Classical solutions of $\lambda$-deformed coset models

We obtain classical solutions of $\l$-deformed $\s$-models based on $SL(2,\mathbb{R})/U(1)$ and $SU(2)/U(1)$ coset manifolds. Using two different sets of coordinates, we derive two distinct classes of solutions. The first class is expressed in terms of hyperbolic and trigonometric functions, whereas the second one in terms of elliptic functions. We analyze their properties along with the boundary conditions and discuss string systems that they describe. It turns out that there is an apparent similarity between the solutions of the second class and the motion of a pendulum.


Introduction
Classical solutions is one of the main subjects in Quantum Field Theory and play an important role in studying its dynamics. A large class of such configurations are the so-called solitons [1]. In general, solitons are solutions which retain their shape, as they propagate at constant velocity. Usually such lumps are topological in nature and saturate a BPS bound. The most studied example, the kinks, interpolate between different classical vacua of the theory. Such solution require the presence of a potential, so that the theory has a non-trivial vacuum structure. These configurations are excitations of the theory, which even though they are not elementary, they exhibit a particle-like behaviour. Kinks can be made to scatter and even create bound states, which are called breathers. An appealing fact about these objects is that the equations describing them are obtained using topological arguments or exploiting integrability and not by attacking the equations of motion straight-forwardly. The reason is the high non-linearity of the equations of motion, which makes the derivation of their solution practically impossible.
Having classical solutions one can utilise them in studying several aspects of QFT. These include semi-classical quantization, the vacuum structure of the theory and the effective theory around a non-trivial background solution.
In this work, focusing on 2-dimensional field theories, we obtain classical solutions of an integrable class of deformations of WZW-models, namely the λ-deformations introduced in [2]. The generalization of these models for symmetric spaces is given in [3], while the asymmetric gauging is constructed in [4], building on the setup of [5]. Recently, a lot of attention has been paid in various aspects of these theories. The reason is that they also possess non-perturbative duality symmetries, enabling the exact calculations of various set of observables among which β-functions, C-functions and anomalous dimensions for a large class of single and composite operators (see [6][7][8][9] and references therein).
It would be interesting though, to move one step further and obtain explicit expressions for classical solutions in order to enlarge the knowledge about the dynamics of such theories. Solving directly the equations of motion is a very hard task due to the high nonlinearity. To overcome this, we study λ-deformations on coset manifolds SL(2, R)/U (1) and SU (2)/U (1). Even though the non-linearity is still present and the equations of motion are not drastically simplified, there is a way to proceed. In the case of non-linear sigma models having 2-dimensional target space, essentially solving the equations cor-responding to the conservation of the Energy-Momentum tensor is equivalent to solving the equations of motion. There is a small caveat it this equivalence but it does not affect our results. In any case, exploiting this remark, along with the fact that these equations are first order and much simpler than the equations of motion, we obtain two different classes of solutions the models under consideration. It is not clear how they fit in the spectrum of the theory, since our results are not based on topological arguments.
The structure of the paper is as follows: in Section 2 we present the specific models under study emphasizing the relations between them via analytic continuations. These

λ-deformations on coset manifolds
In this section we introduce the models we are focused on, namely the SL(2, R)/U (1) and SU (2)/U (1) λ-deformed models. The vectorially gauged WZW SU (2)/U (1) λ-deformed model has been worked out in [2]. In [10] its relation to the non-compact variant, i.e. the SL(2, R)/U (1) λ-deformed model, was presented. The axial gauging of these models was constructed in [4]. Some additional details on the parametrization of the models are given in Appendix A.
The action (2.1) is also written in a new conformally flat form as where the coordinates χ and ψ are defined in term of θ and φ by equations (A.17) and (A.3). These coordinates are valued in an appropriate domain, so that the dilaton is real and the conformal factor e −2Φ positive.

SL(2, R)/U (1) Vector Gauging
Starting with the SU (2)/U (1) model and applying the following transformations Of course, the level k ceases being quantized, but this does not concern our analysis. The transformation of κ is required in order to leave the coupling λ invariant. Also, transformation of θ amounts to exchanging the cosines to hyperbolic cosines. Notice that, the overall sign of the action (due to the transformation of k) is absorbed by the dilaton, so that the later remains manifestly positive. In this case, the action reads where the coordinates y 1 and y 2 are related to the coordinates ρ and φ as The equations of motion coincide with (2.3) and (2.4), since the actions (2.1) and (2.9) as functionals of y 1 and y 2 , differ only by an overall sign. Nevertheless, in this case the coordinates y 1 and y 2 are defined in the complimentary region satisfying y 2 1 + y 2 2 ≥ 1. Following the steps of the previous section, the action (2.9) can be rewritten as (2.11) where the coordinates χ and ψ are defined in terms of ρ and φ by (A.2) and (A.3).

SL(2, R)/U (1) Axial Gauging
Implementing the transformations ρ → ρ + iπ/2, which is equivalent to on SL(2, R)/U (1) V model we obtain theSL(2, R)/U (1) A λ-deformed model. Interestingly enough, the transformation, relating the axial and vector gauged WZW models at full quantum level [11], also relates the λ−deformed theories (at least semi-classically). The corresponding action reads where the coordinates y 1 and y 2 are related to the coordinates ρ and φ as 14) The equations of motion are obtained by applying the transformation (2.12) on the equations of motion (2.3) and (2.4). It amounts to setting −1 to +1 on the denominators of the right-hand-sides. The action (2.13) can be rewritten as where the coordinates χ and ψ are defined in terms of ρ and φ by equations (A.13) and (A.3). Finally, notice that the action (2.15) is obtain by (2.11) by sending χ → χ + iπ/2.

Classical Solutions
In this section we derive two different classes of solutions for each model. The first class of solutions is derived using the actions in terms of the coordinates y 1 and y 2 . For the second class of solutions we use the actions in terms of the coordinates χ and ψ. Even though we could have used a different ansatz to solve the equations of motion of the same action, i.e. the one in terms of y 1 and y 2 , we employ the action in terms of (χ, ψ), which motivates the ansatz more naturally.
The first class of solutions, corresponding to the actions (2.1), (2.9) and (2.13), is expressed in terms of hyperbolic-trigonometric functions, whereas the second one, corresponding to the actions (2.7), (2.11) and (2.15), in terms of Jacobi elliptic functions.
As expected, the relations of the models via analytic continuation, are also reflected to the classical solutions of the same type as well. Recall that (2.12) relates the axially and vectorially gauged SL(2, R)/U (1) models, while the SU (2)/U (1) and the vectorially gauged SL(2, R)/U (1) model merely differ by the fact that the coordinates are valued in complementary regions. Therefore, solving one model is enough for solving all of them and all differences of the solutions are highlighted wherever it is necessary.

A remark on the solution of the equations of motion
It is evident that solving directly the equations of motion (2.3) is quit difficult. To untie our hands, a trick is needed which will allows us to sidestep this obstacle. Let us consider a non-linear sigma model (we do not include B µν field since it is absent in our models) The non-vanishing Energy-Momentum tensor components are given by satisfying ∂ ∓ T ±± = 0 on-shell. Due to the previous equations, the components of the Energy-Momentum tensor T ±± satisfy These are essentially, first integrals of the equations of motion and, as known, solutions of the equations of motion also solve (3.3). In addition, f ± (σ ± ) are functions, which can be set to constants, denoted as C ± . Without loss of generality, we may additionally select C ± = C. Performing a Lorentz boost on the world-sheet we may always restore unequal values of C + and C − . The models under study have a positive definite metric, thus these constants are necessarily positive. Notice that in order to embed this model in string theory, one has to consider the tensor product of this theory with another one, so that the overall Energy-Momentum vanishes, implying that the Virasoro constraints are satisfied. Slightly abusing the terminology and having the previous remark in mind, we will refer to (3.3) as the Virasoro constraints.
As mentioned, the Energy-Momentum tensor is conserved on-shell. Nevertheless, the converse is not generally true, unless the target space is 2-dimensional. In this case it follows that where L is the Lagrangian density and δL/δX µ is the variation of the action with respect to the X µ field. Thus, if the matrix on the right-hand-side is invertible, the solutions of the Virasoro constraints are also solutions of the equations of motion 3 . This approach resembles the Pohlmeyer reduction, where one gauge fixes the world-sheet diffeomorphisms [12].

First class of solutions
Having clarified our methodology, we derive the solutions of the models presented in the previous section.
Vector Gauging As a first example we derive classical solutions of the model described by the action (2.9).
The Virasoro constraints (3.3) constitute the pair of differential equations where we have set C = m 2 /4 so that the right-hand-side is manifestly positive, as the left-hand-side. The factor of 1/4 is introduced for future convenience. In order to obtain non-trivial solutions, it is required that m = 0. We can decouple the equations as follows The equations above have similar form, their only difference is the sign in front of c ± .
Thus, solving one equation suffices to obtain solutions for both equations. Considering 3 Of course it is possible that the solutions of the Virassoro constraint still solve the equations of motion, but this is not the case for the solutions of this work.
where s ± are constants that take independently the value 1 or −1. for c. Finally, one should combine the solutions for y 1 and y 2 , taking into account the requirement that their derivatives are linearly independent as noted in (3.4) . Essentially, this condition implies that the solutions for y 1 and y 2 should not be functions of the same variable. In following we choose y 1 = y 1 (τ ) and also discard the various signs s ± . These can be recovered by parity transformations of the world-sheet coordinates, combined with the discrete symmetries of the action, such as y 1 → −y 1 . We gather the solutions appropriately matched in Table 1. One can verify that these indeed satisfy the equations of motion (2.3) 5 . Notice that τ 0 and σ 0 are integration constants, which may be regarded as collective coordinates of the solutions.

Axial Gauging
In the axial gauging case, writing down the Virasoro constraints corresponding to the action (2.13), one realizes that the solutions are obtained by setting on the coefficients of various functions in Table 1, while neglecting the overall minus sign to compensate the i factor of the transformation (2.12). Note that the range of c should be adjusted appropriately. The solutions are gathered in Table 2.   Table 3. Notice that all solutions in Table 1 and 2 (except the exponential ones) can be cast in the form given above.

Second class of solutions
The second class of solutions is expressed in terms of Jacobi elliptic functions. A short review is given in Appendix B where all the necessary definitions are provided, along with several of their properties. A careful study will help the unfamiliar reader for the better understanding of the material that follows.

Vector Gauging
Let us now derive solutions for the action (2.11). We implement the same strategy that we used for the first class of solutions by solving the Virasoro constraints and combining the solutions appropriately to satisfy the equations of motion. Again, the form of the metric implies that the constants appearing at the Virasoro constraint have to be positive definite. Thus, in the following we obtain solutions of the equations (3.10) Equations (3.10) are decoupled as Positivity of the constants on the right-hand-side of (3.11) is required, due to the manifest positivity of the left-hand-side. Immediately, it follows that χ is given by where α is an integration constant. Notice that the equations (3.11) are incompatible, Moving to the second pair of equations we can factorise it as where κ 2 is the elliptic modulus, which is given by 16) and is defined by the equation As these equations are of the form of (B.2), it follows that their solution is where am(x|m) is the Jacobi amplitude 6 . The reality of the solutions requires 2 ≥ 0, which implies that c is subject to the constraint Choosing χ = χ(τ ) (and necessarily ψ = ψ(σ)) we obtain the solution Of course, the class of solutions χ = χ(σ) and ψ = ψ(τ ) is also admissible.

Axial gauging
For the axial gauging we have to solve the Virasoro constraints case of vector gauging, there are two distinct pairs of equations The solution for ψ is provided by (3.18), where c is subjected to equation (3.19). Of course, c may be taken purely imaginary to provide a solution for the minus sign in (3.23).
In such a case the solution is automatically real and the parameter c is unconstrained.
Putting everything together, we obtain the following class of solutions where the elliptic modulus κ 2 and are defined in (3.16) and (3.17), respectively, as well as, where the elliptic modulusκ 2 and˜ are defined as Notice that this solution is valid for any c. Of course, one can interchange σ and τ to obtain the rest of the solutions belonging to these classes.

SU (2)/U (1) model
As a final step we proceed with the solutions on the SU (2)/U (1). Proceeding in the usual manner, by solving the Virasoro constraints, the decoupled equations read where the right-hand-side of (3.32) is manifestly positive in view of (A.5). The solution of the first equation is while the pair of equations (3.32) can be written as where the elliptic modulusκ 2 is defined as and¯ is defined via the equation¯ The reality of the solutions implies that c is subject to the constraint Considering χ = χ(τ ) and ψ = ψ(σ), the solution reads

Properties of the Solutions
In this section we study the properties of the solutions we derived in the previous one.
First of all, we summarize the second class of solutions and present their basic features.
Then, we specify the boundary conditions. This is required in order to eliminate the surface terms, rising when varying the action. In addition, since σ-models describe string theories 7 , we also mention possible brane configurations, related to the aforementioned boundary conditions. Having specified all admissible cases, we plot the solutions and discuss them. Finally, the effect of the non-perturbative dualities (2.5) and (2.6) is described. Model

Overview of the 2nd Class of Solutions
In this section we summarize the second class of solutions and discuss some of their common features. Before doing so, we present the coordinates y 1 and y 2 for this class of solutions. Taking into account the definitions of y 1 and y 2 for each model, namely equations (2.2), (2.10) and (2.14), along with (A.10), which is common for all models, as well as the equations (A.7), (A.14) and (A. 19), it follows that y 1 and y 2 are given in terms of χ and ψ by In order to do so one has to tensor the models of this work, with another sigma model corresponding to a metric of indefinite signature. The simplest case is to consider a single time dimension. In this case, the equations of motion are solved by t = 1 2 (m + σ + + m − σ − ). This selection also makes the overall Virasoro constraints vanish, implying that the theory is (classically) conformally invariant. Had we considered C + = C − in (3.3), the obtained solutions would depend on this coordinate and on Thus, from the target space perspective, the values of of m + and m − are irrelevant, whereas on the world-sheet their values may be altered by a Lorentz boost.

Model
Similarly, they are characterized as oscillating if the value of the elliptic modulus is greater than 1 and as rotating if it is between zero and one. The expressions for the coordinates χ and ψ, and the elliptic modulus for each of the models of the first column are in Table 4.
, the solution is a rod, whose endpoint(s) lies on ellipses. In both cases, it is straightforward to show that these solutions satisfy the physical time being t = mτ . Thus, there is no momentum flow on strings endpoints.
The exact behaviour of the solution depends on the values of the parameters and the boundary conditions. In Table 4 we gather all equations, which define the second class of solutions for each model. Table 5 presents the classification of all solutions of the second class according to their features. These include the cases ψ = ψ(σ) and ψ = ψ(τ ), as well as whether the value of the elliptic modulus is greater or smaller that one. This classification is analogous to the one performed in [13] regarding elliptic string solutions For the solutions of SL(2, R)/U (1) V model and one of the two solutions of the and is associated to rotating solutions. Similarly, when it satisfies 1 ≤ κ 2 and it is related to oscillating solutions. For these solutions we define The half-period ω 1 is defined in terms of the elliptic modulus via (B.11).
Considering the other solution of the SL(2, R)/U (1) A model, the elliptic modulusκ 2 , defined in (3.30), always satisfies 0 ≤κ 2 ≤ 1 for any value of c. In this case we define Finally, for the solutions of the SU (2)/U (1) model, the elliptic modulusκ 2 , defined in and is associated to oscillating solutions. Finally, in this case we define δσ = ω 1 m¯ . (4.10) The lengths δσ, δσ and δσ will be used when studying the boundary conditions and the periodicity properties.

Special Limits
Solutions of the second class have two interesting limits. The first one is the limit of the vanishing elliptic modulus. This limit is obtained for λ = 0. In this case the Jacobi A far more interesting limit is the one of the diverging real period. This is the case when the elliptic modulus equals to unity. The Jacobi amplitude is given by (B.5). The elliptic functions degenerate to hyperbolic ones. The form of the solutions in this limit is presented in Table 6. Notice that we present only the λ > 0 solutions, while the rest of them are obtained using the duality (2.6) (m is invariant). We also present only the static solutions. The translationally invariant ones are obtained via the σ ↔ τ transformation.

Dirichlet:
Since we can impose Dirichlet conditions for y I,σ and Neumann for y I ,τ for any arbitrary σ.
Combining appropriately the above two cases, we can impose the following boundary conditions on the solutions: • N for y c,σ and y I,τ at σ = σ 0 , corresponding to a space filling p2-brane.
• D for y I,σ and N for y I ,τ at σ = σ D , corresponding to semi-infinite sting ending on a single p1-brane.
• D-D for y I,σ and N-N y I ,τ at σ = σ i and σ = σ f , corresponding to a pair of p1-branes.
• D-N for y c,σ and N-N y I,τ at σ = σ D and σ = σ 0 , corresponding to a p1-brane and a space filling p2-brane.
All these solutions correspond to infinite, semi-infinite or finite moving line segments.
Intestingly enough, the D-D and N-N boundary conditions are integrable [4].

SU (2)/U (1) model
Contrary to the SL(2; R)/U (1) case, the conformal factor e 2Φ , defined in (2.1), does not vanish at σ = ±∞. This class of solutions is naturally periodic, corresponding to closed configurations. We discuss, the y 1 = y 1 (τ ) and y 2 = y 2 (σ) case, but similar conclusions hold for the other case too. In particular, the solutions (Table 3) are periodic under These configurations are folded strings, which as time flows, oscillate in the y 1 direction.
The world-sheet of such solutions is a torus.
One can impose open string boundary conditions in the following cases: we can impose Neumann conditions both for y 1 and y 2 at σ = σ N .

Dirichlet:
Since we can impose Dirichlet conditions for y 2 and Neumann for y 1 for any arbitrary σ D .
Combining appropriately the above two cases, we can impose the following boundary conditions on the solutions: • N-N for both y 1 and y 2 at σ i = σ 0 and σ f = σ 0 + 1−λ 1+λ π m , corresponding to a space-filling p2-brane.
• D-D for y 2 and N-N y 1 at σ i and σ f , which correspond to a pair of p1-branes.
• D-N/N-D for y 2 and N-N y 1 at σ = σ D and σ = σ N , corresponding to a p1-brane and a p2-brane.
These solutions correspond to finite moving line segments.

Boundary conditions -Analysis: 2nd class
The second class of solutions reveals a much larger variety of results including static and translationally invariant configurations. It consists of fourteen distinct types of solutions.
We keep the presentation as short as possible, presenting only basic facts for each of these types of solutions, but the overall presentation is lengthy.
where ψ is given by (3.21) and the corresponding elliptic modulus by (3.16). The worldsheet of such solutions is cylindrical. Static rotating solutions correspond to closed strings. These solutions are ellipses, which, starting from infinity, approach the origin of the y 1 -y 2 plane up to τ = 0, and reflect back to infinity. Their motion, is bounded in the exterior of the ellipsis The background is a deformed version of the well-known trumpet geometry. Intuitively, it has the same features but its cross-section is of elliptic shape. As the cross-section of the trumpet grows towards the origin ρ = 0, the string approaching the origin stretches.
At some point there is no more kinetic energy to be absorbed for the string to keep stretching, thus, it reflects back.
In the case of oscillating solutions, the angle ψ "oscillates" between two limiting values and satisfies where δσ is given by (4.6), just like the angle of an oscillating pendulum depicted on the left panel of Figure (8). Intuitively, their motion is analogous to the rotating case, but the reason which prevents the string from collapsing to a points is not topological, but kinematic. The endpoints of the string move at the speed of light. Indeed, it is easy to show that G y 1 y 1 (∂ t y 1 ) 2 + G y 2 y 2 (∂ t y 2 ) 2 | σ=(2n+1)δσ = 1, ∀t, (4.23) where the time t of the target space, defined as t = mτ . Figure 1 depicts indicative examples of closed static oscillating and rotating strings.
Open Strings As mentioned, in the case of rotating strings the angle ψ is monotonous.
Thus, in order to apply Dirichlet -Neumann (or Neumann -Dirichlet) boundary conditions, the only possibility is to set ψ = nπ/2, where n ∈ N. This is achieved for σ = nδσ.
In this way we obtain strings, which are parts of the closed rotating strings ending on y 1 = 0 or y 2 = 0 axis. One can construct configurations, which extend along one, two or three quadrants and either end on different branes or on the same one.
In the case of oscillating strings, the angle ψ "oscillates" either around 0 or around π/2, depending on whether λ > 0 or λ < 0, see (3.21). The points of the string corresponding to σ = nδσ either lie on an axis, or on the lines which are tangential to the motion of the folded string, like the dashed lines in the right panel of Figure 1.

Translationally Invariant
These solutions are of the form where ψ is given by (3.21) and the corresponding elliptic modulus by (3.16). The worldsheet of such solutions is cylindrical.  These strings satisfy identically where the time t of the target space is defined as t = mτ .
One can consider these configurations either as folded strings, for σ ∈ (−∞, ∞), in order to make the superficial terms vanish, or as open ones. As the σ = 0 point moves at the speed of light, the string is prevented from collapsing. In the case of rotating strings, this point rotates on an elliptic trajectory, whereas in the case of oscillating strings, this point oscillates between two limiting points, see Figure 2. In both cases the motion is periodic with period T = 4δσ.

Static 1
The static solutions of this class are of the form where ψ is given by (3.27) and the corresponding elliptic modulus by (3.16). The worldsheet of such solutions is cylindrical.  [14]. Intuitively, it has the same characteristics but its cross-section is of elliptic shape. As the cross-section of the cigar shrinks towards the origin ρ = 0, the string approaching the origin loosens. The incoming string becomes point-like at the tip of the cigar and then it is reflected back to infinity.
In the case of oscillating solutions, the motion of the strings is analogous to the rotating ones. Again the reason, which prevents the string from collapsing to a points is not the topological, but kinematic. The endpoints of the string move at the speed of light. Figure 3 depicts indicative examples of closed static oscillating and rotating strings.
Open Strings As in the case of the vectorially gauged model, we can enforce Dirichlet -Neumann (or Neumann -Dirichlet) boundary conditions to the rotating solutions, for σ = nδσ. In this way we obtain strings, which are parts of the closed rotating strings and end at the axis y 1 = 0 or y 2 = 0. One can construct configurations, which extend along one, two or three quadrants and either end on different branes or on the same one.
A similar picture emerges in the the case of oscillating strings. The angle ψ "oscillates" either around 0 or around π/2, depending on whether λ > 0 or λ < 0. The points of the string corresponding to σ = nδσ either lie on an axis, or on the lines which are tangential to the motion of the folded string, the dashed lines in the right panel of Figure 3.

Translationally Invariant 1
These solutions are of the form where ψ is given by (3.27) and the corresponding elliptic modulus by (3.16). The worldsheet of such solutions is cylindrical.
These strings satisfy identically where the time t of the target space is defined as t = mτ . One can consider these configurations either as infinite strings, for σ ∈ (−∞, ∞), or semi-inifinite open ones, for σ ∈ (0, ∞). In the second case, the string is considered to end on a D0 brane, which is located at the tip of the cigar. This way the string is prevent from collapsing to a point.
In the case of rotating strings the string rotates freely, whereas in the case of oscillating strings, the string oscillates between two limiting points. The configurations are like the vectorially gauged ones in Figure 2, but either with the strings ending on the origin of the plot, or extending all the way to infinity. They are depicted in Figure 4. In both cases the motion is periodic with period T = 4δσ.

Static 2
These solutions are of the form where ψ is given by (3.29) and the corresponding elliptic modulus by (3.30). The angle  Closed Strings Similar to the static solutions presented so far, these solutions are ellipses. Again, the background is a deformed version of the Witten's cigar geometry [14].
Their motion is analogous to the static rotating solutions of the vectorially gauged model.  Figure 5 depicts indicative examples of such closed static rotating strings.
Open Strings As in the case of the vectorially gauged model, we can enforce Dirichlet -Neumann (or Neumann -Dirichlet) boundary conditions to the rotating solutions, for σ = nδσ. This way we obtain strings, which are parts of the closed rotating strings and end at axis y 1 = 0 or y 2 = 0. One can construct configurations, which extend along one, two or three quadrants and either end on different branes or on the same one.

Translationally Invariant 2
The static solutions of this class are of the form where ψ is given by (3.29) and the corresponding elliptic modulus by (3.30). There are only rotating solutions of this form. The world-sheet of such solutions is toroidal.
These strings satisfy identically where ψ is given by (3.40) and the corresponding elliptic modulus by (3.36). The worldsheet of such solutions is toroidal.
Closed Strings The angle ψ satisfies (4.20) and (4.21) in the case of rotating and oscillating solutions, respectively, but in terms of δσ, defined in (4.10), instead of δσ.
Static rotating solutions correspond to closed strings. They are ellipses, whose scale oscillates. Their motion, is bounded by the ellipsis (4.21), which is expected as the The oscillating solutions are folded strings. They are part of an ellipsis and their endpoints move at the speed of light. Their motion is analogous to the rotating ones, but in this case the reason, which prevents the string from collapsing to a points is not the topological, but kinematic. It is easy to show that where the time t of the target space, defined as t = mτ . Figure 6 depicts indicative examples of closed static oscillating and rotating strings.
Open Strings In the case of rotating strings the angle ψ is monotonous. Thus, in order to enforce Dirichlet -Neumann (or Neumann -Dirichlet) boundary conditions, the only possibility is to set ψ = nπ/2, where n ∈ N. This is achieved for σ = nδσ. This way we obtain strings, which are parts of the closed rotating strings and end at the axis y 1 = 0 or y 2 = 0. One can construct configurations, which extend along one, two or three quadrants and either end on different branes or on the same one.
In the case of oscillating strings, the angle ψ "oscillates" either around 0 or around π/2, depending on whether λ > 0 or λ < 0. The points of the string corresponding to σ = nδσ either lie on an axis, or on the lines which are tangential to the motion of the folded string, like the dashed lines in the right panel of Figure 6.

Translationally Invariant
These solutions are of the form where ψ is given by (3.40) and the corresponding elliptic modulus by (3.36). The worldsheet of such solutions is toroidal. These strings satisfy identically  G y 1 y 1 (∂ t y 1 ) 2 + G y 2 y 2 (∂ t y 2 ) 2 | σ=± 1 mc π 2 = 1, ∀t, (4.36) where the time t of the target space is defined as t = mτ .
One can consider these configurations either as folded strings, or as open ones. As the endpoints moves at the speed of light, the string is prevent from collapsing. In the case of rotating strings, this point rotates on an elliptic trajectory, whereas in the case of oscillating strings, these points oscillate between two limiting ones, see 7. In both cases the motion is periodic with period T = 4δσ.

The effect of the non-perturbative symmetries
Regarding the first class of solutions, let us return for a moment in equations (3.6) and (3.7) and impose invariance under the duality symmetry (2.5). In order to do so, we have to postulate that m 2 is a function of λ transforming as On the contrary, c is not affected in implying that c(1/λ) = c(λ).
As far as the duality (2.6) is concerned, one can easily see from (3.6) and (3.7), that invariance under this duality implies that We conclude that c, is even under (2.5) and odd under (2.6)symmetry. The inverse holds for m 2 10 . One can easily verify that the sets of solutions in Tables 1, 2  Let us turn to the second class of solutions. In the case of the SL(2, R)/U (1) model, the duality symmetry (2.5) acts on the coordinates ρ and φ as (4.40) 10 As an indicative example, one may define As a result, the action (2.11) is invariant under (2.5).
In order to retain the duality symmetry at the level of solutions, equations (3.10), (3.11) and (3.12) imply that m 2 and c 2 are functions of λ satisfying Equation (3.16) implies that κ 2 is invariant, thus (3.21) implies that ψ is indeed invariant.
Last but not least, the symmetry (2.6), in terms of coordinates (θ, φ) is replaced by and one can show that under (4.43) Overall we obtain fourteen distinct solutions of this kind, which are gathered in Table   4. In addition, Table 5 Table 6.  [15] and one expects that the Pohlmeyer avatar is a solitonic object.
On the same line, the derived solutions may be used in order to obtain new ones. This can be achieved via the application of the dressing method [16,17]. The dressing method is a technique, which takes advantage of a known solution, usually refereed to as the seed solution, in order to obtain new ones. The advantage of this method is that in order to derive the solutions one has to solve a system of linear coupled first order PDEs, rather than the equations of motion, which constitute a system of coupled, non-linear, second order PDEs. This method has already been applied in the context of λ−deformations in [18]. In this work the seed solution is analogous to the BMN particle [19], a solution of the undeformed model, which also solves the deformed one. The solutions we derived are much more complicated and the application of the dressing on such seed solutions may reveal interesting phenomena, such as the formation of spikes and memory effect regarding the propagation of the inserted kink on the non-trivial background [20].
It is worth noticing that the dressing method is also related with the stability analysis of the seed solutions [20]. Obviously, it also has the advantage that besides determining the fate of small perturbations, one also obtains the full non-linear solution as well. Of course, even a linear stability analysis is of interest. Its conclusions are expected to match the ones of the dressing method [21].
It would, also be interesting to derive kink configurations i.e time-independent lumps of finite energy, or time independent solutions in general. Unfortunately, for such configurations the approach based on the Energy-Momentum conservation is inapplicable.
This is also the case for models, whose target space is of higher dimension. Regarding models, generalizing the ones of this work, i.e. ones having the groups considered here as subgroups, one can obtain solutions via the dressing method. To do so, one needs to embed the solutions of this work in the higher-dimensional target space and apply the dressing method using this seed solution.
Besides further investigating the solutions themselves or the corresponding models, there are various studies, which are related to them. To begin with, it would be interesting to find classical solutions of other theories having a 2-dimensional target space using the approach based on the Energy-Momentum conservation. For instance, one may obtain solutions of various known integrable deformations, such as Yang Baxter [22] / η [23,24], bi Yang Baxter [25] and asymptotic λ-deformations [26]. Regarding the Yang Baxter and η deformations, it is known that they are related to λ−deformations via Poisson-Lie T-duality and appropriate analytic continuations [27]. Thus, one could test whether the solutions obtained here, also solve the η-deformed models after necessary manipulations.
As far as asymptotic λ-deformations are concerned, it would be interesting to derive solutions for the theory corresponding to the asymptotic limit of the SL(2, R)/U (1) model. In this case we expect a larger variety of results, as a consequence of symmetry enhancement. Such solutions may be relevant to the original FZZ-duality [28,29] or to the study of the effect of the λ−deformation on the duality. Moreover, as we described a lot of open string configurations, another interesting direction, is to investigate the fitting of D-branes in this setup and derive classical solutions for such objects. One can also study whether the boundary conditions preserve the integrability of the theory or not [30].
Finally, the derived solutions enable various field theory calculations. In particular, one could study the effective theory related to a non-trivial classical solution and perform semi-classical quantization.
With some work one can show that defining the new coordinates χ and ψ as the action can be written in a conformally flat form, namely (2.11), where initially the dilaton is defined as Note that the square root in the definition of the χ variable, equation (A.2), should not worry us, because it is well defined for −1 < λ < 1, since Of course, the dilaton field has to be expressed in terms of the new variables (χ, ψ). To invert (A.2) and (A.3) we take advantage of the equation which is a direct consequence of (A.3). Doing so, we obtain following expressions It order to specify uniquely ψ in terms of φ we choose Taking into account (A.6), the inverse transformation is (A.10) These definitions are common for all three models.
In view of (A.5), the quantity under the square root is positive provided that ψ is a real function. Nevertheless, the validity of (A.7) requires This is a constraint that has to be imposed on the solutions. Implementing (A.7) and (A.8) on the dilaton profile (A.4), we can finally express it in terms of χ and ψ as (A.12) Provided the inequality (A.11) is satisfied, the dilatonΦ is real valued as required.

SL(2, R)/U (1) Axial Gauging
The Contrary to the case of vector gauging, the inverse transformation, namely is valid automatically and does not impose any constraint on the parameters. The first new entry here is the expression relating the fields χ and ψ with the new dilaton field.

B The Jacobi Elliptic Functions
In this section we gather some properties of the Jacobi elliptic functions, which are relevant for this work. The fundamental object of Jacobi elliptic functions is the Jacobi amplitude am(x|m). Essentially, is generalizes the linear function f (x) = x. Using the Jacobi amplitude, one defines the elliptic sine and cosine as sn(x|m) = sin (am(x|m)) , cn(x|m) = cos (am(x|m)) . (B.1) The Jacobi amplitude satisfies the differential equation In this work, the elliptic modulus is always positive, so we set m = κ 2 , where κ ∈ R.
Depending on whether 0 < κ 2 < 1 or 1 < κ 2 the Jacobi amplitude is either periodic or quasi-periodic function. In the special case κ 2 = 1 it follows that am(x|1) = 2 arctan (e x ) − π 2 . (B.5) The function is neither periodic or quasi-periodic and interpolates monotonically from −π/2 to π/2. This is the famous kink of the sine-Gordon equation. In this case the elliptic functions degenerate to hyperbolic ones, namely sn(x|1) = tanh(x), cn(x|1) = sech(x). (B.6) In order to proceed, let us consider a simple pendulum. This physical system will reveal all features of the Jacobi amplitude. The energy conservation of the pendulum The energy is normalized so that the (stable) equilibrium point φ = 0 corresponds to 11 The reader should be aware that it is quite common to find this differential equations in the form If E < 2ω 2 , the pendulum oscillates between the angles −φ 0 and φ 0 , where the value of the angle φ 0 is φ 0 = arcsin E 2ω 2 . This behaviour is general, if κ 2 > 1, the Jacobi amplitude is bounded. In particular, it follows that − arcsin(1/κ) ≤ am x|κ 2 ≤ arcsin(1/κ), κ ≥ 1. (B.10) If E > 2ω 2 , the motion of the pendulum is no longer oscillatory. The pendulum rotates, but its motion is modulated by the gravitational force. On average the angle grows linearly with time. Of course, as the energy grows, the modulation becomes less significant.
Finally, if E = 2ω 2 the motion of the pendulum is aperiodic. Starting from the unstable equilibrium point φ = −π, after an infinite amount of time, the pendulum reaches the unstable equilibrium point φ = π. In the main text, solutions corresponding to κ 2 > 1 are referred to as oscillating, whereas solutions corresponding 0 < κ 2 < 1 as rotating.
Elliptic functions are doubly periodic on the complex plane. The elliptic functions appearing in the solutions derived in this work have one real and one imaginary period.
These periods form a lattice on the complex plane. Denoting the real half-period as ω 1 , it turns out that where K(m) is the complete elliptic integral of the first kind, defined as