Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes

We generalize the method of our recent paper on the large-spin expansions of Gubser-Klebanov-Polyakov (GKP) strings to the large-spin and large-winding expansions of finite-size giant magnons and finite-size single spikes. By expressing the energies of long open strings in RxS2 in terms of Lambert's W-function, we compute the leading, subleading and next-to-subleading series of classical exponential corrections to the dispersion relations of Hofman-Maldacena giant magnons and infinite-winding single spikes. We also compute the corresponding expansions in the doubled regions of giant magnons and single spikes that are respectively obtained when their angular and linear velocities become smaller or greater than unity.


Introduction
The exact computation of the full spectrum of AdS/CFT correspondence [1,2,3] and the comparison of scaling dimensions of local operators of N = 4, SU(N ) super Yang-Mills (SYM) theory to the energy spectrum of free string states of type IIB superstring theory in AdS 5 × S 5 , is one first step towards the determination of the precise relationship between these two theories (that are typically treated as identical in AdS/CFT's strongest formulations). It is therefore very important to identify and study the elementary excitations that string theory in AdS 5 × S 5 and N = 4 SYM theory share and are the fundamental building blocks out of which the corresponding spectra may be built.
Giant magnons (GMs) are open single-spin strings rigidly rotating in R × S 2 ⊂ AdS 5 × S 5 found in 2006 by Hofman and Maldacena (HM) [4] and identified as the string theory duals of magnon excitations of N = 4 SYM. They are elementary excitations of the IIB Green-Schwarz superstring on AdS 5 × S 5 out of which closed strings and multi-soliton solutions may be formed. The energy-spin relation of a single giant magnon of angular extent ∆ϕ on a 2-sphere of radius R is: where ∆ϕ = p is the giant magnon's momentum. Superimposing two giant magnons of maximum angular extent ∆ϕ = π gives the Gubser-Klebanov-Polyakov (GKP) closed and folded string that rotates on S 2 [5], the dispersion relation of which is , J, λ → ∞. (1.2) According to the AdS/CFT correspondence, the energy E of a string state in AdS 5 × S 5 should equal the scaling dimension ∆ of its dual N = 4 SYM operator. Despite the finiteness of N = 4 SYM, its operators typically get renormalized and they thus acquire anomalous dimensions γ, which are the eigenvalues of the gauge theory dilatation operator. The anomalous dimensions may also be found at strong coupling by calculating the energy of their dual strings. Although there exists no systematic way by which to assign a certain gauge theory operator to its dual string state, many such heuristic identifications are known. The above GKP string that rotates inside R × S 2 for example is dual to the operator Tr Φ Z m Φ Z J−m + . . . of N = 4 SYM.
It has been known for quite some time that the one-loop dilatation operator of N = 4 SYM theory [6] has the form of an integrable psu (2, 2|4) spin chain Hamiltonian, which can be diagonalized by means of the Bethe ansatz (BA) [7,8]. An all-loop asymptotic Bethe ansatz (ABA) for the su (2) sector of N = 4 SYM 1 has been proposed by Beisert, Dippel and Staudacher (BDS) [9]. The BDS energy for single magnon (the elementary spin chain excitation that is dual to GMs) states in a spin chain of length J + 1 is: where p is the magnon's momentum. One can see that (1.3) reduces to (1.1) at strong 't Hooft coupling, λ → ∞. At weak coupling, the one-loop magnon energy is recovered to lowest order: A very important discovery made by Beisert in 2005 [10] is that the dispersion relation (1.3) can be determined uniquely from the corresponding symmetry algebra, the centrally extended su (2|2) c ⊕ su (2|2) c ⊂ psu (2, 2|4).
The asymptotic Bethe ansatz can lead to the correct form of anomalous dimensions only when the length L of the spin chain is infinite or larger than the loop order L. At and above this critical loop-order L, the range of spin chain interactions exceeds the length of the spin chain (virtual particles start circulating around the spin chain) and wrapping corrections have to be taken into account. The inefficiency of the ABA beyond the critical loop-order has been noted in both gauge [11] and string theories [12]. Conversely, the wrapping effects that appear at critical loop-order have the form of exponentially small corrections to the anomalous dimensions, as noted in [13]. The thermodynamic Bethe ansatz (TBA) [14] and the Y-system [15] are two proposals that correctly account for wrapping corrections.
On the string theory side, one equivalently calculates classical and quantum exponential corrections to the giant magnon dispersion relation (1.1), the general form of which is: (1.5) where (p) ≡ E − J and ∞ = √ λ/π · sin p/2. The first few terms of the classical finite-size expansion δ cl were first derived by Arutyunov, Frolov and Zamaklar (AFZ) in [16]: Astolfi, Forini, Grignani and Semenoff have proven in [17] that the spectrum of finite-size giant magnons in the uniform light-cone gauge is completely independent of the corresponding gauge parameter α.
By using a classical duality between strings on S 2 and solitons of the sine-Gordon model that is known as Pohlmeyer reduction [18], Klose and McLoughlin [19] have obtained the following leading terms of series (1.6): Here L eff ≡ L/ sin p/2 is the effective length and L the spatial periodicity of the spin chain. The leading term of (1.6) has also been obtained by the algebraic curve method in [20], as well as by applying the Lüscher-Klassen-Melzer (LKM) formulae [21,22] at strong coupling [23,24,25].
On the quantum level, it has been shown in [26,27] that the first quantum correction (aka one-loop shift) in infinite volume vanishes: (1.8) In finite volume the calculation of loop shifts proceeds either via the algebraic curve method [25] or by calculating the Lüscher-F and µ-terms [24]. The general form of the one-loop shift is: Formulas which allow the calculation of a n,0 and a 1,m in the above expansion have been given in [25,28]. The first term a 1,0 is given by: (1.10) Other generalizations of the GM include giant magnons on β-deformed backgrounds [29,30], TsTtransformed AdS 5 × S 5 [31,32] and AdS 4 /CFT 3 [33,34].
From the string point of view, the HM giant magnon is a close relative of yet another string sigma model solution on the 2-sphere, the single spike (SS) [35,36]. In the conformal gauge, one may obtain the single spike from the HM ansatz by interchanging the world-sheet coordinates on the 2-sphere, i.e. τ ↔ σ. 2 The corresponding dispersion relation is (∆ϕ = p, T = √ λ/2π): which can be transformed back to the dispersion relation of the giant magnon (1.1) by making the transformations πE/ √ λ − ∆ϕ/2 → p/2 and J → E − J. It has been claimed in [37] that the τ ↔ σ transform carries us from large-spin strings in R × S 2 to large-winding ones, and from the holomorphic sector of N = 4 SYM to its non-holomorphic sector. Furthermore, just as the GMs are the string theory duals of magnons, the elementary excitations above the ferromagnetic ground state TrZ J of the XXX 1/2 spin chain, the SSs are the string theory duals of the corresponding elementary excitations above the anti-ferromagnetic ground state TrS J/2 of an SO(6) SYM spin chain. 3 Being located near the top of the string spectrum, both the hoop string (the string theory dual of the anti-ferromagnetic vacuum) and single spikes are expected to be unstable [38]. They might however be stabilized by a multitude of ways, such as by adding extra angular momenta. Finite momentum effects for single spikes have been considered in [39]. The following result, contains the first, leading finite-momentum correction. In [40] the scattering of single spikes (having infinite momentum) has been studied classically, with the interesting outcome that the phase-shift is identical (up to non-logarithmic terms) with the one that was calculated by Hofman and Maldacena in [4] for giant magnons. An explanation for this fact was provided in [41] by considering single-spike scattering as factorized scattering between infinitely many giant magnons.
In a recent paper [42], we computed the leading, subleading and next-to-subleading series of exponential corrections to the infinite-volume dispersion relation of GKP strings that rotate in R × S 2 and are dual to the long N = 4 SYM operators Tr Φ Z m Φ Z J−m + . . . By the same token, following a program of study initiated in [43], we have computed all the leading, subleading and next-tosubleading coefficients in the large-spin expansion of the anomalous dimensions of twist-2 operators that are dual to long folded strings spinning inside AdS 3 . In [44] the above analysis was applied to strings rotating inside AdS 4 × CP 3 . Crucial to all of these computations was the fact that the corresponding expansions can be expressed in terms of Lambert's W-function. In the discussion section of the aforementioned paper [42], we have also included the corresponding formula for the leading, subleading and next-to-subleading series of finite-size corrections to the dispersion relation (1.1) of giant magnons, but we have not provided a proof for it. The present paper aims, besides studying the finite-size corrections of the elementary excitations of the string sigma model on R × S 2 , to provide the proof of equation (7.3) of [42].
In contrast to [42], where we started from a 2 × 2 system of equations, this time we begin from a 3 × 3 system: where E, J and p are the string's energy, spin and momentum, while x is a parameter depending on the string's angular velocity ω and its velocity v ≡ cos a. d (a, x), h (a, x), c (a, x), b (a, x), f (a, x), g (a, x) are some known power series of x and a, which can be treated as independent variables. We solve system (1.13)-(1.15) as follows. First, we eliminate the logarithm out of equations (1.14)-(1.15), obtaining an analytic expression for the linear momentum p in terms of the angular momentum J and parameters a and x. Next, this expression is inverted for a = a (x, p, J ) which is in turn plugged into equations (1.13)-(1.14) leading to a system analogous to the one encountered in [42]: Proceeding as in [42], we may obtain the dispersion relation γ ≡ E − J = γ (p, J ) as a function of the momenta p and J . The final result for γ (p, J ) is expressed in terms of Lambert's W-function: With slight modifications, our analysis may be repeated for single spikes of large winding p. This time a ≡ cos 1/ω and the logarithm is eliminated from equations (1.14)-(1.15) so as to lead to an expression J = J (a, x, p) for the the angular momentum. The latter is then inverted in terms of a = a (x, p, J ), plugged into equations (1.13), (1.15) and the method of [42] is repeated for the ensuing 2 × 2 system comprised by the energy E = E (x, J ) and the momentum p = p (x, J ).
Let us now summarize our findings. We consider finite-size giant magnons, i.e. open, single-spin strings rotating in R × S 2 . These excitations are dual to N = 4 SYM magnon excitations. By using the method that we have outlined above, we calculate finite-size corrections to the dispersion relation of the HM giant magnon (1.1), the dual N = 4 SYM operator of which is a single-magnon state: The energy minus the spin of giant magnons provide the anomalous scaling dimensions of these operators at strong coupling. The result can be expressed in terms of Lambert's W-function as follows: where the argument of the W-function is W ±16J 2 cot 2 (p/2) e −2J csc p/2−2 in the principal branch and E ≡ π E/ √ λ, J ≡ π J/ √ λ. The minus sign pertains to the branch of the giant magnon for which the linear and angular velocities satisfy 0 ≤ |v| < 1/ω ≤ 1, while the plus sign is for the branch for which 0 ≤ |v| ≤ 1 ≤ 1/ω. In [19], the former has been called "elementary" region of the GM because it corresponds via the Pohlmeyer reduction to a chain of single kinks. The latter is the "doubled" region of the GM corresponding to a kink-antikink chain. For more, see appendix A. Upon expanding Lambert's W-function, the second, third and fourth term on the r.h.s. of (1.20) provide three infinite series of coefficients which completely determine the leading, subleading and next-to-subleading contributions to the large-J finite-size corrections to the dispersion relation of the HM giant magnon: • next-to-leading terms: − 1 16J 3 tan 4 p 2 sin 2 p 2 (3 cos p + 2) W 2 + 1 6 (5 cos p + 11) • next-to-next-to-leading terms: where R ≡ 2J csc p/2 + 2. The general terms in each of these series may be found by using the Taylor expansion of the Lambert W-function (D.2) that is provided in appendix D. The first few terms are given in appendix B, equation (B.4).
We have also worked out the dispersion relations for single spikes in the large winding limit p → ∞. Single spikes are dual to the following operators of N = 4 SYM theory: (1.21) Again, as we outline in appendix A, there exist two branches for single spikes depending on the values of linear and angular velocities v and ω. In the elementary region 0 ≤ 1/ω < |v| ≤ 1 and we obtain · sin 3 q + 45 cos 2q + 148 cos q + 79 W 2 − 16q (11 + 5 cos q) cot q 2 − 37 cos 2q − 172 cos q− −79 W 3 − (11 cos 2q + 64 cos q + 85) W 4 + . . . , (1.22) where J ≡ sin q/2 and the the argument of Lambert's function is W 4p 2 csc 2 (q/2) e −(p+q)·cot q 2 , in the principal branch W 0 . In the doubled region of single spikes 0 ≤ 1/ω ≤ 1 ≤ |v|, the argument of Lambert's function obtains a minus sign, i.e. it becomes W −4p 2 csc 2 (q/2) e −(p+q)·cot q 2 , and the corresponding dispersion relation is: Expanding the W-functions in the above formulas (1.22)-(1.23), we again obtain three infinite series of coefficients which completely determine the leading, subleading and next-to-subleading series of corrections to the corresponding dispersion relations at infinite-winding: • next-to-leading terms: • next-to-next-to-leading terms: 1 64 p 4 sin 4 q 2 tan 3 q 2 2 5 + 7 cos q − 8q cot q 2 where L ≡ (p + q) cot q/2. The general terms in each of these series may be found by using the Taylor expansion of the Lambert W-function (D.2). The first few terms are given in appendix B, equations (B.7), (B.8).
Our paper is organized as follows. Section 2 contains our main result, which consists in computing the leading, subleading and next-to-subleading series of exponential corrections to the large-spin expansion of the energy of giant magnons. In section 3 we briefly present the results for the other branch of giant magnons, as well as the two branches of single spikes. A brief discussion of our results can be found in section 4. In appendix A we outline conformal finite-size giant magnons and single spikes. In appendix B we have collected our symbolic computations of the dispersion relations of finite-size giant magnons and single spikes with Mathematica. In appendix C we briefly revisit single spike scattering and single spike bound states. Appendix D contains some properties of Lambert's W-function. Appendix E contains the definitions and some useful formulae of elliptic integrals and functions.

Large-Spin Expansion of Giant Magnons
The finite-size generalization of the Hofman-Maldacena giant magnon [4] is outlined in appendix A. We shall first consider the elementary region of giant magnons (see appendix A, subsection A.1): where v and ω respectively are the linear and angular velocities of the GM. (2.1) implies where R is the radius of the 2-sphere upon which the GM lives. Setting we obtain the following system of equations for the GM: where (2.7) is obtained by plugging formula (E.11) for the complete elliptic integrals of the third kind into equation (A.16) that gives the momentum of the GM. In what follows, we will obtain the GM dispersion relation E = E (p, J ) in the regime of large (yet not infinite) angular momentum J. This in turn implies x → 0 + .

Inverse Momentum
The first step in obtaining the GM dispersion relation E = E (p, J ) consists in expressing the GM's velocity v in terms of the momenta p and J . For x → 0 + , formulas (2.4)-(2.7) contain logarithmic singularities which are due to the presence of the following pair of elliptic functions: The coefficients that appear in series (2.8) and (2.9) are given by: where n = 0 , 1 , 2 , . . . One may now eliminate the logarithms from equations (2.5), (2.7) as follows: where we have also set v = cos a (arccos 1/ω ≤ a ≤ π/2). This function may be expanded in a double series around both x = 0 and a = p/2, then it can be inverted for a by using a symbolic computations program such as Mathematica. The results of this computation may be found in appendix B (cf. equation (B.2)). The analytic function a = a (x, p, J ) may subsequently be plugged into equations (2.5)-(2.6) and then the method of [42] for inverting equation (2.5) may be used in order to calculate the inverse spin function x = x (p, J ). Inserting the latter into the corresponding formula of the anomalous dimensions (2.6) will provide the wanted answer for the GM's dispersion relation in terms of Lambert's W-function.

Inverse Spin Function
We will now invert the angular momentum series J = J (x, p), that was obtained in the previous subsection by plugging v = cos a (x, p, J ) into equation (2.5), for the inverse spin function x = x (p, J ). This will in turn allow us to obtain γ = γ (p, J ) by substituting given by equation (2.6). Let us first solve (2.5) for ln x: As in [42], equation (2.12) may equivalently be written as a series of the following type: where the coefficients a n = a n (p, J ) are determined from (2.12) and solves (2.12) to lowest order in x. Series (2.13) can be inverted via the Lagrange-Bürmann formula and the result is: where To proceed, one may expand (2.12) and so prove that the a n 's have the following form: where a nm are some known functions of the momentum p. Inserting (2.16) into (2.15) and using one may also show that the inverse spin function x = x (p, J ) has the form: where a nm are again some functions of the momentum p that are determined in terms of the a nm 's in equation (2.16) by using equation (2.15). See also equation (B.3). Specifically one may prove that all leading in J contributions to x (i.e. the terms a n,2n−2 ) are controlled by a 12 , all subleading in J contributions to x (terms a n,2n−3 ) are controlled by a 1 and a 23 , and so on up to the term a nn , i.e. x (J ) has all of its coefficients up to x n 0 J 2n−2−m (0 ≤ m ≤ n − 2) controlled by a 1 , . . . , a m , and a m+1,m+2 . Subleading terms a n0 , . . . , a n,n−1 (multiplying x n 0 J m for 0 ≤ m ≤ n − 1), depend upon the coefficients a 1 , . . . , a n−2 and a n−1,m . The proof of this statement is straightforward but rather lengthy and shall be omitted. One may nevertheless gain insight into it by plugging formula (2.16) into equation (2.15), the first few terms of which are: Having made all of these remarks about the structure of the inverse spin function x, we may now proceed to its actual evaluation. To this end, we calculate coefficients a 1 , a 2 , a 3 from equation (2.12), plug them into (2.19) and keep only the relevant terms by discarding all higher-order contributions. Then, if we use formulas (D.5)-(D.10) to transform the resulting series into Lambert's functions, we're led to the following result for the inverse spin function x = x (p, J ): The argument of Lambert's function is W −16J 2 cot 2 (p/2) e −2J csc p/2−2 in the principal branch W 0 . The structure of our formula for the inverse spin function x = x (p, J ), equation (2.20), is consistent with the observations we have made above. When expanded for J → ∞, it is also found to be in complete agreement with the inverse spin function that we have evaluated with the help of Mathematica in appendix B, cf. equation (B.3). For later purposes, we also define:

Dispersion Relation
Formula (2.20) for the inverse spin function x = x (p, J ) that we have derived, will now be plugged into (2.6) in order to furnish the anomalous dimensions γ = E − J of single-magnon states (1.19) in terms of Lambert's W-function. First, we use series (2.8)-(2.9) to expand (2.6) around x → 0 + : where the coefficients f n and g n are functions of x, p and J defined as Substituting a (x, p, J ) (see equation (B.2) of appendix B) into the above expressions and using equation (2.13) to replace ln x/x 0 , we write the dispersion relation (2.24) as follows: where f n and g n are now just functions of the momentum p and the spin J . The coefficients A n are defined as In particular, A n and f n assume the following forms: where A nm and f nm are some known functions of the momentum p. With this knowledge, one may go on and write down all the terms in the expansion (2.26) that contribute to the anomalous dimensions up to next-to-next-to-leading (NNL) order:  The argument of the W-functions is again W −16J 2 cot 2 (p/2) e −2J csc p/2−2 in the principal branch W 0 . When expanded around J → ∞, (2.30) agrees with the corresponding terms of the large-spin expansion of the anomalous dimensions that were evaluated with the help of Mathematica in appendix B (cf. equation (B.4)). All of our results are in complete agreement with the finite-size corrections to the GM (1.6) that were evaluated by Arutyunov, Frolov and Zamaklar [16], as well as the leading terms (1.7) of Klose and McLoughlin [19]. For p = π, (2.30) becomes: These are the first few terms of the corresponding GKP series, see appendix D of [42].

The Other Branches
The procedure that we have described above may be repeated for the three remaining cases that are outlined in appendix A. The results in each of them are: The doubled region of the GM (dealt with in subsection A.2) is quite similar to the elementary one (subsection A.1). The argument of Lambert's W-function (in the principal branch W 0 ) is the opposite of the previous one, i.e. it's W 16J 2 cot 2 (p/2) e −2J csc p/2−2 , while the first three leading series of terms of the dispersion relation are given by exactly the same expression as before:   [42] is repeated for the 2 × 2 system containing the momentum p = p (x, J ) and the energy E = E (x, J ). The energy minus half the string's momentum is then found to be: · sin 3 q + 45 cos 2q + 148 cos q + 79 W 2 − 16q (11 + 5 cos q) cot q 2 − 37 cos 2q − 172 cos q− −79 W 3 − (11 cos 2q + 64 cos q + 85) W 4 + . . . The arguments of the Lambert W-function are W ±4p 2 csc 2 (q/2) e −(p+q)·cot q 2 in the principal branch W 0 , with J ≡ sin q/2. The plus sign in the argument of Lambert's function corresponds to the elementary region, while the minus sign to the doubled region.

Single Spike -Doubled Region
· sin 3 q + 45 cos 2q + 276 cos q − 256 csc 2 q 2 + 463 W 2 − 16q (11 + 5 cos q) cot q 2 − 37 cos 2q− −172 cos q − 79 W 3 − (11 cos 2q + 64 cos q + 85) W 4 + . . .  where Φ is any scalar field that is not used to build Z and S ∼ X X + YY + ZZ is an SO(6) singlet composite operator of N = 4 SYM. Although giant magnons and single spikes are significantly more complex systems than GKP strings, described by a 3×3 system of equations instead of a 2-dimensional one, the inversion technique of [42] is also applicable here, as the 3 × 3 systems may be reduced to 2 × 2 ones. Again, the final results turn out to be expressible in terms of Lambert's W-function.
It would be interesting to generalize equations (2.20)-(2.30) and (3.2)-(3.3) to all subleading orders by means of general formulas or a recursive process. Just as in the case of GKP strings, we believe that the Lambert functions will keep appearing to all subsequent orders ad infinitum.
Our expressions for the inverse spin function x = x (p, J ) and anomalous dimensions γ = γ (p, J ) have been verified with Mathematica (see appendix B). Closed strings in R × S 2 can be formed as the sum of two giant magnons of maximum momentum p = π and angular momentum J/2. One may check that the magnon large-spin expansion (B.4) reduces to the GKP string with the above substitutions. However the two dispersion relations have rather different structures and the terms that are leading, subleading, etc. in the dispersion relation of the GM are different from the terms that are leading, subleading, etc. in the dispersion relation of the GKP string, with the exception of the first few terms. Thus, for momentum p = π and spin equal to J/2 the anomalous dimensions (2.30) reduce to (2.31), which are only the first few terms of the corresponding Lambert series of the GKP string.
If we expand (2.30) we shall recover (1.6), i.e. formulas (5.14) of Arutyunov-Frolov-Zamaklar [16] and (39) of Astolfi-Forini-Grignani-Semenoff [17]. 5 The Klose-McLoughlin series (1.7) is recovered from the first two terms of (2.30) by letting L eff = 2J csc p/2: for the argument of the W-function W −4 L 2 eff cos 2 (p/2) e −L eff . The Lüscher corrections that were first calculated in [23], completely agree with AFZ and therefore our results agree with both of them too. All of these findings may be further extended to the GMs of ABJM Theory.
It also seems possible that the quantum corrections to the finite-size giant magnon (1.5) may be expressible in terms of Lambert's W-function. This exercise is significantly more challenging and will be left as an open problem for the time being.
Another possible application of the W-function formalism could be the computation of the finitesize corrections to the energy of GMs in γ-deformed backgrounds [29]. 6 The form of the corresponding anomalous dimensions is very reminiscent to those of undeformed backgrounds (1.6): where n 2 is the integer string winding number and β is the real deformation parameter, satisfying |n 2 − β J| ≤ 1/2 [31].
For single spikes, a series of similar remarks applies. Expanding the W-functions in equation (3.2), we recover formulas (1.11) of Ishizeki-Kruczenski [35] and (1.12) of Ahn-Bozhilov [39] to lowest order. In appendix B, our formulas (3.2)-(3.3) have been verified with Mathematica. It could also be worthwhile to extend the W-function formalism to the generalizations of single spikes in ABJM and γ-deformed backgrounds, as well as to their quantum corrections if possible.
Our considerations have been limited to classical strings that live inside R × S 2 . The W-function parametrization should also amply apply to AdS strings. We already know that the dispersion relations of finite-size GKP strings in AdS can be expressed in terms of the W-function [42]. This formalism could also afford generalizations to other stringy AdS configurations, such as spiky Kruczenski strings [45], but most probably also to the correlation functions of sl (2) operators (see e.g. [46,47]). Higher-dimensional extended objects such as membranes may sometimes share many of the nice characteristics of strings (a point of view advocated e.g. in [48,49]) 7 so that they could also merit a more careful study in light of the Lambert W-function formalism.
We shall conclude this section with some thoughts on another possibility that we have investigated. 8 We have considered scattering between giant magnons and single spikes in the infinite-size limit, mainly by examining their solitonic counterparts in sine-Gordon (sG) and complex sine-Gordon 5 In order to compare our results with those of AFZ, we should note the difference between our definition of J ≡ πJ/ √ λ and the one of AFZ, namely J {AFZ} ≡ 2πJ/ √ λ. 6 Aka real Lunin-Maldacena backgrounds. 7 For example magnon-like dispersion relations have been obtained for membranes rotating in AdS4 × S 7 [50]. 8 The authors thank G. Georgiou for suggesting this research topic and for many interesting discussions on it. Some results stemming from these discussions have been placed in appendix C.
(CsG) theories. These excitations seem to belong to disjoint sectors of either theory, which apparently excludes 2-soliton solutions between magnons and single spikes. In the case of sine-Gordon equation, which is dual to classical strings in R × S 2 by means of the Pohlmeyer reduction, GMs correspond to kinks or antikinks that interpolate between consecutive minima of the sG potential. Single spikes correspond to unstable solutions that interpolate between consecutive maxima. We may switch between the two by the τ ↔ σ transform: (4.5) (4.5) similarly transforms all known 2-soliton solutions of sG (corresponding to scattering or bound states of two GMs) into 2-soliton solutions that correspond to scattering between two SSs. In other words, the τ ↔ σ duality effectively transforms a solution of the sine-Gordon equation that obeys the boundary condition of two GMs at ±∞, to a solution that has two SSs at ±∞. We have summarized these results in appendix C.
However, there's no known 2-soliton solution of the sine-Gordon equation that obeys the boundary condition of a GM and a SS at ±∞. Nor could such a solution be obtained by τ ↔ σ transforming another known solution. What is more, we couldn't find any such solution by superposing a GM and a SSà la Hirota [51].
The same result is arrived at by considering the complex sine-Gordon equation, which is the Pohlmeyer dual of classical strings in R × S 3 . This time, giant magnons and single spikes are dual to solutions that correspond to different signs of the CsG coupling constant β. Effectively the initial sG solitons are equipped with a complex phase factor which accounts for the extra U (1) that was added in going from the 2-sphere to the 3-sphere. The CsG solitons can be obtained by applying the Bäcklund transformation to the vacuum solutions of the theory that can be represented with an SU(2) matrix g (see e.g. [52]). The four elements of matrix g are solutions of the β > 0 and β < 0 sectors of the CsG equation, as well as its complex conjugate. One may apply the Bäcklund transformation as many times as wishes and obtain many-soliton solutions. Regrettably, no 2-soliton solution is known to us that describes a β > 0 soliton interacting with a β < 0 one.
An analogous result follows from the dressing method [40,53]. We can neither "dress" a vacuum solution in order to obtain a string solution of a GM interacting with an SS, nor can we find a solution of the string sigma model on R × S 3 that behaves as a GM at plus infinity and a SS at minus infinity. The situation seems more confusing from the dual gauge theory point of view, since what we have effectively been looking for is the possibility of scattering between a ferromagnetic and an anti-ferromagnetic magnon. We are not aware of any such scattering experiments in magnonics, which anyway seem very difficult to set up.
Another possible way to deal with the problem of magnon-spike scattering could involve the picture of single spikes as superposition of infinitely many giant magnons (see e.g. [41]). In this case one has to consider an appropriately formed coherent sum of infinitely many giant magnons (corresponding to a single spike) plus a single magnon, scattering off either in the string or the sine-Gordon picture.

Acknowledgements
We would like to thank Minos Axenides and Stam Nicolis for many illuminating discussions. Most topics of the present paper have been thoroughly discussed with George Georgiou to which the authors are very thankful. The research of E.F. is implemented under the "ARISTEIA" action (Code no.1612, D.654) and title "Holographic Hydrodynamics" of the "operational programme education and lifelong learning" and is co-funded by the European Social Fund (ESF) and National Resources. The research of G.L. supported in part by the General Secretariat for Research and Technology of Greece and from the European Regional Development Fund MIS-448332-ORASY (NSRF 2007-13 ACTION, KRIPIS).

A Finite-Size Giant Magnons and Single Spikes
In this section we outline the finite-size generalizations of giant magnons and single spikes. Let us begin by considering the generic configuration of an open bosonic string in R × S 2 ⊂ AdS × S 5 : where the line element of AdS 5 × S 5 is ds 2 = R 2 − cosh 2 ρ dt 2 +dρ 2 + sinh 2 ρ dθ 2 + sin 2 θ dφ 2 1 + cos 2 θ dφ 2 2 + +dθ 2 + sin 2 θ dφ 2 + cos 2 θ dθ 2 1 + sin 2 θ 1 dφ 2 We perform the change of variables so that z ∈ [−R, R] and φ ∈ [0, 2π). The corresponding embedding coordinates of the string become while all the remaining coordinates are zero. In the conformal gauge (γ ab = η ab ) the string Polyakov action is: 9 If we further impose the static gauge t = τ , we obtain the following set of Virasoro constraints: Now it is known that the classical string sigma model on R × S 2 can be reduced to the classical sine-Gordon model by a procedure that is known as Pohlmeyer reduction [18]. If we define ψ by the formulaẊ it can be shown that ψ solves the sine-Gordon (sG) equation: The giant magnon is an open string of R × S 2 that rotates with angular velocity ω and simultaneously translates with phase velocity v p = v · ω. It can be found by inserting the ansatz into the constraint equations (A.5)-(A.6).
Denoting ±r the open string's world-sheet endpoints, i.e. for σ ∈ [−r, r], we also impose the following boundary conditions where p is known as the string's momentum. Equations (A.5)-(A.7) become: For v · ω = 1 the trivial solution z = ζ v = ζ ω is obtained. This solution is only possible if z = 0 and v = ω = 1. Inserting z = 0 into the equations of motion and the Virasoro constraints stemming from the action (A.4) we obtain either the BMN point-like string (φ = ±τ + φ 0 ) that rotates around the equator of the S 2 , or its dual under τ ↔ σ hoop string (φ = ±σ + φ 0 ) which is wrapped around the equator of the S 2 and remains at rest.
One may prove that constraints (A.11)-(A.12) satisfy the equations of motion that follow from action (A.4), while ψ solves sG equation (A.8). We also obtain: (A.14) There exist four interesting regimes of solutions, depending on the relative values of the open string's velocity v and angular momentum ω. Below we examine each one of them separately.
A.1 Giant Magnon: Elementary Region, 0 ≤ |v| < 1/ω ≤ 1 In this case we have: The magnon's conserved momentum is found as follows: where we have defined, The conserved magnon energy and angular momentum are given by: Some basic limiting cases are worth discussing at this point. The HM [4] solution (1.1) corresponds to taking ω → 1 and J → ∞. One may obtain the closed folded GKP [5] string (1.2) by superimposing two of our GMs with velocity v = 0, maximum momentum p = π and angular momentum J/2. Imposing proper boundary conditions, the two Virasoro constraints for the giant magnon (A.11)-(A.12), admit the following solutions: where y is the floor function of y. One may draw instantanés of giant magnons, by plotting (A.21) on a sphere for various values of the velocities v and ω, −r ≤ σ ≤ r and τ = 0. See figure 1. Using Mathematica one may also animate these magnons, obtaining the worm-like motion that has been described in [16]. According to the Okamura-Suzuki [54] terminology, this is a single-spin helical string of type (i).
The periodic sine-Gordon solitons that are obtained from the corresponding Pohlmeyer reduction can be found by equation (A.13): As described in [19] this solution describes a quasi-periodic series of sG kinks, also known as kink chain/train. The period of the kink train is given by The above solution has been plotted in figure 6 for v = 0.1 and ω = 1.01. It corresponds to a linearly stable subluminal (v · ω < 1) rotational wave, according to [55].
A.2 Giant Magnon: Doubled Region, 0 ≤ |v| ≤ 1 ≤ 1/ω This is the case where The string's conserved momentum is given by the formula where again we have defined, The open string's conserved energy and angular momentum are given by: while the corresponding Virasoro constraints (A.11)-(A.12) with the appropriate boundary conditions are solved by: −vω F arccos z z max , The strings in this case have been plotted for various v's and ω's in figure 3. Their motion is a combination of rotation and translation, initially tangent to the parallel z = z max , shifting gradually towards the parallel z = −z max of the southern hemisphere and all over again. These configurations have also been described by Okamura and Suzuki [54] as type (ii) single-spin helical strings. In figure  2 we have plotted the momentum, energy and spin of the giant magnon in terms of its angular velocity ω for various values of the velocity v, in both the elementary (ω ≥ 1) and doubled (ω ≤ 1) regions.
The Pohlmeyer reduction (A.13) gives a periodic series of sG kinks and anti-kinks (known as kink-antikink chain/train): which we have plotted in figure 6 for v = 0.4 and ω = 0.3. The half-period of the train is It's an spectrally unstable subluminal (v · ω < 1) librational wave [55].

A.3 Single Spike: Elementary
In this case, while the conserved momentum is found to be: with the assignment The energy and the angular momentum of the strings are given by: Equations (A.11)-(A.12) have the following solutions: Single spike strings in the elementary region may be visualized by plotting equation (A.39) on a sphere, giving the shape pictured on the left of figure 4. Apart from the winding, their motion resembles that of elementary giant magnons in A.1. For v → 1, p → ∞ we obtain a single spike that is infinitely wound around the equator of S 2 and is known as the hoop string.
Again, we have defined, The conserved energy and angular momentum of the open string are given by: The Virasoro constraints (A.11)-(A.12) are solved by: The single spike of the doubled region has been plotted on the right of figure 4. The string gradually unwinds from the north pole and starts winding around the south pole. Then the motion repeats. In figure 5, we have plotted the momentum, energy and spin of spiky strings in both the elementary (v ≤ 1) and doubled region (v ≥ 1) in terms of the velocity v, for various values of the angular velocity ω.
The Pohlmeyer reduction leads again to a kink-antikink chain/train, similar to the one of giant magnons in the doubled region A.2: This quasi-periodic solution to the sG equation has been plotted for v = 1.4 and ω = 3 in figure 6.
The half-period of the kink train is given by and it corresponds to a (spectrally) unstable superluminal (v · ω > 1) librational waves [55].

A.5 Symmetries
Although all of the above cases have received a rather independent treatment, one cannot fail to notice that, since these solutions essentially follow from the same basic system of equations (A.11)-(A.13), they ought to be interrelated somehow, or bear some relationship to one another. The first symmetry present in our system is known as τ ↔ σ symmetry aka "2D duality" [37]: 2D duality maps the GM elementary region to the SS elementary region and the GM doubled region to the SS doubled region.
A second intriguing correlation between the four regions was partially revealed in [19], where it was shown that the Pohlmeyer reduction of the finite-size giant magnon has three different branches, depending on the value of the parameter m of the corresponding Jacobi's amplitude function. Roughly speaking, for parameter values m < 1 one is in the "elementary region", where the Pohlmeyer reduction is the familiar kink train of subsections A.1 and A.3, while for parameter values m > 1 we obtain the "doubled region" of kink-antikink trains that we encountered in subsections A.

B Symbolic Computations
This appendix contains some of our symbolic computations on giant magnons and single spikes. We follow the method that was outlined in subsections 2.1-2.2. First we obtain the inverse spin function of giant magnons x = x (p, J ) by inverting the angular momentum series (2.12) with Mathematica. Then x = x (p, J ) is plugged into the corresponding series for the anomalous dimensions γ = γ (x, p), leading to the wanted dispersion relation γ = γ (p, J ). For brevity, only the first few terms of each series are presented here. All of our results agree with the Lambert W-function formulas that were derived in our paper and we summarized in the introduction. Setting we obtain the following results.
• Finite-Size Giant Magnons: Elementary Region, 0 ≤ |v| < 1/ω ≤ 1.  Single spikes are a little different from giant magnons. We may understand this qualitatively by comparing the plots in figures 2 and 5. In the former, the energy E and the spin J are divergent for ω → 1, while the linear momentum p always stays finite and less than π. In the latter, it is the energy E and momentum p that diverge as v → 1, while the spin J is finite and less than 1. This type of behavior signals a different dispersion relation, a different form for the corresponding finite-size corrections and demands a rather modified inversion technique. More has been said in subsection 3.2. Setting the following results are obtained.
• Finite-Size Single Spikes: Elementary Region, 0 ≤ 1/ω < |v| ≤ 1.  • Finite-Size Single Spikes: Doubled Region, 0 ≤ 1/ω ≤ 1 ≤ |v|.  In this appendix we shall briefly revisit the computation of the semiclassical scattering phase-shifts of (infinite-size) giant magnons and single spikes. The giant magnon phase-shift was calculated by Hofman and Maldacena in [4] by considering the kink-antikink solution of the corresponding (via the Pohlmeyer reduction) sine-Gordon equation. The result for the scattering between two giant magnons of linear momenta p 1 and p 2 is: The presence of the last term in (C.1) depends on the choice of the worldsheet gauge and the definition of the worldsheet variable σ. It may be omitted, so that for sin p 1,2 /2 > 0 the phase-shift becomes, This phase-shift is equal to the 2-magnon, strong-coupling "dressing phase" σ 2 12(AFS) = e iδ 12 that was proposed by Arutyunov, Frolov and Staudacher (AFS) in [56] as the string theory-complement of the su (2), all-loop asymptotic Bethe ansatz equations of Beisert, Dippel and Staudacher (BDS) [9].
In [53] the string solution for the scattering between two giant magnons was derived by applying what is known as the dressing method 10 to a point-like string that rotates at the equator of S 2 . By similarly dressing the hoop string 11 one may write down the scattering solution between two single spikes, from which the corresponding phase-shift can be calculated. This has indeed been done in [40], leading to the result where q is defined from J = 1 − 1/ω 2 1/2 ≡ sin q/2 and ω and J are the spike's angular velocity and conserved angular momentum respectively. Obviously, for p ↔ q (C.3) agrees with the phase-shift for giant magnons (C.2) up to the non-logarithmic term q sin q/2. Okamura [41] provided a qualitative explanation for the agreement between the logarithmic terms of the two formulas, by regarding single spike scattering as factorized scattering of infinitely many giant magnons.
We will now provide an alternative derivation of (C.3) by τ ↔ σ transforming the sG solitons that correspond to giant magnons and their scattering solutions. 12 As we have already explained in appendix A, the string sigma model on R×S 2 can be Pohlmeyer-reduced to the following sine-Gordon equation:ψ It can be shown that (C.4) contains the following soliton-soliton scattering solution: (C.5) This solution has topological charge 13 Q = +2 and corresponds to two giant magnons that scatter in their center of mass frame. When it is τ ↔ σ transformed according to continues to satisfy (C.4) and has a topological charge of Q = 0, that is it corresponds to solitonantisoliton scattering. If we further set v = 1/ω < 1, we obtain a solution of sG equation that represents the scattering of a single spike soliton with its corresponding antisoliton: In figures 9 and 10 we have plotted the sG wavefunctions and energy densities for the kink-antikink scattering solutions that correspond to giant magnons (left) and single spikes (right) for v = 0.5 and ω = 2. In a similar fashion we may obtain the solutions of sG equation that correspond to solitonsoliton and antisoliton-antisoliton (Q = ±2) scattering between single spikes.
To obtain the phase-shift for single spikes, we may repeat the analysis of Hofman and Maldacena for the τ ↔ σ transformed solution (C.7). The result is the same if the soliton-soliton or the antisoliton-antisoliton solutions are used instead. In a reference frame where the soliton has velocity v 1 and the antisoliton has velocity v 2 , the corresponding time delay is found to be: where coshθ i ≡ γ i = (1 − v 2 i ) −1/2 = csc q i /2 for i = 1, 2. Just as for giant magnons, this quantity is negative in the center of mass frame, which means that the force between the two solitons is attractive. We find: These results are valid for sin q i /2 > 0. 13 The topological charge Q is defined according to Q = 1/π +∞ −∞ ∂σψ dσ.

C.2 Bound States
One may similarly τ ↔ σ transform any of the N-soliton solutions of sG equation and obtain new (possibly unstable) solutions. For example the breather (Q = 0) solution, tan ψ b 2 = sin aγ a τ a cosh γ a σ , γ a ≡ 1 √ 1 + a 2 (C. 12) becomes under the τ ↔ σ transform: tan ψ b 2 = cosh ωγ ω τ − ω sin γ ω σ cosh ωγ ω τ + ω sin γ ω σ , (C. 13) which again satisfies the sine-Gordon equation. In figure 11, we have plotted the wavefunction (left) and the energy density (right) of this sG solution for ω = 2. The solution is initially constant at ψ = π/2, then between times τ = −τ 0 and τ = 0 its amplitude and energy gradually grow until they become the wiggly periodic curves of figure 11 with extrema at σ = kπ/2γ ω . After that, both curves start decreasing again towards the constant initial value of ψ = π/2 at τ = τ 0 . A stable 3-soliton solution of sG, comprised by a breather and a kink (or antikink), is known as the "wobble" [57,58]: where (C. 16) This solution also exhibits the "flare"-like behavior of the breather that we saw above.

D Lambert's W-Function
Since our paper relies essentially on Lambert's W-function, we shall review here some of its basic properties. The Lambert W-function is defined implicitly by the following relation: where the unsigned Stirling numbers of the first kind, n + m n + 1 are defined recursively from [60]: Using the defining property (D.1), we may obtain simplified expressions for the derivatives and antiderivatives of Lambert's W-function. Here are some useful expressions that we employ in our paper: x W (x) = ∞ n=1 (−1) n+1 n n n! · x n = W (x) 1 + W (x) (D.6) x x W (x) = ∞ n=1 (−1) n+1 n n+1 n! · x n = W (x) (1 + W (x)) 3 (D.7) Figure 12: Lambert's W-function.

E Elliptic Integrals and Jacobian Elliptic Functions
This appendix contains the definitions and some basic properties of elliptic integrals and Jacobian elliptic functions that we use in our paper. Our conventions mainly follow Abramowitz-Stegun [61].

Elliptic Integral of the First Kind
A very useful addition formula for complete elliptic integrals of the third kind, that allows to isolate their logarithmic singularities, can be found in [62]: