the angle ∆ φ (on the r - φ plane) between the two spikes has to be quantizedappropriately. More explicitly, for n -spike solutions, we require∆ φ = 2 πNn . (5.10)Within the ansatz (5.8)-(5.9), an explicit form of ∆ φ reads∆ φ = 2 N (cid:90) r B r A drr (cid:48) = 2 N √ T (cid:90) r B r A drr (1 − r ) | r − r S | (cid:112) ( r − r A ) ( r − r B ) ( r − r C ) . (5.11)In the present setup, we also need to take care of periodicity along the internal S . Forsimplicity, we assume that the string has no winding along the S , which implies0 = (cid:90) π dσψ (cid:48) = 2 n (cid:90) r B r A drr (cid:48) ψ (cid:48) = ± nN r S ν √ T (cid:90) r B r A dr − r λ (cid:0) − ν − r (cid:1) − ω (1 − r ) (cid:112) ( r − r A ) ( r − r B ) ( r − r C ) . (5.12) As we see shortly, the string is smooth everywhere for ν (cid:54) = 0, but we can interpret that spikes arerounded, similarly to the rounded spikes in Sec. 4. Therefore, we use the terminology “spikes” as before. – 21 – A ρ B - - - - - - ρ B ρ A - - - - - - Figure 14 . Shapes of outward spike solutions and inward spike solutions: ρ A and ρ B are defined by r A = sin ρ A (0 ≤ ρ A ≤ π ) and similarly for ρ B . For outward spike solutions, we chose ( ω, ν, C, λ ) (cid:39) (0 . , . , . , . ρ A , ρ B ) (cid:39) (0 . , . ω, ν, C, λ ) (cid:39) (1 . , . , . , . ρ A , ρ B ) (cid:39) (0 . , . Here the plus and minus signs are for outward and inward spike solutions, respectively. Aswe mentioned, there are four parameters of the solutions. If we specify the number of spikes n and the winding number N , there are two constraints originating from the periodicityconditions. Then, we are left with two degrees of freedom characterizing the size of thestring and the internal motion. Shapes.
In Fig. 14, we illustrate outward and inward spike solutions for n = 3 and N = 1.The four parameters ( ω, ν, C, λ ) are chosen such that the two periodicity conditions aresatisfied. In contrast to the case without internal motion, the spikes are indeed rounded. Regge trajectories.
Finally, we study Regge trajectories. First, substituting Eqs. (5.2)-(5.3) into Eqs. (2.17)-(2.19), we find a simplified expression for conserved charges : E = N R πα (cid:48) λC (cid:90) π dσ r (1 − r ) − C − r − λωr (5.13)= ± N R πα (cid:48) nλr S C √ T (cid:90) r B r A dr − r (1 − r ) − C (cid:112) ( r − r A ) ( r − r B ) ( r − r C ) , (5.14) S = N R πα (cid:48) C (cid:90) π dσ r (1 − r ) rλω − C − r − λωr , (5.15)= ± N R πα (cid:48) nr S C √ T (cid:90) r B r A dr − r (1 − r ) rλω − C (cid:112) ( r − r A ) ( r − r B ) ( r − r C ) , (5.16) To derive them, it is convenient to use Eq. (A.1) and Eq. (A.3) provided in Appendix. – 22 – .00 0.05 0.10 0.15 0.20 0.25 0.30 0.350.00.10.20.30.40.50.6 � � � ������ � = ���� = � � = ���� = � � = ����� = � � = ���� � � � ������ � = ���� = � � = ���� = � � = ����� = � � = ���� Figure 15 . Regge trajectories of spiky strings with internal charges. The left and right figures arefor outward and inward spike solutions, respectively. For comparison, we also illustrate the Reggetrajectory of the folded string in the dotted lines. The energy, spin and internal charge are in theunits of
R/α (cid:48) , R /α (cid:48) and R /α (cid:48) , respectively. J = N R πα (cid:48) νC (cid:90) π dσ r (1 − r ) λν − C ( λ − ω )1 − r − λωr (5.17)= ± N R πα (cid:48) nr S νC √ T (cid:90) r B r A dr − r (1 − r ) λν − C ( λ − ω ) (cid:112) ( r − r A ) ( r − r B ) ( r − r C ) , (5.18)where the plus and minus signs are again for outward and inward spikes, respectively.As we mentioned, once we specify the number of spikes n and the winding number N , we are left with two degrees of freedom associated with the size of the string and theinternal motion. If we further specify the internal charge through Eq. (5.18), we are leftwith one degree of freedom characterizing the size of the string. Then, by varying the sizeof the string, we can draw Regge trajectories for fixed n , N and J . See Fig. 15 for Reggetrajectories of outward and inward spike solutions with n = 3, N = 1, and different valuesof J . We find that as the internal charge increases, the Regge trajectory shifts upwards.Also, the maximum spin decreases and the maximum energy increases. In particular, Reggetrajectories for fixed n and N scan a finite region of the energy-spin plane. These propertiesare qualitatively the same as folded strings with internal charges and spiky strings withoutinternal charges, respectively. Besides, we find that Regge trajectories for outward spikestouch the vertical axis S = 0 twice. This explains that in the limit J →
0, outward spikesolutions reduce to both the outward and rounded spike solutions presented in the previoussection. See Appendix A.2 for more details.
In this paper, we studied a class of semiclassical strings in de Sitter space and the corre-sponding Regge trajectories. Within the rigid string ansatz (2.12), there are two classes ofsolutions: folded strings and spiky strings. First, we showed for folded strings that thereexist the maximum spin and energy in each Regge trajectory for a fixed internal charge anda fixed folding number. This means that a single Regge trajectory includes only a finitenumber of higher spin states in sharp contrast to the flat space and AdS ones. Also, asthe internal charge increases, the maximum spin decreases. While this property is helpful– 23 –or the spectrum to be consistent with the Higuchi bound, it implies that Regge trajecto-ries with a fixed folding number (and different internal charges) scan a finite region of theenergy-spin plane. We demonstrated that the same properties hold for spiky strings too.This implies that in order to have infinitely many higher-spin states (within our ansatz),one needs to consider infinitely many Regge trajectories with an increasing folding number.More intuitively, the above mentioned properties are natural consequences of de Sitteracceleration. First, the string can have a large spin if it is long and rotates with a largeangular velocity. On the other hand, causality requires that the string worldsheet cannotpropagate faster than the speed of light, which gives an upper bound on the string lengthin terms of the angular velocity. In flat space and AdS, the string stretches with an infinitelength if the angular velocity approaches to zero. In particular, the large string lengthcompetes against the smallness of the angular velocity, so that strings have larger spins asthey spread more. In sharp contrast, de Sitter space has an acceleration, so that there existsa natural cutoff dictated by causality: the string cannot rotate anymore when touching thehorizon. Therefore, the only way for a string to have a large spin is to shrink inside thehorizon, fold as much as possible, and rotate quickly. This is why string Regge trajectoriesin de Sitter space are qualitatively different from the flat space and AdS ones. Besides,de Sitter acceleration makes spiky strings fatter, leading to several new classes of solutionswhich do not exist in flat space and AdS.As a concluding remark, we would like to present several interesting future directions.The first question to ask is about high-energy behavior of string scattering in de Sitterspace. Recall that in flat space and AdS, string scattering has a mild high-energy behaviordue to existence of infinitely many higher-spin states. In particular, high-energy scatteringis captured by a widely spreading worldsheet, which implies an exponential suppression ofthe amplitudes. On the other hand, in order to have sufficiently many higher-spin states inde Sitter space, one needs to consider strings shrinking and folding inside the horizon. Atleast naively, this would suggest that the way of stringy UV completion could be differentin de Sitter space compared to flat space and AdS. It would be important to study this issuefurther by generalizing developments in holographic correlation functions in AdS [57–63],which would provide cosmological Veneziano amplitudes. A related important questionis to formulate a framework to study consistency of high-energy scattering in de Sitterspace. For example, in the case of AdS, we know what are the AdS analogues of the Reggelimit amplitudes and the hard scattering amplitudes (see, e.g., [67–76]). For de Sitterspace, there is a known flat space limit of late-time correlators corresponding to the hardscattering limit (see, e.g., [70,77,78]). However, to our knowledge, its understanding is stilllimited compared to the AdS case, even at the quantum field theory level before takinginto account stringy effects. It would be important to clarify which kinematics of whichquantities is useful to define consistency of high-energy scattering in de Sitter space. Wehope that this direction would open up a new road toward understanding of de Sitter spacein string theory. – 24 – cknowledgements
We are deeply grateful to Shinji Hirano for fruitful discussion and encouragement since theearly stage of this work. We would like to thank Guilherme Pimentel for the chance ofpresenting this work in the online workshop “Cosmological Correlators” and many helpfuldiscussions during the workshop. We also thank Pablo Soler and Bo Sundborg for usefuldiscussions. M.K. is supported in part by JSPS KAKENHI Grant Number 20K03966. T.N.is supported in part by JSPS KAKENHI Grant Numbers JP17H02894 and 20H01902. T.T.is supported in part by the Iwanami Fujukai Foundation. S.Z. is supported in part by theSwedish Research Council under grant numbers 2015-05333 and 2018-03803.
A Details of spiky strings with internal motion
In this appendix, we summarize details of spiky strings with internal motion.
A.1 Derivation of Eq. (5.5)We begin by providing a derivation of Eq. (5.5). For this, it is convenient to note thefollowing relation which follows from Eq. (5.4): ωN r + νψ (cid:48) = N λr S r (1 − ν ) − (1 + ω ) rr S − r , (A.1)where we defined r S = 11 + λω . (A.2)Substituting this into Eq. (5.2) gives D = ( ωN r + νψ (cid:48) ) (1 − r ) C = N λ r S C · r (1 − r ) ( r S − r ) · (cid:0) (1 − ν ) − (1 + ω ) r (cid:1) . (A.3)On the other hand, we can reformulate Eq. (2.16) using Eq. (5.4) as D = (cid:0) (1 − ν ) − (1 + ω ) r (cid:1) (cid:18) ρ (cid:48) − N r S ν · r (1 − r )( r S − r ) · F ( r ) (cid:19) , (A.4)where F ( r ) is a quadratic polynomial defined by F ( r ) = ( λ − ω ) r + (cid:0) (1 + λ ) ν − ( λ − ω ) (cid:1) r − ν . (A.5)Comparing Eq. (A.3) and Eq. (A.4) gives ρ (cid:48) = N λ r S C · r (1 − r )( r S − r ) · (cid:20) r ( r − (cid:0) (1 + ω ) r − (1 − ν ) (cid:1) + C λ ν F ( r ) (cid:21) . (A.6)Then, we conclude that r (cid:48) = 4 r (1 − r ) ρ (cid:48) = 4 N λ (1 + ω ) r S C · r (1 − r ) ( r S − r ) · (cid:104) r + (cid:101) F ( r ) (cid:105) , (A.7)where (cid:101) F ( r ) is a quadratic polynomial defined by (cid:101) F ( r ) = 11 + ω (cid:20) − (cid:0) (1 + ω ) + (1 − ν ) (cid:1) r + (1 − ν ) r + C λ ν F ( r ) (cid:21) . (A.8)This reproduces Eq. (5.5) by identifying r A,B,C with three solutions for r + (cid:101) F ( r ) = 0.– 25 – igure 16 . The blue curves are y = ( r − r S ) (cid:16) r − r + C ω (cid:17) , which intersect with the r -axis at r = r , ¯ r , r S . The red curves are y = ν r S r (1 − r ). The intersection points of the blue and redcurves are the three solutions r A,B,C of Eq. (A.10). In the limit ν →
0, the three solutions approachto r , ¯ r , r S , and the solutions in Sec. 4 are reproduced. For example, when r < r S < ¯ r , finite ν solutions with r A < r B < r S < r C are reduced to the outward spike solutions. A.2 J = 0 limit Finally, we discuss the limit where the internal charge vanishes J = 0. First, the internalvelocity ν and the internal space dependence ψ (cid:48) of the string have to vanish to reproducethe solutions in Sec. 4. In particular, Eq. (5.4) shows that this is achieved in the limit( λ − ω ) (cid:28) ν (cid:28) . (A.9)Note that ψ (cid:48) diverges if we take the limit ν (cid:28) ( λ − ω ) (cid:28) r A,B,C for λ = ω with a finite ν . Under this assumption, the defining equation r + (cid:101) F ( r ) = 0 of r A,B,C is reduced to( r − r S ) (cid:18) r − r + C ω (cid:19) = ν r S r (1 − r ) , (A.10)where note that r S = (1 + ω ) − in the limit λ = ω . If we further take the limit ν → r A,B,C coincides with r S . Therefore, the double pole at r = r S and one of threezeros collide and form a single pole at r = r S , which is identified with a single pole of ρ (cid:48) at ρ = ρ shown in Eq. (4.4).To see how the limit ν → r = r , ¯ r for r − r + C ω = 0 as r = sin ρ , ¯ r = sin ( π − ρ ) = cos ρ , (A.11)where ρ is identified with that in Sec. 4. Notice that we can employ this parameterizationwithout loss of generality since r + ¯ r = 1. Also, in order for r and ¯ r to be real,0 ≤ r ¯ r = C ω ≤ has to be satisfied, under which we can choose ρ such that 0 ≤ ρ ≤ π .The classification in Sec. 4 is then rephrased as • r < r S < ¯ r : outward spike solutions, • r < ¯ r < r S : rounded spike solutions,– 26 – r S < r < ¯ r : internal spike solutions.As depicted in Fig. 16, the outward spike solutions and rounded spike solutions are obtainedin the limit ν → r A < r B < r S < r C , whereas the internalspike solutions are obtained from those with r S < r A < r B < r C . References [1] U. H. Danielsson and T. Van Riet,
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