Poisson-Lie T-duals of the bi-Yang-Baxter models

We prove the conjecture of Sfetsos, Siampos and Thompson that suitable analytic continuations of the Poisson-Lie T-duals of the bi-Yang-Baxter sigma models coincide with the recently introduced generalized lambda models. We then generalize this result by showing that the analytic continuation of a generic sigma model of"universal WZW-type"introduced by Tseytlin in 1993 is nothing but the Poisson-Lie T-dual of a generic Poisson-Lie symmetric sigma model introduced by Klimcik and Severa in 1995.

T-models are nothing but the analytic continuations of the Poisson-Lie Tduals of the KS-models.
By the way, we find truly remarkable that the T-models and the KSmodels were orbiting around for two decades without "knowing about each other". The reason for this is that the authors of [14] have worked out the target space geometries of the Poisson-Lie T-duals of the KS-models in coordinates natural from the point of view of Poisson-geometry but not natural for the comparison with the T-model. The parametrization of the dual target suitable for this comparison was introduced in [16] and here we use it to establish the announced result. We find also interesting that the KS-models were originally invented as new objects, the reason of existence of which was their T-dualisability of a new kind, and the authors of [14] were not aware that those models were closely related to the T-models already existing on the market which had their independent reason of existence.
Our technical strategy to realize the first purpose of this work will be the following one: First we represent the bi-Yang-Baxter σ-model on the target of the simple compact Lie group G as the so-called E-model of Ref. [14,15,16] which will permit us to dualize it in the sense of the Poisson-Lie T-duality. Then we work out explicitly the resulting dual σ-model on the target G C /G and we then establish that its suitable analytic continuation coincides with the generalized λ-model of Ref. [8]. We then realize the second purpose by repeating the same procedure for the most general E-model based on the same Drinfeld double. We finish our note with a short outlook.
2. E-models. Recall that the E-model, introduced in [14,15,16], is a firstorder dynamical system based on a current algebra of a quadratic 2 Lie algebra D (playing the role of the symplectic structure) and with a Hamiltonian H E being encoded in a choice of a particular linear self-adjoint involution E : D → D. More precisely, the phase space of the E-model is an infinitedimensional symplectic manifold P D with a set of distinguished D-valued coordinates j(σ) (σ is a loop parameter) the Poisson brackets of which are given by Here F AB C are the structure constants of the Lie algebra D in some basis 2 Recall that the quadratic Lie algebra D is by definition equipped with a non-degenerate ad-invariant symmetric bilinear form (., .) D .
T A ∈ D and Recall also that the "linear self-adjoint involution" means that E : D → D verifies Finally the Hamiltonian of the E-model is given by 3. σ-models from the E-models. If there is a Lie subalgebraG of D of dimensionality dimG = 1 2 dimD and such that (u, u) D = 0, ∀u ∈G then for each E there exists a non-linear σ-model on the target D/G, the first order dynamics of which coincides with the E-model (P D , H E ). HereG and D stand for (simply connected) Lie groups corresponding to the Lie algebras G and D. The second order geometrical action of this D/G model is given by [15,16,17]: where f ∈ D parametrizes the right coset D/G (one can choose several local sections covering the whole base space D/G if there exists no global section of this fibration). Most importantly, the symbol P f (E) appearing in (5) denotes a projection operator from D into D, unambiguously defined by the relations and the light-cone variables ξ ± and the derivatives ∂ ± are 4. The E-model for the Yang-Baxter σ-model. The question which is often of interest is in a sense inverse to that answered in 3. That is, given a σ-model on some target, can we associate to it an E-model from which it originates via the formula (5)? For example, let us consider the so-called η-model (or Yang-Baxter σ-model) [1,2] which is the σ-model on the target of a simple compact group G with the second-order action Here g(ξ + , ξ − ) ∈ G is a field configuration, (., .) is the Killing-Cartan form on the Lie algebra G C of G C and R : G → G is the so called Yang-Baxter operator defined, for example, in [2]. It turns out (cf. [1,2,16]) that the model (9) is the E-model for the choice 3 D = G C ,G = AN and E η given by (z * stands for the Hermitian conjugation). The ad-invariant non-degenerate symmetric bilinear form (., .) D is given by the formula 5. The Poisson-Lie T-dual of the Yang-Baxter σ-model. If, given an E-model, there are two different subalgebrasG 1 andG 2 having the properties described in 3. then the E-model gives rise to two σ-models living, respectively, on different 4 targets D/G 1 and D/G 2 . This phenomenon is called the Poisson-Lie T-duality and the models on D/G 1 and D/G 2 are referred to as being Poisson-Lie T-dual to each other. Is there a Poisson-Lie T-dual to the Yang-Baxter σ-model (9)? Yes, there is, if we takeG 2 = G instead ofG 1 = AN. The action of the dual σ-model on the target D/G 2 in the form suitable for our exposition was worked out in [16] and it is given by the formulã Here the standard WZW action S W ZW (g) (based on the ordinary Killing-Cartan form (., .) on G C and not on (., .) D !) is given by and p(ξ + , ξ − ) is a field configuration taking values in the space 5 P of positive definite Hermitian elements of G C which naturally parametrize the space of cosets G C /G. Note that the dual actionS η (p) is real in spite of the occurence of the imaginary units in front of the integrals in the expression (12).
6. The E-model for the bi-Yang-Baxter σ-model. This paragraph 6. interpolates between the review part of this letter presented so far and the original part to follow. In fact, we expose here a result which is new, but could have been extracted without much difficulty from the contents of Ref. [2]. Namely, we construct the E-model corresponding to the twoparametric bi-Yang-Baxter σ-model on the target of a simple compact group G the second-order action of which reads Here R g =Ad g −1 RAd g .
To identify the E-model from which (14) originates we take, of course, the same double D = G C and the same subgroupG 1 = AN as in the case of the Yang-Baxter model (9), however, the crucial involution E η,ρ must now be a one-parametric deformation of the involution E η from the paragraph 4. It turns out (at this is the first new result of this letter) that the correct choice is the following one where the operator R is extended from G to G C by complex linearity: We now parametrize the coset D/G 1 = G C /AN via the Iwasawa decomposition G C = GAN which means that the configuration f in Eq. (5) is G-valued. We set therefore f = g and remark that the term S W ZW,D (g) in (5) vanishes because of the property of the Lie algebraG ofG = AN that (u, u) D = 0, ∀u ∈G. In order to see that the choice (15) gives the bi-Yang-Baxter model (21), it remains to identify the projection operator P 1,g (E η,ρ ) onG 1 . For that, it helps to know that every ζ ∈G can be uniquely written as for some u ∈ G. With this insight, we find that the following expression verifies the conditions (6) and, inserting (18) into (5), we recover the action (14).
7. The Poisson-Lie T-dual of the bi-Yang-Baxter σ-model. This is the central paragraph of this note since here we work out our principal result which is the explicit form of the Poisson-Lie-T-dual of the bi-Yang-Baxter σmodel. Of course, the action of the dual σ-model is derived from the E-model based on D = G C and E η,ρ via the basic formula (5), the thing which changes with respect to 6. is the choice of the Lie subgroupG 2 = G. Identifying the coset D/G with the space P of all positive definite Hermitian elements of the group G C as in [16], we set in (5) f = p ∈ P and find the corresponding dual projection operator P 2,p (E η,ρ ) onG 2 = G : where the operators m ± : G → G are defined by Inserting (19) in (5) and realizing that p is Hermitian (therefore it holds (p −1 ∂ + p, p −1 ∂ − p) D = 0) we find after some work the action of the Poisson-Lie T-dual of the bi-Yang-Baxter σ-model: (21) 8. Generalized λ-model and the analytic continuation. Recently, Sfetsos, Siampos and Thomson introduced in [8] an interesting two-parameter integrable deformation of the WZW model on the simple compact group G which they called the generalized λ-model. The action of this theory is given by the formula where α,ρ are real parameters related to the real parameterst,η originally used in [8] by the formulae We remark also that the terminology λ-model refers to the notation used in [8] where the operator 1+α+ρR 1−α+ρR was denoted as Λ −1 . It is now evident that the action (22) can be transformed into that (21) by performing three operations : 1) replacing the G-valued configuration g(ξ + , ξ − ) by the P -valued p 2 (ξ + , ξ − ), 2) replacing the real parameter α by the purely imaginary one iη, 3) multiplying the action (21) by −i. The two last operations can be clearly interpreted as appropriate analytic continuations and the first one too, if we parametrize g and p 2 in the Cartan way: g as g = kδk −1 and p 2 as p 2 = kak −1 with k ∈ G, δ is unitary diagonal and a is real positive diagonal. Replacing δ by a can be now interpreted as a simple analytic continuation of the coordinates parametrizing the complex Cartan torus of G C . In the case of the target SU (2), the operations 1), 2) and 3) coincide with those carried out in [8] therefore our result generalizes to any G the SU(2) result of Sfetsos, Siampos and Thompson stating that the generalized λ-model is related by an appropriate analytic continuation to the Poisson-Lie T-dual of the bi-Yang-Baxter σ-model.

9.
Poisson-Lie T-duals of the general KS-models. Consider now the most general E-model based on the Drinfeld double D = G C . It is defined by the choice of a linear operator E : G → G, which can be written unambiguously as E = S + A, where (Sx, y) G = (x, Sy) G , (Ax, y) G = −(x, Ay) G , and we require also that the symmetric part S is invertible. We choose the corresponding self adjoint involution E η,E as where E † ≡ S − A. Note that for S equal to the identity and A = ρR we recover the bi-Yang-Baxter involution E η,ρ . It is not difficult to work out the crucial projection operators. Setting E g := Ad g −1 EAd g , we find (26) which, plugged in the fundamental formula (5), yield respectively the generic KS-model and its Poisson-Lie T-dualS 10. T-models and the analytic continuation. Replacing in (28) the P -valued p 2 (ξ + , ξ − ) by the G-valued configuration g(ξ + , ξ − ), replacing the imaginary parameter iη by the real parameter α and multiplying the action (28) by i, we obtain S α,E (g) = S W ZW (g) + k dξ (29) The action (28) is to be compared with the general action of the T-model on the compact group target G: where Λ : G → G is an arbitrary invertible operator. The obvious identification can be generically inverted which confirms our claim that the analytical continuations of the Poisson-Lie T-duals of the KS-models are the T-models.
11. Outlook. In order to relate the η and the λ deformations of the σ-models living on the cosets of G via the Poisson-Lie T-duality and the analytic continuation, it looks promising to use the framework of the dressing cosets generalization of the E-models [20]. We plan to deal with this problem in a near future. Another interesting question to study would be the behavior of the Poisson-Lie symmetries of the models (28) under the analytic continuation yielding the T-models (30). The recent results of Ref. [21] could be of use in tackling this problem.