# Salient features of dressed elliptic string solutions on \(\mathbb {R}\times \hbox {S}^2\)

## Abstract

We study several physical aspects of the dressed elliptic strings propagating on \(\mathbb {R} \times \mathrm {S}^2\) and of their counterparts in the Pohlmeyer reduced theory, i.e. the sine-Gordon equation. The solutions are divided into two wide classes; kinks which propagate on top of elliptic backgrounds and non-localised periodic disturbances of the latter. The former class of solutions obey a specific equation of state that is in principle experimentally verifiable in systems which realize the sine-Gordon equation. Among both of these classes, there appears to be a particular class of interest the closed dressed strings. They in turn form four distinct subclasses of solutions. One of those realizes instabilities of the seed elliptic solutions. The existence of such solutions depends on whether a superluminal kink with a specific velocity can propagate on the corresponding elliptic sine-Gordon solution. Unlike the elliptic strings, the dressed ones exhibit interactions among their spikes. These interactions preserve an appropriately defined turning number, which can be associated to the topological charge of the sine-Gordon counterpart. Finally, the dispersion relations of the dressed strings are studied. A qualitative difference between the two wide classes of dressed strings is discovered.

## 1 Introduction

Classical string solutions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] have enlightened several interesting features of the holographic duality [12, 13, 14, 15] and have provided a framework for non-trivial checks of its validity. String solutions in symmetric spaces, which are relevant to holography, such as the sphere or the AdS spacetime, as well as their tensor product, have been the subject of extensive study in the literature. Furthermore, the study of such solutions facilitates the qualitative understanding of the classical dynamics of the system whose quantum version is the only known consistent quantum theory of gravity.

The sigma models that describe string propagation on symmetric spaces have the additional interesting feature of integrability, which provides several non-trivial tools for the construction of string solutions. Taking advantage of integrability methods, it is possible to find general expressions for string solutions on specific symmetric spaces, in terms of hyperelliptic functions [16, 17]. These suggest a natural classification of the solution in terms of the genus of the relevant algebraic curve. Although this kind of treatment has the advantage of being very generic, the understanding of the physical properties of the solutions in this language is rather limited. This is due to fact that the behaviour of the hyperelliptic functions is much more complicated than that of simpler elliptic or trigonometric/hyperbolic functions. As an exception, the genus one solutions, i.e. elliptic solutions, can be expressed in terms of elliptic functions and their properties have been extensively studied.

Another signature of the system’s integrability is the fact that the sigma models, which describe the propagation of strings in symmetric spaces, are reducible to integrable systems of the family of the sine-Gordon equation, the so called symmetric space sine-Gordon models (SSSG) [18, 19, 20, 21]. This property is formally called *the Pohlmeyer reduction* [22, 23]. It can, in principle, facilitate the study of string propagation, as the SSSGs have been quite extensively studied in the literature. Many non-trivial solutions of those are known. However, the Pohlmeyer reduction is a non-local, many-to-one mapping that is difficult to invert.

The Pohlmeyer reduction sheds new light on another interesting property of the aforementioned sigma models. It is possible to construct new string solutions given an initial seed one, by means of solving a simpler auxiliary system instead of the original equations of motion. This procedure is the so called “dressing method” [24, 25, 26] and it is the analogue of the Bäcklund transformations on the side of the reduced integrable theory. The dressing transformation, as well as the Bäcklund transformations add one extra genus on the initial solution. This is a degenerate one, as one of the related periods is divergent.

Although the Pohlmeyer reduction is difficult to invert, a systematic approach for its inversion in the case of elliptic solutions has been developed recently [27], in the case of strings propagating on \(\hbox {AdS}_3\) or \(\hbox {dS}_3\). We have extended the method for the case of strings propagating on the sphere \(\mathrm {S}^2\) [28]. It was shown that the method leads to a unified and simple description of all elliptic solutions in terms of the Weierstrass elliptic function. In a subsequent work [29], we took advantage of this simple description to construct dressed elliptic solutions through the dressing method, i.e. degenerate genus two solutions. An advantage of this construction is the expression of the solutions in terms of simple elliptic and trigonometric functions, which makes many of their physical properties accessible to study. In the present work we focus on salient aspects of the above solutions, such as spike interactions, implications to the stability of the seed solutions and their dispersion relations.

An interesting feature of the elliptic string solutions is the fact that they have several singular points, which are spikes. These can be kinematically understood, as points of the string that propagate at the speed of light [1] due to the initial conditions. As they cannot change velocity, no matter what forces are exerted on them, they continue to exist indefinitely, as long as they do not interact with each other. In the already studied spiky string solutions [6, 7, 8, 9, 10, 28], the spikes rotate around the sphere with the same angular velocity, and thus, they never interact. Interacting spikes emerge in higher genus solutions. The simplest possible such solutions are those which are constructed via the dressing of elliptic strings [29].

The stability of the elliptic strings is closely related to the stability of their Pohlmeyer counterparts, which are either trains of kinks or trains of kinks–antikink pairs. Although the latter is known [30], it is not easy to construct an explicit non-perturbative solution exposing the instability of the elliptic strings. Naively, such a solution has to be a degenerate genus two solution. In this case, one of the two periods must coincide to the periodicity of the original elliptic solution under study. On the other hand, the degenerate one will describe the infinite evolution which either asymptotically leads to or away from the elliptic solution. Therefore, the dressed elliptic strings are conducive to the determination and study of the instabilities of the elliptic ones.

The structure of the present paper is as follows: in Sect. 2, we review some elements of the construction of the dressed elliptic strings on \(\mathrm {S}^2\) that are necessary for the study of their physical properties. In Sect. 3, we elucidate the properties of the sine-Gordon counterparts of the dressed elliptic string solutions, in order to both facilitate the study of the latter and furthermore establish a mapping between the properties of the string solutions and their counterparts. In Sect. 4, we study the constraints which have to be imposed on the dressed string solutions, so that they are closed. In effect they emerge to belong to four distinct classes. In Sect. 5, we study the time evolution of the string solutions focusing on the interaction of spikes. In Sect. 6, we study a specific class of dressed string solutions that reveals instabilities of a subset of the elliptic string solutions. In Sect. 7, we calculate the energy and angular momentum of the dressed elliptic strings, which have great interest in the context of the holographic dualities. In Sect. 8, we discuss our results. Finally, there is an appendix containing some technical details on the asymptotic behaviour of the dressed strings and the calculation of the conserved quantities.

## 2 Review of dressed elliptic string solutions

*X*, whereas the notation \(\mathbf {X}\) is used for the three-vector composed by the spatial components of

*X*. The inner product of two four-vectors

*A*and

*B*with respect to the Minkowski metric \(g=\mathrm {diag}\{ -1, 1, 1, 1 \}\) is denoted as \(A\cdot B\).

A well-known method for constructing solutions of the equations of motion of (2.1) is the so-called dressing method [24, 25, 26], which connects solutions of the sigma model equations in pairs. Given a solution of the latter, henceforth called the seed solution, one can apply the above method in order to generate a new non-trivial one [31, 32, 33, 34, 35]. Recently, an application of the dressing method appeared [29], where elliptic string solutions [28] were used as seeds. The study of the physical properties of the above dressed solutions is the subject of this work.

*E*is an integration constant that can take any value larger than \(-\mu ^2\). The solutions with \(E<\mu ^2\) are called oscillatory, whereas the ones with \(E>\mu ^2\) are called rotating, in an analogy to the simple pendulum that was established in [28]. Clearly there are four classes of solutions characterized as translationally invariant-oscillating or -rotating and static-oscillating or -rotating.

*a*takes values on the imaginary axis and it is a free parameter of the solution, which reflects the fact that Pohlmeyer reduction is a many to one mapping. The value of

*a*is specified by demanding that the string (2.5) obeys the correct periodicity conditions, so that the solution is a finite closed string. Furthermore,

*a*is connected to the parameter \(\ell \) through

*a*and \(\ell \).

Taking appropriate limits of the parameters involved in (2.5) results in some well known string solutions. In the case of a static Pohlmeyer counterpart, the giant magnons can be recovered in the limit \(E=\mu ^2\) and the GKP strings in the limit \(a=\omega _2\). Had one considered the solutions with translationally invariant counterpart, one would obtain the BMN particle for \(E=-\mu ^2\) and the single spike solution for \(E=\mu ^2\). The reader is referred to [28] for further details on the elliptic string solutions.

*f*is appropriately constrained in the aforementioned coset space. The mapping

*f*of the non linear sigma model, by solving the system (2.14), supplemented by the condition \(\varPsi \left( 0 \right) =f\). Then, one may construct a new solution of the auxiliary system of the form \(\varPsi ' \left( \lambda \right) =\chi (\lambda )\varPsi \left( \lambda \right) \), by utilizing an appropriate

*dressing factor*\(\chi (\lambda )\). Finally, taking \(\lambda \) to zero, one obtains a new solution of the sigma model equations of motion, namely, \(f'=\chi (0)\varPsi \left( 0 \right) \). The latter is called the dressed solution. The dressing factor, which is in general a meromorphic function of the complex parameter \(\lambda \), must obey certain conditions, that ensure that the dressed solution is still an element of the coset \(\mathrm {SO}(3)/\mathrm {SO}(2)\). The simplest possible dressing factor, which fulfils these requirements, has only two poles, complex conjugate to each other, that lie on the unit circle. It reads

*p*is any constant complex vector obeying the conditions \({p^T}p = 0\) and \(\bar{p} = \left( I - 2{X_0}X_0^T \right) p\). The dressing transformation also induces a change in the (left) sigma model charge

^{1}of the seed solution

*U*reads

*a*of the elliptic solution (2.5). The parameter \(\tilde{a}\) is defined through the equations

*U*, via the equations \({U^T}\left( {{\partial _i}U} \right) = k_i^j{T_j}\), where \(T_i\) are the generators of the lie group \(\mathrm {SO}(3)\) defined as usual.

*X*, yields \(X'=U\hat{X}'\). The time component \(t'\) of the dressed solution is the same as the one of the seed solution (2.5), since the dressing transformation acts only on the \(\hbox {S}^2\)-part, i.e. the spatial part of the seed solution. In order to get the corresponding solution, in the case of a seed with translationally invariant Pohlmeyer counterpart, one should simply exchange \(\xi ^0\) with \(\xi ^1\).

*a*is related to the position of the poles of the dressing factor through

*A*and

*B*are completely determined by the seed solution of the sine-Gordon equation as

## 3 Properties of the sine-Gordon counterparts of the dressed elliptic strings

It has been shown that many physical properties of the elliptic strings solutions are directly connected to properties of their sine-Gordon counterparts [28]. The establishment of this mapping enhances the intuitive understanding of the dynamics of string propagation on the sphere via the dynamics of the sine-Gordon equation, which is a much simpler system. For this purpose, in this section, we will study some basic properties of the sine-Gordon counterparts of the dressed elliptic string solutions reviewed in Sect. 2.

The dressed strings, as well as their sine-Gordon counterparts can be classified into two large categories depending on the sign of the constant \(D^2\). When \(D^2 > 0\) (or equivalently when \(\tilde{a}\) lies on the real axis), Eq. (2.36) describes a localized kink travelling on top of an elliptic background. The position of the kink can be identified with the position where the argument of the tanh in Eq. (2.36) vanishes, namely \({{\xi ^1} = - i \varPhi \left( {{\xi ^0};\tilde{a}} \right) }/D\), where it holds that \(\varphi = \hat{\varphi }\). Far away from this region, the solution assumes a form that is determined solely by the seed solution. As we have commented in Sect. 2, a Bäcklund transformation increases the genus of the solution by one, adding a degenerate hole to the relevant torus, which corresponds to a diverging period. This is evident in this case, where the two periods appearing in the solution are the one of the seed solution and the infinite time/space required to accommodate the kink.

The minimum value of the parameter \(D^2\) is \(D_{\text {min}}^2=(\mu ^2-E)/2\). Thus, when a rotating seed is considered, it is possible that \(D^2 < 0\) (or equivalently \(\tilde{a}\) lies on the imaginary axis shifted by the real half period \(\omega _1\)). In such a case, the hyperbolic tangent function appearing in the dressed solution becomes trigonometric tangent. As a result, the effect of the dressing on the solution is not localized in the position where the argument of this function vanishes, but it is rather spread everywhere in a periodic fashion. It follows that these solutions do not describe a kink propagating on an elliptic background. They should be understood as a periodic structure of oscillating deformations on top of a rotating elliptic background. Such solutions contain two periods; one of the seed solution and one imposed by the aforementioned trigonometric tangent. However, it is the imaginary period of the trigonometric tangent that is divergent, and, thus, these solutions are still degenerate genus two solutions, in this manner similar to the solutions of the \(D^2>0\) class.

It follows that a bifurcation of the qualitative characteristics of the dressed solution occurs at \(D^2 = 0\).

### 3.1 \(D^2>0\): kink–background interaction

The translationally invariant background kink solutions for all *a* and *k*

Parity of | \(a \in \left( - \infty , -1 \right) \) | \(a \in \left( -1, 0\right) \) | \(a \in \left( 0, 1 \right) \) | \(a \in \left( 1, \infty \right) \) |
---|---|---|---|---|

| Left moving antikink | Right moving antikink | Right moving kink | Left moving kink |

| Left moving kink | Right moving kink | Right moving antikink | Left moving antikink |

The static background kink solutions for all *a* and *k*

Parity of | \(a \in \left( - \infty , -1 \right) \) | \(a \in \left( - 1, 0 \right) \) | \(a \in \left( 0, 1 \right) \) | \(a \in \left( 1, \infty \right) \) |
---|---|---|---|---|

| Right moving antikink | Left moving kink | Left moving antikink | Right moving kink |

| Right moving kink | Left moving antikink | Left moving kink | Right moving antikink |

These four classes of solutions are the physical depiction of the fact that the same value of \(D^2\) can be obtained for four distinct values of the Bäcklund parameters *a*. The definition of the sign of the function *A* (2.40) has been made so that all four classes of solutions can be accessed with the same formula, simply varying the parameter *a*, in a similar manner to the usual analysis of kinks built using the vacuum as the seed solution. The special case \(a = \pm 1\) corresponds to static kinks/antikinks leading to only two physical distinct cases.

### 3.2 \(D^2>0\): kink velocity

*the specific frame*, where the background is translationally invariant (Fig. 4).

*E*there is a global maximum. Bearing in mind the pendulum picture for the translationally invariant elliptic solution of the sine-Gordon equation, the criterion (3.14) is equivalent to demanding that the mean potential energy of the pendulum vanishes.

*E*. To sum up, only when \(E<E_c\), all kinks moving on the elliptic background are subluminal. When \(E>E_c\), there is always a range of \(\tilde{a}\) corresponding to superluminal kinks.

In the case of the static seed, only when \(E>\mu ^2\) all kinks propagating on the elliptic background are subluminal. When \(E<\mu ^2\), there is always a range of \(\tilde{a}\) which gives rise to superluminal kinks.

### 3.3 \(D^2>0\): periodic properties

The periodic properties of the dressed elliptic solutions have been disturbed due to the presence of the kink, which needs infinite time to complete. However, the new solution still has some interesting periodic properties.

Firstly, in the region far away from the location of the kink \(\left| {D{\xi ^1} + i\varPhi \left( {{\xi ^0};\tilde{a}} \right) } \right| \gg 1\), the solution tends to a shifted version of the elliptic seed solution. Therefore, at this region, the periodic properties (3.19) and (3.20) are approximately recovered.

In a trivial manner, one can obtain the corresponding periodic properties of the dressed elliptic solutions with static seeds, after the interchange \(\xi ^0 \leftrightarrow \xi ^1\).

### 3.4 \(D^2>0\): energy and momentum

The elliptic solutions of the sine-Gordon equation lead to simple expressions for most of the elements of the energy-momentum tensor (see e.g. [28]). Namely, \(T_{\mathrm{ti}}^{00} =T_{\mathrm{st}}^{11} = E\) and \(T_{\mathrm{ti}}^{01} =T_{\mathrm{st}}^{01} = 0\). However, the elements \(T_{\mathrm{ti}}^{11}\) and \(T_{\mathrm{st}}^{00}\) are non-trivial functions of \(\xi ^0\) and \(\xi ^1\) respectively.

*D*. It is now clear why the quantity \(D^2\) is a decreasing function of the energy constant

*E*, since the larger the background energy, the smaller the necessary energy for a kink to jump from the region of one vacuum to the region of the neighbouring one. Furthermore, it is also physically expected that the kink energy is a decreasing function of the background time delay \(2 \tilde{a}\). As the latter gets larger approaching \(\omega _1\), the jump is facilitated and less energy is required for this purpose (see Fig. 2).

*a*, the above imply

*D*and \(\tilde{a}\) [29]. Interestingly, both parameters have a simple physical meaning. The parameter

*D*is directly related to the energy of the kink in the case of a translationally invariant seed solution (Eq. (3.27)) or its momentum in the case of a static one (Eq. (3.36)). The parameter \(\tilde{a}\) directly measures the degree of interaction of the kink with the elliptic background. In the case of a translationally invariant seed, it is directly related to the time delay in the background field oscillation induced by the kink (Eq. (3.4)); in the case of a static seed, it is related to the spatial displacement of the static background (Eq. (3.6)). Bearing in mind that there are not two independent parameters in this class of solutions, but only one (the Bäcklund parameter

*a*), there is a relation connecting the energy/momentum of the kink to the effect that it has on the background. This reads

### 3.5 \(D^2<0\): periodicity

These solutions do not describe a localized kink propagating on top of an elliptic background. They are actually a periodic disturbance propagating on top of a translationally invariant rotating elliptic background. This transition of the qualitative characteristics of the solution is in a sense similar to the well-known behaviour of the solutions that occur after the action of two Bäcklund transformations of the vacuum. These solutions form two classes; one class of two-kink scattering solutions and one class of bound states, the so called breathers. Having this picture in mind, we may understand the Bäcklund transformed elliptic solutions with \(D^2>0\) as the analogue of the scattering solutions, since the kink induced by the Bäcklund transformation propagates on top of the train of kinks that forms the elliptic background, interacting with it, causing a delay/translation. On the contrary, the solutions with \(D^2<0\) are the analogue of the breathers. Of course instead of a single oscillating breather, these solutions are a whole periodic structure of such oscillating formations, a “train of breathers”.

*c*. If the number

*c*is a rational number of the form \(\alpha / \beta \), where \(\gcd \left( {\alpha , \beta } \right) = 1\), then \({{\tilde{\varphi } }}\) will be a quasi-periodic function of \(\xi _0\) with period \(4 \beta \omega _1\) and the quasi-periodicity property \({{\tilde{\varphi } } }\left( {{\xi ^0} + 4\beta {\omega _1},{\xi ^1}} \right) = \hat{\varphi } \left( {{\xi ^0},{\xi ^1}} \right) + 2\pi \left( \alpha + \beta \right) \). On the contrary, if the number

*c*is irrational, then \({{\tilde{\varphi } }}\) will not be periodic in \(\xi _0\). In Fig. 9, a periodic and a non-periodic example are shown.

### 3.6 \(D^2<0\): energy and momentum

### 3.7 The \(D \rightarrow 0\) limit

*a*, namely \(a = \pm \sqrt{E \pm \sqrt{{E^2} - {\mu ^4}} } /\mu \), which set

*D*equal to zero. Half of those correspond to a localized solution that generates an overall jump to the background solution equal to \(- 4\pi \). For the other half, the solution is equal to \(\hat{\varphi }\), thus a periodic, translationally invariant solution. It turns out that in this specific case, \(\hat{\varphi }\) coincides with an elliptic solution, as the corresponding parameter \(\tilde{a}\) is equal to \(\pm \omega _1\), namely,

The total energy and momentum of these solutions exactly match those of the seeds in this limit, not only in the case the dressed solution is a trivial displacement of the seed, but also in the non-trivial cases.

## 4 Asymptotics and periodicity of the dressed elliptic strings

In this and the following three sections, we will study some properties of the dressed elliptic string solutions [29] that we reviewed in Sect. 2 and compare them to the properties of their Pohlmeyer counterpart that we presented in Sect. 3. Here, we determine the appropriate values of the moduli that result in closed string solutions.

The dressed string solutions, similarly to their elliptic seeds, are naturally infinite string solutions. They are parametrized by the spacelike coordinate taking values in the whole real axis. However, the periodic properties of the sine-Gordon counterparts of the elliptic strings (3.19) and (3.20) imply that the string solution obeys appropriate periodicity conditions for specific values of the moduli [28], giving rise to finite string solutions.

In the case of the dressed elliptic strings with \(D^2>0\), the sine-Gordon counterparts cease to obey periodicity conditions of the form (3.19) and (3.20) due to the existence of the extra kink that propagates on the non-trivial elliptic background. However, the above periodic properties are recovered in the region far away from the position of the kink, as the sine-Gordon solution tends to a shifted version of the elliptic seed. This asymptotic behaviour can be used to construct approximate finite closed dressed elliptic string solutions in the same manner as the closed finite elliptic strings. In order to do so, we first need to study the asymptotics of the dressed elliptic string solutions with \(D^2>0\).

Even though the dressed solutions do not have the extended periodicity properties of their elliptic seeds, they still obey the periodic properties (3.21) and (3.22) in the case \(D^2>0\), as well as (3.42) and (3.43) in the case \(D^2<0\). One can take advantage of these periodic properties in order to construct exact finite closed string solutions. It has to be noted that the above equations are expressed in the linear gauge; however, the closed string solution should exhibit appropriate periodicity in their dependence on the spacelike coordinate in the static gauge. In the following, we present all these classes of closed string solutions and derive the appropriate constraints that the moduli should obey for each class.

### 4.1 \(D^2>0\): the asymptotics of the dressed strings

It follows that, in the case of rotating backgrounds, the dressed solutions with Pohlmeyer counterparts, which are kinks or antikinks propagating on top of a train of kinks, have been separated into two classes. Recalling the epicycle description of the action of the dressing on the string solution^{2} [29], their difference is the following: the class with \(\theta _1 < \tilde{\theta }_-\) asymptotically tends to the seed solution rotated around the *z*-axis by an appropriate angle; the class with \(\theta _1 > \tilde{\theta }_+\) asymptotically tends to the seed solution, first inverted with respect to the origin of the enhanced space and then rotated appropriately around the *z*-axis. Finally, notice that \(\varDelta \varphi \) tends to 0 at the limits \(\theta _1 \rightarrow 0\) and \(\theta _1 \rightarrow \pi \) as expected, since the epicycle becomes a point.

### 4.2 \(D^2>0\): approximate finite closed strings

Strictly speaking, it is not possible to fix the parameters of the solution, so that a dressed string with \(D^2 > 0\) satisfies appropriate periodicity conditions (except for very specific cases that we will study in Sect. 4.5). In the elliptic strings case, the functions \({\theta _{\mathrm{seed}}}\) and \({\varphi _{\mathrm{seed}}}\) have the periodic properties (4.7) and (4.8). Therefore, arranging the solution parameters so that \(\delta \varphi = 2\pi /n\) where \(n \in \mathbb {Z}\), in the case of a rotating counterpart, and \(n \in 2\mathbb {Z}\) in the case of an oscillating one, results in a well defined, closed string of finite length, parametrized by \(\sigma ^1 \in \left[ 0, n \delta \sigma _{0/1} \right) \). However, when one considers dressed strings with a Pohlmeyer counterpart that is a kink propagating on an elliptic background, in general these functions are not periodic/quasi-periodic due to the presence of the kink.

*z*-axis and thus, it is possible that they do not contain self-intersection. In general, one could consider a generalization of (4.26) where the left hand side is \(2 \pi m\), where \(m \in \mathbb {Z}\). In such a case, the seed and the dressed solutions are both closed, as long as the ratio \(\varDelta \varphi / \delta \varphi \) is rational; however, they correspond to different ranges of the spacelike parameter \(\sigma ^1\). The simplest case of this kind is the limit \(\tilde{a} \rightarrow \omega _1\) for rotating seeds, where the angle \(\varDelta \varphi \) tends to \(\delta \varphi / 2\).

Figure 12 depicts six such solutions. All solutions of Fig. 12 depict approximate finite closed dressed strings with \(n_2=1\). Two indicative examples of dressed solutions with \(n_2>1\) are depicted in Fig. 13.

Such solutions approximate non-degenerate genus two solutions with appropriate periodicity conditions. Figure 14 clarifies the performed approximation in the language of the sine-Gordon equation.

The performed approximation is analogous to the fact that solutions of the simple pendulum with energies close to that of the unstable vacuum can be well approximated by a series of patches of appropriate segments of the kink solution. This holds for both oscillatory and rotating solutions of the simple pendulum. In our problem, the former case is depicted in the top row of Fig. 14, whereas the latter case is depicted in the bottom row of the same figure.

*n*, the dressed solutions Pohlmeyer counterparts of the kind of the top row of Fig. 14 must have an even value for \(n_2\). The string solution depicted on the left of Fig. 13 has a Pohlmeyer counterpart of the kind of the bottom row of Fig. 14, whereas the one on the right has a Pohlmeyer counterpart of the kind of the top row.

### 4.3 \(D^2>0\): exact infinite closed strings

These exact infinite closed string solutions can be considered as the \(n_1 \rightarrow \infty \) limit of the approximate finite closed strings presented in the previous section, with the additional constraint that the seed solution obeys appropriate periodicity conditions so that the asymptotic behaviour of the infinite dressed string is well-defined. In this limit, the conditions (4.27) and (4.28) are trivially satisfied and the solution ceases being approximate and becomes exact. It follows that the approximate closed strings of the previous section can also play the role of a regularization scheme for the infinite ones of this section. This will become handy in Sect. 7, where we will calculate the energy and momentum of the dressed string solutions.

*n*windings around the circle of the torus that corresponds to the real period \(2 \omega _1\). In the limit that this solution becomes a multi-giant magnon, this period diverges, and, thus the torus is transformed to a cylinder. It follows that appropriate parametrization in this limit, requires the union of

*n*such infinite cylinders, and for this reason these solutions require an infinite range of \(\sigma ^1\) for the parametrization of

*each*hop. The solutions of this section exhibit the same behaviour. They should be understood as the degeneration of genuine genus two solutions, in the limit when one of the two real periods diverges.

### 4.4 \(D^2<0\): exact finite closed strings

When considering dressed string solutions with \(D^2 < 0\), the corresponding Pohlmeyer counterpart is not a kink propagating on an elliptic background, but rather a periodic disturbance of the background. This means that the effect of the dressing on the string (as well as in its Pohlmeyer counterpart) is not localized in some region, as it was in the case \(D^2>0\). This also implies that there is no limit where the dressed solution tends to become similar to the seed. Thus, in this case, it is not possible to construct an approximate genus two solution, similar to those of Sect. 4.2.

Similarly to the exact infinite closed string solutions with \(D^2>0\), these solutions are also the degenerate limit of genuine genus two solutions. The difference between the two classes of solutions is the fact that the divergent period is the real one in the former case and the imaginary one in the latter. In other words, in this case, the \(\sigma ^1\) segment parametrizing the string solution corresponds to winding around the compact direction of the cylinder, which is the degenerate limit of the torus.

### 4.5 \(D^2>0\): special exact finite closed strings

*n*, these conditions can be satisfied, independently of the value of the other parameters. However, there is a special case where this is not possible namely,

The condition (4.46) is not sufficient to ensure appropriate boundary conditions of the solution. Similarly to the \(D^2<0\) case of Sect. 4.4, the worldsheet coordinates appear in three distinct combinations in the solution. The first one is trivially \(\xi ^{0/1}\) or (4.32) in terms of \(\sigma _{0/1}\), which implies that the possible segment of \(\sigma _1\) covering a finite string is given by Eqs. (4.10) and (4.11) for translationally invariant and static seeds respectively. One should remember that in the case under study it holds \(D^2>0\) and thus, the seed may have an oscillating sine-Gordon counterpart. In such a case, \(2\omega _1\) should be substituted with \(4\omega _1\) in these expressions.

Both infinite and finite exact periodic string solutions with \(D^2>0\) can be considered as the analytic continuation of the exact finite string solution with \(D^2<0\). The space or time period of the corresponding sine-Gordon counterparts is equal to \(2\pi / \sqrt{-D^2} \). As \(D\rightarrow 0\) this period diverges. Therefore, naturally the finite strings with \(D^2<0\) of Sect. 4.4 tend to the infinite strings with \(D^2>0\) of Sect. 4.3, unless this vector does not contribute to the \(\sigma ^1\) direction, i.e. \(m^{{\mathrm{dress}}} = 0\), in which case they tend to the finite exact solutions with \(D^2>0\) of this section.

## 5 Time evolution and spike interactions

### 5.1 Shape periodicity

#### 5.1.1 \(D^2>0\): approximate finite and exact infinite strings

*z*-axis with angular velocity equal to [28]

The time evolution of the exact infinite dressed strings with \(D^2 > 0\) is similar to the time evolution of the approximate finite strings.

#### 5.1.2 \(D^2<0\)

*U*, such an angle does not correspond to a variation of the shape of the string. On the contrary, periodicity in time requires appropriate condition for the angle \(\varphi ^{{\mathrm{dressed}}}\). This turns out to be

### 5.2 Spike dynamics

The left part of Fig. 22 depicts the kind of interaction occurring in the top left panel of Fig. 20, whereas the right part depicts the kind of interaction happening in the middle row and the bottom right panel of Fig. 20. Had one considered the case of a kink propagating on a train of kinks, the situation would be rather different. Such a solution is always monotonous (see Fig. 3), and, thus, it is not possible that such phenomena occur. Therefore, although the extra spike corresponding to the kink will overpass all other spikes, as the kink advances in the elliptic background, it is not possible to get in touch and interact with any of those. This is the case of the bottom right panel of Fig. 20.

The same kinds of spike interactions occur in the time evolution of the other classes of closed strings that we developed in Sect. 4.

### 5.3 A conservation law preserved by spike interactions

*N*, being proportional to the difference in the value of the Pohlmeyer field at the endpoints of this interval, obviously being a multiple of \(2\pi \),

In the case of the elliptic strings this has been identified to the number of spikes [28]. However, in this case the spikes never interact with each other, as the time evolution of the elliptic strings is simply a rigid rotation. In the case of dressed elliptic strings, we have seen that spikes may interact in a way that their number is not conserved. Thus, the identification of the topological number in the sine-Gordon equation as the number of spikes cannot be extended beyond the case of the elliptic strings.

The form of these spike interactions guide us to search for a conserved quantity, which receives \(\pm 1\) contributions from each spike and \(\pm 2\) contributions from each loop. Let us consider the turning number of the closed string. This is a difficult task since the string has singular points (the spikes), where the tangent vector is not well defined. However, it is true that the string contains only this kind of non-smooth points, i.e. points where the tangent gets inverted. Other non-smooth points where the tangent is instantly rotated by an arbitrary angle are not allowed. Therefore, the unoriented tangent to the string is continuous, and, thus, an unoriented turning number can be defined. This is an element of the fundamental group of the mappings from \(\mathrm {S}^1\) to the one-dimensional real projective space \(\mathbb {R}P^1\). Notice that possible self intersections of the string should not be treated as the same point, where the tangent would not be well-defined, but as separate points. This way the desired turning number is naturally a member of \(\pi _1 \left( \mathbb {R}P^1 \right) = \mathbb {Z}\) and must be conserved.

This explains the two kinds of interactions we found in Sect. 5.2. Whenever two spikes with opposite contributions to the unoriented turning number get combined, they just disappear. When two spikes with identical contributions to the turning number get combined they disappear and necessarily the conservation of the turning number implies that a loop must take their place. The above imply that the unoriented turning number and the topological charge of the sine-Gordon equation are in correspondence. They do not have to be equal, but they may differ by an even integer.

The above are also in line with the effect of the dressing on the shape of the string that we observe in Fig. 12. In all cases, the action of the dressing procedure on the Pohlmeyer field adds a kink or an antikink to the seed solution, which according to the above should increase or decrease the aforementioned turning number by one. The simplest case is that of a seed solution with a static oscillating counter part (Fig. 12, top-right). In this case the seed solution has no spikes, while the dressed solution has exactly one. In a similar manner, when a seed solution with a translationally invariant oscillating counterpart is considered (Fig. 12, top-left), the seed solution has equal number of spikes that contribute \(+1\) and spikes that contribute \(-1\) to the turning number, having net turning number 0, whereas the dressed string has net turning number equal to 1. In the case of seeds with rotating elliptic counterparts the behaviour is also similar.

## 6 Instabilities of the elliptic strings

When one desires to study the stability of a classical string solution, they usually study the stability of its Pohlmeyer counterpart, as the equations of motion of the reduced system are simpler to study since they contain fewer degrees of freedom and they do not possess any reparametrization symmetry. More specifically, the stability of the elliptic solutions of the sine-Gordon equation has been studied in [30]. It turns out that only the static rotating elliptic solutions of the sine-Gordon equation are stable. Therefore, only one of the four classes of elliptic string solutions on the sphere \(\mathrm {S}^2\) is stable.

However, we should be concerned about the above result. The stability analysis is performed introducing an arbitrary infinitesimal perturbation to the elliptic solutions of the sine-Gordon equation. However, when a closed elliptic string is considered, appropriate periodicity conditions must be applied, and, thus, only perturbations preserving these conditions should be considered in the stability analysis.

This class of string solutions that reveals instabilities of the elliptic strings may contain solutions with various genera. However, the simplest case to consider is a degenerate genus two solution, where only one of the two genera is degenerate. The solution should have a non-degenerate genus, associated with the initial elliptic solution, and furthermore it should have a degenerate one describing the infinite motion that tends asymptotically to the elliptic solution at plus and/or minus infinite time. This is exactly the class of dressed elliptic string solutions.

It turns out that the relevant dressed elliptic solutions are the special finite exact solutions with \(D^2>0\) presented in Sect. 4.5. These solutions have counterparts with \(D^2>0\) being a kink propagating on an elliptic background. Therefore, the sine-Gordon counterparts of these solutions have a specific asymptotic behaviour, namely, far away from the region of the kink they tend to a shifted version of the seed, and similarly the string tends to a rotated version of the seed string solution. In this specific class of solutions, the \(\sigma ^1\) direction is parallel to the direction that the kink moves in spacetime, thus the asymptotic behaviour of the string is never reached at a snapshot of the string, but it is rather reached asymptotically in time. It follows that these specific string solutions evolve from a rotated version of the seed elliptic spiky string solution to another one, rotated by the opposite angle. Notice that these asymptotic string solutions obey appropriate periodicity conditions and thus, they are finite.

The existence of these solutions indicates that their seed elliptic solutions are unstable. They describe a finite disturbance of a spiky string emerging after an infinitesimal perturbation at minus infinity time.

The rigid body rotation of the asymptotic elliptic string has been frozen in the figure so that the time evolution is clearly depicted. In all cases the string finally resettles to the same unstable elliptic string configuration but with a delay proportional to \(2\left| \tilde{a} \right| \) in comparison to the state it would lie had it followed the simple rigid rotation evolution of the elliptic string.

The above are in line with the findings of [30], which support that in general string solutions with sine-Gordon counterparts that can accommodate superluminal kinks are unstable. However, in our case there is a particular difference. The solutions exposing the string instability emerge only when there is a superluminal kink with velocity equal to the inverse of the velocity of the boost connecting the linear and static gauges. This is due to the fact that only such solutions do not disturb the periodicity conditions of the closed seed string solution. Recalling Fig. 5, the above implies that the elliptic strings with oscillating static counterparts always expose this kind of instability, since the kink velocity diverges at the limit \(\tilde{a} \rightarrow \omega _1\), and, thus, any possible superluminal kink velocity can be obtained for some value of \(\tilde{a}\). On the contrary, for elliptic strings with translationally invariant counterparts, even in the case they can accommodate superluminal kinks, there is a maximum velocity of the latter. This means that, depending on the elliptic string moduli *E* and *a*, which determine the velocity of the boost connecting the static and linear gauges, this kind of instability may or may not exist. More specifically, given a value of *E*, there is a minimum value of \(\wp \left( a \right) \), or in other words, there is a minimum number of spikes required for the existence of the instability. This in turn implies that the ”speeding strings” limit of the elliptic strings always exposes this kind of instability (when they have translationally invariant counterparts). Figure 25 shows the subset of elliptic strings that present this kind of instabilities within the moduli space of elliptic string solutions as parametrized by the quantities *E* and \(\wp \left( a \right) \).

In the right panel, the thick black line enclosing the unstable elliptic string solutions with oscillating translationally invariant counterparts tends asymptotically to the \(E = E_c\) vertical line, where the constant \(E_c\) is defined in Eq. (3.14).

Of course the above argument is not a proof of the existence of stable closed elliptic string solutions, with sine-Gordon counterparts that accommodate superluminal kinks; it is possible that more complicated multi-kink generalizations of the above solutions conserve the periodicity conditions and thus give rise to instabilities. These should possess only one non-degenerate genus, thus, they could emerge from the dressing of the elliptic strings with more complicated dressing factors. The latter can be constructed from the solution of the auxiliary system presented in [29] in a straightforward manner. Such solutions should not correspond to multiple kinks travelling on top of an elliptic background, as they would have different velocities and thus, their asymptotic behaviour could not be only temporal. They would rather correspond to a single breather propagating on top of an elliptic background. Nevertheless, the stability issue of the spiky strings requires further investigation concerning the constraints originating from the periodicity conditions.

A simple case to consider in particular is the stability of the GKP strings [1]. These are the elliptic strings with static Pohlmeyer counterparts and modulus \(a = \omega _2\) implying that \(\beta = 0\), i.e. the linear gauge coincides with the static gauge. It follows that a dressed elliptic solution exposing an instability of a GKP string should have a Pohlmeyer counterpart being a superluminal kink on top of an elliptic background with infinite velocity, in other words a translationally invariant kink. As we have shown in Sect. 3.2, the kink velocity on static backgrounds is diverging only in the case of an oscillating seed at the limit \(\tilde{a} = \omega _1\). Therefore, the GKP strings with an oscillating Pohlmeyer counterpart are unstable. This is expected since the latter are great circles rotating around the sphere with subluminal velocities and they tend to shrink due to the string tension.

## 7 Energy and angular momentum

### 7.1 Approximate finite and exact infinite strings with \(D^2>0\)

^{3}Defining as \(E_{0/1}^{\text {hop}}\) the energy of one hop of the seed elliptic string, it follows that the energy of these strings is equal to

We will focus on the calculation of the third component of the angular momentum of the string, which presents a certain interest for holographic applications. Before that, let us argue on the reasons we expect the other two components to vanish, when we consider finite closed dressed strings. In the case of elliptic “naked” solutions obeying appropriate periodicity conditions, \(J_1\) and \(J_2\) vanish as a result of the discrete symmetry that these solutions possess. This is also the case when one considers infinite dressed elliptic strings that obey exact periodicity conditions (see Fig. 16). However, naively this is not the case when we consider the approximate closed finite dressed solutions with \(n_2 = 1\), as the extra spike induced by the dressing breaks this symmetry. Although this symmetry is not present at a given time instant, one should not forget that the dressed strings change shape periodically, while they are simultaneously rotating. Therefore, after a time equal to the period of the string shape, it is expected that the \(J_1\) and \(J_2\) components will have rotated by an arbitrary angle. As the angular momentum is conserved, the latter implies that \(J_1\) and \(J_2\) vanish. In the following *J* denotes the third component of the angular momentum and the indices 0 and 1 refer to whether the seed has a translationally invariant or static Pohlmeyer counterpart.

Although the above relations are expressed in terms of transcendental functions, the properties of the elliptic functions allow the specification of the dispersion relation in a closed form whenever the quantities *a* and \(\tilde{a}\) are a rational fraction of \(\omega _2\) and \(\omega _1\) respectively. This procedure is applied in [28] for the simpler case of elliptic strings and we will not post further details here.

### 7.2 Exact finite strings with \(D^2>0\) and strings with \(D^2<0\)

*n*is equal to \(\mathrm {lcm}\left( n^{{\mathrm {seed}}}, n^{{\mathrm {dress}}} \right) \), in the case of dressed strings with \(D^2 < 0\), as described in Sect. 4.4, and \(n=n^{{\mathrm {seed}}}\) in the case of the exact finite elliptic strings with \(D^2 > 0\) (or equivalently \(n^{{\mathrm {dress}}} = 1\)), as described in Sect. 4.5.

The change of the difference of the energy and angular momentum that is induced by the dressing, is plotted versus the dressing parameter \(\theta _1\) in Fig. 26. In these plots, it is assumed that when the seeds have translationally invariant sine-Gordon counterparts, they also have the instabilities presented in Sect. 6. Had we considered the opposite, the graphs would be identical apart from the inversion of the curve between the two instabilities in the case of an oscillating counterpart and between \(\tilde{\theta }_+\) and the instability in the case of a rotating counterpart, which would be absent.

The full continuum of the curves could be set valid, if the parameters of the seed solution were altered appropriately as one moves on the curve so that appropriate periodicity conditions always apply. Otherwise only a dense discrete subset would be valid.

A region around each instability point would be invalid since the approximation conditions around the instabilities do not hold.

*a*. When considering dressed strings whose seeds have rotating counterparts, the dispersion relation is a rather peculiar function of the angle \(\theta _1\); there is a range for \(\theta _1\) where the dispersion relation does not depend on the latter.

The above is an interesting similarity to the properties of the corresponding solutions of the sine-Gordon equation. As we have seen in Sect. 3.6, the mean energy and momentum density of the dressed solution of the sine-Gordon equation with \(D^2<0\) is identical to those of the seed solution. It would be interesting to interpret this fact on the side of the holographically dual theory. The difference \(E-J\) remains the same after the dressing; however the seed solution is characterised by a single angular opening, i.e. a single quasi-momentum, whereas this is not the case for the dressed solution. A naive interpretation of this solutions could be that they correspond to more complicated excitations, which have formed bound states behaving as a single quasi-momentum state.

There is yet another interesting bifurcation of the form of the dispersion relation of the dressed strings in the case of translationally invariant seeds that has to do with the presence of the instabilities. When the seed is unstable, the quantity \(\varDelta E - \varDelta J\) contains further discontinuities related with the inversion of the sign \(s_\varPhi \). Although the dispersion relations of the dressed strings are too complicated expressions to be directly verifiable in a holographically dual theory, the above discontinuities in the behaviour of the dispersion relation could be in principle detectable.

## 8 Discussion

In the present work, we have carefully studied several physical properties of the dressed elliptic string solutions on \(\mathrm {S}^2\), which were derived in [29]. We have presented these properties in juxtaposition to those of their Pohlmeyer counterparts in an effort to obtain an intuitive understanding of the relativistic string dynamics on the sphere, through the dynamics of the sine-Gordon equation, which can be visualised as a chain of coupled pendulums.

The dressed elliptic solutions have been identified to belong to two large classes depending on the sign of the parameter \(D^2\). The ones with \(D^2>0\) have Pohlmeyer counterparts which describe localised kinks propagating on top of an elliptic background, whereas those with \(D^2<0\) possess Pohlmeyer counterparts which are periodic disturbances on top of an elliptic background. The latter emerge only in the case the seed solution has a rotating Pohlmeyer counterpart.

At first we focused on the necessary conditions that must be obeyed, so that the dressed elliptic strings are closed. We arrived at four specific classes of closed string solutions. One of those is not exact, but these solutions approximate genuine genus two ones, with one of the two genera being almost singular. The other three classes are exact solutions and can be considered as the analytic continuation of one another as \(D^2\) changes sign. One of the latter contains only infinite strings; the approximate class of solutions can serve as a regularization scheme in order to calculate the conserved charges of the infinite ones.

An interesting feature of the dressed elliptic strings is the existence of interactions between their singular points i.e. their spikes. It has been previously noted in [28] that in the case of elliptic strings, the number of spikes is identical to the conserved topological charge on the sine-Gordon equation counterpart. However, these solutions have trivial time evolution and the spikes never interact. In the case of the dressed elliptic strings, the form of the allowed interactions between the spikes suggest that the topological charge should not be connected to the number of spikes. It should rather be connected to a more complicated quantity, which receives a \(\pm 1\) contribution from each spike and a \(\pm 2\) contribution from each loop. This quantity is an appropriately defined turning number, which is the homotopy class of the mapping from each point of the string to the *unoriented* direction of the tangent at this point.

The special class of finite exact solutions with \(D^2>0\) relates in an interesting way to the stability of the seed elliptic strings. Since these solutions asymptotically interpolate in their dynamical evolution between two versions of the seed elliptic string solution, they reveal that the latter is unstable. It is interesting that such solutions emerge only for the classes of elliptic strings whose sine-Gordon counterparts are unstable [30]. However, the opposite is not true; it is not possible to find such a solution for any elliptic string whose Pohlmeyer counterpart is considered unstable. This may be attributed to the fact that the stability analysis for finite closed string should incorporate only the perturbations that preserve the appropriate periodic conditions. This point deserves further investigation.

The conserved charges of the infinite dressed strings are divergent, yet one can define a finite difference with respect to the charges of the elliptic seed. This divergence is not surprising, since these string solutions are a long string limit, similar to that of the giant magnons; the latter correspond to genus one solutions with diverging real period, whereas the former are the genus two generalization. As a consequence, they have a dispersion relation that resembles the one of the giant magnons, with an additional free parameter. The two exact finite classes of solutions have identical energy and angular momenta as their seeds.

The dependence of the conserved charges on the moduli of the dressed string solutions exhibits some discontinuities. One of these is related to the qualitative behaviour of the seed solution, whereas the other one is related to the instabilities of the seeds. Since the dispersion relation is connected to the anomalous dimensions of operators of the boundary theory, it would be interesting to identify these kinds of bifurcations in the spectrum of the dual theory. The same holds true for the sets of operators, which correspond to the exact finite dressed strings and share the same charges with their seeds.

The techniques that were used for the construction of the dressed elliptic strings on \(\mathbb {R} \times \mathrm {S}^2\) have obvious generalizations to other symmetric spaces, such as the AdS, dS, spheres of higher dimensions or tensor products of the latter. Especially the \(\mathrm {AdS}_n \times \mathrm {S}^n\) spaces have obvious interest in the framework of the holographic correspondence. The findings of this work suggest that similar phenomena exist in these more interesting cases and deserve further investigation. Similarly, identical techniques can be applied for the study of minimal surfaces in \(\hbox {AdS}_4\), which are interesting in the context of the Ryu–Takayanagi conjecture, or Wilson loops.

## Footnotes

- 1.
The right charge is not independent in the case of a symmetric space sigma model.

- 2.
The dressed string solutions with the simplest dressing factor, as those presented here, have an interesting geometric relation to their seeds. Every point of the dressed string is connected to the point of the seed solution with the same worldsheet coordinates, via an arc of a maximum circle equal to \(\theta _1\). Therefore, the dressed string can be considered drawn by a point on an epicycle of constant arc radius \(\theta _1\) whose center is running on the seed solution.

- 3.
In Sect. 4.2 we used as \(\bar{\sigma }\) the average position of the kink (see Eqs. (4.30) and (4.31)). One could consider the exact position of the kink, i.e. the \(\bar{\sigma }\) that obeys \(\tilde{\varPhi } \left( \sigma ^0, \bar{\sigma } \right) = 0\). Either selection results in the same values for the energy and the angular momentum of the dressed strings.

## Notes

### Acknowledgements

The research of I.M. and G.P. has received funding from the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT), in the framework of the “First Post-doctoral researchers support”, under grant agreement No 2595. The research of D.K. is co-financed by Greece and the European Union (European Social Fund-ESF) through the Operational Programme “Human Resources Development, Education and Lifelong Learning” in the context of the project “Strengthening Human Resources Research Potential via Doctorate Research” (MIS-5000432), implemented by the State Scholarships Foundation (IKY). The authors would like to thank M. Axenides and E. Floratos for useful discussions.

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