# Elliptic string solutions on \(\mathbb {R}\times \hbox {S}^2\) and their pohlmeyer reduction

## Abstract

We study classical string solutions on \(\mathbb {R}\times \hbox {S}^2\) that correspond to elliptic solutions of the sine-Gordon equation. In this work, these solutions are systematically derived by inverting the Pohlmeyer reduction. A mapping of the physical properties of the string solutions to those of their Pohlmeyer counterparts is established. An interesting element of this mapping is the association of the number of spikes of the string to the topological charge in the sine-Gordon theory. Finally, the adopted parametrization of the solutions facilitates the identification of a dense subset of the moduli space of solutions, where the dispersion relation can be expressed in a closed form, arbitrarily far from the infinite size limit.

## 1 Introduction

Classical string solutions have played an important role in the understanding of the AdS/CFT correspondence [1, 2, 3, 4]. According to the dictionary of the holographic duality, the dispersion relations of classical string solutions are related to the anomalous dimensions of gauge theory operators in the strong coupling limit. Matching the spectra on both sides of the holographic duality was a non-trivial quantitative test [5, 6, 7, 8, 9] of the AdS/CFT correspondence and classical string solutions were necessary in order to perform such calculations. The standard methodology in the literature for this purpose, has been the use of an appropriate ansatz in order to reduce the classical string equations of motion and the Virasoro constraints to a system of equations for a set of unknown functions or parameters [10, 11] (for a review see [12]).

The matching of the spectra of the classical string in \(\hbox {AdS}_5\times \hbox {S}^5\) and the \(\mathscr {N} = 4\) SYM has also been studied with the help of methods from algebraic geometry. The sigma model [13] of the Green-Schwarz superstring possesses a spectral curve, which is a manifestation of integrability [14]. On the field theory side, the anomalous dimensions of operators at strong coupling can be calculated using the Bethe ansatz [15]. It has been shown that at specific limits, the spectra of the dual theories indeed match upon the identification of some parameters [16, 17] (for a review see [18]). In this language, the classical string solutions are provided in terms of abstract hyperelliptic functions, that can be expressed in terms of conventional functions (algebraic or elliptic) only in the genus one case. Thus, although the problem of spectrum matching is formally understood, it is difficult to study and comprehend the generic structure.

A method for the construction of classical solutions in non-linear sigma models (NLSM) with a symmetric target space that is more systematic than the use of an arbitrary ansatz, but yet leads to solutions expressed in terms of functions with well understood properties, was initiated in [19, 20]. In this approach, NLSM solutions are derived through the inversion of the Pohlmeyer reduction. Two-dimensional NLSMs with symmetric target spaces are reducible to integrable systems, the so called symmetric space sine-Gordon systems (SSSGs), which are multi-component generalizations of the sine-Gordon equation. The oldest and most well-known example is the reduction of the O(3) NSLM, which leads to the sine-Gordon equation [21, 22]. The reduced system can always be derived from a local Lagrangian, which is a gauged Wess–Zumino–Witten model with an integrable potential [23, 24, 25, 26]. The Pohlmeyer reduction is equivalent to the Gauss–Codazzi equations for the embedding of the string worldsheet into the target space, which is in turn embedded into a flat enhanced space [27]. In this context, the fact that the target space is a symmetric space is directly connected to the integrability of the reduced model [28, 29, 30].

Even though it is straightforward to calculate the solution of the reduced theory that corresponds to a given solution of the original NLSM, the inversion of the Pohlmeyer reduction is a highly non-trivial process. This can be attributed to the non-local nature of the Pohlmeyer reduction, as well as to the fact that the mapping is many-to-one. Construction of NLSM solutions based on the inversion of the Pohlmeyer reduction has been performed in [19] for strings propagating on \(\hbox {AdS}_3\) and \(\hbox {dS}_3\), and in [20] for minimal surfaces in \(\hbox {H}^3\). These techniques can be applied for a particular class of solutions of the reduced system, which depend on a sole worldsheet coordinate. Given such a solution of the reduced system, the NLSM equations of motion become linear and solvable via separation of variables. Then, the geometric and Virasoro constraints are imposed and NLSM solutions are obtained. This procedure enables a systematic investigation of this class of NLSM solutions. In this work, we apply this method for strings that propagate on \(\mathbb {R}\times \hbox {S}^2\). We argue that this study can be extended in a trivial manner to higher dimensional spheres.

String solutions belonging to this specific sector probe several interesting regimes of the spectrum of the AdS/CFT duality at specific limits. Berenstein et al. [31] studied a particle moving at the equator of \(\hbox {S}^5\) at the speed of light. Gubser et al. [32] studied a closed folded string that rotates around the north pole of the \(\hbox {S}^2\) and its counter part, a string that is a rotating great circle. A few years later, Hofman and Maldacena [33] introduced the giant magnons. These are open strings, whose ends lie at the equator of the \(S^2\) and move at the speed of light. They are the strong coupling, string theory counterpart of infinite size single-trace operators that contain one impurity. In [34, 35, 36, 37, 38, 39] more general spiky string solutions are constructed. All these known solutions emerge naturally in our construction. We give a unified description and classification of all these string solutions in terms of their Pohlmeyer counterpart.

The paper is organized as follows. In Sect. 2, we revisit the Pohlmeyer reduction of the NLSM describing strings propagating on \(\mathbb {R}\times \hbox {S}^2\) that results in the sine-Gordon equation. In Sect. 3, we review the class of solutions of the sine-Gordon equation that can be expressed in terms of elliptic functions. In Sect. 4, it is shown that for these solutions of the sine-Gordon equation, the equations of motion of the NLSM separate into pairs of effective Schrödinger problems. Each pair contains one flat potential, whereas the other one is the \(n=1\) Lamé potential. We obtain the general solution for this system of equations and impose the appropriate constraints to effectively invert Pohlmeyer reduction. In Sect. 5, we study various properties of the elliptic strings, with emphasis to the mapping of their properties to those of their Pohlmeyer counterparts. In Sect. 6, we study the dispersion relations of the string solutions and finally, in Sect. 7, we discuss our results. Throughout the text, various properties of the Weierstrass elliptic and related functions are used. All the necessary formulae can be found in standard mathematical literature, e.g. [40], or in the appendix of [19].

## 2 The Pohlmeyer reduction of strings propagating on \(\mathbb {R}\times \hbox {S}^2\)

The NLSMs that describe string propagation in symmetric spaces, are reducible to integrable systems of the same family as the sine-Gordon equation [41, 42, 43, 44, 45]. In this section, we revisit the Pohlmeyer reduction of strings propagating on \(\mathbb {R}\times \hbox {S}^2\) (\(\mathbb {R}^t\) stands for the time dimension). The main difference of our approach to the original treatment [21] is the implementation of a more general gauge, instead of the static gauge, which will facilitate the construction of the elliptic string solutions via the inversion of the Pohlmeyer reduction, in Sect. 4. This is the main reason we review the well-known Pohlmeyer reduction of strings propagating on the sphere here.

*T*is the tension of the string.

## 3 Elliptic solutions of the sine-gordon equation

In this section, we are going to find the solutions of the sine-Gordon Eq. (2.26) that depend solely on one of the two worldsheet coordinates, i.e. they are either static or translationally invariant. In the following, the dot denotes differentiation with respect to \(\xi ^0\) and the prime denotes differentiation with respect to \(\xi ^1\).

*E*,

*E*, as shown in Fig. 1.

The ordering of the roots

Ordering of roots | |
---|---|

\(E > \mu ^2\) | \(e_1 = x_1 , \quad e_2 = x_2 ,\quad e_3 = x_3\) |

\(\left| E \right| < \mu ^2\) | \(e_1 = x_2 , \quad e_2 = x_1 ,\quad e_3 = x_3\) |

\(E < - \mu ^2\) | \(e_1 = x_2 , \quad e_2 = x_3 , \quad e_3 = x_1\) |

*y*is real, but also it must satisfy

*y*to a real \(\varphi \). The Table 2 shows the range of \(2 y + E / 3\) for each of the two solutions.

The range of \(- \mu ^2 \cos \varphi _0\) for both real solutions of the Weierstrass equation

Range of \(2 \wp ( x ) + E / 3\) | Range of \(2 \wp ( x + \omega _2 ) + E / 3\) | |
---|---|---|

\(E > \mu ^2\) | \(2 y + E / 3 > E\) | \(- \mu ^2< 2 y + E / 3 < \mu ^2\) |

\(\left| E \right| < \mu ^2\) | \(2 y + E / 3 > \mu ^2\) | \(- \mu ^2< 2 y + E / 3 < E\) |

\(E < - \mu ^2\) | \(2 y + E / 3 > \mu ^2\) | \(E< 2 y + E / 3 < - \mu ^2\) |

*E*in Fig. 2.

- 1.
The solutions with \(E < \mu ^2\) are periodic. Their period is equal to \(4 \omega _1\). We will call them the “oscillatory” solutions, inspired by the simple pendulum analogue of Eq. (3.2).

- 2.
The solutions with \(E > \mu ^2\) are quasi-periodic, obeying \(\varphi _{0/1} \left( \xi ^{0/1} + 2 \omega _1 \right) = \varphi _{0/1} \left( \xi ^{0/1} \right) + 2 \pi \). We will call them the “rotating” solutions.

### 3.1 Double root limits

## 4 Elliptic string solutions

### 4.1 The building blocks of elliptic solutions

*linear*differential Eq. (4.1). Using a solution of the reduced system that depends on only one worldsheet coordinate provides an extra advantage; these linear differential equations are solvable using separation of variables [19, 20],

*a*is one of the half-periods. Thus, the corresponding Lamé eigenfunctions degenerate to the form of eigenfunctions lying at the edge of the allowed bands. In general the solution is

### 4.2 Construction of elliptic string solutions

*a*lies on the imaginary axis. Finally, absorbing the \(e_3 - \wp \left( a \right) \) factor of Eq. (4.16) into the definition of \(y_\pm \), the geometric constraint reduces to

*a*, which corresponds to the transformation \(y_\pm \rightarrow y_\mp \) or equivalently interchanging \(m_+\) and \(m_-\).

## 5 Properties of the elliptic string solutions

In this section, we proceed to study the geometric characteristics of the string solutions derived in Sect. 4 and their relation to the features of their Pohlmeyer counterparts. We indicate with index 0, the elliptic string solutions that correspond to a translationally invariant solution of the sine-Gordon equation and with index 1, the solutions with a static sine-Gordon counterpart. It turns out that the natural parametrization of our construction, which is based on the Weierstrass elliptic function, facilitates the study of the properties of the elliptic string solutions.

*t*being an increasing function of the time-like worldsheet coordinate \(\xi ^0\). Having selected one of the two above quantities to be negative, requires taking the opposite value of

*a*according to the Virasoro constraint (4.38). We have restricted

*a*to take values in the segment of the imaginary axis with endpoints \(\pm \omega _2\). Then, equation (4.38) implies that \(\ell = - \mathrm {sgn}({\mathrm{Im}} a) \sqrt{x_1 - \wp ( a )}\). From now on, for simplicity, we make the choice \(\ell > 0\).

### 5.1 Angular velocity

- 1.The solutions with rotating counterparts obey \(x_1 > x_2\). Such solutions do not cross the equator; they lie between two circles, which are parallel to the equator and in the same semi-sphere. For example, in the case this is the north semi-sphere, these solutions obeywhere$$\begin{aligned} \theta _-< \theta < \theta _+ , \end{aligned}$$(5.14)Both subclasses of solutions with rotating counterparts are characterized by \(\omega _{0/1} > 1 / R\). The angles \(\theta _\pm \) that constrain the string on the sphere also depend on the gauge selection, since$$\begin{aligned} \theta _\pm = \arccos \sqrt{\frac{{{x_1} - {x_{2/3}}}}{{{x_1} - \wp \left( a \right) }}} . \end{aligned}$$(5.15)$$\begin{aligned} \sin {\theta _ \mp } = \frac{1}{{R{\omega _{0/1}}}} . \end{aligned}$$(5.16)
- 2.The solutions with oscillating counterparts obey \(x_1 < x_2\). These solutions periodically cross the equator. They lie between two parallel circles, which are symmetrically placed above and below the equator, namely,The angular velocity of solutions with static counterparts obeys \(\omega _1 < 1 / R\). On the contrary, solutions with translationally invariant counterparts have \(\omega _0 > 1 / R\). Smoothness of the solution requires that \(\cos \theta \) changes sign every time the string crosses the equator. Thus, the argument of the Weierstrass elliptic function should be altered by \(4\omega _1\) in order to complete a whole period for \(\theta \), in analogy to the period of the corresponding oscillating solutions of the sine-Gordon equation.$$\begin{aligned} \theta _-< \theta < \pi - \theta _- . \end{aligned}$$(5.17)

### 5.2 Periodicity conditions

It seems that for the hoop string solutions that correspond to translationally invariant solutions of the sine-Gordon equation no periodicity condition is implied. This apparent asymmetry would have been resolved, if we had considered the \(\mathbb {R} \times \mathrm {S}^2\) string target space, as a subspace of \(\mathrm {AdS}_n \times \mathrm {S}^n\), implying that the time direction would be compact, and, thus, the target space would be the fully compact \(\mathrm {S}^1 \times \mathrm {S}^2\). In AdS spaces, it has be shown that hoop solutions have to obey such a time periodicity condition [19], which would be inherited in the \(\mathrm {S}^2\) part of the solution. In general, in such a case the elliptic solutions would be identical and furthermore it would be possible to find solutions that wouldn’t simply correspond to closed strings, but to fully compact toroidal worldsheets. For this purpose, another periodicity condition similar to the above should be imposed, which would effectively select a subspace of the elliptic solutions with appropriate angular velocity.

### 5.3 Spikes

### 5.4 Topological charge and the sine-Gordon/Thirring duality

*n*. We have seen that a spike appears whenever the Pohlmeyer field assumes a value that is an integer multiple of \(2 \pi \). It follows that

The above correspondence naively implies that in the picture of the Thirring model, the string solutions with rotating counterparts can be considered as multi-fermion states. On the contrary, solutions with oscillating Pohlmeyer counterparts have the natural interpretation of bosonic condensates. However, notice that the sine-Gordon/Thirring duality is a full quantum weak to strong duality. Thus, the above statement should be viewed cautiously, since taking the classical limit of a strongly coupled quantum theory is in general non-trivial.

It would be interesting to investigate this duality in the framework of string theory. Type IIB superstring theory in \(\hbox {AdS}_n\times \hbox {S}^n\) is self-S-dual, with the closed strings being S-dual to D1-branes [54, 55]. This hints that the spiky elliptic superstrings should be S-dual to D1-brane configurations, whose Pohlmeyer counterpart has non-trivial fermion number equal to the number of spikes of the original string solutions. The investigation of this correspondence requires the derivation of elliptic superstring solutions propagating on the full \(\hbox {AdS}_n\times \hbox {S}^n\) space and their parallel study in the corresponding supersymmetric Pohlmeyer reduced theory.

### 5.5 Interesting limits and the moduli space of solutions

The elliptic string solutions have some very well known special limits, which are very simple to study in our parametrization. We do so for the completeness of our presentation. At these limits, two of the three roots \(x_1\), \(x_2\) and \(x_3\) coincide, and, thus, the Weierstrass elliptic function degenerates to a simply periodic function, either trigonometric or hyperbolic. There are two such cases:

*E*. For a given value of this constant, the parameter

*a*may take any value on the imaginary axis on the linear segment defined by the origin and the half-period \(\omega _2\). Another interesting limit is the special selection \(a = - \omega _2\) or \(\wp \left( a \right) = x_3\). This is the case where the linear gauge coincides with the static gauge. Had we restricted Pohlmeyer reduction to the static gauge, the method applied in Sect. 4 for the construction of the elliptic string solutions would have resulted to these special solutions only. In this limit, the solution assumes the form

In the case of translationally invariant counterparts, Eqs. (5.57) and (5.58) describe a hoop being always parallel to the equator which shrinks to a point at the pole of the sphere and then extends again. In the case of oscillating solutions it extends further than the equator and then shrinks again to the opposite pole before it starts re-extending; in the case of rotating solutions it extends up to a maximum size and then it shrinks again to the same pole. These solutions, although they have a translationally invariant Pohlmeyer counterpart are spikeless. This is due to the coincidence of the static gauge to the linear one. As there is no need for a worldsheet boost to convert to the static gauge, the singular behaviour characterizes solely the time evolution of the string and not its shape. These solutions satisfy the periodicity conditions with \(n_0 = 0\). The coordinate \(\sigma ^1\) takes values in \(\left[ {0 , 2 \pi / \sqrt{ x_1 - x_3}} \, \right) \) to complete one hoop.

The elliptic string solutions are a two-parameter family of solutions, in our language being the parameters *E* and *a*. The advantage of our parametrization is that only one of the two parameters (the integration constant *E*) affects the corresponding solution of the Pohlmeyer reduced system. The worldsheets of the solutions being characterized by the same constant *E* comprise an associate (Bonnet) family [20]. Demanding appropriate periodicity conditions, restricts one of the two parameters to be discrete, or in other words the moduli space of the elliptic string solutions with appropriate periodicity conditions is a discretely infinite set of one-dimensional curves. Figure 4 depicts the moduli space of elliptic string solutions and visualises their classification according to their Pohlmeyer counterpart.

## 6 Energy and angular momentum

The \(\mathbb {R}\times \hbox {S}^2\) target space has the symmetry of time translations, leading to a conserved energy and that of SO(3) rotations, leading to a conserved angular momentum.

*z*-component of the angular momentum is given by

*R*factors in these definitions is due to the fact that we have considered time as an independent dimension not related to the radius of the sphere. Had we considered \(\mathbb {R}^t \times \hbox {S}^2\) as a submanifold of an \(\hbox {AdS}_n \times \hbox {S}^n\) space with a dual boundary description, the time would have been part of the \(\hbox {AdS}_n\), which has the same radius as that of the sphere, effectively measuring time in units of

*R*. We also recall that the angular opening \(\delta \varphi \), which is associated to the quasi-momentum in the dual theory, is given by

*a*is equal to the imaginary quarter-period \(a = - \omega _2 / 2\). This is a one-dimensional family of solutions, which in the case of static counterparts, contains the giant magnon with angular opening equal to \(\pi / 2\). The Weierstrass functions obey the following quarter period relations

*E*is equal to the algebraic function of the ratio \(({{\delta {\varphi _{0/1}} \pm \pi /2}})/{{{\mathscr {E}_{0/1}}}}\), which solves the equation,

This procedure can be generalized. Consider the more general case \(a = - 2 q \omega _2\), \(q \in \mathbb {Q}\). This is a one-dimensional sector of the moduli space, which, in the case of static counterparts, contains a giant magnon solution obeying appropriate periodicity conditions with \(\delta \varphi = 2 q \pi \) (of course this is going to have self-intersections unless *q* is of the form 1 / *n*, \(n \in \mathbb {Z}\)). The functions \(\wp \left( 2 m z / n \right) \) with \(m,n \in \mathbb {Z}\) and \(\wp \left( z \right) \) are both elliptic functions with periods \(2 n \omega _1\) and \(2 n \omega _2\). Therefore they are algebraically related. The above argument for \(z = \omega _2\) implies that \(\wp \left( 2 q \omega _2 \right) \) is an algebraic function of the root \(e_3\).

The above process cannot be applied in the case of the GKP limit, i.e. the specific selection \(q=1/2\). In this case, the angular opening is not a function of the integration constant *E*, but it simply equals \(\delta \varphi _1 = \pi \), i.e. the algebraic function \(g_q\) in Eq. (6.20) vanishes. Therefore, the integration constant *E* cannot be specified algebraically by an appropriate linear combination of the energy and the angular opening, but it requires the inversion of the elliptic integral that relates it to the string energy. This cannot be performed analytically; usually this inversion is performed perturbatively around the infinite size limit [48, 49, 50, 51].

## 7 Discussion

We applied a systematic method for the construction of classical string solutions propagating on \(\mathbb {R}\times \hbox {S}^2\). Using a specific class of solutions of the Pohlmeyer reduced theory, i.e. the sine-Gordon equation, which are expressed in terms of elliptic functions, we were able to develop a unified description of all known genus one string solutions on \(\mathbb {R}\times \hbox {S}^2\). Our approach is based on a convenient choice of the worldsheet parametrization that leads to equations of motion for the classical string, which are solvable via separation of variables.

The fact that our method can be applied successfully, reproducing all known genus one solutions and providing a unified framework is not accidental. The NLSM is integrable, and, thus, it can be solved using finite gap integration. It is known that any smooth one-gap potential is equivalent to an appropriate \(n=1\) Lamé potential [56]. Thus, in the case of elliptic solutions, the equations of motion are in principle reducible to the \(n=1\) Lamé problem. This is precisely what it is achieved via the application of the Pohlmeyer reduction inversion technique. Since the spiky strings and their various special limits are the most general genus one classical string solutions [57, 58], our approach achieves the inversion of the Pohlmeyer reduction and it is equivalent to the finite gap integration in the case of genus one.

A dictionary between the NLSM and the sine-Gordon model

NLSM | sine-Gordon |
---|---|

Two-parameter family of solutions | Only one of the two parameters affects the solution |

Angular frequency—extremal altitudes | Gauge in which the solution is either static or translationally invariant |

Degenerate one-dimensional worldsheet (BMN particle, hoops) | Vacuum solution |

Strings asymptotically reaching the equator (giant magnons, single spikes) | Kink or instanton solutions |

Rigid rotation/wave propagation | Static/translationally invariant solutions (at some frame) |

Spike | \(\varphi = 2 n \pi , \quad n \in \mathbb {Z}\) |

Number of spikes | Topological charge |

Spiky strings/non-spiky strings or strings with equal number of spikes and anti-spikes | Rotating/oscillating solutions—multi-fermion states/bosonic condensate states in the dual Thirring model |

The Weierstrass elliptic function is the natural parametrization for the study of genus one solutions, since it uniformizes the torus. The manifestation of the latter is the simple unified description of this class of classical string solutions in terms of the “effective energy” *E* of the sine-Gordon reduced system and the purely imaginary parameter *a*. Adopting this parametrization significantly simplifies the expressions for the conserved charges of the string and facilitates the study of the corresponding dispersion relation. In particular, we identify a set of one-dimensional trajectories in the moduli space, where it is possible to express the moduli as an algebraic function of the ratio of the energy and the angular opening, allowing the expression of the dispersion relation in a closed form, arbitrarily far away from the infinite size limit. These trajectories compose a dense subset of the moduli space.

Another interesting feature that emerges from the properties of the sine-Gordon equation has to do with its well known duality with the Thirring model. The duality maps the topological charge of the sine-Gordon theory to the fermion number in the Thirring model. Therefore, the number of spikes has a naive interpretation as a fermion number. In this picture, the strings with rotating Pohlmeyer counterparts have the natural interpretation of fermionic objects of the theory, whereas the strings with oscillating counterparts have the interpretation of bosonic condensates of the latter. The study of elliptic strings in this context would have an enhanced interest in view of the S-duality of the type IIB superstring theory in \(\hbox {AdS}_n \times \hbox {S}^n\) spaces. In this case, such elliptic solutions could provide a quantitative tool to understand the role of the sine-Gordon/Thirring duality as S-duality in the Pohlmeyer reduced theory.

The presented techniques can be directly generalized to higher dimensional spheres and to \(\hbox {AdS}_n\times \hbox {S}^n\) spaces. As long as \(\hbox {S}^n\) is concerned, when *n* is even, the eigenvalues of the problem will have the same structure as in the presented \(\hbox {S}^2\) case: there will be an odd number of enhanced space embedding functions, which will be organised in several pairs, each being associated to a positive eigenvalue connected to a Bloch wave eigenstate of the associated \(n=1\) Lamé problem and a single one that will be associated with a vanishing eigenvalue, and, thus, connected to an eigenstate of the \(n=1\) Lamé problem lying at the margin of a band. When *n* is odd, there will be an even number of enhanced space coordinates, which will be simply organised in pairs each associated with a positive eigenvalue. Such solutions have been constructed with other methods in the literature [39]. Further extending to \(\hbox {AdS}_n\times \hbox {S}^n\), which is of particular interest towards holographic applications, requires the combination of the results presented in this work with those of [19]. The elliptic strings on AdS spaces form some qualitatively distinct classes due to the form of the metric in the enhanced space (which is \(\mathbb {R}^{(2,n-1)}\)). It would be interesting to study how these classes get combined with the elliptic strings on the sphere and how they differ in terms of their dispersion relation or other geometric characteristics.

## Notes

### Acknowledgements

The research of D.Katsinis is funded by the “Strengthening Human Resources Research Potential via Doctorate Research” action of the operational programme “human resources development, education and long life learning, 2014–2020”, with priority axes 6, 8 and 9, implemented by the Greek State Scholarship Foundation and co-funded by the European Social Fund - ESF and National Resources of Greece.

The research of GP is funded by the “Post-doctoral researchers support” action of the operational programme “human resources development, education and long life learning, 2014–2020”, with priority axes 6, 8 and 9, implemented by the Greek State Scholarship Foundation and co-funded by the European Social Fund - ESF and National Resources of Greece.

The authors would like to thank M. Axenides, E. Floratos and G. Linardopoulos for useful discussions.

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