Dressing cosets and multi-parametric integrable deformations

We provide a new construction of the dressing cosets sigma-models which is based on an isotropic gauging of the E-models. As an application of this new approach, we show that the recently constructed multi-parametric integrable deformations of the principal chiral model are the dressing cosets, they are therefore automatically renormalizable and their dynamics can be completely characterized in terms of current algebras.

It turns out that the Hamiltonian dynamics of many integrable models σ-models can be cast in a very transparent way within the formalism of certain specific dynamical systems referred to as the Emodels [51,56]. The E-models are formulated in terms of the current algebras of Drinfeld doubles and were originally introduced in the framework of the Poisson-Lie T-duality [55,56]. However, they turn out to be useful in many respects also in the integrability story, in particular in establishing the relation between the η and λ deformations via the T-duality [42,51,79]. define the symplectic form ω LD on LD by the formula ω LD := − 1 2 (l −1 dl, ∂ σ (l −1 dl)) D . (2.1) Here the symbol expresses the integration over the angle σ.
The (non-degenerate) E-model is a dynamical system the phase space of which is the symplectic manifold (LD, ω LD ) and the Hamiltonian H E of which is given by the formula The knowledge of the symplectic form (2.1) and of the Hamiltonian (2.2) is sufficient to construct the first-order action of the E-model [56] S E (l) =

(2.3)
Every E-model (LD, ω LD , H E ) on the Drinfeld double represents simultaneously the Hamiltonian dynamics of two (or more) σ-models living on geometrically non-equivalent targets. How it comes about? We show this first in a particular case of the so-called perfect Drinfeld doubles. Recall that the Drinfeld double D is perfect if the topological direct product K ×K of its maximally isotropic subgroups is diffeomorphic to D in a way compatible with the multiplication law in D. This means that if Υ : D → K ×K is the diffeomorphism and m : D × D → D is the group multiplication map then the composition map m • Υ is the identity map on D. In particular, every element l(σ) of the loop group LD of the perfect Drinfeld double D can be unambiguously decomposed as the product of one element k(σ) from the loop group LK and one elementh(σ) from the loop group LK as follows l(σ) = k(σ)h(σ), k ∈ LK,h ∈ LK. (2.4) Inserting the decomposition (2.4) into (2.1) and into (2.2), we obtain easily

5)
H E (k,h) = 1 2 (∂ σ kk −1 + k∂ σhh −1 k −1 , E(∂ σ kk −1 + k∂ σhh −1 k −1 )) D . (2.6) The first order action (2.3) of the E-model (LD, ω LD , H E ) in the parametrization k,h is therefore given by the data (2.5) and (2.6): The dependence of S E on ∂ σhh −1 is quadratic, it is therefore easy to eliminate ∂ σhh −1 which gives the second order action of the so called Poisson-Lie σ-model: (2.8) Here ∂ ± ≡ ∂ τ ± ∂ σ , the linear operator E :K → K is such that its graph {x + Ex,x ∈K} coincides with the image of the operator Id+E and the k-dependent operator Π(k) :K → K can be explicitly expressed in terms of the structure of the Drinfeld double as follows Here Ad k stands for the adjoint action on D of the element k ∈ K ⊂ D and J ,J are projectors; J projects to K with the kernelK andJ projects toK with the kernel K.
Recall also that the operator Π(k) :K → K encodes the so called Poisson-Lie bracket of two functions f 1 , f 2 on the group K in the sense of the formula: (2.10) Here ∇ L isK-valued differential operator acting on the functions on K as Of course, every element l(σ) of the loop group LD can be decomposed also in the dual way as Inserting the decomposition (2.12) into (2.1) and into (2.2), and then repeating all the procedure as before leads to the dual σ-model living on the targetK: Of course, the linear operatorẼ : K →K is again such that its graph {x +Ẽx, x ∈ K} coincides with the image of the operator Id+E, which implies that the duality between the models (2.8) and (2.13) holds under the condition, that the operatorẼ is inverse of the operator E.
If the Drinfeld double is not perfect, there exists a generalization of the T-duality between the models (2.8) and (2.13), where the two σ-models live, respectively, on the spaces of cosets D/K and D/K [51,58,59]. If we parametrize (possibly patch by patch) the coset space D/K by a section m ∈ D of the bundle D → D/K and the coset space D/K by a sectionm ∈ D of the bundle D → D/K, then the decompositions l = mh and l =mh generalize those (2.4) and (2.12) and lead respectively to the following dual pair of σ-models with the WZW terms: (2.16) Here the projectors P m (E) : D → D andPm(E) : D → D have the respective imagesK and K and their respective kernels are given by the linear spaces (Id+Ad m −1 EAd m )K and (Id+Adm−1 EAdm)K.
Of course, the Poisson-Lie T-duality relating the models (2.8) and (2.13), or, more generally, relating the models (2.15) and (2.16), is the main result that we review in this Section 2.1, but we add few more formulas about the E-models that will be useful in what follows. First of all, the first order Hamiltonian equations of motions derived from the formulas (2.1) and (2.2) can be written in two useful ways, either as ∂ τ ll −1 = E∂ σ ll −1 (2.17) or as where j(σ) := ∂ σ l(σ)l(σ) −1 .
Moreover, the inversion of the symplectic form ω LD gives the standard current algebra Poisson brackets for the LD-valued variable j(σ): Here and T A ∈ D is some basis of D.

DegenerateÊ-models
We start the exposition of the degenerateÊ-models from the end, that is, we first write to what kind of T-duality they give rise to. To grasp the idea, it is sufficient to consider the perfect Drinfeld doubles and the resulting dual pair of the "dressing cosets" σ-models is then given by the actions  (2.13). What happens is that in the construction of the standard pair (2.8) and (2.13) from the E-model it follows automatically that the symmetric part 1 2 (Ẽ +Ẽ * ) of the operatorẼ : K →K is also invertible (the symbolẼ * stands for the adjoint of the operatorẼ with respect to the bilinear form (., .) D .) In the dressing cosets case, the operator 1 2 (Ê +Ê * ) need not be invertible and the T-duality of the models (2.22) and (2.23) still holds (if some further invariance conditions onÊ are fulfilled). This is not a trivial generalization of the standard Poisson-Lie T-duality, however, because the lack of invertibility of the operatorÊ or of its symmetric part 1 2 (Ê +Ê * ) has drastic consequences on the dynamics of the mutually dual σ-models (2.22) and (2.23). In fact, both models develop a gauge symmetry with respect to some subgroup F of the Drinfeld double D and the common dimension of their targets thus gets effectively diminished by the dimension of the group F .
Let us now give a concrete example of the phenomenon that the common gauge symmetry of the dressing cosets leads to the diminution of the dimension of the targets of the mutually dual dressings cosets σ-models. Consider thus the case where K is a simple compact group and the groupK is the Lie algebra K with the (Abelian) group structure given by the vector space addition. As the manifold, the perfect Drinfeld double D is the topological direct product K × K with the following multiplication law Every element k of the group K is embedded in D as (k, 0) and every element κ ofK as (e, κ), where e is the unit element of K. The bilinear form (., .) D is given by where (., .) K is the standard Killing-Cartan form on the simple Lie algebra K.
For the linear operatorÊ, we pick the orthogonal projector P ⊥ on the subspace T ⊥ ⊂ K perpendicular to the Cartan subalgebra T ⊂ K, moreover, by using the formulas (2.9) and (2.14), we find that the Poisson-Lie bivector Π(k) on K trivially vanishes while the Poisson-Lie bivector onK is given by the adjoint action of the Lie algebra. The actions of the mutually dual σ-models (2.22) and (2.23) in this particular case thus become The σ-models (2.26) and (2.27) have both the gauge symmetry with the gauge group F being the Cartan torus T ⊂ K (the Lie algebra of F is the Cartan subalgebra T ). The element f (τ, σ) of the gauge group acts on the respective fields of the σ-models as In the case of the group K = SU (2), the common dimension of the targets of the models (2.26) and (2.27) is two and the corresponding background geometries were obtained also in the framework of the standard non-Abelian T-duality [3,37,38,73] where the isometry group does not act freely. Therefore the dressing cosets in general can be understood as the Poisson-Lie generalizations of such models. Other examples of the dressing cosets have been studied in [11,14,41,42,54,76,77,79,83].
The first order dynamics of the dressing cosets was described in Ref. [57] and we now review that construction here. The phase space LD F of the mutually dual pair of the dressing cosets is an appropriate symplectic reduction of the non-degenerate phase space (LD, ω LD ). More precisely, consider a subgroup F of D which is isotropic, which means that the restriction of the bilinear form (., .) D to the Lie algebra F vanishes. The set of moment maps generating the left action of the loop group LF on the loop group LD is expressed by the quantity (∂ σ ll −1 , F ) D . We set this quantity to 0 which gives the presymplectic submanifold denoted by LD F . Said in other words, the (pre)phase space LD F of the degenerateÊ-model is the space of the elements l(σ) of the loop group LD, for which it holds for every σ (2.29) Here the orthogonality symbol ⊥ is understood with respect to the non-degenerate bilinear form (., .) D .
The (pre)symplectic form of the degenerate E-model is just the restriction to LD F of the symplectic form The Hamiltonian of the degenerateÊ-model looks the same as in the case of the standard Poisson-Lie T-duality however, the linear operatorÊ has now different properties as its counterpart E in the non-degenerate case. Here are those properties:Ê : F ⊥ → F ⊥ must be self-adjoint, it must commute with the adjoint action of F on F ⊥ and its kernel must contain F . Moreover, the bilinear form (.,Ê.) F ⊥ must be positive semi-definitive and the image of the operatorÊ 2 − Id has to be contained in F .
If the double is perfect, we can decompose the elements l(σ) ∈ LD F either as l = kh or as l =kh and by eliminating respectively the fieldsh and h, we obtain the σ-models (2.22) and (2.23). The linear operatorÊ is obtained via the relation (2.32) After the fixing the gauge symmetry, the target spaces of the models (2.22) and (2.23) become respectively the dressing cosets F \K and F \K. The left dressing action of an element f ∈ F on an element k ∈ K is defined as the K-part of the D-product f k. Said in other words, we decompose f k ∈ D as f k fh , f k ∈ K, fh ∈K and the element f k ∈ K is the result of the dressing action of f on k.
How all this procedure works in detail can be found in the original paper [57], but the reader need not consult it. In fact, we shall present in the next Section 2.3 a new construction of the dressing cosets, which is arguably more straightforward than that of Ref. [57]. We do not know whether the new method permits to derive all dual pairs obtainable by the old one, the new picture is however sufficiently general to underlie the multiparameter integrable deformations of Ref. [17].

New method of producing the dressing cosets
One particular way how to obtain the degenerateÊ-model from non-degenerate one was studied in [77]. The idea described therein is to let a non-degenerate operator E depend on a parameter and study a (singular) limit in which E becomes an operatorÊ characterising the degenerateÊ-model. Here we develop another method, inspired by the procedure of the isotropic gauging described in Ref. [60], that is, we gauge in an isotropic way a non-degenerate E-model to produce from it a degenerateÊ-model. The big advantage of this procedure is its simplicity as well as the rapidity with which the resulting pair of the dressing cosets σ-models is explicitely found. Let us show how it works.
Let (LD, ω LD , H E ) be a non-degenerate E-model. Its first order action can be written as in (2.3) (2.33) Let F be an isotropic subgroup of D (the term "isotropic" means that the restriction of the bilinear form (., .) D to the Lie algebra F vanishes) and let E be such that it commutes with the adjoint action of F on D. The action (2.33) has then a global F -symmetry, where an element f ∈ F acts on the loop l(σ) by the standard left multiplication f l(σ). This global F -symmetry can be gauged by introducing an F -valued gauge field A ≡ A τ dτ + A σ dσ. The gauged action reads It is gauge invariant with respect to the following action of the element f (σ, τ ) of the gauge group F To verify it, the F -invariance of the operator E is needed as well as the Polyakov-Wiegmann formula The basic claim of the present section is the statement: The isotropically gauged non-degenerate E-model (2.34) is a degenerateÊ-model.
Of course, the statement is deliberately short in order to encapsulate the message in the briefest terms, we have therefore explain it in more detail and also to state one more technical condition needed to be verified for the statement to hold. This condition is actually that the restriction of the non-degenerate bilinear form (., E.) D to the subalgebra F remains non-degenerate.
It is not difficult to see that the gauged E-model (2.34) is in fact a degenerateÊ-model in a disguise. Indeed, the component A τ play the role of the Lagrange multiplier which restricts the phase space LD of the non-degenerate E-model to the (pre)phase space LD F = {l ∈ LD, ∂ σ ll −1 ∈ F ⊥ } of the degenerate one. Furthermore, the field A σ appears quadratically in the action (2.34), it can be therefore easily integrated away yielding again a quadratic Hamiltonian of the form (∂ σ ll −1 ,Ê(∂ σ ll −1 ) D for some operatorÊ. We have to show then that this operatorÊ has all the properties to define the degenerateÊ-model as described in the previous Section 2.2.
We perform the elimination of the field A σ by a chain of shortcut arguments, avoiding any "hardline" computation. We first remark that the non-degeneracy of the restriction of the bilinear form (., E.) D to the subalgebra F implies that the linear space EF has trivial (zero) intersection with F . We can therefore set V = F ⊕ EF (2.38) and argue that the vector spaces V and V ⊥ also intersect trivially. Indeed, if the intersection V ∩ V ⊥ contained a non-zero vector x then the vector Ex would be again in V ∩ V ⊥ (because E is self-adjoint and it therefore preserves both V and V ⊥ ). From this as well as from the fact that E squares to Id, we find the following contradiction with the non-degeneracy of the form (., E.) D : We can now represent the Lie algebra D as the direct sum V ⊥ ⊕ F ⊕ EF and every element y ∈ D accordingly as y = y 0 + y 1 + y 2 . (2.40) The action (2.34) can be now rewritten as Note that the component (∂ σ ll −1 ) 2 does not appear in the Hamiltonian part of the action because it is killed by the Lagrange multiplier A τ . Furthermore, the component (∂ σ ll −1 ) 1 lies in F , it can be therefore absorbed into A σ . Integrating away A σ thus means simply the omitting of the last term in Eq.(2.41). At the end, we obtain the degenerateÊ-model where the operatorÊ : It is easy to verify thatÊ has all required properties to define the degenerateÊ-model. It is self-adjoint because E is: its kernel evidently contains F and it commutes with ad φ for every φ ∈ F because both V and V ⊥ are and the form (., E.) D is strictly positive definite.
What is it good for to know that the gauged non-degenerate E-model is in fact the degenerateÊmodel? Well, in some important cases, like those studied in the context of the integrable deformations, it is technically much easier to extract the actions of the dual pair of σ-models from the gauged nondegenerate first order formalism rather than directly from the degenerate one. We give now an example of this situation for the case when D is the perfect double and F is a subalgebra of K.
Instead of eliminating the gauge field A from the gauged first order action (2.41), we first decompose l as the product of one element k(σ) of the loop group LK and one elementh(σ) of the loop group LK as in Eq.
Inserting the decomposition (2.47) into (2.34), we obtain easily It is easy to preform the computation integrating away ∂ σhh −1 from S E (k,h, A), because the only difference with respect to the similar computation leading from (2.7) The result is therefore the following gauged second order action of the standard Poisson-Lie σ-model (2.8): (2.49) Here A ± ≡ A t ± A σ and, as before, the graph {x + Ex,x ∈K} of the operator E :K → K coincides with the image of the operator Id+E. The gauge symmetry k → f k and Remarkably, the integrating away the non-dynamical gauge fields A ± from (2.49) gives the dressing coset action (2.22) (2.50) We now perform in detail the calculation leading from (2.49) to (2.50) which will permit us to identify the operatorÊ : K →K in terms of the operator E :K → K. We start by introducing anK-valued auxiliary 1-form field B ≡ B τ dτ + B σ dσ and by considering an auxiliary action The auxiliary action S E (k, A, B) is dynamically equivalent to the action S E (k, A) because the integrating away the field B from the former yields the latter. Now the gauge field A featuring in the action (2.51) plays the role of the Lagrange multiplier making to vanish some components of the field B. Let us be more precise about that point: LetQ :K →K be the projector with the kernel EF ∩K and the image (F ⊕V ⊥ )∩K and let Q : K → K be the projector with the kernel F ∩ K and the image (EF ⊕ V ⊥ ) ∩ K. Integrating away the gauge field A from the auxiliary action (2.51) then gives and introduce a K-valued field Λ Λ : We can now rewrite (2.55) as It is easy to perform the computation integrating away Λ fromS E (k, Λ, A), because the only difference with respect to the similar computation leading from (2.7) to (2.13) is the replacement of ∂ τkk −1 by The result is therefore the following gauged second order action of the dual Poisson-Lie σ-model (2.13): Integrating away the gauge field A gives  62) or, equivalently, to show that The right-hand-side of (2.63) is equal to {Qx + EQx, x ∈K}, we have to show therefore that But this is evidently true since EQx ∈ V ⊥ ∩ K, hence EQx = (QEQ)x.

DegenerateÊ-models and the multi-parametric deformations
In this section, we study a specific class of σ-models on the simple compact group target K which is fully characterized by a real number κ and by a (2 × 2)-matrix M , the elements of which are linear operators M ab : K → K, a, b = L, R. Given κ and M , the action of any model in this class reads 1) where k is the K-valued field, the standard Wess-Zumino term I WZ is defined as and Q L,R as well as P L,R are field-dependent linear operators on the Lie algebra K constructed out of the field-independent elements M ab of the matrix M as follows Ad k , The multiparametric integrable deformation of the principal chiral model introduced in Ref. [17] belongs to the class of the models (3.1), and it corresponds to the following choice of the matrix M : where

5)
(3.6) The parameters κ, η L , η R are real numbers, while ω : T → T is the linear operator on the Cartan subalgebra T and the matrix elements of ω are considered as the r 2 independent parameters (denoting by r the rank of the Cartan subalgebra). The DHKM model thus depends in total on (3+r 2 ) independent parameters. Finally, R : K → K is the Yang-Baxter operator which annihilates the Cartan subalgebra of K and it is defined as where B α , C α are given in terms of the step generators of K C as The Yang-Baxter operator verifies for every x, y ∈ K two important identities: tr (x Ry) = −tr (Rx y), We observe that the DHKM model (3.1) (which is equivalent to the Lukyanov model [65] for the case of K = SU (2)) is quite a complicated theory and it would be welcome to work with technically more manageable expressions. This goal was achieved in [17] by introducing a K-valued auxiliary gauge field A. Thus the action of the σ-model (3.1) can be written equivalently as Integrating away A ± = A L± = A R± from (3.11) yields back the model (3.1) with k = k L k −1 R . The main concern of the present section is to show that the action (3.11) can be extracted from the isotropically gauged non-degenerate E-model (2.34) by the procedure described in Section 2. We make now an important point, namely, we stress from the very beginning that the ingredients D, E and F (that encode the first order Hamiltonian dynamics defined by the gauged action (2.34)) do not depend on the TsT parameters. In fact, the r 2 TsT parameters only appear when passing from the first order formalism to the second order σ-model one. This passage is not unique and it is the very essence of the dressing coset story [57] that for every choice of the maximally isotropic subgroupK Ω of D the procedure starting by Eq. (2.47) permits to extract from the first order action (2.34) the second order σ-model living on the target F \D/K Ω . The upshot is that the possible choices of the maximally isotropic subgroupK Ω are paramatrized by an arbitrary r × r matrix Ω : T → T called also the TsT matrix. The r 2 TsT parameters Ω in the DHKM model action (3.11) thus come from the parametrization of the double coset F \D/K Ω and it is perhaps worth stressing again that the changing of the value of the matrix Ω from one to another is nothing but the Poisson-Lie T-duality transformation; i.e. the first order Hamiltonian dynamics of the σ-models remains the same and independent of the choice of Ω, however, the target space geometries do depend on Ω.
There is time to specify the non-degenerate E-model the isotropic gauging (2.34) of which leads, upon the choice of the groupK Ω , to the (3 + r 2 )-parametric models (3.11) or (3.1). The Drinfeld double D is the direct product K C × K C , where K C denotes the complexification of the simple compact group K. The invariant bilinear form (., .) D on the Lie algebra D = K C ⊕ K C is given by the formula Here the symbol ℑ means taking the imaginary part of a complex number, the real parameters C L , C R do not vanish and the real parameters ρ L , ρ R range in the open interval ] − π, π[. Moreover, we require the following constraint to be imposed on the parameters C L sin (ρ L ) + C R sin (ρ R ) = 0, (3.13) which can be solved in terms of one arbitrary constant C as follows (3.14) The next step is to specify the operator E : D → D. It is given by the formula 15) where µ L , µ R are real parameters and z * stands for the Hermitian conjugation. It is straightforward to check that the operator E defined by Eq.(3.15) verifies all three properties needed to define the nondegenerate E-model, namely, it squares to the identity, it is self-adjoint with respect to the bilinear form (3.12) and the bilinear form (., E.) D on D is strictly positive definite.
The first order action S E (l) of the non-degenerate E-model defined by the data (D, E) is given by the general expression (2.3). In order to recover the σ-model (3.1) out of it, we have to gauge it isotropically in the sense of Section 2.3, that is, to produce the gauge invariant first order action S E (l, A) given by Eq. (2.34). For the gauge group F ⊂ D, we choose the diagonal embedding of the simple compact group K into D = K C × K C . The elements of F have therefore the form (g, g) ∈ D, g ∈ K and the elements of the Lie algebra F have the form x ⊕ x ∈ D, x ∈ K. The gauge group F is isotropic because of the constraint (3.13). Indeed, it is easy to check that it holds (x ⊕ x, y ⊕ y) D = 0, ∀x, y ∈ K. (3.16) The operator E given by Eq.(3.15) commutes with the adjoint action of the Lie group F on D because it holds (Ad g z) * = Ad g (z * ) for g ∈ K, z ∈ K C . To fit into the general gauging procedure of Section 2.3, the last thing to check is that the restriction of the non-degenerate bilinear form (., E.) D onto the subalgebra F remains non-degenerate. But this is true because this restriction is given by the formula In order to extract the action (3.11) out of the gauged E-model (2.34), we have to choose the maximally isotropic subgroupK Ω ⊂ D. We chooseK Ω ⊂ D to have the form of the semidirect product A Ω ⋉(N ×N ), where N is the nilpotent subgroup of K C appearing in the Iwasawa decomposition K C = KAN and A Ω is an Ω-dependent r-dimensional isotropic subgroup of the group A C ×A C . Actually, the maximally isotropic subgroupK Ω is fully determined by its Lie algebraK Ω which can be conveniently described in terms of the Yang-Baxter operator R as the following half-dimensional subspace of the double It is perhaps interesting to make a small digression and to represent the Lie algebra structure ofK Ω as an alternative commutator [., .] R,ρ,Ω on the vector space K ⊕ K Note that for Ω = 0, the Lie algebraK Ω becomes the direct sum of two Lie algebras K R,ρL ⊕ K R,ρR characterized by the commutators (3.21) Consider now the first order action (2.34) of the isotropically gauged E-model: Following the general procedure described in Section 2.3, we write the field l ∈ K C × K C as where m is K × K-valued field andh Ω takes values inK Ω . It is easy to see that the decomposition (3.23) is global for whatever Ω. Inserting it into the action (3.22), we obtain 1 We wish to integrate away the field ∂ σhΩh −1 Ω . In the case of absence of the gauge field A, the result would be given by the general formula (2.15). In the presence of A, the formula (2.15) has to be modified accordingly: It remains to determine explicitly the projection P Ω,m . We first note that there is no dependence of m since the operator Ad m commutes with the operator E defined by the formula (3.15); therefore P Ω,m ≡ P Ω . Then we find that the kernel (Id + E)K Ω is the half-dimensional subspace of D which can be parametrized by the elements s L ⊕ s R of the Lie algebra K ⊕ K via We have to calculate the action of the projection P Ω on the elements x L ⊕ x R of the Lie algebra K ⊕ K; it is determined from the unambiguous decomposition Said differently: given x L ⊕ x R , we have to find s L , s R , u L , u R ∈ K (they are given unambiguously) such that the relation (3.26) holds. Then we have We find straightforwardly (3.29) By inserting (3.29) and in the left-hand-side of Eq. (3.27), then by substituting the result into the right-hand-side of Eq.(3.27), we proceed to the straightforward evaluation of the action (3.25). Indeed, we set m = (k −1 L , k −1 R ), we change the sign of A and we find that the action (3.25) becomes where It can be directly checked that it holds , a = L, R.
(3.35) in particular, the constraints (3.6) are respected in this case.
We finish this section by noting that the dressing coset (3.30), (3.31) that we have extracted from the gauged E-model (3.22) seems to have (4 + r 2 ) relevant free parameters, that is one more than the (3 + r 2 )-parametric integrable σ-model (3.11), (3.4). Indeed, while the parameter C plays no role at the classical level since it amounts just to the overall rescaling of the σ-model action, the (4 + r 2 ) parameters Ω, µ a , ρ a , a = L, R, are unconstrained. However, the numerical analysis [61] performed in the case of the SU (2) target suggests that the four parameters conspire in the final σ-model Lagrangian (3.1) only into three independent combinations considered already in Ref. [17], which would mean that the fourth parameter appearing in the E-model approach is in fact superfluous.

Integrability of the multi-parametric dressing coset
The purpose of the present section is to prove the integrability of the multi-parametric dressing coset (3.30), (3.31) working directly in the E-model formalism. Our strategy of proof will consist in representing the field equations of the model in terms of two K-valued currents J = J τ dτ +J σ dσ and B = B τ dτ +B σ dσ as follows Here B ± = B τ ± B σ and J ± = J τ ± J σ . As shown in Section 2.1.3 of Ref. [21], the system of the equations (4.1) and (4.2) admits the following Lax pair with the spectral parameter z Indeed, it is straightforward to check that the zero curvature condition is equivalent to the system (4.1) and (4.2).
Coming back to the general dressing coset story described in Section 2.3, it is easy to work out the field equations of the isotropically gauged E-model (2.34). They read where we recall that every element y of the Drinfeld double Lie algebra D can be unambiguously written as y = y 0 + y 1 + y 2 , y 0 ∈ V ⊥ = (F ⊕ EF ) ⊥ , y 1 ∈ F , y 2 ∈ EF . In what follows, we shall need a refinement of the decomposition (4.8) which is obtained by decomposing further y 0 as y 0 = y + + y − , (4.9) where Ey ± = ±y ± . (4.10) We write also (4.11) With this notation, the equations of motion (4.5),(4.6) and (4.7) can be rewritten as Indeed, the relations (∂ ± ll −1 ) 2 = 0 are the direct consequences of the equations (4.6) and (4.7), because Similarly, using the fact that the operator E squares to the identity we can rewrite Eq.(4.5) as and then the changing the roles of σ and τ in the previous chanin of arguments leads to the conclusion that (∂ τ ll −1 ) 1 = A τ . In this way we have proved that (∂ ± ll −1 ) 1 = A ± . Finally, knowing that (∂ ± ll −1 ) 2 = 0 and (∂ ± ll −1 ) 1 = A ± permits to rewrite Eq.(4.5) as 14) or, equivalently, as Adding and substracting Eqs. (4.14) and (4.15) gives (∂ ± ll −1 ) ∓ = 0.
We now specify the equations of motions (4.12) for the gauged E-model constructed in Section 3, which underlies the (4 + r 2 )-parametric dressing coset (3.30), (3.31). Recall that the Drinfeld double D is in this case the direct product K C ×K C , the invariant bilinear form (., .) D on the Lie algebra D = K C ⊕K C is given by the formula (z L ⊕ z R , w L ⊕ w R ) D := C sin (ρ R )ℑtr e iρL z L w L −C sin (ρ L )ℑtr e iρR z R w R , z a , w a ∈ K C , a = L, R, (4.16) the isotropic subalgebra F consists of the elements (x⊕x), x ∈ K and the eponymous self-adjoint operator E : D → D commuting with the adjoint action of F is given by the formula We want to describe the subspaces V ⊥ ± ⊂ D corresponding to those data. We note that V ⊥ ± ⊂ (Id ± E)D, where the half-dimensional subspaces (Id±E)D can be conveniently parametrized in terms of the elements u ± L,R of the Lie algebra K as where we have traded the parameters µ L,R for m L,R according to the formulas µ a = tan (m a ), a = L, R. Note that the subspaces V ⊥ ± are formed by the vectors in (Id ± E)D perpendicular to F which gives (4.20) The equations of motions (4.12) are then solved by where the K-valued fields J ± , A ± must be such that the Bianchi identity be verified: Inserting the expressions (4.21), (4.22) into the left-hand-side of (4.23), we obtain (4.25) (4.27) where (4.29) The equation (4.23) together with Eqs.(4.24),(4.25),(4.26) and (4.27) then imply the validity of the following system of equations The system (4.31) and (4.32) admits the Lax pair (4.3) with spectral parameter because it is equivalent to the system (4.1) and (4.2) upon the identification The (4 + r 2 )-parametric dressing coset (3.30),(3.31) is therefore integrable.

Renormalizability of the multi-parametric dressing coset
The fact that a given σ-model has the first order dynamics which can be expressed in terms of a nondegenerate E-model is very useful for the study of its ultraviolet properties, because such model is automatically renormalisable. Indeed, it was established in [80,82,85], that the renormalisation group flow respects the structure of the E-model, just flowing from one epynomous operator E to another. This flow is described by an elegant formula derived in [82] (and used in an different context already in [84]): Here s is the flow parameter and the capital Latin indices refer to the choice of a basis T A of the Lie algebra D: where the projections P ± are defined as Here we use the Sweedler notation A = A ′ ⊗ A ′′ , B = B ′ ⊗ B ′′ and we view the self-adjoint operators P ± as the elements of S 2 D in the sense of the formula P ± x := P ′ ± (P ′′ ± , x) D , x ∈ D. (5.39) In particular, for the case of the operator E given by the formula (3.15) and the bilinear form given by (3.12), we find where P b L+ , P b R+ and P b L− , P b R− are the respective basis of the subspaces (Id ± E)D (described in Eq. We find straightforwardly where Q 2 is the value of the quadratic Casimir in the adjoint representation of the Lie algebra K. We find easily P ± (e −iρL T b ⊕ 0) = ∓ sin (∓m L + 1 2 ρ L ) sin (2m L ) P b L± , P ± (0 ⊕ e −iρR T b ) = ∓ sin (∓m R + 1 2 ρ R ) sin (2m R ) P b R± (5.47) which permits to evaluate the right-hand-side of Eq.(5.36) we find straightforwardly We observe that the right-hand-sides of the equations (5.48) and (5.50) match perfectly, if the parameters m L and m R flow following the equations dm L ds = Q 2 sin (m L + 1 2 ρ L ) sin (−m L + 1 2 ρ L ) 2C L sin 2 (2m L ) , dm R ds = Q 2 sin (m R + 1 2 ρ R ) sin (−m R + 1 2 ρ R ) 2C R sin 2 (2m R ) . (5.51) We have thus established the renormalisability of the (4 + r 2 )-parametric dressing coset (3.30), (3.31), showing that only the coupling constants m L , m R flow while the parameters ρ L , ρ L and C are RGinvariant.
The flows of the original parameters µ L and µ R can be easily obtained from Eqs. (4.19) and (5.51): dµ a ds = − Q 2 µ 2 a cos 2 ρa 2 − sin 2 ρa 2 (1 + µ 2 a ) 2 8C a µ 2 a , a = L, R. (5.52) This result is in full agreement 2 with the renormalisation group analysis of the Yang-Baxter σ-model with the WZW term which was performed in Ref. [25]. This is not at all surprising because our E-model based on the data (3.15) and (3.12) is nothing but the direct product of two models Yang-Baxter σ-model with the WZW term. The fact that we subsequently gauge this direct product does not change anything because the flow is gauge invariant.

Outlook
The present work solves two from the open problems listed in the outlook of Ref. [17], namely, it provides the E-model formulation of the multi-parametric integrable deformation introduced therein and, also, it settles the issue of its renormalisability. We believe, that the E-model insight should be helpful also for tackling the remaining open question from the list, which is the status of the Hamiltonian integrability of the model.
The dressing coset structure of the DHKM model guarantees that it exhibits a rich T-duality structure well beyond the T-duality corresponding to the changing of the TsT parameters. In particular, this should be the case for the Lukyanov model living on the SU (2) target. We postpone the detailed account of this T-duality story to a forthcoming publication.