Solution of the equations of motion for a super non-Abelian sigma model in curved background by the super Poisson-Lie T-duality

The equations of motion of a super non-Abelian T-dual sigma model on the Lie supergroup $(C^1_1+A)$ in the curved background are explicitly solved by the super Poisson-Lie T-duality. To find the solution of the flat model we use the transformation of supercoordinates, transforming the metric into a constant one, which is shown to be a supercanonical transformation. Then, using the super Poisson-Lie T-duality transformations and the dual decomposition of elements of Drinfel'd superdouble, the solution of the equations of motion for the dual sigma model is obtained. The general form of the dilaton fields satisfying the vanishing $\beta-$function equations of the sigma models is found. In this respect, conformal invariance of the sigma models built on the Drinfel'd superdouble $((C^1_1+A),I_{(2|2)})$ is guaranteed up to one-loop, at least.


Introduction
Non-linear sigma models with target space supermanifolds possess a number of interesting properties. They are of importance both in superstring theory [1] and condensed matter physics [2]. Perhaps the main motivation to study them comes from the AdS/CFT duality. Recently, remarkable progress in exact solution of the maximally supersymmetric case of AdS/CFT duality has been suggested which applies similar integrability-based methods to other AdS/CFT systems [3]. Furthermore, it has been shown that sigma models on the coset supermanifolds can be used for quantizing superstring theory with Ramond-Ramond backgrounds. The best known example is type IIB Green-Schwarz superstring constructed in AdS 5 × S 5 background in terms of supercoset formalism [4] (see, e.g., [5]- [7]). In order to gain a better understanding of how superstring theory might be quantized, the Lie supergroups P SL(n|n) were considered in [5], [6] and [8]. These Lie supergroups, having vanishing Killing form and are therefore Ricci-flat, guarantee that sigma models on them are conformally invariant up to one-loop [9]; this is a result of [6] in which these models are exactly conformal.
On the other hand, duality transformations have played an important role in string theory. Target space duality (T-duality) as a very important symmetry of string theories, or more generally two-dimensional sigma models, has been achieved in revealing the consequences of the symmetry in string theory [10]- [13]. In particular, it states equivalence between string theories propagating on two different target spaces containing some Abelian isometries [10]. Abelian T-duality is possible when the group of isometries of the target manifold is Abelian [14]. In this case, the dual theory has also Abelian isometry and it is possible to reverse the procedure and obtain back the original model from the dual one. Unlike the Abelian case, the non-Abelian T-duality transformation is not invertible since the symmetries of the original theory are not preserved. So, it was not possible to perform the inverse duality transformation to get back to the original model [15]. A solution to this problem was proposed by Klimčík andŠevera [16] where it was argued that the two models are dual in the sense of the called Poisson-Lie T-duality. The Poisson-Lie T-duality does not require existence of isometry in the original target manifold. In this case, the model is not required to be symmetric, but there is still an action of a Lie group G on the target manifold and the Noetherian currents associated with this action are required to be integrable, that is, they satisfy Maurer-Cartan equations with group structure ofG (with the same dimension of G) so that G andG have Poisson-Lie structure and their Lie algebras are dual to each other in the sense that they define a bialgebra structure.
Solution of the sigma models in curved and time-dependent backgrounds is often very complicated. In the contribution [16], the procedure for transforming solutions of a sigma model to those of a dual one has been described. Afterwards this procedure was extended in such a way that by the use of the transformation of group coordinates of the flat model to those for which the metric is constant, a classical solution of the equations of motion for a sigma model in curved background was found [17] (see, also, [18]). Moreover, prior to the procedures done in [17] and [18], conformally invariant three-dimensional sigma models on solvable Lie groups that were Poisson-Lie T-dual or plural to sigma models in the flat background with the constant dilaton were investigated in Ref. [19] (see, also, [20]).
We have recently generalized Piosson-Lie symmetry on manifolds to supermanifolds and have constructed super Poisson-Lie T-dual sigma models on Lie supergroups [21], specially on supermanifolds [22]. We, furthermore, have shown that the super Poisson-Lie T-duality relates the WZW models based on the Lie supergroups to each other [23] (see, also, [24] ). In this paper we are going to solve the equations of motion of a super non-Abelian T-dual sigma model in curved background. To this end, we use the fact that we know these models in curved background that can be transformed to the flat ones. By solving partial differential equations that follow the transformation properties of the components of the connection we find transformation between supergroup coordinates of the model in curved background and Riemannian coordinates (flat supercoordinates) for which the metric is constant. The model is built on the Lie supergroup (C 1 1 + A). The Lie superalgebra (C 1 1 + A) [25] of the Lie supergroup (C 1 1 + A) is a (2|2)-dimensional Lie superalgebra with two bosonic and two fermionic generators. Because of performability of the super Poisson-Lie T-duality transformations, namely finding coordinates of the dual decompositions depending on the complexity of the structure of Drinfel'd superdouble where the sigma models live, we have chosen one of the Lie sub-supergroups of the decomposition of the Drinfel'd superdouble to be Abelian Lie supergroup. In this case, the super Poisson-Lie T-duality reduces to the super non-Abelian T-duality. By reducing flat supercoordinates to the solution of the wave equation we obtain solution of the equations of motion of the original model. We, furthermore, show that flat supercoordinates transformation is a supercanonical transformation. In the following, by using the super Poisson-Lie T-duality transformation that follows two possible decomposition of elements of the Drinfel'd superdouble, we get the solution of the equations of motion for the dual model. The reason why it is very desirable to find the transformation between supergroup coordinates and flat supercoordinates is because in flat supercoordinates the metric becomes constant. Therefore, the vanishing β-function equations become very simple. On the other hand, finding solution for conformally invariant sigma models in curved backgrounds is generally a difficult problem. Thus, with regard to the above discussion we easily solve equations for the dilaton field of the flat sigma model in flat supercoordinates. Then, to get the general dilaton of the original sigma model that satisfies the vanishing β-function equations in curved background, we use the flat supercoordinates transformation. Finally, we obtain the general form of the dilaton field satisfying the vanishing β-function equations for the dual model in terms of the dual Lie supergroup coordinates.
The plan of the paper is as follows. After short review of the super Poisson-Lie symmetry and mutual super Poisson-Lie T-dual sigma models on Lie supergroups, based on the previous papers, in Section 2, we introduce the Lie superalgebra (C 1 1 + A) and the Lie superalgebra of Drinfel'd superdouble (C 1 1 + A) , I (2|2) in Section 3. Moreover, in this Section, the (2|2)-dimensional super non-Abelian T-dual sigma models based on the Drinfel'd superdouble (C 1 1 + A) , I (2|2) are constructed. Section 4 is dedicated to the presentation of a method for finding flat supercoordinates. In Section 5 we find the transformation of supercoordinates that makes the metric constant. Nevertheless, in Section 5, we prove that the flat supercoordinates transformation is indeed a supercanonical transformation. In Section 6, utilizing the solution of the wave equation and using the super Poisson-Lie T-duality transformations we get the solution of the equations of motion for the dual sigma model. Finding the general form of the dilaton fields satisfying the vanishing β−function equations of the sigma models is given in Section 7. Finally, we present our conclusions.
2 A review of the super Poisson-Lie symmetric sigma models on Lie supergroups To set our notation, let us briefly review the construction of the super Poisson-Lie T-dual sigma models by means of Drinfel'd superdoubles. Consider a two-dimensional sigma model action on a supermanifold M as 1 where ∂ ± are the derivatives with respect to the standard lightcone variables σ ± := 1 2 (τ ± σ) and The coordinates Φ Υ include the bosonic and the fermionic coordinates, and the label Υ runs over µ = 0, · · · , d B − 1 and α = 1, · · · , d F where d B and d F indicate the dimension of the bosonic coordinates and the fermionic ones, respectively. Thus, the superdimension of the supermanifold is written as (d B |d F ). We define, respectively, the Christoffel symbols (the components of the connection) and the torsion as 3 Thus, one gets the equations of motion for the action (1) as If the Noether's current one-forms corresponding to the right action of the Lie supergroup G on the target space M are not closed and satisfy the Maurer-Cartan equation [26] on the extremal surfaces, we say that the sigma model (1) has the super Poisson-Lie symmetry with respect to the 1 We note that |Υ| denotes the parity of Υ where |Υ| = 0 for the bosonic coordinates and |Υ| = 1 for the fermionic coordinates; here and in the following we use the notation [26], e.g., (−1) |Υ| := (−1) Υ .
2 Note that G ΥΛ and B ΥΛ are the components of the supersymmetric metric G and the anti-supersymmetric tensor field B, respectively, We will assume that the metric Υ G Λ is superinvertible and its superinverse is shown by G ΥΛ . 3 Notation: If f be a differentiable function on R m c × R n a (R m c are subset of all real numbers with dimension m while R n a are subset of all odd Grassmann variables with dimension n), then, the relation between the left partial differentiation and right one is given by where |f | indicates the grading of f [26].
Lie supergroupG (the dual Lie supergroup to G) [21]. It is a condition that is given by the following relation [21] where L Vi stands for the Lie derivative corresponding to the left invariant supervector fields V i (defined with left derivative) [21] and (Ỹ i ) jk = −f jk i are the adjoint representations of Lie superalgebrã G (the dual Lie superalgebra to G). When L Vi (E ΥΛ ) = 0, the super Poisson-Lie T-duality is replaced by the super non-Abelian T-duality, i.e., the Lie supergroup G is a isometry supergroup of M and we have the conserved currents.
Let us consider the dualizable sigma models defined on the Lie supergroup space (the generalization to the case when a Lie supergroup G acts freely on the target space). The super Poisson-Lie dualizable sigma models can be formulated on a Drinfel'd superdouble D ≡ (G,G) [27], a Lie supergroup whose D admits a decomposition D = G ⊕G into a pair of sub-superalgebras maximally isotropic with respect to a supersymmetric and ad-invariant non-degenerate bilinear form < . , . > which is defined by the brackets Such decomposition is called Manin supertriple (D, G,G). The generators of G andG are, respectively, denoted to X i andX i and satisfy the (anti)commutation relations [28] [ The structure constants f k ij andf ij k are subject to the mixed super Jacobi identities [28] f m Now, we assume ε + is a (d B |d F )-dimensional linear sub-superspace and ε − is its orthogonal complement so that ε + + ε − span the whole Lie superalgebra D. To determine a dual pair of the sigma models with the targets G andG, one can consider the following equation of motion for the mapping l(σ + , σ − ) from the worldsheet into the Drinfel'd superdouble D 5 Then, using equation (9) and the decomposition of an arbitrary element of D in the vicinity of the unit element of D as we obtain where E ± : G →G is a non-degenerate linear mapping and (11) and using relation (6), we can write the field equation (11) in the form [21] (∂ +hh where (∂ ±hh −1 ) i are right invariant one-forms onG and satisfy in the following flat connection equations However, one can show that equations (15) are the field equations of the sigma model with the action in which are left invariant one-forms with left derivative. Comparing the action (16) with the action (1) we Then, using the fact that left invariant one-forms are dual to the supervector fields, i.e., i V Υ Υ L (l) j = i δ j , one can rewrite the equations (13) and (14) in the following form Also, one can rewrite the action (16) in the following form where R ± (l) i 's are right invariant one-forms with left derivative and are defined by The matrix F + ij (g) of the sigma model is of the form where E 0 + (e) is a constant matrix and Π(g) defines the super Poisson structure on the Lie supergroup G, and the sub-matrices a(g) and b(g) are constructed in terms of the bilinear forms as Equivalently, by using the decomposition and by exchanging G ↔G, G ↔G and i E 0j , the dual sigma model is obtained. Furthermore, we note that equations (10), (13), (14) and (25) define the so-called super Poisson-Lie T-duality transformations.
Notice that if we take a dual Abelian Lie supergroup (f ij k = 0) with local coordinatesΦ k characterizing the supergroup elementg, then, in this case we have recovering, thus, the super non-Abelian duality. To continue, we shall present an example of (2|2)−dimensional super non-Abelian T-dual sigma models with a curved background for which the sigma model can be explicitly solved by the super Poisson-Lie T-duality transformations. This model is obtained from the Drinfel'd superdouble (C 1 1 + A) , I (2|2) .

The super non-Abelian T-dual sigma models
Both the original and the dual geometries of the super Poisson-Lie T-dualizable sigma models are derived from the so-called Drinfel'd superdouble which is a Lie supergroup. We shall construct a dualizable sigma model on the (2|2)-dimensional Lie supergroup (C 1 1 + A) when the dual Lie supergroup is the (2|2)-dimensional Abelian Lie supergroup, hence, the super Poisson-Lie T-duality reduces to the super non-Abelian T-duality. To this end, we first introduce the Lie superalgebra (C 1 1 + A) [25] of the Lie supergroup (C 1 1 + A). As mentioned in introduction Section, the Lie superalgebra (C 1 1 + A) possesses two bosonic generators X 1 and X 2 along with two fermionic ones X 3 and X 4 . These four generators obey the following set of non-trivial (anti)commutation relations [25] [X 1 , The Lie superalgebra of the Derinfel'd superdouble which we refer to as (C 1 where {X 1 ,X 2 } and {X 3 ,X 4 } are the respective bosonic and fermionic generators of the Abelian Lie superalgebra.

The original model
The super Poisson-Lie T-dual sigma models are usually expressed in terms of supergroup coordinates. These coordinates are given by the Lie supergroup structure and follow from the possibility to express the elements of the Lie supergroup as a product of elements of one-parametric sub-(super)groups. In order to construct the corresponding original sigma model with the Lie supergroup (C 1 1 + A) as the target space, we use the supergroup coordinates Φ Υ = {x, y; ψ, χ}, for which the elements of the Lie supergroup G are parametrized as g = e χX4 e yX2 e xX1 e ψX3 , where x(τ, σ) and y(τ, σ) are bosonic fields while ψ(τ, σ) and χ(τ, σ) are fermoinic ones. Inserting the above parametrization in (22) and (17), R (l) ± i 's and L (l) ± i 's are, respectively, found to be and In this case, Π(g) = 0 because the dual Lie supergroup is Abelin. Hence, choosing the sigma model matrix E + 0 (e) at the unit element of the Lie supergroup (C 1 1 + A) as where K is a non-zero constant, and using the first equation of (23), the original sigma model action is worked out as follows: This model is not only Ricci flat and torsionless but also is flat in the sense that its Riemann tensor vanishes. Thus, we shall deal with a model having E ΥΛ = G ΥΛ so that

The dual model
In the same way to construct the dual sigma model, we parametrize the corresponding Lie supergroup (Abelian Lie supergroup) with bosonic coordinates {x,ỹ} and fermionic ones {ψ,χ} so that its elements can be written as:g = eχX 4 eỹX 2 exX 1 eψX 3 .
Since the dual Lie supergroup is Abelian, so the components of the right invariant one-forms are simply calculated to be ΥR±i (l) = Υδi ( Υδi is the ordinary delta function). Using the structure constants of the Lie superalgebra (C 1 1 + A) (relation (27)) and then utilizing the second equation of (26), the super Poisson structure on the dual Lie supergroup is found to bẽ It is quite interesting that the super Poisson structure is superinvertible, because one can use the relationω to construct a even supersymplectic two-form 6ω on the dual supergroup supermanifoldG = I (2|2) as follows:ω It is easy to show that dω = 0. Moreover, one can write theω as the exterior derivative of one-form where f (x) is an arbitrary function ofx and C is an even, real constant. Let us, now, write the action of dual sigma model. Substituting (36) in the first equation of (23) (equation (23) in the dual form) and then using the fact that (Ẽ + 0 ) −1 (ẽ) = E + 0 (e), the dual sigma model action is obtained to bẽ (40) 6 Note that in terms of the coordinate basis {dΦ Λ }, a two-formω is defined asω = (−1) ΥΛ 2ωΥΛ dΦ Υ ∧ dΦ Λ (for more detailed description see DEWITT's book [26]).
Comparing the above action with the sigma model action (1), we can elicit the metricG ΥΛ and the fieldB ΥΛ as The investigated dual model has non-zero anti-supersymmetric partB ΥΛ of the tensorẼ ΥΛ , but it is flat in the sense that its scalar curvature vanishes.

Flat supercoordinates
Riemannian coordinates on supermanifolds are one type of coordinates where the sigma models live. In these coordinates, the metric on supermanifold have a special simple form. The Riemannian coordinates of the flat metrics are, here, called flat supercoordinates.
We know that the components of the connection do not transform as the components of a mixed tensor field. Thus, when the transform is from one coordinate basis to another, the Γ Υ Λ∆ takes the form [26] Γ Υ Λ∆ = (−1) For finding the flat supercoordinates Ω A (here, where Ω 1 and Ω 2 are bosonic coordinates while Ω 3 and Ω 4 are fermionic ones), we shall use formula (42). In these supercoordinates the metric becomes constant and, therefore, the elements ofΓ A BC vanish. Then, we get the following system of partial differential equations (PDEs) for Ω(Φ) The possibility to solve the above equation depends on the form of Γ Υ Λ∆ . For the metric given by the relation (34) we find the general explicit solution with the initial condition that will produce the Riemannian coordinates. The initial condition for producing the flat supercoordinates, in which the metric requires the constant formḠ(Ω) = G(Φ = 0), is given by In the next section, we shall present the solution of equations (43) in detail for the metric (34).

Solving the equations for flat supercoordinates
In this section we find the transformation of supercoordinates that makes the metric constant and use it for solving the models. As mentioned in subsection (3.1), the model given by the metric (34) lives in the flat background in the sense that its Riemann tensor vanishes. In spite of the fact that the metric is flat, the Γ Υ Λ∆ are not zero, so, it is not easy to find the coordinates Φ Υ (τ, σ) that solve the equations of motion given by the action (1). We use relation (2) to calculate the components of the connection for the metric (34). Then, equations (43) read where (−1) . The above equations can be solved and the general solution is where the integration constants α 0 and β 0 are real a−numbers (odd real constants) while the rest of constants are real c−numbers (even real constants) [26]. Imposing the initial condition (44), the integration constants will be determined. Then, the solution (51) takes the following form For simplicity, in the next we will assume that α 0 = β 0 = 0. The transformation (52) transforms the metric (34) into constantḠ Thus, the action corresponding to the metric (53) is given bȳ In the following we shall show that the transformation (52) is indeed a classical supercanonical transformation.

Supercoordinates transformation as a supercanonical transformation
This subsection begins with the calculation of the momentums corresponding to the actions (33) and (54) and the transformation between them. To this end, we first write Lagrangians of the actions (33) and (54) in the worldsheet coordinates. Then, defining the conjugate to Φ Υ momentum on supermanifolds as the corresponding moments are found to be andP The transformation (52) is a transformation between supercoordinates Φ Υ and Ω A . Utilizing the transformation (52) and contracting relations (56) and (57), the transformation between momentums is also obtained to be of the form Now, one can use relation (57) to obtain the Hamiltonian corresponding to the action (54). Then Under the transformation (52), the Hamiltonian (59) turns into One can simply show that the Hamiltonian (60) is nothing but the Hamiltonian corresponding to the action (33). Thus, we proved that the Hamiltonians corresponding to the actions (33) and (54) are equal; that is, under the transformation of supercoordinates (52), one goes fromH(Ω) to H(Φ) and vice versa, hence, proving that the transformation (52) is indeed a supercanonical transformation. A bit surprisingly, one can show that under the linear transformation , , , the action (54) reduces to the following action where the indices µ, ν and α, β range over the values 1, 2 and 3, 4, respectively. So, η µν = diag(−1 , 1) denotes the Minkowski metric of the bosonic directions, and the metric of the fermionic directions is The action (62) is nothing but the Polyakov action on a (2|2)-dimensional flat supermanifold [31]. Meanwhile, the linear transformation (61) is also a supercanonical transformation.

Calculation of the dual solution using the super Poisson-Lie T-duality transformations
The relation between the solutions Φ Υ (σ + , σ − ) of the original model (33) andΦ Υ (σ + , σ − ) of the dual model (40) is given by two possible decompositions (10) and (25), in which theh satisfies the equations (19) and (20). Therefore, for a solution Φ Υ (σ + , σ − ) of the sigma model we must find h(σ + , σ − ), i.e., solve the system of PDEs (19) and (20). Then, using the transformations (10) and (25) we find the solutionsΦ Υ (σ + , σ − ) of the dual sigma model. To this end, by introducing we transform the equations of motion to the wave equations. In relations (64) and (65), the W a (σ + ) and the U a (σ − ) are even, arbitrary functions while the η α (σ + ) and the ξ α (σ − ) are odd, arbitrary ones. Thus, functions Φ Υ (σ + , σ − ) follow from (52), (64) and (65) Now one can show that the above functions satisfy the following equations Note that equations (67) are the equations of motion of action (33) which have been obtained by using relation (4). Thus, we could find the solution of the equations of motion of the original model by the use of the flat supercoordinates transformation. In the following to findh(σ + , σ − ), we solve the system of PDEs (19) and (20). To make up the right hand sides of the equations (19) and (20) one must use (31) to calculate i V Υ 's. Then, using (34) the right hand sides read On the other hand, if we parametrize elements of the Abelian Lie supergroupG with bosonic coordinates {h 1 ,h 2 } and fermionic ones {h 3 ,h 4 } as then, the left hand sides are just ∂ ±hi . Substituting the solution Φ Υ (σ + , σ − ) given by (66) into (68), the system of PDEs (19) and (20) can be solved and the general solution is 7 where c 0 and d 0 are even real constants while ζ 0 and ̺ 0 are odd real ones, and the functions Q ± and γ ± solve in which primes denote differentiation. At the end of this section, we get the solution of the equations of motion for the dual sigma model in the curved background (41) by using the duality transformation that follows two possible decompositions of elements of l ∈ D as If we write all elements of the Lie supergroups G andG as a product of elements of one-parametric sub-(super)groups, then, the equation (72) Now, using the (anti)commutation relations of the Derinfel'd superdouble (C 1 1 + A) , I (2|2) obtained in Section 3, the right hand side of the equation (73) is rewritten in the form Hence, we getx and 7 Here, Q± and W a + have been used instead of Q(σ + , σ − ) and W a (σ + ), respectively. The other notations follow this rule, too.
Inserting relations (66) and (70) into (75) we get the solution of the equations of motion for the dual sigma model given by the action (40) as follows: where we have assumed that the constants a 0 , b 0 , c 0 , d 0 and ζ 0 , ̺ 0 are zero. Applying the formula (4) for the dual model, the equations of motion of the sigma model given by the curved background (41) are obtained to be of the form Due to the complexity of the above equations, we are unable to solve them directly. But we have found their solution in the form (77) by using the flat supercoordinates transformation. One can check that solution (77) actually satisfies the equations of motion (78).

Invariance of the dilaton field under the supercoordinates transformation and conformal sigma models
To guarantee the conformal invariance of the sigma models, at least at the one-loop level, one must show that they satisfy the vanishing β-function equations. To this end, we need to add a scalar field ϕ (dilaton field) to the action (1) and rewrite it in the usual standard form [22] where R (2) is the curvature of the worldsheet. It is well known that the connection between the string effective action and the sigma model (79) is through the Weyl anomaly coefficients [22] β (G) where R ΥΛ and R are the Ricci tensor and the scalar curvature, respectively, and H ΥΛ∆ is defined by equation (3). As shown in Ref. [22], these coefficients, which are known as the vanishing β-function equations, can be derived as the equations of motion of the string effective action on supermanifold M where (m, n) = (d B |d F ) and G = sdet( Υ G Λ ). As shown in Section 5, the action (54) has been expressed in terms of the flat supercoordinates Ω A transforming the flat metric G ΥΛ (Φ) into a constant form G AB (Ω). Moreover, the model is torsionless. Therefore,Γ A BC andH ABC are vanished. Consequently, equation (81) is satisfied and equations (80) and (82) are, respectively, read It is clear that for the metric constant (53) and a constant dilaton field, the vanishing β-function equations (equations (84) and (85)) are satisfied. But, since we know the flat supercoordinates of the model, we can easily find the general form of the dilaton field that together with the metric constant (53) satisfy the equations (80)-(82). From the form of equations (84) and (85) it is easy to see that the general form of their dilaton field is whereφ 0 and k 0 are even real constants while λ 1 and λ 2 are odd real ones. Now, inserting the transformation (52) into (86), one gets the general form of the dilaton field of the original sigma model (the model described by the metric (34)) in such a way that the result is where ϕ 0 =φ 0 + λ1λ2 k 0 a 0 + k 0 b 0 + λ 1 α 0 + λ 2 β 0 . On the other hand, as explained in subsection (3.1), the original model is flat (its the Ricci tensor and the scalar curvature are zero) and also torsionless. So, for this model, equation (81) is satisfied and equations (80) and (82) are, respectively, turned into It is interesting to see that the dilaton field (87) is the general solution of equations (88) and (89); that is, the metric (34) together with the dilaton field (87) satisfy the vanishing β-function equations. Thus, we conclude that the models (33) and (54) are conformally invariant up to one-loop. We also showed that under the transformation of supercoordinates (52), one goes from ϕ(Φ) toφ(Ω) and vice versa, i.e., the dilaton field is invariant under the transformation (52). Let us now turn into the dual model. For the dual model with the background (41), we find that the only non-zero component ofR ΥΛ isR 22 = 2K 2 (K 2 −ỹ 2 ) 2 ; asG 22 = 0,R = 0. Also, one quickly finds that the only non-zero component ofH isH 234 = − K 2 (K 2 −ỹ 2 ) 2 . It is then straightforward to verify thatH ΥΛ∆H ∆ΛΥ = 0, and the only non-zero component ofH Υ∆ΞH Ξ∆ Λ isH 234H 43 2 = − K 2 (K 2 −ỹ 2 ) 2 .
Putting these in equations (80)-(82) we arrive at 8 By solving the above equations, one gets the general form of the dilaton field of the dual model to be of the formΦ whereΦ 0 andk 0 are the even constants of integration. Thus, conformal invariance of the dual model is also guaranteed up to one-loop.

Conclusion
We have obtained the solution of the equations of motion for a super non-Abelian T-dual sigma model in curved background. In this way, we have first used the fact that the super Poisson-Lie Tduality transformations can be explicitly performed for the Drinfel'd superdouble (C 1 1 + A) , I (2|2) . Then, we have obtained the explicit transformation between the supergroup coordinates of the model living in the flat background and its flat supercoordinates. Also we have proved that flat supercoordinates transformation is a supercanonical transformation. We, furthermore, have found a linear transformation of the flat supercoordinates which reduces the action of the flat model with the metric constant to the Polyakov action on a (2|2)-dimensional flat supermanifold. By reducing the transformation of supercoordinates to the solution of the wave equation and using the super Poisson-Lie T-duality transformations, we have obtained the solution of the equations of motion of the dual sigma model. Finally, we were able to find an example of the super non-Abelian T-dual conformal sigma models (at least at the one-loop level) for which the vanishing β-function equations are satisfied. By solving the vanishing β-function equations we found the general form of the dilaton field of the models and showed that the dilaton field of the original model is invariant under the flat supercoordinates transformation. That is, one can go from ϕ(Φ) toφ(Ω) and vice versa.